3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 35 Vol. 9 N.º 3 Septiembre - Diciembre
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ALTERNATE FORMULA FOR CALCULATING THE DARCY
COEFFICIENT IN TURBULENT FLOW IN PIPES
Freddy Lizardo Kaseng Solis
National University Federico Villarreal, (Perú).
E-mail: fkaseng@unfv.edu.pe ORCID: https://orcid.org/0000-0002-2878-9053
Remo Bayona Antúnez
National University Federico Villarreal, (Perú).
E-mail: remobayona@yahoo.com ORCID: https://orcid.org/0000-0001-8655-1193
Ciro Rodriguez Rodriguez
National University Mayor de San Marcos, (Perú).
E-mail: crodriguezro@unmsm.edu.pe ORCID: https://orcid.org/0000-0003-2112-1349
Recepción:
04/05/2020
Aceptación:
18/06/2020
Publicación:
14/09/2020
Citación sugerida:
Kaseng, F.L., Bayona, R., y Rodriguez, C. (2020). Alternate formula for calculating the Darcy Coecient in turbulent
ow in pipes. 3C Tecnología. Glosas de innovación aplicadas a la pyme, 9(3), 99-109. https://doi.org/10.17993/3ctecno/2020.
v9n3e35.99-109
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ABSTRACT
The purpose of this research was to determine an alternative formula for calculating the Darcy coecient
in turbulent ow in pipes. The proposed alternate formula is an explicit formula that should be used
to replace the Colebrook-White formula for calculating the Darcy coecient in turbulent ow in pipes
since it has higher precision than the explicit formulas that are currently in use. In this investigation,
the alternate formula was compared with two explicit formulas commonly used in pipe design, the
Swamee-Jain and Pavlov formulas. To determining which formula is better, all of them were compared
with the Colebrook-White formula. For this, the average percentage and maximum percentage errors
of the Darcy coecient values calculated with each of the explicit formulas were determined, with the
values obtained with the Colebrook - White formula. It was determined that the maximum errors in
the calculation of the Darcy coecient concerning the Colebrook-White formula were: 3,104% for the
Swamee-Jain formula, 7,973% for the Pavlov formula and 2,740% for the alternate formula.
KEYWORDS
Pipes, Turbulent Flow, Darcy Coecient, Colebrook-White Formula, Swamee-Jain Formula, Pavlov
Formula, Alternate Formula.
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1. INTRODUCTION
An important part of the design of simple or complex hydraulic systems is the calculation of pressure
pipes. As in all calculations, the designer seeks precision and simplicity, which are opposed, since generally,
the simplicity carries with it the loss of accuracy. That loss of precision must be as little as possible for the
simplication to make sense since a signicant loss of precision would make the proposed simplication
inappropriate.
A well-known formula for calculating the Darcy coecient for turbulent ow in pipes is the Colebrook-
White formula. This formula has been used to prepare graphs for determining the Darcy coecient, as
is the case of the Moody diagram. However, the Colebrook-White formula has the drawback of being
an implicit formula, which has to be solved by successive approximations, which is inconvenient for the
calculation.
There are many explicit formulas to solve this problem that have been proposed that try to approximate
the results obtained with the Colebrook-White formula. Anaya et al. (2014) indicate that Pavlov’s formula
is the most recommended to replace the Colebrook-White implicit formula. According to Mott (2006),
the Swamee-Jain formula produces values for the Darcy coecient, which are within ± 1.0% of the
value of those corresponding to the Colebrook-White equation, within the range of relative roughness
between 0.001 and 1x10-6 and for Reynolds numbers ranging from 5x103 to 1x108.
In the present research, it was demonstrated that an alternative formula, proposed by the author,
has higher precision than the formulas that are mentioned and that are currently used, classication
algorithms could be used as Huapaya et al. (2020), and Levy et al. (2020).
2. MATERIAL AND METHODS
The research design was quasi-experimental; because variables were manipulated to obtain Darcy
coecients by dierent formulas.
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According to Spiegel and Stephens (2009), the sample size as for an innite or unknown population is:
Where:
n : sample size
Zα: value corresponding to the Gaussian distribution
p : expected prevalence of the parameter to be evaluated, if unknown (p = 0.5), which increases the
sample size
i: error
Sampling was carried out at the discretion of the researcher, proposing the values of Reynolds numbers
and relative roughness indicated above. The samples were obtained by calculating through the respective
formula (Colebrook - White, Swamee Jain, Pavlov, and alternate formula) the Darcy coecients
corresponding to predened values of Reynolds numbers and relative roughness. The Reynolds number
and relative roughness values used to obtain the sample were evenly distributed within the limits for
which the Colebrook-White formula is valid, from 4000 to 108 for the Reynolds number and from 0.05
to 10-8 for the relative roughness.
The Darcy coecient depends on the Reynolds number Re and the relative roughness εr.
Values of Reynolds numbers and relative roughnesses within the ranges of application of the Colebrook-
White formula were proposed, and the respective Darcy coecients were determined with the Colebrook-
White, Swamee-Jain, Pavlov formulas and the alternate formula.
Colebrook - White
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Swamee - Jain
Pavlov
Alternate formula
It was worked with a sample of 70007 Darcy coecient values for each of the formulas used, comparing
each of the explicit formulas (Swamee - Jain, Pavlov, and the alternate formula) with the implicit
Colebrook - White formula.
For each of the explicit formulas, the mean percentage error em and the maximum percentage error
Emax were obtained for the Colebrook White formula. Then these errors were compared with each
other to determine which of the formulas is the most appropriate for the calculation of the Darcy
coecient in pipes with turbulent ow.
