An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

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1 International Journal o Hydraulic Engineering 06, 5(): 9-3 DOI: 0.593/j.ijhe An Improved Expression or a Classical Type o Explicit Approximation o the Colebrook White Equation with Only One Internal Iteration Alexandre Beluco *, Edith Beatriz Camano Schettini Instituto de Pesquisas Hidráulicas (IPH), Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil Abstract The solution o the Colebrook White equation continues to attract the attention o engineers and mathematicians. Some solutions proposed in recent years have two or more internal iterations and present relative complexity. At the same time, micro-computers and calculators have shown increased calculation capacity, virtually eliminating approximations that have a certain accuracy but that are too complex. A direct approximation o the equation allowing or quick and accurate solution is still useul. This technical note brings together some explicit approximations with only one internal iteration presented in the 970s and 980s into a single category and proposes a single model with ive parameters to be determined. Moreover, this technical note also presents the values or these ive parameters that minimize the mean square error related to the outcome o the Colebrook White equation, considered as a reerence. The proposed model allows to obtain results with mean square error equal to.4 x 0-8, with maximum relative error o 4.44% and 96.78% o results below.00% o dierence relative to Colebrook White equation, which can be considered quite satisactory. Keywords Friction actor, Energy loss in pipes, Colebrook White equation. Introduction The Colebrook White equation was proposed [] in 937 and its non-algebraic characteristic, hindering its solution, still attracts the attention o engineers and mathematicians. Moody [] proposed a diagram containing all the inormation summarized in the equation and serving as an abacus or determining the riction actor. During the 70s and 80s, the irst approximations to the equation o Colebrook White, basically with only one internal iteration have been proposed. The irst personal computers became popular, but still showing inability to solve the Colebrook White equation in problems involving the calculation o a large number o pipes. During the ollowing decades, several other approximations have been proposed, with increasing complexity, some with two or three [3] or even more internal iterations and others with dierent shapes or dierent methods [4] o the structure proposed by Colebrook and White. Currently, the Colebrook White equation can be solved by personal computers and even an exact although complex * Corresponding author: albeluco@iph.urgs.br (Alexandre Beluco) Published online at Copyright 06 Scientiic & Academic Publishing. All Rights served solution was proposed [5, 6]. Even with great accuracy [7, 8], the evolution o computers reduces the useulness o approximations with high computational costs. Currently, there is a large number o papers proposing alternative equations to the expression proposed by Colebrook White, with dierent shapes and several o them with better results or a particular type o low [9, 0] or or a certain type o luid []. This paper presents an equation with scope and applicability equivalent to the original expression, thereore suitable or Newtonian luids and turbulent lows, or a wide range o roughness values. An approximation o the Colebrook White equation that represents a computational cost equivalent to only one iteration and displays reasonable accuracy could still be applied to calculations o large pipe networks. This technical note includes six classical approximations o the Colebrook White equation with a similar shape in a same category. These approximations have been proposed in the 970s and 980s and have computational cost equivalent to one internal iteration. This technical note also proposes a generic model or this category, with ive parameters to be determined so as to minimize the dierence o their results with the Colebrook White equation, considered as reerence in this study. Finally, this technical note provides values or these ive parameters so that the model proposed to provide better

2 0 Alexandre Beluco et al.: An Improved Expression or a Classical Type o Explicit Approximation o the Colebrook White Equation with Only One Internal Iteration results than those obtained with the six models taken as the starting point.. The Friction Factor and the Colebrook White Equation The riction actor appears in the Darcy equation or energy loss in pipes. The actor is dimensionless and depends on the pipe diameter, roughness o the material o the pipeline and the ynolds number o the low. Darcy's equation or the calculation o the loss o pressure h P [m] is shown in Eq. () where L [m] is the tube length considered in the calculation, D [m] is the pipe diameter, V [m/s] is the speed and g [m/s ] is the acceleration o gravity. L V h P = () D g The riction actor can be calculated or turbulent low by the Colebrook White equation. It is a non-algebraic equation with iterative resolution. The Eq. () shows the typical presentation o this equation, where e [m] is the roughness o the pipe and [] is the ynolds number. e/d [] is the relative roughness. e.5 = log D Eq. (3) directly indicates the value o ater each iteration. The initial value used in the iterations can reduce the amount o calculations, but the process usually converges to a reasonable accuracy ater a ew iterations. 0.5 =.5 e (3) D Good accuracy in result is already obtained with ew iterations. For simple calculations, it is not interesting to adopt approaches to this equation. The use o approximations that directly provide the inal value, even with a loss o precision, will be interesting or the design o networks with large numbers o tubes. 3. Some Classical Basic Approximations o the Colebrook White Equation Several approximations or the Colebrook White equation have been proposed in last decades and they are still being proposed. The trend that can be expected rom new approaches is that they will present lower computational costs. Among them, one interesting class can be established or the approximations with only one internal iteration and similar constitution to the Colebrook White equation. () This class would include the equations proposed by Haaland [], Barr [3, 4], Jain [5], Swamee and Jain [6], Churchill [7] and Eck [8]. Eq. (4) shows the equation proposed by Haaland []. According to Winning and Coole [9], this equation provides results with a maximum percentage relative error o.43% and MSE equal to.67 x0-8. e =.8 log 3.7 D Eq. (5) shows the equation proposed by Barr [3, 4] and Eq. (6) shows the equation proposed by Jain [5]. e = log D e = log D Eq. (7) shows the equation proposed by. [6] and Eq. (8) shows the equation proposed by Churchill [7]. e = log D 5.74 e 7 = log D Eq. (9) shows the equation proposed by Eck [8]. e = log D 5 These equations can be generalized to a model with ive parameters, aiming to optimize these parameters to reduce the error in relation to the Colebrook White equation. The next section discusses a model developed in this way. 4. A Model or an Explicit Approximation with Only One Internal Iteration It is possible to build a generic model or this class o explicit approximations containing ive variables to be optimized, thereby reducing the error in relation to the Colebrook White equation. Eq. (0) shows the proposed model. Comparing the expressions (4) to (9), changes were observed in ive locations on the basic structure o the Colebrook White equation. The ive variables together must compensate or the withdrawal o the actor rom the inside o the logarithm. Z is a actor external to the logarithm unction. A and B respectively integrate the terms including roughness and ynolds number. The exponents a and b do not appear in the original structure proposed by Colebrook and White. (4) (5) (6) (7) (8) (9)

