Proc. Instn Ch. Engrs, Part 2,1985,79, June, 383-389

TECHNICAL NOTE 431 WATER ENGINEERING GROUP

Systematic explicit solutions of the Prandtl and Colebrook-White equations for pipe flow

J. J. J. CHEN*

The Blasius equation was used to provide the initial value in the solution of the Prandtl and Colebrook-White formulae. The accuracy of the resulting equations was compared with the more accurate explicit friction factor-Reynolds number equations available in the literature. Explicit equations suitable for use as a substitute for the implicit Colebrook-White formula were also recommended.

Smooth pipe flow Techo, Tickner and James’ gave an approximation to the Prandtl equation

1 - = 2.0 log - Jf (23

as

f = (0.86859 lnCRel(l.964 In(Re) - 3.8215)]} -’ (2)

wherefis the Darcy friction factor and Re the Reynolds number. Equation (2) was shown to be accurate to less than one-tenth of one percent for 104 < Re 2.5 X 10’.

2. Equation (1) may be written in terms of the natural logarithm as

- = 0.86859 In( 1 Re J Jf

3. Equation (4), which provides a reasonable approximation to equation (l), was attributed by Finniecome’ to Nikuradse, while doubts had been expressed as to whether, in fact, equation (4) was due to N i k u r a d ~ e . ~ . ~ Equation (4) has, however, been given by Colebrook,’ without attribution, as a mathematical approximation to equation (1). Nevertheless, it will be shown later that equation (4) may be derived from equation (1).

1 _ - Jf- 1.8 log(Re/7) (4)

Written discussion closes 15 August 1985; for further details see p.ii. * Department of Chemicals and Materials Engineering, University of Auckland, New Zealand.

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C H E N

Equation (4) may be written as

- = 0.7817 In(Re/7) 1

Jf (5)

4. If equation (5) is taken as the initial value offfor carrying out the iterative solution of equation (3), substitution of equation (5) into the right-hand side of equation (3) and rearranging, yields

1.964 Re In( Re/7) I-' Equation (6) is identical to equation (2) which is given by Techo et al.' as 1.964 In 7 = 3.8218. Equation (2) was simply stated by Techo et al.' without indi- cation of how it was derived, nor was there any mention of equations (4) and (5). It may be inferred, therefore, that Techo et al.' had obtained equation (2) empirically by intuition.

5. An alternative to the analysis presented is given as follows. Consider the Blasius equation as being the initial approximation for use in the iterative solution of equation (1).

f= 0.184Re-"' (7)

Substituting equation (7) into the right-hand side of equation (1) yields

- = 2.0 l0g(O.l7077Re~'~) 1

Jf which may be rewritten as

1

L = 2.0 10g(Re/7.12)''~ = 1.8 log(Re/7.12) Jf Equation (9) bears almost exact resemblance to equation (4) which may, therefore, be viewed as the first approximation to the Prandtl equation, with the Blasius equation providing the initial value for iteration.

6. To obtain greater accuracy, equation (9) may be substituted into the right- hand side of equation (1) to yield, after rearranging

Re 4.518 log(Re/7.12)

This process may be repeated until any desired degree of accuracy is obtained. A somewhat similar procedure has also been carried out by Barr.6

7. Rewriting equation (10) so that there are a smaller number of significant figures in the numerical factors, gives after rearranging

f = {2 log[(4.52/Re)log(Re/7)]} -' (11)

8. The predictions of equations (10) and (1 1 ) are listed in Table 1 to allow comparison with the prediction of the Prandtl formula and the approximation of Techo et al.' It is noted that equation (ll), which has a considerably smaller number of significant figures in the numerical factors in comparison to equation (2), given by Techo et d.,' is just as good as, if not better than, equation (2).

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P R A N D T L A N D C O L E B R O O K - W H I T E E Q U A T I O N S

Table 1. Comparison of the predictions of equations (10) and (11), derived in this work with the Prandtl formula (equation ( 1 ) ) and the approximation of Techo et al.’ (equation (2))

Reynolds number,

Re

10000 20000 5oooo

100000 200000 500000

1000000 2oooo00 5000000

10000000 I Prandtl formula,

0,03089 0.02589 0.02090 0.01 799 0.01564 0.01316 0.01165 0.01037 0.00898 0.00810

Approximation of Techo et al.,

Equation (2)

0.03087 0.02590 0,02091 0.01801 0.01565 0.01317 0.01 165 0.01038 040898 0.008 10

1 Predictions of this work

Equation (10)

0.03084 0.02587 0,02090 0.01800 0-0 1 564 0.01316 0.01 164 0.01037 0,00898 0@0810 l Equation (11)

0.03087 0.02589 0.02091 0.0 1 800 0,01565 0.01316 0.01 165 0.01037 0.00898 0,008 10

Flow in rough pipes

tion 9. Colebrook and White’ combined equation (1) with the von Karman equa-

1 3.7D _ - Jf - log - E

to give the universal formula, now known by their name

where E is the equivalent sand roughness and D the diameter. 10. Equation (13) is not explicit inA asfappears on both sides of the equation

and suffers the same disadvantage as equation (1). As it has already been shown that equation ( 1 1 ) is almost identical to equation ( l ) , one may derive a universal formula by combining equation ( 1 1) and (12) to give

(4,52/Re)log(Re/7) + - 3.70

Equation (14) will be compared with those explicit friction factor formulae which follow, which were claimed to be very accurate and most of which have been reviewed by Chen.’

