COLEBROOKONTURBULENT
133
FLOW IN PIPES.
Paper No. 5204. “ Turbulent Flow in Pipes, with particular reference to the Transition Region between the Smooth and Rough Pipe Laws.”
CYRIL FRANK COL~BROOK, Ph.D., B.Sc. (Eng.), Assoc. M. Inst. C.E. (Ordered by the Council to be published with written.dkcusswn.)l TABLE O F CONTENTS.
. . . . . . . . . Introduction . Theory of turbulent flow inpipes . . . . . . A new theoretical formula for flow in the transition region . Relation between Prandtl-von-Karman and exponential formulas. Analysis of experimental data on smooth pipes . . . . Galvanized,cast-, and wrought-ironpipes . . . . . Old pipes . . . . . . . . . . . Discussion and conclusions . . . . . . . . Appendix-Examplesillustratingthe use of design-Tables . .
. . . . . .
.
. .
.
. . . . . .
PAGE
133
137 139
141 143 145 153 154
. . .
155
INTRODUCTION. The problem offlow in pipes is one which has until recently defied theoretical analysis, owing to its complexity and the absence of a rational basis for its solution. An outstanding contribution t o the knowledge of the subject was made more than half a century ago by Professor OsboTne Reynolds, who succeeded in finding a unifying principle which considerably simplified the analysis of his experimental results. His discovery that the
PUd
change from streamline to turbulent flow depended on the value of P
led later workers to thc corollary that the coefieient X in the well-known hlU2
PUd
pipe-formula h = -- is afunction of the parameter -, which was 2gd P named after him the Reynolds number. His discovery of this criterion led to the formulation of a more general Correspondencc on this Paper can be accepted until the 15th w i l l he published in the Institution Journal for October 1939.-sEC.
May, 1959, and INST.
C.E.
134
COLEBROOK ON TURBULENT FLOW IN PIPES.
“ Principle of Dynamical Similarity,” which determines the conditions for mechanical similarity in the motions in or around geometrically similar bodies. Considerations of dynamical similarity may bereplaced by dimensional reasoning which leads to a grouping of the quantities involved in the problem into anumber of non-dimensional parameters;this enables experimentalresults to be plotted in asystematic manner. Such considerations, however, have definite limitationssince the functional relationship between these groups and their relative importance cannot be determined by dimensional reasoning. It has been suggested as a result of experiments on lead and other smooth pipes that the resistance-coefficienth and the Reynolds number R could be expressed satisfactorily by an exponential ‘formula of the type
h = ARh By re-arrangement of this equation into the form
i t is easy to show that for smooth pipes the sum of the indices of U and d must be 3 for all pipe-sizes and velocities. This equation is widely known and the argument is frequently put forward that the sum of the indices must equal 3 in any exponential formula designed to fit experimental results on a few pipes over a limited range of velocities of flow. Although this relation between the indices is true for smooth pipes, the value of n itself so depends on the Reynolds number that a single value cf n will only give approximatelycorrectresults over alimited range of Reynolds numbers. When the roughness-factor is introduced the relation no longer holds : indeed, it will be shown in a later paragraph that, whatever the roughness, this sum always exceeds 3. F. C. Scobey attempts to justify by dimensional reasoning 1 his formula for riveted steel pipes in which the sum of the indices is 3, but his omission, from the argument, of the roughness-factor, which is particularly important in the case of riveted pipes, seriously affects the value of the formula. In brief, it may be stated that the principle of dynamical similarity determines the non-dimensional parameters governing fluid motion, but fails to determine the functional relationship between them. This has led to a reconsideration of t,he fundamentals of the problem, and therecent success of L. Prandtl and von Karman in Germany, and of G. I. Taylor in Great Brihin, in expressing in mathematical form the mechanism of turbulence, “ Riveted Steel and Analogous Pipes.” Bulletin No. 150, Department of Agriculture, U.S.A., 1930.
COLEBROOK ON TURBULENT FLOW
135
IN PIPES.
linked with the experimental investigations of Nikuradse, have now provided a fundamental basis for the analysis of the problem. They developed a formula of the type
and showed that thelower limit of integration y1 is a function of the wallparticle size k in the case of rough pipes in which the 00w obeys the square resistance-law, and is dependent on the density p, the viscosity p, and theshear stress at thewall T,in thecase of smooth pipes. Substituting appropriate values of y1 in (1) the following resistance laws are obtained for ))
(a) flow in hydraulically smooth pipes :
R 4
1
z
log-2.51
=
. . . . . . . .
(2)
. . . . . .
(3)
(b) flow in rough pipes :
The experimental results of Nikuradse show complete agreement with the above laws provided certain limiting conditions are satisfied. The experiments show that the rough-pipe law is true for values of
PV*k
t
~
P
exceeding 60, whilst for values less than 3 even rough pipes obey the smoothpipe law as the excrescences then cease to contribute to the resistance. Between these values there is a transition from one law to the other. The smooth, rough, and transition laws for Nikuradse's sand roughness in which the grains are of uniform size and closely packed together, are shown in Fig. l (p. 136) together with the transition curvefor a pipe having a roughness composed of isolated particles, the experiments on which are
t I.'=2/?
and is called the
"
shear force " velocity, since it has the dimensions
P
of a velocity. Theroughness Reynolds number
P
may be expanded into
It will be seen that it is the product of three dimensionless numbers, the resistancecoefficient, the relative roughness, and the Reynolds number.
136
COLEBROOK ON TURBULENT FLOW IN PIPES.
described in detail elsewhere.1 It is apparentthat with non-uniform roughness the transition zone extends over a range about 10 times as long as that for uniform sand roughness, and in the case of new commercial pipe8 in which the roughness is non-uniform the whole working range lies within the transition zone. The mean transition curves for galvanized-, cast-, and
wrought-iron pipes, which were determined by an analysis of most of the available reliable data and described later in the Paper, are shown in Fig. 1 for comparison with that for the roughness V . C. F. Colebrook and C. M. White,"Experiments with Fluid-Friction in Roughened Pipes." Proc. Roy. Soc. (A), vol. 161 (1937), pp. 367,351. (See Rough. ness '' V " in this Paper.)
