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A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases
International Journal of Heat and Mass Transfer 88 (2015) 538–543
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases Muhammad Mujtaba Shaikh a,⇑, Shafiq-ur-Rehman Massan b, Asim Imdad Wagan c a
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan Shaheed Zulfikar Ali Bhutto Institute of Science and Technology, Karachi, Pakistan c DHA Suffa University, Karachi, Pakistan b
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 2 May 2015 Accepted 3 May 2015 Available online 19 May 2015 Keywords: Friction factor Colebrook’s equation Explicit approximation Rough pipes Turbulent flow
a b s t r a c t In variety of industrial applications involving complex rough pipe networks under turbulent flow, explicit approximations to friction factor are preferred over implicit approximations. To date many researchers have proposed explicit approximations in this regard. In this study, a new explicit formula to approximate friction in pipes of considerable roughness under highly turbulent regimes is formulated by the help of Colebrook’s equation. The performance of proposed method over recent approximations in literature is tested. Results of proposed and other methods are compared with friction factor approximations from implicit Colebrook’s equation – a widely used standard for computing friction factor – to an error of 1016. Absolute error distributions in friction factor values by proposed and other methods are presented. Statistical analysis of the approximations in terms of mean relative error, maximum positive relative error, maximum negative relative error and correlation quotient are presented. Model selection and Akaike information criterions along with number of parameters and calculations required in explicit equations are also used to demonstrate suitability of used methods. Analysis is based on discretizing the domain into 1000 by 1000 mesh points based on relative roughness from 104 to 0.05 and Reynold’s number from 104 to 108. Comparison shows that the proposed method should be preferred over other methods in highly turbulent cases for explicit approximation of friction factor in rough pipes. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The determination of friction factor in pipes depicting turbulent fluid flow is very important in many engineering applications. While transfer of fluid through chemical reactors and in industrial processes involving single-phase, double-phase and even more complicated pipe flow systems, pipe friction is critical. Nowadays even in medical sciences and biomedical engineering, when high local velocities are attained in blood vessels while transportation of a physiological fluid through catheter tube into the body, analysis of friction factor in catheter is important [1]. To calculate friction factor in such situations, Colebrook–White equation is a widespread standard for pipes (rough or smooth) under turbulent flows. Colebrook formulated an implicit relationship for finding friction factor using equations of smooth and rough pipes [2] along with data from his experiment with White [3]. The Colebrook– White equation – also well known as just the Colebrook’s equation – is a relationship of the form: ⇑ Corresponding author. Mobile: +92 333 2617602. E-mail address: [email protected] (M.M. Shaikh). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.006 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
1 2:51 g pffiffiffi ¼ 2 log pffiffiffi þ b Re b 3:7
ð1Þ
where b is the friction factor in a pipe of diameter D, under influence of a fluid of viscosity l at a velocity of u. The factor g ¼ e=D is known as relative roughness of pipe and Re is the Reynold’s number defined as: Re ¼ Duq=l. Eq. (1) can be used for friction factor in turbulent regime where Re ranges from 4 * 103 to 108, and g, from 0 to 0.05. Moody [4] presented a ready chart of friction factors computed from (1) for a different range of values of Reynold’s number and relative roughness. To find friction factor given implicitly in (1), one needs to use numerical algorithms, which are not as quick as the explicit approximations to solution of (1) for friction factor. Particularly in complex and supercritical pipe-flow systems it becomes difficult to use (1). In such situations for quick estimation of friction factor, reliable explicit approximations are preferred. To date numerous explicit approximations to solution of (1) have been proposed [5–19]. Researchers also attempted to review available explicit formulas from time to time on the basis of absolute errors with reference to Colebrook’s solutions, statistical analysis of the parameters
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M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
like mean relative error, maximum and minimum positive errors, correlation ratio and standard deviation; and through Model selection criterion (MSC) and Akaike information criterion [20,21]. Need of more robust approximations to friction factor have led researchers of every time to propose new explicit methods. In this paper, we propose a new explicit approximation to Colebrook’s friction factor which is well-suited to rough pipes and the cases of highly turbulent pipe-flow systems. The performance of new explicit approximation proposed in this work will be compared – in terms of absolute errors and statistical parameters – with recently available approximations from literature. Rest of the paper is organized as follows: In Section 2algorithms of proposed and some available explicit approximations to Colebrook’s friction factor are presented. Section 3describes the comparison criterions used to test performance of explicit methods. Numerical experiments and discussion on obtained results are given in Section 4. Finally conclusions are provided in Section 5.
2
3
where b ¼ ln 4 1:816 ln
nRe
1:1Re lnð1þ1:1ReÞ
o5
Brkic [20]
2 2:18b g b ¼ 2 log þ Re 3:71 2
ð7Þ
3
where b ¼ ln 4 1:816 ln
nRe
1:1Re lnð1þ1:1ReÞ
o5.
