Znd. Eng. Chem. Res. 1994,33, 2016-2019
2016
CORRELATIONS Improved Correlating Equations for the Friction Factor for Fully Turbulent Flow in Round Tubes and between Identical Parallel Plates, both Smooth and Naturally Rough Stuart W. Churchill' and Christina Chant Department of Chemical Engineering, University of Pennsylvania, 31 1 A Towne Building, 220 South 3 r d Street, Philadelphia, Pennsylvania 19104
Mixed-mean velocities were computed by numerical integration using a comprehensive correlating equation for the velocity distribution, and independently using a corresponding correlating equation for the local turbulent shear stress. As contrasted with earlier analytical integrations for grossly simplified velocity distributions, these calculations take into account the effects of the boundary layer near the wall and the wake in the central region. The resulting sets of numerical values for the mixed-mean velocity are correlated almost exactly in terms of equations with a theoretically derived structure. On the basis of the generally overlooked analogyof MacLeod, the same distributions for the velocity and the turbulent shear stress are used for both round tubes and parallel plates, but of course, the expressions for the mixed-mean velocity differ because of the different areas of integration. The analysis reveals inter alia that the hydraulic-diameter and laminar-equivalentdiameter concepts are fundamentally unsound. The correlating equations for the mixed-mean velocity are extended for the transition engendered by naturally rough surfaces using the speculation of Colebrook, and are reexpressed directly in terms of the friction factor and the Reynolds number for convenience. Table 1. Mixed-Mean Velocities for Smooth Bound T u b s
Introduction The relationship between the mixed-mean velocity and the local velocity distribution in round pipes can be expressed in dimensionless form as follows:
urn+= 2S,lu+( 1 - $ ) d ( $ ) Since
urn+= Urn( ");2
=
(?) 2
U+.
Nikuradse (1932) correlated his own experimental data for the velocity distribution in the "turbulent core near the wall" (30 < y+ < a+/10) with the expression
+ 2.5 ln(y'1
(3)
and used this equation as an approximation for the entire cross section to derive
urn+= 1.75 + 2.5 ln(a+j
(4)
which can also be written as (;)12'
I W2I
= 1.75 + 2.5 In R e
* Author to whom correspondence should be addressed.
Current address: E.I.du Pont Marshall Laboratory, 3500 Grays Ferry Road, Philadelphia, PA 19146. 7
eq4 14.28 15.55 17.29 19.02 21.31 23.04 24.78 26.51
eq6 eqeland7 14.32 13.59 15.57 15.24 17.28 17.27 18.98 19.15 21.24 21.54 22.94 23.30 24.65 25.05 26.35 26.79
e q 8 eqelOand11 eq12 eq13 13.59 12.99 12.99 13.54 15.24 14.81 14.81 15.18 17.27 17.01 17.01 17.21 19.15 19.00 19.00 19.10 21.54 21.44 21.44 21.48 23.30 23.23 23.23 23.26 25.05 24.99 24.99 25.00 26.79 26.73 26.73 26.74
112
the friction factor can be determined by means of eqs 1 and 2 from an expression for u+ as a function of y+ and
u+ = 5.5
a+ 150 250 500 1 000 2500 5000 loo00 20000
Prandtl(1933)plotted the experimentaldata of a number of investigators for the friction factor in the equivalent of the coordinates suggested by eq 5, and fitted these values with a straight line corresponding to ' I 2 = 1.99
+ 2.46 In
It is tempting to rationalize the different constant and coefficient of eq 6 in comparison with those of eq 5 as corrections for the approximation of the entire velocity distribution by eq 3, but in actuality eqs 5 and 6 differ negligibly for all practical values of a+ = Re(f/8)'12. (See Table 1.) Herein, eq 4 or eq 5 rather than eq 6 will be used as a point of reference because of the incompatibility of the coefficient of 2.46 with the subsequent derivations. Equation 6 (or minor modifications thereof) has been widely accepted for general usage even to the present day. However, as will be shown, this success is somewhat fortuitous and perhaps not completely justified. The use of eq 3 for the velocity distribution over the entire cross section of the pipe neglects the downward deviation of the velocity due to the "boundary layer" (the combination of the viscous sublayer and the buffer layer)
