Revista Facultad de Ingeniería, Universidad de Antioquia, No.111, pp. 38-47, Apr-Jun 2024
Explicit pipe friction factor equations:
evaluation, classification, and proposal
Ecuaciones explícitas del factor de fricción de tuberías: evaluación, clasificación y
propuesta
Maiquel López-Silva1*, Dayma Sadami Carmenates-Hernández1, Nancy Delgado-Hernández1,
Nataly Chunga-Bereche1
1Universidad Católica Sedes Sapientiae, Facultad de Ingeniería. Campus: Av. Gonzales Prada s/n. Urb. Villa Los Ángeles,
C.P. 15302. Distrito de Los Olivos, Lima, Perú.
CITE THIS ARTICLE AS:
M. López-Silva, D. S.
Carmenates-Hernández, N.
Delgado-Hernández and N.
Chunga-Bereche. ”Explicit
pipe friction factor equations:
evaluation, classification, and
proposal”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 111, pp. 38-47,
Apr-Jun, 2024. [Online].
Available: https:
//doi.org/10.17533/udea.
redin.20230928
ARTICLE INFO:
Received: August 16, 2022
Accepted: October 19, 2023
Available online: October 19,
2023
KEYWORDS:
Colebrook equation; turbulent
fluid; relative roughness;
Reynolds number
Ecuación de Colebrook; fluido
turbulento; rugosidad relativa;
número de Reynolds
ABSTRACT: The Colebrook equation has been used to estimate the friction factor (f)
in turbulent fluids. In this regard, many equations have been proposed to eliminate
the iterative process of the Colebrook equation. The goal of this article was to
perform an evaluation, classification, and proposal of the friction factor for better
development of hydraulic projects. In this study, Gene Expression Programming
(GEP), Newton-Raphson, and Python algorithms were applied. The accuracy and
model selection were performed with the Maximum Relative Error (∆f /f ), Percentage
Standard Deviation (PSD), Model Selection Criterion (MSC), and Akaike Information
Criterion (AIC). Of the 30 equations evaluated, the Vatankhah equation was the most
accurate and simplest to obtain the friction factor with a classification of very high,
reaching a value of ∆f /f < 0.5% and 1.5 < PSD < 1.6 A new equation was formulated
to obtain the explicit f with fast convergence and accuracy. It was concluded that the
combination of GEP, error theory, and selection criteria provides a more reliable and
strengthened model.
RESUMEN: La ecuación de Colebrook se ha utilizado para estimar el factor de fricción (f) en
fluidos turbulentos. En este sentido, se han propuesto varias ecuaciones para eliminar
el proceso iterativo de la ecuación Colebrook. El objetivo de este artículo fue realizar
una evaluación, clasificación y propuesta del factor de fricción para un mejor desarrollo
de proyectos hidráulicos. En este estudio, se aplicaron los algoritmos de programación
de expresión génica (GEP), Newton-Raphson y Python. La precisión y la selección del
modelo se realizaron con el Máximo Error Relativo (∆f /f ), Porcentaje de Desviación
Estándar (PSD), Criterio de Selección del Modelo (MSC) y Criterio de Información de
Akaike (AIC). De las 30 ecuaciones evaluadas, la ecuación de Vatankhah fue la más
precisa y sencilla para obtener el factor de fricción con una clasificación de muy alta,
alcanzó un valor de ∆f /f < 0.5% y 1.5 < PSD < 1.6 Se formuló una nueva ecuación
para obtener el f explícita con rápida convergencia y precisión. Se concluyó que la
combinación de GEP, teoría del error y criterios de selección proporciona un modelo
más confiable y fortalecido.
1. Introduction
Pipes are used worldwide for the transportation of liquids
with different properties. Non-Newtonian fluids are
transported in pipelines in the mining and metallurgical
industries, such as drilling mud, cementitious composites,
and pastes [1]. In contrast, Newtonian fluids have a
wider field of use, especially in turbulent flow over rough
surfaces, with several engineering applications such
as industrial plants, internal distribution networks in
buildings, hydraulic turbines, irrigation systems, and
drinking water pipelines [2], as well as in open-channel
hydraulics [3]. Head losses are common in pipes or open
channels, an essential parameter that affects the design
and operation of the circulation flow in hydraulic works
[4, 5].
In piping systems, head losses are analyzed by the
38
* Corresponding author: Maiquel López-Silva
E-mail: mlopezs@ucss.edu.pe
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230928
38
Explicit pipe friction factor equations:
evaluation, classification, and proposal
Ecuaciones explícitas del factor de fricción de tuberías: evaluación, clasificación y
propuesta
Maiquel López-Silva1*, Dayma Sadami Carmenates-Hernández1, Nancy Delgado-Hernández1,
Nataly Chunga-Bereche1
1Universidad Católica Sedes Sapientiae, Facultad de Ingeniería. Campus: Av. Gonzales Prada s/n. Urb. Villa Los Ángeles,
C.P. 15302. Distrito de Los Olivos, Lima, Perú.
CITE THIS ARTICLE AS:
M. López-Silva, D. S.
Carmenates-Hernández, N.
Delgado-Hernández and N.
Chunga-Bereche. ”Explicit
pipe friction factor equations:
evaluation, classification, and
proposal”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 111, pp. 38-47,
Apr-Jun, 2024. [Online].
Available: https:
//doi.org/10.17533/udea.
redin.20230928
ARTICLE INFO:
Received: August 16, 2022
Accepted: October 19, 2023
Available online: October 19,
2023
KEYWORDS:
Colebrook equation; turbulent
fluid; relative roughness;
Reynolds number
Ecuación de Colebrook; fluido
turbulento; rugosidad relativa;
número de Reynolds
ABSTRACT: The Colebrook equation has been used to estimate the friction factor (f)
in turbulent fluids. In this regard, many equations have been proposed to eliminate
the iterative process of the Colebrook equation. The goal of this article was to
perform an evaluation, classification, and proposal of the friction factor for better
development of hydraulic projects. In this study, Gene Expression Programming
(GEP), Newton-Raphson, and Python algorithms were applied. The accuracy and
model selection were performed with the Maximum Relative Error (∆f /f ), Percentage
Standard Deviation (PSD), Model Selection Criterion (MSC), and Akaike Information
Criterion (AIC). Of the 30 equations evaluated, the Vatankhah equation was the most
accurate and simplest to obtain the friction factor with a classification of very high,
reaching a value of ∆f /f < 0.5% and 1.5 < PSD < 1.6 A new equation was formulated
to obtain the explicit f with fast convergence and accuracy. It was concluded that the
combination of GEP, error theory, and selection criteria provides a more reliable and
strengthened model.
RESUMEN: La ecuación de Colebrook se ha utilizado para estimar el factor de fricción (f) en
fluidos turbulentos. En este sentido, se han propuesto varias ecuaciones para eliminar
el proceso iterativo de la ecuación Colebrook. El objetivo de este artículo fue realizar
una evaluación, clasificación y propuesta del factor de fricción para un mejor desarrollo
de proyectos hidráulicos. En este estudio, se aplicaron los algoritmos de programación
de expresión génica (GEP), Newton-Raphson y Python. La precisión y la selección del
modelo se realizaron con el Máximo Error Relativo (∆f /f ), Porcentaje de Desviación
Estándar (PSD), Criterio de Selección del Modelo (MSC) y Criterio de Información de
Akaike (AIC). De las 30 ecuaciones evaluadas, la ecuación de Vatankhah fue la más
precisa y sencilla para obtener el factor de fricción con una clasificación de muy alta,
alcanzó un valor de ∆f /f < 0.5% y 1.5 < PSD < 1.6 Se formuló una nueva ecuación
para obtener el f explícita con rápida convergencia y precisión. Se concluyó que la
combinación de GEP, teoría del error y criterios de selección proporciona un modelo
más confiable y fortalecido.
1. Introduction
Pipes are used worldwide for the transportation of liquids
with different properties. Non-Newtonian fluids are
transported in pipelines in the mining and metallurgical
industries, such as drilling mud, cementitious composites,
and pastes [1]. In contrast, Newtonian fluids have a
wider field of use, especially in turbulent flow over rough
surfaces, with several engineering applications such
as industrial plants, internal distribution networks in
buildings, hydraulic turbines, irrigation systems, and
drinking water pipelines [2], as well as in open-channel
hydraulics [3]. Head losses are common in pipes or open
channels, an essential parameter that affects the design
and operation of the circulation flow in hydraulic works
[4, 5].