3. Comparison criterions Two way comparisons, numerical and statistical, of friction factor estimations by proposed method (2) are presented in this section. 3.1. Absolute errors w.r.t. numerical solution
2. Material and methods There are a number of explicit approximations in literature for Colebrook’s friction factor [5–19] and some exhaustive review papers have already investigated the performance of available approximations. So, in this section we present algorithms of proposed explicit approximation along with some recent approximations which are already investigated in [21–22]. 2.1. New explicit approximation The proposed explicit approximation to friction factor in rough pipes under turbulent flow is:
2:51 g 2 b ¼ 0:25 log þ aRe 3:7
ð2Þ
2
where a ¼ ½1:14 2 logðgÞ . Eq. (2) appeared through independent work by first author but later on was investigated jointly by all authors for purpose of comparison. The proposed formula (2) is designed for highly turbulent fluid flows in rough pipes. It will be shown in next sections that solutions by (2) and Colebrook’s equation (1) match quite well for large values of Re and g as compared to other methods.
Numerical solution of Colebrook’s equation can be obtained in many ways. We have used a numerical solution, accurate to 16 decimal places, using Fixed-Point iteration method. Iterative scheme by this method for numerical solution starting with an inif0g
tial assumption b
" fiþ1g
b
¼ 0:25 log
¼ 0:1 is:
2:51 e pffiffiffiffiffiffiffi þ fig 3:7 Re b
!#2 ð8Þ
for, i = 0, 1, 2, . . . A 1000 by 1000 matrix ½Bm n of type (1) equations based on following distribution of Reynold’s number and relative roughness was solved: Re = 104–108 with linear spacing of 99996. g = 104 to 0.05 with linear spacing of 0.00005. Corresponding mesh is shown in Fig. 1.The absolute errors’ matrix between friction factors from used explicit approximations 2–7 and Colebrook’s solution (8) at each node (m, n) of 1000 by 1000 mesh points can be obtained by:
ð2Þð7Þ
ð2Þð7Þ ð8Þ ðA:EÞm n ¼ ½bm n ½bm n
ð9Þ
2.2. Other explicit approximations Following recent explicit schemes from literature will be used for comparison. Manadilli [17]
2 g 95 96:82 b ¼ 2 log þ 0:983 Re 3:7 Re Romeo et al. [18] g 5:0272 b ¼ 2log log Re 3:7065
g 0:9924 5:3326 0:9345 4:567 þ log Re 208:82 þ Re 3:827 7:79
g
ð3Þ
!!#)2 ð4Þ
Fang [22]
2 60:525 56:291 b ¼ 1:613 ln 0:234g1:1007 1:1105 þ 1:0712 Re Re
ð5Þ
Brkic [20]
h g i2 b ¼ 2 log 100:4343b þ 3:71
ð6Þ
Fig. 1. 1000 by 1000 mesh distribution for Reynold’s numbers and Relative roughness.
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M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
3.2. Statistical parameters of explicit approximations
Akaike information criterion (AIK)
If ½bm n is matrix of friction factor approximations by any explið8Þ
cit method and ½bm n is matrix of Colebrook’s friction factor approximations using (8) at each node (m, n) of 1000 by 1000 mesh points, then following statistical parameters are can be defined.
This criterion is based on NP and magnitude of observations and is defined as follows:
" AIC ¼ 1000000 ln
# 2 X ð8Þ ðÞ ½bm n ½bm n þ 2NP
ð15Þ
8 Nodes
Maximum positive relative error The largest relative error in friction factors by a method (⁄) can be obtained by finding maximum positive relative error, defined by:
! ð8Þ ðÞ
½b m n ½b m n
þ R:E ¼ max
ð8Þ
½b m n
ð10Þ
Maximum negative relative error The largest relative error in friction factors by a method (⁄) can be obtained by finding maximum negative relative error, defined by:
! ðÞ ð8Þ
½b m n ½b m n
R:E ¼ max
ð8Þ
½b m n
ð11Þ
Mean relative error Average relative error in friction factor approximations at all nodes of the mesh for a method (⁄) can be computed through mean relative error, defined as:
lR:E:
1 0
ðÞ ð8Þ X ½bm n ½bm n 1 @ A ¼ ð7Þ 1000000 8 Nodes ½bm n
ð12Þ
Correlation quotient Strength of relationship of the friction factor values by a method (⁄) is found using:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi u P ð8Þ ðÞ u ½ b ½ b u m n m n 8 Nodes H¼u 2 t1 P ð8Þ ð8Þ ½b m n 8 Nodes ½bm n
ð13Þ
ð8Þ
where ½bm n is arithmetic mean of the Colebrook’s friction factors through (8). Number of calculations (NC) Number of mathematical operations used in an explicit equation to compute friction factor approximation is used that determines computational cost.