3 International Journal o Hydraulic Engineering 06, 5(): 9-3 Z = a b e B + A D (0) Table shows the values o the ive parameters o the model o Equation (0) corresponding to the approaches presented in previous section. The proposition o this model aims to obtain a model that is more accurate than the six approximations mentioned above. In other words, it is necessary to ind values or the parameters Z, a and b, A and B that achieve values more accurately than those provided by equations (4) to (9). Table. Parameters or the model o Eq (0) Approximation Eq. Z a b A B Haaland (4) Barr (5) Jain (6) Swamee Jain (7) Churchill (8) Eck (9) The accuracy o the approximations can be evaluated by the relative error and the mean square error. Eq. () and () deine respectively the relative error (RE) and the percentage relative error (PER) and Eq. (3) establishes the mean square error (MSE), where cw is the value provided by the Colebrook White equation and est is an estimation obtained with equations (4) to (9) or with the model in Eq. (0). cw est RE = () cw cw est PRE = 00 = RE 00 () cw MSE N i cw = est = (3) this work, the results provided by the Colebrook White equation, solved or a large number o iterations, were considered as reerence. The riction actor cw were obtained with 6 iterations and were considered as reerence or comparison with other models. From the calculations with the Colebrook White equation and the model o Eq. (0), the relative error and mean square error were determined. The calculations were perormed or 46 values o the relative roughness and 5 values o ynolds number. The values o the relative roughness were 0.09, 0.08, 0.07, 0.06, 0.05, 0.04, 0.03, 0.0, 0.0, 0.009, 0.008, 0.007, 0.006, 0.005, 0.004, 0.003, 0.00, 0.00, , , , , , , , 0.000, 0.000, , , , , , , , , N , , , , , , , , , and 0. The values o ynolds number were 3.0x0 3, 4.0x0 3, 5.0x0 3, 6.0x0 3, 7.0x0 3, 8.0x0 3, 9.0x0 3,.0x0 4,.0x0 4, 3.0x0 4, 4.0x0 4, 5.0x0 4, 6.0x0 4, 7.0x0 4, 8.0x0 4, 9.0x0 4,.0x0 5,.0x0 5, 3.0x0 5, 4.0x0 5, 5.0x0 5, 6.0x0 5, 7.0x0 5, 8.0x0 5, 9.0x0 5,.0x0 6,.0x0 6, 3.0x0 6, 4.0x0 6, 5.0x0 6, 6.0x0 6, 7.0x0 6, 8.0x0 6, 9.0x0 6,.0x0 7,.0x0 7, 3.0x0 7, 4.0x0 7, 5.0x0 7, 6.0x0 7, 7.0x0 7, 8.0x0 7, 9.0x0 7,.0x0 8,.0x0 8, 3.0x0 8, 4.0x0 8, 5.0x0 8, 6.0x0 8, 7.0x0 8, 8.0x0 8 and 9.0x0 8. The determination o the values o the ive parameters o the model o equation (0) to minimize the RE and MSE values can be perormed with the Matlab sotware. The ER and MSE values were obtained or the values o Z between and, a between and, A between and, b between and and B between and. The next section presents and discusses the results. 5. sults and Discussion The MSE minimization o the ive parameters resulted in the ollowing values: Z=0.3009, A=.09540, B=6949, a= and b= These values may be rounded to the ourth decimal with no signiicant change in accuracy o the results, as shown in Eq. (4) = e (4) D This equation is the best result o this work, with a maximum relative error o 4.44% and MSE equal to.4 x0-8. Among the values o relative roughness and ynolds numbers used in the calculation, 9.60% are lower than the percentage relative error o 0.08% and 96.78% are lower than PRE o.00%. Figure shows the relative error distribution to the ranges o relative roughness and ynolds number considered. Table shows a comparison between the MSE or the classical equations and the MSE obtained or the equation proposed in this paper. This table also shows maximum values o RE, dierence between negative and positive maximum values o RE and percentage o values o RE lower than 0.08% and lower than.00%. The equation o this paper is the best approximation when compared with the classical equations discussed in previous section. Table shows that the proposed equation has the lowest MSE and the greatest number o results with relative error less than 0.8% and less than.0%. The low MSE value makes the surace appearing in Figure "latter" than the suraces corresponding to other approximations and allows a greater amount o values with less dierence with respect to the Colebrook White equation.

