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Review of explicit approximations to the Colebrook
relation for flow friction
Dejan Brkić
To cite this version:
Dejan Brkić. Review of explicit approximations to the Colebrook relation for flow friction. Journal of
Petroleum Science and Engineering, Elsevier, 2011, 77 (1), pp.34-48. �10.1016/j.petrol.2011.02.006�.
�hal-01586547�

Elsevier Editorial System(tm) for Journal of Petroleum Science and Engineering
Manuscript Draft
Manuscript Number: PETROL2963R2
Title: Review of explicit approximations to the Colebrook relation for flow friction
Article Type: Review Article
Keywords: Colebrook equation; Approximations; Fluid mechanics; Turbulent flow; Friction coefficient;
Darcy friction factor; Hydraulic resistances; Excel spreadsheet
Corresponding Author: Dr Dejan Brkić, PhD in Petroleum Eng.
Corresponding Author's Institution: -
First Author: Dejan Brkić, PhD in Petroleum Eng.
Order of Authors: Dejan Brkić, PhD in Petroleum Eng.
Abstract: Because of Moody's chart as demonstrated applicability of the Colebrook equation over a
very wide range of Reynolds number and relative roughness values, this equation become the accepted
standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being
implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01%
typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive
friction factor directly in one step. Iterative calculus can causes problem in simulation of flow in a pipe
system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is
main reason for attempting to develop a relationship that is a reasonable approximation for the
Colebrook equation but which is explicit in friction factor. Review of existing explicit approximation of
the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy
compared with iterative solution of implicit Colebrook equation is shown for entire range of
turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that
most of available approximations of the Colebrook equation, with few exceptions, are very accurate
with deviation of no more than few percentages.

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Because of Moody’s chart as demonstrated applicability of the Colebrook equation over a very
wide range of Reynolds number and relative roughness values, this equation become the
accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers
from being implicit in unknown friction factor and thus requires an iterative solution where
convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation
cannot be rearranged to derive friction factor directly in one step. Iterative calculus can causes
problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction
factor hundreds or thousands of times. This is main reason for attempting to develop a
relationship that is a reasonable approximation for the Colebrook equation but which is explicit
in friction factor. Review of existing explicit approximation of the implicit Colebrook equation
with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative
solution of implicit Colebrook equation is shown for entire range of turbulence where Moody
diagram should be used as the reference. Finally, it can be concluded that most of available
approximations of the Colebrook equation, with few exceptions, are very accurate with deviation
of no more than few percentages.
Abstract DB
Click here to download Abstract: Abstract DB.doc

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I wish to thank reviewers and editors for valuable comments.
Reviewer 1:
1. I hope that my expression of English is now better.
Reviewer 2:
Missing
Reviewer 3:
Missing
Reviewer 4:
1. Abstract was rearranged now as you suggested (last two sentences). Please, see also Research
highlights. My text has now graphical abstract also….
2. I disagree with your suggestion to reduce number of approximation shown in my review. Honestly, I
have considered your suggestion seriously, but in the meantime I have found new approximations such as
Papaevangelou et al (2010) and Vatankhah and Kouchakzadeh (2008, 2009). They are very accurate.
Therefore, I have added these two approximations...
3. I agree with you that problem of range of applicability of certain approximation is very important. This
problem is very complex. So to avoid any further speculations I have added diagram of accuracy of each
approximation over the entire practical range of Reynolds number and relative roughness. According to
these diagrams, one can choose will he/she use in certain case this particular approximation or not.
4. Detailed discussion with results is now presented in more appropriate way.
5. Conclusion is now rearranged after your suggestions.
Revision Notes DB
Click here to download Revision Notes: Answer reviewers DB.doc

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1
Review of explicit approximations to the Colebrook relation for flow friction
1
Dejan Brkić,
2
Ph.D., Petroleum and Natural Gas Engineer, Strumička 88, 11050 Beograd, Serbia
3
Tel. +381642543668, e-mail: dejanrgf@tesla.rcub.bg.ac.rs
4
5
Abstract: Because of Moody‟s chart as demonstrated applicability of the Colebrook equation
6
over a very wide range of Reynolds number and relative roughness values, this equation become
7
the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation
8
suffers from being implicit in unknown friction factor and thus requires an iterative solution
9
where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook
10
equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can
11
causes problem in simulation of flow in a pipe system in which it may be necessary to evaluate
12
friction factor hundreds or thousands of times. This is main reason for attempting to develop a
13
relationship that is a reasonable approximation for the Colebrook equation but which is explicit
14
in friction factor. Review of existing explicit approximation of the implicit Colebrook equation
15
with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative
16
solution of implicit Colebrook equation is shown for entire range of turbulence where Moody
17
diagram should be used as the reference. Finally, it can be concluded that most of available
18
approximations of the Colebrook equation, with few exceptions, are very accurate with deviation
19
of no more than few percentages.
20
Keywords: Colebrook equation; Approximations; Fluid mechanics; Turbulent flow; Friction
21
coefficient; Darcy friction factor; Hydraulic resistances; Excel spreadsheet
22
23
Text DB
Click here to download Manuscript Including Title Page and Abstract: text DB.doc Click here to view linked References

