FLOW RESISTANCE:
A DESIGN GUIDE FOR ENGINEERS
FLOW RESISTANCE
A DESIGN GUIDE FOR ENGINEERS
Erwin Fried
General Electric Company
I E. Idelchik
Industrial Research Institute, Moscow
Taylor & Francis
Publishers since 1798
USA
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FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
34567890 GPGP 98
This book was set in Times Roman by Hemisphere Publishing Corporation. The editor was Jerry A.
Orvedahl; the production supervisor was Peggy Rote; and the typesetter was Sandra F. Watts.
Cover design by Sharon M. DePass.
Library of Congress Cataloging-in-Publication Data
Idel'chik, I. E.
Flow resistance: a design guide for engineers 1 I.E. Idelchik :
editor, Erwin Fried.
p. cm.
"Based almost exclusively on material presented in the recently
published Handbook of hydraulic resistance, second edition, by I.E.
Idelchik, a translation from the RussianM--Pref.
Includes index.
1. Fluid dynamics. 2. Frictional resistance (Hydrodynamics)
I. Fried, Erwin. 11. Idel'chik, I. E. Spavochnik po
gidravlicheskim soprotivleniiam 111. Title.
TA357.133 1989
620.1 '064--dc20
ISBN 1-56032-487-2 (paperback)
89-7569
CIP
CONTENTS
Preface
Nomenclature
Useful Conversions of Units
1 General Information
1- 1 General Guidelines
General References
2 Flow in Straight Tubes and Conduits: Friction Coefficients and
Roughness
2-1 General Guidelines
2-2 Diagrams of Friction Coefficients
References
3 Flow at the Entrance Into Tubes and Conduits: Resistance Coefficients of
Inlet Sections
3-1 General Guidelines
3-2 Diagrams of Resistance Coefficients
References
4 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 General Guidelines
4-2 Diagrams of Resistance Coefficients
References
vii
ix
...
Xlll
vi CONTENTS
5 Flow with a Smooth Change in Velocity: Resistance Coefficients of
Diffusers and Converging and other Transition Sections
5-1 General Guidelines
5-2 Diagrams of the Resistance Coefficients
References
6 Flow with Changes of the Stream Direction: Resistance Coefficients of
Curved Segments-Elbows and Bends
6-1 General Guidelines
6-2 Diagrams of the Resistance Coefficients
References
7 Merging of Flow Streams and Division into Flow Streams: Resistance
Coefficients of Wyes, Tees, and Manifolds
7-1 General Guidelines
7-2 Diagrams of the Resistance Coefficients
References
8 Flow through Barriers Uniformly Distributed over the Channel Cross
Section: Resistance Coefficients of Grids, Screens, Porous Layers, and
Packings
8- 1 General Guidelines
8-2 Diagrams of the Resistance Coefficients
References
9 Flow through Pipe Fittings and Labyrinth Seals: Resistance Coefficients
of Throttling Devices, Valves, Plugs, and Labyrinth Seals
9- 1 General Guidelines
9-2 Diagrams of the Resistance Coefficients
References
10 Flow Past Obstructions in a Tube: Resistance Coefficients of Sections
with Protuberances, Trusses, Girders, and other Shapes
10-1 General Guidelines
10-2 Diagrams of the Resistance Coefficients
References
11 Flow at the Exit from Tubes and Channels: Resistance Coefficients of
Exit Sections
1 1- 1 General Guidelines
11-2 Diagrams of the Resistance Coefficients
References
PREFACE
This book provides practical, applications-oriented data necessary for the design and
evaluation of internal fluid system pressure losses. It was prepared for the practicing
engineer, consultant, or designer who understands engineering and fluid flow fundamentals, but who needs an easy-to-use compilation of flow resistance coefficients in graphical
or tabular form without the distraction of voluminous theory and text. It is based almost
exclusively on material presented in the recently published Handbook of Hydraulic Resistance, Second Edition, by I. E. Idelchik, a translation from the Russian, which contains
the most extensive compilation of pressure loss coefficients currently available in one
volume.
The material in this book has been arranged in a convenient guidebook format so that
it can be applied easily. The extensive coverage becomes self-evident when one reviews
the hundreds of illustrations of flow passages and flow configurations. Most of these are
sufficiently basic as to allow application to any shape of flow passage encountered in
engineering practice. Each of the illustrations and flow coefficient graphs shows its limits
of applicability. Source references are also shown, to allow for further data verification,
if desired.
In any compilation of empirical data, the accuracy decreases with increasing complexity of the component, due to analysis of experimental uncertainties. This book is no
exception. Thus, a good rule to follow is to check more than one source, if possible.
Since this guidebook is based on a Russian sourcebook, the symbols and nomenclature differ somewhat from U.S. practice. This, however, should provide no impediment
to using the material, because the pictorial representations are quite clear and easy to
follow. The nondimensionality of the pressure loss coefficients and the governing parameters allow their use in any suitable system of units.
Any work of this nature is subject to editorial or translation errors as well as data
viii PREFACE
reporting errors. The publisher and I would be most grateful to the readers and users of
this book for information on such items.
I would like to express my gratitude to Professor I. E. Idelchik, who passed away in
1987 after spending a lifetime in the theoretical and experimental investigation of fluid
mechanics. For myself and many of my colleagues in the engineering profession, his
name is synonymous with the concept of hydraulic resistance.