088&5885/94/2633-2016$04.50/0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 2017
urn+= a+[ - J)l(u")+( 1 - $)2d(
$)I
(10)
[See Churchill (1990) for a derivation of eq 10.1 Equation 10 can be noted to reduce to Poiseuille's law for (u")'= 0; therefore the integral represents the contribution of the turbulent fluctuations to Urn+. A correlating equation for (u")' a, directly measurable quantity, can be used with eq 10 to evaluate urn+independently from eqs 1 and 7. Churchill and Chan (1994) have proposed the following such correlating equation:
H
R : 6100
6 4
""( 1 + 9y)-7/8 (11)
2
'0
a+
0 2 0 4 06 OS
IO
12
14
16
18 2 0 2 2
24 26
28
30 3 2
Figure 1. Experimental data of Lindgren and Chao (1969)for velocity distribution in fully developed turbulent flow of water in a 127-mm Plexiglas8 tubs (R= Re). (Reprinted with permission from Lindgren and Chao (1969). Copyright 1969 American Institute of Physics.)
near the wall and the upward deviation of the velocity due to the "wake" near the centerline as illustrated in Figure 1 by the representative experimental data of Lindgren and Chao (1969). As will be shown, these two effects are neither negligible individually nor completely compensatory.
Round Tubes Churchill and Chan (1994) have proposed the following correlating equation for the velocity distribution over the entire cross section (05 y+ 5 a+) and for the entire regime of fully turbulent flow (a+ I 150):
Equation 7 conforms to the presumed functional behavior in the viscous sublayer, the buffer layer, the region of overlap, and the region of the centerline. Equation 1 can readily be integrated numerically using u+ from eq 7 for a series of values of a+ to obtain the values of urn+listed in the fourth column of Table 1. These values can be represented almost exactly, as indicated in the fifth column of Table 1, by
urn+= 2.042 - 1612 + a+
(47.6 .+) + 2.5 In@+)
(8)
Terms in (a+)-' and (a+)-2are necessary components of any expression for urn+that is based on a velocity distribution incorporating
for the region adjacent to the wall. The leading constant of 2.042 in eq 8 was evaluated analytically from integration of the right-most combining term of eq 7, while the coefficients161.2and 47.6 were evaluated empirically from the computed values of Urn+. The mixed-mean velocity can also be expressed in terms of an integral of the local turbulent shear stress as follows:
The absolute value of the right-most set of terms in eq 11 is specified in order to avoid evaluating an odd power of a negative number for y+ < 0.614. This approximation is quite justified since the set of terms in question are completely negligible in that regime. The results of numerical integration of eq 10 using (u")' from eq 11 are included in Table 1 as the sixth column. These values are closely represented by
urn+= 1.989 - 268 + a+
(y)2 +
2.5 ln(a+)
(12)
as demonstrated by the values in the seventh column. Equation 11 is superior on theoretical grounds to eq 7 in that it incorporates one less empirical constant (the equivalent of 5.5 in eq 3 or 9.025 in eq 7) and fewer structural approximations (only e-2.slY+for 1 - 2.5/y+, and the aforementioned absolute value sign, in both instances to avoid singularities for very small values of y+). On the other hand, eq 7 appears to provide a more accurate representation of the velocity distribution for small values of a+ than that derived from eq 11 (see Churchill (1994), Chapter 6). The values of Urn+ computed from eq 10 using (u")' from eq 11 agree closely for large values of a+ with those computed from eq 1 using u+ from eq 7, but are somewhat lower a t small values of a+. Experimental data of sufficient precision and ostensible reliability to provide a critical test and comparison of the results of the numerically computed values in Table 1 do not appear to exist. Comparison of these predictions with those of eq 6, which presumably represents experimental data on the mean, suggests that eq 8 may be preferable for low values of a+ and eq 12 for high values of a+. As a compromise, the following combination of these two equations is proposed:
urn+= 1.989 --16'
a+
+ (4:f)2 - + 2.5 ln(a+)
(13)
As indicated by the right-most column of Table 1 the predictions of eq 13 approach those of eq 8 as a+ decreases and those of eq 12 as a+ increases. Equations 4,12, and 13 are compared in Figure 2 with the experimentaldata of Nikuradse (19321,Deissler (1950), Abbrecht (1956), and Pate1 and Head (1969). The data of Nikuradse are by far the most comprehensive in the literature but are somewhat suspect in accuracy. The predictions of all three of these expressions (and by inference those of eqs 6 and 8 as well) fall within the band of these data and are thus indistinguishable on experimental grounds. The failure of eq 8,12, or 13 to provide decisively improved representations in terms of Figure 2 may be in part a shortcoming of eqs 7 and 11, which are
2018 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994
Table 2. Mixed-Mean Velocities for Smooth Parallel Plates 30
b+ 150 250 500 1000 2 500 5000 10 000 20 000
25
u,+ 20
eq 19 15.53 16.80 18.54 20.27 22.56 24.29 26.03 27.76
eqe 7 and 16 15.40 16.90 18.79 20.60 22.94 24.69 26.43 28.17
eqs 11 and 17 14.62 16.37 18.49 20.43 22.84 24.62 26.37 28.11
eq 18 14.62 16.40 18.51 20.44 22.85 24.62 26.37 28.11
follow directly. Equation 14can also be expressed directly in terms of the friction factor and Reynolds numbers as
15
(
q)li2
= 1.989 -
161.2 Re(f/8)'/2 2.5 h(
a+
-
= Refll2 2
Figure 2. Mixed-mean velocities in fully developed turbulent flow. Nikuradae (1932)D (mm): 0,lQ0,2Q0,30;A,50; e, 100. Deissler (1950)0; Abbrecht (1956)e; Patel and Head (1969)V. Equation 4 (- - -); eq 12 eq 13 (-). (-e);
only applicable for fully turbulent flow, as well as of the imprecision and uncertainty of the experimental data for the pressure drop. Despite this result, eq 13 is strongly recommended over eqs 4 and 6 on structural grounds. The effects of the wake and the boundary layer on Urn+ and hence on f can be inferred by comparison of eqs 4 and 13. The difference of the first terms on the right-hand sides of these two expressions, namely, 1.989 - 1.750 = 0.239 represents the contribution of the wake to Urn+. This increment, which is independent of a+, results in a maximum increase of only 1.7% in urn+and a maximum decrease of only 3.3% in f (both at a+ = 150). The combination of the second and third terms on the right-hand side of eq 13 represents the decrease in Urn+ due to the boundary layer. This correction corresponds to-6.8% inurn+and+15.2%i n f a t a + = 150andtolesser values at larger values of a+. Although the corrections for the wake and for the boundary layer are in opposite directions, they do not compensate completely or even uniformly since one depends on a+ while the other does not. The net corrections at a+ = 150 are -5.1% in urn+and +11.1%in f , and at a+ = 20 000 they are +0.87% in Urn+ and -1.7% in f. The predicted net effect is thus significant only at low values of a+. Equation 13can be generalized as follows for the gradual transition engendered by the roughness of commercial pipes by adapting the conjecture of Colebrook (1938-1939) as rationalized by Churchill (1973). This results in
(
161 2 + A urn+= 1.989 -2):4; a+
+
(
47.6
)2
+
1
Re(f/8)'l2 (15) 1 + 0.301(8/a)~e(f/8)~/~
Since eq 15 is implicit in f , iterative solution is necessary for specified values of Re and ala, but such calculations are rapidly convergent. (The divisors off in eq 15 could be incorporated in the other constants, but they have been retained to maintain the identification with eq 14.)
