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INDUSTRIAL AND ENGINEERING CHEMISTRY
R = gas constant s = ratio of specifiz heats T = temperature Rankine v = velocity, ft.fsec. w_ = secondary fluid, lb./hr. w = secondary fluid, lb. moles/hr. iL/T& = ratio of molal entrainment of any substance to that of self-entrainment under the same conditions W = primary fluid, lb./hr. W = primary fluid, lb. moles/hr. y = reheat, yo A = difference Subscripts : 6 = boiler conditions e = entrainment conditions 0 = suction conditions at zero entrainment x = exhaust conditions
Literature Cited (1) Bancel, Trans. Am. Inst. Chem. E n g r s . , 30, 136 (1933). (2) Barnard, Ellenwood, and Hirschfeld, “Heat Power Engineering,”Vol. I11 (1933). (3) Copley, Simpson, Tenney, and Phipps, Rev. Sci. Instruments, 6, 265-7 (1935). (4) Edwards, Ibid., 6, 145-7 (1935).
477
Air Conditzoning, 5, 77-81, 147-8 (1933). (6) Foster Wheeler Corp., Catalog 116B (1927). (7) Goodenough, Power, 66, 466-9 (1927). (8) Graham and Kooistra, Heating & Ventzlating, 30, 22-7 (1933). (9) Ho, Rev. Sci. Instruments, 3, 133-5 (1932). (10) Jackson, IXD.ENG.CHEM.,28, 522-6 (1936). (11) Kalustian, Refrig.Eng., 28, 188-93 (1934). (12) Kaula and Robinson, “Condensing Plant,” 1926. (13) Klumb and Glimm, Phvsik, Z.,34, 64-5 (1933). (14) Marks, Mechanical Engineers’ Handbook, 1930. (15) Mellanby, Trans. Inst. Chem. Engrs. (London), 1928, 66-84. (16) Morgan, Power, 78,454-6,506-7 (1934). (17) Natl. Bur. Standards, Circ. 394 (1931). (18) Natl. Bur. Standards, Tech. Paper 193 (1921). (19) Parr and Baumeister, Psychrometric Tables and Charta for Air-Water Vapor Mixtures, 1928. (20) Schutte and Koerting Go., Bull. 5H, Vol. 2 (1930); 4E, Vol. 1 (1931). (21) Sim, “Steam Condensing Plant,” 1925. (22) Stodola (tr. by Lowenstein) “Steam and Gas Turbines,” Vol. (1927). (23) Vivian, Compressed A i r Mag., 38, 4141-6 (1933). (24) Waterfill, Refrig. E n g . , 24, 137-42 (1932). (25) Watson, Engineering, 1933, 230-2, 255-7, 262-5; Proc. Inst. Mech. Engrs. (London), 124,231-300 (1933). (26) Whitney, Refrig’.Eng., 24, 143-6 (1932). (27) Zabel, Rev. Sci. Instruments, 6, 54-5 (1935). ( 5 ) Fleisher, Heating Piping
FLUID RESISTANCE IN PIPES M. P. O’BRIEN, R. G. FOLSOM, AND FINN JONASSEN University of California, Berkeley, Calif.
T
HE theoretical work of von KBrmBn, Prandtl, and G. I. Taylor, and the experiments of Hopf, Fromm, Nikuradse, Schiller, and others have explained many features of the phenomenon of turbulent flow and have led to a number of important practical applications. The original impetus for the study of fluid resistance was the necessity for accurate prediction of the loss of head in pipes, and it is of interest to consider the extent to which the turbulence theory and substantiating experiments have advanced our knowledge of this important engineering problem. Reference is frequently made in the literature to smooth pipes and rough pipes. Originally the terms “smooth” and “rough” referred simply to the physical characteristics of the pipe walls, but in recent years they have acquired a dynamic significance. If a pipe is smooth in the hydraulic sense, the resistance coefficient in turbulent flow is affected by the viscosity but not by small changes in surface roughness. More specifically, a pipe is “smooth” if the friction coefficient plotted as a function of Reynolds number foIlows a certain curve obtained from tests on brass, lead, and other visually smooth surfaces. A pipe is “rough” if its friction coefficient is independent of the viscosity-that is, if the coefficient is a constant on the Reynolds number diagram. This condition of a constant f is also referred to as fully developed turbulence. Thus the same pipe may be smooth under one set of flow conditions and rough under another. Between rough and smooth in this dynamic sense lies almost the entire range of flow conditions that are of engineering importance. This paper will consider the extent to which present concepts of the nature of fluid resistance permit prediction of the head loss in this transition zone.
