Applied mathematics in Engineering, Management and Technology 2 (4) 24:57-578 www.amiemt-journal.com Comparison of Explicit lations of arcy Friction Measurement with Colebrook-White Equation Saeed Kazemi Mohsenabadi (Corresponding Author), Mohammad za Biglari, Mahdi Moharrampour epartment of Civil Engineering, College of Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran. E-mail: s.kazemi@buiniau.ac.ir, biglari.mohammadreza@gmail.com, m62.mahdi@yahoo.com Abstract In order to measure friction drop in flow pipes, arcy- Weisbach relation is usually used, one part of which is arcy friction factor. One precise relation to measure this coefficient is Colebrook-White equation. However, since this relation is implicit and it needs numerical root finding methods to gain friction factor, researchers provide multitude substantial and explicit relations for it, where considering relations accuracy before they are applied is very crucial. In this research, 29 explicit relations collected by different researchers are provided and then they are tested regarding accuracy. sults show that, compared to Colebrook- White equation, Serghides () (984), Vatankhah-Kochakzadeh (28) and Buzzelli(28) relations, with relative error lower than ±.5% for different ynolds numbers and relative roughness provide best responses. Additionally, Goudar- Sonnad (28) and Avci-Kargoz (29) relations show highest error level compared to Colebrook- White relation. relation Keywords: friction factor, Colebrook-White, ynolds numbers, relative roughness, explicit.introduction Estimation of head loss in pipes has always been a major influential issue when designing flow lines. In addition to local drop often happening in connections and section changes, there always exists a friction drop along flow line. To determine friction drop, in most measurements, Hazen- Williams and arcy- Weisbach empirical equations are applied. However, Hazen-Williams equation is dimensionally heterogeneous and has limited number of applications so in order to measure friction drop in pipes, arcy- Weisbach equation is used as followed: P = f L V 2 2g Where P is the head loss (m), f is arcy friction factor (dimensionless), L is pipe length (m), is hydraulic diameter of pipe which is the same circle diameter in round pipes (m), V is average flow velocity ( m ) and g is magnitude s velocity ( m s2). In arcy friction factor for mild flows ( < 2), Hagen- Poisuille relation is used as followed: Where f = 64 = ρv μ is ynolds numbers (dimensionless), ρ is flow density ( kg m 3), V is average flow velocity (m ), is length s index or pipe diameter (m) and μ is dynamic viscosity ( kg ). About (2 < < 4) transition flows and m.s ( > 4) turbulent flow, friction factor using Colebrook- White method is applied as followed: f = log + 2.5 (4) 3.7 f 57 () (2) (3)
Applied mathematics in Engineering, Management and Technology 24 Where f is arcy friction factor (dimensionless), is ynolds number (dimensionless) and is relative roughness. Colebrook- White relation is a valid referent relation (Moody 944), but since this relation is implicit, it is not easily used and numerical methods are needed. There have been many explicit relations used to solve this problem in order to measure friction factor each of which contains some error level compared to Colebrook-White method. One popular relation obtained from Colebrook- White relation is Moody relation (947) and also, Moody s graphic method (Moody diagram) widely used to measure friction factor. It is worth noting that the method s error, compared to Colebrook- White method, is sometimes up to ±5%. In 22, Turhan Coban evaluated explicit relations and introduced Goudar- Sonnad explicit relation, provided in 26, as the best relation replacing Colebrook- White relation in ynolds numbers bigger than 7. In 2, as well, studying more than 5 explicit relations, Srbislav Genic et al recommended Zigrang- Sylvester relation, provided in 982, as most precise relation compared to Colebrook- White relation. 