Parallel Plates The generally overlooked speculation by MacLeod (1951) that u+b+,b+)for parallel platesis identical to u+(y+,a+} for round tubes in the regime of fully turbulent flow as well as in the regime of laminar flow appears to be well substantiated by experimental data, asymptotic analyses, and numerical simulations [see, for example, Churchill (1994), Chapter 101. It follows that (=)+ {y+,b+}must then be identical to (u")+{y+,a+j. This identity also appears to be substantiated. On the other hand, the dimensionless mixed-mean velocity for flow between parallel plates is given by Urn+{b+} =
&+d(
5)
(16)
and
is clearly not the same function as u,+{u+} for round tubes. Values of um+{b+] calculated according to eq 16, using u+ from eq 7, with b+ substituted for a+, and according to eq 17, using (u") from eq 11, again with b+ substituted for a+, are listed in Table 2. The latter set of values may be observed to be well represented by
+
urn+= 3.3618 - - 2.5 ln(b+} b+
+
2.5 In
a+
1
1+ 0.301(8/a)a+
(14)
Equation 14is recommended as an improved expression for the mixed-mean velocity for round pipes, both smooth and naturally rough, for the complete range of fully turbulent flows (a+ > 150 or Re > 3900). urn+is readily calculated for specified values of a+ and 8la. The corresponding values o f f = ( 2 / ~ , + )and ~ Re = 2um+a+
The form of eq 18 was identified by the same process as was used to determine the form of eq 8 for round tubes. The constants 3.3618 and 190.83 were chosen on the basis of the values computed from eq 11. It may be noted that Dukler and Bergelin (1952) derived an expression with a structure equivalent to eq 18 for turbulent, nonrippling flow of a film of liquid down a vertical wall by using a three-part segmental velocity distribution. Because of the relative crudity of their velocitydistribution, the constants differ greatly from those of eq 18even though the predicted values of urn+do not.
Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 2019 Integration of eq 16 using u+ from eq 3 gives
urn+= 3
+ 2.5 ln{b+j
examples of the insight to be gained by the critical assessment of widely accepted approximations. (19)
Nomenclature Equation 19, which is the analog of eq 4, serves as a frame of reference for identifying the effects of the wake and the boundary layer. The correction of urn+due to the wake is seen to be 3.3618 - 3.0 = 0.3618, which is somewhat greater than the corresponding value of 0.239 for round tubes. The correction to urn+due to the boundary layer is -190.83/ b+, which is somewhat greater percentagewise than that for round tubes for b+ = a+. The net corrections are therefore also somewhat greater than for round tubes for b+ = a+. The hydraulic-diameter analogy and the laminarequivalent-diameter analogy have sometimes been proposed for estimation of the friction factor for turbulent flow between parallel plates, but these procedures do not appear to be sound theoretically in view of the term in (a+)-2 in eqs 12-14, which has no analog in eq 18. Equation 18can be generalized speculatively for parallel plates of equal roughness as 190 83 urn+= 3.3618 - - 2.5 In b+
+
(1+
Greek Symbols
0.3&i?/b)b+I
(20)
For parallel plates, the relationship between the dimensionless half-width b and the Reynolds number based on the hydraulic diameter 4b is
Accordingly, eq 19 can be reexpressed as
(;>'I2
= 3.3618 -
190.83 Re (f/ 32)"' 2.5 In
a = radius of round pipe, m a+ = (a/v)(Tw/p)1/2 b = half-spacing between parallel plates, m b+ = (b/v)(Tw/p)1/2 b = effective roughness, m f = 2rw/purn2= Fanning friction factor Re = Reynolds number = 2auJv for round tubes = Q b u d v for parallel plates u = local time-mean velocity, m/s u+ = u(p/rw)'/2 urn = mixed-mean velocity, m/s Urn+ = Urn(P/T,)'/2 -pu'v' = turbulent shear stress (u'v')' = -PU'U'/T, = dimensionless turbulent shear stress y = distance from wall, m Y+ = (I'/V)(TW/P)'/~
+
Re (fl32) 1+ 0.301(i?/b)Re(f/32)'/2
Equations 20 and 22 are recommended, primarily on theoretical grounds because of the lack of experimental data, for prediction of the friction factor for fully turbulent flow (b+ > 150 or Re > 8770) between identical parallel plates, either rough or smooth. This shortage of experimental data is due in part to the difficulty of simulating a cross section of infinite breadth while maintaining uniform spacing. Equation 22 with Re based on the hydraulic diameter is proposed as a first-order approximation for other geometries.