COLERROOK TURBULENT ON
137
FLOW IN PIPES.
Any attempt to express mathematically the transition-function for uniform sand-roughness is rendered difficult owing to the fact that the turbulent motion in the wake behind the grains is complicated by mutual interference, and the resistance mechanism is made up of viscous and mechanical forces which are difficult to separate. In thecase of non-uniform roughness, however, the large isolated grains have a shielding effect on the smaller grains which considerably reduces their effectiveness so far as total resistance is concerned, so that the area of the pipe between the large excrescences may be regarded as behaving as a smooth surface witha coefficient of resistance dependent on the Reynolds number P-.V*d Since the local Reynolds number on the large
P
grains is comparatively large even a t fairly low mean velocities, the local grain co-efficient is practically constant over the entire transition range. Y1 k P In effect, - in (1) is a function of -, the relative roughness, and __ d d P V*d’ and hasdefinite limiting values corresponding a t the one extreme to fullyrough-law flow-conditions in which viscous resistance is negligible, and at the other extremeto smooth-pipe conditions when the resistance mechanism is entirely molecular. The exact form of the function will depend on the distribution of the roughness-elements and is mathematically indeterminate, but it will be shown in the present Paper that it is possible to obtain a particular transition law which is similar to those obtained experimentally for commercial pipes by simply addingtogether 1 the lower limits of integration y1 which satisfy the rough- and smooth-pipe laws. The following general formula is then obtained :
which is in exact agreement with theory a t extreme values of __ and
P
gives results in the transition-zone which approximate very closely to the experimental values. It willbe seen in Pig. I that this transition-curve merges asymtotically into the smooth- and rough-law curves.
THEORYOF TURBULENT FLOWIN PIPES.
In turbulent motion it has been observed that thevelocity-distribution This treatment of the lower limits of integration was suggested by Dr. C. M. White, and the Authordesires to place onrecord his indebtednessto Dr. White for his collaboration in the development of formula (4).
138
COLEBROOK ON TURBULENT FLOW IN PIPES.
may be expressed by the relation
av. &JP
2.5 -y-
. . . . . . . . .
where U denotes the velocity a t a distance y from the wall of the pipe, r, the shear stress at the wall, and p, the density of the fluid. On integration the equation (5) becomes
.-
Since U = 0 when y = y1 the effective hydraulic wall may be regarded as being displaced inwards from the actual wall by an amount yl. The hydraulic wall then represents a plane where the disturbances are theoretically as great as the actualones at the wall. The mean velocity U is numerically equal to the local velocity a t y = 0*113d*,and substitutingthis value of y in (6), the equation becomes d
. . . . .
.
Re-arranging (7) so as to introduce the resistance-coefficient into the equation,
Equation (8) may be regarded as a general formula applicable to all types of turbulent flow in pipes. The shift of the effective hydraulic wall y1 has, however, to be determined in order completely to determine the resistance-law. Since y1 depends on the conditions at the wall it must clearly be a function of (U)the roughness of the wall k, (b) the shear-stress T , ahd
P
(c) the kinematic viscosity of the fluid, v = -
P
.
It has been observed experimentally that providing
PV&
__ exceeds
P
about 60 the resistance is proportional t o the square of the velocity (that is, the resistance-coefficient is independent of the viscosity of the fluid), and in thiscase dimensional reasoning shows that the shift y1 can only be proportioned to k. Nikuradse, experimenting with pipes artscially roughened internally by a uniform layer of sand fmed to the walls, deter-
* For the proof of this expression, see “ The Reduction of Carrying Capacity of Pipes with Age,” by C. F. Colebrook and C. M. White. Journal Inst. C.E.,vol. 7 (1937-38),p.99.(November1937.)
139
UOLEBROOK ON TURBULENT FLOW IN PIPES.
mined a value of
L
Y1
=g'
where k denotes the diameter of the sand grains. Inserting this value of y1 in (S), the resistance-law for rough pipes becomes
(3) In thecase of smooth pipes (or rough pipes when 'Y*k,,,.is
less than
P
3 when the roughness particles cease to shed eddies and contribute to the resistance), the resistance is due entirely to molecular or viscous mixing, and y1 must, by dimensional reasoning, be proportional t o -,CL which is the PV* only combination of 7, p, and p which has the same unit as a length. Other experiments by Nikuradse show that for smooth pipes P y1 =-1 -
10 PV* which on insertion in (8) leads to theresistance-law for smooth pipes
exceeds 3, however, the resistance increases over that
When U
of a smooth 'pipeduetothe protuberances.
shedding of eddies bythe
A NEW THEORETICAL FORMULA ROR FLOW IN
THE
roughness-
TRANSITION REGION.
The value of y1 may be regarded as having two extremes which satisfy the smooth-law and fully rough-law conditions respectively, whilst in the transition range y1 exceeds both of these extreme values due toa combination of mechanical and viscous mixing at thewalls. Thus, Y1
= +PV* L ).
. .
Putting (9) into non-dimensional form, the equation becomes
140
COLEBROOK ON TURBULENT FLOW IN PIPES.
Analytically, equation (10) must take the form
where a and ,!lare numerical constants tobe found by experiment. For pipes having non-uniform roughness k may be regarded as being the roughness of a sanded surface giving the same resistance-coefficient as the non-uniformly roughened surface. 1 l Nikuradse's values for M and ,!l are - and - respectively, and sub33 10 stituting these numerical values in (11) and inserting the resulting value of y1 in (8) the resistance-law becomes
1
0.113d
3= =
log k 33 -2log-
1CL +-.10 p v *
-+- lO'p17,d __
0.113 3%
which may be rewritten as
1 ---2log(:+--)
dX
-
k 37d
2.51
RdA
'
.