2.search method As noted, there have been provided many explicit relations to measure friction factor but, compared to Colebrook-White and for ynolds numbers and relative roughness, their accuracy has never been considered, so the need to designate an explicit relation with a very low error level as a substitution for Colebrook- White relation in most ynolds number and variable roughness is evident. In order for this, the aim of this research, after 29 explicit relations were collected, is to estimate relative error of this relation compared to Colebrook- White and identify best explicit relation to measure friction factor. lative error of these relations are measured using following relation: lative Error = f Coleb f Calc f Coleb (5) Where f calc is computational friction factor and f Coleb is friction factor measured using Colebrook-White equation. In order to increase the accuracy when choosing the most accurate explicit relation, the results from all relations are considered in a wide range of ynolds numbers and relative roughness (4 9 ynolds number and.4. relative roughness). 3.Explicit relations applied In this research, 29 explicit relations were considered which are displayed in table. Table. Explicit relations provided by researchers to measure friction factor Eq. No 6 7 8 9 Equations f =.55 + 2 + 6 f =. + 68.25 f =.53( ) +.94( ).225 + 88( ).44.62.34 f ( = log ) 3.75 + 5 f = log 3.7 + 7.9 3 Authors (year) Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) 57
Applied mathematics in Engineering, Management and Technology 24 f = log 3.7 + 5.74.9 Jain, Swamee (976) 2 3 4 f = 8 8 2 f =.4 2 log + 2.25.9 + A + B 3 2 6 2, A = log 3.7 + 7 = 3753 f = log 3.765 5.452 log ( ).98 2.8257 + 5.856.898.9 6, B Jain (976) Churchill (977) Chen (979) 5 f =.8 log(.35 + 6.5 ) Round (98) 6 = log f 3.7 + 4.58 log 7 +.52 29.7 Barr (98) 7 8 9 = log f 3.7 5.2 log 5.2 log = log f f =.8 log 3.7 5.2 log 3.7. 3.7 + 3 + 6.9 3.7 + 3 Zigrang, Sylvester() (982) Zigrang, Sylvester(2) (982) Haaland (983) 2 2 22 23 A = log A = log + 2 3.7 + 2 3.7, B = log + 2.5A 3.7, B = log A B A 2 = f CB+A + 2.5A 3.7.25, C = log, f + 2.5B 3.7 = 4.78 A 4.78 2 BA+4.78 A =. + 68 If A >.8 ten f = A and if A <.8 ten f = o. oo28 +.85A f = log 3.7 + 95 96.82.983 Serghides () (984) Serghides (2) (984) Tsal (989) Manadilli (997) 572
Applied mathematics in Engineering, Management and Technology 24 f = log 3.765 24 5.272 4.567 log log 3.827 7.79.9924 + 5.3326 28.82 +.9345 Monzon,Rome o, Royo (22) 25 26 27 28 S =.24 + ln.4587, f =.8686 ln S =.24 + ln.4587, f =.8686 ln.744 ln.4 α =, β = +.32 a = 2 ln, b = δcfa = q = s 3.7 f =.4587 S.3.4587 S.3 α α + 2 log β + 2.8 β S S+ S S+.9633 3.7 + 2.5 α, ln, d =, s = b + d + ln d, 5.2 s s+, g = b d ln d q, z = q g, g z δla = g +, 74 + z 2 g + 2 + z 3 + 2g, f = a ln d q + δcfa Goudar,Sonnad (26) Vatankhah, Kouchak zadeh (28) Buzzelli (28) Goudar,Sonnad (28) 29 f = 6.4 ln ln +. + 2.4 Avci,Kargoz (29) 3 f = 4.2479.947 7 log log 3.65 + 7.366.942 2 Papaevanegelo u, Evangleids, Tzimopoulos (2) 3 f =.52 log 7.2.42 + 2.73.952.69 Ghanbari et.al (2) 573
32 33 34 β = ln β = ln Applied mathematics in Engineering, Management and Technology 24 f =.63 ln.234.7 6525 5629 +.5.72..86 ln ln +...86 ln ln +., f = log.4343β + 3.7, f = log 2.8β + 3.7 Fang (2) Brkic () (2) Brkic (2) (2) 4.sults and discussion Once results from empirical relations and relative error were measured, in order to display their accuracy in the best way, two diagrams were used in the form of figures and 2. In figure, relative error curve of relations with their error level more than ±.25% is provided. 35 lative roughness=.4 35 lative roughness=.8 3 3 25 25 2 2 5 5.E+4.E+5.E+6.E+7.E+8.E+9-5 - -5 35 3 25 2 5 5 - -5 ynolds Number Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) Jain and swamee (976) Jain (976) Churchill (977) Round (98) Barr (98) Haaland (983) Tsal (989) Manadilli (997) Goudar and Sonnad (28) Avci,Kargoz(29) Brkic () (2) lative roughness=..e+4.e+5.e+6.e+7.e+8.e+9-5 ynolds Number Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) Jain and swamee (976) Jain (976) Churchill (977) Round (98) Barr (98) Haaland (983) Tsal (989) Manadilli (997) Goudar and Sonnad (28) Avci,Kargoz(29) Brkic () (2) 5 5.E+4.E+5.E+6.E+7.E+8.E+9-5 - -5 5 4 3 2 - ynolds Number Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) Jain and swamee (976) Jain (976) Churchill (977) Round (98) Barr (98) Haaland (983) Tsal (989) Manadilli (997) Goudar and Sonnad (28) Avci,Kargoz(29) Brkic () (2) lative roughness=..e+4.e+5.e+6.e+7.e+8.e+9 ynolds Number Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) Jain and swamee (976) Jain (976) Churchill (977) Round (98) Barr (98) Haaland (983) Tsal (989) Manadilli (997) Goudar and Sonnad (28) Avci,Kargoz(29) Brkic () (2) 574