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1. Introduction
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Difficulty of solving turbulent flow problems in pipes lies in the fact that hydraulic friction factor
2
is a complex function of relative surface roughness and Reynolds number. Precisely, hydraulic
3
resistance depends on flow rate. Similar situation is with electrical resistance when diode is in a
4
circuit. Furthermore, being more complex, widely used empirical Colebrook equation is
5
transcendental which means that it cannot be solved by using only elementary functions and
6
basic arithmetic operations in definitive form. Problem is that, since the Colebrook equation is
7
implicit (i.e. both the right and left-hand terms contain friction factor), containing the unknown
8
friction factor in implicit form, the Reynolds number and the pipe roughness, it has to be solved
9
iteratively. Even today in the era of advance computer technology, explicit approximations of the
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implicit Colebrook relation is very often used for calculation of friction factor in pipes. The
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reason that so many researchers propose approximate solutions to Colebrook equation is that
12
these correlations are necessary to calculate the pressure drop and average velocity in conduits in
13
one step. Friction factor can be derived using logarithmic or power law formulation (Zagarola et
14
al 1997). Colebrook equation belongs to the logarithmic law. The laws of resistances to fluid
15
flow through rough pipe are of great significance. Colebrook equation is valid for turbulent
16
regime in rough pipes including so called rough and smooth turbulent regime with special accent
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on transient regime between them. There is no perfectly smooth inner surface of pipe. All pipe
18
walls have physically rough surfaces (Figure 1). Degree of roughness varies depending on the
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manufacturing process, surface finish, type of pipe material (Hammad 1999), age, conditions of
20
exploitation, etc. In turbulent flow, thin layer of fluid very close to inner pipe surface in which
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flow is laminar is called “laminar sub-layer”. If the pipe roughness (protrusions of inner pipe
22
surface) is completely covered by the sub-layer, the surface is smooth from the hydraulic point of
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view. With increasing of Reynolds number, thickness of laminar sub-layer decreases baring
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protrusions and fluid flow through pipe become consequently rough from the hydraulic point of
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view. At very low Reynolds number, relative roughness does not have influence on friction
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factor and it depends only on value of Reynolds number. But, on the contrary, at very high
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Reynolds number, Reynolds number does not have influence on friction factor and it depends
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only on relative roughness. Between these two opposite regime friction factor depends on both,
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Reynolds number and relative roughness. Colebrook equation was developed to cover this
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transient zone of turbulence, but it also covers completely smooth and completely rough regime.
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In laminar flow, all pipes behave as smooth but Colebrook equation is not valid for this regime.
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One of the presented approximations in this paper includes laminar regime (Churchill 1977).
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Figure 1. Hydraulic regimes; A) Hydraulically “smooth”, B) Partially turbulent, and C)
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Turbulent (rough)
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The Colebrook equation is widely used in the petroleum industry for calculations of oil and gas
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pipelines, in civil engineering for calculation of water distribution systems, for drainage systems,
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ventilation systems, in chemical engineering, and in all fields of engineering where fluid flow
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can be occurred.
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2. On Colebrook equation for flow friction
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Before 1939. when Colebrook equation was published, for turbulent regime in smooth pipes
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widely was used Prandtl equation also implicit in friction factor (Colebrook 1939). Prandtl
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derived a formula from the logarithmic velocity profile and experimental data on smooth pipes
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(1):
2
  8
.
0
Re
log
2
51
.
2
Re
log
2
1
10
10 











 




(1)
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The development of approximate equations for the calculation of friction factor in rough pipes
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began with Nikuradse‟s turbulent pipe flow investigations in 1932 and 1933 (Hager and Liiv
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2008). For turbulent regime in rough pipes widely was used von Karman‟s relation (2):
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




 








 














 D
log
2
14
.
1
D
2
log
2
74
.
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D
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.
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log
2
1
10
10
10 (2)
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Greek ε is the equivalent Nikuradse‟s sand-grain roughness value for the inner surface of pipe (or
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so called uniform roughness). Prandtl‟s and von Karman‟s relations are also known as NPK
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(Nikuradse-Prandtl-Karman) equations. Colebrook later performed experiments on sixteen spun
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concrete-lined pipes and six spun bitumastic-lined pipes ranging in diameter from 101.6 mm to
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1524 mm with average surface roughness values between 0.04318 mm and 0.254 mm (Taylor et
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al 2006).
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In an attempt to classify the data available at the time and those from experiment conducted by
15
himself and his colleague White (Colebrook and White 1937), Colebrook (1939) developed a
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curve fit to describe transitional roughness (3):
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















 D
71
.
3
Re
51
.
2
log
2
1
10 (3)
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Colebrook equation also can be noted as (3a):
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













 Re
35
.
9
D
log
2
14
.
1
1
10
(3a)
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Colebrook equation describes a monotonic change in the friction factor from smooth to fully
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rough (Figure 2). It is valid especially for commercial steel pipes. Strictly mathematically is
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incorrect what Colebrook had done, i.e. log(A+B)≠log(A)+log(B), but physically this relation
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gives good results. Problem can be treated as inverse; according to logarithmic rules equally is
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incorrect to split the Colebrook relation into two pieces.
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Figure 2. Colebrook relation make transitional curve between hydraulically “smooth” regime
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described by Prandtl (1), and turbulent (rough) regime described von Karman (2)
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Colebrook equation is also basis for Rouse (1943) and widely used Moody (1944) chart. Many
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people seem to believe that the Moody diagram has surprisingly good properties. In fact, all it is
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a plot of solutions of the nonlinear transcendental Colebrook equation. In principle, Moody
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diagram is used for solution of three types of problems, i.e. problem in which head loss is
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unknown, in which volume flow rate is unknown and in which diameter is unknown. Solving for
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unknown head loss with Moody diagram is relative straightforward but use of implicit Colebrook
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formula complicate solving all three types of problems.
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Many researchers adopt a modification of the Colebrook equation (4) recommended by the
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American Gas Association (AGA) in case of natural gas pipelines calculations, using 2.825
19
constant instead of 2.51 (Haaland 1983, Coelho and Pinho 2007). This procedure produces
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maximal deviation up to 3.2% (Figure 3).
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