Concluding Remarks Equation 15 is recommended as an improved, theoretically based expression for the prediction of the friction factor in fully turbulent flow (Re > 3900) in round tubes, both smooth and naturally rough and the intervening transition. Equation 22 is recommended as the corresponding expression for fully turbulent flow (b+ > 150 or Re > 8770) between identical parallel plates, including those that are naturally rough. Accounting for the velocity distribution near the wall in the determination of urn+ = l/fI2 by the integration reveals that the hydraulic-diameterand laminar-equivalentdiameter analogies are fundamentally unsound. The net numerical corrections to the friction factor that arise from consideration of the wake and the boundary layer are perhaps too small to be of practical interest, but are in any event of intrinsic interest educationally as
v = kinematic viscosity, m2/s p = specific density, kg/m3 T,
= shear stress on the wall, Pa
Literature Cited Abbrecht, P. H. Effect of Initial Velocity Distribution on Heat Transfer in Smooth Pipes. Ph.D. Thesis, University of Michigan, Ann Arbor, 1956. Churchill, S. W. Empirical Expressions for the Shear Stress in Turbulent Flow in Commercial Pipe. AIChE J. 1973,19, 375376. Churchill, S. W. New and Overlooked Relationships for Turbulent Flow in Channels. Chem. Eng. Technol. 1990,13,264-272. Churchill, S. W.Turbulent Flows. The Practical Use of theory; Notes; University of Pennsylvania: Philadelphia, 1994. Churchill, S. W.; Chan, C. Theoretically Based Correlating Equations for the Local Characteristics of Fully Turbulent Flow in Round Tubes and between Parallel Plates. Submitted for publication in Znd. Eng. Chem. Res. 1994. Colebrook, C. F. Turbulent Flow in Pipes with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws. J. Znst. Ciu. Eng. (London) 1938-1939,11,133-156. Deissler, R. G. "Analytical and Experimental Investigation of Adiabatic Turbulent Flow in Smooth Tubes"; National Advisory Committee for Aeronautics TN 2138;Washington: DC, July 1950. Dukler, S. E.;Bergelin, 0. P. Characteristics of Flow in Falling Films. Chern. Eng. Prog. 1952,48,557-563. Lindgren, E. R.; Chao, J. Average VelocityDistribution of Turbulent Pipe Flow with Emphasis on the Viscous Sublayer. Phys. Fluids 1969,12,1364-1371. MacLeod, A. L. Liquid Turbulence in a Gas-Liquid Absorption System. Ph.D. Thesis, Carnegie Institute of Technology, Pittaburgh, PA, 1951. Nikuradse, J. Gesetzmbsigkeit der turbulenten Stramung in glatten Rohren. Ver. Dtsch. Zng.-Forschungsh. 1932,356.Engl. transl.: Laws of Flow in Smooth Tubes. National Advisory Committee for Aeronautics TM62; Washington, DC, 1950. Patel, C. V.; Head, M. R. Some Observations on Skin Friction and Velocity Profiles in Fully Developed Pipe and Channel Flows. J. Fluid Mech. 1969,38,181-201. Prandtl, L. Neuere Ergebnisse der Turbulenzforschung. 2. Ver. Deutsch. Zng. 1933,77, 105-114. Received for review December 14, 1993 Revised manuscript received March 8, 1994 Accepted May 26, 1994' e
Abstract published in Advance ACS Abstracts, July 1,1994.