Erwin Fried
NOMENCLATURE
Symbol
Name of quantity
a
a
speed of sound
critical speed of sound
sides of a rectangle
specific heats of gases at constant pressure
and constant volume, respectively
coefficient of drag
cross-section diameters
hydraulic or equivalent diameter
(4 X hydraulic radius)
cross-sectional areas
area ratio of a grid, orifice, perforated plate, etc.
mass flow rate of liquid (gas)
gravitational acceleration
height
specific heat ratio
length of flow segment, depth of channel, or
thickness of orifice
Mach number
coefficient of momentum (Boussinesq coefficient)
wetting intensity
exponent
coefficient of kinetic energy
power
area ratio (degree of enlargement or reduction
of cross section); polytropic exponent;
number of elements
static pressure
total pressure or flow stagnation pressure
a, b
c p and c ,
Abridged
notation
in SI units
x NOMENCLATURE
Symbol
Name of quantity
Pex
excess pressure
overall pressure difference
drag force
volumetric flow rate
gas constant
hydraulic radius (4 Dh)
radii of cross sections of a circular pipe or curved
pipe length
Reynolds number
spacing (distance between rods in a bundle
of pipes, between grid holes, etc.) length
of a free jet
surface area
frontal area of a body in a flow
thermodynamic temperature
thermodynamic flow stagnation temperature
(total temperature)
internal energy
specific volume; side discharge (inflow)
velocity
stream velocity
longitudinally fluctuating stream velocity
dust content
dust capacity
central angle of divergence or convergence;
angle of a wye or tee branching; angle of
stream incidence
angle of turning (of a branch, elbow); angle
of valve opening
thickness of a wall, boundary layer, or wall
layer; height of joint
equivalent uniform roughness of walls
mean height of wall roughness protuberances
relative roughness of walls
4'
Pdr
Q
R
Rh
R
7 ( p in US literature)
rln
coefficient of jet contraction
porosity (void fraction)
degree of turbulence
coefficient of fluid resistance (pressure loss
coefficient), K in the US literature
coefficient of local fluid resistance
coefficient of friction resistance of the segment
of length I
dynamic viscosity
cleaning coefficient
Abridged
notation
in SI units
Jlkg
m2/kg;
mls
mls
mls
glm2
kg/m2
degrees
NOMENCLATURE xi
Symbol
Name of quantity
= Sfrl(llDh)
X = f i n US literature
X = w/a,
P
Y
P
n
(b
friction coefficient [friction resistance of
the segment of relative unit length
(l/D, = 11; i.e., friction factor
friction factor
relative (reduced) stream velocity
discharge coefficient: mass concentration
of suspended particles in flow
kinematic viscosity
density of liquid (gas)
cross-sectional (wetted) perimeter
velocity coefficient
Abridged
notation
in SI units
mZ/s
kg/m2
m
-
SUBSCRIPTS
Subscripts listed for the quantities F, f, D, d, II, a, b, w, p, Q, and p refer to the following
cross sections or pipe segments:
con
or
gr
br, st, ch
out
m
governing cross section or minimum area
larger cross section in the case of expansion or contraction of the flow segment
larger cross section after equalization of the stream velocity
intermediate cross section of curved channel (elbow, branch) or the working
chamber of the apparatus
contracted jet section at the discharge from an orifice (nozzle)
orifice or a single hole in the perforated plate or screen
front of the perforated plate, screen, orifice
side branch, straight passage, and common channel of a wye or tee, respectively
outlet
velocity at infinity
Subscripts 0,1, 2, k, and g at 1 refer, respectively, to the straight inlet, straight outlet,
intermediate (for a curved channel), and diffuser pipe lengths, I.
Subscripts at Ap and 5 refer to the following forms of the fluid resistances:
I
fr
ov
tot
out
int
exp
sh
b and st
local
friction
overall
total resistance of an impedance in the network
total resistance of a diffuser or a branch at the outlet from the network
internal resistance of a diffuser
resistance to flow expansion in a diffuser
shock resistance at sudden enlargement of the cross section
resistance of a branch and straight passage of a wye or tee (for the resistance coefficients reduced to the velocity in respective branch pipes)
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
USEFUL CONVERSIONS OF UNITS
Physical quantity
Given in ----, Multiplied by
Gives
r---- Divided by
Length
ft
in
mil
yard
mile (mi)
km
Area
ft2
in2
acre
Volume
ft3
U.S. gal
U.S. gal
L (liter)
Brit gal
U.S. gal
barrel (U.S. pet.)
barrel (U.S. pet.)
Velocity
ftlP
mls
ftlmin
mi/h
km/h
knots
Mass
Ibm
kg
metric ton
ton (2 000 Ib,)
Force
Ibf
Ibf
kgf
kgf
dyne
Amount of substance
ibm-mol
g-mol
kg-moi
mol
Mass flow rate
lbmlh
kgls
Ibmls
Ibm/min
-
0.3048
25.4 (exact)
0.0254
0.9144
1 609.3
0.621388
Gives
Given in
m
mm
mm
m
m
mi
Approximate
or useful
relationship
3tftzzl m
1 in 25 mm
--
9 m2
100 ft"
1 in2 650 mm2
--
m3
m3
liter (L)
U.S. gal
m3
ft3
m3
U.S. gal
35 ft3 1 1 m3
260 gal = 1 m3
1 gal--3$L
1 L =- 0.26 gal
-
--
1 Ib,
.45 kg
1 kg 2.2 Ib,
metric ton = lo3 kg
4.44822
0.45359
2.2046
9.80665
0.00001 (exact)
N = kg m/sZ
kgf
Ibf
N
N
kmol
mol
kmol
kmol
Reprinted from International System of Units (SI), J. Taborek, in Heat Exchanger Design Handbook, pp. xxvii-xxix, Hemisphere, Washington, D.C., 1984.
