A Review of Explicit Approximations of Colebrook's Equation

Srbislav Genić Associate Professor University of Belgrade Faculty of Mechanical Engineering Ivan Arandjelović Associate Professor University of Belgrade Faculty of Mechanical Engineering Petar Kolendić Research Assistant University of Belgrade Faculty of Mechanical Engineering Marko Jarić Research Assistant University of Belgrade Innovation Center of Faculty of Mechanical Engineering Nikola Budimir Research Assistant University of Belgrade Innovation Center of Faculty of Mechanical Engineering A Review of Explicit Approximations of Colebrook’s Equation The most common explicit correlations for estimation of the friction factor in rough and smooth pipes are reviewed in this paper. Comparison of any friction factor equation with the Colebrook’s equation was expressed trough the mean relative error, the maximal positive error, the maximal negative error, correlation ratio and standard deviation. The statistical comparison of different equations was also carried out using the “Model selection criterion” and “Akaike Information Criterion”. It was found that the equation of Zigrang and Sylvester provides the most accurate value of friction factor, and that Haaland’s equation is most suitable for hand calculations. Keywords: Colebrook’s equation, friction factor, approximations, fluid mechanics, turbulent flow. Vojislav Genić Head of Public Health Care and Mobility Siemens IT Solutions and Services, Belgrade 1. INTRODUCTION The determination of a single-phase friction factor of pipe is essential to a variety of industrial applications, such as single-phase flow systems, two-phase flow systems and supercritical flow systems. Typically, the method of choice for computing friction factor is the Colebrook’s equation. This equation is a combination of Prandtl-von Karman-Nikuradse smooth-pipe equation 1 f ( ) = 2 log Re f − 0.08 (1) and rough-pipe equation 1 f = 1.14 − 2 log ( ε ) (2) where Re is the Reynolds number and ε is the relative pipe roughness. Equations (1) and (2) are known as PKN equations [1]. Using these equations and his own data gathered on commercial pipes, Colebrook [2] formed the following equation that covers the whole turbulent flow region ⎛ 2.51 ε ⎞ = −2 log ⎜ + ⎟⎟ ⎜ f ⎝ Re f 3.7 ⎠ 1 (3) Received: April 2011, Accepted: May 2011 Correspondence to: Marko Jaric Innovation Center of Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia E-mail: [email protected] © Faculty of Mechanical Engineering, Belgrade. All rights reserved that became widely accepted design formula for turbulent friction in the range of Re = 4000 – 108 and ε = 0 – 0.05. Due to its demonstrated applicability, the Colebrook’s equation (3) has become the acceptable standard for calculation of the friction factor in turbulent regimes. It should be noted that Rouse [3] was the first to confirm Colebrook’s equation (3) by his own measurements. Equation (3) was plotted in 1944 by Moody [4] into what is now called the Moody chart for pipe friction (this chart is probably the most famous and useful figure in engineering fluid mechanics). The implicit form of (3) disables the quick estimation of friction factor in hand calculations. For this reason, a number of approximate explicit counterparts have been proposed in the last 60 years and a most recent and very good overview of these equations is given in [5-7]. The basic idea of these efforts is to introduce more parameters in equation, in order to obtain as good results as possible, or more precisely as close prediction as possible of a Colebrook’s equation. These explicate equations were compared with Colebrook’s equation as shown in Section 3. 