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*Table of contents : Frontmatter p.i-xxxiichap-1chap-2chap-3chap-4chap-5chap-6chap-7chap-8chap-9chap-10chap-11chap-12*

HANDBOOK OF HYDRAULIC RESISTANCE 4TH REVISED AND AUGMENTED EDITION

I. E. IDELCHIK Moscow

4TH EDITION EDITORS A. S. GINEVSKIY AND A. V. KOLESNIKOV Central Aero-Hydrodynamics Institute (TsAGI) Moscow

TRANSLATED BY GRETA R. MALYAVSKAYA and NATALIA K. SHVEYEVA A. V. Luikov Heat and Mass Transfer Institute Minsk

ENGLISH EDITION EDITOR WILLIAM BEGELL New York

begell house, inc. New York · Connecticut · Wallingford (U.K.)

Handbook of Hydraulic Resistance 4th Edition Revised and Augmented Series Editors: A. S. Ginevskiy and A.V. Kolesnikov Handbook of Hydraulic Resistance I. E. Idelchik This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials for the consequences of their use. All rights reserved. This book, or any parts thereof, may not be reproduced in any form without written consent from the publisher. Direct inquires to Begell House, Inc., 50 Cross Highway, Redding, CT 06896. © 2007 by Begell House, Inc. ISBN: 978-1-56700-251-5 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Library of Congress Cataloging-in-Publication Data Idel_chik, I. E. [Spravochnik po gidravlicheskim soprotivleniiam. English] Handbook of hydraulic resistance / I. E. Idelchik ; editors, A.S. Ginevskiy ... [et al.] ; translated by Greta R. Malyavska. -- 4th ed. rev. and augmented p. cm. Includes bibliographical references and index. ISBN 978-1-56700-251-5 (alk. paper) 1. Fluid dynamics. 2. Frictional resistance (Hydrodynamics) I. Ginevskii, A. S. (Aron Semenovich) II. Title. TA357.I3413 2008 620.1'064--dc22 2008000963

TABLE OF CONTENTS

Editor’s Preface Preface to the English Edition Preface to the Second Russian Edition

i iii v

Preface to the Third Edition

vii

Preface to the Fourth Edition

ix

Nomenclature

xiii

Useful Conversions of Units

xvii

Reader’s Guide and Introduction

xxi

1.

General Information and Elements of Aerodynamics and Hydraulics of Pressure Systems

1

2.

Resistance to Flow in Straight Tubes and Conduits: Friction Coefficients and Roughness

3.

Resistance to Flow at the Entrance into Tubes and Conduits: Resistance Coefficients of Inlet Sections

177

4.

Resistance to Flow through Orifices with Sudden Change in Velocity and Flow Area: Resistance Coefficients of Sections with Sudden Expansion, Sudden Contraction, Orifices, Diaphragms, and Apertures

223

5.

Resistance to Flow with a Smooth Change in Velocity: Resistance Coefficients of Diffusers and Converging and Other Transition Sections

277

6.

Resistance to Flow with Changes of the Stream Direction: Resistance Coefficients of Curved Segments — Elbows, Bends, etc.

391

7.

Resistance in the Cases of Merging of Flow Streams and Division into Flow Streams: Resistance Coefficients of Wyes, Tees, and Manifolds

483

85

8. Resistance to Flow through Barriers Uniformly Distributed Over the Channel Cross Section: Resistance Coefficients of Grids, Screens, Porous Layers, and Packings

575

9. Resistance to Flow through Pipe Fittings and Labyrinth Seals: Resistance Coefficients of Throttling Devices, Valves, Plugs, Labyrinth Seals, and Compensators

617

10. Resistance to Flow Past Obstructions in a Tube: Resistance Coefficients of Sections with Protuberances, Trusses, Girders, and Other Shapes

663

11. Resistance to Flow at the Exit from Tubes and Channels: Resistance Coefficients of Exit Sections

705

12. Resistance to Flow through Various Types of Apparatus: Resistance Coefficients of Apparatus and Other Equipment

779

Index

863

EDITOR’S PREFACE

The first edition of the Handbook of Hydraulic Resistance has been used by knowledgeable engineers in English-speaking countries since 1966, when an English translation sponsored by the U.S. Atomic Energy Commission became available. Although the book was not readily available or publicized, its extensive coverage and usefulness became known through citation, reference, and personal recommendations to a limited body of engineering practitioners in the Western world. Because there exists no English-language counterpart to Professor Idelchik’s book, the translation and publication of the revised and augmented second edition of the Handbook of Hydraulic Resistance has been undertaken. The extensive coverage provided by this book becomes self-evident when one reviews the hundreds of illustrations of flow passages contained herein. Most of these are sufficiently basic to allow application to nearly any shape of flow passage encountered in engineering practice. The editor of this translation has had extensive experience in using the first edition and has learned to appreciate not only the extent of coverage of this book but also its limitations. Based on this experience, the editor has tried to utilize American terminology whenever necessary for clarity while trying to preserve the original manuscript as faithfully as possible. Sometimes this resulted in overly detailed description, and the temptation always existed to rewrite or condense some of the explanatory chapters and sections. However, since this is a translation, the original was followed as faithfully as possible in order to maintain the author’s style and approach. In the text the flow passages of interest are variously described as pipelines, ducts, conduits, or channels — all denoting an internal flow passage or pipe. Similarly, there are references to gas, air, steam, and water, when the term fluid would have been quite adequate in most cases. Since retaining the original translated terms did not affect the technical correctness of the text, changes were made only in isolated cases. Section 1.1 provides general directions for using the book, allowing readers to make their own interpretation. The majority of readers may wish to use this handbook primarily as a source book for pressure loss or hydraulic resistance coefficients, i

applying these coefficients in their own accustomed way. The editor believes that these users may benefit from the few observations that follow. The many sketches, diagrams, and graphs are self-explanatory, with flow directions and areas indicated. The values of pressure loss coefficients may be used over the limits indicated for the particular graph. The nondimensionality of the parameters of most graphs allows them to be used in the English system as well as the metric system. This permits interchangeable use of this book with other sources of pressure loss coefficients. It should be noted that, unless otherwise stated, the data apply to Newtonian fluids considered as incompressible. It is also assumed, unless otherwise stated, that the inlet conditions and exit conditions are ideal; that is, there are no distortions. Very few experimental data exist on the effect of inlet flow distortion on the pressure loss coefficient for most flow devices. Where friction factors are required to find the overall pressure loss coefficient of a component, the values obtained by the favored sources most familiar to the reader may be used in place of the data shown herein. Particular attention should be paid to the limits of applicability of the data provided as well as to the reference flow area used, when there is a flow area change. Much of the data are shown in tabular as well as graphical form. The former allows use of computers in the interpolation of intermediate values. In any compilation of empirical data, the accuracy decreases with increasing complexity of the component, due to analytical and experimental uncertainties. This book is no exception. A good rule to follow is to check more than one source, if possible. Although there will be many flow configurations for which no explicit resistance values are given in this book, it is entirely possible to make up combinations of simple shapes to simulate a complex component, provided suitable engineering judgment is applied. The latter, of course, requires familiarity with the way the data are presented and with the effect of exit conditions from one component on the inlet conditions of the adjacent component. The editor of this translation would be remiss if he did not acknowledge that differences in engineering practice, nomenclature, engineering standards, and training may have an effect on the ability to fully utilize all that is presented in this work. One example is the difficulty in understanding the descriptive terms for some flow system components. However, the graphical presentations of much of the material in this book will help the reader overcome most such difficulties. In a work of this nature, it is very probable that errors of translation or data reporting have occurred. The editor and the publisher would be most grateful to the readers and users of this handbook for information on such items. Erwin Fried ii

PREFACE TO THE ENGLISH EDITION

The present edition of the Handbook of Hydraulic Resistance, translated into English from the second Russian edition of the book (Mashinostroenie Publishing House, Moscow, 1975), differs markedly from its first edition (Gosenergoizdat, Moscow, 1960), translated into English in 1966 (Handbook of Hydraulic Resistance, Israel Program for Scientific Translations, Jerusalem, 1966) and into French in 1969 (Memento des pertes de charge, Eyrolles Editeur, Paris, 1969). The second edition of the book has been substantially augmented by incorporating a considerable body of totally new data on hydraulic resistances obtained as a result of research work in recent years. By and large, as compared with the first, the second edition contains more than 40% new and revised data. When this edition was prepared, all of the misprints and errors discovered in the Russian edition were corrected, and some more precise definitions and changes were made. The book is based on the utilization, systematization, and classification of the results of a large number of studies carried out and published at different times in different countries. A large portion of the data was obtained by the author as a result of investigations carried out by him. It is quite clear that the methods of investigation, the models used, and, consequently, the accuracy of the results obtained and reported by various authors differ markedly in many cases. Such differences in the results could also be due to the fact that the majority of local hydraulic resistance coefficients are greatly influenced not only by the regime of flow but also by the prehistory of the flow, that is, conditions of supply to the section considered, nature of the velocity profiles, and degree of turbulence at the inlet and in some cases by the subsequent history of the flow as well; that is, flow removal from the test section. Many complex elements of pipelines exhibit great instability of flow due to periodic fluid separation from the walls, periodic changes of place and magnitude of separation, and eddy formation resulting in large oscillations of hydraulic resistance. The author was faced with an enormously difficult task: to discover and, where necessary, discard experimental results of questionable validity in that diverse body iii

of data compiled on the hydraulic resistance coefficients; to clear up cases where large variations in the resistance coefficients of the sections are regular and correspond to the essence of the hydrodynamic pattern and those cases where they are due to the experimental uncertainty; and to select the most reliable data and find a successful format for presenting the material so that it is accessible and understandable to nonspecialists in aerodynamics and hydraulics. It had to be taken into account that, in practice, the configurations of sections of various impedances in pipelines, their geometric parameters, the conditions of entry and exit of the flow, and its regimes are so diverse that it is not always possible to find the required reported experimental data necessary to calculate the hydraulic resistances. The author has therefore incorporated in this handbook not only results that have been thoroughly verified in laboratories but also those provided by less rigorous experimental investigations and those predicted or obtained by approximate calculations based on separate experimental studies. In some cases, tentative data are shown and are so noted in the text. We think this approach is justified because the facilities used under industrial conditions, and consequently the conditions of flow passages in them, can greatly differ among themselves and differ from laboratory conditions, under which the majority of hydraulic resistance coefficients have been obtained. In many complex elements of pipelines, these coefficients, as shown above, cannot be constant due to the nature of the phenomena occurring in them; thus, they can vary over wide ranges. The author hopes that the present edition will not only be useful for the further development of engineering science and technology in the English-speaking countries but will also aid in fostering friendly relations between the peoples of these countries and the Soviet people. I. E. Idelchik

iv

PREFACE TO THE 2nd RUSSIAN EDITION

There does not seem to be any branch of engineering that is not somehow involved with the necessity for moving liquids or gases through pipes, channels, or various types of apparatus. The degrees of complexity of hydraulic or fluid systems can therefore be widely different. In some cases these are systems that for the most part are composed of very long straight pipes, such as oil pipelines, gas lines, water conduits, steam pipes, and air ducts of ventilation plants in industrial use. In other cases they are pipelines that are relatively short but that abound in fittings and branches, various impedances in the form of valves, orifices, and adjusting devices, grids, tees, etc. as found in air ducts of complex ventilation systems; gas flues of metallurgical works, chemical and other factories, boiler furnaces, nuclear reactors, and dryers; fuel and oil pipes and various manifolds of aircraft and rockets. Most frequently the system through which a liquid or gas moves constitutes a large single unit (e.g., boilers, furnaces, heat exchangers, engines, air- and gas-cleaning equipment, and chemical, petrochemical, metallurgical, food, textile, and other manufacturing equipment). In all cases, it is essential that the fluid resistance of these systems be properly calculated. Furthermore, the adequate design of sophisticated present-day installations consisting of complex-shaped parts of hydraulic and fluid lines is impossible without insight into the principal physicomechanical processes occurring in them and consideration of suggestions for the improvement of flow conditions and reduction in the local fluid resistance of these elements. The requisite information is given in this handbook. A great body of new data on resistance coefficients accumulated since the first edition of this book has required an extensive revision of the text to account for the results of recent studies. But since it was not practically possible to incorporate all the newly published data on such flow resistance, this gap has been supplemented by an extensive listing of pertinent references. The handbook consists of 12 chapters. Each chapter, except for the first one, contains data on a definite group of fittings or other parts of pipelines and fluid netv

work elements having similar conditions of liquid or gas motion through them. The first chapter is a synopsis of general information on hydraulics of pressure systems and aerodynamics needed for design calculation of the elements of air-gas pipelines and hydraulic networks. All of the subsequent chapters contain: • An explanatory part giving, as a rule, a brief account of the subject matter of the section, an outline of the main physicochemical processes occurring in complex elements of pipelines, additional clarifying remarks and practical recommendations for the calculation and choice of separate network elements, and recommendations on ways to reduce their hydraulic resistance. • A computational part giving the coefficients or, in some instances, the absolute values of the fluid resistances of straight sections and of a wide range of complex-shaped parts of pipelines, fittings, various impedances, and other elements of the fluid networks. In each chapter the data are represented by special diagrams that contain a schematic of the element considered, calculation formulas, graphs, and tables of the numerical values of the resistance coefficients. It is essential for the present-day design analysis of hydraulic (fluid) networks with the use of electronic computers that the resistance coefficients be given in the form of convenient design tormulas. Moreover, it is often practical to represent in a concise form the functional dependence of the resistance coefficient on the main governing parameters. Graphical representation of this dependence is advantageous because, on the one hand, it furnishes a rather vivid illustration of the nature of this dependence and, on the other hand, it makes it possible to obtain intermediate values of the resistance coefficients not listed in tables. The resistance coefficients given in tabular form are the principal values, which can be conveniently used in calculations. The measurement units are given in the SI system. In selected cases, for convenience of usage, some quantities are also given in the meter-kilogram (force)-second system. I. E. Idelchik

vi

PREFACE TO THE 3rd EDITION

The 3rd edition of this Handbook is augmented with the most important results of investigations carried out in recent years. Some of the sections in the book have been refined and changed. The Handbook has been composed on the basis of processing, systematization, and classification of the results of a great number of investigations published at different times. The essential part of the book is the outcome of investigations carried out by the author. The results of investigations (the accuracy with which the models and fittings of pipelines were created, the accuracy of measurements, etc.) carried out by different specialists could differ among themselves. This might also be possible because the majority of local fluid resistances experience the influence of not only the mode of flow, but also the flow "prehistory" (the conditions of its supply to the given section, the velocity profile, and the degree of flow agitation at the inlet, etc.) and in some cases also the subsequent "history" of a flow (flow discharge from the section). All these conditions could be different in the studies undertaken by various authors. In many complex elements of pipeline systems, a great instability of flow is observed due to the periodicity of flow separation from the walls, periodic variation of the place, and magnitude of the zone of flow separation and eddy formation. This results in different values of hydraulic resistances. The author was faced with a difficult problem: when selecting most variegated information on hydraulic resistances, it was necessary to reveal and discard the questionable results of experiments to get a deeper understanding in which cases the great difference between the resistance coefficients of sections is regular, corresponding to the essence of the phenomena that occur during the motion of streams through them, and in which they are not regular; to select the most reliable data and find the most pertinent form of the presentation of information to make it accessible and understandable for engineers and technicians. The configuration of sections and obstacles in pipeline systems, their geometric parameters, conditions of supply and removal, and of the modes of flow are so diverse that one often fails to find out from literature the necessary experimental data vii

for the calculation of their hydraulic resistances. Therefore, the author incorporated not only the data thoroughly verified by laboratory investigations, but also those which were obtained theoretically or by approximate calculations based on separate experimental studies, and in some cases tentative data (specified in the text). This is permissible because the accuracy of fabrication and mounting of the systems of pipes and equipments in industrial conditions and, consequently, the conditions for the flow of streams may greatly differ between separate installations and differ from laboratory conditions at which the majority of fluid resistance coefficients were obtained, and also because of the fact that for many complex elements these coefficients cannot be constant quantities. The present edition of this Handbook should assist in increasing the quality and efficiency of the design and usage of industrial power engineering and other constructions and also of the devices and apparatus through which liquids and gases move.

viii

PREFACE TO THE 4th ENGLISH EDITION

Professor I. E. Idelchik’s Handbook of Hydraulic Resistance has become widely known: its 2nd and 3rd editions were translated into the English, French, Chinese, and Czech languages. Each subsequent edition was enriched with new information and data, as well as with new entries to the bibliography. The present, 4th, English Edition of the Handbook, like the previous one, was prepared after the author’s death, and appears only in its English version. We shall list here the most essential additions and changes that we thought worthwhile to make in some of the chapters of this book. In particular, Chapter 2 dealing with stabilized steady-state flow in channels and tubes was supplemented with the following experimental results: unsteady flows with a sharp change in the turbulent velocity as well as on a smooth change in time and its resulting effect on the hydraulic resistance. This chapter has a new section on the stabilized turbulent flow in plane and annular channels when the flow is induced by longitudinal motion of one of the walls (Couette flow) or when the flow is driven by longitudinal motion of one of the walls and longitudinal pressure gradient (Couette–Poiseuille flow). The computed data and their agreement with experimental results are given. Such flows are typical of the systems of container piping pneumatic- and hydrotransport in which the containers move under the action of forced air or water flow (passive containers) or where a train moves in a tunnel due to the presence of draft (the so-called active containers). The results of computational and experimental studies of the characteristics of a Couette forced turbulent flow in plane and annular channels (concentric and accentric) in the presence and absence of surface roughness are given. Examples of computed dependences needed to determine the velocity of motion of cylindrical passive and self-propelled containers in a tubing for given longitudinal pressure gradients, Reynolds numbers, length and their relative diameters and eccentricity are also furnished. Together with the equation of the balance of forces acting on a container of given length, these dependences can be used to determine the parameters of the container motion. ix

A description is given of the hydrodynamic paradox when the velocity of motion of a sufficiently long enough passive cylindrical container of neutral buoyancy in a turbulent water flow may exceed the maximum water flow velocity along the tube axis. Chapter 3 presents the result of experimental studies of an oblique flow past a frontal air intake with a system of flow controls providing a separationless flow in a channel up to inflow angles of 90o. The problem will be of interest to ground-level transport facilities and ships with frontal air intakes. Chapter 4 describes the technique used to reduce the total pressure losses in channels with an abrupt expansion by breaking down vortices with the aid of transverse partitions as well as by blowing a jet from a slit to create the so-called jet diffuser. In the latter case, the loss coefficient with allowance for pressure losses on injection is decreased 1.5 times. This effect is enhanced by using the Coanda effect in the course of creating a jet diffuser (the phenomenon of adherence of a plane jet to a convex plane surface) when a jet is blown from a curvilinear slit; the loss coefficient is decreased here by a factor of 2–2.5. Chapter 5 devoted to diffuser flows has been thoroughly revised in the present edition. This chapter presents the experimental results of plane and conical diffusers with different area ratios and divergence angles depending on Reynolds and Mach numbers at subsonic velocities and at different parameters of the initial flow nonuniformity and surface roughness. Examples of changes in the geometry of diffusers on replacing rectilinear by curvilinear walls to increase the efficiency of diffusers are given. The means of improving the characteristics of diffusers by installing different kinds of partitions and screens, finning the diffuser surface or installing generators of longitudinal vortices at the inlet to delay flow separation are also described. In contrast to the previous editions of the Handbook, the methods of calculating a turbulent flow in diffuser channels and determining the total pressure losses on the basis of the boundary-layer approximation are briefly reviewed here. Moreover, the use of these methods in solving direct and inverse problems in calculating the diffuser channels is considered. In solving the direct problem, the coefficient of total pressure loss in a diffuser of a given geometry at fixed Reynolds and Mach numbers, initial flow nonuniformity at the inlet to the diffuser, surface roughness up to the section where flow separation occurs are calculated. The solution of an inverse problem for the starting length of diffuser flow is aimed at determining the geometrical parameters of the diffuser at a fixed Reynolds number from the a priori specified velocity distribution along the channel axis or of the surface friction coefficient on its walls. Thus, for example, when specifying a virtually zero surface friction on the walls of a diffuser, the so-called preseparation flow develops in the latter. It appears that such diffusers possess a number of extreme properties. Calculations and experiments have shown that in such diffusers, at a given length, a marked decrease in the total pressure loss is ensured or, at a given area ratio, a substantial decrease in the diffuser length is possible. x

Additional information is also given on aerodynamic methods of controlling the flow characteristics in diffusers with the aid of slit suction or tangential injection — both enabling the increase in the efficiency of a diffuser with allowance for energy losses in such cases. Chapter 6 presents new data on the hydraulic resistance of pipe bends in the presence of cavitation in a stream of water and gas–liquid mixtures. Chapter 8 contains results of calculations and describes experiments aimed at creating the initial flow nonuniformity in a channel with the aid of screens of variable resistance across the flow and of an array of cylinders. It also suggests a technique of creating a high-turbulent flow with a section-uniform turbulence intensity with the aid of a two-row array of cylinders with opposite motion of the rows. Finally, Chapter 12 contains new data on heat transfer and hydraulic resistance in an in-line bank of tubes. It is shown that according to experimental results and of numerical simulation, the finning of their surface as well as indentation of staggered dimples on a smooth surface lead to a substantial enhancement of heat transfer that overtakes an increase in the hydraulic resistance. The chapter also contains data on the enhancement of heat transfer in round and annular tubes with the aid of different kinds of swirlers with continuous twisting along the flow as well as on the hydraulic losses and heat transfer of rotating channels (rotation of a tube around its own axis or around the axis which is perpendicular to that of the tube). These results are of interest in their application to heat transfer problems. By having prepared this edition for publication we are paying tribute to the memory of Professor I. E. Idelchik — the author of this Handbook with whom we had the pleasure of first working in the same laboratory and then remaining all the time in close contact when he took up work at another institute. One of us reviewed the 2nd Russian edition of this book (1975) as well as his monographs "Aerohydrodynamics of Engineering Apparatus" and "Some of the Interesting Effects and Paradoxes in Aerohydrodynamics and Hydraulics" (1982). A. S. Ginevskiy and A. V. Kolesnikov Central Aero-Hydrodynamics Institute (TsAGI), Moscow

xi

NOMENCLATURE

Symbol

Name of quantity

Abridged notation in SI units

a1

speed of sound

m/s

acr

critical speed of sound

m/s

a

speed of sound in frozen flow

m/s

a, b

sides of a rectangle

m

cp and cv

specific heats of gases at constant pressure and constant volume, respectively

J/kg oC

*

cx

coefficient of drag

–

D, d

cross-section diameters

m

Dh = 4F ⁄ Π; dh = 4f ⁄ Π

hydraulic or equivalent diameter (4 × hydraulic radius)

m

F, f _ f = For ⁄ Fgr

cross-sectional areas

m2

area ratio of a grid, orifice, perforated plate, etc.

–

G

mass flow rate of liquid (gas)

kg/s

g

gravitational acceleration

m/s2

h

height

m

k = cp/cv

specific heat ratio

–

l

length of flow segment, depth of channel, or thickness of orifice

m

Ma = w/a1

Mach number

–

M = 1/F ∫ (w ⁄ w0)2dF

coefficient of momentum (Boussinesq coefficient)

–

m0

wetting intensity

m3/m3

m

exponent

–

coefficient of kinematic energy (Coriolis coefficient)

–

Nm

power

W

n

polytropic exponent

–

nar

area ratio (degree of enlargement or reduction of cross section); polytropic exponent; number of elements

–

F

N = 1/F ∫ (w ⁄ w0)3dF F

xiii

Symbol

Name of quantity

Abridged notation in SI units

nel

number of elements

p∗

–

static pressure

Pa

pf

total pressure of flow stagnation pressure

Pa

pex

excess pressure

Pa

∆p

overall pressure difference

Pa

Pdr

drag force

N

Q

volumetric flow rate

m3/s

R

gas constant

J/kg K

Rh

1 hydraulic radius ( Dh) 4

m

R0, r

radii of cross sections of a circular pipe or curved pipe length

m

Re = wDh ⁄ v

Reynolds number

–

S, s

spacing (distance between rods in a bundle of pipes, between grid holes, etc.)

m

Sfr

length of a free jet

m

S0

surface area

m2

Sm

frontal area of a body in a flow

m2

T(t)

thermodynamic temperature

K (oC)

T*

thermodynamic flow stagnation temperature

K

vsp

specific volume

3

v

side discharge (inflow) velocity

m/s

w

stream velocity

m/s

w′

longitudinally fluctuating stream velocity

m/s

z

dust content

g/m3

zd

dust capacity

kg/m2

α

central angle of divergence or convergence; angle of a wye or tee branching; angle of stream incidence

deg

δ

angle of turning (of a branch, elbow); angle of valve opening

δt

thickness of a wall, boundary layer, or wall layer

m

δj

height of joint

m

∆

equivalent uniform roughness of walls

m

∆0 __ __ ∆0 = ∆0 ⁄ Dh; ∆ = ∆ ⁄ Dh

mean height of wall roughness protuberances (absolute roughness) relative roughness of walls

–

ε = Fcon ⁄ F0

coefficient of jet contraction

–

ε′

porosity (void fraction)

–

__ w ′2 ⁄ w0 εt = √ ⎯⎯⎯

degree of turbulence

–

ζ * ∆p ⁄ (ρw2 ⁄ 2)

coefficient of fluid resistance (pressure loss coefficient)

–

xiv

m /kg; m/s

Symbol

Name of quantity

Abridged notation in SI units

ζloc

coefficient of local fluid resistance

ζfr

coefficient of friction resistance of the segment of length l

η

dynamic viscosity

Pa s

ηΠ

cleaning coefficient

–

λ = ζfr ⁄ (l ⁄ Dh)

friction coefficient [friction resistance of the segment of relative unit length (l ⁄ Dh = 1)]

–

λc = w ⁄ acr

relative (reduced) stream velocity

–

µ

discharge coefficient

–

µcon

mass concentration of suspended particles in flow

–

ν

kinematic viscosity

m2/s

ρ

density of liquid (gas)

kg/m3

ρ∗

density of frozen gas flow

kg/m3

ρcr

density of gas at critical velocity

kg/m3

Π

cross-sectional (wetted) perimeter

m

ϕ

velocity coefficient

–

–

SUBSCRIPTS Subscripts listed for the quantities F, f, D, d, Π, a, b, w, ρ, Q, and p refer to the following cross sections or pipe segments: 0

governing cross section or minimum area

1

larger cross section in the case of expansion or contraction of the flow segment

2

larger cross section after equalization of the stream velocity

k

intermediate cross section of curved channel (elbow, branch) or the working chamber of the apparatus

con

contracted jet section at the discharge from an orifice (nozzle)

or

orifice or a single hole in the perforated plate or screen

gr

front of the perforated plate, screen, orifice

br, st, ch

side branch, straight passage, and common channel of a wye or tee, respectively

out

outlet

∞

velocity at infinity

Subscripts 0, 1, 2, k, and d at l refer, respectively, to the inlet, straight outlet, intermediate (for a curved channel), and diffuser pipe lenghths. Subscripts at ∆p and ζ refer to the following forms of the fluid resistances: loc

local

fr

friction

xv

ov

overall

d

total resistance of a diffuser in the network

out

total resistance of a diffuser or a branch at the outlet from the network

int

internal resistance of a diffuser

exp

resistance to flow expansion in a diffuser

sh

shock resistance at sudden enlargement of the cross section

br and st

resistance of a branch and straight passage of a wye or tee (for the resistance coefficients reduced to the velocity in respective branch pipes)

r.br., r.st.

resistance coefficients of the side branch and of the straight passage of a wye or tee reduced to the velocity in a common channel of a wye or tee

xvi

USEFUL CONVERSIONS OF UNITS

Physical quantity

Given in ⎯⎯ > Gives

Length

⎯

Multiplied by ⎯⎯ >

Gives

Gives ⎯

Gives

Gives ⎯ ρa; (b) ρg < ρa.

gz0 +

p0 p1 = gz1 + , ρ ρ

(1.8)

where z0 and z1 are the coordinates of two fluid particles in the given volume relative to the reference plane (the corresponding geometric heights, Figure 1.8), in m, and p0 and p1 are the static pressures (absolute) at the levels of the particles chosen, in Pa. 3. The pressure at any arbitrary point in the liquid or gas volume can be determined if the pressure at some other point of the same volume is known, as well as the difference between the depth of immersion of one point relative to the other, h = z1 – z0 (see Figure 1.8): p1 = p0 – gρ (z1 – z0) = p0 − gρh ⎫ . p0 = p1 + gρ (z1 – z0) = p1 + gρh⎬ ⎭

(1.9)

It is for this reason that, for example, the pressure exerted on the walls of a vessel filled with a stationary hot gas (ρg < ρa) at a level h = zg – za, located above the surface that separates the gas and air (Figure 1.9), is lower, on both the gas side (pg) as well as the air side (ph), than the pressure pa at the surface of separation:

Figure 1.9. Determination of the pressure of a hot gas in a vessel at an arbitrary height in excess of atmospheric pressure at the same level.

Aerodynamics and Hydraulics of Pressure Systems

pg = pa – gρgh

23

(1.10)

and

ph = pa – gρah ,

(1.11)

where ρg and ρa are the densities of the gas and the air (averaged over the height h), in kg/m3. 4. The pressure of a stationary hot gas in a vessel at a level h = zg – za above that of the atmospheric pressure of air at the same level h can be obtained by virtue of Equations (1.10) and (1.11) from:

pg – ph = gh (ρa – ρg) .

1.5 EQUATIONS OF FLUID MOTION Flow Rate Equation and Mean Flow Velocity 1. The flow rate of a fluid (liquid or gas) is defined as the mass or volume of fluid passing a given cross section of the pipe (channel) per time unit. Two types of flow rates are distinguished: the mass flow rate (for example, G kg/s) and the volumetric flow rate (for example, Q m3/s). 2. In a generalized form, at any transverse distribution of the flow velocities, the volumetric flow rate is expressed by the formula

Q = ∫ dQ = ∫ wdF , F

(1.12)

F

where w is the flow velocity at the given point across the pipe (channel) in m/s. The mass flow rate is

G = ρQ = ∫ ρwdF .

(1.13)

F

3. The transverse distribution of the velocities in the pipe is hardly ever uniform. To simplify the solution of practical problems use is made of a fictitious mean velocity of the flow:

∫ w dF wm =

F

F

=

Q F

(1.14)

from which

Q = wmF .

(1.15)

24

Handbook of Hydraulic Resistance, 4th Edition

4. The volumetric flow rate and, consequently, the velocity of the gas flow depend on the fluid temperature, pressure, and humidity.* If under normal conditions (0oC, 101.325 kPa, dry gas) the volumetric flow rate of the gas is Qnc, m3/s, and the mean velocity is wnc, m/s, then under the operating conditions

Qop = Qnc

T pnc ⎛ m ⎞ 1+ 273 pop ⎜⎝ 0.804⎟⎠

(1.16)

and, correspondingly, T pnc ⎛ m ⎞ , 1+ 273 pop ⎜⎝ 0.804⎟⎠

wop = wnc

(1.17)

where m is the content of water vapor in the gas, in kg/m3; pop is the operating gas pressure at the given location F, in Pa, and pnc is the pressure of the gas under normal conditions, in Pa. In the case of a dry gas at a pressure of 101.325 kPa (p = pnc), the volumetric flow rate and, accordingly, the gas flow velocity under operating conditions are determined from the relations:

Qop = Qnc

T T , wop = wnc . 273 273

The density of the gas at operating conditions is

ρop = (ρnc +m)

pop 273 1 , T 1 + m ⁄ 0.804 pnc

(1.18)

where ρnc is the density of the dry gas at normal conditions, in kg/m3. For a dry gas at a pressure of 101.325 kPa ρop = ρnc

273 . T

Equation of Continuity of a Stream 1. The continuity equation is the result of application of the law of the conservation of mass to a moving medium (liquid, gas). In the general case, the continuity equation can be written for any distribution of velocities in two pipe sections 0-0 and 1-1 (Figure 1.10) as

∫ ρ0wdF = ∫ ρ1wdF , F0 ∗

(1.19)

F1

A perfect gas is considered, which obeys the equation pυ = RT and for which the internal energy is a function of the temperature alone.

Aerodynamics and Hydraulics of Pressure Systems

25

Figure 1.10. Scheme of the flow and its basic parameters for two sections of a channel.

where subscripts 0 and 1 refer to the appropriate cross section. In the case of an incompressible homogeneous medium the density across the pipe is always constant, and therefore ρ0 ∫ wdF = ρ1 ∫ wdF . F0

F1

2. On the basis of Equations (1.13)–(1.15), the equation of continuity (the flow rate equation) for a uniform compressible flow and for any incompressible flow can be written as: ρ0w0F0 = ρ1w1F1 = ρwF = G⎫ . ρ0Q0 = ρ1Q1 = ρQ = G⎬⎭

(1.20)

where w0 and w1 are the mean velocities at sections 0-0 and 1-1, respectively, in m/s. If the density of the moving medium does not vary along the flow, that is, ρ0 = ρ1 = ρ, the equation of continuity (flow rate) is of the form

w0F0 = w1F1 = wF or Q0 = Q1 = Q =

G . ρ

Energy Equation (Bernoulli Equation) for Compressible and Incompressible Fluids 1. According to the law of conservation of energy for the medium moving through a pipe (channel), the energy of the liquid (gas) flow passing through section 0-0 per time unit (see Figure 1.10) is equal to the sum of energies of the liquid (gas) flow passing through section 1-1 per time unit plus the internal (thermal) and mechanical energies dissipated along the segment between these sections.

26

Handbook of Hydraulic Resistance, 4th Edition

2. In the general case of an inelastic (liquid) and elastic (gas) flow with nonuniform transverse velocity and pressure distribution,* the corresponding energy equation will have the form ⎛

∫ ⎜⎝p +

F0

⎞ ρw2 + gρz + ρU⎟ wdF 2 ⎠

⎛ ⎞ ρw2 = ∫ ⎜p + + gρz + ρU⎟ wdF + ∆Ntot , 2 ⎝ ⎠ F

(1.21)

1

where z is the geometric height of the centroid of the corresponding section, in m; p is the static pressure (absolute) at the point of the corresponding section, in Pa, U is the internal specific heat energy of the gas flow (which a frictionless flow would have had), in J/kg; and ∆Ntot is the total power lost over the segment between sections 0-0 and 1-1, which characterizes the value of the mechanical energy dissipated into heat, in W. 3. By relating the energy of the flow to the mass flow rate (G =

∫ ρwdF), on the basis of F

Equation (1.21) we obtain ∆etot =

=–

⎞ ∆Ntot 1 ⎛ p w2 = ∫⎜ + + gz + U⎟ ρwdF G G ⎝ρ 2 ⎠ F0 ⎞ 1 ⎛ p w2 + + gz + U⎟ ρwdF = e0 – e1 , ⎜ ∫ G ⎝ρ 2 ⎠ F1

(1.22)

where e0 =

⎞ 1 ⎛ p w2 + + gz + U⎟ ρwdF G ∫ ⎜⎝ ρ 2 ⎠ F0

and e1 =

⎞ 1 ⎛ p w2 + + gz + U⎟ ρwdF ⎜ ∫ G ⎝ρ 2 ⎠ F1

are the specific energies averaged over the mass flow rate through sections 0-0 and 1-1, respectively, in J/kg; ∆etot = ∆Ntot/G is the total loss of the specific energy over the segment between sections 0-0 and 1-1, in J/kg. Having divided Equation (1.22) by g, we obtain

∗

Assuming no heat transfer and shaft work over the given segment.

Aerodynamics and Hydraulics of Pressure Systems

∆Htot = =

27

∆etot g

U⎞ U⎞ 1 ⎛ p w2 1 ⎛ p w2 + z + ⎟ ρwdF – ∫ ⎜ + + + z + ⎟ ρwdF ⎜ ∫ g⎠ g⎠ G ⎝ gρ 2g G ⎝ gρ 2g F0 F1

= H0 – H1 , where H0 =

U⎞ 1 ⎛ p w2 + + z + ⎟ ρwdF g⎠ G ∫ ⎜⎝ gρ 2g F0

and H1 =

U⎞ 1 ⎛ p w2 + + z + ⎟ ρwdF ⎜ ∫ g⎠ G ⎝ gρ 2g F1

are the pressure heads averaged over the mass flow rate in sections 0-0 and 1-1, respectively, in m. 4. By relating the energy of the flow to the volumetric flow rate in a certain section, for instance, 0-0 (Q0 =

∫ wdF), then F0

∆ptot *

⎞ ⎞ ρw2 ρw2 ∆Ntot 1 ⎛ 1 ⎛ dF p + = + + + gρz + ρU⎟ wdF + ρU gρz – w p ⎜ ⎜ ⎟ ∫ ∫ Q0 2 2 Q0 ⎝ Q0 ⎝ ⎠ ⎠ F1 F0

∆ptot =

⎞ ⎞ ρw2 ρw2 Q1 1 ⎛ 1 ⎛ + gρz + ρU⎟ wdF – ⋅ ∫ ⎜p + + gρz + ρU⎟ wdF . ⎜p + ∫ 2 2 Q Q Q0 ⎝ 0 1 ⎝ ⎠ ⎠ F0 F1

or

However, Q1 ρ 0 = Q0 ρ 1 because

ρ0Q0 = ρ1Q1 = G , therefore it is possible to write

28

Handbook of Hydraulic Resistance, 4th Edition

∆ptot *

⎞ ∆Ntot 1 ⎛ ρw2 + gρz + ρU⎟ wdF = ∫ ⎜p + 2 Q0 G ⎝ ⎠ F 0

–

⎞ ρw2 ρ0 1 ⎛ + gρz + ρU⎟ ρwdF = p∗0 – p∗1 , ⎜p + ∫ 2 ρ1 G ⎝ ⎠

(1.23)

F1

where p∗0 =

⎞ ρw2 1 ⎛ p + + gρz + ρU⎟ wdF ⎜ ∫ G ⎝ 2 ⎠ F0

is the total pr essur e aver aged over the mass flow r ate in section 0-0* p∗1 =

⎞ ρ0 1 ⎛ ρw2 ⋅ ∫ ⎜p + + gρz + ρU⎟ ρwdF G 2 ρ1 ⎝ ⎠ F1

is the total pressure averaged over the mass flow rate in section 1-1 and reduced to the volumetric flow rate in section 0-0, that is, to Q0; ∆ptot = ∆Ntot/Q0 are the total losses of the total pressure over the segment between sections 0-0 and 1-1 reduced to the volumetric flow rate Q0. 5. In most practical cases, the static pressure p in straight-line flow is constant across the flow, even when the velocity distribution is greatly nonuniform. The variation of the gas density over the cross section due to varying velocities can then be neglected (with Ma = w/a1 < 1.0). Therefore, in place of Equation (1.21) we can write (p0 + gρ0z0 + ρ0U) w0F0 + ∫

ρw2 dF 2

F0

= (p1 + gρ1z1 + ρ1U) w1F1 + ∫

ρw2 dF + ∆Ntot 2

F1

∗

In the case of segments with nonuniform flow distribution over the cross section (when the stagnation temperature remains constant along the flow and when the energy losses are calculated from the total pressure measured at different points of the cross section) the total pressure logarithms should be averaged rather than the total pressure itself: ln p∗m =

1 G

∫ ln p∗dG . G

However, for moderate nonuniformity of flow and at Ma < 1, the deviation from this rule does not cause appreciable error.41

Aerodynamics and Hydraulics of Pressure Systems

29

or, solving in ∆Ntot and taking into account Equation (1.20), we obtain ⎛ ⎞ ρ0w20 + gρ0z0 + ρ0U0⎟ Q0 ∆Ntot = ⎜p0 + N0 2 ⎝ ⎠ 2 ⎞ ⎛ ρ1w1 + gρ1z1 + ρ1U1⎟ Q1 , – ⎜p1 + N1 2 ⎠ ⎝

(1.24)

where 3

N0 =

1 ⎛w⎞ dF F0 ∫ ⎜⎝ w0 ⎟⎠ F0

and 3

1 ⎛w⎞ dF N1 = F1 ∫ ⎜⎝ w1 ⎟⎠ F1

are the kinetic energy coefficients (the Coriolis coefficients) for sections 0-0 and 1-1, respectively; they characterize the degree of the nonuniformity of kinetic energy and velocity distributions over these sections. 6. By relating the energy of the flow to the mass flow rate, we obtain the generalized Bernoulli equation, which is written for a real fluid accounting for the specific loss of energy (internal and external, that is, mechanical) over the segment considered, p0 + N0 ρ0

p1 w21 w20 + gz0 + U0 = + N1 + gz1 + U1 + ∆etot 2 2 ρ1

(1.25)

and correspondingly w20 U0 p1 p0 w21 U1 + N0 = + ∆Htot + N1 + z1 + + z0 + 2g 2g g gρ1 g gρ0

(1.26)

or ⎞ ⎛ p1 ⎞ w20 ∆Ntot ⎛ p0 w21 = ⎜ + N0 + gz0 + U0⎟ – ⎜ + N1 + gz1 + U1⎟ 2 G 2 ⎝ ρ0 ⎠ ⎝ ρ1 ⎠ = e0 – e1

∆etot *

and ∆Htot *

∆Ntot ⎛ p0 U1 ⎞ w20 U0 ⎞ ⎛ p1 w21 =⎜ + z0 + ⎟ – ⎜ + N1 + z1 + ⎟ + N0 g 2g 2 gG g ⎠ ⎝ gρ1 g⎠ ⎝ gρ0

= H0 – H 1 .

(1.27)

30

Handbook of Hydraulic Resistance, 4th Edition

7. By relating the energy of the flow to the volumetric flow rate (for example, to Q0) we obtain the Bernoulli generalized equation in the form

p0 + N0

ρ0w20 ρ1w21 ∆Ntot + gρ1z1 + ρ1U1 + + gρ0z0 + ρ0U0 = p1 + N1 2 2 Q0

⎛ ⎞ ρ0 ρ1w21 = ⎜p1 + N1 + gρ1z1 + ρ1U1⎟ + ∆ptot 2 ⎝ ⎠ ρ1

(1.28)

or ∆ptot *

⎞ ⎛ ⎞ ρ0 ∆Ntot ⎛ ρ0w20 ρ1w21 + gρ1z1 + ρ1U1⎟ + gρ0z0 + ρ0U0⎟ – ⎜p1 + N1 = ⎜p0 + N0 2 2 Q0 ⎝ ⎠ ⎝ ⎠ ρ1

= p′0∗ – p′1∗ ,

(1.29)

where p′0∗ = p0 + N0

ρ0w20 + gρ0z0 + ρ0U0 2

is the total pressure in section 0-0; p′1∗ = p1 + N1

ρ1w21 + gρ1z1 + ρ1U1 2

is the total pressure in section 1-1 reduced to the volumetric flow rate in section 0-0. All the terms of Equation (1.28) are given in pressure units, that is, in Pa and are gρ0z0, gρ1z1, elevation pressure; p0, p1 static pressure; N0(ρ0w20 ⁄ 2), N1(ρ1w21 ⁄ 2), dynamic pressure; ∆ptot * ∆Ntot ⁄ Q0, total losses of the total pressure (total hydraulic resistance) resulting from the overcoming of the hydraulic resistance of the segment between sections 1-1 and 2-2. 8. A change in the internal energy, U0 – U1, depends on the thermodynamic process that the gas undergoes on its way from sections 0-0 and 1-1. In the case of a polytropic process, the gas parameters change according to p0 ρn0

p1 p = n= n , ρ1 ρ

(1.30)

where n is the polytropic exponent, which in many cases can be considered to be approximately constant for the short local resistance segment in view of the limitation of the section and to be lying within the limits 1 < n < k (k = cp/cv is the isentropic exponent, cf. Table 1.4). 9. Based on the laws of thermodynamics,68 with no heat addition from the outside

Aerodynamics and Hydraulics of Pressure Systems p0

p0

p1

p1

31

p0 p1 dp – –∫ U1 – U0 = ∫ pdv = ρ0 ρ1 ρ =

p0 p1 n – – ρ0 ρ1 n – 1

⎛ p0 p1 ⎞ ⎜ρ – ρ ⎟ , 1⎠ ⎝ 0

(1.31)

where v = 1/ρ is the specific volume of the gas, in m3/kg. On the basis of Equations (1.27), (1.30), and (1.31) we obtain ∆etot = g(z0 – z1) + N0

w20 w21 n ⎛ p0 p1 ⎞ + – – N1 2 n – 1 ⎜ ρ0 ρ1 ⎟ 2 ⎝ ⎠

∆etot = g(z0 – z1) + N0

w20 w21 n p0 ⎡⎛ p1 ⎞ + – N1 2 n – 1 ρ0 ⎢⎜ ρ0 ⎟ 2 ⎣⎝ ⎠

or (n–1 ⁄ n)

⎤ – 1⎥ . ⎦

(1.32)

10. In a number of cases, when performing approximate calculations, the process can be considered isentropic. For this process the polytropic exponent n in Equations (1.31) and (1.32) will be replaced by the isentropic exponent k. 11. In some cases the state of the flow is changed, following an isotherm (constant temperature). There the pressure is proportional to the gas density p0 p1 p = = ρ0 ρ0 ρ

p0

∫ p1

(1.33)

dp p0 p0 = ln . ρ ρ0 p1

(1.34)

Then, on the basis of Equations (1.27) and (1.34), we obtain in the final form ∆etot = g(z0 – z1) + N0

w21 p0 p0 w20 – ln . – N1 2 ρ0 p1 2

(1.35)

12. Gubarev20 in his experiments demonstrated that in parts of the system such as fittings and converging wyes, the state of the gas follows a polytropic relation that is similar to an isotherm. Then, the polytropic exponent for air passing through a converging wye becomes n ≈ 1.0 and for air passing through impedances, n ≈ 1.15. 13. Formulas (1.32) and (1.35) can be used not only in the case of high gas flow velocities, but also in the case of low velocities when they are accompanied by large pressure drops over the segments of local resistance. 14. The basic similarity groups of gas flows are the Mach number or the reduced velocity λc * w/acr.

32

Handbook of Hydraulic Resistance, 4th Edition The Mach number is Ma *

(1.36)

w , a1

where a1 is the speed of sound; a1 =

⎯√⎯k ρ = √⎯⎯⎯kRT⎯ . p

(1.37)

For air ⎯⎯T . a1 + 20.1 √ 15. The flow velocity equal to the local speed of sound and called the critical velocity is acr =

∗

2k p 2k = ⎯⎯⎯⎯ RT∗ ⎯⎯⎯⎯ √ k + 1 ρ∗ √ k + 1⎯

,

(1.38)

where p* is the pressure of the stagnated gas flow (total pressure); ρ* is the density of the stagnated gas flow; T* is the stagnated gas flow temperature (stagnation temperature). The speed of sound in a stagnated medium is a∗ =

∗

kRT ⎯∗ ⎯⎯k ρp∗ = √⎯⎯⎯ √

(1.39)

so that acr = a∗ =

⎯⎯k⎯+2 1 √

.

For air a∗ + 20.1 √ ⎯⎯T∗ , acr + 18.3 √ ⎯⎯T∗ .

(1.40)

The reduced velocity is

λc *

w =w acr

∗

2k p 2k =w%√ RT∗ . ⎯⎯⎯⎯ ⎯⎯⎯⎯ %√ k + 1 ρ∗ k+1⎯

(1.41)

Aerodynamics and Hydraulics of Pressure Systems

33

16. Should an ideal gas jet with velocity w0 = w and having no energy losses (∆etot = 0) and no effect of heat be retarded isentropically (at n = k; p0 = p; ρ0 = ρ; z0 = z1 = 0; N0 = N1 = 1; ap1 = p* is the total or stagnation pressure) up to velocity w2 = 0, then Equation (1.32) will take the form (k–1) ⁄ k

w2 k p ⎡⎛ p∗ ⎞ = ⎢⎜ ⎟ 2 k – 1 ρ ⎣⎝ p ⎠

⎤ – 1⎥ , ⎦

whence k – 1 w2 ⎞ p∗ ⎛ = ⎜1 + p ⎝ k kp ⁄ ρ ⎟⎠

k ⁄ (k–1)

or, taking into account Equations (1.36) and (1.37), k ⁄ (k–1)

k–1 p∗ ⎛ = 1+ Ma2⎞⎟ p ⎜⎝ k ⎠

(1.42)

.

17. There is the following relationship between the numbers Ma and λc: Ma =

⎯⎯k⎯+2 1 √

λc k–1 λc 1– ⎯√⎯⎯⎯⎯ k + 1⎯

(1.43)

or k+1

Ma √ ⎯⎯⎯⎯ 2 ⎯ λc = k–1 1+ Ma2 ⎯⎯⎯⎯⎯⎯ √ 2 ⎯

.

On the basis of Equations (1.42) and (1.43) the following equation is obtained:

π(λc) *

k – 1 2⎞ = ⎛1 – λc ∗ ⎜ k + 1 ⎟⎠ p ⎝ p

k ⁄ (k–1)

.

(1.44)

Taking into account the relation analogous to relation (1.29), i.e., k

∗ p∗ ⎛ ρ ⎞ =⎜ ⎟ , p ⎝ρ⎠

the density of a perfectly stagnated gas will be given by

(1.45)

34

Handbook of Hydraulic Resistance, 4th Edition

Table 1.10 Gasdynamic functions for a subsonic flow and the function χ(λν) at k = 1.4 λν

τ

π

ε

V

y

Ma

χ

0.01

0.99998

0.99994

0.9996

0.01577

0.01577

0.00913

8563.5

0.02

0.99993

0.99977

0.99983

0.03154

0.03155

0.01836

2136.14

0.03

0.99985

0.99948

0.99963

0.04731

0.04733

0.02739

946.367

0.04

0.99973

0.99907

0.99933

0.06306

0.06311

0.03652

530.195

0.05

0.99958

0.99854

0.99896

0.07879

0.07890

0.04565

337.720

0.06

0.99940

0.99790

0.99850

0.09450

0.09470

0.05479

2.33.271

0.07

0.99918

0.99714

0.99796

0.11020

0.11051

0.06393

170.368

0.08

0.99893

0.99627

0.99734

0.12586

0.12633

0.07307

129.599

0.09

0.99865

0.99528

0.99663

0.14149

0.14216

0.08221

101.692

0.10

0.99833

0.99418

0.99584

0.150709

0.15801

0.09136

81.7669

0.11

0.99798

0.99296

0.99497

0.17265

0.17387

0.10052

67.0543

0.12

0.99760

0.99163

0.99401

0.18816

0.18975

0.10968

55.8890

0.13

0.99718

0.99018

0.99297

0.20363

0.20565

0.11884

47.2209

0.14

0.99673

0.98861

0.99185

0.21904

0.22157

0.12801

40.3612

0.15

0.99625

0.98694

0.99065

0.23440

0.23751

0.13719

34.8430

0.16

0.99573

0.98515

0.98937

0.24971

0.25347

0.14637

30.3405

0.17

0.99518

0.98324

0.98300

0.26495

0.26946

0.15556

26.6212

0.18

0.99460

0.98123

0.98655

0.28012

0.28548

0.16476

23.5153

0.19

0.99398

0.97910

0.98503

0.29523

0.30153

0.17397

20.8966

0.20

0.99333

0.97686

0.98342

0.31026

0.31761

0.18319

18.6695

0.21

0.99265

0.97451

0.98173

0.32521

0.33372

0.19241

16.7609

0.22

0.99193

0.97205

0.97996

0.34008

0.34986

0.20165

15.1139

0.23

0.99118

0.96948

0.97810

0.35487

0.36604

0.21089

13.6836

0.24

0.99040

0.99680

0.97617

0.36957

0.38226

0.22015

12.4345

0.25

0.98598

0.96401

0.97416

0.38417

0.39851

0.22942

11.3378

0.26

0.98873

0.96112

0.97207

0.39868

0.41481

0.23869

10.3704

0.27

0.98785

0.95812

0.96990

0.41309

0.43115

0.24799

9.51321

0.28

0.98693

0.95501

0.96765

0.42740

0.44753

0.25729

8.75071

0.29

0.98598

0.95280

0.96533

0.44160

0.46396

0.26661

8.06987

0.30

0.98500

0.94848

0.96292

0.45569

0.48044

0.27594

7.45985

0.31

0.98398

0.94506

0.96044

0.46966

0.49697

0.28528

6.91153

0.32

0.98293

0.94153

0.95788

0.48352

0.51355

0.29464

6.41722

0.33

0.98185

0.93790

0.95524

0.49726

0.53018

0.30402

5.97035

0.34

0.97958

0.93418

0.95253

0.51087

0.54687

0.31341

5.56534

0.35

0.97958

0.93035

0.94974

0.52435

0.56361

0.32282

5.19738

0.36

0.97840

0.92642

0.94687

0.53771

0.58042

0.33224

4.86253

0.37

0.97718

0.92239

0.94393

0.55093

0.59728

0.34168

4.55665

0.38

0.97593

0.91827

0.94091

0.56401

0.61421

0.35114

4.27717

0.39

0.97465

0.91405

0.93782

0.57695

0.63120

0.36062

4.02120

0.40

0.97333

0.90974

0.93466

0.58975

0.64826

0.37012

3.78635

0.41

0.97198

0.90533

0.93142

0.60240

0.66539

0.37963

3.57055

0.42

0.97060

0.90083

0.92811

0.61490

0.68259

0.39917

3.37194

0.43

0.96918

0.89623

0.92473

0.62724

0.69987

0.39873

3.18890

Aerodynamics and Hydraulics of Pressure Systems

35

Table 1.10 (continued) λν

τ

π

ε

V

y

Ma

χ

0.44

0.96773

0.89155

0.92127

0.63943

0.71722

0.40830

3.01999

0.45

0.96625

0.88677

0.91775

0.65146

0.73464

0.41790

2.86393

0.46

0.96473

0.88191

0.91415

0.66333

0.75215

0.42753

2.71957

0.47

0.96318

0.87696

0.91048

0.67503

0.76974

0.43717

2.58590

0.48

0.96160

0.87193

0.90675

0.68656

0.78741

0.44684

2.46200

0.49

0.95998

0.86681

0.90294

0.69792

0.80517

0.45653

2.34705

0.50

0.95833

0.86160

0.89907

0.70911

0.82301

0.46625

2.24032

0.51

0.95665

0.85632

0.89512

0.72012

0.84095

0.47600

2.14113

0.52

0.95493

0.85095

0.89111

0.73095

0.85898

0.48576

2.04889

0.53

0.95318

0.84551

0.88704

0.74160

0.87711

0.49556

1.96305

0.54

0.95140

0.83998

0.88289

0.75206

0.89533

0.50538

1.88313

0.55

0.94958

0.83438

0.87868

0.76234

0.91366

0.51524

1.80866

0.56

0.94773

0.82871

0.87441

0.77243

0.93208

0.52511

1.73926

0.57

0.94583

0.82296

0.87007

0.78232

0.95062

0.53502

1.67454

0.58

0.94393

0.81714

0.86567

0.79202

0.96926

0.54496

1.61417

0.59

0.94198

0.81124

0.86121

0.80152

0.98801

0.55493

1.55783

0.60

0.94000

0.80528

0.85668

0.81082

1.00688

0.56493

1.51525

0.61

0.93798

0.79925

0.85209

0.81992

1.02586

0.57497

1.45676

0.62

0.93593

0.79315

0.84745

0.82881

1.04496

0.58503

1.41033

0.63

0.93385

0.78699

0.84274

0.83750

1.06418

0.59513

1.36753

0.64

0.93173

0.78077

0.83797

0.84598

1.08353

0.60526

1.32757

0.65

0.92958

0.77448

0.83315

0.85425

1.10301

0.61543

1.29025

0.66

0.92740

0.76813

0.82826

0.86231

1.12261

0.62563

1.25541

0.67

0.92518

0.76172

0.82332

0.87016

1.14235

0.63537

1.22289

0.68

0.92293

0.75526

0.81833

0.87778

1.16223

0.64615

1.19254

0.69

0.92065

0.74874

0.81327

0.88519

1.18225

0.65646

1.16423

0.70

0.91833

0.74217

0.80817

0.89238

1.20241

0.66682

1.13783

0.71

0.91598

0.73554

0.80301

0.89935

1.22271

0.67721

1.11321

0.72

0.91360

0.72886

0.79779

0.90610

1.24317

0.68764

1.09029

0.73

0.91113

0.72214

0.79253

0.91262

1.26378

0.69812

1.06894

0.74

0.90873

0.71536

0.78721

0.91892

1.28454

0.70864

1.04909

0.75

0.90625

0.70855

0.78184

0.92498

1.30574

0.71919

1.03064

0.76

0.90373

0.70168

0.77643

0.93082

1.32656

0.72980

1.01351

0.77

0.90118

0.69478

0.77096

0.93643

1.34782

0.74045

0.99762

0.78

0.89860

0.68783

0.76545

0.94181

1.36925

0.75114

0.98291

0.79

0.89598

0.68085

0.75989

0.94696

1.39085

0.76188

0.96931

0.80

0.89333

0.67383

0.75428

0.95187

1.41263

0.77267

0.95675

0.81

0.89065

0.66677

0.74863

0.95655

1.49460

0.78350

0.94518

0.82

0.88793

0.65968

0.74294

0.96099

1.45676

0.79439

0.93455

0.83

0.88518

0.65255

0.73720

0.96519

1.47910

0.80532

0.92479

0.84

0.88240

0.64540

0.73141

0.96916

1.50164

0.81631

0.91588

0.85

0.87958

0.63822

0.72559

0.97289

1.52439

0.82735

0.90775

0.86

0.87673

0.63101

0.71973

0.97638

1.54733

0.83844

0.90037

36

Handbook of Hydraulic Resistance, 4th Edition

Table 1.10 (continued) λν

τ

π

ε

V

y

Ma

χ

0.87

0.87385

0.62378

0.71383

0.97964

1.57049

0.84959

0.89370

0.88

0.87093

0.61652

0.70788

0.98265

1.59386

0.86079

0.88770

0.89

0.86798

0.60924

0.70191

0.98542

1.61745

0.87205

0.88234

0.90

0.86500

0.60194

0.69589

0.98795

1.64127

0.88337

0.87758

0.91

0.86198

0.59463

0.68984

0.99024

1.66531

0.89475

0.87339

0.92

0.85893

0.58730

0.68375

0.99229

1.88959

0.90619

0.86957

0.93

0.85585

0.57995

0.67763

0.99410

1.71411

0.91768

0.86662

0.94

0.85273

0.57259

0.67148

0.99567

1.73887

0.92925

0.86398

0.95

0.84958

0.56522

0.66530

0.99699

1.76389

0.94087

0.86381

0.96

0.84640

0.55785

0.65908

0.99808

1.76389

0.94087

0.86008

0.97

0.84318

0.55046

0.65284

0.99892

1.81469

0.96432

0.85876

0.98

0.83993

0.54307

0.64456

0.99952

1.84049

0.97614

0.85785

0.99

0.83665

0.53568

0.64026

0.99988

1.86657

0.98804

0.85731

1.00

0.83333

0.52828

0.63394

1.00000

1.89293

1.00000

0.85714

ρ∗ ⎛ k – 1 2⎞ ε(λc) * = 1– λc ρ ⎜⎝ k + 1 ⎟⎠

1 ⁄ (k–1)

(1.46)

.

Correspondingly, the stagnation temperature is k–1

∗ T∗ ⎛ ρ ⎞ τ(λc) * =⎜ ⎟ T ⎝ρ⎠

=1–

k–1 2 λc . k+1

(1.47)

The gas dynamic functions (1.44), (1.46), and (1.47) are presented in Table 1.10. This table also contains the functions that characterize the mass flux q(λc) *

1 ⁄ (k–1)

ρw ⎛k + 1 ⎞ =⎜ ⎟ ρcracr ⎝ 2 ⎠

1 ⁄ (k–1)

k – 1 2⎞ λc ⎛⎜1 – λc + 1 ⎟⎠ k ⎝

(this function is called the reduced density of mass flux) and y(λc) *

Fcrp∗ q(λc) ⎛ k + 1 ⎞ = =⎜ ⎟ Fp π(λc) ⎝ 2 ⎠

1 ⁄ (k–1)

λc . k–1 2 λc 1– k+1

The quantity reciprocal to y(λc) characterizes the change in the static momentum in the isentropic flow section depending on velocity. Moreover, Table 1.10 contains also the function

Aerodynamics and Hydraulics of Pressure Systems

χ(λc) =

37

k+1 ⎛1 + 2 ln λc⎞⎟ 2k ⎜ λ2c ⎠ ⎝

_ which makes it possible to calculate friction losses over the segment 0-1 (over the length l = l ⁄ Dh): _ l

_

∫ λfr dx .

χ(λc0) − χ(λc1) =

0

18. The rate of mass flow is expressed in terms of the functions q(χc) and y(λc): G=m

p∗Fq(λc) g√ ⎯⎯T

∗

=m

pFq(λc)

π(λc)g√ ⎯⎯T

∗

=m

pFy(λc) g√ ⎯⎯T∗

,

where m is the coefficient equal for air to 0.3965 K0.5 s–1. 19. Expanding Equation (1.42) in a series by Newton’s binomial rule, the following expression can be obtained for the total pressure: p∗ = p +

ρw2 ⎡ (2 – k) ⎤ 1 1 + Ma2 + Ma4 + ...⎥ 2 ⎢⎣ 24 4 ⎦

=p+

ρw2 (1 + δcom) . 2

(1.48)

The correction for the effect of gas compressibility is δcom =

1 2–k 1 Ma2 + Ma4 + Ma2 . 4 4 24

For a jet of an incompressible fluid the total pressure is p∗ = p +

ρw2 2

(1.49)

If the number Ma * w/a1 is very small, then Relation (1.48) is expressed in the form of Relation (1.49). 20. Table 1.11 presents the values of δcom, δp, and ∆T1 as functions of the number Ma0 and of the air flow velocity w0 (k = 1.41) at 0oC and 101.325 kPa.68 The correction for density is given by 1⁄k

δρ =

ρ1 – ρ0 ⎛ p1 ⎞ =⎜ ⎟ ρ0 ⎝ p0 ⎠

–1.

Ma20 2

Ma0 ⎛ ⎞ ⎜1 + 7 + ...⎟ ⎝ ⎠

and the correction for temperature is

38

Handbook of Hydraulic Resistance, 4th Edition

Table 1.11 Dependence of δcom, δp, and ∆T1 on w0 and Ma0 w0, m/s Ma0 dcom dp, % ∆T1, oC

34 0.1 0.25 0.50 0.59

68 0.2 1.0 2.0 2.4

102 0.3 2.25 4.5 5.4

k–1 ⁄ k

⎡⎛ p 1 ⎞ ∆T1 = T1 – T0 = T0 ⎢⎜ ⎟ ⎣⎝ p0 ⎠

136 0.4 4.0 8.0 9.5

170 0.5 6.2 12.9 14.8

203 0.6 9.0 18.9 21.3

238 0.7 12.8 26.8 29.0

k–1 ⎤ – 1⎥ = T0 Ma20 = 59.2 Ma20 . 2 ⎦

272 0.8 17.3 35.0 37.8

306 0.9 21.9 45.3 48.0

340 1.0 27.5 57.2 59.2

(1.50)

The subscripts 0 and 1 relate to sections 0-0 and 1-1 of the given flow, respectively. 21. For an incompressible liquid, to which gas at small flow velocities (practically up to w ≈ 150 m/s) can also be referred, U0 ≈ U1. Then, on the basis of Equation (1.27) we obtain gρ0z0 + p0 + N0

ρ0w20 ⎛ ρ1w21 ⎞ ρ0 = ⎜gρ1z1 + p1 + N1 + ∆ptot ⎟ 2 ⎠ ρ1 2 ⎝

(1.51)

or ⎛ ρ0w20 ⎞ ⎛ ρ1w21 ⎞ ρ0 ∆ptot = ⎜gρ0z0 + p0 + N0 . ⎟ – ⎜gρ1z1 + p1 + N1 ⎟ 2 ⎠ ⎝ 2 ⎠ ρ1 ⎝

(1.52)

22. In the case of a small pressure drop (practically equal to about 10,000 Pa), ρ0 = ρ1 = ρ; then instead of Equation (1.51) we have gρ0z0 + p0 + N0

ρw20 ⎛ ρw21 ⎞ = ⎜gρz1 + p1 + N1 ⎟ + ∆ptot 2 2 ⎠ ⎝

(1.53)

and with uniform flow velocity, when N0 = N1 = 1, gρz0 + p0 +

ρw21 ⎞ ρw20 ⎛ = ⎜gρz1 + ⎟ + ∆ptot 2 2 ⎠ ⎝

or ⎛ ρw20 ⎞ ⎛ ρw21 ⎞ ∆ptot = ⎜gρz0 + p0 + ⎟ – ⎜gρz1 + p1 + ⎟. 2 ⎠ ⎝ 2 ⎠ ⎝

(1.54)

Buoyancy or Net Driving Head (Self-Draught) 1. If we add to, and subtract the quantities pz0 and pz1 from each side of Equation (1.51), respectively, we obtain

Aerodynamics and Hydraulics of Pressure Systems

39

Figure 1.11. Choice of the "self-draught" (driving head, buoyancy) sign; (a) ρ > ρa; (b) ρ < ρa; (c) ρ > ρa; d) ρ < ρa.

gρz0 + p0 + pz0 – pz0 + N0

ρw20 ρw21 = gρz1 + p1 + pz1 – pz1 + N1 + ∆ptot , 2 2

(1.55)

where pz0 and pz1 are the values of the atmospheric pressure at heights z0 and z1, in Pa. On the basis of Equation (1.11), we get pz0 = pa – gρaz0 ; pz1 = pa – gρaz1 ,

(1.56)

where pa is the atmospheric pressure in the reference plane (Figure 1.11), in Pa, and ρa is the average density of atmospheric air over the height z, in kg/m3. In the present case, the density is considered to be practically equal at the two-heights, z0 and z1, in kg/m3. After performing suitable manipulation on Equation (1.55), we obtain (ρ – ρa)gz0 + (p0 – pz0) + N0

ρw20 2

= (ρ – ρa)gz1 + (p1 – pz1) + N1

ρw21 + ∆ptot . 2

(1.57)

40

Handbook of Hydraulic Resistance, 4th Edition

2. On the basis of Equation (1.57), the loss of total pressure over the segment between sections 0-0 and 1-1 is ∆ptot = (p0 – pz0) – (p1 – pz1) + N0

2

2

ρw0 ρw1 + g(ρa – ρ)(z1 – z0) – N1 2 2

or in a simplified form ∆ptot = (p0,st – p1,st) + (p0d – p1d) + ps = p0,tot – p1,tot + ps ,

(1.58)

where pd = N(ρw2/2) is the dynamic pressure in the given section of the stream (always a positive value), in Pa; pst = p – pz is the excess static pressure, that is, the difference between the absolute pressure p in the section of the stream at height z and the atmoshperic pressure pz at the same height, in Pa; this pressure can be either positive or negative; and ptot = pd + pst is the total pressure in the given section of the stream, in Pa. The excess elevation pressure (net driving head for gases) is ps = g(z2 – z1)(ρa – ρ) .

(1.59)

3. The excess elevation pressure (net driving head) is produced by the fluid, which tends to descend or rise depending on the medium (lighter or heavier) in which the fluid is located. This pressure can be positive or negative depending upon whether it promotes or hinders the fluid flow. If at ρ > ρa the flow is directed upward (Figure 1.11a), and at ρ < ρa downward (Figure 1.11b), the excess pressure ps will be negative and will hinder the flow. If, on the other hand, at ρ > ρa the flow is directed downward (Figure 1.11c), and at ρ < ρa it is upward (Figure 1.11d), the excess pressure ρs will be positive and will enhance the flow. 4. By solving Equation (1.58) in the drop of total pressures ∆ptot = p0,tot – p1,tot which determines the pressure developed by a supercharger, then psup = ∆ptot – g(z1 – z0)(ρa – ρ) = ∆ptot – ps . When ρ > ρa and the flow is directed upward or ρ < ρa and the flow is directed downward, there is a negative driving head (elevation pressure). Then psup = ∆ptot + ps . Otherwise psup = ∆ptot – ps . In the general case, psup = ∆ptot ps .

Aerodynamics and Hydraulics of Pressure Systems

41

5. When the densities of the flowing medium, ρ, and of the surrounding atmosphere, ρa, are equal and the pipes (flow channels) are horizontal, then the elevation pressure (net driving head) is zero. Then Equation (1.58) simplifies to: ∆ptot = p0,tot – p1,tot . 6. In cases where both the static pressure and the velocity are nonuniform over the cross section and this nonuniformity cannot be neglected, the total hydraulic resistance of the segment should be determined as the difference between the total pressures plus (or minus) the net driving head (if it is not zero): ∆ptot =

1 1 (pst + pd) wdF – ∫ (pst + pd) wdF + ps , Q∫ Q F0

F1

where (1/Q)∫ (pst + pd)wdF is the excess total pressure of the liquid (gas) stream passing F0

through given cross section F, in Pa, and pst + pd is the excess total pressure in the given cross section, in Pa.

1.6 HYDRAULIC RESISTANCE OF NETWORKS 1. In each flow system, as well as in its separate segments, that portion of the total pressure which is spent in overcoming the forces of hydraulic resistance is irreversibly lost. The molecular and turbulent viscosity of the moving medium irreversibly converts the mechanical work of the resistance forces into heat. Therefore, the total energy (thermal energy inclusive) of the flow over the given segment of the pipe remains constant in the absence of heat conduction through the walls. However, in this case, the state of the flow undergoes a change because of the pressure drop. The temperature, on the other hand, does not change at constant velocity. This can be attributed to the fact that the work of expansion due to a pressure drop is entirely converted into the work of overcoming the resistance forces and the heat generated by this mechanical work compensates for the expansion-induced cooling. At the same time, the energy acquired by the flow resulting from the work of a compressor, fan, etc., in the form of kinetic or thermal energy, is lost for the given system during the discharge of the fluid into the atmosphere or into another reservoir. 2. Two types of the total pressure (hydraulic resistance) losses in the pipeline are considered: − Pressure losses resulting from friction (frictional drag), ∆pfr. − Local pressure losses (local resistance), ∆ploc. The fluid friction loss is due to the viscosity (both molecular and turbulent) of real liquids and gases in motion, and results from momentum transfer between the molecules (in laminar flow) and between the individual particles (in turbulent flow) of adjacent fluid layers moving at different velocites. 3. The local losses of total pressure are caused by the following: local disturbances of the flow; separation of flow from the walls; and formation of vortices and strong turbulent agitation of the flow at places where the configuration of the pipeline changes or fluid streams

42

Handbook of Hydraulic Resistance, 4th Edition

meet or flow past obstructions (entrance of a fluid into the pipeline, expansion, contraction, bending and branching of the flow, flow through orifices, grids, or valves, filtration through porous bodies, flow past different protuberances, etc.). All of these phenomena contribute to the exchange of momentum between the moving fluid particles (i.e., friction), thus enhancing energy dissipation. The local pressure losses also include the dynamic pressure losses occuring during liquid (gas) discharge from the system or network into another reservoir or into the atmosphere. 4. The phenomenon of flow separation and eddy formation is associated with the difference of velocities over the cross section of the flow and with a positive pressure gradient along the flow. The latter develops when the flow velocity is retarded (for example, in an expanding channel, downstream of a sharp bend, when passing a body) in accordance with the Bernoulli equation. The difference in velocities over the cross section of a negative pressure gradient (e.g., accelerated motion in a contracted channel) does not lead to flow separation. The flow in smoothly contracting segments is even more stable than over segments of constant cross section. 5. The total pressure losses in any complex element of the pipeline are inseparable. However, for ease of calculation they are arbitrarily subdivided, in each element of the pipeline, into local losses (∆ploc) and frictional losses (∆pfr). It is also assumed that the local losses (local resistance) are concentrated in one section, although they can occur virtually throughout the entire length, except, of course, for the case of flow leaving the system, when its dynamic pressure becomes immediately lost. 6. The two kinds of losses are summed according to the principle of superposition of losses and consist of the arithmetic sum of the frictional and local losses: ∆pov = ∆pfr + ∆ploc . In fact, the value of ∆pfr should be taken into account only for relatively long fittings or only for elements (branch pipes, diffusers with small divergence angles, etc.), or when this value is commensurable with ∆ploc. 7. Present-day hydraulic calculations use the dimensionless coefficient of fluid resistance, which conveniently has the same value in dynamically similar flows, that is, flows over geometrically similar regions and with equal Reynolds numbers or other pertinent similarity criteria, irrespective of the kind of fluid or of the flow velocity (at least up to Ma = 0.8–0.9) and transverse dimensions of the segments being calculated. 8. The fluid resistance coefficient is defined as the ratio of the total energy (power) lost over the given segment (0-0)–(1-1) to the kinetic energy (power) in the section taken (for example, 0-0) or (which is the same) the ratio of the total pressure lost over the same segment to the dynamic pressure in the section taken, so that on the basis of Equations (1.21) and (1.23) for the general case, that is, for the case of nonuniform distribution of all the flow parameters over the section and of variable density along the flow, it is possible to write

ζ*

∆Ntot ρ0F0w30 ⁄ 2

=

∆Ntot Q0ρ0w20 ⁄ 2

=

∆ptot ρ0w20 ⁄ 2

=

p∗0 – p∗1 ρ0w20 ⁄ 2

Aerodynamics and Hydraulics of Pressure Systems

=

43

⎡1 ⎛ ⎞ ρw2 ⎢ ∫ ⎜p + + gρz + ρU⎟ ρwdF 2 ⎢G 2 ρ0w0 ⎢ F ⎝ ⎠ ⎣ 0 2

⎞ ⎤ ρ0 1 ⎛ ρw2 + gρz + ρU⎟ ρwdF⎥ . ⎜p + ∫ ⎥ 2 ρi G ⎝ ⎠ ⎥ F1 ⎦

–

(1.60)

For the case of uniform distribution of static pressure and density over the section, but which are variable along the flow, the resistance coefficient based on Equation (1.29) will acquire the form ζ*

∆ptot ∆Ntot ∆Ntot p′0∗ – p′1∗ = = = ρ0F0w3 ⁄ 2 Q0ρ0w20 ⁄ 2 ρ0w20 ⁄ 2 ρ0w20 ⁄ 2 =

⎡⎛ ⎞ ρ0w20 p + ρ + + N gρ z U ⎢ ⎟ ⎜ 0 0 0 0 0 0 2 ρ0w20 ⎣⎝ ⎠ 2

⎞⎤ ⎛ ρ1w21 + gρ1z1 + ρ1U1⎟⎥ . – ⎜p1 + N1 2 ⎠⎦ ⎝

(1.61)

If the density is invariable along the flow (ρ0 = ρ1 = ρ = const) ζ*

∆ptot ρw20 ⁄ 2

.

9. The value of ζ depends on the velocity, and consequently on the flow cross section. In a general case (ρi is variable along the flow), the resistance coefficient ζi * (∆ptot)/(ρiw2i ⁄ 2) based on the flow velocity wi in the ith section (Fi) is calculated for another section (for example, F0) using the formula ζ0 *

∆ptot ρ0w20 ⁄ 2

2

= ζi

ρi ⎛ wi ⎞ , ρ0 ⎜⎝ w0 ⎟⎠

(1.62)

since ∆ptot = ζ0

ρ0w20 ρiw2i = ζi . 2 2

Taking into account the flow rate equation ρ0w0F0 = ρiwiFi, we obtain 2

ζ0 = ζi

ρ0 ⎛ F0 ⎞ . ρi ⎜⎝ Fi ⎟⎠

When ρ0 = ρi = ρ ,

(1.63)

44

Handbook of Hydraulic Resistance, 4th Edition 2

⎛ F0 ⎞ ζ0 = ζi ⎜ ⎟ . ⎝ Fi ⎠

(1.64)

10. The overall fluid resistance of any network element is ∆pov = ∆ploc + ∆pfr = (ζloc + ζfr)

ρw2 ρw2 = ζov , 2 2

or ∆pov = ζov

ρop ρopw2op = ζov 2 2

2

⎛ Qop ⎞ ⎜ F ⎟ . ⎝ ⎠

(1.65)

In accordance with the arbitrarily accepted principle of superposition of losses we have ζov = ζloc + ζfr . Here, ζfr * ∆pfr ⁄ (ρopw2op ⁄ 2) is the friction loss coefficient in the given element of pipe (channel); ζloc * ∆ploc ⁄ (ρopw2op ⁄ 2) is the coefficient of local resistance of the given element of pipe (channel); wop is the mean flow velocity at section F under the operating conditions, in m/s [see Equation (1.17)]; Qop is the volumetric flow rate of a liquid or a working gas, m3/s [see Equation (1.16)]; ρop is the density of a liquid or a working gas, in kg/m3 [see Equation (1.18)]; and F is the cross-sectional area of the pipe (channel) element being calculated, in m2. 11. The friction loss coefficient of the element considered is defined through the friction factor of hydraulics λ as: ζfr =

λ⋅l . Dh

The coefficients λ and, hence, ζfr at the constant value of __l/Dh and incompressible flow is __ a function of Re and of the roughness of the channel walls, ∆0 = ∆0/Dh or ∆ = ∆/Dh. 12. The local resistance coefficient ζloc is mainly a function of the geometric parameters of the pipe (channel) element considered and also of some general factors of motion, which include: − the velocity distribution and the degree of turbulence at the entrance of the pipe element considered; this velocity profile, in turn, depends on the flow regime, the shape of the inlet, the shape of various fittings and obstacles, and their distance upstream from the element considered, as well as the length of the preceding straight pipe; − the Reynolds number; and − the Mach number, Ma * w/a1. 13. The principle of superposition of losses is used not only for calculation of a separate element of the pipe (channel), but also in the hydraulic calculation of the entire network. This means that the sum of the losses in separate elements of the pipe (channel) yields the total resistance of the system. Here it is understood, of course, that the mutual enhancement or interference effect of the adjacent elements is taken into account.

Aerodynamics and Hydraulics of Pressure Systems

45

14. The principle of superposition of losses can be realized by two methods: (1) by summing the total pressure losses in separate sections (elements) of the system; or (2) by summing the resistance coefficients of separate sections (elements), which were first normalized to a certain velocity and then expressing the total resistance of the system through its total coefficient of resistance. In the first method, it should be taken into account that in the case of a great difference between the densitites of liquid (gas) over different sections (elements), the values of the total pressure losses, taken as the losses of energy (power) which are related to the volumetric flow rate ∆Ntot/Q = ∆ptot through a formula analogous to Equation (1.23), depend on the fact to which section of the channel this volumetric flow rate is related. Therefore, the losses in different sections should be summed only after their normalization to the same volumetric flow rate. Thus, when these losses are normalized to the flow rate Q0 in section 0-0, then the total losses of the total pressure in the entire system will be n

∆psys = ∑

n

∆Ni Qi ∆Ni = Q0 ∑ Qi Q0

i=1 n

= ∑ ∆pi i=1

i=1 n

n

i=1

i=1

ρ0 ρiw2i ρ0 ζiρ0w2i = ∑ ζi =∑ , 2 ρi 2 ρi

(1.66)

where i is the number of the network section (element) being calculated; n is the total number of such sections (elements); ∆pi = ∆Ni/Qi are the total (overall) losses of total pressure (resistance) in the ith section (element) of the system,* normalized to the volumetric flow rate of the medium Qi through this section (element); ζi * 2∆pi/((ρiw2i ) is the resistance coefficient of the given section (element) of the network normalized to the velocity wi. In the second method, the general resistance coefficient of the network is ζ0,sys *

∆psys ρ0w20 ⁄ 2

n

n

= ∑ ζφi = ∑ ζi i=1

i=1

2

ρ0 ⎛ F0 ⎞ , ρi ⎜⎝ F1 ⎟⎠

(1.67)

where ζφi *

∆pi ρ0w20 ⁄ 2

is the resistance coefficient of the given (ith) element of the network normalized to the velocity w0 in the adopted section of the network F0 [see Equation (1.63)]; ζi is the resistance coefficient of the given (ith) section (element) of the network normalized to the velocity wi in the section Fi of the same section (element). Generally, the coefficient ζi includes also the correction for the mutual effect of adjacent elements of the network. ∗

Here, the subscripts "tot" or "ov" at ∆p and ζ for individual sections (elements) of the network are omitted.

46

Handbook of Hydraulic Resistance, 4th Edition The total losses of the total pressure over the entire network are given by ∆psys = ζ0,sys

n

2

n

2

ρ0w0 ρ0w0 ρ0 = ∑ ζi = ∑ ζφi 2 2 ρi i=1

n

= ∑ ζi i=1

i=1

2

2

2

⎛ F0 ⎞ ρ0w0 ⎜ Fi ⎟ 2 ⎝ ⎠

2

ρ0 ⎛ F0 ⎞ ρ0 ⎛ Q0 ⎞ ρi ⎜⎝ Fi ⎟⎠ 2 ⎜⎝ F0 ⎟⎠

or n

2

ρop ⎛ F0⎞ ρop ⎛ Qop ⎞ ∆psys = ∑ ζi ρip ⎜⎝ Fi ⎟⎠ 2 ⎜⎝ F0 ⎟⎠

2

(1.68)

i=1

and at ρi = ρ0 = ρ n

2

2

⎛ F0 ⎞ ρp ⎛ Q0 ⎞ ∆psys = ∑ ζi ⎜ ⎟ . Fi ⎠ 2 ⎜⎝ F0 ⎟⎠ ⎝ i=1

(1.69)

1.7 DISTRIBUTION OF STATIC PRESSURE OVER THE SECTIONS OF A NETWORK OF RATHER HIGH RESISTANCE 1. The loss of specific energy over any (ith) segment in a network can be defined through the resistance coefficient of the given segment: ∆ei,tot *

∆Ni,tot ∆Ni,tot w2i = = ζi G Gw2i ⁄ 2 2

w2i , 2

where i = 1, 2, 3, ... . From this, the equation analogous to Equation (1.23) for two sections (i-1)–(i-1) and i-i takes the form gzi–1 +

pi–1 w2i–1 pi + Ni–1 + Ui–1 = gzi + + Ui + (Ni + ζi) 2 ρi–1 ρi

w2i . 2

The latter equation, together with Equations (1.30), (1.31), and (1.32) for the ith and (i1)th sections, leads to the following r elation allowing the calculation of the static pr essur e in section i-i if it is known for section (i-1)–(i-1):

Aerodynamics and Hydraulics of Pressure Systems pi ⎧⎛ pi–1 ⎞ =⎨ pa ⎩⎜⎝ pa ⎟⎠

(n–1) ⁄ n

+

47

n–1⎡ ⎢gρi–1(zi–1 – z) n ⎣ –1 ⁄ n

ρi–1w2i ⎤ ⎛ pi–1 ⎞ ρi–1w2i–1 + Ni–1 – (Ni + ζi) ⎥ 2 2 ⎦ ⎜⎝ pa ⎟⎠

n ⁄ (n–1)

1⎫ ⎬ pa⎭

.

(1.70)

In this case, all of the quantities with the subscript i-1 as well as ζi, Ni, zi, and wi are known (assigned or calculated). Only the quantity pi/pa is unknown. 2. In the majority of cases, the process can be regarded to be isentropic. Then, the exponent n in Equation (1.70) can be replaced by k. For locking devices n ≈ 1.15,20 and Equation (1.70) acquires the form

pi ⎧⎛ pi–1 ⎞ =⎨ pa ⎩⎜⎝ pa ⎟⎠

0.13

⎡ ρi–1w2i–1 + 0.13 ⎢gρi–1(zi–1 – zi) + Ni–1 2 ⎣ –0.87

ρi–1w2i ⎤ ⎛ pi–1 ⎞ – (Ni + ζi) ⎥ 2 ⎦ ⎜⎝ pa ⎟⎠

7.67

1⎫ pa⎬ ⎭

.

(1.71)

For T-joints and other analogous shaped elements, when n ≈ 1 and the pressure is proportional to the gas density [see Equation (1.33) in which, in a general case, the subscript will be i-1 and i, respectively],

ln

ρi–1w2i–1 pi 1 ⎧ = ⎨gρi–1(zi–1 – zi) + Ni–1 2 pi–1 pi–1 ⎩ – (Ni + ζi)

ρi–1w2i ⎤ ⎥=A . 2 ⎦

(1.72)

Then

pi = eA and pi = pi–1eA pi–1 or

pi pi–1 A = e . pa pa

(1.73)

3. Pressure distribution along the network is calculated in the following order: using the quantitites which are given for initial section 0-0 (i = 1) and which enter, in the case of n > 1, into the right-hand side of Equation (1.70), the value of pressure p1/pa in section 1-1 is calculated. On the basis of Equations (1.20) and (1.30) the values of w1 and ρ1 are calculated

48

Handbook of Hydraulic Resistance, 4th Edition

and, correspondingly, the pressure p2/pa from Equation (1.71) for the sections 2-2 [Equations (1.20) and (1.30) are used with subscripts i-1 and i]. Analogously, calculations are also made for the case n = 1, using Equations (1.20), (1.33), and (1.73).

1.8 GENERALIZED FORMULAS OF RESISTANCE FOR HOMOGENEOUS AND HETEROGENEOUS SYSTEMS9,10 1. The total resistance to the motion of a Newtonian fluid can be considered as a sum of resistance forces: 1. Viscous forces that hinder the irrotational (laminar) motion of fluid. 2. Those opposing the change in the momentum of the system when secondary fluid flows originate in it under the influence of some external forces. 3. A group of driving forces that involve the projections of external forces onto the axis of motion, so that it is possible to write that the resistance force per volume unit of the system is

∆p k1ηw0 nρw20 = 2 + + Σ F1 Σ F2 , L l l where k1ηw0 ⁄ l2 is the viscous resistance force per volume unit of the system; w0 is the flow velocity averaged over the channel section; nρw20 ⁄ l are the additional resistance forces per volume unit of the system that oppose the motion of fluid in the case of turbulent mode of flow and also during the flow through individual obstacles (local resistances); Σ F1 = Σ miwi ⁄ V is the resistance force which is numerically equal to the sum of external forces per volume unit of the system that develop and suppress internal flows in it; this force follows from the momentum conservation law of the system (mi and wi are the mass and velocity of the volume element inside of which no internal motions originate any longer; V is the volume of region B [Figure 1.12]; Σ F2 is the sum of the projections, onto the pipe axis, of the potential part of external forces which act on the fluid and which are related to the volume unit of the system; this sum of forces can be either a driving force (minus sign) or the resistance force (plus sign); k1 is the coefficient of the shape (for a pipe of circular cross section k1 = 32); l is the characteristic dimension (it is the diameter for a pipe and l = Dh = 4F/II for a

Figure 1.12. Scheme of internal eddy motion of fluid and of the effect of external forces on it.

Aerodynamics and Hydraulics of Pressure Systems

49

channel), L is the length of the system segment considered; n is the proportionality factor; when the fluid flows through obstacles, it is equal to the coefficient of local resistances ζloc. 2. The internal flows can be induced by the influence of Archimedean forces on a fluid under the condition of heat (ρgβt∆T) or mass (∆ρg) transfer (where βt and ∆T are the termal coefficient of fluid expansion and the temperature head, respectively; ∆ρ is the difference of densities). In electromagnetic fields, the internal flows in a system can originate under the action of a group of forces; these are induction electromagnetic forces that suppress the internal flows; conduction electromagnetic forces that result from the interaction of electric current with a current-conducting fluid and from the interaction of the magnetic field of the current with the external magnetic field; electromagnetic forces that originate during the interaction of an electric layer at the phase interfaces with the external electric and magnetic fields.16,57 Internal forces can also originate, for example, during the flow of a fiuid in a straight pipe rotating around its axis.66 3. In heterogeneous (nonuniform) systems, the phases of which have substantially different densities, the internal flows originate due to a relative motion of phases. In this case, the force per volume unit of the system that drives individual local particles is defined as F = (ρp – ρ)g , where ρp is the density of a particle, in kg/m3. This motion is hindered by the viscous forces k1ηw0 ⁄ l2 and by the forces Σ F1 that follow from the momentum conservation law. Therefore, for one local particle these forces are

Σ F1 = (ρp – ρ)g – k1ηw0 ⁄ l2 . In the case of the volumetric concentration of the dispersed phase µcon, the forces that induce the internal flows in a volume unit of the system are ⎡ k1ηw0 ⎤ Σ F1 = ⎢(ρp – ρ)g – 2 ⎥ µcon . l ⎦ ⎣ 4. Simple transformations of the expression ∆p ⁄ L make it possible to obtain the friction coefficient

λ=

∆p 2l 2k1 = , B L ρw20

(1.74)

where B=

Re Σ F1l2 n 1 + Re + k1 k1ηw0

Σ F2l2 k1ηw0

.

(1.75)

50

Handbook of Hydraulic Resistance, 4th Edition

Table 1.12 The function n = f(Re)

0–2 × 10

n 0

Re 105

n 0.0087

2.5 × 103

0.0042

106

0.006

4 × 10 104 2 × 104

0.0120

107

0.004

0.0128 0.0098

8

0.003

Re 3

3

10

Equation (1.75) is a generalized criterion of hydrodynamic similarity. It follows from Equation (1.74) that the relationship between the resistance coefficient and the above-indicated criterion should be linear under any conditions of fluid motion in the system. In particular, for a turbulent flow in straight circular pipes (Σ F1 = 1 and Σ F2 = 0) Equation (1.74) takes the form λ=

64 Re

⎛1 + n Re⎞ = 64 + 2n , ⎜ 32 ⎟⎠ Re ⎝

where n can be found by equating the values of λ from the latter relation to its values from Diagram 2.1. The function n = f(Re) is presented in Table 1.12. 5. When, during the fluid flow through pipes and channels, the external forces simultaneously contribute to and hinder the development of internal flows in the system (for example, during the motion of fluids having appreciable electrical conductivity, in a longitudinal magnetic field,5,11,14,32,61 the generalized criterion of hydrodynamic similarity is Re

B=

n Re – 1+ 32

,

⎯⎯ √

n Ha 32

σ ⁄ η is the Hartman number (B0 is the magnetic field induction; σ is the ⎯⎯⎯⎯ where Ha = B0l√ electrical conductivity of the fluid; l = Dh is the hydraulic diameter). In this case, the resistance coefficient is

λ=

64 . B

(1.76)

6. During the flow of conducting fluids in pipes or channels in the transverse magnetic field, two cases are considered: (a) a plane-parallel flow in a channel when the magnetic field induction vector is normal to the large side of the magnetohydrodynamic (MHD) channel;11,14,37,61 for this case, B=

Re n n 1 + Re – ⎛⎜ ⎞⎟ k k1 ⎝ 1⎠

0.5 0.25

β

, Ha

Ha + 0.25 0.5 β k1

Aerodynamics and Hydraulics of Pressure Systems λ=

51

2k1 , B

(1.77)

where β = a/b is the aspect ratio of the channel; when β = 1:15 and β = 1:17, it was obtained that k1 = 44 and for β = 1:25 that k1 = 32.7; (b) the flow in the ϕ-field, when the magnetic induction vector is parallel to the large side of the MHD channel;5,64 for this case, B=

Re n n 1 + Re – ⎛⎜ ⎞⎟ k k1 ⎝ 1⎠

,

0.5 0.25

β

Ha Ha + 0.5 βk1

whereas λ can be found from Equation (1.77); when β = 14.5, k1 = 44; when β = 32, k1 = 48. The intermediate case is the MHD flow in a channel at β = 1 or in a circular pipe when λ is determined from Equation (1.76). 7. When a fluid flows in a bent pipe, the system experiences the centrifugal inertia forces. These forces bring about the redistribution of pressures over the section due to which transverse (secondary) flows originate. In this case, λ is found from Equation (1.77) and B=

Re

m n 1 + Re + 32 k1

,

⎯⎯ √

D Re 2R

where D is the diameter of the pipe cross section; R is the mean radius of the pipe bent rounding; m = 1.76 × 10–1 for the laminar regime of flow; m = 1.57 × 10–2 for the turbulent regime of flow. In coiled pipes the fluid flow varies simultaneously in two directions with the rounding radii R1 and R2. For this case, the values of λ are found from Equation (1.77) and B=

Re ⎡ m ⎢⎛ n ⎜ Re + 1+ 32 ⎣⎝ 32

2

⎞ ⎛ D Re⎟ + ⎜ 2R1 ⎠ ⎝

⎯⎯ √

2

⎞ ⎤ D Re⎟ ⎥ 2R2 ⎠ ⎦

0.5

.

⎯⎯ √

For a fluid moving in a pipe the axis of which is normal to the rotation axis of this pipe, the fluid experiences the action of the Coriolis inertia forces that redistribute the pressure in the fluid and induce internal flows.66 For this case, the values of λ are determined from Equation (1.77) and B=

Re , ωD ω2D2 m R n Re + Re 1+ Re w0 32D 16 32 w20

where R is the mean radius of pipe rotation and ω is the angular speed of pipe rotation. 8. In the case of nonisothermal flow in pipes and channels, in the flow core and near the wall, the viscosities and densities of the fluid in the flow core and near the wall can be sub-

52

Handbook of Hydraulic Resistance, 4th Edition

stantially different due to the difference of temperatures of the fluid in these zones, and this leads to the origination of internal flows (heat convection). For this case, B=

Re 2δ ρgβt∆Tl2 n 1 + Re + k1l ηw0 k1

,

where l = Dh , δ + 2 × 10–4 – 5 × 10–4 m . During the pipe flow of different oils the coefficient λ is found from Equation (1.77) and for the channel flow λ = 77.4/B. In the case of nonisothermal flow of low-viscous fluids (for example, water), the thermal convection even at small temperature differences can substantially influence the resistance, and, in this case, B=

Re . ρgβt∆TD2 n Re 1+ 32 32ηw0

9. The forces that induce internal flows in a heterogenous system due to the relative motion of phases depend on both the difference of the densitities of the fluid and dispersed particles and the characteristic dimension and shape of these particles and the velocity of their motion in the fluid. In the case of the flow of suspensions in straight hydraulically smooth pipes, the generalized hydrodynamic criterion is defined as B=

Re [(ρp – ρ)g – k2ηw2 ⁄ l22]µD2 n Re + 1+ 32 32ηw0

,

where w2 is the velocity of the relative motion of a disperse particle in the fluid; k2 is the coefficient of the shape of the disperse particle (for a sphere, k2 = 12); l2 is the characteristic dimension of the particle (for a sphere, l2 = dloc). For the pipe flow of dusted streams, when the density of solid particles ρp is much smaller than the gas density ρ, the latter quantity can be neglected. Then B=

Re [ρpg – k2ηw2 ⁄ l22]µD2 n Re + 1+ 32 32ηw0

.

Aerodynamics and Hydraulics of Pressure Systems

53

For the pipe of gas–liquid mixtures, ρ >> ρ2 (where ρ2 is the density of gas bubbles); therefore B=

Re [ρg – k2ηw2 ⁄ l22]µD2 n Re + 1+ 32 32ηw0

.

In all three cases, the resistance coefficient λ can be taken according to Equation (1.77).

1.9 LIQUID AND GAS FLOW THROUGH AN ORIFICE Flow of an Incompressible Fluid 1. The velocity wcon of jet discharge for an incompressible fluid passing from the exit section of a submerged nozzle or orifice in the wall of vessel A into vessel B (Figure 1.13), is expressed, based on the Bernoulli and continuity equations, by the following formula:*

wcon = ϕ

⎯ √

2 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ gρ(z1 – z2) + (p1 – p2) , ρ

where the velocity coefficient ϕ has the form: ϕ=

1 ζcon(1 – 2) + ε [N2(F0 ⁄ F2)2 – N1(F0 ⁄ F1)2] ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ 2

.

(1.78)

Here, z1 and z2 are the depths to which the center of gravity of the orifice (nozzle) is submerged relative to the free liquid level in vessels A and B, respectively, in m; p1 and p2 are the pressures on the free surface in respective vessels (sections 1-1 and 2-2), in Pa; N1 and N2 are the kinetic energy coefficients in sections 1-1 and 2-2; F1 and F2 are the areas of these cross sections, in m2; ε = Fcon/F0 is the coefficient of contraction of the nozzle exit section (for an orifice in a thin wall it is the coefficient of jet contraction in the narrowest cross section of the jet); Fcon is the area of the jet (not of the nozzle) cross section at the exit from the nozzle; in the area of an orifice in a thin wall (Figure 1.14), Fcon is the area of the contracted cross section of the jet, in m2; F0 is the area of the exit cross section of the nozzle (office), in m2; and ζ(1–2) = ∆p(1–2)/(ρw2con/2) is the resistance coefficient of the entire flow path from section 1-1 to section 2-2 reduced to the velocity wcon. 2. In the general case of the flow from vessel A into vessel B (see Figure 1.13), the pressure losses consist of the losses over the stretch from section 1-1 to the exit from the nozzle or orifice (section c-c) and shock losses at the jet expansion from the narrow section c-c at the exit from the nozzle (orifice) to section 2-2, that is, ∗

In the case of a gas flow, the quantities z and l are nealected.

54

Handbook of Hydraulic Resistance, 4th Edition

Figure 1.13. Discharge from a submerged orifice.

2

F0 ⎞ ⎛ ζcon(1–2) = ζcon,noz – 1 + ζsh = ζcon,noz – 1 + ⎜1 – ε ⎟ F 2⎠ ⎝ 2

= ζcon,noz – 2ε

F0 ⎛ F0 ⎞ + , F2 ⎜⎝ F2 ⎟⎠

where ζcon,noz is the total resistance coefficient of the nozzle or orifice, which also includes the dynamic pressure losses at the exit, reduced to the velocity wcon. The resistance coefficient ζcon(1–2) can be expressed in terms of the resistance coefficient ζ0(1–2) = ∆p1–2/(ρw20 ⁄ 2), reduced to the mean velocity w0 at the exit from the nozzle (orifice): 2

2 ⎡ F0 ⎛ F0 ⎞ ⎤ 2 ⎛ w0 ⎞ 2 ⎥ε , ⎢ ζcon(1–2) = ζ0(1–2) ⎜ ε = ζ = ζ ε + – 2 , 0(1–2) 0 noz ⎟ F2 ⎜⎝ F2 ⎟⎠ ⎦ ⎣ ⎝ wcon ⎠

whence, having substituted into Equation (1.78), we obtain

Figure 1.14. Discharge from a vessel through an orifice in the bottom or wall.

Aerodynamics and Hydraulics of Pressure Systems ϕ=

1 ε√ ζ0,noz – 2εF0 ⁄ F2 + (F0 ⁄ F2)2 + N2(F0 ⁄ F2)2 – N1(F0 ⁄ F1⎯)2 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

55

.

3. In the case of the liquid flow from vessel A into vessel B of large volume, that is, at F0 4–5

104 < Ret < 105

300 < Ret ≤ 104

40 < Ret ≤ 300

10 < Re0 ≤ 40

Re0 ≤ 10

Orifice in a thin (l/D0 ≤ 0.1) wall (bottom) of a vessel2,27 (Figure 1.17a) w0D0 Re0 = ≥ 105 v

Shape of the orifice, nozzle

Table 1.13 Values of the discharge coefficients µ

µ=

µ=

1 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ 1.07 – 0.07F0 ⁄ F⎯1

–

–

–

–

–

1 1 + 0.707 √ ⎯⎯⎯⎯⎯⎯⎯⎯ 1 – F0 ⁄ F⎯1 –

At different F0/F1

0

Re ⎯⎯⎯⎯⎯ √ 25.2 + Re

0.97

Ret µC 10 + 1.5 Ret Ret µC 5 + 1.5 Ret 0.27 µ + 0.59 + Re1t ⁄ 6 B1 µ + 0.59 + ⎯⎯⎯ √ Ret B1 = 5.5 for a circular section;2 B1 = 8.9 for a rectangular section60 0.925

µ+

0

At F0/F1 → 0

µ + 0.59

Formulas for calculation of µ

58 Handbook of Hydraulic Resistance, 4th Edition

27

27

3

Ret > 10 l ⁄ D0

2

3

10 l ⁄ D0 < Ret < 10 l ⁄ D0

10 l ⁄ D0 < Ret < 102 l ⁄ D0

Ret < 10 l ⁄ D0

F0 ⁄ F1 > 4–5

Rounded inlet edge (r/D0 > 0; Figure 1.17d);

Ret ≥ 105

3 × 10–3 l ⁄ D0 < Ret < 105 0.336 l ⁄ D0 Ret0.25

1 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ 1.5 – 0.5F0 ⁄ F⎯1

1.5 +

√ ⎯⎯⎯⎯⎯⎯⎯⎯

25 7.4 l ⁄ D0 + Ret Ret 1

16 Ret

(0.25r ⁄ D0 + l0 ⁄ D0) +

0

0.33

1 0.25

N + ζr + ⎯√⎯⎯⎯⎯⎯⎯ Ret

µ by formula (b)

µ by formula (a) µ=

6.3 Ret

(0.25r ⁄ D0 + l ⁄ D0)

30.4 90 (0.25r ⁄ D0 + l0 ⁄ D0) + Ret Ret

at l0 ⁄ D0 > 0.5

b2 =

b22 + 0.714 – ⎯b2 (b) µ=√ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

for l0 ⁄ D0 < 0.5

a2 =

a22 + 0.5 – a2 (a) µ=√ ⎯⎯⎯⎯⎯⎯⎯⎯⎯

µ=

µ=

b0 =

µ=√ b20 + 0.588 – ⎯b0 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Ret < 102 l ⁄ D0

102 l ⁄ D0 < Ret < 3 × 10–3 l ⁄ D0

a0 =

5.8 14.8 l ⁄ D0 + Ret Ret

µ=√ a20 + 0.463 – ⎯a0 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

F0 ⁄ F1 > 4–5; l ⁄ D0 = 1–7

External cylindriacl nozzle;31 sharp inlet edge (Figure 1.17a,d);

Figure 1.17c) Re ≥ 105

0.82

–

Aerodynamics and Hydraulics of Pressure Systems 59

60l ⁄ D0 < Ret < 10 l ⁄ D0

3

10l ⁄ D0 < Ret < 60l ⁄ D0

Ret ≤ 10l/D0

F0 ⁄ F1 > 4–531

Outer cylindrical nozzle, conical inlet (Figure 1.17e)

Shape of the orifice, nozzle

Table 1.13 (continued)

1.56

c...1.70

3

1.41

c...1.70

(b)

1.48

µ — from formula (b)

µ — from formula (a)

n and c — from the table At l0 ⁄ D0 > 0.5l ⁄ D0

60

0.26 0.13 At l0/D0 < 0.05 1.42 1.45 At l0/D0 > 0.05 1.41 1.43

40

85 ⎛ 25.2 ⎞ nl1 ⁄ D0 + l0 ⁄ D0 + 2c Ret ⎜⎝ 2c Ret ⎟⎠

2 3

25.2 –b b + ⎯⎯⎯⎯⎯⎯⎯ √ 2c Ret

b3 =

µ=

At l0 ⁄ D0 ≤ 0.5l ⁄ D0

0.46

0.63 1.40

20

10

n — from the table

15.2 6.0 (nl1 ⁄ D0 + l0 ⁄ D0) + (a) Ret Ret

α0...0 n

a3 =

2

a3 + 0.476 – ⎯a3 µ=√ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

N0 from Diagrams 4.2 and 4.3: ζr from Diagram 3.4

At different F0/F1

80

1.45

1.48

0.04

Formulas for calculation of µ

1.50

1.53

0.02

100

At F0/F1 → 0

1.54

1.56

0.01

120

60 Handbook of Hydraulic Resistance, 4th Edition

External smoothly converging-diverging nozzle (Venturi tube) (α = 6–8o; F2/F0 = 4–5 (Figure 1.17k)27 Internal cylindrical nozzle: inlet edge of different thicknesses (δ/D0 > 0); F0/F1 > 4–5; l/D0 = 3 (Figure 1.17l) Ret < 2 × 104 48

External conical diverging nozzle; sharp inlet edge F2/F0 = 2; α = 15o (Figure 1.17i) Ret ≥ 105

5,27

Ret ≥ 10

External conical converging nozzle (α = 13o; Figure 1.17e)

Ret > 2 × 103 0

k

0.25

0.33

N +ζ + ⎯√⎯⎯⎯⎯⎯⎯ Ret

1

... ... ... ...

µ µ µ

... ...

Ret

µ

Ret

µ

5

0.05

0.11

0.21

50

4

100 0.34

2.50

At certain values of δ/D0 500 103 2 × 103 At δ/D0 = 0.004–0.006 0.57 0.64 0.69 At δ/D0 = 0.02–0.03 0.59 0.66 0.72 At δ/D0 = 0.04 0.62 0.70 0.75

0.03

2.43 At any δ/D0 10 20

2 2.32

1 2.15

1 1.2 – 0.2F0 ⁄ F⎯1 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √

Re × 10–5

µ=

3

(nl1 ⁄ D0 + l ⁄ D0)

N0 — from Diagrams 4.2 and 4.3 ζk – from Diagram 3.7

µ=

0.80

0.75

0.70

104

0.46

200

2.52

≥6

0.65–0.7

0.92

Aerodynamics and Hydraulics of Pressure Systems 61

27

For a compressible fluid45

Stalling flow (Figure 1.17m)48

Internal cylindrical nozzle. Inlet edge is of different thicknesses (δ/D0 > 0); l/D0 < 3; F0 ⁄ F1 > 4–5

Sharp inlet edge ( δ/D0 F 0); l/D0 = 3 (Figure 1.17l) Re >105

Shape of the orifice, nozzle

Table 1.13 (continued)

1 2 – F0 ⁄ F⎯1 ⎯⎯⎯⎯⎯⎯⎯⎯ √

Macomp =

a

+

Ma4 80

is the Mach number

2 Maco mp

8 wcomp

µcomp - µ +

µ = 0.495 + 4δ ⁄ D0

µ=

Formulae for calculation of µ At different F0/F1

0.7

At F0/F1 → 0

62 Handbook of Hydraulic Resistance, 4th Edition

Aerodynamics and Hydraulics of Pressure Systems

63

Discharge of a Compressible Gas 1. When a gas (vapor, air) issues at high pressure into the atmosphere, a significant change occurs in its volume. Therefore, it is necessary to take into account the compressibility of the gas. Neglecting the nozzle losses for an ideal gas and the effect of its mass, the velocity of the adiabatic discharge can be determined from the Saint–Wantzel formula as:

w0 =

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ 2

(k–1) ⁄ k ⎤ k p1 ⎡ ⎛ p0 ⎞ ⎢1 – ⎜ ⎟ ⎥ k – 1 ρ1 ⎣ ⎝ p1 ⎠ ⎦

(1.79)

and the mass discharge G, with allowance for losses in the nozzle (µ = 1/√ ⎯⎯ζ ):

G = µF0

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √ (k+1) ⁄ k 2⁄k ⎡⎛ p0 ⎞ ⎤ 2k ⎛ p0 ⎞ ρ1p1 ⎢⎜ ⎟ – ⎜ ⎟ ⎥ p p k–1 1 1 ⎝ ⎠ ⎣⎝ ⎠ ⎦

,

(1.80)

where the subscript 1 indicates that the respective quantities refer to the section of the pipe (vessel) upstream of the constricted nozzle section, and 0, to the smallest section of the nozzle or to the medium into which the gas issues. 2. At the given pressure p1 and density ρ1 in the vessel, the discharge velocity and the mass discharge at the given F0 depend on the pressure of the medium into which the gas issues, i.e., on the ratio p0/p1. With decrease in p0/p1, the discharge velocity w0 increases until this ratio becomes equal to the critical pressure ratio: k ⁄ (k+1)

p0 ⎛ p0 ⎞ 2 ⎞ = = ⎛⎜ p1 ⎜⎝ p1 ⎟⎠ k + 1⎟⎠ cr ⎝

.

When p0/p1 = (p0/p1)cr, the velocity in the nozzle throat F0 is equal to the speed of sound in the given medium. With a further decrease in p0/p1, the velocity in the smallest cross section remains equal to the local speed of sound wcr = acr =

1 RT1 = √ √ ⎯⎯⎯⎯ ⎯⎯⎯⎯ k+1⎯ k + 1 ρ1

2k p

2k

.

Thus, with decrease of the pressure ratio below the critical value the mass flow rate of the gas does not increase at constant values of p1, ρ1, and F0: 1 ⁄ (k–1)

2 ⎞ G = µF0ρ0w0 = µF0 ⎛⎜ k + 1⎟⎠ ⎝

ρ1p1 √ ⎯⎯⎯⎯⎯ k+1 2k

.

