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Estimate pressure drop throughout petroleum production and transport system has an important role to properly sizing the various parameters involved in those complex facilities. One of the most challenging variables used to calculate the pressure drop is the friction factor, also known as Darcy-Weisbach's friction factor. In this context, Colebrook's equation is recognized by many engineers and scientists as the most accurate equation to estimate it. However, due to its computational cost, since it is an implicit equation, several explicit equations have been developed over the decades to accurately estimate friction factor in a straightforward way. This paper aims to investigate accuracy of 46 of those explicit equations and Colebrook implicit equation against 2397 experimental points from single-phase and two-phase flows, with Reynolds number between 3000 and 735000 and relative roughness between 0 and 1.40x10-3. Applying three different statistical metrics, we concluded that the best explicit equation, proposed by Achour et al. (2002), presented better accuracy to estimate friction factor than Colebrook's equation. On the other hand, we also showed that equations developed by Wood (1966), Rao and Kumar (2007) and Brkić (2016) must be used in specifics conditions which were developed, otherwise can produce highly inaccurate results. The remaining equations presented good accuracy and can be applied, however, presented similar or lower accuracy than Colebrook's equation.
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FRICTION FACTOR EQUATIONS ACCURACY FOR SINGLE
AND TWO-PHASE FLOWS
Germano Scarabeli Custódio Assunção1,2,*, Dykenlove Marcelin2, João Carlos Von
Hohendorff Filho1, Denis José Schiozer1, Marcelo Souza De Castro1
1University of Campinas, Campinas, Brazil
2Univel University Center, Cascavel, Brazil
ABSTRACT
Estimate pressure drop throughout petroleum production
and transport system has an important role to properly sizing the
various parameters involved in those complex facilities. One of
the most challenging variables used to calculate the pressure
drop is the friction factor, also known as Darcy–Weisbach’s
friction factor. In this context, Colebrook’s equation is
recognized by many engineers and scientists as the most
accurate equation to estimate it. However, due to its
computational cost, since it is an implicit equation, several
explicit equations have been developed over the decades to
accurately estimate friction factor in a straightforward way. This
paper aims to investigate accuracy of 46 of those explicit
equations and Colebrook implicit equation against 2397
experimental points from single-phase and two-phase flows, with
Reynolds number between 3000 and 735000 and relative
roughness between 0 and 1.40x10-3. Applying three different
statistical metrics, we concluded that the best explicit equation,
proposed by Achour et al. (2002), presented better accuracy to
estimate friction factor than Colebrook’s equation. On the other
hand, we also showed that equations developed by Wood (1966),
Rao and Kumar (2007) and Brkić (2016) must be used in
specifics conditions which were developed, otherwise can
produce highly inaccurate results. The remaining equations
presented good accuracy and can be applied, however, presented
similar or lower accuracy than Colebrook’s equation.
Keywords: friction factor, Colebrook’s equation, explicit
equations, petroleum production system.
*E-mail addresses:
germano@univel.br (G.S.C. Assunção), marcelin.dykens@gmail.com (D. Marcelin), hohendorff@cepetro.unicamp.br (J.C.v. Hohendorff Filho),
denis@unicamp.br (D.J. Schiozer), mcastro@fem.unicamp.br (M.S.d. Castro).
1. INTRODUCTION
Methodologies to solve coupling of petroleum reservoir and
production systems have generated great interest in industry,
because such approach can improve production forecasts and
optimize projects. Discoveries in the Brazilian pre-salt
intensified this interest since old and new fields can share the
same production system. In these projects, estimation of pressure
drop has a key role to correctly sizing the production system and
correct analyze production forecasts.
For accurately calculate pressure drop throughout
production system, it is well known that, at least, seven
parameters may be known: (1) internal pipe diameter; (2) pipe
length and fittings; (3) fluids velocities, (4) fluids properties; (5)
flow regimes, (6) void fraction and (7) stress at the pipe wall,
which determines the so-called friction factor.
The first four parameters can be easily obtained by direct
measurement, differently of the last three. Evaluation of flow
regime and void fraction are totally related to the development
of multiphase flow models from the mid-20th century onward,