A model with smallest values of AIC is considered most appropriate. 4. Results and discussion The numerical solution of (1) for 1000 by 1000 nodes, obtained using fixed point iterations (8) correct to 16 decimal places, was used as comparison criterion to test performance of proposed and available explicit approximations from literature. Absolute errors in friction factor approximations by available approximations 3–7 and in the proposed explicit approximation (2) for the defined mesh points in Fig. 1 are shown in Figs. 2–7. Absolute errors in Figs. 2–7 were represented as third dimension with values in reverse scale. More the height of the error region w.r.t. this dimension, better the accuracy and smaller errors. The data in Fig. 2 show that the Mandalli’s method [17] for both- the highly turbulent and rough pipe regimes- is subject to absolute errors close to 104. However the method has slightly higher accuracy than 104 in partial regions for small values of g or Re. The approximation by Romeo et al. [18] through Fig. 3 shows absolute errors slightly higher than 104 in both- highly turbulent and rough pipe sections. However only at few points on the other regimes errors are close to 105. For Fang’s method [22], absolute error distribution (Fig. 4) for most of the part in the solution mesh is also close to 104. Errors are close to 106 only for a fixed value g over all Reynold’s numbers. Both approximations proposed by Brkic [20] in view of Figs. 5 and 6 illustrate better performance for partial regions from small Re or for small g. But in the main part, the highly turbulent as well as rough regimes, this method showed errors close to 104. The proposed method (2), as it was designed for rough pipes and also highly turbulent regimes, performed better than all other
Number of parameters (NP) This parameter calculates the number of degrees of freedom required to fit an explicit equation to friction factor values from Colebrook’s equation. The computation of NP also helps in finding coefficients in MSC and AIC. Model selection criterion (MSC) This criterion helps in attempting to maximize the information content in a model by relating NP and coefficient of determination as follows:
2P 3 ð8Þ ð8Þ ½ b ½ b m n m n 8 Nodes NP 6 7 MSC ¼ ln 4P 5 ð8Þ ðÞ 500000 ½b m n 8 Nodes ½bm n A model with largest MSC is considered most appropriate.
ð14Þ
Fig. 2. Absolute errors in Manadilli [17] versus Colebrook’s friction factors.
M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
541
methods used in comparison. It appears from Fig. 7 that the new approximation converges rapidly to the 16 decimal places’ accurate solution (8) when values of Re and g increase. Unlike other methods – which behaved better in terms of accuracy in smooth pipe and low turbulent regimes as compare to the rough pipe and highly turbulent regimes – the proposed method showed smaller errors in the latter part. The parameters used for statistical analysis of the results by methods 2–7 are given in Table 1. The solution matrices by all methods correlate well – upto 99% – with the Colebrook’s numerical solution matrix. The mean relative error in proposed method is better than Manadilli [17] and Brkic [20] solutions and compares well with that of Rome et al. [18] and Fang [22]. The mean relative error also depends on the smooth pipe values of g and less turbulent parts, so in the case of proposed method it appears to be slightly high. However eliminating these regimes absolute relative error in proposed method can be made smaller. Methods 3–7
Fig. 3. Absolute errors in Romeo et al. [18] versus Colebrook’s friction factors.
Fig. 6. Absolute errors in Brkic [20] versus Colebrook’s friction factors. Fig. 4. Absolute errors in Fang [22] versus Colebrook’s friction factors.
Fig. 5. Absolute errors in Brkic [20] versus Colebrook’s friction factors.
Fig. 7. Absolute errors in proposed versus Colebrook’s friction factors.
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M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
Table 1 Calculated statistical parameters for methods 2–7.
+
R.E R.E
lR:E: H NC NP MSC AIC (⁄106)
Manadilli [17]
Romeo et al. [18]
Fang [22]
Brkic [20]
Brkic [20]
Proposed
0.001245 0.019632 0.001003 0.99 10 6 27.82 5.44
0.000815 0.001319 0.000678 0.99 20 11 27.48 6.50
0.004223 0.002895 0.000516 0.99 11 8 25.46 7.16
0.030347 0.011101 0.001000 0.99 16 9 27.85 5.60
0.001245 0.021474 0.001005 0.99 16 9 27.81 5.42
0.000009 1.861135 0.000962 0.99 13 7 – 1.38
Table 2 Statistical parameters for proposed method for other meshes of size 1000 by 1000. Mesh I II III
g
Re 4
8
10 –10 105–108 106–108
4
10 –0.05 103–0.05 102–0.05
R.E+
R.E
lR:E:
H
MSC
AIC ⁄ 106
0.000008 0.000008 0.000008
1.861135 0.428114 0.006627
0.000962 0.000399 0.000083
0.99 0.99 0.99
– – 24.57
1.38 4.62 8.97