xiv
- -
USEFUL CONVERSIONS O F UNITS
Physical quantity
Volume flow rate
Given in
Gives
U.S. gal/min
U.S. bbl/day
U.S. bbl/day
ft3/s
ft3lmin
Multiplied by
Divided by
Gives
4--- Given in
6.309 x lo-*
0.15899
1.84 x
0.02832
0.000472
Mass velocity
(mass flux)
Energy (work)
(heat)
Approximate
or useful
relationship
m3/s
m3/day
m3/s
m31s
m3/s
kg/s m'
Ibm/h ft'
~ t u ~
Btu
Btu
kcal
ft Ibf
Wh
1 055.056
0.2520
778.28
4 186.8
1.3558
3 600
I
I
Power
Btu/h
W
kcal/h
ft Ibf/s
hp (metric)
Btulh
tons refrig.
0.2931
3.41 18
1.163
1.3558
735.5
0.2520
3 516.9
W = l/s
Btu/h
W
W
W
kcal/h
W
Heat flux
Btu/h ft'
W/m2
kcal/cm2 s
3.1546
0.317
41.868
W/m2
Btu/h ft'
W/ma
Heat transfer coefficient
Btu/h ftl 'I:
W/ml K
kcal/cmZ s "C
5.6784
0.1761
41.868
W/m2 K
Btu/h ft'
W/m2 K
Heat transfer resistance
(Btulh ft' OF)-'
(Wlm' K)-I
Pressure
Ibf/in2 (psi)
kPa
bar
Ibf/ftz
mm Hg (torr)
in Hg
mmH,O
inH,O
at (kgf/cm2)
atm (normal)
0.1761
5.6784
J=Nm=Ws
kcal
ft Ibf
1 B t u z 1 0001
1 kcai 4 Btu
--
J
O F
1 000 Btu/h ft' O F =
5 600 W/ml K
(W/m2 K)-'
(Btu/h ft' OF)-'
0.001 (Btu/h ft2 OF)-' z
0.000 18 (W/m2 K)-'
kN/mz = kPa
psi
kPa
kPa
kPa
kPa
Pa
Pa
kPa
kPa
1 psi 7 kPa
14.5 psi =. 100 kPa
--
1 000 kPa = 1 MPa
150 psi
-
atm = 760 mmHg
Mass flux
Physical and Transport Properties
Thermal conductivity
Btu/ft h 'F
W/m K
kcal/m h "C
Density
Ibm/ft3
kg/m3
lbm/U.S. gal
Specific heat capacity
Btu/lbm "F
kcallkg ' C
Enthalpy
Btu/lbm
kcal/k&,
Dynamic (absolute)
viscosity
centipoise (cP)
poise (P)
CP
CP
Ib,/ft h
lbm/ft h
CP
Ibm/ft s
steel =- 50 W/m K
water (20°C) 4: 0.6 W/m K
air (STP) * 24 mW/m K
water (10O0C), 0.31 cP
air (lOO°C), 0.021 CP
Physical quantity
Kinematic vtscosity
Given in
Gives
- Multiplied by
Divided by
USEFUL CONVERSIONS OF UNITS xv
* Gives
Given in
stoke (St), cm2s
cent~stoke(cSt)
ft2/s
0.0001
0.092903
m2 1s
mz/s
m2 1s
Diffusivity
ft2/s
0.092903
m' /s
Thermal diffusivity
ml/h
ft'ls
ft2/h
Surface tension
dynelcm
dynejcm
lbf/ft
Temperature relations: "C = $ ["F - 32)
"F=pcC)+32
Miscellaneous:
= (OF + 40)g - 40
" F = ("C + 40); - 40
OC
Acceleration of gravity (standard):
Gas constant:
Stefan-Boltzmann constant:
AT(OC)= ;aT(OF)
aT("F) = $ AT("C)
Approximate
or useful
relationship
K = "C + 273.15
R = " F + 459.67
g = 9.806 65 m/s2
R = 8 314.3 m N/K kmol
5.669 7 X
Wlm' K4
1.714 x lo-' Btu/ft2 h R4
i ~ v e though
n
the abbreviations s and h were introduced only with the SI, they are used here throughout for consistency.
Note: the calorie and Btu are based on theInternationalStandard Table values. The thermochemical calorie equals 4.184 J (exact) and
is used in some older texts.
CHAPTER
ONE
GENERAL INFORMATION
1-1 GENERAL GUIDELINES
This design guide is intended to enable the practicing engineer to analyze and evaluate the
flow resistance or pressure loss coefficient for most flow passage types, devices, and
components. In keeping with the assumption that the user of this design guide has some
understanding of engineering fundamentals, only that material necessary to use the charts
and graphs presented herein will be provided. Should the user want to delve into the subject
in greater depth, the source book for this design guide, Handbook of Hydraulic Resistance
by I. E . Idelchik, published by Hemisphere Publishing Corporation, 1986, should be consulted. This source book will be cited throughout this text as "Idelchik." All references
cited in Idelchik are shown herein, to allow the user access to the original sources of the
data. For basic fluid mechanics information the user is referred to any convenient or
familiar fluid mechanics text. For completeness and convenience, a list of recent fluid
mechanics texts is included in the bibliography of this chapter.
Following are some general guidelines to get the most usefulness from this book.