'
'
In order to represent (12) graphically it is convenient to separate the independent variable
_ -1
' 3 from the remainder. CL 3.7d 3.28 2log= 2log k
Thus,
(13)
P
This function is shown as a heavy line in Fig. 1. (p. 136). It will be noticed that the theory indicates aslight increase in resistance over that for P V*k purely rough-law flow at -- 60, butthis
P
discrepancy against
pv k experiment is very small and diminishes with increasing values of 2. CL The curve approaches the smooth- and rough-laws asymtoticallyin accordance with experimental observation. The formula for flow in smooth pipes (2)
COLEBROOK ON TURBULENT FLOW IN PIPES.
141
is rather inconvenient for practical use since the resistance-coefficient appears on both sides of the equation. This difficulty is overcome by using the formula
which is a mathematical approximation to the exact formula (2) but gives numerical results within & $ per cent. over a range of Reynolds numbers of from 5,000 to 100,000,OOO.
RELATION BETWEEN PRANDTI~VON-KARMAN AND EXPONENTIAL FORMULAS.
It is of interest to compare the results obtained by the modern rational method of analysis of the problem of fluid-flow with the earlier empirical formulas of the exponential type, The Prandtl-von-Karman rough pipe1 a law -- 08 3.7-may be converted to the exponential type dj- 2 1 l;
by taking logarithm and differentiating.
Thus
Formula (17) may be extended into theusual form
or
It is clear that the exponent, n, is itself a function of the resistancecoeficient, so that a single value will only give approximatelycorrect results over a limited range of d/k values. In order to illustrate the argument, suppose it is necessary to develop exponential formulas to cover a
142
COLEBROOK ON TURBULENT FLOW IN PIPES.
range of dlk = 10 to 40,000 so as to give results to within f 24 per cent. 1 (1 of the correct value. It will be found by plotting log - against log -
2/h
X:
that it is necessary to divide up therange into two components of dlk = 10 to 200 and d/k = 200 to 40,000. The values of A and n then become
dlk dlk
= 10 to
200, A = 2.03, and n = 0.20 40,000, A = 3.25, and n = 0.111
= 200 to
It is to be noted that thesum of the indices of U and d always exceeds 3 in thecase of rough pipes by 1-74.\/ri. An exponential formula of the type 1
=ARn
. . . .
. . . .
may be developed from the Prandtl-von-Karman smooth-pipe law 1 Rdi _ - 2 log -by taking logarithms and differentiating. The exponent 2.51 dX log-n is given by d( :A) , which becomes d(log R )
Thus
Equation (20) on extension becomes
or
where m is given by (19). Here again it is seen that the exponent m is a function of the resistance-coefficient, but in this case the sum of the indices of U and d equal8 3 as predicted by dimensional analysis. This development of the relationship between rational and exponential formulas shows quite clearly that single values of the exponent, n, can only give approximately correct results over a limited range of pipe-sixes, and
COLEBROOK ON TURBULENT FLOW IN PDES.
143
velocities and exponential formulas are not, therefore, capable of universal application.
ANALYSISOF EXPERIMENTAL DATAON SMOOTH PIPES. A number of commerical pipes may be regarded as hydraulically smooth, at least for all ordinary velocities of flow. Among these may be included good commercial drawn-brass pipes, lead, glass, or tin pipes, centrifugallyspun lined (with bitumen or concrete) cast-iron pipes, and concrete-lined pipes which have been deposited against oiled steel forms and carefully rubbed down to remove any imperfections. The results of an analysis of much of the available experimental data are shown in Figs. 2 (p. l44), and are seen to be in close agreement with the Prandtl-von-Karman smooth-pipe law. The data include the experimental results on only one brass pipe of 0-5 inch diameter,obtained a t the National Physical Laboratory by Stanton and Pannell in 1915, although the results for a large number of brass pipes of other diameters tested by them also agree very closely with theory. The results on sixteen spun concrete-lined pipes and on six spun bitumastic-lined pipes ranging in size from 4inches to 60 inches in diameter are included. Of these, the laboratory tests by M. L. Enger on 4-inch, 6-inch and 8-inch pipes were probably subject to the least experimental error, and the result.s exhibit only slight scatter from the theoretical law. I n analyzing the dat,a obtained by B. W. Bryan on the Stour Supply, Danbury to Herongate main, which included one hundred and ninety-two lobster-back bends of radius 3 4 4 and having a total change in direction U2
of 2,987 degrees, an allowance of 2 0 - was made in the calculations for 2g bends. The results on the 216-inch diameter Ontario tunnel, the biggest of its kind in the world, are especially interesting, as particular care was taken in its construction and therange of test-velocities was large. The concrete used in the construction of the tunnel was deposited against oiled steel forms which resulted in a smooth and even surface. All defects were then removed andthe surface rubbed down withcarborundum brick. I n analyzing the test-data1, it was found that an arithmetical error had been h1 1JZ made in calculating the resistance-coefficientsin h = --. 2@ The correct values, which are considerably lower than those given by &obey, are shown in Table I (p. 145) together with the t’est-results from which they were computed. Despite an appreciableexperimental scatterthe test-results are in very satisfactory agreement with theory. 1
F. C. Scobey,
No. 852.
“
Concrete Pipes.” Department of Agriculture, U.S.A., Bulletin
144
COLEBROOKONTURBULENTFLOW
IN PIPES.
I45
C'OLEBROOK ON TURBULENT FLOW IN PIPES.
TABLEI.
1 zGi! 1
I
1
I
1,018 2,036 3,045 4,063 5,091
in 1,000feet :per
I
4 8 12 16 20
feet,
Coemclent of friction, X .