approximate the friction factor values in like manner throughout the mesh, so maximum positive and negative relative errors are close to the corresponding mean relative errors. The proposed method, designed for the highly turbulent regimes, has very small maximum positive relative errors as compared to other methods; implying that the friction factor approximations by (2) are comparatively more close to that of the Colebrook’s friction values (8). The number of computations (NC) in proposed approximation is 13: which is less than the recent methods by Romeo et al. [18] and Brkic [20] but higher than those in Mandilli [17] and Fang [22]. The number of parameters in proposed method being 7 can is acceptable in comparison to others methods with large NP. In summary the proposed method uses only 13 mathematical operations with 7 degrees of freedom in corresponding model and approximates well the friction factor values in highly turbulent and rough pipe sections. The values of MSC and AIC in other methods seem closer to appropriate values as compared to the proposed method. This is due to the fact that corresponding mesh also involves less turbulent and smooth pipe regimes. While the proposed method was designed for highly turbulent and rough pipe regimes, we also investigated the performance of proposed method in view of MSC and AIC by varying the mesh parameters (but keeping same number of mesh points) in order to exclude the smooth pipe and less turbulent regimes. The performance of proposed method in other meshes that involve most of the turbulent and rough pipe regimes is shown in Table 2. It can be seen that AIC considerably decreases as highly turbulent and rough pipe sections are focused and other statistical parameters have also improved; implying that the proposed method in suitable for these regimes. In some explicit equations due to selective distribution of mesh parameters (Re and g), the MSC does not result in real values – same can also be seen in some methods used in [21] – so was the case in proposed method for meshes I and II. However for mesh-III proposed method does result in appropriate value of MSC coefficient in view of the similar values by other methods in Table 1. The overall analysis and values in Tables 1 and 2 indicate that for computing friction factor in rough pipes under highly turbulent regimes, the new explicit approximation proposed in this work (2) should be preferred over other methods. 5. Conclusion The friction factor estimation in pipe-flow networks in industries is presented in this paper through explicit approximations
to Colebrook’s equation. A new explicit formula (2) to approximate Colebrook’s friction factors (1) is proposed and compared with some recent explicit formulas 3–7 by means of absolute errors versus Colebrook’s friction factor estimations (8). Error distribution obtained for a mesh of 1000 by 1000 points depends on relative roughness from 104 to 0.05 and Reynold’s number from 104 to 108. Statistical analysis of the results in terms of maximum positive relative error, maximum negative relative error, mean relative error, and correlation quotient along with the number of computations, number of parameters and coefficients in MSC and AIC; is also presented. Analysis shows that the proposed method behaves better than the methods [17,18,22,20] for fluid flow in rough pipes under highly turbulent regimes and should be preferred over used methods. Acknowledgement Authors are highly indebted to the referee for important suggestions that have led to improvements in this paper. References [1] Stanley Dunn, Alkis Constantinides, Prabhas V. Moghe, Numerical Methods in Biomedical Engineering, Academic Press, 2005. [2] Cyril Frank Colebrook, Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, J. ICE 11 (4) (1939) 133–156. [3] C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes, Proc. R. Soc. London Ser. A Math. Phys. Sci. (1937) 367–381. [4] Lewis F. Moody, Friction factors for pipe flow, Trans. ASME 66 (8) (1944) 671– 684. [5] G.A. Gregory, Maria Fogarasi, Alternate to standard friction factor equation, Oil Gas J. 83 (13) (1985). [6] Lewis F. Moody, An approximate formula for pipe friction factors, Trans. ASME 69 (12) (1947) 1005–1011. [7] A.L. Akniykm, O,c,ieyyaz aopvyka cogponbdkeybz npy,ogpodolod, Ublpadkbxecrbe cnpobnekmcndo 6 (1952). [8] Don J. Wood, An explicit friction factor relationship, Civil Eng. 36 (12) (1966) 60–61. [9] Stuart W. Churchill, Empirical expressions for the shear stress in turbulent flow in commercial pipe, AIChE J. 19 (2) (1973) 375–376. [10] Akalank K. Jain, Accurate explicit equation for friction factor, J. Hydraul. Div. 102 (5) (1976) 674–677. [11] Prabhata K. Swamee, Akalank K. Jain, Explicit equations for pipe-flow problems, J. Hydraul. Div. 102 (5) (1976) 657–664. [12] Ning Hsing Chen, An explicit equation for friction factor in pipe, Ind. Eng. Chem. Fundam. 18 (3) (1979) 296–297. [13] G.F. Round, An explicit approximation for the friction factor-reynolds number relation for rough and smooth pipes, Can. J. Chem. Eng. 58 (1) (1980) 122–123. [14] D.J. Zigrang, N.D. Sylvester, Explicit approximations to the solution of Colebrook’s friction factor equation, AIChE J. 28 (3) (1982) 514–515.