Applied mathematics in Engineering, Management and Technology 24 6 lative roughness=. 5 4 3 2.E+4.E+5.E+6.E+7.E+8.E+9 - ynolds Number Moody (947) Altshul (952) Wood (966) Eck (973) Churchill (973) Jain and swamee (976) Jain (976) Churchill (977) Round (98) Barr (98) Haaland (983) Tsal (989) Manadilli (997) Goudar and Sonnad (28) Avci,Kargoz(29) Brkic (2) (2) Fig - relative error of explicit relations compared to Colebrook- White relation As is displayed in figure, with a decrease in relative roughness, relative error of relations decreases in most cases as well and as ynolds number amounts to over 7, error curve gradient for all relations faces a falling pace. Among relations considered in this research and in the scope studied, Goudar- Sonnad (28) and Avci- Kargoz (29), in most cases, contain an error more than percent and provide the least appropriate results compared to Colebrook- White relation. In the following, in figure 2, the error curve of relations with relative error less than ±.25% is considered..25 lative roughness=.4.25 lative roughness=.8.75.75.5.5.25.25.e+4.e+5.e+6.e+7.e+8.e+9 -.25.E+4.E+5.E+6.E+7.E+8.E+9 -.25 -.5 -.5 -.75 -.75 - - -.25 ynolds Number Chen (979) Zigrang and sylvester() (982) Zigrang and sylvester(2) (982) Serghides () (984) Serghides (2) (984) Monzon,Romeo,Royo(22) Goudar,Sonnad (26) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Papevangelou et.al (2) Ghanbari et.al (2) Fang (2) Brkic (2) (2).25 lative roughness=. -.25 ynolds Number Chen (979) Zigrang and sylvester() (982) Zigrang and sylvester(2) (982) Serghides () (984) Serghides (2) (984) Monzon,Romeo,Royo(22) Goudar,Sonnad (26) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Papevangelou et.al (2) Ghanbari et.al (2) Fang (2) Brkic (2) (2).25 lative roughness=..75.75.5.5.25.25.e+4.e+5.e+6.e+7.e+8.e+9 -.25.E+4.E+5.E+6.E+7.E+8.E+9 -.25 -.5 -.5 -.75 -.75 - - -.25 ynolds Number Chen (979) Zigrang and sylvester() (982) Zigrang and sylvester(2) (982) Serghides () (984) Serghides (2) (984) Monzon,Romeo,Royo(22) Goudar,Sonnad (26) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Papevangelou et.al (2) Ghanbari et.al (2) Fang (2) Brkic (2) (2) -.25 ynolds Number Chen (979) Zigrang and sylvester() (982) Zigrang and sylvester(2) (982) Serghides () (984) Serghides (2) (984) Monzon,Romeo,Royo(22) Goudar,Sonnad (26) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Papevangelou et.al (2) Ghanbari et.al (2) Fang (2) Brkic (2) (2) 575
Applied mathematics in Engineering, Management and Technology 24.25 lative roughness=..75.5.25.e+4.e+5.e+6.e+7.e+8.e+9 -.25 -.5 -.75 - -.25 ynolds Number Chen (979) Zigrang and sylvester() (982) Zigrang and sylvester(2) (982) Serghides () (984) Serghides (2) (984) Monzon,Romeo,Royo(22) Goudar,Sonnad (26) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Papevangelou et.al (2) Ghanbari et.al (2) Fang (2) Brkic () (2) Fig 2: lative error for explicit relations compared to Colebrook-White Considering figure 2, one can say that in this set of relations, as well, for ynolds number bigger than 7, curve gradient of relative error decreases while most relations encounter highest error for ynolds numbers smaller than 7. All relations considered in figure 2 have a low relative error and recommended to be applied. But in order to study these relations more deeply and suggest precise relation(s), relations with relative error lower than ±.5% are considered which are displayed in figure 3.. lative roughness=.4. lative roughness=.8.e+4.e+5.e+6.e+7.e+8.e+9.e+4.e+5.e+6.e+7.e+8.e+9 -. -. -.2 -.2 -.3 -.3 -.4 -.4 -.5 ynolds Number Serghides () (984) Vatankhah,Kouchak zadeh (28) Buzzelli (28) -.5 ynolds Number Serghides () (984) Vatankhah,Kouchak zadeh (28) Buzzelli (28) 576