Reynolds number.
l
l
I
0.108 0.448
0.990 1.701 2,397
5,550,000
11,100,000 16,650,000
22,2oo,oO0 27,700,000
0.00782 0.00812 0.00798 0.00773 040697
GALVANIZED,CAST- AND WROUGHT-IRON PIPES. In analyzing the dataon the various types of iron pipes it was necessary mean to determine boththe mean hydraulic-roughness, k , andthe transition law for each class. The problem is complicated by the fact that in practice there are variations of roughness due to non-uniformity in the method of manufacture so that ineach class there is considerable variation both in the size and type of roughness. It was necessary, therefore, to determine the transition law and roughness k for each individual pipe-a t,ask which is rendered difficult by the fact that with one or two exceptions the experimental results donot cover a wide enough range and rarely reach square-law. However, the experiments on pipe V t indicate fairly rapid transition to the square-law at the higher values of
' 9 and, thus II
with many of the test-results it is possible to extend them with very little error so as toreach square-law and enable the determination of the k values, and thuslocate the test-results in thetransition-range, The experimental results for each class of pipe are plotted in Pigs. 3, 1 5, and 7 (pp. 146 et s q . ) with - as ordinate against log RdX as abscissa.
dA
This arrangement gives a sloping straight line for the smooth-law flow and a series of parallel horizontal lines in the square-law region which extends to the right of the dotted line representing the lower limit of rough-law flow. The results may be brought to a single line in the rough-law region by 3.7d l P V& This has been plotting 2 log -- -as afunction of log
X: di P carried out in Pigs. 4 , 6, and 8 (pp. 147 el seq.), and a mean transition curve drawn in for each class of pipe. The k-values determined for all pipes are shown in Pys. 9 (p. 152), and using the mean Ic-value for each class together with the corresponding mean transitioncurve, a number of transition curves have been drawn in Pigs. 3, 5, and 7 for direct comparison with the p-.
-f Footnote (l),p. 136.
10
146
COLEBROOK ON TURBULENT FLOW IN PIPES.
COLEBROOK “IJRRULENT ON
FLOW IN PIPES.
147
148
COLEBROOK ON TURBULENT FLOW
IN PIPES.
test-results. It is seen that although some of the pipes do not agree very closely with the mean curves, some having too rapid transition andothers too slow, there appears to be sufficient positive evidence to justify the adoption of the given mean transition laws together withthe mean k-values. It is to be expected that these will enable the prediction of resistancecoefficients in pipes of sizes other thanthose tested and at velocities beyond the normal range with less uncertainty than with any existing empirical Fig. 5.
l? ,/T
EXPERIMENTAL DATAON TAR-COATED CAST-IRON PIPES.
formula. With regard to the experimental data itself, space prohibits a detailed description of all the data available, so remarks will be confined to a few observations with regard to the most accurate data. The experiments made by F. Heywood on new galvanized-iron pipes of 2 inches and 4 inches diameter were carefully conducted and are most valuable, as therange of velocities waB very wide, being from 0.5 to 21 feet per second. Referring to Fig. 3 it will be noticed that the resistancecoefficient for the 2-inch pipe becomes constant a t high velocities, thus enabling the determination of k and the major portion of the transition curve.
At the lower values of -the 2-inch and 4-inch pipes diverge P
COLEBROOK ON TURBULENT FLOW
IN PIPES.
149
in opposite directions from the mean curves, and the 2-inch pipe is somewhat rougher than the 4-inch. The remaining data on galvanized pipes was obtained by Saph and Schoder, but in thedetermination of the mean value of k for this class the
Author has neglected pipe XVIII (0.85 inch in diameter) as the experimenters makethe following statement concerning this pipe-" Pipe XVIII (0.85 inch in diameter) seems to be an exceptional pipe, but it has to be remembered that a slight silt-like deposit had occurred on the inner walls which was entirely sufficient to relieve the roughness."
150
COLEBROOK ON TURBULENT FLOW IN PIPES.
Very reliable data, used in the present analysis on tar-coated cast-iron pipes, was obtained by J. Freeman and H. Mills a t Lawrence, Massachussetts on pipes of 4 inches, 8 inches, and 12 inches diameter, and another carefully made experiment was that on a 6-inch pipe described in the Report on “ Pipe Line Coefficients,” 1 in which the range of velocities was
I L \‘X
E XPERIMENTAL DATAON
WROUGHT-IRON PIPES.
15 to 1. Ot,her carefully conducted experiments include those on the Manchester, Thirlmere siphons (44 inches indiameter), theSudbury conduit (48 inches in diameter), and the 61-inch diameter siphon experimented on by Fitzgerald. Practically all of the available data on wrought-iron pipes were obtained by J. R. Freeman. Extreme care was exercised in making the experiments which covered a wide range of velocities. The pipes were considered to be fairly representative of ordinary lap-welded wrought-iron pipes used in the U.S.A. The remaining experiments by J. B. Francis and H. Smith, Jr., indicate that their pipes were considerably smoother than those used by Issued by a Committee of the New England Water Works Association in 193.5: Journal New England Water Works Aasoc., vol. 49 (1935).
Fig. 8.
Y
2
152
COLEBROOK ON TURBULENT FLOW TN PIPES.
Freeman, although Freeman’s results are remarkably consistent among themselves. Some experiments 1 on asphalted wrought-iron pipes are also
001
0.W6 W
-
$0034
X
0 002
0.001 DIAMETER: INCHES, GALVANIZED-IRON PIPES.
0.006
2
0-004
U
r
a 0.002
0.00I 0
10
20
30 40 DIAMETER: INCHES. ASPHALTEDCAST-IRON
50
60
70
1 0
I2
14
PIPES
0 004
vi o’w2
2 5:
., 0 . 0 0 1
k .!
0-006 0.004 0
2
4
6 8 DIAMETER: INCHES.
WROUGHT-IRON PIPES.
included, but these pipes appear to have a capacity averaging about 5 per cent. greater than that of uncoated pipes. Pipes Nos. 302, 304, and 310 in “ The Flow of Water in Riveted Steel and Anagolous Pipes,” by F. C. Scobey (U.S. Dept. Agriculture-Tech. Bul. No. 50, Jan. 1930). Denoted in Fig. 9 of the present Paper by d.
153
COLEBROOK ON TURBULENT FLOW IN PIPES.
The mean values of k are : Galvanized-iron pipes , Asphalted cast-iron pipes ‘Uncoated cast-iron pipes Wrought-iron pipes . .
. . . k = 0.006 inch.
. . .
I% = 0.005 inch.
. . . k = 0.01 inch. . . . k = 0.0017 inch.