In piping systems, head losses are analyzed by the
38
* Corresponding author: Maiquel López-Silva
E-mail: mlopezs@ucss.edu.pe
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230928
38
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
universal Darcy-Weisbach equation. However, the implicit
friction factor (f ) intervenes in the equation. In this sense,
Colebrook [6] proposes an equation that is currently the
best approximation of the friction factor, especially for
turbulent flow [7]. Nevertheless, its calculation is complex
and cumbersome because the friction factor is present
at both ends of the equation. In addition, its solution
needs more time and processing in calculators. Therefore,
its solution requires using iterative methods such as
the Newton-Raphson approximation method. Although
diagnostic and control algorithms are implemented in
the mathematical modeling of hydraulic systems, precise
parameter tuning is necessary.
Several authors [8–15] have developed explicit
approximations of the friction factor as an alternative to
the Colebrook equation, but the explicit models developed
differ in their accuracy and computational efficiency
[16–19]. The work presented by [20] highlighted that the
equation by [21] was more accurate than the Colebrook
equation for the experimental data in their research. On
the other hand, [22] cite that the equations by [16] and [23]
are the most efficient, with a maximum-recorded error of
0.18% and 0.54%, respectively. Likewise, [24] propose that
the equations available in the literature lead to a deviation
of between 2% and 3% for a turbulent flow with a Reynolds
number of 2300. In turn, they suggest a new equation
based on the relationship between friction forces and
viscous forces to determine f with a maximum standard
deviation of 0.25% with respect to the Colebrook equation.
There have been significant contributions in recent
years to predicting the friction factor value with artificial
intelligence approaches such as Gene Expression
Programming (GEP), Evolutionary Polynomial Regression
(EPR), Adaptive Neuro-Fuzzy Inference System (ANFIS),
Artificial Neural Network (ANN), and physical and
numerical models that manage to predict the fluid
behavior in different media [25–28]. In particular, [26]
estimated f using Bayesian learning neural networks and
reached a relative error of 0.0035%. Furthermore, [29],
using some artificial intelligence approaches, reached
mean absolute errors of 0.001%. In this sense, [30] cite
some gaps in the artificial intelligence technique, such
as the data set, the layers of predesignated neurons, the
percentage of training, and the test in the model tree.
However, increasing the number of variables and implicit
functions of the friction factor is necessary. Likewise,
there is still a need to insert model selection criteria.
Many authors tend to use the Mean Squared Error (MSE),
Mean Relative Error (MRE), Mean Absolute Error (MAE),
Standard Deviation (SD), and Relative Error (RE) [7, 22]
and [31]. This has several disadvantages when compared
to other models since the value of R is more significant
when the number of variables in the mathematical model
increases [32]. Therefore, the value R can be increased,
and the models can become more complex.
There are several techniques to adjust the training
error for model sizes, such as Model Selection Criteria
(MSC), Akaike Information Criterion (AIC) [33], Bayesian
Information Criterion (BIC) [34], and Mallows’ Cp Criterion
[35]. The MSC and AIC have applied for the best prediction
model, but there have been limits: 4000 < Re < 108 and
10−6 < e/D < 5 · 10−2 [12, 36], discrepancies in the
results. The selection is important because the decision of
the criterion could affect the interpretation of the variable
as well as its prediction. Thus, the following hypothesis is
proposed in the present study: the explicit friction factor
equations can be classified, and the GEP can provide a new
equation with a minimum error. In this sense, the goal of
this work was to perform an evaluation, classification, and
a new suggested explicit pipe friction factor equation with
the least amount of error.
2. Materials and methods
The Colebrook equation is the most cited, accepted, and
validated equation in fluid dynamics studies for obtaining
friction losses in pipes.
It relates, in its implicit form, to the unknown friction factor
(f ), the relative roughness (e/D), the known pipe inner
surface area, and the known Reynolds number (Re). Valid
for 4000 < Re < 108 y 0 < e/D < 5 · 10−2, as
shown in Equation 1. However, Equation 1 requires some
mathematical iterations to get the optimal solution.
1
√f = −2 log
( ε
3.71D + 2.51
Re √f
)
(1)
Where f is the implied friction factor (f ), e is the absolute
roughness of the pipe’s inside wall, D is the pipe diameter,
and Re is the Reynolds number.
Nonetheless, there are several explicit approaches
reported in the scientific literature to calculate the friction
factor, as shown in Equations 2 to 36.
[21].
f =
[
−2 log
( ε/D
3.7 + 4.5
Re log Re
6.97
)]−2
Re ≥ 104 and 0 < ε/D < 5 · 10−2
(2)
[24]
f =
[
−2 log
( ε
3.7D + 10.04
R∗
)]−2
(3)
R∗ = 2 Re
[
− log
( ε/D
3.7 + 5.45
Re0.9
)]−1
(4)
R* the dimensionless number;
Limit: Re ≥ 2300 and 0 < ε/D < 0.05
39
universal Darcy-Weisbach equation. However, the implicit
friction factor (f ) intervenes in the equation. In this sense,
Colebrook [6] proposes an equation that is currently the
best approximation of the friction factor, especially for
turbulent flow [7]. Nevertheless, its calculation is complex
and cumbersome because the friction factor is present
at both ends of the equation. In addition, its solution
needs more time and processing in calculators. Therefore,
its solution requires using iterative methods such as
the Newton-Raphson approximation method. Although
diagnostic and control algorithms are implemented in
the mathematical modeling of hydraulic systems, precise
parameter tuning is necessary.
Several authors [8–15] have developed explicit
approximations of the friction factor as an alternative to
the Colebrook equation, but the explicit models developed
differ in their accuracy and computational efficiency
[16–19]. The work presented by [20] highlighted that the
equation by [21] was more accurate than the Colebrook
equation for the experimental data in their research. On
the other hand, [22] cite that the equations by [16] and [23]
are the most efficient, with a maximum-recorded error of
0.18% and 0.54%, respectively. Likewise, [24] propose that
the equations available in the literature lead to a deviation
of between 2% and 3% for a turbulent flow with a Reynolds
number of 2300. In turn, they suggest a new equation
based on the relationship between friction forces and
viscous forces to determine f with a maximum standard
deviation of 0.25% with respect to the Colebrook equation.
There have been significant contributions in recent
years to predicting the friction factor value with artificial
intelligence approaches such as Gene Expression
Programming (GEP), Evolutionary Polynomial Regression
(EPR), Adaptive Neuro-Fuzzy Inference System (ANFIS),
Artificial Neural Network (ANN), and physical and
numerical models that manage to predict the fluid
behavior in different media [25–28]. In particular, [26]
estimated f using Bayesian learning neural networks and
reached a relative error of 0.0035%. Furthermore, [29],
using some artificial intelligence approaches, reached
mean absolute errors of 0.001%. In this sense, [30] cite
some gaps in the artificial intelligence technique, such
as the data set, the layers of predesignated neurons, the
percentage of training, and the test in the model tree.
However, increasing the number of variables and implicit
functions of the friction factor is necessary. Likewise,
there is still a need to insert model selection criteria.
Many authors tend to use the Mean Squared Error (MSE),
Mean Relative Error (MRE), Mean Absolute Error (MAE),
Standard Deviation (SD), and Relative Error (RE) [7, 22]
and [31]. This has several disadvantages when compared
to other models since the value of R is more significant
when the number of variables in the mathematical model
increases [32]. Therefore, the value R can be increased,
and the models can become more complex.
There are several techniques to adjust the training
error for model sizes, such as Model Selection Criteria
(MSC), Akaike Information Criterion (AIC) [33], Bayesian
Information Criterion (BIC) [34], and Mallows’ Cp Criterion
[35]. The MSC and AIC have applied for the best prediction
model, but there have been limits: 4000 < Re < 108 and
10−6 < e/D < 5 · 10−2 [12, 36], discrepancies in the
results. The selection is important because the decision of
the criterion could affect the interpretation of the variable
as well as its prediction. Thus, the following hypothesis is
proposed in the present study: the explicit friction factor
equations can be classified, and the GEP can provide a new
equation with a minimum error. In this sense, the goal of
this work was to perform an evaluation, classification, and
a new suggested explicit pipe friction factor equation with
the least amount of error.