 D
71
.
3
Re
825
.
2
log
2
1
10 (4)
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Figure 3. Distribution of estimated deviation of implicit Colebrook equation modified by AGA
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(4) compared with standard implicit Colebrook equation (3)
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3
Figure 3 is three-dimensional. Similar figures in further text are two-dimensional.
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Also, some researchers use Fanning factor which is not the same as the Darcy friction factor
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(here noted as λ). Darcy friction factor is 4 times greater than the Fanning friction factor, but
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physical meaning is equal. Darcy, Darcy-Weisbach and Moody friction factors are synonyms.
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Colebrook equation is somewhere known as Colebrook-White equation (CW equation). White
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was not actually a co-author of the paper in which this equation was presented (Colebrook 1939).
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But, Colebrook made a special point of acknowledging important contribution of White to the
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development of the equation (Colebrook and White 1937, Colebrook 1939). Letter W has
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additional symbolic value because alternate explicit reformulations of Colebrook equation with
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Lambert W-function involved exist (Brkić 2011a, Clamond 2009, Goudar and Sonnad 2003,
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Keady 1998, More 2006, Nandakumar 2007, Sonnad and Goudar 2004, 2005, 2006, 2007).
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Approximations proposed by Brkić (2011a) were also developed using Lambert W-function and
17
its solution proposed by Barry et al (2000). Further about Lambert-W function, readers can see in
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paper of Hayes (2005).
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3. Available explicit approximations of the Colebrook equation with analysis of their
21
estimated accuracy
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How well Colebrook equation fits the experimental data is beyond the scope of presented
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approximations. Perhaps one of these equations even fits the available data better than the
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Colebrook equation. Until the comparison is made with real, measured values, however, this will
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not be known. According to Cipra (1996), some of the key formulas of turbulence are off by as
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much as 65%. Yoo and Singh (2004, 2010) found that the Colebrook equation produced an
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average error of more than 11% while the roughness height of commercial pipes varied quite
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significantly, depending on the pipe size and type.
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As it will be shown, there were some early expressions of Colebrook equation in explicit form
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which were not particularly accurate, but in the years 1973-1984 there was a flurry of activity
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obtaining more accurate approximations that appeared mainly in the chemical engineering
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literature. Note that some of presented approximations exist in several versions. Here has to be
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very careful because typographical errors are always possible (Concha 2008, Brkić 2009a).
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Approximations will be presented starting from the oldest. Estimated accuracy compared with
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iterative solution of implicit Colebrook equation will be shown in figures 4-23 for entire range of
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turbulence where Moody diagram should be used as the reference.
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17
3.1 Moody approximation
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Approximation proposed by Moody (1947) is the oldest approximation of implicit Colebrook
19
relation (5):
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



























3
1
6
4
Re
10
D
10
2
1
0.0055 (5)
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8
With error up to 21.49% compared with implicit Colebrook equation (Figure 4), it has today only
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historical value.
2
3
Figure 4. Distribution of estimated error of Moody approximation compared with implicit
4
Colebrook equation
5
6
3.2 Wood approximation
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Approximation proposed by Wood (1966) is, equal as those by Moody (1947), power-law type
8
equation (6). Its accuracy is not improved compared with Moody approximation.
9
V
44
.
0
225
.
0
Re
D
88
D
53
.
0
D
094
.
0 






 







 







 


 (6)
10
Where V is (7):
11
134
.
0
D
62
.
1
V 




 

 (7)
12
Estimated error of Wood approximation is up to 23.79% compared with implicit Colebrook
13
equation (Figure 5).
14
15
Figure 5. Distribution of estimated error of Wood approximation compared with implicit
16
Colebrook equation
17
18
3.3 Eck approximation
19
Approximation proposed by Eck (1973) is most simple but not very accurate (8), but better than
20
those by Moody (1947) and Wood (1966).
21












 Re
15
D
715
.
3
log
2
1
10 (8)
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Estimated error of Eck approximation is up to 8.2% compared with implicit Colebrook equation
1
(Figure 6).
2
3
Figure 6. Distribution of estimated error of Eck approximation compared with implicit
4
Colebrook equation
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6
3.4 Churchill approximation (only for turbulent regime)
7
Approximation proposed by Churchill (1973) is very similar with approximations proposed by
8
Swamee and Jain (1976) and Jain (1976). It is first approximation with improved accuracy (9):
9





















9
.
0
10
Re
7
D
71
.
3
log
2
1
(9)
10
Estimated error of Churchill approximation (valid only for turbulent regime) is up to 2.18%
11
compared with implicit Colebrook equation (Figure 7).
12
13
Figure 7. Distribution of estimated error of approximation by Churchill (valid only for turbulent
14
regime), Swamee and Jain, approximation by Jain and Churchill approximation (valid for full
15
range of flow) compared with implicit Colebrook equation
16
17
3.5 Swamee and Jain approximation
18
Approximation proposed by Swamee and Jain (1976) (10) with error up to 2.04% is almost
19
identical as those proposed by Churchill (1973). Distribution of estimated error of Swamee and
20
Jain approximation over turbulent part of Moody‟s chart is shown in figure 7.
21












 9
.
0
10
Re
74
.
5
D
7
.
3
log
2
1
(10)
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
10
Some further details on this approximation readers can see in paper of Swamee and Rathie
1
(2007).
2
3
3.6 Jain approximation
4
Approximation proposed by Jain (1976) (11) with error up to 2.05% is comparable with those
5
proposed by Churchill (1973) and Swamee and Jain (1976):
6





















9
.
0
10
Re
943
.
6
D
715
.
3
log
2
1
(11)
7
Distribution of estimated error of Jain approximation over turbulent part of Moody‟s chart is
8
shown in figure 7.
9
10
3.7 Churchill approximation (full range of turbulence including laminar regime)
11
Approximation proposed by Churchill (1977) covers entire laminar and turbulent regime
12
(including unstable zone between them) with error up to 2.19% (12):
13
 
12
1
5
.
1
2
1
12
C
C
1
Re
8
8


















 (12)
14
Where C1 is (13):
15
16
9
.
0
1
D
27
.
0
Re
7
1
ln
457
.
2
C
























 (13)
16
And C2 is (14):
17
16
2
Re
37530
C 





 (14)
18

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
11
Distribution of estimated error of Churchill approximation (full range of turbulence including
1
laminar regime and unstable zone between them) over turbulent part of Moody‟s chart is shown
2
in figure 7.
3
4
3.8 Chen approximation
5
First, really accurate approximation (15) was developed by Chen (1979).
6






















 








 8981
.
0
1098
.
1
10
10
Re
8506
.
5
D
8257
.
2
1
log
Re
0452
.
5
D
7065
.
3
log
0
.
2
1
(15)
7
Estimated error of Chen approximation is up to 0.35% compared with implicit Colebrook
8
9
10
Figure 8. Distribution of estimated error of Chen approximation compared with implicit
11
Colebrook equation
12
13
For some details on Chen (1979) approximation readers can see discussion by Schorle et al
14
(1980) and closure of Chen (1980).
15
16
3.9 Round approximation
17
Approximation proposed by Round (1980) is relative simple but not very accurate (16):
18




















 




 5
.
6
D
Re
135
.
0
Re
log
8
.
1
1
10 (16)
19

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
12
Estimated error of Round approximation is up to 10.92% compared with implicit Colebrook
1
2
3
Figure 9. Distribution of estimated error of Round approximation compared with implicit
4
Colebrook equation
5
6
3.10 Barr approximation
7
Approximation proposed by Barr (1981) was very accurate for the time when it was developed
8
(17). It does not require internal iterative calculus.
9





