OLD PIPES. The deterioration of pipes with age has already been discussed a t solne length in a previous Paper 1 so only brief reference to thisproblem will be made here. The hydraulic resistance of water-mains increases after themains have been in service for some time due togrowths or deposits upon the internal surfaces. By making various simplifying assumptions it has been possible to develop a formula 1 which gives the relation between the age of a pipe and its carrying capacity, which may be written as
Q
$10-0)
. . . . . .
where Q denotes the discharge at theend of T years, Q. denotes the initial discharge, p. = Co/22/& (where CO is the initial Chezy coefficient) and a is the average rate of growth of roughness. If in any district the growth-ratea is required this may be computed from the results of experimental observations by means of the equation
a
3-7a
= -(lO*-
T
10-0)
. . .
. .
(22)
where p = C/22/89and C denotes the final Chezy coefficient. The diameterof a proposed pipe may be determined from the formula
where i denotes the hydraulic gradientandkodenotesthe original roughness size, say 0.01 inch. Alternatively, the appended design-Tables 11-VI may be used to determine Chezy coefficientsand values of A C d m corresponding to various values of k and d. The roughness k is readily obtained from
k
= ko
+ uT,
and u may be computed from experimental observation using formula (22). Where no experimental data is available for calculating the growth-rate 1 C. F. Colebroolr and C. M. White, “The Reduction of Carrying Capacity of Pipes with Age.” Journal Inst. C.E., vol. 7 (1937-38), p. 99. (November1937).
I54
COLEBROOK ON TURBULENT FLOW IN PIPES.
this may be estimated for asphalted cast-iron pipes from the pH value of the water, using t.hc interpolation formula
CI
2 l o g a = 3.8 - p H
. . . . . .
(24)
which gives the growth-rate in inches per year.
DISCUSSION AND CONCLUSIONS. The present analysis of the problem offlow in commercial pipes has been based on the premise that transition from smooth-law to rough-law flow in commercial pipes takes place in a gradual manner, as shown in Fig. l (p. 136). By an extensionof the Prandtl-von-Karmanlaws for smooth and rough pipes, a theoretical transition law (12) has been developed by the Author, in collaboration with D r . C. M. White, which gives favourable support to this assumption. A l t ~ T h ~ ~ i Z i i i iexperimental ble data is so incomplete and limited in range that fully rough conditions were only reached in a few cases, a collection of data on old mains shown in Pig. 2 of a previousPaper 1 proves conclusively that in thecase of non-uniformly roughened pipes (which include most commercial pipes), the resistancecoefficient falls with decreasing rapidity as thevelocity increases, and once having reached square-law it remains constant at all higher velocities. The fact that there are considerable variations in the roughness and transition curves in each class of pipe must not be considered a defect in the method of analysis. Such variations are to be expected, since manufacturing conditions are not identical in different plants. For design purposes a series of transition curves for each class is obviously impracticable, so mean curves corresponding to average conditions have been determined. The scatter of the k-values in Pig. 9 is too great t o be able to ascertain any possible dependence of k on pipe-size, so a single value for each class seems justified especially as pipes of all sizes in any particularclass are made by the same process. In thecase of built-up pipes, such as riveted steel pipes, a variation of k with pipe-size would be expected, and in a later Paper it will be shown that thisoccurs in thecase of a certain class of riveted pipe. Where it is not possible to determine by experiment the transition curve for any particular type of pipe, the theoretical transition curve (12) may be used with verylittle error provided that theroughness can be determined, and this is not difficult since some reliable experimental data on a few pipes over at least a small range of velocities is usually available. All formulas in the Paper are non-dimensional throughout and it is possible,therefore, to use the results inany system of units. Since the transition curves are somewhat complex and are not, therefore, easy to use, five design-Tables (Tables 11-VI) based on these functions are included Footnote ( l ) ,p. 163.
COLEBROOK ON TURBULENT FLOW IN PIPES.
158
in order t o facilitatecalculations on the flow of water.The Chezy coefficient C in U = C 4 2 is given for various pipe-sizes, velocities, and gradientsinEnglishunits a t atemperature of 55" F., as calculations involving the Chezy formula are easily and rapidly made by slide-rule. Similar tables for gas, air and other fluids may be compiled by means of the transition curve determined by the Author. The work was carried out in the Civil Engineering Department of the Imperial College of Science and Technology, London, and the Author isindebted tothe generosity of the Clothworkers Company, who, in supporting another researchof purely academic nature, indirectly inspired the present work. The Paper is accompanied by nine sheetsof drawings and five designTables-from which the Figures in the text and thefollowing Appendixes have been prepared.
APPENDIX. '
Examples illustrating the we of t h deaign-Tabks. Problem ( I ) . To find the discharge of a new asphalted cast-iron pipe, 48 inches diameter, with a gradient of 1 in 6,000. The dischargeis determinedfrom Q = (AGdiijda and from Table IV the value of A C d G corresponding to a gradient of the order 1 in 6,000 is A C d m = 1,710. Hence Q
=
1,710 X 4-63
= 22.1 cu8ecB.
Problem (2). To find thc diameter of a new asphalted cast-iron pipe to discharge 10 cuseca with a gradient of 1 in 4 00. The sizeof pipe is determined by the value of
156
COLEBR.OOK ON TURBULENT FLOW IN PIPES.
From Table IV it is seen that a 21-inch diameter pipe hacrav;dueofACl/ii- ,008 at.,zpproximately the given gradient. The actual dischargeof this pipe at thegiven gradient is
Q = 208 .d-=
10.4 cusecs
Problem (3). To find the diameter of an asphalted cast-iron pipewhich will discharge 36 cusecs 30 years hence witha gradient of 1 in 100 and apH value of 7.2. The required pipe must have a value of
and by interpolation in Table VI for a pH value of 7.2 it is seen that a 33-inch diameter pipe has a valueof A C d g = 365 approximately a t this pH value.
TABLETT.-SMOOTH
T
7
D: inrhes.
