Proc. Instn Ciu. Engrs, Part 2, 1984,77, Mar., 49-55

TECHNICAL NOTE 400

A simple explicit formula for the estimation of pipe friction factor J. J. J. CHEN*

Explicit equations for the prediction of pipe flow friction factors are reviewed. A simple and explicit equation is proposed, which is suitable for most practical preliminary engineering calculations and on-site estimations.

Available explicit equations With the need to predict the friction factor necessary for pipe line design calcu- lations, Blasius, in 1912,’ was the first to give an empirical formula for smooth pipes which is valid in the range 103 < Re < 105

f = 0.3164Re-0’25 (1)

where f is the Darcy friction factor, and Re the Reynolds number. Equation (1) is a convenient form of expression as it also represents laminar flow conditions if 64 were substituted for the coefficient 0.3164 and 1.0 for the exponential index 0.25. Another common form of the Blasius equation has 0.184 for the coefficient and 0.2 for the exponent.

2. Based on extensive experimental data Nikuradse, in 1932,’ introduced the formula

1 = C1.S log(Re/?)]’

but the most widely accepted formula for smooth pipes is that due to Prandtl

- = 2 log - j, [?if] and for the fully rough regime, where the Reynolds number has no effect, the von Karman equation is the most used

- 1 = 2 log[3.7D] Jf

where E is the effective roughness height and D the diameter.

combining equations (3) and (4) 3. Colebrook and White’ subsequently introduced their universal formula by

Written discussion closes 15 May 1984; for futher details see p. (ii). * Department of Mechanical Engineering, University of Hong Kong.

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Unfortunately, equation ( 5 ) is not explicit in asfappears on both sides of the equation. To obtainffrom equation ( 9 , an iterative procedure, which may some- times be undesirable and time-consuming, will need to be performed. To overcome this inconvenience, Moody3 prepared a chart based on equation ( 9 , which gives a plot of the friction factor against the Reynolds number with the roughness ratio as the parameter.

4. However, Moody4 recognized the need to be able to obtain f directly without having to use a chart, and proposed the following expression, which differed from equation (5) by - 16% and + 13% in the range 4 X 103 Re 5 10' and0 < ( & / D ) < 5 X 10-'

f = 0.0055[1 + (2 X 104(&/~) + 1 0 6 / ~ e ) l q (6)

5. An explicit correlation had also been provided by Wood,' which is valid in the range Re > 10000 and 10' < (&/D) < 0.04.

f = a + bRe-' where

a = 0 . 0 9 4 ( ~ / D ) ~ ' ~ ~ ' + 0,53(~/D) b = 88.0(~/D)O'~~

C = 1.62(~/D)O"~~

The accuracy of equation (7) in the specified range is between -4% and + 5%. Its application in regions of low (&/D), where the value of (&/D) approaches zero, with a, b and c also tending to zero, inevitably results in a greater margin of error.

6. Churchill6 proposed the following equation for a developing and fully tur- bulent flow, incorporating equation (2), which he attributed incorrectly to Cole- brook, with equation (4)

which is explicit in$ 7. Swamee and Jain' obtained the formula

0.25 = [Iog(~/3.7D) + 5.74/Re0'9]2

by combining

0.25 = [log(~/3.7D)]~

Equations (10) and ( 1 1) were obtained by curve-fitting to the results of equation ( 5 ) for the smooth-turbulent and rough-turbulent regions respectively. Equation (lO), however, is exactly the same as the Nikuradse equation given here as equation ( 2 )

50

ESTIMATION OF PIPE FRICTION FACTOR

and may be rewritten as

1 = C2 10g(5.74/Re~'~)]~ (10a)

which is the same as

1 = [l43 l0g (5~74 /Re~ '~ )~~ ' ' ' ]~ ( 1 Ob)

Equation (lob) is exactly equal to equation (2) as 5.742/1'8 = 6.97 and (Re0'9)2/1'8 = Re.

8. By curve-fitting to the smooth turbulent region, Jain' obtained

fo'"= 1.8 log Re - 1.5146 (12)

Equation (12) is the same as equation (2) because 1.8 log 7 = 1.52. By combining equation (12) with equation (13) for rough pipes due to Prandtl, which is really the same as equation (4) as 2 log 3.7 = 1.14,

(13) 1

= 1.14 - 2 log 5 D

Jain obtained

-- f0'5 - 1.14 - 2 log(; + S)

which is really

1 - = 2 log 3.72 - 2 log f 0 ' 5

= - 2 log[& (; +S)] Equation (14b) is the same as equation (9); the latter equation being the one given by Swamee and Jain.'

9. In addition to equation (S), Churchill9 gave a formula which was claimed to hold for all Re and €/D as

f = 8[(8/Re)" + 1/(A + B)3/2]1112 (15)

where

B = (37 530/Re)16

An accurate formula, as claimed by Ning Hsing Chen," was given as

1 5.0452 5.8506

Jf 51

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10. The following equation was a modification of equation (2), quoted by Nekrasov"

Round" subsequently modified it to

Round,I2 however, attributed equation (2) to the Russian work maintaining that it was little known outside of the latter, in which the expression was written as f = (1 .8 log Re - 1.5)2. Equation (18) approximates the Colebrook-White equa- tion reasonably well.