 
















 7
.
0
52
.
0
10
10
D
29
Re
1
Re
7
Re
log
518
.
4
D
7
.
3
log
2
1
(17)
10
Estimated error of Barr approximation is up to 0.27% compared with implicit Colebrook
11
12
13
Figure 10. Distribution of estimated error of Barr approximation compared with implicit
14
Colebrook equation
15
16
3.11 Zigrang and Sylvester approximations
17
Approximations proposed by Zigrang and Sylvester (1982) use internal iterative procedure to
18
achieve high accuracy (18, 19):
19
























 Re
13
D
7
.
3
log
Re
02
.
5
D
7
.
3
log
2
1
10
10 (18)
20

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
13




































 Re
13
D
7
.
3
log
Re
02
.
5
D
7
.
3
log
Re
02
.
5
D
7
.
3
log
2
1
10
10
10 (19)
1
Form of approximation by Zigrang and Sylvester (1982) (18) is less accurate than (19), since the
2
first one is based on two internal iterations while the second one uses three internal iterations.
3
Estimated error of more complex but also more accurate approximation by Zigrang and Sylvester
4
(1982) is up to 0.13% compared with implicit Colebrook equation (Figure 11). For simpler form
5
of Zigrang and Sylvester approximation error is up to 1% (Figure 11).
6
7
Figure 11. Distribution of estimated error of Zigrang and Sylvester approximations compared
8
with implicit Colebrook equation
9
10
3.12 Haaland approximation
11
Approximation proposed by Haaland (1983) is very accurate and simple. It was first one,
12
designed equally for calculation of friction factor for liquid and gaseous flow (20):
13



























 n
n
11
.
1
10
Re
9
.
6
D
7
.
3
log
n
8
.
1
1
(20)
14
For n=1, Haaland equation (20) is valid for flow of liquid. Haaland (1983) suggested that n=3
15
yields friction factors in consonance with those recommended for use in gas transmission lines.
16
17
Estimated error of approximation by Haaland (1983) valid for liquid flow is up to 1.4%
18
compared with implicit Colebrook equation (Figure 12).
19
20
Figure 12. Distribution of estimated error of Haaland approximation for liquid flow compared
21
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
14
1
Same comparison is done for Haaland approximation for gaseous flow (Figure 13). This was
2
done in comparisons with standard implicit Colebrook equation and with modified implicit
3
Colebrook equation rearranged by AGA.
4
5
Figure 13. Distribution of estimated error of Haaland approximation for gaseous flow compared
6
with implicit Colebrook equation (upper) and with modified implicit Colebrook equation
7
rearranged by AGA (lower)
8
9
3.13 Serghides approximations
10
Approximations proposed by Serghides (1984)1
. are accurate but with internal iterative calculus
11
(21, 22):
12
 
2
1
2
3
2
1
2
1
S
S
2
S
S
S
S















 (21)
13
 
2
1
2
2
1
781
.
4
S
2
S
781
.
4
S
781
.
4















 (22)
14
Where S1 is (23):
15












Re
12
D
7
.
3
log
2
S 10
1 (23)
16
Where S2 is (24):
17
1
Cronologically, after approximation by Serghides (1984) are approximations proposed by Chen
(1984, 1985). But these approximations are similar with Altshul, Russian power-law equation
from Soviet practice and therefore they will be presented with this equation later in the text. Note
also that author of Chen approximation from 1979 is Chen N.H. (Chen 1979), while the author of
Chen approximations from 1984 is Chen J.J.J. (Chen 1984).

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
15





 






Re
S
51
.
2
D
7
.
3
log
2
S 1
10
2 (24)
1
Where S3 is (25):
2





 






Re
S
51
.
2
D
7
.
3
log
2
S 2
10
3 (25)
3
Form of approximation by Serghides (1984) (21) is more accurate than (22), since the first one is
4
based on three internal steps while the second one use two internal steps. More complex version
5
is with accuracy up to 0.13% while less complex one is up to 0.35% (Figure 14).
6
7
Figure 14. Distribution of estimated error of Serghides approximations compared with implicit
8
Colebrook equation
9
10
3.14 Manadilli approximation
11
Approximation by Manadilli (1997) contains signomial terms (26).
12













 Re
82
.
96
Re
95
D
7
.
3
log
2
1
983
.
0
10 (26)
13
A special group of functions appearing in mathematical models of many processes is the
14
signomial functions. A signomial function is defined as the sum of signomial terms, which in
15
turn are products of power functions multiplied with a real constant. Estimated error of
16
approximation by Manadilli (1997) is up to 2.06% compared with implicit Colebrook equation
17
(Figure 15).
18
19
Figure 15. Distribution of estimated error of Manadilli approximation compared with implicit
20
Colebrook equation
21
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
16
3.15 Romeo, Royo and Monzón approximation
1
Approximation proposed by Romeo et al (2002) has three internal iterations (27):
2






















































9345
.
0
9924
.
0
10
10
10
Re
815
.
208
3326
.
5
D
7918
.
7
log
Re
567
.
4
D
827
.
3
log
Re
0272
.
5
D
7065
.
3
log
2
1 (27)
3
The calculation of the parameters of Romeo et al (2002) approximation was done through non-
4
linear multivariable regression. Estimated error of approximation by Romeo, Royo and Monzón
5
is up to 0.13% compared with implicit Colebrook equation (Figure 16).
6
7
Figure 16. Distribution of estimated error of approximation by Romeo, Royo and Monzón
8
compared with implicit Colebrook equation
9
10
3.16 Sonnad and Goudar approximation
11
Approximation by Sonnad and Goudar (2006) was developed using Lambert W-function (28):
12
  




 