(1.81)

Equations (1.79) and (1.80) can therefore be used for the calculation of the velocity and, correspondingly, of the flow rate only when p0/p1 ≥ (p0/p1)cr. When p0/p1 ≥ (p0/p1)cr, Equa-

64

Handbook of Hydraulic Resistance, 4th Edition

tion (1.81) should be used. In this case, the mass discharge is independent of the external pressure p0 and is controlled by the pressure p1 in the vessel, increasing with its rise.

1.10 WORK OF THE SUPERCHARGER* IN A SYSTEM** 1. To set a liquid or gas medium at the ends of a given piping system in motion, it is necessary to create a difference of the total pressure means of a pressure-boosting device (pump, fan, flue-gas fan, compressor). 2. In the most general case, the total pressure developed in the supercharger is spent: (1) to overcome the difference of pressures in the intake and discharge volumes; (2) to overcome excessive elevation pressure (negative buoyancy) that is, to raise a liquid or gas, heavier than the atmoshperic air, a height z from the initial to the final section of the system (in the case of positive buoyancy [self-draught***] the height z is subtracted from the supercharger pressure); and (3) to create a dynamic pressure at the exit of the liquid (gas) (Figure 1.20) from the system (not from a supercharger); that is, the total pressure ptot (Pa),**** developed in the supercharger is comprised of ptot = (pinj – psuc) ps + (∆psuc + ∆pinj) +

ρw2ex , 2

(1.82)

Figure 1.20. A supercharger in the system.

*

Supercharger refers to a pressure enhancement device such as a booster pump; supercharge refers to pressurization. **

The case of incompressible fluid is considered.

***

The term "self-draught" can be considered as the net driving head or buoyancy.

****

In what follows, the quantity ptot will be called simply pressure instead of total pressure.

Aerodynamics and Hydraulics of Pressure Systems

65

where psuc is the excess pressure in the suction volume, pinj the excess pressure in the injection volume, ps the excess elevation pressure (buoyancy), ∆psuc the pressure losses (resistance) over the suction stretch of the system, ∆pinj the pressure losses (resistance) over the injection stretch, and wex the flow velocity at the exit from the system, in m/s. 3. In the case where the pressures of the suction and injection volumes are equal (psuc = pinj), we have ptot = ∆psuc + ∆pinj +

ρw2ex ps = ∆psys , 2

(1.83)

where ∆psys is calculated from Equation (1.66) or (1.68) [or (1.69)] for the entire system as a sum of the losses over the suction and injection stretches of the system (including the dynamic pressure losses at the exit from the system), while the buoyancy ps is calculated from Equation (1.59). 4. Since at ps = 0 the sum of all the losses in the system is equal to the difference between the total pressures upstream and downstream of the supercharger, then ⎛ ρw2inj ⎞ ⎛ ρw2suc ⎞ ptot = ⎜pst,inj + ⎟ – ⎜pst,suc + ⎟ 2 ⎠ ⎝ 2 ⎠ ⎝ = p∗inj – p∗suc ,

(1.84)

where p∗suc – p∗inj are the excess total pressures respectively upstream and downstream of the supercharger, in Pa; pst,suc and pst,inj are the excess static pressures respectively upstream and downstream of the supercharger, in Pa; and wsuc and winj are the mean stream velocities upstream and downstream of the supercharger, respectively, in m/s. 5. Under normal operating conditions of the supercharger, ptot is positive, that is, ptot,inj > ptot,suc At the same time both the static and the dynamic pressure downstream of the supercharger can be smaller than upstream of it. 6. In a specific case of equal cross-sectional areas of the suction and injection orifices, ρw2suc ρw2inj = 2 2 and, therefore, the pressure created by the supercharger will be ptot = pst,inj – pst,suc ,

(1.85)

that is, the pressure created by the supercharger is equal to the difference between the static pressures immediately downstream and upstream of the supercharger. 7. The power on the supercharger shaft is determined from Nsup =

Qopptot,op Qop∆psys , = ηtot ηtot

(1.86)

66

Handbook of Hydraulic Resistance, 4th Edition

where Qop is the volumetric flow rate of the medium being displaced under operating conditions, in m3/s, taken for that section to which all the pressure losses are reduced by a formula similar to Equations (1.66)–(1.69); ptot,op is the pressure created by a supercharger under operating conditions, in Pa, equal to the total pressure losses ∆psys reduced to the same volumetric flow rate. 8. Usually the volumetric flow rate of the medium displaced is a specified quantity, while the pressure created by a supercharger is calculated from Equations (1.82)–(1.85) for the prescribed conditions in the system, that is, for given difference of pressures in the suction and injection volumes (pinj – psuc), excess elevation pressure (±pel). The resistance coefficients ζfr and ζloc, the flow velocity in each element, and, consequently, the value of ptot depend on the shape and dimensions of the system. 9. To determine whether a given supercharger meets the required predictions of Qop and ptot,op, it is necessary first to reduce these quantities to those conditions (density) of the medium for which the supercharger performance is given. Then if the flow rate of the medium being displaced is given in m3/s for normal conditions, it is recalculated for the operating conditions from Equation (1.16). 10. The reduced pressure of the supercharger is pred = pcal

ρper Tw pper , ρnc Tper psup

(1.87)

where pcal is the calculated pressure of the supercharger, in Pa; ρper is the density of the medium at which the performance of the supercharger is determined under normal conditions (t = 0oC; atmospheric pressure B = 101.325 kPa), in kg/m3, ρnc is the density of the medium for which the supercharger is chosen at normal conditions, in kg/m3; Tw is the working temperature of the displaced medium in the supercharger, in K; Psup is the working pressure (absolute) of the displaced medium in the supercharger, in Pa; Tper, pper are the temperature (K) and pressure (Pa) of the medium at which the supercharger performance was determined; in the case of fans, pper = 101.325 kPa. 11. In the case of high-head superchargers, the density of the medium being displaced is related to the mean pressure on the rotor. Then psup in Equation (1.87) is replaced by the mean absolute pressure on the rotor: pm = psup + (∆pinj – 0.5∆psys) , where ∆pinj are the pressure losses in the injection section of the system, in Pa, and ∆psys are the total pressure losses in the whole system, in Pa. 12. The rated power on the supercharger shaft is Nsup =

Qoppcal Qopptot,perρncTperpinj = ηtot ηtotρperTwpper

= Nper

ρnc Tper pinj , ρper Tw pper

where ptot,per is the pressure created by the supercharger according to specification, in Pa, and Nper is the power on the supercharger shaft according to specification, in W.

Aerodynamics and Hydraulics of Pressure Systems

67

1.11 EXAMPLES OF THE METHOD OF CALCULATION OF THE FLUID RESISTANCE OF SYSTEMS Example 1.1. Forced Ventilation System A schematic diagram for the calculation of the ventilation system network is presented in Figure 1.21. Given are: 1. Total quantity of the air sucked from the atmosphere, Q = 0.89 m3/s, under normal conditions; 2. Flow rate through lateral branches, Q = 0.22 m3/s, under normal conditions; 3. Temperature of the external (atmospheric) air, t = –20oC; 4. Air temperature downstream of the heater, t = 20oC, and 5. Material from which the ducts are made: sheet steel (oil coated), roughness ∆ ≈ 0.15 mm (see Table 2.5). Since the gas temperature in the system varies (due to a heater), we shall use the first method of the superposition of losses, that is, summation of the absolute losses in the separate elements of the network, reduced in this case to the volumetric flow rate of air through the entry section of the fan (Figure 1.21, 7): ∆psys = ∑ ∆pi = ∑ i

i

∆Ni ρ7w2i , = ∑ ζi 2 Q7 i

where ρ7 = 1.4 kg/m3. The calculation of the resistance is given in Table 1.14. According to this table, for the fan to be selected, we have Qopi = 0.955 m3 ⁄ s and ∆psys = ∆ptot = 225 Pa . The power on the fan rotor at a fan efficiency of ηtot = 0.6 is Nsup =

Qopptot Q7∆psys 0.955.255 W = 0.36 kW . = = 0.6 ηtot ηtot

Figure 1.21. Scheme of calculation of the ventilation system network.

3 Elbow bend

2 Straight stretch (vertical)

1 Supply vent

Element of the system

Diagram and basic dimensions of the element

∆=0.0003

r = 0.2; _Dp _

δ = 90o;

= 0.0003

l = 8.0 D0 __ ∆ ∆= D0

h = 0.6 D0

Parameters

0.825

0.825

0.825

Qop,i, m3 ⁄ s

–20

–20

–20

ti, o C

1.40

1.40

1.40

ρi, kg/m3

1.17

1.17

1.17

vi × 105, m2/s

4.27

4.27

4.27

wi, m/s

10.94

10.94

= 10.94

1.2(4.27)/2z

ρtw2i , Pa 2

1.80

1.80

1.80

Re = wiDhi × 10−5 v

Table 1.14 Calculation of the resistance in the forced ventilation system (see Figure 1.21)

0.44

–

0.30

ζloc,i

0.018

0.018

–

λi

0.024

0.144

–

Dhi

ζfr,i = li λi ,

0.464

0.144

0.30

5.08

1.58

3.28

6.9

2.5

3.1

Basis for ∆pi = determination ζi = ζloc,i ρiwi2 of ζi ζi , + ζfr,i, 2 (reference Pa, to figure)

68 Handbook of Hydraulic Resistance, 4th Edition

7 Straight stretch (horizontal)

6 Sudden sharp contraction

5 Air heater with three rows of smooth pipes

4 Straight stretch (horizontal) ∆=

∆=

0.0003

l = 2.0; D0 __

F0 = 0.5 F1

kg/m2 s

ρmw0 = 3.86

0.0003

l = 2.0; D0 __

0.955

0.955

–

0.825

+20

+20

–

–20

1.20

1.20

–

1.40

1.5

1.5

–

1.17

4.95

4.95

–

4.27

14.7

14.70

–

10.94

1.64

1.64

–

1.80

–

0.25

–

–

0.0185

–

–

0.018

0.037

–

–

0.036

0.037

0.25

–

0.036

0.55

3.68

9.90

0.40

2.5

4.9

12.26

2.5

Aerodynamics and Hydraulics of Pressure Systems 69

12 Symmetrical smooth wye (dovetail) in discharge region (division of flow)

11 Straight stretch (horizontal)

10 Flow divider (Passage with division of flow)

9 Straight stretch (horizontal)

8 Pyramidal diffuser (rectangular cross section)

Table 1.14 (continued)

F1 F0

0.478

0.955

0.955

r = 1.5 Dt

Fbr =0.50; Ft

Qbr = 0.5; Qt

0.00056

∆=

l = 18.8 D0 __

0.239

0.478

wst =1.0; α = 15o wt

Qst = 0.5; Qt

Fst = 0.5; Ft

0.0004

∆=

l =10.7; D0 __

α = 10o

= 2.25

n=

+20

+20

+20

+20

+20

1.20

1.20

1.20

1.20

1.20

1.5

1.5

1.5

1.5

1.5

8.0

8.6

8.6

8.6

19.5

38.4

44.5

44.5

44.5

23.8

1.04

1.5

1.5

2.15

3.25

0.25

–

0

–

0.19

0.019

0.019

–

0.018

–

0.05

0.36

–

0.193

–

0.30

0.36

–

0.193

0.19

11.5

16.0

0

8.57

43.0

7.30

2.5

7.2

2.5

5.77

70 Handbook of Hydraulic Resistance, 4th Edition

17 Intake nozzle at the exit from the bend

16 Butterfly valve

15 Straight stretch (horizontal)

14 90o bend

13 Straight stretch (horizontal)

0.00077

∆=

0.239

l = 2.0; D0 r =0.20; D0 __

0.239

0.239

0.239

0.239

δ = 5o

0.00077

l = 20.5; D0 __ ∆=

0.00077

∆=

R0 = 2.0; D0 __

δ = 90o

0.00077

l = 20.5; D0 __ ∆=

+20

+20

+20

+20

+20

1.20

1.20

1.20

1.20

1.20

1.5

1.5

1.5

1.5

1.5

8.0

8.0

8.0

8.0

8.0

38.4

38.4

38.4

38.4

38.4

1.04

1.04

1.04

1.04

1.04

1.70

0.25

–

0.24

–

0.02

–

0.02

0.02

0.02

0.06

–

0.41

0.065

0.41

67.6

10.8

15.7

11.7

15.7

11.8

9.17

2.5

6.1

2.5

Σ ∆pi = ∆psus ≈ 225 Pa i=1

17

1.76

0.28

0.41

0.305

0.41

Aerodynamics and Hydraulics of Pressure Systems 71

72

Handbook of Hydraulic Resistance, 4th Edition

Example 1.2. Installation for the Scrubbing of Sintering Gases The schematic of the installation is shown in Figure 1.22. Given are: Total volumetric flow rate of the gas (at t = 20oC and p = 101.325 kPa), Q = 278 m3/s; Density of the gas at normal conditions, ρ = 1.3 kg/m3; Kinematic viscosity of the gas at normal conditions, v = 1.32 × 10–5 m2/s; Internal coating of the gas mains (comparatively long): sheet steel, roughness taken to be the same as for seamless corroded steel pipes (after several years of service), ∆ ≈ 1.0 mm (see Table 2.5); 5. Gas cleaning, done in a wet scrubber, rate of spraying A ≈ 0.014 m3/m2⋅s (see Diagram 12.11). 1. 2. 3. 4.

Figure 1.22. Scheme of calculation of the installation for scrubbing sintering gases: (a) plan of an installation; (b) side view.

4 Lateral branch of the distribution header

3 Distributing header

2 Compound bend

1 Right angle elbow t1 2500 = 0.5 b0 5000

l0 = 5.6 b0

__ ∆ ∆ = = 0.0002 b0

= 1.0 ;

Fch Fst4 7.5 =1– = 0.75 =1– Fsup F0 30

Fs 2 × 4 = Ft 7.5

wc

ws

. 1.0

= 1.0

Qs Qs = = 1.0 Qt Qst4

430

430

430

430 273 + 120 = 100 4 273 + 150

_ Σ F Σ Fs 4 × 2 × 4 = = f= = 1.07; α = 90o Fst F0 30

k=1–

__ ∆ 1.0 = = 0.00024 ∆= Dh,av 41.00

l0 6000 = = 14.5 Dh,av 410

w1n w2n w3n w4n wn = = = = 1.0 = w0 w0 w0 w0 w0

__ ∆ = 0.0002; α = 30o

F0

F1

120

150

150

150

0.90

0.84

0.84

0.84

Table 1.15 Calculation of the resistance in the system of installation for wet scrubbing of sintering gases (see Figure 1.22)

2.7

3.0

3

3

Aerodynamics and Hydraulics of Pressure Systems 73

9 Inlet to the flue

8 Straight horizontal stretch

7 Exit stretch of the scrubbersymmetric tee

6 Wet scrubber

5 Butterfly valve (at 10% closing, δ = 5o)

(see Diagram 12.11); F = 32 m2

Table 1.15 (continued) Fh

F0

= 0.9 (δ = 5o)

2

F0 4.2 = F1 15.8

+ 0.27

l0 10.5 = = 5.8 Dh 1.800 __ 1.0 = 0.0006 ∆= 1.800

Qs = 0.5 Qt

Ft 4.2 = = 0.33 2Fs 2 × 6.4

0.014 m /m s

3

At the entrance t = 120oC; at the exit t = 50oC; rate of liquid spraying A =

vi × 105 m2 ⁄ s

273 + 85 273 + 120

273 + 50 273 + 85

2 × 40 = 80

2 × 40 = 80

= 45.5 × 0.9 = 41

45.5

= 50 × 0.91 = 45.5

50

100 = 50 2

2

50

50

50

120 + 50

120

= 85

1.10

1.10

1.0

1.0

0.90

1.8

1.8

–

–

2.7

74 Handbook of Hydraulic Resistance, 4th Edition

Driving head in the entire flue

Exit from the flue

12 Second straight stretch of the flue

11 Transition passage-converging section

10 First straight stretch of the flue

l0 41.500 + 14 D0 3000 __ 1.0 = 0.00033 ∆ 3000 – –

F0 7.05 = 0.45 = F1 15.8 l0 2400 = 0.8 D0 3000 __ 1.0 α = 34o ∆ = 0.00033 3000

l0 22.0 = 4.9 D0 4.5 __ 1.0 = 0.00022 ∆= 4500

313 = 77.5 323

313 = 77.5 323

–

313 = 77.5 323 313 = 77.5 80 323 80

80

80

–

40

40

40

40

1.13

1.13

1.13

1.13

1.13

–

1.7

1.7

1.7

1.7

Aerodynamics and Hydraulics of Pressure Systems 75

11

10

9

8

7

11.0 –

Exit from the flue

Driving head in the entire flue

converging section

Transition passage11.0

4.9

of the flue

19.0

First straight stretch

19.0

9.8

Inlet to the flue

stretch

Straight horizontal

wye

scrubber-symmetric

Exit stretch of the

Wet scrubber

δ = 5o) 1.42

12.5

6

80

wst ≈ wt=13.3

Lateral branch

Butterfly valve

4

5

(at 10% closing,

86

14.3

Distributing header

3

–

51.0

540

10.1

152

152

40.4

0.85

65.6

86

14.3

Compound bend

86

14.3

Right angle elbow

ρiw2i , Pa 2

2

wi, m/s

1

Type of element

Table 1.15 (continued)

–

2.15

2.15

1.3

1.9

1.9

–

–

–

–

~2.0

2.6

2.6

Re = wiDh × 10–6 vi

–

1.0

–

–

0.53

–

≈2.0

960

0.28

–

2.6

0.20

0.72

ζloc,i

–

–

0.014

0.015

–

0.018

–

–

–

–

–

0.014

–

λi

–

–

0.20

0.074

–

0.10

–

–

–

–

–

0.079

–

ζfr,i = li λi Dh

–

1.0

0.2

0.074

0.53

0.10

2.0

960

0.28

–

2.6

0.28

0.72

ζi = ζloc,i + ζfr,i

∆pi =

–72.3

51.0

10.2

0.75

80

16.0

80.8

816

18.4

–

224

24.0

62.0

ρ1w2i ζi , Pa 2

i=1

Pa

ρ1 0.84 pst = pst = 72.3 1.13 ρ12 (reduced to Q1)

∑ ∆pi = ∆psys + 1320

12

pst,1 =

Equation (1.59): pc = z(ρa – ρg)g, where ρa = 1.29 at t = 0oC

2.5

2.5

4.1 (sudden expansion)

2.5

and a symmetric tee 90o (Diagram 7.29)

Tentatively as in Diagram 4.9 (sudden contraction)

12.11

9.17

7.40

6.13

6.10

Basis for determination of ζi (reference to diagram)

76 Handbook of Hydraulic Resistance, 4th Edition

Aerodynamics and Hydraulics of Pressure Systems

77

In the present case, the gas temperature in the system varies due to cooling; therefore, as done in Example 1.1, we use the first method of superposition of losses: summation of the absolute losses in the separate elements of the network, reduced to the volumetric flow rate, for example through section 0-0, that is, through the section of entry into an elbow 1 (Figure 1.22), where ρ = 0.18 kg/m3. The calculation of the resistance is given in Table 1.15. The self-draught created by the exhaust flue is equal to: ps = Hfl(ρa – ρg)g , where Hfl = 62 m is the height of the flue; ρa is the density of the atmospheric air, in kg/m3; ρg is the density of the gas at the inlet to the exhaust flue, in kg/m3; g is the gravitational acceleration, assumed to be 9.8 m/s2. At the temperature of atmospheric air tair = 0oC we have

ρa = 1.29 kg ⁄ m3 . At the temperature tg = 40oC, the mean density of the gas is

ρg = 1.13 kg ⁄ m3 , whence ps = 62(1.29 – 1.13) × 9.81 + 98 Pa . This positive self-draught (buoyancy) favors the motion of the stream, therefore it should be subtracted from the total losses (see Table 1.15). The power on the shaft of the flue blower intended only for this installation at ηtot = 0.6 is Nsup = Qopptot/ηtot = Q1∆psys/0.6 = 430.13200.6 W ≈ 946 kW.

Example 1.3. Low-Velocity, Closed-Circuit, Open-Throat Wind Tunnel A layout of the wind tunnel (aerodynamic circuit) is shown in Figure 1.23. Given are: 1. Diameter of the working section (nozzle exit section), D0 = 5000 mm; 2. Length of the working section, lws = 8000 mm; 3. Flow velocity in the working section (at the exit from the nozzle), w0 = 60 m/s; 4. Air temperature, t ≈ 20oC; ρ = 1.22 kg/m3; 5. Kinematic viscosity, ν = 1.5 mm2/s; and 6. Material from which the tunnel is made: concrete with roughness of the internal surface, ∆ = 2.5 mm (see Table 2.5). At low velocities, changes in the pressure and temperature along the tunnel can be neglected in hydraulic calculations. Therefore, it is convenient here to use the second method of superposition of losses: summation of the reduced resistance coefficients of the separate elements of the system (see Section 1.6).

78

Handbook of Hydraulic Resistance, 4th Edition

Table 1.16 Calculation of the resistance of the wind tunnel (Figure 1.23) Type of element 1 Circular open throat

2 First diffuser

Diagram and basic dimensions of the element

Parameters lws 8.0 = = 1.6 D0 5.0

2

F2 ⎛ 8.0 ⎞ = = 2.24 F1 ⎜⎝ 5.35 ⎟⎠ > 1.12 ; kd + 1.8

α = 7o ; n =

wmax w0 __ ∆ ∆ + 0.0004 D

_ 3 Adapter (from an annular section to a square

df =

_

df =

_

df F3 = 0.5 ; n = D2 Ff F_3 (1 – d2f )Ff

8.02 = 1.7 1 – 0.25(π ⁄ 4)8.02 __ α3 + 15o ; ∆ + 0.0004 wmax > 1.2 ; k1 + 1.8 w0

df =

4 Elbow 1 with reduced number of guide vanes

b4 = 1.0 b3 __ r = 0.2 ; ∆ = 0.0003 b3

5 Cylindrical stretch

__ l4 6.0 = = 0.75 ; ∆ = 0.0003 b4 8.0

6 Elbow 2; guide vanes as for elbow 1

b5 = 1.0 b4 r = 0.2 D5

Aerodynamics and Hydraulics of Pressure Systems

79

Table 1.16 (continued) Type of element 7 Reverse channel (second diffuser)

Diagram and basic dimensions of the element

Parameters α2 = 5.5o ; 2

F6 ⎛ 12.0 ⎞ = = 2.25 F5 ⎜⎝ 8.0 ⎟⎠ wmax > 1.1 ; k1 + 1.8 w0

n=

8 Elbow 3; same conditions as for elbow 1

9 Elbow 4; same conditions as for elbow 3, but number of guide vanes is normal

10 Honeycomb (coated sheet iron lacquered)

11 Nozzle (curvilinear converging section)

b7 = 1.0 b6 r = 0.13 D6

b8 = 1.0 b7 r = 0.13 D8

_ Fx lx = 7.5 ; f = = 0.9 dx F8 __ 0.2 = 0.001 ∆= 200

α3 + 35o ; n=

12.0 = 7.35 (π ⁄ 4)5.02

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Handbook of Hydraulic Resistance, 4th Edition

Table 1.16 (continued) Type of element Area ratio

Diagram and basic dimensions of the element ζloc.i

λi

ζfr,i = li λi Dh

ζi = ζloc,i + ζfr,i

20

0.13

–

–

0.13

⎛ F0 ⎞ ζi ⎜ ⎟ ⎝ Fi ⎠ 0.13

52.5

19

0.05

0.01

0.16

0.066

0.051

5.2

0.27

31

18

0.19

0.011

0.01

0.20

0.054

5.22

0.096

18

10

0.155

0.011

0.015

0.17

0.016

0.096 0.096 0.096

18 18 18

10 10 10

– 0.12 0.046

0.011 0.011 0.011

0.008 0.015 0.011

0.008 0.135 0.057

0.001 0.015 0.006

6.21; increased by 1.2 in order to allow for influence of diffuser placed before it 2.5 6.21 5.4

0.0181

8.2

6.5

0.15

0.011

0.015

0.165

0.003

6.21

6.5 widx = 0.12 v

Re =

2

F0 Fi

Parameters ζ0 =

⎛ F0 ⎞ ⎜ Fi ⎟ ⎝ ⎠

wi, m/s

1.0

60

0.77

1.0

wiDh v

× 10–6

2

Basis for determination of ζi (reference to figure) 4.25

2

⎛ 5.0 ⎞ ⎜ 5.35 ⎟ = 0.875 ⎝ ⎠

F0 Ff

F0 Ff

=

5.02 0.75 × 8.02

= 0.52

(π ⁄ 4)5.02 8.0

2

= 0.306

0.306 0.306 0.306

π 5.02

= 0.137 4 1202 0.137 0.137 = 0.152 0.9

0.0181

8.2

0.0232

9.1

54

60

7.35

20

0.17

0.011

0.015

0.185

–0.004

0.11

0.060

0.45

0.565

0.013

–

0.008

0.003

0.003

0.003

6.21 ζi = ζin + ζex + ζ_fr, where ζin + 1 – f (see Diagram _ 3.1); ζex = (1 – f)2 (see Diagram 4.1) ζfr = λ(lx ⁄ dx) 5.23 11

∑ = ∑ ζ0i + 0.30 0sys

i=1

The calculation of the tunnel resistance is given in Table 1.16. According to this table the total resistance of the tunnel is 11

∆psys = ∑ ζ0i i

ρw20 1.22 ⎞ 2 = 0.30 ⎛⎜ ⎟ 60 + 660 Pa . 2 ⎝ 2 ⎠

The volumetric air flow rate through the working section (nozzle) is Q = w0F0 = 60(19.6) = 1175 m3 ⁄ s . The power on the fan shaft at a fan efficiency ηtot ≈ 0.7 is: Nsup =

Q∆psys 1175(660) W + 1100 kW . = 0.7 ηtot

Aerodynamics and Hydraulics of Pressure Systems

81

Figure 1.23. Schematic diagram of a closed-circuit, open-throat wind tunnel (dimensions in m): D0 = 5; D1 = 5.35; D2 = 8; din = 4; b3 = 8; b4 = 8; b5 = 8; b6 = 12; b7 = 12; b8 = 12; t1 = 2.2; t2 = 1.5; lwork,sect = 8; ld = 13.5; lf = 2; ltr = 5; lcyl = 6; lel = 43.5; lh = 1.5; lch = 13.5; r = 1.6; α1 = 7o.

The aerodynamic calculations use the concept of the "quality" of a wind tunnel K, which is defined as the ratio of the velocity pressure in the working section of the tunnel to its total resistance. For the present case, K=

0.5ρw20 11

0.5ρw20

∑ ζ0i

=

1 0.30

+ 3.3 .

i

REFERENCES l. 2. 3. 4. 5. 6. 7. 8. 9.

Abramovich, G. N., Applied Gas Dynamics, Nauka Press, Moscow, 1969, 824 p. Altshul, A. D., Hydraulic Resistance, Nedra Press, Moscow, 1982, 224 p. Altshul, A. D. and Kiselyov, P. G., Hydraulics and Aerodynamics, Stroiizdat Press, Moscow, 1975, 327 p. Mochan, S. I. (Ed.), Aerodynamic Calculation of Boiler Equipment, Energiya Press, Leningrad, 1977, 255 p. Branover, G. G., Gelfgat, Yu. M., and Vasiliyev, A. S., Turbulent flow in a plane perpendicular to the magnetic field, Izv. Akad. Nauk Latv. SSR, Ser. Fiz.-Tekh. Nauk, no. 4, 78–84, 1966. Burdukov, A. P., Valukina, N. V., and Nakoryakov, V. E., Specific features of gas–liquid bubble mixture flow at small Reynolds numbers, Zh. Prikl. Mekh. Tekh. Fiz., no. 4, 137–139, 1975. Burdum, G. D., Handbook of the International System of Units, Izd. Standartov Press, Moscow, 1971, 231 p. Vakina, V. V., Discharge of viscous fluids at high pressure drop through throttling washers, Vestn. Mashinostr., no. 8, 93–101, 1965. Vitkov, G. A. and Orlov, I. I., Hydraulic calculations of systems from their overall characteristics (heterogeneous systems). Deposited at VINITI 28.01.1980 under No. 337–80, Moscow, 1980, 16 p.

82

Handbook of Hydraulic Resistance, 4th Edition

10. Vitkov, G. A. and Orlov, I. I., Hydraulic calculations of systems from their overall characteristics (homogeneous systems). Deposited at VINITI 28.01.1980 under No. 338–80, Moscow, 1980, 30 p. 11. Vulis, L. A., Paramonova, T. A., and Fomenko, B. A., Concerning the resistance to liquid metal flow in magnetic field, Magn. Gidrodin., no. 1, 68–74, 1968. 12. Hartman, U. and Lazarus, F., Experimental study of mercury flow in a homogenous magnetic field, in MHD-Flows in Channels, Garris, L. (Ed.), Moscow, 1963, 262 p. 13. Geller, Z. I., Skobeltsyn, Yu. A., and Mezhdivo, V. Kh., Influence of rings on flow discharge from nozzles and orifices, Izv. VUZ, Neft Gaz, no. 5, 65–67, 1969. 14. Genin, L. G. and Zhilin, V. G., Influence of the longitudinal magnetic field on the coefficient of resistance to mercury flow in a round tube, Teplofiz. Vys. Temp., vol. 4, no. 2, 233–237, 1966. l5. Guizha, E. A., Stabilization of Forced Turbulent Flows Downstream of Local Resistances, Thesis (Cand. of Tech. Sci.), Kiev, 1986, 186 p. 16. Gil, B. B., An approximate method for calculating the velocity field in the MHD-separation problems, in New Physical Methods for the Separation of Mineral Raw Materials, pp. 59–68, Moscow, 1969. 17. Guinevskiy, A. S. and Solodkin, E. E., Hydarulic resistance of annular channels, in Prom. Aerodin., no. 20, pp. 202–215, Oborongiz Press, Moscow, 1961. 18. Guinevskiy, A. S. and Solodkin, E. E., Aerodynamic characteristics of the starting length of a circular tube with a turbulent boundary layer flow, in Prom. Aerodin., no. 12, pp. 155–168, Oborongiz Press, Moscow, 1959. 19. Grabovsky, A. M. and Kostenko, G. N., Bases of the use of SI units, in Thermal and Hydraulic Calculations, Tekhnika Press, Kiev, 1965, 106 p. 20. Gubarev, N. S., Local resistance of the high-pressure air-pipeline fittings, Sudostroenie, no. 3, 41– 46, 1957. 21. Gukhman, A. A., Introduction to the Similarity Theory, Vysshaya Shkola Press, Moscow, 1963, 254 p. 22. Deich, M. E. and Zaryankin, A. E., Hydrogasdynamics, Energoatomizdat Press, Moscow, 1984, 284 p. 23. Elovskikh, Yu. P., Concerning the calculation of the parameters of gas in a pipeline, in Pneumatics and Hydraulics, no. 6, pp. 132–141, Moscow, 1979. 24. Zelkin, G. G., Hydraulic induction in discharging incompressible fluid into a full and empty pipeline with local resistances, Inzh.-Fiz. Zh., vol. 47, no. 5, 856–857, 1984. 25. Zelkin, G. G., Unsteady-State Flows in Local Resistance, Minsk, 1981, 141 p. 26. Zelkin, G. G., The phenomenon of hydraulic induction in unsteady-state motion of incompressible viscous fluid, Inzh.-Fiz. Zh., vol. 21, no. 6, 1127–1130, 1971. 27. Idelchik, I. E., Nozzles, in Large Soviet Encyclopedia, vol. 29, pp. 184–185, Sovetskaya Entsiklopediya Press, Moscow, 1953. 28. Idelchik, I. E., Fluid Resistances (Physical and Mechanical Fundamentals), Gosenergoizdat Press, Moscow, 1954, 316 p. 29. Idelchik, I. E., Some notes concerning hydraulic losses in motion of a real fluid in forced systems, Izv. VUZ, Energetika, no. 9, 99–104, 1975. 30. Kiselyov, P. G., Hydraulics, Fundamentals of the Mechanics of Liquid, Energoizdat Press, Moscow, 1980, 360 p. 31. Komlev, A. F., Skobeltsyn, Yu. A., and Geller, Z. I., Influence of the shape and dimensions of the entrance on the discharge coefficient of outer cylindrical nozzles, Izv. VUZ, Neft Gaz, no. 11, 59– 61, 1968. 32. Levin, V. B. and Chenenkov, A. I., Experimental investigation of the turbulent flow of an electrically conducting liquid in a tube in the longitudinal magnetic field, Magn. Gidrodin., no. 4, 147– 150, 1966. 33. Loitsyanskiy, L. G., Mechanics of Liquids and Gases, 5th ed. revised, Nauka Press, Moscow, 1978, 736 p.

Aerodynamics and Hydraulics of Pressure Systems

83

34. Lyatkher, V. M. and Prudovskiy, A. M., Hydrodynamic Modeling, Energoatomizdat Press, Moscow, 1984, 392 p. 35. Makarov, A. N. and Sherman, M. Ya., Calculation of Throttling Devices, Metalloizdat Press, Moscow, 1953, 283 p. 36. Malkov, M. P. and Pavlov, K. F., Handbook of Deep Cooling, Gostekhizdat Press, Moscow, 1947, 411 p. 37. Mergertroid, V., Experimental MHD-flows in channels, in MHD-Flows in Channels, Garis, L. (Ed.), pp. 196–201, Moscow, 1963, p. 262. 38. Mikheev, M. A., Filimonov, S. S., and Khrustalyov, B. A., Convective and Radiative Heat Transfer, Moscow, 1960. 39. Monin, A. S. and Yaglom, A. M., Statistical Hydromechanics, Part I, Fizmatizdat Press, Moscow, 1965, 640 p.; Part II, Nauka Press, Moscow, 1967, 720 p. 40. Nevelson, M. I., Centrifugal Ventilators, Gosenergoizdat Press, Moscow, 1954, 335 p. 41. Petukhov, B. S. and Krasnoshchekov, E. A., Hydraulic resistance in viscous nonisothermal motion of fluid in tubes, Zh. Tekh. Fiz., vol. 28, no. 6, 1207–1209, 1958. 42. Petukhov, B. S., Heat Transfer and Resistance in Laminar Liquid Flow in Tubes, Energiya Press, Moscow, 1967, 412 p. 43. Pisarevskiy, V. M. and Ponomarenko, Yu. B., Concerning variations in the gas density and pressure in local resistances of pipelines, Izv. VUZ, Mashinostroenie, 66–70, 1979. 44. Prandtl, L., Fundamentals of Hydro- and Aerodynamics, McGraw-Hill, 1934, Russian translation — GIIL Press, Moscow, 1953, 520 p. 45. Industrial Aerodynamics (Trudy TsAGI), no. 7 (Air Conduits), Moscow, 1954, 154 p. 46. Rikhter, G., Hydraulics of Pipelines, ONTI Press, Moscow, 1936, 340 p. 47. Sedov, L. I., Self-Similar and Dimensional Methods in Mechanics, Nauka Press, Moscow, 1967, 428 p. 48. Skobeltsyn, Yu. A., Mezhidov, V. Kh., and Geller, Z. I., Flow discharge from inner cylindrical nozzles with incomplete contraction due to a baffle or tapering, Izv. VUZ, Neft Gaz, no. 9, 71–74, 1967. 49. Skobeltsyn, Yu. A., Bashilov, E. B., and Geller, Z. I., Flow discharge from external cylindrical capillary nozzles, Izv. VUZ, Neft Gaz, no. 10, 80–84, 1971. 50. Solodkin, E. E. and Guinevskiy, A. S., Turbulent Flow of Viscous Fluid over the Starting Lengths of Axisymmetric and Plane Channels, Oborongiz Press, Moscow, 1957 (Trudy TsAGI No. 701). 51. Kiselev, P. G. (Ed.), Handbook of Hydraulic Calculations, 4th ed., Moscow, 1972, 312 p. 52. Handbook of Chemistry, Vol. 1, Goskhimizdat Press, Moscow, 1951, 1072 p. 53. Vargaftik, N. B. (Ed.), Handbook of the Thermal Properties of Liquids and Gases, Nauka Press, Moscow, 1972, 720 p. 54. Stepanov, P. M., Ovcharenko, I. Kh., and Skobeltsyn, Yu. A., Handbook of Hydraulics for Land Reclaimants, Kolos Press, Moscow, 1984, 207 p. 55. Stochek, N. P. and Shapiro, A. S., The Hydraulics of Liquid-Propellant Rocket Engines, Moscow, 1978, 127 p. 56. Tananayev, A. V., The Flow in the MHD-Equipment Channels, Atomizdat Press, Moscow, 1979, 364 p. 57. Blum, E. Ya., Zaks, M. V., Ivanov, U. I., and Mikhailov, Yu. A., Heat and Mass Transfer in the Electromagnetic Field, Riga, 1967, 223 p. 58. Fabrikant, N. Ya., Aerodynamics, Gostekhizdat Press, Moscow, 1964, 814 p. 59. Filippov, G. V., On turbulent flow over starting lengths of straight circular tubes, Zh. Tekh. Fiz., vol. 28, no. 8, 1823–1828, 1958. 60. Frenkel, V. Z., Hydraulics, Gosenergoizdat Press, Moscow, 1956, 456 p. 61. Khozhainov, A. I., Turbulent liquid metal flow in the MHD-channels of round cross section, Zh. Tekh. Fiz., vol. 36, no. 1, 147–150, 1966. 62. Jen, P., Stalling Flows, vol. 1, 298 p.; vol. 2, 280 p.; vol. 3, 300 p., Mir Press, Moscow, 1972.

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Handbook of Hydraulic Resistance, 4th Edition

63. Shiller, L., Flow of Liquids in Pipes, Tekhizdat Press, Moscow, 1936, 230 p. 64. Schlichting, H., Boundary Layer Theory, Nauka Press, Moscow, 1974, 711 p. 65. Shcherbinin, E. V., An electrically conducting liquid in the intrinsic magnetic field of the electric current, Magn. Gidrodin., no. 1, 68–74, 1975. 66. Shchukin, V. K., Heat Transfer and Hydrodynamics of Internal Flows in the Fields of Body Forces, Mashinostroenie Press, Moscow, 1970, 331 p. 67. Elterman, V. M., Air Screens, Mashinostroenie Press, Moscow, 1966, 164 p. 68. Yuriev, B. N., Experimental Aerodynamics, ONTI Press, Moscow, 1936, 315 p. 69. Barach, A. L., The flow of heavy gases through small orifices, including comparison between oxygen and perfluoropropane, C3F8 perfluorobutane, C4F10, Am. J. Med. Sci., vol. 243, no. 1, 30–34, 1962. 70. Benedict, P. and Carlucci, A., Handbook of Specific Losses in Flow Systems, Plenum Press, Data Division, New York, 1970, 30 p. 71. Boussinesq, I., Memoir sur l’influence des frottements dans les mouvements reguliers des fluides, J. Math. Pur Appl., no. 13, 377, 1868. 72. Forst, T. H., The compressible discharge coefficient of a Borda pipe and other nozzles, J. R. Aeronaut. Soc., no. 641, 346–349, 1964. 73. Iversen, H. W., Orifice coefficients for Reynolds numbers from 4 to 50,000, Trans. ASME, vol. 78, no. 2, 359–364, 1956. 74. Jackson, R. A., The compressible discharge of air through small thick plate orifices, Appl. Sci. Res., vol. A13, nos. 4–5, 241–248, 1964. 75. Kolodzie, P. A., Jr. and Van Winkle, M., Discharge coefficients through perforated plates, AIChE J., vol. 3, 305–312, 1959. 76. Maa Yer., Ru., Gas flow through an annular gap, J. Vac. Sci. Technol., vol. 5, 153–154, 1968. 77. Murakami, M. and Katayama, K., Discharge coefficients of fire nozzles, Trans. ASME, vol. D88, no. 4, 706–716, 1966. 78. Wielogorski, J. W., Flow through narrow rectangular notches, Engineer, vol. 221, 963–965, 1966.

CHAPTER

TWO RESISTANCE TO FLOW IN STRAIGHT TUBES AND CONDUITS FRICTION COEFFICIENTS AND ROUGHNESS

2.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS 1. The pressure losses along a straight tube (conduit) of constant cross section (linear or friction losses) are calculated from the Darcy–Weisbach equation: ∆pfr =

2 Π0 ρw20 λ l ρw20 λ s0 ρw0 =λ l = 4F0 2 4 F0 2 4 Rh 2

(2.1)

or ∆pfr = λ

2 ρw20 l ρw0 =ζ , 2 Dh 2

(2.2)

where Π0 is the perimeter; Rh is the hydraulic radius; s0 is the area of the friction surface. 2. The use of the hydraulic (equivalent) diameter Dh as the characteristic length in resistance Equations (2.1) and (2.2) is permissible only in cases where the thickness δ0 of the boundary layer (within which the velocity changes from zero to nearly a maximum value) is very small over the entire or almost the entire perimeter of the cross section compared with the dimensions of the channel cross section (δ0 ∆; (b) δl < ∆.

thickness of the viscous sublayer is larger than roughness protuberances (δl > ∆, Figure 2.la), the latter are entirely covered with this layer. At low velocities, typical of a laminar sublayer, the fluid moves smoothly past surface irregularities and they have no effect on the character of the flow. In this case, λ decreases with a rise in Re. 11. With an increase in the Reynolds number, the laminar sublayer becomes thinner and, at Re attaining a certain value, it can become smaller than the height of the asperities (δl > ∆, Figure 2.lb). The asperities enhance the formation of vortices and hence increase the pressure losses, which result in the rise of λ with increasing Re. Thus, tubes can be considered smooth as long as the height of asperities is smaller than the thickness of the laminar sublayer. 12. The equivalent roughness ∆ depends on: • The material of tubular products and the method by which they were manufactured. For example, iron pipes manufactured by centrifugal casting are smoother than welded tubes. Tubes manufactured by the same method have, as a rule, the same equivalent roughness irrespective of their diameter. • The properties of the fluid flowing in a tube; liquids may cause corrosion on the inner surface of the tube, resulting in formation of protuberances and deposition of scale. • The service life and history of the tubes. 13. In the dependence of the resistance coefficient (λ) on Reynolds number (Re) for smooth tubes, several characteristic values of Re can be separated. At Re + 1000 the values of the resistance coefficient λ for a steady-state (stabilized) laminar flow and steady-state turbulent flow coincide if they are calculated from the Hagen–Poiseuille formula for a laminar flow: λ = 64%Re

(2.3)

and from the Prandtl formula for a turbulent flow: 1%√ ⎯⎯λ = 2 log √ ⎯⎯λ Re – 0.8. Therefore, when Re < 1000, a steady-state flow can only be laminar. When Re > 1000, at a certain Reynolds number, which depends on perturbations at the tube inlet, a steady-state laminar flow is replaced by a steady-state turbulent one. On increase in the intensity of perturbations at the inlet, the minimum Reynolds number at which a steady-state turbulent flow was observed is decreased to a certain value. Different values are presented for this quantity in the literature by different authors: from 1900 to 2320.268 It was shown by A. A. Paveliev et al.269 that the value of this quantity depends not only on the intensity but also on the structure of perturbations at the tube inlet. Based on the available experimental data, this prevents the statement that a steady-state turbulent flow cannot be realized at Reynolds values smaller that 1900.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 2.2. Dependence of the resistance coefficient λ on Re for tubes with uniform-grain roughness.190

As the intensity of perturbations at the tube inlet decreases, the maximum Reynolds number at which a steady-state laminar flow can be observed increases. With the flow at the tube inlet being specially organized, this Reynolds number can reach a value of 105. At a high intensity of perturbations at the inlet with Re > 2000, the formation of a steadystate laminar flow requires the starting length of the tube to be equal to about 200 tube diameters.269 Over this length the initial perturbations damp out, and a velocity profile typical of a laminar flow is formed. 14. The dependence of the resistance coefficient λ on Re and roughness has been established in Nikuradse’s experiments for a stabilized flow (see Sec. 1.3) in tubes with uniformgrain roughness* (Fig. 2.2). In Nikuradse’s experiments with a laminar steady-state flow, roughness does not exert its effect on the value of λ. In the range of Re numbers corresponding to a transient region between a laminar and a turbulent flow λ increases with Re. The greater the roughness, the less valid is the Blasius formula for a steady-state turbulent flow in smooth tubes λ=

0.3164 Re0.25

.

(2.4)

With increases in Re, the dependence of λ on Re deviates from the dependence which is described by the Blasius formula, and λ tends to a constant value which is the higher the greater the relative roughness. ∗ A form of artificial sand uniform-grain roughness is meant here, as obtained by Nikuradse. The curves for other forms of artificial roughness can differ somewhat.152

Flow in Straight Tubes and Conduits

89

15. It has been found experimentally that with the wall roughness being somewhat organized, the resistance coefficient for a turbulent flow can be lower than that calculated by the formula for smooth tubes. As an example of such an ordered roughness mention can be made of longitudinal depressions in the tube walls called riblets.270 16. The third regime is called the quadratic or square-law regime, the regime of rough walls, and sometimes the regime of turbulent self-similarity. It is characterized by the resistance coefficients for each value of the relative roughness becoming constant, independent of Re. 17. It follows from Nikuradse’s87 resistance equations for rough tubes [see Equation (2.5)] and Filonenko171 and Altshul’s6 resistance equation for smooth tubes [see Equation (2.8)] that tubes with uniform-grain roughness can be considered hydraulically smooth provided that __ __ ∆ ≤ ∆lim , where __ 181 log Re − 16.4 ⎛∆⎞ . ∆lim = ⎜ ⎟ = Re D ⎝ 0 ⎠ lim For the range of Reynolds numbers up to Re = 105, the Blasius foimula gives __ ∆lim + 17.85 Re−0.875 . From this, the boundary (limiting) values of the Reynolds number, at which roughness begins to be important, can be defined as 26.9 Re′lim = __ 1.143 . ∆ 18. For tubes with uniform-grain roughness the limiting value of the Reynolds number, for which the quadratic law of resistance will hold, is determined from __ 217 382 log − ∆ __ Re′′lim = ∆ which follows from Nikuradse’s87 formula for a stabilized flow in the transitional and quadratic regions, i.e., within the limits __ __ log ∆ . __26.9 ≤ Re ≤ 217 − 382 ∆ ∆ 1.143 This formula has the following form: λ=

1

__ , [a1 + b1 log (Re √ ⎯⎯λ + c1 log ∆)]2

(2.5)

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Handbook of Hydraulic Resistance, 4th Edition

__ where at 3.6 ≤ ∆ Re √ ⎯⎯λ ≤ 10 a1 = −0.8, b = +2.0, c1 = 0 (smooth walls) ; __ at 10 ≤ ∆ Re √ ⎯⎯λ ≤ 20 a1 = +0.068, b = 1.13, c1 = −0.87 ; __ at 20 ≤ ∆ Re √ ⎯⎯λ ≤ 40 a1 = +1.538, b1 = 0, c1 = −2.0 ; __ at 40 ≤ ∆ Re √ ⎯⎯λ ≤ 191.2 a1 = +2.471, b1 = −0.588, c1 = −2.588 ; __ at ∆ Re √ ⎯⎯λ > 191.2 a1 = +1.138, b1 = 0, c1 = −2.0 , which* 1

(2.6) . 2 3.7 ⎛2 log __ ⎞ ⎜ ∆ ⎟⎠ ⎝ 19. Proceeding from the assumption that laminar and turbulent flows occur simultaneously and applying the normal law of distribution to determine the probability of occurrence of respective regimes, Kerensky50 suggested a single formula for the friction resistance of tubes with uniform-grain roughness for the zone of the replacement of regimes** λ=

∗

Tep1ov,128 having made a more rigorous treatment of Nikuradse’s87 experiments, has shown that for the quadratic region the following formula is more appropriate: λ=

1

. 2 ⎛1.8 log 8.3 ⎞ __ ⎜ ∆ ⎟⎠ ⎝ Closely coinciding with this is the formula suggested by Altshul:6 λ=

1 __ . (1.8 log 10 ⁄ ∆)2

However, taking into account that Equation (2.6) deviates from the experiment data by not more than 5%, but to the side of adding to the safety margin, it can be used in practical calculations of pipelines including the nonlined power conduits.7 ∗∗

Treatment of turbulent flows in a boundary layer and tubes utilizing the principle of superposition of molecular and turbulent viscosities allowed Millionshchikov77,78 to find a general formula for the friction resistance which is applicable for all flow regimes. About the same approach was used in the work of Svirsky and Platon107 and also in the work of Adamov.4

Flow in Straight Tubes and Conduits λ = λlam plam + λsm psm + λr pr ,

91 (2.7)

where λlam, λsm, and λr are the friction resistance coefficients in laminar (2.3) and turbulent flows in smooth (2.4) and rough tubes (2.6); plam = 1 − pt psm = (1 − pr.t)pt pr = pr.tpt are the probabilities for the appearance of laminar and turbulent regimes of flow in smooth and rough tubes at the given Reynolds number; in this case, pr.t = 2φ(U) , where U=

275 1 Re , σr = __ , pt = + φ(U) , 2 σr ∆

where

Figure 2.3.__ Dependence of the resistance coefficient λ on the Reynolds number Re and the relative roughness ∆ for nonuniform roughness.100,106

92

Handbook of Hydraulic Resistance, 4th Edition

U=

Re − Rer Re − 2850 1 ; φ(U) = = 600 σt ⎯⎯2π √ ⎯

U

∫

exp (−t2 ⁄ 2)dt

0

is the normalized Laplace function (the integral of probabilities; see Reference 56, Table 18.8-9). __ 20. The resistance curves λ = f(Re, ∆) for a stabilized flow in tubes with nonuniform roughness (commercial tubes) show that for this case there also exist three main flow regimes: laminar, transition, and quadratic (Figure 2.3). However, unlike the case of flow in tubes with uniform-grain roughness, here one should take into account the following two specific features: • For different degrees of roughness the resistance coefficient is not the same over the section related to the transition region between the laminar and turbulent flows (critical zone or zone of change of regime) the resistance coefficient depends on the relative roughness and on the Reynolds number; pressure losses in this zone are proportional to the velocity raised to a power greater than two.100 • The transition segment of a purely turbulent regime is free of a deflection typical of the curves of λ in tubes with uniform-grain roughness (see Figure 2.2). In this case, there is a progressive and smooth decrease in the resistance curves with increase in Re, with the lowest position being attained in the quadratic regime.82,171 21. The__curves of the friction coefficients for commercial tubes with relative equivalent roughness ∆ > 0.007 will, at some value of Re, depart from the Hagen–Poiseuille law to the side of higher λ’s, and the greater the relative roughness, the earlier this departure will occur (see Figure 2.3). The Reynolds number corresponding to the onset of this departure can be determined from the equation suggested by Samoilenko:106 0.0065 Re0 = 754 exp ⎛⎜ __ ⎞⎟ . ⎝ ∆ ⎠ 22. There __ is a transition curve with boundaries Re1 and Re2 which corresponds to each value of ∆ in the region where __ a change in flow regime occurs in the range Re1 and Re2 (see

Figure 2.3). For tubes with ∆ > 0.007 the following equation is obtained, based on the data of Samoilenko:106 0.11

1 Re1 = 1160 ⎛⎜ __ ⎞⎟ , ⎝∆⎠ __ that is, Re1 decreases with an increase in the relative roughness ∆. The Reynolds number that determines the Re2 boundary for tubes with any roughness is 1 Re2 = 2090 ⎛⎜ __ ⎞⎟ ⎝∆⎠

0.0635

.

23. At Re1 > Re2, tubes with nonuniform roughness (commercial tubes) can be considered hydraulically smooth (with an accuracy up to 3–4%) provided that

Flow in Straight Tubes and Conduits

93

__ __ 15 . ∆ < ∆lim + Re From this, the limiting Reynolds number at which commercial tubes cease to be hydraulically smooth is 15 Re′lim + __ . ∆ 24. In the case of nonuniform roughness, the limiting Reynolds number at which the quadratic law of resistance will hold can, within 3–4% accuracy, be written as (see Diagram 2.4): 560 Re′′lim + __ . ∆ 25. For a stabilized laminar flow (up to Re ≈ 2000) the resistance coefficient λ for circular tubes, which is independent of the relative roughness of walls, is determined from Equation (2.3) or from curve a of Diagram 2.1. 26. For the critical region of a stabilized flow (Re = 2000–4000), the friction coefficient λ of a circular tube with hydraulically (commercially) smooth walls is determined from curve b of Diagram 2.1. 27. For the region of purely turbulent stabilized flow (Re > 4000) the friction coefficient λ of circular tubes with hydraulically (commercially) smooth walls is determined from curve b of Diagram 2.1 or is calculated from Filonenko141 and Altshul’s6 formula* λ=

1 (1.8 ln Re − 1.64)2

.

(2.8)

28. The friction coefficient λ for stabilized flow in the transition region where there occurs __ a change of flow regime is determined from curve λ = f(Re, ∆) of Diagram 2.3 or from formulas suggested by Samoilenko:106 __ at Re0 < Re < Re1 and ∆ ≥ 0.007 λ = 4.4 Re−0.595 exp −

0.00275 __ ; ∆

at Re1 < Re < Re2 ⎧ ⎫ λ = (λ2 − λ∗) exp ⎨⎩−[0.0017 (Re2 − Re)]2⎬⎭ + λ∗ , __ __ where at ∆ ≤ 0.007, λ∗ = λ1, and at ∆ = 0.0007, λ∗ = λ1 – 0.0017. The coefficients λ1 and λ2, which correspond to Re1 and Re2, are

∗

This formula is very similar to the formulas of Konakov54 and Murin.82

94

Handbook of Hydraulic Resistance, 4th Edition __ at ∆ ≤ 0.007

λ1 + 0.032 ,

__ at ∆ > 0.007

0.0109 λ1 = 0.0775 − __ 0.286 , ∆

__ at ∆ ≤ 0.007

λ2 = 7.244 (Re2)−0.643 ,

__ at ∆ > 0.007

0.145 , λ2 = __ ∆ −0.244

0.0065 Re0 = 754 exp ⎛⎜ __ ⎞⎟ , ⎝ ∆ ⎠ 0.11

1 Re1 = 1160 ⎛⎜ __ ⎞⎟ ⎝∆⎠

,

0.0635

1 Re2 = 2090 ⎛⎜ __ ⎞⎟ ⎝∆⎠

.

29. For a stabilized flow and the region of purely turbulent flow (Re > Re2), the friction coefficient λ of all commercially circular tubes* (with nonuniform roughness of walls), except for special cases for which the values of λ are given separately, can be determined from the curves of Diagram 2.4 plotted on the basis of the Colebrook–White** formula:171 λ=

1 __ [2 log (2.51 ⁄ Re √ ⎯⎯λ ) + ∆ ⁄ 3.7]2

(2.9)

or for engineering calculations, from Altshul’s6 approximate formula*** __ 68 0.25 . λ = 0.11 ⎛∆ + ⎞ ⎝ Re ⎠

(2.10)

30. For the region with a change of regime of a stabilized flow in commercial tubes, a single formula can also be used to calculate the friction coefficient (as suggested by Adamovich), viz. ∗

Including steel, concrete, and iron-concrete pressure tunnels.7

∗∗

The Colebrook–White curves lie somewhat above (by 2–4%) similar curves of Murin82 and hence provide some safety margin for the calculations. Analogous formulas were obtained by Adamov,3 Filonenko,141 and Frenkel.144 The interpolation formula of Colebrook has now been theoretically substantiated6. __ ∗∗∗

4 The formula close to Equation (2.10) __ was also obtained by Adamov ; at 68/Re < ∆ it coincides 0.25 with the formula of Shifrinson: λ = 0.11(∆) . There is another simple formula convenient for application __ in the transient region (within λ = 0.0001–0.01) which was suggested by Lobaev: λ = 1.42/(log Re/∆)2.

Flow in Straight Tubes and Conduits

95

λ = λlam (1 − p) + λtp , where λlam is determined from Equation (2.3), λt from Equation (2.9) or (2.10), and ⎛ Re0 ⎞⎤ 1 ⎡ ⎛ Re − Re0 ⎞ erf ⎟ + erf ⎜ ⎥ 2⎢ ⎜ √ ⎯⎯2 σ ⎟⎠⎦ ⎣ ⎝ ⎯⎯2 σ ⎠ ⎝√ __ in which Re0 = 1530(∆)–0.08 and σ = 540. Here the tabulated function of errors is used in the form p=

2 erf (z) = ⎯⎯π √

z

∫

exp (−t2)dt

0

(see Reference 56, Table 18.8–10). 31. A single formula for calculating the friction coefficient in the zone with the change of regimes was also suggested by Slisskiy110 λ = λlam (1 − γ) + λtγ , where γ is the intermittency factor; Re − Relow ⎞ ⎛ γ = sin3 ⁄ 2 ⎜π ⁄ 2 ⎟, Reup − Relow ⎠ ⎝ 0.00465 Relow = 1000 exp ⎛⎜ __ ⎞⎟ , ∆ ⎠ __ ⎝ Reup = 1600(∆) −0.16 , where Relow and Reup are the lower and upper boundaries of the transition zone. The coefficients λlam and λt are calculated respectively from Equation (2.3) and from Teplov’s128 formula 8.25 __ ⎞ λt = ⎛⎜1.8 log Re + ∆ ⎟⎠ 56 ⁄ ⎝

−2

.

32. The friction coefficients λ for circular tubes, except in special cases for which the values of λ are given separately with any kind of roughness (both uniform __ and nonuniform) for stabilized flow in the quadratic region, i.e., virtually when Re > 560/∆, are determined from the graphs of Diagram 2.5 plotted on the basis of Equation (2.6). The specific feature of flow in channels with the complex geometry of cross sections is the presence of convective transfer across the flow due to the motion of large-scale vortices and secondary flows (Figure 2.4).* This fact and also the variable roughness of the channel walls ∗

It is imperative to distinguish between the secondary flows observed in straight channels of complex cross section and those originating for other reasons in curvilinear channels.

96

Handbook of Hydraulic Resistance, 4th Edition

Figure 2.4. Schemes of secondary flows: (a) in a rectangular tube; (b) in an equilateral triangular tube.

are responsible for the nonuniform distribution of shear on the flow boundaries. Therefore, the friction coefficients can be calculated most accurately when replacing the flow characteristics averaged over the channel cross section (mean velocity, Reynolds number, mean relative roughness, mean shear stress) by the local characteristics (local relative roughness, local Reynolds number, local friction factors of hydraulics, local shear stresses).133 As the local governing flow parameters, it is recommended to use the local characteristic dimension of flow, the flow velocity averaged over this dimension, and the local roughness of walls. The other local characteristics of the flow are expressed in terms of these determining quantities. 33. The local shear stress τw at the point of the wetted perimeter is expressed in terms of the local velocity ww averaged over the normal to the wall: τw = λloc

ρw2w , 2

where λloc is the local friction coefficient, being a function of the local Reynolds number and local relative roughness ⎛ wwl ∆ ⎞ , ⎟, λloc = f ⎜ l⎠ ⎝ v l is the characteristic local dimension of the flow, depending on the shape of the channel cross section (for example, for a square channel l is the distance from the wall to the corner bisector). 34. In the specific case of a rectangular channel, for which it is assumed that the shear stresses on the long and short sides of it differ, but their distribution over the walls is uniform, Skrebkov112,113 and Skrebkov and Lozhkin114 suggested a formula which relates the friction coefficient of the channel with its shape and roughness: λ=4

b⁄h 1+b⁄h

⎛ λsh ⎜1 + λlong ⎝

h⎞ , λ b ⎟ long ⎠

Flow in Straight Tubes and Conduits

97

where λsh and λlong are the friction coefficients respectively on the short and long walls of the channel; b and h are the halves of the width and of the height of the channel, respectively. The coefficients λsh and λlong are calculated by the laws of resistance of a plane wall (λpl) depending on the characteristic Reynolds number and wall roughness:112–114 __ (λpl)sh = f ⎡(Repl)b, ∆sh⎤ , ⎦ ⎣ __ (λpl)long = f ⎡(Repl)h, ∆long⎤ , ⎦ ⎣ where (Repl)b =

Re =

Re Re ⎛ 1 + b ⁄ h ⎞ , (1 + b ⁄ h) , (Repl)h = 4 4 ⎜ b⁄h ⎟ ⎠ ⎝

w0Dh . v

For smooth walls λpl =

1 (3.6 log Repl − 2)2

,

for commercial walls ∆ 54 λpl = 0.024 ⎛⎜ + ⎞ Repl l ⎟ ⎠ ⎝

0.25

,

for rough walls λpl =

1 4 log l ⁄ ∆ + 3.48)2

.

35. In many cases it is easier to determine the resistance coefficient of noncircular tubes by the introduction into the formulas for circular tubes the corresponding correction factors λnonc = knoncλ, where λ is the friction coefficient of circular tubes at the same Reynolds number, Re = w0Dh/v = w0D0/v; λnonc is λ for noncircular tubes; knonc is the correction factor allowing for the effect of tube cross-sectional shape.* 36. For tubes with nearly circular cross sections (for example, a circle with one or two notches, starlike shapes, see Diagram 2.6), it can be assumed, according to the data of Nikuradse87 and Shiller,158 that knonc ≈ 1.0 for all flow regimes. ∗

A. G. Temkin125,126 suggests calling the correction factor knonc the criterion of Leibenson (Le), who made an important contribution to the hydraulics of pipelines. ln the works cited, Temkin gives corresponding formulas to calculate the number Le.

98

Handbook of Hydraulic Resistance, 4th Edition

For rectangular tubes for laminar flow (Re ≤ 2000), the correction factor, which depends on the aspect ratio a0/b0, lies in the range knonc = krec = 0.89–1.50. When a0/b0 = 1.0 (square), krec = kquad = 0.89 or λquad =

57 Re

and when a0/b0 → 0 (plane slot), krec = kpl = 1.50 or λpl =

96 . Re

For turbulent flow (Re > 2000), krec = 1.0–1.1. When a0/b0 = 1.0, kquad ≈ 1.0, and when a0/b0 → 0, kpl ≈ 1.1.40,180 37. For elliptical tubes in laminar flow (Re ≤ 2000) the correction factor, which depends on the ratio of the ellipse axes (see Petukhov95), is determined as 2 ⎛ b0 ⎞ 1 ⎛ Dh ⎞ ⎡ knonc * kell = ⎜ ⎟ ⎢1 + ⎜ ⎟ 8 bo ⎣ a ⎝ ⎠ ⎝ 0⎠

2

⎤ ⎥, ⎦

where a0 and b0 are the major and minor semiaxes of the ellipse. For turbulent flow this factor can be approximated as kell ≈ 1.0. 38. For a circular annular tube (a tube within a tube) the correction factor, which is a function of the diameter ratio d/D0, (see Leibenson68 and Petukhov95), can be found for laminar flow (Re ≤ 2000) from knonc = kann =

(1 − d ⁄ D0)2 1 + (d ⁄ D0)2 + [1 − (d ⁄ D0)2] ⁄ ln d ⁄ D0

,

where d and D0 are the diameters of the inner and outer cylinders of the annular tube. In the case of turbulent flow, kann depends only slightly on d/D0 and lies in the range 1.0–1.07.29 The resistance coefficient λann of such a tube can also be calculated from the following formula:39 1 d d λann = ⎛⎜0.02 + 0.98⎞⎟ ⎛⎜ − 0.27 + 0.1⎞⎟ . D0 D0 λ ⎝ ⎠⎝ ⎠ 39. The inner cylinder of a circular annular tube is centered by means of longitudinal or spiral fins (see Diagram 2.7). A narrow annular tube (d/D0 ≈ 0.9) with three longitudinal fins is approximately equivalent to a rectangular channel with aspect ratio a0/b0 ≈ 0.06, for which, in the case of laminar flow, the correction factor (based on the experiments of Subbotin et al.120), is k′ann = krec ≈ 1.36. For turbulent flow, the correction factor can be taken the same as that for an annular tube with no fins (according to paragraph 38).

Flow in Straight Tubes and Conduits

99

40. For an annular tube with spiral fins, the correction factor, which depends on the relative pitch of the winding of fins, T/d (see Diagram 2.7) can approximately be determined for all the flow regimes from the following formula:120 k′′ann = 1 +

20 (T ⁄ d)2

k′ann ,

where k′ann is the correction factor for an annular tube with longitudinal fins. 41. The friction coefficient of an eccentric annular tube (see Diagram 2.7) for both laminar and turbulent flows depends on the eccentricity and the relative width of the annular channel. 42. The correction factor for laminar flow is calculated from the approximate formula of Gostev and Riman30 knonc = kell =

1

_ kann , (1 + B1e)2

_ where e = 2e/D0 – d is the eccentricity (e is the distance between the centers of the inner and outer cylinders); B1 is a coefficient that depends on the ratio d/D0, obtained on the basis of the data of Johnston and Sparrow178 (see Diagram 2.7, graph c); and kann is the correction factor for a concentric ring. 43. The correction factor for turbulent flow is kell = k′ellkann , where k′ell = λ/λann is the ratio of the resistance coefficient of an eccentric annular tube to the resistance coefficient of a concentric annular tube. The coefficient k′ell for narrow annular channels (d/D0 ≥ 0.7) is nearly independent of d/D0 and is a function only of eccentricity (see graph d of Diagram 2.7 for d/D0 = 0.5 and d/D0 ≥ 0.7). When d/D0 ≥ 0.7, the correction factor can be determined from the formula of Kolesnikov:21 _ _ k′ell = 1 − 0.9 (1 − 2 ⁄ 3e) e 2 . 44. The correction factor knonc for laminar flow in tubes with a cross section in the form of an isosceles triangle (see Migai76) is knonc = ktr =

(1 − tan2 β)(B + 2) 3 , 4 (B − 2)(tan β + √ 1 + tan2 β ) 2 ⎯⎯⎯⎯⎯⎯⎯

where B=

4 + ⎛⎜ 2 ⎯ − 1⎞⎟ √⎯⎯⎯⎯⎯⎯ ⎯ 2 tan β ⎝ ⎠ 5

1

100

Handbook of Hydraulic Resistance, 4th Edition

is a parameter, and β is half the apex angle of the isosceles triangle, in degrees. For an equilateral triangle (β = 30o) k′′tr = 0.833 . For a right triangle k′′tr =

(1 − 3 tan2 β)(B + 2) 3 2 (3 ⁄ B − 4)(tan β + √ 1 + tan2 β )2 ⎯⎯⎯⎯⎯⎯⎯

and for an equilateral rectangular triangle (β = 45o) k′′tr = 0.825 . 45. In the case of turbulent flow, the correction factor knonc for an equilateral triangle varies in the range knonc = 0.75–1.0 depending on the angle β: the larger the angle, the higher ktr.170 For an equilateral triangle we may assume ktr = 0.95.158 46. The correction factor for laminar flow for a tube with a cross section in the form of a circular sector is knonc = ksec = 0.75–1.0, depending on the angle β;5 for turbulent flow ksec can be assumed the same as for an equilateral triangle (paragraph 45). 47. The resistance of the starting length of tubes (immediately downstream of a smooth inlet), which are characterized by a nonstabilized flow (see Section 1.3), is higher than in the sections with stabilized flow. The closer to the inlet, the higher is the friction coefficient λnonst of the section of a nonstabilized flow. This is due to the fact that with a smooth entrance the boundary layer in the initial sections is much thinner than in subsequent ones, and consequently the shear forces at the walls in these sections are higher. This refers to both nonstabilized laminar and nonstabilized turbulent flow if it is already entirely agitated at the inlet to the tube. 48. In the case of a very smooth entrance, when at Re > Recr a "mixed" flow regime sets in, the coefficient knonst of short tubes (whose length is much shorter than the starting length) is, within certain ranges of the Reynolds number, much smaller for a stabilized turbulent flow which is due to the laminar behavior of the boundary layer in the inlet section of the tube (see Section 1.3). At Re = 2 × 105, the average friction coefficient for a short tube of length l/D0 = 2.0 is seven- to eightfold lower than λ for a stabilized flow (Figure 2.5, see also Filippov138). 49. Creation of conditions under which the flow becomes turbulent in the boundary layer at the inlet into the tube leads to an increase in the coefficient λnonst for short lengths as well (see Figure 2.5). Therefore, for short tubes in real devices (in which the flow at the inlet is very much perturbed as a rule), the local value of the friction coefficient λ′nonst should be determined, for example, from the formula of Sukomel et al.122 for the conditions of turbulent boundary layer flow λ′nonst *

∆p ρw20 ⁄ 2

× ∆x ⁄ D0

=

0.344 (Re × x ⁄ D0)0.2

= k′nonstλ ,

(2.11)

101

Flow in Straight Tubes and Conduits

Figure 2.5. Dependence of the friction coefficient λ on Re for a short starting length (lst/D0 = 2) with smooth walls: (1) test section is installed immediately downstream of a smooth inlet (l0/D0 = 0); (2) upstream straight section of length l0/D0 = 0.4 is installed between the smooth inlet and the test section; (3) relative length of the upstream section is l0/D0 = 4.3; (4) the resistance curve is according to Blasius; (5) Hagen–Poiseuille curve.

where k′nonst + 1.09

Re0.05 (x ⁄ D0)0.2

,

λ is the friction coefficient of a stabilized flow; ∆x = x1 – x2 is a small portion of the tube length from x1 to x2. The average value of the friction coefficient λ′′nonst over the entire given length l of the starting section can be calculated for the conditions of a turbulent boundary layer flow from another equation of the same authors λnonst * where

∆p ρw20 ⁄ 2

× ∆x ⁄ D0

=

0.43 (Re × x ⁄ D0)0.2

= k′′nonstλ ,

(2.13)

102 k′′nonst + 1.36

Handbook of Hydraulic Resistance, 4th Edition Re0.05 (x ⁄ D0)0.2

,

(2.14)

Equations (2.11) to (2.14) are correct at least within the range 1.7 × 104 ≤ Re ≤ 106. For practical calculations these equations can also be used in the case of noncircular channels; moreover, the upper limit of Re can be raised. The values of λ′nonst and λ′′nonst are listed in Table 1 of Diagram 2.21. 50. In the case of high sub- and supersonic velocities of a gas flow, that is, in the case of a compressible gas, both under the conditions of cooling and in an adiabatic flow, the friction coefficient for the conditions of a turbulent boundary layer flow is122 ~ λ′com = λ′nonst [τ(λ)]0.4 and accordingly ~ λ′′com = λ′′nonst [τ(λ)]0.4 , ~ where τ(λ) is the gas dynamic function determined from Equation (1.47); λ′nonst and λ′′nonst are found from Equations (2.11) and (2.13), respectively. 51. The friction coefficient of the starting length for nonstabilized laminar flow is calculated from the equation similar to Equation (2.13) in which knonst is a function of the parameter x/(DhRe). It is determined from Table 2 of Diagram 2.21 obtained on the basis of Frenkel’s data.44 52. Flow channels made of bundles of circular cylinders (tubes or rods), such as are widely used in many heat-exchanging systems (e.g., fuel elements of atomic reactors or tubes in conventional heat exchangers), have flow cross sections of shapes other than circular. Usually, the cylinders (rods) in a bundle are placed either in an equilateral triangle or a rectangular pattern (Figure 2.6). The correction factor for the cross-sectional shape of a longitudinal tube bundle depends on both the relative pitch of the cylinders s/d (s is the distance between the axes of the cylinders) and the shape of the tube array and number of cylinders. 53. For laminar liquid flow along the bundle without support plates (i.e., an infinite space) the correction factor knonst68 is:

Figure 2.6. Arrangement of cylinders or tubes in the array of (a) an equilateral triangle and (b) a square.

Flow in Straight Tubes and Conduits

103

_ 2 (d −_ 1)3 _ , knonst = kbun = _ 4 _ 4d ln d − 3d 4 + d 2 − 1 _ where d = d*/d; • With cylinders located in the corners of the equilateral triangle (triangular array) ⎛2 √ ⎯⎯3 ⎞ d∗ = ⎜ ⎟ π ⎠ ⎝

1⁄2

s,

6 Dh = d ⎡⎢ π ⎯⎯3 ⎣ √

2

⎛ s ⎞ − 1⎤ . ⎜d⎟ ⎥ ⎝ ⎠ ⎦

In this case, the correction factor can be determined with the limits 1.0 ≤ s/d ≤ 1.5 from an approximate formula kbun ≈ 0.89s/d + 0.63. • With cylinders located in the corners of a square with side s d∗ = 2s ⁄ √ ⎯⎯π , Dh = 4s2 ⁄ (πd) − d and kbun + 0.96

s + 0.64 . d

54. In the case of turbulent liquid flow through a bundle of loosely arranged cylinders (without baffle plates) in a triangular or square array with s/d = 1.0, the correction factor kbund = 0.64 (see Ibraguimov et al.40). For an array with a small number of cylinders held by a baffle plate the correction factor increases and can exceed unity. A relative spacing s/d between the cylinders has different effects on the resistance coefficient, depending on the form of the array (see Diagram 2.9). For a baffled bundle of finned cylinders at s/d = 1.05 the correction factor may be taken the same as that for annular finned tubes (see paragraphs 39 and 40). 55. The shape (bulging) of the cross section of flat-rolled tubes (made from metallic strips) depends on the extent to which they expand under internal pressure and is characterized by the ratio of the cross-sectional semiaxes a0/b0. The friction coefficient of flat-rolled aluminum and steel tubes (see Maron and Roev74) is: at 4 × 103 < Re < 4 × 104 λ=

A1 Re0.25

104

Handbook of Hydraulic Resistance, 4th Edition

and at 4 × 104 < Re < 2 × 105 λ=

A2 Re0.12

,

where the coefficients A1 and A2 depend on the ratio of the tube semiaxes a0/b0 and are determined from the graphs of Diagram 2.10. 56. The resistance of steel tubes with welded joints on which there are metal upsets or burrs is higher than_ the resistances of seamless tubes. When the weld joints are separated by a relative distance l = lj/δj ≥_ 50, the additional resistance of welded tubes may be assumed constant and independent _ of l. Within the limits l ≤ 50, the effect of a single joint decreases with decreasing distance between them, so that 0

ζj = k4ζj ,

_ 0 _where ζj and ζj are the coefficients of resistance of one joint at a distance l and at a distance l ≥ 50, respectively; k4 is the correction factor for the interaction effect of joints. This correction factor can be determined approximately from the dependence of the resistance _ coefficient for a longitudinal row of cylinders placed in a tube on the relative distance l = lcyl/dloc = lj/dloc between the cylinders in the form18 _ ζcyl = nj [2 log l + 1] (dloc ⁄ D0) 1.4 , (2.15) where nj is the number of cylinders or, in the given case, the number of joints over the tube segment of given length. _ 57. The interaction effect of cylinders in a longitudinal row manifests itself up to about l = 50. The interaction effect of the joints is analogous to the same effect of the cylinders in a longitudinal row. _ Therefore, the correction factor k4 can be approximately determined as the ratio ζcyl/(ζcyl)l = 50. This means that after corresponding cancellations Equation (2.15) will yield _ k4 = 0.23 [2 log l + 1] . The coefficient ζ0j is determined depending on δj/D0 from plot a of Diagram 2.1 or from the formula6 ζj = 13.8(δj ⁄ D0)3 ⁄ 2 = k5(δj ⁄ D0)3 ⁄ 2∗ . 0

The overall resistance of the segment of tubes with joints is ⎛ lj ⎞ + ζj⎟ , ζ = nj ⎜λ D0 ⎝ ⎠ ∗ According to the experiments of Altshul,6 the coefficient k5 = 8.26 for rectangular joints and k5 = 4.14 for rounded joints.

Flow in Straight Tubes and Conduits

105

__ where λ is determined as a function of Re and ∆ from the graphs of Diagrams 2.1 through 2.5. 58. Arc and resistance welded joints have less effect on the flow resistance than joints with backing rings, since the height of the joint is then smaller. On the average, it is possible to take the "equivalent height of the electrical arc and resistance of welded joints"* to be δeq = 3 mm, while the height of a joint with a backing ring is δ = 5 mm. 59. In practice, the resistance of steel tubes with coupled joints can be considered equal to the resistance of welded tubes. In calculations of cast-iron piping, one may neglect the additional resistance caused by the presence of bell and spigot joints. 60. Annular grooves on the inner surface of a tube also increase its resistance. The overall resistance of the segment with grooves is ζ*

∆p ρw20 ⁄ 2

⎛ lgr ⎞ = ngr ⎜λ + ζgr⎟ , D0 ⎝ ⎠

where ngr is the number of grooves over the considered segment of the tube; lgr is the distance between the grooves; ζgr is the resistance coefficient of one groove; when lgr/D0 ≥ 4,133 ζgr = 0.046b ⁄ D0 , where b is the width of the groove; at lgr/D0 = 2 ζgr = 0.059b ⁄ D0 , at lgr/D0 < 4 ζgr = f (b ⁄ D0, lgr ⁄ D0) is determined from the graph of Diagram 2.12. 61. The water conduits withdrawn from operation at State Electric Stations have the roughness of walls which varies substantially. To take into account this factor7 it is recommended to introduce into Equation (2.10) the additional parameter αr (correction for the local roughness), so that the indicated formula takes the form 0.25 __ 68 ⎞ ⎛ 0.11 . λ= ⎜∆ + αr Re ⎟ ⎝ ⎠

(2.16)

The parameter αr can vary within wide ranges (see Table 2.5). 62. The surfaces of concrete pipelines differ from the surfaces of other tubes by the presence of longitudinal and transverse seams, shuttering marks, cavities, and other irregularities. The state of the concrete surfaces of pipelines varies in the process of service, that is, their roughness increases. In calculations of the resistance of such pipelines the effect of the joints, local resistances, blockings, and other complicating factors can also be taken into account by ∗

This expression is understood to refer to the height of a joint with a backing ring, the flow resistance of which is equivalent to the arc (and contact) welded joints.

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Handbook of Hydraulic Resistance, 4th Edition

Equation (2.16), which involves the correction factor αr, the values of which are listed in Table 2.5. 63. Formation of deposits in pipelines is a complex process which depends on the physicochemical properties of the liquid transported (with regard to the method and the scale of its cleaning), the pipeline material, and the coating characteristics, as well as on hydraulic parameters such as mean flow velocity, liquid pressure, and tube diameter. 64. Taking into account the tendency of water to form deposits in pipelines, Kamershtein suggests that for water-supplying pipelines the natural waters be divided into the following groups, each determining the character and intensity of the reduction in the transporting capacity of pipelines: • Group I. Weakly saline, noncorrosive water with a stability index from –0.2 to 0.2; water with a moderate content of organic substances and free iron. • Group II. Weakly saline, corrosive water with a stability index up to –1.0; water containing organic substances and free iron in quantities below 3 g/m3. • Group III. Very corrosive water with a stability index from –1.0 to 2.5, but with a small content of chlorides and sulfates (less than 100–150 g/m3); water with an iron content above 3 g/m3. • Group IV. Corrosive water with a negative stability index, but with a high content of sulfates and chlorides (above 500–700 g/m3); nontreated water with a high content of organic substances. • Group V. Water distinguished for appreciable carbonate and low constant density with a stability index above 0.8; heavily saline and corrosive water. 65. The dependence of the roughness asperity height ∆t (mm) on number of years of service is determined from a formula derived by Mostkov on the basis of Kamershtein’s experiments: ∆t = ∆ + αyty ,

(2.17)

where ∆ is the initial height of the roughness asperities (see Table 2.5); αy is the rate of increase in the number of asperities (millimeters per year), which is dependent on the physicochemical properties of water (see Table 2.1). 66. The dependence of the fluid transport capacity of water-supplying pipelines on the time of their service, properties of the transported water, and pipeline diameter is expressed as Qt = Q (1 − 0.01ntm y) , where Q is the predicted transporting capacity of a pipeline, ty is the duration of service (years), and n and m are parameters that depend on physicochemical properties of the transported water (see Table 2.1).* 67. Because they have higher flow rates, gas pipelines are less subjected to mechanical contamination than water pipelines. Dry gases that do not cause corrosion of the inner surface of the tube may even somewhat reduce the roughness as the tubes are slightly abraded by the dry gas. ∗

The increase in the resistance of water-supplying pipelines in the process of service has been refined in Reference l28.

107

Flow in Straight Tubes and Conduits Table 2.1 Values of the parameters αy, n, and ma Water quality group I

II

III

IV

V

Pipeline diameter D0, mm

αy, mm/year

n

m

150–300

0.005−0.055 0.025

4.4

0.5

2.3

0.5

0.055−0.18 0.07

6.4

0.5

2.3

0.5

0.18−0.40 0.20

11.6

0.4

6.4

0.5

18.0

0.35

400–600

0.40−0.60 0.51

11.6

0.40

150–300

0.60–3.0

32.0

0.25

18.0

0.35

400–600 150–300 400–600 150–300 400–600 150–300

The value of the parameter αy increases with a decrease in the pipeline diameter. The numerator contains the limits of variation of αy and the denominator — the most probably average value. a

68. Moisture and hydrogen sulfide, carbonic acid, and oxygen, which are contained in gases, cause corrosion of the metal of tubes, which is accompanied by changes in the size, shape, and distribution of asperities on the inner surface of the duct. The transporting capacity of gas conduits is sometimes reduced with time by 15% or more due to corrosion and contamination. 69. Growth of asperities on the inner surface of ventilation air ducts during service may be taken into account through a formula similar to Equation (2.17):62 ∆t = ∆ + αmtm , where αm is the rate of growth of asperities, in millimeters per month (see Table 2.2) and tm is the duration of service, in months. Table 2.2 Growth of surface asperities in air pipelines during service62 Limits of variation of αm, mm/month

Region of use of air pipelines or ducts

Kind of local suction

Conveyor soldering of small ratio components with application of the KST flux

Aspirating (sucking) funnel or hood

Impregnation of abrasive disks with bakelite

Bakelitization chamber

0.92–1.36

Cooking on a kitchen range

Circumferential suction

0.34–0.49

Chrome-plating of articles in a galvanic bath

Suction from two sides of the bath

0.49–0.80

Exhaust section of air pipeline installed outside a building

–

2.3–4.4

0.03

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Handbook of Hydraulic Resistance, 4th Edition

70. According to Datochnyi,31 motion of gas in low-pressure gas pipelines is possible in all flow regimes, except the quadratic one, while in gas pipelines with moderate and high pressures it occurs under transition and quadratic conditions. All gas pipelines operate principally under transition conditions. For refined formulas for calculating gas pipelines of low and high pressure, see Altshul.6 71. The resistance of flexible tubes made of metallic strips (metallic hoses; see Diagram 2.12) substantially (by a factor of 2–2.5) exceeds that of smooth tubes. At Re = 5 × 106 to 4⋅105, the friction coefficient of such tubes changes only slightly (λ = 0.025–0.0285). It depends on the direction of the flow along the hose; at places where the flow runs off the edges of the inner strip it is somewhat smaller than at places where the flow impinges on the edges.146 72. High hydraulic resistance is also exhibited by a flexible air conduit made by winding a glass fabric strip round a steel-wire framework. The resistance of such air conduits is primarily determined by the crimpness of their surfaces (rather than conventional roughness). The friction coefficient of glass-fabric air conduits with a regular crimpy surface can be determined from an approximate formula of Klyachko and Makarenkova,53 which reflects the structural dependence of λ on the air conduit diameter and the strip width b: ′

λ + λ0(D0 ⁄ D0′) D0 ⁄ D0 (b ⁄ b0)m , where λ0, D′0, and b0 are the friction coefficient, diameter, and width of the strip of an air conduit; λ0 = 0.052; D′0 = 0.1 m; b0 = 0.02 m; m is the coefficient which takes into account the change in the winding pitch; for the construction considered m = 1/5. Air conduits of diameters D0 ≤ 0.2 m have glass-fabric strip width b = 0.02 m and those with the diameters D0 > 0.2 m, b = 0.03 m. More accurate values of λ obtained experimentally for glass-fabric air conduits53 are presented as a function of the diameter D0 and the Reynolds number in the respective table of Diagram 2.13. 73. The resistance of flexible corrugated tubes to turbulent flow depends on the ratio of the height of the crimp crest h to its length lcr and depends only slightly on the Reynolds number. 74. The friction coefficient λ of reinforced rubber hoses, whose characteristics are given in Diagram 2.14, does not depend on the Reynolds number in the range ≥4000, owing to the appreciable roughness of such hoses. The value of λ increases with increasing diameter of hoses since the height of the inner seams is then also increased.131,132 When determining pressure losses from Equation (2.2) it is necessary that the nominal hose diameter dnom be replaced by dcal determined from curve b of Diagram 2.14, depending on the mean inner pressure. 75. The friction coefficient λ of smooth rubber hoses, whose characteristics are given in Diagram 2.15, can be determined from the Toltsman–Shevele132 formula: λ=

A Re

0.265

,

where, at Reynolds numbers (Re = w0dnom/v) from 5000 to 120,000, the value of A = 0.38 to 0.52 (depending on the quality of the hoses).

Flow in Straight Tubes and Conduits

109

If pressure losses are determined from Equation (2.2), the diameter should be calculated based on the mean internal pressure (according to curve b of Diagram 2.14). 76. The friction coefficient λ of smooth reinforced rubber hoses is determined from the curves of Diagram 2.16, depending on the average internal pressure and dnom. In determining pressure losses from Equation (2.2) it is necessary that the calculated rather than a nominal diameter of the hose be multiplied by the correction factor k, which is found from curves c and d of Diagram 2.17, depending on the average internal pressure. 77. For large-diameter (300–500 mm) tubes made from rubberized material, such as may be used for ventilation of shafts, and the connections made with wire rings closed at the ends by pipe sockets (see Diagram 2.17), the total resistance is composed (according to Adamov) of the friction resistance and the resistance of joints ζ=

∆p ρw20 ⁄ 2

⎛ lj ⎞ = nc ⎜λ + ζc ⎟ , D0 ⎝ ⎠

where nc is the number of connections; λ (see Diagram 2.17) is determined for different degrees of tension: small (with extensive crimping and fractures), medium (with minor crimping), and large (without crimping); lj is the distance between the joints, in m; and ζc is the resistance coefficient of one connection (see Diagram 2.16). 78. The friction coefficients λ of plywood tubes (made from birch plywood with fibers running lengthwise) are determined according to the data of Adamov and Idelchik1 given in Diagram 2.18. 79. The friction coefficients of tubes made from polymers (plastic) can be determined from formulas of Offengenden,91,92 which are given in Diagram 2.19. Indicated there also are the regions of the applicability of these formulas. As a rule, plastic tubes relate to tubes with slight roughness (∆ ≤ 30 µm). Tubes made from fluoroplastic have the smallest absolute roughness and those made from glass-reinforced plastic, and from phaolete the greatest roughness. The plastic tubes also have micro- and macrowaviness.92 When 5 × 104 ≤ Re ≤ 3 × 105, to make hydraulic calculation of plastic tubes, it is possible to use in the first approximation (with an error up to 25% and above) the formula of Colebrook–White (2.9) or similar formulas (see above) with the substitution of the values of ∆ given in Table 2.5. For polyethylene (nonstabilized), fluoroplastic, and polypropylene tubes the value of ∆ is not determined, as the coefficient λ for them can be determined from formulas for smooth tubes.92 80. The local resistance coefficients for different types of joints of plastic tubes can be determined from corresponding formulas92 given in Diagram 2.20. 81. All the values of λ recommended above refer to an incompressible fluid. In order to approximate the effect of gas compressibility for a section of very large length, one may use the formula derived by Voronin:22 k − 1 2⎞ ⎛ M ⎟ λcom = λ ⎜1 + 2 ⎠ ⎝

−0.47

,

where λ and λcom are the friction coefficients, respectively, for incompressible and compressible liquid (gas).

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Handbook of Hydraulic Resistance, 4th Edition

Figure 2.7. Diagram of flow with a change in the flow rate along the path: (a) collector with perforated_ _ walls; (b) collector with a longitudinal slot; (c) collector with side branches. (1) v = 1.0 = const. (2) v with variable discharge.

The formula shows that up to Ma = 0.6, the effect of compressibility may be neglected within 3% or less error. An appreciable decrease in the coefficient λcom is observed only in a narrow transonic region and also at supersonic flow velocities (by about 15%).121,122 82. When a liquid (gas) moves in a pipeline (conduit) of constant cross section and there is an outflow or inflow through porous side walls, slots, or side branches (Figure 2.7), the resistance coefficient λ varies along this path due to a change of the average flow velocity (Re) along the flow path. 83. The local resistance coefficient λloc of a cylindrical tube with porous walls and uniform and circular (over the whole perimeter) outflow, that is, when _ __ _ v * v ⁄ vs = 1 and w * w ⁄ w0 = 1 − α0x in the case of laminar flow, is calculated from the formula of Bystrov and Mikhailov.18* ∗

The coefficient α0 introduced by the author into this formula extends it also to the case of transit flow rate (α0 < 1).

Flow in Straight Tubes and Conduits

111

_ ~ λloc = 32 (3 + α0 ⁄ [Re0 (1 − α0x)] , _ _ Here vs = α0w0/f is the average velocity of outflow (inflow) through side orifices; f = Σf ⁄ F0 is the ratio of the overall (branches) of the porous segment of the _ area of the side surfaces ~ tube; α0 = 1 – w1/w0; x = x/l; Re0 = w0D0/v; α0 is determined by the velocity profile at the ~ inlet to the discharge collector (for the parabolic profile α0 = –0.17; for the cosinusoid profile ~ α0 ≈ –0.33); w0 and w1 are the average velocity in the initial (x = 0) and final (x = l) sections of the porous segment of the tube. The resistance coefficient of the porous tube segment of length l44 is ζ*

a) ⎤ ⎡ 32 (3 + ~ l D ⁄ = (1 − 0.5α0)⎥ . 0 ⎢ Re 2 0 ρw0 ⁄ 2 ⎦ ⎣ ∆p

84. The local friction coefficient λloc under the same conditions as those in paragraph 83 is calculated in the case of turbulent flow and 20 ≤ L/D ≤ 125 from the formula:18 at ε′ ≤ 0.2 λloc = λ + 5.54ε′v ⁄ w ; when ε′ > 0.2 Nv ⎛ v ⁄ w0 ⎞ (2.19) ⎟. ⎜1 − v⁄w ⎠ ⎝ Here λ is the friction coefficient of a smooth tube determined from the graphs of Diagram 2.1: λloc = λ + 5.54ε′v ⁄ w +

ε′v ⁄ w

Nv = 0.0256B(ε′v ⁄ w)0.435 , B=

λε − λ

,

0.2 − λ

λε is determined from the expression log λε = log λ exp (−6.63ε′3) , where ε′ is the porosity factor of the tube walls. Within the range 20 ≤ l/D ≤ 125, the resistance coefficient of the porous segment of the tube of length l is at ε′ ≤ 0.2 ζ*

∆p ρw20 ⁄ 2

=

at ε′ > 0.2

_ ⎤ α20 ⎞ l ⎡ ⎛ ⎢λ ⎜1 − α0 + ⎟ + 5.54εα0 ⁄ f (1 − 0.5α0)⎥ . 3⎠ D0 ⎣ ⎝ ⎦

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Handbook of Hydraulic Resistance, 4th Edition

ζ*

+

∆p ρw20 ⁄ 2

=

l ⎧ ⎛ ⎨λ ⎜1 − α0 + D0 ⎩ ⎝

_ 0.00157α0.435 f 0.565 0 (ε′)0.565

_ α20 ⎞ ⎟ + 5.54ε′α0 ⁄ f (1 − 0.5α0) 3⎠

⎫ ⎡1 − 4.565 (1 − α0)3.565 + 3.565 (1 − α0)4.565⎤⎬ . ⎦ ⎣ ⎭

85. The local friction coefficient λloc of a discharging collector with one-sided and uniform outflow and with turbulent flow18 is λloc = λ + 8ε′v ⁄ w , whereas the resistance coefficient of the entire section of the collector of length l44 is ζ*

∆p ρw20 ⁄ 2

=

_ ⎤ α20 ⎞ l ⎡ ⎛ ⎢λ ⎜1 − α0 + ⎟ + 8ε′α0 ⁄ f (1 − 0.5α0)⎥ . 3⎠ D0 ⎣ ⎝ ⎦

86. With a circular and uniformly variable outflow from a_ cylindrical tube, _ _ when the rela_ _ = 1 – to = 1 + and accordingly tive velocity of the outflow varies linearly from v v v ∆v ∆v 0 1 __ __ __ w * w/w0 = 1 – α0(1 – ∆v)x – α0∆v x2, _ __ _ v * v ⁄ vs = α0 (1 − ∆v + 2∆v x) , _ where ∆v * ∆v/vs is the departure of the relative velocity from its average value (from unity, see Figure 2.7). The local resistance coefficient in the case of laminar flow is λlam =

~ 32 (3 + α0) __ . __ Re0 [1 − α0 (1 − ∆v)x − α0∆v x 2]

The resistance coefficient of a porous segment of length l44 is ζ*

∆p ρw20 ⁄ 2

=

~ _ 32 (3 + α0) l ⁄ D0 [1 − 0.5α0 + 1 ⁄ 6α0∆v] . Re0

87. In the case of turbulent flow under identical conditions as those in paragraph 86, the local friction coefficient λloc is determined approximately from Equations (2.18) and (2.19). The resistance coefficient of the porous segment of length l44 is at ε ≤ 0.2 ζ*

∆p ρw20 ⁄ 2

⎧ ⎡ α0 = l ⁄ D0 ⎨λ ⎢1 − α0 + 3 ⎩ ⎣

113

Flow in Straight Tubes and Conduits Table 2.3 The values of J1 – J2 v

α0

0

0.1

0.2

0.3

0.4

0.6

0.8

1.0

0.1

0.042

0.040

0.038

0.036

0.035

0.031

0.028

0.027

0.2

0.070

0.067

0.064

0.061

0.059

0.053

0.048

0.042

0.3

0.086

0.083

0.080

0.077

0.074

0.068

0.061

0.54

0.4

0.093

0.091

0.088

0.085

0.083

0.077

0.070

0.061

0.5

0.094

0.092

0.090

0.088

0.086

0.081

0.074

0.065

0.6

0.090

0.089

0.088

0.087

0.086

0.082

0.076

0.067

0.7

0.084

0.084

0.084

0.084

0.083

0.080

0.075

0.067

0.8

0.076

0.077

0.078

0.079

0.079

0.078

0.074

0.066

0.9

0.068

0.070

0.072

0.073

0.074

0.074

0.072

0.065

1.0

0.061

0.064

0.066

0.068

0.070

0.071

0.070

0.063

_ _ _ _ ⎫ × ⎛∆v + α0 − 0.5α0∆v + 0.1α0∆v 2⎞⎤ + 5.54ε′α0 ⁄ f (1 − 0.5α0)⎬ ; ⎝ ⎠⎦ ⎭ at ε > 0.2 ζ*

∆p ρw20 ⁄ 2

⎧ ⎡ α0 = l ⁄ D0 ⎨λ ⎢1 − α0 + 3 ⎩ ⎣

_ _ _ _ ⎫ × ⎛∆v + α0 − 0.5α0∆v + 0.1α0∆v 2⎞⎤ + 5.54ε′α0 ⁄ f (1 − 0.5α0) + ∆λ⎬ , ⎝ ⎠⎦ ⎭ where 0.0256B _ ∆λ = (εα ⁄ f)0.565

⎧1 ⎪ ⎨ ⎪ ⎩0

∫

__ __ [1 − α0 (1 − ∆v)x − α0∆v x 2]2.565 _ _ __ dx (1 − ∆v + 2∆v x)0.565

__ __ [1 − α0 (1 − ∆v)x − α0∆v x 2]3.565 _ ⎫ 0.0256B _ _ _ _ 0.565 dx = (J − J2)⎬ , −∫ 0.565 1 ⎭ (1 − ∆v + 2∆v x) (εα0 ⁄ f) 0 1

J1 and J2 are the first and second integrals in the expression for ∆λ. The value of ∆λ can be determined numerically on a computer. The calculated values of the difference J1 – J2 are presented in Table 2.3. 88. In the case of turbulent flow and one-sided nonuniform outflow (see Figure 2.7), the local friction coefficient is determined according to Reference 18 as: at ε′ ≤ 0.2

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Handbook of Hydraulic Resistance, 4th Edition

λloc = 6.5ε′v ⁄ w , at ε′ > 0.2 λloc = λ +Ndiscε′v ⁄ w , where _ Ndisc + 7.4ε′(l ⁄ D0)0.5 [1 − exp (−0.016l ⁄ D0ζbr ⁄ f 2)0.6] (ζbr * ∆p ⁄ (ρv2s ⁄ 2) is the overall resistance coefficient of the side branch of the collector reduced to the velocity vs). The resistance coefficient of the porous segment of length l 44 is at ε′ ≤ 0.2 ζ*

∆p ρw20 ⁄ 2

⎧

= l ⁄ D0 ⎨⎩λ [1 − α0 + α0 ⁄ 3

_ _ _ _ ⎫ × (∆v + α0 − 0.5α0∆v + 0.1α0∆v 2)] + 6.5ε′α0 ⁄ f (1 − 0.5α0)⎬⎭ ; at ε′ > 0.2 ζ*

∆p ρw20 ⁄ 2

⎧

= l ⁄ D0 ⎨⎩λ [1 − α0 + α0 ⁄ 3

_ _ _ _ ⎫ × (∆v + α0 − 0.5α0∆v + 0.1α0∆v 2)] + Ndiscε′α0 ⁄ f (1 − 0.5α0)⎬⎭ . 89. In the case of a turbulent flow and uniform inflow (injection), the local friction coefficient is λloc = 1.5ε′v ⁄ w .

(2.20)

Then the resistance coefficient of the porous segment of length l 44 is ζ*

∆p ρw20 ⁄ 2

_ = 1.5ε′α0 ⁄ f l ⁄ D0 (1 − 0.5α0) .

90. For a turbulent flow and uniformly variable inflow (injection), veloc_ _ _when the relative _ ity of __inflow varies according _ _linear law from v0 = 1 + ∆v to v1 = 1 – ∆v and accord_ _ to the ingly w = 1 – α0(1 + ∆v)x + α0∆v x 2, _ __ _ v = α0 (1 + ∆v − 2∆v x) .

Flow in Straight Tubes and Conduits

115

The local friction coefficient is determined approximately from Equation (2.20), whereas the resistance coefficient of the porous segment of length l is ζ*

∆p ρw20 ⁄ 2

_ _ = 1.5ε′α0 ⁄ f l ⁄ D0 (1 − 0.5α0 − 0.17α0∆v) .

91. The resistance coefficient of the side branching of collectors15,16 is: in the case of outflow (discharging collector) ζbr *

∆p ρv2s ⁄ 2

_ = 0.25f 2 + (fs ⁄ fa)2 + ζap + ζseg ,

in the case of injection (intake collector) _ 1 1 ζbr = 1.5f 2 ⎛⎜ − 2 − 0.125⎞⎟ + 0.75 + (fs ⁄ fa)2 + ζap + ζseg , ns ns ⎝ ⎠ for paired collectors (Π-like or Z-like) _ ζbr = 0.2f 2 + 1.75 + ζap + ζseg . Here fs and fa are the areas of the side orifice and of the final cross section of the entire branching (exit into an infinite space); ζap is the resistance coefficient of any apparatus involved in the system of the side branching and reduced to the velocity vs; ζseg is the resistance coefficient of all the segments of the side branching upstream and downstream of the apparatus reduced to the velocity vs; ns is the number of side branchings. 92. The introduction of macroscopic particles into the flow of a liquid or gas, or the addition of polymer molecules with a very large molecular mass relative to the liquid, substantially reduces the friction coefficient in tubes (Thomas effect).264 Addition of polymers to a liquid or solid particles to a gas leads to a notable decrease of the transverse velocity pulsations and of the turbulent friction expressed in terms of the Reynolds stresses, and as a result the resistance coefficient decreases. These additives do not decrease the resistance coefficient of laminar flow and do not contribute to its preservation. The maximum decline in the resistance coefficient is observed in the region of low Reynolds numbers of a fully developed turbulent flow (Figure 2.8). 93. The friction coefficient also varies depending on the concentration and kind of polymer (in water) and, correspondingly, on the size of suspended solid particles (in an air flow). The higher the concentration of the polymer (polyacrilamide, PAA) in water at the given Reynolds number (Figure 2.9), the more appreciable is the decrease in the coefficient λ (similar results can be observed also from the data of other works (see References 11, 12, 97, 98, 111). The coefficient λ is determined210 from the formula ⎡⎛ Re∗thr ⎞ 1 = −2 log ⎢⎜ Re∗ ⎟ ⎯⎯λ √ ⎠ ⎣⎝

αpol ⁄ 5.75

__ ∆ ⎞⎤ 2.51 ⎛ + ⎟⎥ , ⎜ Re √ 3.7 ⎠⎦ ⎝ ⎯⎯λ

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Handbook of Hydraulic Resistance, 4th Edition

Figure 2.8. Friction coefficient of a smooth plate in a dust-laden air flow (Gm = 3.7 g/s):214 (1) pure air; (2) 1680 µm; (3) 840 µm; (4) 200 µm; (5) 100 µm.

where Re* = (w*/D0)/v is the dynamic Reynolds number; Re*thr = (w*thrD0)/v is the threshold Reynolds number corresponding to the start of the decrease in hydraulic resistance; [w* = ⎯⎯⎯⎯ √ τ0 ⁄⎯ρ is the dynamic velocity (τ0 is the shear stress on the wall); w*thr is the threshold dynamic velocity]; αpol is the parameter, depending on the kind and concentration of polymer (it can be determined from experimental data). 94. With the rise of concentration of solid particles µ (Figure 2.10) the friction coefficient λ first falls very sharply or, which is the same, the ratio (λ0 – λ)/λ0 reaches a maximum, after which it begins to decrease until at µ = 2–3 it becomes equal to zero. The smaller the fraction of suspended particles, the larger is the maximum of (λ0 – λ)/λ0 and the earlier this maximum occurs, but at the smaller values of µ the friction coefficient starts to decrease. 95. In the case of pneumatic transport, when density and dimensions of solid particles suspended in the flow are nearly always substantial, the effect of the cross-current velocities of turbulent flow on the mechanism of particle suspension and the friction drag becomes negligible. In this case, such additional factors as the drag of particles, the lift exerted on them, and the gravitational force and other factors, which increase resistance to the transporting flow motion, are of prime importance (see the list of references to Chapter 2). 96. When the flow in a horizontal tube is steady (far from the inlet, absence of transported material effects), the difference between the densities of suspended particles and air is substantial and the dimensions of particles are such that individual particles periodically strike the tube wall and bounce off it, thus executing a continuous bouncing motion. 97. The loss of energy during impact on the wall is responsible for a decrease in the translational velocity of particles, which subsequently recovers again due to interaction of particles with the flow. This causes an additional expenditure of energy by a transporting flow. 98. In the presence of heat transfer through the tube walls the liquid (gas) temperature varies over both its length and cross section; the latter leads to a change in the fluid density and viscosity and as a result, in the velocity profile and fluid resistance.51 99. The friction coefficient of a nonisothermal flow of a liquid is calculated from the equation n

λnon ⎛ ηw ⎞ =⎜ ⎟ , λis ⎝ ηfl ⎠

(2.21)

Flow in Straight Tubes and Conduits

117

Figure 2.9. Function 1/√ ⎯⎯λ = f(Re√ ⎯⎯λ ) for water with surfactants of different concentrations:210 (1) for smooth tubes; (2), (3), and (4) by formula of paragraph 93 at different concentrations of surfactants at αpol = 11.5, 7.1, and 4.2, respectively: +) tap water; ¤) water + PAA (c = 0.0053%); À) water + PAA (c = 0.008%); ∆) water + PAA (c = 0.012%).

where λnon and λis are the friction coefficients in the case on nonisothermal and isothermal motion, respectively (in calculations of λis the density and viscosity are taken for the average fluid temperature); ηw and ηfl are the dynamic viscosities, respectively, at the temperature of the tube wall Tw and average fluid temperature Tfl; n = f(ηw/ηfl, Pe⋅d/l), see Table 2.4; Pe = wl/at is the Peclet number; at is the thermal diffusivity, m2/s. When the fluid is cooled, ηw/ηfl > 1; it follows from Equation (2.21) that the friction coefficient increases. When the fluid is heated, ηw/ηfl < 1; λnon becomes smaller than λis. 100. To determine the friction coefficient of hydraulically smooth tubes with turbulent fluid flow the following formula can be used:75

Figure 2.10. Friction resistance on the surface of a circular tube at different ratios of the mass flow rates:214 (1) 60 µm; (2) 15 µm; (3) 100 µm; (4) 200 µm; (5) 840 µm; (6) 1680 µm; λ0 is the value of λ at µ = 0. G and Gm is the mass flow rate without and with particles, respectively.

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Handbook of Hydraulic Resistance, 4th Edition

Table 2.4 The values of n in Equation (2.21) ηw ⁄ ηfl

Pe⋅d ⁄ l 0.1

1.0

ηw ⁄ ηfl

Pe⋅d ⁄ l

10

100

1000

0.1

1.0

10

100

1000

60

0.78

0.67

0.58

0.51

0.44

1000

0.33

0.29

0.25

0.22

0.19

100

0.67

0.58

0.50

0.44

0.38

15000 0.36

0.26

0.22

0.19

0.17

150

0.59

0.51

0.45

0.39

0.33

2500

0.28

0.25

0.21

0.18

0.16

200

0.54

0.47

0.41

0.35

0.31

5000

0.26

0.23

0.20

0.17

0.15

400

0.44

0.38

0.33

0.29

0.25

10,000

0.25

0.21

0.19

0.16

0.14

600

0.39

0.34

0.29

0.25

0.22

30,000

0.22

0.19

0.17

0.14

0.13

λnon =

1 ηw ⁄ ηfl ) − 1.64] 2 [1.82 log (Re √ ⎯⎯⎯⎯⎯⎯

.

101. The friction coefficient for a nonisothermal turbulent flow of a heated gas can be calculated from the approximate formula of Kutateladze–Leontiyev which is valid within the ranges Re = 105–6 × 106 and Tw/Tg = 1–3: λnon ⁄ λis =

4 Tw ⁄ T⎯g + 1) 2 (√ ⎯⎯⎯⎯⎯

,

where Tg is the mean-mass gas temperature. 102. In determining the equivalent roughness of the walls of the calculated segment of the tubes (channel), one may use the data given in Table 2.5. 103. A change in the flow rate and, correspondingly, in the Reynolds number in time exerts a direct effect on the characteristics of turbulent transfer in a tube. Therefore, the effect of the hydrodynamic instability on the resistance turns out to be different from the case of laminar flow. Below the cases are considered where the Reynolds number over the entire range of the flow rates lies substantially above the critical value that characterizes transition from the laminar to the turbulent mode. The review of the state-of-the-art of experimental and theoretical investigations of the friction drag in stabilized unsteady turbulent liquid flow in circular tubes is given in References 271 and 272. Consider some of the results of the corresponding experimental investigations. The friction coefficient in unsteady turbulent flow in a circular tube ζ is compared with the quasi-stationary value ζ* determined from the well-known dependences on the Reynolds number for steady flow at a given time. Figure 2.11 presents the results of an experimental study of the friction coefficient273 on the stepwise increase in the flow rate in time on Re1 = 5 × 103 to Re2 = 104 (a) and on the decrease in the flow rate from Re1 = 2 × 104 to Re2 = 104 (b). Hence it follows that an increase in the flow rate causes the curve depicting the dependence of the friction coefficient on time to pass through a minimum at which the friction coefficient ζ is lower than the quasi-stationary value ζ*, whereas on stepwise decrease in the flow rate ζ is larger than ζ*. With time stationary values of the friction coefficient are established.

Flow in Straight Tubes and Conduits

119

Figure 2.11. Experimental values of the coefficient ζ/ζ* on jumpwise change in the flow rate in time t: a) increasing flow rate; b) decreasing flow rate.

Figure 2.12. Friction coefficient during retardation. Experimental data. Initial, section-average velocity is w = 2.92 m/s, dw/dt = 1.46 m2/s.

Figure 2.13. Friction coefficient during acceleration.

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Handbook of Hydraulic Resistance, 4th Edition

Table 2.5 Equivalent roughness of the surfaces of tubes and channels Group

I

II

Type of tubes, material Seamless tubes made from brass, copper, lead Aluminum tubes Seamless steel tubes (commercial)

State of tube surface and conditions of use A. Metal tubes Commercially smooth

New, unused

∆, mm 0.0015–0.0100 0.015–0.06

0.02–0.10a

Ref.