2. EXPLICIT EQUATIONS FOR CALCULATION OF THE FRICTION FACTOR IN TURBULENT FLOW The most widely used explicit approximations for the Colebrook’s equation postulated since 1947 are synthesized in Table 1, in the order of publication. Additionally, this table contains the range of validity for each approximation cited as defined in the original paper. Most of these approximations are typically valid over only a limited range of the Re and ε values encountered in practice. FME Transactions (2011) 39, 67-71 67 Table 1. Various approximations of the Colebrook’s equation Eq. num. Equation 1/ 3 ⎤ ⎡ ⎛ 106 ⎞ ⎥ ⎢ f = 0.0055 1 + ⎜ 20000ε + ⎟ ⎢ ⎜ Re ⎟⎠ ⎥ ⎣ ⎝ ⎦ (4) (5) ⎛ 68 ⎞ +ε ⎟ f = 0.11⎜ ⎝ Re ⎠ (6) f = 0.53ε + 0.094ε 0.225 + 88ε 0.44 Re−1.62ε ⎡ 5.74 ⎞ ⎤ ⎛ ε f = ⎢ −2 log ⎜ + ⎟⎥ ⎝ 3.7 Re0.9 ⎠ ⎦ ⎣ (9) (10) −2 −2 ⎛ ε 1.1098 5.8506 ⎞ ⎤ ⎫⎪ 5.0452 ⎧ ⎡ ε − + f = ⎨−2 log ⎢ log ⎜ ⎟⎥ ⎜ 2.8257 Re0.8981 ⎟ ⎥ ⎬ Re ⎣ 3.7065 ⎩ ⎝ ⎠ ⎦ ⎭⎪ ⎡ 6.5 ⎞ ⎤ ⎛ f = ⎢ −1.8log ⎜ 0.135ε + ⎟ Re ⎠ ⎥⎦ ⎝ ⎣ (11) (12) 0.134 −2 ⎡ 21.25 ⎞ ⎤ ⎛ f = ⎢1.14 − 2 log ⎜ ε + ⎟⎥ Re0.9 ⎠ ⎦ ⎝ ⎣ (8) −2 −2 ⎧ 5.02 13 ⎞ ⎞ ⎤ ⎫⎪ ⎡ ε 5.02 ⎛ ⎛ ε f = ⎨−2 log ⎢ log ⎜ ε − log ⎜ − + ⎟ ⎟⎥ ⎬ Re ⎝ ⎝ 3.7 Re ⎠ ⎠ ⎦ ⎪⎭ ⎣ 3.7 Re ⎩ ⎧⎪ ⎡⎛ ε ⎞1.11 6.9 ⎤ ⎫⎪ f = ⎨−1.8log ⎢⎜ + ⎥⎬ ⎟ Re ⎥ ⎪ ⎢⎣⎝ 3.7 ⎠ ⎪⎩ ⎦⎭ (13) Ref. Authors (year) Re = 4000 – 5 · 108 ε = 0 – 0.01 [8] Moody (1947) Not specified [9] Altshul (1952) Re = 4000 – 5 · 107 ε = 0.00001 – 0.04 [10] Wood (1966) Not specified [11] Churchill (1973) Re = 5000 – 107 ε = 0.00004 – 0.05 [12] Jain (1976) Re = 5000 – 108 ε = 0.000001 – 0.05 [13] Swamee, Jain (1976) Re = 4000 – 4 · 108 [14] Chen (1979) Re = 4000 – 4 · 108 ε = 0 – 0.05 [15] Round (1980) Re = 4000 – 108 ε = 0.00004 – 0.05 [16] Zigrang, Sylvester (1982) Re = 4000 – 108 ε = 0.000001 – 0.05 [17] Haaland (1983) Re = 4000 – 108 ε = 0 – 0.05 [18] Tsal (1989) Re = 4000 – 108 ε = 0 – 0.05 [19] Manadilli (1997) Re = 3000 – 1.5 · 108 ε = 0 – 0.05 [20] Romeo, R yo, onzon (2002) Re = 3000 – 108 ε = 0 – 0.05 [21] Fang (2011) Not specified [7] Brkić (2011) Not specified [7] Brkić (2011) 0.25 ⎡ 7 ⎞⎤ ⎛ ε f = ⎢ −2 log ⎜ + ⎟⎥ ⎝ 3.7 Re0.9 ⎠ ⎦ ⎣ (7) Range −2 −2 ⎛ 68 ⎞ A = 0.11⎜ +ε ⎟ ⎝ Re ⎠ If A ≥ 0.018 then f = A and if A < 0.018 then f = 0.0028 + 0.85 A 0.25 (14) ⎡ 95 96.82 ⎞ ⎤ ⎛ ε + − f = ⎢ −2 log ⎜ ⎟⎥ 0.983 Re ⎠ ⎦ ⎝ 3.70 Re ⎣ (15) ⎧ ⎡ ε 5.0272 4.567 ⎞ ⎞ ⎪ ⎛ ε f = ⎨−2 log ⎢ log ⎜ − − ⋅ ⎟ ⎟ Re Re ⎢⎣ 3.7065 ⎝ 3.827 ⎠ ⎟⎠ ⎪⎩ ⎛ ⎛ ε ⎞0.9924 ⎛ 5.3326 ⎞0.9345 ⎞ ⎞ ⎤ ⎫⎪ ⎟ ⎟⎥ ⋅ log ⎜ ⎜ +⎜ ⎟ ⎜ ⎝ 7.79 ⎟⎠ ⎟ ⎟ ⎥ ⎬⎪ ⎝ 208.82 + Re ⎠ ⎝ ⎠ ⎠⎦ ⎭ (16) (19) −2 ⎡ ⎛ 60.525 56.291 ⎞ ⎤ + f = 1.613 ⎢ln ⎜ 0.234ε 1.1007 − 1.1105 1.0712 ⎟ ⎥ Re Re ⎝ ⎠⎦ ⎣ (17) (18) −2 β = ln −2 , f = ⎡⎢ −2 log ⎛⎜10−0.4343β + ε ⎞⎟ ⎤⎥ 3.71 ⎠ ⎦ ⎝ ⎣ ⎛ ⎞ 1.1Re 1.816 ln ⎜⎜ ⎟⎟ ⎝ ln (1 + 1.1Re ) ⎠ β = ln Re , f = ⎡⎢ −2 log ⎛⎜ 2.18β + ε ⎞⎟ ⎤⎥ 3.71 ⎠ ⎦ ⎝ Re ⎣ ⎛ ⎞ 1.1Re 1.816 ln ⎜⎜ ⎟⎟ ⎝ ln (1 + 1.1Re ) ⎠ 68 ▪ VOL. 39, No 2, 2011 Re −2 −2 FME Transactions o ∆av, standard deviation 3. STATISTICAL COMPARISON OF THE EQUATIONS ⎛ f C,i − f pred,i ∑ ⎜⎜ f C,i i =1 ⎝ n n The statistical comparison of any friction factor