1. All sketches, diagrams, and graphs are self-explanatory, with flow direction, areas,
and other features indicated.
2. Particular attention should be paid to the limits of applicability shown on each of the
tables and graphs. These are usually expressed in terms of Reynolds number or in
terms of geometric parameters.
3. It is assumed that the inlet and exit conditions are ideal, i.e., there are no flow profile
distortions, unless otherwise indicated. There exists only a very limited amount of data
on the effect of the inlet flow distortion or inlet swirl for most flow devices. Since each
2 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
4.
5.
6.
7.
8.
application involving distorted flow is unique, it is recommended that experimental
methods be considered when such conditions exist and pressure loss is of importance.
Unless otherwise indicated, the data shown herein apply to Newtonian fluids considered homogeneous, incompressible, and involving neither work nor energy addition.
The pipe or duct walls are considered rigid.
For graphs dealing with components involving a change in area, particular attention
should be paid to the graph, whether the value of the pressure loss coefficient is based
on the inlet, minimum, or exit area.
The nondimensionality of the parameters of most of the graphs allows their use in any
convenient system of units.
The basic reference data given in this book are the static pressure loss coefficients, or
K-factor as used in the US literature. This term can be considered the overall static
pressure loss coefficient for the component of interest. It includes the nonrecoverable
losses within the component as well as the frictional and the recoverable losses. The
frictional losses are usually considered negligible when compared to the nonrecoverable losses and generally are neglected unless stated otherwise in the graphs.
If one considers how the pressure loss coefficient 3- is evaluated experimentally, this
becomes evident. It is the measured static pressure drop Ap, divided by the dynamic or
velocity head, p 4 , for the component. Thus,
9. The basic pressure loss equation to be used with the data given in this book is
Ap
=
~2, in consistent units
3- 2
10. The overall static pressure drop is considered a positive quantity if the sign convention
used in this book is followed. Therefore, a static pressure rise, such as in a diffuser,
will show up as negative quantity.
11. The effect of Reynolds number on the pressure loss coefficient is most pronounced at
low values (Re < 10'). At higher values of Re it can be assumed as independent of Re,
unless otherwise stated.
12. When there is no indication of the Reynolds number at which the value of 3- was
obtained, it may be assumed that the given value of !:is virtually independent of Re.
However, in the case of purely laminar flow (Re < 2 . lo3),the value of 3- is only an
approximation.
13. For the determination of Reynolds numbers in noncircular ducts, an equivalent or
hydraulic diameter must be used. It is defined as four times the cross-sectional flow
area divided by the wetted perimeter n, with both measured in a direction perpendicular to the flow. If as usual, the fluid fills the entire cross-section of the duct, this
definition is equivalent to the relation
14. For a few simple configurations we have the following hydraulic diameter D,,
GENERAL INFORMATION 3
Circle of Diameter D
Square with side a
Rectangle with sides a, b
Parallel plates separated a distance a
Annular duct of cylinders, Dl, D2
D
a
2abl(a + b)
2a
01- D2
15. Property data, such as viscosity, density, etc., can be obtained from any consistent
source available to the user. It is purposely omitted here to keep the size of this book to
a minimum.
16. For gases and steam, the variation of density is sometimes very important. If the
calculation shows that the resulting pressure drop is such as to change the density, then
the piping system can be subdivided and the calculation can be done on a section-bysection basis. In that method, the exit conditions of one section become the inlet
conditions of the next section. For condensing steam the density can change quite
rapidly and the segmentation method becomes important. It should be noted that the
segmentation method is only an approximation, but with judicious selection of segments it can provide acceptable engineering results.
17. Most values of the pressure loss coefficient shown in this book are valid for Mach
numbers of less than 0.3 unless otherwise stated.
18. The value of the overall pressure losses in a piping network can be evaluated by use of
electrical resistance network methods or by use of one of the several computer programs currently available. This book will provide the necessary pressure loss coefficients 3., or K-factors.
19. In a piping or ducting network, the pressure losses in each segment can be calculated as
if the others did not exist and the pressure losses added. However, if the components
are close to one another, the exit conditions of one may affect the entry conditions of
the following component. Engineering judgement must be applied in such a case.
20. When a system is analyzed for pressure losses, it is often convenient to use the entry or
similar dimension as the reference dimension, because the loss coefficient 5- depends on
the velocity, which is a function of the cross section.
In general, with variable density along the flow, the resistance coefficient 3., based
on the velocity in any given section (area F,), is calculated for another section (area F2)
using the relation
For the case of no change in density, the usual case, this is simplified and becomes a
most useful relation, which can be used to normalize any system.
21. A few comments need to be made about the calculation of friction losses in a system.
When the straight runs of pipe are significant in relation to the flow obstructions or
components, then it is advisable to calculate the friction losses for these straight runs
and add them to the other section losses. The friction loss, or K,, can be treated like
another loss coefficient, or K-factor, by use of the following relation.
r
rfr
4 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
In American practice this becomes
Ap
=
(C K,
+ C K;) 2
where Kfr = 3;, = f(1lD)is the friction loss coefficient, and Ki = 5;. is the static
pressure loss coefficient from this book.
22. When friction factors are required for solution of an overall system, the graphs and
tables allow the use of any friction factor sources familiar to the user, such as Moody
or Fanning charts. It should be noted that the value of the Moody friction factor is 4
times that of the Fanning friction factor. This is due to the way the hydraulic diameter
is defined. A convenient way to tell which of these two friction factors is given is by
inspection of the laminar friction factor. Iff is 1 6 / ~ e ,then it is Fanning. If it is 6 4 / ~ e , it
is Moody.