187

Cleaned after many years of use

Up to 0.04

42, 152, 185, 197 187

Bituminized

Up to 0.04

187

Superheated steam pipes of heating systems and water pipes of heating systems with deaeration and chemical treatment of running water After one year of use in gas pipelines

0.10

82

0.12

42

After several years of use as tubing in gas wells under various conditions After several years of use as casings in gas wells under different conditions Saturated steam ducts and water pipes of heating systems with minor water leakage (up to 0.5%) and deaeration of water supplied to balance leakage

0.04–0.20

7

0.06–0.22

7

0.20

82

Pipelines of water heating systems independent of the source of supply Oil pipelines for intermediate operating conditions Moderately corroded

0.20

~0.4

197

Small depositions of scale

~0.4

197

Steam pipelines operating periodically and condensate pipes with the open system of condensate Compressed air pipes from pistonand turbocompressors After several years of operation under different conditions (corroded or with small amount of scale) Condensate pipelines operating periodically and water heating pipes with no deaeration and chemical treatment of water and with substantial leakage from the system (up to 1.5–3%)

0.5

82

0.8

82

0.15–1.0

7

1.0

32

0.20

82

121

Flow in Straight Tubes and Conduits Table 2.5 (continued) Group

Type of tubes, material

State of tube surface and conditions of use

∆, mm

Ref.

A. Metal tubes Water pipelines previously used Poor condition III

Welded steel tubes

New or old, but in good condition

179, 187

~0.05

186

Used previously, corroded, bitumen partially dissolved Used previously, uniformly corroded

~0.10

197

~0.15

197

Without noticeable unevenness at joints; lacquered on the inside layer (10 mm thick); adequate state of surface Gas mains after many years of use

0.3–0.4

182

~0.5

197

With simple or double transverse riveted joints; lacquered 10 mm thick on the inside or with no lacquer, but not corroded Lacquered on the inside, but rusted; soiled when transporting water, but not corroded Layered deposits; gas mains after 20 years of use With double transverse riveted joints, not corroded, soiled during transport of water Small deposits

0.6–0.7

179

0.95–1.0

179

1.1

197

1.2–1.5

152, 197

1.5

197

2.0

179

2.0–4.0

197

2.4

197

Used for 25 years in municipal gas mains, nonuniform deposits of resin and naphthalene Poor condition of the surface Riveted steel tubes

≥5.0 0.04–0.10

New, bituminized

With double transverse riveted joints, heavily corroded Appreciable deposits

IV

1.2–1.5

Lateral and longitudinal riveting with one line of rivets; 10 mm thick lacquered on the inside; adequate state of the surface With double longitudinal riveting and simple lateral riveting; 10 mm thick lacquered on the inside, or without lacquer, but not corroded

≥5.0

179

0.3–0.4

179

0.6–0.7

179

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Handbook of Hydraulic Resistance, 4th Edition

Table 2.5 (continued) Group

Type of tubes, State of tube surface and conditions of use material

∆, mm

Ref.

1.2–1.4

179

2.0

179

4.0

179

≥5.0

179

A. Metal tubes With simple lateral and double longitudinal riveting; from 10 to 20 mm thick lacquered or torred on the inside With four to six longitudinal rows of rivets; long period of use With four lateral and six longitudinal rows of rivets; joints overlapped on the inside Very poor condition of the surface; V

Roofing

uneven overlapping of joints Oiled

1.10–0.15

steel sheets

Not oiled

0.02–0.04

VI

Galvanized

Bright galvanization; new

0.07–0.10

steel tubes

Ordinary galvanization

0.1–0.15

197

VII

Galvanized

New

0.15

185

sheet steel

Used previously for water

0.18

171

VIII

Steel tubes

Coated with glass enamel on both sides

Cast-iron tubes

New

0.25–1.0

171

New, bituminized

0.10–0.15

197

Asphalt-coated

0.12–0.30

185

1.4

152

IX

Water pipelines, used previously

X

Water conduits of

197

0.001–0.01

Used previously, corroded

1.0–1.5

197

With deposits

1.0–1.5

185, 197

Appreciable deposits

2.0–4.0

197

Cleaned after use for many years

0.3–1.5

171

Up to 3.0

179

0.015–0.04

7, 26

Heavily corroded New, clean Seamless (without joints), well fitted

electric power Welded lengthwise, well fitted

0.03–0.012

stations, steel

0.08–0.17

Same, with transverse welded joints New, clean, coated on the inside Bituminized when manufactured

0.014–0.018

Same, with transverse welded joints

0.20–0.60

Galvanized

0.10–0.20

Roughly galvanized

0.40–0.70

Bituminized, curvilinear in plan Used, clean Slightly corroded or with incrustation

0.10–1.4 0.10–0.30

123

Flow in Straight Tubes and Conduits Table 2.5 (continued) Group

Type of tubes, State of tube surface and conditions of use material

∆, mm

Ref.

A. Metal tubes Moderately corroded or with slight deposits

0.30–0.70

Heavily corroded

0.80–1.5

Cleaned of deposits or rust

0.15–0.20

Formerly used (mounting in industrial conditions) All welded, up to 2 years of service, without deposits Same, up to 20 years of service, without deposits With iron-bacterial corrosion (heavily rusted) Heavily corroded, with incrustation (deposits from 1.5 to 9 mm thick) Same, with deposits from 3 to 25 mm thick Used, coated on the inside Bituminized (coal-tar varnish, coal tar), up to 2 years of service Note: For new water conduits αr = 1.3–1.5 For new bituminized water conduits αr = 1.3 For used water conduits the value of αr may vary within wide limits (up to 85), depending on the time

0.12–0.24 0.6–5.0 3.0–4.0 3.0–5.0 6.0–6.5 0.1–0.35

of service, properties of water, kind of deposits, etc. B. Concrete, Cement, and Other Tubes and Conduits I

Concrete

Water conduits without surface finish

tubes

New, plaster finish, manufactured with

7, 26 0.05–0.15

the aid of steel formwork with excellent quality (sections are mated thoroughly, joints are prime coated and smoothed) (αr = 1) Used, with corroded and wavy surface; wood framework (αr > 3.0) Old, poorly manufactured, poorly fitted; the surface is overgrown and has the deposits of sand, gravel, clay particles (αr > 3)

1.0–4.0

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Handbook of Hydraulic Resistance, 4th Edition

Table 2.5 (continued) Group

Type of tubes, State of tube surface and conditions of use material

∆, mm

B. Concrete, Cement, and Other Tubes and Conduits Very old with the surface heavily damaged and overgrown for years of service (αr > 3) Water conduits with subsequent finish of the surface (plastered, smoothed) New, with a very smooth surface, manufactured with the aid of steel or oiled steel casing with excellent quality of work; hand-smoothing with trowels; joints are prime coated and smoothed (without asperities) (αr = 1) New or previously used, smooth; also monolithic (steel casing) or sectional tubes with sections up to 4 m long of good quality; cement finish handsmoothed; joints are smoothed (αr > 1, but 1.5, but ≤2.5) Prefabricated and monolithic (fabricated on site), previously used, with cement plaster wood floated, joints are rough (αr > 2.5, but ≤30) Water conduits with concrete sprayed surface or of sprayed concrete Carefully smoothed air-placed concrete

5.0 and above

0.10–0.20

0.15–0.35

0.30–0.60

0.50–1.0

0.50

or sprayed concrete on concrete surface (αr ≈ 2.5) Brushed air-placed concrete or sprayed concrete on concrete surface (αr > 3.0) Nonsmoothed air-placed concrete or

2.30 3.0–6.0

sprayed concrete on concrete surface (αr > 3.0) Smoothed air-placed concrete or sprayed concrete on concrete surface (αr > 3.0)

6.0–17.0

Ref.

125

Flow in Straight Tubes and Conduits Table 2.5 (continued) Group

Type of tubes, State of tube surface and conditions of use material

∆, mm

Ref.

B. Concrete, Cement, and Other Tubes and Conduits II

Reinforced concrete

New Nonprocessed

0.25–0.34

26

2.5

187

tubes III

Asbestoscement

New

0.05–0.10

Average

0.60

tubes IV

Cement tubes Smoothed

0.3–0.8

Nonprocessed

1.0–2.0

Joints not smoothed V

VI

Conduit with

Good plaster made of pure cement with

a cement-

smoothed joints; all asperities removed;

mortar

metal casing

plaster

Steel-troweled

Plaster over a

187

1.9–6.4

179

0.05–0.22

179

0.5 –

10–15

–

1.4

–

1.5

metallic screen VII

Ceramic saltglazed conduits

VIII

Slag-concrete

IX

Slag and

slabs Carefully made slabs

1.0–1.5

171

alabasterfilling slabs C. Wood, Plywood, and Glass Tubes I

II

Wood tubes

Plywood tubes

Boards very thoroughly dressed

0.15

Boards well dressed

0.30

Boards undressed, but well fitted

0.70

Boards undressed

1.0

Staved

0.6

Of good-quality birch plywood with

0.12

197 1

transverse grain Of good-quality birch plywood

0.03–0.05

with longitudinal grain III

Glass tubes

Pure glass

0.0015–0.010

185

126

Handbook of Hydraulic Resistance, 4th Edition

Table 2.5 (continued) Group

I

II

Type of tubes, State of tube surface and conditions of use material D. Tunnels Tunnels in Rocks (Rough) Blast-hewed in rock mass with little jointing Blast-hewed in rock mass with appreciable jointing Roughly cut with highly uneven surfaces Tunnels Unlined Rocks: gneiss (D = 3–13.5 m) granite (D = 3–9 m) Shale (D = 9–12 m) quartz, quartzite (D = 7–10 m)

∆, mm

Ref.

100–140 130–500 500–1500

300–700 200–700 250–650 200–600

sedimentary rocks (D = 4–7 m)

400

nephrite-bearing (D = 3–8 m)

200

Figure 2.12 presents experimental data of Reference 274 on the friction coefficient for a slow motion of water in a tube with d = 25 mm. It follows then that on retardation ζ/ζ* > 1. Figure 2.13 presents experimental dependences of the ratio ζ/ζ* on time in an accelerated flow274. This dependence passes through a minimum at which ζ/ζ* < 1 and it is the more substantial the higher the acceleration. Here d = 25 mm, dν/dt = 0.097 m2/s (1) and 1.46 m2/s (2). The trends noted should be kept in mind when analyzing an unsteady turbulent flow in tubes. 104. In a number of practically important applications one has to deal with the motion of a liquid or a gas in a pipeline inside which there are passive cylindrical containers moving under the action of a pressure drop in a carrying medium and self-propelled ones (container pipeline pneumo- and hydrotransport, pneumopost, motion of trains in a tunnel, subway) (see Figure 2.14). When a passive container moves in a flow of liquid (or gas), the pressure drop that acts in the pipeline on the end surfaces of the container is directed to the side of motion; corresponding to the positive values of velocity are positive pressure drops (negative pressure gradients). Conversely, during motion of self-propelled containers the pressure before the

Figure 2.14. Scheme of container pneumotransport:282 1) container; 2) undercarriage; 3) pneumodriver with collars.

Flow in Straight Tubes and Conduits

127

Figure 2.15. Scheme of an annular tube with an inner and outer moving cylinders.

container increases and that behind it decreases. Therefore, the pressure drop does not accelerates but rather decelerates the container and together with the sliding friction on the tube wall (rails, guides) or rolling friction of the wheel undercarriage it create the resistance force. A self-propelled container, which acts like a piston, creates a flow of a liquid or gas in a pipeline and plays the role of a kind of a pumping plant. The pressure drop appearing in the forward and hinder parts of the container is counterbalanced by the aerodynamic drag of the pipeline walls offered to the induced flow. Equating these drops, one obtains an equation of the balance of pressures in a transport pipeline. Thus, in calculating the motion of self-propelled containers the pressure balance equation plays the same role as that of the characteristics of the system and of the pumping plant in calculating pipeline transport systems with passive containers. 105. In the case of passive or self-propelled cylindrical containers moving in a pipeline, the character of liquid or gas flow differs substantially from a forced flow in a round or annular tube (Poiseuille flow). Here another type of flow is realized, that is, flow in an annular channel, one of the cylindrical surfaces of which moves along the main flow or apposite to it — the so-called Couette flow. In the absence of the longitudinal pressure gradient one deals with a Couette flow in a plane or annular concentric or eccentric channel. In the presence of a longitudinal pressure gradient the so-called generalized (forced) Couette flow is realized or, in other words, the Couette–Poiseuille flow. Figure 2.15 presents velocity profiles in an annular Couette flow for the cases where the inner cylindrical surface (a) or outer one (b) is mobile . The variants a and b correspond to the cases of passive and self-propelled containers. Here, rp and rc are the radii of the outer and inner cylindrical surfaces, h is the height of the annular channel, τ(y) and u(y) are the profiles of the tangential shear stress and velocity, uc and um are the velocity of motion of the cylindrical surface and mean velocity of liquid in the annular channel. At uc = 0 the pattern of a stabilized Poiseuille flow in an annular tube obtains, and with rc → ∞ — a plane Couette flow. 106. With the aid of the simplest algebraic models of turbulence basic characteristics of a 275 forced turbulent in the form of the _ Couette flow in_annular concentric tubes were calculated dependences uc(λ, θ, Rem) and τc(λ, θ, Rem) in application to the motion of passive and self-

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Handbook of Hydraulic Resistance, 4th Edition

propelled containers. As an example, Figure 2.16 presents such dependences for one Reynolds number Rem at different values of the ratio of cylinders radii θ = rc/rp and Figure 2.17 — at a fixed value of θ and at different values of the Reynolds number Rem and of the relative eccentricity ε (the ratio of the distance between the centers of the cylinders to the difference between the radii h = rp – rc). The solid curves correspond to passive containers, the dashed curves — to the self-propelled ones. Similar dependences for other parameters Rem, θ, and ε are given in Reference 275 in application to the motion of passive containers. Also considered there are cases where one of the cylindrical concentric surfaces or both are rough. The above-mentioned dependences can be used for calculating the parameters of motion of passive and self-propelled finite-length containers on the basis of the balance of the forced acting on a container with allowance for the influence of intercontainer gaps and mechanical friction.275 The monograph also presents a method of numerical investigation of laminar and turbulent motion of a succession of passive concentric cylindrical containers on the basis of

_ _ Figure 2.16. Dependences uc(λ, θ) and τc(λ, θ) for an annular forced Couette flow at Rem = 104 and Re′m = 104.

Flow in Straight Tubes and Conduits

129

_ _ Figure 2.17. Dependences uc(λ, Rem ε) and τc(λ, Rem ε) at θ = 0.75 for a turbulent forced flow in an eccentric annular tube.

stationary Navier–Stokes equations (laminar flow) or Reynolds equations closed with the aid of the two-parameter model of turbulence (turbulent flow). 107. Below the results of experimental investigation are presented for the hydrodynamic characteristics of turbulent flow in an annular tube (Poiseuille flow) (Figure 2.18), plane Couette flow (Figures 2.19 and 2.20), in a plane forced Couette flow (Figure 2.21), as well as in an annular forced Couette flow (Figure 2.22a and b). The figures also present the corresponding results of calculation based on _ and _ the use of _the simplest algebraic _ _differential models of turbulence. Here Rem = umh/ν, p = 2p/ρu2m, τ = 2τ/ρu2m, λ = −dp ⁄ dx, x = x/h, θ = rc/rp. The results of calculation agree satisfactorily with experimental data.

Figure 2.18. Dependence λ(Rem) for a turbulent flow in an annular tube: a, b) experimental data; c) results of calculations with the use of various models of turbulence.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 2.19. Comparison between experimental and calculated velocity profiles in a plane forceless Couette flow: 1) calculation; 2) experiment.289

Figure 2.20. Comparison between experimental and calculated values of the coefficients of friction in a plane turbulent Couette flow: 1) laminar flow; 2) calculation for a turbulent flow using various models of turbulence; 3) experiment.284,285

108. Attention should be paid to one, at first glance paradoxical, fact which follows from the calculations of motion of a long passive container in an annular concentric channel with a turbulent mode of flow and which is supported by experiments. Investigation of the hydrodynamic characteristics of a container of neutral buoyancy moving in an axisymmetric turbulent flow of water in the pipeline has shown that if the container has a sufficient length, when the influence of the end effects is insignificant, at rc/rp ≤ 0.7 the velocity of the container exceeds the maximum velocity of the liquid in the pipeline at the same Reynolds number Rem but which is obtainable in the absence of a cylindrical container.

Flow in Straight Tubes and Conduits

131

Figure 2.21. a) Comparison between experimental and calculated velocity profile in a plane forced Couette flow: 1) calculation with the use of various models of turbulence; 2) experiment;283 I) λ = –0.028; II) λ = –0.1; III) λ = –1.006; IV) λ = 0; V) λ = 0.005; b) comparison between the results of calculations of the parameters of a plane forced Couette flow with experimental data; 3) experiment;284 4) calculation with the use of various models of turbulence for Rem = (1.4–10) × 104; 5) calculation at Rem = const.

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286 Figure 2.22. Scheme of an experimental setup for studying_ an annular _ forced Couette flow:286 b) comparison between experimental and calculated dependences uc(λRem, ucRem): 1) experiment; 2) calculation.

This effect is well described by calculations275 and is due to the weakening of turbulent mixing in the vicinity of the solid body surface as compared to the case where the container immersed in a fluid is replaced by the fluid. This explanation was given by L. Prandtl when he analyzed a similar effect arising on the surface of a river-bed stream after motion of a ship (or of any other streamlined body) which outruns the carrying stream — the velocity of the ship with a cut-out engine is approximately 1.5 times higher than that of the stream on the water surface.279,280 Note that in a hypothetical laminar bed stream a ship does not outruns the flow but rather lags behind it.

133

Flow in Straight Tubes and Conduits

2.2 DIAGRAMS OF FRICTION COEFFICIENTS Circular tube with smooth walls; stabilized flow6,175,193

Diagram 2.1

1. Laminar regime (Re ≤ 2000): λ=

∆p [(ρw20 ⁄ 2)(l ⁄ D0)]

=

64 = f (Re) see graph a. Re

2. Transition regime (200 ≤ Re ≤ 400): λ = f (Re) see graph b. 3. Turbulent regime (4000 < Re < 105); Re =

w0D0 v

0.3164 see graph c. Re0.25 4. Turbulent regime (Re > 4000): λ=

λ= Re

λ Re

λ

Re

λ Re λ Re λ Re λ

100 0.640 1100 0.058

200 0.320 1200 0.053

300 0.213 1300 0.049

400 0.160 1400 0.046

1 see graph c. (1.8 log Re − 1.64)2

500 0.128 1500 0.043

600 0.107 1600 0.040

700 0.092 1700 0.038

800 0.080 1800 0.036

900 0.071 1900 0.034

2 × 103 2.5 × 103 3 × 103 0.032 0.034 0.040

4 × 103 0.040

5 × 103 0.038

6 × 103 0.036

8′′103 0.033

104 0.032

1.5 × 104 0.028

2 × 104 0.026

3 × 104 0.024

0.022

0.021

0.020

0.019

0.018

0.017

0.016

4 × 105 0.014

5 × 105 0.013

0.013

0.012

0.012

0.011

0.011

0.010

0.010

5 × 106 0.009

8 × 106 0.009

0.008

0.008

0.008

0.007

0.007

0.006

0.006

1000 0.064 2000 0.032

0.015

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Handbook of Hydraulic Resistance, 4th Edition

Circular tube with walls of uniform roughness; stabilized flow; Re > 200087,190

Diagram 2.2 ζ* λ*

∆p ρw20 ⁄ 2

=λ

l D0

∆p (ρw20 ⁄ 2)(l ⁄ D0)

=

1

__ [a1 + b1 log (Re √ ⎯⎯λ ) + c1 log ∆]2

λ * f (Re) see graph; the values of a1, b1, and c1 are given below:

For the single formula for calculating λ, see paragraph 19 of Section 2.1. __ ∆Re√ ⎯⎯λ 3.6–10 10–20 20–40 40–20 >191.2

a1

b1

c1

–0.800 0.068 1.538 2.471 1.138

2.000 1.130 0.000 –0.588 0

0 –0.870 –2.000 –2.588 –2.000

__ ∆ ∆= D0 __ __ for ∆, see Table 2.5 (Section 2.1); for v, see Section 1.2. At ∆ < ∆limD0, for the values of λ, see Diagram 2.1, where ∆lim ≈ 17.85Re–0.875

135

Flow in Straight Tubes and Conduits Circular tube with walls of uniform roughness; stabilized flow; Re > 200087,190

Diagram 2.2

Values of λ __ ∆ ∆=

D

0.05 0.04 0.03 0.02 0.015 0.010 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 0.00005

Re 2 × 10

3 × 10

4 × 10

6 × 10

10

2 ×104

4 × 104

6 × 104

105

2 × 105

0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032

0.052 0.044 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040

0.060 0.052 0.044 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040

0.063 0.055 0.046 0.041 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038

0.069 0.060 0.050 0.042 0.037 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033

0.072 0.065 0.056 0.044 0.039 0.032 0.030 0.028 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027

0.072 0.065 0.057 0.048 0.042 0.035 0.032 0.028 0.025 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023

0.072 0.065 0.057 0.049 0.044 0.036 0.033 0.029 0.025 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.030 0.026 0.021 0.018 0.018 0.018 0.018 0.018 0.018 0.018

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.021 0.017 0.016 0.016 0.016 0.016 0.016 0.016

3

3

3

3

4

Values of λ __ ∆ ∆=

D

Re 5

4 × 10

5

6 × 10

6

10

2 × 10

6

0.05

0.072

0.072

0.072

0.072

0.04 0.03 0.02 0.015 0.010 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 0.00005

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.022 0.018 0.016 0.015 0.014 0.014 0.014 0.014

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.018 0.017 0.016 0.014 0.013 0.013 0.013

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.018 0.017 0.014 0.012 0.012 0.012

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.015 0.012 0.011 0.011

4 × 106

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.013 0.011 0.010

6 × 106

107

2 × 107

>108

0.072

0.072

0.072

0.072

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.011 0.010

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.010

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.010

0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011

136

Handbook of Hydraulic Resistance, 4th Edition

Circular tube with walls of uniform roughness; stabilized flow; critical zone (Re0 < Re < Re2)100,106 ζ* λ*

∆p

=λ

ρw20 ⁄ 2

Diagram 2.3

l D0

∆p (ρw20 ⁄ 2)(l ⁄ D0)

__ 1. Re0 < Re < Re1 ; ∆ ≥ 0.007

__ 0.00275 __ ⎞ = f (Re, ∆) λ = 4.4 Re−0.595 exp ⎛⎜− ⎟ ∆ ⎠ ⎝ 2. Re1 < Re < Re2

__ ⎧ ⎫ λ = (λ2 − λ∗) exp ⎨⎩− [0.0017 (Re2 − Re)] 2⎬⎭ + λ∗ = f (Re, ∆) __ at ∆ ≤ 0.007 , λ∗ = λ1 + 0.032 , λ2 = λ′2 = 7.244 Re−0.643 2 __ 0.0109 ∗ at ∆ > 0.007 , λ = λ1 = 0.0017 = 0.0758 − __ 0.286 , ∆ 0.145 and λ2 = λ′′2 = __ 0.244 ∆ __ at ∆ > 0.007: 0.0065 Re0 = 754 exp ⎛⎜ __ ⎞⎟ ∆ ⎠ ⎝ __ at any ∆:

1 Re1 = 1160 ⎛⎜ __ ⎞⎟ ⎝∆⎠

0.11

0.0635

1 Re2 = 2090 ⎛⎜ __ ⎞⎟ ∆ ⎝ ⎠ For the values of Re0, Re1, Re2, λ1, λ′2, and λ′′2, see the table; __ ∆ w0D0 Re = ∆= , v D0 where for ∆, see Table 2.5, Section 2.1; for v, see Section 1.2. 3. For the single formula to calculate λ, see paragraph 30 of Section 2.1.

Values of λ __

Re × 10−3

∆ 0.025 0.017 0.0125 0.0100 0.0080 0.0070 0.0060 0.0050 0.0040 0.0030 0.0024 0.0020

1 0.65 0.064 – – – – – – – – – –

1.1 0.061 0.068 – – – – – – – – – –

1.2 0.058 0.055 0.053 – – – – – – – – –

1.3 0.056 0.053 0.050 0.049 – – – – – – – –

1.4 0.053 0.050 0.048 0.046 – – – – – – – –

1.5 0.051 0.048 0.046 0.044 0.043 – – – – – – –

1.6 0.049 0.046 0.044 0.042 0.040 – – – – – – –

1.7 0.046 0.043 0.040 0.039 0.037 0.036 – – – – – –

137

Flow in Straight Tubes and Conduits Circular tube with walls of uniform roughness; stabilized flow; critical zone (Re0 < Re < Re2)100,106

Diagram 2.3

Values of λ __

Re × 10−3

∆ 0.025 0.017 0.0125 0.0100 0.0800 0.0070 0.0060 0.0050 0.0040 0.0030 0.0024 0.0020

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

0.049 0.044 0.040 0.037 0.035 0.033 0.033 0.033 0.032 0.032 0.032 0.032

0.053 0.047 0.043 0.039 0.037 0.035 0.035 0.035 0.034 0.033 0.033 0.033

0.057 0.051 0.046 0.043 0.040 0.038 0.038 0.037 0.036 0.035 0.035 0.034

0.059 0.053 0.049 0.046 0.043 0.041 0.041 0.039 0.039 0.038 0.037 0.036

0.059 0.054 0.050 0.047 0.045 0.044 0.043 0.042 0.041 0.040 0.039 0.037

0.059 0.054 0.050 0.048 0.046 0.045 0.044 0.043 0.042 0.041 0.040 0.038

0.059 0.054 0.050 0.048 0.046 0.045 0.044 0.043 0.043 0.042 0.041 0.040

0.059 0.054 0.051 0.049 0.047 0.045 0.044 0.043 0.043 0.043 0.042 0.041

0.060 0.054 0.051 0.048 0.047 0.046 0.045 0.044 0.044 0.043 0.043 0.042

0.060 0.054 0.051 0.050 0.048 0.046 0.045 0.044 0.044 0.044 0.043 0.042

0.060 0.054 0.051 0.050 0.048 0.046 0.045 0.044 0.044 0.044 0.043 0.042

Intermediate values of Re and λ __

∆

Re0

Re1

Re2

λ1

λ′1

λ′′2

0.00125 0.00197 0.0028 0.0036 0.0063 0.0072 0.0185 0.0270 0.0450 0.0600

2000 2000 2000 2000 2000 1850 1070 960 870 830

2000 2000 2000 2000 2000 1995 1799 1725 1633 1575

3190 3100 3029 2987 2880 2860 2690 2630 2548 2500

0.032 0.032 0.032 0.032 0.032 0.0329 0.0437 0.0469 0.0510 0.0532

0.0406 0.0412 0.0417 0.0420 0.0431 – – – – –

– – – – – 0.0436 0.0547 0.0600 0.0673 0.0730

138

Handbook of Hydraulic Resistance, 4th Edition

Circular tube with walls of nonuniform roughness; stabilized flow; Re > Re210,171 Re =

w0D0 v

Diagram 2.4

For Re2, see Diagram 2.3 ζ* λ*

∆p l0 =λ D0 ρw20 ⁄ 2 ∆p

1 __ [2 log (2.51 ⁄ Re √ ⎯⎯λ + ∆ ⁄ 3.7)]2 __ or within the limits of ∆ = 0.00008–0.0125: 0.25 __ 68 see graph a λ + 0.11 ⎛⎜∆ + ⎞⎟ Re ⎠ ⎝ __ ∆ ∆= D0 (ρw20 ⁄ 2)(l ⁄ D0)

=

for ∆, see Table 2.5, Section 2.1; for v, see Section 1.2. __ At ∆__ < ∆limD0, for λ, see Diagram 2.1; for ∆lim, see graph b as a function of Re. The manner in which the roughness of the tube walls during their use is taken into account is considered under paragraphs 63–69 of Section 2.1 For Re2, see Diagram 2.3.

139

Flow in Straight Tubes and Conduits Circular tube with walls of nonuniform roughness; stabilized flow; Re > Re210,171

Diagram 2.4

Values of λ __

∆=

∆ Dh

0.05 0.04 0.03 0.02 0.015 0.010 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 0.00005 0.00001 0.000005

Re 3 × 10

4 × 10

6 × 10

10

2 × 104

4 × 104

6 × 104

105

2 × 105

0.077 0.072 0.065 0.059 0.055 0.052 0.050 0.049 0.048 0.045 0.044 0.043 0.040 0.036 0.036 0.036 0.036 0.036 0.036

0.076 0.071 0.064 0.057 0.053 0.049 0.047 0.046 0.044 0.042 0.042 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040

0.074 0.068 0.062 0.054 0.050 0.046 0.044 0.042 0.040 0.038 0.037 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036

0.073 0.067 0.061 0.052 0.048 0.043 0.041 0.039 0.036 0.034 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032

0.072 0.065 0.059 0.051 0.046 0.041 0.038 0.036 0.033 0.030 0.028 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027

0.072 0.065 0.057 0.050 0.045 0.040 0.037 0.034 0.031 0.027 0.025 0.024 0.023 0.023 0.022 0.022 0.022 0.022 0.022

0.072 0.065 0.057 0.049 0.044 0.039 0.036 0.033 0.030 0.026 0.024 0.023 0.022 0.022 0.021 0.021 0.021 0.021 0.021

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.033 0.030 0.026 0.023 0.022 0.021 0.020 0.019 0.019 0.019 0.019 0.019

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.024 0.021 0.020 0.018 0.018 0.017 0.017 0.016 0.016 0.016

5

5

3

3

3

4

Values of λ __ ∆ ∆=

Dh

0.05 0.04 0.03 0.02 0.015 0.010 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 0.00005 0.00001 0.000005

Re 4 × 10

6 × 10

10

2 × 10

4 × 106

6 × 106

107

2 × 107

>108

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.024 0.021 0.020 0.018 0.017 0.016 0.015 0.014 0.014 0.014

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.018 0.017 0.015 0.014 0.013 0.013 0.013

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.015 0.013 0.013 0.012 0.012

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.012 0.011 0.011

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011 0.010 0.009

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011 0.009 0.009

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011 0.009 0.009

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011 0.009 0.008

0.072 0.065 0.057 0.049 0.044 0.038 0.035 0.032 0.028 0.023 0.020 0.019 0.017 0.016 0.014 0.012 0.011 0.009 0.008

6

6

140

Handbook of Hydraulic Resistance, 4th Edition

Circular tube with rough walls; stabilized flow; __ regime of quadratic resistance law (Relim > 560/∆)99,190 ζ* λ*

∆p ρw20 ⁄ 2

=λ

Diagram 2.5

l D0

∆p (ρw20 ⁄ 2)(l ⁄ D0)

=

__ __ = f (∆) [2 log (3.7 ⁄ ∆)]2 1

__ ∆ ∆= D0 for ∆, see Table 2.5 of Section 2.1; for v, see Section 1.2.

The manner in which increase in the asperities on tube walls during use is taken into account is considered under paragraphs 63–69.

__ ∆ ∆=

D0

λ __ ∆ ∆=

D0

0.00005

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.010

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.018

0.0009

0.001

0.002

0.003

0.004

0.005

0.006

0.008

0.010 0.038

λ __ ∆ ∆=

0.019

0.020

0.023

0.026

0.028

0.031

0.032

0.035

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

λ

0.044

0.049

0.053

0.057

0.061

0.065

0.068

0.072

D0

141

Flow in Straight Tubes and Conduits Tubes of rectangular, elliptical, and other types of cross section; stabilized flow.87,158 ζ*

∆p ρw20 ⁄ 2

Dh =

4F0 Π0

Diagram 2.6

= λnonc Re =

1 Dh

w0Dh v

∆p

λnonc *

(ρw20 ⁄ 2)(l ⁄ Dh)

= knoncλ ,

where λ is determined as for circular tubes from Diagrams 2.1 through 2.5

Shape of tube (conduit) cross section and schematic

Correction factor knonc Laminar regime (Re < 2000, curve 1)

Rectangle:

a0 b0 knonc = krec

0

1.0

0.2

0.4

0.6

0.8

1.0

1.50 1.34 1.20 1.02 0.94 0.90 0.89 Turbulent regime (Re > 2000, curve 2)

knonc = krec

Dh =

1.10 1.08 1.06 1.04 1.02 1.01

1.0

2a0b0 a0 + b0

Trapezoid:

knonc is determined in approximately the same way as for a rectangle

h ⎛ 1 1 ⎞⎤ Dh = 2h % ⎡⎢1 + ⎜ sin ϕ1 + sin ϕ2 ⎟⎥ a + a 1 2 ⎠⎦ ⎣ ⎝

142

Handbook of Hydraulic Resistance, 4th Edition

Tubes of rectangular, elliptical, and other types of cross section; stabilized flow.87,158

Diagram 2.6

Shape of tube (conduit) cross section and schematic

Correction factor knonc

Circle with one or two recesses. Star-shaped circle

knonc = krec = kst + 1.0

Laminar regime (Re ≤ 2000): knonc = kell =

Ellipse

Dh +

2 1 ⎛ Dh ⎞ ⎡ ⎛ b0 ⎞ ⎢1 + ⎜ ⎟ ⎜ ⎟ 8 ⎝ b0 ⎠ ⎣ ⎝ a0 ⎠

2

⎤ ⎥ see graph b ⎦

b0 a0

0.1

kell

1.21 1.16 1.11 1.08 1.05 1.03 1.02 1.01 1.01 1.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4a0b0 a0b0 ⎯⎯⎯⎯ 1.5 (a0 + b0) − √

More precisely: Dh +

πa0b0 0.983a0 + 0.311b0 + 0.287b20 ⁄ a0 Turbulent regime (Re > 2000); kell ≈ 1.0

1.0

Dh = D0

(1 + d ⁄ D0) + (3 ⁄ π)(1 − d ⁄ D0) − 6b ⁄ πD0

(1 − d ⁄ D0)(1 + d ⁄ D0 − 6b ⁄ πD0)

Concentric narrow annulus (d/D0 ≈ 0.9) with longitudinal fins

Dh = D0 − d

Concentric annulus:

Shape of the tube (conduit) cross section

Π0

4F0 Re =

w0Dh v

ζ*

∆p ρw20 ⁄ 2

= λnonc

l Dh

λnonc *

∆p (ρw20 ⁄ 2)(l ⁄ Dh)

= knoncλ ,

Diagram 2.7

Schematic

k1r

1.40

0.1

1.44

0.2

1.47

0.3

1.48

0.4

1.49

0.5

1.49

0.6

1.50

≥0.7

1 + (d ⁄ D0)2 + [1 − (d ⁄ D0)2] ⁄ (ln d ⁄ D0)

1 − (d ⁄ D0)2

1.0 1.0

105 106

1.02

1.02

1.03

0.1

1.03

1.03

1.04

0.2

1.04

1.04

1.05

0.3

1.04

1.05

0.5

1.05

1.05

1.06

d ⁄ D0 1.05

0.4

Longitudinal fins (d/D0 ≈ 0.9) Laminar regime at Re ≤ 3 × 103 knonc = k′′r = 1.36 Turbulent regime at Re > 3 × 103; for k′′r , see k2r or a concentric annulus without fins

1.0

0 104

Re

Values of k2r

1.05

1.06

1.06

0.6

1.05

1.06

1.07

0.7

1.05

1.06

1.07

0.8

1.06

1.06

1.07

1.0

0.02d 1 d Turbulent regime (Re > 2000): λnonc * λr = ⎛⎜ + 0.98⎞⎟ ⎛⎜ − 0.27 + 0.1⎞⎟ D0 D ⎝ 0 ⎠ ⎠ ⎝λ knonc = k2r, see curves of graph a

0 1.0

d ⁄ D0

see curve k1r of graph a.

Laminar regime (Re < 2000): knonc = k1r =

Correction factor, knonc

where λ is determined in the same way as for circular tubes from Diagrams 2.1 through 2.5.

Dh =

Dh = D0 − d

Circular tubes; stabilized flow29,30,39,65,95,120,205

,

Flow in Straight Tubes and Conduits 143

144

Handbook of Hydraulic Resistance, 4th Edition

Circular tubes; stabilized flow29,30,39,65,95,120,205

Diagram 2.7

Shape of the tube (channel) cross section

Schematic

Spiral fins ⎧ 6b ⎤ d ⎡ 2 (T ⁄ πd)(d ⁄ D0) Dh = D0 ⎨⎛⎜1 − ⎞⎟ ⎢ (A − B) − ⎥ D0 1 ⁄ πD D − d 0 0⎦ ⎩⎝ ⎠⎣ ⎫ d⁄D % Dd T ⎛⎜ A1 + B 0 ⎞⎟ + 3 ⎛⎜1 − Dd ⎞⎟ − 6b ⎬ π πD 0 πd ⎝ 0 0 ⎠ ⎭ ⎠ ⎝

A=

d T⎞ 1 + ⎛⎜ ⎯√⎯⎯⎯⎯ ⎯⎟⎠ D πd ⎝

2

T 1 + ⎛⎜ ⎞⎟ ⎯√⎯⎯⎯ ⎝ πd ⎠

2

0

B=

d D0

Eccentric annulus Dh = D0 − d _ 2e e= D0 − d

Correction factor knonc

Spiral fins for all values of Re 20 ⎞ k′′r + knonc = ⎛⎜1 + (T ⁄ d)2 ⎟⎠ ⎝

k′r = A1k′r

for A1, see graph b; for k′r, see concentric annulus with fins

T⁄d

3.5

4.5

6.0

8.0

A1

2.63

1.98

1.56

1.31

10 1.20

25 1.03

Flow in Straight Tubes and Conduits

145

Circular tubes; stabilized flow29,30,39,65,95,120,205

Diagram 2.7

Correction factor knonc

Laminar regime (Re ≤ 2000):

knonc = kell =

1

_ k1r ,

(1 + B1e)2

where for B1 = f(d/D0), see graph c; for k1r, see concentric annulus without fins

d ⁄ D0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8

1.0

B1

0

0.65

0.90

1.10

1.22

1.30

1.39

1.45

1.48

Turbulent regime (Re > 2000): ′ = k ′k ; for k′ at d/D = 0.5, k′ell ell 2r ell 0 see curve 1 of graph 2; at d/D0 ≥ 0.7, see curve 2 of graph 2 _ _ or k′ell = 1 – 0.9(1 – 2/3e)e 2

Values of k′ell _ e

d ⁄ D0 0

0.2

0.4

0.6

0.5 (curve 1)

1.0

1.0

0.95

0.87

0.80

0.77

≥0.7 (curve 2)

1.0

0.98

0.90

0.80

0.73

0.70

Tubes of triangular (and similar) cross section; stabilized flow76,95,158

0.8

Diagram 2.8

Dh =

4F0 Π0

Re =

w0Dh , v

1.0

146

Handbook of Hydraulic Resistance, 4th Edition

Tubes of triangular (and similar) cross section; stabilized flow76,95,158 Shape of the tube (channel) cross section

Diagram 2.8 Schematic

Isosceles triangle:

B=

5 1 − 1⎞⎟ 4 + ⎛⎜ ⎯√⎯⎯⎯⎯⎯⎯ 2 tan β ⎝ ⎠ 2

Dh =

2h 1 ⁄ tan β +⎯1 ⎯⎯⎯⎯⎯⎯⎯⎯⎯ 1+√ 2

Right triangle: for Dh, see isosceles triangle

Equilateral triangle (β = 30o): for Dh, see isosceles triangle

Sector of a circle: Dh =

2πD0β ⁄ 180o 1 + πβ ⁄ 180o

(β in o)

ζ*

∆p ρw20 ⁄ 2

= λnonc

1 Dh

λnonc *

∆p (ρw20 ⁄ 2)(l ⁄ Dh)

= knoncλ

where λ is determined in the same way as for circular tubes from Diagrams 2.1 through 2.5.

147

Flow in Straight Tubes and Conduits Tubes of triangular (and similar) cross section; stabilized flow76,95,158

Diagram 2.8

Correction factor knonc Laminar regime (Re ≤ 2000): knonc = k′tr =

1 − tan2 β (B + 2) 3 4 (B − 2)(tan β + √ ⎯⎯⎯⎯⎯⎯⎯⎯ 1 + tan2 β) 2

β, deg

0

k′tr

0.75

10

see curve 1.

20

0.81

30

40

60

80

90

0.82

0.83

0.82

0.80

0.75

0.78

0.89

0.93

0.96

0.98

0.90

1.0

Turbulent regime (Re > 2000), see curve 2 k′tr

0.75

0.84

Laminar regime: knonc = k′tr =

(1 − 3 tan2 β)(B + 2) 3 ⎧ ⎫ 2 4 ⎨(3 tan β) ⁄ [2√ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯ 4 tan β + 5 ⁄ 2 (1 − tan2 β) ] − 2⎬⎭ (tan β + √ 1 + tan2⎯β )2 ⎩

β, deg

0

k′tr

0.75

10

20

0.78

Turbulent regime: for

k′′tr,

see

k′tr

30

0.80

40

0.81

see curve 4.

60

0.82

80

90

0.81

0.77

0.75

60 0.95

80 0.98

90 1.0

of isosceles triangle (curve 2)

Laminar regime: knonc * k′′tr = 0.835 Turbulent regime: knonc * k′′tr = 0.95 Laminar regime: ksec = ktr; see curve 3 β, deg knonc = ksec

0 0.75

10 0.82

20 0.86

30 0.89

40 0.92

Turbulent regime: for ksec, see k′tr of isosceles triangle (curve 2)

Bundles of tubes, rods; parallel interstitial flow40,41,68,119,120,157 Dh =

4F0 Π0

λbun *

Re =

Diagram 2.9 w0Dh v

∆p (ρw20 ⁄ 2)(l ⁄ Dh)

ζ*

∆p ρw20 ⁄ 2

= λbun

l Dh

= kbunλ

where λ is determined in the same way as for circular tubes from Diagrams 2.1 through 2.5.

148

Handbook of Hydraulic Resistance, 4th Edition

Bundles of tubes, rods; parallel interstitial flow40,41,68,119,120,157 Shape of channel cross section and schematic

Diagram 2.9

Correction factor kbun

Triangular, loose array (without shroud, equilateral triangle) Laminar regime (Re ≤ 2000); 1.0 ≤ s/d ≤ 1.5; kbun ≈ 0.89s/d + 0.63 or see curve 1 of graph a (Table 1). s⁄d kbun

1.0

1.05

1.10

1.20

1.30

1.40

1.50

1.52

1.56

1.61

1.70

1.79

1.88

1.97

Turbulent regime at s/d = 1.0: kbun = 0.64

2

6 ⎛s⎞ ⎤ Dh = d ⎡⎢ ⎜ d ⎟ − 1⎥ ⎯ √ ⎯ ⎯ 3π ⎠ ⎝ ⎣ ⎦

Rectangular loose array (without shroud) Laminar regime; at 1.0 ≤ s/d ≤ 1.5; kbun ≈ 0.96s/d + 0.63 or see curve 2 of graph a (Table 2). s⁄d kbun

1.0

1.05

1.10

1.20

1.30

1.40

1.50

1.59

1.64

1.68

1.78

1.88

1.98

2.07

Turbulent regime at s/d = 1.0: kbun = 0.64

4 s1s2 Dh = d ⎛⎜ − 1⎞⎟ 2 π ⎝ d ⎠

Turbulent regime (Re > 2000) a) Array without fins: kbun = f(s/d), see graph b.

149

Flow in Straight Tubes and Conduits Bundles of tubes, rods; parallel interstitial flow40,41,68,119,120,157 Shape of channel cross section and schematic

Diagram 2.9

Correction factor kbun

Triangular array with shroud; z is the number of rods (cylinders) in a bundle; for Dh, see triangular loose array

Values of kbun s⁄d Curve

1.0

1.05

1.10

1.20

1.30

1.40

1.29

1.30

1.13

1.15

0.72

0.57

Shaped shroud, z = 19 and 37 1

0.85

1.0

1.12

1.25

Hexahedral shroud, z = 37 2

0.70

0.84

0.95

1.06

Triangular z = 3 3

–

1.30

1.25

0.95

20 ⎤ kbun = A1kbun, b) Array with helical fins: k′bun = ⎡⎢1 + ⁄ d)2 ⎥⎦ (T ⎣ where for kbun, see graph b; for A1, see graph c; for T, see Diagram 2.7

T⁄d

3.5

4.5

6.0

10

A1

2.63

1.98

1.56

1.20

Rectangular four-tube array; for Dh, see rectangular loose array

Turbulent regime; a) Array without fins: 1) At s/d = 1/45, kbun = 0.97 2) When a rod is in contact with walls, kbun = 0.71; 3) When rods and walls are in contact, kbun = 0.68. b) Array with helical fins: for kbun, see triangular array with shroud.

150

Handbook of Hydraulic Resistance, 4th Edition

Tubes made from aluminum or steel strips (plane-welded). Stabilized flow74 ζ*

∆p ρw20 ⁄ 2

=λ

Diagram 2.10 l Dh

a) 4 × 103 < Re < 4 × 104 λ*

∆p (ρw20 ⁄ 2)(l ⁄ Dh)

=

A1 Re0.25

,

where A1 = f(a0/b0), see graph a; b) 4 × 104 < Re < 2 × 105 Dh =

4F0 Π0

Re =

w0Dh v

Dh is determined in the same way as for an ellipse (see Diagram 2.6)

λ=

A2 Re0.12

,

where A2 = f(a0/b0), see graph a. Re0.25 and Re0.12, see graph b.

a0 b0 A1 A2

Re × 10−5

0.4

0.5

0.6

0.7

0.8

0.250 0.165

0.275 0.17

0.300 0.18

0.310 0.185

0.316 0.185

0.4

0.6

0.8 1

1.2

1.4

1.6 1.8

2

0.25

14.1 15.7 16.8 17.8 18.6 19.3 20.0 2.06 21.1

Re0.12

3.57 3.75 3.88 3.98 4.07 4.15 4.21 4.27 4.33

Re

151

Flow in Straight Tubes and Conduits Welded tube with joints; stabilized flow6,194

Diagram 2.11 ∆p

⎞ ⎛ lj = n0 ⎜λ + ζj⎟ , ⎠ ⎝ D0 where n0 is the number of joints over the section; for λ, see Diagrams 2.2 through 2.6; ζj is the resistance coefficient of one joint: ζ*

ρw20 ⁄ 2

1) at

lj δd

< 50 3⁄2

⎛ δj ⎞ 0 0 = k4ζj where ζj = 13.8 ⎜ ⎟ ρw20 ⁄ 2 ⎝ D0 ⎠ see graph a; k4 = 0.23 (2 log lj/δj + 1), see graph b; ζj *

2) at

lj δd

ζj *

∆p

≥ 50 ∆p

ρw20 ⁄ 2

, see Table 3.

Table 1 δj D0

0.01

0.02

0.03

0.04

0.05

0.06

ξ0j

0.017

0.039

0.075

0.115

0.15

0.20

Table 2 lj δj k4

4

8

12

16

20

24

30

50

0.51 0.65 0.73 0.78 0.83 0.86 0.91

1.0

Table 3 Values of ζj for welded joints of different types Tube diameter D0, mm

Type of joint 200 With backing rings (δj = 5 mm) Made by electroarc and contact (resistance) welding (δj = 3 mm)

400

500

600

700

800

900

0.06

0.03

300

0.018

0.013

0.009

0.007

0.006

0.005

0.026

0.0135

0.009

0.006

0.004

0.0028

0.0023

0.002

Tubes with rectangular annular recesses; stabilized flow; Re ≤ 105 133

p = nr (λlr ⁄ D0 + ζr) , ρw20 ⁄ 2

0.06 0.0019 0.0027 0.0028 0.0020 0.0017

lr ⁄ D0 0.35 1.0 2.0 3.0 4.0

Values of ζr

0.0033

0.0039

0.0047

0.0043

0.0033

0.10

0.0052

0.0064

0.0073

0.0062

0.0048

0.14

b ⁄ D0

at lr/D0 ≥ 4: ζr ≈ 0.046 b/D0; at lr/D0 = 2: ζr = ζmax ≈ 0.059 b/D0; at lr/D0 < 4: ζr = f(b/D0, lr/D0); see the curves.

0.0070

0.0084

0.0094

0.0081

0.0065

0.18

0.0089

0.0113

0.0120

0.0105

0.0075

0.22

0.0110

0.0133

0.0142

0.0127

0.0090

0.26

0.0120

0.0148

0.0156

0.0137

0.0097

0.28

where nr is the number of recesses over the section; for λ, see Diagrams 2.2 through 2.6; ζr is the resistance coefficient of a recess:

ζ*

Diagram 2.12

152 Handbook of Hydraulic Resistance, 4th Edition

153

Flow in Straight Tubes and Conduits Flexible circular tubes; stabilized turbulent flow53,146,194 ζ*λ

Diagram 2.13

l D0

1. Tube made of metallic strip (metallic hose);146 for λ, see graph a.

Values of λ Re × 10−4

Curve 0.5 1 2

0.8

1.2

1.6

4

a) Flow running over the edges 0.0250 0.0254 0.0256 0.0257 0.0257 b) Flow entering the edges 0.0250 0.0262 0.0275 0.0284 0.0285

2. Corrugated tube;194 for λ, see graph b.

c = corrugation

Values of λ Re × 10−5

h lc

0.4 0.150 0.082 0.022

0.421 0.083 0

0.6 0.155 0.088 0.023

0.8 0.162 0.090 0.024

1 0.168 0.092 0.025

1.4 0.175 0.098 0.026

2 0.180 0.103 0.027

2.5 0.185 0.105 0.028

3 0.190 0.110 0.029

3. Tube made from glass cloth53 (see paragraph 72 of Section 2.1) λ ≈ 0.052 (10D0)0.1/D0 (50b)0.2, see the Table b is the width of the band wound around the wire tramwork of the tube made from glass cloth (when D0 ≤ 0.2 m, b = 0.02 m; when D0 > 0.2 m, b ≥ 0.03 m).

Values of λ D0, m

1.0

1.1

1.4

1.6–1.7

1.8

1.9

2.3

2.6–2.7

2.8

2.9

3.3

0.100 0.155 0.193 0.250

– – 0.070 –

0.053 0.063 – –

0.53 – – –

0.051 – – –

0.05 – – –

– 0.063 0.072 –

– 0.064 0.072 –

– 0.064 0.073 –

– – – 0.085

– – – 0.077

– – – 0.82

154

Handbook of Hydraulic Resistance, 4th Edition

Steel reinforced rubber hoses; stabilized flow132 w0dnom Re = > 4 × 104 v

Diagram 2.14

ζ*

∆p ρw20 ⁄ 2

=λ

l , dcal

where λ is determined from graph a as a function of the nominal diameter dnom; dcal is the calculated diameter determined as a function of the internal excess pressure pex at different dnom; see graph b; for v, see Section 1.2.

Characteristics of the hose Internal nominal diameter dnom, mm Diameter of the helix, mm Pitch, mm Cloth insert 1.1 mm thick, nos. 1 Rubber layer, mm Diameter of cotton helix, mm Rubber layer, mm Cloth insert 1.1 mm thick, nos. 2 dnom,, mm

λ

25 0.051–0.057

25 2.8 15.6 1 1.5 1.8 1.5 2 32 0.053–0.066

32 2.8 15.6 1 1.5 1.8 1.5 2 38 0.072–0.090

38 2.8 17.6 1 2.0 1.8 1.5 2 50 0.083–0.094

50 3.0 20.0 1 2.0 1.8 1.5 2

65 3.4 20.8 2.0 1.8 1.5 3 65 0.085–0.100

155

Flow in Straight Tubes and Conduits Smooth-rubber hoses; stabilized flow132

Diagram 2.15

ζ*

∆p

ρw20 ⁄ 2

=λ

l , dcal

where λ = A/Re0.265, see curves λ = f(Re) of graph a: A = 0.38–0.52 within the limits of Re = w0dcal/v = 5000– 120,000 and depending on the hose quality; dcal is the calculated diameter determined as a function of the internal excess pressure pex, see graph b; for v, see Section 1.2.

Characteristics of the hose Internal nominal diameter dnom, mm Rubber layer (internal), mm Cloth insert 1.1 mm thick, nos. Rubber layer (external), mm

25 2 2 0.9

32 2 2 0.9

38 2 2 0.9

50 2.2 3 1.2

65 2.2 3 1.2

Values of λ Re × 10−4

A 0.52 0.38

0.4

0.6

1

2.0

4.0

6.0

10

20

0.057 0.042

0.052 0.038

0.046 0.033

0.038 0.028

0.031 0.023

0.028 0.020

0.025 0.018

0.020 0.015

156

Handbook of Hydraulic Resistance, 4th Edition

Steel reinforced smooth-rubber hoses; stabilized flow132

Diagram 2.16 ∆p

lout , δ* 2 =λ dcal ρw0 ⁄ 2 where λ = f(Re, dnom, pex), see graphs a and b, dcal is the calculated diameter, which is determined as a function of the average internal pressure pex, see graph c; lout = kl; k is determined as a function of the average internal excess pressure pex, see graph d; Re = w0dnom/v; for v, see Section 1.2.

Values of λ at dnom = 65 mm Re × 10−5

pex, MPa 0.025 0.05 0.10 0.015 0.20 0.25

0.4 0.03 0.04 0.05 0.07 0.09 0.11

0.6 0.03 0.03 0.05 0.07 0.09 0.11

0.8 0.03 0.03 0.05 0.07 0.09 0.11

1 0.03 0.03 0.04 0.07 0.09 0.11

1.4 0.03 0.03 0.04 0.07 0.09 0.11

2 0.03 0.03 0.04 0.06 0.08 0.11

2.5 – 0.03 0.04 0.06 0.08 0.11

4 – 0.03 0.03 0.05 0.07 –

Values of λ at dnom = 100 mm Re × 10−5

pex, MPa 0.025 0.05 0.10 0.15 0.20 0.25

0.25 0.03 – – – – –

0.4 0.03 0.03 – – – –

0.6 0.03 0.03 0.03 0.03 0.05 –

0.8 0.02 0.03 0.03 0.03 0.05 0.06

1 0.02 0.02 0.03 0.03 0.04 0.06

1.4 – 0.02 0.02 0.03 0.04 0.05

2 – 0.02 0.02 0.03 0.04 0.05

2.5 – 0.02 0.02 0.03 0.04 0.05

4 – – 0.02 0.03 0.04 0.05

6 – – – 0.02 0.03 –

157

Flow in Straight Tubes and Conduits Tube made from tarpaulin-type rubberized material; stabilized flow [according to Adamov]

Diagram 2.17

⎞ ⎛ lj = nj ⎜λ + ζj⎟ , where nj is the number of D 0 ⎠ ⎝ pipes (joints); lj is the distance between joints;

ζ*

λ*

∆p

ρw20 ⁄ 2

∆p

(ρw20 ⁄ 2)(lj ⁄ D0)

= f1(Re), see graph a for different

degrees of the tube tension; ζj = f2(Re), see graph b; w0D0 ; for v, see Section 1.2. Re = v

Values of λ Degree of tube tension Good Moderate Poor

Re × 10−5 1

2

3

4

5

6

7

8

9

0.024 0.064 0.0273

0.020 0.042 0.0195

0.018 0.034 0.139

0.016 0.028 0.110

0.014 0.025 0.091

0.013 0.023 0.074

0.012 0.021 0.063

0.011 0.020 0.054

0.011 0.019 0.048

Re × 10−5

1

2

3

4

5

6

7

8

10

ξj

0.20

0.17

0.14

0.12

0.11

0.10

0.09

0.08

0.08

158

Handbook of Hydraulic Resistance, 4th Edition

Tube made from birch plywood with longitudinal grain; stabilized flow1 λ* __

∆=

Diagram 2.18 ∆p

(ρw0 ⁄ 2)(l ⁄ Dh) 2

__ , see curves λ = f(Re) for different ∆;

w0Dh ∆ ; for ∆, see Table 2.5; Re = , for v, v Dh

see Section 1.2.

Dh =

4F0 Π0

Values of λ __

Re × 10−5

∆ 0.00140 0.00055 0.00030 0.00015 0.00009

0.2

0.3

0.4

0.6

0.8

1

1.5

0.030 – – – –

0.028 – – – –

0.027 – – – –

0.025 0.021 – – –

0.024 0.21 – – –

0.023 0.019 0.018 0.018 0.018

– 0.018 0.017 0.017 0.017

(continued) __

Re × 10−5

∆ 0.00140 0.00055 0.00030 0.00015 0.00009

2

3

4

6

8

10

20

– 0.017 0.017 0.016 0.016

– 0.018 0.016 0.015 0.014

– 0.018 0.016 0.014 0.014

– – 0.016 0.014 0.013

– – – 0.014 0.012

– – – 0.013 0.012

– – – – 0.011

159

Flow in Straight Tubes and Conduits Plastic tubes; stabilized flow91,92

Diagram 2.19 ζ*

∆p ρw20 ⁄ 2

= λl ⁄ D0

1. Polyethylene (stabilized), rigid-vinyl plastic at 40 mm ≤ D0 ≤ 300 mm and 8 × 103 ≤ Re = λ=

w0D0

v

≤ 7.5 × 105

0.29 − 0.00023D0 Re0.22

, see Table 1

(D0, in mm; the coefficient at D0, in mm–1) 2. Glass cloth at 100 mm ≤ D0 ≤ 150 mm and 104 ≤ Re ≤ 3 × 105 λ=

0.282 − 0.000544D0 Re0.19

, see Table 2

3. Faolite at 70 mm ≤ D0 ≤ 150 mm and 104 ≤ Re ≤ 2 × 105 λ=

0.274 − 0.000662D0 Re0.2

, see Table 2

Table 1 Values of λ for polyethylene and rigid-vinyl plastic Re × 10−4 0.8 2 5 10 50 80

D0, mm 40

100

160

200

250

300

0.039 0.031 0.026 0.022 0.016 0.014

0.037 0.030 0.025 0.021 0.015 0.013

0.035 0.029 0.024 0.020 0.014 0.013

0.034 0.028 0.023 0.020 0.014 0.012

0.032 0.026 0.022 0.019 0.013 0.012

0.031 0.025 0.021 0.018 0.012 0.011

120

140

160

0.036 0.026 0.023

0.034 0.025 0.022

Table 2 Values of λ for glass cloth and Faolite Re × 10−4

D0, mm 60

80

100 Glass cloth

1 5 10

0.043 0.032 0.028

0.041 0.031 0.027

0.040 0.030 0.026

0.038 0.028 0.024

30

0.023

0.022

0.021

0.020

1 5 10 20

0.037 0.027 0.023 0.020

0.035 0.025 0.022 0.019

0.019

0.018

Faolite 0.033 0.024 0.021 0.018

0.031 0.022 0.019 0.017

0.029 0.021 0.018 0.016

0.027 0.019 0.017 0.015

160

Handbook of Hydraulic Resistance, 4th Edition

Tubes made from plastics (polyethylene or rigid vinyl) with joints; stabilized flow91,92

No.

Diagram 2.20

Joint

a b c d

Material

Welded Funneled Coupled Flanged

Polyethylene Vinyl plastic "–" Polyethylene

1, funnel; 2, circular recess; 3, coupling; 4, flange; 5, flanged end of the tube; 6, gasket (rubber ring) 15 × 4 mm ζ*

∆p ρw20 ⁄ 2

= nj(λlj ⁄ D0 + ζj) ,

where nj is the number of joints over the section; for λ, see Diagrams 2.1 through 2.5; ζj is the resistance coefficient of one joint; at 50 ≤ D0 ≤ 300 mm: a) welded joint at 1.8 × 105 = Re ≤ 5 × 105 ζj =

0.0046 ; see the Table D1.75 0

(D0 is in m; the coefficient at D0 is in m–1); b) welded joint at 2.4 × 105 ≤ Re ≤ 5.6 × 105 ζj = (0.113 − 0.225)D0 (see the Table); c) joint with the help of coupling at 1.8 × 105 ≤ Re ≤ 6 × 105 ζj = (0.045 − 0.156)D0 (see the Table); d) joint with the help of flanges at 2.8 × 105 ≤ Re ≤ 5 × 105 ζj = 0.148 − 0.344D0

(see the Table);

Values of ζ j for different types of joints and D0 D0, m

Joint Welded Funned Coupled Flanged

0.05

0.075

0.10

0.15

0.20

0.25

0.30

0.411 0.102 0.044 0.131

0.224 0.096 0.033 0.130

0.046 0.091 0.029 0.114

0.079 0.079 0.022 0.096

0.057 0.068 0.014 0.079

0.037 0.570 0.006 0.062

0.028 0.046 0.002 0.045

2.5 4.73

2.34 1.87

k′nonst

k′′nonst

k′nonst

1

′ k′non st

Parameter

Table 1

1.63

2.04

1.50

1.88

2

Dh =

1.42

1.78

1.31

1.64

4

Π0

4F0

1.31

1.63

1.21

1.51

6

Tube of any cross section behind a smooth inlet (starting length); nonstabilized flow22,2-144

1.23

1.54

1.14

1.43

8

1.18

1.47

=λ

λnonst = (ρw20 ⁄ 2)(x ⁄ Dh)

∆p

l Dh

=

0.43 = κnonstλ , (Re x ⁄ Dh)0.2

∆p

1.10

1.38

1.02

1.27

14

1.03

1.28

1.0

1.18

20

1.0

1.18

1.0

1.09

30

1.0

1.12

1.0

1.03

40

1.0

1.0

50

1.0

1.07

0.344 = κ′nonst , (Re x ⁄ Dh)0.2 Re0.05 , see Table 1; (x ⁄ Dh)0.2

(ρw20 ⁄ 2)(∆x ⁄ Dh)

where k′nonst + 1.09

λ′nonst =

=

Re0.05 , see Table 1; (x ⁄ Dh)0.2

see Diagrams 2.1 to 2.20

′′ + 1.36 whereknonst

where for λ see Diagrams 2.1 through 2.17

x ⁄ Dh

Re = 5 × 105

1.09

1.36

Re = 104

10

∆p ρw20 ⁄2

Turbulent flow:

ζ*

Diagram 2.21

1.0

1.0

1.0

1.0

70

Flow in Straight Tubes and Conduits 161

2.17

k′nonst

3.05

′ k′non st

knonst

x 1 ⋅ × 103 Dh Re

2.35

knonst

′

knonst

2.94

2.71

′ k′non st

′′

2.10

2.62

1.94

2.43

1

knonst

′

knonst

′′

knonst

′

′ k′non st

Parameter

Table 2

1.95

2

2.65

2.05

2.56

1.89

2.36

1.83

2.28

1.69

2.11

2

1.64

5

2.31

1.78

2.23

1.64

2.05

1.59

1.99

1.47

1.84

4

2.12

1.64

2.05

1.52

1.90

1.47

1.84

1.36

1.70

6

1.37

10

Tube of any cross section behind a smooth inlet (starting length); nonstabilized flow22,2-144

1.94

Re = 107

1.48

1.65

Re = 5 × 106

1.37

1.71

Re = 106

1.32

1.65

Re = 5 × 10

1.22

1.52 5

x ⁄ Dh Re = 105

10

1.80

1.39

1.74

1.28

1.60

1.24

1.55

1.14

1.43

14

1.25

15

20 1.17

x 1⎞ knonst = f ⎛⎜ see Table 2 Dh Re ⎟ ⎠ ⎝

Laminar flow (Re ≤ 2000):

2.02

1.55

1.94

1.43

1.78

1.39

1.74

1.28

1.60

8

1.68

1.29

1.61

1.19

1.49

1.15

1.44

1.06

1.32

20

1.12

25

1.55

1.19

1.49

1.10

1.37

1.06

1.33

1.0

1.23

30

Diagram 2.21

30 1.08

1.47

1.12

1.40

1.04

1.30

1.0

1.25

1.0

1.16

40

1.39

1.08

1.35

1.04

1.24

1.0

2.00

1.0

1.11

50

1.0

40

1.30

1.0

1.26

1.16

1.0

1.12

1.0

1.03

70

162 Handbook of Hydraulic Resistance, 4th Edition

Flow in Straight Tubes and Conduits

163

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284. El Telbany, M. M. M. and Reynolds, A. J., Turbulence in plane channel flows, J. Fluid. Mech., vol. 111, 283–318, 1981. 285. El Telbany, M. M. M. and Reynolds, A. J., The structure of turbulent plane Couette flow, Trans. ASME, J. Fluids Eng., vol. 104, 367–372, 1982. 286. Polderman, H. G., Analytical and experimental studies on horizontal and vertical capsule transport, in Hydrotransport 6, Proc. 6th Int. Conf. Hydraul. Transp. of Solids in Pipes, Canterbury, vol. 1, pp. 169–186, 1979. 287. Oleinik, A. Ya., Karasik, V. M., Kril, S. I., Berman, V. P., and Ocheretko, V. F., Hydraulic Pipeline Transport of Containers (Theory and Experiment), Naukova Dumka Press, Kiev, 1983, 124 p. 287. Kroonenberg, H. H., A mathematical model for concentric horizontal capsule transport, Can. J. Chem. Eng., vol. 56, no. 5, 538–543, 1978. .. .. 289. Reichardt, H, Gesetzmaβigkeiten der geradligen turbulenten Couettestromung, in Mitt. Aus dem .. .. Max-Plank-Institut fur Stromungsfoschung, Gottingen, Nr. 22, 1959, 45 S.

CHAPTER

THREE RESISTANCE TO FLOW AT THE ENTRANCE INTO TUBES AND CONDUITS RESISTANCE COEFFICIENTS OF INLET SECTIONS

3.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS 1. Resistance of the flow at the entrance into a straight tube or conduit of constant cross section (Figure 3.1) is governed by two parameters: the relative thickness δ1/Dh of the inlet tube wall edge and the relative distance b/Dh from the tube edge to the wall in which the tube is installed. 2. The resistance coefficient ζ of the straight inlet section is a maximum when the edge is very sharp (δ1/Dh = 0) and when the tube edge is at an infinite distance from the wall in which the tube is mounted (b/Dh = ∞). In this case, ζ = 1.0. 3. The minimum value of the resistance coefficient ζ is 0.5, and it can be attained by thickening the inlet edge. The coefficient ζ has this same value when the tube is mounted flush with the wall (b/Dh = 0). 4. The effect of the wall on the inlet resistance coefficient virtually ceases at b/Dh ≥ 0.5. This case corresponds to the condition where the tube entrance inlet edge is at an infinite distance from the wall. 5. When the flow enters a straight tube or conduit, it separates by inertia from the inner surface close behind the entrance if the inlet orifice edge is insufficiently rounded. This separation of the flow and the induced formation of eddies constitute the major sources of the inlet pressure losses. Flow separation from the tube walls leads to a decrease in the jet cross section (jet contraction). For a straight inlet orifice with a sharp edge, the jet contraction coefficient ε = Fcon/F0 is equal to 0.5 for turbulent flow. 6. When the inlet wall is thickened, beveled, or rounded or when the edge of the tube or conduit is adjacent to the wall into which the tube is mounted, the flow passes the inlet edge more smoothly and the flow separation zone becomes shorter, thus decreasing the inlet resistance. 177

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Figure 3.1. Flow pattern at the inlet into a straight tube from an infinite space.

7. A substantial decrease in the resistance occurs when the flow enters the tube through a smooth inlet, the cross section of which forms an arc of a circle, etc. (Figure 3.2a). Thus, for example, for a circular nozzle or collector having a relative value of the radius of curvature of r/Dh = 0.2, the resistance coefficient ζ decreases to 0.04–0.05 as compared to a value of ζ = 1.0 at r/Dh = δ1/Dh = 0, a sharp edge.* 8. A relatively low resistance is also observed when the flow enters the tubes through inlets with straight boundaries shaped like a truncated cone (Figure 3.2b and c) or in the form of contracting sections with a transition from rectangular to circular, or from circular to rectangular (Figure 3.2d). The resistance coefficient of such inlets depends on both the contraction angle α and the relative length l/Dh of the contracting section. For each length of the conical inlet there exists an optimum value of α at which the resistance coefficient ζ is a minimum. Practically, the optimum value of α for a fairly wide range of l/Dh (of order 0.1 to 1.0) lies within the limits of 40–60o. For such angles and, for example, the relative length l/Dh = 0.2, the resistance coefficient is equal to 0.2. 9. Pressure losses in a conical inlet are mainly associated with flow separation in two locations: directly downstream of the inlet section and over the straight section following this entry (Figure 3.2b and c). The losses in the first location dominate when the contraction angle α of the conical entry is relatively small (Figure 3.2b). The losses in the second location start to dominate at large values of α and increase with increases in this angle (Figure 3.2c). For α = 0o, this reverts to the conventional case of a straight inlet for which ζ = 1. For α = 180o, the inlet channel is flush mounted into the wall and ζ = 0.5. ∗ When the tube inlet is smooth, the only losses are those of the total pressure in the boundary layer. These are not observed in the core of the flow. Therefore, the resistance coefficient of a favorable or smooth inlet (collector) can be determined most accurately by experiment, through measurement of the total pressure distribution and the velocity in the outlet section of the inlet collector. The measurements in the boundary layer should then be made with the use of a microprobe. In this case, the resistance coefficient is

ζ*

∆p ρw20 ⁄ 2

∫ (p0 − p′0)wdF =

F0

(ρw30 ⁄ 2)F0

,

where w is the velocity in the outlet section of the collector; p0, p′0 is the total pressure upstream of the collector entrance and at its outlet. Editor’s note: The term "collector" is used to denote an entry to a pipe or conduit, such as a nozzle.

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179

Figure 3.2. Schematic diagrams of smooth inlet sections.

10. When the inlet section is mounted into the end-face wall at an angle (see Diagrams 3.2 and 3.3), the inlet resistance increases. The resistance coefficient in the case of circular or square cross sections and w∞ = 0 can be calculated from Weisbach’s formula49

ζ*

∆p ρw20 ⁄ 2

+ 0.5 + 0.3 cos δ + 0.2 cos2 δ .

For other shapes of the channel cross sections the resistance coefficients are given on Diagram 3.2 (rounded off to 10%).20 11. When a flow with velocity w∞ passes the wall into which a tube is mounted (see scheme on Diagram 3.3), the behavior is, in most respects, similar to that occurring when the flow discharges through an orifice in the wall under the same conditions (see Chapter 4, paragraphs 40–47). However, there are some differences. Thus, when the flow is sucked into a straight channel, there are no velocity pressure losses in the aspirated jet, with the result that the resistance coefficient is much lower than in the case of discharge from an orifice. Moreover, when the angles of inclination δ of straight sections are greater than 90o, the coefficient ζ takes on negative values, owing to an increase in the ram effect at certain velocity ratios w∞/w0 > 0 (see Diagram 3.3). 12. A baffle or wall (Figure 3.3) placed in front of the inlet section at a relative distance h/Dh < 0.8–1.0 will increase the inlet resistance. This increase becomes greater the closer the baffle is to the inlet orifice of the tube, that is, the smaller h/Dh is. 13. The resistance coefficient of inlet sections, which are not flush mounted with the wall and which have different thicknesses of the rounded or beveled entrances with a baffle placed before the entrance, is determined from the author’s approximate formula12,13

ζ*

∆p ρw20 ⁄ 2

+ ζ′ +

σ1 n2

,

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Handbook of Hydraulic Resistance, 4th Edition

Figure 3.3. Inlet section with a baffle in front of the entrance.

where ζ′ is the coefficient which accounts for the effect of the inlet edge shape and which is determined in the same way as ζ in Diagrams 3.1, 3.4, and 3.6; σ1 is the coefficient accounting for the effect of the baffle or screen, where σ1 = f(h/Dh), shown in the curve of Diagram 3.8. The resistance coefficient of smooth collectors mounted flush with the wall, with a baffle placed before the entrance, is determined from the curves ζ = f(h/Dh, r/Dh) of Diagram 3.5. 14. In inlet sections with a sudden transition from a larger cross section of area F1 to a smaller one of area F0 (Borda mouthpiece, Figure 3.4), the resistance coefficient at large Reynolds numbers (Re = w0Dh/v > 104) depends on the area ratio F0/F1 and can be calculated from the formula

ζ*

∆p ρw20 ⁄ 2

=

ζ′

F0 ⎞ ⎛ ⎜1 − F ⎟ 1⎠ ⎝

m

,

(3.1)

where ζ′ is a coefficient which depends on the inlet edge shape of the smaller channel (see Diagram 3.9) and which is determined in the same way as ζ from Diagrams 3.1, 3.2, and 3.6; m is an exponent depending on the inlet conditions; for values of b/Dh = 0–0.01 it varies from 0.75 to 1.0, while for b/Dh > 0.01 it can be assumed equal to 1.0.12,13 When the inlet edge of a narrow channel is mounted flush with the end-face wall of a wider channel (b/Dh = 0), this represents a typical case of sudden contraction, which is considered in Chapter 4, paragraphs 22–24.

Figure 3.4. Schematic diagram of flow with sudden contraction.

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181

Figure 3.5. Entrance through a circular bellmouth.

15. The resistance coefficient of inlet sections depends also on the location and the manner in which they were mounted into the wall. Thus, a lowered resistance coefficient can be attained by installing an annular lip or an annular step ahead of the inlet section enclosing the orifice (Figure 3.5). If the edge of the lip or of the step is sharp, the flow, upon entering this section, separates from its surface. The resulting recirculation favors smooth inflow of the fluid into the main inlet section of the tube without separation. As a result, the inlet resistance decreases considerably. 16. The optimum dimensions of the enlarged section, over which the recirculation bellmouth is formed, must correspond to the vertical dimensions of the region adjacent to the contracted cross section of the stream at the straight tube with sharp edges and, similarly, in a tube mounted flush with the wall. In fact, Khanzhonkov30 has shown experimentally that the lowest resistance coefficient ζ = 0.10–0.12 is obtained with the use of a lip of l/D0 ≈ 0.25 and D1/D0 ≈ 1.2 and with the use of a step of l/D0 ≈ 0.2 and D1/D0 ≈ 1.3. With a rounded inlet edge, the lowest resistance coefficient in these cases would be 0.07– 0.08. 17. The values of ζ in the case of other modes of mounting the inlet sections (in the endface wall or between the walls) are given in Diagrams 3.10 and 3.11. 18. The resistance coefficient for the flow entering into a straight section through a single orifice or an orifice grid (entry with sudden expansion, F1 = ∞; see Diagram 3.12) at Re = wordh/v > 10 for the general case (inlet edges of any shape and of any thickness) is calculated from the author’s approximate formula13,14 ζ*

_ _ ⎡ζ ′ + (1 − f) 2 + τ (1 − f) + λ l ⎤ _1 , = ⎢ dh ⎥ f 2 ρw20 ⁄ 2 ⎣ ⎦ ∆p

(3.2)

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where ζ′ is the coefficient which accounts for the shape of the inlet and which is determined in the same way as ζ for inlet sections with the end-face wall from Diagrams 3.1, 3.2, 3.4, and 3.7; τ is the coefficient accounting for the effect of the grid (orifice) wall thickness, inlet edge shape, and conditions of flow passage through the orifice; λ is the friction __ coefficient over the length (depth) of the _ grid orifices, determined as a function of Re and ∆ = ∆/dh from Diagrams 2.2 through 2.6; f = For/F0 = For/Fgr is the area ratio. 19. The general case of the flow entry through an orifice or an orifice grid consists of a number of particular cases: _ a) Sharp edges of orifices (l = l/dh ≈ 0), for which ζ′ = 0.5 and τ = 1.41; in this case, Equation (3.2) is reduced to the following formula derived by the author:12,13 ζ*

2 _ 2 _1 ⎛ 1.707 _ − 1⎞ . 1.707 f = − ) = ( ⎜ ⎟ f2 ⎝ f ρw20 ⁄ 2 ⎠

∆p

(3.3)

b) Thickened edges of orifices for which the coefficient ζ′ = 0.5 and _ _ _ −ϕ(l) , (l = l ⁄ Dh) , τ = (2.4 − l) × 10

(3.4)

where _ _ _ ϕ(l) = 0.25 + 0.535 l 8 ⁄ (0.05 + l 7) .

(3.5)

c) Beveled or rounded (in the flow direction) edges of orifices for which it is assumed that λl/Dh = 0 and τ ≈ 2√ ⎯⎯ζ′ ; the case in which ζ*

_ 2 ⎛1 − √ ′ − f⎞ _1 . ζ = ⎯ ⎯ ⎠ f2 ρw20 ⁄ 2 ⎝

∆p

(3.6)

In the case where the edges of orifices are beveled in the flow direction, ζ′ is determined similarly to ζ for a conical collector with an end-face wall from Diagram 3.7 as a function _ of the contraction angle α and the relative length l = l/Dh. For the values of α = 40–60o, it is determined from the formula ζ′

= 0.13 + 0.34 × 10

_ _ 2.3 −(3.45l+88.4l )

.

(3.7)

In the case of orifices with rounded edges, the coefficient ζ′ is determined in the same _ way as ζ for a circular collector with the end-face wall as a function of r = r/Dh from Diagram 3.4 or from the formula _

ζ = 0.03 + 0.47 × 10 −7.7r . *

(3.8)

∗

Calculations according to paragraphs b and c can be performed virtually starting from the values Re = 104 and higher.5

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183

20. For transition and laminar flow regions (Re = w0Dh/v < 104–105) and conventional entrance of flow (without orifices or grids), the resistance coefficient can be determined from the formula analogous to Equation (1.3): ζ*

∆p 2

ρw0 ⁄ 2

A + Re + ζquad ,

where ζquad is taken as ζ for the quadratic region (Re > 104–105), A = 30.2 21. For transition and laminar regions with entrance of flow through an orifice or a grid, the resistance coefficient can be calculated from the following approximate formulas (according to paragraphs 30–36 of Chapter 4): at 30 < Re < 104–105 ζ*

ζϕ _ = _ 2 + ε0Reζquad , ρw0 ⁄ 2 f ∆p 2

at 10 < Re < 30 ζ*

33 _1 _ + ε0Reζquad , and Re f 2

at Re < 10 ζ=

33 _1 , Re f 2

_ where ζϕ = f1(ReF0/F1), as shown in _the graph of Diagram 4.19 (it is postulated that f = For/F0 corresponds to the ratio F0/F1); ε0Re = f2(Re), in Diagram 4.19. ζquad is the resistance coefficient of the inlet section with an orifice (grid) of the given shape which is determined similarly to ζ from Equations (3.2)–(3.8). 22. With lateral (transverse) entrance into the end section of the tube (Figure 3.6) the resistance is much higher than_ that with straight entrance and sudden expansion (through an orifice, grid), particularly at f > 0.2, since more complicated conditions for the flow of liquid (air) are observed in the case of lateral entrance. 23._On the basis of visual observations, Khanzhonkov and Davydenko31 showed that at small f the jet, which enters through an orifice into the tube, moves to the opposite wall, over which it spreads in all directions. Part of the jet moves toward the closed end of the tube, rotates through 180o, and flows into the other end of the tube in the form of two rotating streams (Figure 3.6a). _ At some ("critical") values of f air inflow into the closed space of the tube nearly ceases, while the jet in the form of two rotating streams flows completely into the opposite end of the tube (Figure 3.6b). 24. This type of a flow is not only responsible for the increased resistance of the side inlet, but is the reason for the complex dependence of the resistance coefficient ζ on the area ratio

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Handbook of Hydraulic Resistance, 4th Edition

Figure 3.6. Schematic diagrams of flow entering into a side orifice in the end section of a tube: (a) at _ _ small values of f and (b) at large values of f.

_ _ f (Figure 3.7). A sharp decrease in ζ corresponds to the "critical" value of f at which the above rearrangement of the flow occurs after entrance into the tube. 25. According to the author’s data, flow entrance into the tube through two side orifices, located one opposite _ the other, increases the inlet resistance, which becomes greater the higher the value of f. 26. Entrance through side orifices is often utilized in ventilating shafts of rectangular cross section. Such orifices are furnished with louvers to prevent the entry of particulate matter. The resistance coefficient of such shafts also depends not only on the relative area of the orifices, but on their relative arrangement. Diagram 3.17 presents the resistance coefficients of intake shafts with differently positioned side orifices. The values of ζ are given here for orifices with and without fixed louvers. 27. The resistance of intake shafts with straight entrances, but provided with canopies (see Diagram 3.18), is similar to the resistance of conventional inlet sections with baffles. For ventilating shafts of circular cross section, for which the relative thickness δ1 of the inlet edges lies within 0.01–0.002, the effect of this parameter may be neglected and the resistance coefficient ζ assumed to have values similar to those for shafts with sharp inlet edges. The relative distance h/Dh between the canopy hood and the inlet edge of the shaft can be assumed equal to 0.4. Larger distances would require extremely large canopy hoods, otherwise atmospheric contaminants might enter the shaft. Of all the available constructions of intake shafts one should recommend the one with a conical entry section at the inlet. This shaft has the minimum resistance coefficient, ζ = 0.48.28 28. When the flow enters the tube through a screen, the total resistance coefficient can be approximated as the sum of the separate resistance coefficients of the screen and of the inlet, that is, ζ*

∆p

ρw0 ⁄ 2 2

+ ζ′+

ζsc

n2

,

Flow at the Entrance into Tubes and Conduits

185

Figure 3.7. Dependence _ of the inlet resistance coefficient of a side orifice in the end section of the tube on the relative area f: solid lines, experiments of Khanzhonkov and Davydenko31 with one orifice; ∆p dashed lines, experiments of the author15 with two side orifices opposite each other: ζ = . ρw20 ⁄ 2

where ζ′ is the resistance coefficient of the inlet, without a screen, determined as ζ for the given shape of the inlet edge from the corresponding graphs of Diagrams 3.1 and 3.4 through 3.8; ζsc is the resistance coefficient of the screen, determined as ζ from the corresponding graphs of Diagram 8.6; n = F1/F0 is the area ratio of the cross section in the place where the screen is mounted to the minimum cross section of the inlet length. 29. The _ resistance coefficient of a fixed louver grating depends on both its open area co-

efficient f = For/Fgr and the relative depth of the channels l/b′1. For each open area coefficient

of the grating there is an optimum value of the relative depth (l/b′1)opt at which the resistance coefficient is minimal. It is therefore recommended _ that, as a rule, gratings be used which

have the optimal values of l/b′: (l/b′1)opt ≈ 11(1 – f).* 30. In the case of standard gratings with fixed louvers, the inlet edges of the fins are cut along the vertical (see scheme of a Diagram 3.19). However, it is more beneficial that the inlet edges be cut along the horizontal (see scheme b). This provides a 40% decrease in the resistance. ∗

This formula was obtained by the author on the basis of Bevier’s37 data.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 3.8. Inlet pipes of axial stationary turbo machines: (a) circular annular entry (collector) formed by two curviliner surfaces; (b) the same with a sloping bottom of the annular entry; (c) circular collector formed by a curvilinear outer surface and a center body.

31. The resistance coefficient of gratings with fixed louvers at the entrance to the channel* is determined as: at l/b′1 ≥ (l/b′1)opt 2

ζ*

_F ⎡ ⎤ 1 ⎞ ⎛ ⎢0.85 + ⎜1 − f gr ⎟ + ζfr⎥ _ k = 2 F0 ρw20 ⁄ 2 ⎣ ⎦f ⎠ ⎝ ∆p

2

⎛ F0 ⎞ ⎜F ⎟ ; ⎝ gr ⎠

at l/b′1 < (l/b′1)opt ∗

The formulas agree satisfactorily with the experimental data of Bevier37 and Cobb.40

Flow at the Entrance into Tubes and Conduits

187

2

_F ⎡ ⎤ 1 ⎛F ⎞2 ⎛ gr ⎞ 0 _ + ζ ζ * 2 = k ⎢0.85 + ⎜1 − f fr⎥ 2 ⎜ ⎟ ⎟ + ∆ζ , F F 0 gr ρw0 ⁄ 2 ⎣ ⎦f ⎝ ⎠ ⎠ ⎝ _ _ where ∆ζ = 0.5[11(1 – f) – l/b′1]; ζfr = λl ⁄ b′1; f = For/F0, see Diagram 3.19; k = 1.0 for a standard grating (inlet edges cut vertically); k = 0.6 for an improved grating (inlet edges cut horizontally); and λ is the coefficient of hydraulic friction along the length (depth) of the louver channels, determined depending on Re = worb′1 ⁄ v from Diagrams 2.1 through 2.5. 32. The primary requirement for inlet pipes to axial flow turbo machines (Figure 3.8) is that the total pressure losses should be minimal and the velocity profile in the outlet section of the inlet collector, which supplies air directly to the blade rims of the machines, should be almost undistorted. 33. As demonstrated by the experiments of Dovzhik and Kartavenko,10 for inlet pipes designed on the basis of the use of a collector with two curvilinear surfaces (Figure 3.8a), these conditions are best fulfilled for a high degree of pipe constriction (np ≥ 3.5, where np = Fin/F0, Fin = HB is the area of entrance into a scroll or volute). In this case, the degree of collector constriction should be close _ to the_ degree of pipe constriction (ncol = np, where ncol = Fcol/F0 = 2hcol/h0[Dcol/D0(1 + d)] and d = d/D0), while the radial dimension of the pipe should be large (Dscr = Dscr/D0 > 1.3). Sloping of the back wall in the bottom of the scroll (Figure 3.8b) insignificantly, within certain limits, decreases the pressure losses in the pipe. With the above optimization parameters, the resistance coefficient of the pipe is ζ * ∆p ⁄ (ρ0w20 ⁄ 2) = 0.12–0.15 (where w0 = ca is the average axial velocity in the outlet section of the circular collector [in section F0] and ρ0 is the gas density in the same cross section). 34. It is advisable to use the above collector (Figure 3.8a) in cases where the pipe has a large degree of contraction (axial compressors, turbines). When the degree of contraction needs not be large (fans) and the available radial dimensions of the pipe are substantially limited, it is advisable to use a pipe in which the circular collector is formed by the one curvilinear surface (Figure 3.8c). In this case, __ the pipe will have the minimum __ resistance coefficient at np ≥ 3.5, H/D0 ≥ 0.95 and Dsn = 1.15–1.25. At larger values of Dsn(>1.0) it is ∆p

advisable that the front wall of the scroll be inclined up to a/H ≈ 0.4. This inclination of the wall provides additional reduction in the resistance coefficient. 35. Nonuniform velocity distribution both in the radial direction and circumferentially about the outlet section of the collector, obtained at the above optimum parameters of the pipes (departure from the average velocity ca of the order of 15–20%), does not influence the characteristics of the compressor stages. However, velocity nonuniformity leads to a periodic change in the aerodynamic forces acting on the blades of the rotor, which adversely affects the fatigue strength of the machine.19 36. Fans or engines of ground transport facilities and vessels are usually installed in channels that are furnished with forward intakes that ensure uniform velocity fields and total pressure with a low total pressure loss coefficient at the inlet to a fan or an engine.6 During slip motion or in the presence of a side wind at the inlet to the fan or engine, circumferential or radial nonuniformity of flow is formed resulting in the occurrence of aerodynamic losses.50 There were attempts to increase the efficiency of an air intake by installing guide vanes at the inlet to the channel. Testing of these vanes at angles of flow incidence of 90o 51 has

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Handbook of Hydraulic Resistance, 4th Edition

Figure 3.9. Schematic diagram of an experimental setup.

shown that the range of working regimes which is determined by the ratio of the incidence flow velocity V to the channel flow velocity U2 can here be enlarged only up to V/U2 = 0.5. A more effective way to increase the efficiency of air intakes consists of controlling a boundary layer to prevent its separation at the inlet to the channel.52,53 An experimental investigation of an air intake with a jet inlet device employing the Coanda effect, i.e., adherence of a fine jet to a convex curvilinear surface, plane or axisymmetric, was carried out in Reference 54. The schematic diagram of an experimental setup is presented in Figure 3.9. The experiments were carried out in a wind tunnel, the velocity V varied within 10–30 m/s, the Reynolds number Re = U2 d/ν was equal to (0.8–2.4) × 104. The experiment was aimed at determing the velocity fields inside the channel (at different angles α and at V/U2 = 0.7 and 1.0) as well as the total pressure loss coefficient ζt with _ allowance for energy lost in blowing and the blowing intensity q = q/(Q – q), where Q is the air flow rate in the channel and q is the injected air flow rate.

Flow at the Entrance into Tubes and Conduits

189

Figure 3.10. Velocity profiles in the horizontal section of the air intake: a) with oblique flow incidence without injection; b) at inflow angle α = 75o and radial injection; c) at inflow angles α = 60o and 90o and radial injection.

Figure 3.10a presents the velocity profiles in the horizontal section of the air intake with oblique flow incidence (α = 45o–90o at V/U2 = 0.7 and 1.0. The figure_ illustrates formation of a separating flow. Similar velocity profiles in radial air injection at q = 0–0.11, α = 75o, V/U2_ = 0.7 and 1.0 are given in Figure 3.10b and at α = 60o and 90o, V/U2 = 0.7 and 1.0, and q = 0–0.12 — in Figure 3.10c. It follows from Figure 3.10b and c that flow separation _ in the channel is entirely eliminated when q = 0.1. This is evidenced also by the flow spectra shown in Figure 3.11. 37. In the engines of aircraft, ships, and also of subway cars, air intakes are installed (intake pipes, Figure 3.12). The aerodynamic characteristics of these devices depend on the operational and constructional parameters. Detailed investigations of the aerodynamic characteristics of intake pipes of aircraft engines are described in Reference 11. The results of the investigations of the aerodynamics of air intakes of gas-turbine ships are given in Reference 6. 38. The inlet conditions into an intake pipe, the inlet section of which is arranged on a solid surface (wing of an aircraft, hood of an aircraft engine, fuselage of a helicopter, body of a ship, top of a car, etc.), depend on the velocity ratio win at the entrance to the pipe or, which is the same, on the velocity w0 at the exit from the pipe to the velocity w∞ of the free stream (flight velocity, ship motion velocity, car velocity). When the inlet area is selected so that at the given flow rate through the pipe the ratio win/w∞ is smaller than unity, one observes retardation (expansion) of the jet accompanied by an increase in the static pressure. The formation of the positive pressure gradient along the jet in the presence of a relatively thick boundary layer on the solid surface leads (as in a conventional thick-walled diffuser) to flow separation from this surface (Figure 3.12a). With an increase in the pressure gradient and, consequently, with a decrease of the ratio win/w∞, the separation becomes more intensive, and the inlet pressure losses increase.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 3.11. Spectra of flow past objects with oblique inflow toward an air intake.

39. The pipe can have such an area of the inlet orifice at which the velocity ratio win/w∞ for the given flow rate will be equal to, or higher than, unity. When win/w∞ = 1, the crosssectional area and correspondingly the velocity, and, consequently, the static pressure along the jet remain virtually constant up to the entrance into the pipe. In this case, no flow separation from the surface can occur (Figure 3.12b) and air enters the pipe virtually without loss.

Figure 3.12. Different cases of flow inlet into the pipe:11 a, at small discharge coefficients (Win/Win is much smaller than unity); b, at large discharge coefficients; c, at Win/W∞ ≥ 1.

Flow at the Entrance into Tubes and Conduits

191

40. At win/w∞ > 1, the flow enters a pipe with acceleration (the jet is contracted) and, consequently, this is accompanied by a fall in the pressure. Therefore, flow separation from the solid surface is even more impossible. However, in the case of a very appreciable jet contraction, such inflow may lead to flow separation from the inner edge of the pipe (Figure 3.12c). This separation can be eliminated by using rather a smooth (thick) inlet edge. 41. The values of the resistance coefficients of the entrances to the intake pipes (ζin * ∆p ⁄ ρw20 ⁄ 2) with different versions of the location of inlet sections with respect to the solid surface (in the given case the surface of the aircraft engine hood) and at different velocity ratios w0/w∞ are given in Diagram 3.22. This diagram also contains the schemes of the versions of testing of pipes. The pressure losses associated with the entrance of flow into an intake pipe are smallest when the pipe is located directly near the front edge of the hood (version 1). In this case, there is no flow separation before the entrance, whereas the substantial increase in loss with a decreasing velocity ratio at w0/w∞ < 0.3 is due to the flow separation after its entrance to the pipe (see paragraph 39). 42. The influence of the flow separation from the solid surface before the entrance to the pipe on the inlet resistance can be considerably decreased or entirely eliminated by increasing the distance h of the protruding portion of the pipe from the solid surface, especially if the neck of the pipe could be streamlined for the overflow of the boundary layer (see version 6 on Diagram 3.22). However, it is necessary here to take into account the increase in the drag of the pipe with an increase in the indicated distance from the solid surface. 43. The total energy losses in the intake pipe (air-intake device) are composed of the inlet pipe energy losses and internal losses over the entire pipe from the entrance to exit of flow from it. The general (total) resistance coefficient of the intake pipe is

ζp *

∆p ρw20 ⁄ 2

= ζin + ζex ,

where ζin * ∆p ⁄ ρw20 ⁄ 2 is the resistance coefficient of the entrance depending on the velocity ratio w0/w∞ and on the location of the pipe (air-intake device) on the given object; ζex * ∆p ⁄ ρw20 ⁄ 2 is the coefficient of internal resistance of the entire section of the air-intake device from the entrance to the exit of flow from it. 44. The drag of the pipe is composed of two values: the "hydraulic" cxh and external frontal resistance cxo. The hydraulic frontal resistance originates due to the loss of momentum by the flow entering the pipe. The external frontal resistance is induced by the external flow past the pipe and its interference onto the adjacent part of the aircraft (helicopter, ship, car). 45. Diagram 3.23 depicts some schemes of the inlet elements of industrial axial fans. This diagram also gives the resistance coefficients of the inlet elements calculated according to the recommendations of Bychkova3,4 for different inlet and operational conditions of the fans. 46. Diagram 3.24 presents the schemes of the inlet elements of radial (centrifugal) fans and the values of the resistance coefficients of these elements according to the same recommendations as given in paragraph 45.

192

Handbook of Hydraulic Resistance, 4th Edition

3.2 DIAGRAMS OF RESISTANCE COEFFICIENTS Entrance into a straight tube of constant cross section; Re = w0Dh/v > 104 12,13

Diagram 3.1 1) Entrance into a tube at a distance (b/Dh < 0.5) from the wall in which it is mounted. 2) Entrance into a tube mounted flush with the wall (b/Dh = 0). 3) Entrance into tube at a distance from the wall (b/Dh < 0.5) in which it is mounted.

Dh = hydraulic diameter F0 = area Π0 = wetted perimeter of cross section 1) and 2) ζ * 3) ζ *

⎛ δ1 ⎞ ∆p b , see curve ζ = f ⎜ ⎟ at the given . 2 Dh ρw0 ⁄ 2 ⎝ Dh ⎠

⎛ δ1 ⎞ ∆p b , see curve ζ = f ⎜ ⎟ at ≥ 0.5. 2 Dh ρw0 ⁄ 2 ⎝ Dh ⎠

For computer calculations at δ/Dh < 0.05 and 0.01 < b/Dh < 0.05. ζ*

∆p ρw20 ⁄ 2

3

⎧3 ⎪

⎫ ⎪

∑ ⎨⎪∑ [ai,j(b ⁄ Dh) j]⎬⎪ (δ ⁄ Dh) , i=0 ⎩i=0

⎭

where for ai,j, see the Table.

Values of ai,j j

i

0 0.549356 –4.93702 160.273 1,650.38

0 1 2 3

1 9.22856 –681.756 17,313.6 –139,018

2 –79.0065 7,189.72 –212,416.0 1,930,080

3 258.742 –24,896.6 766,932 –7,239,530

4 –268.925 26,416.2 –827,816 795,042

Values of ζ ζ1%Dh 0 0.004 0.008 0.012 0.016 0.020 0.024 0.030 0.040 0.050

b%Dh 0 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.002 0.57 0.54 0.53 0.52 0.51 0.51 0.50 0.50 0.50 0.50

0.005 0.63 0.58 0.55 0.53 0.51 0.51 0.50 0.50 0.50 0.50

0.010 0.68 0.63 0.58 0.55 0.53 0.52 0.51 0.51 0.51 0.50

0.020 0.73 0.67 0.62 0.58 0.55 0.53 0.25 0.52 0.51 0.50

0.050 0.80 0.74 0.68 0.63 0.58 0.55 0.53 0.52 0.51 0.50

0.100 0.86 0.80 0.74 0.68 0.64 0.60 0.58 0.54 0.51 0.50

0.200 0.92 0.86 0.81 0.75 0.70 0.66 0.62 0.57 0.52 0.50

0.300 0.97 0.90 0.85 0.79 0.74 0.69 0.65 0.59 0.52 0.50

0.500 1.00 0.94 0.88 0.83 0.77 0.72 0.68 0.61 0.54 0.50

∞ 1.00 0.94 0.88 0.83 0.77 0.72 0.68 0.61 0.54 0.50

193

Flow at the Entrance into Tubes and Conduits Entrance into a straight tube of constant cross section; Re = w0Dh/v > 104 12,13

Diagram 3.1

Entrance from an infinite space (w∞ = 0) into a tube mounted flush into a wall at any angle δ; Re = w0Dh/v ≥ 104 20,49

Diagram 3.2

For circular and square orifices

ζ*

∆p ρw20 ⁄ 2

= 0.5 + 0.3 cos δ + 0.2 cos2 δ .

For orifices of any shapes

ζ*

Dh =

∆p ρw20 ⁄ 2

= f (δ) .

4F0 Π0

Values of ζ (rounded up to 10%) δ, deg

l a

20

30

45

60

70

80

90

1.0 0.2–0.5 2.0 5.0

0.96 0.85 1.04 1.58

0.90 0.80 1.00 1.45

0.80 0.70 0.90 1.20

0.70 0.62 0.80 0.95

0.63 0.56 0.70 0.78

0.56 0.50 0.58 0.60

0.50 0.45 0.45 0.45

194

Handbook of Hydraulic Resistance, 4th Edition

Entrance from an infinite space (w∞ = 0) into a tube mounted flush into a wall at any angle δ; Re = w0Dh/v ≥ 104 20,49

Diagram 3.2

Entrance into a tube mounted flush into a wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 20

Diagram 3.3

ζ*

⎛ w∞ ⎞ ∆p , see curves ζ = f ⎜ ⎟ 2 ρw0 ⁄ 2 ⎝ w0 ⎠

Values of ζ (rounded up to 10%) for circular and square cross sections, i.e., at l/a = 1.0 (see graph a) w∞ w0

δ, deg 30 45 60 90 120 150

0

0.5

1.0

1.5

2.0

2.5

0.90 0.80 0.65 0.50 0.65 0.85

1.55 1.30 1.04 0.56 0.15 0.15

2.18 1.72 1.35 1.62 –0.15 –0.60

2.85 2.08 1.58 0.66 –0.30 –1.22

3.50 2.30 1.70 0.70 –0.50 –1.70

4.00 2.60 1.86 0.70 –0.60 –2.0

195

Flow at the Entrance into Tubes and Conduits Entrance into a tube mounted flush into a wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 20

Diagram 3.3

Values of ζ (rounded up to 10%) at l/a = 0.2– 0.5 (see graph b) w∞ w0

δ, deg 30 45 60 90 120 150

0

0.5

1.0

1.5

2.0

2.5

0.80 0.67 0.58 0.45 0.53 0.80

1.30 1.10 0.92 0.45 0.15 0.13

1.85 1.43 1.25 0.60 –0.10 –0.50

2.20 1.65 1.45 0.67 –0.30 –1.00

2.50 1.83 1.60 0.75 –0.40 –1.35

2.75 2.0 1.75 0.85 –0.50 –1.70

Values of ζ at l/a = 2.0 (see graph c) w∞ w0

δ, deg 30 45 60 90 120 150

0

0.5

1.0

1.5

2.0

2.5

1.00 0.88 0.60 0.45 0.60 1.00

1.68 1.46 1.02 0.55 0.10 0.15

2.22 1.90 1.35 0.75 –0.13 –0.60

2.78 2.30 1.60 0.87 –0.20 –1.30

3.32 2.77 1.75 0.95 –0.23 –2.00

3.80 3.20 1.87 0.95 –0.30 –2.5

196

Handbook of Hydraulic Resistance, 4th Edition

Entrance into a tube mounted flush into a wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 20

Diagram 3.3

Values of ζ at l/a = 5.0 (see graph d) w∞ w0

δ, deg 45 60 90 120 135

0

0.5

1.0

1.5

2.0

2.5

1.20 0.90 0.45 0.80 1.20

2.40 1.72 0.60 0.12 0.12

3.30 2.47 1.18 –0.23 –0.53

4.12 3.08 1.78 –0.10 –1.05

4.85 3.60 1.88 –0.35 –0.88

5.50 4.10 2.10 –0.80 –0.45

Circular bellmouth inlet (collector) without baffle; Re = w0Dh/v ≥ 104 12,13

Dh =

ζ*

Diagram 3.4

4F0 Π0 ∆p

ρw20 ⁄ 2

, see curves a and b as a function of _

For case c: ζ = 0.03 + 0.47 × 10 −7.7r

_

r = r ⁄ Dh

r Dh

197

Flow at the Entrance into Tubes and Conduits Circular bellmouth inlet (collector) without baffle; Re = w0Dh/v ≥ 104 12,13

Diagram 3.4

Values of ζ r Dh

Bellmouth (collector) characteristics a) Free Standing b) Wall mounted

0

0.01

0.02

0.03

0.04

0.05

0.06

0.08

0.12

0.16

≥ 0.20

1.0 0.5

0.87 0.44

0.74 0.37

0.61 0.31

0.51 0.26

0.40 0.22

0.32 0.20

0.20 0.15

0.10 0.09

0.06 0.06

0.03 0.03

Circular bellmouth, wall mounted (collector) with a facing baffle; Re = w0Dh/v ≥ 104 18

ζ*

Dh =

∆p ρw20 ⁄ 2

Diagram 3.5

h r⎞ , see curves ζ = f ⎛⎜ D D ⎟ ⎝ h h⎠

4F0 Π0

Values of ζ h Dh

r Dh 0.10

0.125

0.15

0.20

0.25

0.30

0.40

0.50

0.60

0.80

0.2

–

0.80

0.45

0.19

0.12

0.09

0.07

0.06

0.05

0.05

0.3

–

0.50

0.34

0.17

0.10

0.07

0.06

0.05

0.04

0.04

0.5

0.65

0.36

0.25

0.10

0.07

0.05

0.04

0.04

0.03

0.03

198

Handbook of Hydraulic Resistance, 4th Edition

Circular bellmouth, wall mounted (collector) with a facing baffle; Re = w0Dh/v ≥ 104 18

Diagram 3.5

Converging conical nozzle (collector) without wall mounting; Re = w0Dh/v ≥ 104 12,13

Diagram 3.6

ζ*

Dh =

∆p ρw20 ⁄ 2

, see curves ζ = f(α) for different

l Dh

4F0 Π0

Values of ζ (approximate) α, deg

l Dh

0

10

20

30

40

60

100

140

180

0.025 0.050 0.075 0.10 0.15 0.25 0.60 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.96 0.93 0.87 0.80 0.76 0.68 0.46 0.32

0.93 0.86 0.75 0.67 0.58 0.45 0.27 0.20

0.90 0.80 0.65 0.55 0.43 0.30 0.18 0.14

0.86 0.75 0.58 0.48 0.33 0.22 0.14 0.11

0.80 0.67 0.50 0.41 0.25 0.17 0.13 0.10

0.69 0.58 0.48 0.41 0.27 0.22 0.21 0.18

0.59 0.53 0.49 0.44 0.38 0.34 0.33 0.30

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

199

Flow at the Entrance into Tubes and Conduits Converging conical nozzle (collector) without wall mounting; Re = w0Dh/v ≥ 104 12,13

Diagram 3.6

Converging conical nozzle (collector) wall mounted; Re = w0Dh/v > 104 12,13

Diagram 3.7

ζ*

∆p ρw20 ⁄ 2

, see curves ζ = f(α) for different

l Dh

Values of ζ (approximate) α, deg

l Dh

0

10

20

30

40

60

100

140

180

0.025 0.050 0.075 0.10 0.15 0.60

0.50 0.50 0.50 0.50 0.50 0.60

0.47 0.45 0.42 0.39 0.37 0.27

0.45 0.41 0.35 0.32 0.27 0.18

0.43 0.36 0.30 0.25 0.20 0.13

0.41 0.33 0.26 0.22 0.16 0.11

0.40 0.30 0.23 0.18 0.15 0.12

0.42 0.35 0.30 0.27 0.25 0.23

0.45 0.42 0.40 0.38 0.37 0.36

0.50 0.50 0.50 0.50 0.50 0.50

200

Handbook of Hydraulic Resistance, 4th Edition

Various inlets with facing baffle; Re = w0Dh/v > 104 12,13

Diagram 3.8

Dh =

4F0

ζ*

∆p

Π0

,

ρw20 ⁄ 2

+ ζ′ +

σ1 n2

,

where 1) for ζ′, see curve ζ = f(δ1/Dh) at b/Dh ≥ 0.50 on Diagram 3.1; 2) for ζ′, see curves ζ = f(r/Dh) on Diagram 3.4; 3) for ζ′, see curve ζ = f(α, l/Dh) on Diagram 3.6; for σ1, see curve σ1 = f(h/Dh).

h Dh

0.20

0.30

0.40

0.50

0.60

0.70

0.80

1.0

∞

σ1

1.60

0.65

0.37

0.25

0.15

0.07

0.04

0

0

Entry from a limited volume (F0/F1 > 0) at b/Dh > 0; Re = w0Dh/v > 104 12,13 Dh = Inlet edge

4F0 Π0

ζ*

∆p ρw20 ⁄ 2

Scheme

Diagram 3.9 F0 ⎞ ⎛ = ζ ′ ⎜1 − ⎟ F1 ⎠ ⎝ Coefficient ζ′

Sharp or thick

⎛ δ1 b ⎞ From curves ζ = f ⎜ ⎟ of Diagram 3.1 ⎝ Dg Dh⎠

Rounded (bellmouth)

r From curves ζ = f ⎛⎜ ⎞⎟ of Diagram 3.4 Dh (graphs a and c) ⎝ ⎠

Beveled (conical)

l From curves ζ = f ⎛⎜α, ⎞⎟ of Diagram 3.6 Dh ⎠ ⎝

201

Flow at the Entrance into Tubes and Conduits Inlets with different types of mounting of a straight tube to an end wall; inlet thickness δ1 = 0.03–0.04a0; Re = w0a0/v > 104 12,13

Inlet conditions

Configuration

Diagram 3.10 Resistance coefficient ∆p

ζ*

ρw20 ⁄ 2

Entrance with the end wall on one side of the tube (conduit)

0.58

Entrance with end walls on two opposite sides of the tube (conduit)

0.55

Entrance with end walls on two adjacent sides of the tube (conduit)

0.55

Entrance with end walls on three sides of the tube (conduit)

0.52

Entrance with end walls on four sides of the tube (conduit)

0.50

202

Handbook of Hydraulic Resistance, 4th Edition

Inlets with different mounting of the straight condiut between the wall; inlet edge thickness δ1 = 0.03–0.04a0; Re = w0a0/v > 104 12,13

Inlet conditions

Entrance into a tube (channel) with a visor projection on one side at l/a0 = 0.5

Diagram 3.11 Resistance coefficient ∆p

ζ*

Configuration

l a0 ζ

0

ρw20 ⁄ 2

0.10 0.20 0.30 0.40 0.50

0.60 0.63 0.65 0.67 0.68 0.68

Entrance into a tube (channel) with visor projection on two sides at l/a0 = 0.5

0.82

Entrance into a tube (channel) mounted on top of a wall

0.63

Entrance into a tube (channel) mounted between two walls

0.71

Entrance into a tube (channel) mounted in an L-shaped angle (between two walls)

0.77

Entrance into a tube (channel) clamped between three walls making a U-shape

0.92

203

Flow at the Entrance into Tubes and Conduits Entrance into a straight tube through an orifice or a perforated plate (grid); with sharp-edged orifices (l/dh = 0–0.015); Re = wordh/v > 105 12,13

_ f

0.05

0.10

dh =

4for

ζ*

∆p

Πor 2

ρw0 ⁄ 2

0.15

Diagram 3.12

_ F For Σ for or = = f= , Fgr F0 F0

_

+ (1.707 − f )2 _12 f

0.20

0.25

_ , see curve ζ = f (f)

0.30

0.35

0.40

0.40

ζ _ f

1100

258

98

57

38

24

15

11

7.8

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.90

1.0

ζ

5.8

4.4

3.5

2.6

2.0

1.7

1.3

0.8

0.5

204

Handbook of Hydraulic Resistance, 4th Edition

Entrance into a straight tube through an orifice plate or a perforated plate (grid); with differently shaped orifice edges; Re = wordh/v ≥ 104 12,13 dh =

Characteristics of plate, grid, or orifice edge

Thick orifices

4for Πor

Diagram 3.13

_ F Σfor or = f= F0 F0 Resistance coefficient ∆p ζ* 2 ρw0 ⁄ 2

Configuration

_ _ l 1 ζ + ⎡⎢0.5 + (1 − f) 2 + τ (1 − f) + λ ⎤⎥ × _ 2 , dh ⎦ ρ ⎣ where for τ, see graph a; _ _ _ ϕ(l_) = 0.26 _+ 0.535l8/(0.05 + l 7; or τ = 2.4 – l) × 10−ϕ(l), see Diagrams 2.2 through 2.6; _ _ ζ0 = 0.5 + (1 − f) 2 + τ (1 − f)

_ l * l%dh

0

0.2

0.4

0.6

0.8

τ _ l * l%dh

1.35

1.22

1.10

0.84

0.42

1.0

1.2

1.6

2.0

2.4

τ

0.24

0.16

0.07

0.02

0

_

2 ζ = ⎛1 + √ ⎯⎯ζ ′ − f⎞⎠ _12 , where at α = 40–60o ⎝ f for ζ′, see graph b or ζ′ = 0.13 + 0.34 × _

Orifices with beveled edges

_

−2.3

10−(3.45l + 88.4l) , at other d’s, ζ′ is determined as ζ from Diagram 3.7

_ l * l%dh

0.01

0.02

0.03

0.04

ζ′ _ l * l%dh

0.46

0.42

0.38

0.35

0.06

0.08

0.12

0.16

ζ′

0.29

0.23

0.16

0.13

205

Flow at the Entrance into Tubes and Conduits Entrance into a straight tube through an orifice plate or a perforated plate (grid); with differently shaped orifice edges; Re = wordh/v ≥ 104 12,13

Diagram 3.13

_ 2 1 ζ′ − f⎞ _ 2 , where for ζ′, see ζ = ⎛1 + √ ⎯ ⎯ ⎝ ⎠ f _

graph c or ζ′ = 0.03 + 0.47 × 10−7.7r Orifices with rounded edges

_ r * r%dh 0

0.01 0.02 0.03 0.04 0.05

0.50 0.44 0.37 0.31 0.26 0.22 ζ′ _ l * l%dh 0.06 0.08 0.12 0.16 0.20 ζ′

0.20 0.15 0.09 0.06 0.02

Entrance into a straight tube (conduit) through an orifice plate or a perforated plate (grid) with differently shaped orifice edges; transition and laminar flow regions (Re = worDh/v ≥ 104–105)16

1) 30 < Re < 104 − 105 : 2) 10 < Re < 30 : 3) Re < 10 :

ζ=

ζ=

ζ*

Diagram 3.14

∆p

ρw20 ⁄ 2

1 _ = ζφ _ 2 + ε0 Re ζquad , f

33 _1 _ + ε0 Re ζquad , Re f 2

33 _1 , Re f 2

4.19 (it should be kept in mind that _where ζφ = f1(Re, F0/F1), see Diagram _

f = For/F0 corresponds to F0/F1; ε0Re = f2(Re), see the same graph; ζquad is determined as ζ at Re > 104–105, see Diagrams 3.12 and 3.13.