 1
G
/
G
G
Re
4587
.
0
ln
8686
.
0
1
(28)
13
Where G is (29):
14
 
Re
4587
.
0
ln
D
Re
124
.
0
G 




 (29)
15
For some details on Sonnad and Goudar (2006) approximation readers can see discussions by
16
Vatankhah and Kouchakzadeh (2008, 2009) and Yıldırım (2008). Paper of Sonnad and Goudar
17
(2007) should be also recommended as reference. Approximation by Sonnad and Goudar (2006)
18
is not suitable for all range of relative roughness and Reynolds numbers (Sonnad and Goudar
19
2004). Estimated error of approximation by Sonnad and Goudar (2006) is up to 0.8% compared
20
with implicit Colebrook equation (Figure 17).
21
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
17
Figure 17. Distribution of estimated error of Sonnad and Goudar approximation compared with
1
implicit Colebrook equation
2
3
Vatankhah and Kouchakzadeh (2008, 2009) rearranged approximation proposed by Sonnad and
4
Goudar (2006) as (29a, 29b), to increase its accuracy with error up to 0.15% (Figure 18). These
5
two improved equations here will be noted as approximations by Vatankhah and Kouchakzadeh
6
(2008, 2009).
7
    











  9633
.
0
G
/
G
31
.
0
G
Re
4587
.
0
ln
8686
.
0
1
(29a)
8
   
2
98
.
0
G
/
G
28
.
0
G
Re
4587
.
0
ln
8686
.
0

 




















(29b)
9
10
Figure 18. Distribution of estimated error of Vatankhah and Kouchakzadeh approximation
11
12
13
Parameter G in Vatankhah and Kouchakzadeh approximation is actually parameter G from
14
Sonnad and Goudar approximation
15
16
3.17 Rao and Kumar approximation
17
Approximation by Rao and Kumar (2007) cannot be recommended to be used because of its
18
inaccuracy (30). Of course this inaccuracy is valid only apropos standard Colebrook equation.
19
 




















 






 





Re
Re
Re
135
.
0
444
.
0
D
2
log
2
1
1
10 (30)
20
Where Φ(Re) is (31):
21

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2
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5
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7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
18
 
2
5
.
6
Re
ln
33
.
0
e
55
.
0
1
Re


















 (31)
1
Φ(Re) can be neglected in most cases. Some additional details can be seen in paper of Rao and
2
Kumar (2009). Estimated error of Rao and Kumar approximation is up to 82% compared with
3
implicit Colebrook equation (Figure 19).
4
5
Figure 19. Distribution of estimated error of Rao and Kumar approximation compared with
6
7
8
3.18 Buzzelli approximation
9
Buzzelli (2008) proposed one up to date among most accurate and also relatively simple
10
approximation (32):
11


























2
2
10
1
1
B
18
.
2
1
Re
B
log
2
B
B
1
(32)
12
Where B is (33):
13
 







 





D
32
.
1
1
41
.
1
)
Re
ln
774
.
0
(
B1 (33)
14
and B2 is (34):
15
1
2 B
51
.
2
Re
D
7
.
3
B 




 (34)
16
Estimated error of approximation by Buzzelli (2008) is up to 0.13% compared with implicit
17
Colebrook equation (Figure 20).
18
19

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5
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7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
19
Figure 20. Distribution of estimated error of Buzzelli approximation compared with implicit
1
Colebrook equation
2
3
3.19 Avci and Karagoz approximation
4
Formula by Avci and Karagoz (2009) have been recently developed from the experimental
5
Princeton super-pipe data (35):
6
 
4
.
2
D
10
1
D
Re
01
.
0
1
ln
Re
ln
4
.
6























 









 (35)
7
Estimated error of approximation by Avci and Karagoz (2009) is up to 4.7% compared with
8
implicit Colebrook equation (Figure 21).
9
10
Figure 21. Distribution of estimated error of Avci and Karagoz approximation compared with
11
12
13
3.20 Papaevangelou, Evangelides and Tzimopoulos approximation
14
Papaevangelou et al (2010) noticed that error values tended to “bend” to negative values in an
15
exponential way for Re lower than 106
. According to that they chose parameters in their equation
16
(36):
17
 
2
9142
.
0
10
4
10
Re
366
.
7
D
615
.
3
log
Re
log
7
0000947
.
0
2479
.
0





















 (36)
18
Estimated error of approximation by Papaevangelou, Evangelides and Tzimopoulos is up to
19
0.85% compared with implicit Colebrook equation (Figure 22).
20
21

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4
5
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7
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9
10
11
12
13
14
15
16
17
18
19
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22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
20
Figure 22. Distribution of estimated error of approximation by Papaevangelou, Evangelides and
1
Tzimopoulos compared with implicit Colebrook equation
2
3
3.21 Brkić approximation
4
Brkić (2011a) approximations were developed using Lambert W-function (37, 38):
5
















D
71
.
3
10
log
2
1 4343
.
0
10 (37)
6














 D
71
.
3
Re
18
.
2
log
2
1
10 (38)
7
Where β is (39):
8
 













Re
1
.
1
1
ln
Re
1
.
1
ln
816
.
1
Re
ln (39)
9
Estimated error of Brkić approximation is up to 2.3% compared with implicit Colebrook
10
11
12
Figure 23. Distribution of estimated error of Brkić approximation compared with implicit
13
Colebrook equation
14
15
Additionally (Barry et al. 2000), error can be reduced using (39a) or (39b):
16
 
















917365
.
0
Re
1
ln
917365
.
0
Re
ln
4586887
.
0
4586887
.
1
1 (39a)
17





























1
1
Re
ln
Re
488
.
0
ln
(39b)
18
For parameter β (39), solution for Lambert W-function by Barry et al. (2000) is used. Procedure
19
by Winitzki (2003) gives parameter β with similar accuracy (40):
20

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3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
21
   
 
  



















Re
458
.
0
1
ln
2
Re
458
.
0
1
ln
1
ln
1
Re
458
.
0
1
ln
(40)
1
Parameter β calculated by using procedures by Barry et al. (2000) and by Winitzki (2003) gives
2
similar results in accuracy.
3
4
4. Comparative analysis and complexity of available explicit approximations of the
5
Colebrook equation
6
Although the Colebrook formula itself is not very accurate (Cipra 1996), its accurate resolution
7
is nonetheless an issue for numerical simulations because a too crude resolution may affect the
8
repeatability and comparisons of calculation (Clamond 2009).
9
10
There is no special explanation for different coefficients in some equations (e.g. 3.7065 or 3.707
11
instead of 3.71 etc). Only reasonable explanation can be that this changed coefficients maybe
12
better fit experimental data. Examples for this are e.g. Churchill (1973) approximation (9) and
13
Jain (1976) approximation (11) with slightly different coefficients.
14
15
Churchill (1977) relation (12) holds for all values of Reynolds and relative roughness, including
16
laminar regime (Figure 24).
17
18
Figure 24. Churchill (1977) approximation includes laminar and highly unstable transient zone
19
from laminar to turbulent
20
21
Since Churchill (1977) relation (12) is a continuous function for Reynolds numbers above 0, it
22
also lets one calculate a friction factor in the transient zone, between laminar and smooth
23