M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543 [15] S.E. Haaland, Simple and explicit formulas for the friction factor in turbulent pipe flow, J. Fluids Eng. 105 (1) (1983) 89–90. [16] R.J. Tsal, Altshul–Tsal friction factor equation, Heat. Pip. Air Condition. 8 (1989) 30–45. [17] G. Manadili, Replace implicit equations with signomial functions, Chem. Eng. 104 (8) (1997) 129–132. [18] Eva Romeo, Carlos Royo, Antonio Monzón, Improved explicit equations for estimation of the friction factor in rough and smooth pipes, Chem. Eng. J. 86 (3) (2002) 369–374. [19] C.T. Goudar, J.R. Sonnad, Comparison of the iterative approximations of the Colebrook–White equation: here’s a review of other formulas and a
543
mathematically exact formulation that is valid over the entire range of Re values, Hydrocarbon Process. 87 (8) (2008) 79–83. [20] Dejan. Brkic´, Review of explicit approximations to the Colebrook relation for flow friction, J. Petrol. Sci. Eng. 77 (1) (2011) 34–48. [21] Srbislav Genic´ et al., A review of explicit approximations of Colebrook’s equation, FME Trans. 39 (2) (2011) 67–71. [22] Xiande Fang, Xu Yu, Zhanru Zhou, New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations, Nucl. Eng. Des. 241 (3) (2011) 897–902.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases Muhammad Mujtaba Shaikh a,⇑, Shafiq-ur-Rehman Massan b, Asim Imdad Wagan c a
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan Shaheed Zulfikar Ali Bhutto Institute of Science and Technology, Karachi, Pakistan c DHA Suffa University, Karachi, Pakistan b
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 2 May 2015 Accepted 3 May 2015 Available online 19 May 2015 Keywords: Friction factor Colebrook’s equation Explicit approximation Rough pipes Turbulent flow
a b s t r a c t In variety of industrial applications involving complex rough pipe networks under turbulent flow, explicit approximations to friction factor are preferred over implicit approximations. To date many researchers have proposed explicit approximations in this regard. In this study, a new explicit formula to approximate friction in pipes of considerable roughness under highly turbulent regimes is formulated by the help of Colebrook’s equation. The performance of proposed method over recent approximations in literature is tested. Results of proposed and other methods are compared with friction factor approximations from implicit Colebrook’s equation – a widely used standard for computing friction factor – to an error of 1016. Absolute error distributions in friction factor values by proposed and other methods are presented. Statistical analysis of the approximations in terms of mean relative error, maximum positive relative error, maximum negative relative error and correlation quotient are presented. Model selection and Akaike information criterions along with number of parameters and calculations required in explicit equations are also used to demonstrate suitability of used methods. Analysis is based on discretizing the domain into 1000 by 1000 mesh points based on relative roughness from 104 to 0.05 and Reynold’s number from 104 to 108. Comparison shows that the proposed method should be preferred over other methods in highly turbulent cases for explicit approximation of friction factor in rough pipes. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The determination of friction factor in pipes depicting turbulent fluid flow is very important in many engineering applications. While transfer of fluid through chemical reactors and in industrial processes involving single-phase, double-phase and even more complicated pipe flow systems, pipe friction is critical. Nowadays even in medical sciences and biomedical engineering, when high local velocities are attained in blood vessels while transportation of a physiological fluid through catheter tube into the body, analysis of friction factor in catheter is important [1]. To calculate friction factor in such situations, Colebrook–White equation is a widespread standard for pipes (rough or smooth) under turbulent flows. Colebrook formulated an implicit relationship for finding friction factor using equations of smooth and rough pipes [2] along with data from his experiment with White [3]. The Colebrook– White equation – also well known as just the Colebrook’s equation – is a relationship of the form: ⇑ Corresponding author. Mobile: +92 333 2617602. E-mail address: [email protected] (M.M. Shaikh). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.006 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
1 2:51 g pffiffiffi ¼ 2 log pffiffiffi þ b Re b 3:7
ð1Þ
where b is the friction factor in a pipe of diameter D, under influence of a fluid of viscosity l at a velocity of u. The factor g ¼ e=D is known as relative roughness of pipe and Re is the Reynold’s number defined as: Re ¼ Duq=l. Eq. (1) can be used for friction factor in turbulent regime where Re ranges from 4 * 103 to 108, and g, from 0 to 0.05. Moody [4] presented a ready chart of friction factors computed from (1) for a different range of values of Reynold’s number and relative roughness. To find friction factor given implicitly in (1), one needs to use numerical algorithms, which are not as quick as the explicit approximations to solution of (1) for friction factor. Particularly in complex and supercritical pipe-flow systems it becomes difficult to use (1). In such situations for quick estimation of friction factor, reliable explicit approximations are preferred. To date numerous explicit approximations to solution of (1) have been proposed [5–19]. Researchers also attempted to review available explicit formulas from time to time on the basis of absolute errors with reference to Colebrook’s solutions, statistical analysis of the parameters
539
M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
like mean relative error, maximum and minimum positive errors, correlation ratio and standard deviation; and through Model selection criterion (MSC) and Akaike information criterion [20,21]. Need of more robust approximations to friction factor have led researchers of every time to propose new explicit methods. In this paper, we propose a new explicit approximation to Colebrook’s friction factor which is well-suited to rough pipes and the cases of highly turbulent pipe-flow systems. The performance of new explicit approximation proposed in this work will be compared – in terms of absolute errors and statistical parameters – with recently available approximations from literature. Rest of the paper is organized as follows: In Section 2algorithms of proposed and some available explicit approximations to Colebrook’s friction factor are presented. Section 3describes the comparison criterions used to test performance of explicit methods. Numerical experiments and discussion on obtained results are given in Section 4. Finally conclusions are provided in Section 5.