2. Materials and methods
The Colebrook equation is the most cited, accepted, and
validated equation in fluid dynamics studies for obtaining
friction losses in pipes.
It relates, in its implicit form, to the unknown friction factor
(f ), the relative roughness (e/D), the known pipe inner
surface area, and the known Reynolds number (Re). Valid
for 4000 < Re < 108 y 0 < e/D < 5 · 10−2, as
shown in Equation 1. However, Equation 1 requires some
mathematical iterations to get the optimal solution.
1
√f = −2 log
( ε
3.71D + 2.51
Re √f
)
(1)
Where f is the implied friction factor (f ), e is the absolute
roughness of the pipe’s inside wall, D is the pipe diameter,
and Re is the Reynolds number.
Nonetheless, there are several explicit approaches
reported in the scientific literature to calculate the friction
factor, as shown in Equations 2 to 36.
[21].
f =
[
−2 log
( ε/D
3.7 + 4.5
Re log Re
6.97
)]−2
Re ≥ 104 and 0 < ε/D < 5 · 10−2
(2)
[24]
f =
[
−2 log
( ε
3.7D + 10.04
R∗
)]−2
(3)
R∗ = 2 Re
[
− log
( ε/D
3.7 + 5.45
Re0.9
)]−1
(4)
R* the dimensionless number;
Limit: Re ≥ 2300 and 0 < ε/D < 0.05
39
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
[37]
f = 0.11
[ ε
D + 68
Re
]0.25
(5)
Limit: not specified
[38]
f = 6.4
[ln(Re) − ln (1 + 0.01 Re ε
D
(1 + 10√ ε
D
))]2.4 (6)
Limit: Re ≤ 4000
[39]
f =
−2 log
( ε
3.7D
)
+
4.518 log ( Re
7
)
Re
(
1 + 1
29 (Re0.52) ( ε
D
)0.7)
−2
(7)
Limit: 5000 < Re < 108 and 10−2 < ε/D < 10−6
[14]. Model I
f =
[
−2 log
(
10−0.4343β + ε/D
3.71
)]−2
(8)
Limit: not specified
[14]. Model II
f =
[
−2 log
( 2.18β
Re + ε/D
3.71
)]−2
(9)
Where β is:
β = ln
Re
1.816 ln
{ 1.1Re
ln(1+1.1Re)
}
(10)
Limit: not specified
[17]
f =
[
0.8686
(
B +
( 1.038 ln(B + A)
(0.332 + B + A)
)
− ln(B + A)
)]−2
(11)
A =
( Re(ε/D)
8.0878
)
(12)
B = ln
( Re
2.18
)
(13)
[40]
f =
[
−2 log
( ε
3.7065D − 5.0452
Re log
[ (ε/D)1.1098
2.8257 + 5.8506Re−0.8981
])]−2 (14)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[41]
f =
[
−2 log
( ε
3.71D + 7
Re0.9
)]−2
(15)
Limit: 4000 < Re < 108 and 10−6 < E/D < 5 · 10−2
[42]
f = 8
[( 8
Re
)12
+ [[2.457 ln
(
1
70.9
Re + 0.27 ( ε
D
)
)]16
+
( 37530
Re
)16] −3
2
1
12 (16)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[43]
f =
[
−2 log
( ε
3.715D + 15
Re
)]−2
(17)
Limit: not specified
[13]
f = 0.2479 − 0.0000947(7 − log Re)4
[log ( ε
3.615D + 7.366
Re0.9142
)]2 (18)
Limit: not specified
[23]
f = 1.613
{
ln
[
0.234
( ε
D
)1.1007
− 60.525
Re1.1105 + 56.291
Re1.0712
]}−2
(19)
Limit: 3000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[44]
f =
{
−1.52 log
[( ε
7.21D
)1.042
+
( 2.731
Re
)0.9152]}−2.169
(20)
Limit: 2100 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[45]
f =
{
−1.8 log
[( ε
3.7D
)1.11
+ 6.9
Re
]}−2
(21)
Limit: 400 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[46]
f =
[
−2 log
( ε
3.7D + 95
Re0.983 − 96.82
Re
)]−2
(22)
Limit: 5235 < Re < 109
[47]
f =
{
−1.8 log
[ 7.35 − 1200(ε/D)1.25
Re +
( ε
3.15D
)1.115]}−2
(23)
40
[37]
f = 0.11
[ ε
D + 68
Re
]0.25
(5)
Limit: not specified
[38]
f = 6.4
[ln(Re) − ln (1 + 0.01 Re ε
D
(1 + 10√ ε
D
))]2.4 (6)
Limit: Re ≤ 4000
[39]
f =
−2 log
( ε
3.7D
)
+
4.518 log ( Re
7
)
Re
(
1 + 1
29 (Re0.52) ( ε
D
)0.7)
−2
(7)
Limit: 5000 < Re < 108 and 10−2 < ε/D < 10−6
[14]. Model I
f =
[
−2 log
(
10−0.4343β + ε/D
3.71
)]−2
(8)
Limit: not specified
[14]. Model II
f =
[
−2 log
( 2.18β
Re + ε/D
3.71
)]−2
(9)
Where β is:
β = ln
Re
1.816 ln
{ 1.1Re
ln(1+1.1Re)
}
(10)
Limit: not specified
[17]
f =
[
0.8686
(
B +
( 1.038 ln(B + A)
(0.332 + B + A)
)
− ln(B + A)
)]−2
(11)
A =
( Re(ε/D)
8.0878
)
(12)
B = ln
( Re
2.18
)
(13)
[40]
f =
[
−2 log
( ε
3.7065D − 5.0452
Re log
[ (ε/D)1.1098
2.8257 + 5.8506Re−0.8981
])]−2 (14)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[41]
f =
[
−2 log
( ε
3.71D + 7
Re0.9
)]−2
(15)
Limit: 4000 < Re < 108 and 10−6 < E/D < 5 · 10−2
[42]
f = 8
[( 8
Re
)12
+ [[2.457 ln
(
1
70.9
Re + 0.27 ( ε
D
)
)]16
+
( 37530
Re
)16] −3
2
1
12 (16)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[43]
f =
[
−2 log
( ε
3.715D + 15
Re
)]−2
(17)
Limit: not specified
[13]
f = 0.2479 − 0.0000947(7 − log Re)4
[log ( ε
3.615D + 7.366
Re0.9142
)]2 (18)
Limit: not specified
[23]
f = 1.613
{
ln
[
0.234
( ε
D
)1.1007
− 60.525
Re1.1105 + 56.291
Re1.0712
]}−2
(19)
Limit: 3000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[44]
f =
{
−1.52 log
[( ε
7.21D
)1.042
+
( 2.731
Re
)0.9152]}−2.169
(20)
Limit: 2100 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[45]
f =
{
−1.8 log
[( ε
3.7D
)1.11
+ 6.9
Re
]}−2
(21)
Limit: 400 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[46]
f =
[
−2 log
( ε
3.7D + 95
Re0.983 − 96.82
Re
)]−2
(22)
Limit: 5235 < Re < 109
[47]
f =
{
−1.8 log
[ 7.35 − 1200(ε/D)1.25
Re +
( ε
3.15D
)1.115]}−2
(23)
40
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
Limit: 4000 ≤ Re ≤ 35.5 · 106
[12]
f =
{
−2 log
[ ε
3.7065D − 5.0272
Re log
( ε
3.827D − 4.657
Re log
(( ε
7.7918D
)0.9924
+
( 5.3326
208.815 + Re
)0.9345))]}−2
(24)
Limit: Re > 4000
[9]
f = 0.0055
(
1 +
(
2 · 104 ( ε
D
)
+
( 106
Re
)) 1
3
)
(25)
Limit: 4000 < Re < 108 and 0 < ε/D < 10−2
[31]
f =
{
−2 log
[( ε
3.71D
)
− 1.975
Re
(
ln
(( ε
3.93D
)1.092
+
( 7.627
395.9 + Re
)))]}−2 (26)
Limit: not specified
[48]
f =
{
−2 log
[
ε
7D +
( 6.81
Re
)0.9]}−2
(27)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 10−2
[49]
f =
{
−1.8 log
[
0.27(ε/D) +
( 6.5
Re
)]}−2
(28)
Limit: 4000 < Re < 107 and 10−6 < ε/D < 10−2
[50]
f =
{
−2 log
[ ε
3.7D − 5.02
Re log
( ε
3.7D + 14.5
Re
)]}−2
(29)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[51]
f =
[
−2 log
(( ε
3.715D
)
+
( 6.943
Re
)0.9)]−2
(30)
Limit: 5000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[10]
f = 0.25
[
log
( ε
3.7D + 5.74
Re0.9
)]−2
(31)
Limit: 5000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[16]
f = 0.8686 ln
[
0.3984Re
(0.8686S) S−0.645
S+0.39
]−2
(32)
Where S is:
S = 0.12363Re(ε/D) + ln(0.3984Re) (33)
Limit: not specified
[52]
f = 0.094(ε/D)0.225+0.53(ε/D)+88(ε/D)0.44 Re−1.62(ε/D)0.134
(34)
Limit: Re > 4000 and 10−5 < ε/D < 5 · 10−2
[11] Model I.