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3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
22
turbulent regime. Of course the accuracy of such a friction factor probably cannot be determined,
1
but it will be reasonable. Regarding this issue, readers also can see paper of Swamee and
2
Swamee (2007).
3
4
In his recent paper, Yıldırım (2009, 2011) compared 12 of here presented more than 20
5
approximations of the implicit Colebrook equation. Ouyang and Aziz (1996) made similar
6
research for the approximations available in that time. Data of Ouyang and Aziz (1996) are also
7
available from paper of Abdolahi et al (2007). Similar comparisons are available from the papers
8
of Gregory and Fogarasi (1985), Zigrang and Sylvester (1985) and papers of Goudar and Sonnad
9
(2007, 2008).
10
11
For the analysis of relative error distribution, 20 points (16 for relative comparisons shown in
12
figures 25-30) are used for relative roughness (ε/D) and 37 points for Reynolds number (Re).
13
Points for relative roughness (ε/D) used for comparative study are shown in figures 25-30. Points
14
for Reynolds number are between 1·104
-1·108
. This means that grid with 592 check points is
15
formed for comparative study (Figures 25-30) and with 740 check points for error distribution
16
study (Figures 4-23). This comparative study is based on some particular conditions of test grid
17
points which means that presented relative error could be different using different check points.
18
But here presented analysis with 592 and 740 check points give good general picture of accuracy
19
for the presented approximations. Relative error is not distributed systematically over the entire
20
range of Reynolds number and relative roughness which means that real maximal relative error
21

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
23
can be slightly above here reported. MS Excel file is available as electronic appendix2
to this
1
paper and in this file any other values for Reynolds number and for relative roughness can be
2
used as input parameters. Modern software, especially in the case of spreadsheets, can be
3
conveniently programmed to solve any implicit equation, with a minimum of programming and
4
details. Some of MS Excel effectiveness for numerical computations comes from a module
5
„Solver‟. It was originally designed for optimization problems, where one has to find values of a
6
number of different parameters such that some quantity is minimized, usually the sum of errors
7
of a number of equations. With this tool one can find such optimal solutions, or solutions of one
8
or many equations, even if they are nonlinear. In more details, to allow iterative computations in
9
MS Excel 2007, one has to choose „Excel options‟, and then in „Formulas‟ to tick box „Enable
10
iterative calculation‟. In this case, Excel is set to terminate the calculation after maximum 3·104
11
iterations or 1·10-7
difference or less between the values of two successive iterations.
12
Consequently, deviations involving explicit equations have been reported to 1·10-5
%. This means
13
that real relative error is presented by sum of calculated relative error and deviation. So deviation
14
in „Excel options‟ has to be set to be significantly smaller compared to estimated error of
15
observed approximations („Maximum Change‟ in „Formulas‟).
16
17
Maximal percentage (relative) error of presented approximations over the entire range of
18
applicability of Colebrook equation is shown in table 1.
19
20
Table 1. Maximal relative error for available approximations for test check points
21
2
MS Excel can be also successfully used in other engineering fields; see electronic appendix in
Brkić and Tanasković (2008)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
24
1
Approximation by Rao and Kumar (2007) is extremely inaccurate compared with standard
2
Colebrook equation (Figure 25). Approximations proposed by Moody (1947), Wood (1966), Eck
3
(1973) and Round (1980) should not be used because they produce significant relative error
4
(Figure 26). Moderate accurate approximations such as proposed by Jain (1976), Swamee and
5
Jain (1976), Churchill (1973, 1977), Manadilli (1997), Brkić (2011a) and Avzi and Karagoz
6
(2009), can be used since they made maximal relative error up to 5% (Figure 27).
7
Approximations by Zigrang and Sylvester (1982) (18), Haaland (1983), Sonnad and Goudar
8
(2006) and Papaevangelou et al (2010) produce maximal error up to 1.5% (Figure 28). Very
9
accurate (Figure 29), with estimated error up to 0.5%, are approximations by Chen (1979), Barr
10
(1981), Zigrang and Sylvester (1982) (19), Serghides (1984), Romeo et al (2002), Buzzelli
11
(2008), and approximations proposed by Vatankhah and Kouchakzadeh approximations (2008,
12
2009). Note that approximations proposed by Vatankhah and Kouchakzadeh (2008, 2009) are
13
actually very successfully improved approximation by Sonnad and Goudar (2006).
14
15
Figure 25. Inaccuracy of approximation by Rao and Kumar (2007)
16
17
Figure 26. Non-advisable approximations
18
19
Figure 27. Approximations with estimated error up to 5% (less accurate approximations)
20
21
Figure 28. Approximations with estimated error up to 1.5% (moderate accurate approximations)
22
23

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
25
Figure 29. Approximations with estimated error up to 0.5% (very accurate approximations)
1
2
Extremely accurate approximations from figure 29 can be seen in figure 30 in better resolution.
3
4
Figure 30. Extremely accurate approximations presented in higher resolution
5
6
Numerical solutions for friction factors based on the Colebrook equation can be obtained by to
7
any desired degree of precision. For many applications, the simpler but less accurate explicit
8
equation will be sufficed. Sometime, simplicity is sacrificed for excessive accuracy. To find
9
balance between these two extremes it is appropriate to introduce concept of complexity or
10
complexity index of explicit approximations. Zigrang and Sylvester (1985) defined complexity
11
as the number of algebraic notation calculator key strokes required to solve the equation for
12
Re=105
and ε/D=0.001 (Table 2). Complexity index is defined as quotient of key strokes
13
required for an observed approximation and the least complex one (Table 2).
14
15
Table 2. Complexity and complexity index of available explicit approximations
16
17
Eck (1973) approximation is the least complex with only 27 key strokes required for solution and
18
hence this equation has complexity index 1 (Table 2).
19
20
5. A note on power-law formulas valid for the same range of turbulent flow as standard
21
Colebrook equation
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
26
Derivation of relations for the friction factor is mostly based on the logarithmic or power law
1
formulation of velocity profiles in boundary layers (Zagarola et al 1997). As mentioned in
2
introduction, Colebrook equation is based on the logarithmic formulation. Example of the power-
3
law formula, valuable for the same flow regimes as Colebrook‟s is old Altshul equation from the
4
Soviet era (Figure 31) (41, 42):
5
4
1
Re
68
D
11
.
0 