Smooth Pipes Von KArmBn’s theory of the local similarity of the flow pattern in the central core of any turbulent flow (7, 8, 9) leads to an equation for the relative velocity distribution which has been found to agree with experiment. A further assumption regarding the effect of the traction a t the wall of a smooth pipe gives the form of the equations for both the absolute velocity throughout the cross section and the friction
Formulas based on the theory of fully developed turbulent flow and on experiments using artificially roughened pipes are applied to extrapolated data for commercial pipes in order to obtain the equivalent roughness. Using this roughness, it is found that clean pipes do not follow a curve similar to that of Nikuradse in the transition zone between rough and smooth flow conditions. The conclusion is drawn that the turbulence theory has not yet provided a reliable generalized treatment of pipe resistance in the region important in engineering problems.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
factor. Prandtl evaluated the constants in the equation for smooth pipes, obtaining:
1= 2.0 loglo (Re .t/j;> - 0.8 v7
(1)
Data on the resistance offered by small brass, lead, and other apparently smooth pipes are in agreement with this equation up to very high Reynolds numbers. Experiments by Nikuradse (1.9)show agreement to Reynolds numbers of 3 X lo6.
Equation 2 may be rewritten as:
V* = Re*
(3)
The equation obtained by von KBrmBn for the velocity distribution in a smooth pipe outside of the laminar sublayer is:
V* = 5.5 + 5.75 loglo Re*
(4)
Solving Equations 3 and 4 for Re* gives as the point of intersection :
1;
Re*
=
vi
11.5 = B -
(5)
The value B refers to the average thickness of the laminar layer. I n terms of the usual friction coefficient and Reynolds number, the thickness of the laminar layer is then:
B=- 3 2 5 D Re
l/f
The average velocity at the boundary of the laminar layer is: I-
FIGURE1. NIKURADSE’S DATA(fa) A practical difficulty affecting the use of this equation lies in the decision as to whether a particular pipe under known flow conditions is to be classed as smooth. There R i reason to doubt whether any pipe could follow Equation 1 to indefinitely high Reynolds numbers. Some pipes follow this law only below a certain Reynolds number; others show higher friction coefficients over the whole turbulent range. The smooth pipe equation must be supplemented by a quantitative criterion showing whether a pipe of certain surface character and diameter is smooth a t a specified Reynolds number. Otherwise it is necessary to test each type of pipe over the whole working range in order to know whether it is smooth. Laminar Sublayer Surrounding the central core of fully developed turbulent flow, a layer in laminar motion is attached to the solid boundaries. Little quantitative information is available regarding the nature of the flow in this laminar sublayer, but it is generally thought of as having a thickness variable with both time and position. Presumably the laminar layer continually sheds the vortices which make up the turbulent core, and its average outer boundary is indefinite in position. However, the thickness of this layer may a t least be approximated. If the laminar layer is thin, the tractive force will be approximately constant across it, and the velocity gradient will be linear. Then 7
u
= p-
2
where
tractive force at wall absolute viscosity u = av. velocity at distance z from wall 7
=
p =
Introducing a local Reynolds number and a local velocity ratio:
These equations hold for smooth pipes only and therefore are limited in application by the necessity for a quantitative definition of “smooth.” If the average or effective size of the asperities which form the roughness of commercial pipes can be specified as a linear dimension, then it is reasonable to assume that the criterion defining smooth and rough pipe can be expressed as a ratio of this length to B. Such a criterion was suggested by von KBrmBn (8, IO) and will be discussed subsequently.
