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Explicit Friction Factor Correlation for Turbulent Flow in Smooth Pipes Chetan T. Goudar*,† and Jagadeesh R. Sonnad‡ Biological Products Division, Bayer HealthCare, 800 Dwight Way, B56-A, Berkeley, California 94710, and Department of Radiological Sciences, University of Oklahoma Health Sciences Center, Oklahoma City, Oklahoma 73190
We present a truly explicit representation of the Nikuradse, Prandtl, von Ka´rma´n (NPK) friction factor equation for turbulent fluid flow in smooth pipes. Specifically, the Lambert W function is used to relate the friction factor explicitly to the Reynolds number. In addition to the elegance of an explicit expression, this new representation allows for the computation of friction factor values that are accurate to machine precision. As this is an exact explicit representation, it is far more accurate than current methods such as graphical estimation of friction factors from friction factor versus Reynolds number charts or several other explicit approximations of the NPK equation. We believe that this is the first truly explicit representation of the NPK equation and the elegance and high accuracy of this representation should make it the method of choice for estimating turbulent flow friction factors in smooth pipes. Introduction The friction factor for turbulent fluid flow in smooth pipes is usually described by the expression
1 ) 4.0 log10(Rexf) - 0.4 xf
(1)
where f is the friction factor and Re is the Reynolds number. Equation 1 is also referred to as the universal law of friction and is attributed to the work of Nikuradse, Prandtl, and von Ka´rma´n1-3 (NPK equation). Equation 1 is implicit in f, requiring the need for iterative estimation of the friction factor. Alternatively, f vs Re charts have been constructed4 that allow for the graphical estimation of f when the Reynolds number is known. Several explicit approximations to eq 1 have also been proposed1,5-12 that provide friction factor estimates of varying accuracy. Recently, a neural-network-based approach was also proposed for the estimation of f from eq 1.13 Although eq 1 is based on the logarithmic velocity distribution profile, a power-law formulation of the velocity profile in turbulent flow has also been proposed,14,15 resulting in an alternative expression for the friction factor. The complication of using eq 1 arises because it is implicit in f, and there have been no reports to date that present eq 1 as truly explicit in f. However, an explicit form of the Colebrook-White equation6 that is used to describe fluid flow in rough pipes was derived recently through the use of computer algebra.16 In the present study, we present an alternate form of eq 1 that is truly explicit in the friction factor, f. Specifically, the friction factor is expressed as a function of the Reynolds number alone through use of the Lambert W function. This new representation of eq 1 provides estimates of f that * To whom correspondence should be addressed. Tel.: 510705-4851. Fax: 510-705-5451. E-mail: chetan.goudar.b@ bayer.com. † Bayer HealthCare. ‡ University of Oklahoma Health Sciences Center.
are accurate to machine precision. We believe that this is the first truly explicit representation of eq 1, and its high accuracy along with its explicit nature should make it the preferred method for computing friction factors for turbulent flow in smooth pipes. Theory In this section, we derive an explicit form of eq 1 through simple algebraic techniques. To simplify the derivation, eq 1 can be rewritten as
1 ) a ln(Rexf) - b xf
(2)
where a ) 4/ln(10) and b ) 0.4. Equation 2 can be simplified to
( )
1 1 ) a ln(Re) - b + a ln xf xf
(3)
Substituting φ ) 1/axf into eq 3 results in the expression
φa + a ln(φa) ) a ln(Re) - b
(4)
Dividing eq 4 by a and rearranging then gives
(Rea) - ab
φ + ln(φ) ) ln
(5)
The left-hand side of eq 5 is analogous to the Lambert W function, which is defined as17
W(x) + ln[W(x)] ) ln(x)
(6)
From eqs 5 and 6, φ can be written as
φ)W
[Rea exp(- ab)]
(7)
Substituting for φ ) 1/axf in eq 7 results in the desired explicit expression for the friction factor
10.1021/ie0300676 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/17/2003
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2879
1 Re b ) aW exp a a xf
[
( )] (8)
4 , b ) 0.4 a) ln(10) Substituting for a and b in eq 8 gives
1 ) 1.7372W[0.4573Re] xf
(9)
where the value of a is accurate to four decimal places. If additional accuracy in f estimates is desired, such a truncation should not be made. It is clear from eq 8 that this representation of eq 1 is truly explicit in the friction factor, f. Several approaches that allow for the highly accurate computation of W in eq 8 exist.17-20 Thus, eq 8 offers a simple and highly accurate method of estimating the friction factor for turbulent flow in smooth pipes. Also, unlike the original implicit form, it does not involve iterative estimation of f through root-solving techniques such as the Newton-Raphson method.
Figure 1. Error associated with computing the friction factor using eq 8. A total of 1000 f values were computed for 1000 Re values that were logarithmically spaced in the range 4000 < Re < 108.