 (41)
6
4
1
Re
100
D
46
.
1
1
.
0 










 (42)
7
8
Altshul formula was eliminated from the last wording of the Russian norms. However, it is used
9
as before since other recommendations were not proposed (Sukharev et al 2005).
10
11
Figure 31. Distribution of deviation of Altshul formula (41) compared with implicit Colebrook
12
equation
13
14
Similar are formulas proposed by Chen (1984) (43, 44):
15
3
.
0
83
.
0 D
11
.
0
Re
1
3164
.
0 












 




 (43)
16
3
.
0
67
.
0 D
7
.
0
Re
1
184
.
0 












 




 (44)
17
Deviation of presented power-law formulas from Russian practice and by Chen (1984) is shown
18
in figure 32.
19
20

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
27
Figure 32. Power law formulas as substitution for implicit Colebrook equation
1
2
Readers also can see paper by Chen (1985). In general, approximations by Moody (1947) and by
3
Wood (1966) also belong to power-law formulas.
4
5
6. Some remarks on further developments
6
Today, different approach can be used for determination of friction factor. Good example for the
7
era of computerization is approach of Özger and Yildlrim (2009). They use adaptive neuro-fuzzy
8
computing technique for determination of turbulent flow friction coefficient. In the paper of Yoo
9
and Singh (2005) are shown two new methods for the computation of commercial pipe friction
10
factor. Today, main problem is not how to calculate friction factor. Problem is how to measure or
11
estimate roughness of pipe (Farshad et al 2001). Most pipes usually have rough inner pipe
12
surface. Resistance to fluid flow offered by rough boundaries is larger than that for smooth one
13
due to the formation of eddies behind protrusions. Colebrook equation is valid for both, smooth
14
and rough turbulent regime including transient zone between them. In principle, a system of
15
partial differential equations known as Navier-Stokes equations describes the exact behavior of
16
the fluid flow in so-called boundary layer, but solving these equations remains beyond current
17
theory and computations. Sletfjerding and Gudmundsson (2003) proposed also methodology for
18
determination of friction factor directly from roughness measurements. In that way they
19
eliminated roughness as a parameter in Colebrook equation (only Reynolds number and pipe
20
diameter are necessary as input parameters). Using a similar approach to that of in Nikuradse‟s
21
experiment, Sletfjerding and Gudmundsson (2003) related measured roughness values with
22

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
28
friction factor, but their equation is implicit in friction factor. In formulation given by
1
Sletfjerding and Gudmundsson (2003) equation is implicit and valid for average steel pipe (45):
2





















03
.
1
10
D
4
Re
55
.
1
log
89
.
1
1
(45)
3
For other materials of pipes readers can consult paper of Sletfjerding and Gudmundsson (2003).
4
5
7. Conclusion
6
Maybe, it is difficult for many to recall for the time as recently as the 1970‟s where there were no
7
personal computers or even calculators that could do much more than add or subtract. In that
8
environment an implicit relationship such as Colebrook (1939), which was well-known then, was
9
impractical and some simplification was essential. Today, it is not difficult to solve single
10
Colebrook equation by iteration. But solution of complex looped pipeline problem in such case
11
requires double iterative procedure where first is for the standard implicit Colebrook equation
12
while second one is for Hardy Cross method or similar iterative method used to solve simulation
13
problem in a looped pipe network (Brkić 2009b, 2011b). This double procedure can be serious
14
burden even for today very powerful computers.
15
16
All shown approximate equations give the friction factor explicitly as a function of Reynolds
17
number and relative roughness. Comparative analysis indicates that almost all approximate
18
equations give a very good prediction of the friction factor and can reproduce the Colebrook
19
equation and its Rouse and Moody plot. Therefore, these approximations for the friction factor
20
provide a rational, accurate, and practically useful method over the entire range of the Moody
21
chart in terms of Reynolds number and relative roughness. Apropos relative complexity at first
22
sight, these approximations can be very easily implemented in a computer code.
23

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1
Most available approximations of the Colebrook equation are very accurate. Exceptions are
2
Round (1980), Eck (1973), Moody (1947), Wood (1966), and Rao and Kumar (2007)
3
approximations. The average error of almost all explicit approximations of the Colebrook (1939)
4
relation is up to 3%.
5
6
Acknowledgment
7
Author received his PhD in Petroleum and Natural Gas engineering in Serbia in March 2010. He
8
is currently finishing master degree studies in Mechanical engineering in Serbia financed by
9
TEMPUS. Author will appreciate future comments from readers. Approximations presented in
10
this paper are sometimes very complex and hence typographical errors are possible. These
11
potential errors are by author himself and not by original authors of approximations. Therefore,
12
author regrets because of any future inconvenience regarding this matter.
13
14
Appendix. Supporting Information
15
Supplementary data associated with this article can be found in the online version at doi:
16
17
MS Excel spreadsheet file is accompanied as electronic annex with on-line version of this paper.
18
Note that you have to install Office 2007 (Enterprise edition) to inspect this file. File contains all
19
formulas presented in the text. Readers can change values of Reynolds number and relative
20
roughness to calculate Darcy friction factor. To allow necessary implicit calculation in MS Excel
21
2007 which is used for calculation of implicit Colebrook‟s relation, the „Office button‟ at the
22
upper-left corner of the Excel screen have to be pressed, and in the „Excel options‟, „Formulas‟
23