The calculations were carried out on an Excel spreadsheet, resulting in Table 1.
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Table 1. Calculation of the percentage errors of the explicit formulas to the Colebrook - White formula.
Colebrook Swamee - Jain Pavlov Alternate formula
Nº Re ε
r
f f
1
e
1
|e
1
| e
m
f
2
e
2
|e
2
| e
m
f
3
e
3
|e
3
| e
m
1 4000 0,05 0,07699 0,0794 3,104 3,104 0,551 0,08079 4,936 4,936 1,683 0,07885 2,416 2,416 0,236
2 4000 1,00E-03 0,04091 0,0417 1,931 1,931 e
max
0,04415 5,875 7,920 e
max
0,04073 -0,440 0,440 e
max
3 4000 1,00E-04 0,04001 0,0407 1,650 1,650 3,104 0,0432 6,221 7,973 7,973 0,03967 -0,850 0,850 2,416
4 4000 1,00E-05 0,03992 0,0406 1,603 1,603 0,0431 6,262 7,966 0,03956 -0,902 0,902
5 4000 1,00E-06 0,03991 0,0406 1,604 1,604 0,04309 6,264 7,968 0,03955 -0,902 0,902
6 4000 1,00E-07 0,03991 0,0406 1,604 1,604 0,04309 6,264 7,968 0,03955 -0,902 0,902
7 4000 1,00E-08 0,03991 0,0406 1,604 1,604 0,04309 6,264 7,968 0,03955 -0,902 0,902
8 10000 0,05 0,07178 0,072 0,306 0,306 0,07208 0,111 0,418 0,07197 0,265 0,265
9 10000 1,00E-03 0,02217 0,0223 0,767 0,767 0,02277 1,926 2,706 0,02218 0,045 0,045
10 10000 1,00E-04 0,01851 0,0185 -0,324 0,324 0,01913 3,686 3,350 0,01818 -1,783 1,783
11 10000 1,00E-05 0,01804 0,0179 -0,665 0,665 0,01865 4,074 3,381 0,01763 -2,273 2,273
12 10000 1,00E-06 0,018 0,0179 -0,722 0,722 0,0186 4,085 3,333 0,01757 -2,389 2,389
13 10000 1,00E-07 0,01799 0,0179 -0,723 0,723 0,01859 4,087 3,335 0,01757 -2,335 2,335
14 10000 1,00E-08 0,01799 0,0179 -0,723 0,723 0,01859 4,087 3,335 0,01757 -2,335 2,335
15 20000 0,05 0,07167 0,0718 0,167 0,167 0,07183 0,056 0,223 0,07177 0,140 0,140
Source: authors’ own elaboration.
Table 1 shows the following percentage errors:
• e1 (percentage error of the Swamee - Jain formula to the Colebrook - White formula)
• e2 (percentage error of Pavlov’s formula to Colebrook-White’s formula)
• e3 (percentage error of the alternate formula to the Colebrook - White formula)
These errors are determined using the following formulas:
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Where:
f is the value of the Darcy coecient calculated with the Colebrook - White formula.
f1 is the value of the Darcy coecient calculated with the Swamee - Jain formula.
f2 is the value of the Darcy coecient calculated with the Pavlov formula.
f3 is the value of the Darcy coecient calculated with the alternative formula.
3. RESULTS
The testing of the hypothesis was performed by comparing the mean and maximum percentage errors,
obtained from the comparison between the Darcy coecients calculated with each of the explicit
formulas, and the Darcy coecients obtained by the Colebrook-White formula.
• emed1 and emax1 the mean percentage and maximum percentage errors obtained when calculating
the Darcy coecients with the Swamee-Jain formula, compared to those obtained using the
Colebrook-White formula.
• emed2 and emax2 the mean percentage and maximum percentage errors obtained when calculating
Darcy coecients with the Pavlov formula, compared to those obtained using the Colebrook –
White formula.
The summary of the errors is shown in Table 2:
Table 2. Percentage errors of explicit formulas.
Swamee - Jain Pavlov Alternate
e
m
0,551 1,683 0,236
e
max
3,104 7,973 2,416
Source: authors’ own elaboration.
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As can be seen, for the sample used, the alternate formula presents a mean percentage error of 0.236%
and a maximum percentage error of 2.416% on the Colebrook-White formula. Both values are
signicantly smaller than the errors in the Swamee - Jain, and Pavlov formulas.
4. DISCUSSION
The results do not agree with the results obtained by Anaya et al. (2014), who propose Pavlov’s formula
for calculating the Darcy coecient. The use of a single relative roughness value of 0.001 in that
investigation may have led to less than exact conclusions.
The statement of Mott (2006) is conrmed in that the Swamee-Jain formula is a good alternative for
calculating the Darcy coecient for turbulent ow in pipes.
The alternate formula outperforms the other formulas. It has an average error equal to 42.7% of the
average error of the Swamee - Jain formula and equivalent to 14% of the error of the Pavlov formula.
As for the maximum error, this is 77.8% of the maximum error of the Swamee-Jain formula and 30.3%
of the maximum error of the Pavlov formula.
It is concluded that the alternative formula is the best option for calculating the Darcy coecient for
turbulent ow in pipes.
5. CONCLUSION
The alternate formula outperforms the other formulas. It has an average error equal to 42.7% of the
average error of the Swamee - Jain formula and equivalent to 14% of the error of the Pavlov formula.
As for the maximum error, this is 77.8% of the maximum error of the Swamee-Jain formula and 30.3%
of the maximum error of the Pavlov formula.
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It is concluded that the alternative formula is the best option for calculating the Darcy coecient for
turbulent ow in pipes.
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