GENERAL REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Idelchik, I. E., Handbook of Hydraulic Resistance, 2nd Ed., Hemisphere Publ. Corp., 1986.
White, F. M., Fluid Mechanics, 2nd Ed., McGraw-Hill, New York, 1985.
Streeter, V. L., and Wylie, E. B., Fluid Mechanics, 8th Ed., McGraw-Hill, New York, 1985.
Blevins, R. D., Applied Fluid Dynamics Handbook, Van Nostrand Renhold, New York, 1984.
Miller, D. S., I n t e m l Flow Systems, British Hydromech. Research Assn., Cranfield, U.K., 1978.
Flow of Fluids Through 'Valves, Finings, and Pipe, Crane Co. Technical Paper No. 410, Chicago, 1957.
Ward-Smith, A. J., I n t e m l Fluid Flow, Clarendon Press, Oxford, U.K., 1980.
Olson, R. M., Essentials of Engineering FZuid Mechanics, 4th Ed., Harper & Row, New York, 1980.
Panton, R. L., Incompressible Flow, Wiley, New York, 1984.
White, F. M., Viscous Fluid Flow, McGraw-Hill, New York, 1974.
Marks Mechanical Engineers Handbook, Ed. by Th. Baumeister, McGraw-Hill, New York, 1978.
Rouse, H., Elementary Mechanics of Fluids, Wiley, New York, 1946.
Binder, R. C., Fluid Mechanics, 2nd Ed., Prentice Hall, 1950.
Kays, W. M., and London, A. L., Compact Heat Exchangers, McGraw-Hill, New York, 1964.
CHAPTER
TWO
FLOW IN STRAIGHT TUBES AND CONDUITS
Friction Coefficients and Roughness
2-1 GENERAL GUIDELINES
1. The pressure losses along a straight tube of constant cross section are calculated from
the Darcy-Weisbach equation:
where sois the area of the friction surface. Note that X corresponds to the friction factor f in
US literature. This book uses the Moody [I271 friction factor.
2. The hydraulic diameter D, was discussed in Chapter 1 briefly. It is equal to the pipe
diameter for circular pipe and varies for other shapes of ducts as shown in the tables and
diagrams of this chapter. It should be noted that the concept of hydraulic diameter will
provide values of pressure losses of acceptable engineering accuracy for most shapes,
where there are no secondary flows of significance. Triangular ducts, and shapes where
laminar flows can persist locally, are the exception. Empirically derived values are best for
such cases, where available.
For parallel plates and flat ducts it has been shown that the concept of effective laminar
diameter, D,, [151, 1521, is a method that will yield slightly better pressure loss predic-
6 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
tions. Typical values of Deff/Dhfor concentric annuli are shown below.
Inside Diam.
Outside Diam.
0
0.01
0.1
1.o
Reference [I521 provides a useful discussion of the above concept. Also see Diagram
2-7.
3. In laminar flow, due to the overriding effects of viscosity, even flow past surface
asperities appears to be smooth. Therefore the roughness of the wall, unless it is very
significant, does not affect the flow resistance. Under these conditions of flow the friction
coefficient is always a function of the Reynolds number alone.
4. As the Reynolds number increases, the inertia forces, which are proportional to the
velocity squared, begin to dominate. Turbulent motion is then initiated, which is characterized by the development of transverse velocity components giving rise to agitation of the
fluid throughout the entire stream and to momentum exchange between randomly moving
masses of fluid. All this causes a significant increase in the resistance to motion in turbulent
flow as compared with the case for laminar flow.
When the surface of the walls is rough, separation occurs in the flow past roughness
asperities and the resistance coefficient becomes a function not only of the Reynolds number but also of the relative roughness
-
A,
=
E
- , or - in U.S. practice
D h
Dh
5. Pipes and channels can be either smooth or rough, with the roughness being either
uniform or nonuniform. These two types of roughness differ according to the shape of such
protuberances, their dimensions, the spaces between them, etc. The majority of commercial
pipes and tubes have nonuniform roughness.
6. The averaged height A, of asperities, in terms of the absolute length units, is called
the absolute geometric roughness. The ratio of the average height of asperities to the tube
diameter, that is, i,= A,/Dh or dDh, is called the relative roughness. In view of the fact
that the geometric characteristics of the roughness cannot adequately determine the flow
resistance of the tube, the concept of a derived hydraulic or equivalent roughness A is
introduced, which is determined by measuring the resistance.
7. Although the resistance coefficient for smooth tubes should decrease with increasing Re, rough tubes show an increase in the coefficient A with increase of this number with
constant geometric roughness. This is explained by the effect of a viscous sublayer. When
the thickness of the viscous sublayer is larger than roughness protuberances (6 > A, Fig. 2la), the latter are entirely covered with this layer. At low velocities, typical of a laminar
Figure 2-1 Flow past roughness asperities for different modes of flow: (a) 6 > A; (b) 6 < A.
FLOW IN STRAIGHT TUBES AND CONDUITS 7
Figure 2-2 Dependence of the resistance coefficient X on Re for tubes with uniform grain roughness [132].
sublayer, the fluid moves smoothly past surface irregularities and they have no effect on the
character of the flow. In this case X decreases with a rise in Re.
8. 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
(6 < A, Fig. 2-lb). The asperities enhance the formation of vortices and hence increase the
pressure losses, which result in the rise of X 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.