f =
{
−2 log
[( ε
3.7D
)
− 5.02
Re log
( ε
3.7D − 5.02
Re log
( ε
3.7D + 13
Re
))]}−2 (35)
Limit: 4000 < Re < 108 and 10−5 < ε/D < 5 · 10−2
[11] Model II.
f =
{
−2 log
[( ε
3.7D
)
− 5.02
Re log
( ε
3.7D + 13
Re
)]}−2
(36)
Limit: 4000 < Re < 108 and 10−5 < ε/D < 5 · 10−2
The Colebrook equation and the 30 explicit equations
found in the scientific literature were evaluated for
different conditions of relative roughness (e/D) from 10−6
to 5 · 10−2 and the Reynolds number from 4000 to 108,
which implied a base of 47601 data points. The analysis
interval integrates the onset of turbulence and complete
turbulence to test the best behavior of the correlations in
the mathematical formulations.
In this study, the Newton-Raphson method was used
in Colebrook Equation 1 by the Python algorithm. The
method has been generalized due to its simplicity and
speed of convergence to solve nonlinear problems,
systems of equations, and nonlinear differential and
integral equations [23].
Similarly, Gene Expression Programming (GEP),
implemented in GeneXpro software, was applied, after
obtaining the evaluation, classification, and generation
of the most suitable equations. Initially, the database
composed of 47,601 variables was used to select the
best adjustment according to their fitness and introduce
genetic variation using genetic operators.
Additionally, the procedure for estimating the pipeline
friction coefficient using GEP involved fitness function
selection, choice of T-termini and F-functions to create
chromosomes, choice of chromosome architecture, choice
of linkage function, and choice of genetic operators.
41
Limit: 4000 ≤ Re ≤ 35.5 · 106
[12]
f =
{
−2 log
[ ε
3.7065D − 5.0272
Re log
( ε
3.827D − 4.657
Re log
(( ε
7.7918D
)0.9924
+
( 5.3326
208.815 + Re
)0.9345))]}−2
(24)
Limit: Re > 4000
[9]
f = 0.0055
(
1 +
(
2 · 104 ( ε
D
)
+
( 106
Re
)) 1
3
)
(25)
Limit: 4000 < Re < 108 and 0 < ε/D < 10−2
[31]
f =
{
−2 log
[( ε
3.71D
)
− 1.975
Re
(
ln
(( ε
3.93D
)1.092
+
( 7.627
395.9 + Re
)))]}−2 (26)
Limit: not specified
[48]
f =
{
−2 log
[
ε
7D +
( 6.81
Re
)0.9]}−2
(27)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 10−2
[49]
f =
{
−1.8 log
[
0.27(ε/D) +
( 6.5
Re
)]}−2
(28)
Limit: 4000 < Re < 107 and 10−6 < ε/D < 10−2
[50]
f =
{
−2 log
[ ε
3.7D − 5.02
Re log
( ε
3.7D + 14.5
Re
)]}−2
(29)
Limit: 4000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[51]
f =
[
−2 log
(( ε
3.715D
)
+
( 6.943
Re
)0.9)]−2
(30)
Limit: 5000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[10]
f = 0.25
[
log
( ε
3.7D + 5.74
Re0.9
)]−2
(31)
Limit: 5000 < Re < 108 and 10−6 < ε/D < 5 · 10−2
[16]
f = 0.8686 ln
[
0.3984Re
(0.8686S) S−0.645
S+0.39
]−2
(32)
Where S is:
S = 0.12363Re(ε/D) + ln(0.3984Re) (33)
Limit: not specified
[52]
f = 0.094(ε/D)0.225+0.53(ε/D)+88(ε/D)0.44 Re−1.62(ε/D)0.134
(34)
Limit: Re > 4000 and 10−5 < ε/D < 5 · 10−2
[11] Model I.
f =
{
−2 log
[( ε
3.7D
)
− 5.02
Re log
( ε
3.7D − 5.02
Re log
( ε
3.7D + 13
Re
))]}−2 (35)
Limit: 4000 < Re < 108 and 10−5 < ε/D < 5 · 10−2
[11] Model II.
f =
{
−2 log
[( ε
3.7D
)
− 5.02
Re log
( ε
3.7D + 13
Re
)]}−2
(36)
Limit: 4000 < Re < 108 and 10−5 < ε/D < 5 · 10−2
The Colebrook equation and the 30 explicit equations
found in the scientific literature were evaluated for
different conditions of relative roughness (e/D) from 10−6
to 5 · 10−2 and the Reynolds number from 4000 to 108,
which implied a base of 47601 data points. The analysis
interval integrates the onset of turbulence and complete
turbulence to test the best behavior of the correlations in
the mathematical formulations.
In this study, the Newton-Raphson method was used
in Colebrook Equation 1 by the Python algorithm. The
method has been generalized due to its simplicity and
speed of convergence to solve nonlinear problems,
systems of equations, and nonlinear differential and
integral equations [23].
Similarly, Gene Expression Programming (GEP),
implemented in GeneXpro software, was applied, after
obtaining the evaluation, classification, and generation
of the most suitable equations. Initially, the database
composed of 47,601 variables was used to select the
best adjustment according to their fitness and introduce
genetic variation using genetic operators.
Additionally, the procedure for estimating the pipeline
friction coefficient using GEP involved fitness function
selection, choice of T-termini and F-functions to create
chromosomes, choice of chromosome architecture, choice
of linkage function, and choice of genetic operators.
41
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
The 30 Chromosomes were executed, with a
head size of 8 and the number of genes 1,
2, 3, and 6; linking functions (+, -, *, /); and
mathematical functions divided into GEP1, GEP2,
GEP3, and GEP4 in +, −, /, ·, √x, ex, log10, 10xx1/3,
x1/4, x1/5, x2, x3, x4, x5, 1/x.
In the investigation, percent standard deviation (PSD)
and Equation 37 Maximum Relative Error (∆f /f ) were
used as criteria for the accuracy of the explicit models.
∆f
f =
( fCW − fproposed
fCW
)
100% (37)
Additionally, efficient methods of model comparison and
selection based on model complexity were applied. Model
Selection Criteria (MSC) [29] and Akaike’s Information
Criteria (AIC) were used [26]. These criteria expressed by
Equations 38 and 39 are based on the greatest likelihood
and smallest parameters, and the variables follow a
normal distribution.
M SC = ln
[ ∑n
i=1
(fCW − ¯fProposed
)2
∑n
i=1
(fCW − fProposed
)2
]
− 2p
n (38)
AIC = n ln
[
1
n
n∑
i=1
(fCW − fProposed
)2
]
+ 2p (39)
Where fCW is the true value of the Colebrook-White (CW)
friction factor, fproposed is the value of the proposed friction
factor, p is the number of equation parameters including
constants, i = 1,… n is the number of friction factor values,
and n is the sample size.