equation with the Colebrook’s equation can be done by the following procedure: • Divide the range of possible Re and ε using appropriate pitch into n nodes. • Calculate the friction factor fpred,i by the individual approximate equation. • Calculate friction factor value fC,i calculated with the Colebrook’s equation (fC,i was calculated numerically within the range of error ± 10–8). • Calculate the following parameters: o the mean relative error 1 n f C,i − f pred,i ∑ n i =1 f C,i meanRE = o the maximal positive error ⎛ f C,i − f pred,i maxRE + = max ⎜ ⎜ f C,i ⎝ (20) ∆av = ⎛ f pred,i − fC,i maxRE − = max ⎜ ⎜ f C,i ⎝ o Θ, correlation ratio ∑ ( fC,i − f pred,i ) n Θ = 1 − i =n1 ∑ ( fC,i − fC,av ) 2 (24) where fC,av is the average value of fC for complete set of nods ∑ fC,i n f C,av = i =1 n . (25) In this paper, we will use the range of Re = 4000 – 108 and ε = 0 – 0.05 and a net will be formed using linear scale with 106 nods. Three ways were used to produce the number of nods, presented in Table 2. 6 Table 2. Three ways for forming the net with 10 nods ⎞ ⎟ ⎟ ⎠ (21) I II o the maximal negative error ⎞ ⎟⎟ ⎠ ⎞ ⎟⎟ ⎠ (22) 2 (23) 2 i =1 III Range Nods Re = 4000 – 108 ε = 0 – 0.05 1000 99996 1000 50 · 10–6 Re = 4000 – 108 10000 9999.6 100 500 · 10–6 100 999960 10000 5 · 10–6 ε = 0 – 0.05 Re = 4000 – 108 ε = 0 – 0.05 Linear step It should be noted that similar analysis covering the observed range (Re = 4000 – 108 and ε = 0 – 0.05) with a much lesser number of points (about 500 points in [20], 1000 points in [21], 10000 points in [5] and [22], 740 points in the recent one [7]). The statistical comparison of different equations was also carried out using the “Model selection criterion” (MSC) and “Akaike Information Criterion” (AIC). Table 3. Statistical parameters for observed equations Eq. num. meanRe [%] maxRe+ [%] maxRe– [%] Θ [%] ∆av [%] MSC AIC · 10–6 NP NC (4) 7.517 15.90 – 12.532 84.22 8.853 – 29.92 3.493 4 5 (5) 16.42 46.83 – 2.622 30.26 18.34 – 30.72 4.864 3 4 (6) 3.647 100 – 6.241 99.02 10.37 – 1.040 7 11 (7) 0.0818 0 – 0.00121 100 0.685 – – 1.882 5 8 (8) 0.181 0.790 – 3.185 100 0.335 – 25.95 – 3.212 5 8 (9) 0.0406 0.708 – 3.358 100 0.315 – – 3.305 5 8 (10) 0.0676 0.316 – 0.324 100 0.0686 – 25.16 – 6.514 8 14 (11) 90.21 94.45 0 0 90.33 – 32.33 7.857 4 7 (12) 0.000612 0.114 – 0.0496 100 0.00615 – – 14.087 7 16 (13) 0.207 1.420 – 1.314 100 0.222 – – 4.393 5 8 (14) 16.16 27.30 – 2.622 30.26 17.99 – 30.71 4.864 4 5 (15) 0.0324 0.00404 – 2.729 100 0.245 – – 3.755 6 10 (16) 0.0680 0.0815 – 0.146 100 0.069 – 25.00 – 6.511 11 20 (17) 0.0550 0.441 – 0.491 100 0.077 – 22.96 – 6.769 8 11 (18) 0.118 3.374 – 1.655 100 0.220 – 25.37 – 4.590 9 16 (19) 0.123 0.124 – 2.856 100 0.280 – 25.33 – 3.530 9 16 FME Transactions VOL. 39, No 2, 2011 ▪ 69 The MSC and AIC attempt to represent the “information content” of a given set of parameter estimates by relating the coefficient of determination to the NP (or equivalently, the number of degrees of freedom) that were required to obtain the fit. When comparing two models (equation) with different numbers of parameters, this criterion places a burden on the model with more parameters not only to have a better coefficient of determination, but quantifies how much better it must be for the model to be deemed more appropriate. MSC criterion is given in the form ( ⎡ n ⎢ ∑ fC,i − f C,av ⎢ MSC = ln ⎢ i =1 n ⎢ ∑ f C,i − f pred,i ⎢⎣ i =1 ( )⎥ ⎤ ) ⎥ 2 NP ⎥− n ⎥ ⎥⎦ (26) where NP is the number of parameters in proposed equation. For this criterion, the most appropriate model will be that with the largest MSC, because we want to maximize information content of the model. AIC is defined by the following expression ( ⎡n AIC = n ln ⎢ ∑ f C,i − f pred,i ⎢⎣ i =1 ) 2⎤ ⎥ + 2 NP . ⎥⎦ (27) The AIC as defined above is dependent on the magnitude of the data points as well as the number of observations. According to this criterion, the most appropriate model is the one with the smallest values of the AIC. Statistical comparison of equations (4) – (19) with Colebrook’s equation (3) is given in Table 3, where NC is the number of mathematical calculations in a given equation. The numbers from Table 3 speak for themselves. Equation (12) is the best one according to most important criterions ∆av and Θ, and maximal relative errors are quite low. The only shortcoming of the (12) is the number of calculations (mathematical operations) that have to be done in order to obtain the result. It is interesting to compare, for example, (10) and (16). They have almost the same standard ∆av and Θ, as well as other statistical parameters. Equation (10) should be given the advantage, in hand calculations, because it has much lesser NP and NC compared to (16). Another interesting equation is (13). Although it is published 28 years ago, it provides very fine statistical parameters and needs only NC = 8 mathematical operations. Altshul’s equation (5) and Tsal’s correction (14) of Altshul’s equation is cited in one of the most significant engineering handbooks [22]. The citation from [22] is interesting: “Friction factors obtained from the AltshulTsal equation are within 1.6 % of those obtained by Colebrook’s equation.” Our analysis shows that both equations do not predict friction factor well. Maximal relative error of (14) is 27.30 %, standard deviation is about 18 %. Alshul’s, equation shows even worse parameters: maximal error 46.83 % is highly unacceptable. 70 ▪ VOL. 39, No 2, 2011 Although NC is small, these equations cannot be recommended for engineering practice. Equation (11) is the worst one among the cited equations. 4. CONCLUSION As stated by many engineers and scientists, famous Colebrook’s equation is still the best equation that provides a link between the friction factor, Reynolds number and relative roughness. Its only disadvantage is the implicit form of equation, and many authors reported their explicit approximations. After the statistical analysis given in this paper, two equations can be recommended: • equation (12) of Zigrang and Sylvester [16] provides the most accurate value of friction factor using 16 calculations to obtain the result; • equation (13) of Haaland [17] provides reasonably good statistical parameters but needs only 8 calculations, which is more convenient for hand calculation. Equations (4) – (6), (11) and (14) should be avoided in engineering practice. REFERENCES [1] Nikuradse, J.: Strömungsgesetze in rauhen Rohren, VDI-Verlag, Berlin, 1933. [2] Colebrook, C.F.: Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, Journal of the Institution of Civil Engineers, Vol. 11, No. 4, pp. 133-156, 1939. [3] Rouse, H.: Evaluation of boundary roughness, in: Proceedings of