9. The equivalent roughness A 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.
10. The dependence of the frictional resistance coefficient X on Re and &, as determined by the experiments of Nikuradse [56] for a stabilized flow in tubes with uniform
roughnesst (Fig. 2-2), suggests the existence of three principal regimes of flow.
11. The first regime, called the laminar regime, involves small values of the Reynolds
number (up to Re = 2000) and is characterized by X being independent of roughness. From
the Hagen-Poiseuille law [I171
+A form of artificial sand roughness is meant here, as obtained by Nikuradse. The curves for other forms can
differ somewhat [loo].
8 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
12. The second regime, called the transition regime, consists of three segments of the
resistance curves for uniform roughness:
The segment related to the transition region between laminar and turbulent flow (approximately within Re = 2000-4000). The resistance coefficient X in this region increases
rapidly with Re. However, this coefficient remains independent of the value of relative
roughness.
The segment for which the resistance curves of tubes with different roughness coincide
with the Blasius curve for smooth tubes
According to this equation, the resistance law is valid for the lower range of Reynolds
numbers. For the larger values the relative roughness dominates.
The segment for which the resistance curves of tubes with different roughness diverge from
each other, departing from the straight line obtained from Eq. (2-4). Here, the resistance
coefficients for certain ranges of Re increase with increasing relative roughness.
13. The third regime is called turbulent or square-law region. It is characterized by the
resistance coefficients for each value of the relative roughness becoming constant, independent of Re.
14. For a stabilized flow and the region of purely turbulent flow, the friction coefficient X of commercial circular tubes (with nonuniform roughness of walls), except for
special cases for which the values of A are given separately, can be determined from the
curves of Diagram 2-4 plotted on the basis of the Colebrook-White formula [114]:
or for engineering calculations, from Altshul's approximate formula [6]
15. Using Colebrook's formula [114], Moody [I271 developed the now widely used
Moody Chart (Fig. 2-3), which covers laminar as well as turbulent flow and relative
roughness. It can be used for circular as well as noncircular ducts, provided the proper
hydraulic diameter is used. The Moody chart may be used in preference to Diagrams 2-1
through 2-4.
16. The resistance coefficient of noncircular tubes depends on the shape of the cross
section. It can be expressed in terms of the resistance coefficient of circular tubes through
the use of a correction factor which allows for the effect of the shape of the tube cross
section:
a
3
8 3 %
SS3NH9nOtl 3AIlWl3ki
-88 B
s s s s
s
V)
5888 8 8 8
ssss s s s
cow*
N
-
0
0-
0
8
0
00
10 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
-
-
Figure 2-4 Dependence of the friction coefficient X on Re for a short starting length ([,,/Do 2) with smooth
walls: (1) test section is installed immediately downstream of a smooth inlet (lo/Do 0); (2) upstream straight
section of length lo/Do = 0.4 is installed between the smooth inlet and the test section; (3) relative length of the
upstream section is lo/Do = 4.3; (4) three rows of paper bands are passed on the inner surface at the end of the
upstream section of length lo/Do = 3.4; ( 5 ) the resistance curve is according to Blasius; (6) Hagen-Poiseuille
curve.
where X is the friction coefficient of circular tubes at the same Reynolds numbers Re =
w,,llh/v = w,,DO/v;&on., is X for noncircular tubes; and k,,., is the correction factor allowing for the effect of tube cross-sectional shape (see Diagram 2-6).
17. When a fluid enters a straight duct, the newly formed boundary layer is quite thin
and requires some distance before the boundary layer thickens and the flow becomes fully
developed. This entrance length or nonstabilized flow region depends on the inlet shape,
turbulence level, pre-existing conditions, etc. and results in higher flow resistance than is
the case in the developed or stabilized region. This entrance length is usually expressed in
terms of duct diameters.
18. 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 Lon,,for short lengths
as well (see Fig. 2-4). Therefore, at relatively small Reynolds numbers (Re,, < Re < 5 x
10~-10~)
for short tubes in real devices (in which the flow at the inlet is very much perturbed as a rule), one may, with a certain factor of safety, assume that &,,, = A, until more
detailed data are obtained. For a nonstabilized turbulent flow at larger values of Re
where knon,, > 1.0 is the correction factor which compensates for the nonstabilized behavior of the flow and which is determined from the curve know,,= f(x/Dh) of Diagram 2-16.
FLOW IN STRAIGHT TUBES AND CONDUITS 11
For nonstabilized laminar flow, the friction coefficient of the starting length is calculated from Eq. (2-7), in which n,,,,, a function of the parameter Re (x/D,),
is determined
by use of graph b of Diagram 2-16.
Table 2-1 Equivalent roughness of tubes and channels
Group
Type of tubes, material
State of tube surface and conditions of use
A. Metal tubes
I
Seamless tubes made
Commercially smooth [122,129, 1391
from brass, copper, lead
Aluminum tubes
The same
I1
Seamless steel tubes
(commercial)
I11
Welded steel tubes
(See foomote on p. 13.)