3. Results and discussion
Figure 1 shows the accuracies of the explicit models
according to the Maximum Relative Error (∆f /f ) and
percent standard deviation (PSD). Figure 1 a) shows that
the (∆f /f ) values ranged from 0.082% to 38.435%, and
43% of the equations had values lower than 2.0% of the
Maximum Relative Error. Group I is the most efficient
approximation where the Maximum Relative Error is less
than 1%; therefore, those are recommended for precision
engineering work. In Group I, the results are outstanding,
presenting values ∆f /f < 0.5% by the equations of [12],
model I [11, 16, 31] and [17, 23, 24, 40]. In particular, the
equations by [13, 39] and [50] have 0.5 < ∆f /f < 1%.
Other authors have formulated new, noteworthy, accurate
equations; these are classified in group II because they
have a Maximum Relative Error of less than 2%, which are
those proposed for model II by [11] and [45].
Group III was classified as having a lower approximation
to Colebrook’s with a Maximum Relative Error between
2.587 2.587 ≤ ∆f /f ≤ 8.303, as equations cited by
[21, 46], model II by [10, 14, 38, 44, 51], and model I
by [14]. However, the equation by [21], according to
[20] in their research, was the most accurate. Possible
causes were that [20] only used 2397 experimental points,
3000 ≤ R e ≤ 735 − 103, and 0 0 < ε/D < 1.4 − 10−3.
Nonetheless, group IV had to be rejected because they
exceeded ∆f /f > 10%, as are [9, 42, 47–49, 52], and [37].
In particular, the equation proposed by [9], at the time
provided significant results for solving problems, but it
is shown that new and more accurate formulations have
been developed.
Results that agree with those obtained by [22], who
evaluated 33 equations in a range of the Moody diagram
with 2300 ≤ Re ≤ 108, 0 < ε/D < 5 − 10−2 and in
relation to the equation proposed by [9] the error test
was high, exceeding 10%. Similarly, it agrees with the
results by [29] on the mathematical models analyzed using
Machine Learning tools in which [9] and [42] had the most
unfavorable equations.
Regarding Figure 1 b) and the Percent Standard Deviation
(PSD), it is observed that, in general, the 30 equations
analyzed presented a deviation between 1.2%<PSD<2%.
However, 81% of the equations had a stable standard
deviation between 1.5% and 1.6%. Nevertheless, there
are three equations of approximations with the lowest
standard deviation, such as [9, 48] and [37], but they
presented a high relative error for which they were
rejected.
The 30 equations analyzed in this article have two
perspectives: firstly, the equations with a high number
of parameters tend to be more accurate, and secondly,
the equations with the least number of parameters are
less accurate. On the other hand, the engineer needs
the easiest and most accurate equation for friction factor
calculation, according to [24]. In summary, as a result
of the increasing digitization of work, educational and
economic environments, the equations must be formulated
with the highest precision and best computational
performance.
For this reason, the MSC and AIC Model Selection
Criteria have been implemented using a Ranking
because it considers a decisive variable as the number
of parameters, including the constants in the equations (p).
Based on the accuracies of the models, a preliminary
model ranking (Rk) was proposed for each evaluation
criterion p, ∆f /f, PSD, MSC, and AIC, and finally, a
Global Ranking. Table 1 shows the results of the models.
It is observed that the error theory and theoretical
functions show results that differ in their rank order
42
The 30 Chromosomes were executed, with a
head size of 8 and the number of genes 1,
2, 3, and 6; linking functions (+, -, *, /); and
mathematical functions divided into GEP1, GEP2,
GEP3, and GEP4 in +, −, /, ·, √x, ex, log10, 10xx1/3,
x1/4, x1/5, x2, x3, x4, x5, 1/x.
In the investigation, percent standard deviation (PSD)
and Equation 37 Maximum Relative Error (∆f /f ) were
used as criteria for the accuracy of the explicit models.
∆f
f =
( fCW − fproposed
fCW
)
100% (37)
Additionally, efficient methods of model comparison and
selection based on model complexity were applied. Model
Selection Criteria (MSC) [29] and Akaike’s Information
Criteria (AIC) were used [26]. These criteria expressed by
Equations 38 and 39 are based on the greatest likelihood
and smallest parameters, and the variables follow a
normal distribution.
M SC = ln
[ ∑n
i=1
(fCW − ¯fProposed
)2
∑n
i=1
(fCW − fProposed
)2
]
− 2p
n (38)
AIC = n ln
[
1
n
n∑
i=1
(fCW − fProposed
)2
]
+ 2p (39)
Where fCW is the true value of the Colebrook-White (CW)
friction factor, fproposed is the value of the proposed friction
factor, p is the number of equation parameters including
constants, i = 1,… n is the number of friction factor values,
and n is the sample size.
3. Results and discussion
Figure 1 shows the accuracies of the explicit models
according to the Maximum Relative Error (∆f /f ) and
percent standard deviation (PSD). Figure 1 a) shows that
the (∆f /f ) values ranged from 0.082% to 38.435%, and
43% of the equations had values lower than 2.0% of the
Maximum Relative Error. Group I is the most efficient
approximation where the Maximum Relative Error is less
than 1%; therefore, those are recommended for precision
engineering work. In Group I, the results are outstanding,
presenting values ∆f /f < 0.5% by the equations of [12],
model I [11, 16, 31] and [17, 23, 24, 40]. In particular, the
equations by [13, 39] and [50] have 0.5 < ∆f /f < 1%.
Other authors have formulated new, noteworthy, accurate
equations; these are classified in group II because they
have a Maximum Relative Error of less than 2%, which are
those proposed for model II by [11] and [45].
Group III was classified as having a lower approximation
to Colebrook’s with a Maximum Relative Error between
2.587 2.587 ≤ ∆f /f ≤ 8.303, as equations cited by
[21, 46], model II by [10, 14, 38, 44, 51], and model I
by [14]. However, the equation by [21], according to
[20] in their research, was the most accurate. Possible
causes were that [20] only used 2397 experimental points,
3000 ≤ R e ≤ 735 − 103, and 0 0 < ε/D < 1.4 − 10−3.
Nonetheless, group IV had to be rejected because they
exceeded ∆f /f > 10%, as are [9, 42, 47–49, 52], and [37].
In particular, the equation proposed by [9], at the time
provided significant results for solving problems, but it
is shown that new and more accurate formulations have
been developed.
Results that agree with those obtained by [22], who
evaluated 33 equations in a range of the Moody diagram
with 2300 ≤ Re ≤ 108, 0 < ε/D < 5 − 10−2 and in
relation to the equation proposed by [9] the error test
was high, exceeding 10%. Similarly, it agrees with the
results by [29] on the mathematical models analyzed using
Machine Learning tools in which [9] and [42] had the most
unfavorable equations.
Regarding Figure 1 b) and the Percent Standard Deviation
(PSD), it is observed that, in general, the 30 equations
analyzed presented a deviation between 1.2%<PSD<2%.
However, 81% of the equations had a stable standard
deviation between 1.5% and 1.6%. Nevertheless, there
are three equations of approximations with the lowest
standard deviation, such as [9, 48] and [37], but they
presented a high relative error for which they were
rejected.
The 30 equations analyzed in this article have two
perspectives: firstly, the equations with a high number
of parameters tend to be more accurate, and secondly,
the equations with the least number of parameters are
less accurate. On the other hand, the engineer needs
the easiest and most accurate equation for friction factor
calculation, according to [24]. In summary, as a result
of the increasing digitization of work, educational and
economic environments, the equations must be formulated
with the highest precision and best computational
performance.
For this reason, the MSC and AIC Model Selection
Criteria have been implemented using a Ranking
because it considers a decisive variable as the number
of parameters, including the constants in the equations (p).
Based on the accuracies of the models, a preliminary
model ranking (Rk) was proposed for each evaluation
criterion p, ∆f /f, PSD, MSC, and AIC, and finally, a
Global Ranking. Table 1 shows the results of the models.