ACdG
C
1 2 3
4 5
6 7 8 9 10 11 12 15 18
21 24 27 30 33 36 40 44 48 54 60 66 72 78 84
107.5 110.5 112.5 114,:i 116
0.061 0.385 1.11 2.37 4.25 6.87 10.3 14.6 20 26.3 33.6 42.2 76 121.5 182 2.57
U=l
PIPES : VALVESOF C
l
c
A4cd/m
c
0.0713 94
0.141
U
103
ACdG
0474 0.46 1.32
105
106.5 108 109.5
l1 0.5 112 114.5
4.45
7.15 10.7 15.2 20.7 27.3 34.9
-l 124
44
78.8
85.3 136 203 288 392 515 660 825 1085 l390 1740 2360 3940 4980 6150 7420
1 -
100,000
127.5
130 131.5 133.5 l35 l36 137.5 134.5 140 141 142 143.5 145 146.5 147.5 148.5 150
G ~ ~ A QR =D( A C ~ ~ ~ VARIOUS T A T VELOCITIES.
-
l
5 8.05 1245 17.1 23.2 30.5 39.2 49.1 87.7 40.5
209 296 402 527 678 848
1115 142.5 1785 2420 3180 4090 5130 62S0 7620
GRADIENT =
-I
C
99 107.5 112.5
0.078
l
7
0.48
1.38 2.93
119 121 123 125 126 127.5 129 130 132.5 135 136.5 138.5
5.25 8.4 12.55 17.8 24.1 31.8 40.7 51.1 91 146 217 307
140.5
419
141 142.5 143.5 145 146.5 147.5 149 150 151.5 153 154 154.5
547
--
1 10,000 ~
-1
0=3
1l 6
2.8
101 103
=
U=Z
1 -1-
JIOO
GRADIENT =
I
U=1.5
IN
703 880 1153 1480 1855 2510 3290 4220 5310 6.520 7850
u=5 C
u=10
U=;
ACd\/m
C
ACdm
0.096F 0.585 14ij
_105 114 l19 122 l25 127.5 130 131
132.5 134 135.5 136.5 139.5 141 143 145 146.5 148 149
150 151.5 153 154.5 l56 157 158 159.5 160.5 161
114 122 128 131 I 134.5
0.0828 0.508
1.46 3.08 5.5 8.85 13.25
15.4 13.4 12.8
53.6 16 152 227 322 437 575 73.5 920 1206 1545 1940 2630 3440 4400 5550 6800 8200
123 126.5 129.5 131.5 133.6 135.5 137 138.5 139.5 140.5
143.3 146 148 150 150.5 152 153.5 154.5 156 157 158.5 l60 161.5 162 163.5 164.5 165.5
-
1.51
3.19 5.7 9.12 13.6 19.3 E6.2
136.5 138 140 141.5 142.5
!4.5
0.09
119
0.545
1'X
1.57 5.92
132..5 I36 139.5
9.45
141.5
14.05 20 27.1 35.5
143 145 146.5 138 149 150 153.5 1 55 l57..5 139.<5 161 162 163 164.5 165.5 167 168 169.5 171 172 I73 174 175
3.31
14.1 i5.3
57.5 23.5 333 450 590
757 94.7
1215 1585 l990 2700 3540 4520 5680 6970 8420
GRADIENT
162
155.5 156.5 158 159.5 160.5
161.5 163 164.5 165.5 167 168
l69 170
1 1,000
=-
465
607 780 975 1280 1630 2050 2780 3630 4660 5830 7150 8650
l
7
3.53 6.3 10.05 15.0 21.3 28,s
GR,\DIENT= 1 148 150 151.5 153 155 156.5 158 159.5 162 I 161..j 166 168 170 170.5 172 173 174.9 176 177 179 180 181.5 I82-.5 183.5 1 83
l
48.3
l
60.5 07.5 I72 256 362 492 642
823 1030 1350 1720 2160 2920 3830 4910
6I U ,;
7550 9100
6.5 10.4 15.45 21.8 29.7 39 50 62.7 ]L11.5
-l
17.4
264 373 507 661 818
I060 l390
>R .\DIEXI'=
--
l 10
1780 2220 3020 3940 5070 6340 7770 9400
1L
GRADIEhT = 100
[TABLE111.
--
D: inches
c
1 1.5
2 2.5 3 3.5 4 4.5 5 5.5 6 7 8 9 10 11 12
0.00136 0.00546 0.01228 0.0218 04341 0.0491 04668 0.0874 0.1 104 0.1362 0.165 0.1965 0,267 0,349 0,442 0.546 0.66 0.786
ACdK
0.102 0.1444 70.5 75 0.1772 79 0.2042 0.2282 I_ H:! 0.25 84 0.27 86 0.289 87.5 0.306 89 92 0.323 90.5 0.339 92 0-353 93 0.382 94.5 0.408 96 0.433 97.5 0,457 99 3,478 100 3.50 101 62
-10.0086 0.0555 0.163 0.352 0.638
3.01 3.97 5.13 6.45 9.62 13.6 18.7 24.7 31.5 39.6
lGRADIENT=
I
64.5 73 78 82 84.5
93.5 94.5 95.5
I
97.5 99.5 100.5 102 103 104
1 lo,ooo
T
-l
C
--
-0.5
GALVAIVIZTD-’IRON PIPES: VALUESOF C m U
TABLE111.-NEW
=
Cdgi AND Q
u=3
I
= ( A C d n ) d i AT
u=5
U=7
A C G
l
I
-l
73.5
1
0.0102
0.39
1
U=20
I
U=30
I
I
-I
-l-l
76
‘34.5
0.705 1.14 1.71 2.44 3.31 4.36 5.62 7.05
2.55
10.55 15.0
103.5
111
117 11.8 16.8 22.9 30.0 38.8 123 38.5
20.5
1 7 34.5 43.5
GRADIENT =
U=lO
C
-1-1 04097 0,062 0. I82
VARIOUS VELOCITIES.
114
1 -
1000
117 118.5 GRADIENT =
1 ,m
119 120 121.5
11.9 17 23 30.2
I-
GYADTT?\”l’=
I
1
D: inrhes.