because both parameters only make sense to be studied when
more than a single-phase exists, which usually happens in
petroleum production system, due to natural or artificial
phenomena. However, although the evaluation of both
parameters is difficult, it can still be carried out, using, for
example, cameras and sensors to observe flow regimes or to
obtain void fraction in laboratory applications and then scale it
up to real production system through models.
The seventh and last parameter, friction factor, is the most
complex to be evaluated, because it is a variable and not a
“measurable” parameter. It should always be estimated,
considering Reynolds number, wall roughness and pipe cross
section shape. Its analysis has been considered since the
OMAE2020-18682
1
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Proceedings of the ASME 2020 39th International
Conference on Ocean, Offshore and Arctic Engineering
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August 3-7, 2020, Virtual, Online
development of the first models to calculate pressure drop in
hydraulic studies, back to 18th century.
One of the first and notable works addressing this issue was
developed by Prony (1804 apud Rennels and Hudson, 2012),
which expressed water pressure drop in internal conduits using
two empirical coefficients. Later, Weisbach (1845) proposed the
use of a dimensionless group called “friction factor () in a
pressure drop equation. Then, Darcy (1857) developed another
equation using three empirical coefficients. Although he had not
used the dimensionless group proposed by Weisbach (1845),
Darcy (1857) had an important role, identifying that pressure
drop depends on type and condition of the boundary material.
For these reasons, it is traditional to call friction factor as Darcy
friction factor or Darcy–Weisbach’s friction factor, although
Darcy (1857) had not proposed it, neither Weisbach (1845) had
a theorical basis for friction factor meaning.
As suggests Brown (2002), the first to put together concepts
developed by Weisbach (1845) and Darcy (1857) was Fanning
(1877), who studied a large compilation of values as a function
of pipe material, pipe dimension and velocity. However, instead
of diameter, he used radius in his analysis, which returned
values equal to 1/4th of Darcy–Weisbach’s . To give credit to
his work, we can find frequently in literature (mainly American
works) applications with the so-called Fanning friction factor.
Those pioneering works were the basis for several
subsequent publications in beginning of 20th century. At this
time, friction factor for laminar flows was already known, due to
works of Hagen (1839) and Poiseuille (1841). Thus, friction
factor estimations remained challenging for turbulent flows.
Aiming to solve this challenge, several authors proposed
equations as the ones developed by Blasius (1913) and Von
Kármán (1930). Blasius (1913) proposed an explicit equation to
be used in smooth pipes under fully turbulent flow, which is still
used today. Von Kármán (1930) proposed an explicit equation
for applications in smooth and rough pipes, also for fully
turbulent flows.
Nikuradse (1933) confirmed the equation developed by Von
Kármán (1930) testing against several experimental points. He
concluded that that equation could be used in smooth and rough
pipes, in laminar, turbulent and transition flows. However, in his
work, he used “artificial roughened surfaces” — uniform layers
of sand grains, with known size, glued on the inner surfaces of
the pipes.
In this context, Colebrook and White (1937) used
experimental data from air flow to show that variation in size and
pattern of individual protuberances (non-uniform layers of sand
grains), generates different friction factor responses, mainly for
flows in transition zone (laminar to turbulent). From that
conclusion, Colebrook (1939) performed a comprehensive
study of friction factors in laminar, turbulent and transition flow
conditions using commercial pipes (galvanized, cast and
The equation proposed by Colebrook (1939) is also well known as Colebrook-White’s equation, due to the collaboration of Dr. White in the development of
formula, as mentioned by Colebrook in this original paper.
Although extensively cited in the literature as an equation developed by Eck, in fact, the equation presented by Eck (1973) was developed by the Yugoslav engineer
Pecornik, in 1963, on the paper entitled “Determination of the coefficient of friction in pipes at stationary uniform turbulent flow in a transitional region”. A note was
presented in Eck (1973) about the work of Pecornik.
wrought iron pipes). He proposed a theorical transition equation
from laminar to turbulent, which is an implicit equation and since
then, has been largely used as response for friction factor under
different flow conditions:



(1)
where is the friction factor (or Darcy–Weisbach’s friction
factor),  is Reynolds number, is the surface roughness and
is the internal diameter.
One of the main causes of the popularity of Colebrook’s
equation is due to the work of Moody (1944), who summarized
the results of the equation 1 in a diagram. Equation and diagram
have been tested against several data over the decades,
presenting outstanding accuracy. It is common both to be studied
in introductory courses of fluid mechanics.
However, since the development of Colebrook’s equation an
effort has been made to develop an explicit equation that
provides such accurate results, because to find friction factor
from equation 1, numerical algorithms are required. Some
examples of those explicit equations were developed by Moody
(1947), Atshul (1952), Wood (1966), Churchill (1973), Eck
(1973), Jain (1976), Swamee and Jain (1976) and several others,
until the most recent ones, proposed by Azizi et al. (2018) and
Brkić and Praks (2019).
Thus, the objective of this work is comparing friction factor
estimates from several explicit equations and from Colebrook’s
equation against experimental data. We can find in literature
several review papers that already investigated the performance
of some of those explicit equations against Colebrook equation.
We highlight recent works of Génic et al. (2011), Asker et al.
(2014), Pimenta et al. (2018) and Zeghadnia et al. (2019), whose
studies compared several explicit equations to Colebrook
equation. Génic et al. (2011) concluded that Zigrang and
Sylvester (1982) presented the most accurate explicit equation.
Asker et al. (2014) obtained that Sonnard and Goudar (2006) and
Serghides (1984) are the most accurate explicit equations.
Zeghadnia et al. (2019) showed that Vatankhah and
Kpuchakzadeh (2009) is the better one and Pimenta et al. (2018)
concluded that all 29 explicit equations analyzed in their work
presented accurate results, however, recommend application of
equation developed by Offor and Alabi (2016).
Although these works are extremely relevant, a drawback is
the fact that accuracy of explicit equations was compared to
estimates from Colebrook’s equation, in other words, estimates
from Colebrook equation were used as real response
(“measured” data). Nonetheless, Colebrook’s equation is also an
approximation for the variable , therefore, such analysis can
generate some bias in the results and consequently in conclusions
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taken. A proof of this claim is that none of the four works had the
same conclusion about the most accurate explicit equation.
Therefore, we propose here a comprehensive comparison
between estimated friction factors () and “measured§
friction factors (). To perform this analysis, we used
real observed data acquired from literature and from an
experimental facility.
This paper is organized as follows: in section 2, a description
of experimental data from literature is presented, as well as the
facility to obtain experimental data acquired in this work.
Section 3 describes methodologies used to obtain experimental
data and comparison criterions used to test performance of
explicit equations and Colebrook’s equation against
experimental data. Section 4 shows the results, including a
scrutiny analysis of the most accurate explicit equation found
and finally section 5 the main conclusions obtained.
2. DATA DESCRIPTION
Table 1 presents a summary of experimental data used in this
work. A total of 47 datasets under different conditions were used,
which gives 2397 experimental points. We divided our data in
four main conditions: (1) horizontal single-phase flow, (2)
horizontal two-phase flow, (3) upward two-phase flow and (4)
downward two-phase flow. Since the focus here is applications
in petroleum industry, most of data used are from two-phase
flows, which is most common to observe in production pipelines
than single-phase flows. Most of data obtained uses water-air and
kerosene-air mixtures, but some oil-air mixtures were also
analyzed. Internal diameter varies among 21.6 mm and 149.6
mm, while pipe inclination varies between -90º and +90º.
For single-phase flow analysis, an experimental facility was
built to obtain datasets 1 and 2 shown in Table 1 — section 2.1
presented more details about this facility. Other experimental
data were acquired from work developed by Silva (2018).
2.1 Experimental facility to acquirer datasets 1 and 2
The experimental setup consisted of a water reservoir, a
centrifugal pump, a flow meter, a test section with a digital
differential pressure manometer and a return line of PVC pipe.
The flow meter is a rotameter able to accurately measure water
flow rates from 0 to 5000 liters/hour. Water flows in a closed
loop following the layout presented in Figure 1 and the flow rate
is set using a variable-frequency drive in the centrifugal pump.
Two PVC pipes of internal diameter 21.6 mm and 27.8 mm
were used in test section, therefore, two experimental pipelines
between Valve 1 and Valve 2 (Figure 1) were used. Roughness
of both pipes were measured using a portable roughness tester,
§In fact, it is not measured, but obtained from measured data, using, for instance, equation 4.
applying an amplitude parameter Ra — the most common
parameter.
Pressure drop between P1 and P2 (Figure 1) was measured
using a digital differential pressure manometer with range of
operation among 0 to 250 mm H20 resolution 0.25 mm H20
— coupled to a software and data acquisition interface. This
software/interface is able to record measured data for every 5
millisecond or more, thus, for every set velocity, several data
were recorded and then statistically analyzed, in order to obtain
more accurate data. Details about those steps are shown in
section 3.1.
Note that the manometer was 1000 mm way from both ends
of the test section, to decrease the inlet and outlet effects. To
obtain this value, we used three literature equations, as shows
Table 2. Since in our experiment Reynolds number ranged
between 12600 to 50600, taking into account the internal
diameter of 27.8 mm and water flow rate of 4000 liters/hour
which give us the longer entrance length — we conclude, from
results presented in Table 2, that 1000 mm would be a safe value
for our analyses.
Water temperature was also measured by thermometer to
assure values within 23 ± 3 ℃ during experiment.
3. METHODOLOGY
In this section we present methodologies used in this work.
In section 3.1, we present the steps followed to collect
experimental pressure drop of datasets 1 and 2 (Table 1), while
section 3.2 regards the means to measure the agreement between
 and .
3.1 Methodology to measure pressure drop of
datasets1 and 2
The follow steps were applied for both pipe diameters
setups:
1. For each diameter, we set the flow rates in liters/hour
(lph):
a. For D = 21.6 mm: (1) 1000 lph, (2) 1500 lph, 
2000 lph and 2500 lph;
b. For D = 27.6 mm: (1) 1000 lph; (2) 1500 lph; (3)
2000 lph, 2500 lph, (5) 3000 lph, (6) 3500 lph and (7) 4000
lph.
2. After set each of the flow rates from step 1, we wait flow
stabilization for 60 seconds. Then, we started to record pressure
drop between P1 and P2 (Figure 1) for each second, during 200
seconds. After that, we stopped the line. So, in the end, we had
200 pressure drop points measured.
3
Copyright © 2020 ASME
Figure 1. Schematic of experimental setup for pressure drop measurement.
3. To give more accuracy in our measurements, we repeated
step 2. five times. Thus, for every flow rate, we obtained 1000
points.
4. Using these 1000 points, we identified outliers using
Tukey’s method. In such method, one point can be identified as
outlier if it is outside lower and upper quartiles. More details
about Tukey’s method and other methods to identify outliers can
be found in Seo (2002) and Wilcox (2012).
5. Subsequently, error (e) of measurements were calculated
using the following equation:


(2)
where is the sample standard deviation, is the sample size
and is the sample average. Note that we used “10” in equation
2 due to Chebyshev's inequality with confidence of 99% (values
must lie within 10 standard deviation of the mean), while
 is due to pressure sensor error, which is, according to
manufacturer, 2% of measured value.
6. Finally, pressure drop estimates for every flow rate were
obtained: .
Values of viscosity and density of water were assumed from
literature (taking into account measured temperature), while
roughness was obtained using a roughness tester. In both pipes,
we measured roughness in 4 points 90° spaced. The cross-section
area used to measure roughness was located in the half of the
pipe used in test section (Figure 1). We measured the roughness
for both pipes, thus, we obtained 8 values, which were used to
calculate the mean value. To calculate error of measured
roughness, we applied equation 2, using , as
recommend by the manufacturer.
3.2 Methodology to compare  and 
To perform the comparison between  and
, for every point of 2397 available, first step was to
calculate frictional pressure drop per unit length, which is related
to the wall shear stress. In inclined flows, it is necessary to
remove gravitational pressure gradient, as presents equation 3.
In horizontal flows, it is not necessary, since .

 

(3)
where 
 is the pressure drop experimentally
measured, is mixture density for two-phase flow used in
homogenous non-slip model; is gravitational acceleration and
is pipe inclination.
After that,  can be obtained using the following
equation:


(4)
where is the pipe internal diameter and is the mixture
velocity (equation 5).
Note that we applied homogenous no-slip model for two-
phase flow analysis. As explained by Shoham (2006), in the
homogeneous no-slip model, two-phase mixture is considered a
pseudo single-phase flow with fluids proprieties based on the no-
slip void fraction (). Thus, flow velocity and fluid proprieties
are calculated as follows:

(5)
where is the mixture velocity,  and  are the liquid and
gas superficial velocities, respectively.
In single-phase flows, mixture velocity is equal to liquid
velocity. These superficial velocities represent the phase
volumetric flow rate per unit area, it is the velocity which would
be observed if only that specific phase flows throughout the pipe.
4
Copyright © 2020 ASME
Table 1: Summary of experimental data used for performance evaluation of friction factor equations.
Reference
Fluids
(deg.)
D (mm)
Nº of
points
Reynolds range
(x103)
Horizontal single-phase flow
Authors
Water
0
21.6
4
12 50
Authors
Water
0
27.8
7
24 41
Silva (2018)
Water
0
53.55
64
7 – 126
Silva (2018)
Water
0
82.25
186
4 70
Silva (2018)
Water
0
106.47
24
34 400
Horizontal two-phase flow
Silva (2018)
Water-Air
0
53.55
70
14 163
Silva (2018)
Water-Air
0
82.25
129
13 104
Silva (2018)
Water-Air
0
106.47
78
21 451
Mukherjee (1979)
Kerose-Air
0
38.1
59
4 – 124
Magrini (2009)
Water-Air
0
76.2
20
238 – 604
Johnson (2005)
Water-Air
0
100
206
65 735
Cheremisinoff (1977)
Water-Air
0
63.5
174
57 232
Fan (2005)
Water-Air
0
50.8
51
34 232
Fan (2005)
Water-Air
0
149.6
87
84 545
Brito (2012)
Oil-Air
0
50.8
8
4 – 7
Brill et al. (1995)
Kerose-Air
0
77.9
48
28 151
Kouba (1986)
Kerose-Air
0
76.2
18
35 287
Meng (2001)
Oil-Air
0
50.8
31
26 141
Zheng (1989)
Kerose-Air
0
76.2
6
65 141
Upward two-phase flow
Mukherjee (1979)
Kerose-Air
90º
38.1
39
7 – 128
Yuan (2011)
Kerose-Air
30º
76.2
22
9 – 521
Yuan (2011)
Kerose-Air
60º
76.2
43
68 482
Yuan (2011)
Kerose-Air
75º
76.2
44
70 482
Yuan (2011)
Kerose-Air
90º
76.2
44
39 482
Fan (2005)
Water-Air
50.8
28
70 241
Fan (2005)
Water-Air
50.8
28
71 245
Fan (2005)
Water-Air
149.6
33
204 – 538
Meng (2001)
Oil-Air
50.8
45
32 143
Meng (2001)
Oil-Air
50.8
36
32 143
Zheng (1989)
Kerose-Air
76.2
3
67 217
Alsaadi (2013)
Water-Air
76.2
56
44 525
Alsaadi (2013)
Water-Air
76.2
56
42 525
Alsaadi (2013)
Water-Air
10º
76.2
56
40 524
Alsaadi (2013)
Water-Air
20º
76.2
56
42 526
Alsaadi (2013)
Water-Air
30º
76.2
64
40 526
Downward two-phase flow
Mukherjee (1979)
Kerose-Air
-
38.1
33
38 482
Mukherjee (1979)
Kerose-Air
-20º
38.1
49
3 – 435
Mukherjee (1979)
Kerose-Air
-30º
38.1
43
10 377
Mukherjee (1979)
Kerose-Air
-50º
38.1
56
3 – 332
Mukherjee (1979)
Kerose-Air
-70º
38.1
50
3 – 338
Mukherjee (1979)
Kerose-Air
-80º
38.1
43
3 – 305
Mukherjee (1979)
Kerose-Air
-90º
38.1
38
4 – 334
Fan (2005)
Water-Air
-
50.8
35
34 233
Fan (2005)
Water-Air
-
50.8
25
34 218
Fan (2005)
Water-Air
-
149.6
34
116 – 538
Meng (2001)
Oil-Air
-
50.8
32
23 146
Meng (2001)
Oil-Air
-
50.8
36
29 143
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Table 2: Definition of entrance length
Author
Equation
Entrance length for D = 27.8 mm and Re ~ 50600
Bhatti and Shah (1987)

548 mm
White (2011)

744 mm
Çengel and Boles (2013)

278 mm
Fluids proprieties such as density and viscosity of the
pseudo single-phase flow are calculated as follows:

(6)
and

(7)
where is the mixture density, is liquid density and is
gas density, while and are mixture, liquid and gas
viscosity, respectively.
More details about homogenous non-slip model can be
found in Shoham (2006). For two-phase flow analysis,
application of homogeneous no-slip model increases absolute
errors, since it is well known that such approach is an
approximation of real multiphase flow. However, using such
model we can have an overall picture of friction factor accuracy
equations in a standard way, which is the focus of the present
work. A next step would be to observe how can accuracy of
friction factor affect accuracy of other multiphase flow models
such as correlations, two-fluid model, drift-flux and others.
Then, absolute percentage error ( between  and
 for every equation studied was calculated by equation
8. Table 3 presents all explicit equations used to calculate
. Beyond these equations, Colebrook equation was
used to estimate friction factors in this study.



 
(8)
where index represents each data studied, presented in column
“Data” in Table 1, while is the size of each of these data,
presented in column “Nº of points” in Table 1.
The cumulative absolute percentage error ( can be
calculated by equation 9. The aim in calculating was to
measure performance of each equation against the entire
databank.



(9)
where  represents Data 47 from Table 1.
Similarly, percentage error with sign ( between
 and  was also calculated. The reason to
apply this statistical metric was observing whether equations
were overestimating or underestimating .