1)
2)
3)
4)
New, unused (22,99,127]
Cleaned after many years of use [I291
Bituminized [I201
Superheated steam pipes of heating systems
and water pipes of heating systems with
deaeration and chemical treatment of
running water [53]
5) After one year of use in gas pipelines [22]
6) After several years of use as tubing in gas
wells under various conditions [4]
7) After several years of use as casings in gas
wells under different conditions [4]
8) 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 [53]
9) Pipelines of water heating systems independent of the source of supply [ 131
10) Oil pipelines for intermediate operating
conditions 1531
11) Moderately corroded [139]
12) Small depositions of scale [I391
13) Steam pipelines operating periodically and
condensate pipes with the open system of
condensate [53]
14) Compressed air pipes from piston- and
turbocompressors [53]
15) After several years of operation under
different conditions (corroded or with
small amount of scale) [4,84,129]
16) 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%) [53]
17) Water pipelines previously used [99]
18) With large depositions of scale [129]
19) Poor condition; nonuniform overlapping of
joints [I191
1) New or old, but in good condition; welded
or riveted joints [122, 1391
2) New, bituminized [I281
A, mm
12 FLOW RESISTANCE: A DESIGN GUIDE FOR ENGINEERS
TPable 2-1 Equivalent roughness of tubes and channels (Continued)
Group
Type of tubes, material
State of tube surface and conditions of use
A, mm
A. Metal tubes (Cont.)
3) Used previously, corroded, bitumen
-0.10
partially dissolved [I391
4) Used previously, uniformly corroded [139]
-0.15
5) Without noticeable unevenness at joints
0.3-0.4
[ 1391 ;lacquered on the inside layer (10 mm
thick); adequate state of surface [125]
-0.5
6) Gas mains after many years of use [139]
0.6-0.7
7) With simple or double transverse riveted
joints; lacquered 10 mm thick on the inside
or with no lacquer but not corroded [I221
8) Lacquered on the inside but rusted; soiled
when transporting water but not corroded I1221
9) Layered deposits; gas mains after 20 years
of use [I391
10) With double transverse riveted joints, not
corroded; soiled during transport of water
[99, 1391
11) Small deposits [ 1391
12) With double transverse riveted joints, heavily
corroded [l22]
13) Appreciable deposits [I391
14) Used for 25 years in municipal gas mains,
nonuniform deposits of resin and naphthalene [I391
15) Poor condition, nonuniform overlapping of
joints [122]
IV
Riveted steel tubes
1) Lateral and longitudinal riveting with one
0.3-0.4
line of rivets; 1 0 mm thick lacquered on the
inside; adequate state of the surface [122]
2) With double longitudinal riveting and simple
0.6-0.7
lateral riveting; 1 0 mm thick lacquered on the
inside, or without lacquer but not corroded [I221
3) With simple lateral and double longitudinal
1.2-1.3
riveting; from 1 0 to 20 mm thick lacquered or
torred on the inside [I221
4) With four to six longitudinal rows of rivets;
2.0
long period of use [122]
5) With four lateral and six longitudinal rows of
4.0
rivets; joints overlapped on the inside [I221
6) Very poor condition; uneven overlapping of
b5.0
joints [122]
V
Roofing steel sheets
1) Oiled
2) Not oiled
VI
Galvanized steel tubes
1) Bright galvanization; new [I391
2) Ordinary galvanization [ 1391
VII
Galvanized sheet steel
1) New [127]
2) Used previously [139]
VIII
Cast-iron tubes
1) New [114]
2) New, bituminized [139]
3) Asphalt-coated 0 2 7 1
(See foolnote on p. 13.)
0.07-0.10
0.1-0.15
FLOW IN STRAIGHT TUBES AND CONDUITS 13
'Pable 2-1 Equivalent roughness of tubes and channels (Continued)
Group
Type of tubes, material
State of tube surface and conditions of use
A, mm
-
A. Metal tubes (Cont.)
4)
5)
6)
7)
8)
9)
Water pipelines, used previously [99]
Used previously, corroded [I391
With deposits [127,139]
Appreciable deposits [129, 1391
Cleaned after use for many years [I391
Heavily corroded
B. Concrete, Cement, and Other Tubes and Conduits
Concrete tubes
Reinforced concrete
tubes
1) Good surface, plaster finish [I391
2) Average conditions [139]
3) Coarse (rough) surface [139]
[841
Asbestos-cement tubes
1) New [34]
2) Average [84]
Cement tubes
1) Smoothed [84]
2) Nonprocessed [84, 1291
3) Mortar at joints not smoothed [I221
Conduit with a cementmortar plaster
1) Good plaster made of pure cement with
smoothed joints; all asperities removed;
metal casing I1221
2) Steel-troweled [84]
Plaster over a metallic
screen
VII
Ceramic salt-glazed
conduits
VIII
Slag-concrete slabs
IX
Slag and alabasterfilling slabs
[I31
[I31
Carefully made slabs [13, 1141
- -
C. Wood, Plywood, and Glass Tubes
I
Wooden tubes
1)
2)
3)
4)
5)
I1
Plywood tubes
1) Of goodquality birch plywood with transverse grain [ 11
2) Of good-quality birch plywood with longitudinal grain [ 11
111
Glass tubes
Boards very thoroughly dressed
Boards well dressed
Boards undressed but well-fitted
Boards undressed [ 1391
Staved [84]
Pure glass [I271
a ~ e p e n d i n go n how long these were stored.
1.4
1.0-1.5
1.0-1.5
2.0-4.0
0.3-1.5
Up to 3.0
2.2 DIAGRAMS OF FRICTION COEFFIClENTS
Circular tube with smooth walls; stabilized flow
[6,118,135]
Diagram
2-1
1. Laminar regime (Re 4 2000):
A=
=
64
=/'(Re)
see graph a.
2. Transition regime (2000 4 Re < 4000):
A = f (Re)
see graph b.
3. Turbulent regime (4000 < Re < l o * ) :
0.3164
A =ReO.w
See graph c.