11. It should be pointed out that the explicit form given as equation (14) is more convenient to use for determining the pressure drop, given the flow rate and the pipe size. However, for the case where the flow rate is the unknown to be determined, Re J f may be readily evaluated and hence the use of equation (13) is to be preferred.

12. Churchill,8 in 1973, proposed the following equation

1 0 .9

- Jf = -2 log[(&) + &] 385

CH EN

which was later also given by Swamee and Jaing and Jain," in 1976, and has often been attributed inappropriately to these latter writers. It should, however, be pointed out that Barr," in 1972, gave equation (16) which is very similar to equation (15)

Equation (16) was later" modified to

- 1 = -2 ,,,(F + 5) Jf

Barr," therefore, preceded Churchill' in arriving at this form of equation. 13. Haaland13 recently gave

which is really one variation of the form of equation as given by equations (1 5), (16) or (17).

14. Barr,14 in 1980, gave

1 5.02 log[Re/4.518 log(Re/7)] Re(1 + Reo'52(~/D)o.7/29)

while a slightly less cumbersome form was given in 1981 as

1 4.518 log(Re/7) E -- Jf - -2 log[ Re(1 + Reo'52(~/D)o'7/29) + G]

Chen Ning Hsing16 gave

while Churchill17 also gave a rather complicated equation as

f = 8[(8/Re)" + 1/(A + 8)312]1/12

where

and

B = (37530/Re)16

It should be noted that equation (22) also encompasses the critical and laminar flow regimes. Zigrang and Sylvester" derived the following equations

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PRANDTL AND COLEBROOK-WHITE EQUATIONS

1 -- Jf - -2.0 log [(C - - - ye2 log (E - + - E)] -- 1 5.02 [&/D 5.02 (E Fe)]} Jf

- -2.0 log - - - log - - - log - + -- (:L: Re 3.7 Re (24)

where equation (24) is a better approximation than equation (23), obtained by substituting equation (23) into the right-hand side of equation (13). This is really just writing out a full iteration procedure with a finite limit to the number of loops.

15. The values offare calculated to four decimal places using equations (13)- (15) and (18H24) for the test points formed by combining Re = 4 X 103, 104, 10s, 106, 10’ and 10’ with (&/D) = 0, 10-5, 10-4, IO-’, IO-’ and 5 X IO-’. Deviations from equation (13), the Colebrook-White formula, have also been calculated. Discrepancies of 0.0001 in absolute value are taken to have zero percent deviation on account of the rounding-off of figures. For those with discrepancies of greater than 0.0001, the absolute percent deviation is calculated to two decimal places. It should be pointed out that all these equations give virtually identical results. Furthermore, the Colebrook-White formula given as equation (13) has in itself an error of up to 5% when compared to experimental data.

16. Nevertheless, the various equations are listed in Table 2 in the order of increasing average percentage deviation. It should be noted that equations (24), (21), (19), (20) and (23) match the Colebrook-White formula very closely, with equation (24), given by Zigrang and Sylvester,’* providing essentially exact match- ing. Equations (15) and (22), both due to Churchill, are almost equivalent in terms of accuracy, with equation (22), despite its rather complex form, in the range of Re and (&/D) considered here, offering only very slight improvement over the much simpler equation (15). There is really little to choose between equations (14) and (18), both of which are of about the same degree of simplicity as the Colebrook- White formula.

Conclusions 17. An equation, namely equation (1 l), has been derived which is equivalent to

the explicit formula of Prandtl given by equation (l). Equation (4), which relatesf explicitly with Re, has been shown to be equivalent to a first order approximation

Table 2. Explicit friction factor equations arranged in order of increasing average absolute deviation

Equation Maximum absolute Average absolute Author deviation, % deviation, ‘X,

(24)

3.10 0.58 Churchill* (15) 3.05 0.45 Churchill” (22) 1.61 0.33 Haaland” (18) 2.60 0.30 This work (14) 1.11 0.10 Zigrang and Sylvester” (23) 0.73 0.07 Barr’ (20) 0.62 0.04 Barr14 (19) 0.50 0.04 Chen Ning HsingL6 (21) 0.00 0.00 Zigrang and Sylvester”

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C H E N

to the Prandtl formula when the Blasius equation was used to provide the initial value of f in the iterative solution of the Prandtl formula. Following the method of Colebrook and White,5 and using equation (ll), an accurate formula was derived, namely equation (14), which is a good representation of the Colebrook-White formula in the range 4000 < Re < 10' and 0 < (&/D) < 0.05.

18. A comparison was made with the Colebrook-White formula using some of the more accurate explicit formulae for friction factor. Equation (15), the Barr- Churchill equation,'"' and equations (18) and (14) due respectively to Haaland13 and this work, which have maximum absolute deviations of 3.1%, 1.61% and 2.60% respectively, and are of about the same degree of complexity as the Colebrook-White formula, are a suitable substitute for the implicit Colebrook- White fornlula. Formulae that match exactly the Colebrook-White formula are available. However, these formulae are rather cumbersome in form, and the requirement for exact matching with the Colebrook-White formula is really not necessary as the latter formula is itself 3-5y0 in error when compared with actual experimental data.

Acknowledgements 19. The Author is grateful to Professor D. I. H. Barr, University of Strathclyde,

who commented on an initial draft and brought to the Author's attention a number of matters, including references 11 and 12; to Dean N. D. Sylvester, University of Tulsa, for bringing reference 18 to his attention; and to Professor S. W. Churchill, University of Pennsylvania, for his comments which are incorpo- rated in the text. This work was initiated while the Author was at the University of Hong Kong.

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P R A N D T L A N D C O L E B R O O K - W H I T E E Q U A T I O N S

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