 
(10)
where is the size of each data, presented in column “Nº of
points” in Table 1. We used for the sum of points with
positive errors, while  for the sum of points with negative
errors.
Cumulative percentage error with sign ( was also
calculated, as follows:



(11)
To observe average percentage errors, we used the average
percentage error with sign (, calculated by equation 12:



(12)
where is the size of the entire sample (from Data 1 to Data
47) with positive or negative error.
We highlight that equations from (8) to (12) were applied
for each explicit equation presented in Table 3 and for
Colebrook’s equation, as well.
In literature there are several others statistical metrics that
can be used to perform such comparison. It is commonly found:
(1) percentage error, (2) percentage standard deviation, (3)
average error, (4) absolute average error or (5) standard deviation
error, as presents Shoham (2006) for example. In general, theses
metrics present the same trend and for this reason only two (
and ) were applied here.
In addition, other statistical index that is not usually applied
in those analyses was also used, the refined Willmott index ().
This metric was proposed by Willmott et al. (2012) and evaluate
index of agreement for meteorological and agricultural studies,
called Willmott index (), proposed by Willmott (1981).
The refined Willmott index is a dimensionless number,
bounded by -1.0 and 1.0. Values of near to 1.0 means that both
data are in agreement and near to -1.0 the converse.
To calculate , equation 13 must be used when
 
  
 :
6
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 

 

(13)
However, when 
  
 equation
14 must be applied:
 

 

(14)
where is the estimates or predicted values and the pair-
wise-matched observations.
We can interpret refined Willmott index as modified Pearson
correlation coefficient, widely used in regression analyses. More
details about and other statistical metrics related to Willmott
indices can be found in Willmott et al. (2012) and Pereira et al.
(2018).
3.2.1 Methodology to compare the best explicit
equation to Colebrook’s equation
After definition of the best explicit equation among all
studied, we performed an additional comparison using scatter
plots, which is a two-dimensional data graph that uses dots to
represent the values obtained for two different sources — one
plotted along the x-axis (estimated data) and the other plotted
along the y-axis (measured data). Examples of studies that used
scatter plots in this context are the works developed by Ouyang
and Aziz (1996), Shoham (2006), Babajimopoulos and Terzidis
(2013) and Azizi et al. (2018).
Instead of the simple visualization of those scatter plots, we
defined three intervals (±5%, ±25% and ±50%) to quantify
percentage of estimated data within each of those intervals. Thus,
we measured the accuracy of the chosen explicit equation and
Colebrook’s equation in predicting friction factor under different
“error intervals”.
4. RESULTS AND DISCUSSION
4.1 Pressure drop from datasets 1 and 2
Using experimental facility described in section 2.1 and
applying methodology from section 3.1, we obtained average
values and uncertainties presented in Table 4. As a whole, 11,000
points were collected.
Roughness values measured for 21.6 mm and 27.8 mm pipes
are presented in Table 5. The average value found following
methodology proposed in section 3.1 was (0.9 ± 0.3) μm.
Table 4: Pressure drop measured for Data 1 and Data 2.
Data
D (mm)


Data 1
21.6




Data 2
27.8






)
Table 3: Friction factor explicit equations.
Eq. number
Reference
Equation
15
Moody (1947)



16
Altshul (1952)



17
Wood (1966)





18
Churchill (1973)



19
Pecornik (1963) apud
Eck (1973)



20
Jain (1976)



21
Swamee and Jain (1976)





22
Chen (1979)


 


Continues on the next page
7
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Continued from Table 3
23
Churchill (1977)


where:



24
Round (1980)



25
Shacham (1979)


 


26
Barr (1981)



where:



27
Zigrang and
Sylvester I (1982)


 


28
Zigrang and
Sylvester II (1982)


 

 


29
Haaland (1983)



30
Serghides I (1984)
 

where:









31
Serghides II (1984)


where:






32
Robaina (1992)



33
Manadilli (1997)



 
34
Sousa, Cunha and
Marques (1999)





Continues on the next page
8
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Continued from Table 3
35
Romeo, Royo and
Monzon (2002)


 
where:

 


36
Achour et al. (2002)


where:




37
Sonnard and
Goudar (2006)



where:


38
Rao and
Kumar (2007)



 
where:


39
Buzzelli (2008)




where:




40
Vantankah and
Kouchakzadeh (2008)



where:


Continues on the next page
9
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Continued from Table 3
41
Cheng (2008)




where:


 

42
Avci and
Karagoz (2009)




43
Papaevangelou et al.
(2010)




44
Brkić I (2011)


where:
 


45
Brkić II (2011)



where:
 


46
Fang et al. (2011)




47
Ghanbari et al. (2011)


 
48
Ćojbašić and
Brkić I (2013)


where:









Continues on the next page
10
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Continued from Table 3
49
Ćojbašić and
Brkić II (2013)




 



50
Achour and Bedjaoui
(2012)



where:



51
Brkić (2016)






52
Offor and
Alabi (2016)


 


53
Azizi, Hojjati and
Homayoon (2018)


 

54
Brkić and
Praks (2019)


where:




55
Brkić and
Praks II (2019)



where:




56
Blasius (1913)

57
Ward-Smith (1980)

58
Ward-Smith (1980)

59
Knudsen and Katz (1958)

60
Ward-Smith (1980)

11
Copyright © 2020 ASME
Table 5: Measured roughness of Data 1 and Data 2.
D (mm)
( ± 10%
21.6
0.80
1.01
0.98
0.97
27.8
0.89
0.90
0.89
0.96
Mean
0.9 ± 0.3
4.2 Comparison between  and 
Figure 2 presents stacked bar plot for overall cumulate
absolute percentage error. Each color represents absolute
percentage error () for each data shown in Table 1. We used
stacked bar instead of simple bar plot because we were interested
in observing if one of the datasets affected cumulative error more
than others. Nonetheless, as shown in Figure 2, there is not a
notable percentage error for a specific data. We did not plot
legend in Figure 2 for each data once we were interested in
overall performance.
Observing Figure 2, equations 36, 41, 56, 57, 58, 59 and 60
produced better results than others equations studied. Equation
36, developed by Achour et al. (2002), exhibited the lowest
cumulate error, 1951%. The lowest percentage error between
 (using equation 36) and  was observed for
Data 2, with , while the greatest one was observed
for Data 28, .
Zeghadnia et al. (2019) has already shown that equation
developed by Achour et al. (2002) brings accurate results,
however, they highlighted that this equation is not very known
and it is not, indeed — it can be partially explained by the fact
that Achour’s original paper was written in French. In their
work, Zeghadnia et al. (2019) tested Achour’s equation against
Colebrook’s equation. Here, we tested this equation against real
data, and we showed its good accuracy in comparison to other
equations. Note that we were interested in relative error, that is,
we considered Achour’s equation more accurate in comparison
to others 46 equations studied. With a cumulative percentage
error of 1951%, we can considerer that it is not that accurate, but
it is important to take into account that most of data studied in
this work is from two-phase flows, thus, errors related to
homogenous no-slip model must be considered as main source
of error in friction factor estimates. However, this error is equal
for all 47 equations studied, therefore, from a relative
perspective, Achour’s equation can be considered accurate.
Equation 41 developed by Cheng (2008) also had good
accuracy, with the second lowest cumulative percentage error,
1976 % (Figure 2). In his work, he compared equation 41 against
Nikuradse (1933) experimental data and obtained accurate
results. This paper reinforces the accuracy of the equation
developed by Cheng (2008). However, from literature review, we
concluded that Cheng’s equation has not been used frequently, as
also observed for Achour’s equation. As example, none of the
recent works developed by Genic et al. (2011), Asker et al.
(2014), Zeghadnia et al. (2019) and Pimenta et al. (2018)
considered Cheng’s equation in their studies.
Figure 3 presents results from analysis of average
percentage error with sign (). We note that Achour’s equation
can estimate friction factor within the lowest error interval,
between [-27%; +34%]. Cheng’s equation presented the second
lowest error, with errors between [-26%; +36%]. From this
perspective, most of explicit equation (33 equations out of 47
studied) presented similar trend, with interval between [-30%;
+37%], including Colebrook’s equation.
Equations from 56 to 60, called Blasius derivative
equations, had presented good performance in both analyses, as
we can observe in Figures 2 and 3. We measured their accuracy
in this work because several handbooks and articles recommend
them and we concluded that they present better accuracy than
most equations — including Colebrook’s equation. Nonetheless,
we believe that those equations will not be that accurate whether
pipes with greater roughness would be applied, since they do not
include roughness to estimate . Thus, we can recommend
Equations 56 to 60 only for flows under low roughness pipes,
which is the case here studied, once relative roughness in this
work varies from 0 to 1.40x10-3.
On the other hand, equations 17, 38 and 51 presented the
worst results, with values of cumulate absolute percentage error
greater than 3000%. Cumulative percentage errors observed in
equations 17 and 51 can be explained by the fact that they were
applied in intervals that were not recommended by authors:
Wood (1966) recommend applications for relative roughness
from 10-5 to 4x10-2, while Brkić (2016) recommended
applications of his equation for relative roughness from 10-2 to
5x10-2. Despite that, we used both equations to access their
accuracy, since we are looking for an equation that can be
employed under different conditions.
Asker et al. (2014) compared 27 explicit equation to
Colebrook’s equation and also concluded that Wood (1966) was
one of the worst equations to predict friction factor. Same
conclusions were shown by Genic et al. (2011) and Zeghadnia et
al. (2019). The equation developed by Brkić (2016) was only
systematically analyzed by Pimenta et al. (2018), because it is a
relatively recent equation. Pimenta et al. (2018) concluded that
Brkić equation have low accuracy, as shown here. In their work,
they compared explicit equations to Colebrook’s.
Differently of equations 17 and 51, equation 38 developed
by Rao and Kumar (2007) do not have an applicable range. They
proposed this equation to cover the whole turbulent range flow