4. Turbulent regime (Re > 4000):
A=
1
(1.8 1g Re - 1.64)'
See graph
''
Diagram
Circular tube with walls of uniform roughness;
stabilized flow; Re > 2000 [56,132]
2-2
A=
Ap
( p w ~ / 2 ) ( 1 / ~ , ) - la,
A = f (Re)
for A see Table 2-1. At A <
& l i m ~ o , for
+ b,
1
lg (Re
6)
+ c, lg iil
see graph; the values of a, ,b , ,and c, are
given below:
the values of A, see Diagram 2-1, where ilim
= 17.85
Circular tube with walls of uniform roughness;
stabilized flow; Re > 2000 [56, 1321
Values of h
Values of h
Diagram
2-2
Circular tube with walls of nonuniform roughness; stabilized flow;
critical zone (Re, < Re < 4000) [66,69]
< Re < Re, ; A
1. Re,
2 0.007
(-
h = 4.4 Re-0-59Jexp -
< Re < Re,
2. Re,
A
0.,75>
h)
= f(Re,
+ A* = f(Re,
= (A2 - A*) exp {-[0.0017 (Re, -Re)]
at h
< 0.007,
at h
> 0.007, A* = A, - 0.0017 = 0.0758 - 0 0109 and
A* = h, = 0.032, and h, =
Re, = 2090
t ~ ) (krl
where for A, see Table 2-1.
Values of A
Re, = 1160
(iy'0635
B)
= 7.244 Re-0.643
%,
Re, = 754 exp
Diagram
2-3
;4 = A; =
0.145
Circular tube with walls of nonuniform roughness; stabilized flow;
critical zone (Reo < Re < 4000) [66, 691
Diagram
2-3
Values of h
-
-
-
-
-
-
Rex
-
A
2
2.2
to
2.4
1.4
Intermediate values of Re and h
2.6
2.8
22
3
3.2
2.6
3.4
3.0
3.6
3,4
3.8
4
R ~ X I O - ~
Diagram
2-4
Circular tube with walls of nonuniform roughness; stabilized flow;
Re > R q (for R q , see Diagram 2-3) [114]. Also see Fig. 2-3
A=
AP
(pw;/2)(//Dh)
or within the limits of
A = 0.11 (Z
-
1
[2 ig (2.51/Re Ji
= 0.00008-0.0125:
+325
see graph a
for A, see Table 2-1.
At A
< BlimDn,for A see Diagram 2-1; for slim see graph b as a function o f Re.
+ A/3.7)]
Diagram
2-4
Circular tube with walls of nonuniform roughness; stabilized flow;
Re > Re2 (for Rez, see Diagram 2-3) [114]. Also see Fig. 2-3
Values of h
A=-
Re
A
Dh
3 x lo3
Values of h
4 x lo3 6 x l o 3
lo4
2 X lo4 4 X lo4 6
X
lo4
los
2 X lo5
Circular tube with rough walls; stabilized flow; regime
of quadratic resistance law (Reli, > 5601s) [65, 1321
for A, see Table 2-1.
Diagram
2-5
Diagram
2-6
Tubes of rectangular, elliptical, and other types of cross section;
stabilized flow [56, 1051
where A is determined as for circular tubes from Diagrams 2-1
through 2-5
Shape of tube (conduit)
cross section and schematic
Correction factor k,,
( Laminar regime (Re < 2000, curve 1)
Rectangle:
Turbulent regime (Re > 2000, curve 2)
knon~=k,,,l.10
1.08 1.06
1.04 1.02 1.01 1.0
Trapezoid:
k,,,
is determined in approximately the same way as
for a rectangle
Diagram
2-6
Tubes of rectangular, elliptical, and other types of cross section;
stabilized flow [56, 1051
--
Shape of the tube (conduit)
cross section and schematic
-
Correction factor knon+
Circle with one or two recesses.
Star-shaped circle
-@-@-
knon.,=
-@+
-.
/
k,,,
= kSt = 1.0
/
ey
Laminar regime (Re i2000):
knOn-. = k1.1 =
8
+
($11
see graph b
Ellipse
3
- e
Dh
4a0bo
1.5(a,, + b , ) -
a
More precisely:
Dh =
a0
0.1 0.2 0.3
0.4
0.5
0.6
0.7
0.8 0.9
1.0
kell 1.21 1.16 1.11 1.08 1.05 1.03 1.02 1.01 1.01 1.0
kell
fl
naobo
0.983a0
+ 0.311b0 + 0.287bi/ao
Turbulent regime (Re > 2000); kell = 1.0
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6P'T
9.0
-
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1-0
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C(
'alp)
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0.1
o
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'alp
a m aas
remure?
JO
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S-z y 2 n 0 1 ~1-2 smBe!a
s a q q 1ern3q3
WOIJ
Yaj-uouy =
1
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IOJ
se hem aures ayl tq pau!urIalap s! Y alaym
ed
=
L-a
dv
ya"M
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OdP
[ L P I '9L ' £ 9 ' O P ' 6 1 ' S I ' P I ] MOI3 p a z ~ q e l s f s a q nEIn3Q3
l
L-2 weBela
v
N
Circular tubes; stabilized flow
[14, 15, 19,40,63, 76, 1471
Diagram
2-7
I
Shape of the tube (channel) cross section
Spiral fins
Dh = DO
-
a[
Eccentric annulus
?I,/
2(Tlnd)(d/Do
1 - d / ~ . (A -B) - nDo
Schematic