Rough Pipes From the standpoint of fluid resistance, a rough pipe is one in which the Reynolds number is sufficiently high so that the friction coefficient is not affected by viscosity. If we consider two pipes with identical surfaces carrying the same fluid a t the same mean velocity, one may be rough and the other smooth, depending on the diameter. The same pipe may be smooth a t low Reynolds numbers and rough a t much higher values. A quantitative criterion is needed if equations for rough pipes are to be applied. The theory developed by von KBrmBn also gives the form of the equation for rough pipes. He determined the constants from the data of Fromm (4) and Nikuradse ( I 3 , 1 4 ) , and proposed the equation: 1 v9= 2.07 log,, Ek + 1.50;
1 r or - = 2.07 log10 5.3
df
Nikuradse’s data alone followed the equation: 1
77
= 2.0 loglo
.\/J 1 = 2.0 log,, 7.4 5 + 1.74; or IC
(9)
I n both formulas k is the diameter of rounded sand grains forming the roughened surface used by Nikuradse. Equations 8 and 9 specify a definite relation which should be followed by all rough pipes, provided an equivalent value of k can be determined.
Experiments of Nilruradse Artificially roughened pipes offer the advantage of a known roughness pattern. A number of experiments of this kind are
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INDUSTRIAL AND ENGINEERING CHEMISTRY
available; the best known are those of Nikuradse (13, 14) who used circular pipes coated with a layer of sand grains. Similarity of the roughness pattern was approximately achieved by using building sand which had been sieved so as to restrict the size range within narrow limits. The technique of attaching the grains was carefully developed to obtain a reproducible roughness. The results of Nikuradse’s experiments are shown in Figure 1 where each curve represents a constant ratio of sand size to pipe diameter. The curves are those appearing in Nikuradse’s report. Line A-A indicates the lower limit of Reynolds number shown in the tabulated data, whereas experimental points (not tabulated) are shown in the laminar region as well as throughout the transition from laminar to turbulent flow. Considering these curves as drawn, roughness did not affect either the resistance in laminar flow or the transition from laminar to turbulent flow. In the turbulent region the general trend with increasing Reynolds number is for the friction coefficient first to shift to the smooth pipe curve and then to diverge and approach a constant value. I n other words, each one of the artificially roughened pipes became hydraulically rough a t Reynolds numbers which decreased with increasing roughness. Equation 9 is based on these values of f for rough surfaces. Nikuradse generalized his data by plotting
479
An equivalent representation of the data is obtained by plot1 against Re, as shown in Figure 2.
-) ( v7 In the original paper Nikuradse plotted more experimental ting 2.0 log 7.4
points in his Figure 9 (corresponding to our Figure 1) than were tabulated in the paper. The solid lines of our Figure 2 correspond to the lines and points plotted by Nikuradse in his Figure 11 ; the points he used in this figure included tabulated data only. If the points plotted by Nikuradse in his Figure 9 correspond to experimental values, the dotted lines of our Figure 2 show the characteristics for the points which were not tabulated. The dotted lines show that the curve plotted by h’ikuradse cannot be considered a generalization of all of his own data in the transition region between smooth pipe and fully developed turbulent flow. As mentioned previously, a plausible criterion for the upper limit of smooth pipes and the lower limit of rough pipes is the ratio k / B , where k is a linear dimension characterizing the roughness and B is the thickness of the laminar sublayer which would exist in a perfectly smooth pipe a t the same average flow conditions. Nikuradse’s experiments gave some information on the quantitative values. Taking the values of Re, from the curve drawn in Figure 2: Smooth: Rek = 2.7; k / B = 0.24 Rough: Rek = 70; k/B = 6.3
Von K4rm4n gave 0.27 and 5.5 as the values of this ratio a t the limits of smooth and rough pipes, respectively. The latter quantity is a Reynolds number based on the roughness size and the friction velocity which can be conveniently expressed in more familiar terms as follows:
Experiments of Colebroolr and White ( 1 ) Circular 2-inch pipes artificially roughened with both uniform and nonuniform sand were tested by Colebrook and White with results as shown in Figure 3. The five surfaces used had the following characteristics:
FIGURE2. GENERALIZED ROUGHNESS FUNCTION OF NIKURADSE DATA(13) Full lines = tabulated data: dashed lines = data read from curves of Nikuradse but not tabulated by him.
INDUSTRIAL AND ENGINEERING CHEMISTRY
480 Surface
No. I I1 I11 IV V
Description
Equivalent k from Equation 9, Cm.