based on experimental data from Nikuradse (1933) and did not
specify any limitations concerning relative roughness. Similar
results found in our work were also observed by Brkić (2011)
and Pimenta et al. (2018), whose conclusions showed that errors
from equation 34 were the highest in relation to others explicit
equations analyzed.
From Figure 3 we can also note a trend to underestimate
values, which means that usually . This
trend is predominant in equations that showed greater
cumulative absolute percentage errors in Figure 2, namely,
12
Copyright © 2020 ASME
equations 17, 38 and 51. The best results were again observed
for equations 36, 41, 56, 57, 58, 59 and 60.
We also used the refined Willmott index () to measure the
agreement between  and . Table 6 presents
results from this statistical metric. Equations 17, 38 and 51
presented the worst results, with dr of 0.11, 0.01 and 0.11,
respectively. Therefore, from three different statistical metrics,
equations 17, 38 and 51 shown to be inaccurate. This conclusion
was already observed in previous works, but here we proved
their inaccuracy using real data.
On the other hand, Achour’s equation presented the most
accurate result again, with dr = 0.40, followed by Cheng’s
equation with dr = 0.39, the same conclusion observed from
Figures 3 to 5. However, it is interesting to note that using refined
Willmott index, equations 57, 58, 59 and 60 did not presented
greater accuracy than others equations, following the trend of dr
between 0.36 and 0.37. This result complement previous
observations about applicability of Blasius derivative equations.
Thus, although they presented a good accuracy, Achour’s
equation can generate better results with the same mathematical
cost, once all are straightforward equations.
Note that taking into account only data from single-phase
flows, errors decreased significantly (Figure 4), since errors in
modelling multiphase flows are not included. However, the same
trend from two-phase flow can be observed for this one: the best
explicit equation was the proposed by Achor et al. (2002), with
an average confidence interval of [-12 %; +10 %], while
equations 17, 38 and 51 presented the worst results.
4.3 Comparing Achour’s and Colebrook’s equations
In previous section we showed that Achour’s equation
presented the most accurate estimates from all 47 equations
studied, including Colebrook’s equation. Here, we perform a
comparative study between both equations.
Figure 5 presents scatter plot for dataset 1. As we can see,
both equations presented roughly the same results. Using the
seven points analyzed, two were within an error interval ±5%,
which gives the value of 28.5% points within this interval as
observed in Table 7. Within interval ±25%, we can observe six
points in Figure 5, that is, 85.7% and 100% of the points are
within ±50% interval.
Figure 6 shows scatter plot for dataset 2 and again the same
trend was observed for both equations: 50% of both estimates are
within an error interval of ±5% and 100 % of estimates within an
error interval of ±25%.
Both figures show error intervals from data measured in this
study — datasets 1 and 2. Table 7 summarizes this analysis for
the entire data bank. Observing overall performance (weighted
mean), it is possible note that Achour’s equation gives slightly
better results than Colebrook’sthe same conclusion taken
from section 4.2.
We must highlight that our study covers relative roughness
from 0 to 1.40x10-3, thus we cannot guarantee that accuracy in
rough pipes.
Colebrook’s equation is frequently presented as the most
accurate equation to predict friction factor, usually followed by
discussions about computational cost involving its applications
and subsequent presentation of a “less accurate” explicit
equation that can be applied. We showed here that, under some
conditions, Colebrook’s equation can be exchanged by an
explicit, low computational cost, equation without losing any
accuracy in estimates of .
5. CONCLUSION
Comparing 46 explicit equations and Colebrook’s equation
for friction factor against 2897 experimental points, we
concluded that estimates from Achour’s equation was the most
accurate, stood out from the others.
We highlight that, under conditions here considered, Achour
(2002) performed better than Colebrook’s equation. An
innovation of the present work in comparison to recent similar
works was the application of experimental data to measure
accuracy of explicit equations instead of use Colebrook’s
estimates as a benchmark. This approach allowed us to observe
that, under certain conditions, Colebrook’s equation is not the
most accurate one.
The equation developed by Cheng (2008) also presented
outstanding performance, while equations developed by Wood
(1966), Rao and Kumar (2007) and Brkić (2016) must be used in
specifics conditions that were developed, otherwise can produce
highly inaccurate results. The remaining 41 equations presented
good accuracy and can be applied.
The next step is to study how uncertainty of the friction
factor equation can affect application of the most accurate
friction factor equation can affect mechanistic and empirical
correlations for two-phase flows to predict pressure drop in oil
production systems and how these results can affect the coupled
simulation with petroleum reservoirs.
13
Copyright © 2020 ASME
Table 6: Refined Willmott index (dr)
