Index Copernicus Value (2015): 78.96 | Impact Factor (2015 ...

1 International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064. Index Copernicus Value (2015): | Impact Factor (2015): Error Analysis of friction Factor formulae with Respect to Colebrook-White Equation Bharati Medhi Das1, Bibhash Sarma2, Madan Mohan Das3. 1. Assistant Professor, Civil Engineering Department, Assam Engineering College, Jalukbari, Guwahati, Assam, India 2. Associate Professor, Civil Engineering Department, Assam Engineering College, Jalukbari, Guwahati, Assam, India 3. Formerly Professor of Civil Engineering Department, Assam Engineering College Emeritus Fellow, of AICTE, Retd. Director of Technical Education, Govt. of Assam, India Abstract: An important and integral part of pressure drop in a pipe involves the determination of friction Factor . The darcy Weisbach friction Factor formula is used for calculation of pressure loss in pipes. The Colebrook White (C-W) equation gives the best approximation to darcy Weisbach friction Factor for turbulent flow.

2 C-W equation cannot be solved directly due to its implicit form. Several numbers of approximate explicit equations have been proposed by many investigators. A brief review on friction Factor formulae is presented. The study is to select a suitable friction Factor formula in order to use it in pipe flow study for the calculation of friction Factor . Thus an analysis is done to compare the percentage error of friction Factor correlations. Based on applicability range of Reynolds number and pipe roughness, few equations are chosen. Relative percentage error of these selected approximate resistance equations are evaluated against the full range of flow conditions at different roughness sizes and found that average percentage error by Fang's(2011). and Romeo's(2008) equations are quite low in comparison to the others. Moreover it gives significantly better result than the commonly used equation of Barr (1981).Thus field engineer may use any one of the above explicit equations in the computation of friction Factor Value .

3 Keywords: Colebrook-White, explicit approximations, friction Factor formula, error analysis. 1. Introduction has to be made to a more modern concept. Prandtl (1932). had deduced a formula for friction Factor f in smooth pipe Pipe flow under pressure is used for a lot of purposes. as a function of Reynolds number R . In a smooth pipe Therefore it has become necessary to select a correct flow, the viscous sub-layer completely submerges the effect resistance equation for friction Factor evaluation. The various of average roughness of the pipe k on the flow. In this case, C-W explicit approximations taken are: Barr (1981), Zigrang- f is a function of Reynolds s number R and is independent of Sylvester (1982), Halland (1983), Romeo (2002), Brkic (2011) and the effect of k on the flow. Fang (2011). Comparative error analysis is made to assess the validity in using these six equations to evaluate friction In 1933, Nikuradse verified the Prandtl s boundary layer Factor values.

4 Full range of Reynolds numbers from x 10 3 theory and proposed the universal resistance equation for to 107 are taken to cover the turbulent flow stage up to fully fully developed turbulent flow in smooth pipe with empirical turbulent. Relative roughness [k/D] values are also changed constants 2 and , where pipe diameter is considered as from 10-7 to 10-2. friction Factor values by these six uniform and is expressed as: equations and C-W equation for above flow stages and 1 R f roughness are calculated. Percentage error is calculated by = 2log10. (2). comparing the friction Factor of selected equations to that of fsmooth friction Factor by C-W equation using the following equation: In case of rough pipe flow, the viscous sub-layer thickness is very small when compared to roughness height and thus the Percentage error flow is dominated by the roughness of the pipe and hence f friction Factor of explicit equation ( friction Factor by C W ) becomes independent of R and depends only on relative = x 100.

5 ( friction Factor by C W ) roughness values. Nikuradse evaluated a formula for f in terms of ratio of diameter D of the pipe to the diameter size k 2. Reviews on friction Factor Formula of the sand grains that he had used to roughen the inside of the pipe. The formula is re-arranged as: The darcy -Weisbach friction Factor is used to assess the 1 = 2log10 (3). resistance of flow that is frough k expressed as: 1 V Nikuradse s data have been served as the basis for many = (1). f sgD subsequent analysis of frictional resistance in pipes and in open channels for smooth, transitional and fully rough Equation (1) does not consider overtly the roughness and turbulent flow. However, Nikuradse used uniform sand viscosity of the liquid. To obtain a more general formula grains for roughness which produced an increase in f with which considers these two parameters explicitly, recourse Reynolds s number R over a particular range of partly rough Volume 6 Issue 3, March 2017.

6 Licensed Under Creative Commons Attribution CC BY. Paper ID: ART20172097 2105. International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064. Index Copernicus Value (2015): | Impact Factor (2015): flow. In 1937, Colebrook and White investigated the same 1 . range using non-uniform roughness. It was found that f is = 2 log10.. decreased throughout the partly rough flow regions. In 1939, . R = 400 4 . Chen . combining equations (2) and (3), Colebrook gave the 108. equation: . following friction Factor formula for commercial pipe 7. (1979) Where, . 1 D. = 2log10 + (4) = not specified = . where, k/D is the relative pipe roughness which is the ratio of R= 4000 4 . the mean height of roughness of the pipe to the pipe Round 108. equation: diameter. Thus the equation (4) is called Colebrook-White 8. (1980). 2.. = log +. (C-W) transition formula for smooth to rough turbulent flow, D. and is one of the important and popular formulae in = 0 resistance to flow in pipes.