4 Alexandre Beluco et al.: An Improved Expression or a Classical Type o Explicit Approximation o the Colebrook White Equation with Only One Internal Iteration Figure. lative error o the proposed model shown in Eq. (3) Table. Comparison o results with Eq. (4) Approx. Eq. MSE Max RE Δ RE RE<0.8% RE<.0% Haaland (4) 3.33x Barr (5).8x Jain (6).47x Sw.Jain (7).46x Churchill (8).68x Eck (9) 7.90x this.note (4).4x Conclusions This technical note included six classical approximations o the Colebrook White equation in a same category and also proposed a generic model or this category, with ive parameters to be determined. These ive parameters were determined, presenting a new alternative equation with only a single iteration with RMSE equal to.5x0-8. REFERENCES [] Colebrook, C.F.; White, C.M.; 937. Experiments with luid riction in roughened pipes. Proceedings o the Royal Society o London, Series A, 6, p [] Moody, L.F.; 947. An approximate ormula or pipe riction actors. Transactions o the ASME, 69 (), p [3] Romeo, E.; Royo, C.; Monzón.; 00. A. Improved Explicit equations or estimation o the riction actor in rough and smooth pipes. Chemical Engineering Journal, 86, p [4] Ozger, M.; Yildirim, G Determining turbulent low riction coeicient using adaptive neuro uzzy computing technique. Advances in Sotware Engineering, 40, p [5] Sonnad. J.R.; Goudar, C.T.; 004. Constraints or using Lambert W unction based explicit Colebrook White equation. Journal o Hydraulic Engineering, 30, p [6] Sonnad, J.R.; Goudar, C.T.; 007. Explicit reormulation o the Colebrook White equation or turbulent low riction actor calculation. Industrial & Engineering Chemistry search, 46, p [7] Yildirim, G.; 009. Computer based analysis o explicit approximations to the implicit Colebrook White equation in turbulent low riction actor calculation. Advances in Sotware Engineering, 40, p

5 International Journal o Hydraulic Engineering 06, 5(): [8] Brkic, D.; 0. view o explicit approximations to the Colebrook relation or low riction. Journal o Petroleum Science and Engineering, 77, p [9] Shaikh, M.M.; Massan, S.T.; Wagan, A.I.; 05. A new explicit approximation to Colebrook s riction actor under rough pipes under highly turbulent cases. International Journal o Heat and Mass Transer, 66, p [0] Brkic, D.; 06. A note on explicit approximations to Colebrook s riction actor in rough pipes under highly turbulent cases. International Journal o Heat and Mass Transer, 93, p [] Dosunmu, I.T.; Shah, S.N.; 03. Evaluation o riction actor correlations and equivalent diameter deinitions or pipe and annular low o non-newtonian luids. Journal o Petroleum Science and Engineering, 09, p [] Haaland, S.E Simple and explicit ormulas or riction actor in turbulent pipe low. Journal o Fluids Engineering (ASME), 05 (), p [4] Barr, D.I.H.; 98. Solutions o the Colebrook White unction to resistance to uniorm turbulent low. Proceedings o the Institution o Civil Engineers, 7 (), p [5] Jain, A.K.; 976. Accurate explicit equations or riction actor. Journal o the Hydraulic Division (ASCE), 0 (5), p [6] Swamee, P.K.; Jain, A.K.; 976. Explicit equation or pipe low problems. Journal o the Hydraulic Division (ASCE), 0 (5), p [7] Churchill, S.W.; 973. Empirical expressions or the shear stressing turbulent low in commercial pipe. AIChE Journal, 9 (), p [8] Eck, B. Technische Stromungslehre. Springer, New York, United States, 973. [9] Winning, H.K.; Coole, T.; 03. Explicit riction actor accuracy and computational eiciency or turbulent low in pipes. Flow, Turbulence and Combustion, 90, p.-7. [3] Barr, D.I.H.; 97. New orms o equations or the correlation o pipe resistance data. Proceedings o the Institution o Civil Engineers, 53 (), p

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