1 1 . Recently, HaalandI3 proposed the following generalized equation

by combining equation (2), the von Karman equation and

- 1 = 1.8 l o g ( 5 ) Jf which he attributed to Colebrook. It is interesting to note that, apart from HaalandI3 and Churchil1,'j BarrI4 also attributed this equation to Colebrook while, in fact, the equation was first used by Nikuradse' in 1932 to correlate his extensive results; while Round" attributed a similar equation to the Russian work (Nekrasov").

12. Haaland showed that by taking n = 1.0, equation (19) predicts friction factors to f 1.5% of that predicted by the Colebrook-White equation. It should be noted that equation (19) is of the same form as that proposed originally by Churchill6 and is identical to equation (8) if n = 0.9.

Proposed explicit equation 13. An explicit equation is now developed which is simpler than any of those

already reviewed. Although it is not expected to be of high accuracy, the proposed correlation has to be reasonably accurate to render it suitable for most practical engineering purposes. Starting with the Blasius equation, a check of the variation off with (&/D) shows that f varies approximately with Therefore the general equation for the friction factor over the entire range of conditions may be written as

14. For equation (21), in order to reduce to the Blasius equation in the case of turbulent smooth pipe flow, c is taken as 0.3164 and a as 0.83. For the simplest case, when b equals 1.0, K = 0.11 gives quite a good prediction of the friction factor for the rough-transition and rough-turbulent regions. Equation (21) then becomes

52

ESTIMATION O F P I P E FRICTION FACTOR

--- -v- m o o ---

-11 rdo U

--e -v- m o o

m

S v)

2 '0

S

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0 .3

and its predictions are listed in Table 1. Also listed in Table 1 are the predictions of equation (23) in which c = 0,184, a = 0.67, K = 0.7 and b = 1.0.

15. It may be seen from Table 1 that the predictions of equation (22) in the range outside of the region described by (Re > 10’; (&/D) < 10-3) has an error of the order of less than f 8%, while equation (23) has an error of less than f 12% in the range except that described by (Re > 10’; (&/D) < lO-’). The larger error in the excluded range is only to be expected as the equations are derived from the Blasius equation which is itself not valid in the high Reynolds number regions.

Conclusions 16. A literature survey has been presented which shows that explicit equations,

varying in degrees of complexity and accuracy, for the prediction of friction factors in pipe flow are available. The simplest form of equation capable of accurate predictions resembles equation (S), which, as far as can be established, was first given by ChurchilL6 A number of proposed equations, such as those put forward by Swamee and Jain7 and Jain,’ are similar in essence and are no other than simple arithmetical variations of the Churchill equation.

17. In our work, an explicit equation for predicting the friction factor, which is so simple that it may be as easily committed to memory as the Blasius equation, was obtained. Although its accuracy is not as high as some of the more accurate equations given in the review, it nevertheless provides sufficient accuracy for most practical preliminary engineering calculations and on-site predictions of pipe fric- tion factors for pressure drop calculations.

References 1 .

2.

3. 4.

5.

6.

7.

X .

9.

10.

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FINNIECOME J. R. The friction coefficient for circular pipes at turbulent flow. Mech. Wld Engny Rec., Manchester and London, 1950,127, No. 331,725-739.

COLEBROOK C. F. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J . Instn Ciu. Engrs, 1939,11, 133-156.

MOODY L. F. Friction factor for pipe flow. Trans. Am. Soc. Mech. Engrs., 1944,671-684. MOODY L. F. An approximate formula for pipe friction factors. Mech. Engng, New York,

WOOD D. J. An explicit friction factor relationship. Ciu. Engng , Am. Soc. Ciu. Engrs, 1966,

CHURCHILL S. W. Empirical expressions for the shear stress in turbulent flow in com-

SWAMEE P. K. and JAIN A. K. Explicit equations for pipe flow problems. J . Hydraul. Diu.

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CHURCHILL S. W. Friction factor equation spans all fluid flow regimes. Chem. Engng,

CHEN NINC HSING. An explicit equation for friction factor in pipe. Ind. Engng Chem.

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E S T I M A T I O N O F P I P E F R I C T I O N F A C T O R

11. NEKRASOV B. Hydraulics. Peace Publishers, Moscow, 1968 (Transl. by V. Talmy), 95-101.

12. ROUND G. F. An explicit approximation for the friction factor-Reynolds number rela- tion for rough and smooth pipes. Can. J . Chem. Engng, Ottawa, 1980,58,122-123.

13. HAALAND, S. E. Simple and explicit formulas for the friction factor in turbulent pipe flow. J . Fluids Engng, ASME, New York, 1983,105,89-90.

14. BARR D. I. H. Solutions of the Colebrook-White function for resistance to uniform turbulent flow. Proc. Instn Ciu. Engrs, Part 2,1981,71, June, 529-535.

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