206

Handbook of Hydraulic Resistance, 4th Edition

Entrance into a tube with a screen at the inlet

Characteristics of plate, grid, or orifice edge

Entrance with sharp inlet edge (δ1/Dh = 0)

Configuration

Diagram 3.15 Resistance coefficient ∆p ζ* 2 ρw0 ⁄ 2

ζ + 1 + ζsc , where ζsc is determined as ζ for a screen from Diagram 3.6

ζ = ζ ′ + ζsc , Entrance with thickened inlet edge (δ1/Dh > 0)

where for ζ′, see curves ζ = f(δ1/Dh, b/Dh) of Diagram 3.1; for ζsc, see above

ζ = ζ′ +

Bellmouth (collector) entry

n2 where for ζ′, see curves ζ = f(r/Dh, b/Dh) of Diagram 3.4; for ζsc, see above

ζ = ζ′ +

Conical nozzle (collector)

ζsc

ζsc

, n2 where for ζ′, see curves ζ = f(α, l/Dh) of Diagrams 3.6 and 3.7, respectively; for ζ′sc, see above

207

Flow at the Entrance into Tubes and Conduits Entrance into a straight circular tube through the first side orifice; Re = worb/v ≥ 104 15 ζ*

∆p ρw20 ⁄ 2

Diagram 3.16

_ , see curves ζ = f (f)

Values of ζ (graph a) _ f Number of orifices One (curve 1) Two (curve 2)

Number of orifices One (curve 1) Two (curve 2)

0.2 64.5 65.5

0.9 2.70 4.54

0.3 30.0 36.5

1.0 2.28 3.84

0.4 14.9 17.0

0.5 9.0 12.0 _ f 1.2 1.60 2.76

0.6 6.27 8.75

0.7 4.54 6.85

0.8 3.54 5.50

1.4

1.6

1.8

2.01

1.40

1.10

208

Handbook of Hydraulic Resistance, 4th Edition

Entrance into a straight circular tube through the first side orifice; Re = worb/v ≥ 104 15

Diagram 3.16

Values of ζ (graph b) _ f

b D0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6

0.13 0.26 0.38 0.48 0.62 0.7

0.13 305 280 260 235 230

335 85.0 79.0 75.0 61.0 63.0

850 42.2 38.3 36.3 32.5 30.2

– 22.5 23.2 22.0 20.0 18.4

– 15.6 16.0 15.2 13.8 12.8

– 11.6 11.7 11.3 10.2 9.40

– – 9.30 8.80 8.00 7.35

– – 6.40 6.85 6.50 5.95

Curve

b D0

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1 2 3 4 5 6

0.13 0.26 0.38 0.48 0.62 0.7

– – 5.40 4.20 4.00 4.85

– – – 3.40 3.30 2.95

– – – 3.80 2.82 2.50

– – – – 2.50 2.22

– – – – 2.30 2.02

– – – – 2.15 1.83

– – – – 2.05 1.70

– – – – – 1.56

Curve

_ f

209

Flow at the Entrance into Tubes and Conduits Intake shafts of rectangular cross section; side orifices with and without fixed louver grating19

Diagram 3.17

_ nbh Fgr = f′= F0 F0

h = 0.5 B

Straight shafts

Layout of orifices No. of orifices Without louvers

_ f′

Resistance coefficient ∆p ζ* ρw20 ⁄ 2

b h

With louvers

Without louvers

α = 30o; b1′ ⁄ h = 0.029;

α = 45o; b1′ ⁄ h = 0.024;

l ⁄ b1′ = 1.6; δ ⁄ b1′ = 0.058 l ⁄ b1′ = 1.4; δ ⁄ b1′ = 0.07

1

0.44

1.5

12.6

17.5

–

2

0.88

1.5

3.60

5.40

–

2

0.88

1.5

4.20

6.30

–

3

1.30

1.5

1.80

3.20

–

4

1.74

1.5

1.20

2.50

3.80

4

1.16

1.0

2.00

3.60

6.00

4

0.58

0.5

8.00

13.7

21.5

210

Handbook of Hydraulic Resistance, 4th Edition

Intake shafts of rectangular cross section; side orifices with and without fixed louver grating19

Diagram 3.17

Shafts with bends

Layout of orifices No. of orifices Without louvers

_ f′

Resistance coefficient ∆p ζ* 2 ρw0 ⁄ 2

b h

With louvers

Without louvers

α = 30o; b1′ ⁄ h = 0.029;

α = 45o; b1′ ⁄ h = 0.024;

l ⁄ b1′ = 1.6; δ ⁄ b1′ = 0.058 l ⁄ b1′ = 1.4; δ ⁄ b1′ = 0.07

1

0.44 1.5

14.0

18.6

–

1

0.44 1.5

16.0

19.0

–

1

0.44 1.5

16.7

20.0

–

2

0.88 1.5

4.50

6.50

–

2

0.88 1.5

5.20

7.00

–

2

0.88 1.5

5.30

7.20

–

3

0.88 1.5

5.30

7.50

–

3

1.30 1.5

2.60

3.90

–

3

1.30 1.5

3.00

4.50

–

3

1.30 1.5

3.40

5.10

–

4

1.74 1.5

2.70

4.00

5.60

4

1.16 1.0

3.10

4.70

6.90

4

0.58 0.5

9.00

14.4

22.0

211

Flow at the Entrance into Tubes and Conduits Straight intake circular shafts; Re = w0D0/v > 104 28

Diagram 3.18

Shaft characteristic

Schematic

Resistance coefficient

1. With a plane top

2. With a cone top

ζ*

∆p ρw20 ⁄ 2

h , see curves ζ = f ⎛⎜ ⎞⎟ D0 ⎝ ⎠

3. With a canopy top and sharp inlet edge

4. With a canopy top and thickened inlet edge

5. With a canopy top and a cone

6. With a convergent entry and a canopy top

Values of ζ h%D0

Scheme 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

∞

– – 2.63 2.13 2.90 1.32

4.40 48.0 1.83 1.30 1.90 0.77

2.15 6.40 1.53 0.95 1.59 0.60

1.78 2.72 1.39 0.84 1.41 0.48

1.58 1.73 1.31 0.75 1.33 0.41

1.35 1.47 1.19 0.70 1.25 0.30

1.23 1.26 1.15 0.65 1.15 0.29

1.13 1.16 1.08 0.63 1.10 0.28

1.10 1.07 1.07 0.60 1.07 0.25

1.06 1.06 1.06 0.60 1.06 0.25

1.06 1.06 1.06 0.60 1.06 0.25

212

Handbook of Hydraulic Resistance, 4th Edition

Entrance into _ a straight channel through a fixed louver grating at f = For/Fgr = 0.1–0.9

Diagram 3.19

l ≥ ⎛ ⎞ ⎡where b′1 ⎜⎝ b′1 ⎟⎠ ⎢ opt ⎣ l

⎛l⎞ ⎜ b′ ⎟ ⎝ 1⎠

_

+ 11 (1 − f) opt

⎤: ⎥ ⎦

2

_F⎞ ⎡ ⎤ 1 ⎛F ⎞ 2 ⎛ p 0 ζ * 2 * k ⎢0.85 + ⎜1 − f + ζfr ⎥ × _ 2 ⎜ ⎟ = kζ ′ ⎟ F0 ⎠ ρw0 ⁄ 2 ⎣ ⎦ f ⎝ Fp ⎠ ⎝ ∆p

Inlet edges of fins cut vertically

1 b′

l < ⎛⎜ ⎞⎟ : ′ ⎝ b1⎠ opt

ζ*

∆p ρw20 ⁄ 2

+ kζ ′ + ∆ζ ,

where k = 1.0 for scheme a: k = 0.6 for scheme b; 1 ∆ζ + 0.5 ⎡⎢11 (1 − f) − ⎤⎥ ; ′ b 1⎦ ⎣ for λ, see Diagrams 2.1 through 2.6. Inlet edges of fins cut horizontally At

l

b′1

l = ⎛⎜ ⎞⎟ , ′ b ⎝ 1 ⎠opt

For For = and Fgr F0

ζfr = λ

l b′1

;

λ = 0.064

_ (at Re = w∞b1′ ⁄ v ≈ 103); for values of ζ′, see curve ζ′ = f(f).

_ f

0.1

ζ′

235 52.5 20.5 10.5 6.00 3.60 2.35 1.56 1.18 0.85

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

213

Flow at the Entrance into Tubes and Conduits Entrance into a straight channel through stamped or cast and shaped perforated plates

Diagram 3.20

∆p Q ; F — clear area of the grid ζ * 2 , For or ρw 0⁄2 _ see curve ζ = f(f) of Diagram 3.12 (approximately) wor =

Inlet pipes of axial stationary turbo-machines10

Diagram 3.21

Converging annular collector (scheme a) formed by two curvilinear surfaces at nar ≥ __ 3.5; ncol ≈ nar; Dsn ≥ 1.3 (optimal parameters) ζ*

∆p + 0.07 . ρw20 ⁄ 2

Annual collector (scheme b) formed __ by one surface at nar ≥ 3.5; H ≥ 0.95; _curvilinear _ Dsn ≥ 1.15–1.25 (optimal parameters) ζ*

nar =

Fin F0

_ d d= D0

ncol =

Dcol _ Fcol hcol =2 F0 h0 D0 (1 + d)

__ Dsn Dsn = D0

__ H H= D0

∆p + 0.08 . ρw20 ⁄ 2

ζex * ρw20 ⁄ 2

∆p

∆p

ρw20 ⁄ 2

⎛ w0 ⎞ = f ⎜ ⎟ see the graph (w∞ is the free stream velocity); w ⎝ ∞⎠

= ζin + ζex ,

∆p

ρw20 ⁄ 2

where ζ *

ζ*

Diagram 3.22

is determined depending on the shape and geometric parameters of the entire air-intake device from the data of the Handbook.

Air-intake devices (intake pipes) (at velocities much below sonic ones)11

214 Handbook of Hydraulic Resistance, 4th Edition

5.0 – – – – – –

2

3

4

5

6

7

0.1

1

Version

Values of ζin

3.5

4.3

5.3

–

–

–

1.5

0.2

2.6

2.8

3.2

5.4

6.0

6.0

1.0

0.25

1.9

1.9

2.3

3.2

5.0

5.0

0.7

0.3

1.2

0.9

1.2

1.5

2.5

2.5

0.4

0.4

w0%w∞ 0.5

0.9

0.5

0.70

0.70

1.5

1.5

0.35

0.6

0.7

0.25

0.40

0.45

0.8

0.8

0.25

0.7

0.5

0.20

0.20

0.25

0.45

0.45

0.15

0.8

0.4

0.10

0.10

0.20

0.25

0.25

0.03

0.15

0.05

0.05

0.05

0.10

0.10

0.3

0.9

Flow at the Entrance into Tubes and Conduits 215

216

Handbook of Hydraulic Resistance, 4th Edition

Inlet elements of axial fans3,4,26

ζ*

∆p ρw20 ⁄ 2

Diagram 3.23

,

zbl is the number of blades of the fan wheel; F0 =

π (D 2 − d 2) ; w0 = Q ⁄ F0 ; 4 _ _ l = l⁄D ; d = d⁄D Values of ζ of the elements (d = 0.3–0.45; zbl = 3–4)

Operational conditions of the fan Inlet element Inlet box (a): _a = 0.75D; b = 2D; c = 0.2D; l = 0 c=a _

c = 0.2D; l = l%D = 0.1 c = 0DD; 0.1 < l ≤ 0.3

Maximum full pressure

Maximum flow rate

pmax

Qmax

0.15

0.07

0.34 0.03

0.2 0.08

0.03

0.06

α = 40o _Confuser, cone (b, c): _l = 0.1 _l = 0.2

o _l = 0.3; α = 60 l = 0.1; α = 80o

0.07

0.09

0

0.02

0.03

0

0.07

0.06

Flow separation

0.35

l = 0.2; α = 80o Step (d) _

D1%D = 1; l = _0 D1%D_= 1.25; l = 0.1

0.07

0.15

0.1 < l ≤ 0.3 Diffuser (e):

0.3

0.10

α = 8–12o; nar = 2

0.12

0.15

Note: The fan is of type K-121.

Flow at the Entrance into Tubes and Conduits

217

Inlet elements of cetrifugal fans3,4,25,33

Diagram 3.24

ζ*

∆p ρw20 ⁄ 2

η′ is the fan efficiency F0′ = BC;

w0 = Q ⁄ F0′ π Fcol = ab ; F0 = D02 ; 4

_ l = l ⁄ D0

Values of ζ of the elements (the blades of the fan are bent backward) Operational conditions∗ Inlet element

Angle of element installation β 0

Q < Qn nominal Q > Qn

Type of fan

Q = Qn f f f η f ≥ 0.9ηmax η f ≥ 0.9ηmax η f ≥ ηmax

Inlet box (a): Fcol%F0 = 1.7 b%a = 2.3; α = 12o Fcol%F0 = 1.2 b%a = 2.3; α = 12o Fcol%F0 = 1−1.5 b%a = 2.3; α = 0o Composed elbow (b): R%D0 = 1.5 _Diffuser (c): _l = 0.8; nar = 1.5 _l = 0.8; nar = 2 _l = 1.4; nar = 1.5 l = 1.4; nar = 2 Simple elbow (d): _Conical confuser (e): _l = 1; nar = 0.67 _l = 1.2; nar = 0.5 l = 1.4; nar = 0.4 ∗

0

0.3

0.3

0.3

Ts4-76

90 180 270 0–270

0.5 0.6 0.5 0.07

0.5 0.6 0.5 0.7

0.5 0.7 0.3 0.7

0–270

0.15

0.15

0.15

Ts4-70

– – – – 0–270

0.5 0.5 0.2 0.2 1.0

0.5 0.8 0.3 0.3 1.0

0.5 0.8 0.3 0.65 1.0

Ts4-76

– – –

0.7 0.8 0.5

0.3 0.4 0.1

0.2 0.3 0.1

Ts4-70 Ts4-76

f The operational conditions of the fan that correspond to the maximum efficiency ηmax are called nominal, with the flow rate Qn. The working region of the fan characteristic is that for which f . η f ≥ 0.9ηmax

218

Handbook of Hydraulic Resistance, 4th Edition

Inlet elements of cetrifugal fans3,4,25,33

Diagram 3.24

Values of ζ of the elements (the blades of the fan are bent forward) Operational conditions∗ Inlet element

Angle of element installation β 0

Q < Qn nominal Q > Qn f η f ≥ 0.9ηmax

Inlet box (a): Fcol%F0 = 1.3

Type of fan

Q = Qn f η f ≥ ηmax

f η f ≥ 0.9ηmax

0

0.3

0.3

0.35

180

0.45

0.45

0.5

270

0.2

0.2

0.3

b%a = 2.3; α = 12o

0

0.5

0.5

0.5

Fcol%F0 = 1.2−1.8

0–270

0.85

0.85

0.85

0 90 180 270

0.3 0.4 0.5 0.3

0.3 0.4 0.5 0.3

0.4 0.4 0.4 0.35

–

0

0.2

0.2

b%a = 2.4; α = 12o

Ts9-55

Fcol%F0 = 1.1

b%a = 2.3; α = 0o Composed elbow (b): R ≥ 1.5D0

_Diffuser (c): l = 0.5; nar = 1.5 _ l = 0.5; nar = 2.0 _ l = 0.8; nar = 1.5 _ l = 0.8; nar = 2.0 _ l = 0.4; nar = 1.5 _ l = 0.4; nar = 2.0 Simple elbow (d): _Conical confuser (e):

–

0.5

0.8

0.7

–

0.1

0.15

0.1

–

0.3

0.3

0.2

–

0.2

0.2

0.15

–

0.4

0.5

0.4

0–270

2.0

2.0

2.0

l = 1.5 nar = 0.4–0.7 Step (eddy collector) (f):

–

0

0

0

–

0.8

0.4

0.3

nar ≥ 0.7

–

0

0

0

∗

Ts9-55

Ts14-46

Ts4-70

Ts14-16 f ηmax

The operational conditions of the fan that correspond to the maximum efficiency are called nominal, with the flow rate Qn. The working region of the fan characteristic is that for which f η f ≥ 0.9ηmax .

Flow at the Entrance into Tubes and Conduits

219

REFERENCES l. 2. 3. 4. 5. 6. 7. 8.

9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Averiyanov, A. G. et al., Ventilation of Shops of Shipbuilding Works, Sudostroenie Press, Moscow, 1969, 268 p. Altshul, A. D., Hydraulic Resistance, Nedra Press, Moscow, 1970, 216 p. Bychkova, L. A., Entrance elements of installations with axial fans, Vodosnabzh. Sanit. Tekh., no. 5, 29–31, 1977. Bychkova, L. A., Recommendations for Calculating the Hydraulic Resistances of the Complicated Elements of Ventilation Systems, Moscow, 1981, 32 p. Guinevskiy, A. S. (Ed.), Introduction to the Aerohydrodynamics of the Container Pipeline Transport, Nauka Press, Moscow, 1986, 232 p. Zakharov, A. M., Bulygin, P. A., Raikin, L. I., et al., Air-Intakes and Gas-Dischargers of FastGas-Turbine Ships, Leningrad, 1977, 207 p. Gretsov, N. A., Hydraylic resistances and a rational shape of rectiaxial converging tubes with a baffle before the inlet, Tr. Mosk. Selkhoz. Akad., vyp. 87, 37–42, 1963. Davydov, A. P., Investigation of the operation of the suction orifice with an inner screen, in Investigations in the Fields of Heating, Ventilation and Air Conditioning (Tr. LISI), no. 110, pp. 27–34, 1975. Dzyadzio, A. M., Pneumatic Transport at Grain-Processing Factories, Zagotizdat Press, Moscow, 1961, 250 p. Dovzhik, S. A. and Kartavenko, V. M., Experimental investigation of inlet nozzles of axial stationary turbomachines, in Prom. Aerodin., vyp. 29, pp. 56–73, Mashinostroenie Press, Moscow, 1973. Idelchik, I. E., The aerodynamics of the intake branch pipes of aircraft engines, Tekh. Vozdush. Flota, nos. 5–6, 1–10, 28, 1944. Idelchik, I., E., Hydraulic resistance during flow entrance in channels and passage through orifices, Prom. Aerodin., no. 2, pp. 27–57, BNT, NKAP, 1944. Idelchik, I. E., Hydraulic Resistances (Physical and Mechanical Fundamentals), Gosenergoizdat Press, Moscow, 1954, 316 p. Idelchik, I. E., Determination of the resistance coefficients in discharge through orifices, Gidrotekh. Stroit., no. 5, 31–36, 1953. Idelchik, I. E., Handbook of Hydraulic Resistances (Local Resistance Coefficients and Friction Resistances), Gosenergoizdat Press, Moscow, 1960, 464 p. Idelchik, I. E., Account for the effect of viscosity on the hydraulic resistance of diaphragms and grids, Teploenergetika, no. 9, 75–80, 1960. Levin, B. M., Local inlet losses during ground suction under water, Tr. Mosk. Inst. Inzh. Zheleznod. Transp., vyp. 122, 281–312, 1959. Nosova, M. M., Resistance of inlet and exit bellmouths with baffles, in Prom. Aerodin., no. 7, pp. 95–100, Oborongiz Press, Moscow, 1956. Nosova, M. M. and Tarasov, N. F., Resistance of intake ventilating shafts, in Prom. Aerodin., no. 12, pp. 197–215, Oborongiz Press, Moscow, 1959. Nosova, M. M. and Barnakova, T. S., Resistance of inlet and exit orifices in the presence of the passing stream, in Prom. Aerodin., no. 15, pp. 20–37, Oborongiz Press, Moscow, 1959. Oslyansky, Ya. L., Pressure head losses in the intake pipe of a dredger during suction of a waterground mixture, Tr. Leningr. Inst. Vodn. Transp., vyp. 119, 135–142, 1968. Staroverov, I. G. (Ed.), Handbook for a Designer of Industrial, Living and Communal Buildings and Structures, Stroiizdat Press, Moscow, 1969, 536 p. Stemenko, V. A., Study of the resistance coefficients of inlet boxes of fans of the kinematics of air flow in them, Tr. Inst. Gorn. Mekh. Tekh. Kibern., no. 17, 32–43, 1967.

220

Handbook of Hydraulic Resistance, 4th Edition

24. Stepanov, P. M., Ovcharenko, I. Kh., and Skobeltsyn, Yu. A., Handbook of Hydraulics for Land Reclaimers, Kolos Press, Moscow, 1984, 207 p. 25. Steshenko, V. A. and Pak, V. V., Shaping of the inlet boxes of centrifugal double-suction guns, Vopr. Gorn. Mekh., no. 17, 43–47, 1967. 26. Surnov, N. V., Inlet devices of axial fans, in Prom. Aerodin., no. 9, pp. 28–34, Oborongiz Press, Moscow, 1957. 27. Temnov, V. K., Coefficient of the hydraulic resistance of a smooth entrance during turbulent fluid flow, Izv. VUZ, Energetika, no. 4, 89–93, 1963. 28. Khanzhonkov, V. I., Resistance of inflow and outflow shafts, in Prom. Aerodin., no. 3, pp. 210– 214, Oborongiz Press, Moscow, 1947. 29. Khanzhonkov, V. I., Aerodynamic characteristics of collectors, in Prom. Aerodin., no. 4, pp. 45– 62, Oborongiz Press, Moscow, 1953. 30. Khanzhonkov, V. I., Reduction of the aerodynamic resistance of orifices by means of annular fins and recesses, in Prom. Aerodin., no. 4, pp. 45–62, Oborongiz Press, Moscow, 1953. 31. Khanzhonkov, V. I. and Davydenko, N. I., Resistance of side orifices of the terminal section of a pipeline, in Prom. Aerodin., no. 15, pp. 38–46, Oborongiz Press, Moscow, 1959. 32. Shepelev, I. A. and Tyaglo, I. G., Suction patterns in the vicinity of outflow orifices (based on reported data), in Local Exhaust Ventilation, pp. 81–90, 1969. 33. Bruk, A. D., Matikashvili, T. I., Nevelson, M. I., et al., Centrifugal Fans, Moscow, 1975, 415 p. 34. Ashino, I., On the theory of the additional loss at the pipe entrance in viscous fluid. 1st rept. On the influence of rounded entrance, Bull. JSME, vol. 14, no. 45, 463–468, 1969. 35. Ashino, J., On the theory of the additional loss at the pipe entrance in viscous fluid. 2. When an entrance is tapered type, Bull. JSME, vol. 12, no. 51, 522–529, 1969. 36. Basavarajaiah, B. S., Exit loss in a sharp edged pipe, J. Inst. Eng. (India) Civ. Eng. Dep., vol. 43, no. 11, part 6, 549–563, 1963. 37. Bevier, C. W., Resistance of wooden louvers to fluid flow, in Heating, Piping and Air Conditioning, pp. 35–43, 1955. 38. Bossel, H. H., Computation of axisymmetric contractions, AIAA J., vol. 7, no. 10, 2017–2020, 1969. 39. Campbell, W. D. and Slattery, I. C., Flow in the entrance of a tube, Trans. ASME, vol. D85, no. 1, 41–45, Discuss., pp. 45–46, 1963. 40. Cobb, P. R., Pressure loss of air flowing through 45-degree wooden louvers, in Heating, Piping and Air Conditioning, pp. 35–43, 1953. ∨ 41. Kubicek, L., Ssaci nástavce, Strojirehstvi, no. 4, 427–433, 1954. 42. Gibbings, J. C., The throat profile for contracting ducts containing incompressible irrotational flows, Int. J. Mech. Sci., vol. 11, no. 3, 29–301, 1969. 43. Hebans, G. G., Crest losses for two-way drop inlet, J. Hydraul. Div., Proc. Am. Soc. Civ. Eng., vol. 95, no. 3, 919–940, 1969. 44. Lundgren, T. S., Sparrow, E. N., and Starr, J., Pressure drop due to the entrance region in ducts of arbitrary cross section, Trans. ASME, vol. D86, no. 3, 620–626, 1964. 45. Oosthuizen, P. H., On the loss coefficient for a sharp-edged pipe entrance, Bull. Mech. Eng. Educ., vol. 7, no. 2, 157–159, 1968. 46. Rimberg, D., Pressure drop across sharp-end capillary tubes, Ind. Eng. Chem. Fundam., vol. 6, no. 4, 599–603, 1967. 47. Unger, J., Stromung in zylindrischen Komalen mit Versperrungen bei hohen Reynolds-zahlen, Forsch. Ingenieurwes., Bd. 45, No. 3, 69–100, 1979. 48. Webb, A., Head loss of a sudden expansion, Int. J. Mech. Eng., vol. 8, no. 4, 173–176, 1980. 49. Weisbach, G., Lehrbuch der Ingenieur und Maschinentechnik, 11 Aufl., 1850, 320 p. 50. Ushakov, K. A. and Bushel, A. R., Investigation of the operation of an axial fan sucking from a passing stream, Tr. TsAGI, vyp. 976, 216–242, 1965.

Flow at the Entrance into Tubes and Conduits

221

51. Stockman, N. J. Potential and viscous flow in VTOL, STOL or CTOL propulsion system inlets. AIAA J., no. 1186, 11, 1975. 52. Burley, R. R. and Hwang, D. P., Experimental and analytical results of tangential blowing applied to subsonic V/STOL inlet, AIAA Pap. no. 1984, 1982, 11 p. 53. Miller, B. A., A novel concept for subsonic inlet boundary layer contol, J. Aircraft, vol. 14, no. 4, 403–404, 1977. 54. Volostnykh, V. N. and Frankfurt, M. O., Investigation of the efficiency of an jet air-intake device, in Prom. Aerodin., vyp. 3(35), pp. 55–64, Mashinostroenie Press, Moscow, 1988.

CHAPTER

FOUR RESISTANCE TO FLOW THROUGH ORIFICES WITH SUDDEN CHANGE IN VELOCITY AND FLOW AREA RESISTANCE COEFFICIENTS OF SECTIONS WITH SUDDEN EXPANSION, SUDDEN CONTRACTION, ORIFICES, DIAPHRAGMS, AND APERTURES

4.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS 1. An abrupt enlargement of a tube (channel) cross-sectional area gives rise to so-called shock losses. In the case of uniform velocity distribution over the cross section of the smaller upstream channel in turbulent flow (Re = w0Dh/v > 104), the local resistance coefficient of the "shock" depends only on the cross-sectional area ratio F0/F2 (measure of expansion n = F2/F0) and is calculated from the Borda–Carnot formula as ζloc *

2

∆p

F0 ⎞ ⎛ = ⎜1 – ⎟ . 2 F2 ρw0 ⁄ 2 ⎝ ⎠

(4.1)

The total resistance coefficient of the section with an abrupt expansion* is ζ*

ζfr′ ∆p = ζloc + ζfr = ζloc + 2 , 2 ρw0 ⁄ 2 nar

(4.2)

where ζ′fr *

∆pfr ρw22 ⁄ 2

=λ

l2 . D2h

∗

The additional coefficient ζfr is incorporated if it was disregarded when friction losses throughout the piping system were determined.

223

224

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.1. Schematic diagram of flow at an abrupt expansion.

2. In an abruptly expanded section a jet is formed which is separated from the remaining medium by a bounding surface that disintegrates into strong vortices (Figure 4.1). The length l2 of the section over which the vortices develop and gradually disappear while the flow completely speads over the cross section ranges from 8 to 12D2h (D2h is the hydraulic diameter of the larger section). The shock losses at an abrupt expansion are associated with this formation of vortices over the length l2. 3. When an abrupt expansion of the tube cross section occurs only in one plane (Figure 4.2), the shock losses decrease with an increase in the aspect ratio B/H (B is the width of the larger cross section; H is the constant height of the channel); in this case, the local resistance coefficient is 2

F0 ⎞ ⎛ ζloc = k1 ⎜1 – ⎟ , F2 ⎝ ⎠ where k1 ≤ 1 is the correction factor which depends on the aspect ratio B/H.

Figure 4.2. Dependence of k1 on B/H.

225

Flow through Orifices with Change in Velocity and Flow Area

4. For practical conditions, the velocity distribution over the conduit length upstream of an abrupt expansion is, as a rule, never uniform. This substantially contributes to the losses as compared with those predicted by Equation (4.1). In order to calculate the local resistance coefficient of a shock for a flow with nonuniform velocity distribution at large Re, it is necessary to use a generalized formula that allows for this nonuniformity, provided the velocity distribution over the channel cross section13,15 is ζloc =

∆p ρw20 ⁄ 2

1 M . = 2 +N–2 nar nar

(4.3)

The total resistance coefficient is calculated from a formula similar to Equation (4.2). In Equation (4.3), M = (1/F0)

∫

(w/w0)2dF is the flow momentum coefficient (the Boussi-

F0

nesq coefficient) at the exit from the smaller channel into the larger one; N = (1/F0)

∫

(w/w0)3 dF is the coefficient of the kinetic energy of the flow (the Coriolis coefficient) in

F0

the same section. An approximation can be made that N ≈ 3M – 2. The approximation is more correct the nearer M and N are to unity. The last expression leads to the following approximate formula for determining the local resistance coefficient ζloc =

∆p ρw20 ⁄ 2

+ N ⎛⎜1 – 3n2 ⎝

⎞+ 1 – 4 . ⎟ n2ar 3nar

ar ⎠

5. If the velocity distribution over the cross section is known, the coefficients M and N can be easily calculated. However, if this distribution is unknown, it must be determined experimentally. Then the coefficients M and N can be determined by graphic integration from the curves obtained for the velocity distribution. 6. In diffusers with divergence angles up to α = 8–10o and over long straight sections of constant cross section with a developed turbulent velocity profile (see Section 1.3) the distribution of velocities over the cross section is close to the power function law 1⁄m

y , = ⎛1 – ⎞⎟ R0 ⎠ wmax ⎜⎝ where w and wmax are the velocity at the given point and the maximum velocity over the cross section, respectively, m/s; R0 is the section radius, m; y is the distance from the tube axis to the given point, m; and m is an exponent which can vary from 1 to ∞. 7. At m = 1, the velocity profile acquires the shape of a triangle (Figure 4.3). At m = 8, it takes on the shape of a rectangle, that is, the velocity distribution over the section is completely uniform. The velocity profile is already almost rectangular at m = 8–10. This value of m can be used for long straight sections with turbulent flow. The values m = 2–6 can be used for long diffusers (n1 = F1/F0 > 2): w

226

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.3. Velocity distribution in plane diffusers with divergence angles up to 8o and comparison with the power law.

at α = 2o m + 6 , at α = 6o m + 3 , at α = 4o m + 4 , at α = 8o m + 2 . 8. With the power -law distr ibution, the values of M and N in Equation (4.3) can be calculated from the author’s formulas:12,13 for circular and square tubes M=

N=

(2m + 1)2(m + 1) 4m2(m + 2)

,

(2m + 1)3(m + 1)3 4m4(2m + 3)(m + 3)

,

for a rectangular tube or diffuser (with the aspect ratio of the rectangular cross section a0/b0 = 0.3–3.0) M=

N=

(m + 1)2 , m(m + 2) (m + 1)3 m2(m + 3)

.

9. Over long straight sections of tubes and channels (usually at a distance over 10Dh from the inlet) for laminar flow, a parabolic velocity profile is developed

Flow through Orifices with Change in Velocity and Flow Area

227

Figure 4.4. Velocity distribution resembling a sinusoidal function (downstream of perforated plates and guide vanes).15

2

y = 1 – ⎛⎜ ⎞⎟ . R0 wmax ⎝ ⎠ w

In this case, for a circular or square tube M = 1.33 and N = 2, and for a plane (rectangular) tube M = 1.2 and N = 1.55. 10. In tubes and channels directly downstream of perforated plates, in elbows behind guide vanes, and in other similar cases, the velocity profile resembles a trigonometric function (Figure 4.4), which for a plane channel is calculated from the author’s formula:13,15

∆w 2y w sin 2k1π , =1+ b0 w0 wmax where b0 is the width of the plane channel, m; ∆w is the departure of velocity at the given point of the narrow channel cross section from the section-average velocity w0, m/s; and k is an integer. In this case, 2

2

1 ⎛ ∆w ⎞ 3 ⎛ ∆w ⎞ M=1+ ⎜ ⎟ , N=1+ ⎜ ⎟ . 2 w0 2 ⎝ w0 ⎠ ⎝ ⎠ 11. A nonsymmetrical velocity field (Figure 4.5) is established downstream of diffusers with divergence angles at which flow separation occurs (α ≥ 14o), elbows, branches, and so on. In particular, in plane diffusers with divergence angles α = 15–20o and in straight elbows (δ = 90o), the velocity distribution is governed by:13,15 2y w = 0.585 + 1.64 sin ⎛⎜0.2 + 1.95 ⎞⎟ . w0 b0 ⎠ ⎝

228

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.5. Asymmetric velocity distribution downstream of an elbow and in a diffuser with the divergence angle at which flow separation takes place.15

In this case, M = 1.87 and N = 3.7. 12. When a nonuniform velocity field develops in a tube (channel) of constant cross section (n = 1), subsequent equalization of the flow is also accompanied by irreversible pressure losses (losses for flow deformation), which are calculated by a formula obtainable from Equations (4.2) and (4.3): ζ*

∆p ρw20 ⁄ 2

= 1 + N – 2M + ζfr ,

(4.4)

or accordingly from ζ*

∆p ρw20 ⁄ 2

+ 13 (N – 1) + ζfr ,

where M and N are determined in accordance with the nonuniformity pattern obtained. These losses are taken into account only in the case where they were disregarded during determination of the local resistance of fittings and obstructions which resulted in a nonuniform velocity distribution over the straight section. 13. The coefficients M and N for the inlet section of the ejector mixing chamber, when the "main"* portion of the free jet enters it (Figure 4.6), are calculated from the author’s formulas:13,15

∗

The "main" portion of a free jet is defined in Chapter 11.

Figure 4.6. Velocity distribution over the main section of the free jet and after its entrance into the mixing chamber of the ejector: dashed lines, 2 12 a theoretical curve for the free jet; solid lines, experimental curve for the jet in the channel. g = (p – pa)/(ρw1 /2).

Flow through Orifices with Change in Velocity and Flow Area 229

230

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.7. Sections with abrupt expansions: (a) with baffles; (b) with "pockets".

2 1 ⎛ F2 ⎞ _ 1 F2 , N = _3 ⎜ ⎟ e , M = _2 q ⎝ F0 ⎠ q F0

where F2/F0 is the area _ ratio of the given free jet (mixing chamber) section to the inlet jet (inlet nozzle) section; q = Q/Q0, is the dimensionless flow rate through the given section, that is, the ratio between the flow rate of the medium passing through the _tube (mixing chamber) and the initital flow rate of the jet (at the exit from the inlet nozzle); e = E/E0 is the dimensionless kinetic energy of the jet at the given cross section, that is, the ratio between the energy of the jet at the entrance to _ the tube _ (mixing chamber) and the initial energy of the jet. The values of F2/F0, Fj/F0, q, and e depend on the relative length of the free jet s/Dh and are determined from the corresponding curves of Diagrams 11.24 and 11.25. 14. The resistance of the section with an abrupt expansion can be substantially reduced by installing baffles (Figure 4.7a). When the baffles are correctly installed,* the losses are reduced by 35–40%, so that the local resistance coefficient of such a section can be approximated from ζloc *

∆ploc

ρw20 ⁄ 2

+ 0.6 ζ′loc ,

where ζ′loc is the resistance coefficient of the section with an abrupt expansion without baffles, which is determined as ζ from the data given in Diagram 4.1. 15. A substantial decrease in the resistance of the section with an abrupt expansion is also attainable by arranging "pockets" immediately downstream of the narrow cross section (Figure 4.7b) which favor the formation of a steady recirculation ring (for circular tubes) or two steady vortices (for a plane channel) which act like pumps.44 16. Pressure losses in sections with sudden expansion can be considerably reduced by shattering vortices in this section with the aid of transverse partitions (Figure 4.8)21,46. The upper edges of these partitions should be located at the level of the upper boundary of the recirculation zone and must not extend into the active flow. As an example Figure 4.8 shows the dependence of the loss coefficients in a straight channel on the dimensions of single vortices downstream of a one-sided abrupt expansion. In these experiments the height of the partitions remained constant and equal to the step height (a = 35 mm), but their number n amounted to 1, 2, 3, 4, 5, and 10 at a constant length of *

The general rules to be followed when installing the baffles are given in Chapter 5 (paragraph 65).

Flow through Orifices with Change in Velocity and Flow Area

231

Figure 4.8. Dependence of hydraulic losses in a straight channel with a one-sided abrupt expansion in the sizes of single vortices.

the finned section (400 mm) and, consequently, the distance between the partitions b and ratio b/a was changed. Here, the point b/a = 0 (ζ = 0.03) corresponds to a smooth channel without an abrupt expansion. As follows from Figure 4.8, a decrease in the size of vortices (in the parameter b/a) leads to smaller losses in the system. 17. A substantial increase in the degree of static pressure recovery and decrease in losses in channels with abrupt expansions can be attained by employing jet diffusers.47 A jet diffuser is formed when a jet is blown from a slot in the exit section of the narrow portion of the channel. The jet is deflected from the channel axis to the outer wall of the broad portion of the channel by the angle α. The deflection can be attained either by orienting the nozzle axis (Figure 4.9a) or rounding its outer edge at the nozzle exit (Figure 4.9b) αk/2 = 30–60o (using the Coanda effect). The investigations carried out by Frankfurt47 showed that in circular channels with the expansion ratio nar = 4.2 and 11 the use of a nozzle with α/2 = 15–30o between the channel axis and jet direction allows one to increase the degree of pressure recovery almost up to the limiting value for an ideal diffuser. Here, the loss coefficient with account for the energy lost on injection decreases by a factor of 1.5. A still greater positive effect was obtained when a

Figure 4.9. Schematic diagram of a step diffuser (with an abrupt expansion of the cross-sectional area).

232

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.10. Schematic diagram of step diffuser (with an abrupt expansion of the cross-sectional area).

jet diffuser was arranged with the aid of a device operating on the Coanda effect. In an optimum case (at αcol/2 = 60o and fs/F0 + 0.05, fs is the slit area, F0 is the cross sectional area of the narrow portion of the channel) the loss coefficient was decreased by a factor of 2–2.5. This was also the case when the end wall between the narrow and broad portions of the channel with an abrupt expansion was permeable. The characteristics of jet diffusers are considered in more detail in paragraph 65 of Chapter 5. 18. When the gas flow velocity in the section b-b (Figure 4.10) is close to the speed of sound and remains subsonic over the jet section between cross sections c-c and n-n, then7 the shock losses can be determined with a sufficient accuracy from the above-given formulas for an incompressible fluid (in the case of the relative (reduced) velocity λex = w/acr ≤ 0.75 the error is practically equal to zero; at λex = 1, the error is 8%). 19. In the general case, a stepwise channel with a flow can have a supersonic nozzle at the inlet and then the geometric shape of the channel will be characterized by the dimensions of three sections: by the area of the critical section Fcr, by the area of nozzle cross section at the exit Fex, and by the area of the cross section of a cylindrical channel Ftot. In the specific case, Fcr = Fex and the supersonic nozzle is absent. 20. If in any cross section of the jet over the section c-n the jet velocity is higher than the speed of sound, then in this case compression shock losses should be taken into account. Thus, the total pressure losses are composed of the straight compression shock losses and shock losses (according to Borda–Carnot) originating when a subsonic jet expands from section 2-2 to section n-n.7 21. The relative losses of total pressure in a stepwise channel can be determined as ∆ptot

=1–σ , p∗ where σ is the ratio of total pressures in cross sections n-n and 0-0: σ=

p∗tot p∗0

+

ptot p∗0

+

2 ρtotwto t

2p20

,

or after corresponding transformations

2 ⎞ k σ= + ⎛⎜ ⎟ ∗ 2 p0 ⎝k + 1⎠ ptot

(k+1) ⁄ k

2

1 ⁄ (k–1)

1 ⎛ k – 1 2⎞ ⎛ Fcr ⎞ ⎜ Ftot ⎟ × 2 ⎜1 – k + 1 λ1⎟ ⎝ ⎠ λ1 ⎝ ⎠

,

(4.5)

Flow through Orifices with Change in Velocity and Flow Area

233

Figure 4.11. Dependence of the pressure ratio ptot/p∗0 on λ1 (a) and of the pressure recovery coefficient σ on p∗0/ptot (b).7

where λ1 = w1/acr is the reduced velocity in section 1-1; it is determined from the relation 1 ⁄ (k–1)

k – 1⎞ ⎛ k – 1 2⎞ = ⎛⎜λ21 – ⎟ ⎜1 – k + 1 λ1⎟ ∗ 1 k + p0 ⎝ ⎠⎝ ⎠

ptot

⎡ m ⎛k + 1⎞ × ⎢1 – ⎜ ⎟ λ 1⎝ 2 ⎠ ⎣

1 ⁄ (k–1)

ptot p∗0

m2

+ 0.2344

2.5

λ21

km ⎛ k + 1 ⎞ ⎜ ⎟ λ1 ⎝ 2 ⎠

1 ⁄ (k–1)

⎛1 – k – 1 λ2⎞ 1 ⎜ k + 1 ⎟⎠ ⎝

where m = Fcr/Ftot. For air (k = 1.41) σ=

k ⁄ (k–1)

+

⎛1 – 1 λ2⎞ ⎜ 6 1⎟ ⎠ ⎝

,

⎤ ⎥, ⎦

(4.6)

234

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.12. Schematic diagram of flow with an abrupt contraction of the cross-sectional area.

and Equation (4.6) acquires the form 2.5 ⎛ λ21 ⎞ m m ⎡ 2 1⎞ ⎤. ⎛ × ⎢1 – = ⎜λ1 – ⎟ ⎜1 – ⎟ + 0.7396 ⎥ 2.5 ∗ 6 6 λ ⎠ 1 ⎢ p0 ⎝ ⎠⎝ ⎥ 1 ⎢ 1.5774 ⎛⎜1 – λ21⎞⎟ λ1 ⎥ ⎣ ⎦ 6 ⎠ ⎝

ptot

22. The dependence of ptot/p∗0 on λ1 and m at k = 1.41 is presented in Figure 4.11a and the relation σ = (p∗0/ptot, m) in Figure 4.11b. At small values of λ1 for the given values of ptot/p∗0 and m, two values of λ1 are obtained. However, as σ weakly depends on λ1 at small values of λ1, the choice of λ1 virtually does not influence the value of σ.7 The above-given formulas apply for the values 1 ≤ λ1 ≤ λlim, where λlim corresponds to the full expansion of the supersonic jet up to F1 = Ftot. 23. When the cross section abruptly contracts, the phenomenon is basically similar to that observed when shock losses occur during an abrupt expansion. But now such losses occur mainly when the jet, compressed during the entry from a broad channel into a narrow one (section c-c, Figure 4.12), expands until it fills the entire section of the narrow channel (section 0-0). 24. The coefficient of local resistance to an abrupt contraction at large Reynolds numbers (Re > 104) can be approximately determined from the author’s formula12,13

ζloc =

∆ploc (ρw20 ⁄ 2)

F0 ⎞ ⎛ = 0.5 ⎜1 – ⎟ , F 1⎠ ⎝

or, more exactly, from the formula which the author derived by processing the experimental results obtained by other research workers: ∆ploc

3⁄4

F0 ⎞ ⎛ = 0.5 ⎜1 – ⎟ ζloc = 2 F 1⎠ (ρw0 ⁄ 2) ⎝

.

In this case, the total resistance coefficient is

ζ*

∆p = ζloc + ζ′fr , ρw20 ⁄ 2

Flow through Orifices with Change in Velocity and Flow Area

235

where

ζ′fr *

∆pfr ρw20 ⁄ 2

=λ

l0 D0h

(l0 is the length of the straight section downstream of the contraction). 25. The resistance of the contracting section can be substantially reduced by arranging a smooth transition from a wide section to the narrow one with the aid of a nozzle (collector) with curvilinear or rectilinear boundaries (see Diagram 4.9). The author recommends determining the local resistance coefficient of such a contracting section at Re > 104 from the formula*:

ζloc =

∆ploc

3⁄4

F0 ⎞ ⎛ = ζ′ ⎜1 – ⎟ 2 F1 (ρw0 ⁄ 2) ⎝ ⎠

,

where ζ′ is the coefficient which depends on the shape of the inlet edge of the narrow channel mounted flush with the wall (see Diagrams 3.1, 3.4, and 3.7). 26. In the general case of the flow passing from one volume into another through an opening in the wall, the following phenomena are observed and are illustrated in Figure 4.13. The flow passes from channel 1, located before the partition A with an opening of diameter D0 into channel 2, located behind the partition. The cross sections of both channels may be larger than, or equal to, the cross section of the opening. Flow passage through the opening is accompanied by distortion of the trajectories of particles with the result that they continue their motion by inertia toward the axis of the opening. This reduces the initial area of the jet cross section F1 until the area Fcon (section c-c) is smaller than the area of the cross section of the opening F0. Starting with section c-c, the trajectories of the moving particles are straightened and thereafter an abrupt jet expansion takes place. 27. In the general case the resistance coefficient of the flow passage through an opening with sharp edges in the wall (l/Dh = 0, Figure 4.13a) is calculated for the self-similar (quadratic) flow region (Re = w0D0/v ≥ 105) by the author’s refined formula:

ζloc =

⎡ = ⎢1 + 0.707 2 ρw0 ⁄ 2 ⎣ ∆p

√ ⎯⎯⎯⎯⎯ ⎛ ⎜1 – ⎝

F0 ⎞ 3 ⁄ 4 F0 ⎤ – ⎥ F1 ⎟ F2 ⎠ ⎦

2

2

0.375 ⎡ ⎛ F0 ⎞ ⎛ F0 ⎞⎤ = ⎢0.707 ⎜1 – ⎟ + ⎜1 – ⎟⎥ . F1 ⎠ ⎝ ⎝ F2 ⎠⎦ ⎣

(4.7)

28. Thickening (Figure 4.13b), beveling (Figure 4.13c), or rounding (Figure 4.13d) of the orifice edges reduces the effect of the jet contraction in the opening (increases the jet contrac′ ≥F tion coefficient ε), that is, decreases the jet velocity in its smallest section (Fcon con and *

Equation (7.25) recommended in Reference 3 gives close agreement with experiment at large values of F0/F1 and considerable discrepancy (up to 20%) at low values of F0/F1.

236

Handbook of Hydraulic Resistance, 4th Edition

Figure 4.13. Flow passage through an orifice in the wall from one volume into another: a) sharp-edged orifice (l/Dh ≈ 0); b) orifice with thick edges (l/Dh > 0); c) orifice with edges beveled in the flow ditection; d) orifice with edges rounded in the flow direction.

w′j < wj). And since it is this velocity which determines the shock losses at discharge from the orifice, the total resistance of the passage through it is decreased. 29. The resistance coefficient of the flow passing through orifices in the wall, with edges of any shape and of any thickness, is calculated at great Reynolds numbers (virtually for Re ≥ 105) (in the general case considered under paragraph 25) from the author’s generalized and refined formula:

ζ*

∆p

F0 ⎞ ⎛ = ζ′ ⎜1 – ⎟ 2 F1 ⎠ ρw0 ⁄ 2 ⎝ 0.75

F0 ⎞ ⎛ = ⎜1 – ⎟ F1⎠ ⎝

3⁄4

2

⎛ F0⎞ + ⎜1 – ⎟ + τ ⎝ F2⎠ 0.375

⎛ F0 ⎞ + τ ⎜1 – ⎟ ⎝ F1⎠

⎯⎯⎯⎯⎯ √

F0 ⎞ 3 ⁄ 4 ⎛ – 1 ⎜ F1 ⎟ ⎝ ⎠ 2

F0 ⎞ ⎛ ⎜1 – F ⎟ + ζfr 2⎠ ⎝

F0 ⎞ ⎛ F0 ⎞ ⎛ ⎜1 – F ⎟ + ⎜1 – F ⎟ + ζfr , 2⎠ ⎝ 2⎠ ⎝

(4.8)

where ζ′ is a coefficient which depends on the shape of the orifice inlet edge and is determined as ζ from Diagrams 3.1 through 3.4 and 3.7; τ is the coefficient representing the effect of the wall thickness, the inlet edge shape of the opening, and conditions of flow passage through the opening; in the case of thick edges, it is determined from the formulas similar to

237

Flow through Orifices with Change in Velocity and Flow Area

Equations (3.4) and (3.5) or from the curve τ = f(l/Dh) in Diagram 4.12, while for beveled or rounded edges, it is approximated by τ ≈ = 2√ ⎯⎯ζ′ , where ζ′ is determined from formulas similar to Equations (3.7) and (3.8) or from Diagram 4.13; ζfr = λ(l/Dh) is the friction coefficient over the entire depth of the orifice opening; and λ is the hydraulic friction factor of the opening depth determined from diagrams in Chapter 2. In the case of beveled or rounded edges, ζfr is assumed to be zero. 30. The general case of flow passage through an opening in the wall can be divided into a number of particular cases: • F1 = F0, a sudden expansion of the cross section (see Figure 4.1); for this Equation (4.8) reduces to Equation (4.1). • F2 = F0, a sudden contraction of the cross section (see Figure 4.12); Equation (4.8) is then reduced to the form of Equation (3.1) at m = 3/4. • F1 = ∞, entrance with a sudden expansion (entrance through an orifice plate or a perforated plate at the entrance of a tube); in this case, Equation (4.8) has the following form (if ζ is expressed in terms of the velocity w2 downstream of the entrance)*: 2

2

⎤ ⎛ F2 ⎞ ⎡ F0 ⎞ F0 ⎞ ⎛ ⎛ ζ* = ⎢ζ′ + ⎜1 – ⎟ + τ ⎜1 – ⎟ + ζfr⎥ ⎜ ⎟ , 2 F F F 2⎠ 2⎠ ρw0 ⁄ 2 ⎣ ⎝ ⎝ ⎦ ⎝ 0⎠ ∆p

(4.9)

• F2 = ∞, discharge from an opening into an infinite space (flow discharge through an orifice or a perforated plate at the end of the tube, see Diagram 11.22); in this case Equation (4.8) has the form (if ζ is expressed in terms of the velocity w1 upstream of the opening)** ζ*

3⁄4

⎡ ⎛ F0 ⎞ = ⎢1 + ζ′ ⎜1 – ⎟ 2 ρw0 ⁄ 2 ⎣ ⎝ F1 ⎠ ∆p

F ⎞ ⎛ ⎯√⎯⎯⎯ ⎜1 – F⎯ ⎟ ⎝ ⎠

⎤ ⎛ F1 ⎞ + ζfr⎥ ⎜ ⎟ F ⎦ ⎝ 0⎠

0.375

2

3⁄4

+τ

0.75

0

1

2

⎤ ⎛ F1 ⎞ ⎡ F0 ⎞ ⎛ ⎛ F0 ⎞ = ⎢1 + ζ′ ⎜1 – ⎟ + τ ⎜1 – ⎟ + ζfr⎥ ⎜ ⎟ . (4.10) F F F 1 1 ⎝ ⎠ ⎠ ⎝ ⎦ ⎝ 0⎠ ⎣ • F1 = F2, restriction orifice, perforated plate (see Diagrams 4.14 to 4.17); in this case, Equation (4.8) reduces to the following form (if ζ is expressed in terms of the velocity w1 before the opening):

ζ*

3⁄4

⎡ ⎛ F0 ⎞ = ⎢ζ′ ⎜1 – ⎟ 2 ρw1 ⁄ 2 ⎣ ⎝ F1⎠

∆p

+τ

√ ⎯⎯⎯⎯⎯ ⎛ F0⎞ ⎜1 – F1 ⎟ ⎠ ⎝

0.375

0.75 ⎡ ⎛ F0 ⎞ F0 ⎞ ⎛ + τ ⎜1 – ⎟ = ⎢ζ′ ⎜1 – ⎟ F1 ⎠ F1 ⎠ ⎝ ⎣ ⎝

3⁄4

⎛ F0⎞ + ⎜1 – ⎟ ⎝ F1 ⎠

F0 ⎞ ⎛ F0 ⎞ ⎛ × ⎜ 1 – ⎟ + ⎜1 – ⎟ F1 ⎠ F 1⎠ ⎝ ⎝ 2

2

⎤ ⎛ F1 ⎞2 + ζfr⎥ ⎜ ⎟ ⎦ ⎝ F0⎠

⎤ ⎛ F1 ⎞2 + ζfr⎥ ⎜ ⎟ . ⎦ ⎝ F0 ⎠

*

Subscript 0 corresponds to subscript "or" and subscript 2 to subscript 0 in Chapter 3.

**

Subscript 0 corresponds to subscript "or" and subscript 1 to subscript 0 in Chapter 11.

(4.11)

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Handbook of Hydraulic Resistance, 4th Edition

• F1 = F2 = ∞, an opening in the wall between infinite flow areas (passage through an opening from one large volume into another, see Diagram 4.18); in this case, Equation (4.8) reduces to the form ζ*

∆p ρw20 ⁄ 2

= ζ′ + τ + 1 + ζfr .

(4.12)

31. The resistance coefficient of a restriction having orifice edges of different shapes and at Re ≥ 105 is expressed as follows: • With sharp-edged orifices, ζ′ = 0.5, τ = 1.41, and ζfr = 0, so that Equation (4.8) is reduced to the author’s formula of the form ζ*

⎡ = ⎢0.707 2 ρw1 ⁄ 2 ⎣ ∆p

⎯⎯⎯⎯⎯ √ ⎛ ⎜1 – ⎝

3⁄ 4

F0 ⎞ F1 ⎟ ⎠

2

2

0.37

2

F0 ⎤ ⎛ F1 ⎞ +1– ⎥ ⎜ ⎟ F F1 ⎦ ⎝ 0⎠ 2

⎡ F0 ⎤ ⎛ F1 ⎞ F0 ⎞ ⎛ = ⎢0.707 ⎜1 – ⎟ +1– ⎥ ⎜ ⎟ . F F1 ⎝ F0 ⎠ 1⎠ ⎝ ⎦ ⎣ • With thick-edged orifices, ζ′ = 0.5, resulting in ζ*

3⁄4

⎡ ⎛ F0 ⎞ ∆p = ⎢0.5 ⎜1 – ⎟ 2 ρw1 ⁄ 2 ⎣ ⎝ F1 ⎠ 2

F0 ⎞ ⎛ + ⎜1 – ⎟ F 1⎠ ⎝

+τ

(4.13)

⎯⎯⎯⎯⎯ √ F0 ⎞ ⎛ ⎜1 – F ⎟ 1⎠ ⎝

3⁄4

⎛ F0 ⎞ ⎜1 – F ⎟ 1⎠ ⎝

0.75 1.375 ⎤ ⎛ F1 ⎞2 ⎡ ⎛ F0 ⎞ ⎛ F0 ⎞ + ζfr⎥ ⎜ ⎟ = ⎢0.5 ⎜1 – ⎟ + τ ⎜1 – ⎟ F1 ⎠ ⎦ ⎝ F0 ⎠ ⎝ F1 ⎠ ⎣ ⎝

2 ⎤ ⎛ F1 ⎞2 ⎛ F0 ⎞ + ⎜1 – ⎟ + ζfr⎥ ⎜ ⎟ , ⎦ ⎝ F0 ⎠ ⎝ F1 ⎠

(4.14)

where _ _ τ = (2.4– l) × 10–ϕ ⁄ l ,

(4.15)

_ _ _ _ ϕ(l) = 0.25 + 0.535l8 ⁄ (0.05 + l 7) , (l = l ⁄ Dh) .

(4.16)

• With beveled or round-edged orifices τ ≈ 2√ ⎯⎯ζ′ and ζfr = 0, then ζ*

⎡ = ⎢1 + 2 ρw1 ⁄ 2 ⎣ ∆p

⎡ F0 = ⎢1 – +√ ⎯⎯ζ′ F1 ⎣

⎛ F ⎞ ζ′ ⎜1 – ⎟ √ ⎯⎯⎯⎯⎯ ⎝ F ⎠

3⁄4

0 1

0.375 2

F0 ⎞ ⎛ ⎜1 – F ⎟ 2⎠ ⎝

2

–

2

F0 ⎤ ⎛ F1 ⎞ F2 ⎥ ⎜ F0 ⎟ ⎦ ⎝ ⎠ 2

⎤ ⎛ F1 ⎞ ⎥ ⎜F ⎟ . ⎦ ⎝ 0⎠

(4.17)

Flow through Orifices with Change in Velocity and Flow Area

239

In the case of orifices with beveled edges, at α = 40–60o ζ′ = 0.13 + 0.34 × 10

_ _ 2.3 –(3.45l+88.4l )

,

(4.18)

or see Diagram 4.13; at other values of α, ζ′ is determined as ζ from Diagram 3.7. For round-edged orifices ζ′ is determined as ζ for a circular nozzle with an end face wall, i.e., _ _ ζ′ = 0.03 + 0.47 × 10–7.7r, (r = r ⁄ Dh) ,

(4.19)

or from the curve of Diagram 4.13. 32. The resistance coefficient of an aperture in the wall of an infinite area having opening edges of different shapes and at Re ≥ 105 is expressed as follows: • With sharp-edged orifices, ζ′ = 0.5, τ = 1.41, and ζfr = 0, so that, on the basis of Equation (4.12), ζ*

∆p ρw20 ⁄ 2

+ 2.9 ,

and, according to the author’s experiments,12 ζ = 2.7–2.8 . • With thick-edged orifices ζ′ = 0.5 and Equation (4.8) takes on the form ζ*

∆p ρw20 ⁄ 2

= 1.5 + τ + ζfr = ζ0 + ζfr ,

(4.20)

where ζ0 = 1.5 + τ was obtained experimentally by the author and presented in the form ζ0 = f(l/Dh) (curve a) in Diagram 4.18, _ _ –ϕ(l) , ζ0 = 1.5 + (2.4 – l) × 10 (4.21) _ where ϕ(l) is determined from Equation (4.16). • With beveled or rounded (in the flow direction) edges of the opening it is assumed that ⎯⎯ζ′ ; then ζfr = 0 and τ ≈ 2√

ζ*

∆p ρw20 ⁄ 2

= (1 + √ ⎯⎯ζ′ )2 ,

(4.22)

where ζ′ is determined in the manner described in paragraph 30, Chapter 3. 33. The resistance coefficient of a flow with an abrupt change in the cross section depends (Figure 4.14) not only on the geometric parameters of the section, but also on the flow re-

240

Handbook of Hydraulic Resistance, 4th Edition

gime (a function of the Reynolds number Re = w0Dh/v).20 In the case considered, as in the case of friction, three specific flow regions can be distinguished: • The laminar regime, in which ζ depends linearly on Re (in logarithmic coordinates); • The transition regime, in which the linear dependence ζ = f(Re) is violated; and • The self-similar turbulent regime (the region of the quadratic resistance law), in which an effect of the Reynolds number on the resistance coefficient is virtually absent. The limiting values of Re beyond which the laminar pattern of the flow ceases, as well as the limiting values of Re at which the transition regime terminates, depend on the geometry of the section. 34. The resistance coefficient with an abrupt change in the cross section can be expressed in general form for all the flow regions from Equation (1.3) at k3 = 1 ζ=

(4.23)

A + ζquad , Re

where A is a coefficient depending on the geometry of the section considered. 35. For the case of an abrupt expansion of the flow area, the resistance coefficient for the transition region (10 < Re < 104) can be determined from the experimental curves ζ = f(Re, F0/F1) obtained by Altshul,3 Karev,17 and Veziryan6 (see Diagram 4.1). For the laminar region (Re < 10) the resistance coefficient is ζ*

∆p ρw20 ⁄ 2

30 + Re

.

(4.24)

Figure 4.14. Dependence of the resistance coefficient of orifices on the Reynolds number for different values of F0/F1.20 1) 0.05; 2) 0.16; 3) 0.43; 4) 0.64.

Flow through Orifices with Change in Velocity and Flow Area

241

36. For the case of an abrupt contraction of the flow area, the resistance coefficient in the transition region (10 < Re < 104) can be determined (see Reference 18) from the curves ζ = f(Re, F0/F1) of Diagram 4.10 and in the laminar region (Re < 6–7) from Equation (4.24). 37. For the case of flow passage through openings in a wall (the general case of passage is shown in Figure 4.13; restriction, aperture) the resistance coefficient in the transition and laminar regions can be found: • within 30 < Re < 104–105* from the author’s formula16 ζ*

∆p ρw20 ⁄ 2

+ ⎛⎜ 12 – 1⎞⎟ + 0.342 2 ⎝ϕ

⎡ × ⎢1 + 0.707 ⎣

ε0 Re

⎠

F ⎞ ⎛ √ ⎯⎯⎯⎯ ⎜1 – F⎯ ⎟ ⎝ ⎠

3⁄4

0 1

–

2 _ F0 ⎤ = ζϕ + ε0 Reζ0quad , ⎥ F2 ⎦

(4.25)

where ϕ is the velocity coefficient of discharge from the sharp-edged orifice, which depends on Re and the area ratio F0/F1; ε0Re = Fcon/F0 is the fluid jet area ratio of the sharp-edged orifice section at F0/F1 = 0 (F1 = ∞), which depends on the Reynolds number; ζϕ = (1/ϕ2 – 1) is determined from the curves ζϕ = f1(Re, F0/F1) of Diagram 4.19; ε0Re = 0.342/ε20Re is _ determined from the curve ε0Re = f(Re) of the same diagram; ζ′0quad is the resistance coefficient of the given type of flow restriction for the self-similar (quadratic) region, which is determined as ζ from Equations (4.7)–(4.22), where ζ0 = ∆p ⁄ ρw20; • Within 10 < Re < 30 from the approximation suggested by the author ζ*

∆p 2

ρw0 ⁄ 2

_

A + Re + ε0 Reζquad ,

(4.26)

while at Re < 10 ζ*

∆p A , = 2 Re ρw0 ⁄ 2

(4.27)

where A = 33.3 38. If the resistance coefficient is expressed in terms of the velocity w1 in section F1 upstream of the orifice (and not w0 in the orifice itself), then Equations (4.25)–(4.27) are replaced by ζ1 *

*

2 _ ⎛ F1 ⎞ = (ζφ + ε0 Reζquad) ⎜ ⎟ , F ρw21 ⁄ 2 ⎝ 0⎠

∆p

For sharp-edged orifices the upper limit of Re is taken as 105, while for other shapes it is of the order of 104.

242

ζ1 *

ζ1 *

Handbook of Hydraulic Resistance, 4th Edition 2

∆p

⎛ F1 ⎞ 33 = ⎛⎜ + ε0 Reζquad⎞⎟ ⎜ ⎟ , 2 Re ρw1 ⁄ 2 ⎝ ⎠ ⎝ F0 ⎠ ∆p ρw21 ⁄ 2

2

=

33 ⎛ F1 ⎞ . Re ⎜ F0 ⎟ ⎝ ⎠

39. At low values of the area coefficients F0/F1 of the restriction, the flow through the orifice attains high velocities (high Mach numbers), even at relatively low velocity in the pipeline upstream of the restriction. Here, the compressibility effect, which sharply increases the resistance coefficient of the restriction, becomes noticeable: ζM *

∆p ρ1w21 ⁄ 2

= kMζ ,

where ζM is the resistance coefficient of the restriction at small Mach numbers determined as given under 25–31; kM is the coefficient which considers the effect of compressibility in the contracted section of the jet during its passage through the orifice (see Diagram 4.20); and Ma1 = w1/a1 is the Mach number upstream of the restriction. 40. A significant decrease in the resistance of the orifice can be attained by installing an annular rib at the inlet into the straight channel or a ledge at the inlet into the orifice (Figure 4.15). Thus, for example, according to Khanzhonkov’s27 experiments, the installation of an annular rib with D1/D0 ≈ 1.22 and l/D0 ≈ 0.25 reduces the resistance coefficient of the orifice in the wall of an infinite upstream area from ζ ≈ 2.7–2.8 (obtained without a rib) to ζ = 1.15. 41. When the flow enters through a smooth inlet nozzle (collector) installed in the wall of an infinite surface area (see Diagram 4.21), the flow resistance is made up of the resistance of entrance into the nozzle, the frictional resistance over the straight section, and the exit resistance. The resistance coefficient of such a section is determined from

Figure 4.15. Entrance into the orifice: a) through an annular rib; b) through a ledge.

Flow through Orifices with Change in Velocity and Flow Area ζ*

∆p ρw20 ⁄ 2

243

= ζ′ + ζfr ,

where ζ′ is the coefficient representing the inlet and outlet losses, which is determined from the curves ζ = f(l/Dh, r/Dh) of Diagram 4.21, and ζfr = λl/Dh is the friction coefficient over the straight section of the nozzle. 42. When the flow discharges through an orifice in the wall in the presence of the passing stream* (see Diagram 4.22), the resistance coefficient in the case of both entry from the stream (suction, aspiration) and discharge into the stream (influx) is a function of the velocity ratio w∞/w0, as shown in Reference 27. 43. In the absence of the passing stream (w∞ = 0), the fluid approaches the orifice from all sides, while the flow discharges symmetrically into the stream, with the smallest contraction of the jet section. In the presence of the passing stream, the fluid approaches the orifice from one side, while the flow discharges at an angle with a more contracted section of the jet downstream of the orifice. The jet contraction causes an increase of the velocity pressure, which is lost at the exit for the given system. 44. At small velocities of the passing stream (w∞ w0) and reducing the dimensions of the recirculation zone, and the coefficient ζ increases. 46. In the cases of both suction and discharge, the resistance coefficients ζ remain practically the same for square and circular apertures, as well as for orifices with rounded edges. At the same time, their values depend substantially on the orientation of orifices of elongated (rectangular) shape. The largest values of ζ are obtained for the orifices with their longer sides placed perpendicular to the flow. 47. In the case of suction, the larger resistance coefficients ζ for elongated orifices with their larger side perpendicular to the flow are due to a major portion of the flow entering *

The passing stream (w∞) moves independently of the flow discharging through the orifice under the influence of its own driving force — for example, wind, opposing stream on an airplane, etc.

244

Handbook of Hydraulic Resistance, 4th Edition

these apertures from the wall region. The flow entering the orifice therefore has a small amount of kinetic energy, and the additional blowing (diffusion) effect is small. When the elongated orifices are arranged with their larger side parallel to the flow, the portion of the perimeter facing the flow is small and the prevailing portion of the flow enters the orifice from its upper layers, which have a higher velocity. This increases the blowing effect and correspondingly decreases the resistance coefficient. 48. In the case of discharging flow (influx), for elongated orifices placed with their longer side normal to the flow, an increase in ζ is explained by the fact that the throttling effect exerted by the passing stream on the jet escaping from the orifice is more pronounced than with the larger side placed parallel to the flow, since the front surface of the jet in the former case is larger than in the latter. 49. Baffles installed at the edges of orifices (see schemes of Diagram 4.22) have a substantial effect on the values of ζ in the case of suction as well as discharge. An inclined baffle increases and a straight one decreases the value of the resistance coefficient. In the first case, the baffle compresses the flow passing through the orifice; consequently, the velocity pressure, lost on escape from the orifice, increases. In the second case the baffle weakens the effect of flow contraction, which correspondingly decreases the velocity pressure losses at the exit from the orifice. 50. When the fluid passes through the apertures in a wall, fitted with various flaps (projections), the resistance is higher than in the absence of flaps, since they complicate the flow trajectory. In this case, the resistance coefficient becomes a function of the angle of opening of the flaps, α, and of the relative length of the flaps lfl/bfl. 51. The open working section of a wind tunnel (see Diagram 4.25) can also be regarded as a section with abrupt expansion. The main source of losses in this section is ejection dissipation of energy. The second source of losses is cutting off of the "added masses" from the surrounding medium by the wind tunnel diffuser. The kinetic energy of the portion of the jet that was cut off turns to be lost for the wind tunnel and, therefore, constitutes a part of the resistance of the open working section. The coefficient of total resistance of the open working section (w.s.) is calculated from Abramovich’s1 formula. For a circular (or rectangular) cross section:

ζ*

∆p ρw20 ⁄ 2

2

= 0.1

lw.s. ⎛ lw.s. ⎞ – 0.008 ⎜ , Dh D ⎟ ⎝ h⎠

where Dh = 4F0 ⁄ Π0 is the hydraulic diameter of the exit section of the tunnel nozzle, m. For an elliptical cross section: ζ*

where

∆p 2

ρw0 ⁄ 2

= 0.145

lw.s. lw.s. – 0.0017 , Dh a0b0

Flow through Orifices with Change in Velocity and Flow Area Dh +

245

4a0b0 , 1.5(a0 + b0) – ⎯√⎯⎯⎯ a0b0

lw.s. is the length of the open working section of the wind tunnel, m; and a0 and b0 are the ellipse semiaxes, m.

246

Handbook of Hydraulic Resistance, 4th Edition

4.2 DIAGRAMS OF RESISTANCE COEFFICIENTS Sudden expansion of a flow having a uniform velocity distribution13,15,17

Diagram 4.1

w0Dh ≥ 3.3 × 103: v a) Without baffles

1. At Re =

ζ*

2

ζfr ζfr F0 ⎞ ⎛ = ⎜1 – ⎟ + 2 = ζloc + 2 , 2 F 2⎠ ρw0 ⁄ 2 ⎝ nar nar ∆p

where ζloc = f(F0/F2), see graph a; ζfr = λ(l2/D2h); for λ, see Chapter 2. b) With baffles ζ*

Dh =

4F0 Π0

, D2h =

2

ζfr F0 ⎞ ⎛ = 0.6 ⎜1 – ⎟ + 2 . 2 F 2⎠ ρw0 ⁄ 2 nar ⎝ ∆p

Relative losses with a sudden expansion at supersonic velocities are considered in paragraphs 20 and 21 of Section 4.1

4F2 Π2

F2 , Π is the perimeter nar = F0

Values of ζ′

0

0.1

0.2

0.3

F0%F2 =

1 nar

0.4

0.5

0.6

0.7

0.8

1.0

0.09

0.04

0

0.05

0.02

0

Without baffles (curve 1) 1.00

0.81

0.64

0.50

0.36

0.25

0.16

With baffles (curve 2) 0.60

0.49

0.39

0.30

0.21

0.15

0.10

2. When 500 ≤ Re < 3.3 × 103, for ζloc, see graph b or ζloc is determined from the formula ζloc = –8.44556 – 26.163(1 – F0/F2)2 – 5.38086(1 – F0/F2)4 + log Re [6.007 + 18.5372(1 – F0/F2)2 + 3.9978(1 – F0/F2)4] + (log Re)2[–1.02318 – 3.091691 – F0/F2)2 – 0.680943(1 – F0/F2)4]

Values of ζ Re

F0 1 = F2 nar

10

15

20

30

40

50

102

2 × 102

5 × 103

103

2 × 103

3 × 103

≥ 3 × 103

0.1

3.10

3.20

3.00

2.40

2.15

1.95

1.70

1.65

1.70

2.00

1.60

1.00

0.81

0.2

3.10

3.20

2.80

2.20

1.85

1.65

1.40

1.30

1.30

1.60

1.25

0.70

0.64

0.3

3.10

3.10

2.60

2.00

1.60

1.40

1.20

1.10

1.10

1.30

0.95

0.60

0.50

0.4

3.10

3.00

2.40

1.80

1.50

1.30

1.10

1.00

0.85

1.05

0.80

0.40

0.36

0.5

3.10

2.80

2.30

1.65

1.35

1.15

0.90

0.75

0.65

0.90

0.65

0.30

0.25

0.6

3.10

2.70

2.15

1.55

1.25

1.05

0.80

0.60

0.40

0.60

0.50

0.20

0.16

247

Flow through Orifices with Change in Velocity and Flow Area Sudden expansion of a flow having a uniform velocity distribution13,15,17

Diagram 4.1

3. At 10 < Re < 500, ζloc is determined from graph b or from the formula: ζloc = 3.62536 + 10.744(1 – F0 ⁄ F2)2 – 4.41041(1 – F0 ⁄ F2)4 +

1 ⎡ –18.13 – 56.77855(1 – F0 ⁄ F2)2 + 33.40344(1 – F0 ⁄ F2)4⎤ ⎦ log Re ⎣

+

1 ⎡30.8558 + 99.9542(1 – F0 ⁄ F2)2 – 62.78(1 – F0 ⁄ F2)4⎤ ⎦ (log Re)2 ⎣

+

1 ⎡–13.217 – 53.9555(1 – F0 ⁄ F2)2 + 33.8053(1 – F0 ⁄ F2)4⎤ . ⎦ (log Re)3 ⎣

4. At Re < 10 ζloc +

30 . Re

Sudden expansion of a long straight section, diffusers, and so on, with exponential velocity distribution; circular or rectangular cross section; Re = w0Dh/v > 3.5 × 103 13,15 ζ*

Diagram 4.2

ζfr ∆p 1 2M ζfr = 2 +N– + 2 = ζloc + 2 , 2 n nar ar ρw0 ⁄ 2 nar nar

⎫ , ⎪ ⎪ where ⎬ see graph b , 3 ⎪ 3 (2m + 1) (m + 1) N= 4 ⎪ 4m (2m + 3)(m + 3) ⎭ M=

(2m + 1)2(m + 1) 4m2(m + 2)

ζloc = f(m, F0/F2), see graph a, ζfr = λl2/D2h; for λ, see Chapter 2.

Dh =

4F0

nar =

F2

Π0 F0

; D2h =

4F2 Π2

, Π is the perimeter; 1⁄m

;

y w = ⎛1 – ⎞⎟ R0 wmax ⎜ ⎠ ⎝

,

m≥1

248

Handbook of Hydraulic Resistance, 4th Edition

Sudden expansion of a long straight section, diffusers, and so on, with exponential velocity distribution; circular or rectangular cross section; Re = w0Dh/v > 3.5 × 103 13,15

Diagram 4.2

Values of ζ F0 = 1%nar F2 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

1.0 1.35 2.0 3.0 4.0 7.0

2.70 2.00 1.50 1.25 1.15 1.06

2.42 1.74 1.28 1.04 0.95 0.86

2.14 1.51 1.08 0.84 0.77 0.69

1.90 1.29 0.89 0.68 0.62 0.53

1.66 1.00 0.72 0.53 0.47 0.41

1.45 0.93 0.59 0.41 0.35 0.29

1.26 0.77 0.46 0.30 0.25 0.19

1.09 0.65 0.35 0.20 0.17 0.12

0.94 0.53 0.27 0.14 0.11 0.06

0.70 0.36 0.16 0.07 0.05 0.02

∞

1.00

0.82

0.64

0.48

0.36

0.25

0.16

0.09

0.04

0

m N M

1.0 2.70 1.50

1.35 2.00 1.32

2.0 1.50 1.17

3.0 1.25 1.09

4.0 1.15 1.05

7.0 1.06 1.02

∞ 1.0 1.0

249

Flow through Orifices with Change in Velocity and Flow Area Sudden expansion of a long straight section, diffusers, and so on, with exponential velocity distribution; circular or rectangular cross section; Re = w0Dh/v > 3.5 × 103 13,15 ζ*

∆p ρw20 ⁄ 2

=

Diagram 4.3

ζfr 1 2M ζfr +N– + = ζloc + 2 , nar n2ar na2r nar

(m + 1)2 ⎫ ,⎪ m(m + 2) ⎪ where ⎬ see graph b; (m + 1)3 ⎪ N= 2 ⎪ m (m + 3) ⎭ M=

ζloc = f(m, F0/F2), see graph a, ζfr = λl2/D2h; for λ see Chapter 2 Dh =

4F0 Π0

; D2h =

4F2 Π2

;

Π is the perimeter; 1⁄m F2 2y w ; = ⎛⎜1 – ⎞⎟ , F0 wmax b0 ⎠ ⎝ m≥1

nar =

Values of ζloc F0 = 1%nar F2 m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

1.0 1.35 2.0 3.0 4.0 7.0

2.00 1.65 1.35 1.19 1.12 1.04

1.74 1.40 1.14 0.98 0.92 0.85

1.51 1.20 0.94 0.80 0.74 0.64

1.28 1.00 0.77 0.64 0.60 0.54

1.19 0.83 0.62 0.49 0.46 0.41

0.92 0.67 0.48 0.37 0.33 0.28

0.77 0.53 0.36 0.24 0.23 0.18

0.64 0.41 0.26 0.18 0.14 0.08

0.51 0.32 0.19 0.12 0.09 0.05

0.34 0.20 0.10 0.05 0.04 0.02

∞

1.00

0.81

0.64

0.49

0.36

0.25

0.15

0.08

0.04

0

m N M

1.0 2.00 1.33

1.35 1.64 1.22

2.0 1.35 1.13

3.0 1.18 1.07

4.0 1.12 1.04

7.0 1.04 1.02

∞ 1.0 1.0

250

Handbook of Hydraulic Resistance, 4th Edition

Sudden expansion of a plane channel downstream of perforated plates, guide vanes in elbows, and so on, with sinusoidal velocity distribution; Re = w0Dh/v > 3.5 × 103 13,15 Dh =

4F0 Π0

; D2h =

4F2 Π2

Diagram 4.4 ; nar =

F2

F0

;

∆w 2y w =1+ sin 2k1π ; w0 b0 w0

k is an integer; Π is the perimeter

ζ*

∆p ρw20 ⁄ 2

=

ζfr 1 2M ζfr +N– = ζloc + 2 , + nar n2ar na2r nar

where M = 1 +

1 2

2

2

⎛∆w⎞ 3 ⎛ ∆w⎞ ⎜ w ⎟ ; N = 1 + 2 ⎜ w ⎟ , see graph b; 0 ⎝ 0⎠ ⎝ ⎠

see graph a; ζfr =

λl2 ; for λ, see Chapter 2. D2h

Values of ζloc ∆w w0

F0 = 1%nar F2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

0.1 0.2 0.4 0.6 0.8 1.0

1.01 1.06 1.24 1.54 1.96 2.50

0.83 0.88 1.04 1.31 1.70 2.21

0.66 0.70 0.84 1.18 1.47 1.95

0.50 0.54 0.68 0.92 1.27 1.70

0.38 0.40 0.54 0.75 1.07 1.46

0.26 0.29 0.41 0.61 0.89 1.25

0.17 0.20 0.30 0.48 0.75 1.05

0.10 0.13 0.22 0.39 0.60 0.88

0.06 0.07 0.16 0.29 0.49 0.74

0.01 0.02 0.08 0.18 0.32 0.50

∆w w0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

N M

1.0 1.00

1.06 1.02

1.13 1.04

1.24 1.08

1.37 1.12

1.54 1.18

1.73 1.24

1.96 1.32

2.22 1.40

2.50 1.50

251

Flow through Orifices with Change in Velocity and Flow Area Sudden expansion downstream of a plane diffuser with α > 10o, elbows, and so on, with asymmetrical velocity distribution; Re = w0Dh/v > 3.5 × 103 13,15 Dh =

4F0 Π0

; D2h =

4F2

Π2

Diagram 4.5

; nar =

F2 ; Π is the perimeter F0

2y w = 0.585 + 1.64 sin ⎛⎜0.2 + 1.95 ⎞⎟ , w0 b0 ⎝ ⎠ ζ*

∆p ρw20 ⁄ 2

=

ζfr 1 3.74 ζfr + + 3.7 – =ζ + , nar n2ar loc n2ar n2ar

where ζloc = f(F0/F2), see graph; ζfr = λl2/D2h, for λ, see Chapter 2.

F0 = 1%nar F2

ζ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

3.70 3.34 2.99 2.66 2.36 2.09 1.82 1.58 1.35 0.96

Sudden expansion downstream of sections with parabolic velocity distribution; Re = w0Dh/v > 3.5 × 103 13,15 Dh =

4F0 Π0

; D2h =

Diagram 4.6 4F2 Π2

; nar =

F2 F0

; Π is the perimeter.

1. Circular tube:

ζ* w y = 1 – ⎛⎜ ⎞⎟ wmax R ⎝ 0⎠

∆p ρw20 ⁄ 2

=

ζfr 1 2.66 ζfr +2– + = ζloc + 2 . nar n2ar n2ar nar

2

2. Plane channel:

ζ*

∆p ρw20 ⁄ 2

=

ζfr 1 2.4 ζfr + 1.55 – + = ζloc + 2 , nar n2ar n2ar nar

where ζloc = f(F0/F2), see graph; ζfr = λl2/D2h, for λ, see Chapter 2.

252

Handbook of Hydraulic Resistance, 4th Edition

Sudden expansion downstream of sections with parabolic velocity distribution; Re = w0Dh/v > 3.5 × 103 13,15 F0 = 1%nar F2

0

Diagram 4.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

1. Circular tube

ζloc

2.00 1.75 1.51 1.30 1.10 0.92 0.78 0.63 0.51 0.34 2. Plane channel

ζloc

1.55 1.32 1.11 0.92 0.75 0.60 0.47 0.36 0.27 0.15

Flow deformation in a straight tube (channel); nar = 1; Re = w0Dh/v > 3.5 × 103 13,15

Diagram 4.7

Exponential law of velocity distribution: 1⁄m

w y = ⎛1 – ⎞⎟ m≥1 , wmax ⎜ R0 ⎠ ⎝ ∆p ζ * 2 = 1 + N – 2M + ζfr = ζloc + ζfr , ρw0 ⁄ 2 ζloc = f(1/m); for M and N, see graph b of Diagrams 4.2 and 4.3; ζfr = λl0/D0, for λ, see Chapter 2. Dh =

4F0 Π0

Π is the perimeter

Parabolic velocity distribution: 2

u w = 1 – ⎛⎜ ⎞⎟ . R0 wmax ⎝ ⎠

1. Circular tube ζloc *

∆p ρw20 ⁄ 2

= 0.34 .

2. Plane channel ζloc *

∆p ρw20 ⁄ 2

= 0.15 .

253

Flow through Orifices with Change in Velocity and Flow Area Flow deformation in a straight tube (channel); nar = 1; Re = w0Dh/v > 3.5 × 103 13,15

m

1.0

1.35

Diagram 4.7

2.0

4.0

7.0

∞

0.05

0.02

0

0.04

0.02

0

1. Circular tube ζloc

0.7

0.36

0.16 2. Plane channel

ζloc

0.31

0.19

0.10

Flow deformation in a straight tube with a free jet entering it (ejector); nar = 1; Re = w0Dh/v > 3.5 × 103 13,15

Diagram 4.8

Dh =

4F0

ζ*

∆p

Π0

; D2h =

ρw22 ⁄ 2

4F2 Π2

;

= 1 + N – 2M + ζfr = ζloc + ζfr ;

2 1 ⎛ F2 ⎞ _ 1 ⎛ F2⎞ M = _2 ⎜ ⎟ ; N = _3 ⎜ ⎟ e . q ⎝ F0 ⎠ q ⎝ F0 ⎠

__ The values of_ ζloc, M, and N are determined from the graph as a function of the free jet length S/Dh; F = F2/F0 = _ Fj/F0, q and e are determined as functions of the free length S/Dh from Diagrams 11.28 and 11.29; ζfr = λl2/D2h; for λ, see Chapter 2.

254

Handbook of Hydraulic Resistance, 4th Edition

Flow deformation in a straight tube with a free jet entering it (ejector); nar = 1; Re = w0Dh/v > 3.5 × 103 13,15

S

Diagram 4.8

0.5

1.0

1.5

2.0

2.5

3.0

ζloc

0.16

0.46

0.84

1.43

2.02

2.54

N

1.65

2.89

3.90

4.85

5.65

6.35

M

1.25

1.71

2.00

2.20

2.30

2.40

4.0

5.0

6.0

8.0

10

ζloc

3.26

3.65

3.80

3.81

3.81

N

7.20

7.55

7.68

7.70

7.70

M

2.45

2.45

2.45

2.45

2.45

Dh

S

Dh

Sudden contraction at b/Dh = 0; Re = w0Dh/v > 3.5 × 104 12,13

N

Kind of inlet edge

Diagram 4.9 Resistance coefficient ∆p ζ* 2 ρw0 ⁄ 2

Configuration 3⁄4

F0 ⎞ ⎛ ζ = 0.5 ⎜1 – ⎟ F1 ⎠ ⎝

+ ζfr = 0.5a + ζfr ,

where for a, see curve a = f(F0 ⁄ F1) ; ζfr = λ see Chapter 2.

l0 ; for λ, Dh

255

Flow through Orifices with Change in Velocity and Flow Area Sudden contraction at b/Dh = 0; Re = w0Dh/v > 3.5 × 104 12,13

Diagram 4.9

1. Sharp edges

Dh =

F0 F1

0

a

1.0

0.2

0.4

0.6

0.8

0.9

0.850 0.680 0.503 0.300 0.178

1.0 0

4F0 Π0

3⁄4

F0 ⎞ ⎛ ζ = ζ′ ⎜1 – ⎟ + ζfr = f′a + ζfr, where for ζ′, F1 ⎠ ⎝ see curve ζ = f(b/Dh) of Diagram 3.4 (curve c); for a, see curve a = f(F0/F1) (para. 1); ζfr = λl0/Dh; for λ, see Chapter 2.

2. Rounded edges

3⁄4

F0 ⎞ ⎛ ζ = ζ′′ ⎜1 – ⎟ + ζfr = ζ′′a + ζfr, where for ζ′′, F1 ⎠ ⎝ see curve ζ = f(α, l/Dh) of Diagram 3.7; for a, see curve a = f(F0/F1) (para. 1); ζfr = λl0/Dh;

3. Beveled edges

for λ, see Chapter 2.

Sudden contraction in transition and laminar regions; Re = w0Dh/v < 104 3,18 ζ*

∆p

ρw20 ⁄ 2

Diagram 4.10

= ζloc + ζfr .

1. At 10 ≤ Re < 104 ζloc is determined from curves ζloc = f(Re, F0/F1) or from the formula ζloc = A⋅B(1 – F0 ⁄ F1) , 7

Dh =

4F0 Π0

where A = ∑ ai(log Re)i ; i=0

a0 = –25.12458; a1 = 118.5076; a2 = –170.4147; a3 = 118.1949; a4 = –44.42141; a5 = 9.09524; a6 = –0.9244027; a7 = 0.03408265 ⎫ ⎡2 ⎤ ⎢ a (F ⁄ F ) j⎥ (log Re)i⎪ ; ⎬ ij 0 1 ∑ ⎢ ⎥ ⎪ ⎢ j=0 ⎥ ⎭ ⎣ ⎦ the values of aij are given below. ⎧ ⎪ ⎨ ⎪ i=0 ⎩ 2

B=∑

2. Re < 10 , ζloc +

30 Re

256

Handbook of Hydraulic Resistance, 4th Edition

Sudden contraction in transition and laminar regions; Re = w0Dh/v < 104 3,18

Diagram 4.10

Values of aij 10 ≤ Re ≤ 2 × 103 i%j 0 1 2

0 1.07 0.05 0

2 × 103 ≤ Re ≤ 4 × 103

1 1.22 –0.51668 0

2 2.9333 0.8333 0

0 0.5443 –0.06518 0.05239

1 –17.298 8.7616 –1.1093

2 –40.715 22.782 –3.1509

Values of ζloc Re

F0 F1

10

20

30

40

50

102

2 × 102

5 × 102

103

2 × 103

4 × 103

5 × 103

104

>104

0.1

5.00

3.20

2.40

2.00

1.80

1.30

1.04

0.82

0.64

0.50

0.80

0.75

0.50

0.45

0.2

5.00

3.10

2.30

1.84

1.62

1.20

0.95

0.70

0.50

0.40

0.60

0.60

0.40

0.40

0.3

5.00

2.95

2.15

1.70

1.50

1.10

0.85

0.60

0.44

0.30

0.55

0.55

0.35

0.35

0.4

5.00

2.80

2.00

1.60

1.40

1.00

0.78

0.50

0.35

0.25

0.45

0.50

0.30

0.30

0.5

5.00

2.70

1.80

1.46

1.30

0.90

0.65

0.42

0.30

0.20

0.40

0.42

0.25

0.25

0.6

5.00

2.60

1.70

1.35

1.20

0.80

0.56

0.35

0.24

0.15

0.35

0.35

0.20

0.20

Sharp-edged orifice (l/Dh = 0–0.015) installed in a transition section; Re = w0Dh/v ≥ 104 13,14

ζ*

∆p

F0 ⎞ ⎡ ⎛ = ⎢0.707 ⎜1 – ⎟ F ρw20 ⁄ 2 ⎣ 1⎠ ⎝

0.375

F0 ⎞⎤ ⎛ + ⎜1 – ⎟⎥ F2 ⎠⎦ ⎝

Diagram 4.11

2

⎛ F0 F0 ⎞ =f⎜ ; ⎟ ⎝ F1 F2 ⎠

Dh =

4F0 Π0

257

Flow through Orifices with Change in Velocity and Flow Area Sharp-edged orifice (l/Dh = 0–0.015) installed in a transition section; Re = w0Dh/v ≥ 104 13,14

Diagram 4.11

Values of ζ F0%F1

F0%F2 0 0.2 0.4 0.6 0.8 1.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

2.91 2.27 1.71 1.23 0.82 0.50

2.82 2.19 1.64 1.17 0.77 0.46

2.72 2.10 1.56 1.10 0.72 0.42

2.61 2.01 1.48 1.03 0.67 0.38

2.51 1.91 1.40 0.97 0.61 0.34

2.39 1.81 1.31 0.89 0.56 0.30

2.25 1.69 1.21 0.81 0.49 0.25

2.10 1.56 1.10 0.72 0.42 0.20

1.92 1.41 0.97 0.62 0.34 0.15

1.68 1.20 0.80 0.48 0.25 0.09

1.00 0.64 0.36 0.16 0.04 0

Thick-edged orifice (l/Dh > 0.015) installed in a transition section; Re = w0Dh/v > 105 13,14 Dh =

4F0

Π0

ζ*

Diagram 4.12

, ∆p

ρw20 ⁄ 2

0.75

+ 0.5 ⎛⎜1 – F0 ⎞⎟ F

⎝

F ⎞ ⎛ l × ⎜1 – 0 ⎟ + λ , Dh F 2⎠ ⎝ _ _ τ = (2.4 – l) × 10–ϕ(l) ,

1⎠

0

0.2

0.4

0.6

0.8

τ _ l l*

1.35

1.22

1.10

0.84

0.42

1.0

1.2

1.6

2.0

2.4

τ

0.24

0.16

0.07

0.02

0

Dh

Dh

0.375

where τ = f(l/Dh); for λ, see Chapter 2;

_ _ _ ϕ(l) = 0.25 + 0.535l8 ⁄ (0.05 + l 7) . _ l l*

2

F0 ⎞ F0 ⎞ ⎛ ⎛ + ⎜1 – ⎟ + τ ⎜1 – ⎟ F F1 ⎠ 2⎠ ⎝ ⎝

258

Handbook of Hydraulic Resistance, 4th Edition

Orifice with beveled and rounded (in flow direction) edges installed in a transition section; Re = w0Dh/v ≥ 104 13,14 Orifice

Configuration

Diagram 4.13

Resistance coefficient ∆p

ζ*

F

ρw20 ⁄ 2

With beveled edges

0.75

+ ζ′ ⎛⎜1 – F0 ⎞⎟ 1⎠

⎝

2

F0 ⎞ ⎛ + ⎜1 – ⎟ + 2 F 2⎠ ⎝

⎯⎯⎯⎯⎯ √ F0 ⎞ ⎛ ζ′ ⎜1 – ⎟ F1 ⎠ ⎝

0.375

F0 ⎞ ⎛ + ⎜1 – ⎟ , F2 ⎠ ⎝

where ζ′ at α = 40–60o, see graph a, or ζ′ = 0.13 + 0.34 _

Dh =

_

2.3

× 10−(34l+88.4l ). At other values of α, ζ′ is determined as ζ from Diagram 3.7

4F0 Π0

l%Dh

0.01

0.02

0.03

0.04

0.06

0.08

0.12 ≥0.16

ζ′

0.46

0.42

0.38

0.35

0.29

0.23

0.16

ζ*

⎛

∆p ρw20 ⁄ 2 2

F0 ⎞ ⎛ + ⎜1 – ⎟ + 2 F2 ⎠ ⎝

With rounded edges

0.75

= ζ′ ⎜1 –

⎝

0.13

F0 ⎞ F1⎟⎠

⎯⎯⎯⎯⎯ √

0.375

F0 ⎞ ⎛ ζ′ ⎜1 – ⎟ F1 ⎠ ⎝

F0 ⎞ ⎛ × ⎜1 – ⎟ , F2 ⎠ ⎝

r where ζ′ = f ⎛⎜ ⎞⎟ , see graph (b) or D ⎝ h⎠ _ ζ′ = 0.03 + 0.47 × 10–7.7r

r%Dh ζ′

0

0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.12 0.16 ≥0.2

0.50 0.44 0.37 0.31 0.26 0.22 0.19 0.15 0.09 0.06 0.03

259

Flow through Orifices with Change in Velocity and Flow Area Sharp-edged orifice (l/Dh = 0–0.015) in a straight tube; Re = w0Dh/v ≥ 105 13,14

ζ*

∆p ρw20 ⁄ 2

⎡⎛

Diagram 4.14

= ⎢⎜1 – ⎣⎝

F0 ⎞ F0 ⎞ ⎛ + 0.707 ⎜1 – ⎟ F1 ⎟⎠ F1 ⎠ ⎝

0.375

2

2 ⎤ ⎛ F1 ⎞ ⎥ ⎜F ⎟ , ⎦ ⎝ 0⎠

_ F ⎛ 0⎞ see curve ζ1 = f ⎜ ⎟ ⎝ F1 ⎠

Dh =

4F0 Π0

F0 F1

0.02

0.03

0.04

0.05

0.06

0.08

0.10

0.12

0.14

ζ1

7000

3100

1670

1050

730

400

245

165

117

F0 F1

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

ζ1 F0 F1

ζ1

86.0 0.34 13.1

65.6 0.36 11.6

51.5

40.6

32.0

26.8

22.3

18.2

15.6

0.38

0.40

0.43

0.47

0.50

0.52

0.55

9.55

8.25

6.62

4.95

4.00

3.48

2.85

F0 F1

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

ζ1

2.00

1.41

0.97

0.65

0.42

0.25

0.13

0.05

0

260

Handbook of Hydraulic Resistance, 4th Edition

Thick-edged orifice (l/Dh > 0.015) in a straight tube (channel); Re = w0Dh/v > 103 13,14 ζ1 *

F0⎞ ⎡ ⎛ = ⎢0.5 ⎜1 – ⎟ F1 ⎠ ρw21 ⁄ 2 ⎣ ⎝ ∆p

0.75

2

Diagram 4.15 F0 ⎞ ⎛ + τ ⎜1 – ⎟ F1 ⎠ ⎝

1.375

2

⎛ F ⎞ l ⎤ ⎛F ⎞ + ⎜1 – 0⎟ + λ ⎥ ⎜ 1 ⎟ , D F 1⎠ h ⎦ ⎝ F0 ⎠ ⎝ where for τ, see the table below or graph a of Diagram 4.12 or _

τ = (2.4 – l) × 10–ϕ(l) ;

_ _ _ ϕ(l) = 0.25 + 0.535l 8 ⁄ (0.05 + l 7 ) , see Chapter 2;

4F0 Dh = Π0 _ l = l ⁄ Dh

At λ = 0.02 for the values of ζ1 = f(l/Dh, F0/F1) see the graph.

Values of ζ1 at λ = 0.02 _ l = l Dh

F0%F1

%

τ

0 0.2

0.02

0.04

0.06

0.08

0.10

0.15

0.20

0.25

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.35 1.22

6915 1676 6613 1602

716 684

394 376

244 233

99.5 95.0

51.4 49.0

30.0 28.6

18.8 18.0

8.56 8.17

4.27 4.08

2.19 2.09

1.11 1.07

0.53 0.51

0.19 0.19

0 0

0.4 0.6

1.10 0.84

6227 5708

1533 1382

655 591

360 324

223 201

91.0 81.9

47.0 42.3

27.4 24.6

17.2 15.5

7.83 7.04

3.92 3.53

2.01 1.82

1.03 0.94

0.50 0.46

0.19 0.18

0.01 0.01

0.8 1.0 1.4

0.42 0.24 0.10

4695 4268 3948

1137 1033 956

485 441 408

266 242 224

165 150 139

67.2 61.0 56.4

34.6 31.4 29.1

20.2 18.3 17.0

12.7 11.5 10.7

5.77 5.24 4.86

2.90 2.64 2.45

1.50 1.37 1.29

0.78 0.72 0.68

0.39 0.37 0.36

0.16 0.16 0.16

0.02 0.02 0.03

2.0

0.02

3783

916

391

215

133

54.1

27.9

16.3

10.2

4.68

2.38

1.26

0.68

0.36

0.17

0.04

3.0 4.0

0 0

3783 3833

916 929

391 397

215 218

133 135

54.3 55.2

28.0 28.6

16.4 16.7

10.3 10.6

4.75 4.82

2.43 2.51

1.30 1.35

0.71 0.75

0.39 0.42

0.20 0.22

0.06 0.08

5.0 6.0

0 0

3883 3933

941 954

402 408

221 224

137 139

56.0 56.9

29.0 29.6

17.0 17.4

10.8 11.0

5.00 5.12

2.59 2.67

1.41 1.46

0.79 0.83

0.45 0.48

0.24 0.27

0.10 0.12

7.0

0

3983

966

413

227

141

57.8

30.0

17.7

11.2

5.25

2.75

1.52

0.87

0.51

0.29

0.18

8.0 9.0

0 0

4033 4083

979 991

419 424

231 234

143 145

58.7 59.6

30.6 31.0

18.0 18.3

11.4 11.6

5.38 6.50

2.83 2.91

1.57 1.63

0.91 0.95

0.54 0.58

0.32 0.34

0.16 0.18

10.0

0

4133

1004

430

237

147

60.5

31.6

18.6

11.9

5.62

3.00

1.68

0.99

0.61

0.37

0.20

261

Flow through Orifices with Change in Velocity and Flow Area Orifice with beveled edges facing the flow (α = 40–60o) in a straight tube; Re = w0Dh/v > 104 13,14 ζ1 *

Diagram 4.16

F0 F0 ⎞ ⎡ ⎛ = ⎢1 – +√ ⎯⎯ζ′ ⎜1 – ⎟ 2 F F ρw1 ⁄ 2 ⎣ 1⎠ 1 ⎝ ∆p

0.375 2

2

⎤ ⎛ F1 ⎞ ⎥ ⎜ F ⎟ , see the graph; ⎦ ⎝ 0⎠

ζ′ = f(l/Dh), see the table below or graph a of Diagram 4.13, or _

Dh =

_

ζ′ = 0.13 + 0.34 × 10–(3.4l + 88.4l _ l = l ⁄ Dh .

4F0 Π0

2.3

)

,

Values of ζ1 _ l = l Dh

%

F0%F1 ζ′

0.02

0.04

0.06

0.08

0.10

0.15

0.20

0.25

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.0

0.01

0.46

6840

1656

708

388

271

98.2

50.7

29.5

18.5

8.39

4.18

2.13

1.08

0.51

0.18

0

0.02

0.42

6592

1698

682

374

232

94.5

48.7

28.4

17.8

8.05

4.00

2.03

1.02

0.48

0.17

0

0.03

0.38

6335

1535

655

360

223

90.6

46.7

27.2

17.0

7.69

3.80

1.93

0.97

0.45

0.16

0

0.04

0.35

6140

1488

635

348

216

87.7

45.2

26.2

16.4

7.40

3.66

1.84

0.92

0.43

0.15

0

0.06

0.29

5737

1387

592

325

201

81.5

41.9

24.4

15.2

6.83

3.35

1.68

0.83

0.38

0.13

0

0.08

0.23

5297

1281

546

300

185

75.0

38.5

22.3

13.9

6.20

3.02

1.51

0.74

0.33

0.11

0

0.12

0.16

4748

1147

488

267

165

66.7

34.1

19.7

12.2

5.40

2.61

1.29

0.62

0.27

0.09

0

0.16

0.13

4477

1.81

460

251

155

62.7

32.0

18.4

11.4

5.02

2.42

1.18

0.56

0.24

0.08

0

262

Handbook of Hydraulic Resistance, 4th Edition

Orifice with rounded inlet edges in a straight tube; Re = w0Dh/v > 104 13,14 ζ1 *

Diagram 4.17

F0 ⎡ = ⎢1 – +√ ⎯⎯ζ′ F1 ρw21 ⁄ 2 ⎣ ∆p

0.75 2

F0 ⎞ ⎛ ⎜1 – F ⎟ 1⎠ ⎝

2

⎤ ⎛ F1 ⎞ ⎥ ⎜ F ⎟ , see the graph. ⎦ ⎝ 0⎠

r ζ′ = f1 ⎛⎜ ⎞⎟ , see the table below, graph b of Diagram 4.13 or Dh ⎝ ⎠ _ _ ζ′ = 0.03 + 0.47 × 10–7.7r ; r = r ⁄ Dh .

Dh =

4F0 Π0

Values of ζ1 F0%F1 r%Dh

ζ′

0.02

0.04

0.06 0.08 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90

1.0

0.01 0.44 6717

1628

595 382 236 96.4 49.7 29.0 18.2 12.0 8.24 5.75 4.10 2.91 2.08 1.49 1.05 0.73 0.49 0.18

0

0.02 0.37 6273

1520

648 356 221 89.7 46.2 26.9 16.8 11.1 7.59 5.29 3.75 2.65 1.90 1.35 0.95 0.66 0.44 0.15

0

0.03 0.31 5875

1421

607 332 206 83.6 43.0 25.0 15.6 10.3 7.01 4.87 3.45 2.43 1.74 1.23 0.86 0.59 0.40 0.14

0

0.04 0.26 5520

1336

570 312 193 78.3 40.2 23.4 14.6 9.54 6.51 4.51 3.19 2.24 1.60 1.13 0.79 0.54 0.36 0.12

0

0.06 0.19 4982

1206

513 281 174 70.3 36.0 20.8 12.9 8.46 5.76 3.97 2.79 1.96 1.38 0.97 0.67 0.46 0.30 0.10

0

0.08 0.15 4657

1125

479 262 162 65.3 33.4 19.3 12.0 7.80 5.29 3.63 2.55 1.78 1.25 0.88 0.60 0.41 0.26 0.08

0

0.12 0.09 4085

986

420 229 141 56.8 29.0 16.6 10.2 6.65 4.48 3.06 2.14 1.48 1.03 0.71 0.48 0.33 0.21 0.06

0

0.16 0.06 3745

902

384 210 129 51.8 26.3 15.0 9.26 5.99 4.02 2.73 1.90 1.31 0.91 0.62 0.42 0.28 0.17 0.05

0

263

Flow through Orifices with Change in Velocity and Flow Area Orifice with various edges in a wall with infinite surface area13,14 Orifice edges

Diagram 4.18

Configuration Dh =

Resistance coefficient 4F0 Π0

Re =

Sharp (l/Dh = 0 → 0.015)

Thick-walled (deep orifice) (l/Dh > 0.015)

ζ = 2.7–2.8

ζ = ζ0 + λ(l ⁄ Dh) , Re ≥ 104 _ where ζ0 = f(l) _or ζ =_ 1.5 + (2.4 – l) ×_10–ϕ(l) + λl_ ⁄ Dh , ϕ(l) = 0.25 + 0.535l 8 ⁄ (0.05 + l 7) , for λ, see Chapter 2.

_ l * l%Dh

Beveled facing flow direction

w0Dh ≥ 104 , v

0

0.2

0.4

0.6

0.8

1.0

ζ0 _ l * l%Dh

2.85

2.72

2.60

2.34

1.95

1.76

1.2

1.4

1.6

1.8

2.0

4.0

ζ0

1.67

1.62

1.60

1.58

1.55

1.55

Re ≥ 104 _ ζ0 = f(l), see graph b or ζ = (1 + √ ⎯⎯ζ′ )2 , where

_

_

ζ′ = 0.13 + 0.34 × 10–(3.4l + 88.4l

_ l * l%Dh

2.3

)

.

0

0.01

0.02

0.03

0.04

0.05

ζ _ l * l%Dh

2.85

2.80

2.70

2.60

2.50

2.41

0.06

0.08

0.10

0.12

0.16

0.20

ζ

2.33

2.18

2.08

1.98

1.84

1.80

264

Handbook of Hydraulic Resistance, 4th Edition

Orifice with various edges in a wall with infinite surface area13,14 Orifice edges

Diagram 4.18

Configuration

Resistance coefficient Re ≥ 104 _ ζ0 = f(r), see graph c or

Rounded facing flow direction

ζ = (1 + √ ⎯⎯ζ′ )2, where

_

ζ′ = 0.03 + 0.47 × 10–7.7r .

_ r = r%Dh 0.01 ζ

2.72

0.02

0.03

0.04

0.06

0.08

0.12

0.16

0.20

2.56

2.40

2.27

2.06

1.88

1.60

1.38

1.37

Orifice with any edges for different conditions of flow in the transient and laminar regions (Re = w0Dh/v < 104–105)16

Diagram 4.19

1. At 30 < Re < 104–105 : ζ * and correspondingly ζ1 *

2. 10 < Re < 30 : ζ * and ζ1 *

∆p

∆p ρw20 ⁄ 2

_ = ζφ + ε0Reζ0quad ,

⎛ F1 ⎞ _ = ζφ ⎜ ⎟ + ε0Reζ1quad . 2 ρw0 ⁄ 2 ⎝ F0 ⎠ 2

∆p

⎛ F1⎞ 33 _ = ζφ ⎜ ⎟ = + ε0Reζ0quad F Re 0 ⎝ ⎠ 2

∆p ρw20 ⁄ 2

33 ⎛ F1 ⎞ _ ζ + ε0Reζ1quad , Re φ ⎜⎝ F0 ⎟⎠ 2

=

ρw21 ⁄ 2 _ where for ε0Re, see below.

