--

J1 Jlq"'---'

Estimatefriction factor

accurately--

Here is a very accurate equationfor calculating the friction factordirectly, without trial-and-error.

T. K. Serghides, Ken-McCee Chemical Corp.

The Colebrook equation is a widely used method forpredictingf, the friction factor for fluid flow:*'-

1(

EID 2.51)VT = -2.0 lag 3.7 + ReVJ (1)

where E is the absolute pipe roughness (ft), D is theinternal diameter ofthe pipe (ft), and Re is the Reynoldsnumber for flow.

\ You cannot salve the Colebrook equation directly, asthe friction factor appears on both sides. To salve itrequires trial-and-error or a graph-which are difficult toprogramo Or you can use one of several approximatesolutions that are explicit in f, and therefore can besolved directly.

This anide presents two new approximations, basedon numerical solution of the Colebrook equation. Botharrear to be more accurate than any of the other pub-lished approximations.

The friction-factor equationsThis explicit friction-factor equation is valid for transi-

tional and turbulent flow (Re> 2,100) at any relativeroughness (EID):

- ( (B -.

A)2 )-2

f - A - C - 2B + A

(EID 12 )A = -2.0 lag 3.7 + Re

B = -2,Olog(~~ + 2,~:A)"-'" .Note thal thefused here is Ihe Darey faCtor, wilh whieh frinional head loss (h.

ti) mav be eakulaled as /¡ = f(UD) (1,212g). Here, Lis lenglh (fl), l' is veloeity (n/s),and g is the gravitational constant (n/s').

where:

-----

C = -2.0 lag (EID + 2,51B )3.7 Re

(2)

Eq. 2 is derived by applying Steffenson's accelerated-convergence technique to an iterative, r'lUmerical solu-tion of Eq. l. The constants A, B and C are approxima-tion's of Eq. 1 obtained by three iterations of thedirect-substitution method. Eq. 2 is the approximationobtained by combining those constants according toSteffenson's formula.

lt happens that the numerical solution ofEq. 1 by thistechnique converges very fast, so Eq. 2 is very accurate-i.e., it approximates Colebrook's equation within a smallfraction of a percent.

Henrici [1] explains this behavior in discussing thebounds of error on a numerical solution after a finite

number ofiterations. A previous anide [2] discusses theconditions for convergence in solving the Colebrookequation by direct substitution, and explains Steffen-son's method.

A simpler version of Eq. 2 is nearly as accurate, andperhaps easier to use for hand calculations. Like Eq. 2,this equation is obtained by applying Steffenson'smethod to a numerical solution of Eq. 1. lt is valid forRe > 2,100 and any value of El D:

( (A - 4.781)2 )-2

f = 4.781 - B - 2A + 4.781

(EID 12)where: A = -2.0 lag 3.7 + Re

B = _2 O1 (EID 2.51A ). og 3.7 + Re

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63CHEMICAL ENGINU:RING MARCH 5,1984

FRICTION FACTOR

3.95

6.02

1.19

55.6

0.537

0.859

0.138

.Only for points within equation's limits.

How accurate are the equations?The table shows the accuracy ofEq. 2 and 3 over a test

matrix of 70 points, and compares these accuracies withthose of seven other explicit approximations of theColebrook equation:

. Moody equation [3], which is valid for 4,000 :<;;Re :<;;107 and E/D :<;;0.01:

r1= 5.5 X 10-3 O + (2 X 104E/D + 1O6/Re)I/3) (4)

. Wood equation [4], which is valid for Re ¿ 4,000and all E/D:

1= 0.094(E/D)O225+ 0.53(E/D) + 88(E/D)O44Re" (5)where: a= -1.62(E/D)o.134

. Jain equation [5], which is valid for 5,000 < Re <107 and 0.00004 < E/D < 0.05:

~ = 1.14 - 2.0 lag (E/D + 21.25/Re09)

. Churchill equation [6], which is valid for all values ofRe and E/D:

1 = 8 ((8/Re)12 + lI(A + B)I5)Yí2

where: A = (-2.457 In ((7/Re)09 + 0.27E/D))I6

B = (37,530/Re)I6

. Chen equation [5], which is valid for all values of Reand E/D:

1

v¡=- 2 O l (~ - 5.0452A ). og 3.7065 Re

where: - ( (E/D)IIO98 5.8506 \A - lag 2.8257 + Re 0.8981/

. Zigrang and Sylvester equations [5], which are validfor 4,000 < Re < 108and 0.00004 < E/D < 0.05:

1

v¡= -2.0A

1

v¡= -2.0 log (E/D - 5.02A )3.7 Re (10)

(E/D 5.02 ) (E/D 13)where: A = lag 3.7 - 7fe lag 3.7 + ReThe test matrix is 70 points: 10 relative roughness

values by 7 Reynolds numbers. The values of E/D are:0.00004,0.00005,0.0002,0.0006,0.0015,0.004,0.008,0.015, 0.03 and 0.05. The values of Re are: 2,500, 4,000,30,000, 105, 106, 107 and 108.

The measure of deviation (E) is the fractional differ-ence between the equation's friction-factor value and thenumerical solution of the Colebrook equation:

' /

E = l(j' - 1)/11 01)

where f is the Colebrook friction factor as calculatednumerically, andj' is the approximation. This compari-son is similar to the one performed by Zigrang andSylvester [5], but it covers the critical zone (2,100 < Re< 4,000) in addition. --"

The result? As the table shows, the maximum devi-ation ofEq. 2 from the numerical solution ofEq. 1 is only0.0023%, and the average deviation is a hundred ormore times smaller than that ofEq. 4 - 10. For Eq. 3, themaximum deviation is only 0.2%, and while the averagedeviation is not as low as that of Eq. 2, it is lower thanthose of the other equations.

Note that the figures for Eq. 4 - 6 in volved only thosepoints for which those equations were claimed to bevalido Eq. 9 and 10 are numerical solutions of Eq. 1, sothey were evaluated ayer all 70 points in the matrix.

Conclusion

Eq. 2 appears to approximate the Colebrook equation '-'"

more accurately than other explicit friction-factor equa-tions. Eq. 3 is not quite as accurate, but is stil1 better thanthe other equations looked at here, and is easier to use.

Mark Lipowicz, Editor

(6)References " /

1. Henrici. P.."Elernents ofNurnericalAnalysis,"john Wiley&Sons, NewYork, 1964.Serghides, T. K.. Iterative solution by direct substitution, Chem. Eng.,Ser!. 6, 1982.Daugherty, R. L, and Ingersoll, A. C., "Fluid Mechanics with Engineer-ing Applications," McGraw-Hill, New York, 1954.jeppson, R. W., "Analysis of Flow in Pipe Networks," Ann ArborSéience, Ann Arbor, Mich., 1977.Zigrang, D.J., and Sylvester, N. D., Explicit Approxirnations to theSolution ofColebrook's Friction Factor Equation, AJChE]., Vol. 28,No. 3, 1982.Churchill, S. W., Friction-factor equation spans all fluid-flow regirnes,Chem. Eng., Nov. 7, 1977.

2.

3.

(7) 4.

5.

6.

The author

(8)

' /

T. K. Serghides is a productionsuperintendent with Kerr-McGeeChernical Corp., P.O. Box 367, Trona,CA 93562, where he supervisesproduction of the South atea plants.Previously, he worked as a seniorprocess engineer at the Argus andTrona facihties. MI. Serghides earnedbis B.S. and M.E. degrees in chernicalengineering at lowa State University.Registered in California, he belongs toAIChE and the Instmrnent Soco ofArnerica.

(9)

64 CHEMICAL ~:NGINEt:RING MARCH 5, 1984....

4* 1.71

5* 2.67

6* 0.383

7 5.16

8 0.138

9 0.208

10 0.027