TARLX IV-NEW ASPRALTEDCAST-IRONPIPES : VALUESOF C u=1.0
A:
U = 1.5
l
1
C
ACd\/m
c
c
5
6 7 8 9
10 11 12 l5
40 44 48
54 60
U
=
C ~ ~ ~ F AQI =T ( D A C d m ) d ; AT T I
T V E L~O ~ ~ ~~ E S . ~
u=m
I
-__.
sqllnre feert.
3 4
18 21 24 27 30 33 36
u=2.0
IN
181
1240
I560 2120 2780
116.5 118.5
120 121
122 123.5 125 126 127.5 l29 130
1Mi 358 470
602 756 993 I270 l605 21 80 2860
I .25
136 137.5 139 1
= lo,ooO
AC&i
2.71 4.87 7.90
135
GRADIENT
C
99.5
l25 127 128.5 130 131 I32 133.5
263
ACdK
103 106.5 108.5 110.5 112 114 115.5 116.5 118 121 163
255 347 458 587 735 967
c
c
ACl’\/m
C
~
U = 30.0
ACV%
C
107.5 1.31 2.78
111
-~
117
1.32 2.7!1 5.03 8.1 3 12.2
120.5
16.7 22.8 30 38.3
139 208 294 400 524 672 842 1105 1415 l780 241 5 3180
~
--
11.8
48.2 86.7
~
130 132.5 134.5 136.5 137.5 139
88.5
142 212 300 407 4x15
F89 860 l130 1440 1820 2470 32.50
140.5
142 143 144.5 146 148 149
GRADTEXT =-
17.1 23.4 30.9 39.6 49.8 X!I.T,
144
214 303 410 540 694 870 1136 14-55
1835 2490 3280
c
AC&i
1.32
107..5 111.5
2.80
114.5
5.03
117 119 121 123 124.5 125.5 127 130 132.5 134.5 136.5
8.13 12.2 17.2 23.6
31 39.6 49.8 89.5 144
214 303
138
416
139.5 141 142 143.5 14.5 146 147.5 149.5
542 G96 870 1140 1460
I835 2190 3290
l
100
[TABLEV.
TABLE V.-NEW WROUQHT-IRON PWES: VALUESOF C IN U D : inches
A: qnarr feet.
U =0.5
dGi : (feet)&.
U=0.7
c
1
c
C
1 1.5
2 2.5 3 3.5 4 4.5 5
5.5 6 7 8 9 10 11 12
0.001362 0,0054fi 0,01228 0.02182 0.0341 0.0491 0.0668 0.0874 0.1104 0.1332 0.165 0.1965 0.267 0.349 0.442 0,546 0.66 0.786
0.102 0.144c5 0.1772 0.204 0.228 0.25 0.27 0.289 0.306 0.323 0.339 0.353 0.383 0.408 0.433 0.457 0.478 0.5
67.5 76 80.5 84 87 89.5 91.5 93 94.5 96 97 98 100 101.5 103
104.5 105.5 106.5
0.0094
0.060 0.175 0.374 0.677 1.10 1.64 2.35 3.20 4.20 5.42 6.82 10.2 14.4 19.7 26.0 33.2 41.7
GRADIENT =
70.5 7'3 84 87.5 90 92.5 94.5 96 98 99.5 100.5 101.5 103 104.5 106 107.5 109 112.5 110
1 10,000
2.42
99.5
0.0103 0.065 0.19 0.405 0.727 1.18 1.76 2.51 3.41 4.49 5.78
10.5 14.8 20.3 26.8 34.3 43.1
Gdz
I
AD
Q
u=5
= (ACdm)d/7AT VARIOUS VELOCTRES.
I
u=7
1 -I
U=IO
I
c
ACdK
C
u=30
i
U=15
c
--
-l 0.5
u=3
U=2
U=1.5
U=l
=
106.5 109.5 113.5
7.30 10.85 15.4 21.0 27.7 35.4 44.5
77 85.5 91 94.5 97.5 100 101.5 103 104.5 106 107.5 108.5 110.5 112 113.5 1l 5 116 117
0.0107 0.0675 0.197
0.01l l
79.5
0.421
0.758 1.23 1.83 2.60 3.53 4.65 6.00 7.55 11.3 15.9 21.7 28.6
36.6 45.8
GRADIENT =
100 102 104 105.5 107 108.5 110 111 113 114.5 116 117.5 119 120
1 -
1,000
,
0.778 1.25 1.87 2.66 3.62 4.75 6.15 7.72 11.5 16.3 22.2 29.3 37.5 47.0
83 91.5 96.5 100.5 103 105.5
0.012 88.5 0.075 97.5 0.218 102.5 104 04ti:I 106.5 I07 0.833 ' 109.5 109.5 1.34 112 111 2.00 113.5 2.85 115.5 113 114.5 3.87 117 118.5 5.08 6.55 120 8.23 121 122.5 12.25
0.072
107.5
0.448 0.802 1.30 1.93
109
2.75
110.5
112 113.5 114.5 116.5 l18 119.5 121 122.5 123.5
48.4
j
128.5
1
GRADIENT = 100
0.077 0.222 0.47.5 0.852
2.04 2.92 3.96 5.20 6.70 8.40 12.5 17.6 24.1 31.8 40.5 50.8
!)9.5
105 109 111.5 114 I16 118 119.5 121 122.5 123.5 125 126.5 128.5 130 131 132
94
95.5
103
104.5
108.5
110 114 117 119.5 121.5 123.5 125
112.5
I
I20
I
3.03
11.5.5 118 120 122 123.5 123 126 127.5 1-09 130.R 132.5 131
126..i
128 129 130.5 132.5 134.5 136 137 138
135 136
GRBDIENT
1 10
=-
k=n.n75 inch.
k y 0 . 1 inch
k = @ 3 inrh
k=0.15 inch
k-0.5 inch.
k=0.7.5 inch.
6
7 8
9 10 11 12 15 18 21 24 27 30 33 36 40 44 48 54 60 66 72 78 84
0.0491 0.0873 0.13F 0.196 0.267 0.349 0.442 0.545 0.66 0.785 1.227 1.767 2.405 3.14 3.98 4.91 5.94 7.07 8.73 10.56 12.57 15.90 19.63 23.76 28.27 33.15 3848
k-3.0inch.