3. At Re < 10 : ζ *

∆p ρw20 ⁄ 2

2

=

∆p 33 33 ⎛F1⎞ and ζ1 * 2 = ⎜F ⎟ , Re Re ρw1 ⁄ 2 ⎝ 0⎠

ζ0quad * ∆p ⁄ (ρw20 ⁄ 2), ζ1quad * ∆p ⁄ (ρw20 ⁄ 2) are determined as ζ at Re > 104 from the corresponding Diagrams 4.11 to 4.18. ζϕ is determined from Table 1 or from the formula ζϕ =%[18.78 – 7.768F1 ⁄ F0 + 6.337(F1 ⁄ F0)2] exp [–0.942 – 7.246F0 ⁄ F1 – 3.878(F0 ⁄ F1)2] log Re ; ⎧ ⎨ ⎩

⎫ ⎬ ⎭

265

Flow through Orifices with Change in Velocity and Flow Area Orifice with any edges for different conditions of flow in the transient and laminar regions (Re = w0Dh/v < 104–105)16

Diagram 4.19

5 _ _ ε0Re is determined from Table 2 or from the formula ε0Re = ∑ ai(log Re)i, where

i=0

a0 = 0.461465; a1 = –0.2648592; a4 = 0.01325519;

a2 = 0.2030479;

a3 = –0.06602521;

a5 = –0.001058041

Values of ζϕ Re

F0 F1

30

40

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95

1.94 1.78 1.57 1.35 1.10 0.85 0.58 0.40 0.20 0.03

1.38 1.36 1.16 0.99 0.75 0.56 0.37 0.24 0.13 0.03

60

2

10

2 × 10

4 × 10

10

1.14 1.05 0.88 0.79 0.55 0.30 0.23 0.13 0.08 0.02

0.89 0.85 0.75 0.57 0.34 0.19 0.11 0.06 0.03 0

0.69 0.67 0.57 0.40 0.19 0.10 0.06 0.03 0.01 0

0.64 0.57 0.43 0.28 0.12 0.06 0.03 0.02 0 0

0.39 0.36 0.30 0.19 0.07 0.03 0.02 0.01 0 0

2

2

3

2 × 103 4 × 103 104 2 × 104 0.30 0.26 0.22 0.14 0.05 0.02 0.01 0 0 0

0.22 0.20 0.17 0.10 0.03 0.01 0 0 0 0

0.15 0.13 0.10 0.06 0.02 0.01 0 0 0 0

0.11 0.09 0.07 0.04 0.01 0 0 0 0 0

105

2 × 105

106

0.04 0.03 0.02 0.02 0.01 0 0 0 0 0

0.01 0.01 0.01 0.01 0.01 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

_ 2. Values of ε0Re Re _

ε 0Re

10 0.34 20 0.35 30 0.36 40 0.37 60 0.40 80 0.43 102 0.45 2 × 102 0.52 4 × 102 0.58 6 × 102 0.62 103 0.65

Re _

ε 0Re 2 × 103 0.69 4 × 103 0.74 6 × 103 0.76 104 0.80 2 × 104 0.82 4 × 104 0.85 6 × 104 0.87 105 0.90 2 × 105 0.95 3 × 105 0.98 4 × 105 1.0

266

Handbook of Hydraulic Resistance, 4th Edition

Perforated plates or orifice in tubes at large subsonic velocities (high Mach numbers)34

Diagram 4.20

Orifices with sharp edges: ζM *

∆p ρw20 ⁄ 2

= kMζ ,

where for ζ, see Diagrams 4.11 and 4.12; kM = f(Ma1) ; Ma1 = a1 =

⎯⎯ √ k

p1

ρ1

w1 ; a

is the velocity of sound; for k, see Table 1.4.

For beveled or rounded edges of orifices, see Diagram 8.7.

Values of kM _ f 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ma1 0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.09 1.03 0.00 1.00 0.00 1.00 1.00 1.00

1.30 1.13 1.03 1.00 1.00 1.00 1.00 1.00

– 1.51 1.14 1.03 1.00 1.00 1.00 1.00

– – 1.41 1.10 1.12 1.03 1.01 1.00

– – – 1.27 1.30 1.08 1.03 1.00

– – – 1.85 1.77 1.18 1.07 1.02

– – – – – 1.35 1.12 1.04

– – – – – 1.68 1.20 1.07

– – – – – – 1.37 1.13

– – – – – – 1.63 1.21

– – – – – – 2.01 1.33

– – – – – – – 1.50

– – – – – – – 1.75

Bellmouth nozzle installed in a wall of infinite surface area, Re = w0Dh/v > 104 27

ζ*

∆p = ζ′ + ζfr, ρw20 ⁄ 2

where ζfr = λ

Dh =

4F0 Π0

Diagram 4.21

l l r ⎞ ; ζ′ = f ⎛⎜ , ; for λ, see Chapter 2. Dh D Dh ⎟ ⎠ ⎝ h

267

Flow through Orifices with Change in Velocity and Flow Area Bellmouth nozzle installed in a wall of infinite surface area, Re = w0Dh/v > 104 27

Diagram 4.21

Values of ζ ′ l Dh

r Dh 0.02 0.04 0.06 0.08 0.10 0.12 0.20 0.50

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.0

2.5

3.0

3.5

4.0

2.64 2.20 1.90 1.44 1.12 1.08 1.04 1.03

2.25 1.70 1.30 1.19 1.10 1.08 1.04 1.03

1.89 1.42 1.23 1.16 1.10 1.07 1.04 1.03

1.68 1.37 1.22 1.15 1.10 1.07 1.04 1.03

1.60 1.34 1.22 1.15 1.10 1.07 1.04 1.03

1.56 1.33 1.21 1.15 1.10 1.07 1.04 1.03

1.54 1.33 1.21 1.15 1.10 1.07 1.04 1.03

1.53 1.32 1.21 1.15 1.10 1.07 1.04 1.03

1.51 1.32 1.21 1.15 1.10 1.07 1.04 1.03

1.50 1.32 1.21 1.15 1.10 1.08 1.05 1.03

1.49 1.31 1.20 1.15 1.10 1.08 1.05 1.03

1.48 1.30 1.20 1.15 1.10 1.08 1.05 1.03

Orifices in a thin wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 28

Diagram 4.22

Without baffles (schemes 1 and 2):

ζ*

∆p , see graphs a–c. With baffles ρw20 ⁄ 2

at a circular orifice (scheme 3): ∆p ζ * 2 , see graphs b and d. ρw0 ⁄ 2 A. Suction orifices (intake; w0, dashed arrows)

268

Handbook of Hydraulic Resistance, 4th Edition

Orifices in a thin wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 28

Diagram 4.22

Values of ζ for scheme 1 (graph b) w∞%w0

l a

0

0.5

1.0

2.0

3.0

4.0

5.0

6.0

0.17 0.5 1.0 2.0 6.0

2.70–2.80 2.70–2.80 2.70–2.80 2.70–2.80 2.70–2.80

2.75–2.85 2.65–2.75 2.65–2.75 2.65–2.75 2.55–2.65

2.95 2.85 2.85 2.85 2.65–2.75

4.00 3.35 3.35 3.20 3.15

5.20 4.15 4.15 3.80 3.55

6.65 5.00 5.00 4.50 4.15

8.05 6.00 6.00 5.20 4.75

9.50 7.00 7.00 5.95 5.45

Values of ζ for scheme 2 (graph a) w∞%w0

Arrangement of orifices

0

0.5

1.0

2.0

3.0

4.0

5.0

6.0

No. 1 No. 2

2.70–2.80 2.70–2.80

2.70–2.80 2.55–2.65

2.80–2.90 2.60–2.70

3.50 3.40

4.10 4.05

4.95 4.95

5.75 5.75

6.70 6.70

Values of ζ for scheme 3 (graph b) w∞ w0

Curve a b c

0

0.5

1.0

2.0

3.0

4.0

5.0

6.0

4.95 2.73–2.85 2.16–2.20

5.75 4.00 2.60–2.70

6.60 5.00 3.20

8.45 6.50 4.20

10.0 7.80 5.20

– 8.95 6.20

– 10.0 7.20

– – 8.20

269

Flow through Orifices with Change in Velocity and Flow Area Orifices in a thin wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 28

Diagram 4.22

B. Discharge orifices (exit; w0, solid arrows)

Values of ζ for scheme 1 (graph c) w∞ w0

l a 0

0.5

1.0

1.5

2.0

3.0

4.0

5.0

0.17

2.70–2.80

2.50–2.60

2.45–2.55

2.55–2.65

3.05

0.5

2.70–2.80

2.40–2.50

2.25–2.35

2.45–2.55 2.80–2.90

1.0

2.70–2.80

2.25–2.35

2.20–2.30

2.0

2.70–2.80

2.25–2.35

2.05–2.15

6.0

2.70–2.80

2.25–2.35

2.00–2.10

1.90–2.00 1.90–2.00 2.25–2.35

6.0

4.75

7.0

9.00

–

4.10

5.70

7.30

9.00

2.25–2.35 2.60–2.70

3.65

5.00

6.50

8.00

2.05–2.10 2.40–2.50

3.35

4.50

5.80

7.25

2.75–2.85

3.30

3.90

Values of ζ for scheme 2 (graph c) Arrangement of orifices

w∞ w0 0

0.5

1.0

1.5

2.0

3.0

4.0

5.0

6.0

3.50

4.95

6.45

7.90

3.00

3.60

4.20

No. 1

2.70–2.80 2.25–2.35

2.00–2.10

2.05–2.15 2.50–2.60

No. 2

2.70–2.80 2.40–2.50

2.10–2.20

2.05–2.15 2.10–2.20 2.50–2.60

270

Handbook of Hydraulic Resistance, 4th Edition

Orifices in a thin wall in the presence of a passing flow (w∞ > 0); Re = w0Dh/v ≥ 104 28

Diagram 4.22

Values of ζ for scheme 3 (graph d) w∞ w0

Curve 1 2 3

0

0.5

1.0

1.5

2.0

3.0

4.0

5.0

6.0

4.75 3.00 2.16–2.20

4.40 3.00 2.05–2.10

4.05 3.00 2.10–2.20

3.85 3.00 2.35–2.45

3.85 3.15 2.65–2.75

4.40 4.00 3.50

5.35 5.65 4.75

6.55 6.45 6.20

7.75 7.40 7.55

Movable flaps5

Diagram 4.23

Exhaust, single top-hinged flap

Values of ζ (graph a)

lfl is the flap length; Q w0 = ; F0

ζ*

∆p ρw20 ⁄ 2

.

α, deg

lfl bfl

15

20

25

30

45

60

90

1.0 2.0

11 17

6.3 12

4.5 8.5

4.0 6.9

3.0 4.0

2.5 3.1

2.0 2.5

∞

30

16

11

8.6

4.7

3.3

2.5

271

Flow through Orifices with Change in Velocity and Flow Area Movable flaps5

Diagram 4.23

Values of ζ (graph b) α, deg

lfl bfl

15

20

25

1.0 2.0

16 21

11 13

8.0 5.7 3.7 3.1 2.6 9.3 6.9 4.0 3.2 2.6

30

45

60

90

Single center-hinged flap

Values of ζ (graph c) α, deg

lfl bfl

15

1.0

46

26

16

11

5.0 3.0 2.0

∞

59

35

21

14

5.0 3.0 2.4

20

25

30

45

60

90

Double, top-hinged flaps

Values of ζ (graph d) α, deg

lfl bfl

15

20

1.0 2.0

14 31

9.0 6.0 4.9 3.8 3.0 2.4 21 14 9.8 5.2 3.5 2.4

25

30

45

60

90

272

Handbook of Hydraulic Resistance, 4th Edition

Movable flaps5

Diagram 4.23

Double flaps (one top- and the other bottom-hinged)

Values of ζ (graph e)

Grating with _adjustable louvers in a wall infinite surface area (f ≈ 0.8 for completely open louvers)

α, deg

lfl bfl

15

20

25

30

45

60

90

1.0 2.0

19 44

13 24

8.5 15

6.3 11

3.8 6.0

3.0 4.0

2.4 2.8

∞

59

36

24

17

8.6

5.7

2.8

Diagram 4.24

∆p + 1.6, where w1 is the mean velocity ρw21 ⁄ 2 over the total area of the grating in the wall. ζ*

_ F 0 f = ; F0 is the open flow area of the grating. F1

273

Flow through Orifices with Change in Velocity and Flow Area Working section (open) of a wind tunnel1

Diagram 4.25 For a rectangular section Dh =

2a0b0 a0 + b0

,

for an elliptical section Dh =

4a0b0 ⎯⎯⎯⎯ a0b0 1.5(a0 + b0) – √

,

where a0 and b0 are the sides of the rectangle or semiaxes of the ellipse. Circular (or rectangular) cross sections:

ζ*

∆p

ρw20 ⁄ 2

2

= 0.1

lw.s. ⎛ lw.s. ⎞ ⎛ lw.s. ⎞ – 0.008 ⎜ ⎟ , see curve ζ = f ⎜ D ⎟ . Dh D h ⎠ ⎝ ⎝ h⎠

lw.s. Dh

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

ζ

0

0.04

0.08

0.12

0.15

0.18

0.21

0.23

0.25

0.27

0.29

Elliptical cross section:

ζ*

∆p ρw20 ⁄ 2

= 0.145

lw.s. lw.s. – 0.0017 . Dh a0b0

274

Handbook of Hydraulic Resistance, 4th Edition

REFERENCES 1. Abramovich, G. N., Theory of Turbulent Jets, Fizmatgiz Press, Moscow, 1960, 715 p. 2. Altshul, A. D., Arzumanov, E. S., and Veziryan, R. E., Experimental study of the dependence of the resistance coefficient in the course of sudden expansion of flow on the Reynolds number, Neft. Khoz., no. 4, 64–70, 1967. 3. Altshul, A. D., Hydraulic Resistance, Nedra Press, Moscow, 1982, 224 p. 4. Balanin, V. V. and Vasilevskiy, V. P., Determination of the flow resistance coefficient by the method of the theory of turbulent jets, Tr. Leningr. Inst. Vodn. Transp., vyp. 158, pp. 10–16, 1977; vyp. 162, pp. 5–10, 1978. 5. Bromblei, M. F., Discharge coefficients of orifices covered by flaps, in Current Problems of Ventilation, pp. 40–65, Stroiizdat Press, Moscow, 1941. 6. Veziryan, R. E., Investigation of the mutual effect of locking and regulating devices on the hydraulic resistance, Tr. NIIAvtomat., vyp. 2, pp. 25–29, 1974. 7. Volkova, L. P. and Yudelovich, M. Ya., Shock losses in stepwise tubes at supersonic pressure ratios, Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, no. 4, 68–72, 1958. 8. Garkusha, A. V. and Kucherenko, S. I., Specific features of flow through an annular channel with a step at n = 1.7, Energ. Mashinostr., vyp. 31, 13–18, 1981. 9. Glotov, G. F. and Moroz, E. K., Investigation of the flow of gas in a cylindrical channel with an abrupt expansion of the sonic flow, Uch. Zap. TsAGI, vol. 1, no. 2, 53–59, 1970. 10. Dudintsev, L. M., Discharge coefficient of an orifice in the wall with a parallel ditected flow, Izv. VUZ, Stroit. Arkhit., no. 5, 97–103, 1969. 11. Zhukovsky, I. E., Variation of the Kirchhoff method to determine liquid flow in two dimensions at constant velocity prescribed at the unknown streamline, in Collected Works, vol. 2, pp. 130– 240, Gosizdat Press, Moscow, 1949. 12. Idelchik, I. E., Hydraulic resistances during entry of flow into channels and in passage through orifices, in Prom. Aerodin., no. 2, pp. 27–57, Oborongiz Press, Moscow, 1944. 13. Idelchik, I. E., Hydraulic Resistances (Physical and Mechanical Fundamentals), Gosenergoizdat Press, Moscow, 1954, 316 p. 14. Idelchik, I. E., Determination of the resistance coefficients during discharge through orifices, Gidrotekh. Stroit., no. 5, 31–36, 1953. 15. Idelchik, I. E., Shock losses in a flow with a nonuniform velocity distribution, Tr. TsAGI, vyp. 662, 2–24, MAP, 1948. 16. Idelchik, I. E., Account for the viscosity effect on the hydraulic resistance of diaphragms and grids, Teploenergetika, no. 9, 75–80, 1960. 17. Karev, V. N., Pressure head losses with an abrupt expansion of the pipeline, Neft. Khoz., no. 11/12, 13–16, 1952. 18. Karev, V. N., Pressure head losses with an abrupt contraction of the pipeline, and the effect of local resistances on flow disturbances, Neft. Khoz., no. 8, 3–7, 1953. 19. Levin, A. M. and Malaya, E. M., Investigation of the hydrodynamics of flow with an abrupt expansion, Tr. Gos. Proektno-Issled. Inst. Vostokgiprogaz, vyp. 1, pp. 41–47, 1969. 20. Levkoyeva, N. V., Investigation of the Effect of Fluid Viscosity of Local Resistances, Thesis (Cand. of Tech. Sci.), Moscow, 1959, 186 p. 21. Migai, V. K. and Nosova, I. S., Reduction of eddy losses in channels, Teploenergetika, no. 7, 49–51, 1977. 22. Morozov, D. I., The optimum degree of a sudden enlargement of the channel, Tr. Khark. Univ., Gidromekhanika, vyp. 4, 53–55, 1966. 23. Panchurin, N. A., Extension of the Borda–Carnot theorem on the pressure head loss during an abrupt expansion to the case of unsteady-state flow, Tr. Leningr. Inst. Vodn. Transp., vyp. 51, 34–39, 1964.

Flow through Orifices with Change in Velocity and Flow Area

275

24. Fedotkin, I. M., Hydraulic resistance of throttling diaphragms to a two-phase flow, Izv. VUZ, Energetika, no. 4, 37–43, 1969. 25. Frenkel, N. A., Hydraulics, Gosenergoizdat Press, Moscow, 1956, 456 p. 26. Khanzhonkov, V. I., Aerodynamic characteristics of collectors, in Prom. Aerodin., no. 4, pp. 45– 64, Oborongiz Press, Moscow, 1953. 27. Khanzhonkov, V. I., Reduction of the aerodynamic resistance of orifices by means of annular fins and recesses, in Prom. Aerodin., no. 12, pp. 181–196, Oborongiz Press, Moscow, 1959. 28. Khanzhonkov, V. I., Resistance to discharge through an orifice in the wall in the presence of passing stream, in Prom. Aerodin., no. 15, pp. 5–19, Oborongiz Press, Moscow, 1959. 29. Chjen, P., Separation Flows, pt. 1, 300 p.; pt. 2, 280 p., Mir Press, Moscow, 1972. 30. Shvets, I. T., Repukhov, V. M., and Bogachuk-Kozachuk, K. A., Full pressure losses during air injection into a stalling air flow through orifices in the wall, Teploenergetika, 1976. 31. Alvi, Sh. H., Contraction coefficient of pipe orifices, Flow Meas., Proc. FLUMEX 83 I MeCO Conf., Budapest, nos. 20–22, pp. 213–218, 1983. 32. Astarita, G. and Greco, G., Excess pressure drop in laminar flow through sudden contraction, Ind. Eng. Chem. Fundam., vol. 7, no. 1, 27–31, 1968. 33. Ball, J. W., Sudden enlargements in pipelines, J. Power Div., Proc. Am. Soc. Civil Eng., vol. 88, no. 4, 15–27, 1962. 34. Cornell, W. G., Losses in flow normal to plane screens, Trans. ASME, no. 4, 145–153, 1958. 35. Dewey, P. E. and Vick, A. R., An Investigation of the Discharge and Drag Characteristics of Auxiliary Air Outlets Discharging into a Transonic Stream, NACA Tech. Note, no. 3466, 1955, 38 p. 36. Dickerson, P. and Rice, W., An investigation of very small diameter laminar flow orifices, Trans. ASME, vol. D91, no. 3, 546–548, 1969. 37. Forst, T. H., The compressible discharge coefficient of a Borda pipe and other nozzles, J. R. Aeronaut. Soc., no. 641, 346–349, 1964. 38. Geiger, G. E. and Rohrer, W. M., Sudden contraction losses in two-phase flow, Trans. ASME, Ser. C, vol. 88, no. 1, 1–9, 1966. 39. Hebrard, P. and Sananes, F., Calcul de l’ecoulement turbulent decolle en aval de l’elargissement brusque dans une veine de section circulaire, C.R. Acad.. Sci., vol. 268, no. 26, A1638–A1641, 1969. 40. Iversen, H. W., Orifice coefficients for Reynolds numbers from 4 to 50,000, Trans. ASME, vol. 78, no. 2, 125–133, 1956. 41. Johansen, F., Flow through pipe orifices of low Reynolds numbers, Proc. R. Soc. London, Ser. A., vol. 126, no. 801, 125–131, 1930. 42. Kolodzie, P. A. and Van Winkle, M., Discharge coefficients through perforated plates, AIChE J., vol. 3, no. 9, 23–30, 1959. 43. Reichert, V., Theoretische-experimentelle Untersuchungen zur Widerstanscharakteristik von Hy.. draulikventilen, Wassenschaffliche Zeitschrift der Technischen Universitat, Dresden, Bd. 3, Heft 2, 149–155, 1982. 44. Ringleb, T., Two-dimensional flow with standing vortexes in diffusers, Trans. ASME, Ser. D, no. 4, 130–135, 1960. 45. Pearson, H. and Heutteux, B. M., Losses at sudden expansions and contractions in ducts, Aeronaut. Q., Bd. 14, no. 1, 63–74, 1963. 46. Migai, V. I. and Gudkov, E. I., Design and Calculation of the Exit Diffusers of Turbomachines, Mashinostroenie Press, Leningrad, 1981. 47. Frankfurt, M. O., Experimental investigation of jet diffusers, Uch. Zap. TsAGI, vol. 13, no. 2, 78– 86, 1982.

CHAPTER

FIVE RESISTANCE TO FLOW WITH A SMOOTH CHANGE IN VELOCITY RESISTANCE COEFFICIENTS OF DIFFUSERS AND CONVERGING AND OTHER TRANSITION SECTIONS

5.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS; DIFFUSERS IN A NETWORK 1. A smoothly expanding tubular section, a diffuser, is used in order to make the transition from a tube (channel) of smaller cross section to a larger one (to convert the kinetic energy of flow into the potential energy or of velocity pressure into static pressure) with minimum total pressure losses (Figure 5.1).* Due to the fact that an increase in the cross-sectional area of the diffuser causes a drop in the average flow velocity with an increase in the divergence angle α, the total resistance coefficient of the diffuser, expressed in terms of the velocity in the smaller (initial) section, becomes smaller up to certain limits of α, than for the equivalent segment of a tube of constant cross section, the area of which is equal to the initial area of the diffuser. Starting from this limiting divergence angle of the diffuser, a further increase in this angle considerably increases the resistance coefficient, so that it becomes much larger than that for a straight tube of equivalent length. 2. The increase of the resistance coefficient of a diffuser of a given length with further increases in the divergence angle is caused by enhanced turbulence of the flow, separation of the boundary layer from the diffuser wall, and resultant violent vortex formation.

∗

The main geometric characteristics of diffusers with straight walls are the divergence angle α, the area ratio nar1 = F1/F0, and the relative length ld. These quantities are connected by the following relanar1 ⎯⎯⎯ ⎯ – 1)/(2 tan α/2), for a plane diffuser ld/a0 = (nar1 – 1)/(2 tionships: for a conical diffuser l0/D0 = (√ tan α/2).

277

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Figure 5.1. FIow patterns in diffusers with different divergence angles at nar1 = F1/F0 = 3.3.48

The boundary layer separates from the walls (see Figure 5.1) due to the adverse pressure gradient along the diffuser walls, resulting from the velocity drop as the cross-sectional area increases (according to the Bernoulli equation). 3. Under constant flow conditions at the entrance and for constant relative length ld or at the area ratio nar1 = F1/F0, an increase in the divergence angle α, starting from α = 0o, will result in a successive achievement of the four main flow regimes: • Stable regime, nonseparating flow ("separation-free" diffusers); • Regime with a large nondeveloped flow separation, where the size and intensity of separation change with time (regime of strongly oscillating flows, diffusers with local flow separation); • Regime of fully developed flow separation, where the major portion of the diffuser is occupied by an extensive zone of reverse circulation (diffusers with substantial flow separation); • Regime of jet flow, where the main flow is separated from the diffuser walls over the whole perimeter (diffusers with complete flow separation). 4. The inception of flow separation in a diffuser is a function of both its geometric parameters and the flow regime at its inlet (Reynolds numbers Re = w0Dh/v and Mach numbers Ma0 = w0/a1), as well as of the condition of the flow at the inlet (displacement thickness δ*

279

Flow with a Smooth Change in Velocity

of the boundary layer or the "momentum loss" thickness δ**, the level of turbulence, etc.).* Experiments carried out by Idelchik and Ginzburg54 show that for a conical diffuser (α = 4o) installed both immediately behind the smooth inlet (collector) without an insert (l0/D0 = 0 and the boundary-layer displacement thickness at the inlet δ∗0 ≈ 0) and far behind it with a straight insert (l0/D0 ≠ 0 and δ∗0 ≠ 0), there is no flow separation along the entire length of the diffuser even when the length corresponds to the section with area ratio nx = Fx/F0 = 16 (Figure 5.2). "Blurring" of the potential core (the core of constant velocities), the presence of which determines the "starting length" of the diffuser, that is, the section with a nonstabilized flow and a corresponding "extension" of the whole velocity profi1e at l0/D0 = 0, terminates at about nx = 6–8. Downstream of this section, that is, over the length of stabilized flow (where the boundary layer fills the whole section), a noticeable equalization of the elongated velocity profile is observed. 5. If there is a straight insert (l0/D0 ≠ 0) the starting length of the diffuser (with the core of constant velocities) becomes shorter. For example, at l0/D0 = 20 and α = 4o, the core is retained only up to nx = 4 (see Figure 5.2). As a result, the velocity profiles in the first sections of the starting length are much more extended than at l0/D0 = 0. In subsequent sections downstream of the starting length (nx ≥ 6) the velocity profiles at l0/D0 ≠ 0 become more equalized than at l0/D0 = 0 and this can be attributed to intensification of flow turbulence. 6. According to the experiments mentioned above, with an increase of α up to 10–14o, the value of nx at which the core of constant velocities is still preserved increases (since the length of the diffuser at the same nx is reduced). At the same time, at the divergence angles cited and at certain ratios l0/D0 the flow starts to separate despite the presence of the core of constant velocities (Figures 5.3 to 5.5). 7. For practical purposes the regions without separation both in spatial and plane diffusers can, with limited accuracy, be determined with the help of Figure 5.6. Curves 1 and 2 in Figure 5.6 are the result of generalization of numerous experimental data.36,54,129 The curves separate the whole region of α = f(nx) into two parts: separation-free diffusers (region I) and separation-prone diffusers (region II). Curve 1 is related to more favorable inlet conditions (l0/D0 ≈ 0, δ∗0 ≈ 0). Curve 2 is related to the case where the diffuser is installed downstream of the long inlet section at which δ∗0 ⁄ D0 >> 0. 8. As a rule, flow starts to separate from the walls of diffusers with divergence angles up to about α = 40o, not over the whole perimeter of the section, but in the region where, because of asymmetry of the diffuser, asymmetric velocity profile at the entrance, and so on, the flow velocity in the wall layer is lower than in other regions of the section. As soon as the flow has separated from one side of the diffuser, the static pressure along the diffuser does not increase further or even becomes weaker, with the result that the flow does not separate from the opposite side of the diffuser. This phenomenon is responsible for asymmetric velocity distributions over the sections of diffusers (see Figures 5.1 and 5.5).

δ

∗

w δ ∗ = ∫ ⎛⎜1 − ⎞⎟ dy , w c⎠ 0⎝

δ

δ ∗∗ =

∫ wc ⎛⎜1 − wc⎞⎟ dy w

0

w

⎝

⎠

,

where wc is the velocity in the potential core along the flow axis, δ is the boundary-layer thickness in the plane diffuser walls.

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Figure 5.2. Velocity fields over the diameter II–II of different sections (different nx) of a conical diffuser at α = 4o and Re = (4–5) × 105:54 a) l0/D0 = 0; b) l0/D0 = 20.

Figure 5.3. Velocity fields over the diameter I–I of different sections (different nx) of a conica1 diffuser at α = 8o and Re = (4–5) × 105:54 a) l0/D0 = 0; b) l0/D0 = 10.

Flow with a Smooth Change in Velocity

281

Figure 5.4. Velocity fields in a conical diffuser at α = 10o over the section nx = 4 at Re = (4–5) × 105 and different l0/D0:54 a) diameter I–I; b) diameter II–II.

Figure 5.5. Velocity fields in a conical diffuser at α = 10o over the section nx = 4 at Re = (4–5) × 105 and different l0/D0:54 a) diameter I–I; b) diameter II–II.

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Figure 5.6. Regions of flow separation in diffusers: 1) l0/D0 ≈ 0; 2) l0/D0 ≥ 0.54,129

9. In a symmetrical diffuser with a symmetrical velocity profile at the entrance, the separation of flow from the wall occurs alternately on one side of the diffuser and on the other (Figure 5.7), which leads to substantial oscillations of the whole flow. 10. The profiles of reduced velocities λci = wi/acr at the exit from plane diffusers with divergence angles α equal to 4, 6, and 8o and with l0/D0 = 5.8 are given in Figure 5.8 for

Figure 5.7. Velocity fields in a conical diffuser at α = 20o over the section nx = 4 at Re = (4–5) × 105 and different l0/D0:54 a) diameter I–I; b) diameter II–II; 1) l0/D0 = 0; 2) l0/D0 = 5; 3) l0/D0 = 10; 4) l0/D0 = 20.

Flow with a Smooth Change in Velocity

283

Figure 5.8. Fields of reduced velocity in the exit sections of plane diffusers at β = 0 and α = 4, 6, and 8o; λc is the reduced velocity at the inlet section of the diffuser.

both sonic and supersonic flows over the starting length of diffusers (according to experimental data of Bedrzhitskiy).6 Up to a certain value of p∗ch in the blowing chamber (upstream of the inlet into a straight entrance section), which corresponds to the formation of the local supersonic zone in the initial section of the diffuser, no flow separation from the diffuser walls is observed (separation "from under the shock"), and the velocity field at the exit from the diffuser remains uniform. However, starting from a certain position of the compression shock that brings up the rear of the local supersonic zone, separation occurs, as well as a steep increase in the velocity field nonuniformity in the exit section of the diffuser. 11. The resistance coefficients of the diffusers ζd = ∆p/(ρw20 ⁄ 2), as well as the flow structure in them and the separation phenomena, depend on many parameters, such as the divergence angle α (for diffusers with rectilinear walls); the area ratio nar1 = F1/F0; the shape of the cross section; the shape of the boundaries; the boundary-layer thickness (momentum loss thickness) at the entrance; the shape of the velocity profile at the entrance; the degree of flow turbulence at the entrance; the flow regime (Reynolds number Re) both in the boundary layer and in the main flow; and the flow compressibility (Mach number Ma0). 12. The effect of the Reynolds number on the resistance coefficients of the diffusers is different for different divergence angles. In the case of separation-free diffusers, the character of the relationship ζd = f(Re) is close to the character of the relationship λ = f(Re) for straight tubes; the values of ζd decrease monotonically with increases in Re (Figure 5.9).

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Figure 5.9. Dependence of the resistance coefficient ζd of a conical diffuser on Re at nar1 = 4 and different values of α and l0/D0:54 1) α = 4o; 2) α = 10o; 3) α = 30o.

With increases in the divergence angle of the diffusers the character of the dependence of ζd on Re becomes complicated (see Figure 5.9), since flow separation from the channel wall becomes important. 13. When the diffusers (not only with small, but also with sufficiently large angles α) are installed directly after a smooth inlet nozzle (collector) (l0/D0 = 0), the flow in the boundary layer of the diffuser remains laminar over some distance downstream of the inlet even though the Reynolds numbers of the main flow substantially exceed the critical value Recr. Just as for λ of straight tubes, this causes a sharper decrease (with increase of Re) in the resistance coefficient of separation-free diffusers and of diffusers with local separation of the flow (α < 14o), than might have occurred, had there been a fully developed turbulent flow in the boundary layer over the whole length of the diffuser. 14. In the presence of a straight, sufficiently long insert between a smooth inlet nozzle (collector) and a diffuser: (1) the boundary layer in the beginning of the diffuser is additionally agitated (Figure 5.10) and (2) the boundary layer thickness increases (and, accordingly, the velocity profiles "extend") as early as the entrance into the diffuser (see graph b of Diagram 5.1). These factors exert directly opposite effects on the diffuser resistance. 15. When the straight insert has a length up to about l0/D0 ≈ 10, the first factor is predominant. At larger values of l0/D0, the influence of the first factor is stabilized, while the influence of the second factor continues to increase to some extent. As a result, with further increases in l0/D0, the constant influence of this parameter is established (the constant ratio kd = ζdl0>0 ⁄ ζdl0=0, which takes into account the effect of straight or curved sections upstream of the diffuser) or even some decrease in its effect on the resistance of separation-free diffusers.

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285

__ Figure 5.10. Variation of the longitudinal fluctuation of velocity w′ = w′/w0 in the inlet section of the diffuser with a change in the relative length of the straight starting section l0/D0.172,173

16. A thicker boundary layer at the entrance to the diffuser causes somewhat earlier appearance of the wall layer instability and occasional stalling of vortices. The larger the divergence angle of the diffuser, the stronger this phenomenon becomes, until the flow completely separates from the walls at certain values of α. These combined effects increase the total resistance of the diffuser. 17. For diffusers with large divergence angles, at which the flow completely separates from the walls (α > 14o), the effect of the Reynolds number and the inlet conditions on the change in the resistance coefficient is due to slightly different factors, namely, to the displace-

Figure 5.11. The functions ζtot = f1(Re) (a) and the velocity field over the section nx = 2 in a conical diffuser at α = 30o, nar1 = 2, and l1/D1 = 0 (b):54 I, Re = 0.3 × 105; II (α), Re = 1.2 × 105; III (β), Re = 3.3 × 105; IV (γ), Re = 4 × 105; 1) l0/D0 = 0; 2) l0/D0 = 2; 3) l0/D0 = 3 with a turbulizer; 4) l0/D0 = 20.

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ment of the separation point along the walls of the diffuser and to a change in the stalling zone thickness together with a change in the mode of the boundary-layer flow. 18. These factors are responsible for the complex character of the resistance curve of separation-prone diffusers installed directly downstream of the smooth collector, that is, at l0/D0 = 0. As is evident from Figure 5.11, at very small Re an increase in this number first leads to a sharp drop of the coefficient ζ∗tot until it reaches a certain minimum (segment A, Figure 5.11a), and then ζtot starts to increase sharply up to its maximum value at Re = (0.8–1.4) × 105 (segment B). This maximum is followed by a new sharp decrease in ζtot (the crisis of the resistance) until the second minimum of the values of ζtot (segment C) is attained at Re = 3.3 × 105. Following this minimum, the coefficient ζtot increases again, first rather sharply (segment D) and then slightly (segment E) with increases in Re. 19. Segment A of curve 1 (see Figure 5.11a) corresponds to a nonseparating laminar flow, where the resistance coefficient is inversely proportional to the Reynolds number, and segment B corresponds to the development of separation of the laminar boundary layer. The maximum of ζtot corresponds to complete laminar separation occurring close to the inlet section of the diffuser. This is due to the separation zone in laminar flow being most extensive both in its lateral dimensions and its extent (Figure 5.11b, region α) with the clear area of the main flow being the lowest one — whence the maximum of the pressure losses. 20. A sharp drop in ζtot over segment B of curve 1 (see Figure 5.11a) corresponds to the onset of the crisis when the separated laminar layer becomes turbulent. The layer becomes thinner and, as a result of vigorous turbulent agitation, the flow attaches to the wall again. The separation point (now of the turbulent flow) is thus displaced downstream. In this case, the separation zone greatly diminishes while the clear area of the flow correspondingly increases (see Figure 5.11b, region β), which leads to a sharp drop in the diffuser resistance coefficient. 21. Further increase in the resistance coefficient ζtot in the postcritical region (segments D and E, Figure 5.11a) is explained by some reverse displacement of the turbulent separation point upstream of the flow (Figure 5.11b, region γ). Such a displacement in the diffuser may occur under the action of the inertia forces that increase with the Reynolds number. 22. The character of the curves ζtot = f(Re) for separation-prone diffusers changes, depending on the inlet conditions. In particular, even a short, straight insert (l0/D0 = 2) installed upstream of the diffuser agitates the flow and simultaneously thickens the boundary layer at the entrance into the diffuser even at relatively small Reynolds numbers. Under these conditions, the maximum of ζtot decreases within 0.4 × 105 < Re < 2.3 × 105, while at Re > 2.3 × 105 the values of ζtot, as a whole, turn out to be higher (see curves 2, 3, and 4 in Figure 5.l1a). The latter is attributed to some displacement of the turbulent separation point upstream of the flow (in the direction of the diffuser inlet) due to the thickening of the boundary layer. The same effect can be achieved by artificial flow turbulence creation upstream of the diffuser inlet. 23. When the divergence angle of the diffuser α > 30o, the effect of a straight insert installed upstream of the diffuser decreases sharply and, at α ≥ 60o, becomes practically negligible. The explanation is that at very large values of α the flow separates so close to the ∗

Here a diffuser is considered, for example, which is installed at the exit from the network and for which ζtot is the coefficient of total resistance of the diffuser (which also accounts for the exit velocity pressure losses). A similar phenomenon is also observed in diffusers installed in the network, that is, for the coefficient ζd.

Flow with a Smooth Change in Velocity

287

Figure 5.12. Dependence of ζtot on the divergence angle of a conical diffuser at different area ratios nar1.36

diffuser inlet section that any further backward displacement of the separation point is, naturally, impossible. 24. The influence of the divergence angle α on the resistance coefficient of a conical diffuser can be judged from Figure 5.12.36 At small divergence angles, the diffuser resistance first decreases slightly with an increase in α and then increases. When α < 15o, experimental dependences separate for different area ratios (ζtot decreases with an increase in nar1) and, with a further increase in the divergence angle, tend to a single dependence on α. This fact is attributed to the development of a separating flow in the diffuser. The increase in the re-

Figure 5.13. Dependence of the total resistance coefficient ζtot on the area ratio of a conical diffuser at constant values of the angle α.36

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sistance coefficient ζtot with α is due to the displacement of flow separation to the inlet of the diffuser. The area ratio of the diffuser determines its capabilities for converting the kinetic energy of a flow into a potential one. The higher the nar1, the lower the kinetic energy of flow (provided the latter is separationless) at the outlet from the diffuser. However, as it has already been noted, an increase in nar1 on the occurrence of separation does not lead to a decrease in losses at the outlet from the diffuser. These losses may even increase in connection with the increase in the nonuniformity of the velocity profile at the outlet. Consequently, for such diffusers the optimum area ratios nar1 can be determined that correspond to minimum values of ζtot. This is clearly illustrated by the experimental data of Reference 36 given in Figure 5.13. The larger the divergence angle α, the lower the optimum value nar1 and the higher the minimum level of resistance ζtot. Qualitatively, the above-considered trends are preserved also for diffuser channels of a more complex shape (see also Diagrams 5.2, 5.4, and 5.5). 25. The influence of compressibility on a subsonic fluid flow in a diffuser is characterized by the dimensionless velocity λ1 or by the Mach number M1. The compressibility exerts its influence on the velocity and pressure distributions along the channel. Here, the greatest changes (as against an incompressible liquid) are observed near the inlet section where dimensionless velocities λ1 are appreciable. Also, one observes here an increase in the longitudinal pressure gradients, and this, in turn, influences the development of a boundary layer on the diffuser walls. Figure 5.14a and b presents the experimental data of Reference 36 illustrating the change in the resistance coefficient of the diffuser depending on the dimensionless velocity λ1 at different Re numbers. In these experiments, high subsonic velocities were attained by installing nozzles ahead of the diffuser and this ensured the needed channel constriction. As a result, first the velocity profiles became more peaked (curve 2 in Figure 5.14a) because of the increase in the favorable pressure gradient in the nozzle, but this was followed by attenuation of turbulence, and the velocity distribution in the boundary layer acquired the form (curve 3 in Figure 5.14a) typical of a laminar boundary layer. This was accompanied by flow separation and by a sharp increase in the resistance coefficient (Figure 5.14b). Experiments show that at high Reynolds numbers and moderate divergence angles of the diffuser (α < 10o) a separationless flow with a high coefficient of pressure recovery can be ensured over

Figure 5.14. Velocity distribution in a boundary layer near the inlet section of the diffuser at Re = (2.1–3.7) × 105 and different values of λ1 (a); dependence of the total resistance coefficient of the diffuser on λ1 at different values of Re (b): 1) λ1 = 0.48; 2) λ1 = 0.65; 3) λ1 = 0.81.

Flow with a Smooth Change in Velocity

289

the entire subsonic range of inlet velocities λ1. The calculations of the diffuser characteristics at subsonic flow velocities are considered in more detail in para. 31 and the experimental data at subsonic and supersonic velocities in paragraphs 52–54. 26. A straight insert ahead of the diffuser produces a symmetrical velocity profile at the diffuser inlet with the maximum velocity at the center and reduced velocities at the walls ("convex" shape). When the diffuser is preceded by a curved part of the pipeline or any barrier producing a nonuniform velocity profile at its inlet with reduced velocities at the center and elevated velocities at the walls ("concave" shape), the effect of such a profile on the diffuser resistance will be opposite to the effect of the convex profile; that is, at small angles α the diffuser resistance will increase, while at large angles α it will probably decrease somewhat compared with the resistance when there is a uniform velocity distribution at the inlet. 27. The structure of the f1ow in rectangular diffusers and the character of the resistance curves are basically the same as for conical diffusers. However, the flow conditions in rectangular diffusers are additionally influenced by the corners due to the cross sections, which are conducive to earlier flow separation from the wall. As a result, the resistance in such diffusers is always higher than in conical diffusers. On the other hand, the effect of the upstream straight section insert somewhat decreases, so that a relative rise in the resistance coefficient with l0/D0 turns out to be lower in such diffusers than in conical ones. 28. The resistance of plane diffusers (section expansion in one plane only) at the same divergence angles and area ratios is noticeably lower than in diffusers with the section expansion in two-dimensional planes and in many cases is even smaller than in conical diffusers. This is attributed to the fact that at the same divergence angles and area ratios the plane diffusers are correspondingly larger than the conical or rectangular diffusers, with expansion occurring in two planes. Thus, with a smoother change in the cross section, a lower pressure gradient along the flow and a weaker flow separation from the wall result. 29. When there is a nonseparating flow in diffusers, then all of its characteristics, including the resistance coefficient, can be calculated with the help of boundary-layer theory. With the use of these methods, the most extensive results were obtained by Guinevskiy et al.,5,19–25 Solodkin and Guinevskiy,77–81 Voitovich and Emeliyanova,14 Deich and Zaryankin,36 and Emeliyanova.43 30. In order to calculate the hydraulic resistance of diffuser channels, contemporary numerical methods of simulating laminar, transient, and turbulent flows of an incompressible liquid and a gas can be used. The methods of the boundary-layer theory are the most simple and effective. As applied to diffuser channels — planar and axisymmetric ones — two sections are distinguished: the initial one, with a comparatively thin boundary layer in the presence of a potential core, and a section of a stabilized flow with closed boundary layers.27,28,92,93 For the initial section of a diffuser channel the coefficient of total pressure losses averaged over the discharge velocity is expressed by the formula

ζ =

∆∗∗∗ n2(1 − ∆∗)3

−

∆∗∗∗ 0 (1 − ∆∗0)3

,

(5.1)

where ∆*** and ∆* represent the ratio of the thicknesses or areas of energy loss and displacement to the cross-sectional area of the channel. For a plane channel ∆*** = 2δ***/h and ∆* =

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Handbook of Hydraulic Resistance, 4th Edition δ

δ __ __ __ *** 2δ /h, h is the channel height, δ = ∫(1 − w)dy, δ = ∫(1 − w2)wdy, δ* and δ*** are the *

*

0 0 __ thicknesses of displacement and loss of energy, w = w/wc; for an axisymmetric channel ∆*** δ ⁄ rw

= ϑ

__ /πrw and ∆ = ϑ /πrw, rw is the cross-section radius, ϑ = 2∫(1 − w)(1 − y ⁄ rw)dy ⁄ rw, 2

***

*

2

*

*

0 δ ⁄ rw

__ __ ϑ*** = 2∫(1 − w2)w(1 − y ⁄ rw)dy ⁄ rw, δ is the boundary-layer thickness, ϑ* and ϑ*** are the 0

displacement and energy loss areas. The second term in formula (5.1) represents the total pressure loss over the channel section in front of the diffuser. The total pressure loss coefficient ζtot, which includes losses on exhaust from the channel, is determined by summing up the coefficient ζ and the supplementary coefficient ∆ζ that represents kinetic energy losses of the flow in the exit section of the diffuser: ∆ζ =

1 − ∆∗ − ∆∗∗∗ ρw2 , Q = ∫ wdF . wdF ⁄ (Q ρ w20 ⁄ 2) = ∗ 3 2 2 (1 − ∆ ) n F F

∫

(5.2)

Thus, for the initial portion of the diffuser channel ζtot = ζ + ∆ζ =

1 n2(1 − ∆∗)2

−

∆∗∗∗ 0 (1 − ∆∗0)3

(5.3)

.

In order to characterize the initial nonuniformity of the flow in the inlet section of a diffuser, the ratio of the maximum velocity to the mean one, wmax/wm, is used as well as the coefficient of the nonuniform momentum M or of the kinetic energy N with wmax 1 , = wm 1 − ∆∗0

M =

1 − ∆∗0 − ∆∗∗ 0 (1 − ∆∗0)2

,

N =

1 − ∆∗0 − ∆∗∗∗ 0 1 − ∆∗0)3

.

Here ∆∗∗ 0 is the ratio of the thickness (area) of momentum loss to the width (area) of the inlet cross section of the diffuser. The same technique is used to determine the total pressure loss coefficient also for the section of the diffuser channel with closed boundary layers.93 The above formulas allow one to calculate the losses in a diffuser at different Reynolds numbers, area ratios, divergence angles, and initial nonuniformities up to the section of boundary-layer separation. It is only required to calculate the parameters of the boundary layer — laminar, transient, and turbulent. Usually, a direct or an inverse problems of the boundary layer is solved: the direct problem consists of determining the parameters of a boundary layer in a diffuser of a specified geometry for a fixed Reynolds number Re and an initial flow nonuniformity; the inverse problem is reduced to determining the geometric pa-

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291

Figure 5.15. Pressure coefficients in the section of closing or separation of boundary layers in a plane diffuser (a); a comparison between the experimental91 and calculated (∆∗0_ = 0.005) dependences of the divergence angle of a plane separationless diffuser on its limiting length x (b).

rameters of the diffuser at the fixed Reynolds number and initial nonuniformity of the flow from the given streamwise distribution of the velocity over the channel axis or the friction factor along the diffuser surface. The following is an example which shows the solution of a direct problem for plane separation-free diffusers with fully turbulent boundary layers. Figure 5.15a presents the static pressure recovery factor cp = (p − p0) ⁄ (ρw20 ⁄ 2) in the section with closing or separating boundary layers in a plane diffuser depending on the Reynolds number Re, divergence angle α, and the initial flow nonuniformity ∆∗0 = 2δ∗0 ⁄ h0, where h0 is the width of the inlet cross section of the diffuser. Whence it follows that in the considered range of the values of the initial nonuniformity ∆∗0 at divergence angles not exceeding α = 7o, the closing of the boundary layers in a diffuser occurs earlier than their separation. At divergence angles exceeding 8o, vice versa, the separation of the boundary layers on the opposite walls precedes their closing. Here, in the

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first region (before the closing of the boundary layers) the static pressure recovery coefficient increases with the divergence angle, whereas in the second case it decreases. Figure 5.15b presents a comparison of experimental91 and calculated27,92,93 dependences of the divergence angle of a plane diffuser on its limiting length corresponding to flow separation in its outlet section. The open experimental points correspond to the small initial turbulence, whereas the black dots — to an enhanced one. Since an increase in the initial turbulence favors boundary layer turbulization, the case of enhanced turbulence corresponds more to the case of an entirely turbulent layer adopted in the calculation. The agreement between the calculation and experiment appeared to be satisfactory (here xs = x/h0). 31. We will consider the application of the above method to the calculation of pressure losses in conical diffusers at subsonic flow velocities. In a channel with impermeable walls the condition of mass flow rate constancy should hold:

Figure 5.16. Coefficient of total pressure recovery in conical diffusers depending on λ1 and Re: 1) l0/r1 = 4; 2) l0/r1 = 10; 3) l0/r1 = 20; 4) l0/r1 = 40.

293

Flow with a Smooth Change in Velocity rw

G = 2π ∫ ρwrdr = const

(ρ is the gas density).

0

To characterize the magnitude of total pressure losses in a diffuser channel in the absence of heat transfer between a gas and the walls, either the overall pressure recovery coefficient σ = p02/p01 is used or the total pressure loss factor: ζ =

p01 − p02 G(ρ1w21 ⁄ 2)

(5.3a)

,

where ri

p0i =

∫ p0ρwrdr 0

ri

= π(ri − δi) ρδiwδip00 + 2π ∫ p0ρwrdr , (i = 1, 2) . 2

ri − δi

Here ρδi and wδi are the density and velocity of the gas in the flow core, p0 is the total pressure of the gas flow in the boundary layer, p00 is the total pressure in the flow core; the subscripts 1 and 2 correspond to the inlet and outlet cross sections of the diffuser. Calculations were carried out for a series of conical diffusers with divergence angles α = 4, 6, 10, and 20o for the values of the preinserted cylindrical segment l0/r1 = 4–40 and area ratio n = (r2/r1)2 = 4.95 The values of the total pressure coefficient σ calculated as a function of the reduced velocity λ1 = wm1/acr are compared with the experimental data of Reference 211. The characteristic feature of these experiments is that an increase in the reduced velocity at the inlet λ1 = wm1/acr was accompanied by a corresponding increase in the Reynolds number. As follows from Figure 5.16, the results of the calculation — the σ(λ1) curves — agree satisfactorily with the experiment (symbols). This agreement is observed until the velocity in the flow core in the inlet section becomes equal to the local speed of sound, i.e., a "choking" regime sets in.*

Figure 5.17. Influence of the symmetric initial nonuniformity of flow on a streamwise change in the total pressure loss factor in a conical diffuser at Re = 5 × 107 and α = 2o: 1) ∆∗0 = 0.01; 2) ∆∗0 = 0.02; 3) ∆∗0 = 0.04; 4) ∆∗0 = 0.1; 5) place of closure of the boundary layers. *

Formula (5.3a) was derived by averaging the total pressure over the mass flow rate. The theory of averaging nonuniform gas flows in channels has been developed by L. I. Sedov abd G. G. Chernyi.212

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32. Figure 5.17 presents the calculated values of the total pressure loss factors in a conical diffuser ζ(x/rw1) at Re = 5 × 107 and α = 2o at varying degrees of flow nonuniformity, i.e., ∆∗0 = 0.01, 0.02, 0.04, and 0.1. They illustrate the increase in ζ in a nonuniform flow.187 33. At different lengths of the preinserted cylindrical segment l0/D0 = 1–9 and area ratios nar1 = 1.5, 2, 2.36, 3, and 4 at Re = w0D0/ν = (0.35–0.45) × 106 calculations and experi__ ments94 yielded the dependences of the total loss factor at different values of the height of roughness protrusions ∆ = ∆/D0 = 0.002 and 0.005 as a function of nar1, α, l0/D0, and ∆/D0. In all of the cases considered the surface roughness considerably increased hydraulic losses in the diffuser (Figure 5.18a), with the relative increment in the hydraulic losses because of the

__ __ 14 Figure 5.18. Dependence of the diffuser resistance __ on roughness ∆: a) ζtot = f(nar1, α, l0/D0, ∆); b, (ζtot rough – ζtot smooth)/ζtot smooth = f(l0/D0, nar1, ∆).

Flow with a Smooth Change in Velocity

295

Figure 5.19. Distribution of the static pressure along the diffuser generatrix at α = 10o depending on nar1, 2k/D0, and l0/D0.

roughness (ζr – ζsm)/ζsm being the highest at a divergence angle of α = 4o and it increases with the area ratio nar1 (Figure 5.18b). With increase in the boundary-layer thickness upstream of the diffuser, i.e., with increase in l0/D0, a relative increase in ζtot due to roughness decreases sharply. The distribution of the static pressure Cp = (p1 – p2)/(ρw2/2) along the diffuser generatrix at α = 10o depending on nar1 = 1–3.5, 2k/D0, and l0/D0 is shown in Figure 5.19. In all the cases considered, the surface roughness made the increase in the static pressure smaller. The corresponding computational relations are also plotted (dashed curves) in Figures 5.18 and 5.19. Calculations showed that the influence of roughness is substantial only when it is indented in sections with a thin boundary layer where the height of roughness protrusions exceeds the viscous sublayer thickness. 34. In the foregoing, the possibilities of the use of the boundary-layer theory methods for solving a direct problem — calculation of the loss factors of plane and axisymmetric diffusers

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of prescribed geometry — were demonstrated. In a number of cases it is of interest to solve an inverse problem,95 i.e., when the streamwise distribution of some of the aerodynamic parameters (e.g., velocity or surface friction) is given, and it is required to calculate the geometric parameters of the diffuser. The diffusers thus determined appear to be more efficient as compared to plane (wedge-like) or conical ones. This primarily relates to the so-called preseparation diffusers with zero surface friction,21,189–192 in which, at a given length, a noticeable decrease in the total pressure losses is ensured or, at a given value of the area ratio nar, a marked decrease in the diffuser length is possible. Thus, as applied to a preseparation diffuser of circular cross section with prescribed conditions in the inlet section (behind the preinserted cylindrical segment), it is needed to calculate the law governing the change in the flow sections to obtain, at a certain distance from the inlet to the diffuser, a boundary layer with a very small or zero surface friction. To attain this, one has to prescribe the law of surface friction distribution over the short inlet length, according to which this friction decreases over the length ∆x∗ from the given value at the end of the preinserted cylindrical segment cf to zero or to a very small value. The diffuser obtained in this way, over the length x > ∆x∗, acquires a bell-like shape which depends on the given value of the initial nonuniformity of the velocity profile in the inlet section of the diffuser, i.e., on ∆∗0. In this sense, the calculated diffuser is a single-mode one. However, experiments showed that a certain small change in ∆∗0 does not have a strong influence on the efficiency of the diffuser. Figure 5.20 presents the laws governing a change in the flow sections of a series of preseparation axisymmetric diffusers nar1(x/r0) in the case of different extensions of the section of formation of zero surface friction Ax = ∆x∗/r0 = 0.5, 1, 2, and 4 at Re = 2.42 × 105 and ∆∗0 = 0.02. This figure also contains straight lines for conical diffusers with divergence angles α = 3, 5, and 7o. Figure 5.21 presents the corresponding dependences ζtot(nar1) for preseparation diffusers at different values of Ax = ∆x∗/r0 and for conical diffusers obtained by solving a direct problem. Figure 5.22 gives a comparison between the data of calculation and experiment for a preseparation axisymmetric diffuser at Re = 1.4 × 105, ∆∗0 = 0.2.5,14,21–23,27,189–191

Figure 5.20. Area ratio of preseparation axisymmetric and conical diffusers (Re = 2.42 × 105): 1) preseparation diffusers; 2) conical diffusers.

Flow with a Smooth Change in Velocity

297

Figure 5.21. Total pressure losses in preseparation and conical diffusers. The symbols are same as in Figure 5.20.

35. Diagrams 5.1 through 5.5 contain the total resistance coefficients ζd based on the experiments of Idelchik and Ginzburg51–55 with diffusers installed in a system or network that had a variety of cross-sectional shapes (conical, square, plane), which in turn depend on the basic geometric parameters (α, nar1), inlet conditions (l0/D0 ≥ 0), and flow regime (Re).

Figure 5.22. Geometric and aerodynamic characteristics of an axisymmetric preseparation diffuser (Re = 1.4 × 105; ∆∗0 = 0.02): 1) experiment of Bychkova,189 2) contour of the diffuser,189 3) calculation.190

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36. The total resistance coefficient of a diffuser installed in the network is in the general case (under any inlet conditions)

ζdl0>0 *

∆p = kdζdl0=0 , ρw20 ⁄ 2

(5.4)

where ζdl0=0 is the total resistance coefficient of a diffuser at l0/D0 = 0 (see Diagram 5.1); ζdl0>0 is the resistance coefficient of a diffuser with a straight section or a curved part installed upstream. 37. In the case of an asymmetrical velocity distribution downstream of elbows, throttling devices, and other fittings, it is possible for practical applications to utilize the values of kd given in Diagrams 5.1 (para. 3) and 5.19 (para. 2). The data shown under para. 3 of Diagram 5.1 were obtained using the results of investigations of a conical diffuser installed downstream of branches with different geometric parameters.180 Those under para. 2 of Diagram 5.19 have been obtained on the basis of studies of circular diffusers upstream of which different velocity distributions were produced with the aid of special screens.127 38. The data of Diagrams 5.1 through 5.5 consider the simultaneous effect of the parameters Re and λ0 = w0/acr. In general, these parameters exert a combined effect on the characteristics of the diffusers.36 However, in the absence of separation and large Reynolds numbers this combined effect is of no importance. Flow compressibility exhibits itself most strongly at small Reynolds numbers in the region of a critical drop in the resistance. Since there is a lack of data to evaluate the combined influence of the above parameters, this effect can be neglected in practical application calculations, particularly since in many practical cases Re and λ0 vary simultaneously. 39. It is sometimes convenient in engineering calculations to resort to a conventional method of dividing the total losses in a diffuser ∆p into two parts:* ∆pfr, the friction losses along the length of the diffuser, and ∆pexp, the local losses associated with expansion of the cross section. The total resistance coefficient of the diffuser ζd is accordingly composed of the friction resistance coefficient ζfr and the expansion resistance coefficient ζexp: ζd =

∆p ρw20 ⁄ 2

= ζfr + ζexp .

(5.5)

40. It is convenient that the "expansion" losses be expressed in terms of the shock coefficient,47,49 which is the ratio of the actual expansion losses in the diffuser to the theoretical shock losses due to a sudden expansion of the cross section (α = 180o), that is, ϕexp =

∗

∆pexp (ρ ⁄ 2) (w0 − w1)2

.

(5.6)

Since the method has not strict foundation, the formulas that will given further should be considered as interpolational ones convenient for practical calculations. An attempt at improving the considered empirical method for calculating the resistance of diffuser channels was made by Chernyavskii and Gordeev.97,98

Flow with a Smooth Change in Velocity

299

Figure 5.23 presents the dependence ϕexp = f(αo) for conical diffusers with the area ratio nar1 = 2–16 that are installed in the system; these were obtained based on the experimental data of Idelchik and Ginzburg54 at the Reynolds numbers Re = (1–9) × 105. For a uniform velocity profile at the inlet section (kd = 1.0) and large Reynolds numbers (Re ≥ 2⋅105), the shock coefficient of diffusers with divergence angles 0 < α < 40o can be calculated from the author’s formula47,49

Figure 5.23. Dependence ϕexp = f(αo) for conical diffusers in the system at l0/D0 = 0 and different Re numbers.54

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Handbook of Hydraulic Resistance, 4th Edition (5.7)

ϕexp = 3.2k tan1.25 α ⁄ 2 , where, based on the experiments,26,52,54 for conical diffusers k with expansion in two planes:

C

1; for pyramidal diffusers

k = 0.66 + 0.12α , 4o < α < 12o ; k = 3.3 − 0.03α , 12o < α < 30o ; for plane diffusers k = 2.0 − 0.03α , 4o < α < 12o ; k = 2.0 − 0.04α , 12o < α < 20o , where α is given in degrees. The expansion resistance coefficient is expressed in terms of the shock coefficient as ζexp B

2

2

α 1 ⎞ 1 ⎞ . = ϕexp ⎛⎜1 − = 3.2k tan1.25 ⎛⎜1 − ⎟ 2 2 nar1 nar1⎟ ρw0 ⁄ 2 ⎠ ⎠ ⎝ ⎝ ∆p

(5.8)

41. The friction resistance coefficient for conical diffusers47,49 is ∆pfr

ζfr B

ρw20 ⁄ 2

=

λ

⎛1 − 1 ⎞ . α⎜ n2ar 1⎟⎠ 8 sin ⎝ 2

(5.9)

For pyramidal diffusers with the sides of the inlet section a0 and b0 and with identical divergence angles in both planes the friction coefficient ζfr is calculated from Equation (5.9). For a pyramidal diffuser with different divergence angles (α A β) in both planes47, 49 ζfr

B

∆ρ ρw20 ⁄ 2

=

λ ⎛ 1 1 ⎞ 1 1 − 2 ⎞⎟ ⎛⎜ + ⎟. 16 ⎜ nar⎠ ⎜ sin α sin β ⎟ ⎝ ⎜ 2 2 ⎟⎠ ⎝

(5.10)

For a plane diffuser with the sides a0 and b0 (where b0 is constant in length)47,49 ζfr B

∆pfr ρw20 ⁄ 2

=

λ 4

⎡ a0 1 ⎢b ⎢ 0 tan α ⎢ 2 ⎣

⎛1 − 1 ⎞ + 1 ⎜ nar 1⎟ ⎠ 2 sin α ⎝ 2

⎛1 − 1 ⎞⎤ . ⎥ ⎜ n2ar1 ⎟⎠⎥ ⎝ ⎥ ⎦

(5.11)

Practically, it is possible to assume that ζfr =

λ

⎡ a0 ⎛ 1 ⎞ 1 ⎤ 1− + 0.5 ⎛⎜1 − 2 ⎞⎟⎥ . nar1 ⎟ α ⎢ b0 ⎜⎝ nar 1⎠⎦ ⎠ ⎝ 4 sin ⎣ 2

(5.12)

Flow with a Smooth Change in Velocity

301

42. When α ≤ 40–50o, the shock coefficient ϕexp is smaller than unity (see Figure 5.23). This indicates that the losses in a diffuser are lower than the shock losses at a sudden expansion (α = 180o). For angles α = 50–90o the value of ϕexp becomes somewhat larger than unity, that is, the losses in a diffuser increase compared with the shock losses. Starting from α = 90o and up to α = 180o, the value of ϕexp decreases and approaches unity; this means that the losses in the diffuser become close to the sudden expansion losses and therefore, unless a uniform velocity is expected downstream of the diffuser, it is not advisable to use diffusers with the divergence angles α > 40–50o. When a very short transition piece is needed because of the requirement of limited overall size, then, from the point of view of the resistance, this piece may have the divergence angle α = 180o. 43. When a uniform velocity profile is required downstream of the transition piece and guide vanes, dividing walls, or perforated plates (screens, nozzles) are to be installed for this purpose, then any diffuser, even one with a very large divergence angle (α > 50o), should be preferred to a sudden expansion (α = 180o). 44. Since a smooth expansion of the cross section of a tube with rectilinear walls at small divergence angles (a diffuser) leads first to a decrease in the pressure losses compared with the losses in a tube of the same length, but of constant cross section, while larger divergence angles lead again to an increase in these losses, then there probably exists an optimum divergence angle at which the losses are reduced to minimum (see curves ζd = f(α) of Diagrams 5.2, 5.4, and 5.5). 45. The minimum value of the resistance coefficient ζmin of conical diffusers exists within the region αopt = 4–12o and depends mainly on the area ratio nar1 and the relative length l0/D0: the smaller nar1, the larger αopt at which this minimum is attained (see graph α of Diagram 5.2); conversely, the parameter l0/D0 decreases the value of αopt. For rectangular (square) diffusers, the upper limit of αopt is much smaller (7o). The effect of l0/D0 on decrease of αopt is more substantial in this case (see Diagram 5.4). For plane diffusers, the optimum divergence angle, at which the miminum of the pressure losses is attained, exists within the range αopt = 6–12o (see Diagram 5.5). 46. It is very important for many practical applications to recover the maximum possible static pressure over a minimum length of the diffuser even at the cost of greater energy losses in it. Theoretically, the larger the area ratio nar1 or the relative length ld/D0 at the given divergence angle, the larger the coefficient of the static pressure recovery ηd of the diffuser: ηd =

p1 − p0

ρw20 ⁄ 2

(5.13)

.

47. Based on the Bernoulli and continuity equations and Equation (5.13), the following relationship exists between the pressure recovery coefficient and the resistance coefficient of the diffuser installed in the network 2

⎛ F0 ⎞ ηd = N0 − N1 ⎜ ⎟ − ζd , ⎝ F1 ⎠

(5.14)

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Handbook of Hydraulic Resistance, 4th Edition

where 3

N0 =

3

1 ⎛w⎞ 1 ⎛w⎞ dF N1 = dF . F0 ∫ ⎜ w0 ⎟ F1 ∫ ⎜ w1 ⎟ ⎝ ⎝ ⎠ ⎠ F0 F1

If nonuniformity in the velocity distribution at the inlet and outlet sections of the diffuser is neglected, that is, if it is assumed that N0 = N1 = 1.0 (which is often admissible for practical calculations), then

ηd = 1 −

1 n2ar1

− ζd .

(5.15)

An analogous relation is obtained for the total resistance coefficient of the diffuser installed at the outlet from the network: ηtot = 1 − ζtot . 48. The efficiency of the diffuser is sometimes characterized by the efficiency coefficient, which is the ratio of the actual to the ideal (without losses) increment in the static pressure: η1d =

p1 − p0 p1 − p0 , = 2 (p1 − p0)id N0(ρw0 ⁄ 2) − N1(ρw21 ⁄ 2)

(5.16)

where (p1 – p0)id is the difference between the static pressures in sections 1–1 and 0–0 for an ideal diffuser (without losses). The coupling between the efficiency coefficient and the resistance coefficient of a diffuser, installed in the network, is expressed by η1d = 1 −

ζd N0 − (N1 ⁄ n2ar 1)

,

at N0 = N1 = 1 η1d = 1 −

ζd 1 − (1 ⁄ n2ar1)

.

A similar relationship is obtained for the total resistance coefficient of the diffuser installed at the exit from the network: η1tot =

1 − ζtot 1 − (1 ⁄ n2ar 1)

.

49. Owing to flow separation from the diffuser walls with a large area ratio and with a substantial nonuniformity of the velocity distribution over the section, the effective area ratio

303

Flow with a Smooth Change in Velocity

nar1 at which the maximum possible static pressure recovery is attainable (due to a decrease in the flow velocity) is considerably lower than it might have been in an ideal diffuser (without separation and losses and with uniform velocity distribution over the cross section). In cases where the geometric dimensions of the diffuser (area ratio nar1 and length ld) are not limited by any requirement (not prescribed), the low values of nar1 allow the use of diffusers with the optimum area ratio [(nar1)opt and (ld/D0)opt] at which ηd attains the absolute maximum* possible for the given inlet conditions (boundary-layer thickness of length l0/D0). 50. The values of ηdmax, ζd, (nar1)opt, and (ld/D0)opt for circular and rectangular diffusers, as well as for plane diffusers, obtained from Equation (5.13) and from Diagrams 5.1 through 5.5 are listed in Table 5.1. The limits of the geometric parameters of the diffusers are listed in the same sequence as ηdmax and ζd. 51. The static pressure recovery coefficients in diffusers with the prescribed geometric parameters can be determined from the curves of ηd vs. nar1 for different divergence angles α and inlet conditions (l0/D0) given in Figures 5.24 through 5.26 (the curves are based on the data of Diagrams 5.1 through 5.5 for Re > 4 × 105). 52. Figures 5.27 and 5.28 give the data144 for conical diffusers with the divergence angle ′ vs. the diffuser efficiency calculated from the formula similar to α = 10o in the form of η1d Equation (5.14) and, correspondingly, of the resistance coefficient ζd′ calculated as the ratio of the difference of total pressures in sections 0–0 and 1–1 to the difference of velocity pressures in the same sections, that is Table 5.1. Optimum characteristics of the diffusers l0

D0

η 1max

ζd

(nar1)opt

(l0 ⁄ D0)opt

0.13–0.08

6–10

5.8–12.3

Conical diffusers (α = 14–10o) 0

0.84–0.91

2

0.69–0.82

0.29–0.17

6–10

5.8–12.3

5

0.64–0.77

0.30–0.20

4–6

4.1–8.2

10

0.58–0.71

0.17–0.27

2–6

1.7–8.2

20

0.57–0.70

0.19–0.27

2–6

1.7–8.2 o

Rectangular diffusers (of square cross section at α = 10–6 ) 0

0.74–0.84

0.18–0.13

10

0.66–0.76

0.28–0.18

6

8.2–13.5

4

5.7–9.40

o

Plane diffusers (α = 14–10 )

∗

0

0.78–0.80

0.16–0.14

4

12.2–17.0

10

0.71–0.75

0.23–0.17

4–2

12.0–5.70

Since the finite equation of velocities and pressures over the section of interest occurs not directly downstream of the diffuser, but at some distance along the straight section behind it, the maximum of the static pressure recovery is attained at the distance downstream of the diffuser (for practical purposes at the distance up to 2D1, where D1 is the diameter of the exit section; for a plane diffuser D1 is replaced by the larger side of the exit section, i.e., 2a1).

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.24. Dependence of ηd on nar1 for a conical diffuser.

Figure 5.25. Dependence of ηd on nar1 for a rectangular (square) diffuser.

Figure 5.26. Dependence of ηd on nar1 for a plane diffuser.

Flow with a Smooth Change in Velocity

305

Figure 5.27. Dependence of the efficiency n1d of the diffuser with α = 10o on the Mach number (Ma0), inlet conditions (δ*/D0), and divergence ratio nar1.144

Figure 5.28. Dependence of the resistance coefficient ζ′d of the diffuser with α = 10o on the Mach number (Ma0), inlet conditions (δ*/D0), and divergence ratio nar1.144

306

ζd′ B

Handbook of Hydraulic Resistance, 4th Edition p∗0 − p∗1

ρ0w20 ρ1w21 − N1′ 2 2 vs. the parameters nar1, δ*/D0, and Ma0. Here, N0′ and N1′ are the coefficients of the nonuniformity in the distributions of flow parameters over the cross sections 0–0 and 1–1; Ma0 is the Mach number in section 0–0. The coupling between the Mach number and the reduced velocity λ0 is given by Equation (1.41). ′ and ζd′ were obtained at the Reynolds number Re = 2 53. The data on the coefficients η1d 5 6 × 10 –1.7 × 10 , Mach number at subsonic velocities from Ma0 = 0.2 till the regime of flow choking and at supersonic velocities within the range Ma0 = 1.2–1.4. ′ : ζd′ = 1 – η1d ′ . There is the following relationship between the quantities ζd′ _and η1d ∗ ∗ 54. The most detailed data in the form of the relationship p0 = p1 ⁄ p0 of the coefficient p∗1 of the total pressure recovery at the exit from conical diffusers in terms of the total pressure (stagnation pressure) p∗0 in the smallest _ cross section (0–0) on the numbers λ0 and Re are given in Diagram 5.3. The relationships p0 = f(λ0, Re) (according to the experiments of Idelchik and Ginzburg27,51–54) are given for the divergence angles α = 4–14o, area ratios narl = 2–16, and relative lengths l0/D0 = 0–10. _ At velocities close to the sonic one, the dependence of p0 on λ0 degenerates into vertical straight lines (see Diagram 5.3). This is explained by the onset of flow choking regime in the diffuser when a compression shock occurs. The larger the relative length of the straight inlet section, the earlier, that is, at the smaller values of λ0, the choking regime occurs. 55. The relationship between the resistance coefficient of the diffuser and the total pressure coefficient may be obtained on the basis of the following formula:19

ζd B

N0′

∆p ρ∗0w20 ⁄ 2

=

k+1 1 1 ln _ , k λ20 p0

where ρ∗0 is the density of the stagnated flow in the inlet section of the diffuser. For diffusers with small divergence angles at which the pressure losses are small,2

ζd B

∆p ρ∗0w20 ⁄ 2

=

_ k+1 1 (1 − p0) , 2 k λ0

whence _ p0 = 1 −

k λ20ζd , k+1

where

λ0 B

w0 , acr

acr =

2k RT ⎯⎯⎯⎯ √ k+1

∗ 0

,

Flow with a Smooth Change in Velocity

307

ρ∗0 is the density of the stagnated flow in the inlet section of the diffuser; T0∗ is the stagnated flow temperature in the same section. _ 56. In Diagram 5.6 the total pressure recovery coefficients p0 and the hydraulic resistance coefficient ζd of a plane five-channel subsonic _ diffuser are given for the following geometric parameters: at α′ equal to 8, 12, and 16o; at l0 equal to 3.23, 6.45, and 9.68; narl = 6.45; Re = (0.6–4) × 105. 57. At very low Reynolds numbers (at least within 1 < Re < 30–50) the resistance coefficient of the diffusers is described by the same equation as in the case of a sudden expansion:4 ζB

∆p 2

ρw0 ⁄ 2

=

A . Re

Here, the quantity A is a function of both the angle and the area ratio A = f (α, nar1) . At α ≤ 40o , A=

20n0.33 ar1 (tan α)0.75

.

58. At high gas flow velocities it is more convenient to operate not on the resistance coefficient, but on the total pressure recovery coefficient at the end of the diffuser p∗1, expressed in terms of the total pressure (stagnation pressure) p∗0 in its smallest cross section (0–0): _ p∗1 . p0 = p∗0 59. A resistance located downstream of the diffuser and uniformly distributed over the cross section (provided by a screen, perforated plate or grid, nozzle, air heater, etc.) regulates the flow in both the diffuser and the channel following it. The losses in the diffuser decrease somewhat. However, the total losses in the diffuser and in the grid (screen, etc.) vary only slightly. For rectilinear diffusers with divergence angles α up to 40–60o, and especially for curvilinear diffusers, these losses remain equal to the sum of losses separately in a diffuser and in a perforated plate or grid,47–49 that is,

ζB

∆p ρw20 ⁄ 2

ζgr = ζw.gr + 2 , nar1

where ζw.gr = ∆pw.gr ⁄ (ρw20 ⁄ 2) is the resistance coefficient of the diffuser without a grid, determined as ζ from corresponding diagrams of Chapter 5; ζgr = ∆pgr ⁄ (ρw2gr ⁄ 2) is the resis-

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.29. Different means of improving operation of short diffusers.

tance coefficient of the grid (screen, nozzle, etc.), based on the flow velocity at its face, determined as ζ from corresponding diagrams of Chapter 8. 60. The flow conditions in short diffusers (with larger divergence angles) can be greatly improved, and the resistance lowered, by preventing flow separation with them or by attenuating vortex formation. The main measures that improve flow conditions in the diffuser (Figure 5.29) include boundary-layer suction (Figure 5.29a) and blowing (renewal) (Figure 5.29b); installation of guide vanes (baffles, Figure 5.29c) and dividing walls or splitters (over the whole length of the diffusers, Figure 5.29d, or over part of it, Figure 5.29e); use of curvilinear walls (Figure 5.29f, g, h), stepped walls (stepped diffusers, Figure 5.29j), and preseparation diffusers (Figure 5.29i); and transverse ribs or fins (Figure 5.29k); use of the generators of longitudinal vortices (Figure 5.29l). 61. When there is sucking through slits (see Figure 5.29a), the thickness of the boundary layer on the diffuser wall decreases, as a result of which the separation zone moves down-

Flow with a Smooth Change in Velocity

309

wards, the flow becomes smoother, the degree of pressure recovery in the diffuser increases and its resistance decreases. The blowing off of the boundary layer (see Figure 5.29b) increases the flow velocity near the walls, as a result of which the separation zone moves downstream. 62. The effectiveness of boundary-layer suction depends on the ratio of the flow rate q of the medium sucked through the slits in the side walls of the diffuser to the total flow rate Q _ q of this medium through the diffuser (on the flow rate coefficient q = ) and on the relative Q _ distance from the inlet section of the diffuser to a slit. At q = 0.02–0.04 the diffuser resistance decreases by 30–40%. Here, the intrinsic losses in the suction system for the values of _ q indicated are relatively low.91,93 63. Figure 5.30b presents the results of Frankfurt’s experiments93 on pressure distribution o along the _ diffuser with a divergence angle α = 10 at the flow rate coefficients of the air sucked q from zero to 0.1. Here, the suction through the slit located in the preseparation zone turned out to be most efficient. An increase in the flow rate coefficient leads to a marked increase in the degree of the static pressure recovery for diffusers of different lengths. Figure 5.30c presents the data on the resistance coefficients ζtot of conical diffusers operating in discharge with divergence angles of α = 30o and 60o and an area ratio of nar1 = 2–8 at Re =

Figure 5.30. Schematic of boundary layer suction (a), pressure _ distribution over the length of the diffuser with a_ divergence angle α = 10o at different values of q (b), dependence of ζtot on the flow _ rate of the diffuser (x coefficient q (c):93 solid lines, flow suction through a slit in the_ initial section _ _= 0); o dashed lines, simultaneous suction through two slits in sections x = 0 and x = 0.78 at α = 30 ; x=0 i _ and xi = 0.35 at α = 60o; 1) nar1 = 8; 2) nar1 = 4; 3) nar1 = 3; 4) nar1 = 2.

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Handbook of Hydraulic Resistance, 4th Edition

_ _ (3.7–4.8) × 105 depending on the flow rate coefficient q and position of a sucking slit xi = xi . Here, the total pressure losses of the diffuser with sucking (ζtot) were determined with D0 account for the additional energy spent on suction which were taken equal to the product of the amount sucked (reduced to normal atmospheric conditions) by the difference of pressures above the slit and in the suction chamber (see Figure 5.30a). The efficiency of the sucking unit was taken equal to unity. _ For α = 30o the optimal degree of suction lies within q = 0.02–0.03. The losses are the _ _ xi smallest on application of combined suction through slits: at distances x = 0 and xi = = D0 0.78. For a diffuser with the angle α = 60o the optimum value of the flow rate coefficient is _ q_ = 0.04. The smallest losses result when a slit is located in the initial section of the diffuser (x0 = 0). 64. Investigations of the effect of the boundary layer blowoff on the static pressure recovery ηd and aerodynamic resistance ζtot were carried out on models of plane and conical diffusers95,211 with a separational and separationless models of flow, divergence angles α = 10, 30, and 60o and area ratios nar1 = 3–8 at the Reynolds number Re C 2.7 × 105. The abovementioned works contain a detail bibliography on boundary layer blowoff in diffusers. Boundary-layer blowoff was arranged through annular slits located around the periphery of the inlet section of the diffuser.187 To blow a boundary layer, an air jet was directed either along the wall or along the axis of the diffuser.187 The investigations have shown that the most efficient is blowing through a slit located at the inlet to the channel.211 The boundary layer blowoff with the flow velocity exceeding the

Figure 5.31. Dependence of the degree of static pressure recovery on air flow rate through a slit and on jet momentum for a diffuser with a divergence angle α = 30o.

311

Flow with a Smooth Change in Velocity

_

Figure 5.32. Dependence of the resistance coefficient ζtot on q for diffusers with divergence angles 30 and 60o:95 _ _ _ a) α = 30o, nar1 = 3–8: 1) fs _= 0.03; 2) fs _= 0.06; 3) fs _= 0.08; b) α = 60o, nar1 = 4 and 8: 1) fs = 0.03, 2) fs = 0.06; 3) fs = 0.08.

Figure 5.33. Dependence of the minimal losses of a diffuser with boundary-layer blowoff on divergence angle α at different values of the slit area.

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flow velocity in the channel leads to an increase in the static pressure recovery coefficient ηd of the diffuser. In this case, the results practically do not depend on the area ratio nar1. In diffusers with unsteady separation, the flow pulsations decrease.187 Figure 5.31 presents the results of measurements95 of the coefficient ηd in a diffuser with the divergence angle _α = 30o and area ratio nar1 = 3–8 at different values of the relative width of the annular slit fs = (fs/ns)F0, where ns = Fs/F0, fs is the cross-sectional area of the slit; Fs is the cross-sectional area of the diffuser at_ the place of boundary-layer blowoff. The coefficient ηd is based _ on_ the flow rate coefficient q and the momentum coefficient of the jet cµ = 2qνs/(F0w20) = 2q2/(fsns), where ns is the area ratio of the diffuser in the section with the slit. In the latter case, the results obtained for different widths of the slit practically coincide. Similar conclusions were obtained187 in studying blowing through some of discrete nozzles installed around the perimeter of the inlet section of the diffuser. The dependence of the coefficient ζtot (that takes into account the expenditures of energy in the blowing system which were taken equal to the product of the flow rate through the slit by the difference of total pressures _ in the chamber of the blowing system and at the exit of the slit) on the relative flow rate q at different values of fs and area ratio of the diffuser nar1 for α = 30o and _60o is presented in Figure 5.32. The optimum efficiency of blowoff lies within the limits q = 0.04–0.12 and, just as ηd, it also does not depend on the value of nar1. The value of ζtot can be reduced by a factor of 2–3 by boundary layer blowoff, or at the identical values of ζtot, the length of the diffuser can be diminished the same number of 95 times. This is confirmed _ by Figure 5.33 which presents the dependence of the minimum coefficient ζtot,min on fs at different values of α. Here, the symbol q designates the parameters of a diffuser with suction (length, divergence angle) with the same coefficients of resistance ζtot as a diffuser without suction at nar1 = 8. 65. A thin annular jet issuing from a peripheral slit located in the outlet section of the channel must produce a diffuser effect and thus decrease the total pressure losses at the exit of the stream into an open space. In this case, flow retardation is ensured without application of rigid walls at the exit from the channel. This provides the possibility of using such jet diffusers in installations with high-temperature jets or aggressive media as well as of reducing the axial clearance limits of various facilities.

b a Figure 5.34. Schematic diagrams of setups for investigation of jet diffusers: a) conical nozzle; b) nozzle with a rounded edge.

Flow with a Smooth Change in Velocity

313

Figure 5.35. Dependence of the static pressure recovery coefficient in a jet diffuser on jet momentum.

Frankfurt94 cites the data of experimental investigations of the degree of recovery of the static pressure and energy characteristics of jet diffusers for circular (diameter D = 122 mm), annular (ratio of the inner diameter d to the outer one, D, being equal to 0.5 and 0.75), and plane (315 × 80 mm) channels. He also performed estimation of the optimal parameters of such facilities for different shapes of the channel. Moreover, the application of jet diffusers for improving the efficiency of channels with stepwise expansion of the cross-sectional area was considered. Two type of nozzle units shown in Figure 5.34 were used to create a jet: a conical (a) and with a rounded edge (b) which deviates an annular jet in conformity with the Coanda effect near a curvilinear surface. The width of the slit was changed from 0.8 to 2.4 mm, the diverge angle of a conical nozzle was equal to 30o and 60o, the angle of the nozzle with the rounded-off edge varied from 30 to 120o. In designing the nozzle with the roundedoff edge the curvature radius of the convex surface, Rw/D = 0.18, and the slit width, ts/Rw = 0.04–0.11, were selected from the condition of a separationless flow in the wall jet on change in the angles of deviation in the range αcon/2 = 15o–90o. Figure 5.35 presents the dependence of the degree of static pressure recovery ηjd on the jet _ 2_ _ momentum coefficient = 2q /f for channels of circular and annular cross section. Here q c _ µ s = q/Q, fs = fs/F, fs is the cross-sectional area of the slit, F is the cross-sectional area of the channel, q is the flow rate through the slit, Q is the flow rate through the channel, and α is the divergence angle _ of the conical annular nozzle. The points correspond to different values of the slit width fs (from 0.03 to 0.17). The results confirm that the degree of static pressure recovery in a jet diffuser is determined by the angle of jet blowing α, geometric parameters of the channel (d/D), and by the coefficient of jet momentum cµ. As the jet momentum increases, the degree of pressure recovery increases constantly. The total pressure loss factor ζtot determined with allowance for energy losses_ on jet blow_ ing depends on the angle α, relative area of the slit fs, flow rate coefficient q, and on the geometric parameters of the channel. Investigations showed that for _ channels of circular cross section the resistance coefficient ζtot can be reduced to 0.55 at q = 0.08–0.12 and α = 30o

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.36. Dependence of the minimum coefficients of losses in jet diffusers on the slit area: a) round channel; b) plane channel; c) stepwise diffuser with area ratio 4.2; d) stepwise diffuser with area ratio 11; 1) slit nozzle; 2) nozzle with a rounded edge.

_ and to 0.65 at q = 0.14 and α = 60o. When a nozzle with a rounded edge is used, total o pressure losses decrease with increase in the angles_ of deviation of _ the jet up to αcon = 120 and in the optimum case (for a circular channel at fs = 0.05 and q C 0.09) can be reduced to ζtot C 0.4. Figure 5.36a shows the dependence of the minimum values of the coefficient of losses ζtot _ on the relative width of the slit fs for a channel of circular cross section with a conical nozzle (α = 30 and 60o) and a nozzle in which the Coanda effect is used (αcon = 30–120o). In the latter case, the efficiency of a jet diffuser is higher approximately by a factor of 1.5. Similar results for a plane channel are presented in Figure 5.36b and for channels with a stepwise change in the cross-sectional area (nar1 = 4.2 and nar1_= 11) are presented in Figures 5.36c and 5.36d. Here, in an optimum case (αcon = 120o and fs D 0.05) the resistance coefficient with allowance for energy losses on jet blowing is 2–2.5 times smaller that in the initial channel. 66. Comparison of the efficiency of the enumerated means of aerodynamic effect on the characteristic of diffusers with large divergence angles are given in Figure 5.37. Here, the

Flow with a Smooth Change in Velocity

315

Figure 5.37. Dependence of the minimum total pressure losses and of the equivalent length of diffusers with suction and tangential injection on the divergence angle: 1) diffuser without suction and injection; 2) diffuser with injection; 3) diffuser with suction; 4) jet diffusor with a conical nozzle.

minimum values of the total pressure loss factor ζtot,min are given (with allowance of the energy spent on suction or blowing) depending on the divergence angle of the diffuser (or on the angle α that characterizes jet orientation for a jet diffuser) as well as the values of the equivalent length of the diffuser lq with suction or blowing that has the same coefficient of resistance ζtot,min as in the initial diffuser at nar1 = 4. From this it follows that the tangential blowoff of the boundary layer is a more efficient means of the reduction of losses in diffuser channels with large divergence angles than suction. The efficiency of a jet diffuser with angle α = 60o corresponds approximately to the efficiency of a diffuser with boundary layer suction. 67. Guide vanes (baffles) deflect a portion of the flow with higher velocities from the central region of the diffuser to its walls into the separation zone (Figure 5.29c); as a result, the separation zone diminishes or vanishes completely. The baffles produce their greatest effect at large divergence angles. Thus, at α1 = 90–180o the resistance coefficient decreases by almost a factor of 2. Several general rules can be given when installing baffles (guide vanes) in a diffuser: • The vanes should be placed ahead of the entrance angle to the diffuser and behind it (Figure 5.29c); the number of vanes should be increased with an increase of the divergence angle. The channels between the vanes and the walls should, as a rule, contract; however, at • large divergence angles satisfactory results can also be obtained with expanding chan-

316

• • • •

Handbook of Hydraulic Resistance, 4th Edition nels. It is necessary to allow the flow to expand in the peripheral channels just as in the central channel. For divergence angle α = 90o, the relative distance h1/h2 = 0.95; for α = 180o, h1/h2 = 1.4 (Figure 5.29c). The vanes should have a small curvature and can be made of sheet metal having constant curvature and chord. The chord of the vanes can constitute 20–25% of the diameter or the height of the diffuser section. The most advantageous angle of inclination of the vanes can be selected by first placing them close behind one another and then rotating each vane through some angle until the minimum resistance of the diffuser is attained.

68. Splitters divide the diffuser with large divergence angles into several diffusers with smaller angles (Figure 5.29d). This provides both a decrease in the resistance and a more uniform velocity distribution over the section.50 The efficiency of splitters is greater, the larger the total divergence angle of the diffuser. The dividing walls or splitters are selected and installed along the whole length of the diffuser with large divergence angles in the following way: • The number z of splitters is chosen depending on the divergence angle α: α, deg

30

45

60

90

120

z

2

4

4

6

8

• The splitters are placed so that the distances a0′ between them at the inlet to the diffuser are strictly equal while the distances a1′ at the exit from it are approximately the same. • The splitters protrude before and after the diffuser parallel to the diffuser axis. The length l of the protruding parts should not be smaller than 0.1a0 and 0.1a1, respectively. 69. The rules of arranging the diffusers with shortened walls (guide vanes) following the scheme of Figure 5.29e96 are • From Figure 5.38, one finds ∆θopt (the angle between the extension of the line of the outer wall of the diffuser and the line of the displacement of the "source" M*, that is, the point at which the prolongations of the lines of all the vanes are crossed, Figure 5.39). • A fictitious divergence angle of the diffuser is calculated as

Figure 5.38. Dependence of the angle ∆θoopt, on the area ratio nar1.96

317

Flow with a Smooth Change in Velocity

Figure 5.39. Arrangement of guide vanes in a diffuser.96

α∗ = α + 2∆θopt , and the arc a–b of the circle is drawn which connects the angles of the bends of the diffuser walls (the line of transition of the flow in the throat to the flow that "radially" escapes from the source M*) over the radius r=

3a0 2α∗div

,

where a0 is the diffuser throat width and α∗div = 0.01745α* is the fictitious divergence angle of the diffuser, in radians. • The number of vanes is determined so that the divergence angle of the channels between them is approximately β=

α∗ z+1

+ 7−10o,

whence z=

α∗ −1 . 7−10

• The relative length l ′/a0′ of the vanes is determined as a function of β (Figure 5.29e): β, deg

7

8

9

10

12

l ′/a0′

20

16

12

10

9

• The width of the entrance into the diffuser is divided into z + 1 equal parts and the vanes are placed radially beginning from the points of intersection of the dividing lines with the transition line a–b; the length of the vanes is laid off from the transition line (Figure 5.29e). • The vanes in the zone of the forward edges are curved to provide a smooth transition from the throat into the expanding part of the channel.

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Handbook of Hydraulic Resistance, 4th Edition

• If the diffusers are relatively short and the length of the vanes exceeds the length of the diffuser, the vanes can be shortened to have l ′/ld = 0.6. • When it is necessary to reduce the number of vanes — for example, when the width of the inlet section is small and there is the possibility of a compressibility effect, the vane should be made shorter, since in this case the divergence angle β increases. 70. The variation of the pressure gradient is smoother in a diffuser with curved walls (Figure 5.29f), in which the rate of increase of the cross-sectional area is lower in the initial section than in the end section. This reduces the main cause of flow separation and, consequently, diminishes the main source of losses. A most advantageous diffuser, from this point of view, is one in which the pressure gradient remains constant (dp/dx = const) along the channel in potential flow. For divergence angles α = 25–90o, the losses in such diffusers can be reduced to 40% as compared with rectilinear diffusers, the reduction increasing with an increase of the divergence angle within the limits mentioned.47 When the divergence angles are small (α < 15–20o), the losses in curved diffusers become even larger than in rectilinear diffusers. Therefore, the use of curved diffusers is advisable only when the divergence angles are large. The equation of the boundary of a curved wall diffusers of a circular (or square) cross section for dp/dx = const (Figure 5.29f) has the form y=

y1 4√ 1 + [(y1 ⁄ y0)4 − 1]x ⁄ ld ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

.

The equation of the bounding wall for a plane diffuser is y=

y1 1 + [(y1 ⁄ y0)2 − 1]x ⁄ ld √ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

.

The resistance coefficient of a curved wall diffuser at dp/dx = const within the limits 0.1 < F0/F1 < 0.9 can be calculated from an approximate formula based on data from the author’s experiments:47 ζB

∆p 1.3 ⎞ ⎛ 1 ⎞ 1− = ϕ0 ⎛⎜1.43 − ⎟ ⎟ ⎜ 2 n n ar1 ⎠ ⎝ ar1⎠ ρw0 ⁄ 2 ⎝

2

where ϕ0 is a coefficient that depends on the relative length of the curved diffuser (see Diagram 5.8). 71. A marked reduction in the resistance is also attained in "radius" diffusers, where the bounding walls completely100 or partially55 follow the shape of a circular arc (Figure 5.29g, h). The resistance coefficients of diffusers with partially circular walls and equivalent angles α = 45 and 60o are similar to ζ for a longer diffuser with α = 30o without rounding. This means that it is expedient to replace the rectilinear diffuser with α = 30o by a shorter diffuser with the equivalent angles α = 45–60o, but having circular arc walls. The length of these diffusers is smaller than that of a diffuser with α = 30o by C60–100%.

Flow with a Smooth Change in Velocity

319

72. Diffusers with a preseparated turbulent boundary layer ("preseparation" diffusers) are also very efficient. An approximate calculation method has been developed by Bychkova10 and Ginevsky and Bychkova.21 Initially (just behind the entrance) they are of a bell shape, which transforms into a segment with straight walls (Figure 5.29i). Diffusers of circular cross section have a total divergence angle over this segment equal to α = 4o, while in plane diffusers this angle is α = 6o. A preseparation diffuser is a diffuser with a nonseparating flow of the minimum length. See also para. 34 of this Chapter. 73. The combination of blowing with a profiled preseparation section (the Griffith diffuser, Figure 5.29i) further decreases pressure losses and the length of the diffuser. 74. A considerable reduction in the resistance (by a factor of 2 or more) is attained by arranging transverse ribs or fins in the diffuser (Figure 2.29k).61,62 The reduction of the resistance is accompanied by equalization of the velocity profile over the diffuser section. All this is due to the fact that macroseparation of the flow from the walls is replaced by a system of minor separations (Figure 5.40), with the largest effect for circular diffusers being attained at α = 40–45o. The optimum parameters of finning are given in Figure 5.40. The lateral fins can be made flexible. The resulting vorticity and recirculation due to separation of the boundary layer rotate these fins through an angle, thus varying the efficient cross

Figure 5.40. Schematic diagram of flow in a diffuser61,62 without fins (a) and with fins (b).

Figure 5.41. Efficiency of finned diffusers: a) smooth; b) finned; 1) conical, nar1 = 5.85; 2) annular, nar1 = 6.85; 3) annular, nar1 = 8.68.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.42. Distribution of the static pressure in a diffuser involving vortex generators: 1) position of the first row of the generators; 2) calculated curve; 3) position of the second row of vortex generators; 4) outlet section of the diffuser; circular points, experiments without generators; triangular, square, and rhombic points, with vortex generators of different geometries.

section of the diffuser and preventing the distribution of reverse stream of the diffuser mouth.32 The increase in the efficiency of conical and annular diffusers with a large area ratio attained by finning the surface can be illustrated by Figure 5.29m. The corresponding experimental dependences of the pressure recovery coefficient for such diffusers with a smooth and finned surface on the mean velocity in the inlet section are presented in Figure 5.41.65 75. An efficient method of reducing total pressure losses in diffuser channels is based on using of generators of longitudinal vortices (Figure 5.29l).65,203–207 They consist of a system of plane plates with the height commensurable with the boundary layer thickness in the inlet section of the diffuser or somewhat exceeding it. These plates are located around the entire perimeter of the inlet section of the channel at a certain angle of attack α to the free stream flow, and in this position they generate a system of longitudinal vortices of the same sign. In a number of cases, the generators that pairwise form a system of plates with angles of attack ±α turn to be more efficient: they pairwise produce a system of vortices with opposite rotation (Figure 5.29l). The interaction of a system of vortices with the main stream leads to the enhancement of exchange in the boundary layer and to its delayed separation. Generators of longitudinal vortices ensure a substantial decrease in the total pressure losses in subsonic diffusers with large area ratios and, correspondingly, an increase in the static pressure in the diffuser (Figure 5.42). 76. In a stepped diffuser (Figure 5.29j), in which a smooth change in the cross-sectional area is followed by a sudden expansion, the main losses (shock losses) occur even at relatively low velocities.

321

Flow with a Smooth Change in Velocity

As a result, the losses in the diffuser are greatly reduced (by a factor of 2–3). The coefficient of total resistance of a stepped diffuser of circular or rectangular cross section can be approximated47 from: ζB

∆p

⎛ λ =⎜ 2 ρw0 ⁄ 2 ⎜ 8 sin α ⎜ 2 ⎝

q2 + 1

α⎞ + k1 tan1.25 ⎟ × (1 − 1 ⁄ q2)2 + (1 ⁄ q2 − 1 ⁄ nar)2 , 2⎟ q −1 ⎟ ⎠ 2

where q = 1 + 2(l/Dh) tan α/2; k1 = 3.2 for circular diffusers; k1 C 4–6 for rectangular diffusers;* nar = F2/F0 is the total area ratio of the stepped diffuser (the ratio of the widest part of the diffuser to its smallest part, see Figure 5.29j). 77. The total resistance coefficient of a plane stepped diffuser can be calculated approximately:47 ζB

∆p ρw20 ⁄ 2

q2 ⎞ ⎡ λ ⎛ a0 q1 1.25 =⎢ + ⎜b ⎟ + 3.2 tan α α α 0 ⎢ ⎜ ⎟ ⎢ 8(ld ⁄ a0) tan 2 ⎜ tan 2 sin 2 ⎟ ⎣ ⎝ ⎠

α⎤ 2 ⎥⎥ ⎥ ⎦

× (1 − 1 ⁄ q1)2 + (1 ⁄ q1 − 1 ⁄ nar)2 , where q1 = 1 + 2(ld ⁄ a0) tan q2 = 1 + (ld ⁄ a0) tan

α , 2

α , 2

(b0 is constant along the diffuser length). 78. For each area ratio n and each relative length ld/Dh (or ld/a0) of the stepped diffuser there is an optimum divergence angle αopt at which the total resistance coefficient is minimum (see Diagrams 5.9 through 5.11). It is recommended that stepped diffusers be used with the optimum divergence angles. The resistance coefficient of such diffusers is

ζ*

∆p ρw20 ⁄ 2

= ζmin ,

where ζmin is the minimum resistance, which depends on the relative length of the smooth part of the diffuser ld/Dh (or ld/a0) and the total area ratio n of the stepped diffuser (see Diargams 5.9 through 5.11).

∗ The curves in Diagram 5.10 were ca1culated for k1 = 6.0 which gives a certain safety factor in the calculation.

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Handbook of Hydraulic Resistance, 4th Edition

79. The limiting divergence angle αmin of the smooth part of the stepped diffuser, that is, the angle at which the steps cease to have influence at the given overall area ratio nar and relative length ld/Dh (or ld/a0) of the smooth part, is given by tan

αlim ⎯√⎯⎯ n⎯ − 1 , = ar1 2 2ld ⁄ Dh

and correspondingly tan

αlim nar1 − 1 . = 2 2ld ⁄ a0

When the relative length ld/Dh (ld/a0) of the stepped diffuser is selected in practice, it is advisable to use not the minimum value ζmin, but a value about 10% higher, which allows a considerable reduction in the length of the diffuser without noticeably increasing the losses in it. The optimum values of ld/Dh (ld/a0) are given in graphs a of Diagrams 5.9 through 5.11 as a dashed line. 80. In addition to the methods of increasing the efficiency of diffuser channels (Figure 5.29) mention should also be made of other methods based on the use, for the purpose, of solid or permeable screens and nets installed at a certain optimal distance from the outlet section of the diffuser as well as permeable screens installed inside a diffuser.128,162,197–201 The nets can prevent separation and ensure flow reattachment and can increase the stability of flow in a diffuser. 81. When a diffuser is installed behind a fan, one should take into account that there is a greate difference between the flow patterns at the fan exit and at the entrance into an isolated diffuser preceded by a straight segment of constant cross section. As a rule, the velocity profile downstream of a centrifugal fan is asymmetric due to a certain deflection of the flow in the direction of fan rotation. The velocity profile depends on both the type of the fan and the mode of its operation characterized by the relative flow rate Q/Qopt, where Qopt is the flow rate at maximum efficiency of the fan. 82. The flow deflection to the periphery of fan rotation allows the use of diffusers with larger divergence angles than conventionally used behind centrifugal fans. In this case, it is advisable that the plane diffusers with divergence angles α > 25o be made asymmetric so that the outer wall either would be a continuation of the housing or would deviate no more than 10o to the side of the housing, while the inner wall follows along the side of the impeller. Displacement of the diffuser axis toward the side of the fan housing is not expedient, since the resistance of such diffusers at α > 15o is 2–2.5 times higher than that of symmetrical diffusers whose axis is displaced toward the impeller axis.58 83. The resistance coefficient of plane diffusers with divergence angles α < 15o and of pyramidal diffusers with α < 10o, installed downstream of centrifugal fans of any type under any working conditions, can be calculated approximately from the data given above for isolated diffusers with the following velocity ratio taken for their inlet section: wmax + 1.1 . w0

323

Flow with a Smooth Change in Velocity

Figure 5.43. Schematic diagram of an axial-annular diffuser.

When the divergence angles of the diffusers are larger than 10–15o, the values of ζ obtained for isolated diffusers should not be used; the values of ζ should be determined from Diagrams 5.13 through 5.18. These data are applicable in practice for Q = Qopt and Q ã Qopt. 84. When there is a lack of space following a centrifugal fan, a stepped diffuser can be installed which is much shorter than a straight diffuser for the same resistance. The optimum divergence angle of the diffuser at which the minimum resistance coefficient is obtained can be determined from the corresponding Diagram 5.18. 85. Axial-annular diffusers are installed in the outlet section of axial compressors and fans. Figure 5.43 shows the schematic diagram of tested axial-annular diffusers and their main geometric parameters. Depending on a change in the conditions of operation of a compressor or a fan, the velocity profiles in their outlet sections change and, correspondingly, they change in the inlet section of the diffuser. The above-adopted schematic representation of flow in a diffuser, viz., a potential core with a constant velocity across the flow and a boundary layer on the wall, cease to be valid, because here, outside the boundary layer on the convex and concave walls of the diffuser, the velocity profiles in the "flow core" are variable along the radius under the modes of operation of the compressor or the fan that do not allow of calculation. Figure 5.44 presents distributions of the velocity _ downstream of the model of a compressor at different values of the flow rate coefficient ca. The figure also contains the corresponding values of the coefficient of radial nonuniformity ϕ and of the coefficient of kinetic energy α _ for each velocity profile, with ϕ =

1−

r2

⎛ ca 2 _ ⎜ cam d r ⎝ 1

2_

∫

_ _ ⎞_ _ − 1⎟ rdr, where r1 and r2 correspond to the ⎠

points on the velocity profile; ca/cam = 1, and cam is the average value for the given profile. The diffusers indicated were tested on a rig when they had been connected to a plenum chamber with wire nets inside that created a nonuniform velocity profile in the inlet section of the diffuser.192,193 Figure 5.45 presents the dependences of the efficiencies of axial-annular diffusers installed downstream of a compressor on the coefficient of radial nonuniformity ϕ.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.44. Distribution of velocities downstream of a compressor model.

The dashed curves in the figure show the corresponding dependences obtained in testing the same diffusers but under the conditions when the inlet nonuniformity was created by the resistance from wires. The character of the curves obtained by both methods is approximately the same. The works indicated also cite dependences of the coefficients of total pressure losses ζd and of the efficiency of diffusers ηd1 on the coefficient of nonuniformity ϕ derived in tests on a rig in which a wire net of variable resistance was used to create radial nonuniformity of the flow.

Flow with a Smooth Change in Velocity

325

Figure 5.45. Dependence of the efficiency of axial-annular diffusers installed downstream of a compressor on the coefficient of radial nonuniformity.

The above data should be kept in mind when designing units for the cases where the efficiency of transformation of a dynamic pressure in a diffuser exerts a notable effect on the economical efficiency of the unit. 86. To convert the dynamic pressure downstream of the outlet vane ring of axial turbomachines (fans, compressors, turbines), wide use is made of annular diffusers, which are made with rectilinear boundaries (axial-annular diffuser, Figure 5.46), with curvilinear boundaries (radial-annular diffuser, Diagram 5.20), or combined (axial-radial annular diffuser, Diagram 5.20). The area ratio of the axial-annular diffuser is determined from the formula given in Diagram 5.19, and that of the radial-annular diffuser is determined from the formulas of Diagram 5.20. 87. The internal resistance coefficients* ζin B ∆p/(ρw20/2) = ∆p(ρca/2) of the axial-annular diffuser with positive angles α1, depend on the area ratio nar1 at the given d0, and have in ∗ ′ , here and later, we mean the ratio of the difference By the internal resistance coefficients ζin and ζin of total pressures at the entrance and directly at the exit from the diffuser to the inlet velocity pressure, regardless of the additional losses that might have occurred in the straight exit section behind the diffuser due to equalization of the velocity profile distorted during the passage through the diffuser.

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Handbook of Hydraulic Resistance, 4th Edition

_ practice only one curve for _ _ each value of ld = L/D0. This kind of dependence of ζin on nar1 ′ at d0 = 0.650–0.688 and ld = 0.5–2.0 is shown in graph a of Diagram 5.19. The curves ζin _ = f(nar1) for ld = 0.5 and 1.0 were constructed from the experimental data of Dovzhik and Morozov,40 while the remaining values were constructed to approximately account for both 40 the experimental data of Dovzhik and Morozov and the experimental data of Bushel.9 _ Within the limits 2 < nar1 < 4 and 0.5 < ld < 2.0 the following interpolation formula can be used: ζin′ C

2 0.25n _ ar1 . ld0.5

When the velocity at the inlet to the axial-annular diffuser is nonuniform or when the diffuser is installed downstream of the operating axial machine, the internal resistance coefficient is determined as ζin B

∆p

ρw20 ⁄ 2

C kd ⁄ ζin′ ,

where kd is the correction factor (see Diagram 5.1 or 5.19). 88. The resistance coefficient of an axial-annular diffuser with a converging back fairing (see Diagram 5.19) can be determined from ζin B

∆p

2

F0 ⎞ ⎛ = kdϕd ⎜1 − ⎟ , 2 F 1⎠ ρw0 ⁄ 2 ⎝

where ϕd is the total* shock coefficient determined, depending on the divergence angle α, from graph b of Diagram 5.19.

Figure 5.46. Axial-annular diffuser. ∗

The total shock coefficient takes into account the total losses in a diffuser.47,49

Flow with a Smooth Change in Velocity

327

89. The experimental investigations of twisting a flow at inlets to conical diffusers that were carried out in wide ranges of divergence angles (α = 4–30o) and area ratios nar1 = 1.3– 8.3 have shown that the twisting exerts a double effect on the diffuser characteristics and depends strongly on the mode of flow in a diffuser in the absence of twisting. The flow twisting at the inlet influences the characteristics of the diffuser if the original untwisted flow is either separationless or contains minor separation zones. However, in conical diffusers, the flow in which (with an axial flow at the inlet) contains large separation zones, twisting considerably improves the characteristics and suppresses separation.210 90. In annular diffusers installed downstream of axial compressors or fans the presence of twisting should sometimes be taken into account. A twisted flow is characterized by the presence of a circumferential (tangential) velocity cu in addition to the axial one ca (Figure 5.47). In References 162, 197, 198, data were obtained on the influence of flow twisting on the properties of the axial-annular diffuser channels connected to an axial compressor. The tests in Reference 197 were carried out at an axial velocity cam = 60–70 m/s, with the Reynolds number being equal here to Re = cam⋅h/ν = (6–7) × 105, where h is the height of the axial channel in the inlet section of the diffuser. Flow twisting was produced by blading (No. 1) with straight _blades and blading (No. 2) with the blades profiled following the constant cir_ culation law r⋅cu = const; in both cases, the angle of twisting α1 was practically constant along the radius. Figure 5.48 presents the results of tests of a compressor with an axial-annular diffuser installed downstream of it. The figure contains the dependences of the coefficients of losses in the diffuser ζ and ζtot, flow twisting angle in the outlet section of the diffuser α2, and parameter ψ = tan α2/tan α1 on the angle of twisting α. 91. The present chapter considers one type of radial-annular diffuser with the outline of the curvilinear part constructed following a circular arc with R1/h0 = 1.5 and R0/h0 = 2.0 (see Diagram 5.20) and one type of axial-radial-annular diffuser with the outline of the curvilinear part constructed following an elliptical arc (see Diagram 5.20 and Figure 5.49) having semiaxes:

Figure 5.47. Axial-annular diffuser with a swirled flow.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.48. Efficiency of the axial-annular diffuser installed downstream of the compressor.

a = Ls − (Lin − b sin α) and b=

D1 − r0 − Lin tan α , 2

where α = (α1 + α2)/2. The axial line was assumed to be the locus of centers of the circles inscribed in the diffuser outline while the diameters of these circles varied along the axial line from the initial

Flow with a Smooth Change in Velocity

329

Figure 5.49. Schematic diagrams of construction of an axial-radial-annular diffuser.

diameter h0 to the final diameter h1, linearly. The relative diameter of the hub at the entrance to diffusers of both types is d0 = 0.688. 92. The internal resistance coefficients ζin of the above types of diffusers_*_ are given in Diagram 5.20 as functions of the area ratio n1_ at different values _ of "radiality" __ D = D1/D0 for two cases: with an operating compressor at ca = 0.5 (where ca = ca0/u = w = 4Q/[π(D20 – d02)u]; u is the circumferential velocity of the compressor blades at the external radius, m/s; and Q is the flow rate, m3/s) and with an idle compressor. The value of ζin for an operating compressor exceeds the corresponding value of ζin for an idle compressor (turbomachine) by 15–20%. The resistance coefficient of the diffusers under consideration depends on the mode of compressor operation, that is, on the discharge coeffi_ cient ca0 (see Dovzhik and Ginevsky).39 93. A combined diffuser, that is, an axial-radial-annular diffuser in which a radial bend follows a short annular diffuser is somewhat better. In this diffuser, the radial turn is achieved at lower stream velocities, and therefore the pressure losses are somewhat lower. At the same time, the axial dimensions of such a diffuser are much larger than those of a radial-annular diffuser. 94. The resistance of an annular diffuser, like that of the conventional ones, can be noticeably reduced by installing one or several splitters or guiding surfaces, which would divide the diffuser with large α into several diffusers with smaller values of α and would generally regulate the flow in the diffuser. Just as with conventional diffusers, these guiding surfaces are efficient only at large divergence angles and at definite combinations of the angles α1 and α2, that is, combinations at which the resistance coefficients of the diffusers without these surfaces are largest.36,39,40 95. Different mechanical systems involving such machines as pumps, turbines, and compressors require, besides velocity retardation and turning of the flow, that the supplying channels be of small overall size. This is achieved in diffuser elbows or (which is the same) in ∗

For the values of the total resistance coefficients ζfr, see Chapter 11.