the 2nd Hydraulics Conference, 2226.06.1943, Lowa City, USA, Paper 27. [4] Moody, L.F.: Friction factors for pipe flow, Transactions of the ASME, Vol. 66, No. 8, pp. 671684, 1944. [5] Yıldırım, G: Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation, Advances in Engineering Software, Vol. 40, No. 11, pp. 1183-1190, 2009. [6] Fang, X., Xua, Y. and Zhou, Z.: New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations, Nuclear Engineering and Design, Vol. 241, No. 3, pp. 897-902, 2011. [7] Brkić, D.: Review of explicit approximations to the Colebrook relation for flow friction, Journal of Petroleum Science and Engineering, Vol. 77, No. 1, pp. 34-48, 2011. [8] Moody, L.F.: An approximate formula for pipe friction factors, Transactions of the ASME, Vol. 69, pp. 1005-1006, 1947. ь, А. .: я [9] А я , ь , No. 6, 1952. [10] Wood, D.J.: An explicit friction factor relationship, Civil Engineering, Vol. 36, No. 12, pp. 60-61, 1966. FME Transactions [11] Churchill, S.W.: Empirical expressions for the shear stress in turbulent flow in commercial pipe, AIChE Journal, Vol. 19, No. 2, pp. 375-376, 1973. [12] Jain, A.K.: Accurate explicit equation for friction factor, Journal of the Hydraulics Division, Vol. 102, No. 5, pp. 674-677, 1976. [13] Swamee, P.K. and Jain, A.K.: Explicit equations for pipe-flow problems, Journal of the Hydraulics Division, Vol. 102, No. 5, pp. 657-664, 1976. [14] Chen, N.H.: An explicit equation for friction factor in pipe, Industrial and Engineering Chemistry Fundamentals, Vol. 18, No. 3, pp. 296-297, 1979. [15] Round, G.F.: An explicit approximation for the friction factor – Reynolds number relation for rough and smooth pipes, The Canadian Journal of Chemical Engineering, Vol. 58, No. 1, pp. 122-123, 1980. [16] Zigrang, D.J. and Sylvester, N.D.: Explicit approximations to the solution of Colebrook’s friction factor equation, AIChE Journal, Vol. 28, No. 3, pp. 514-515, 1982. [17] Haaland, S.E.: Simple and explicit formulas for the friction factor in turbulent pipe flow, Transactions of the ASME, Journal of Fluids Engineering, Vol. 105, No. 1, pp. 89-90, 1983. [18] Tsal, R. J.: Altshul-Tsal friction factor equation, Heating, Piping and Air Conditioning, No. 8, pp. 30-45, 1989. [19] Manadilli, G.: Replace implicit equations with signomial functions, Chemical Engineering, Vol. 104, No. 8, pp. 129-132, 1997. [20] Romeo, E., Royo, C. and Monzón, A.: Improved explicit equations for estimation of the friction factor in rough and smooth pipes, Chemical Engineering Journal, Vol. 86, No. 3, pp. 369-374, 2002. [21] Goudar, C.T. and Sonnad, J.R.: Comparison of the iterative approximations of the Colebrook-White equation, Hidrocarbon Processing, Vol. 87, No. 8, pp. 79-83, 2008. ASHRAE Handbook: Fundamentals, [22] 2001 American Society of Heating Refrigerating and Air-conditioning Engineers, Atlanta, 2001. FME Transactions NOMENCLATURE ε relative pipe roughness friction factor number of nodes (points) number of mathematical calculations number of parameters Reynolds number f n NC NP Re Greek symbols β ∆ Θ nondimensional parameter standard deviation correlation ratio Subscripts av C pred average Colebrook predicted Ј Ј и ић, ић, ји Ј Ј ић, ђ и ић Њ ић, и и , У ђ . К , , , . ђ e „Model selection criterion“ Information Criterion“ (AIC). К З С , , (MSC) „Akaike Х . VOL. 39, No 2, 2011 ▪ 71