Ind. Eng. Chem. Fundam. 1980, 19, 228-230
228
CORRESPONDENCE Comments on: “An Explicit Equation for Friction Factor in Pipe” Sir: The recent claim by Chen (1979) that the explicit equation for the friction factor in pipe that he has proposed is good for all values of Reynolds number (Re)is not true; it definitely does not apply in the laminar zone, and it probably does not apply in the critical zone. Furthermore, when doing a comparison such as was done by Chen, it must be remembered that the proposed equations are being compared to a previously determined equation and not to actual data. Therefore, when the deviations from the widely used Colebrook-White equation are no greater than those obtained by Chen for the Chen and Churchill equations, it is hard to justify the selection of either equation based solely on the deviations. Perhaps one of these equations even fits the available data better than the Colebrook-White equation; until the comparison is made, however, this will not be known. One advantage of the Churchill equation (1977) is that it does give reasonable values for the friction factor within the practical roughness range for all Reynolds numbers (above 0) up to lo8 for all pipes and to even higher Reynolds numbers for all but very smooth pipes. Since it is a continuous function for Reynolds numbers above 0, it also lets one (or a machine) calculate a friction factor in the critical zone; of course the accuracy of such a friction factor probably cannot be determined, but it will be reasonable. Also, Chen gives a validity criteria for the Colebrook-
White and von Karman equations that is incorrect; moreover, the von Karman equation is incorrect since the right-most constant should be 1.14 rather than 1.74. The line represented by the relation (D/E)/(Re& = 0.005, where D is the diameter, t is the effective roughness, and f is the Darcy friction factor, is the boundary line between the transition zone and the fully turbulent zone, where ( D / t ) / ( R e & is less than 0.005. The Colebrook-White equation is valid for Reynolds numbers greater than about 3000 to 4000, the generally accepted boundary between the critical and transition zones. The Colebrook-White and von Karman equations become identical when (D/E)/ (Re& becomes sufficiently small (about 0.005). Knudsen and Katz (1958) made a similar mistake regarding the validity criteria. Finally, it should be noted that Jain (1976) has also proposed an explicit equation to replace the ColebrookWhite equation.
Sir: Chen (1979) presents an empirical equation for the friction factor in turbulent flow which he contends has maximum deviations of +0.4651 to -0.2286% from the Colebrook (1939) equation as compared to maximum deviations of +0.6041 to -2.6096% for the empirical equation of Churchill (1977). It should be noted that the criterion Chen has chosen for this comparison is itself an empirical representation for the experimental data of Colebrook. In view of the scatter and misfit of that data, and of subsequent changes in methods of manufacture of pipe, hence in the character of the roughness, a maximum difference of +0.6 to -2.6% between the three equations is not significant. A meaningful comparison a t that level would require more precise experimental data for modern pipe. If attention were to be confined to Re > 4000 and if the data of Colebrook were accepted as a standard, the equation of Colebrook would appear to be preferable to the other two since the trial and error required for a
specified pressure gradient is not onerous. The merit of the equation of Churchill is that it provides a single, continuous representation for the laminar, transitional, and turbulent regimes, whereas the equations of Colebrook and Chen are limited to Re > 4000 and hence to the turbulent regime alone. (Transition from laminar to turbulent flow occurs from -1800 to -4000, not above Re = 4000 as stated by Chen.)
Sir: Development of explicit equations for friction factor became lately a popular subject in the engineering literature. One such equation has just recently been proposed in this journal by Chen (1979). The objective of those explicit equations is to correlate the following implicit equation, which was originated by Colebrook (1939), for the friction factor calculation
-2-= -2.0
6
where
fD
--)
log (2+ 2.51 3.7D R e d
Literature Cited Chen, N. H., Ind. Eng. Chem. Fundam., 18, 296 (1979). Churchill, S. W., Chem. Eng.. 84, 91 (Nov 7, 1977). Jain, A. K., J. Hyd. Div., Roc. ASCE, 102,674 (1976). Knudsen, J. G.,Katz, D. L., “Fluid Dynamics and Heat Transfer”, McGraw-Hill. New York, N.Y., 1958.
Literature Cited Chen, N. H., Ind. Eng. Chem. Fundam., 18, 296 (1979). Coiebrook, C. F., J. Inst. CivilEng., (London), 11, 133 (1938-1939). Churchill, S. W., Chem. Eng.. 84 (24),91 (Nov 7, 1977).
is Darcy’s friction factor, Re is the Reynolds 0 196-4313/80/1019-0228$01 .OO/O
Stuart W. Churchill
Department of Chemical & Biochemical Engineering University of Pennsylvania Philadelphia, Pennsylvania 19104
number, t is the roughness of the pipe, and D is its diameter. The explicit equation which was proposed by Chen is c
1 - = -2.0 log I/% 13.7665D ~
5.0452 (1)
Bernard J. Schorle
Chicago Bridge & Iron Company Oak Brook, Illinois 60521
Re log
(A( 6)
1.1098
+ 5.8506)]
(2)
Re0.8981
According to Chen this equation correlates a slightly modified form of Colebrook’s equation (where the con-
0 1980 American Chemical Society