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Figure 5.50. The values of η1d and wmax/w1 for curved-axis diffusers at nar1 = 4; ld/D0 = 7.15 (α = 8o); β = 15 and 30o; l0/D0 = 0.35; 2δ∗0 ⁄ D = 0.51%; Re = 5.2 × 105 and for straight-axis diffusers with flow-straightening elbows.172,173

curved axis diffusers (Diagram 5.21). The flow in such diffusers is much more complex than in straight axis diffusers and is composed of (1) the flow in a straight-axis diffuser and (2) the flow in a curved channel of constant cross section. The latter is accompanied by secondary currents due to nonuniformity of the velocity and pressure field in the direction normal to the plane of the turn and to the existence of boundary layers at the channel walls (see Chapter 6). These factors contribute to an earlier flow separation and cause pressure losses different from those in straight diffusers. Besides the parameters mentioned in paragraph 11, the resistance of the curved-axis diffuser is also affected by the angle of curvature of the axis β and the relative curvature radius of the axis R/D0 (r/b0). 96. The internl resistance coefficients ζ′in of plane curved-axis diffusers of constant length (ld/b0 = 8.3), most often encountered in multistage pumps, are given in Diagram 5.21 as a function of the divergence angle α* for four values of the relative curvature radius of the inner side wall of the diffuser: r/b0 = ∞, 22.5, 11.6, and 7.5. These data were obtained by Polotsky69,70 for diffusers installed direct1y behind a smooth inlet collector, that is, at l0/b0 = 0. 97. The internal resistance coefficients of spatial curved-axis diffusers with exit sections of different shapes (circle, ellipse with a larger axis in the curvature plane, ellipse with a smaller axis in the curvature plane; see Diagram 5.29) at constant length (ld/D0 = 7.15) and area ratio (nar1 = 4)** and different bending angles (β = 0, 15, and 30o; R/D0 = ∞, 27.30, and 13.65 are given in Diagram 5.22.*** Some data are given for constant values of the Reynolds number (Re = 5.2 × 105), and some as a function of Reynolds number. In all the cases the dif∗

For a circular diffuser the divergence angle α = 8o.

∗∗

By the divergence angle or a curved-axis diffuser we mean an angle made by the side walls of a straight diffuser obtained by "unbending" a curved axial diffuser. ∗∗∗ These data were obtained on the basis of an approximate recalculation of the values of the efficiency taken from Shiringer’s experimental work.173

Flow with a Smooth Change in Velocity

331

fusers were tested when they were installed downstream of a smooth inlet collector with a small straight section (l0/D0 = 0.35). 98. In some types of curved-axis diffusers secondary currents can also exert a positive effect as they transport a portion of the moving medium from the region with larger kinetic energy to the boundary layers which are affected by separation. In this case the resistance coefficient of a curved diffuser becomes noticeably smaller than that of a straight diffuser with the same parameters [compare the curves ζ = f(Re) for diffusers 9 and 10 of Diagram 5.22]. 99. In some cases, a curved-axis diffuser can be replaced by a straight diffuser with an elbow having guide vanes. The effect on the resistance is evident from the data given in Figure 5.50.

Converging Nozzles in the System 1. Transition from a larger section to a smaller one through a smoothly converging section — converging nozzles — is also accompanied by comparatively large irreversible losses of total pressure. The resistance coefficient of a converging nozzle with rectilinear boundaries (Diagram 5.23) depends on the convergence angle α and the area ratio n0 = F0/F1 (and correspondingly on the relative length lcon/D0), while at small Reynolds numbers it also depends on Re. 2. At sufficiently large angles (α > 10o) and area ratios (n0 < 0.3), the flow, after passing from the contracting section of a rectilinear converging nozzle to the straight part of the tube, separates from the walls, which is the main source of the local losses of total pressure. The larger α and the lower n0, the stronger is the flow separation and the greater the resistance of the converging nozzle. The resistance is naturally highest at α = 180o, that is, when there is a sudden contraction in the cross section (see Figure 4.12). The friction losses occur along the length of the contracting section. 3. For engineering calculations it is convenient to represent the general resistance coefficient of converging nozzles as

ζB

∆p ρw20 ⁄ 2

= ζloc + ζfr .

The local resistance coefficient of a converging nozzle136 is

ζloc B

∆p ρw20 ⁄ 2

= ⎛−0.0125n40 + 0.0224n30 − 0.00723bn20 + 0.00444n0 − 0.00745⎞ ⎝ ⎠

× ⎛α3ρ − 2πα2p − 10αp⎞ , ⎠ ⎝ where αp = 0.01745α rad (α in degrees). The friction resistance coefficient ζfr of a contracting section is determined from Equations __ sec(5.9) and (5.9)–(5.12), where λ is assumed to be approximately constant along the entire tion, but dependent on Re at the entrance and on the relative roughness of the walls ∆. Diagram 5.23 also contains the values of the total resistance coefficient ζ obtained experimentally by Yanshin100 at Re = 5 × 105.

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Handbook of Hydraulic Resistance, 4th Edition

4. Within the limits 10 < α < 40o, the general resistance coefficient of the converging nozzle with rectilinear boundaries has a minimum which, at least at Re ≥ 105, remains practically constant and is equal to ζ ≈ 0.05. 5. The resistance of converging nozzles can be greatly diminished by providing a smooth transition from a larger section to a smaller one with the help of curvilinear boundary walls (following the arc of a circle or any other curve; see Diagram 5.23), as well as by bending rectilinear walls of the converging nozzles at the exit into the straight section (thin lines in scheme a of Diagram 5.23). With smooth contraction of the cross section, when the contraction angle is very small (α < 10o) or when the contracting section has very smooth curvilinear walls (see scheme b of Diagram 5.23), the flow does not separate from the walls at the place of transition into a straight section, and the pressure losses reduce to only friction losses in the contracting portion:

ζ*

∆p ρw20 ⁄ 2

B ζfr .

6. At very small Reynolds numbers (1 < Re < 50) the resistance coefficient of converging nozzles, like that of diffusers,4 is

ζ*

∆p ρw20 ⁄ 2

=

A . Re

Within the limits 5 ≤ α ≤ 40o, tan−0.5 . A = 20.5n−0.5 0

Transition Sections 1. There are two kinds of transition sections: (1) those with a variable cross section along the flow, with the shape of the cross section kept constant and (2) those in which both the cross section and its shape vary. 2. The first type includes, in particular, converging-diverging transition pieces (Diagram 5.25). According to experiments carried out by Yanshin100 the optimal parameters of transition pieces in the form of a converging nozzle with rectilinear walls are as follows: αcon = 30−40o and αd = 7−10o . For a converging nozzle with curvilinear walls the optimal bending radius is Rcon = 0.5– 1.0D0. 3. The resistance coefficient of converging-diverging transition pieces of annular cross section, like that of conventional diffusers, depends on the relative length of the intermediate straight section l0/D0 and on the area ratio F1/F0 and can be determined from ζB

∆p ρw20 ⁄ 2

= A (k1k2ζ1 + ∆ζ) ,

(5.17)

Flow with a Smooth Change in Velocity

333

where ζ1 is the resistance coefficient of the transition piece with a smooth (curvilinear) converging section at l0/D0 = 1.0, which is determined at Re = w0D0/v ≥ 2 × 105 from curves ζ1 = f1(αd) of Diagram 5.25 plotted on the basis of Yanshin’s data.100 At Re = 2 × 105, for coefficient ζ1 of an annular diffuser, see ζ in Diagram 5.2; k1 = ζ1n/ζ1n≥4 is the ratio of the coefficient ζ1 at n1 = F1/F0 < 4 to its value at n1 ≥ 4, see curves k1 = f2(αd, F1/F0) of Diagram 5.5.25; k2 is the correction for the effect of the relative length l0/D0; within the limits 0.25 ≤ l0/D0 ≤ 5.0 k2 C 0.66 + 0.35

l0 ; D0

(5.18)

∆ζ is an additional term which considers the effect of l0/D0; A = 1.0 is for transition with a smooth converging diffuser; A = f(αd) for transition with a converging diffuser having rectilinear generatrices (see Diagram 5.2.5). 4. For transition pieces of rectangular (square) cross section and plane transition pieces (for which both contraction and divergence of the cross section occur in one plane), the resistance coefficient can be roughly determined from Equations (5.17) and (5.18), but with ζ1 replaced by ζd of the diffuser at l0/D0 = 0 from Diagrams 5.4 and 5.5, respectively. 5. In transition pieces that connect tubes of circular and rectangular cross sections (see Diagram 5.27), conversion of an axisymmetric flow into a plane one (and vice versa) is accompanied by its deformation in two mutually perpendicular planes — by expansion in one plane and contraction in the other.84 Such a complex flow can simultaneously exhibit the characteristic effects of the diffusers and converging sections. If a longer side of the rectangular section exceeds the diameter of a circular tube (b1 > D0), stalling phenomena can occur, leading to large pressure losses. Therefore, the transition piece of the type considered should have such a length and a shape as to prevent the possibility of flow separation or to displace it into the region having lower flow velocities. This can be achieved by proper selection of the geometric shape and overall dimensions. 6. As to the shape of the walls forming the transition pieces, the latter can be divided into three characteristic types (Figure 5.51). Type A is obtained when the truncated circular cone (with rectilinear boundaries) is intersected by planes. Type B is constructed on the basis of a linear change of the cross-sectional areas along the length of the transition pieces; in this case, in the plane of symmetry, parallel to the longer side of the rectangle, the boundaries of the transition pieces are rectilinear. Type C, like type B, retains a linear change of variation of areas over the greater part of the length of the transition pieces, but at the same time provides a more uniform distribution of the average velocity at each location. All the boundary walls in these transition pieces are curvilinear. 7. In the transition pieces of type A, provided b1 > D0, a nonseparating flow can be produced near the diverging walls at α = 20–30o. In this case, the length of the converging transition segment should be asssumed to be equal to: at b1 > 1.5D0, lcon ≈ 1.8(b1 > D0), and at b1 ≤ 1.5D0, lcon = 1.5b1. The length of the converging transition pieces of types B and C can be reduced by a factor of 1.5–2 as compared with the length of the transition piece of type A. Tentatively, the length of these transition pieces is

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Handbook of Hydraulic Resistance, 4th Edition

Figure 5.51. Transition pieces having different shapes of the boundary walls.84

lcon C (1 − 1.5)b1 . In cases where b1/D0, the walls of the converging transition pieces taper and then their length should correspond to the optimum angle of convergence, as in annular converging nozzles, that is, α = 40–50o. In this case, lcon C 1.1 (D0 − a1) C D0 . 8. An increase in the length of converging transition pieces leads to an increase in friction losses, while a decrease in their length causes an increase in the resistance due to flow separation from diverging walls. 9. Diverging transition pieces, like plane or conical diffusers, also have an optimal length for which there is a minimum hydraulic resistance. It is most important here to prevent separation at the inlet portion of the unit. To achieve this, the complete angle between the diverging walls at the beginning of the transition piece should not exceed 8–10o. Compliance with this condition in transition pieces of type A leads to their comparatively large length, corresponding at α = 10o to ld C 5.7 (D0 − a1) C 6D0 . Therefore, to make the length of the diverging transition pieces shorter, type B or C sshould be used, and their length should be equal to ld C (3 − 4)D0 .

Flow with a Smooth Change in Velocity

335

10. When the relative width of the rectangular cross section is small (b1/D0 < 2), transition pieces of type B should be used. The walls adjacent to the longer side of the rectangular section should in this case be made curvilinear, while the walls adjacent to a shorter side should be left rectilinear. At b1/D0 > 2, diverging transition pieces of type C should be used. 11. Dimensions of any cross section along the length of the transition piece of type B can be determined analytically from Fx = F1 − (F1 − F2)

x , l

Fx = 4axbx − (4 − π) a2x ,

(5.19)

x bx = D0 + (b − D0) , l where Fx is the cross-sectional area at a distance x from the entrance. 12. The cross-sectional dimensions of transition pieces of type C can be determined from Equation (5.19), using them separately for each of the three characteristic parts of the transition piece shown in Figure 5.51.* For example, when the middle part I is calculated (see Figure 5.51), the width bxI = D0 is known, while the size axI is determined from the relation F = f(x/l). When parts II are calculated, the size axII = axI is known, while the size bxII is determined from the relation F = f(x/l). The total width of any cross section should be equal to bx = bxI + bxII . 13. As for conventional (plane and axisymmetric) diffusers and converging nozzles, the hydraulic resistance of the transition pieces considered depends on the geometric parameters (area ratio nar and relative lengths of the transition pieces ld/D0 and lcon/D0), on the flow pattern (Reynolds number) and inlet conditions. Moreover, the ratio of the sides of the rectangular cross section b1/a1, the shape of the walls of the transition pieces, and the method of variation of the cross-sectional areas over the length are important parameters for the above transition pieces. 14. The resistance coefficient of the transition pieces considered can be determined from the interpolation formula of Tanaev:84

ζ*

∆p

Re ⎞ = ζsim + A exp ⎛⎜−k2 ⎟, Re sim ⎠ ρw0 ⁄ 2 ⎝ 2

(5.20)

where the numerical coefficients A and k2 depend on the method of variation of the values and shape of cross-sectional areas over the length of the transition piece and on the ratio b1/a1; ζsim is the resistance coefficient of the similar flow regime (Re ≥ Resim = 5 × 105); w0 is the average flow velocity in a circular cross section of the transition piece; A = Ad ≈ 0.5 and k2 = kd = 5.0 for a diverging transition piece; A = Acon ≈ 0.3 and k2 = kcon ≈ 5.0 for the converging transition piece. ∗

In important cases, specification and final selection of the optimal shapes and dimensions should be made on the basis of experiments.

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Handbook of Hydraulic Resistance, 4th Edition

The first term on the right-hand side of Equation (5.20) is b1 ⎞ ⎛ ζsim = ⎜c0 + c1 ⎟ n20 , a1 ⎠ ⎝ where c1 depends on the length and shape of the transition piece. For transition pieces with a linear method of variation of the areas, values of c1 are given in graph b of Diagram 5.27. The coefficient c0 is the resistance of the tube portion of constant cross section, the length of which is equal to the length of the transition piece. Its value is c0 =

λl , Dh

where Dh is the average value (over the length of the transition piece) of the hydraulic diameter. At the length of the transition pieces l/D0 ≈ 3.5, it can be tentatively assumed that c0 ≈ 0.06, c1d ≈ 0.01 — for the diverging transition piece, and c1con = 0.002 — for the converging transition piece. The quantity n0 = F0/F1, where F0 is the area of the circular cross section of the transition piece and Fl = a1/b1 is the area of the rectangular cross section. 15. The resistance coefficients of transition pieces in which a rectangular cross section with small aspect ratio (a1/b1 ≤ 2.0) changes into an annular cross section, or vice versa (see scheme of Diagram 5.28), can be determined from the data for diffusers of rectangular cross section with equivalent divergence angles. The equivalent angle αe is determined from the following expressions: for transition of a circle into a rectangle tan

a1b1 ⁄⎯π − D0 ⎯⎯⎯⎯⎯⎯ αe 2 √ ; = 2 2ld

for transition of a rectangle into a circle tan

a0b0 ⁄⎯π ⎯⎯⎯⎯⎯⎯ αe D1 −2 √ . = 2 2ld

337

Flow with a Smooth Change in Velocity

5.2 DIAGRAMS OF THE RESISTANCE COEFFICIENTS Diffusers. Determination of the inlet conditions (kd)51–55,127

Dh =

Diagram 5.1

4F 0 Π0

1. When wmax/w or 2δ∗0 ⁄ Dh is known in the symmetrical velocity field upstream of a diffuser (scheme 1), the relative length l0/Dh is determined from the curves wmax/w0 = f1(l0/Dh) (graph a) or alternatively from the curves 2δ∗0 ⁄ Dh = f2(l0/Dh) (graph b); then, from these values of l0/Dh, the value of kd is determined from the appropriate diagrams. 2. For a free jet (working section of a wind tunnel (scheme 2), wmax/w0 = f3(lw.s./Dh) (graph c) is determined using the known length lw.s./Dh, then l0/Dh is determined from graph a, and finally kd = f(l0/Dh) is determined from the appropriate diagrams. 3. When diffusers (of any shape) with α = 6–14o are installed downstream of a branch pipe (scheme 3) or any other curved parts with similar velocity profiles upstream of the diffuser (graph d) kd = f(w/w0, R0/Dh, l0/Dh, z) is taken from the table below.

Parameters of the branching pipe

Shape of velocity profiles (graph d)

R0%Dh

l0%Dh

1 2 – – – –

0.8–1.0 0.8–1.0 0.8–1.0 2.0 2.0 ≥3.0

0 0 0 0 1.0 0

Number of concentric splitters z 0 2 3 0 0 0

kd 6.8 2.1 1.9 2.6 1.0 1.0

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Handbook of Hydraulic Resistance, 4th Edition

Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

1. Uniform velocity field at the entrance into the diffuser (wmax/w0 = 1.0 or l0/D0 ≈ 9):* ζ*

nar1 =

F1 , F0

Re =

∆p ρw20 ⁄ 2

= ζd = f (α, n ar 1, Re) ,

see the table and the curves of graph a (for approximating formulas see para. 38–40 of Section 5.1). Calculation of the preseparation diffusers is considered in para. 34 of Section 5.1).

w0D0 v

2. Nonuniform velocity field at the entrance into the diffuser (wmax/w0 > 1.0, 2δ∗0 ⁄ D0 > 0 or l0/Dh > 0):

⎛ 2δ0∗ l0 ⎞ = kdζd; for diffuser downstream of a straight section: kd = f ⎜α, , , nar1, Re⎟ , see the tables and ⎝ D0 D0 ⎠ graphs b and c; for diffusers behind a free jet (working section of a wind tunnel): ζ*

∆p

ρw20 ⁄ 2

2δ0∗ wmax ⎛ wmax ⎞ ⎛ l0 ⎞ ⎛ l0 ⎞ , nar1, Re⎟ , see the tables and graphs b and c, where = f1 ⎜ ⎟ or = f1 ⎜ ⎟ , kd = f ⎜α, w w D D 0 0 0 ⎝ ⎠ ⎝ 0⎠ ⎝ D0 ⎠ see Diagram 5.1; for diffusers with α = 6–14o downstream of a shaped (curved) piece kd = f(w/w0), see the table of Diagram 5.1.

ζd at l0%D0 = 0 Re × 10−5

α, deg 3

4

6

0.148 0.120 0.093 0.079

0.135 0.106 0.082 0.068

0.121 0.090 0.070 0.056

8

10

12

14

0.112 0.083 0.068 0.048

0.107 0.080 0.062 0.048

0.109 0.088 0.062 0.048

0.120 0.102 0.063 0.051

45

60

90

120

0.331 0.297 0.279 0.271

0.326 0.286 0.268 0.272

0.315 0.283 0.268 0.272

0.308 0.279 0.265 0.268

nar1 = 2 0.5 1.0 2

≥4 Re × 10−5

α, deg 16

20

30 nar1 = 2

0.5 1.0 2

≥4

*

0.141 0.122 0.073 0.051

0.191 0.196 0.120 0.068

0.315 0.298 0.229 0.120

Here and later on l0/D0 = 0 means that the diffuser is installed directly following a smooth collector (inlet nozzle).

339

Flow with a Smooth Change in Velocity Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

α, deg

Re × 10−5

3

4

6

8

10

12

14

0.151 0.119 0.096 0.079 0.089

0.157 0.120 0.096 0.082 0.080

0.174 0.131 0.107 0.090 0.107

0.197 0.155 0.120 0.107 0.135

nar1 = 4 0.5 1.0 2 4

≥6

Re × 10−5

0.197 0.154 0.120 0.101 0.101

0.180 0.141 0.112 0.091 0.091

0.165 0.126 0.101 0.085 0.085

α, deg 16

20

30

0.225 0.183 0.146 0.124 0.169

0.298 0.262 0.180 0.172 0.240

0.461 0.479 0.360 0.292 0.382

45

60

90

120

180

0.680 0.628 0.586 0.562 0.560

0.643 0.600 0.585 0.582 0.582

0.630 0.593 0.580 0.577 0.577

0.615 0.585 0.567 0.567 0.567

nar1 = 4 0.5 1.0 2 4

≥6

0.606 0.680 0.548 0.462 0.506

α, deg

Re × 10−5

3

4

6

8

10

12

14

0.168 0.126 0.101 0.084 0.079

0.179 0.132 0.101 0.087 0.080

0.200 0.159 0.118 0.104 0.098

0.240 0.193 0.151 0.151 0.137

nar1 = 6 0.5 1.0 2 4

≥6

Re × 10−5

0.182 0.153 0.128 0.106 0.092

0.170 0.144 0.118 0.095 0.090

0.168 0.131 0.109 0.090 0.080

α, deg 16

20

30

45

60

90

120

180

0.766 0.755 0.700 0.660 0.690

0.742 0.731 0.710 0.696 0.707

0.730 0.720 0.708 0.695 0.700

0.722 0.707 0.690 0.680 0.695

nar1 = 6 0.5 1.0 2 4 ≥6

0.268 0.218 0.185 0.160 0.160

0.330 0.286 0.280 0.224 0.286

0.482 0.488 0.440 0.360 0.456

0.640 0.680 0.640 0.510 0.600

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Handbook of Hydraulic Resistance, 4th Edition

Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

α, deg

Re × 10−5

3

4

6

8

10

12

14

0.190 0.156 0.123 0.100 0.085

0.220 0.162 0.134 0.106 0.086

0.227 0.184 0.151 0.128 0.114

0.256 0.212 0.167 0.160 0.160

nar1 = 10 0.5 1.0 2 4

≥6

Re × 10−5

0.195 0.160 0.123 0.100 0.085

0.181 0.156 0.120 0.097 0.084

0.184 0.155 0.120 0.097 0.084

α, deg 16

20

30

45

60

90

120

180

0.800 0.800 0.800 0.735 0.760

0.834 0.820 0.806 0.804 0.825

0.840 0.820 0.807 0.805 0.840

0.827 0.815 0.808 0.809 0.825

nar1 = 10 0.5 1.0 2 4 ≥6

0.290 0.240 0.195 0.195 0.212

0.380 0.332 0.240 0.254 0.332

0.585 0.572 0.426 0.407 0.520

0.760 0.812 0.760 0.605 0.600

α, deg

Re × 10−5

3

4

6

0.179 0.148 0.118 0.120 0.094

0.174 0.146 0.120 0.098 0.085

0.176 0.147 0.120 0.095 0.084

8

10

12

14

0.185 0.147 0.120 0.094 0.085

0.196 0.151 0.120 0.095 0.094

0.224 0.179 0.140 0.118 0.118

0.270 0.233 0.176 0.160 0.160

nar1 ≥ 16 0.5 1.0 2 4

≥6

Re × 10−5

α, deg 16

20

30

0.306 0.275 0.208 0.191 0.212

0.378 0.340 0.280 0.264 0.342

0.600 0.600 0.520 0.480 0.560

45

60

90

120

180

0.880 0.905 0.868 0.778 0.790

0.880 0.877 0.868 0.847 0.853

0.880 0.986 0.868 0.868 0.874

0.880 0.876 0.868 0.869 0.886

nar1 ≥ 16 0.5 1.0 2 4 ≥6

0.840 0.840 0.760 0.700 0.720

341

Flow with a Smooth Change in Velocity Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

342

Handbook of Hydraulic Resistance, 4th Edition

Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

Values of kd at n1 = 2 l0 D0

α, deg 3

4

6

8

10

12

14

16

20

30

45

60

>90

1.16

1.05

1.00

0.01

1.01

1.01

Re = 0.5 × 105 2

1.00

1.10

1.20

1.25

1.26

1.26

1.23

5

1.45

1.62

1.75

1.83

1.86

1.80

1.70

1.53

1.10

1.02

1.02

1.02

1.02

10

1.88

1.96

2.05

2.07

2.07

2.05

2.00

1.93

1.60

1.12

1.11

1.10

1.10

≥20

1.68

1.83

1.96

2.00

1.99

1.93

1.85

1.74

1.45

1.03

1.01

1.01

1.01

1.13

Re = 1 × 105 2

1.00

1.10

1.20

1.27

1.43

1.60

1.67

1.60

1.10

0.85

0.96

1.11

5

1.63

1.83

2.00

2.11

2.20

2.19

2.11

1.88

1.20

1.00

1.13

1.15

1.15

10

1.93

2.13

.241

2.75

2.93

3.00

3.05

2.99

1.40

1.100

1.13

1.15

1.15

≥20

1.86

2.07

2.31

2.60

2.68

2.60

2.45

2.13

1.45

1.00

1.13

1.13

1.15

1.13

Re = (3–4) × 10

5

2

1.31

1.45

1.60

1.80

2.05

2.33

2.40

2.40

2.20

1.56

1.20

1.15

5

1.53

1.70

1.90

2.14

2.54

2.90

3.02

3.00

2.60

1.56

1.20

1.15

1.13

10

2.20

2.33

2.55

3.00

3.80

4.00

4.07

4.00

3.30

2.00

1.33

1.20

1.25

≥20

1.91

2.07

2.25

2.46

3.20

3.70

3.83

3.73

3.03

1.56

1.20

1.15

1.13

1.11

Re = (2–5) × 10

5

2

1.18

1.33

1.50

1.67

1.95

2.20

2.31

2.13

1.60

1.27

1.14

1.13

5

1.15

1.75

2.05

2.30

2.60

2.70

2.80

3.58

1.85

1.33

1.15

1.14

1.11

10

2.06

2.25

2.54

2.91

3.40

3.70

3.82

3.73

2.27

1.50

1.26

1.20

1.12

≥20

1.75

1.93

2.28

2.60

3.00

3.22

3.36

3.20

2.10

1.43

1.20

1.16

1.11

1.10

Re > 6 × 10

5

2

1.00

1.14

1.33

1.65

1.90

2.00

2.06

1.90

1.53

1.26

1.10

1.07

5

1.15

1.33

1.60

1.90

2.06

2.10

1.20

1.90

2.20

1.62

1.30

1.23

1.10

10

1.73

1.90

2.15

2.45

2.93

3.13

3.25

3.15

2.20

1.62

1.30

1.23

1.10

≥20

1.46

1.65

1.95

2.86

2.54

2.65

2.70

2.60

1.70

1.33

1.13

1.12

1.10

343

Flow with a Smooth Change in Velocity Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55

Diagram 5.2

Values of kd at 4 ≤ nar1 ≤ 16 l0 D0

α, deg 3

4

6

8

10

12

14

16

20

30

45

60

≥90

1.14

1.07

1.05

1.05

1.06

1.05

Re = 0.5 × 105 2

1.00

1.04

1.07

1.20

1.33

1.28

1.05

5

1.00

1.25

1.47

1.60

1.66

1.65

1.60

1.58

1.43

1.23

1.08

1.06

1.05

10

1.50

1.65

1.85

1.90

2.10

2.10

2.05

1.93

1.70

1.38

1.26

1.20

1.05

≥20

1.30

1.43

1.65

1.85

1.98

1.74

1.75

1.66

1.48

1.23

1.10

1.06

1.05

Re = 1 × 10

5

2

1.05

1.10

1.14

1.26

1.47

1.40

1.28

1.18

1.06

0.95

0.95

0.95

1.02

5

1.30

1.46

1.68

1.93

2.15

2.15

2.05

1.90

1.60

1.07

1.00

1.00

1.02

10

1.67

1.83

2.08

2.28

2.60

2.50

2.43

2.20

1.83

1.30

1.10

1.03

1.02

≥20

1.50

1.63

1.93

2.15

2.60

2.50

2.27

2.07

1.73

1.20

1.05

1.07

1.02

344

Handbook of Hydraulic Resistance, 4th Edition

Conical diffuser (in the system with l1/D1 > 0) at α = 3–180o 51–55 l0 D0

Diagram 5.2

α, deg 3

4

6

8

10

12

14

16

20

30

45

60

>90

2.12 2.40 3.00 2.75

1.90 2.20 2.65 2.40

1.53 1.60 1.80 1.67

1.25 1.26 1.30 1.30

1.10 1.15 1.15 1.15

1.05 1.06 1.06 1.06

1.95 2.27 2.80 2.50

1.68 1.95 2.40 2.10

1.32 1.40 1.53 1.50

1.15 1.19 1.26 1.23

1.13 1.13 1.20 1.15

1.07 1.07 1.07 1.07

1.70 1.90 2.20 1.98

1.50 1.55 1.83 1.60

1.23 1.25 1.33 1.30

1.13 1.15 1.22 1.20

1.10 1.10 1.18 1.15

1.07 1.07 1.07 1.07

Re = (3–4) × 105 2 5 10 ≥20

1.07 1.30 1.90 1.52

1.25 1.47 2.05 1.73

1.40 1.67 2.30 2.13

1.60 2.00 2.70 2.50

2.14 2.45 3.38 3.27

2.25 2.53 3.30 3.13

2.20 2.47 3.13 2.93

Re = (2–5) × 105 2 5 10 ≥20

1.00 1.30 1.80 1.54

1.20 1.47 2.00 1.73

1.40 1.69 2.25 2.12

1.63 2.00 2.60 2.43

2.05 2.27 3.30 3.20

2.13 2.35 3.20 3.00

2.07 2.37 3.00 2.75

Re ≥ 6 × 105 2 5 10 ≥20

1.00 1.05 1.60 1.35

1.13 1.23 1.82 1.63

1.42 1.60 2.15 2.10

1.73 1.95 2.55 2.43

1.98 2.25 3.20 3.05

1.93 2.20 3.02 2.70

1.83 2.08 2.53 2.23

345

Flow with a Smooth Change in Velocity Conical diffusers at large subsonic velocities (in the system with l1/D1 > 0 (coefficients of total pressure recovery)27

Diagram 5.3

_ p∗1 l0 ⎞ ⎛ p0 * ∗ = f ⎜λ0, α, n1, ⎟ D 0⎠ p0 ⎝ is determined from the curves of graphs a–e; ζd *

∆p ρw20 ⁄ 2

=

k+1 1 1 ln _ ; k λ20 p0

cp , see Table 1.4; cv w0 2k , acr = RT0∗ . λ0 = a∗ k+1

k=

⎯⎯⎯⎯ √

_ p0 at α = 4o (graph a) λ0 nar1

0.1 1.7

0.2 3.2

0.3 4.6

0.4 6.0

2–6 10–16

0.999 0.999

0.998 0.997

0.995 0.994

9.991 0.990

2–6 10–16

0.999 0.999

0.998 0.007

0.995 0.993

0.991 0.989

2 4–6 10–16

0.999 0.999 0.998

0.998 0.997 0.996

0.995 0.994 0.993

0.991 0.990 0.988

2–4 6–16

0.999 0.999

0.997 0.995

0.992 0.990

0.985 0.983

0.5

0.6

0.7

0.8

0.9

0.94

0.95

7.3

Re × 10−5 8.6

9.7

10.8

11.7

11.9

12.0

0.975 0.973

0.971 0.968

0.964 0.961

0.961 0.958

0.930 0.920

0.975 0.973

0.969 0.967

0.962 0.961

0.960 0.958

– –

0.974 0.971 0.970

0.965 0.930 0.960

– – –

– – –

– – –

0.959 0.955

– –

– –

– –

– –

l0 =0 D0 9.987 9.983 0.985 0.980 l0 =2 D0 0.986 0.981 0.984 0.978 l0 =5 D0 0.986 0.980 0.985 0.978 0.983 0.977 l0 ≥ 10 D0 0.978 0.969 0.975 0.966

346

Handbook of Hydraulic Resistance, 4th Edition

Conical diffusers at large subsonic velocities (in the system with l1/D1 > 0, coefficients of total pressure recovery)27 _ p0 at α = 6o (graph b)

Diagram 5.3

λ0 nar1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.94

0.95

8.6

9.7

10.8

11.7

11.9

12.0

0.984 0.983

0.980 0.977

0.975 0.973

0.970 0.966

0.960 0.958

0.930 0.930

0.975 0.970 0.967

0.965 0.960 0.958

– – 0.946

– – –

– – –

0.968 0.966 0.962

0.958 0.955 0.950

– – –

– – –

– – –

0.958 0.956 0.952

– – –

– – –

– – –

– – –

−5

Re × 10 1.7

3.2

4.6

6.0

7.3 l0 =0 D0

2–4 6–16

0.999 0.999

0.999 0.998

0.996 0.995

0.993 0.991

0.989 0.987

l0 =2 D0 2 4 10–16

0.999 0.999 0.999

0.998 0.997 0.996

0.995 0.993 0.992

0.992 0.989 0.987

0.988 0.984 0.982

0.983 0.977 0.975

l0 =5 D0 2 4 10–16

0.999 0.999 0.999

0.998 0.996 0.995

0.995 0.992 0.991

0.990 0.987 0.986

0.985 0.981 0.980

0.977 0.975 0.972

l0 ≥ 10 D0 2 4 10–16

0.999 0.999 0.998

0.995 0.996 0.995

0.993 0.991 0.989

0.987 0.985 0.982

0.980 0.977 0.974

0.970 0.967 0.964

347

Flow with a Smooth Change in Velocity Conical diffusers at large subsonic velocities (in the system with l1/D1 > 0, coefficients of total pressure recovery)27

Diagram 5.3

_ p0 at α = 8o (graph c) λ0 nar1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.95

−5

Re × 10 1.7

3.2

4.6

6.0

7.3

8.6

9.7

10.8

11.7

12.0

l0 =0 D0 0.987 l0 =2 D0

0.982

0.976

0.970

0.962

0.930

2–16

0.999

0.998

0.995

0.992

2 4 6–16

0.999 0.999 0.999

0.997 0.997 0.996

0.995 0.993 0.992

0.992 0.989 0.985

0.987 0.984 0.979 l0 =5 D0

0.982 0.978 0.970

0.975 0.971 0.960

– – 0.950

– – 0.948

– – –

2 4 6–16

0.999 0.999 0.999

0.997 0.996 0.995

0.995 0.992 0.989

0.991 0.986 0.983

0.980 0.971 0.966

0.970 0.961 0.955

– 0.948 0.943

– – –

– – –

2 4 6–16

0.999 0.999 0.999

0.996 0.995 0.993

0.993 0.990 0.987

0.989 0.984 0.980

0.987 0.979 0.975 l0 ≥ 10 D0 0.984 0.974 0.970

0.972 0.962 0.959

– – –

– – –

– – –

– – –

348

Handbook of Hydraulic Resistance, 4th Edition

Conical diffusers at large subsonic velocities (in the system with l1/D1 > 0, coefficients of total pressure recovery)27 _ p0 at α = 10o (graph d)

Diagram 5.3

λ0 nar1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.94

0.95

9.7

10.8

11.7

11.9

12.0

Re × 10−5 1.7

3.2

4.6

6.0

7.3

8.6 l0 =0 D0

2 4 6–16

0.999 0.999 0.999

0.998 0.998 0.998

0.997 0.997 0.996

0.995 0.994 0.992

0.991 0.987 0.990 0.985 0.987 0.982 l0 =2 D0

0.984 0.980 0.975

0.981 0.975 0.970

– 0.971 0.963

– 0.960 0.959

– 0.950 0.940

2 4 6 10–16

0.999 0.999 0.999 0.999

0.998 0.997 0.996 0.995

0.995 0.993 0.991 0.989

0.991 0.988 0.984 0.981

0.986 0.981 0.977 0.972

0.968 0.963 0.958 0.953

– 0.950 0.947 0.940

– – – –

– – – –

– – – –

0.7

0.8

0.9

0.94

0.95

9.7

10.8

11.7

11.9

12.0

– 0.955 0.950

– – –

– – –

– – –

– – –

– – –

– – –

– – –

– – –

– – –

0.978 0.972 0.968 0.963

λ0 nar1

0.1

0.2

0.3

0.4

0.5

0.6 Re × 10−5

1.7

3.2

4.6

6.0

7.3

8.6 l0 =5 D0

2 4 10–16

0.999 0.999 0.999

0.998 0.996 0.996

0.995 0.992 0.989

0.990 0.985 0.981

0.985 0.978 0.978 0.967 0.973 0.962 l0 ≥ 10 D0

2 4 10–16

0.999 0.998 0.998

0.996 0.995 0.993

0.992 0.989 0.985

0.988 0.982 0.976

0.982 0.971 0.965

0.975 0.959 0.954

349

Flow with a Smooth Change in Velocity Conical diffusers at large subsonic velocities (in the system with l1/D1 > 0, coefficients of total pressure recovery)27 _ p0 at α = 14o (graph e)

Diagram 5.3

λ0 nar1

0.1 1.7

2 4 6–16

0.2 3.2

0.3 4.6

0.4

0.5

0.6

0.7

0.8

0.9

0.95

6.0

Re × 10−5 7.3 8.6

9.7

10.8

11.7

12.0

0.986 0.974 0.966

0.982 0.965 0.956

0.976 0.957 0.945

– 0.948 0.934

– 0.945 0.924

0.6

0.7

0.8

0.9

0.95

9.7

10.8

11.7

12.0

0.999 0.999 0.999

0.998 0.997 0.996

0.996 0.994 0.990

0.993 0.990 0.984

l0%D0 = 0 0.990 0.982 0.974

0.1

0.2

0.3

0.4

0.5

λ0 nar1

−5

1.7

3.2

4.6

6.0

Re × 10 7.3 8.6

2 4 6–16

0.999 0.999 0.998

0.997 0.995 0.992

0.993 0.988 0.983

0.988 0.979 0.972

l0%D0 = 2 0.982 0.970 0.960

2 4 6–16

0.999 0.998 0.998

0.997 0.994 0.991

0.993 0.987 0.981

0.988 0.978 0.968

l0%D0 = 5 0.982 0.966 0.953

0.974 0.952 0.938

– 0.938 0.920

– – –

– – –

– – –

2 4 6–16

0.999 0.998 0.997

0.995 0.992 0.990

0.991 0.984 0.972

0.985 0.974 0.963

l0%D0 = 10 0.978 0.961 0.933

0.969 0.948 0.922

– – –

– – –

– – –

– – –

0.975 0.957 0.945

0.966 0.941 0.930

– – –

– – –

– – –

0.140 0.10 0.095 0.085 0.170 0.145 0.115 0.106 0.185 0.155 0.130 0.120 0.180 0.160 0.130 0.120

≥4

0.5 1 2

≥4

0.5 1 2

≥4

0.5 1 2

≥4

4

0.5 1 2

Re × 10−5

ζd at l0%D0 = 0

0.135

0.195 0.175 0.155

0.125

0.190 0.165 0.140

0.188

0.185 0.155 0.135

0.085

0.136 0.110 0.090

6

0.160

0.240 0.205 0.180

0.145

0.205 0.185 0.165

0.130

0.200 0.180 0.150

0.090

0.135 0.105 0.095

8

0.235

0.300 0.265 0.240

0.230

0.295 0.250 0.235

0.195

0.245 0.225 0.200

0.112

0.153 0.130 0.116

10

0.320

0.375 0.340 0.320

0.300

0.370 0.320 0.320

0.260

0.300 0.280 0.260

0.145

0.175 0.160 0.150

12

Diffuser of rectangular cross section (in the system with l1/D1h > 0)26

0.370

0.430 0.400 0.370

0.360

0.420 0.380 0.360

0.335

0.335 0.335 0.335

0.175

0.200 0.185 0.175

14

0.420

0.470 0.440 0.420

nar1 ≥ 10

0.400

0.460 0.420 0.420

nar1 = 6

0.360

0.380 0.360 0.360

nar1 = 4

0.185

0.235 0.200 0.180

16 nar1 = 2

α, deg

0.490

0.530 0.550 0.490

0.465

0.525 0.485 0.465

0.430

0.450 0.430 0.420

0.220

0.250 0.230 0.216

20

0.590

0.635 0.615 0.590

0.580

0.625 0.600 0.580

0.500

0.520 0.500 0.500

0.250

0.300 0.270 0.250

30

0.700

0.750 0.725 0.700

0.675

0.715 0.695 0.675

0.560

0.580 0.560 0.560

0.285

0.325 0.300 0.285

45

0.795

0.840 0.815 0.795

0.720

0.775 9.750 0.720

0.605

0.620 0.605 0.605

0.310

0.326 0.315 0.310

60

0.870

0.890 0.880 0.870

0.760

0.790 0.775 0.760

0.530

0.640 0.630 0.630

0.315

0.325 0.310 0.315

90

0.850

0.890 0.880 0.850

0.760

0.790 0.770 0.760

0.630

0.640 0.630 0.630

0.325

0.320 0.310 0.325

120

0.860

0.880 0.865 0.860

0.750

0.785 0.760 0.750

0.625

0.640 0.625 0.625

0.310

0.300 0.300 0.300

180

2. Nonuniform velocity field at the entrance into the diffuser (wmax/w0 > 1.0, 2δ∗0 ⁄ Dh > 1.0% or l0/Dh ≥ 10): ζ * ∆p ⁄ (ρw20 ⁄ 2) = ζd = f(α, nar1, Re), see the table and graph b. 3. For diffusers with α = 6–14o downstream of the shaped piece (elbow) ζ * ∆p ⁄ (ρw20 ⁄ 2) = kdζd, where for ζd, see the table and graph a of Diagram 5.4, for kd = f(w/w0), see the table of Diagram 5.1; for wmax/w0 = f(l0/Dh), see Diagram 5.1; α ≥ β.

ζ * ∆p ⁄ (ρw20 ⁄ 2) = ζd = f(α, nar1, Re), see the table and graph a. Approximate formulas are given in para. 40 of Section 5.1).

1. Uniform velocity field at the entrance into the diffuser (wmax/w0 ≈ 1.0 or l0/Dh ≈ 0):

Diagram 5.4

350 Handbook of Hydraulic Resistance, 4th Edition

0.315 0.265

0.180 0.130

0.310 0.250 0.190 0.140

0.300 0.240 0.185 0.130

2

≥4

0.5

1

2

≥4

0.5

1

2

≥4 0.200

0.360

0.205

0.255

0.305

0.360

0.180

0.230

0.270

0.320

0.125

0.260

0.150

≥4

0.160

0.220

0.140

2

0.200

1

0.175

1

0.240

6

0.5

0.200

4

0.5

Re × 10−5

ζd at l0%D0 ≥ 10

0.270

0.325

0.370

0.415

0.255

0.305

0.375

0.400

0.220

0.275

0.320

0.360

0.140

0.180

0.215

0.280

8

0.345

0.400

0.455

0.470

0.320

0.370

0.405

0.450

0.270

0.320

0.365

0.400

0.160

0.195

0.235

0.280

10

0.400

0.460

0.490

0.520

0.380

0.420

0.455

0.490

0.320

0.365

0.400

0.430

0.200

0.210

0.250

0.298

12

Diffuser of rectangular cross section (in the system with l1/D1h > 0)26

0.460

0.515

0.540

0.570

0.425

0.460

0.500

0.530

0.350

0.400

0.435

0.455

0.195

0.225

0.260

0.305

14

0.500

0.550

0.580

0.600

nar1 ≥ 10

0.460

0.495

0.530

0.560

nar1 = 6

0.380

0.430

0.460

0.480

nar1 = 4

0.210

0.240

0.275

0.315

nar1 = 2

16

α, deg

0.570

0.610

0.640

0.670

0.520

0.545

0.580

0.615

0.430

0.470

0.495

0.510

0.235

0.260

0.290

0.325

20

0.680

0.715

0.730

0.760

0.615

0.635

0.650

0.685

0.500

0.530

0.550

0.565

0.265

0.280

0.310

0.340

30

0.790

0.810

0.830

0.850

0.695

0.710

0.720

0.750

0.580

0.590

0.600

0.610

0.300

0.310

0.330

0.355

45

60

0.855

0.860

0.880

0.900

0.740

0.745

0.775

0.775

0.620

0.620

0.630

0.635

0.320

0.320

0.340

0.355

Diagram 5.4

0.930

0.930

0.940

0.960

0.770

0.775

0.780

0.795

0.650

0.650

0.650

0.655

0.335

0.335

0.340

0.350

90

0.910

0.910

0.910

0.920

0.775

0.75

0.775

0.785

0.650

0.650

0.650

0.650

0.320

0.320

0.320

0.340

120

0.880

0.880

0.880

0.880

0.760

0.760

0.760

0.760

0.640

0.640

0.640

0.640

0.310

0.310

0.310

0.310

180

Flow with a Smooth Change in Velocity 351

352

Handbook of Hydraulic Resistance, 4th Edition

Flow with a Smooth Change in Velocity

353

Diffuser with expansion in one plane (in the system with l1/D1h > 0)26

Diagram 5.5

1. Uniform velocity field at the entrance into the diffuser (wmax/w0 ≈ 1.0 or l0/Dh ≈ 0):

ζ*

Dh =

4F0 Π0

; n1 =

F1

F0

;

Re =

∆p ρw20 ⁄ 2

= ζd = f (α, nar1, Re) ,

see the table and graph a. Approximate formulas are given in para. 41 of Section 5.1. Calculation of preseparation diffusers is made in para. 34 of Section 5.1. 2. Nonuniform velocity field at the entrance into the diffuser (wmax/w0 > 1.0, 2δ∗0 ⁄ Dh > 1.0% or l0/Dh ≥ 10):

w0Dh v

ζ*

∆p = ζd = f (α, nar1, Re) , ρw20 ⁄ 2

see the table and graph b of Diagram 5.5. 3. For diffusers α = 6–20o downstream of a shaped (curved) piece ζ*

∆p = kdζd , ρw20 ⁄ 2

where for ζd, see the table and graph a; for kd = f(w/w0) and for wmax/w0 = f(l0/Dh), see Diagram 5.1.

ζd at l0%Dh = 0 Re × 10−5

α, deg 4

6

8

10

14

20

30

45

60

90

120

180

0.235 0.200 0.200 0.200

0.350 0.335 0.335 0.335

0.370 0.370 0.370 0370

0.380 0.380 0.380 0.380

0.370 0.370 0.370 0.370

0.350 0.350 0.350 0.350

0.420 0.420 0.420 0.420

0.600 0.600 0.600 0.600

0.680 0.680 0.680 0.680

0.700 0.700 0.700 0.700

0.700 0.700 0.700 0.700

0.660 0.660 0.660 0.660

0.480 0.480 0.480 0.480 0.480

0.650 0.650 0.650 0.650 0.650

0.760 0.760 0.760 0.760 0.760

0.830 0.830 0.830 0.830 0.830

0.830 0.830 0.830 0.830 0.830

0.800 0.800 0.800 0.800 0.800

nar1 = 2 0.5 1 2 ≥4

0.200 0.180 0.163 0.150

0.165 0.145 0.125 0.115

0.142 0.125 0.110 0.100

0.135 0.115 0.100 0.096

0.125 0.105 0.093 0.83

0.154 0.120 0.115 0.115 nar1 = 4

0.5 1 2 ≥4

0.275 0.230 0.210 0.165

0.225 0.182 0.162 0.150

0.285 0.160 0.142 0.133

0.170 0.163 0.140 0.135

0.5 1 2 4 ≥6

0.310 0.250 0.235 0.215 0.200

0.250 0.205 0.190 0.165 0.150

0.215 0.175 0.160 0.143 0.130

0.205 0.170 0.158 0.143 0.130

0.182 0.180 0.162 0.162

0.250 0.250 0.250 0.250 nar1 = 6 0.210 0.300 0.190 0.300 0.190 0.300 0.190 0.300 0.190 0.300

354

Handbook of Hydraulic Resistance, 4th Edition

Diffuser with expansion in one plane (in the system with l1/D1h > 0)26

Diagram 5.5

ζd at l0%Dh ≥ 10 α, deg

Re × 10−5 4

6

8

10

14

0.5

0.260

0.225

0.210

0.210

0.220

1

0.225

0.200

0.190

0.190

2

0.150

0.130

0.125

0.125

4

0.125

0.110

0.100

≥6

0.125

0.110

0.100

20

30

45

60

90

120

180

0.240

0.300

0.360

0.370

380

0.370

0.350

0.200

0.220

0.270

0.340

0.370

380

0.370

0.350

0.150

0.185

0.245

0.340

0.370

380

0.370

0.350

0.105

0.120

0.155

0.250

0.340

0.370

380

0.370

0.350

0.105

0.120

0.155

0.205

0.340

0.370

380

0.370

0.350

nar1 = 2

Flow with a Smooth Change in Velocity

355

Diffuser with expansion in one plane (in the system with l1/D1h > 0)26

Diagram 5.5

α, deg

Re × 10−5 4

6

8

10

14

20

30

45

60

90

120

180

nar1 = 4 0.5

0.300

0.280

0.270

0.275

0.320

0.420

0.570

0.660

0.690

0.700

0.700

0.660

1

0.280

0.250

0.240

0.240

0.295

0.400

0.560

0.650

0.690

0.700

0.700

0.660

2

0.210

0.190

0.195

0.200

0.260

0.380

0.520

0.640

0.680

0.700

0.700

0.660

4

0.185

0.160

0.160

0.170

0.230

0.375

0.520

0.640

0.680

0.700

0.700

0.660

>6

0.170

0.155

0.150

0.160

0.210

0.360

0.520

0.640

0.680

0.700

0.700

0.660

nar1 = 6 0.5

0.335

0.310

0.300

0.305

0.360

0.500

0.650

0.760

0.810

0.830

0.830

0.800

1

0.280

0.260

0.255

0.270

0.350

0.490

0.640

0.750

0.800

0.830

0.830

0.800

2

0.215

0.200

0.205

0.220

0.320

0.475

0.610

0.730

0.790

0.830

0.830

0.800

4

0.190

0.180

0.185

0.210

0.300

0.460

0.610

0.730

0.790

0.830

0.830

0.800

≥6

0.180

0.165

0.165

0.180

0.280

0.440

0.590

0.710

0.780

0.830

0.830

0.800

356

Handbook of Hydraulic Resistance, 4th Edition

Plane five-channel subsonic diffusers in the system;55 nar1 = 6.45; Re = (0.6–4) × 105

_ p∗1 p0 = , p∗0 ζd *

Diagram 5.6

see graph a;

∆p ′ − 0.024 ; − 0.024 = ζtot ρw20 ⁄ 2

for ζ′tot, see graph b

_ _ Values of p0 at various ld λ0

α, deg 0.1

0.2

0.3

0.4

_

0.5

0.6

0.7

0.8

0.9

0.95

ld = ld%D0 = 3.23 8

0.999

0.995

0.990

0.998

0.981

0.975

0.967

0.960

0.950

0.900

12

0.999

0.995

0.990

0.989

0.975

0.963

0.950

0.938

0.870

–

16

0.999

0.996

0.987

0.979

0.968

0.934

0.938

0.920

0.870

–

8

0.999

0.992

0.988

0.979

0.945

0.926

0.907

0.88

12

0.998

0.991

0.984

0.976

0.965

0.950

0.930

0.904

0.850

–

16

0.997

0.991

0.983

0.959

0.954

0.954

0.913

0.887

0.82

–

_

ld = ld%D0 = 6.45

_

0.969

0.956

ld = ld%D0 = 9.68 8

0.998

0.990

0.983

0.975

0.963

0.950

0.931

0.913

0.84

–

12

0.996

0.990

0.982

0.970

0.957

0.940

0.917

0.888

0.83

–

16

0.995

0.988

0.978

0.963

0.948

0.927

0.900

0.868

–

–

Flow with a Smooth Change in Velocity

357

Plane five-channel subsonic diffusers in the system;55 nar1 = 6.45; Re = (0.6–4) × 105

Diagram 5.6

_ Values of ζd at various ld λ0

α, deg 0.1

0.2

0.3

0.4

_

0.5

0.6

0.7

0.8

0.9

0.95

ld = ld%D0 = 3.23 8

0.10

0.10

0.11

0.12

0.13

0.12

0.11

0.10

0.10

0.18

12

0.20

0.20

0.21

0.22

0.22

0.21

0.20

0.18

0.18

0.28

16

0.23

0.23

0.23

0.24

0.24

0.24

0.23

0.23

0.33

–

8

0.16

0.16

0.17

0.17

0.18

0.17

0.16

0.15

0.20

–

12

0.23

0.23

0.23

0.24

0.24

0.24

0.24

0.24

0.31

–

16

0.28

0.28

0.28

0.29

0.30

0.30

0.29

0.28

0.38

–

_

ld = ld%D0 = 6.45

_

ld = ld%D0 = 9.68 8

0.22

0.22

0.23

0.23

0.23

0.22

0.21

0.20

0.29

–

12

0.30

0.30

0.30

0.30

0.31

0.31

0.30

0.29

0.38

–

16

0.36

0.36

0.37

0.37

0.37

0.36

0.35

0.35

0.43

–

358

Handbook of Hydraulic Resistance, 4th Edition

Plane five-channel subsonic diffusers in the system;55 nar1 = 6.45; Re = (0.6–4)⋅105

Diagram 5.6

Diffusers of circular cross section (in the system with l1/D1 > 0);4 laminar flow (Re = w0Dh/v ≥ 50)

Diagram 5.7

ζ*

∆p ρw20 ⁄ 2

=

A , Re

where at α ≤ 40o

A=

20n0.33 ar1 (tan α)0.75

,

see the curves A = f(α, nar1).

Values of A α, deg

nar1 4

6

8

10

14

20

25

30

35

40

1.5

178

130

104

87.5

67.2

50.1

41.4

35.1

30.3

23.3

2

197

144

115

96.8

74.4

55.4

45.8

38.8

33.5

29.1

3

227

166

133

112

85.7

63.8

52.8

44.7

38.6

33.5

4

251

184

147

123

94.8

70.6

58.4

49.5

42.7

37.1

6

290

212

169

142

109

81.4

67.3

57.0

49.2

42.7

Flow with a Smooth Change in Velocity

359

Diffusers of circular cross section (in the system with l1/D1 > 0);4 laminar flow (Re = w0Dh/v ≥ 50)

Diagram 5.7

Diffusers with curvilinear boundaries (in the system with l1/D1 > 0);47–49 Re = w0Dh/v ≥ 105

ζ*

∆p ρw20 ⁄ 2

0.1 ≤

+ ϕ0σ0d ∗

Diagram 5.8

(the formula is applicable within

1.3F0 F0 ⎛ F0 ⎞ ≤ 0.9), where σ0 = 1.43 − =f⎜ ⎟ F1 F1 ⎝ F1 ⎠ 2

F0 ⎞ ⎛ ⎛ F0 ⎞ and d ∗ = ⎜1 − ⎟ = f2 ⎜ ⎟ , see graph a; F 1 ⎠ ⎝ ⎝ F1 ⎠ ⎛ ld ⎞ ⎛ ld ⎞ ⎟ or ϕ0 = f ⎜ a ⎟ , see graph b. D h ⎝ ⎠ ⎝ 0⎠

ϕ0 = f ⎜

Plane diffuser Dh =

4F0 Π0

;

360

Handbook of Hydraulic Resistance, 4th Edition

Diffusers with curvilinear boundaries (in the system with l1/D1 > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.8

Values of σ0 and d* F0 F1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

σ0

1.30

1.17

1.04

0.91

0.78

0.65

0.52

0.39

0.26

d∗

0.81

0.64

0.49

0.36

0.25

0.16

0.09

0.04

0.01

Values of ϕ0 ld ld Dh a0

0

0.5

1.0

1.5

2.0

1.01

0.75

0.62

0.53

0.47

1.02

0.83

0.72

0.64

0.57

2.5

3.0

3.5

4.0

4.5

5.0

6.0

0.38

0.37

–

–

–

0.45

0.43

0.41

0.39

0.37

1. Diffuser of annual or rectangular cross section 0.43

0.40

2. Plane diffuser 0.52

0.48

Diffusers of circular cross section with stepped walls (in the system with l1/D1 > 0);47–49 Re = w0Dh/v ≥ 105 ζ*

∆p ρw20 ⁄ 2

Diagram 5.9

+ ζmin

This formula can be applied for the selection of the optimal angle αopt from graph b; for ζmin, see graph a as a function of ld/D0 and nar.

2

nar1 =

F1 ⎛ F2 ld F2 α⎞ tan ⎟ ; nar2 = , nar = = 1+2 F0 ⎜⎝ 2⎠ D0 F1 F0

Flow with a Smooth Change in Velocity

361

Diffusers of circular cross section with stepped walls (in the system with l1/D1 > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.9

Values of ζmin ld ⁄ Dh nar

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

0.03 0.08 0.13 0.17 0.32 0.30 0.34 0.36 0.39 0.41

0.02 0.06 0.09 0.12 0.17 0.22 0.26 0.28 0.30 0.32

0.03 0.04 0.06 0.09 0.12 0.16 0.18 0.20 0.22 0.24

0.03 0.04 0.06 0.07 0.10 0.13 0.15 0.16 0.18 0.20

0.04 0.04 0.06 0.07 0.09 0.12 0.13 0.14 0.16 0.17

0.05 0.05 0.06 0.06 0.08 0.10 0.12 0.13 0.14 0.15

0.06 0.05 0.06 0.06 0.08 0.10 0.11 0.12 0.13 0.14

0.08 0.06 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.12

0.10 0.08 0.07 0.07 0.08 0.09 0.09 0.10 0.10 0.11

0.11 0.09 0.08 0.08 0.08 0.09 0.09 0.09 0.10 0.11

0.13 0.10 0.09 0.08 0.08 0.08 0.09 0.09 0.10 0.11

Values of αopt, deg ld ⁄ Dh nar

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

17 21 25 27 29 31 32 33 33 34

10 17 16 17 20 21 22 23 23 24

6.5 8.5 10 11 13 14 15 15 16 16

4.5 6.2 7.4 8.5 9.8 11 12 12 13 13

3.5 5.0 6.0 7.0 8.0 9.4 10 11 11 11

2.8 4.3 5.4 6.1 7.2 8.2 8.8 9.4 9.6 9.8

2.2 3.8 4.8 5.6 6.6 7.4 8.0 8.4 8.7 9.0

1.7 3.0 4.0 4.8 5.8 6.2 6.6 7.0 7.3 7.5

1.2 2.3 3.5 4.2 5.2 5.6 5.8 6.2 6.3 6.5

1.0 2.0 3.0 3.8 4.8 5.2 5.4 5.5 5.6 6.0

0.8 1.6 2.5 3.2 4.4 4.7 5.0 5.2 5.4 5.6

362

Handbook of Hydraulic Resistance, 4th Edition

Diffusers of circular cross section with stepped walls (in the system with l1/D1 > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.9

Diffusers of rectangular cross section with stepped walls (in the system with l1/D1h > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.10

ζ*

∆p ρw20 ⁄ 2

+ ζmin

This formula can be applied for the selection of the optimal angle αopt from graph b; ζmin is determined from graph a as a function of ld/Dh and nar (with ample safety margin). Dh =

4F0 Π0

F1 ⎛ ld α⎞ tan ⎟ = 1+2 2⎠ F0 ⎜⎝ Dh F2 F2 nar2 = nar = F1 F0

2

nar1 =

Values of ζmin ld ⁄ Dh nar

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

0.04 0.11 0.16 0.21 0.27 0.36 0.41 0.44 0.47 0.49

0.03 0.08 0.13 0.17 0.22 0.28 0.32 0.35 0.37 0.40

0.03 0.06 0.09 0.12 0.17 0.21 0.24 0.26 0.28 0.30

0.04 0.06 0.08 0.10 0.14 0.18 0.21 0.22 0.24 0.26

0.05 0.06 0.08 0.09 0.12 0.16 0.18 0.20 0.21 0.23

0.05 0.06 0.07 0.09 0.11 0.15 0.17 0.18 0.20 0.21

0.06 0.07 0.08 0.09 0.11 0.14 0.16 0.17 0.18 0.19

0.08 0.07 0.07 0.09 0.11 0.13 0.14 0.15 0.16 0.17

0.10 0.08 0.08 0.09 0.11 0.12 0.13 0.14 0.15 0.16

0.11 0.09 0.08 0.09 0.10 0.12 0.12 0.13 0.14 0.15

0.13 0.10 0.09 0.09 0.10 0.11 0.12 0.13 0.14 0.14

Flow with a Smooth Change in Velocity

363

Diffusers of rectangular cross section with stepped walls (in the system with l1/D1h > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.10

Values of αopt, deg ld ⁄ Dh

nar 1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

14 18 20 21 22 24 25 25 26 26

9.0 12 14 15 16 17 17 18 18 19

5.3 8.0 9.0 10 11 12 12 12 13 13

4.0 6.3 7.2 7.8 8.5 9.4 9.7 10 10 11

3.3 5.2 6.1 6.5 7.1 8.0 8.3 8.7 9.0 9.2

2.7 4.5 5.4 5.8 6.2 6.9 7.3 7.6 7.8 8.1

2.2 3.8 4.8 5.2 5.5 6.2 6.5 6.9 7.1 7.3

1.7 3.0 4.0 4.4 4.8 5.2 5.5 5.8 6.1 6.4

1.2 2.3 3.2 3.6 4.0 4.5 4.8 5.0 5.2 5.5

1.0 2.0 2.9 3.3 3.8 4.3 4.6 4.8 5.0 5.2

1.0 1.8 2.4 2.9 3.5 4.0 4.2 4.5 4.7 4.9

364

Handbook of Hydraulic Resistance, 4th Edition

Diffusers with expansion in one plane with stepped walls (in the system with l1/D1h > 0);47–49 Re = w0Dh/v ≥ 105

Diagram 5.11

ζ*

∆p ρw20 ⁄ 2

+ ζmin .

This formula can be applied for the selection of the optimal angle αopt from graph b; ζmin is Dh =

determined from graph a as a function of ld ⁄ Dh and nar.

4F0 Π0

nar1 =

a1 ld α = 1 + 2 tan 2 a0 a0

nar2 = a0nar1

nar =

F2 a2 = F0 a0

Values of ζmin ld ⁄ Dh

nar 1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

0.04 0.12 0.18 0.23 0.30 0.38 0.43 0.46 0.50 0.53

0.04 0.09 0.14 0.18 0.24 0.31 0.36 0.38 0.41 0.44

0.04 0.07 0.11 0.14 0.19 0.25 0.28 0.30 0.33 0.35

0.04 0.07 0.10 0.12 0.16 0.21 0.25 0.26 0.29 0.31

0.05 0.06 0.09 0.11 0.15 0.19 0.22 0.24 0.26 0.28

0.06 0.07 0.09 0.11 0.14 0.18 0.20 0.22 0.24 0.25

0.06 0.07 0.09 0.10 0.13 0.17 0.19 0.21 0.22 0.24

0.08 0.07 0.09 0.10 0.12 0.16 0.17 0.19 0.20 0.22

0.10 0.08 0.09 0.10 0.12 0.15 0.16 0.18 0.19 0.20

0.11 0.10 0.10 0.10 0.12 0.14 0.16 0.17 0.18 0.19

0.13 012 0.10 0.11 0.12 0.14 0.15 0.16 0.18 0.19

Flow with a Smooth Change in Velocity

365

Diffusers of reduced resistance, with a screen (in the system with l1/D1 > 0);50,55 Re = w0Dh/v ≥ 105

Diagram 5.11

Values of αopt, deg ld a0

nar 1.5 2.0 2.5 3.0 4.0 6.0 0.08 10 14 20

0.5

1.0

2.0

3.0

4.0

5.0

6.0

8.0

10

12

14

25 33 37 39 42 45 47 48 49 50

18 23 26 27 30 31 32 33 34 35

11 15 18 20 21 23 23 24 25 25

8.0 12 14 16 17 18 19 20 20 21

6.4 9.7 12 13 15 16 17 17 17 18

5.4 8.4 10 12 13 14 15 15 16 16

4.7 7.5 9.4 11 12 13 14 14 14 15

3.5 6.0 8.0 9.1 10 11 12 12 13 13

2.8 5.2 7.0 8.0 9.0 10 11 11 12 12

2.4 4.7 6.3 7.2 8.2 9.4 10 10 11 11

2.0 4.3 5.6 6.4 7.4 8.5 9.1 9.5 9.9 10

Diffusers of reduced resistance

Diagram 5.12

Internal arrangement of diffuser

Scheme

Dividing splitters, number of splitters, z1

Resistance coefficient ∆p . ζ* ρw20 ⁄ 2 ζ + 0.65ζd ,

α, deg

30

45

60

90

120

z1

2

4

6

6

6–8

where ζd is determined as ζ from Diagrams 5.2, 5.4, and 5.5.

366

Handbook of Hydraulic Resistance, 4th Edition

Diffusers of reduced resistance

Diagram 5.12

ζ + 0.65ζd ,

Baffles

where ζd is determined as ζ from Diagrams 5.2, 5.4, and 5.5

Rounded insert R in the inlet section; nar1 = F1/F2 = 2–4

l ′ ⁄ D0 =

R sin α D0

ζ = kζd , where for ζd see Diagrams 5.1 through 5.5: (a) at l ′⁄ D0 ≈ 0.5 and α = 45 and 60o, k ≈ 0.72; (b) at l ′⁄ D0 ≈ 0.8 and α = 60o, k ≈ 0.67

(a) at α = 0–60o ζgr ζ = ζ0 + 2 , nar1 (b) at α > 60o Screen or grid (perforated plate) at the exit of diffuser

⎛ ζgr ⎞ ζ = (1.2−1.3) ⎜ζ0 + 2 ⎟ , n ar1 ⎠ ⎝ where ζ0 is determined as ζ from Diagrams 5.2, 5.4 and 5.5, and ζgr is determined as ζ of the screen or grid from Diagrams 8.1 to 8.7; nar1 = F1/F0.

Flow with a Smooth Change in Velocity

367

Diffuser with symmetric expansion in one plane, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

∆p

ρw20 ⁄ 2

Diagram 5.13

⎛ F1 ⎞ =f⎜ ⎟ , ⎝ F0 ⎠

see the curves at different α values

Values of ζ F1 F0

α, deg

1.4

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35

0.05 0.06 0.07 0.08 0.16 0.24

0.07 0.09 0.10 0.13 0.24 0.34

0.09 0.11 0.13 0.16 0.29 0.39

0.10 0.13 0.15 0.19 0.32 0.44

0.11 0.13 0.16 0.21 0.34 0.48

0.11 0.14 0.16 0.23 0.35 0.50

Diffuser with asymmetric expansion (at α1 = 0) in one plane, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

∆p

ρw20 ⁄ 2

Diagram 5.14

⎛ F1 ⎞ =f⎜ ⎟ , ⎝ F0 ⎠

see the curves at different α values

Values of ζ F1 F0

α, deg

1.4

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35

0.08 0.10 0.12 0.15 0.18 0.21

0.09 0.11 0.14 0.18 0.25 0.31

0.10 0.12 0.15 0.21 0.30 0.38

0.10 0.13 0.16 0.23 0.33 0.41

0.11 0.14 0.17 0.25 0.35 0.43

0.11 0.15 0.18 0.26 0.35 0.44

368

Handbook of Hydraulic Resistance, 4th Edition

Diffuser with asymmetric expansion (at α1 = 10o) in one plane, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

∆p

ρw20 ⁄ 2

Diagram 5.15

⎛ F1 ⎞ =f⎜ ⎟ , ⎝ F0 ⎠

see the curves at different α values

Values of ζ F1 F0

α, deg

1.4

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35

0.05 0.06 0.07 0.09 0.13 0.15

0.08 0.10 0.11 0.14 0.18 0.23

0.11 0.12 0.14 0.18 0.23 0.28

0.13 0.14 0.15 0.20 0.26 0.33

0.13 0.15 0.16 0.21 0.28 0.35

0.14 0.15 0.16 0.22 0.29 0.36

Diffuser with asymmetric expansion (at α1 = –10o) in one plane, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

Diagram 5.16

⎛ F1 ⎞ ∆p =f⎜ ⎟ , ρw20 ⁄ 2 ⎝ F0 ⎠

see the curves at different α values

Values of ζ F1 F0

α, deg

1.5

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35

0.11 0.13 0.19 0.29 0.36 0.44

0.13 0.15 0.22 0.32 0.42 0.54

0.14 0.16 0.24 0.35 0.46 0.61

0.14 0.17 0.26 0.37 0.49 0.64

0.14 0.18 0.28 0.39 0.51 0.66

0.14 0.18 0.30 0.40 0.51 0.66

Flow with a Smooth Change in Velocity

369

Diffuser of rectangular cross section installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

∆p

ρw20 ⁄ 2

⎛ F1 ⎞ =f⎜ ⎟ , ⎝ F0 ⎠

see the curves at different α values

Diagram 5.17

Values of ζ F1%F0

α, deg

1.5

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30

0.10 0.23 0.31 0.36 0.42

0.18 0.33 0.43 0.49 0.53

0.12 0.38 0.48 0.55 0.59

0.23 0.40 0.53 0.58 0.64

0.24 0.42 0.56 0.62 0.67

0.25 0.44 0.58 0.64 0.69

Diffuser with stepped walls, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

ζ*

∆p

ρw20 ⁄ 2

,

⎛ F2⎞ ζmin = f ⎜ ⎟ , ⎝ F0⎠

see the curves at different ld/b0 in graph a; αopt = f(F2/F0), see the curves at different ld/b0 in graph b

Diagram 5.18

Values of ζmin ld b0 1.0 1.5 2.0 3.0 4.0 5.0

F2%F0 2.0 0.16 0.13 0.12 0.09 0.08 0.06

2.5 0.25 0.20 0.17 0.13 0.12 0.10

3.0 0.33 0.26 0.22 0.18 0.15 0.13

3.5 0.38 0.31 0.26 0.21 0.18 0.15

4.0 0.43 0.34 0.29 0.24 0.20 0.17

4.5 0.47 0.38 0.33 0.26 0.22 0.18

5.0 0.50 0.41 0.35 0.28 0.24 0.20

6.0 0.56 0.46 0.38 0.31 0.26 0.22

370

Handbook of Hydraulic Resistance, 4th Edition

Diffuser with stepped walls, installed downstream of a centrifugal fan operating in a system with l1/D1h > 058

Diagram 5.18

Values of αopt, deg F2%F0

ld b0

2.0

2.5

3.0

3.5

4.0

4.5

5.0

6.0

1.0 1.5 2.0 3.0 4.0 5.0

9 8 7 6 4 3

10 9 8 7 5 4

10 9 8 7 6 5

11 10 9 7 6 6

11 10 9 7 7 6

11 10 9 8 7 6

11 10 9 8 7 6

12 10 9 8 8 7

Annular diffusers_ with inner fairing, in a system with l1/D1h > 0; d0 = 0.68840,128

Diagram 5.19

1. Inner diverging fairing (α1 = 8–16o):

ζin *

∆p = kdζin′ , ρw20 ⁄ 2

where _ for ζ′in see graph a, or within 2 < nar1 < 4 and at ld = 0.5–1.0 determine from the formula _ ld ; ld = D0

_ d0 d0 = D0 _ 4l_ ⎛ 2 tan α2 − tan2 α1⎞ nar 1 = 1 + ⎠ 1 − d02 ⎝ _ _ 4l_ ⎛ tan α2 − d0 tan α1⎞ + 2 ⎝ ⎠ 1 − d0

ζin′ +

0.25n2ar1 _ , ld1.5

for kd, see Diagram 5.1 or graph b (when installed downstream of an operating axial flow machine).

′ Values of ζin

_ ld 0.5 0.75 1.0 1.5–2.0

nar1 1.5

2.0

2.5

3.0

3.5

4.0

0.06 0.06 0.06 0.05

0.22 0.15 0.10 0.07

0.50 0.24 0.15 0.10

– 0.35 0.23 0.15

– – 0.35 0.18

– – 0.46 0.25

Flow with a Smooth Change in Velocity

371

Annular diffusers_ with inner fairing, in a system with l1/D1h > 0; d0 = 0.68840,128

Diagram 5.19 2. Inner converging fairing (α1 < 0): 2

F0 ⎞ ⎛ = kdϕd ⎜1 − ⎟ , F1 ⎠ ρw20 ⁄ 2 ⎝ where for ϕd, see graph c as a function of the divergence angle α; for kd, see graph b as a function of the divergence angle α2 for different velocity profiles shown in graph d.

ζ*

∆p

Values of kd α2, deg 7 8 10 12 14

Velocity profile (graphs b and d) 1 1.0 1.0 1.0 1.0 1.0

2 1.40 1.60 1.60 1.45 1.40

3 2.00 2.10 2.10 2.00 1.86

4 1.16 1.21 1.20 1.10 1.08

5 0.90 1.15 1.36 1.42 1.50

6 2.74 2.98 3.02 2.70 2.48

α2, deg

ϕd

7

0.25

8

0.25

10

0.30

12

0.37

14

0.44

372

Handbook of Hydraulic Resistance, 4th Edition

Turbomachinery diffusers (radial-axial and axial-radial-annular) _ in a system with l1/D1 > 0; d0 = 0.68839

Diagram 5.20

1. Radial-axial

ζin *

∆p

= f(nar, α1) , __ where ζin = f1(nar, D1); ζin = f(nar, α1), see graphs a–c

ρw20 ⁄ 2

1. Values of ζin 2. Axial-radial-annular

__ D1

nar 1.4

1.8

2.2

2.6

3.0

3.4

3.8 4.2 _ (a) Diffuser downstream of operating compressor at ca0 = 0.5 1.5 1.7 1.9 2.2

– – – –

0.45 0.34 – –

0.55 0.48 0.37 –

0.62 0.56 0.49 0.35

0.65 0.61 0.56 0.45

– 0.64 0.62 0.52

1.4 1.6 1.8 2.0

0.31 0.25 0.19 –

– – 0.65 0.60

– – – 0.65

(b) Diffuser downstream of idle compressor

__ _ D1 = 2.06 α2 = 8o ca0 = 0.5 __ D h1 __ 1_ 1 D1 = nar = 2 D1 h0 D0 1 + d0 _ _ Q ca0 = w0 = ca0 = 2 2 π (D0 − d0) ⁄ 4

0.41 0.33 0.26 0.20

0.48 0.40 0.33 0.25

0.55 0.46 0.39 0.30

0.60 0.52 0.44 0.35

– 0.55 0.48 0.40

– – 0.51 0.43

– – – –

_ d0 d0 = D0

ca0 , u

where Q is the fluid discharge, m3/s; u is the circumferential velocity on the outer radius, m/s

2. Values of ζin α1, deg –2 +2 +4

nar 1.8

2.2

2.6

3.0

3.4

3.6

4.0

0.28 0.14 0.08

0.31 0.22 0.13

0.35 0.27 0.18

0.38 0.31 0.24

0.40 0.35 0.29

0.41 0.37 0.32

0.43 0.41 0.39

Flow with a Smooth Change in Velocity

373

Diffusers with curved axis (and with expansion in one plane in the system with ld/b0 = 8.3; l0 ⁄ b0 = 069,70

ζin *

∆p

ρw20 ⁄ 2

= k0ζin′ ;

r ζin′ = f ⎛⎜α, ⎞⎟ ; b0 ⎠ ⎝

Diagram 5.21

for kd, see Diagram 5.1

′ Values of ζin

α, deg

r b0

2

4

6

8

10

12

∞

0.037

0.068

0.088

0.106

0.123

0.138

0.150

0.160

0.42

0.72

0.97

0.113

0.130

0.144

0.155

0.163

0.043

0.077

0.103

0.124

0.140

0.154

0.163

0.168

0.043

0.081

0.113

0.136

0.153

0.163

0.170

0.175

14

16

(straight, β = 0) 22.5 (β = 21o 15′) 11.6 o

(β = 40 5′) 7.5 (β = 63o 42′)

374

Handbook of Hydraulic Resistance, 4th Edition

Diffusers of circular cross section with a curved axis in the system with nar1 = 4; ld/D0 = 7.15; α = 8o; l0/D0 = 0.35172,173

Diagram 5.22

∆p

ζin *

ρw20 ⁄ 2

= kdζin′ ,

for ζ′in, see the table at Re ≥ 5 × 105 and the curves ζ′in = f(Re) in the graph; for kd, see Diagram 5.1 nar1 = Re =

Diffuser no. β, deg R ⁄ D0 ′ at Re ≥ 5 × 105 ζin

F1 F0

,

w0D0 . v

1

2

3

4

5

6

7

8

9

10

11

12

0

15

30

0

15

30

15

30

0

30

30

30

∞

27.30

13.65

∞

27.30

13.65

27.30

13.65

∞

13.65

13.65

13.65

0.081

0.131

0.092

0.087

0.108

0.145

0.154

0.220

0.131

0.115

0.265

0.118

Values of ζ′in Diffuser no.

Re × 10–5 0.10

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1. (β = 0; r ⁄ D0 = ∞)

0.244

0.216

0.178

0.160

0.150

0.140

0.120

0.100

0.088

0.075

2. (β = 15o; r/D0 = 27.30)

0.410

0.366

0.290

0.240

0.200

0.180

0.160

0.150

0.132

0.127

o

3. (β = 30 ; r ⁄ D0 = 13.65)

–

0.385

0.338

0.250

0.230

0.216

0.210

0.200

0.198

0.184

9. (β = 0; r ⁄ D0 = ∞)

0.132

–

–

–

0.340

0.280

0.240

0.180

0.136

0.132

10. (β = 15o; r ⁄ D0 = 27.30)

–

0.375

0.265

0.220

0.185

0.175

0.140

0.122

0.113

0.103

11. (β = 30o; r ⁄ D0 = 13.65)

–

–

–

–

–

0.375

0.300

0.275

0.253

0.244

Flow with a Smooth Change in Velocity

375

Converging nozzles of circular cross section in the system with l0/D0 > 047,49,100,136

Diagram 5.23 1. Rectilinear boundary walls (scheme a, 1)

ζ*

∆p ρw20 ⁄ 2

, see graph a

(Re = w0D0/v ≥ 105) or use the approximate formula

ζ = (−0.0125n40 + 0.0224n30 − 0.00723n20 + 0.00444n0 − 0.00745) × (α3r − 2παr2 − 10αr) + ζfr , where n0 = F0/F1 ≤ 1.0, αr = 0.01745α. 2. Curvilinear boundary walls — completely over the radius R of the circle (scheme a, 2): ζ*

∆p ρw20 ⁄ 2

, see graph b (Re ≥ 105) .

3. Rectilinear boundary walls with rounding at the exit over the radius r (scheme a, 3) at α = 90o: ζ*

∆p ρw20 ⁄ 2

, see graphs c and d (Re ≥ 105) .

4. Curvilinear boundary walls of double curvature (nozzle scheme b): ζ*

∆p ρw20 ⁄ 2

= ζfr ,

where for ζfr, see Equations (5.9)–(5.12).

Values of ζ n0 0.64 0.45 0.39 0.25 0.16 0.10

α, deg 3 0.072 0.076 0.098 0.100 0.108 0.118

5 0.067 0.064 0.070 0.071 0.084 0.093

10 0.054 0.052 0.051 0.047 0.048 0.053

15–40 0.040 0.050 0.046 0.044 0.044 0.050

50–60 0.058 0.072 0.064 0.068 0.074 0.079

76 0.076 0.104 0.110 0.127 0.136 0.142

90 0.094 0.138 0.162 0.174 0.184 0.190

105 0.112 0.170 0.210 0.220 0.232 0.237

120 0.131 0.202 0.250 0.268 0.278 0.285

150 0.167 0.246 0.319 0.352 0.362 0.367

180 0.190 0.255 0.364 0.408 0.420 0.427

376

Handbook of Hydraulic Resistance, 4th Edition

Converging nozzles of circular cross section in the system with l0/D0 > 047,49,100,136

Diagram 5.23

Values of ζ n0 0.64 0.45 0.33 0.25

R ⁄ D0 0 0.190 0.255 0.364 0.408

0.1 0.055 0.076 0.062 0.070

0.2 0.046 0.065 0.056 0.068

0.3 0.044 0.060 0.054 0.066

0.5 0.044 0.054 0.052 0.062

1.0 0.044 0.052 0.048 0.053

1.5 0.044 0.049 0.045 0.052

2.0 0.045 0.047 0.048 0.052

0

0.02

0.04

0.06

0.08

0.10

0.15

0.20

0.061 0.064 0.092 0.071

α = 900 0.060 0.060 0.077 0.056

0.059 0.058 0.066 0.053

0.058 0.057 0.059 0.052

0.055 0.057 0.058 0.049

0.052 0.057 0.057 0.045

0.064 0.090 0.115 0.120

α = 1200 0.062 0.067 0.085 0.083

0.060 0.065 0.072 0.063

0.059 0.064 0.065 0.055

0.057 0.062 0.055 0.054

0.054 0.060 0.053 0.053

Values of ζ n0

0.64 0.45 0.33 0.25 0.64 0.45 0.33 0.25

r ⁄ D0

0.097 0.138 0.150 0.160 0.130 0.196 0.237 0.250

0.063 0.074 0.113 0.108 0.087 0.138 0.165 0.170

Flow with a Smooth Change in Velocity

377

Converging nozzles of circular cross section in the system with l0/D0 > 0; laminar flow Re = w0Dh/v ≤ 504

Diagram 5.24

ζ*

∆p ρw20 ⁄ 2

=

A , Re

where at 5o ≤ α ≤ 40o A=

Dh =

4F0 Π0

;

n0 =

20.5 0.75 n0.5 0 (tan α)

,

see curves A = f(α, n0) .

F0 F1

Values of A α, deg

n0 0.15 0.25 0.33 0.5 0.6

5 333 255 221 178 162

Converging-diverging transition pieces in the system with l1/Dh > 0100

10 197 151 131 105 95.7

15 144 110 95.5 77.0 70.0

20 114 87.6 75.8 61.1 55.5

25 95.0 72.8 63.0 50.8 46.2

30 80.8 61.9 53.6 43.2 39.3

35 69.9 53.6 46.4 37.4 34.0

Diagram 5.25

1. Circular cross section (a) Curvilinear converging section (scheme a): ζ*

∆p ρw20 ⁄ 2

= k1k2ζ1 + ∆ζ ,

where at Re = w0Dh/v ≥ 2 × 105, ζ1 = f1(αd), see the graph; at Re < 2 × 105, ζ1 is determined as ζd from Diagram 5.2; k1 = f2(αd, F1/F0) see the graph; k2 ≈ 0.66 + 0.35l0/D0 at 0.25 ≤ l0/D0 ≤ 5; for ∆ζ, see the table.

40 61.0 46.8 40.5 32.6 29.7

378

Handbook of Hydraulic Resistance, 4th Edition

Converging-diverging transition pieces in the system with l1/Dh > 0100

Diagram 5.25

(b) Rectilinear converging section (scheme b): ζrec *

∆p ρw20 ⁄ 2

= Aζcur ,

where ζcur is determined as ζ for a curvilinear converging section; A = f(αd), see the graph. 2. Square cross section (tentatively): ζ*

∆p see paragraph 1, but ζ1 is ρw20 ⁄ 2

determined as ζd at l0/D0 from Diagram 5.4; 3. Rectangular cross section with expansion in one plane (tentatively): ζ*

∆p see paragraph 1, but ζ1 is ρw20 ⁄ 2

determined as ζd at l0/D0 = 0 from Diagram 5.5. αd, deg

Value ζ A

5

7

10

12.5

15

0.10 1.08

0.10 1.09

0.11 1.13

0.13 1.16

0.16 1.15

Values of k1 F1 ⁄ F0

D1 ⁄ D0

1.5–1.6

≈1.25 ≈1.50 ≈1.75 ≥2.0

2.2–2.3 3.0–3.2

≥4.0

αd, deg 5

7

10

12.5

15

0.59

0.55

0.48

0.40

0.33

0.81 0.90 1.0

0.81 0.89 1.0

0.78 0.85 1.0

0.77 0.81 1.0

0.66 0.77 1.0

Values of ∆ζ l0 ⁄ D0 F1 ⁄ F0

D1 ⁄ D0

0.25

0.50

0.75

1.00

12.5

1.50

1.5–1.6 2.2–2.3 3.0–3.2

–0.012 –0.020 –0.022

–0.08 –0.014 –0.016

–0.004 0 –0.010

0 0 0

0.004 0 0.010

0.008 0.014 0.014

≥4.0

≈1.25 ≈1.50 ≈1.75 ≥2.0

–0.028

–0.020

–0.010

0

0.010

0.016

F1 ⁄ F0

D1 ⁄ D0

1.75

2.0

2.5

3.0

3.5

4.0

1.5–1.6 2.2–2.3

≈1.25 ≈1.50 ≈1.75 ≥2.0

0.012 0.020

0.016 0.026

– 0.038

– 0.048

– 0.06

– 0.072

0.022 0.028

0.027 0.030

0.038 –

0.050 –

0.062 –

0.073 –

l0 ⁄ D0

3.0–3.2

≥4.0

Flow with a Smooth Change in Velocity

379

Transition pieces with sharply changing cross sections in the system with l1/D1 > 0

Diagram 5.26

where at Re = w0Dh/v > 104 F0 ⎞ ⎛ ζ1* 0.5 ⎜1 − ⎟ F1 ⎠ ⎝

3⁄4

= ζsc + ζgr + λ

ζ*

∆p ρw20 ⁄ 2

2

F0 ⎞ l0 ⎛ + ⎜1 − ⎟ + λ F D 1⎠ 0 ⎝

l0 D0

,

ζsc * 0.5(1 – F0/F1)3/4, see Diagram 4.9, paragraph 1; ζgr = (1 – F0/F1); at Re < 104, ζsc is determined as ζ from Diagram 4.10 and ζgr is determined as ζ from Diagram 4.1; at all the Re numbers k1 = f(l0/D0, F1/F0) see the graph. For λ, see Diagrams 2.1 through 2.6.

= k1ζ1

Values of k1 F1 F0

l0 D0

D1 D0 0.5

0.6

0.7

0.8

1.0

1.4

≥2.0

1.5–1.6

(≈1.25)

1.02

1.01

1.0

1.0

1.0

1.0

1.0

2.2–2.3

(≈1.50)

1.06

1.03

1.02

1.01

1.0

1.0

1.0

3.0–3.2

(≈1.75)

–

1.10

1.06

1.04

1.01

1.0

1.0

≥4.0

(≥2.0)

–

1.15

1.10

1.08

1.04

1.03

1.0

380

Handbook of Hydraulic Resistance, 4th Edition

Transition from rectangular to circular cross sections in the system with l0/D0 > 0;7,84 Re = w0D0/v > 104

Diagram 5.27

1. Diverging transition piece (F0 > F1): ζd *

∆p ρw20 ⁄ 2

= ζsim + 0.5 exp (−Re × 10−5) = ζsim + ∆ζd ,

∆ζd = 0.5 exp (−Re × 10−5) see graph a. 2. Converging transition piece (F0 < F1): ζcon *

∆p ρw20 ⁄ 2

= ζsim + 0.3 exp (−Re × 10−5) = ζsim + ∆ζcon ,

2

b1 ⎞ ⎛ F0 ⎞ ⎛ l ∆ζcon = 0.3 exp (−Re × 10−5), see graph a; ζsim = ⎜c0 + c1 ⎟ ⎜ ⎟ ; c1 = f ⎛⎜ ⎞⎟ , see graph b (c1d, for the diverging a F D 1 1 ⎝ ⎠⎝ ⎠ ⎝ 0⎠ transition piece; c1con, for the converging transition piece); c0 = λ(l/Dh): Dh = D1h + D0/2 = [2a1b1/(a1 + b1) + 0.5D0]; for λ, see Diagrams 2.1 through 2.6. The choice of the shape and optimal dimensions of the transition pieces is described under para. 1 through 15 (transition pieces). Re × 10–4

1

2

4

6

8

10

20

40

50

∆ζcon

0.272

0.245

0.201

0.165

0.135

0.111

0.041

0.005

0.002

∆ζd

0.453

0.409

0.335

0.275

0.225

0.185

0.068

0.009

0.003

l D0 c1d c1con

1.0

1.5

2.0

2.5

0.055 0.002

0.030 0.002

0.023 0.002

0.018 0.002

3.0

4.0

0.015 0.008 0.0015 0.0010

5.0 0.006 0

Flow with a Smooth Change in Velocity

381

Diffuser with transition from a circle into a rectangle or from a rectangle to a circle, in the system with l1/D1 > 0

Diagram 5.28

ζ*

∆p ρw20 ⁄ 2

,

see Diagram 5.4 for a pyramidal diffuser of rectangular cross section with an equivalent divergence angle which is determined from the relations: tan

⎯⎯⎯⎯⎯⎯ a1b1 ⁄ π − D0 α 2√ , = 2ld 2

for transition from a circle into a rectangle tan

⎯⎯⎯⎯⎯⎯ a0b0 ⁄ π α D1 − 2√ . = 2ld 2

382

Handbook of Hydraulic Resistance, 4th Edition

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.. .. .. 113. Donch, T., Divergente und konvergente turbulente Stromungen mit kleinen Offnungswinkeln, VDI Forschungsarb., no. 282, 1929, 70 p. 114. Eiffel, G., Souffleries aerodynamiques, Resume des principaux travaux executes pendant la guerre au laboratorie aerodynamique, pp. 135–175, 1918. 115. Fernholz, H., Eine grenzschichttheoretische Untersuchung optimalen Unterschalldiffusoren, Ing. Arch., vol. 35, no. 3, 192–201, 1966. 116. Furuya, Y. and Sato, T., Pressure recovery efficiency of short conical diffusers and of roughened diffusers, Bull. JSME, vol. 3, no. 12, 437–443, 1960. 117. Gardel, A., Perte de charge dans un etranglement conique, Bull. Tech. Suisse Romande, vol. 88, no. 21, 313–320, 1962. 118. Gibbings, I. C., Flow in contracting ducts, AIAA J., vol. 2, no. 1, 191–192, 1964. 119. Gibson, A., On the flow of water through pipes and passages having converging or diverging boundaries, Proc. R. Soc., London, vol. 83, pp. 27–37, 1910. 120. Gibson, A., On the resistance to flow of water through pipes or passages having diverging boundaries, Trans. R. Soc., vol. 48, 123–131, 1911. 121. Chose, S. and Kline, S. J., The computation of optimum pressure recovery in two-dimensional diffusers, J. Fluids Eng., vol. 100, 419–426, 1978. 122. Goldsmith, E. L., The effect of internal contraction, initial rate of subsonic diffuser and coil and center body shape on the pressure recovery of a conical center body intake of supersonic speed, Aeronaut. Res. Counc. Rep. Mem., no. 3204, pp. 131–140, 1962. .. 123. Hackeschmidt, M. and Vogelsang, E., Versuche an Austrittsgehause mit neuartigen Diffusoren, Maschinenbautechnik, vol. 15, no. 5, 251–257, 1966. .. 124. Hofmann, A., Die Energleumsetzung in saugrohrahnlicherweiterten Dusen, Mitteilungen, no. 4, 75–95, 1931. .. 125. Imbach, H. E., Beitrag zur Berechnung von rotationssymetrischen turbulenten Diffusorstromungen, Brown Boveri Mitt., vo1. 51, no. 12, 784–802, 1964. 126. Jahn, K., Ein Beitrag zum Problem der Siebdiffusoren, Maschienbautechnik, vol. 19, no. 2, 35–45, 1970. 127. Jezowiecka-Kabsch, K., Wplyw ksztaltow dyfuzorow na wysokosc stratthydraulicznych, Pr. Nauk Inst. Tech. Cieplnej i Aparatury Przen. PW, sv. 3, 51, 1971. 128. Johnston, J. H., The effect of inlet conditions on the flow in annular diffusers, C.O.N. 178, Memo. M 167, no. 1, pp. 21–30, 1953. 129. Johnston, J. P. and Powers, C. A., Some effects of inlet blockage and aspect ratio on diffuser performance, Trans. ASME, vol. D91, no. 3, 551–553, 1969. 130. Kline, S. J., On the nature of stall, Trans. ASME, Ser. D, vol. 81, no. 3, 305–320, 1969. 131. Kline, S. J., Moore, C. A., and Cochran, D. L., Wide-angle diffusers of high performance and diffuser flow mechanisms, J. Aeronaut. Sci., vol. 24, no. 6, 469–470, 1957. 132. Kline, S. J., Abbott, D. E., and Fox, R. W., Optimum design of straight-walled diffusers, Trans. ASME, Ser. D, vol. 81, no 3, 321–331, 1959. 133. Kmonicek, V., Scurgerea subsonica in difusoare conice, Stud. Cercet. Mec. Apl. Acad. RPP, Sv. 12, No. 2, 383–390, 1961. 134. Kmonicek, V., Ovlivenini cinnosti prostych kuzelovyck difusori vlozenymi telesy, Strojnicky casop, sv. 14, no. 5, 484–498, 1963. 135. Kmonicek, V. and Hibs, M., Vysledki experimentalniho a teoretickeho vyzkumu mezikruhovych difusorovych kanalu, Lake, probl. ve stavbe spalov turbin, pp. 371–397, Praha, SCAU, 1962. 136. Toshisuke, K. and Tatsuhiro, U., On the characteristics of divided flow and confluent flow in headers, Bull. JSME, no. 52, 138–143, 1969. 137. Levin, L.. and Clermont, F., Etude des pertes de charge singulieres dans les convergents coniques, Le Genie Civil, vol. 147, no. 10, 11–20, 1970. .. 138. Liepe, F., Experimentale untersuchungen uber den Einfluss des Dralles auf die Stromung in Schlanken Kegeldiffusoren, Wiss. Z. TH, Dresden, vol. 8, no. 2, 330–335, 1962.

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139. Liepe, F. and Jahn, K., Untere Wirkungsgrade von Kegel diffusoren, Maschinenbautechnik, no. 11, 41–52, 1962. l40. Limberg, H., Scurgerea de intrare laminara intr-un canal convergent, Stud. Cercet. Mec. Apl. Acad. RPR, vol. 12, no. 1, 3–10, 1961. .. 141. Linneken, H., Betrachtungen uber Wirkungsgrade gasdurchstromter Diffusoren, Konstruktion, vol. 15, no. 7, 10–17, 1963. 142. Livesey, J. L. and Turner, J. T., The dependence of diffuser performance upon inlet flow conditions, J. R. Aeronaut. Soc., vol. 69, no. 6159, 794–795, 1965. 143. Livesey, J. L. and Hugh, T., Some preliminary results for conical diffusers with high subsonic entry Mach numbers, J. Mech. Eng. Soc., vol. 8, no. 4, 384–391, 1966. 144. Lau, W. T. F., An analytical method for the design of two-dimensional contractions, J. R. Aeronaut. Soc., vol. 68, no. 637, 59–62, 1964. 145. Markland, E. and North, F., Performance of Conical Diffusers Up the Choking Condition, in Proc. 6th Conf. Fluid Machinery, Budapest, vol. 2, pp. 703–713, 1979. 146. Mathieson, R. and Lee, R. A., Diffusers with boundary layer suction, Int. Assoc. Hydraul. Res. 10th Congr., London, vol. 4, no. 249, pp. 81–88, 1964. 147. McDonald, A. T. and Fox, R. W., An Experimental Investigation of Incompressible Flow in Conical Diffusers, Pap. Am. Soc. Mech. Eng., no. FE-25, 1966. 148. McDonald, A. T., Fox, R. W., and Dewoestine, R. V., Effects of swirling inlet flow on pressure recovery in conical diffusers, AIAA J., vol. 9, no. 10, 2014–2018, 1971. 149. Naumann, Efficiency of diffusers on high subsonic speeds, Reports and Transactions, no. 11, A, June, pp. 1–20, 1964. .. 150. Nikuradse, I., Untersuchungen uber die Stromungen des Wassers in konvergenten und divergenten Kanalen, VDI Forschungsarb., no. 289, 1929, 60 p. 151. Patterson, G., Modern diffuser design, Aircraft Eng., pp. 1–15, 1938. 152. Peryez, S., Der Einfluss des Diffusorwirkungsgarde auf den Austrittverlust im Dampfturbinen, Brennsr. Waerme Kraft, vol. 13, no. 9, 9–15, 1961. 153. Peters, H., Energieumsetzung in Querscnittserweiterung bei verschiedenen Zulaulbedingungen, Ing. Arch., no. 1, 7–29, 1931. .. 154. Pohl, K., Stromungverhaltnisse in einen Diffusor mit vorgeschalteten Krummer, Ing. Arch., no. 29, 21–28, 1960. .. 155. Polzin, J., Stromungsuntersuchungen an einem ebenen Diffusor, Ing. Arch., vol. 5, 30–49, 1950. 156. Prechter, H. P., Gesichtpunkte sur Auslegung von Diffusoren unter Berucksichtigung neuerer Forschungser-gebnisse, Der Maschinenmarkt, vol. 13, no. 82, 31–39, 1961. 157. Raghunathan, S. and Kar, S., Theory and Performance of Conical Diffuser Exit Duct Combinations, Pap. Am. Soc. Mech. Eng., no. NWA/FE-45, 1968, 8 p. 158. Rao, D. M., A method of flow stabilization with high pressure recovery in short conical diffusers, Aeronaut. J., vol. 75, no. 725, 336–339, 1971. 159. Rao, P. V. and Dass, H. S., Design and testing of streamline shapes for axisymmetric diffuser, J. Inst. Eng. India, vol. 62, pt. ME 2, pp. 39–46, 1981. 160. Rao, D. M. and Raju, K. N., The Use of Splitters for Flow Control in Wide-Angle Conical Diffusers, Tech. Note Nat. Aeronaut. Lab. Bangalore, no. AE-26, 1964, 19 p. 161. Rao, D. H. and Raju, K. N., Experiments on the Use of Screens and Splitters in a Wide-Angle Conical Diffuser, Tech. Note Nat. Aeronaut. Lab. Bangalore, no. AE-24, 1964, p. 23. 162. Rao, P., Samba, S., Vyas, B. D., and Raghunathan, S., Effect of inlet circulation on the performance of subsonic straight conical diffusers, Ind. J. Technol., vol. 9, no. 4, 135–137, 1971. 163. Ringleb, F. O., Two-dimensional flow with standing vortexes in ducts and diffusers, Trans. ASME, Ser. D, vol. 82, no. 4, 921–927, 1960. 164. Robertson, J. M. and Fraser, N. R., Separation prediction for conical diffusers, Trans. ASME, Ser. D, vol. 82, no. 1, 135–145, 1960.

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165. Robertson, J. M. and Fraser, H. R., Investigation of the boundary layer stall in a conical diffuser, Trans. ASME, Ser. D, vol. 81, no. 1, 35–43, 1961. 166. Runstadler, P. W. and Dean, R. C., Straight channel diffusor performance at high inlet Mach numbers, no. Pap. Am. Soc. Mech. Eng., no. NWA/FE-19, 1968, 16 p. 167. Sagi, J., The design and performance of two-dimensional curved subsonic diffusers, Diss. Abstr., vol. 28, no. 7, 1968. .. l68. Schlichting, P. W. and Gersten, K., Berechnung der Stromung in rotationssymmetrischen Diffusoren mit Hilfe der Grenzschichttheorie, Z. Flugwiss., vol. 9, no. 4, 5, 18–27, 1961. 169. Sharan, W. Kr., Improving diffuser performance by artificial means, AIAA J., vol. 10, no. 8, 1105–1106, 1972. .. 170. Siedschlag, H. J., Die Stromung in Diffusoren Verschiedener Querschnittsformen, Wiss. Z. Tech. Univ. Dresden, vol. 12, no. 1, 85–96, 1963. l71. Sisojev, V. O stepenu korisnog dejstva nadzvue difozora aerotunela, Tehnika, sv. 16, no. 3, 100– 104, 1961. 172. Sokwan, L., Vortex phenomena in a conical diffuser, AIAA J., vol. 5, no. 6, 1072–1078, 1967. l73. Sprenger, H., Messungen an Diffusoren, Z. Angew. Math. Phys., vol. 7, no. 4, 372–374, 1957. 174. Sprenger, H., Experimentelle Untersuchungen an geraden und gekrummten Diffusoren, Mitt. Inst. Aerodyn (Zurich), vol. 84, no. 27, 1959. 175. Stock, H. W., Compressible turbulent flows in long circular cross-section diffusers of large area ratio, Z. Flugwiss. Weltraumforsch, Bd. 9, Heft 3, 143–155, 1985. l76. Stull, F. D. and Velkoff, H. R., Effects of Transverse Ribs on Pressure Recovery in Two-Dimensional Subsonic Diffusers, AIAA Pap., no. 1141, 1972, 11 p. 177. Squire, H. B., Experiments on conical diffuser, Reports and Memoranda, no. 2751, November, pp. 41–60, 1950. 178. Stevens, S. G. and Markland, E., The Effect of Inlet Conditions on the Performance of Two Annular Diffusers, Pap. Am. Soc. Mech. Eng., no. NWA/FE-38, 1968, p. 15. 179. Stratford, B. S. and Tubbs, H., The maximum pressure rise attainable in subsonic diffusers, J. R. Aeronaut. Soc., vol. 69, no. 652, 275–278, 1965. .. 180. Szablewski, W., Turbulente Stromungen in divergenten Kanalen mittlerer und starker Druckanstieg, Ing. Arch., vol. 22, no. 4, 268–281, 1954. .. 181. Winter, H., Stromungsverhtiltnisse in einem Diffusor mit vorgeschalteten Krummer, Maschinenbau .. Warmenwirtschaft, no. 2, 38–49, 1953. 182. Winternitz, F. A. L. and Ramsay, W. J., Effect of inlet boundary layer on pressure recovery, energy conversion and losses in conical diffusers, J. R. Aeronaut. Soc., vol. 61, no. 554, 15–23, 1957. 183. Wolf, S. and Johnson, J. P., Effects of Nonuniform Inlet Velocity Profiles on Flow Regimes and Performance in Two-Dimensional Diffusers, Pap. Am. Soc. Mech. Eng., no. WA/FE-25, 1969, 13 p. 184. Wu, J. H. T., On a two-dimensional perforated intake diffuser, Aerosp. Eng., vol. 21, no. 7, 13– 19, 1962. 185. Van Dewoestine, R. V. and Fox, R. W., An experimental investigation of the effect of subsonic inlet Mach number on the performance of conical diffusers, Int. J. Mech. Sci., vol. 8, no. 12, 759– 769, 1966. 186. Villeneuve, F., Contribution a l’etunde de l’ecoulement dans on diffuseur a six degres, Publ. Sci. Tech. Minist. Air, no. 397, 1963, 9 p. 187. Kwong, A. H. M. and Dowling, A. P., Active boundary-layer control in diffusers, AIAA J., vol. 32, no. 12, 2409–2414, 1994. 188. Frankfurt, M. O. A compararive estimate of the efficiency of diffuser channels with suction and tangential injection, in Prom. Aerodin., no 4(36), pp. 191–195, Mashinostroenie Press, Moscow, 1991. 189. Bychkova, L. A., Experimental investigation of diffuser channels with preseparation state of a boundary layer, Uch. Zap. TsAGI, vol. 1, no. 6, 89–93, 1970.

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190. Molochnikov, V. M., Determination of the aerodynamic characteristic of preseparation diffusers based on solution of the inverse boundary-layer problem, Inzh.-Fiz. Zh., vol. 62, no. 1, 808–813, 1992. 191. Kozlov, A. P. and Molochnikov, V. M., Structure of flow in a profiled diffuser with a preseparation state of a turbulent boundary layer, Aviats. Tekh., no. 2, 84–88, 1994. 192. Kozlov, A. P., Mikheev, N. I., Molochnikov, V. M., and Nilov, G. A., Structure of flow in the wall region of a profiled preseparation diffuser, Izv. Ross. Akad. Nauk, Energetika, no. 3, 83–88, 2001. 193. Mazo, A. S., Calculation of the geometric and aerodynamic characteristics of conical diffusers with a boundary layer in a preseparation state, Inzh.-Fiz. Zh., vol. 64, no. 5, 827, 1983. 194. Dovzhik, S. A. Influence of the radial nonuniformity of flow on the efficiency of annular diffusers, in Prom. Aerodin., no. 32, pp. 5–21, Mashinostroenie Press, Moscow, 1975. 195. Sovran, G. and Komp, E. D., Experientally determined optimum geometries for rectilinear diffusers with rectangular, conical or annular cross section, in Proc. of the Symposium on the Fluid Mech. of Internal Flow, General Motors Research Laboratories, Warren, Michigan, 1965, Amsterdam and others, pp. 270–319, 1967. 196. Adenubi, S. O. Performance and flow regime of annular diffusers with axial turbomachine discharge inlet conditions, J. Fluid Eng., Trans. ASME, Ser. I, vol. 98, no. 2, 236–243, 1976. 197. Dovzhik, S. A. and Kartavenko, V. M., Experimental investigation of the effect of flow twisting on the efficiency of annular channels and exit branches of axial turbines, in Prom. Aerodin., no. 31, pp. 94–103, Mashinostroenie Press, Moscow, 1974. 198. Lohmann, R. P., Markowski, S. T., and Brookman, E. N., Swirling flow through annular diffusers with conical walls, Trans. ASME, J. Fluids Eng., vol. 101, no. 2, 1979. 199. Verevkin, N. N. and Lashkov, A. I., Concerning the means of reducing pressure losses in diffusers with large divergence angles, in Prom. Aerodin., no. 7, pp. 81–94, Oborongiz Press, Moscow, 1956. 200. Khanzhonkov, V. I., Improvement of the efficiency of diffusers with large divergence angles by using plane screens, in Prom. Aerodin., no. 3, pp. 210–217, Oborongiz Press, Moscow, 1947. 201. Schubauer, G. B. and Spangenberg, W. G., Effect of Screens in Wide Angle Diffusers, NACA Rep., 949, 1949. 202. Gibbings, J. C., The pyramid gauze diffuser, Ing.-Archiv, vol. 42, no. 4, 225–233, 1973. 203. Chang, P. K., Control of Flow Separation, Hemisphere Publishing Corporation, New York–Washington, 1976. 204. Senoo, K. and Nishi, M., Improvement of the performances of conical diffusers by vortex generators, Trans. ASME, Ser. D, no. 1, 96–103, 1974. 205. Ledovskaya, N. N., Some ways of increasing the efficiency of annular diffusers with large divergence angles, Trudy TsIAM, no. 1112, 1–13, 1984. 206. Ledovskaya, N. N., Experimental investigation of the three-dimensional structure of a separation flow in an axisymmetrical annular diffuser, Inzh.-Fiz. Zh., vol. 51, no. 2, 321–328, 1986. 207. Brown, A. C. et al., Subsonic diffusers designed integrally with vortex generators, Paper 67–464, in Proc. AIAA Third Propulsion Joint Specialist Conf., Washington, D.C., 1967. 208. Dovzhik, S. A., Investigations into the aerodynamics of an axial subsonic compressor, Trudy TsAGI, No. 1099, 1968. 209. Guinevskiy, A. S., On calculation of the hydraulic resistance with separation flows, Inzh.-Fiz. Zh., vol. 8, no. 4, 540–545, 1965. 210. Khalatov, A. A., Theory and Practice of Twisted Flows, Naukova Dumka Press, Kiev, 1989, 192 p. 211. Frankfurt, M. O., Efficiency of tangential blowoff of a boundary layer in conical diffusers, Uch. Zap. TsAGI, vol. 4, no. 5, 50–55, 1973. 212. Sedov, L. I. and Chernyi, G. G., in Collection of Papers no. 12, L. I. Sedov (Ed.), vyp. 4, pp. 17–30, Oborongiz Press, Moscow, 1954.