7 The equation can be solved by . trial and error methods. White gave approximations to the Shacham = 2 log10.. logarithmic smooth turbulent element in the Colebrook 9. equation: . not specified White functions which was compatible in form with the (1980) 2. original, if Reynolds number to an Index is accepted as a . log10.. +.. substitute for R f Barr 10. equation 1 not specified 1 R R = 2 log10 + ~ log10 = 2log10 (5) 1981 . f 1 . = 2 log 10 +. This is one of the important and popular formulae in Zigrang- R= 2300 108. resistance of flow in pipes. Sylvester . Where, 11. equation D. Explicit approximations for Colebrook-White equations are :(1982) A = log10 + = .. shown in tabular form as: 13. B = log10 +.. Table 1: Explicit approximation for Colebrook-White R= 2300 108. equation Halland . equation: 1 Sl. Applicability 12. no Authors Mathematical expression range (1983) = log10 + D. = . Moody R = 4000 equation: 1. 1 10 6 3 5 108. (1944) f = 1 + 2000 / + R= 4000 108.

8 K = 0 Tsal . equation: (C ) . Altshul 13. =. (1989) + (C < ). equation: 68 D. 2 = + not specified 68 = 0 (1966) = +.. Manadilli R= 4000 108. f = a + bR-c 95. correlatio = 2 log + . 14 . R= 4000 5 n:(1997) 2. where D. Wood 107 = 0 . equation: a = + . 3 . (1966) 1 . D = 2 log10. b = 88. = .. c = + R = 2000 . Monzon- . Romeo- 108. 1 B = log10 +. 8 12 royo . 15. 3 12. = 8 + + 2 , where equation: .. Churchill 16 (2002). equation: D. 4 = 2log10 + not specified = not specified (1973) . = log10. 16.. 37530 B= +. + . Eck 1 . equation: 1 15 = ln + . 5 not specified (1973) = 2 log10 +.. Goudar - 2. Sonnad Where, =. Swamee- R= 5000 108 ln (10). 17. equation: . ln 10 Not specified Jain (2008). =. ln 10. =.. 6 equations 1 = 2 log 10 D = + ln . : (1976) = . = +1 = + ln . Volume 6 Issue 3, March 2017. Licensed Under Creative Commons Attribution CC BY. Paper ID: ART20172097 2106. International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064.

9 Index Copernicus Value (2015): | Impact Factor (2015): = ln = ln( . Fang . equation: R= 2000 108. = 1+ 2 21.. +1 2 (2011) 2. + 1 . + . (2 1) Ghanbari . 3 . = Farshad = log ( + 1) 22. Rieke s Not specified Avci and equation: (2011) Karagoz +. 18. = Not specified . correlatio n:(2009) . ln ln 1 + 1 + 10 .. Papaevan Frictiona Factor plays an important role (Medhi Das and gelou Sarma 2016, Medhi Das et , Medhi Das et ) in 19 correlatio =. (7 log )4 Not specified unsteady flow equations. n:(2010) 2. log +. 3. Results and Analysis = 2 log . Brkic 2. The results of the Numerical percentage error of explicit C- correlatio . 20. n:(2011). +.. Not specified W equations are shown in different figures for full range of flow conditions at different roughness sizes. The results are = ln ln also shown in tabular form only for roughness size, k/D = 10- 7. ln 1 + just as illustration. Numerical percentage error assessment of C-W. equations: Table 2: Values of friction Factor for the following formulae , when k/D = 10 -7.)

10 Reynolds No. k/D Barr Fang Romeo Brk ic Halland Zigrang Sylvester C-W. 2500 1E-07 4000 1E-07 6000 1E-07 8000 1E-07 10000 1E-07 30000 1E-07 50000 1E-07 80000 1E-07 100000 1E-07 300000 1E-07 700000 1E-07 1000000 1E-07 3000000 1E-07 7000000 1E-07 10000000 1E-07 Table 3: Percentage error in the formulae , when k/D = 10-7. Reynolds No. k/D Barr Fang Romeo Brkic Halland Zigrang-Sylvester C-W. 2500 1E-07 4000 1E-07 6000 1E-07 8000 1E-07 10000 1E-07 30000 1E-07 50000 1E-07 80000 1E-07 100000 1E-07 300000 1E-07 700000 1E-07 1000000 1E-07 3000000 1E-07 7000000 1E-07 10000000 1E-07 Average Volume 6 Issue 3, March 2017. Licensed Under Creative Commons Attribution CC BY. Paper ID: ART20172097 2107. International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064. Index Copernicus Value (2015): | Impact Factor (2015): Figure 1: Percentage error in different formulae , Figure 4: Percentage error in different formulae , when k/D=10-7 when k/D=10-4.