It is observed that the error theory and theoretical
functions show results that differ in their rank order
42
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
Figure 1 Comparative assessment flowchart
for each equation, with a discrepancy in optimal model
selection. Equation 5, proposed by [37] is the simplest and
has the least number of steps to obtain the friction factor.
Nevertheless, in the previous analysis, it was rejected
because of its high relative error, which is positioned at
number 30. Meanwhile, Equation 11 by [17] is classified
as the most complex for its solution due to the number
of steps and parameters it includes. However, it was
classified in group I with a relative error of less than 0.5%
and an acceptable deviation of less than 1.6%, with a
ranking of 8.
In this sense, MSC and AIC contributed to the selection
of the best model. However, in both cases, they present
discrepancies with respect to the function of greater
likelihood and entropy. The MSC value indicates that by
[49] equation occupies rank 1, while the MSC value of the
[11] equation model I occupies rank 30. In relation, the AIC
reached inversely proportional values, the [49] equation
reached rank 30 and [11] equation model I has rank 1.
On the other hand, in contrast to the previous equations,
the number of parameters by [11] equation model I is
47% higher than by [49] equation. Consequently, it can
be pointed out that the AIC criterion does not follow the
parsimony principle because the smaller the number of
parameters, the smaller the AIC tends to be.
contradict the theories for which the AIC criterion
was defined. In finite samples, the AIC value is only
approximate [33]. Therefore, difficulties could arise
regarding the validity and applicability of the method for
this purpose.
Additionally, the MSC criterion also showed
inconsistencies between the models due to the number
of parameters; however, this coincides with the results of
the AIC criterion. This trend in the results corresponds
with those results obtained by [36].
The global ranking obtained in Table 1 integrates
the positions of the most accurate and inaccurate
approximation models with their degrees of complexity.
The explicit Equation 32 proposed by [16] leads the Global
Ranking in the first position as the most accurate, followed
in second place by Equations 29, 26, and 22 by [31, 50],
and [46]. The least accurate and most complex to solve
are Equations 34, 23, 28 by [47, 52], and [49], which in turn
belong to the rejected group IV.
Consequently, in Table 1 an easier classification has
been established, according to the level of precision
and simplicity for the first five global rankings. It
was established from a very high level, which indicates
excellent precision and simplicity, to a very low level, which
is interpreted as an inaccurate and complex equation to
solve due to the number of operations and parameters
present.
As a new proposal for explicit friction factor approximation
equations, 64 models were analyzed in Gene Expression
Programming (GEP). The theoretical and experimental
databases were developed as a training process to train
the GEP algorithm. Twenty percent of the data was
reserved for validation and the rest for calibration. Only
the most efficient results of GEP1, GEP2, GEP3, and GEP4
according to the performance criteria are reflected in
Table 3.
Table 3 shows that the most significant models had Linking
Functions + and ∗, a Number of Chromosomes of 30, a
Head Size of 8, and a Number of Genes of 2 and 6. The
best-performing model was GEP1, with the lowest number
of functions (4), and 7 parameters including constants.
43
Figure 1 Comparative assessment flowchart
for each equation, with a discrepancy in optimal model
selection. Equation 5, proposed by [37] is the simplest and
has the least number of steps to obtain the friction factor.
Nevertheless, in the previous analysis, it was rejected
because of its high relative error, which is positioned at
number 30. Meanwhile, Equation 11 by [17] is classified
as the most complex for its solution due to the number
of steps and parameters it includes. However, it was
classified in group I with a relative error of less than 0.5%
and an acceptable deviation of less than 1.6%, with a
ranking of 8.
In this sense, MSC and AIC contributed to the selection
of the best model. However, in both cases, they present
discrepancies with respect to the function of greater
likelihood and entropy. The MSC value indicates that by
[49] equation occupies rank 1, while the MSC value of the
[11] equation model I occupies rank 30. In relation, the AIC
reached inversely proportional values, the [49] equation
reached rank 30 and [11] equation model I has rank 1.
On the other hand, in contrast to the previous equations,
the number of parameters by [11] equation model I is
47% higher than by [49] equation. Consequently, it can
be pointed out that the AIC criterion does not follow the
parsimony principle because the smaller the number of
parameters, the smaller the AIC tends to be.
contradict the theories for which the AIC criterion
was defined. In finite samples, the AIC value is only
approximate [33]. Therefore, difficulties could arise
regarding the validity and applicability of the method for
this purpose.
Additionally, the MSC criterion also showed
inconsistencies between the models due to the number
of parameters; however, this coincides with the results of
the AIC criterion. This trend in the results corresponds
with those results obtained by [36].
The global ranking obtained in Table 1 integrates
the positions of the most accurate and inaccurate
approximation models with their degrees of complexity.
The explicit Equation 32 proposed by [16] leads the Global
Ranking in the first position as the most accurate, followed
in second place by Equations 29, 26, and 22 by [31, 50],
and [46]. The least accurate and most complex to solve
are Equations 34, 23, 28 by [47, 52], and [49], which in turn
belong to the rejected group IV.
Consequently, in Table 1 an easier classification has
been established, according to the level of precision
and simplicity for the first five global rankings. It
was established from a very high level, which indicates
excellent precision and simplicity, to a very low level, which
is interpreted as an inaccurate and complex equation to
solve due to the number of operations and parameters
present.
As a new proposal for explicit friction factor approximation
equations, 64 models were analyzed in Gene Expression
Programming (GEP). The theoretical and experimental
databases were developed as a training process to train
the GEP algorithm. Twenty percent of the data was
reserved for validation and the rest for calibration. Only
the most efficient results of GEP1, GEP2, GEP3, and GEP4
according to the performance criteria are reflected in
Table 3.
Table 3 shows that the most significant models had Linking
Functions + and ∗, a Number of Chromosomes of 30, a
Head Size of 8, and a Number of Genes of 2 and 6. The
best-performing model was GEP1, with the lowest number
of functions (4), and 7 parameters including constants.
43
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
Table 1 Preference models
Authors
No. equations
p Main Model selection Global
statistics criteria Ranking
Parameter
∆f/f
PSD
MSC
AIC
Total
Global
No Rk Rk Rk Rk ∑ Rk GR
[21] 2 11 14 20 17 14 76 14
[24] 3 17 6 14 28 3 68 7
[37] 5 6 30 3 3 28 70 9
[38] 6 14 18 25 10 21 88 20
[39] 7 19 9 13 23 8 72 11
[14]I 8 18 21 18 11 20 88 20
[14]II 9 19 16 22 15 15 87 19
[17] 11 39 5 8 25 6 83 18
[40] 14 16 7 11 24 7 65 5
[41] 15 9 23 27 8 23 90 22
[42] 16 19 24 5 5 27 80 17
[43] 17 8 22 28 9 22 89 21
[13] 18 14 10 16 21 10 71 10
[23] 19 13 8 17 22 9 69 8
[44] 20 10 17 4 12 19 62 3
[45] 21 10 13 19 18 13 73 13
[46] 22 10 15 2 16 16 59 2
[47] 23 12 28 29 6 25 100 24
[12] 24 21 1 12 29 2 65 5
[9] 25 8 26 2 4 26 66 6
[31] 26 15 3 10 27 4 59 2
[48] 27 7 27 1 2 29 66 6
[49] 28 8 25 30 1 30 94 23
[50] 29 11 11 6 20 11 59 2
[51] 30 9 19 23 14 17 82 15
[10] 31 8 20 24 13 18 83 16
[16] 32 13 4 9 26 5 57 1
[52] 34 16 29 26 7 24 102 25
[11]I 35 17 2 15 30 1 65 5
[11]II 36 14 12 7 19 12 64 4
Table 2 Model classification
GR Authors Precision Simplicity
1 [16] Very high Very high
2 [50] Very high High
2 [31] Very high High
3 [46] Medium Medium
3 [44] Medium Medium
4 [11]II High Low
5 [11]I Very high Very Low
5 [12] Very high Very Low
5 [40] Very high Very Low
The Root Mean Square Error (RMSE) was 0.078%, the
Mean Absolute Error (MAE) was 0.055%, the Pearson
correlation coefficient (R) was 0.99873, the ∆f /f was
6.22%, and the PSD was 1.86%.