Results and Discussion Verifying the Accuracy of f Values Obtained from Eq 8. To verify the accuracy of f values obtained from eq 8, 1000 logarithmically spaced Re values were generated in the range 4000 < Re < 108, and the corresponding f values were calculated using eq 8. These 1000 sets of Re and f values were substituted into a rearranged form of eq 1 that resulted in error estimates for f values computed from eq 8
error )
1 - 4.0 log10(Rexf) + 0.4 xf
(10)
All computations were performed using double-precision arithmetic, and the Lambert W function was evaluated using the Maple implementation (Waterloo Maple Inc.) that is based on the method presented in Corless et al. (1996).17 Figure 1 shows a plot of the error in the f values computed using eq 8 for 1000 Re values between 4000 and 108. The maximum error in f was 5.7 × 10-15, suggesting that machine-precision f values could be readily obtained from eq 8. Comparison of f Values from Eq 8 with other Friction Factor Correlations. The f values obtained from eq 8 were compared with some of the more accurate explicit approximations of the NPK equation10,12 and the Barenblatt formula.14,15 The percentage difference in friction factor estimates was computed as
(
)
|fPNK - fcorrelation| × 100 percentage f difference ) fPNK (11) where fPNK is the value computed using eq 8 and fcorrelation is the friction factor obtained from the other correlations. Figure 2 shows the percentage difference in friction factor estimates as a function of Reynolds number for the correlations examined in this study. The approximations presented in Jain (1976)10 and Techo et al. (1965)12 were very good representations of the PNK equation, with maximum absolute differences of 1.93 and 0.45%, respectively. Friction factor estimates
Figure 2. Percentage differences in friction factor estimates computed using eq 11 as a function of Reynolds number for some of the friction factor correlations examined in this study.
from the Barenblatt formula14,15 that is based on the power-law distribution of turbulent flow velocity were characterized by a maximum difference of 3.88%. Conclusions The implicit nature of eq 1 has required the use of graphical or numerical solution techniques to compute the friction factor for turbulent fluid flow in smooth pipes. Although several explicit approximations of eq 1 have been proposed, there has never been, to our knowledge, a representation of eq 1 that is truly explicit in the friction factor. In this study, we have presented a truly explicit form of eq 1 through the use of the Lambert W function. In addition to its explicit nature, this new expression (eq 8) provides estimates of friction factor that are accurate to machine precision. The convenience of the explicit solution coupled with its high accuracy should make this new expression the method
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of choice for estimating turbulent flow friction factors in smooth pipes. Literature Cited (1) Nikuradse, J. Gesetzma¨ssigkeit der turbulenten stro¨mung in glatten ro¨hren. Ver. Dtsch. Ing.-Forschungsh. 1932, 356. (2) Prandtl, L. Neuere ergebnisse der turbulenzforschung. Z. Ver. Deutsch. Ing. 1933, 77, 105. (3) von Ka´rma´n, T. Turbulence and skin friction. J. Aerospace Sci. 1934, 7, 1. (4) Moody, L. F. Friction factors for pipe flow. Trans. ASME 1944, 66, 671. (5) Blasius, H. Das a¨hnlickhkeitsgesetz bei reibungsvorga¨ngen in flu¨ssigkeiten. Forsch. Arb. Ingenieurwes. 1913, 131. (6) Colebrook, C. F. Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civil Eng. 1938-1939, 11, 133. (7) Drew, T. B.; Koo, R. C.; McAdams, W. H. The friction factor for clean round pipes. Trans. AIChE 1932, 28, 56. (8) Filonenko, G. K. Hydraulic resistance in pipes. Teploenergetika 1954, 1, 40 (in Russian). (9) Gulyani, B. B. Simple equations for pipe flow analysis. Hydrocarbon Process. 1999, 67. (10) Jain, A. K. Accurate explicit equations for friction factor. J. Hydraul. Div. 1976, 102, 674. (11) McAdams, W. H. Heat Transmission; 3rd ed.; McGrawHill: New York, 1954. (12) Techo, R.; Tickner, R. R.; James, R. E. An accurate equation for the computation of the friction factor for smooth pipes from the Reynolds number. J. Appl. Mech. 1965, 32, 443.
(13) Sablani, S. S.; Shayya, W. H.; Kacimov, A. Explicit calculation of the friction factor in pipeline flow of Bingham plastic fluids: A neural network approach. Chem. Eng. Sci. 2003, 58, 99. (14) Barenblatt, G. I.; Chorin, A. J. Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 6749. (15) Barenblatt, G. I.; Chorin, A. J.; Prostokishin, V. M. Scaling laws for fully developed turbulent flow in pipes: Discussion of experimental data. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 773. (16) Keady, G. Colebrook-White formula for pipe flows. J. Hydraul. Eng. 1998, 124, 96. (17) Corless, R. M.; Gonnet, G. H.; Hare, D. E.; Jeffrey, D. J.; Knuth, D. E. On the Lambert W function. Adv. Comput. Math. 1996, 5, 329. (18) Barry, D. A.; Barry, S. J.; Culligan-Hensley, P. J. Algorithm 743: A Fortran routine for calculating real values of the W-function. ACM Trans. Math. Software 1995, 21, 172. (19) Barry, D. A.; Culligan-Hensley, P. J.; Barry, S. J. Real values of the W-function. ACM Trans. Math. Software 1995, 21, 161. (20) Fritsch, F. N.; Shafer, R. E.; Crowley, W. P. Algorithm 443: Solution of the transcendental equation wew ) x. Commun. ACM 1973, 16, 123.
Received for review January 24, 2003 Revised manuscript received April 25, 2003 Accepted April 30, 2003 IE0300676