Uniform sand, 0.035-cm. diam. in 2-in. pipe Uniform sand with large 0.35-om. grains covering 2.5% of area Uniform sand with large 0.35-cm. grains covering 5 % of area 48y0 of area smooth, 47% area uniformly covered fine grains, 5% area large grains 95% area smooth, 5% area covered large grains
0.048
0.073 0.093 0.066
0.038
An interesting feature of Figure 3 is that the curve for surface I, made up of uniform sand grains, does not coincide with Nikuradse's generalized curve. Furthermore, the equivalent roughness is 0.048 cm. as compared with the actual
?-&
average sand grain diameter of 0.035 cm. At = 200, all of the curves had become tangent to thc line for rough pipes. This limit corresponds to k / B = 17.4.
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rough joints. I n order to obtain the approximate value of k for common surfaces such as galvanized and wrought iron, resistance curves covering a wide range of Reynolds numbers were extrapolated, with results as shown in Table I. The amount of the extrapolation is indicated by comparison of the extrapolated value of Re a t f = constant with Re,,,. The original data were replotted, and the roughness was computed only for data showing a small scattering of the experimental points and an almost constant friction coefficient a t the maximum test Reynolds number. Since no tests on commercially smooth pipes were found which extended into the region of constant friction coefficient, there was no pattern on which to base the extrapolated curves, and the assumption that the friction coefficient ultimately becomes constant may be unwarranted. Some of Nikuradse's curves which fitted into the generalized diagram nevertheless did not converge towards the smooth pipe curve, a behavior characteristic of commercial pipes. This phenomenon suggested the possibility that a single characteristic curve might be found for all pipes or perhaps a different characteristic for
to be accounted for by errors
FIGURE 3. GENERALIZED ROUGHNESS FUNCTION OF COLEBROOK AND WHITE'SDATA(1)
Colebrook and White concluded that with nonuniform roughness, the transition from smooth to rough, resistance is more gradual than for the type of roughness used by Nikuradse; they suggested that the resistance curves of commercial pipes should be regarded as the result of nonuniformity of roughness rather than of waviness of the surface, as a number of other experimenters have also suggested. Colebrook and White also concluded that the deviation from the smooth pipe equation depends upon the largest grain sizes used in roughening the surface, but this conclusion is based upon extrapolated curves.
SURFACES TABLEI. APPROXINIATEVALUESOF k FOR COMMON Ranee of ---Expt.F
Investigator
Type of Pipe
Size
f
Inches Kessler ( I f )
New wrought iron, with oouplings
'/P
'/a
s/r
1 1' / t 2 3 4 6
8 Wrought iron, with Corp and smooth joints Ruble (9)
a/("
1" 2"
6b 6b
Roughness of Commercial Pipes
8E
Representation of resistance data on a generalized diagram (Figure 2) such as that of Nikuradse requires that the equivalent roughness be known. Equation 9, from which k is to be computed, applies only in the region in which f is constant. The majority of resistance data of this type either do not reach fully developed turbulence or apply to pipes with heavy incrustations or
Extrauo-latioh-
Remax. Re X X 10-4 X 10-4 10-4 Remin.
FreemanMills (3) Heywood (6)
New wrought iron
N e w galvanized wrought iron, w i t h smooth joints
5
2 Greve and Martin (6) Spiral riveted 6 a New and clean. b Used, clean, and good condition. C Old, rusted, but clean.
2.0
0.0332 0.0252 0.0249 0.0236 0.0217 0.0210 0.0202 0.0175 0.0150 0.0145 0.0260 0.0259 0.0222 0.0220 0.0148 0.0192 0.0176 80.0 200.0 0.0170
0.18
7.1
5.86
70.0
k X
k X
104 108 Feet Cm. 1.88 5.72 1.32 4.01 1.71 5.21 1.80 5.48 1.98 6.04 2.24 6.82 2.86 8.71 2.04 6.21 1.57 4.78 1.76 5.35 1.44 4.39 1.96 5.97 1.45 4.41 2.65 8.07 1.51 4.60 4.65 14.18 4.27 13.02 2.31 7.04
7.1 0,0309 8.89 70.0 0.0244 11.6
27.1 35.4
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481
in the computed values of k , and it does not appear that a single generalized curve can be made to represent the data on one type of surface. None of the curves resemble the generalized curve of Nikuradse and many of them tend to diverge from the smooth pipe curve at the lower Reynolds numbers. Extension of the experimental data to the region of constant friction coefficient as refinement of the method of extrapolation might result in values of k which would produce less divergence between the curves in the transition region. However, it does not appear that a single value of k can be found for each experimental pipe which would make the curves coincide. The divergence of the curves and the small variation i n the computed value of k seem to eliminate the possibility that a single roughness characteristic can be found t h a t will apply to all diameters of the same type of pipe.