In contrast to the groups made in Figure 1 due to the
maximum relative error, GEP1 was classified in group III
because it was within the interval 2.5 ≤ ∆f/f ≤ 8.3, this
being an alternative to obtain the friction factor quickly
and easily.
Although GEP4 has the highest R and a lower ∆f /f , PSD,
it is shown to be more significant for having a greater
number of functions, according to [24]. In addition, the
GEP4 model has a greater number of operations for its
solution, making it less simple. Regarding the increase
of functions, the Number of Chromosomes, Head Size,
and Number of Genes showed a partial relationship to the
results obtained by [51] that the GEP models increase with
increasing functions.
Equation 40 is proposed as a new nonlinear model to
determine the explicit friction factor coefficient with the
lowest error without the existence of logarithmic functions,
speed of calculation, or more accurate approximation in
the turbulent flow regime. The Limit: 4000 < Re < 108
and 10−6 < ε/D < 10−2.
GEP 1 = 0.219[(0.028ε/D) + (0.896/R)]0.25 (40)
4. Conclusions
Thirty explicit friction factor equations were analyzed on a
base of 47601 theoretical and experimental data points and
according, to the maximum relative error (∆f /f ), were
classified into 4 groups: group I of 0.5% < ∆f /f , group
II of 0.5% < ∆f /f < 1%, group III of 1% < ∆f /f < 2%
and group IV ∆f /f > 2%. Group I includes the most
accurate explicit friction factor equations, developed by
[12], model I [11, 16, 17, 24, 31, 40] and [23]. In general,
the Percentage Standard Deviation (PSD) was acceptable
and comprised between 1.2% < PSD ≤ 1.9%.
The MSC and AIC selection criteria contributed to the
selection of the most accurate equations to estimate
the friction factor, but they presented a discrepancy
in likelihood and entropy. However, the number of
parameters and operations of the equations (p) was
a decisive variable in obtaining the global ranking of
the 30 friction factor equations explicit in Table 2. In
summary, the first five global rankings were classified
by the most accurate and simple equations. Therefore, it
was concluded that the estimates of the equation by [16]
ranked very high in accuracy and simplicity for obtaining
explicit friction factors. The [50] and [31] equations also
presented very high performance. In contrast, the use
44
Table 1 Preference models
Authors
No. equations
p Main Model selection Global
statistics criteria Ranking
Parameter
∆f/f
PSD
MSC
AIC
Total
Global
No Rk Rk Rk Rk ∑ Rk GR
[21] 2 11 14 20 17 14 76 14
[24] 3 17 6 14 28 3 68 7
[37] 5 6 30 3 3 28 70 9
[38] 6 14 18 25 10 21 88 20
[39] 7 19 9 13 23 8 72 11
[14]I 8 18 21 18 11 20 88 20
[14]II 9 19 16 22 15 15 87 19
[17] 11 39 5 8 25 6 83 18
[40] 14 16 7 11 24 7 65 5
[41] 15 9 23 27 8 23 90 22
[42] 16 19 24 5 5 27 80 17
[43] 17 8 22 28 9 22 89 21
[13] 18 14 10 16 21 10 71 10
[23] 19 13 8 17 22 9 69 8
[44] 20 10 17 4 12 19 62 3
[45] 21 10 13 19 18 13 73 13
[46] 22 10 15 2 16 16 59 2
[47] 23 12 28 29 6 25 100 24
[12] 24 21 1 12 29 2 65 5
[9] 25 8 26 2 4 26 66 6
[31] 26 15 3 10 27 4 59 2
[48] 27 7 27 1 2 29 66 6
[49] 28 8 25 30 1 30 94 23
[50] 29 11 11 6 20 11 59 2
[51] 30 9 19 23 14 17 82 15
[10] 31 8 20 24 13 18 83 16
[16] 32 13 4 9 26 5 57 1
[52] 34 16 29 26 7 24 102 25
[11]I 35 17 2 15 30 1 65 5
[11]II 36 14 12 7 19 12 64 4
Table 2 Model classification
GR Authors Precision Simplicity
1 [16] Very high Very high
2 [50] Very high High
2 [31] Very high High
3 [46] Medium Medium
3 [44] Medium Medium
4 [11]II High Low
5 [11]I Very high Very Low
5 [12] Very high Very Low
5 [40] Very high Very Low
The Root Mean Square Error (RMSE) was 0.078%, the
Mean Absolute Error (MAE) was 0.055%, the Pearson
correlation coefficient (R) was 0.99873, the ∆f /f was
6.22%, and the PSD was 1.86%.
In contrast to the groups made in Figure 1 due to the
maximum relative error, GEP1 was classified in group III
because it was within the interval 2.5 ≤ ∆f/f ≤ 8.3, this
being an alternative to obtain the friction factor quickly
and easily.
Although GEP4 has the highest R and a lower ∆f /f , PSD,
it is shown to be more significant for having a greater
number of functions, according to [24]. In addition, the
GEP4 model has a greater number of operations for its
solution, making it less simple. Regarding the increase
of functions, the Number of Chromosomes, Head Size,
and Number of Genes showed a partial relationship to the
results obtained by [51] that the GEP models increase with
increasing functions.
Equation 40 is proposed as a new nonlinear model to
determine the explicit friction factor coefficient with the
lowest error without the existence of logarithmic functions,
speed of calculation, or more accurate approximation in
the turbulent flow regime. The Limit: 4000 < Re < 108
and 10−6 < ε/D < 10−2.
GEP 1 = 0.219[(0.028ε/D) + (0.896/R)]0.25 (40)
4. Conclusions
Thirty explicit friction factor equations were analyzed on a
base of 47601 theoretical and experimental data points and
according, to the maximum relative error (∆f /f ), were
classified into 4 groups: group I of 0.5% < ∆f /f , group
II of 0.5% < ∆f /f < 1%, group III of 1% < ∆f /f < 2%
and group IV ∆f /f > 2%. Group I includes the most
accurate explicit friction factor equations, developed by
[12], model I [11, 16, 17, 24, 31, 40] and [23]. In general,
the Percentage Standard Deviation (PSD) was acceptable
and comprised between 1.2% < PSD ≤ 1.9%.
The MSC and AIC selection criteria contributed to the
selection of the most accurate equations to estimate
the friction factor, but they presented a discrepancy
in likelihood and entropy. However, the number of
parameters and operations of the equations (p) was
a decisive variable in obtaining the global ranking of
the 30 friction factor equations explicit in Table 2. In
summary, the first five global rankings were classified
by the most accurate and simple equations. Therefore, it
was concluded that the estimates of the equation by [16]
ranked very high in accuracy and simplicity for obtaining
explicit friction factors. The [50] and [31] equations also
presented very high performance. In contrast, the use
44
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
Table 3 Efficient model of the GEP
Model Functions Performance criteria
Training and Validations
RMSE MAE R ∆f /f PSD
(%) (%)) (%)
GEP1 +, −, /, ∗, √x, 0.078 0.055 0.99873 6.22 1.86
GEP2 +, −, /, ∗√x, ex, log10, ln 0.114 0.093 0.99729 6.49 1.92
GEP3 +, −, /, ∗, √x, ex, log10, ln, 10x, x1/3, x1/4, x1/5 0.080 0.064 0.99868 6.31 1.89
GEP4 +, −, /, ∗, √x, ex, log10, 10xx1/3, x1/4, x1/5, x2, x3, x4, x5, 1/x 0.089 0.067 0.99893 6.20 1.84
of the equations developed by [52], [47], and [49] is not
recommended, and in the case of their use, they should
be under specific conditions because they can produce
inaccurate results. This new approach made it possible
to observe that, under certain conditions, the Colebrook
equation is not the most accurate at present.
With the GEP, it was possible to provide a new model
to determine the explicit friction factor f (R, ε/D) with the
lowest degree of complexity in the turbulent flow regime.
It has an RMSE of 0.078%, an MAE of 0.055%, and an R
of 0.99873. Compared with the Colebrook equation, it has
more simplicity, fast convergence, less computational
time, and a good relationship between accuracy and
computational efficiency. From the analyzed equations
of the explicit friction factor for turbulent flow, it was
found that there are new equations with optimal efficiency
indicators for the original equations that are cited, such as
those by [9, 10] and [51]. In this regard, it is recommended
to consider the mathematical models’ new functions as
more accurate explicit approximations.