2
3
where b ¼ ln 4 1:816 ln
nRe
1:1Re lnð1þ1:1ReÞ
o5
Brkic [20]
2 2:18b g b ¼ 2 log þ Re 3:71 2
ð7Þ
3
where b ¼ ln 4 1:816 ln
nRe
1:1Re lnð1þ1:1ReÞ
o5.
3. Comparison criterions Two way comparisons, numerical and statistical, of friction factor estimations by proposed method (2) are presented in this section. 3.1. Absolute errors w.r.t. numerical solution
2. Material and methods There are a number of explicit approximations in literature for Colebrook’s friction factor [5–19] and some exhaustive review papers have already investigated the performance of available approximations. So, in this section we present algorithms of proposed explicit approximation along with some recent approximations which are already investigated in [21–22]. 2.1. New explicit approximation The proposed explicit approximation to friction factor in rough pipes under turbulent flow is:
2:51 g 2 b ¼ 0:25 log þ aRe 3:7
ð2Þ
2
where a ¼ ½1:14 2 logðgÞ . Eq. (2) appeared through independent work by first author but later on was investigated jointly by all authors for purpose of comparison. The proposed formula (2) is designed for highly turbulent fluid flows in rough pipes. It will be shown in next sections that solutions by (2) and Colebrook’s equation (1) match quite well for large values of Re and g as compared to other methods.
Numerical solution of Colebrook’s equation can be obtained in many ways. We have used a numerical solution, accurate to 16 decimal places, using Fixed-Point iteration method. Iterative scheme by this method for numerical solution starting with an inif0g
tial assumption b
" fiþ1g
b
¼ 0:25 log
¼ 0:1 is:
2:51 e pffiffiffiffiffiffiffi þ fig 3:7 Re b
!#2 ð8Þ
for, i = 0, 1, 2, . . . A 1000 by 1000 matrix ½Bm n of type (1) equations based on following distribution of Reynold’s number and relative roughness was solved: Re = 104–108 with linear spacing of 99996. g = 104 to 0.05 with linear spacing of 0.00005. Corresponding mesh is shown in Fig. 1.The absolute errors’ matrix between friction factors from used explicit approximations 2–7 and Colebrook’s solution (8) at each node (m, n) of 1000 by 1000 mesh points can be obtained by:
ð2Þð7Þ
ð2Þð7Þ ð8Þ ðA:EÞm n ¼ ½bm n ½bm n
ð9Þ
2.2. Other explicit approximations Following recent explicit schemes from literature will be used for comparison. Manadilli [17]
2 g 95 96:82 b ¼ 2 log þ 0:983 Re 3:7 Re Romeo et al. [18] g 5:0272 b ¼ 2log log Re 3:7065
g 0:9924 5:3326 0:9345 4:567 þ log Re 208:82 þ Re 3:827 7:79
g
ð3Þ
!!#)2 ð4Þ
Fang [22]
2 60:525 56:291 b ¼ 1:613 ln 0:234g1:1007 1:1105 þ 1:0712 Re Re
ð5Þ
Brkic [20]
h g i2 b ¼ 2 log 100:4343b þ 3:71
ð6Þ
Fig. 1. 1000 by 1000 mesh distribution for Reynold’s numbers and Relative roughness.
540
M.M. Shaikh et al. / International Journal of Heat and Mass Transfer 88 (2015) 538–543
3.2. Statistical parameters of explicit approximations
Akaike information criterion (AIK)
If ½bm n is matrix of friction factor approximations by any explið8Þ
cit method and ½bm n is matrix of Colebrook’s friction factor approximations using (8) at each node (m, n) of 1000 by 1000 mesh points, then following statistical parameters are can be defined.
This criterion is based on NP and magnitude of observations and is defined as follows:
" AIC ¼ 1000000 ln
# 2 X ð8Þ ðÞ ½bm n ½bm n þ 2NP
ð15Þ
8 Nodes
Maximum positive relative error The largest relative error in friction factors by a method (⁄) can be obtained by finding maximum positive relative error, defined by:
! ð8Þ ðÞ
½b m n ½b m n
þ R:E ¼ max
ð8Þ
½b m n
ð10Þ
Maximum negative relative error The largest relative error in friction factors by a method (⁄) can be obtained by finding maximum negative relative error, defined by:
! ðÞ ð8Þ
½b m n ½b m n
R:E ¼ max
ð8Þ
½b m n
ð11Þ
Mean relative error Average relative error in friction factor approximations at all nodes of the mesh for a method (⁄) can be computed through mean relative error, defined as:
lR:E:
1 0
ðÞ ð8Þ X ½bm n ½bm n 1 @ A ¼ ð7Þ 1000000 8 Nodes ½bm n
ð12Þ
Correlation quotient Strength of relationship of the friction factor values by a method (⁄) is found using:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi u P ð8Þ ðÞ u ½ b ½ b u m n m n 8 Nodes H¼u 2 t1 P ð8Þ ð8Þ ½b m n 8 Nodes ½bm n
ð13Þ
ð8Þ
where ½bm n is arithmetic mean of the Colebrook’s friction factors through (8). Number of calculations (NC) Number of mathematical operations used in an explicit equation to compute friction factor approximation is used that determines computational cost.