CHAPTER

SIX RESISTANCE TO FLOW WITH CHANGES OF THE STREAM DIRECTION RESISTANCE COEFFICIENTS OF CURVED SEGMENTS — ELBOWS, BENDS, ETC.

6.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS 1. Bending of a flow in curved tubes and channels (elbows, bends, and bypasses*) results in the appearance of centrifugal forces directed from the center of curvature to the outer wall of the tube. This causes an increase of the pressure at the outer wall and its decrease at the inner wall, when the flow passes from the straight to the curved section of the pipe (until it is completely turned). Therefore, the flow velocity will correspondingly be lower at the outer wall and larger at the inner wall (Figure 6.1). Thus, in this bend a diffuser effect occurs near the outer wall and a bellmouth effect occurs near the inner wall. The passage of flow from the curved into the straight section, after turning, is accompanied by these effects in the reverse order: a diffuser effect near the inner wall and a bellmouth effect near the outer wall. 2. The diffuser phenomena lead to corresponding flow separation from both walls. In this case, separation from the inner wall is intensified by inertial forces acting in the curved zone in the direction toward the outer wall. An eddy zone, formed as a result of flow separation ∗

Bypasses are meant to be curved sections in which the inner and outer walls represent arcs of concentric circles with inlet and outlet cross sections being equal: r0 ≥ 0 and r1 = r0 + b0 , where r0 is the radius of curvature of the inner wall and r1 is the radius of curvature of the outer wall. Since the two walls have the same center of curvature, the bend is characterized by the radius of curvature R0 of its axis when R0/b0 ≥ 0.5. Elbows are meant to be curved sections, the curvatures of the inner and outer walls of which are not arcs of concentric circles.

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Figure 6.1. Variation of velocity and pressure profiles in an elbow and a slraight section downstream.

from the inner wall, propagates far ahead and across, greatly reducing the cross section of the main stream. 3. The appearance of centrifugal forces and the presence of boundary layers at the walls explain the occurrence of a secondary (transverse) flow in a curved tube, i.e., the formation of the so-called vortex pair, which is superimposed on the main stream parallel to the channel axis and imparts a helical shape to the streamlines (Figure 6.2). 4. The main portion of pressure losses in curved tubes is due to formation of eddies at the inner wall, and this, together with the secondary flows, determines the nature of the velocity distribution downstream of the bend. The magnitude of the resistance coefficient of curved tubes and the flow structure within them vary under the influence of both the factors governing the degree of flow turbulence __ shape. These factors are the Reynolds number Re = w0Dh/v, and the inlet velocity profile relative roughness of walls ∆ = ∆/Dh, inlet conditions, relative length of the straight starting section l0/D0,* relative distance from the preceding shaped piece, etc., and geometric parameters of the tube, i.e., the angle of bend δ, the relative radius of curvature r/D0 or R0/D0 (R0/b0) (Figure 6.3), the aspect ratio (relative elongation) a0/b0, the ratio of the inlet area to the exit area F1/F0, etc. 5. Other conditions being equal, the curved tube offers the largest resistance in the case when the curvature at the inner wall is a sharp corner; the flow separates from this wall most vigorously. At the angle of bend δ = 90o, the region of flow separation at the inner wall downstream of the bend amounts to 0.5 of the tube width. Hence, the intensity of eddy (vortex) formation and the resistance of the curved tube (channel) increase, with an increase in the angle of bend. Rounding of the elbow corners (especially of the inner corner) makes the flow separation much smoother and, consequently, lowers the resistance. ∗

Here l0/D0 is the length of the straight section downstream of a smooth inlet (collector).

Flow with Changes of the Stream Direction

393

Figure 6.2. Vortex pair in an elbow: (a) longitudinal section; (b) cross section (rectangular channel); (c) cross section (of a circular tube).

6. If the outer edge of the elbow is left sharp (radius of the outer curvature r1 = 0) and only the inner corner is rounded, an increase in the radius of inner curvature r0, then the minimal resistance of the elbow with a 90o bend will be attained at r0/b0 = 1.2–1.5. With a further increase in r0/b0, the resistance will grow noticeably. Such an increase in the resistance is due to the fact that if the inner corner is rounded, the cross-sectional area at the place of bending increases considerably, and hence the velocity decreases. This intensifies the diffuser separation of flow, originating at the place of transition from the starting length into the elbow. 7. Rounding the outer corner and keeping the inner corner sharp (r0 = 0) do not lead to a noticeable decrease in the elbow resistance. A significant increase in the radius of curvature of the outer wall even causes an increase in the elbow resistance. This indicates that it is undesirable to round the outer wall alone (with the inner corner kept sharp), since then the cross-sectional area of the flow decreases at the place of flow turning and increases the diffuser losses, which originate during flow passage from the elbow to the exit section of the pipe.

Figure 6.3. Scheme of rounding of an elbow and the dependence of the resistance coefficient of an elbow on the curvature radius r/b0.

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The minimum resistance is achieved by an elbow for which r1/b0 = r0/b0 + 0.6 (an elbow of the optimum shape), while the resistance close to the minimum is offered by a bend or by a "normal" elbow for which r1/b0 = r0/b0 + 1.0. Since a bend is technically more easily achieved, it can supplant an optimal elbow in the majority of cases. 8. The resistance of right-angle elbows can be greatly reduced by installing a fairing on the inner corner (see Diagram 6.10). The optimum value of the relative curvature radii of the fairing amounts to r0/b0 = 0.45. If such a fairing is installed, the resistance coefficient of the elbow bend (δ = 90o) diminishes from ζ = 1.15 to ζ = 0.55.30 Rounding of the outer corner of the elbow with the radius r1/b0 = 0.45 additionally reduces the losses by 5%, i.e., up to ζ = 0.49. Reduction in the elbow resistance can also be attained by beveling (along the chord) sharp corners of the bend (especially the inner corner; see Diagram 6.10). 9. A change in the ratio of the areas F1/F0 of the entrance to and exit from the elbow changes its resistance. With an increase in the cross-sectional area downstream of the bend, the diffuser effect increases, which intensifies the flow separation and the formation of vortices (increases the eddy zone). At the same time, when the discharge is constant, the flow velocity in the exit section decreases. The effect of the velocity decrease, expressed as a decrease in the pressure losses, is greater (with the increase in the ratio F1/F0 up to certain limits) than the effect of the increasing eddy zone, which leads to higher losses. As a result, the total losses in the elbow of enlarged cross section decrease within certain limits. 10. The resistance of straight elbows (δ = 90o) with sharp corners is minimum for the ratio F1/F0 lying within 1.2–2.0. In elbows and bends with smooth turns the optimal ratio F1/F0 is closer to unity; in some cases, it is even less than unity, which is clearly seen in Figure 6.4. The internal resistance coefficient* ζin of plane bends with δ = 90o and height-to-width ratio a0/b0 = 2.4 depends on the relative curvature radius of the outer wall r1/b0 at different values of the relative curvature radius of the inner wall r0/b0. The envelope of the curves ζ = f(r0/b0, r1/b0), over the whole range of values of r0/b0 and r1/b0, is higher for the diffuser channel when F1/F0 = 1.3 and lower when F1/F0 = 0.5. The intermediate position is occupied by a channel of constant cross section (F1/F0 = 1.0). Figure 6.4 can provide guidelines for the choice of optimal relationships between r0/b0 and r1/b0 of plane branches with δ = 90o. In the absence of data on the resistance of elbows and expanding bends, it is possible to neglect a decrease in the pressure losses within the above limits of F1/F0 and to assume that the resistance coefficient is the same as for F1/F0 = 1. An increase in the resistance at values of F1/F0 differing substantially from the optimal values should not be neglected. 11. The resistance of curved tubes (channels) decreases with an increase in the relative elongation of the cross-sectional area of the elbow a0/b0 and increases when a0/b0 decreases within the limits of less than unity. 12. In the majority of cases, for the convenience of engineering calculations, the total resistance coefficient of elbows and bends is determined as the sum of the coefficient of local resistance of the bend ζloc and the friction coefficient ζfr: ∗

The internal resistance coefficient ζin is determined as the ratio of the difference of total pressures at the entrance to and exit from the bend to the velocity pressure at the entrance. It does not consider the additional losses that would have occurred in a straight exit section downstream of the bend due to further equalization of the velocity profile distorted when the flow turns in the branch.

395

Flow with Changes of the Stream Direction

Figure 6.4. Dependence of ζin of elbows with δ = 90o on r1/b0 at different values of r0/b0.77

ζ = ζloc + ζfr , where ζfr = λ(l/Dh) is calculated as ζ for straight sections with λ taken from Diagrams 2.1 __ through 2.6 as a function of the Reynolds number and of the relative roughness ∆ = ∆/Dh; l is the length of the elbow or bend measured along the axis. The ratio R0 o δ o R0 l =π = 0.0175 δ . o Dh Dh 180 Dh Then R0 δλ . Dh 13. The coefficient of local resistance of branches is calculated from the formula suggested by Abramovich:1* ζfr = 0.0175

ζloc *

∆p ρw20 ⁄ 2

− A1B1C1 ,

(6.1)

where A1 is the coefficient that allows for the effect of the angle of the bend δ; B1 is the coefficient that allows for the effect of the relative curvature radius of the bend R0/D0 (R0/b0); and C1 is the coefficient that allows for the effect of the relative elongation of the bend cross section a0/b0. The value of A1 can be determined from the data of Nekrasov:31 ∗

Abramovich’s formula contains a numerical factor 0.73, which is included here in the quantity B1.

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at δ = 90o A1 = 1.0 , at δ < 70o A1 = 0.9 sin δ , at δ > 100o A1 = 0.7 + 0.35

δ , 90

or from graph a of Diagram 6.1. The value of B1 can be calculated from the approximate formulas:* at

R0 ⎛ R0 ⎞ 0.21 < 1.0 B1 = , 0.25 D0 ⎜ b0 ⎟ (D 0 ⁄ D0) ⎝ ⎠

at

R0 ⎛ R0 ⎞ 0.21 , ≥ 1.0 B1 = D0 ⎜ b0 ⎟ R √ ⎯ ⎯⎯⎯⎯⎯ 0 ⁄ D0 ⎝ ⎠

or from graphs b and c of Diagram 6.1. The value of C1 is determined from graph d of Diagram 6.1. 14. The total resistance of very smooth curvilinear tubes and channels (bends, coils) (R0/D0 ≥ 3.0) can be considered as an increased friction coefficient at which the resistance coefficient is dependent not only on the Reynolds number and roughness, but also on the relative curvature radius R0/D0 (R0/b0) or on the parameter Re√ 2R0 ⁄ D0 : ⎯⎯⎯⎯⎯⎯⎯ ⎛ __ R 0 ζ = f ⎜Re, ∆, , Re D0 ⎝

⎞ 2R0 ⎟ . D0 ⎠

⎯⎯ √

Here ζ = λcur(l/Dh) = 0.0175(R0/D0)δλcur, where λcur is the coefficient of hydraulic friction of a curved channel (bend). __ 15. The dependence of the hydraulic friction coefficient of smoothly curved tubes (bends), 2R0 ⁄ D0 , R0/D0, and ∆, established by various authors (Figure 6.5), points to the λcur, on Re√ ⎯⎯⎯⎯⎯⎯⎯ existence of a close analogy for such tubes with straight tubes (see Chapter 2). In this case, four flow regimes are possible. The first regime, within the limits up to Re = 6.5 × 103, is laminar. It is __characterized by the condition where the straight lines of resistance for different R0/D0 and ∆ are parallel to each other and form a sharp angle with the abscissa ln Re√ 2R0 ⁄ D0 . ⎯⎯⎯⎯⎯⎯⎯ The second regime, with 6.5⋅103 < Re < 4⋅104, is a transition regime. The coefficient λcur under these conditions is practically independent of the Reynolds number. The third regime is turbulent, which corresponds to 4 × 104 < Re < 3 × 105. The resistance curves of smooth bends exhibit behavior similar to that of the resistance curves of straight commercial pipes (with nonuniform roughness) in the transition region (see Diagram 2.4): they decline __smoothly with an increase in the parameter Re√ 2R0 ⁄ D0 . Besides, for dif⎯⎯⎯⎯⎯⎯⎯ ferent R0/D0 and ∆ these curves are also parallel to each other. ∗

For a rectangular cross section R0/D0 should be replaced by R0/b0.

Flow with Changes of the Stream Direction

397

Figure 6.5. Resistance coefficient λel of smooth 90o bends as a function of the dimensionless parameter Re√ 2R0 ⁄ D0 ;56 1) Ito, D0 = 35 mm, brass; 2) Goffman, D0 = 43 mm, brass; 3) Goffman, D0 = 43 mm, ⎯⎯⎯⎯⎯⎯⎯ rough brass; 4) Tsimerman, D0 = 50 mm, steel; 5) Gregorik, D0 = 89.3 mm, steel; 6) Idelchik, smooth; 7) Lee, D0 = 26 mm, steel.

The fourth regime is observed at Re > 3⋅105; the curves λcur = f(Re√ 2R0 ⁄ D0 ) are parallel ⎯⎯⎯⎯⎯⎯⎯ __ to the abscissa so that λcur practically ceases to depend on Re and remains a function of __ R0/D0 and ∆ alone. 16. For smooth curved tubes (made of glass, brass, lead, rubber, steel at ∆ < 0.0002, etc.) at any δ, including spirals (coils), the value of λcur up to Re ≈ 105 can be calculated from the following equation* (see also Diagram 6.2): λcur =

a . Re (2R 0 ⁄ D0)m n

(6.2)

17. Analogous formulas have been obtained for curvilinear channels of square cross section84 (see Diagram 6.2). Formulas for the rectangular cross section of different orientation differ somewhat: the value of λcur can be calculated from the formulas suggested by Dementiyev and Aronovl2 (see also Diagram 6.2): at Re = (0.5−7) × 103 λcur = ⎡1.97 + 49.1(Dh ⁄ 2R0)1.32(b ⁄ h)0.37⎤ Re−0.46 = Alam Re−0.46 , ⎦ ⎣ or ∗

Equation (6.2) was derived by Aronov3,4 on the basis of his experiments and those of Adler51 and White.95 Dala close to the values of λcur obtained by Aronov are given in the works of Kvitkovsky,24 Koshelev et al.,27 Mazurov and Zakharov,28 and Shchukin et al.48

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Handbook of Hydraulic Resistance, 4th Edition

Figure 6.6. Schematic representation of the winding of a helical tube.

λcur ⁄ Alam = Re−0.46 at Re = (7−38) × 103 λcur = ⎡0.316 + 8.65(Dh ⁄ 2R0)1.32(b ⁄ h)0.34⎤ Re−0.25 = Atur Re−0.25 , ⎦ ⎣ or λcur ⁄ A tur = Re−0.25 . 18. Within the limits of the subcritical Dean number which is determined approximately from the formula89 (De)cr = 2 × 104(Dh ⁄ 2R 0)0.82 {where R0′ = R0[1 + tp/(2πR0)2]; tp is the spiral pitch (Figure 6.6)}, it is possible to use the following single formula for calculating the hydraulic friction coefficient λcur which will be valid for any cross-sectional shape of a curvilinear channel (circular, rectangular, quadratic, and elliptical):89 Dh ⁄ 2R0 ) λcur = 0.1008f (γ) (Re √ ⎯⎯⎯⎯⎯⎯⎯ Dh ⁄ 2R0 ) + 7.782f (γ) (Re √ ⎯⎯⎯⎯⎯⎯⎯

−1

0.5

[1 + 3.945f (γ) (Re √ Dh ⁄ 2R0 ) ⎯⎯⎯⎯⎯⎯⎯

Dh ⁄ 2R0 ) + 9.097f (γ) (Re √ ⎯⎯⎯⎯⎯⎯⎯

−0.5

−1.5

−2

Dh ⁄ 2R0 ) ] λ , + 5.608f (γ) (Re √ ⎯⎯⎯⎯⎯⎯⎯ where λ = f(Re) is the hydraulic friction coefficient determined for the given cross-sectional shape of the channel from the corresponding diagrams of Chapter 2; γ = b0/a0 is the ratio of the channel cross section axes;

Flow with Changes of the Stream Direction

399

for the rectangular cross section f (γ) = Dh ⁄ 2 at γ < 1 , f (γ) = 2 ⁄ Dh at γ > 1 , for the elliptical cross section f (γ) = 2γ ⁄ (γ + 1) at γ < 1 , f (γ) = (γ + 1) ⁄ (2γ) at γ > 1 . 19. The coefficient of local resistance of elbows with sharp corners can be calculated for the entire range of angles 0 ≤ δ ≤ 180o from the equation ζloc *

∆p ρw20 ⁄ 2

= C1Aζ ′ ,

where ζ′ is determined by Weisbach’s equation:92 ζ ′ = 0.95 sin2

δ δ + 2.05 sin4 , 2 2

A is the correction factor, which is obtained from the experimental data of Richter79 and Schubart81 and is determined from the curve A = f(δ) of Diagram 6.5. 20. The local resistance coefficients of any elbows and bends can be considered constant and independent of the Reynolds number only when Re > (2–3) × 105. At lower values of this number, it influences the resistance value and this influence increases with decreasing values of Re. This is particularly true of bends, as well as of elbows with smooth inner curvature. 21. The dependence ζ = f(Re) is complex and, according to the author’s data,16,17 its character is mainly determined by a change in the flow regime in the boundary layer. In particular, in bends with R0/b0 = 0.55–1.5, especially when the bends are installed close to the smooth entrance, the phenomenon is similar to that observed for a flow around a cylinder or a sphere. 22. Starting with very small values of the Reynolds number, the coefficient of total resistance* ζtot of the bend drops at R0/b0 ≈ 0.5–1.5, reaching the first minimum at about Re = 5⋅104 (Figure 6.7). Following this, there is a slight increase in ζtot until it reaches the value corresponding to Recr (in this case at about 105), at which there occurs a sharp drop in the resistance coefficient (the resistance crisis in the transition regime) until the second minimum at Re = (0.2–2.5) × 105 (developed or postcritical regime), whereupon there follows a slight increase in the resistance coefficient. ∗ The coefficient ζtot also includes the velocity pressure losses at the exit from the bend into the atmosphere.

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Handbook of Hydraulic Resistance, 4th Edition

_ total resistance coefficient ζtot on the Reynolds number for a bend with Figure 6.7. Dependence of _the δ = 90o and smooth walls (∆ = 0.00003).17

23. At relatively small Reynolds numbers (approximately up to Re = 105), the boundary layer is laminar in a bend installed close to a smooth entrance; therefore, at moderately large R0/b0 the separation of the flow from the walls of the inner curvature is laminar. The critical Reynolds number, at which ζtot starts to decrease, is characterized by transition from laminar to turbulent flow. Turbulization of the separated boundary layer, which leads to an intensified momentum exchange between separate fluid particles, causes the jet expansion in this layer resulting in the contraction of the inner eddy zone (Figure 6.8).

Figure 6.8. Regions of flow separation from the inner wall and distribution of velocities over the mean line of the cross section of a bend with smooth walls at different flow regimes:17 1) point of laminar separation; 2) "dead" zone; 3) transition point; 4) point of reattachment of the separated layer; 5) turbulent expansion of the separated layer; 6 and 7) lower boundary of the separated laminar and turbulent layers, respectively; 8) point of turbulent separation.

Flow with Changes of the Stream Direction

401

24. As the Reynolds number increases, the transition point moves progressively upstream, while the separated boundary layer expands until it again clings to the inner wall of the bend. However, the centrifugal forces in the turning region prevent the layer from adhering to the entire curvature of the bend, causing the flow to separate again from the wall, but this time it is the separation of a turbulent layer at a larger distance from the inner curvature (see Figure 6.8). 25. In the first instant after the reattachment of the layer, a closed eddy zone is formed between the point of laminar separation and the point of flow attachment. With further increase in Re, this zone disappears completely, when the transition point coincides with the point of laminar separation. This moment corresponds to completion of the transition flow regime, subsequent to which the resistance coefficient does not decrease any further and takes on almost a constant value. In the case considered, it occurs at Re = (2–2.5) × 105. 26. Separation of the laminar layer at the point closest to the beginning of the curvature of the bend naturally produces the most extensive eddy zone at the inner wall (see Figure 6.8). As the transition point approaches the point of laminar separation, this zone compresses. The zone has the smallest dimensions during turbulent separation at the point furthest removed from the beginning of the curvature. 27. The effect of the Reynolds number on the local resistance coefficient of bends and elbows at Re ≥ 104 is accounted for by the coefficients kRe in the expressions for the local resistance coefficients ζloc on respective diagrams. The values of kRe are represented by the curves of kRe vs. Re,17,88 which, pending further refinement, are taken tentatively for all angles δ. 28. The resistance coefficient ζloc at Re < 2 × 103 can be determined from the formula suggested by Zubov:13 ζloc = (k 1 + 1) ζsim.loc + A ⁄ Re , where ζsim.loc is the value of ζloc at Re > 2 × 105 (similarity region); k1 is the correction factor; for an elbow with a recess δo k1

...

30

45

60

3.6

75 1.5

90 1.3

A is a coefficient that depends on the geometric parameters of the elbow (bend), on R0/D0 in particular; according to some data (see, e.g., Reference 2), for an elbow of 90o, A ≈ 400; for a 135o elbow, A ≈ 600; at Re ≤ 103 for a 90o elbow, r/D0 = 2.6, A ≈ 1300; for a 180o elbow, r/D0 = 1.5–2.0, A ≈ 1200. 29. The influence of fluid compressibility at large subsonic flow velocities on the resistance of curved channels can be taken into account by the coefficient kλ, which can be determined from the following empirical formula obtained74 by processing the results of experimental investigations of some types of elbows and bends: kλ * ζλ ⁄ ζ = 1 + α1λc , where λc * wav ⁄ acr is the reduced flow velocity at the entrance into a curved channel; wav = (1/2)(w0 + w1); kλ and ζ are the resistance coefficients of the curved channel, respectively, at

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Handbook of Hydraulic Resistance, 4th Edition

Figure 6.9. Resistance curves ζtot = f(Re) of the bend with different relative roughness of the surface of the entire inner wall:17 1) smooth walls; 2) rolling paper; 3) poster stamped paper; 4) emery paper no. 140; 5) emery paper no. 60.

the given subsonic value of λc and at its small value determined from the graphs in the corresponding diagrams of the present chapter; α1 and β are constants the values of which are given in Diagram 6.4. It is evident that ζλ * kλζ. 30. The condition of the inner surface (uniform or local roughness, roughness over the whole surface or over a part of it) of elbows and bends, just before the curvature at large Reynolds numbers,17 has a greater effect on the local resistance coefficient than on the friction coefficient. At small Re the resistance coefficient of a bend, the inner wall of which has a different degree of roughness, barely differs from ζtot of a bend with a smooth inner wall (Figure 6.9). As Re increases, the resistance coefficient decreases sharply; at some value of this number, ζtot attains the minimum and then increases again. __ 31. The critical Reynolds number, at which the minimum value of ζtot is achieved, and the Reynolds number, at __which ζtot starts to increase again, depends on the relative roughness ∆ = ∆ ⁄ Dh. The larger ∆, the smaller are the above values of Re and the larger are the corresponding minimum values of ζtot and values of ζtot, which are achieved at large Re (under selfsimilar conditions). 32. As long as the Reynolds number is small, the laminar boundary layer is so thick that it almost completely covers the asperities of roughness (Figure 6.10a) and they negligibly influence the state of the flow. As a result, the boundary layer, which has separated from the inner curvature of the bend, remains laminar, while the value of the resistance of the bend with rough walls virtually resembles the value of ζtot of the bend with smooth walls. 33. As the value of Re increases, the boundary layer becomes thinner and the asperities of roughness partially start to protrude (Figure 6.10b) and agitate the flow. When compared with a smooth wall, the point of laminar-to-turbulent boundary-layer flow transition is displaced closer to the beginning of the bend curvature and turbulent separation occurs earlier, i.e., both the critical Reynolds number, at which the resistance coefficient starts to decrease, and the value of Re, at which the minimum value of ζtot is attained, decrease. 34. As the Reynolds number increases further, the thickness of the boundary layer progressively diminishes and the asperities protrude even more from the layer, causing local stalls of the flow (Figure 6.10c), owing to which the turbulent point of separation from the inner wall moves upstream. This displacement of the separation point naturally leads to expansion of the

Flow with Changes of the Stream Direction

403

__ Figure 6.10. Regions of flow separation and velocity distributions over the mean line of the cross section of a bend with rough wall (∆ = 0.001) under different flow conditions:17 (a and c) laminar and turbulent flow, respectively, over asperities of roughness; (b) transition regime: 1) lower boundary of the separated laminar layer at Re < Recr; 2) laminar separation; 3) turbulent expansion of the separated layer at Recr < Re < Redev; 4) turbulent separation at Re ≥ Redev; 5) lower boundary of the separated turbulent layer at Recr < Re < Redev; 6) roughness asperities.

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Handbook of Hydraulic Resistance, 4th Edition

__ Figure 6.11. Resistance curves ζtot = f(Re) for different walls (sides) of a bend covered with rough paper:17 1) all four walls are rough (stamped paper, ∆ = 0.0005); 2) inner wall is rough, the remaining __ smooth; 3) inner wall is smooth, the remaining three walls rough; 4) all the walls are three walls smooth (∆ = 0.00003).

eddy region and again to an increase in the resistance coefficient of the bend. The greater the relative roughness, the earlier and more vigorous is its effect. Consequently, the earlier the minimum of resistance is reached, the larger is the value of this minimum and the value of ζtot at large Reynolds numbers. 35. The resistance coefficient of the bend is affected primarily by the state of the surface of the inner wall. The roughness of the remaining three walls has virtually no influence on the value of ζtot (Figure 6.11). 36. In case of partial (local) roughness or local asperities on the inner wall of the bend, the curves of the resistance coefficient of bends are smoother (without a marked minimum). The value of ζtot at large Reynolds numbers is larger, the closer the asperity is to the start of the curvature of the bend and the larger its dimensions are (Figure 6.12).

Figure 6.12. Resistance curves ζtot = f(Re) of a bend with local __roughness and different asperities on __ (∆ = 0.002); 2) wire step at a distance the inner wall:17 1) the whole inner surface of the wall is rough from the curvature x/b0 = 0.13; 3 and 4) rough pasted strip (∆ = 0.002) at __distances x/b0 = 0.13 and 0.63, respectively; 5) flute layer at a distance x/b0 = 1.45; 6) smooth walls (∆ = 0.00003).

Flow with Changes of the Stream Direction

405

37. When elbows and bends are not smoothly rounded, i.e., when there are very small relative radii of internal rounding within the limits 0 < r/D0 < 0.05 (0.5 < R0/D0 < 0.55), then the effect of the general roughness ∆ (and not of local asperities) is considerably weaker than for smoothly curved elbows and bends. In this case, the place of flow separation is fixed near the corner of the bend. The effect of the general roughness in such elbows and bends can tentatively be calculated by the following equation (until refined experimentally): ζ*

∆p ρw20 ⁄ 2

= k∆ζsm ,

(6.3)

__ where at Re > 4 × 10 and ∆ < 0.001 __ k∆ + ⎛1 + 0.5 × 103 ∆⎞ , ⎝ ⎠ __ 4 while at Re > 4 × 10 and ∆ > 0.001 4

k ∆ + 1.5 ,

__ ζsm is determined as ζloc for smooth walls (∆ ≈ 0). 38. The effect of general roughness in elbows and bends with relative radii of inner cur__ for by the coefficient vature within 0.05 < r/D0 < 1.0 (0.55 < R0/D0 < 1.5) can be accounted k∆ in Equation (6.3), which at 4 × 104 < Re < 2 × 105 and ∆ < 0.001 is given tentatively, until refined experimentally, by the equation of Abramovichl k∆ =

λ∆ λsm

,

__ and at Re > 2 × 105 and ∆ < 0.001 is given tentatively by the equation17 __ k∆ + 1 + ∆ × 103 , __ and at Re > 4 × 104 and ∆ > 0.001, tentatively by k∆ + 2 . Here λsm is the coefficient of hydraulic friction of a smooth tube determined as λ at the given __ Re > 4 × 104 from Diagrams 2.1 and 2.6; λ∆ is the coefficient of hydraulic friction of a rough tube, determined as λ, at the given Re > 4 × 104 and ∆ = 0–0.001, from Diagrams 2.2 through 2.6. 39. The effect of the general roughness on bends with R0/D0 > 1.5 can be accounted for __ based on the author’s data17 and those of Reference approximately by the following equations 4 64 in Chapter 4: at Re > 4 × 10 and ∆ < 0.001 __ k∆ + 1 + ∆ 2 × 106 ,

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Handbook of Hydraulic Resistance, 4th Edition

__ and at Re > 4 × 10 and ∆ > 0.001 4

k∆ + 2.0 . 40. At Re < 4 × 104, the resistance coefficient of all the elbows and bends can be considered practically independent of the degree of general roughness, being a function of the Reynolds number alone. It is therefore calculated according to paragraph 28 of this chapter. 41. The resistance coefficients of elbows with rounded corners of the bend and with diverging or converging exit cross section (n2ar1 = F1/F0 = b1/b0 ≠ 1.0) can be approximated from the following equation:37 ζ*

∆p ρw20 ⁄ 2

= A1C1 exp (−k1 ⁄ nar) ,

where A1 = f(δ) and C1 = f(a0/bcon) are determined in the same way as above; k1 = –2.3 log ζ0; ζ0 is the resistance coefficient of the elbow at nar = F1/F0 = 1.0 and δ = 90o; wcon is the mean velocity in the contracted section of the elbow; bcon is the width of the contracted section of the elbow. 42. All other conditions being equal, the coefficients of local resistances of welded bends are higher than those of smooth bends, since welding seams on their inner surfaces increase the local roughness. With increase in the diameter of the bend, the relative value of the local roughness (seams) decreases, and the resistance coefficient decreases accordingly. All other conditions being equal, the coefficient of local resistance of corrugated elbows is higher than for bent and welded elbows. Since the absolute dimensions of the corrugations increase with the bend diameter, the resistance coefficient also increases. Elbows made of sheet material, fabricated from several interlocked links or corrugated, also result in curved sections with an increased resistance coefficient. 43. In the case of cast iron (steel) branches with threaded joints, a projection is formed at the junction between the straight and curved sections, which sharply changes the cross section (Figure 6.13), creating additional pressure losses. The smaller dimensions of such bends, the larger is the relative magnitude of the projection. Therefore, the resistance coefficient of standard gas fittings, which usually have small dimensions, is much higher than ζ of ordinary turns with a flanged joint. The values of the resistance coefficients of gas pipe fittings given in Diagram 6.3 can be extended to standard bends and fittings with dimensions which are close to those given in these diagrams.

Figure 6.13. Threaded cast-iron elbows.

Flow with Changes of the Stream Direction

407

Figure 6.14. __Characteristics of a turn of the "gooseneck"-type branch at δ = 30o, R0/D0 = 1.0, Re = 1.6⋅105, and ∆ = 0.0003.8,10 (a) Scheme of flow distribution along the exit section of the turn; (b) dependence of the coefficient ζloc on lel/D0.

44. The resistance of combined (joint) bends and elbows depends greatly on the relative distance lel/D0 between the two elbows; the total resistance coefficient ζ for sharply bent channels can be larger or smaller than the sum of the resistance coefficients of two separate bends, while for smoothly bent channels it can be smaller than the resistance coefficient of even one isolated (single) bend. 45. The difference between the local resistance coefficients of smoothly connected bends is mainly determined by the position of the maximum velocities (the "core" of the flow) before the entrance into the second bend and by the direction of inertia forces in it. 46. According to Goldenberg8,10 and Goldenberg and Umbrasas,9 different situations are possible. Thus, for the "gooseneck"-type branch with δ = 30o and R0/b0 = 1.0, these are (Figure 6.14): • The insert between the bends is small (in the present case lel/D0 < 2.5); the inertia forces in the second bend hinder the development of the transverse (secondary) flow caused by the first bend. The net velocity of the transverse flow is smaller than it would have been downstream of a separate bend and the resistance coefficient ζloc of the gooseneck-type channel is smaller than the resistance coefficient (ζis) of a separate (isolated) branch with the same geometric parameters (δ and R0/b0), i.e., ζloc < ζis. • The insert is increased up to the value lel/D0 = 5.0. In this case, the inertia forces in the second bend, acting on the core, increase the intensity of the transverse circulation. Thus the losses increase and attain the maximum when the core of the flow at the entrance into the second bend occupies a postion which corresponds to positions I and II on the scheme of Figure 6.14. Hence, ζis < ζloc < 2ζis.

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Handbook of Hydraulic Resistance, 4th Edition

• The insert is increased up to lel/D0 = 11.0. The inertia forces have a lesser effect on the flow (this is also favored by a simultaneous transverse equalization of the flow). The point (minimum) of the curve ζloc has a corresponding position in the second bend, where the magnitude of transverse circulation is not appreciably affected. The resistance coefficient of the gooseneck-type channel is approximately equal to the resistance coefficient of one isolated bend: ζloc = ζis. • A further increase in the length of the insert leads to higher losses, due to more complete flow equalization over the cross section downstream of the first bend and to secondary excitation of transverse circulation and losses in the second bend. As a result, the total resistance coefficient increases, approaching ζloc = 2ζis in the limit. Similar situations are also observed at other parameters of combined bends. 47. The resistance coefficients of the paired branches (of the gooseneck type), of three bends joined in one plane and in space (see Diagrams 6.18 to 6.21) are determined from ζ*

∆p ρw20 ⁄ 2

′ +λ = Aζloc

l0 , Dh

where ζ′loc is determined as ζloc of a single bend and A = f(lel/Dh) is determined from corresponding curves obtained on the basis of the experimental data of Goldenberg,8,10 Goldenberg and Umbrasas,9 Klyachko et al.,26 and Chun Sik Lee.56 48. In the case of sharply bent channels, the interaction between the paired elbows is mainly determined by the position and the magnitude of the separation zones downstream of the bend. Thus, for a Π-shaped elbow made from a couple of 90o elbows with sharp corners and small relative distance (lel/b0 ≈ 0), the flow separates from the inner wall only after the complete turn by the angle δ = 180o. At such a large turning angle the flow separation is most intense and as a result the resistance coefficient is highest. 49. A significant increase in the relative distance lel/b0 (to lel/b0 = 4–5 and above) leads to almost complete speading of the flow over the linear segment after the first 90o turn, and the conditions of the subsequent 90o turn come to be nearly the same as those for the first turn. Thus, the total resistance coefficient of such a Π-shaped elbow will be close to twice the resistance coefficient of a right-angle elbow (δ = 90o).

Figure 6.15. Flow pattern in a Z-shaped bend.

Flow with Changes of the Stream Direction

409

Figure 6.16. Flow in a combined elbow with a 90o turn in two mutually perpendicular planes.

50. At some intermediate value of lel/b0 on the order of 1.0, the separation zone behind the first 90o turn has insufficient time to develop completely and, being concentrated at the inner wall before the second 90o turn, it creates the conditions for smooth rounding of the main flow. Under these conditions the second turn of the flow occurs almost without separation, and therefore the total resistance coefficient of such Π-shaped elbow is a minimum. 51. Rounding of the corners of Π-shaped elbows decreases the difference between the values of ζ corresponding to different lel/b0, but the flow and the character of the resistance curves remain similar to those for elbows with sharp corners. 52. In the case of a pair of 90o elbows joined Z-shaped (Figure 6.15), the increase in the relative distance lel/b0 between the axes of two single elbows first leads to a sharp increase of the total resistance coefficient and then, when a certain maximum is reached, to its gradual decrease to a value roughly equal to twice the resistance coefficient of a single right-angle elbow (δ = 90o). 53. The resistance coefficient of a Z-shaped elbow reaches the maximum in the case where the second of the two single elbows is placed near the widest section of the eddy zone formed after the first 90o turn (see Figure 6.16). A maximum reduction of the stream cross section is then obtained at the second turn. 54. In the case of a combined elbow with flow turning in two mutually perpendicular planes (Figure 6.16), the total resistance coefficient increases with an increase in the relative distance lel/b0 between the axes of the two constituent right-angle elbows. This increase from an initial value equal to the resistance coefficient of a single right-angle elbow (δ = 90o) reaches a maximum at some small relative distance lel′ ⁄ b0. With a further increase in lel′ ⁄ b0,

Figure 6.17. Dependence of the resistance coefficient ζ of a smooth bend on the relative length of the starting (inlet) section l0/b0.1

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Handbook of Hydraulic Resistance, 4th Edition

Figure 6.18. Different velocity profiles at the entrance into the elbow:32 1) wmax near the inner corner of the turn; 2) wmax near the outer corner of the turn; 3) wmax at the left wall of the elbow; 4) wmax at the right wall of the elbow.

the total resistance coefficient begins to decrease again, approaching a value approximately equal to twice the resistance coefficient of a right-angle elbow (δ = 90o). 55. The resistance coefficient of curved channels can vary with the character of the velocity profile at the entrance. In turn, the shape of the velocity profile can also differ according to the conditions of flow supply to these channels. 56. In particular, when the flow is supplied via a straight section placed downstream of a smooth inlet nozzle, the resistance coefficient ζ of the bends and elbows increases up to a certain limit together with an increase in the relative length l0/b0 of the straight inlet section (Figure 6.17). There is no increase in ζ when the length of the inlet section is nearly the same as the length of the starting section, i.e., of the section over which the velocity profile develops, which corresponds to this particular mode of flow. 57. An increase in the resistance coefficient of a curved channel with the development of the velocity profile, i.e., with thickening of the boundary layer, is apparently due to the influence of the latter on both the intensity of flow separation from the walls and the formation and development of the secondary flows (vortex pair). 58. A velocity profile that has been strongly distorted by a barrier or fitting before the flow entered a curved channel can have a more significant effect on the resistance coefficient of the curved channel, than a straight inlet section. The resistance coefficient can either increase or decrease depending on the character of the velocity profile. If the velocity has its maximum near the inner corner of the turn (Figure 6.18), then the resistance coefficient of the curved channel can become even smaller than in the case of uniform velocity distribution. With other positions of the velocity maximum, the resistance coefficient increases. 59. The resistance of an elbow can be lowered not only by rounding or beveling its corners, but also by installing guide vanes. In the former case, the overall size of the channel becomes larger, while in the latter the compact form of the channel is preserved. The guide vanes can be aerodynamically shaped (Figure 6.19a), simplified, and bent along the surface of

411

Flow with Changes of the Stream Direction

a cylinder (Figure 6.19b and c), and thin concentric vanes (Figure 6.19d). The shape of the guide vane is chosen according to the following tabulation: Symbols

Relative dimensions

Symbols

Relative dimensions

t1

1.0

y2

0.215t1

x1

0.519t1

z1

0.139t1

x2

0.489t1

z2

0.338t1

r1

0.663t1

z3

0.268t1

r2

0.553t1

ρ

0.033t1

y1

0.463

Turning vanes of identical shape and dimensions are usually mounted within the elbows and generally are installed along the line of bend of the channel (Figure 6.19a, b, and c). 60. To achieve smooth turning of the flow, a centrifugal fan is followed by bends.3,4 The resistance coefficients of such bends depend on the operating conditions of the fan and on the angle of installation β, that is, on the angle between the velocity vectors at the inlet to the fan and at the exit from the bend reckoned in the direction of the rotation of the impellar (see Diagram 6.4). Under all the operating conditions of the fan, the resistance of the bend, installed downstream of it, is much higher than that under conventional flow conditions. 61. An aerodynamic grid, composed of guide vanes and placed in the elbow, deflects the flow to the inner wall due to the aerodynamic force developed in it. When the dimensions, number, and angle of the vanes are appropriately chosen, this flow departure prevents jet

Figure 6.19. Guide vanes in elbows and turns: (a) shaped; (b) thin, along a 95o arc; (c) thin, along a 107o arc; (d) concentric; (e) slotted.

412

Handbook of Hydraulic Resistance, 4th Edition

Figure 6.20. Distribution of dimensionless velocities (velocity pressures) in an elbow.15 (a) Without vanes; (b) with a normal number of vanes; (c) with a reduced number of vanes.

separation from the walls and formation of the eddy zone. The velocity distribution over the cross section downstream of the turn (Figure 6.20) is improved and the resistance of the elbow is decreased. 62. Since the main factor in decreasing the resistance and equalization of the velocity field is elimination of the eddy zone at the inner wall of the channel, the largest effect will be produced by the vanes placed closer to the inner curvature. This suggests the possibility of reducing the number of vanes located near the outer wall of the elbow.5,15

Flow with Changes of the Stream Direction

413

63. In the case where it is especially important to obtain a uniform velocity distribution directly after the turn, the "normal" number of vanes is used in the elbow, which is determined by the equation* −1

r nnor = 2.13 ⎛⎜ ⎞⎟ D0 ⎝ ⎠

−1 .

(6.4)

In the majority of practical cases it is sufficient to use a reduced number of vanes ("most advantageous" or minimal):15 r nadv + 1.4 ⎛⎜ ⎞⎟ D0 ⎝ ⎠

−1

−1 ,

(6.5)

−1 .

(6.6)

or r nmin + 0.9 ⎛⎜ ⎞⎟ D0 ⎝ ⎠

−1

In ordinary elbows, lower resistance and better distribution of the velocities are achieved with the optimum number of vanes determined by Equation (6.5). The chord t1 of the shaped vane is taken as the chord of a 90o arc of a circle, i.e., of the arc of the inner curvature of the elbow, and therefore t1 = r √ ⎯⎯2 ,

(6.7)

r t1 = D0 ⎛⎜ ⎞⎟ √ ⎯⎯2 . D0 ⎝ ⎠

(6.8)

or

Equations (6.4) to (6.6) are correct only for this relationship between the dimensions of the vane chord and the radius of curvature of the elbow. 64. If the curvature in the elbow is not smooth (there are sharp or beveled corners), then = t1 0.15–0.60D0. Then the number of vanes can be determined by the following equations:15 n nor =

3D0 −1 , t1

(6.9)

D0 , t1

(6.10)

nadv + 2

nmin + 1.5 ∗

D0 . t1

For a right-angle elbow, D0 in Equations (6.4) to (6.11) is replaced by b0.

(6.11)

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Handbook of Hydraulic Resistance, 4th Edition

65. For elbows with a diverging section, in which the cross section past the turn is larger than that before it (b1 > b0), the numbers of vanes are determined, respectively, from nnor = 2.13

n adv

S −1 , t1

+ 1.4 tS

,

1

nmin + 0.9

S , t1

where b20 + b21 . S=√ ⎯⎯⎯⎯⎯ 66. When the "normal" number of vanes is used, they are uniformly placed along the bending line of the elbow, so that the distance between the vane chords is ai = S/(n + 1). If a reduced number of vanes is used, the author15 recommends that the distance a between the chords be taken as varying according to an arithmetic progression, such that in the case of the optimum number of vanes an+l/a1 = 2, and in the case of the minimum number of vanes an+1/a1 = 3. Here, a1 is the distance from the arc chord of the inner curvature of the elbow to the chord of the first vane (see Figure 6.19); an+1 is the distance between the chords of the last vane and the outer curvature. The intermediate distances between the vanes are determined by the following equations:16 with an advantageous number of vanes ai = 0.67

i − 1⎞ S ⎛ ⎜1 + n ⎟ , n+1 ⎝ ⎠

with a minimum number of vanes ai =

i − 1⎞ S ⎛ ⎜0.5 + n ⎟ . n+1⎝ ⎠

67. In the majority of practical cases, the vanes used in elbows are simply thin shaped vanes selected for a 90o turn, on the average, along the arc of a circle ϕ1 = 95o independent of the elbow parameters (the relative curvature radius, the area ratio, etc.). The position and the angle of installation of such vanes are selected according to the same criteria used for profiled vanes. The resistance coefficient of elbows with such vanes is markedly higher than for elbows with profiled vanes. 68. A low value of resistance is obtained, coming close to the resistance of elbows with profiled vanes, when thin vanes are selected by Yudin’s method.50 The optimum angle of the vane arc and the vane angle depend on both the relative curvature radius of the elbow and its area ratio (see Diagram 6.30).

415

Flow with Changes of the Stream Direction

69. The installation of guide vanes in elbows is expedient as long as the relative curvature radius is comparatively small. For elbows of constant cross section, the installation of vanes is justifiable as long as r/b0 ≤ 0.4–0.5. For diffuser elbows (i.e., with diverging exit cross section) the limiting value of r/b0 increases up to about 1.0. In the case of converging elbows (with converging exit cross section), the value of r/b0 is decreased to about 0.2. 70. The action of concentric vanes installed in turns is evidenced mainly in that they divide this bend into a number of bends with more elongated cross sections, which leads to a decrease in the pressure losses. The normal number, nch, of thin optimally installed concentric vanes in a bend is determined on the basis of the data of Khanzhonkov and Taliev:46 r ⁄ b0

0–0.1

0.1–0.4

0.4–1.0

1.0

nch

3–4

2

1

0

The optimum position of the vanes in a bend is reached (see Figure 6.19d) when ri = 1.26ri−1 + 0.07b0 . 71. The resistance coefficient of a rectangular bend with a normal number of optimally installed concentric vanes can be determined, approximately,46 by the following equation: ζ*

∆p ρw20 ⁄ 2

R0 ⎛ ⎞ = ⎜0.46 + 0.04⎟ ζw.v , b0 ⎝ ⎠

where ζw.v is the resistance coefficient of the bend without vanes. 72. The normal number of vanes in a circular bend, according to the experiments of Ito and Imai,68 is r0 ⁄ D0

0–0.5

0.5–1.0

1.0

nch

2

1

03

With one vane installed, its optimum distance should be (see Figure 6.19d): 1 + D0 ⁄ ⎯r0 . r1 = r0 √ ⎯⎯⎯⎯⎯⎯⎯⎯ In the case of two vanes, this distance is equal to r1 = r0

3

1 + D0 ⁄ ⎯r0 and r2 = r0 ⎯√⎯⎯⎯⎯⎯⎯⎯

3

(1 + D0 ⁄ r0)2 . ⎯⎯⎯⎯⎯⎯⎯⎯⎯ √

The values of the resistance coefficients of circular bends with guide concentric vanes and without them are given in Diagram 6.27. 73. When guide vanes are installed in combined elbows, the resistance coefficient is determined as the sum of the resistance coefficients of single elbows with vanes ζ = nisζυ ,

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Handbook of Hydraulic Resistance, 4th Edition

where ζυ is the resistance coefficient of a single elbow with vanes and nis is the number of turns in a combined elbow. 74. The coefficient of the local hydraulic resistance of a spatial (annular) turn through 180o depends on the relative distance h/D0 from the entry edge of the inner tube up to the end cover (hood) of the outer circular tube; on the area ratio nar = F1/F0 between the outer and inner tubes; on the relative thickness δel/D0 or the curvature radius r/D0 of the cut edge of the inner tube, as well as on the relative curvature radius R/D1 of the cover (Figure 6.21). 75. The coefficient of local resistance ζloc is most strongly affected by the parameter h/D0. When the ratio a increases from zero, the coefficient ζloc first decreases very sharply, reaching a minimum at some value of a, then increases somewhat sharply. Then it remains constant in some cases, and in others, it decreases to a certain value or smoothly increases. 76. Within certain limits of h/D0 (following the first minimum of ζloc) and at some values of nar and r/D0 (δel/D0), one can observe marked oscillations (in time) of the coefficient ζloc caused by flow instability. This instability, which is typical of turns with small values of r/D0 (δel/D0) can be explained by periodic blowing-off and entrainment by the flow of the separating (eddy) zones 1 at the outer wall and zones 2 at the inner wall of the annular turn (Figure 6.22I) under certain conditions. This occurrence corresponds to a sharp decrease in the resistance. Following this, the vortices reappear and the resistance increases sharply in a contracted section behind the turn.

Figure 6.21. Dependence of the resistance coefficient ζloc of an annular turn on h/D0 at R/D1 = 0.3:19 (a) pumping at r/D0 = 0.1; (b) suction at r/D0 = 0.2: 1) nar = 0.80; 2) nar = 1.07; 3) nar = 2.1; (c) suction at r/D0 = 0.1: 1) nar = 0.76; 2) nar = 1.06; 3) nar = 2.07.

Flow with Changes of the Stream Direction

417

Figure 6.22. Flow patterns in a 180o turn without guide vanes (I) and with guide vanes and fairings (II):47 I (a) pumping; (b) suction: 1) eddy zones at the outer wall; 2) eddy zones at the inner wall; 3) divider. II (a) pumping; (b) suction: 1) vane; 2) fairing.

The curves ζloc = f(a, r/D0 or δel/D0, nar) of Diagrams 6.31 and 6.32 correspond to the time-average experimental values of ζloc. 77. Table 6.1 contains the values of ζmin of annular turns corresponding to the first minimum of the resistance coefficient and to the optimum values of (h/D0)opt for different r/D0 (δel/D0), R/D, and nar. The table also contains the values of (h/D0)tr within the limits of which the flow is markedly unstable. 78. The relative thickness of the corner δel/D0 of the inner tube of an annular turn, together with h/D0, is also an important factor which influences the value of ζmin, decreasing it noticeably especially at nar < 2. At the same time, the rounding of this corner within r/D0 = 0.05–0.2 barely reduces ζmin. Therefore, in those cases where rounding of the corner is difficult, it may be left as is. 79. In the case of suction (entry through an annular tube), the optimum value of the parameter m = F1/F0 lies at all values of r/D0 (δel/D0) within 1.0–2.0; in the case of pumping (exit through an annular tube), it is different for different parameters. At r/D0 < 0.2, an annular turn with the ratio nar < 1.0 is expedient. At r/D0 ≥ 0.2 (δel/D0 ≥ 0.4), the optimum value of nar = 1.0–1.5. 80. In the case of suction, the optimum curvature radius of the end cover R/D1 lies within 0.18–0.35, and in the case of pumping, within 0.2–0.45. 81. In order to better stabilize the flow in an annular turn, divider 3 can be used (see Figure 6.22I), which does not influence the losses appreciably. The resistance of the annular turn can be decreased by installing guide vanes 1 in the vicinity of inner corners of the turn (Figure 6.22II). 82. A symmetrical 180o turn of the flow can also be achieved in a plane channel.47 Plane symmetrical turns are often used, for example, in heating furnaces with closed-cycle circula-

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Handbook of Hydraulic Resistance, 4th Edition

Table 6.1. Characteristics of the spatial (annular) turn through 180o Suction (inlet) Characteristics of the turn

r ⁄ D0

δel ⁄ D0 0.1

0.2 0.4 F2 = 0.75–0.80 nar =

0.5

0.1

0.2

F1

ζmin (h ⁄ D0)opt (h ⁄ D0)tr

1.70 0.23–0.27 0.28–1.4

1.10 0.22–0.28 0.20–1.4

0.72 0.03–0.38 Stable

1.90 0.18–0.22 0.25–1.0

0.96 0.18–0.23 0.40–1.0

0.70 0.22–0.30 0.60–1.4

ζmin (h ⁄ D0)opt (h ⁄ D0)tr

1.05 0.7–0.34 0.38–1.40

nar = 1.0–1.10 0.45 0.40 0.23–0.33 0.26–0.36 0.77–1.45 Stable

1.05 0.24–0.33 0.26–1.40

0.40 0.20–0.29 0.60–1.0

0.32 0.18–0.28 Stable

ζmin (h ⁄ D0)opt (h ⁄ D0)tr

0.55 0.35–0.45 0.50–1.80

nar = 2.0–2.1 0.50 0.40 0.22–0.48 0.26–0.40 0.50–1.40 1.1–1.30

0.50 0.33–0.60 0.45–1.60

0.20 0.28–0.40 0.40–1.60

0.16 0.17–0.50 Stable

Pumping (exit) 0.5

r ⁄ D0 0.1

0.2

0.19 0.40–0.80 0.75–2.0

0.16 0.25–0.50 0.40–2.0

0.30 0.23–0.45 Stable

(h ⁄ D0)opt (h ⁄ D0)tr

0.24 0.40–0.62 0.60–20

δel ⁄ D0 0.2 0.4 nar = 0.75–0.80 0.22 0.36 0.40–0.50 0.32–0.45 0.55–2.0 Stable

ζmin (h ⁄ D0)opt (h ⁄ D0)tr

0.40 0.50–0.60 0.55–2.0

nar = 1.0–1.1 0.26 0.26 0.35–0.55 0.30–0.40 0.90–2.0 0.75–1.0

0.40 0.47–0.83 0.80–2.0

0.23 0.30–0.50 0.60–2.0

0.20 0.25–045 Stable

0.34 0.75–1.0 1.0–2.0

nar = 2.0–2.1 0.32 0.30 0.65–0.93 0.50–0.90 0.55–2.0 0.30–1.8

0.34 0.65–0.95 0.30–2.0

0.32 0.60–1.0 1.1–2.0

0.40 0.20–1.0 1.0–2.0

Characteristics of the turn

ζmin

ζmin (h ⁄ D0)opt (h ⁄ D0)tr

0.1

tion of gas flow. The resistance coefficient of such a turn depends on the same parameters as in an annular turn (paragraph 74). 83. Table 6.2 contains the values of ζmin obtained at optimal geometric parameters of a plane 180o turn both in the absence and in the presence of divider 3 (Figure 6.22I) in the place of merging (division) of flows. At optimal values of (h/a0)opt the divider moderately decreases the resistance coefficient of the turn.* However, the principal function of the divider ∗

At small values of h/a0, the divider may moderately increase the resistance of the turn.

419

Flow with Changes of the Stream Direction Table 6.2. Characteristics of a symmetric plane turn through 180o Type of device in the turning region

(h ⁄ a0)opta

(h0 ⁄ a0)opta

ζmina

Without divider, without fairing and guide vanes

0.40–0.60

–

4.0–4.2

(0.55–0.70)

–

(4.0–4.2)

0.40–0.60

–

3.4–3.5

Without divider, guide vanes, but with fairing installed on the inside of the channel Without divider, but with fairing and with guide vanes Without divider, without fairing, but with guide vanes Without fairing, guide vanes, but with a plane divider With divider, fairing, but without guide vanes on the inside of the channel With divider, fairing, and guide vanes With divider, without fairing, but with guide vanes With divider, fairing, guide vanes, but with a burden

(0.45–0.60)

–

(2.3–2.5)

0.35–0.50

0.76

1.70–1.75

(0.35–0.50)

(0.76–0.127)

(0.90–1.0)

0.40–0.55

0.127

1.75–1.80

(0.45–0.57)

(0.127)

(1.30–1.35)

0.53–0.65

–

3.6–3.7

(≥0.60)

–

(3.9–4.0)

0.50–0.65

–

3.3–3.4

(0.55–0.70)

–

(2.2–2.3)

0.35–0.55

0.76

1.2–1.3

(0.40–0.65)

–

(0.90–1.0)

0.45–0.60

0.127

1.2–1.3

(0.50–0.70)

–

(1.30)

0.40–0.50

0.076

3.1–3.2

(>0.40)

(0.076–0.127)

(2.6)

a

Numbers not in parentheses refer to suction (inlet); those in parentheses — to pumping (exit).

is its stabilizing effect on the flow. Only a plane divider should be used in this case, as during suction it provides a somewhat greater reduction in the resistance than a profiled divider. During pumping, the effects of the plane and profiled dividers are practically the same. The resistance of the turn can also be reduced by installing fairing 2 on one of the sides of the inner channel (see Figure 6.22II). A still greater reduction in the resistance of a plane turn is achieved by the use of guide vanes (Figure 6.22II) installed in the vicinity of inner corners of the turn. The minimum values of the resistance coefficients of turns with guide vanes are obtained at noticeably lower ratios (h/a0)min than without such vanes. 84. The right-angle turn with fairing and guide vanes is the best one among those studied. In the case of suction (h/a0)opt ≈ 0.45 and (h0/a0)opt ≈ 0.076, and in the case of pumping (h/a0)opt = 0.5–0.6 and (h0/a0)opt = 0.076–0.175. 85. Bent flexible glass-fabric air conduits, just as straight air conduits (see paragraph 72 of Section 2.1) have an elevated resistance. Some of the experimental data53 on the resistance coefficients of such bends are given in Diagram 6.25. 86. During the pneumatic transport of pulverized materials, the highest resistance is produced in the places where the flow alters its direction, that is, in bent channels (elbows, bends, etc.).69 The overall resistance coefficient of bent channels with pulverized material in the flow is calculated as

420 ζ*

Handbook of Hydraulic Resistance, 4th Edition ∆p ρw20 ⁄ 2

= ζ + κ (ζ1 − ζ0) ,

(6.40)

where ζ0 and ζ1 are the resistance coefficients of the bent channel without (κ = 0) and with (κ = 1) pulverized materials in the flow, respectively; κ = mtot/mh is the coefficient of dust content (the ratio of the mass flow rate of the pulverized material to the mass flow rate of the gas flow). 87. When 2.5 × 105 ≤ Re ≤ 4.5 × 105 and 20 ≤ Fr ≤ 36, the overall resistance coefficient ζ is independent of either the Reynolds number Re = w0D0/vc or the Froude number Fr = w0/√ gD⎯0 , where vc is the average value of the kinematic coefficient of viscosity of the gas ⎯⎯⎯ flow laden with pulverized material. Pressure losses in the dust-laden flow are determined as ∆p = ζρw20 ⁄ 2 , where ρ is the average value of the density of a dust-laden gas flow. 88. The resistance coefficient of curved channels with dust-laden flows is practically independent of whether the transported flow moves in the horizontal plane or changes its direction to the vertical, and vice versa. The values of ζ are also independent of the size of particles of the pulverized material. 89. Rectangular bends differ from circular bends by a smaller local wear in motion of dust particles. Elbows with sharp turning angles and without guiding devices are inapplicable for pneumatic systems, since dust is accumulated in outer corner elements and periodically returns to the main stream. In this case, the resistance and the wear of the system increase sharply. Combined elbows occupy intermediate position between the elbow with a sharp turn and smooth bends. 90. Guide vanes or plates in elbows and bends not only decrease the resistance, but also diminish the wear, since the latter is distributed uniformly over these units. During the pneumatic transport of not very hard material (e.g., sawdust) in large-diameter tubes, combined circular elbows can be used. When the material is transported in large-diameter tubes which produces appreciable wear, elbows with guide vanes should be used. 91. During flow of a liquid in pipelines with local resistance the pressure is decreased to the saturation vapor pressure and cavitation caverns are formed; this leads to a change in the hydraulic characteristics. Peshkin96 called attention to the possibility of the appearance of cavitation in the local resistances of pipelines, which leads to a sharp increase in the loss coefficient. He carried out experimental investigations of the cavitation characteristics of certain local resistances (elbows, branches) to determine the limiting cavitation numbers. Figure 6.23 presents the dependences of the local resistance coefficient ζ of two elbows of circular cross section with a turning angle of 60 and 90o on the cavitation number σ = (p1 – p2)/0.5ρw2 at the tube diameter d = 13.5 mm, water temperature t = 30oC and Re = (9–18) × 104. Here ps is the saturation vapor pressure. The local resistance coefficient of both elbows on attainment of a certain limiting cavitation number is increased sharply. This is due to the formation of an extensive cavitation zone downstream of the inner corner. Similar results were obtained in investigation of a branch of

Flow with Changes of the Stream Direction

421

Figure 6.23. Dependence of the coefficient of local resistance of an elbow ζ on the cavitation number σ. Tube diameter 13.5 mm, water with t = 30oC, Re = (9–18) × 104. 1) elbow with a turning angle of 60o; 2) elbow with a turning angle of 90o.

rectangular cross section with a turning angle of 90o at different radii of curvature (Figure 6.24). It was established that with an increase in the curvature radius of the branch the cavitation cavern is decreased. For elbows and branches, a simple empirical dependence of the critical cavitation number on the local resistance coefficient of a cavitation-free flow was obtained: σcr = 2ζ. This dependence can be used for elbows and branches with ζ = 0.5–2.0. Thus, in hydraulic calculations of pipelines one should take into account the possibility of the occurrence of cavitation in local resistances and check whether or not the average velocity of flow exceeds the admissible value from the condition of a cavitation-free flow.

Figure 6.24. Dependence of the coefficient of local resistance of branches ζ with a turning angle of 90o on the cavitation number σ. The branches are of rectangular cross section of size 16 × 8 mm; water with t = 30oC, Re = (9–16) × 104.

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Handbook of Hydraulic Resistance, 4th Edition

Figure 6.25. Dependence of the coefficient of local resistance of a branch ζ with a turning angle of 90o on the volumetric flow rate gas content Cvfr (section 16 × 8 mm, water–air mixture, Frmix = 880–1010): 1) spatial position of the branch; 2) rin = 0 mm, rout = 16 mm; 3) rin = 3.5 mm, rout = 19.5 mm; 4) rin = 8 mm, rout = 24 mm.

92. The hydraulic resistance in the flow of gas–liquid mixtures was studied in References 97 and 98. The experimental results presented in Figure 6.25 show that in all branches tested the value of the local resistance coefficient ζ increases substantially with an increase in the volumetric gas content in a liquid flow and in a gas–liquid mixture flow it depends for each branch on its orientation in space. Thus, if a mixture flow is directed from a vertical channel

Figure 6.26. Dependence of the reduced resistance coefficient of a branch ψ with a turning angle of 90o on the volumetric flow rate gas content Cvfr (section 16 × 8 mm, water–air mixture, Frmix = 880– 1010): 1) spatial position of the branch; 2) rin = 0 mm, rout = 16 mm; 3) rin = 3.5 mm, rout = 19.5 mm; 4) rin = 8 mm, rout = 24 mm.

Flow with Changes of the Stream Direction

423

into a vertical one upwards, the resistance coefficient will be smaller than in the case of it being directed into a vertical channel downwards. This is explained by the different direction of the action of gravitational forces on the gas phase. In the first case, air bubbles tend to move upwards, i.e., in the direction of the flow, whereas in the second case, they move opposite to the flow, and a higher amount of energy is to be spent to move the bubbles. Figure 6.26 presents the dependence of the reduced resistance coefficient ψ, i.e., the ratio of the local resistance coefficient in flow of a gas–liquid mixture to the coefficient of local resistance for water flow on gas content for two versions of the location of a branch (with a mixture moving upwards and downwards).

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Handbook of Hydraulic Resistance, 4th Edition

6.2 DIAGRAMS OF THE RESISTANCE COEFFICIENTS Bends at

R0 D0

⎛ R0 ⎞ o l0 *1,17,31,61,64,77,78,91 ⎜ b0 ⎟ < 3.0 and 0 < δ ≤ 180 , Dh ≥ 10 ⎝ ⎠

Diagram 6.1

1) Smooth walls (∆ = 0) and Re = w0Dh/v ≥ 2 × 105; ∆p

ζB

ρw20 ⁄ 2

= ζloc + ζfr = ζloc + 0.0175 δλ

R0 , Dh

ζloc = A1B1C1 , A1 = f(δ), see graph a or, tentatively, the corresponding formulas:

δ, deg

≤70

90

≥100

A1

0.9 sin δ

1.0

0.7 + 0.35

δ 90o

B1 = f(R0/D0) or f(R0/b0), see graphs b and c or, tentatively, the corresponding formulas: R0 ⎛ R0⎞ D0 ⎜⎝ b0 ⎟⎠ 0.21(R0/D0)–2.5

B1

⎯⎯⎯⎯⎯⎯ (R0 ⁄ D⎯0 ) 0.21(√

−0.5

C1 = f(a0/b0), see graph d for a circular or square cross section C1 = 1.0 or, tentatively: a0 ⁄ b0

≤4

c1

0.125 0.85 + a0 ⁄ b0 ζfr = 0.0175 δλ

≥4

R0 . Dh

2. Rough walls (∆ > 0) and Re ≥ 104; R0 , ζ = k∆kReζloc + 0.0175 δλ D h __ k = f(∆ = ∆ ⁄ Dh), see the tables; kRe = f(Re), see graph e or, tentatively: 0.50 – 0.55 >0.70 R0 ⁄ D0(R0 ⁄ b0) ≥0.55 to 0.70 kRe 5.45/Re0.131 1 + 4400 ⁄ Re 1.3–0.29 ln (Re × 10−5) __ λ = f(Re and ∆), see Diagrams 2.2 through 2.6; at λ ≈ 0.02, ζfr = 0.00035δ(R0/Dh). 3) 3 × 103 < Re < 104 , A2 + ζqu + ζfr , Re where for A2, see the table (tentatively); ζqu is determined as ζloc at Re > 2 × 105. ζ=

R0 ⁄ D0(R0 ⁄ b0) r0 ⁄ D0 A2 × 10−3 ∗

0–0.05 4.0

>0.05–0.20 6.0

>0.2–0.5 4.0–2.0

>0.5–1.5 1.0

>1.5–2.0 0.6

From here onward l0/Dh = 0 means that an elbow (turn) is installed directly behind a smooth collector, while l0/Dh > 0 means that it is installed downstream of a straight section (insert) behind the collector.

425

Flow with Changes of the Stream Direction

Bends at

R0 ⎛ R0 ⎞ l0 ≥ 10*1,17,31,61,64,77,78,91 < 3.0 and 0 < δ ≤ 180o, D0 ⎜⎝ b0 ⎟⎠ Dh

Diagram 6.1

Values of k∆ R0 ⁄ D0

(R0 ⁄ b0) __

Re

∆

3 × 103– 4 × 104

0 0–0.001 >0.001

1.0 1.0 1.0

>4 × 104 1.0 __ 1 + 0.5 × 103∆ ~1.5

3 × 103– 4 × 104 1.0 1.0 1.0

>4 × 104– 2 × 105 1.0 λ∆ ⁄ λsm ~2.0

>2 × 105 __1.0 1 + ∆ × 103 ~2.0

where for __ λsm, see λ of commercially __ smooth tubes at the given Re in Diagrams 2.5 and 2.6; for λ∆, see λ of rough tubes (∆ > 0) at the given Re and ∆ in Diagrams 2.2 through 2.6.

δ, deg A1

0

20

30

45

60 0.78

0

0.31

0.45

0.60

δ, deg

75

90

110

130

150

180

A1

0.90

1.00

1.13

1.20

1.28

1.40

0.15

0.11

0.09

0.07

0.07

R0 ⁄ D0(R0 ⁄ b0) B1

1.18

0.77

0.51

0.37

0.28

0.21

0.19

0.17

R0 ⁄ D0(R0 ⁄ b0) B1

R0 ⁄ D0(R0 ⁄ b0) B1 R0 ⁄ D0(R0 ⁄ b0) B1

a0 ⁄ b0 C1 a0 ⁄ b0 C1

0.25 1.30 3.0 0.85

0.50 1.17 4.0 0.90

0.75 1.09 5.0 0.95

1.0 1.00 6.0 0.98

1.5 0.90 7.0 1.00

0.15

2.0 0.85 8.0 1.00

0.11

>40

426

Bends at

Handbook of Hydraulic Resistance, 4th Edition R0 D0

⎛ R0 ⎞ o l0 *1,17,31,61,64,77,78,91 ⎜ b0 ⎟ < 3.0 and 0 < δ ≤ 180 , Dh ≥ 10 ⎝ ⎠

Diagram 6.1

Values of kRe R0 ⁄ D0 (R0 ⁄ b0) 0.5–0.55 >0.55–0.70 >0.70 R0 ⁄ D0 (R0 ⁄ b0) 0.5–0.55 >0.55–0.70 >0.70

0.1 1.40 1.67 2.00

0.14 1.33 1.58 1.89

0.2 1.26 1.49 1.77

0.3 1.19 1.40 1.64

0.4 1.14 1.34 1.56

0.6 1.09 1.26 1.46

0.8 1.06 1.21 1.38

1.0 1.04 1.19 1.30

1.4 1.0 1.17 1.15

2.0 1.0 1.14 1.02

3.0 1.0 1.06 1.0

4.0 1.0 1.0 1.0

⎛ R0 ⎞ Tubes and channels (smooth), smoothly curved ⎜ ≥ 3⎟ with any D 0 ⎝ ⎠ angle of the turn (coils);3,4,12,24,28,51,79,84 l0/D0 ≥ 10

Diagram 6.2

1. Circular cross section ζB

∆p

ρw20 ⁄ 2

= 0.0175 λelδ

R0 , Dh

where λel = f(Re, R0/D0), see curves, or for a circular cross section: at 50 < Re

⎯⎯ √ ⎯⎯ √ ⎯⎯ √

at 600 < Re

at 1400 < Re Re =

w0Dh v

D0 < 600 2R0

λel =

20 ⎛ D0 ⎞ ⎜ ⎟ Re0.65 ⎝2R0 ⎠

0.175

,

10.4 ⎛ D0 ⎞ ⎜ ⎟ Re0.55 ⎝ 2R0 ⎠

0.225

D0 5 ⎛ D0 ⎞ < 5000 λel = 0.45 ⎜ ⎟ 2R0 Re ⎝ 2R0 ⎠

0.275

D0

2R0

< 1400 λel =

,

.

427

Flow with Changes of the Stream Direction ⎛ R0 ⎞ Tubes and channels (smooth), smoothly curved ⎜ ≥ 3⎟ with any D 0 ⎝ ⎠ angle of the turn (coils);3,4,12,24,28,51,79,84 l0/D0 ≥ 10

Diagram 6.2

Values of λel (graph a) Re × 10−3 R0 ⁄ D0(R0 ⁄ b0)

0.4

0.6

0.8

1

2

4

6

3.0–3.2 3.8–4.0 4.3–4.5 5.0–8.0 10–15 20–25 30–50 >50

0.34 0.30 0.28 0.26 0.24 0.22 0.20 0.18

0.26 0.23 0.22 0.20 0.18 0.16 0.15 0.135

0.22 0.19 0.18 0.16 0.15 0.14 0.13 0.105

0.19 0.17 0.16 0.14 0.13 0.12 0.11 0.090

0.12 0.11 0.10 0.09 0.08 0.075 0.070 0.052

0.078 0.070 0.065 0.060 0.055 0.048 0.045 0.040

0.063 0.060 0.056 0.052 0.043 0.040 0.038 0.035

Re × 10−3 R0 ⁄ D0(R0 ⁄ b0)

8

10

20

30

50

100

3.3–3.2 3.8–4.0 4.3–4.5 5.0–8.0 10–15 20–25 30–50 >50

0.058 0.055 0.052 0.049 0.040 0.037 0.035 0.032

0.055 0.052 0.049 0.047 0.038 0.035 0.033 0.030

0.050 0.047 0.045 0.043 0.034 0.030 0.028 0.025

0.048 0.045 0.043 0.042 0.033 0.029 0.027 0.023

0.046 0.044 0.041 0.040 0.030 0.027 0.025 0.022

0.044 0.042 0.040 0.038 0.028 0.026 0.023 0.020

2. Square cross section: ⎯⎯⎯⎯⎯⎯⎯ a0 ⁄ (2R⎯0) = 100 − 400 at Re √ ⎯⎯⎯⎯⎯⎯⎯ a0 ⁄ (2R⎯0) ) λel = 16.5 × (Re √

0.35

or

Re =

100 − 400 ⎯√⎯⎯⎯⎯⎯⎯ a0 ⁄ (2R⎯0)

⁄ Re, see graph b ;

a0 ⁄ (2R⎯0) > 400, for λel, see graph b . ⎯⎯⎯⎯⎯⎯⎯ at Re √

Values of λel (graph b) Re × 10−3 R0 ⁄ a0

0.4

0.6

0.8

1.0

1.5

2.0

2.5

3

4

6

8

10

20

1.70 3.40 6.85 13.7

0.272 0.240 0.212 0.188

0.210 0.180 0.160 0.142

0.172 0.152 0.136 0.120

0.160 0.132 0.116 0.104

0.140 0.112 0.092 0.080

0.148 0.112 0.080 0.068

0.140 0.108 0.072 0.060

0.136 0.104 0.068 0.056

0.132 0.096 0.061 0.048

0.120 0.088 0.052 0.044

0.112 0.080 0.048 0.040

0.108 0.076 0.044 0.038

0.092 0.072 0.040 0.034

428

Handbook of Hydraulic Resistance, 4th Edition

⎛ R0 ⎞ Tubes and channels (smooth), smoothly curved ⎜ ≥ 3⎟ with any D 0 ⎝ ⎠ angle of the turn (coils);3,4,12,24,28,51,79,84 l0/D0 ≥ 10

Diagram 6.2

3. Rectangular cross section 1. Re = (0.5–6) × 103 (laminar regime): λel = ⎡1.97 + 49.1(Dh ⁄ (2R0)) 1.32 (b0 ⁄ a0)0.37⎤ Re−0.46 = Alam Re−0.46 or λel ⁄ Alam = Re−0.46 . ⎦ ⎣ 2. Re = (7–38) × 103 (turbulent regime): λel = ⎡0.316 + 8.65(Dh ⁄ (2R0)) 1.32(b0 ⁄ a0) 0.34⎤ Re−0.25 = Aturb Re−0.25 or ⎣ ⎦

λel ⁄ Aturb = Re−0.25

Turns and bends;22,90 Re = w0D0/v ≥ 2 × 105

Diagram 6.3 Resistance coefficient ζ B

Characteristics of the bend

D0 __ ∆ = ∆ ⁄ D0 L, mm ζ

L, mm ζ

L, mm ζ

1⁄ 2

∆p

ρw20 ⁄ 2

in.

1 in.

11⁄2 in.

2 in.

0.02

0.01

0.0075

0.0050

30

44

56

66

0.81

0.52

0.32

0.19

36

52

68

81

0.73

0.38

0.27

0.23

30

40

55

65

2.19

1.98

1.60

1.07

429

Flow with Changes of the Stream Direction Turns and bends;22,90 Re = w0D0/v ≥ 2 × 105 δ = 90o; R0 ⁄ D0 = 1.36−1.67

δ = 90o; R0 ⁄ D0 = 2−2.13

Diagram 6.3 L, mm ζ

L, mm ζ

δ = 180o

L, mm

Branch δ = 90o; furrowed; R0/D0 = 2.5

D0, mm

ζ

ζ

ζλ B

ρmw2m ⁄ 2

where λc B wm/acr; wm = 0.5(w0 + w1); ζm = 0.5(ρ0 + ρ1), acr is the critical velocity of sound.

98

1.20

0.80

0.81

0.58

55

85

116

140

0.82

0.53

0.53

0.35

38

102

102

127

1.23

0.70

0.65

0.58

150

200

250

300

350

0.25

0.30

0.33

0.37

0.42

0.45

0.50

where for ζ, see corresponding diagrams of Chapter 6 for small velocities. 1. Small-diameter elbows and bends (25 mm), pure (not rusted) at λc < 0.9 and

kλ = 1 + α1λβc ,

85

100

= kλζ ,

105 < Re < 7 × 105 ,

63

50

Circular section bends and elbows at high subsonic flow velocities41,74 ∆p

45

Diagram 6.4

430

Handbook of Hydraulic Resistance, 4th Edition

Circular section bends and elbows at high subsonic flow velocities41,74

Diagram 6.4

Values of α1 and β δ, deg

R0 ⁄ D0

α1

β

∆, µm

90 90 90 90 90 90

0.90 0.90 1.34 1.34 0.62 0.62

5.84 6.86 6.57 6.76 1.52 2.56

3.17 3.17 3.17 3.17 1.95 1.95

15 1.5 15 1.5 120 1.5

90

1.34

3.40

1.95

15

180 180 45 45 45

1.34 1.34 1.34 1.34 1.20

3.88 5.02 7.34 7.53 3.14

3.17 3.17 3.17 3.17 1.85

15 1.5 15 1.5 120

45 89 89 90 91 180

3.25 2.48 8.36 29.29 15.57 4.80

4.45 13.47 9.33 8.24 4.39 4.45

3.18 3.17 3.17 3.17 3.17 3.17

2.5 2.5 2.5 2.5 2.5 2.5

2. Standard bends and elbows:

kλ = f(λ0), see the graph

Note Elbows Butt–welding Same Same Same Threaded connection Same Butt-welding; transition from a 32-mm diameter elbow to a straight 25-mm-diameter section of the tube Butt–welding Same Same Same Threaded connection Bends

431

Flow with Changes of the Stream Direction Circular section bends and elbows at high subsonic flow velocities41,74

Diagram 6.4

Values of kλ No. 1 2 3 4 5

λc

Characteristics Bend; δ = 45–90o; R0 ⁄ D0 > 1 Bend; δ = 45o; R0 ⁄ D0 = 1 Bend; δ = 90o; R0 ⁄ D0 = 1 Bend; δ = 90o; 0.75 ≤ R0 ⁄ D0 ≤ 1 Elbow; δ = 90o; rin/D0 = 0; rout/D0 = 0–0.5

0.1

0.2

0.3

0.4

0.45

0.5

0.52

0.55

0.60

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.02

1.04

1.05

1.08

1.15

1.0

1.02

1.08

1.17

1.22

1.28

1.30

1.34

1.40

1.0

1.03

1.10

1.19

1.24

1.30

1.32

1.35

1.42

1.0

1.0

1.05

1.16

1.26

1.41

1.50

–

–

Bends located downstream of centrifugal fans (Chapter 3, References 3, 4, 26)

ζB

Diagram 6.5

∆p ρw20 ⁄ 2

Values of ζ

Turn

Rectangular cross section (a), R0 = Dh Circular cross section, R0 = 2Dh Rectangular cross section, R0 = 1.5Dh with a pyramidal diffuser (b) Rectangular cross section (a), R0 = Dh Circular cross section, R0 = 2Dh Rectangular cross section, R0 = 1.5Dh with a pyramidal diffuser (b)

Angle of element installment, deg

Nominal operational conditions of the fan Q ⁄ Qn;

Q = Qn;

Q > Qn;

ηf ≥ 0.9ηfmax

ηf = ηfmax

ηf ≥ 0.9ηfmax

Vanes bent backward 90–270 0.6 90–360 0.5 90–360 0.2

0.2 0.5 0.2

0.3 0.4 0.2

Ts4–76 Ts4–76

Vanes bent forward 90–180 0.2 270–360 0.7 90–360 0.3 90–180 0.4

0.3 0.5 0.4 0.2

0.3 0.5 0.4 0.2

Ts4–76

Type of fan

Note: ηf and ηfmax are the efficiency and the maximum efficiency of the fan, respectively.

432

Handbook of Hydraulic Resistance, 4th Edition

Elbows with sharp corners (r/b0 = 0) at δ = 90o; l0 ⁄ Dh ≥ 0∗36

Diagram 6.6 w0Dh ≥ 2 × 105; v

1. Smooth walls (∆ = 0), Re =

ζB

∆p ρw20 ⁄ 2

⎛ b1 a0⎞ at l0 ⁄ Dh = 0−2; ζ = ζloc = f ⎜ , ⎟ , see graph a; ⎝ b0 b0⎠ at l0 ⁄ Dh ≥ 10, ζ C 1.05ζloc . Dh =

2a0b0 a0 + b0

Values of ζloc a0 ⁄ b0 0.25 1.0 4.0 ∞

__ where k∆ = f(Re and ∆ = D/Dh), see the table; kRe, see graph b or, tentatively,

∗

0.8

1.0

1.2

1.4

1.6

2.0

1.43 1.36 1.10 1.04

1.24 1.15 0.90 0.79

1.14 1.02 0.81 0.69

1.09 0.95 0.76 0.63

1.06 0.90 0.72 0.60

1.06 0.84 0.66 0.55

__

ζ = k∆kReζloc ,

Re × 10−4 kRe

0.6 1.76 1.70 1.46 1.50

Values of k∆

2. Rough walls (∆ > 0) and Re ≥ 104:

kRe C 4.06 ⁄ Re

b1 ⁄ b0

Re × 10−3

∆

0.118

3–40

>40

0

1.0

1.0

0.001

1.0

__ 1 + 0.5 × 103 ∆

0.001

1.0

≈1.5

1

1.4

2

3

4

6

8

1

14

≥

1.40

1.33

1.26

1.19

1.14

1.09

1.06

1.04

1.0

1.0

Here and subsequently l0 ⁄ Dh = 0 means that the elbow (bend) is installed directly downstream of the smooth collector inlet.

433

Flow with Changes of the Stream Direction Elbows with sharp corners (r/b0 = 0) at 0 < δ ≤ 180o; l0 ⁄ Dh ≥ 101,17,77,79,81,92

Diagram 6.7

1. Elbow without recess: 1) Smooth walls (∆ = 0), Re = w0Dh ⁄ v ≥ 2 × 105:

ζB

∆p ρw20 ⁄ 2

= C1Aζloc(ζfr C 0) ,

C1 = f(a0/b0), see graph a (in the case of a circular or square section C1 = 1.0) or, tentatively, C1 = 0.97 – 0.13 ln (a0/b0), ζloc = 0.95 sin2 (δ/2) + 2.05 sin4 (δ/2) = f(δ), see graph b; A = f(δ), see graph b or, tentatively, A ≈ 0.95 + 33.5/δ; 2) Rough walls (∆ = 0) and Re ≥ 104: ζ = k∆kReC1Aζloc ,

__ k∆ and kRe are determined as a function of ∆ = ∆/Dh and Re, respectively; see Diagram 6.6.

δ, deg

0

20

30

45

60

75

90

110

130

150

180

ζloc A

0 –

0.05 2.50

0.07 2.22

0.17 1.87

0.37 1.50

0.63 1.28

0.99 1.20

1.56 1.20

2.16 1.20

2.67 1.20

3.00 1.20

a0 ⁄ b0 C1

0.25

0.50

0.75

1.0

1.5

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1.10

1.07

1.04

1.00

0.95

0.90

0.83

0.78

0.75

0.72

0.71

0.70

II. Elbow with recess: ζB

∆p ρw20 ⁄ 2

C 1.2ζw.r ,

where for ζw.r, see ζ for elbow without recess.

434

Handbook of Hydraulic Resistance, 4th Edition

Elbows with rounded and diverging or converging exit37 (F1/F0 ä 1.0); 0 < δ ≤ 180o; l0 ⁄ Dh ≥ 10

Diagram 6.8

bcon is the width of the contracted section. 1) Smooth walls (∆ = 0) and Re =

ζB

∆p

ρw2con ⁄ 2

wconbcon ≥ 2 × 103; v

⎛ k1 ⎞ ⎟ + ζfr = AC1ζ ′ + ζfr , ⎝ nar ⎠

= A1C1 exp ⎜−

⎛ F1 r ⎞ ⎛ k1 ⎞ where ζ ′ = exp ⎜− ⎟ = f ⎜ ⎟ , see graph a; ⎝ F0 bcon ⎠ ⎝ nar ⎠ r ⎞ r δ λ; for λ, see Diagrams 2.1 and 2.6; at λ ≈ 0.025ζfr = 0.02 + 0.00035δ ; Dh Dh ⎟ ⎠ a ⎞ ⎛ a0 A1 = f (δ), see graph b; C1 = f ⎜ = 0⎟ , see, tentatively, graph d of Diagram 6.1; b b0 ⎠ con ⎝

ζfr = ⎛⎜1 + 0.0175 ⎝

1 ; ζ0 is the resistance coefficient of the elbow at nar = F1/F0 = 1.0 and δ = 90o; wcon is the mean ζ0 velocity of flow in the contracted section.

k1 = 2.3 log

2) Rough walls (∆ > 0) and Re ≥ 104: ζ = k∆kReA1C1ζ ′ + ζfr , for k∆ and kRe, see Diagram 6.1.

Values of ζ′ r

F1 ⁄ F0

bcon

0.2

0.5

1.0

1.5

2.0

3.0

4.0

5.0

0.10 0.15 0.20 0.30 0.40 1.00

0.20 0.13 0.08 0.06 0.04 0.04

0.45 0.32 0.20 0.13 0.10 0.09

0.69 0.57 0.45 0.30 0.25 0.21

0.78 0.68 0.58 0.45 0.40 0.35

0.83 0.76 0.67 0.56 0.51 0.47

0.88 0.83 0.76 0.67 0.64 0.59

0.91 0.87 0.81 0.74 0.70 0.67

0.93 0.89 0.85 0.79 0.76 0.73

δ, deg

0

20

30

45

60

75

90

110

130

150

180

A1

0

0.31

0.45

0.60

0.78

0.90

1.00

1.13

1.20

1.28

1.40

435

Flow with Changes of the Stream Direction Elbows with rounded corners at 0.05 < r/D0 ≤ 0.5 and

Diagram 6.9

0 < δ ≤ 180o; l0 ⁄ Dh ≥ 10

1) Smooth walls (∆ = 0) and Re =

ζB

∆p ρw20 ⁄ 2

w0Dh ≥ 2 × 105: v

= ζloc + ζfr ,

where ζloc = A1B1C1; ζfr = (1 + 0.0175δr/Dh)λ; for λ, see Diagrams 2.1 and 2.6; at λ ≈ 0.02, ζfr = 0.02 + 0.00035δr/Dh; A1 = f(δ) and C1 = f(a0/b0), see Diagram 6.1; B1 = f(r/D0), see the table or, tentatively, B1 ≈ 0.155(r0/D0)–0.595. 2) Rough walls (∆ > 2) and Re

ä 104:

ζ = k∆kReζloc + ζfr , for k∆ and kRe, see Diagram 6.1.

r ⁄ D0 (r ⁄ b0) B1

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.87

0.70

0.44

0.31

0.26

0.24

0.22

436

Handbook of Hydraulic Resistance, 4th Edition

Elbows of rectangular cross section with different shapes of inner and outer corners of the turn at δ = 90o; l0 ⁄ Dh = 0–25,30,37 No.

Resistance coefficient ζ B

Characteristics of the elbow

∆p ρw20 ⁄ 2

1) Smooth walls (∆ = 0) and Re = w0b0 ⁄ v > 2 × 105:

1. Rounded inner corner, sharp outer corner: Dh =

Diagram 6.10

4F0

ζ = C1ζloc + ζfr ,

__ where ζfr = [1 + 1.57(r0/b0)]λ; λ = f(Re and ∆), see Diagrams 2.1 and 2.6; at λ = 0.02, ζfr = 0.02 + 0.031(r0/b0); C1 = f(a0/b0), see graph d of Diagram 6.1. 2) Rough walls (∆ > 0) and Re > 104:

Π0

ζ = k∆kReC1ζloc + ζfr , where for k∆ and kRe, see Diagram 6.1; ζloc = f(r0/b0), see graph a or, tentatively, ζloc ≈ 0.39(r0/b0)–0.352 r0 ⁄ b0

0.05

0.1

0.2

0.3

0.5

0.7

1.0

ζloc

1.10

0.88

0.70

0.56

0.48

0.43

0.40

ζ is the same as under No. 1 at ζloc = 0.20.

2. Rounded inner corner (r0/b0 = 1.0), beveled outer corner

⎛ t1 ⎞ ζ is the same as under No. 1, but ζloc = f ⎜ ⎟ , ⎝ b0 ⎠ 1 see graph b or ζloc C 0.72 + 1.85t1 ⁄ b0

3. Beveled inner corner, sharp outer corner

t1 ⁄ b0

0.1

0.2

0.3

0.4

0.5

ζloc

1.10

0.90

0.88

0.69

0.60

437

Flow with Changes of the Stream Direction Elbows of rectangular cross section with different shapes of inner and outer corners of the turn at δ = 90o; l0 ⁄ Dh = 0–25,30,37 No.

Diagram 6.10

Resistance coefficient ζ B

Characteristics of the elbow

∆p ρw20 ⁄ 2

ζ is the same as under No. 1, but ζloc = 0.47

4. Inner corner cut by two chords; the outer corners is sharp

5. Inner and outer corners are beveled

ζ is the same as under No. 1, but ζloc = 0.28

6. Right-angle elbow (δ = 90o) of rectangular cross section with circular fairing

a) r1/b0 = 0 ζ is the same as under No. 1, but ζloc is determined from graph c

r0 ⁄ b0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ζloc

1.13

0.88

0.69

0.57

0.55

0.58

0.65

b) r0/b0 = 0.45; r1/b0 = 0.45 ζ is the same as under No. 1, but ζloc = 0.49

438

Handbook of Hydraulic Resistance, 4th Edition

Elbows composed of separate elements at different angles δ; circular cross section; l0/D0 ≥ 1022,6–71,6–81 No.

Resistance coefficient ζ B

Characteristics of the elbow

1. δ = 45 ; three elements at the angle 22.5 o

Diagram 6.11

o

∆p ρw20 ⁄ 2

1) Smooth walls (∆ = 0) and Re = w0D0 ⁄ v ≥ 2 × 105: ζ = ζloc + ζfr , where ζloc = 0.11; ζfr = λlel ⁄ D0; at λ C 0.02, ζfr = 0.02lel ⁄ D0. 2) Rough walls (∆ > 0) and Re ≥ 104: ζ = k∆kReζloc + ζfr , for λ, k∆, kRe, see Diagram 6.1.

2. δ = 60o; three elements at the angle 30o

ζ is the same as under No. 1, but ζloc = 0.15

3. δ = 60o; four elements at the angle 20o

ζ is the same as under No. 1, but ζfr = 2(lel/D0); at λ ≈ 0.02, ζfr = 0.04(lel/D0)

4. δ = 90o; three elements at the angles 60 and 30o

5. δ = 90o; three elements at the angles 45o welded with welding seams; Re > 10 × 105

ζ is the same as under No. 1, but ζloc = 0.40

D0, mm

50 0.80

100 0.60

150 0.45

200 0.38

250 0.32

300 0.30

350 0.30

Elbow characteristics

1. Made from five elements at the angle 22.5o

No.

Segmented elbows of circular cross section at δ = 90o; l0/D0 > 1071,68 ∆p ρw20 ⁄ 2

ζloc = 0.14

a1 = −0.5625683; a3 = −0.5663924 × 10−1; a5 = −0.4796866 × 10−3; ni = 6;

a0 = 0.6408985; a2 = 0.2448837; a4 = 0.7245266 × 10−2; a6 = 0.1279164 × 10−4;

0.2 0.50 0.75

lel ⁄ D0 R0 ⁄ D0

ζloc

0.45

0.98

0.4

0.34

1.47

0.6

for k∆ and kRe, see Diagram 6.1;

ζ = k∆kReζloc + ζfr

0.15

1.90

0.8

2) Rough walls (∆ > 0) and Re ä 104:

0.12

2.50

1.0

0.16

5.0

2.0

0.42

7.50

3.0

0.14

10.0

4.0

0.14

12.5

5.0

a1 = −0.2413919; a3 = 3.920123; a5 = 1.464781; ni = 6;

(1) a0 = 1.100609; a2 = −2.257211; a4 = −3.270671 a6 = −0.2737305;

ζfr = (nelem – 1)lel/D0; nelem is the number of elements in the elbow.

at r0 ⁄ D0 ≥ 10,

at 1.9 < R0/D0 < 10

at R0/D0 ≤ 1.9

i=0

ζloc = ∑ ai(R0 ⁄ D0) ;

ni

where ζloc = f(lel/D0), see graph a or

ζloc = f (lel ⁄ D0) ,

1) Smooth walls (∆ = 0) and Re = w0D0 ⁄ v ≥ 2 × 105:

Resistance coefficient ζ B

Diagram 6.12

0.14

15.0

6.0

Flow with Changes of the Stream Direction 439

Elbow characteristics

2. Made from four elements at the angle 30o

No.

Resistance coefficient ζ B ρw20 ⁄ 2

∆p

Diagram 6.12

1.10

0

R0 ⁄ D0

ζ

0

lel ⁄ D0

at R0 ⁄ D0 ≥ 7.5

at R0/D0 < 7.5

0.92

0.37

0.2

0.70

0.75

0.58

1.12

0.40

1.50

0.8

0.30

1.85

1.0

0.16

3.70

2.0

ni = 7;

a7 = 0.144361 × 10−3;

0.6

a6 = 0.339646 × 10−2;

a5 = −0.3035488 × 10−3;

0.4

a3 = −0.2609621;

a2 = 0.3342034;

ζloc = 0.2

a1 = −0.6822401;

a0 = 1.110851;

0.19

5.55

3.0

0.20

7.40

4.0

0.2

9.25

5.0

a4 = 0.127691;

ζ is the same as under No. 1, but ζloc = f(lel/D0) is determined as a function of lel/D0 from graph b from formula (1):

Segmented elbows of circular cross section at δ = 90o; l0/D0 > 1071,68

0.20

11.0

6.0

440 Handbook of Hydraulic Resistance, 4th Edition

3. Made from three elements at the angle 45o

ζ

1.10

0.95

0.24

0

R0 ⁄ D0

0.72

0.48

0.4

0.2

0.60

0.70

0.6

0.42

0.97

0.8

0.38

1.20

1.0

0.32

2.40

2.0

0.38

3.60

3.0

ni = 7;

a7 = −0.1241125 × 10−2;

0

a6 = 0.1058802 × 10−1;

a5 = −0.6968263 × 10−3;

lel ⁄ D0

a3 = 0.7030898;

a2 = −0.481815;

ζloc = 0.4

a1 = −0.697757;

a0 = 1.118112;

at R0 ⁄ D0 ≥ 4.5

at R0/D0 < 4.5

0.41

4.80

4.0

0.4

6.0

5.0

a4 = −0.2244795;

ζ is the same as under No. 1, but ζloc = f(R0/D0), see graph c or from formula (1):

0.41

7.25

6.0

Flow with Changes of the Stream Direction 441

Elbow characteristics

1. Made from two 90o elbows; flow turning in one plane; l0/b0 = 0–2; rectangular cross section

No.

Z-shaped elbows with sharp corners (r0/b0 = 0)36,68,71

Π0

4F0

Resistance coefficient ζ *

∆p ρw20 ⁄ 2

0 0 2.4 3.65

lel′ ⁄ b0 ζloc lel′ ⁄ b0 ζloc

3.30

2.8

0.62

0.4

for k∆ and kRe, see Diagram 6.4

ζ = k∆kReC1ζloc + ζfr ,

3.20

3.2

0.90

0.6

2) Rough walls (∆ > 0) and Re ≥ 104:

3.08

4.0

1.61

0.8

2.92

5.0

2.63

1.0

2.92

6.0

3.61

1.2

2.80

7.0

4.01

1.4

2.70

9.0

4.18

1.6

2.45

10

4.22

1.8

2.30

∞

4.18

2.0

where ζloc = f(lel/b0), see graph a; ζfr = λlel′ /b0; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr = 0.02lel′ /b0; for C1, see, tentatively, graph a of Diagram 6.5.

ζ = G1ζloc + ζfr ,

1) Smooth walls (∆ = 0) and Re = w0b0 ⁄ v ≥ 2 × 105:

Dh =

Diagram 6.13

442 Handbook of Hydraulic Resistance, 4th Edition

Elbow characteristics

2. Made from two 90o elbows; flow turning in two mutually perpendicular planes; l0/b0 = 0–2; rectangular cross section

No.

Π0

4F0

Resistance coefficient ζ *

∆p ρw20 ⁄ 2

0 1.15 2.4 3.16

lel′ ⁄ b0 ζloc lel′ ⁄ b0 ζloc 3.18

2.8

2.40

0.4

3.15

3.2

2.90

0.6

3.00

4.0

3.31

0.8

2.89

5.0

3.44

1.0

2.78

6.0

3.40

1.2

2.70

7.0

3.36

1.4

ζ is the same as under No. 1, but at ζloc = f(lel′ /b0), from graph b

Dh =

2.50

9.0

3.28

1.6

2.41

10

3.20

1.8

2.30

∞

3.11

2.0

Flow with Changes of the Stream Direction 443

3. Made from two 30o elbows; turning of the flow in one plane; l0/b0 > 10; circular cross section

Z-shaped elbows with sharp corners (r0/b0 = 0)36,68,71

4

0 0 0

lel′ ⁄ D0 R0 ⁄ D0

ζloc 0.15

1.85

1.0

0.15

3.70

2.0

0.16

5.55

3.0

0.16

7.40

4.0

0.16

9.25

5.0

0.16

11.3

6.0

where a0 = 0.0095; a1 = 0.22575; a2 = –0.1177083; a3 = 0.02475; a4 = –0.1791667 × 10–2. The formula is valid at lel/D0 < 3; at lel/D0 ≥ 3, ζloc = 0.16.

i=0

ζ is the same as under No. 1, but ζloc = f[R0/D0(lel/D0)], from graph c or ζloc = ∑ ai(lel ⁄ D0)i,

Diagram 6.13

444 Handbook of Hydraulic Resistance, 4th Edition

Flow with Changes of the Stream Direction

445

Z-shaped elbows with rounded corners32 r/Dh ≤ 0; l0 ⁄ D0 ≥ 10

Diagram 6.14 Dh =

No. 1.

Characteristics of the elbow Made from two 90o elbows; turning of the flow in one plane

4F0 Π0

Resistance coefficient ζ B

∆p ρw20 ⁄ 2

Normal turbulent velocity profile at the entrance at Re =

w0Dh v

≥ 104 ,

′ +ζ , ζ B ζloc fr ′ = k k ζ , see Diagram 6.1; where ζ = f(lel/Dh), see graph a; ζloc ∆ Re loc ζfr ≈ (5.0r/Dh + lel/Dh)λ; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02,

ζfr = 0.1r ⁄ Dh + 0.02lel ⁄ Dh . Nonuniform velocity profile at the entrance _ ′ +ζ , ζnon =kζζloc fr where for k, see below No. of velocity profile from Fig. 6.18 k1 (at all lel ⁄ Dh)

_ Values of ζ r ⁄ D0

1

2

3

4

0.8

1.05

1.2

1.2

lel ⁄ Dh

(r0 ⁄ b0)

0.4

0.6

1.0

1.5

2.0

3.0

5.0

10

∞

0.2

1.20

1.11

1.05

1.10

1.10

1.09

1.09

1.05

1.0

0.5

–

–

0.94

0.82

0.81

0.81

0.81

0.85

1.0

Normal turbulence velocity profile at the entrance at 2.

Made from 90o elbows; flow turning in two mutually perpendicular planes

Re ≥ 104: _ ′ +ζ , ζ = ζζloc fr ′ = k k ζ , see Diagram 6.1; where ζ = f(lel/Dh), see graph b; ζloc ∆ Re loc for ζfr, see No. 1. Nonuniform velocity profile at the entrance:

_ Values of k1ζnon = k ζloc + ζfr lel ⁄ Dh

No. of velocity profile from Fig. 6.18

1–3 0.85 1.10 1.05 1.15

1 2 3 4

4 0.87 1.15 1.13 1.17

≥7 0.94 1.32 1.34 1.26

5 0.89 1.20 1.18 1.20

_ Values of ζ r ⁄ D0

lel ⁄ Dh

(r0 ⁄ b0)

0.4

0.6

1.0

1.5

2.0

3.0

5.0

10

∞

0.2

1.20

1.11

1.05

1.10

1.10

1.09

1.09

1.05

1.0

0.5

–

–

0.94

0.82

0.81

0.81

0.81

0.85

1.0

446

Handbook of Hydraulic Resistance, 4th Edition

Π-shaped elbows (180o) with sharp corners (r/b0 = 0);

Diagram 6.15

rectangular cross section;36 F1/F0 ä 1.0; l0/b0 = 0–2

The resistance coefficient ζ B 1.

F1 b1 = = 0.5 F0 b0

∆p ρw20 ⁄ 2

1) Smooth walls (∆ = 0) and Re = w0bh/v ≥ 2 × 105:

ζ = C1ζloc + ζfr , where ζloc = f(lel/b0), see graph a; ζfr ≈ λ(1 + lel ⁄ b0); for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr = 0.02 + 0.02lel/b0; C1, tentatively, see graph a of Diagram 6.7. 2) Rough walls (∆ > 0) and Re ≥ 104; ζ = k∆kReC1ζloc + ζfr, where for k∆ and kRe, see Diagram 6.6.

Values of ζloc lel ⁄ b0

bel ⁄ b0

0.5 0.73 1.0 2.0

0

7.5 5.8 5.5 6.3

0.2

5.2 3.78 3.5 4.2

0.4

3.6 2.4 2.1 2.7

0.6

0.8

3.4 1.9 1.7 2.1

1.0

4.5 2.2 1.9 2.1

2.

1.2

6.0 2.7 2.1 2.2

1.4

6.7 3.3 2.3 2.2

7.1 3.7 2.4 2.0

1.6

7.8 4.0 2.6 2.0

1.8

7.5 4.3 2.7 1.8

2.0

7.6 4.7 2.7 1.6

F1 b1 = = 1.0 : F0 b0

ζ is the same as under No. 1, but ζloc = f(lel/b0) from graph b.

Values of ζloc bel ⁄ b0

0.5 0.73 1.0 2.0

lel ⁄ b0 0

7.9 4.5 3.6 3.9

0.2

6.9 3.6 2.5 2.4

0.4

6.1 2.0 1.8 1.5

0.6

5.4 2.5 1.4 1.0

0.8

4.7 2.4 1.3 0.8

1.0

4.3 2.3 1.2 0.7

1.2

4.21 2.3 1.2 0.7

1.4

4.3 2.3 1.3 0.6

1.6

4.4 2.4 1.4 0.6

1.8

4.6 2.6 1.5 0.6

2.0

4.8 2.7 1.6 0.6

2.4

5.3 3.2 2.3 0.7

Flow with Changes of the Stream Direction

447

Π-shaped elbows (180o) with sharp corners (r/b0 = 0); rectangular cross section;36 F1/F0 ä 1.0; l0/b0 = 0–2

Diagram 6.15

F1 b1 = = 1.4: F0 b0

3.

ζ is the same as under No. 1, but ζloc = f(lel/b0), from graph c.

Values of ζloc lel ⁄ b0

bel b0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.5 0.73 1.0 2.0

7.3 3.9 2.3 1.7

6.6 3.3 2.1 1.4

6.1 3.0 1.9 1.2

5.7 2.9 1.8 1.0

5.4 2.8 1.7 0.9

5.2 2.8 1.7 0.8

5.1 2.8 1.8 0.8

5.0 2.9 1.9 0.7

4.9 2.9 1.9 0.7

4.9 3.0 2.0 0.8

5.0 3.2 2.1 0.8

4.

F1 b1 = = 2.0: F0 b0 ζ is the same as under No. 1, but ζloc = f(lel/b0), from graph d.

Values of ζloc lel ⁄ b0

bel b0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.5 0.73 1.0 2.0

8.4 4.1 2.5 1.2

7.8 3.9 2.5 1.1

7.3 3.8 2.4 1.0

6.8 3.6 2.3 1.0

6.3 3.5 2.2 0.9

5.9 3.4 2.1 0.9

5.6 3.2 2.0 0.8

5.3 3.1 2.0 0.8

5.2 3.0 1.9 0.8

5.0 3.0 1.9 0.9

4.9 2.9 1.9 0.9

448

Handbook of Hydraulic Resistance, 4th Edition

¤-shaped elbows (180o) with rounded corners (r/Dh = 0);32

Diagram 6.16

F1/F0 = 1.0; l0/Dh ≥ 10 Normal turbulent velocity profile at the entrance Re =

w0Dh

ζB

∆p

v

ρw20 ⁄ 2

≥ 104 , _ ′ +ζ , = ζζloc fr

_ ′ = k k ζ , see Diagram 6.1; where ζ = f(lel/Dh), see the graph; ζloc ∆ Re loc ζfr = (5.0r/Dh + lel/Dh)λ; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr = 0.1r/Dh + 0.02lel/Dh.

_

′ +ζ . Nonuniform velocity profile at the entrance ζnon = kζζloc fr

Values of k lel ⁄ Dh

No. of velocity profile (Fig. 6.18) 1–3

≥4

1

0.80

0.80

2

1.15

1.05

3

1.20

1.15

4

1.20

1.15

Values of ζ lel ⁄ Dh

r ⁄ D0 (r ⁄ b0)

10

∞

0.4

0.6

1.0

1.5

2.0

3.0

5.0

0.2

0.93

0.75

0.57

0.60

0.67

0.77

0.86

0.97

1.0

≥ 0.5

–

–

0.63

0.58

0.58

0.63

0.74

0.85

1.0

Flow with Changes of the Stream Direction

449

U-shaped elbows (180o); rectangular cross section;36 F1/F0 ä 1.0; l0/b0 = 0–2

Diagram 6.17

Resistance coefficient ζ B

∆p ρw20 ⁄ 2

.

F1 b1 = = 0.5 , 1. F0 b0 1) Smooth walls (∆ = 0) and Re = w0b0/v ≥ 2 × 105: ζ = C1ζloc + ζfr, where ζloc = f(lel/b0), see graph a; ζfr ≈ (1.5 + 2lel/b0)λ; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr = 0.03–0.04lel/b0; C1, tentatively, see graph a of Diagram 6.7. 2) Rough walls (∆ = 0) and Re

ä 104:

ζ = k∆kReC1ζloc + ζfr for k∆ and kRe, see Diagram 6.1; Values of ζloc lel ⁄ b0

bel b0

0.2 0.5 2.6 0.75 1.1 1.0 1.8 2.0 2.1

0.4 1.3 0.8 1.1 1.9

0.6 0.8 0.7 0.9 1.7

0.8 0.7 0.7 0.8 1.5

1.0 0.7 0.6 0.8 1.4

1.2 0.8 0.6 0.7 1.3

1.4 0.9 0.6 0.6 1.1

1.6 1.0 0.7 0.6 1.0

1.8 1.1 0.7 0.6 0.9

2.0 1.2 0.7 0.5 0.8

F1 b1 = = 1.0 , F0 b0

2.

ζ is the same as under No. 1, but ζloc = f(lel/b0) from graph b. Values of ζloc lel ⁄ b0

bel b0

0.2 0.5 4.5 0.75 2.5 1.0 1.6 2.0 1.6

0.4 2.6 1.5 0.9 1.0

0.6 1.9 0.9 0.5 0.8

0.8 1.7 0.7 0.3 0.7

1.0 1.5 0.5 0.3 0.6

1.2 1.3 0.5 0.3 0.5

1.4 1.2 0.4 0.2 0.5

1.6 1.1 0.4 0.2 0.4

1.8 1.0 0.4 0.2 0.4

2.0 0.9 0.3 0.3 0.4

F1 b1 = = 1.4 , F0 b0

3.

ζ is the same as under No. 1, but ζloc = f(lel/b0) from graph c. Values of ζloc lel ⁄ b0

bel b0

0.2 0.5 4.2 0.75 2.8 1.0 1.9 2.0 1.2 4.

F1 = F0

0.4 3.1 1.8 1.3 0.9

0.6 2.5 1.4 0.9 0.8

0.8 2.2 1.1 0.7 0.7

1.0 2.0 0.9 0.5 0.6

1.2 1.9 0.8 0.4 0.5

1.4 1.9 0.8 0.3 0.4

1.6 1.8 0.7 0.3 0.4

1.8 1.8 0.7 0.2 0.4

2.0 1.8 0.7 0.2 0.4

b1 = 2.0 , b0

ζ is the same as under No. 1, but ζloc = f(lel/b0) from graph d. Values of ζloc bel b0

0.2 0.5 6.0 0.75 2.9 1.0 2.0 2.0 1.0

lel ⁄ b0 0.4 3.5 2.1 1.6 0.9

0.6 2.8 1.7 1.2 0.8

0.8 2.5 1.5 1.0 0.7

1.0 2.4 1.3 0.9 0.7

1.2 2.3 1.2 0.8 0.7

1.4 2.2 1.1 0.8 0.7

1.6 2.1 1.0 0.8 0.8

1.8 2.1 1.0 0.9 0.9

2.0 2.0 0.9 0.9 0.9

450

Handbook of Hydraulic Resistance, 4th Edition

Doubly curved turns at different values of δ; l0/Dh ≥ 108–10,26 S-shaped bend ("gooseneck"-type); flow in one plane

Diagram 6.18 R0 > 1.0 , D0

1)

ζB

∆p ρw20 ⁄ 2

′ +ζ , = Aζloc fr

′ , see ζ where for ζloc loc of a single branch on Diagrams 6.1 and 6.2;

R ⎞ ⎛ lel + 0.035 0 δ⎟ ; for λ, see Diagrams 2.1 Dh ⎠ ⎝ Dh

ζfr = λ ⎜

⎛ lel

through 2.6; at λ C 0.02, ζfr = 0.02 ⎜

⎝ Dh

+ 0.0007

R0 ⎞

δ ; Dh ⎟⎠

A = f(lel/Dh), see Table 1 and graph a (correct at Re = 2 × 104):

Values of A lel ⁄ D0

δ, deg

0

1

2

3

4

6

8

10

12

14

16

18

20

25

40–50

15 30 45 60 75 90

0.20 0.40 0.60 1.05 1.50 1.70

0.42 0.65 1.06 1.38 1.58 1.67

0.60 0.88 1.20 1.37 1.46 1.40

0.78 1.16 1.23 1.28 1.30 1.37

0.94 1.20 1.20 1.15 1.27 1.38

1.16 1.18 1.08 1.06 1.30 1.47

1.20 1.12 1.03 1.16 1.37 1.55

1.15 1.06 1.08 1.30 1.47 1.63

1.08 1.06 1.17 1.42 1.57 1.70

1.05 1.15 1.30 1.54 1.68 1.76

1.02 1.28 1.42 1.66 1.75 1.82

1.0 1.40 1.55 1.76 1.80 1.88

1.10 1.50 1.65 1.85 1.88 1.92

1.25 1.70 1.80 1.95 1.97 1.98

2.0 2.0 2.0 2.0 2.0 2.0

2) R0/D0 = 0.8 (circular cross section):

ζ = Aζloc + ζfr , where ζloc = f(δ), see Table 2; A = f(lel/D0), see Table 2 and graph b or 0.33

1.25 ⎛ 1.25⎞ − 3 − ζloc ⎛λlel ⁄ D0⎞ ζ = 3.5ζloc

⎝ for ζfr, see para. 1.