The main finding of the research developed is the
integration of statistical tools, Python algorithms, Genetic
Expression Programming, and the new model proposed for
obtaining the level of complexity and effectiveness of the
explicit friction factor equations of the Colebrook equation.
Likewise, novel information would ease the elaboration
and decision-making of hydraulic engineering projects. In
response to the previous conclusion, it is recommended
to extend the analysis methods with artificial intelligence
and new criteria for the selection of mathematical models.
5. Declaration of competing interest
We declare that we have no significant competing interests,
including financial or non-financial, professional, or
personal interests, interfering with the full and objective
presentation of the work described in this manuscript.
6. knowledgements
Acknowledgements for the collaboration of the Applied
Hydraulics Research Group of the Universidad Catholica
Sedes Sapientiae.
7. Funding
The authors received no financial support for the research,
authorship, and/or publication of this article.
8. Author contributions
Maiquel López Silva: Conceived and designed the analysis,
the scientific literature review, the statistical analysis, and
interpretation of data, the formulation of gene expression
programming algorithms, prepared the text. Dayma
Carmenates Hernández: Conceived and designed the
analysis, assisted with the scientific literature review,
statistical analysis, and interpretation of data, and
prepared the text and edited the manuscript. Nancy
Delgado Hernández: Scientific literature review, digital
processing of data, formulation of gene expression
programming algorithms, and Newton-Raphson
programming in Python. Nataly Chunga Bereche:
Scientific literature review, digital processing of data,
formulation of gene expression programming algorithms,
and Newton-Raphson programming in Python.
9. Data availability statement
The origin of the data is from the turbulent regime, with
different conditions of relative roughness (/D) from 110−6
to 5 × 10−2 and the Reynolds number from 4000 to 108,
which implied a base of 47601 data points. The authors
confirm that the data supporting the findings of this study
are available within the article and its supplementary
materials.
References
[1] L. F. Ospina-H., M. E. López-G., C. A. Palacio, and J. F. Jiménez-M.,
“Dispositivo de reynolds para el estudio reológico de fluidos no
newtonianos independientes del tiempo: Diseño, construcción
y realización de pruebas preliminares,” Revista Colombiana de
Materiales, no. 3, Oct. 19, 2012. [Online]. Available: https:
//revistas.udea.edu.co/index.php/materiales/article/view/13227
45
Table 3 Efficient model of the GEP
Model Functions Performance criteria
Training and Validations
RMSE MAE R ∆f /f PSD
(%) (%)) (%)
GEP1 +, −, /, ∗, √x, 0.078 0.055 0.99873 6.22 1.86
GEP2 +, −, /, ∗√x, ex, log10, ln 0.114 0.093 0.99729 6.49 1.92
GEP3 +, −, /, ∗, √x, ex, log10, ln, 10x, x1/3, x1/4, x1/5 0.080 0.064 0.99868 6.31 1.89
GEP4 +, −, /, ∗, √x, ex, log10, 10xx1/3, x1/4, x1/5, x2, x3, x4, x5, 1/x 0.089 0.067 0.99893 6.20 1.84
of the equations developed by [52], [47], and [49] is not
recommended, and in the case of their use, they should
be under specific conditions because they can produce
inaccurate results. This new approach made it possible
to observe that, under certain conditions, the Colebrook
equation is not the most accurate at present.
With the GEP, it was possible to provide a new model
to determine the explicit friction factor f (R, ε/D) with the
lowest degree of complexity in the turbulent flow regime.
It has an RMSE of 0.078%, an MAE of 0.055%, and an R
of 0.99873. Compared with the Colebrook equation, it has
more simplicity, fast convergence, less computational
time, and a good relationship between accuracy and
computational efficiency. From the analyzed equations
of the explicit friction factor for turbulent flow, it was
found that there are new equations with optimal efficiency
indicators for the original equations that are cited, such as
those by [9, 10] and [51]. In this regard, it is recommended
to consider the mathematical models’ new functions as
more accurate explicit approximations.
The main finding of the research developed is the
integration of statistical tools, Python algorithms, Genetic
Expression Programming, and the new model proposed for
obtaining the level of complexity and effectiveness of the
explicit friction factor equations of the Colebrook equation.
Likewise, novel information would ease the elaboration
and decision-making of hydraulic engineering projects. In
response to the previous conclusion, it is recommended
to extend the analysis methods with artificial intelligence
and new criteria for the selection of mathematical models.
5. Declaration of competing interest
We declare that we have no significant competing interests,
including financial or non-financial, professional, or
personal interests, interfering with the full and objective
presentation of the work described in this manuscript.
6. knowledgements
Acknowledgements for the collaboration of the Applied
Hydraulics Research Group of the Universidad Catholica
Sedes Sapientiae.
7. Funding
The authors received no financial support for the research,
authorship, and/or publication of this article.
8. Author contributions
Maiquel López Silva: Conceived and designed the analysis,
the scientific literature review, the statistical analysis, and
interpretation of data, the formulation of gene expression
programming algorithms, prepared the text. Dayma
Carmenates Hernández: Conceived and designed the
analysis, assisted with the scientific literature review,
statistical analysis, and interpretation of data, and
prepared the text and edited the manuscript. Nancy
Delgado Hernández: Scientific literature review, digital
processing of data, formulation of gene expression
programming algorithms, and Newton-Raphson
programming in Python. Nataly Chunga Bereche:
Scientific literature review, digital processing of data,
formulation of gene expression programming algorithms,
and Newton-Raphson programming in Python.
9. Data availability statement
The origin of the data is from the turbulent regime, with
different conditions of relative roughness (/D) from 110−6
to 5 × 10−2 and the Reynolds number from 4000 to 108,
which implied a base of 47601 data points. The authors
confirm that the data supporting the findings of this study
are available within the article and its supplementary
materials.
References
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“Dispositivo de reynolds para el estudio reológico de fluidos no
newtonianos independientes del tiempo: Diseño, construcción
y realización de pruebas preliminares,” Revista Colombiana de
Materiales, no. 3, Oct. 19, 2012. [Online]. Available: https:
//revistas.udea.edu.co/index.php/materiales/article/view/13227
45
M. López-Silva et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 111, pp. 38-47, 2024
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46
[2] W. H. Alawee, Y. A. Almolhem, B. Yusuf, T. Mohammad, and H. A.
Dhahad, “Variation of coefficient of friction and friction head losses
along a pipe with multiple outlets,” Water, vol. 12, no. 3, Mar. 17,
2020. [Online]. Available: https://doi.org/10.3390/w12030844
[3] F. J. Mejía, “Relación de las curvas de energía específica y pendiente
de fricción con las zonas de flujo libre en canales,” Revista Escuela
de Ingeniería de Antioquia, vol. 9, pp. 69–75, Jan-Jun. 2008.
[4] J. A. Gómez-Camperos, P. J. García-Guarín, and C. Nolasco-Serna,
“Modelo numérico de detección de fugas para sistema de tuberias,”
AiBi Revista De Investigación, Administración E Ingeniería, vol. 8,
no. 2, Apr. 22, 2020. [Online]. Available: https://doi.org/10.15649/
2346030X.723
[5] C. A. García-Ubaque and E. O. L.-M. nd M. C. García-Vaca,
“Determination of the inside diameter of pressure pipes for drinking
water systems using artificial neural networks,” Revista Facultad
de Ingeniería, vol. 31, no. 59, Mar. 25, 2022. [Online]. Available:
https://doi.org/10.19053/01211129.v31.n59.2022.14037
[6] C. F. Colebrook, “Turbulent flow in pipe with particular reference
to the transition region between the smooth and rough pipe laws,”
Journal of the Institution of Civil Engineers, vol. 11, no. 4, Jun. 05,
2015. [Online]. Available: https://doi.org/10.1680/ijoti.1939.13150
[7] R. T. de A. Minhoni, F. F. S. Pereira, T. B. G. da Silva, E. R.
Castro, and J. C. C. Saad, “The performance of explicit formulas
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agriambi.v25n7p439-445-v2
47