Conclusions On the basis of the data and graphs presented, it seems justifiable t o conclude that: 1. The generalized resistance curve suggested by Nikuradse is not typical o f commercial pipes in the transition region between smooth and rough pipes. Furthermore, the discrepancy between ROUGHNESS FUNCTION OF COMMERCIAL P I P E S t h e experiments of Nikuradse and those FIGURE 4. GENERALIZED .of Colebrook and White indicate that Data extrapolated t o f = constant (Table I) this curve is not representative of all surfaces roughened with uniform sand grains. 2. The transition curve for commercial pipes follows an esRe* = local Reynolds number sentially different course from the generalized curve of NikuReh = roughness Reynolds number radse. Errors in the values of k computed from extrapolated r = pipe radius, ft. curves could not account for the change in shape of these curves. V* = local velocity ratio 3. The ratio k / B is not an adequate criterion of the limits of U = av. velocity at any point, ft./sec. smooth and rough pipes, if k is the equivalent roughness computed Urn = mean velocity in pipe, ft./sec. from Equation 9. uo = velocity at edge of laminar sublayer, ft./sec. 4. Experiments extending into the region of fully developed 2 = distance from wall, f t . turbulence (f = constant), supplemented by direct measurements B = thickness of laminar sublayer, ft. of the roughness of commercial pipe surfaces, are needed to pro!J = viscosity, 1b.-sec./ft. sq. vide a more accurate check on the theory than can be obtained Y = kinematic viscosity, ft. sq./sec. from extrnpolated curves. P = density, slugs per cu. ft. 5. Although the theory of turbulent flow has been useful in 7 = shearing stress, lb./sq. ft. explaining the nature of pipe resistance and has produced useful results in a number of related fields, it has not provided the enLiterature Cited gineer with an improved method of computing the head losses in pipes. Colebrook, C. F., and White, C. M., Proc. Roy. SOC.(London),
Acknowledgment The authors wish to thank the National Youth Administ>ration for assistance in preparing preliminary graphs and computations.
Nomenclature D = pipe diameter, ft. LUm2 f = friction factor in Weisbach equation, h = f D 28 k = size of sand grain (defined by Equation 9) forming roughness, ft. L = length of pipe, ft. Re = Reynolds number
A161, 367-81 (1937). Corp and Ruble, Univ. W i s . Eng. Series, 9 (1922-26). Freeman-Mills, M e m . Am. Acad. Arts Sci., 15, 126 (1924-26). Fromm, K . , 2. angew. Math. Mech., 3, 339-58 (1923). Greve and Martin, Purdue Eng. Expt. Sta.,Bull. 8 (1921). Heywood, Proc. Inst. Civil Engrs. (London), 219, 174-210 (1924-25). K&rm&n,Th. yon, J. Aeronaut. Sci., 1, No. 1, 1-20 (1934). K&rm&n,Th. von, Nachr. Ges. W i s s . Gottingen, 1930,58-76. K&rm&n,Th. von, Proc. 3rd Intern. Congr. Applied Mech., Stockholm, pp. 85-93 (1930). Ibzd., 4th Congr., Cambridge, pp. 54-91 (1935). Kessler, Univ. Wis. Eng. Expt. Sta.,Bull. 82 (1935). Nikuradse, J., Forsch. Gebiete Ingenieumu., Forschungsheft 356 (1932). Ibid., 361 (1933). Nikuradse, J . , Proc. 3rd Intern. Congr, Applied Mech., Stockholm, pp. 239-48 (1930).