⎠⎝

⎠

ζloc + ζfr

Values of A lel ⁄ D0 No. of curve

δ, deg

ζloc

0

5

10

15

1 2

45 90

0.23 0.35

2.39 2.66

2.26 2.20

2.13 2.11

2.0 2.02

Flow with Changes of the Stream Direction

451

Doubly curved turns at different values of δ; l0/Dh ≥ 108–10,26

Diagram 6.18 3) δ = 90o (rectangular cross section) ζloc = f(R0/b0, b0/a0), see Table 3; A = f(lel/Dh), see Table 3 and graph c or formula (1) (D0 is replaced by Dh); for ζfr, see para. 1.

Values of A lel ⁄ D0 No. of curve

R0 ⁄ b0

1 2 3

0.75 0.70 0.60

b0 ⁄ a0 1.25 1.0 1.0

ζloc

0

6

12

18

0.75 0.52 0.45

2.87 2.98 3.20

2.60 2.50 2.33

2.33 2.11 2.26

2.0 2.11 1.93

S-shaped joined bends; spatial (flow in two mutually perpendicular planes)8–10,26

1.

Diagram 6.19

R0 ⁄ D0 ≥ 1.0:

ζB

∆p ρw20 ⁄ 2

′ +ζ , = Aζloc fr

′ , see ζ where for ζloc loc of a single bend on Diagrams 6.1 and 6.2; ζfr = λ(lel/Dh + 0.035 δR0/Dh); for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζ = 0.02lel/Dh + 0.0007 δR0/Dh; A = f(lel/Dh), from Table 1 and graph a (valid at Re ≥ 2 × 104).

2.

R0/D0 = 0.8 (circular cross section):

ζ = Aζloc + ζfr , where ζloc = f(δ), see Table 2; A = f(lel/Dh), see Table 2 and graph b or 1.25 0.5 2 − 3.3 ⎛ζloc + 2D0lel⎞ ⎛λlel ⁄ D0⎞ ζloc + ζfr , ζ = 3.0ζloc

for ζfr, see para. 1.

⎝

⎠⎝

⎠

452

Handbook of Hydraulic Resistance, 4th Edition

S-shaped joined bends; spatial (flow in two mutually perpendicular planes)8–10,26

Diagram 6.19

1. Values of A l0 ⁄ Dh

δ, deg

0

1

2

3

4

6

8

10

12

14

20

25

40

60 90

2.0 2.0

1.90 1.80

1.50 1.60

1.35 1.55

1.30 1.55

1.20 1.65

1.25 1.80

1.50 1.90

1.63 1.93

1.73 1.98

1.85 2.0

1.95 2.0

2.0 2.0

2. Values of A lel ⁄ D0 No. of curve

δ, deg

ζloc

0

5

10

15

1 2

45 90

0.23 0.35

2.09 2.28

2.04 2.23

1.95 2.20

2.0 2.03

3.

δ = 90o (rectangular cross section): ζloc = f(R0/b0, b0/a0), see Table 3; A = f(lel/Dh), see Table 3 and graph c or formula (1) (D0 is replaced by Dh); for ζfr, see para. 1.

3. Values of A lel ⁄ D0 No. of curve

R0 ⁄ b0

b0 ⁄ a0

ζloc

0

6

12

18

1 2 3

0.75 0.70 0.60

1.25 1.0 1.0

0.75 0.52 0.45

2.33 2.50 0.67

2.21 2.27 2.34

2.11 2.11 2.20

2.06 2.11 2.06

0 1.50 1.37

δ, deg

60

90

1. Values of A

U-shaped in one plane; smooth (R0/D0 ≥ 1.0); 0 < δ < 180o

0.95

1.15

1 1.10

1.05

2 1.25

1.10

3

U-shaped joined bends and turns in one plane;8–10,26 l0 ⁄ Dh ≥ 10

1.35

1.20

4

1.45

1.30

6

1.45

1.35

8

lel ⁄ Dh

∆p ρw20 ⁄ 2

′ +ζ , = Aζloc fr

⎛ lel

1.50

1.46

10

1.50

1.57

12

1.60

1.73

14

A = f(lel/Dh), see Table 1 and graph a

20 1.70

1.85

⎝ Dh

+ 0.0007

1.90

1.95

25

δ ; Dh ⎟⎠

R0 ⎞

⎞ δ⎟ ; for λ, see Diagrams 2.1 ⎠

through 2.6; at λ + 0.02, ζfr = 0.02 ⎜

R0 ⎛ lel ζfr = λ ⎜ + 0.035 Dh ⎝ Dh

2.0

2.0

40–50

′ , see ζ where for ζloc loc of a single branch on Diagram 6.1 and 6.2;

ζ*

1. R0/D0 ≥ 1.0

Diagram 6.20

Flow with Changes of the Stream Direction 453

45 90

1 2

0.75 0.70 0.60

1 2 3

R0 ⁄ b0 No. of curve

3. Values of A

δ, deg

No. of curve

2. Values of A

1.29

1.30

0

1.49

1.61

5

1.77

2.0

10

2.0

2.0

15

(1)

1.0

1.0

1.25

b0 ⁄ a0

0.45

0.52

0.75

ζloc

1.20

1.35

1.20

0

1.45

1.73

1.67

6

lel ⁄ D0

1.80

1.83

1.78

12

2.0

1.93

2.0

18

3. δ = 90o (rectangular cross section) ζloc = f(R0/b0, b0/a0), see Table 3; A = f(lel/Dh), see Table 3 and graph c or formula (1) (D0 is replaced by Dh); for ζfr, see para. 1.

0.35

0.23

ζloc

lel ⁄ D0

0.5 0.25 ζ = 1.2ζloc + 1.87 ⎛ζloc + 2D0 ⁄ lel⎞ ⎛λlel ⁄ D0⎞ ζloc + ζfr ⎠⎝ ⎝ ⎠ for ζfr, see para. 1.

where ζloc = f(δ), see Table 2; A = f(lel/D0), see Table 2 and graph b or

ζ = Aζloc + ζfr ,

2. R0/D0 = 0.8 (circular cross section):

454 Handbook of Hydraulic Resistance, 4th Edition

455

Flow with Changes of the Stream Direction Joined gate-like bends, turns;26 l0/Dh

Diagram 6.21 1. 0 < δ < 180o, R0/D0 ≥ 1.0 (scheme 1): ζB

∆p ρw20 ⁄ 2

C 2ζ1 + ζfr′

(tentatively) ,

where ζ1 is determined as ζ from para. 1 of Diagram 6.18; ζfr′ = λlel′ /Dh; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr′ = 0.02lel′ /Dh. 0 < δ < 180o, R0/D0 ≥ 1.0 (scheme 2):

2.

ζB

∆p ρw20 ⁄ 2

= ζ1 + ζ2 + ζfr′ (tentatively) ,

where ζ1 is determined as ζ from para. 1 of Diagram 6.18; ζ2 is determined as ζ from para. 1 of Diagram 6.19; for ζfr′ , see para. 1. 3. δ = 90o, R0/D0 = 0.8 (circular cross section; scheme 1): ζ = 1.55 + ζfr , where ζfr = λ(2lel/D0 + lel′ /D0 + 5.04); at λ ≈ 0.02 ζfr = 0.1 + 0.04lel/D0 + 0.02lel′ /D0 . 4. δ = 90o, R0/D0 = 0.6 (square cross section; scheme 1): ζ = 4.12 + ζfr, where for ζfr, see para. 3.

Joined curved turns and elbows, 3 × 90o and 4 × 90o of rectangular cross section at a0/b0 = 0.5;8,10,88 lel/b0 ≥ 10 Turns (R0/b0 = 0.75)

Diagram 6.22

1) Smooth walls (∆ = 0) and Re = w0b0 ⁄ v ≥ 4 × 105: ζB

∆p ρw20 ⁄ 2

= ζloc + ζfr .

2) Rough walls (∆ > 0) and 104 < Re < 4 × 105 : ζ = k∆kReζloc + ζfr ,

ζloc = 1.5A ,

where A = f(lel/b0), see graph a; ⎛ lel1 lel2 ⎞ + 3.5⎟ λ , ζfr = ⎜ + b b 0 ⎝ 0 ⎠ for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02 ⎛ lel1 lel2 ⎞ + 3.5⎟ + 0.07 , ζfr = 0.02 ⎜ + b b 0 ⎝ 0 ⎠ for k∆, see Diagram 6.1; kRe = f(Re), see graph b.

lel ⁄ b0

0

1

2

3

4

5

A

1.63

1.53

1.16

1.07

1.03

1.0

456

Handbook of Hydraulic Resistance, 4th Edition

Joined curved turns and elbows, 3 × 90o and 4 × 90o, of rectangular cross section at a0/b0 = 0.5;8,10,88 l0/b0 ≥ 10 3. Turns, spatial (sharply curved)

Diagram 6.22

Without guide vanes: flow direction a-a ζ = 12.5kRe ; flow direction b-b ζ = 8.7kRe ; With guide vanes: ζ = 0.4kRe . Here for kRe, see graph b.

4. Turns, spatial (sharply curved)

Without guide vanes: flow direction a-a

ζ = 6.9kRe ; flow direction b-b ζ = 8.3kRe . With guide vanes ζ = 0.4kRe ; Here for k, Re see graph b.

Re × 10–4 kRe

1

1.4

2

3

4

6

8

2.20

2.03

1.88

1.69

1.56

1.34

1.14

10 1.02

14 0.89

20 0.80

30 0.83

40 1.0

457

Flow with Changes of the Stream Direction Joined curved turns and elbows, 3 × 90o and 4 × 90o, of rectangular cross section at a0/b0 = 0.5;8,10,88 l0/b0 ≥ 10

Diagram 6.22

5. Elbows at r/b0 = 0 without vanes

ζ = f(Re), see the table and graph b

6. Elbows at r/b0 = 0.25 with guide vanes

ζ = f(Re), see the table and graph d

Values of ζ Re × 10–4

Scheme 5. Graph c 6. Graph d

2

3

4

6

10

20

40

9.70 1.0

9.70 1.0

9.55 0.77

9.00 0.61

9.25 0.53

8.75 0.46

8.75 0.38

458

Handbook of Hydraulic Resistance, 4th Edition

By-passes 4 × 90o (sharply curved) of rectangular cross section at a0/b0 = 0.5;88 l0/b0 ≥ 10

By-pass characteristics

Scheme

Diagram 6.23 Resistance coefficient ∆p

ζB

ρw20 ⁄ 2

r1 r2 r3 r4 = = = =0 b0 b0 b0 b0

6.77kIRe

r1 r2 r3 = = = 0; b0 b0 b0

r4 = 1.5 b0

6.38kIRe

r2 r4 = =0 b0 b0

5.30kII Re

r1 r3 = = 0.07; b0 b0

r1 r2 r3 = = 0; = 0.5; b0 b0 b0

r1 r2 = = 0.25; b0 b0

r4 = 1.5 b0

r3 = 0; b0

r4 = 1.5 b0

3.80kII Re

1.65kIII Re

with guide vanes in elbows No. 1 and No. 4*

r1 r2 = = 0.25; b0 b0

r3 = 0.5; b0

r4 = 1.5 b0

0.60kIV Re

with guide vanes in elbows No. 1 and No. 2*

r1 r2 r3 r4 = = = = 0.25 b0 b0 b0 b0 with guide vanes in all the elbows*

∗

Position and construction of vanes are described under para. 64–72.

0.60kIV Re

Flow with Changes of the Stream Direction

459

By-passes 4 × 90o (sharply curved) of rectangular cross section at a0/b0 = 0.5;88 l0/b0 ≥ 10

Diagram 6.23

Re × 10–4 kRe

1

2

3

4

6

8

10

20

30

≥ 60

kIRe kII Re kIII Re kIV Re

1.28

1.15

1.10

1.06

1.04

1.02

1.01

1.0

1.0

1.0

1.40

1.26

1.19

1.14

1.09

1.06

1.04

1.0

1.0

1.0

1.86

1.60

1.46

1.37

1.24

1.15

1.10

1.0

1.0

1.0

–

2.65

2.20

1.95

1.65

1.52

1.40

1.23

1.11

1.0

Joined elbows made from zinc-coated sheet at R0/D0 = 1.0; D0 = 100 mm and corrugated elbows at r0/D0 = 0.7; D0 = 100 mm; Re = w0D0/v ≥ 1.5 × 105; l0/D0 ≥ 1058

Elbow characteristics

Elbow; δ′ = 45o

Scheme

Diagram 6.24

Resistance coefficient ∆p

ζ*

0.60

ρw20 ⁄ 2

460

Handbook of Hydraulic Resistance, 4th Edition

Joined elbows made from zinc-coated sheet at R0/D0 = 1.0; D0 = 100 mm and corrugated elbows at r0/D0 = 0.7; D0 = 100 mm; Re = w0D0/v ≥ 1.5 × 105; l0/D0 ≥ 1058

Elbow characteristics

Elbow; δ = 90o

"Gooseneck," 2δ = 2 × 90o

Scheme

Diagram 6.24

Resistance coefficient ∆p

ζ*

0.92

2.16

1.50 "Gooseneck" (turning in two planes); δ + δ′ = 90o + 45o

1.60 "Gooseneck" (turning in two planes); 2δ = 2 × 90o

Turns, 4δ′ = 4 × 45o

Elbow; δ′ = 45o

Elbow; 2δ′ = 2 × 45o

2.65

0.53

0.82

ρw20 ⁄ 2

461

Flow with Changes of the Stream Direction Joined elbows made from zinc-coated sheet at R0/D0 = 1.0; D0 = 100 mm and corrugated elbows at r0/D0 = 0.7; D0 = 100 mm; Re = w0D0/v ≥ 1.5 × 105; l0/D0 ≥ 1058

Elbow characteristics

Elbow; δ = 90o

"Gooseneck", 2δ′= 2 × 45o

"Gooseneck", 2δ = 2 × 90o

"Gooseneck" (turning in two planes); δ + δ′ = 90o + 45o

"Gooseneck" (turning in two planes); δ = 2 × 90o

Turn, 4δ′ = 4 × 45o

Scheme

Diagram 6.24

Resistance coefficient ∆p

ζ*

ρw20 ⁄ 2

1.33

1.00

3.30

1.93

2.56

2.38

462

Handbook of Hydraulic Resistance, 4th Edition

Flexible glass-cloth bends with furrowed surfaces;53 Re ≥ 105

ζB

Diagram 6.25

∆p ρw20 ⁄ 2

= 0.9nbζloc + ζfr ,

where for ζloc, see the tables; ζfr = λ(lel/D0 + 0.035R0/D0), λ = 0.052(10D0)0.1 ⁄ D0(0.05b)0.2 ; b is the width of the tape wound on the wire framework of a glass-cloth tube (see para. 72 of Section 2.1); D0 is the diameter of the tube, m; nb is the number of bends.

Values of ζloc at R0 ⁄ D0 = 1.5 (scheme a) δ, deg D0, m

30

45

60

90

0.100 0.155 0.193 0.250

0.69 – 0.43 0.26

1.18 1.07 0.50 0.34

1.48 – 0.73 0.41

1.78 1.30 0.86 0.56

At D0 < 0.3 m ζloc = 1.05a exp (−cD0) sin δ , where a = 3.86; c = 7.8 m–1; at D0 ≥ 0.3 ζloc C 0.4

Flow with Changes of the Stream Direction

463

Flexible glass-cloth bends with furrowed surfaces;53 Re ≥ 105

Diagram 6.25

Values of ζloc at δ = 90o (numerator) and δ = 45o (denominator) (scheme a) R0 ⁄ D0 D0, m 0.100 0.155 0.193 0.250

0.75

1.5

3.0

2.28 1.25 1.30 1.12 1.12 – 0.90 0.44

1.78 1.18 1.30 1.07 0.86 – 0.71 0.39

1.70 1.04 1.18 1.05 – 0.52 0.25

Values of ζloc at R0 ⁄ D0 = 1.5 and δ = 90o D0, m Scheme of the turn In one plane a b Spatial c In one plane d e f

Number of bends nb

0.100

0.193

1 2

1.78 3.55

0.73 1.29

2

3.11

1.40

2 3 4

– 5.06 6.03

1.33 1.89 2.40

nb

Elbows and turns (δ = 90o) of rectangular cross section with guide vanes*5,15 1. Elbow (r0 = r1 = r; ⎯⎯2 ) with profiled guide vanes; t1 = r√

Re =

ζB

w0b0 = 2 × 105 , v

∆p ρw20 ⁄ 2

= ζloc + ζfr ,

Re ä 2 × 105 , ζ = kReζloc + ζfr ,

∗

Disposition and design of vanes are described under para. 64–72.

∑ ζloc = 0.9nbζloc

Diagram 6.26

464

Handbook of Hydraulic Resistance, 4th Edition

Elbows and turns (δ = 90o) of rectangular cross section with guide vanes*5,15

Diagram 6.26

where ζloc = f(r/b0), see graph a; ζfr = (1 + 1.57r/b0)λ; for λ, see Diagrams 2.1 through 2.6; at λ ≈ 0.02, ζfr = 0.02 + 0.31r/b0; kRe = f(Re), see, tentatively, graph b, or the formula kRe = 0.8 + 4.02 × 104/Re.

Normal number of vanes −1

r nnor = 2.13 ⎛⎜ ⎞⎟ − 1 b ⎝ 0⎠ S = 2.13 − 1 . t1 Reduced number of vanes

Values of ζloc

−1

r S nadv C 1.4 ⎛⎜ ⎞⎟ − 1 = 1.4 . b0 t1 ⎝ ⎠ Minimal number of vanes r nmin C 0.9 ⎛⎜ ⎞⎟ b ⎝ 0⎠

−1

− 1 = 0.9

2. The same as under No. 1, but guide vanes are thin at ϕ1 = 90–95o, ζ is the same as under No. 1, but ζloc = f(r/b0), according to graph c or the formulas.

r ⁄ b0

Number of vanes (see graph a)

S . t1

Normal (curve 1) Reduced (curve 2) Minimal (curve 3) Re × 10–4 kRe

3

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.33 0.33 0.45

0.23 0.23 0.33

0.17 0.15 0.27

0.16 0.11 0.22

0.17 0.13 0.17

0.22 0.19 0.15

0.31 0.30 0.17

30

≥60

5

6

8

10

14

20

2.10 1.80 1.60 1.50 1.35 1.23 1.12 1.0 0.90 0.80

Values of ζloc r ⁄ b0

Number of vanes (see graph c)

0

0.05

0.10

0.15 0.20 0.25

0.30

ζloc

Normal (curve 1)

0.42

0.35

0.30

0.26 0.23 0.21

0.20

1 ⁄ (8.39r ⁄ b0 + 2.58)

Reduced (curve 2)

0.42

0.35

0.30

0.24 0.20 0.17

0.14

0.4 × 0.037r ⁄ b0

Minimal (curve 3)

0.57

0.48

0.43

0.39 0.35 0.31

0.28

1 ⁄ (5.43r ⁄ b0 + 1.85)

465

Flow with Changes of the Stream Direction Turns (δ = 90o) with concentric guide vanes46,68

Diagram 6.27

1. Turn of rectangular cross section (r0/b0 = R0/b0 – 0.5) with vanes at Re = 105: ζB at Re

∆p ρw20 ⁄ 2

= ζloc + ζfr ,

ä 105

ζ = kReζloc + ζfr , where ζloc = (0.46R0/b0 – 0.04)ζw.v., see graph a; for ζw.v. see ζ R0 without vanes from Diagram 6.1; ζfr = 1.57λ ; for λ, see Diagrams 2.1 b0 R0 through 2.6; at λ ≈ 0.02, ζfr = 0.03 ; for kRe, see, tentatively, graph e b0 of Diagram 6.1; distance between vanes: r1 = 1.26ri–1 + 0.07b0.

r0 ⁄ b0

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.3

1.5

ζloc

0.24

0.15

0.12

0.10

0.09

0.08

0.07

0.06

0.07

2. Turn of circular cross section with vanes: ζB ζB

Values of ζ at different R0 ⁄ b0

∆p ρw20 ⁄ 2

∆p ρw20 ⁄ 2

= f (r1 ⁄ b0, Re), see graph c.

Re × 10–4 R0 ⁄ b0

3

1 (one vane) 1 (two vanes) 1.8 (one vane)

4

6

8

10

15

20

30

0.32 0.3 0.29 0.25 0.24 0.23 0.22 0.2 0.31 0.29 0.28 0.24 0.23 0.21 0.20 0.20 0.3 0.27 0.24 0.23 0.22 0.20 0.20 0.19

Values of ζ at R0 ⁄ b0 = 75 and one vane (i = 1) r1 ⁄ b0 Re

0

5 × 10

–4

–5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

= f (Re, R0 ⁄ b0), see graph b;

0.9

1.0

0.60 0.42 0.35 0.31 0.32 0.35 0.39 0.44 0.49 0.55

0.6

10

0.54 0.34 0.29 0.27 0.28 0.30 0.35 0.39 0.44 0.49 0.54

2 × 10–5

0.48 0.29 0.26 0.23 0.24 0.26 0.30 0.35 0.39 0.43 0.48

466

Handbook of Hydraulic Resistance, 4th Edition

Elbows of rectangular cross section at δ = 90o and with thin guide vanes* (ϕ1 = 95o)5,15

No.

Elbow characteristics

Scheme

1. Inner corner is sharp (t1 = 0); α = 45o; normal number of vanes: nv = 2.13

S −1 . t1

2. The same as under No. 1, but α = 50o 3. The same as under No. 1, but with reduced (most advantageous) number of vanes: nv C 1.4

S . t1

4. The same as under No. 1, but the inner corner is beveled (t1 = 0.25b0)

5. Diverging elbow (F1/F0 = 1.35) r/b0 = 1.18; α ≈ 53o, normal number of vanes: nv = 2.13

S −1 . t1

6. The same as under No. 5, but with reduced (minimal) number of vanes: nv C 0.9

S t1

∗

Disposition of vanes is described under para. 64–72.

∗∗

For kRe, see, tentatively, Diagram 6.26.

Diagram 6.28 Resistance coefficient ∆p

ζB

ρw20 ⁄ 2

ζ = 0.45kRe + λ∗∗ , at λ ≈ 0.02, ζ ≈ 0.47kRe; for λ, see Diagrams 2.1 through 2.6

ζ = 0.40kRe + λ , at λ ≈ 0.02, ζ ≈ 0.42kRe ζ = 0.36kRe + λ , at λ ≈ 0.02, ζ ≈ 0.38kRe

ζ = 0.32kRe + 1.28λ , at λ C 0.02, ζ C 0.35kRe

ζ = 0.40kRe + 1.28λ , at λ C 0.02, ζ C 0.43kRe ,

ζ = 0.60kRe + 1.28λ , at λ C 0.02, ζ C 0.63kRe .

467

Flow with Changes of the Stream Direction Smooth elbows of rectangular cross section at δ = 90o and with thin guide vanes50

1.

F1 r = 0.5; = 0.2; ϕ1 = 103o; F0 b0

ζB

∆p ρw20 ⁄ 2

Diagram 6.29

= kReζloc + ζfr ,

r where ζfr = ⎛⎜1 + 1.57 ⎞⎟ λ; for λ, see Diagrams 2.1 through 2.6; at b0 ⎠ ⎝ r λ C 0.02 ζfr = 0.02 + 0.031 , b0

r0 = r1 = r , number of vanes (most advantageous) nadv = 11

kRe = f(Re), see, tentatively, graph b of Diagram 6.26; ζloc = f(θ), see graph a.

2.

F1 r = 1; = 0.2; ϕ1 = 107o; F0 b0 r0 = r1 = r ,

number of vanes (most advantageous) nadv = 5

3.

F1 = 2; r0 = r1 = r F0

θ, deg

106

108

110

112

114

116

118

ζloc

0.52

0.46

0.43

0.42

0.44

0.48

0.52

ζ is the same as under No. 1, but ζloc = f(θ), see graph b.

θ, deg

82

84

86

88

90

92

94

96

98

ζloc

0.50

0.30

0.22

0.17

0.14

0.12

0.11

0.12

0.14

ζ is the same as under No. 1, but ζloc = f(θ), see graph c.

(a) r ⁄ b0 = 0.2; ϕ1 = 154o;

Values of ζloc θ, deg

nadv = 5 (b) r ⁄ b0 = 0.5; ϕ1 = 138 ; o

nadv = 2 (c) r ⁄ b0 = 1.0; ϕ2 = 90o; nadv = 5

Curve 1 2 3

68 0.39 0.32 0.40

70 0.36 0.29 0.26

72 0.34 0.27 0.21

74 0.33 0.26 0.21

76 0.34 0.26 0.25

78 0.37 0.25 0.32

80 0.40 0.25 0.52

82 0.44 0.25 0.67

468

Handbook of Hydraulic Resistance, 4th Edition

Elbows of circular cross section at δ = 90o with profiled guide vanes*15

Elbow characteristics

Smooth turn (r/D0 = 0.18); normal number of vanes: nv =

3D0 −1 t1

Smooth turn (r/D0 = 0.18); reduced number of vanes: nv =

2D0 t1

Diagram 6.30

Scheme

Resistance coefficient ∆p

ζB

ρw20 ⁄ 2

ζ = 2.3kRe + 1.28λ , at λ C 0.2; ζ C 0.26kRe; for λ, see Diagrams 2.1 through 2.6

ζ = 0.15kRe + 1.28λ , at λ C 0.2, ζ C 0.18kRe

Vanes are installed according to arithmetic progression at an+1 =2 a1 Beveled corners of the turn (t1/D0 = 0.25); normal number of vanes nv =

ζ = 0.30kRe + 1.28λ ,

at λ C 0.2, ζ C 0.33kRe

3D0 −1 t1

Beveled corners of the turn (t1/D0 = 0.25); reduced number of vanes nv = 2

ζ = 0.23kRe + 1.28λ ,

at λ C 0.2, ζ C 0.26kRe

D0 t1

Vanes are smoothly embedded and installed according to arithmetic progression at an+1 =2 a1 Beveled corners of the turn (t1/D0 = 0.25); reduced number of vanes (1st and 3rd vanes are removed from the outer wall)

∗

ζ = 0.21kRe + 1.28λ ,

at λ C 0.2, ζ C 0.24kRe

Disposition and design of vanes are described under paragraphs 64–72. For kRe, see, tentatively Diagram 6.26.

469

Flow with Changes of the Stream Direction Spatial (circular) turn through 180o (during suction);19 R/D1 = 0.2–0.5; Re = w0D0/v ≥ 4 × 104

Diagram 6.31

A. Rounded corners at the turn (r/D0 > 0): ζB

∆p ρw20 ⁄ 2

=f

h r , ,n , D0 D0 ar

see graph a. nar = F1 ⁄ F0

Values of ζ h ⁄ D0 r ⁄ D0 0.05

0.10

0.20

nar

0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0 1.20 1.40 1.60 1.80 2.00

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

5.80 4.70 3.45 4.10 3.30 2.20 2.70 1.40 0.50

2.90 2.60 1.50 1.40 1.12 0.60 1.05 0.50 0.23

1.90 1.35 1.00 0.98 0.42 0.40 0.80 0.33 0.20

2.40 1.10 0.73 1.17 0.40 0.30 0.70 0.32 0.18

2.80 1.08 0.60 1.33 0.47 0.22 0.80 0.40 0.16

3.00 1.30 0.50 1.60 0.90 0.23 1.03 0.68 0.16

3.25 1.77 0.50 1.80 1.10 0.35 1.26 0.90 0.18

3.55 1.80 0.70 2.00 1.17 0.52 1.43 1.00 0.28

4.00 1.73 1.20 2.15 1.20 0.66 1.58 1.06 0.43

2.80 1.66 1.37 2.20 1.25 0.75 1.72 1.08 0.55

2.10 1.55 1.37 1.20 1.43 0.77 1.84 1.00 0.58

1.95 1.48 1.30 1.00 0.90 0.75 1.90 1.00 0.53

1.90 1.33 0.80 0.96 0.60 0.63 1.55 1.05 0.45

1.95 1.23 0.60 0.48 0.60 0.46 1.27 0.90 0.32

2.00 1.30 0.55 1.03 0.60 0.40 1.15 0.55 0.27

2.03 1.30 0.55 1.07 0.60 0.40 1.15 0.53 0.26

2.05 1.30 0.55 1.10 0.60 0.40 1.25 0.55 0.25

470

Handbook of Hydraulic Resistance, 4th Edition

Spatial (circular) turn through 180o (during suction);19 R/D1 = 0.2–0.5; Re = w0D0/v ≥ 4⋅104

Diagram 6.31

B. Thickened corners at the turn without rounding (r/D0 = 0). For ζ, see graphs b–d.

Values of ζ h ⁄ D0 δ ⁄ D0 0.10

0.20

0.40

nar

0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.0

1.20 1.40 1.60 1.80 2.00

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

8.70 – 4.20 8.26 5.75 4.40 8.26 3.90 2.00

2.00 1.30 1.87 2.60 2.63 1.13 3.65 2.65 1.14

1.90 1.20 1.20 1.60 1.10 1.35 2.42 1.70 1.36

3.90 3.90 2.20 3.00 1.60 0.75 2.40 2.00 0.60

2.20 1.75 0.80 1.50 0.80 0.60 1.25 0.70 0.43

1.70 1.20 0.62 1.10 0.50 0.53 0.90 0.46 0.40

2.80 1.00 0.56 1.90 0.46 0.50 0.78 0.40 0.40

3.10 1.40 0.45 2.50 0.90 0.52 2.50 0.60 0.43

3.40 2.10 0.50 2.85 1.30 0.55 3.40 1.40 0.50

3.70 2.66 0.70 3.20 1.67 0.65 3.90 2.00 0.60

4.25 2.66 1.00 3.45 1.98 0.78 4.30 2.46 0.73

3.40 2.10 1.38 3.55 2.26 0.90 4.25 2.66 0.87

2.30 1.60 1.60 3.30 2.53 1.03 4.05 2.72 1.00

1.95 1.16 0.93 1.30 0.83 0.70 2.00 1.30 1.20

2.00 1.17 0.73 1.25 0.80 0.56 1.83 1.13 0.65

2.00 1.18 0.60 1.27 0.83 0.60 1.77 1.03 0.58

2.00 1.20 0.57 1.30 0.85 0.63 1.75 0.95 0.57

471

Flow with Changes of the Stream Direction Spatial (circular) turn through 180o (with pumping);19 R/D1 = 0.2–0.5; Re = w0D0/v ≥ 4 × 104

Diagram 6.32

A. Rounded corners of the turn (r/D0 > 0): ∆p

h r = f ⎛⎜ , , nar⎞⎟ , D0 D0 ⎠ ⎝ see graph a.

ζB

ρw20 ⁄ 2

nar = F1 ⁄ F0 .

Values of ζ h ⁄ D0 r ⁄ D0 0.05

0.10

0.20

nar

0.10

0.15

0.20

0.25

0.30

0.40

0.50

0.60

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

5.70 7.60 – 1.95 2.80 3.40 1.15 1.20 1.35

2.40 2.60 3.16 0.62 1.20 1.28 0.60 0.50 0.70

1.18 1.45 2.05 0.35 0.40 0.85 0.40 0.32 0.45

0.70 0.90 1.48 0.26 0.30 0.70 0.33 0.23 0.40

0.40 0.70 1.15 0.20 0.25 0.60 0.32 0.20 0.40

0.20 0.52 0.72 0.17 0.23 0.50 0.32 0.20 0.40

0.18 0.42 0.55 0.20 0.28 0.43 0.40 0.20 0.40

0.18 0.40 0.43 0.60 0.80 0.36 1.15 0.30 0.40

h ⁄ D0 r ⁄ D0 0.05

0.10

0.20

nar

0.70

0.80

0.90

1.00

1.20

1.40

1.60

1.80

2.00

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

0.19 0.42 0.38 0.90 1.15 0.35 1.53 0.73 0.40

0.20 0.42 0.35 1.00 1.37 0.33 1.70 1.30 0.40

0.75 0.45 0.38 1.10 1.40 0.33 1.76 1.45 0.40

1.08 0.80 0.60 1.18 1.27 0.35 1.55 1.45 0.40

1.10 0.77 0.88 1.25 1.18 0.70 1.37 1.40 0.20

1.00 0.67 0.72 1.20 1.15 0.75 1.37 1.30 0.15

0.80 0.56 0.70 1.00 1.14 0.77 1.37 1.30 0.10

0.60 0.50 0.88 0.80 1.10 0.80 1.36 1.27 0.10

0.40 0.40 0.43 0.60 0.80 0.36 1.15 0.30 0.10

472

Handbook of Hydraulic Resistance, 4th Edition

Spatial (circular) turn through 180o (with pumping);19 R/D1 = 0.2–0.5; Re = w0D0/v ≥ 4 × 104

Diagram 6.32

B. Thickened corners of the turn without rounding (r/D0 = 0). For ζ, see graph b.

Values of ζ h ⁄ D0 δ ⁄ D0 0.10

0.20

0.40

nar

0.10

0.15

0.20

0.25

0.30

0.40

0.50

0.60

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

7.70 5.70 6.60 – – 4.10 – – 2.40

2.25 2.10 3.90 2.90 1.80 3.00 3.10 2.45 0.80

1.20 1.60 2.50 1.35 0.85 1.60 1.45 1.00 0.56

0.60 1.10 2.60 0.60 0.46 1.10 0.70 0.50 0.48

0.40 0.83 1.32 0.40 0.35 0.90 0.50 0.33 0.45

0.25 0.60 0.80 0.22 0.28 0.65 0.38 0.27 0.40

0.23 0.48 0.56 0.24 0.27 0.50 0.60 0.40 0.36

0.24 0.46 0.45 0.70 0.50 0.45 1.60 0.77 0.35

h ⁄ D0 δ ⁄ D0 0.10

0.20

0.40

nar

0.70

0.80

0.90

1.00

1.20

1.40

1.60

1.80

2.00

0.75 1.08 2.05 0.76 1.06 2.07 0.80 1.07 2.10

0.30 1.10 0.40 1.27 1.00 0.40 1.85 1.20 0.33

0.50 1.35 0.35 1.52 1.40 0.40 1.80 1.60 0.30

1.20 1.30 0.34 1.68 1.50 0.40 1.75 1.60 0.33

1.40 1.20 0.35 1.77 1.50 0.60 1.70 1.55 0.56

1.50 1.00 0.82 1.85 1.43 0.75 1.80 1.60 0.80

1.40 0.83 0.92 1.78 1.40 0.75 1.77 1.67 0.88

0.90 0.70 0.90 1.60 1.30 0.73 1.75 1.73 0.93

0.60 0.60 0.87 1.40 1.28 0.72 1.73 1.76 1.00

0.50 0.57 0.88 1.25 1.25 0.70 1.70 1.75 1.00

473

Flow with Changes of the Stream Direction Symmetric turn through 180o in one plane (during suction);47 Re = w0a0/v ≥ 0.8 × 105

A. Without dividers. ζ B

∆p ρw20 ⁄ 2

Diagram 6.33

h = f ⎛⎜ ⎞⎟ , see graph a. a ⎝ 0⎠

Values of ζ h ⁄ a0

Scheme and curve

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

1 2 3 4 5

– 10.5 7.9 – 3.8

9.5 7.5 6.3 4.2 2.3

7.9 5.7 5.0 2.6 1.7

5.5 4.7 4.4 1.8 1.5

4.5 3.9 4.0 1.6 1.4

4.1 3.5 3.8 1.5 1.5

4.0 3.4 3.9 1.5 1.6

4.0 3.7 4.0 1.6 1.8

4.2 4.5 5.0 1.8 2.2

5.2 – – 2.1 –

B. With plane dividers. For ζ, see graph b.

Values of ζ h0 ⁄ a0

Scheme and curve

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

1 2 3 4 5

– 10.5 8.6 – 3.0

9.5 8.0 6.7 3.6 2.1

7.5 6.0 5.3 2.3 1.6

5.6 4.6 4.3 1.7 1.3

4.6 4.0 3.8 1.4 1.2

4.1 3.5 3.6 1.3 1.2

3.8 3.3 3.5 1.3 1.3

3.6 3.2 3.5 1.3 1.3

3.6 3.3 3.6 1.4 1.5

3.6 3.3 3.8 1.5 1.6

474

Handbook of Hydraulic Resistance, 4th Edition

Symmetric turn through 180o in one plane (during suction);47 Re = w0a0/v ≥ 0.8 × 105

A. Without dividers. ζ B

∆p ρw20 ⁄ 2

Diagram 6.34

h = f ⎛⎜ ⎞⎟ , see graph a. a ⎝ 0⎠

Values of ζ h ⁄ a0 Scheme 1 2 3 4 5

0.20 – 8.8 7.0 – 3.0

0.25 10 6.6 4.7 3.8 1.7

0.30 7.3 5.2 3.7 2.3 1.2

0.35 6.0 4.4 3.2 1.7 1.0

0.40 5.2 3.9 2.7 1.4 0.9

0.45 4.6 3.6 2.5 1.3 0.9

0.50 4.3 3.4 2.4 1.3 0.9

0.55 4.2 3.3 2.4 1.3 1.0

0.60 4.0 3.4 2.5 1.4 1.1

0.65 4.0 3.4 2.6 1.5 1.2

0.70 4.0 3.0 2.7 1.7 1.4

B. With plane dividers. For ζ, see graph b.

Values of ζ h ⁄ a0 Scheme

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

1 2 3 4 5

– 8.7 6.6 – 3.0

9.7 6.5 4.6 3.6 1.7

7.3 5.0 3.6 2.3 1.3

6.0 4.3 3.1 1.7 1.1

5.2 3.8 2.7 1.5 1.0

4.7 3.5 2.5 1.4 1.0

4.4 3.3 2.3 1.3 0.9

4.2 3.2 2.3 1.3 0.9

4.0 3.2 2.2 1.3 0.9

4.0 3.1 2.2 1.3 1.0

3.9 3.0 2.2 1.3 1.0

Elbow of circular cross section Composed elbow of circular cross section (nele, number of elements)

Turn of circular cross section

Name

Scheme

Turns and elbows in the system of pneumatic transport;69 Re = w0D0/vc > 2 × 105

ρw20 ⁄ 2

∆p = ζ0 + κ (ζ1 − ζ0) ,

1.96 1.34 1.67

3.28

2.20

2.05 1.94 1.92

0.15 0.13

1.14

0.33

0.22 0.20 0.20

3.33 5.0

0

1.5

1.5 1.64 3.0

ζ1

0.17

ζ0

1.44

R0 ⁄ D0

transportation of material with κ = 1; κ = md/mg is the dust content factor, kg/kg

where ζ0 is the resistance coefficient without the material transported; ζ1 is the same during

ζ*

Diagram 6.35

Flow with Changes of the Stream Direction 475

4

πD02 .

Turn of variable rectangular cross section with transition from square to inscribed circle

A×B=

Turn of rectangular cross section A × B with transition to equidimensional circle:

Name

Turn of square cross section with transition to an inscribed circle

Name

Scheme

Scheme

Turns and elbows in the system of pneumatic transport;69 Re = w0D0/vc > 2 × 105

R2 = 2D0

R1 = D0

R1 = R2 = 2D0

B = 1.8 A

B = 1.0 A

0.18

1.57

1.50

0.61

0.15

0.15

0.51

0.15

ζ1

ζ0

Geometric characteristics

1.57

0.09

3.0

1.98

1.84

0.23

0.15

3.0

2.05

ζ1

1.5

0.19

ζ0

1.5

R0 ⁄ D0

Diagram 6.35

476 Handbook of Hydraulic Resistance, 4th Edition

The same, but with guide plates

Elbow with guide vanes in transition from inscribed circle to square

Elbow with transition from inscribed circle to square

0.33

Four plates

0.20

0.24

0.35

D0 3

0.56

0.84

Two plates

Two vanes Five vanes

R1 = R2 =

D0 R1 = R2 = 3

R1 = R2 = 0

1.82

1.87

1.48

1.80

3.17

3.66

Flow with Changes of the Stream Direction 477

478

Handbook of Hydraulic Resistance, 4th Edition

REFERENCES 1. Abramovich, G. N., Aerodynamics of local resistances, in Prom. Aerodin., vyp. 211, pp. 65–150, Oborongiz Press, Moscow, 1935. 2. Agureikin, S. S., Spodyryak, N. T., and Ustimenko, B. P., Experimental investigation of the turbulent flow hydrodynamics in curvilinear channels, Probl. Teploenerg. Prikl. Teplofiz., vyp. 5, pp. 35–45, Nauka Press, Alma-Ata, 1969. 3. Aronov, I. Z., Heat Transfer and Hydraulic Resistance in Curved Tubes, Thesis (Cand. of Tech. Sci.), Kiev Polytechnic Institute, Kiev, 1950, 130 p. 4. Aronov, I. Z., On the hydraulic similarity in fluid motion in curved tubes-coils, Izv. VUZ, Energetika, no. 4, 52–59, 1962. 5. Baulin, K. K. and Idelchik, I. E., Experimental investigation of air flow in elbows, Tekh. Zametki TsAGI, no. 24, 1934, 24 p. 6. Berkutov, I. S. and Pakhmatuin, Sh. I., Experience in the reduction of hydraulic losses of the channels, Neft. Khoz., no. 1, 46–47,1964. 7. Volkov, V. G., Khorun, S. P., and Yakovlev, A. I., Hydraulic resistance of plane channels with the reverse symmetric turning, in Aerodynamics and Heat Transfer in Electrical Machines, vyp. 1, pp. 98–105, Kharkov, 1972. 8. Goldenberg, I. Z., Study of the field of the axial flow velocity component in the ship pipeline behind a side branch, Tr. Kaliningr. Tekh. Inst. Rybn. Prom., vyp. 22, 125–134, 1970. 9. Goldenberg, I. Z. and Umbrasas, M.-R. A., Relationship between the hydraulic losses and the secondary flow velocity in pipeline bends, Tr. Kaliningr. Tekh. Inst. Rybn. Prom., vyp. 58, pp. 36–42, 1975. 10. Goldenberg, I. Z., Experimental investigation of the effect of interaction of flow turns on hydraulic losses in pressure channels, Tr. Kaliningr. Tekh. Inst. Rybn. Prom., vyp. 19, 29–34, 1966. 11. Gontsov, N. G., Marinova, O. A., and Tananayev, A. V., Turbulent flow over the section of circular tube bend, Gidrotekh. Stroit., no. 12, 24–28, 1984. 12. Dementiyev, K. V. and Aronov, I. Z., Hydrodynamics and heat transfer in rectangular curvilinear channels, Inzh.-Fiz. Zh., vol. 34, no. 6, 994–1000, 1978. 13. Zubov, V. P., Study of Pressure Losses in Wyes During the Separation and Merging of Flows, Thesis (Cand. of Tech. Sci.), Moscow, 1978, 65 p. 14. Ivanov, K. F. and Finodeyev, O. V., Concerning certain aspects of the process of flow stabilization downstream of the bend, Izv. VUZ, Energetika, 1985. 15. Idelchik, I. E., Guide vanes in elbows of aerodynamic tubes, Tekh. Zametki TsAGI, no. 133, 1936, 35 p. 16. Idelchik, I. E., Hydraulic Resistances (Physical and Mechanical Fundamentals), Gosenergoizdat Press, Moscow, 1954, 316 p. 17. Idelchik, I. E., About the effect of the Re number and roughness on the resistance of curved channels, in Prom. Aerodin., no. 4, pp. 177–194, BNI MAP, 1953. 18. Idelchik, I. E., On the separation flow modes in shaped parts of pipelines, in Heat and Gas Supply and Ventilation, pp. 43–49, Budivelnik Press, Kiev, 1966. 19. Idelchik, I. E. and Ginzburg, Ya. L., Hydraulic resistance of annular 180o-bends, Teploenergetika, no. 4, 87–90, 1968. 20. Ito and Nanbu, Flow in a rotating straight tube of circular cross section, Trans. ASME (Russian translation), no. 3, 46–56, Mir Press, Moscow, 1971. 21. Kazachenko, V. S., Local resistances of rectangular elbows, Vodosnabzh. Sanit. Tekh., no. 2, 7–11, 1962. 22. Kamershtein, A. G. and Karev, V. N., Investigation of hydraulic resistance of bent, welded, sharply bent and corrugated elbows-compensators, VNIIStroineft i MIIGS, pp. 52–59, 1956.

Flow with Changes of the Stream Direction

479

23. Karpov, A. I., Resistance of elbows of a small curvature radius under the conditions of pneumatic transport, Izv. VUZ, Energetika, no. 8, 93–95, 1962. 24. Kvitkovsky, Yu. V., Hydraulic resistance of smoothly bent tubes, Tr. Mosk. Inst. Inzh. Zheleznod. Transp., vyp. 176, pp. 6–63, 1963. 25. Klyachko, L. S., Refinement of the method of theoretical determination of the resistance coefficients of side branches of different outline, Tr. Nauchn. Sess. LIOT, vyp. 1, 79–137, 1955. 26. Klyachko, L. S., Makarenkova, T. G., and Pustoshnaya, V. F., Correlating formulas for determining the resistance coefficients of arbitrary assemblies of units from bends in ventilation systems, in The Problems of the Design and Mounting of Sanitary-Technical Systems (Tr. VNIIGS), pp. 3–8, Leningrad, 1980. 27. Koshelev, I. I., Eskin, N. B., and Abryutina, N. V., On the hydraulic resistance of bent small-diameter tubes of stainless steel with isothermal liquid flow, Izv. VUZ, Energetika, no. 2, 64–69, 1967. 28. Mazurov, D. Ya. and Zakharov, G. V., Study of some problems of the aerodynamics of tubular coils, Teploenergetika, no. 2, 39–42, 1969. 29. Maksimenko, A. V., Toward the problem of the regard for the effect of shaped parts in hydraulic calculation of ventilation systems, Sudostroenie, no. 8, 35–40, 1959. 30. Migai, V. K. and Gudkov, E. I., Some means for reducing losses in the elements of boiler gas-air pipelines, Tr. TsKTI, vyp. 110, pp. 40–46, 1971. 31. Nekrasov, B. B., Hydraulics, Voenizdat Press, Moscow, 1960, 264 p. 32. Novikov, M. D., Aerodynamic resistance of twin turns of boiler gas-air pipelines, Tr. TsKTI, vyp. 110, pp. 53–60, 1971. 33. Paraska, D. I., Technique for Improving the Hydraulic Characteristics of Curved Forced Channels by Visualizing the Flows of Two-Beam-Refracting Fluid, Thesis (Cand. of Tech. Sci.), Leningrad, 1982, 146 p. 34. Permyakov, B. A., The influence of the number of turns on the aerodynamics of coils made of helical tubes, Prom. Teplotekh., vol. 6, no. 2, 21–22, 1984. 35. Polotsky, N. D., On inception of secondary flows during liquid motion along a curved channel, Tr. Vses. Nauchno-Issled. Inst. Gidromashin., vyp. 29, 60–70, 1961. 36. Industrial Aerodynamics, Collected Papers No. 7, 1956, 154 p. 37. Rikhter, L. A., Thrust and Blasting at Steam Electric Stations, Gosenergoizdat Press, Moscow, 1962, 200 p. 38. Rozovsky, I. L., Water motion at the turn of an open channel, Izv. Akad. Nauk Ukr. SSR, Kiev, pp. 41–47, 1957. 39. Tatarchuk, G. T., Resistance of rectangular side branches, Problems of Heating and Ventilation (Tr. TsNIIPS), pp. 17–28, Gosstroiizdat Press, Moscow, 1951. 40. Topunov, A. M., Rubtsov, Yu. V., and Izmailovich, V. V., Reduction of hydraulic resistances in the elements of gas pipelines of power engineering equipment, Teploenergetika, no. 11, 43–46, 1981. 41. Trofimovich, V. V., Energy losses during turbulent motion of fluid in side branches, Sanit. Tekh., vyp. 5, pp. 156–164, 1967. 42. Uliyanov, I. E., Krumilina, N. N., and Vokar, N. V., The Design of Air Conduits of the Aeroplane Power Units, Moscow, 1979, 96 p. 43. Umbrasas, M.-R. A. and Goldenberg, I. Z., The influence of roughness on the magnitude of hydraulic losses in bends, in Hydraulics, Hydraulic Transport of Fish and Pertinent Technical Means, vyp. 69, pp. 62–69, Kaliningrad, 1977. 44. Umbrasas, M.-R. A., Evaluation of the Failure-Free Performance of Ship Pipelines with Bends when Designing Sea Water Systems, Thesis (Cand. of Tech. Sci.), Sevastopol, 1984, 155 p. 45. Heckestad, Flow in a plane rectangular elbow, Trans. ASME (Russian translation), no. 3, 86–97, 1971. 46. Khanzhonkov, V. I. and Taliev, V. N., Reduction of resistance in square side branches by means of guide vanes, Tekhn. Otchyoty, no. 110, 1947, 16 p.

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Handbook of Hydraulic Resistance, 4th Edition

47. Khanzhonkov, V. I., Aerodynamic resistance of plane channels with a reverse symmetrical turn, in Prom. Aerodin., vyp. 21, pp. 151–166, Oborongiz Press, Moscow, 1962. 48. Shchukin, V. K, Flow regimes and hydraulic resistance of radially rotating channels, Izv. Akad. Nauk SSSR, Energ. Transp., no. 3, 152–159, 1980. 49. Shchukin, V. K., Heat Transfer and Hydrodynamics of Internal Flows in the Fields of Body Forces, Mashinostroenie Press, Moscow, 1970, 331 p. 50. Yudin, E. Ya., Elbows with thin guide vanes, in Prom. Aerodin., no. 7, pp. 55–80, Oborongiz Press, Moscow, 1956. .. .. 51. Adler, M., Stromung in gekrummten Rohren, Z. Angew. Math. Mech., vol. 14, 30–50, 1934. 52. Ajukawa, K., Pressure drop in the hydraulic conveyance of solid materials through a bend in vertical plane, Bull. JSME, vol. 12, no. 54, 1969, 57 p. .. 53. Bambach, Plotzliche Umlenkung (Stoss) von Wasser in geschlossenen unter Druck durchstromten .. Kanalen, VDI, no. 327, 1–60, 1930. 54. Benson, R. S. and Wollatt, D., Compressible fIow loss coefficients at bends and T-junctions, Engineer, vol. 221, no. 5740, 70–85, 1966. 55. Blenke, M., Bohner, K., and Mirner, W., Drucklust bei der 180o-Stromungsumlenkung in Schlaufenreaktor, Verfarenstechnik, vol. 3, no. 10, 444–452, 1960. .. 56. Chun Sik Lee, Strumungeswiderstande in 90o-Rohrkrummern, Gesund. Ing., no. 1, 12–15, 1969. 57. Cross, Ph. and Pemes, P., Etude des pertes de charge singulieres dans les coudes brusques a 90o en polychlorure di vinyle, Bull. Tech. Genie Rural, no. 111, I–VII, pp. 1–33, 1971. 58. Conn, H. G., Colborne, H. G., and Brown, W. G., Pressure losses in 4-inch diameter galvanized metal duct and fittings, Heating, Piping and Air Conditioning, no. 1, 30–35, 1953. 59. Decock, P. and Pay, A., Mesure des pertes de charge localisus dans des accessoires de tuyauteriecoudes arrondis de 90o, Chal. Clim., vol. 34, no. 398, 27–31, 1969. 60. Eastwood, W. and Sarginson, E. J., The effect of a transition curve on the loss of head at a bend in a pipeline, Proc. Inst. Civ. Eng., vol. 16, no. 6, 129–142, 1960. .. .. 61. Fritzche, and Richter, H., Beitrag zur Kenntnis des Stromungswiderstandes gekrummter rauher Rohrleitung, Forsch. Geb. Ingenieurwes., vol. 4, no. 6, 40–90, 1933. .. 62. Haase, D., Stromung in einem 90o-Knie, Ing. Arch., vol. 22, no. 4, 282–292, 1954. 63. Hassoon, H. M., Pressure drop in 180o pipe bends, Build. Serv. Eng. Res. Technol., vol. 3, no. 2, 70–74, 1982. .. 64. Hofmann, A., Der Verlust in 90o, Rohkrummern mit gleichbleibenden Kreisquerschnitt, Mitt. Hy.. draul. Inst. Tech. Hochschule Munchen, no. 3, 30–45, 1929. 65. Iguchi, M., Ohmi, M., and Nakajima, H., Loss coeffjcient of screw elbows in pulsatile flow, Bull. JSME, vol. 27, no. 234, 2722–2729, 1984. 66. Idelchik, I. E., Some Amazing Effects and Paradoxes in Aerodynamics and Hydraulics, Mashinostroenie Press, Moscow, 1989, 97 p. 67. Ito, H., Trans. JSME, Ser. D, vol. 82, no. 1, 131–136, 1963. 68. Ito, H. and Imai, K., Pressure losses in vaned elbows of a circular cross section, Trans. ASME, vol. D88, no. 3, 684–685, 1966. .. 69. Jung, R., Die Stromungsverluste in 90o-Umlenkungen beim pneumatischen Staubtransport, Brennst. Waerme Kraft, vol. 19, no. 9, 430–435, 1967. 70. Kamiyama, S., Theory of the flow through bends with turning vanes, Sci. Rep. Res. Inst. Tohoku Univ., Ser. B, High-Speed Mech., no. 20, 1–14, 1969. .. .. 71. Kirchbach, Der Energieverlust in Kniestucken, Mitt. Hydraul. Inst. Tech. Hochschule Munchen, no. 3, 25–35, 1929. 72. Markowski, M., Wspolczynniki oporow przeplywu dwufozowego czynnika przez luki przenolnikov powie-trznych, Arch. Budowy Maszyn., part 14, no. 2, 339–370, 1967. 73. Machne, G., Perdite di carico prodotte da curve isolate on cambiamento di diresione di 90o in tubazioni a serione circolare costante in moto turbolento, Tec. Ital., vol. 22, no. 2, 77–91, 1957.

Flow with Changes of the Stream Direction

481

74. Morikawa, L., Druckverlust in pneumatischen Forderungen von kornigen Guten bei grossen Gutbelagen, Bull. JSME, vol. 11, no. 45, 469–477, 1968. 75. Morimune, T., Hirayama, N., and Maeda, T., Study of compressible high speed gas flow in piping system, Bull. JSME, vol. 23, no. 186, 1997–2012, 1980. 76. Murakami, M., Shimuzu, Y., and Shiragami, H., Studies on fluid flow in three-dimensional bend conduits, Bull... JSME, vol. 12, no. 54, 1369–1379, 1969. .. .. 77. Nippert, H., Uber den Stromungsverlust in gekrummten Kanalen, Forsch. Geb. Ingenieurwes., VDI, no. 320, 1922, p. 85. 78. Padmarajaiah, T. P., Pressure losses in 90o-bends in the region of turbulent flow, J. Inst. Eng. (India), Civ. Eng. Div., vol. 45, part 1, no. 1, 103–111, 1964. .. 79. Richter, H., Der Druckabfall in gekrummten glatten Rohrleitungen, Forschungsarb. Geb. Ingenieurwes, VDI, no. 338, 30–47, 1930. 80. Richter, H., Rohrhydraulik, Berlin, 1954, 328 p. .. 81. Schubart, Der Verlust in Kniestucken bei glatter und rauher Wandung, Mitt. Hydraul. Inst. Tech. Hoschschule Munchen, no. 3, pp. 13–25, 1929. 82. Sharma, H. D., Varshney, D. V., and Chaturvedi, R. N., Energy loss characteristics in closed conduit bends (in air model study), in Proc. 42nd Ann. Res. Sess. Madras, vol. 2, pp. 11–18, Jamil Nadu, 1972. 83. Shimizu, Y. and Sugino, K., Hydraulic losses and flow patterns of a swirling flow in U-bends, Bull. JSME, vol. 23, no. 183, 1443–1450, 1980. 84. Shiragami, N. and Inoue, I., Pressure losses in square section bends, J. Chem. Eng. Jpn., vol. 14, no. 3, 173–177,1981. 85. Smith, A. T. and Ward, The flow and pressure losses in smooth pipe bends of constant cross section, J. R. Aeronaut. Soc., vol. 67, no. 631, 437–447, 1963. .. .. 86. Spalding, D. B., Versuche uber den Stomungsverlust in gekruten Leitungen, VDI, no. 6, pp. 1–17, 1933. 87. Spychala, F. A. S., Versuche zur Ermittlung von Druckverlusten in Rohrleitungen und .. Formstucken von Luftungsanlagen, Schiffbauforschung, vol. 7, no. 5–6, 216–222, 1968. .. .. 88. Sprenger, H., Druckverluste in 90o Krummern fur Rechteckrohre, Schweiz. Bauztg. (SBZ), vol. 87, no. 13, 223–231, 1969. 89. Takami, T. and Sudou, K., Flow through curved pipes with elliptic sections, Bull. JSME, vol. 27, no. 228, 1176–1181, 1984. .. .. 90. Vuskovic, G., Der Stromungswidersland von Formstucken fur Gasroh leitungen (Fittings), Mitt. .. Hydraul. Inst. Tech. Hochschule Munchen, no. 9, 30–43, 1939. .. 91. Wasilewski, J., Verluste in glatten Rohrkrummern mit kreisrundem Querschnitt bei weniger a1s .. o 90 Ablenkung, Mitt. Hydraul. Inst. Tech. Hochschule Munchen, no. 5, 18–25, 1932. 92. Weisbach, J., Lehrbuch der Ingenieur und Maschinenmechanik, II Aufl., 1850 u. Experimentalhydraulik, 1855, 320 p. 93. Werszko, D., Badania iloseiowego wplywu chropowatosci i liczby Reynoldsa nu wspolczynnik strat hydrauliznych 90o krzywakow kolowych, Lesz. Nauk Politech. Wroclawski, no. 173, pp. 57– 78, 1968. 94. Wolf, S. and Huntz, D. M., Losses in compact 180-deg. return flow passage as a function of Reynolds number, Trans. ASME, vol. D92, no. 1, 193–194, 1970. 95. White, C. M., Streamline flow through curved pipes, Proc. R. Soc. London, Ser. A, vol. 123, 20– 31, 1929. 96. Peshkin, M. A., Cavitation characteristics of local resistaces of pipelines, Teploenergetika, no. 12, 59–62, 1960. 97. Peshkin, M. A., On the hydraulic resistance of branches with a gas–liquid mixture flow, Teploenergetika, no. 6, 79–80, 1961. 98. Kutateladze, S. S. and Styrikovich, M. A., Hydraulics of Gas–Liquid Mixtures, Gosenergoizdat Press, Moscow, 1958.

CHAPTER

SEVEN RESISTANCE IN THE CASES OF MERGING OF FLOW STREAMS AND DIVISION INTO FLOW STREAMS RESISTANCE COEFFICIENTS OF WYES, TEES, AND MANIFOLDS

7.1 EXPLANATIONS AND PRACTICAL RECOMMENDATIONS 1. Different types of wyes are considered in the present handbook: nonstandard wyes when Fs + Fst = Fc (Figure 7.1a and b) and when Fs + Fst > F (Figure 7.1c); normalized wyes of ordinary design (Figure 7.ld); and normalized wyes with branching assemblies of industrial construction (Figure 7.1e). 2. A wye is characterized by a branching angle α and the ratios of the cross-sectional areas of its branches Fs/Fc, Fst/Fc, and Fs/Fst. A wye can have different ratios of flow rates Qs/Qc and Qst/Qc and velocity ratios ws/wc and wst/wc. Wyes can be installed to merge or converge the flows or to diverge or separate the flows from the main passage. 3. The resistance coefficients of converging or merging wyes depend on the parameters named above, while those of diverging wyes of standard shape (without smooth rounding of the side branch and without divergence or convergence of both branches) depend only on the branching angle α and the velocity ratios ws/wc and wst/wc. The resistance coefficients of wyes of rectangular cross section are assumed to be nearly independent of the aspect ratio of their cross section, unless such coefficients are refined later. 4. When two streams moving in the same direction, but with different velocities, merge (Figure 7.la), turbulent mixing of streams (a shock) usually occurs, which is accompanied by nonrecoverable total pressure losses. In the course of this mixing, momentum exchange takes place between the particles of the medium moving with different velocities. This exchange favors equalization of the flow velocity field. In this case, the jet with higher velocity loses a part of its kinetic energy by transmitting it to the slower moving jet. 5. The total pressure difference between sections before and after mixing is always a large and positive quantity for a jet moving with a higher velocity. This difference increases the is 483

484

Handbook of Hydraulic Resistance, 4th Edition

Figure 7.1. Schemes of wyes: (a) with the same direction of flows, Fs + Fst = Fc, (b) and (c) with flow at an angle α at Fs + Fst = Fc and Fs + Fst > Fc, Fst = Fc, respectively; (d) normalized; (e) with branching assemblies of industrial-type construction.

the greater the larger the part of the energy which it transmits to a jet moving at a lower velocity. Therefore, the resistance coefficient, which is defined as the ratio of the difference of total pressures to the mean velocity pressure in the given section, also is always a positive quantity. The energy stored in the jet moving with a lower velocity also increases as a result of this mixing. Consequently, the difference between the total pressures, and, accordingly, the resistance coefficient of the branch in which the flow moves with a lower velocity, can also have negative values (see paragraph 2 of Section 1.1). 6. In practice, the branch is connected to the common channel on the side (side branching) at a certain angle α (see Figure 7.1b and c). In this case, losses due to turning of the stream are added to losses in a wye. The losses due to the turning of the flow are mainly due to the flow separation from the inner wall, flow contraction at the point of the turn, and its subsequent expansion (see Figure 7.1b). The contraction and expansion of the jet occur in the region of merging of streams and therefore influence the losses not only in a side branch, but also in the straight common passage. 7. When the branches are conical rather than cylindrical in shape or when there is a sudden expansion, there are losses due to flow expansion (diffuser or "shock" losses). If a side branch has a smooth turn, losses in this turn are also added. In general, the principal losses in a converging wye are composed of (1) the losses due to turbulent mixing of two streams moving with different velocities (shock), (2) the losses due

Merging of Flow Streams and Division into Flow Streams

485

Figure 7.2. Flow patterns in intake diverging wyes: (a) Qs < Qst; (b) Qs ≥ Qst; (c) Qs = 0.37

to flow turning when it passes from the side branch into the common channel, (3) the losses due to flow expansion in the diffuser part, and (4) the losses in a smooth branch. 8. The flow pattern in the diverging wye during flow separation into two jets (side branching and straight passage) varies with the ratio of velocities ws/wst or of flow rates Qs/Qst.37 9. When Qs < Qst, a large eddy zone is formed after the turn of the flow into the side branch (much larger than in the place of flow turning). This is due to the diffuser effect, that is, formation of a large positive pressure gradient at the place of wye branching, where the cross-sectional area increases sharply as compared with the main channel area. This large pressure gradient also causes partial flow separation from the opposite straight wall of the straight passage (Figure 7.2a). Both zones of flow separation from the wall create local jet contraction in both the side branch and the straight common passage. Flow contraction is followed by flow expansion. 10. When Qs ≥ Qst, the flow separates more vigorously from the outer wall of the straight passage as well as from the wall of the side branch after turning (Figure 7.2b). 11. At Qs = 0, an eddy zone forms at the entrance of the side branch (Figure 7.2c), which causes local contraction with subsequent expansion of the jet into the straight passage. 12. The distributions of velocities in side branches and in straight passages of the diverging wye with α = 90o and Fs = Fst = Fc for the cases of Qs/Qc = 0.5 and Qs/Qc = 1.0 obtained by Aslaniyan et al.1 are shown in Figures 7.3 and 7.4, respectively. These characteristics are given as the profiles and the fields of the axial velocity components in sections at different relative distances from the intersection of the wye axes. 13. The losses in the diverging wye are composed mainly of shock losses on sudden expansion at the place of flow division, losses due to flow turning into the side branch, losses

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Handbook of Hydraulic Resistance, 4th Edition

Figure 7.3. The profiles and fields of axial velocity components in a side branching of a straight, equally discharging wye:1 (a) Qs = Qc; Qst = 0; (b) Qs = 0.5Qc; Qst = 0.5Qc.

in the smooth passage of the side branch, and losses due to a sudden contraction of the passage (an industrial wye). 14. At certain flow rate ratios Qs/Qc, the resistance coefficient of the straight passage can have a negative value, that is, the energy of the flow can rise in this passage. Due to flow division a portion of the slowly moving boundary layer adjacent to the wall passes into the side branch and the energy per volume unit of the fluid medium moving in the straight passage becomes higher than that in the side branch. In addition, at the time of passing into the side branch, a part of the momentum is transferred to the flow in the straight passage. An increase in energy in the straight passage is accompanied by an increase of losses in the side branch, so that the whole flow process results in irreversible pressure losses. 15. The resistance coefficients of nonstandard converging wyes of normal shape (without roundings and divergence or convergence of the side branch or of the straight passage) can be calculated by formulas of Levin31 and Taliev.43 These are obtained by comparing the predicted results with the experiments of Levin,31 Gardel,54 Kinne,60 Petermann,66 and Vogel.78 For the side branch:

ζc.s *

2 2 2 ⎡ ⎤ Fst ⎛ wst ⎞ Fs ⎛ wst ⎞ ⎛ ws ⎞ ⎢ 2 cos α⎥ + Ks , 1 2 A − − = + ⎜ ⎟ ⎟ ⎜ wc ⎟ ⎜ 2 Fc ⎝ wc ⎠ Fc ⎝ wc ⎠ ρwc ⁄ 2 ⎣ ⎦ ⎝ ⎠

∆ps

487

Merging of Flow Streams and Division into Flow Streams

Figure 7.4. The profiles and fields of axial velocity components in the passage of a straight, equally discharging wye:1 (a) Qs = 0.5Qc; Qst = 0.5Qc; (b) Qs = 0.27Qc; Qst = 0.73Qc.

or

ζc.s *

2 2 ⎡ Q⎞ F ⎛ ⎛Q F ⎞ ⎢1 + ⎜ s c ⎟ − 2 c ⎜1 − s ⎟ A = Q F Fst Qc ρw2c ⁄ 2 ⎣ ⎝ c s⎠ ⎠ ⎝

∆ps

F c ⎛ Qs ⎞ −2 Fs ⎜⎝ Qc ⎟⎠

2

⎤ cos α⎥ + Ks ⎦

(7.1)

Table 7.1 Values of A Fs ⁄ Fc

≤0.35

>0.35

>0.35

Qs ⁄ Qc

≤1.0

≤0.4

>0.4

A

1.0

Qs ⎞ ⎛ 0.9 ⎜1 − ⎟ Q c⎠ ⎝

0.55

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Handbook of Hydraulic Resistance, 4th Edition

′′ for wyes of the type F + F = F Table 7.2 Values of Ks and Kst c s st

Fs ⁄ Fc

α, deg

0.10

0.20

0.33

Ks

Kst′′

Ks

Kst′′

15

0

30

0

0

0

0

0

45

0

0.5

0

0.14

60

0

0

0

0

90

0

0

0.10

0

0.20

0.5

Ks

Kst′′

Ks

Kst′′

0

0

0.14

0

0.40

0

0

0.17

0

0.35

0

0.14

0

0.30

0

0.10

0.10

0.25

0

0.25

0

For wyes of the type Fs + Fst > Fc, Fst = Fc at all values of A is given in Table 7.1 compiled by Zubov on the basis of Gardel’s54 experiments. In all cases, the value of Ks is zero. For wyes of the type Fs + Fst = Fc, the quantity A = 1 while the value of Ks is taken from Table 7.2.

Straight Passage For wyes of the type Fs + Fst > Fc, Fst = Fc ζc.st *

2

2

Qs ⎞ ⎛ Qs ⎞ Qs ⎞ ∆p ⎛ ⎛ = 1 − ⎜1 − ⎟ − ⎜1.4 − ⎟ ⎜ ⎟ sin α 2 Q Q c ⎠ ⎝ Qc ⎠ c ρwc ⁄ 2 ⎝ ⎝ ⎠

− 2Kst′

Fc cos α , Qc

(7.2)

where the values of Kst′ are given in Table 7.3. For wyes of type Fs + Fst = F0 2

2

2

Fs ⎛ ws ⎞ Fst ⎛ wst ⎞ ∆p ⎛ wst ⎞ ζc.st * 2 = 1 + ⎜ ⎟ − 2 ⎜ w ⎟ − 2 F ⎜ w ⎟ cos α + Kst′′ , w F c ⎝ c⎠ c ⎝ c⎠ ρwc ⁄ 2 ⎝ c⎠ or

′ Table 7.3 Values of Kst

Fs ⁄ Fc

≤0.35

Qs ⁄ Qc

0–10

≤0.6

>0.6

Kst′′

0.8Qs ⁄ Qc

0.5

0.8Qs ⁄ Qc

>35

489

Merging of Flow Streams and Division into Flow Streams

ζc.st *

2

2

Qs ⎞ Qs ⎞ Fc ⎛ ∆p ⎛ Fc ⎞ ⎛ 1− ⎟ = 1 + ⎜ ⎟ ⎜1 − ⎟ − 2 ⎜ 2 Qc ⎠ F Qc Fst ρwc ⁄ 2 ⎝ st ⎠ ⎝ ⎝ ⎠

2

2

Fc ⎛ Qs ⎞ cos α + Kst′′ , −2 Fs ⎜ Qc ⎟ ⎝ ⎠ where for the values of Kst′′ see Table 7.2. 16. The resistance coefficients of nonstandard diverging wyes of normal shape with a turbulent flow can be calculated from the formulas of Levin28 and Taliev43 with correction factors obtained by comparing the predicted results with the experimental data of Levin,28 Gardel,54 Kinne,60 Petermann,66 and Vogel.78 For the side branch

ζc.s *

∆ps ρw2c ⁄ 2

=

⎡

⎛ ws ⎞ +⎜ ⎟ ⎣ ⎝ wc ⎠

2

A ′ ⎢1

2 ⎤ ws ⎛ ws ⎞ ′ cos α⎥ − Ks ⎜ ⎟ , −2 wc w ⎦ ⎝ c⎠

or

ζc.s *

2 2 ⎡ ⎤ Q F Q F Q F ′ ⎢1 + ⎛ s s ⎞ − 2 s c cos α⎥ − K ′ ⎛ s c ⎞ , A = st ⎜ ⎜Q F ⎟ Qc Fs Q F⎟ ρw2c ⁄ 2 ⎣ ⎦ ⎝ c s⎠ ⎝ c c⎠

∆ps

where Kst′′ is the coefficient of flow compressibility. For wyes of the type Fs + Fst > Fc, Fst = Fc the values of A′ are given in Table 7.4, whereas the values of Ks′ are taken to be equal to zero. For wyes of the type Fs + Fst = Fc, A′ = 1.0 and the values of Ks′ are given in Table 7.5. For wyes of the type Fs + Fst > Fc, Fst = Fc (within the limits wst/wc ≤ 1.0)

ζc.st *

∆pst ρw2c ⁄ 2

= τst(Qs ⁄ Qc)2 ,

Table 7.4 Values of A ′ ≤0.35

Fs ⁄ Fc

6

A′

1.1–0.7Qs ⁄ Qc

0.85

1.0–0.6Qs ⁄ Qc

0.6

Table 7.5 Values of Ks′ α, deg

15

30

45

60

90

Ks′

0.4

0.16

0.36

0.64

1.0

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Handbook of Hydraulic Resistance, 4th Edition

where τst is given in Diagram 7.20. For wyes of the type Fs + Fst = Fc, ζc.st = f(wst/wc), see Diagram 7.20. 17. Since the resistance coefficient of diverging wyes of normal shape is independent of the area ratios Fs/Fc and Fst/Fc, generalized curves can be plotted for this coefficient as a function of ws/wc and, respectively, of wst/wc rather than of Qs/Qc and Qst/Qc. Therefore, in some cases in Chapter 7, the curves of the resistance coefficient are given in the form ζc.s = f(wst/wc) and ζc.st = f(wst/wc), respectively, though most curves are given as ζc.s = f(Qs/Qc) and ζc.st = f(Qst/Qc). 18. The resistance coefficients of standard wyes and wyes with branches of industrial design can be calculated for turbulent flow from extrapolation formulas of Klyachko and Uspenskaya21 (see Diagrams 7.15 through 7.17, 7.25, and 7.26). 19. There is a simple relationship* between the resistance coefficients of wyes based on the average velocity in the common channel and on the average velocity in branches:

ζs *

∆ps ρw2c ⁄ 2

=

ζc.s

(wst ⁄ wc)

2

=

ζc.s [(Qs ⁄ Qc)(Fc ⁄ Fs)]2

and

ζst *

∆pst 2

ρwst ⁄ 2

=

ζc.st (wst ⁄ wc)

2

=

ζc.st (1 − Qs ⁄ Qc)2(Fc ⁄ Fst)

.

The total resistance coefficient of a wye based on the kinetic energy in the common channel17,66 is

ζtot =

Qs Qst ζ . ζ + Qc s Qc st

20. The resistance of wyes of normal shape can be markedly reduced by rounding the junction between the side branch and the main passage. In the case of converging wyes, only the outside corner has to be rounded (r1, Figure 7.5). In the case of diverging wyes, both corners have to be rounded (r2, Figure 7.5); this makes the flow more stable and reduces the possibility of flow separation from the inner corner. Virtually, the rounding of the inner corners between the side branch and the main passage is sufficient when r/Dc = 0.2–0.3.14 21. The above formulas suggested for calculating the resistance coefficients of wyes and the corresponding graphic and turbulated data in Diagram 7.2 relate to very carefully manufactured (turned) wyes. Industrial defects occurring in wyes during their production ("depressions" in the side branch and the "blocking" of its section by the miscut wall in the straight passage [common channel; main tube, to which a side branch is attached]) become the source of a sharp increase in the fluid resistance. The increase in the resistance of side branches is especially significant if the diameter of the cut-out in the main tube for the side branch is smaller than the diameter of the latter. ∗

For an incompressible fluid.

Merging of Flow Streams and Division into Flow Streams

491

Figure 7.5. Schematic of an improved wye.

22. An increased resistance is also observed in wyes made of sheet steel with the flat seamed parts (see Diagrams 7.22 and 7.36). 23. The resistance of both the diverging and converging wyes is very effectively reduced if there is a gradual enlargement (diffuser) of the side branch. This noticeably reduces the losses both due to relative decrease in the flow velocity in the diverging section and due to decrease of the true angle of the flow turn at the same nominal angle of wye branchings (α1 < α, Figure 7.5). Combination of rounding and beveling of corners and widening of the side branch gives a still larger reduction of the wye resistance. The least resistance is achieved in wyes where the branch is smoothly bent (Figure 7.6); such branches with small branching angles ( 0.8, the reverse phenomenon is observed: the resistance coefficient of the machined wye is 10–15% higher than in the case of a threaded connection of branches.8,9 Probably, this is due to the fact that the enlargement of the section at the place of threaded connection (Figure 7.7) creates the condition like that in a stepped diffuser when a decrease in the resistance is observed as compared to a diffuser with straight walls (see Chapter 5). In diverging wyes with threaded connection of branches the values of the coefficients ζc.s remain practically the same as for machined wyes. The values of ζc.st are correspondingly higher.8,9 25. The values of the resistance coefficients of wyes increase with the growth of the reduced velocity of the flow in the common channel λc = wc/acr. The dependence of ζc.s and ζc.st on λc, presented in the work by Uliyanov et al.47 for certain wyes, are given in Diagram 7.24.

Figure 7.6. Schematic of a wye with a smoothly rounded branch.

492

Handbook of Hydraulic Resistance, 4th Edition

Figure 7.7. Schematic of an annular step in a standard wye: (a) welded wye; (b) wye with screwed pipes.

26. In the case of a turbulent flow (Rec = wc/v ≥ 4000), the resistance coefficients of wyes depend little on the Reynolds number. A slight decrease of ζc.s with an increase in Rec is observed only in converging wyes.8,9 27. In the course of transition from a turbulent to a laminar flow within the range Rec = 2 × 103 to 3 × 103 there occurs a sudden increase in the coefficient ζc.s for both converging and diverging wyes (Figure 7.8). The same occurs with the coefficient ζc.st of a diverging wye. For a converging wye, such a jump in ζc.st occurs at α > 60o and Fs/Fc = 1; when α = 45o and Fs/Fc = 1, the coefficient ζc.st does not increase; in this case, it is independent of Rec. When α = 30o and Fs/Fc = 1.0, even a sudden decrease in ζc.st occurs in transition from turbulent to laminar flow.8,9 28. In the case of laminar flow, the values of the resistance coefficients of wyes depend substantially on the relative length of the inlet section l0/D0 and increases with the increase in this length within the velocity profile stabilization as is the case for branches (see Chapter 6).8,9 29. The expression for the resistance coefficient of wyes with laminar flow has generally the form suggested by Zubov8,9

ζl *

∆p 2

ρwc ⁄ 2

= [(N − 1) k1 + 1] ζ t +

A , Rec

(7.5)

3

1 ⎛w⎞ where N = dF is the coefficient of kinetic energy (of the Coriolis energy) of the Fc ∫ ⎜wc ⎟ Fc ⎝ ⎠ flow in section c-c; k1 is a correction factor; superscripts l and t refer to a laminar and a turbulent flow, respectively. 30. For a converging wye with a laminar flow N = 2, and for a side branch k1 = 1, so that, according to Equation (7.5), ζcl.s *

∆p

ρwc ⁄ 2 2

t

= 2ζc.s +

A , Rec

(7.6)

493

Merging of Flow Streams and Division into Flow Streams

Figure 7.8. Dependence of the resistance coefficient ζs of wyes on data in Reference 8: (a) converging wyes; (b) diverging wyes. t

where ζc.s = ζc.s, see Equation (7.1); A is the quantity which depends on the parameters α, Qs/Q0, and Fs/Fc but its numerical value has not as yet been established; tentatively A ≈ 150. According to the data of Zubov,8,9 for a straight passage ζcl.st

=

l 2ζc.s + a0

⎛ Fc Qs ⎞ (1 − Qs ⁄ Qc) − (1.6 − 0.3Fs ⁄ Fc) × ⎜ F Q ⎟ ⎝ s c⎠ 2

2

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Handbook of Hydraulic Resistance, 4th Edition

Table 7.6 Values of a0 Fs ⁄ Fc

≤0.35

Qs ⁄ Qc

0–1.0

≤0.2

>0.2

a0

1.8–Qs ⁄ Qc

1.8–4Qs ⁄ Qc

1.2–Qs ⁄ Qc

>0.35

l

where for ζc.s, see Equation (7.6); for a0, see Table 7.6. 31. According to Equation (7.5), for a diverging wye with laminar flow (N = 2), the resistance coefficient of a side branch is l ζc.s *

∆ps ρw2c ⁄ 2

= (k1 + 1) ζtc.s +

A , Rec

t

where ζc.s = ζc.s, see Equation (7.3); tentatively A ≈ 150; for k1 see Table 7.7 (at Fs/Fc = 1).* For a straight passage N = 2 and k = 2, so that, according to Equation (7.5), we have l * ζc.st

∆ps ρw2c ⁄ 2

t = 3ζc.st +

A , Rec

t

where ζc.st = ζc.st, see Equation (7.4); tentatively A ≈ 33. 32. In equilateral wyes which are used for joining two opposite streams (merging) (scheme of Diagram 7.29), the resistance coefficients of the two branches are practically equal. 33. When a partition is installed at the place of junction of a symmetrical wye, the two flows are independent of each other before converging into the common channel. This is followed by conventional turbulent mixing of two flows moving with different velocities. The

Table 7.7 Values of k1 α, deg Qs ⁄ Qc

30

45

60

90

At Fs ⁄ Fc ≤ 1 0 0.2 0.4 0.6 0.8 1.0

0.9 1.8 3.4 6.1 7.2 6.0

0.9 1.8 2.9 4.3 4.3 3.0

0.9 1.5 2.2 3.0 2.7 2.3

0.9 1.1 1.3 1.5 1.4 1.3

0.9 0

0.9 0

At Fs ⁄ Fc ≥ 1 Up to 0.4 Above 0.4 ∗

0.9 0

0.9 0

For other values of Fs /Fc the coefficient k1 is as yet unknown.

Merging of Flow Streams and Division into Flow Streams

495

losses in the wye in this case are made up of (1) the losses on mixing (shock losses) and (2) losses when the flow turns through 90o. 34. The resistance coefficient of a flow stream that moves in one of the branches with smaller velocity can have a negative value just as in a conventional diverging wye (due to the additional energy from the higher-velocity flow). Without a partition the flow pattern in a symmetrical wye is less clearly defined. The pressure drops before and after merging of the flows mainly reflect the losses common to both branches. The value of these is positive at any velocity ratio (flow rate) between the branch and the common channel ws/wc(Qh/Qc) and is approximately equal to the losses in a diverging elbow. 35. The resistance coefficient of each branch of a symmetrical wye at a junction can be calculated with the following formula of Levin:32 ζc.s *

2

∆ps

2

2

Qs ⎤ ⎛ Fc ⎞ ⎛ Fc ⎞ ⎡⎛ Qs ⎞ = 1 + ⎜ ⎟ + 3 ⎜ ⎟ ⎢⎜ ⎟ − ⎥ . 2 F Qc ρwc ⁄ 2 ⎝ s⎠ ⎦ ⎝ Fs ⎠ ⎣⎝ Qc ⎠

36. If a symmetrical wye is used for the division of flow, the conditions of flow passage in it are about the same as with a conventional turn. Therefore, the losses in this wye can be approximately determined from the data for elbows with different aspect ratios b1/b0. The resistance coefficient of this wye can also be calculated with Levin’s formula:32 ζc.s *

2

∆ps

⎛ ws ⎞ = 1 + k1 ⎜ ⎟ , 2 w ρwc ⁄ 2 ⎝ c⎠

where k1 ≈ 1.5 for standard threaded malleable-iron wyes; k1 ≈ 0.3 for welded wyes. 37. A symmetrical wye can be fabricated to have smooth branches ("swallowtail"), reducing its resistance appreciably. 38. The resistance coefficient of converging symmetrical wyes at α < 90o and Fc = 2Fs can be calculated from Levin’s formula:32 ζc.s *

∆ps ρw2c ⁄ 2

=4

Qs ⎛ Qs ⎞ (0.9 + cos2 α) + ⎜ ⎟ Qc ⎝ Qc ⎠

4

4

⎡ ⎞ ⎤ ⎛ Qc × ⎢1 + ⎜ − 1⎟ ⎥ ⎛1 − cos2 α⎞ ⎠ ⎝ ⎣ ⎠ ⎦ ⎝ Qs 2

⎛ Qs ⎞ − 4 ⎜ ⎟ cos2 α − 4 ⎛0.2 + 0.5 cos2 α⎞ . ⎝ ⎠ Q ⎝ c⎠ 39. The resistance coefficient of diverging symmetrical wyes at α < 90o and Fc = 2Fs can be calculated, approximately, just as for a side branch of a conventional wye of the type Fs + Fst = Fc from Diagram 7.16. 40. Diagram 7.31 contains the values of the resistance coefficients of symmetrical wyes of the Fs = Fst = Fc at α = 45o. These data were obtained experimentally57 for wyes with

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Handbook of Hydraulic Resistance, 4th Edition

Figure 7.9. Symmetrically shaped wyes of the type Fs = Fst = Fc;57 (a) butt jointed branching; (b) threaded branching at δ = 0; (c) threaded branching with d/D0 ≈ 0.13.

branches both butt jointed (Figure 7.9a) and threaded (Figure 7.9b and c). Two versions of the latter were tested: complete screw threading of branches (Figure 7.9b, δ = 0) and incomplete threading (Figure 7.9c, d/D0 ≈ 0.13). In the case of butt jointing, the same work suggests approximating formulas for calculating the coefficients (see the diagram indicated). 41. The flow pattern in crosses is basically similar to that in wyes. The resistance coefficients of double wyes of area Fst = Fc at stream junction (see Diagrams 7.31 through 7.35) can be determined approximately.29,30 For one of the side branches (for example, No. 1): ζ1c.s *

2

∆p1s

⎛ Q1s Fc ⎞ ⎛ Q1s ⎞ =1+⎜ ⎟ −8⎜Q ⎟ 2 Q F s c 1 ρwc ⁄ 2 ⎝ ⎝ c⎠ ⎠

2

2 [(Qc ⁄ Q1s) − (1 + Q2s ⁄ Q1s)]2 ⎛ Q1s ⎞ Fc ⎡ ⎛Q ⎞ ⎢1 − ⎜ 2s ⎟ × −2⎜ ⎟ Q F1s ⎣ Q 4 − (1 + Q2s ⁄ Q1s)(Q1s ⁄ Qc) ⎝ c⎠ ⎝ 1s ⎠

2

⎤ ⎥ cos α . ⎦

For the other side branch (ζ2c.s) the subscripts 1 and 2 are interchanged. For a straight passage: ζc.st *

2

2

1 + Qst ⁄ Qc ⎛ Qst ⎞ ⎛ Qst ⎞ =1+⎜ ⎟ −⎜ ⎟ 2 2 Q Q ρwc ⁄ 2 ⎝ c⎠ ⎝ c ⎠ (0.75 + 0.25Qst ⁄ Qc) ∆pst

2

2

2 ⎛ Q1s ⎞ Fc 1 + (Q2s ⁄ Q1s) ⎛ Qc ⎞ −2⎜ − 1⎟ cos α . ⎟ Qc F1s (1 + Q s ⁄ Q )2 ⎜ Qst 2 1s ⎝ ⎝ ⎠ ⎠

42. In order to determine the resistance coefficients of welded inflow (converging) crosses in cylindrical pipelines for steam, water, etc., with α = 90o, the following formulas are recommended:29,30 for one of the side branches (for example, No. 1) ζ1c.s *

∆p1s ρw2c ⁄ 2

⎛ Q1s Fc ⎞ = 1.15 + ⎜ ⎟ ⎝ Qc F1s ⎠

497

Merging of Flow Streams and Division into Flow Streams 2

2 ⎛ Q1s ⎞ [(Qc ⁄ Q1s) − (1 + Q2s ⁄ Q1s)] , −8⎜ Q ⎟ ⎝ c ⎠ 4 − (1 + Q2s ⁄ Q1s)(Q1s ⁄ Qc)

for a straight passage ζc.st *

2

∆pst

2

1 + Qst ⁄ Qc ⎛ Qst ⎞ ⎛ Qst ⎞ . = 1.2 + ⎜ ⎟ − ⎜ ⎟ 2 2 Q Q ρwc ⁄ 2 ⎝ c⎠ ⎝ c ⎠ (0.75 + 0.25Qst ⁄ Qc)

For standard crosses fabricated of malleable cast-iron at Qst/Qc > 0.7, the following quantity is added to the values of ζc.st: ⎞ ⎛Qst ∆ζc.st = 2.5 ⎜ − 0.7⎟ . Qc ⎠ ⎝ 43. The resistance coefficients of crosses for flow division are determined approximately just as for diverging wyes from Diagrams 7.18 through 7.20. For straight diverging double wyes (α = 90o) fabricated of sheet steel (with the parts flat seamed), the values of the resistance coefficients obtained experimentally by Sosin and Neimark40 for a turbulent flow are given in Diagrams 7.36 and 7.37. 44. When one side branch of the wye is close to the other, they influence each other. This especially refers to converging wyes. The extent of this effect exerted by one branch on another branch depends on both the relative distance between them and the flow rate ratio Qs/Qc. 45. To date, there exist insufficient data on the correction factors for this interaction effect for all types of wyes; in order to approximately determine this effect in the case of suction, one can use the experimental results of Bezdetkina.2 The values of the correction factor k2, expressed as the ratio of the resistance coefficient ζs2 of the second side branch to the resistance coefficient ζs1 of the first side branch, are summarized in Table 7.8 for different relative flow rates Qs/Qc and different relative distances lel/Dc between adjacent branches. 46. At small relative flow rates (Qs/Qc ≤ 0.1) the mutual effect of the branches of the wye is negligible (k2 ≈ 1.0, see Table 7.8). Therefore, if there are many side branches in which the ratios Qs/Qc for each single branch are small, the mutual effect can practically be neglected and the values of the resistance coefficients for each of them can be assumed such as for a single wye. Table 7.8 Values of k2 Qs ⁄ Qc lel ⁄ Dc 0–3 4 6 8 9

≤0.1 1.0 1.0 1.0 1.0 1.0

0.2

0.3

0.4

0.75 0.83 0.96 1.0 1.0

0.70 0.77 0.88 0.91 1.0

0.66 0.74 0.83 0.93 1.0

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Handbook of Hydraulic Resistance, 4th Edition

Some additional data on the mutual effect of a wye will be given in Chapter 12 (in the section "Mutual Effect of Local Fluid Resistances"). 47. Outlet and inlet headers (collectors) also fall into the wye-type sections (see Diagrams 7.39 through 7.44). The primary requirements of such collectors are uniform outflow and, correspondingly, uniform inflow through all of their branches. 48. The degree of uniformity of the flow rate through the side branches of headers (collectors) depends, as the theory shows,15–19 on a basic governing criterion, that is, "the characteristic of the header or collector" A1,2, which is a function of a number of parameters: A1,2 = ϕ (f, µ or ζbr, ζcoll) , _ where subscripts 1 and 2 refer, respectively, to supply and intake headers; f

* Σs ⁄ Fi

≈ nsfs ⁄ Fin is the relative cross-sectional area ns of the side branches; fs is the area of the inlet section of one side branch, m2; Fin is the cross-sectional area of the header (initial for supply and final for intake), m2; µ = 1/√ (fs ⁄ fa)2 + ζbr is the coefficient of discharge through a side ⎯⎯⎯⎯⎯⎯⎯⎯⎯ branch; fa is the area of the outlet section of the final length of the side branch, m2; ζbr

*

∆pbr ⁄ (ρw2s ⁄ 2)

is the resistance coefficient of the whole branch, which includes the resistance of all the adjacent sections, apparatus, or other devices; it is reduced to the velocity ws in the side branch ζbr = ζs + ζsec + ζapp , where ζs * ∆ps ⁄ (ρw2s ⁄ 2) is the resistance coefficient of the side branch alone; ζsec * ∆psec ⁄ (ρw2s ⁄ 2) is the resistance coefficient of all the sections of the side branch after flow division (up to its merging) except for the resistance of an apparatus (device); ζapp * ∆papp ⁄ (ρw2s ⁄ 2) is the resistance coefficient of the apparatus (device) installed in the side branch; ζcoll * ∆pcoll ⁄ (ρw2in ⁄ 2) is the resistance coefficient of the supplying (intake) part of the collector (header) reduced to the average velocity of the flow win in the section Fin of the header. If there are no additional barriers along the channels, it is assumed for practical calculations that ζcoll + 0.5λ

L , Dh.in

where L is the total length of the header, m; Dh.in = 4Fin/Πin is the hydraulic diameter of the initial cross section of the supplying channel, m. 49. With constant cross section of the header and with other conditions being equal, the degree of uniformity in the _ flow rate is the higher the greater is Fin (the condition of completely uniform supply is f → 0). In order to obtain uniform distribution of the flow rate without an increase in the crosssectional area of the header, it should converge in the direction of the flow (header of variable cross section). This can be done in different ways: a linear change of the cross section (see dash-dotted lines in Diagrams 7.40 through 7.43), a stepwise change of the cross section (Figure 7.10a) or an appropriate shaping of one of the side walls (Figure 7.10c).

Merging of Flow Streams and Division into Flow Streams

499

Figure 7.10. Headers of variable cross section: (a) stepwise change in the section with sharp branching; (b) stepwise change in the section with smooth branching; (c) with a shaped side wall.

When A2 > 0.3, the intake header should not have a variable cross section, since the flow distribution among the branches would be worse rather than better.* When A2 < 0.3, the intake header can be of variable cross section (to save metal), since it will not worsen uniformity of flow distribution among the branches. 50. The total resistance coefficient of isolated (single) headers of constant cross section and of variable cross section with a linear change in the cross section along the flow is determined from interpolation formulas based on experimental data50 and given in Diagrams 7.40 through 7.43. 51. In the majority of cases, the outlet and inlet headers operate jointly (joint headers). The flow in them can have opposite directions (Π-shaped collector, Diagram 7.42) or the same direction (Z-shaped collector, Diagram 7.49). When the resistance coefficients of both (outlet and inlet) headers are the same and ζcoll < 1, a Π-shaped collector provides a more uniform distribution of flow than a Z-shaped one. At ζcoll > 1, the situation is the reverse. 52. It is desirable that the outlet part of a Z-shaped header be of variable cross section (contraction in the flow direction) and the intake part of constant cross section at A4 > 0.3.** (See paragraph 49.) In some cases, a more uniform division of flow in a Π-shaped header may be achieved by contracting the cross section of the intake header in the direction toward its inlet and keeping the cross section of the outlet part constant. 53. The total resistance coefficient of joint Π- and Z-shaped headers with constant cross sections of both channels or with variable cross section of the outlet channel and constant cross section of the inlet channel is determined from the interpolation formulas50 given in Diagrams 7.42 through 7.47. 54. In order to decrease the resistance of the side branches of the header, the transition sections for them can be made as shown in Diagram 7.39. Their design is simple and their resistance coefficients are minimal. These may be taken as standard.

∗

In the case of a dust-laden flow, the flow velocity along the length of the intake collector should not be smaller than a certain limiting value (10–15 m/s) to prevent dust from settling. In this case, the intake collector should be of variable cross section, though it entails worsening of flow distribution among the branches. ∗∗

A4 is the characteristic of a Z-shaped collector.

500

Handbook of Hydraulic Resistance, 4th Edition

Figure 7.11. Π-shaped joint header with smooth branches of the supplying header.

The resistance of the side branches of a header decreases sharply if branching is smooth (see Figures 7.10b and 7.11). 55. The resistance coefficient of the ith branch ζis * ∆pis ⁄ (ρw2(i−1)s ⁄ 2) of the outlet (inlet) box with transition sections made according to the scheme of Diagram 7.39 depends only on the velocity ratio wis/w(i–1)s. This coefficient is virtually independent of the Reynolds number, at least from Re = 104, the aspect ratio of the cross section of the outlet box (within h/b = 0.5–1.0), and the area ratio Fs/Fc. The resistance coefficient of the branch installed in the side of the outlet box is smaller than the resistance coefficient of the branch installed above or below this box, since in the latter case the flow successively turns twice through 90o in two mutually perpendicular directions (see Diagram 7.39).

501

Merging of Flow Streams and Division into Flow Streams

7.2 DIAGRAMS OF THE RESISTANCE COEFFICIENTS Converging wye of the type Fs + Fst > Fc; Fst = Fc; α = 30o 31,43

Diagram 7.1

Values of ζc.s Fs ⁄ Fc

Qs

Qc

Side branch ⎡ ⎛ Qs Fc ⎞ ζc.s * 2 = A ⎢1 + ⎜ ⎟ ρwc ⁄ 2 ⎣ ⎝ Qc Fs ⎠ ∆ps

2

2

Qs ⎞ Fc ⎛ Qs ⎞ ⎤ ⎛ ′ −2 ⎜1 − ⎟ − 1.7 ⎜ ⎟ ⎥ = Aζc.s Qc ⎠ Fs ⎝ Qc ⎠ ⎦ ⎝ For ζc′.s, see the table and curves ζc′.s = f(Qs ⁄ Qc) at different Fs ⁄ Fc (graph a); A = f(Fs ⁄ Fc, Qs ⁄ Qc), see Table 7.1;

ζs*

∆ps ρw2s ⁄ 2

=

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1 0.2 –1.00 –1.00 0.21 –0.46 3.02 0.37 7.45 1.50 13.5 2.89 21.2 4.58 30.4 6.55 41.3 8.81 53.8 11.5 67.9 14.2 83.6 17.3

0.3 –1.00 –0.57 –0.07 0.50 1.15 1.83 2.60 3.40 4.32 5.30 6.33

0.4 –1.00 –0.60 –0.20 0.20 0.58 0.97 1.37 1.77 2.14 2.52 2.90

0.6 –1.00 –0.62 –0.28 –0.01 0.26 0.47 0.64 0.76 0.84 0.88 0.88

0.8 –1.00 –0.63 –0.30 –0.04 0.18 0.35 0.46 0.52 0.53 0.48 0.39

1.0 –1.00 –0.63 –0.31 –0.05 0.16 0.32 0.41 0.46 0.45 0.38 0.26

ζc.s [QsQc)(FcFs)]

Values of ζc.st∗ Qs Qc Passage 2

∆pst

2

Qs ⎞ Fc ⎛ Qs ⎞ ⎛ = 1 − ⎜1 − ⎟ −1.74 Q Fs ⎜⎝ Qc ⎟⎠ ρwc ⁄ 2 c ⎠ ⎝ see the table and curves ζc.st = f(Qs ⁄ Qc) at different Fs/Fc (graph b);

ζc.st *

ζst * ∗

2

∆pst ρw2st ⁄ 2

=

ζc.st (1 − Qs ⁄ Qc)2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fs ⁄ Fc 0.1

0.2

0.3

0.4

0.6

0.8

1.0

0 +0.02 –0.34 –1.06 –2.15 –3.60 –5.40 –7.60 –10.2 –13.0 –16.4

0 0.10 +0.01 –0.27 –0.75 –1.43 –2.29 –3.35 –4.61 –6.06 –7.70

0 0.13 +0.13 –0.01 –0.30 –0.70 –1.25 –1.95 –2.74 –3.70 –4.80

0 0.15 0.19 +0.12 –0.06 –0.35 –0.73 –1.20 –1.82 –2.55 –3.35

0 0.16 0.24 0.22 0.17 0.00 –0.20 –0.50 –0.90 –1.40 –1.90

0 0.17 0.27 0.30 0.29 0.21 +0.06 –0.15 –0.43 –0.77 –1.17

0 0.17 0.29 0.35 0.36 0.32 0.21 +0.06 –0.15 –0.42 –0.75

When a stream moves in the passage of a side branch past the free surface formed when Qs/Qc = 0, some loss of energy always takes place, therefore, in real conditions at Qs/Qc = 0 the coefficient ζc.s is not equal to zero. Here and in subsequent tables ζc.s = 0 is obtained by calculation from the formulas given.

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Handbook of Hydraulic Resistance, 4th Edition

Converging wye of the type Fs + Fst > Fc; Fst = Fc; α = 45o 31,43

Diagram 7.2

Values of ζc.′ s Qs Qc

Side branch ⎡ ⎛ Qs Fc ⎞ ζc.s * = A ⎢1 + ⎜ ⎟ 2 2 ρwc ⁄ ⎣ ⎝ Qc Fs ⎠ ∆ps

2

Qs ⎞ 2 Fc ⎛ Qs ⎞ 2⎤ ⎛ ⎥ = Aζc′.s − 2 ⎜1 − ⎟ − 1.41 Fs ⎜⎝ Qc⎟⎠ ⎦ Qc ⎠ ⎝

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fs ⁄ Fc 0.1 –1.00 0.24 3.15 8.00 14.0 21.9 31.6 42.9 55.9 70.6 86.9

0.2 –1.00 –0.45 0.54 1.64 3.15 5.00 6.90 9.20 12.4 15.4 18.9

0.3 –1.00 –0.56 –0.02 0.60 1.30 2.10 2.97 3.90 4.90 6.20 7.40

0.4 –1.00 –0.59 –0.17 0.30 0.72 1.18 1.65 2.15 2.66 3.20 3.71

0.6 –1.00 –0.61 –0.26 0.08 0.35 0.60 0.85 1.02 1.20 1.30 1.42

0.8 –1.00 –0.62 –0.28 0 0.25 0.45 0.60 0.70 0.79 0.80 0.80

1.0 –1.00 –0.62 –0.29 –0.03 0.21 0.40 0.53 0.60 0.66 0.64 0.59

0.6 0 0.17 0.27 0.28 0.26 0.16 –0.04 –0.25 –0.55 –0.88 –1.35

0.8 0 0.17 0.29 0.32 0.36 0.30 0.20 0.08 –0.17 –0.40 –0.77

1.0 0 0.17 0.31 0.40 0.41 0.40 0.33 0.25 0.06 –0.18 –0.42

′ , see the table and curves ζ ′ = f(Q ⁄ Q ) for ζc.s c.s s c

at different Fs ⁄ Fc (graph a); A = f(Fs ⁄ Fc, Qs ⁄ Qc), see Table 7.1;

ζs *

∆ps ρw2s ⁄ 2

=

ζc.s [(Qs ⁄ Qc)(Fc ⁄ Fs)]2

Straight passage

ζc.st *

∆pst

Q⎞ ⎛ = 1 − ⎜1 − s ⎟ Q ρwc ⁄ 2 c⎠ ⎝

2

2

− 1.41

Fc Fs

⎛ Qs ⎞ ⎜Q ⎟ ⎝ c⎠

2

see the table and curves ζc.st = f(Qs ⁄ Qc) at different Fs ⁄ Fc (graph b);

ζst *

∆pst ρw2st ⁄ 2

=

ζc.st (1 − Qs ⁄ Qc)2

Values of ζc.st Qs Qc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fs ⁄ Fc 0.1 0 0.05 –0.20 –0.76 –1.56 –2.77 –4.30 –6.05 –8.10 –10.00 –13.20

0.2 0 0.12 0.17 –0.13 –0.50 –1.00 –1.70 –2.60 –3.56 –4.75 –6.10

0.3 0 0.14 0.22 0.08 –0.12 –0.49 –0.87 –1.40 –2.10 –2.80 –3.70

0.4 0 0.16 0.27 0.20 0.08 –0.13 –0.45 –0.85 –1.30 –1.90 –2.55

503

Merging of Flow Streams and Division into Flow Streams Converging wye of the type Fs + Fst > Fc; Fst = Fc; α = 60o 31,43

Diagram 7.3

Values of ζc.′′ s Qs Qc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Side branch ⎡ ⎛ Qs Fc ⎞ ζc.s * = A ⎢1 + ⎜ ⎟ 2 ρwc ⁄ 2 ⎣ ⎝ Qc Fs ⎠ ∆ps

2

Qs ⎞ ⎛ − 2 ⎜1 − ⎟ Q c⎠ ⎝

F ⎛Q ⎞ − c ⎜ s⎟ Fs ⎝ Qc ⎠

2

2

⎤ ⎥ = Aζc′.s ⎦

Fs ⁄ Fc 0.1 –1.00 0.26 3.35 8.20 14.7 23.0 33.1 44.9 58.5 73.9 91.0

0.2 0.3 –1.00 –1.00 –0.42 –0.54 0.55 0.03 1.85 0.75 3.50 1.55 5.50 2.40 7.90 3.50 10.0 4.60 13.7 5.80 17.2 7.65 21.0 9.70

0.4 –1.00 –0.58 –0.13 0.40 0.92 1.44 2.05 2.70 3.32 4.05 4.70

0.6 –1.00 –0.61 –0.23 0.10 0.45 0.78 1.08 1.40 1.64 1.92 2.11

0.8 –1.00 –0.62 –0.26 0 0.35 0.58 0.80 0.98 1.12 1.20 1.35

1.0 –1.00 –0.62 –0.26 –0.01 0.28 0.50 0.68 0.84 0.92 0.99 1.00

0.6 0.8 0 0 0.17 0.18 0.29 0.31 0.32 0.41 0.37 0.44 0.33 0.44 0.25 0.40 0.08 0.28 –0.11 0.16 –0.38 –0.08 –0.68 –0.28

1.0 0 0.18 0.32 0.42 0.48 0.50 0.48 0.42 0.32 0.18 0

A = f(Fs ⁄ Fc, Qs ⁄ Qc), see Table 7.1; for ζc′.s, see the table and curves ζc.s = f(Qs ⁄ Qc) at different Fs ⁄ Fc (graph a); ζs *

∆ps ρw2s ⁄ 2

=

ζc.s (QsFc ⁄ QcFs)2

.

Straight passage ζc.st *

∆pst 2

⎛

ρwc ⁄ 2

= 1 − ⎜1− ⎝

2

Qs ⎞ F ⎛Q ⎞ − c ⎜ s⎟ Qc ⎟⎠ Fs ⎝ Qc ⎠

2

Values of ζc.st see the table and curves ζc.st = f(Qs ⁄ Qc) at different Fs ⁄ Fc (graph b); ζst *

∆pst ρw2st ⁄ 2

=

ζc.st (1 − Qs ⁄ Qc)2

Qs Qc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fs ⁄ Fc 0.1 0 0.09 0 –0.40 –1.00 –1.75 –2.80 –4.00 –5.44 –7.20 –9.00

0.2 0 0.14 0.16 0.06 –0.16 –0.50 –0.95 –1.55 –2.24 –3.08 –4.00

0.3 0 0.16 0.23 0.22 0.11 –0.08 –0.35 –0.70 –1.17 –1.70 –2.30

0.4 0 0.17 0.26 0.30 0.24 0.13 –0.10 –0.30 –0.64 –1.02 –1.50

504

Handbook of Hydraulic Resistance, 4th Edition

Converging wye of the type Fs + Fst > Fc; Fst = Fc; α = 90o 31,43

Diagram 7.4

Side branch 2

⎡

2

Qs ⎞ ⎤ ⎛ Qs Fc ⎞ ⎛ − 2 ⎜1 − ⎟ ⎥ = Aζc′.s , ⎟ 2 ρwc ⁄ 2 ⎣ ⎝ Qc Fs ⎠ ⎝ Qc ⎠ ⎦ where ζc.s = f(Qs ⁄ Qc), see the table and graph a; A = f(Fs ⁄ Fc, Qs ⁄ Qc), see Table 7.1. ζc.s *

ζs *

∆ps ρw2s ⁄ 2

=

∆ps

= A ⎢1 + ⎜

ζc.s (QsFc ⁄ QcFs)2 ′′ Values of ζc.s

Qs Qc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fs ⁄ Fc 0.1 –1.00 0.40 3.80 9.20 16.3 25.2 36.7 42.9 64.9 82.0 101.0

0.2 –1.00 –0.37 0.72 2.27 4.30 6.75 9.70 13.0 16.9 21.2 26.0

0.3 –1.00 –0.51 0.17 1.00 2.06 3.23 4.70 6.30 7.92 9.70 11.90

0.4 –1.00 –0.54 –0.03 0.58 1.30 2.06 2.98 3.90 4.92 6.10 7.25

0.6 –1.00 –0.59 –0.17 0.27 0.75 1.20 1.68 2.20 2.70 3.20 3.80

0.8 –1.00 –0.60 –0.22 0.15 0.55 0.89 1.25 1.60 1.92 2.25 2.57

1.0 –1.00 –0.61 –0.30 –0.11 0.44 0.77 1.04 1.30 1.56 1.80 2.00

Qs ⁄ Qc

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ζc.st

0

0.16

0.27

0.38

0.46

0.53

0.57

0.59

0.60

0.59

0.55

Straight passage

ζc.st *

∆pst

Q

ρw2c ⁄ 2

+ 1.55 Qs

c

2

⎛ Qs ⎞ −⎜ ⎟ ⎝ Qc ⎠ see the table and curve ζc.st = f(Qs ⁄ Qc), which is virtually correct for all values of Fs ⁄ Fc (graph b); ζc.st *

∆pst ρw2c ⁄ 2

=

ζc.st

(1 − Qs ⁄ Qc)2

505

Merging of Flow Streams and Division into Flow Streams Converging wye of the type Fs + Fst = Fc; α = 15o 31,43

Diagram 7.5 Side branch

ζc.s *

2