A model with smallest values of AIC is considered most appropriate. 4. Results and discussion The numerical solution of (1) for 1000 by 1000 nodes, obtained using fixed point iterations (8) correct to 16 decimal places, was used as comparison criterion to test performance of proposed and available explicit approximations from literature. Absolute errors in friction factor approximations by available approximations 3–7 and in the proposed explicit approximation (2) for the defined mesh points in Fig. 1 are shown in Figs. 2–7. Absolute errors in Figs. 2–7 were represented as third dimension with values in reverse scale. More the height of the error region w.r.t. this dimension, better the accuracy and smaller errors. The data in Fig. 2 show that the Mandalli’s method [17] for both- the highly turbulent and rough pipe regimes- is subject to absolute errors close to 104. However the method has slightly higher accuracy than 104 in partial regions for small values of g or Re. The approximation by Romeo et al. [18] through Fig. 3 shows absolute errors slightly higher than 104 in both- highly turbulent and rough pipe sections. However only at few points on the other regimes errors are close to 105. For Fang’s method [22], absolute error distribution (Fig. 4) for most of the part in the solution mesh is also close to 104. Errors are close to 106 only for a fixed value g over all Reynold’s numbers. Both approximations proposed by Brkic [20] in view of Figs. 5 and 6 illustrate better performance for partial regions from small Re or for small g. But in the main part, the highly turbulent as well as rough regimes, this method showed errors close to 104. The proposed method (2), as it was designed for rough pipes and also highly turbulent regimes, performed better than all other
Number of parameters (NP) This parameter calculates the number of degrees of freedom required to fit an explicit equation to friction factor values from Colebrook’s equation. The computation of NP also helps in finding coefficients in MSC and AIC. Model selection criterion (MSC) This criterion helps in attempting to maximize the information content in a model by relating NP and coefficient of determination as follows:
2P 3 ð8Þ ð8Þ ½ b ½ b m n m n 8 Nodes NP 6 7 MSC ¼ ln 4P 5 ð8Þ ðÞ 500000 ½b m n 8 Nodes ½bm n A model with largest MSC is considered most appropriate.
ð14Þ
Fig. 2. Absolute errors in Manadilli [17] versus Colebrook’s friction factors.
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methods used in comparison. It appears from Fig. 7 that the new approximation converges rapidly to the 16 decimal places’ accurate solution (8) when values of Re and g increase. Unlike other methods – which behaved better in terms of accuracy in smooth pipe and low turbulent regimes as compare to the rough pipe and highly turbulent regimes – the proposed method showed smaller errors in the latter part. The parameters used for statistical analysis of the results by methods 2–7 are given in Table 1. The solution matrices by all methods correlate well – upto 99% – with the Colebrook’s numerical solution matrix. The mean relative error in proposed method is better than Manadilli [17] and Brkic [20] solutions and compares well with that of Rome et al. [18] and Fang [22]. The mean relative error also depends on the smooth pipe values of g and less turbulent parts, so in the case of proposed method it appears to be slightly high. However eliminating these regimes absolute relative error in proposed method can be made smaller. Methods 3–7
Fig. 3. Absolute errors in Romeo et al. [18] versus Colebrook’s friction factors.
Fig. 6. Absolute errors in Brkic [20] versus Colebrook’s friction factors. Fig. 4. Absolute errors in Fang [22] versus Colebrook’s friction factors.
Fig. 5. Absolute errors in Brkic [20] versus Colebrook’s friction factors.
Fig. 7. Absolute errors in proposed versus Colebrook’s friction factors.
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Table 1 Calculated statistical parameters for methods 2–7.