Applied mathematics in Engineering, Management and Technology 24. lative roughness=.. lative roughness=..e+4.e+5.e+6.e+7.e+8.e+9.e+4.e+5.e+6.e+7.e+8.e+9 -. -. -.2 -.2 -.3 -.3 -.4 -.4 -.5 -.5 ynolds Number ynolds Number Serghides () (984) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Serghides () (984) Vatankhah,Kouchak zadeh (28) Buzzelli (28) lative roughness=...e+4.e+5.e+6.e+7.e+8.e+9 -. -.2 -.3 -.4 -.5 ynolds Number Serghides () (984) Vatankhah,Kouchak zadeh (28) Buzzelli (28) Fig 3: lative error of best explicit relations compared to Colebrook-White Considering figure 3, in these relations, as well, for ynolds number bigger than 7, curve gradient of relative error, and in some cases, the amount of error decreases. Additionally, as relative roughness increases, relations error decreases where most error decline happens with a change in relative roughness from. to.. Considering the very low relative error of these three relations in all states, using these relations as the optimal explicit ones, is recommended. It is worth noting that Serghide () (984) with an error level less than ±.2% compared to Colebrook-White, is the best explicit relation and an appropriate replace for Colebrook- White implicit relation. 5.Conclusion Studies show that Goudar-Sonnad (28) and Avci-Kargoz (29) sometimes have a 2% relative error, so using all explicit relations provided by researchers is not right and much care is needed when determining them. sults show that for ynolds numbers bigger than 7, the error curve gradient or relations error decrease. Generally speaking, as relative roughness decreases, the explicit relations error increases. Considering figure 3, Serghides () (984) and Vatankhah- Kochakzadeh (28) and Buzzelli (28) with a relative error smaller than ±.5% is the best explicit relation provided to measure friction factor in flow pipes. After some deep studies it was concluded that Serghides() (984) with an error lower than ±.2% has the least relative error for all percentages and all ynolds numbers, so applying this relation is recommended in all states. ferences 577
Applied mathematics in Engineering, Management and Technology 24 ) Barr IH (98). Solutions of the Colebrook While function for resistance to uniform turbulent flow. Proc. Inst. Civil. Eng., Part 27, p. 529. 2) Brkić, (2). view of explicit approximations to the Colebrook relation for flow friction, Journal of Petroleum Science and Engineering, Vol. 77, No., pp. 34-48. 3) Chen NH (979). An explicit equation for friction factor in pipe. Ind. Eng. Chem. Fundamentals, 8(3): 296. 4) Churchill SW (977). Friction factor equations spans all fluid-flow regimes. Chem. Eng., 84(24): 9. 5) Colebrook CF (938-939). Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civil. Eng. (London), : 33. 6) Fang X, XuaY. and Zhou Z (2). New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations. Nuclear Engineering and esign, Vol. 24, No. 3, pp. 897-92. 7) Genić s, Arandjelović I, Kolendić P, Jarić M, Budimir N, Genić V (2). A view of Explicit Approximations of Colebrook s Equation. Faculty of Mechanical Engineering, Belgrade. Vol 39, pp 67-7. 8) Haaland SE (983). Simple and explicit formulas for the friction factor in turbulent pipe flow. Trans. ASME, 66(8): 67-684. 9) Goudar C T. and Sonnad J R (28). Comparison of the iterative approximations of the Colebrook-White equation. Hydrocarbon Processing, August 28, pp 79-83. ) Manadilli G (997). place implicit equations with signomial functions. Chem. Eng., 4(8): 29. ) Moody L F(944).Friction factors for pipe flows.trans. ASME, 66:64. 2) Romeo E, Royo C, Monzón A (22). Improved explicit equations for estimation of the friction factor in rough and smooth pipes. Chem. Eng. J, p. 369-374. 3) Round, G F (98), An explicit approximation for the friction factor-ynolds number relation for rough and smooth pipes, Can. J. Chem. Eng., 58,22-23. 4) Serghides T K (984), Estimate friction factor accurately, Chem. Eng. 9, pp. 63-64. 5) Turhan ÇOBAN M (22), Error Analysis of Non-Iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes. Journal of Naval science and Engineering. Vol8. No. pp -3. 6) Wood J (966). An explicit friction factor relationship. Civil Eng. ASCE 6. 7) Zigrang J, Sylvester N (982). Explicit approximations to the Colebrook s friction factor. AIChE J., 28(3): 54. 578