Equation
dr
Equation
dr
Equation
dr
Equation
dr
Equation
dr
1
0.37
20
0.36
30
0.37
40
0.36
50
0.37
11
0.37
21
0.37
31
0.37
41
0.37
51
0.37
12
0.36
22
0.37
32
0.40
42
0.37
52
0.38
13
0.11
23
0.37
33
0.37
43
0.37
53
0.37
14
0.37
24
0.37
34
0.01
44
0.37
54
0.37
15
0.35
25
0.37
35
0.37
45
0.37
55
0.37
16
0.37
26
0.37
36
0.37
46
0.37
56
0.36
17
0.37
27
0.37
37
0.39
47
0.11
18
0.37
28
0.36
38
0.38
48
0.37
19
0.37
29
0.37
39
0.37
49
0.37
Figure 2. Cumulative absolute percentage error ( between observed and estimated friction factor for the entire data bank.
14
Copyright © 2020 ASME
Figure 3. Average percentage error with sign ( between “measured” and estimated friction factor for the entire data
bank.
Figure 4. Average percentage error with sign ( between “measured” and estimated friction factor using data from single-
phase flows.
15
Copyright © 2020 ASME
Figure 5. “Measured” friction factor vs. estimated friction factor for Data 1.
Figure 6. “Measured” friction factor vs. estimated friction factor for Data 2.
16
Copyright © 2020 ASME
Table 7: “Measured” vs. estimated friction factors within intervals of ±5%, ±25% and ±50%.
Data
Achour et al. (2002)
Colebrook (1939)
± 5%
± 25%
± 50%
± 5%
± 25%
± 50%
Horizontal single – phase flow
Data 1
28,6 %
85,7 %
100 %
28,6 %
85,7 %
100 %
Data 2
50,0 %
100 %
100 %
50,0 %
100 %
100 %
Data 3
51,5 %
100 %
100 %
10,9 %
62,2 %
100 %
Data 4
29,0 %
87,6 %
96,2 %
32,3 %
86,0 %
96,2 %
Data 5
83,3 %
100 %
100 %
0 %
75 %
100 %
Horizontal two-phase flow
Data 6
21,4 %
75,7 %
85,7 %
8,6 %
28,6 %
72,9 %
Data 7
7,8 %
43,4 %
78,3 %
10,0 %
49,6 %
79,8 %
Data 8
30,8 %
69,2 %
89,7 %
5,1 %
50,0 %
82,0 %
Data 9
8,5 %
37,3 %
67,8 %
8,5 %
39,0 %
57,6 %
Data 10
0 %
0 %
90 %
0 %
0 %
95 %
Data 11
15,0 %
55,8 %
99,5 %
18,9 %
92,7 %
99,0 %
Data 12
9,8 %
51,7 %
77,0 %
9,8 %
52,3 %
75,9 %
Data 13
9,8 %
52,9 %
84,3 %
9,8 %
52,9 %
84,3 %
Data 14
19,5 %
66,7 %
88,5 %
19,5 %
66,7 %
88,5 %
Data 15
0 %
50,0 %
75,0 %
0 %
50,0 %
75,0 %
Data 16
8,3 %
32,25 %
85,4 %
8,3 %
31,2 %
85,4 %
Data 17
5,6 %
5,6 %
11,1 %
5,6 %
5,6 %
11,1 %
Data 18
6,5 %
25,8 %
77,4 %
6,5 %
25,8 %
77,4 %
Data 19
0 %
0 %
50 %
0 %
0 %
50 %
Upward two-phase flow
Data 20
7,7 %
33,3 %
59,0 %
0 %
43,6 %
69,2 %
Data 21
0 %
0 %
27,3 %
0 %
0 %
27,3 %
Data 22
0 %
2,3 %
4,7 %
0 %
2,3 %
4,7 %
Data 23
0 %
0 %
0 %
0 %
0 %
0 %
Data 24
0 %
0 %
0 %
0 %
0 %
0 %
Data 25
3,6 %
28,6 %
100 %
3,6 %
28,6 %
100 %
Data 26
3,6 %
32,1 %
100 %
3,6 %
32,1 %
100 %
Data 27
21,2 %
100 %
100 %
21,2 %
100 %
100 %
Data 28
0 %
11,1 %
60,0 %
0 %
11,1 %
60,0 %
Data 29
0 %
16,7 %
52,8 %
0 %
16,7 %
52,8 %
Data 30
0 %
0 %
60,0 %
0 %
0 %
60,0 %
Data 31
0 %
3,6 %
35,7 %
0 %
3,6 %
35,7 %
Data 32
0 %
0 %
28,6 %
0 %
0 %
28,6 %
Data 33
0 %
0 %
17,9 %
0 %
0 %
17,9 %
Data 34
0 %
0 %
3,6 %
0 %
0 %
3,6 %
Data 35
0 %
0 %
0 %
0 %
0 %
0 %
Downward two-phase flow
Data 36
24,2 %
54,5 %
63,6 %
3,0 %
27,3 %
63,6 %
Data 37
22,4 %
71,4 %
71,4 %
4,1 %
42,9 %
71,4 %
Data 38
25,6 %
55,9 %
69,8 %
0 %
32,6 %
62,8 %
Data 39
12,5 %
44,7 %
51,8 %
7,1 %
25,0 %
51,8 %
Data 40
14,0 %
50,0 %
62,0 %
0 %
30,0 %
60,0 %
Data 41
14,0 %
51,2 %
65,1 %
7,0 %
30,2 %
67,4 %
Data 42
13,2 %
44,8 %
60,5 %
7,9 %
39,5 %
60,5 %
Data 43
11,4 %
28,6 %
80,0 %
11,4 %
28,6 %
80,0 %
Data 44
8,0 %
28,0 %
92,0 %
8,0 %
28,0 %
92,0 %
Data 45
17,7 %
64,7 %
91,2 %
17,7 %
64,7 %
91,2 %
Data 46
0 %
21,9 %
78,1 %
0 %
21,9 %
78,1 %
Data 47
2,8 %
16,7 %
58,3 %
2,8 %
16,7 %
58,3 %
Weighted mean
13,5 %
44,1 %
68,7 %
9,2 %
41,4 %
67,8 %
17
Copyright © 2020 ASME
ACKNOWLEDGEMENTS
The authors would like to thank the following institutions
for supporting this work: UNISIM Research Group, School of
Mechanical Engineering and Center for Petroleum Studies
(CEPETRO) both at the University of Campinas — UNICAMP,
Brazil. Acknowledgements are extended to UNIVEL University
Center (Brazil) and ALGETEC.
The authors also acknowledge Petrobras for the Financial
Support and Energi Simulation.
Finally, thanks to Profs. Cem Sarica and Eduardo Pereyra
from University of Tulsa (TUFFP) for providing part of the
datasets used in this work.
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... In general, it was found to be a function of the Reynolds number (Re) and the pipe material roughness coefficient (e). The Swamee-Jain equation can be used to describe the friction factor for a pipe of circular section (ð) as follows [34,35]: ...
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This revised book provides a thorough explanation of the foundation of robust methods, incorporating the latest updates on R and S-Plus, robust ANOVA (Analysis of Variance) and regression. It guides advanced students and other professionals through the basic strategies used for developing practical solutions to problems, and provides a brief background on the foundations of modern methods, placing the new methods in historical context. Author Rand Wilcox includes chapter exercises and many real-world examples that illustrate how various methods perform in different situations. Introduction to Robust Estimation and Hypothesis Testing, Second Edition, focuses on the practical applications of modern, robust methods which can greatly enhance our chances of detecting true differences among groups and true associations among variables. * Covers latest developments in robust regression * Covers latest improvements in ANOVA * Includes newest rank-based methods * Describes and illustrated easy to use software.