+
R.E R.E
lR:E: H NC NP MSC AIC (⁄106)
Manadilli [17]
Romeo et al. [18]
Fang [22]
Brkic [20]
Brkic [20]
Proposed
0.001245 0.019632 0.001003 0.99 10 6 27.82 5.44
0.000815 0.001319 0.000678 0.99 20 11 27.48 6.50
0.004223 0.002895 0.000516 0.99 11 8 25.46 7.16
0.030347 0.011101 0.001000 0.99 16 9 27.85 5.60
0.001245 0.021474 0.001005 0.99 16 9 27.81 5.42
0.000009 1.861135 0.000962 0.99 13 7 – 1.38
Table 2 Statistical parameters for proposed method for other meshes of size 1000 by 1000. Mesh I II III
g
Re 4
8
10 –10 105–108 106–108
4
10 –0.05 103–0.05 102–0.05
R.E+
R.E
lR:E:
H
MSC
AIC ⁄ 106
0.000008 0.000008 0.000008
1.861135 0.428114 0.006627
0.000962 0.000399 0.000083
0.99 0.99 0.99
– – 24.57
1.38 4.62 8.97
approximate the friction factor values in like manner throughout the mesh, so maximum positive and negative relative errors are close to the corresponding mean relative errors. The proposed method, designed for the highly turbulent regimes, has very small maximum positive relative errors as compared to other methods; implying that the friction factor approximations by (2) are comparatively more close to that of the Colebrook’s friction values (8). The number of computations (NC) in proposed approximation is 13: which is less than the recent methods by Romeo et al. [18] and Brkic [20] but higher than those in Mandilli [17] and Fang [22]. The number of parameters in proposed method being 7 can is acceptable in comparison to others methods with large NP. In summary the proposed method uses only 13 mathematical operations with 7 degrees of freedom in corresponding model and approximates well the friction factor values in highly turbulent and rough pipe sections. The values of MSC and AIC in other methods seem closer to appropriate values as compared to the proposed method. This is due to the fact that corresponding mesh also involves less turbulent and smooth pipe regimes. While the proposed method was designed for highly turbulent and rough pipe regimes, we also investigated the performance of proposed method in view of MSC and AIC by varying the mesh parameters (but keeping same number of mesh points) in order to exclude the smooth pipe and less turbulent regimes. The performance of proposed method in other meshes that involve most of the turbulent and rough pipe regimes is shown in Table 2. It can be seen that AIC considerably decreases as highly turbulent and rough pipe sections are focused and other statistical parameters have also improved; implying that the proposed method in suitable for these regimes. In some explicit equations due to selective distribution of mesh parameters (Re and g), the MSC does not result in real values – same can also be seen in some methods used in [21] – so was the case in proposed method for meshes I and II. However for mesh-III proposed method does result in appropriate value of MSC coefficient in view of the similar values by other methods in Table 1. The overall analysis and values in Tables 1 and 2 indicate that for computing friction factor in rough pipes under highly turbulent regimes, the new explicit approximation proposed in this work (2) should be preferred over other methods. 5. Conclusion The friction factor estimation in pipe-flow networks in industries is presented in this paper through explicit approximations
to Colebrook’s equation. A new explicit formula (2) to approximate Colebrook’s friction factors (1) is proposed and compared with some recent explicit formulas 3–7 by means of absolute errors versus Colebrook’s friction factor estimations (8). Error distribution obtained for a mesh of 1000 by 1000 points depends on relative roughness from 104 to 0.05 and Reynold’s number from 104 to 108. Statistical analysis of the results in terms of maximum positive relative error, maximum negative relative error, mean relative error, and correlation quotient along with the number of computations, number of parameters and coefficients in MSC and AIC; is also presented. Analysis shows that the proposed method behaves better than the methods [17,18,22,20] for fluid flow in rough pipes under highly turbulent regimes and should be preferred over used methods. Acknowledgement Authors are highly indebted to the referee for important suggestions that have led to improvements in this paper. References [1] Stanley Dunn, Alkis Constantinides, Prabhas V. Moghe, Numerical Methods in Biomedical Engineering, Academic Press, 2005. [2] Cyril Frank Colebrook, Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, J. ICE 11 (4) (1939) 133–156. [3] C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes, Proc. R. Soc. London Ser. A Math. Phys. Sci. (1937) 367–381. [4] Lewis F. Moody, Friction factors for pipe flow, Trans. ASME 66 (8) (1944) 671– 684. [5] G.A. Gregory, Maria Fogarasi, Alternate to standard friction factor equation, Oil Gas J. 83 (13) (1985). [6] Lewis F. Moody, An approximate formula for pipe friction factors, Trans. ASME 69 (12) (1947) 1005–1011. [7] A.L. Akniykm, O,c,ieyyaz aopvyka cogponbdkeybz npy,ogpodolod, Ublpadkbxecrbe cnpobnekmcndo 6 (1952). [8] Don J. Wood, An explicit friction factor relationship, Civil Eng. 36 (12) (1966) 60–61. [9] Stuart W. Churchill, Empirical expressions for the shear stress in turbulent flow in commercial pipe, AIChE J. 19 (2) (1973) 375–376. [10] Akalank K. Jain, Accurate explicit equation for friction factor, J. Hydraul. Div. 102 (5) (1976) 674–677. [11] Prabhata K. Swamee, Akalank K. Jain, Explicit equations for pipe-flow problems, J. Hydraul. Div. 102 (5) (1976) 657–664. [12] Ning Hsing Chen, An explicit equation for friction factor in pipe, Ind. Eng. Chem. Fundam. 18 (3) (1979) 296–297. [13] G.F. Round, An explicit approximation for the friction factor-reynolds number relation for rough and smooth pipes, Can. J. Chem. Eng. 58 (1) (1980) 122–123. [14] D.J. Zigrang, N.D. Sylvester, Explicit approximations to the solution of Colebrook’s friction factor equation, AIChE J. 28 (3) (1982) 514–515.
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