ACdk
C
C
-l C
C
ACl/Z
C
C
C
0.25 0.288 0.322 0.354 0.383 0.408 0,433 0.457 0.478 0.5 0.56 0.613 0.662 0.707 0.75 316 0.79 0.83
0.02 2.0 3.63 5.90 8.90 12.7
75.3 79.3 52.5 85 87.2 89
1.86 3.38 5.52 8.30 11.8 16.3 21.5 27.6 3'4.9 63.3 102 154 219 299
89
105.9 107.5
417
0.866
0.912 0.955 1.0 1.06 1.12 1.17 1.225 1.275 1,325
0.85
69.6 73.7 76.8 79.4 81.5 83.3 85 86.4 87.8
114 115.6 118.5 120.8 121.8
I
1
9.5
1432 1950 2570 3290 4150 5120 6200
92.1 94.6 96.7 98.7 100.2 101.7 103 104.4 105.8 107 108.3 110 111.4 112.9 114 115. 116
39.5 508 640 843 1080 1360 1860 2450 3 140 3960 48S0 5900
9.1
0.81 1.76 3.2
65.7 69.6 72.8 75.3 77.4 79.3 80.8 82.5 83.8 85 88.2 90.7 92.8 94.7 96.3 97.8 99 100.2 101.6 103 104.4 106 107.4 108.8 111.1 lI2'l
5.23 7.9 11.3 15.5 20.5 26.4 33.2 60.7 98.0 148 210 287 379 488 614 810 1040 1310
l
17nn 2360 3020 3820 4710 5710
60 64 67.2 69.7 71.5 73.7 75.3 76.8 78.2 79.3 82.5 85 87.1 89 90.6 92.1 93.5 94.7 96.2 97.5 98.6 100.4 101.9
103.2 104.4 105.5 10F.5
0.74 1.61 2.96 4.85 7.33 10.5 14.4 19.1 24.7 31.1 56.8 92.0 139 19s 270 R37 461 580
765 953 1040 1700 2240 2870 3620 4470 5430
60
65.7 67.8 69.7 71.2 72.8 74.2 56.3
85 86.7 88 89.3 90.6 92.1 93.3 94.6 96.2 2150 97.7 99 low2 101.4 102.4
i
0.69 1.51 4.57 6.92 9.9 13.6 18.1 23.4 29.5
189
258 341 440 555 733 940 l190 l624 2750 3480
4300 5220
50.3 54.4 57.5 60 62.2 64.1 65.7 67.2 68.5 69.7 72.8 7R.4 77.5 79.3 81 82.5
83.8 85 86.5 87.8 89
90.7 92.2 93.5 94.8 95.8 96.8
0.62 1.37 2.53 4.17 6.35 9.12 12.6 16.7 21.6 27.3 50.0 81.5 123 176 241 320 413 520 688 885 1120 1530 2030 2600 3290 4060 4930
43.2 47.2 50.3 52.9 55 56.9 58.5 60 61.3 62.5 65.7 68.2 70.3 72.2 73.7 75.3 76.7 77.8 79.3 80.6 81.9 83.5 85 86.3 874 88.7 89.7
0.53 1.19 2.21 3.68 5.61 8.1 11.2 14.9 19.4 24.5 45.2 74.0 112 160 220 292 378 477 630 813 1030 1410 1870 2400 3040 3760 4570
37.6
0.46 1.05 1.96 3.28 5.06 7.3 10.1 13.5 17.6
41.6 44.7 47.2 494
51.3 52.9 54.3 55.7 56.9 60 62.5 62.7 66.7 68.2 69.7 71 72.2 73.7 75 76.2 77.8 79.3 80.7 81.9 83 84
22.3 41.3 67.5 103 148 203 271 360 442 587 757 957 1314 1745 2245 2840 3 520 4270
pH* value of vater to give above vaIues of k aftor 30 years' grovt: n cast-. In pipes 7.0 8.4 8.2 7.8 7.4
8.8
*
AC.\/m
C
ACd/na
G
C
ACdK
18.2 22.3 25.4 27.9 30.1 314 33.6 35.1 36.4 37.6 40.7 43.2 45.3 47.2 48.8 50.3 51.7 52.9 54.3 55.7 564 58.5 60 61.3 62.6 63.7 64.7
0.22 0.56 1.11 1.94 3.07 4.54 6.45 8.75 11.5 14.75 28.0 46.7 72.2 105 14.5 195 2.55 324 432 562 715 987 1320 1705 21 70 2700 3290
-~ -~
.-
3 4 5
k = 1 4 inch.
.-
D: A: inchcs. square fret.
Based on 2 1og a = 3.8 - pH, There a denotes the growth-rate in inches per year =
k" - 0405' 30
33.6 37.6 40.7 43.2 45.4 47.2 48.8 50.3 51.7 52.9 56 58.6 60.7 62.6 64.2 65.7 67 68.2 60.7 71 72.2 73.8 75.3 76.7 774 79 80
0.41 0.95 1.79 3.00 4.65 6.72 9.35 12.5 16.3 20.7 38.5 63.3 96.7 139 191 255 330 418 555 716 907 1245 1660 2130 2700 3350 4070
6.8
27.0 31.9 3.i.l 37.6
2.61 4.08 5.92 8.30 11.1
23.9 0.29 274 0.7 31 1.54 1.36 33.5 2.33 35.7 3.62 37.6 5.35 7.50 39.2 40.7 10.1 13.3 16.9
57.2 87.7
I
0.34 0.8
39.8 41.6 43.2 44.7 46.1 47.2 50.3 52.9 55 5F.9
31.9 52.7 81.5 117 I63 217 283 358 477 619 785 1084 1445 1860 2370 2940 3580
48.8 51.1
68.6
60 61.3 62.6 64.1 65.3 66.5 68.3 69.7 71 72.3 73.3 74.3
302 67.3 383 58.5 510 60 658 61.3 837 62.5 l152 64.2 1532 65.7 1975 67
6.4
6.2
5.5