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has to be chosen and finally box „Enable iterative calculation‟ have to be ticked. This allows
1
implementation of so called „Circular references‟ into a calculation.
2
3
References:
4
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Resour. ASME 107(2), 280–283.
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Nomenclature:
1
D-inner diameter of pipe (m)
2
Re-Reynolds number (-)
3
ε-absolute roughness (m)
4
λ-Darcy (i.e. Moody or Darcy-Weisbach) friction factor (-)
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δ-relative error (%)
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V-auxiliary term in Wood approximation
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C1, C2-auxiliary terms in Churchil approximation
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S1, S2, S3-auxiliary terms in Serghides approximations
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G-auxiliary term in Sonnad-Goudar and Vatankhah-Kouchakzadeh approximation
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Φ(Re)-auxiliary term in Rao-Kumar approximation
11
B1, B2-auxiliary terms in Buzzelli approximation
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β-auxiliary term in Brkić approximations
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n-explained with Haaland approximation (can be n=1 or n=3, n=1 is valid for flow of liquid
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while n=3 is recommended for use in gas transmission lines)
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List of figures
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Figure 1. Hydraulic regimes; A) Hydraulically “smooth”, B) Partially turbulent, and C)
2
Turbulent (rough)
3
Figure 2. Colebrook relation make transitional curve between hydraulically “smooth” regime
4
described by Prandtl (1), and turbulent (rough) regime described von Karman (2)
5
Figure 3. Distribution of estimated deviation of implicit Colebrook equation modified by AGA
6
(4) compared with standard implicit Colebrook equation (3)
7
Figure 4. Distribution of estimated error of Moody approximation compared with implicit
8
Colebrook equation
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Figure 5. Distribution of estimated error of Wood approximation compared with implicit
10
Colebrook equation
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Figure 6. Distribution of estimated error of Eck approximation compared with implicit
12
Colebrook equation
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Figure 7. Distribution of estimated error of approximation by Churchill (valid only for turbulent
14
regime), Swamee and Jain, approximation by Jain and Churchill approximation (valid for full
15
range of flow) compared with implicit Colebrook equation
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Figure 8. Distribution of estimated error of Chen approximation compared with implicit
17
Colebrook equation
18
Figure 9. Distribution of estimated error of Round approximation compared with implicit
19
Colebrook equation
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Figure 10. Distribution of estimated error of Barr approximation compared with implicit
21
Colebrook equation
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Figure 11. Distribution of estimated error of Zigrang and Sylvester approximations compared
1
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Figure 12. Distribution of estimated error of Haaland approximation for liquid flow compared
3
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Figure 13. Distribution of estimated error of Haaland approximation for gaseous flow compared
5
with implicit Colebrook equation (upper) and with modified implicit Colebrook equation
6
rearranged by AGA (lower)
7
Figure 14. Distribution of estimated error of Serghides approximations compared with implicit
8
Colebrook equation
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Figure 15. Distribution of estimated error of Manadilli approximation compared with implicit
10
Colebrook equation
11
Figure 16. Distribution of estimated error of approximation by Romeo, Royo and Monzón
12
13
Figure 17. Distribution of estimated error of Sonnad and Goudar approximation compared with
14
15
Figure 18. Distribution of estimated error of Vatankhah and Kouchakzadeh approximation
16
17
Figure 19. Distribution of estimated error of Rao and Kumar approximation compared with
18
19
Figure 20. Distribution of estimated error of Buzzelli approximation compared with implicit
20
Colebrook equation
21
Figure 21. Distribution of estimated error of Avci and Karagoz approximation compared with
22
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40
Figure 22. Distribution of estimated error of approximation by Papaevangelou, Evangelides and
1
Tzimopoulos compared with implicit Colebrook equation
2
Figure 23. Distribution of estimated error of Brkić approximation compared with implicit
3
Colebrook equation
4
Figure 24. Churchill (1977) approximation includes laminar and highly unstable transient zone
5
from laminar to turbulent
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Figure 25. Inaccuracy of approximation by Rao and Kumar (2007)
7
Figure 26. Non-advisable approximations
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Figure 27. Approximations with estimated error up to 5% (less accurate approximations)
9
Figure 28. Approximations with estimated error up to 1.5% (moderate accurate approximations)
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Figure 29. Approximations with estimated error up to 0.5% (very accurate approximations)
11
Figure 30. Extremely accurate approximations presented in higher resolution
12
Figure 31. Distribution of deviation of Altshul formula (41) compared with implicit Colebrook
13
equation
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Figure 32. Power law formulas as substitution for implicit Colebrook equation
15

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41
List of tables
1
2
3

Research highlights
-Empirical Colebrook equation is an accepted standard for calculation of flow friction factor.
-Colebrook equation is transcendental function implicit in unknown friction coefficient.
-Implicit Colebrook equation cannot be rearranged to derive friction factor directly.
-Colebrook equation has to be solved iteratively or using approximations.
-Iterative calculus can cause problem in simulation of flow in a pipe system.
-Error of almost all explicit approximations of the Colebrook relation is up to 3%.
-An explicit approximation of the Colebrook relation can be very complex but also it can be
easily implemented in a computer code.
*Research Highlights DB

Figure 1 DB
Click here to download high resolution image

Figure 2 DB

Figure 3 DB

Figure 4 DB

Figure 5 DB

Figure 6 DB

Figure 7 DB

Figure 8 DB

Figure 9 DB

Figure 10 DB

Figure 11 DB

Figure 12 DB

Figure 13 DB

Figure 14 DB

Figure 15 DB

Figure 16 DB

Figure 17 DB

Figure 18 DB

Figure 19 DB

Figure 20 DB

Figure 21 DB

Figure 22 DB

Figure 23 DB

Figure 24 DB

Figure 25 DB

Figure 26 DB

Figure 27 DB

Figure 28 DB

Figure 29 DB

Figure 30 DB

Figure 31 DB

Figure 32 DB

δmax (%)
Relative
roughness (ε/D)
Reynolds number
(Re)
Romeo, Royo and Monzon (27) -0.1345 1·10-6
1·104
Buzzelli (32) -0.1385 7.5·10-2
1·108
Serghides (21) -0.1385 7.5·10-2
1·108
Zigrang and Sylvester (19) -0.1385 7.5·10-2
1·108
Vatankhah and Kouchakzadeh (29a) 0.1472 7.5·10-2
Barr (17) +0.2775 7.5·10-4
1·104
Serghides (22) +0.3544 1·10-6
2·106
Chen (15) -0.3556 7.5·10-4
8·104
Sonnad and Goudar (28) -0.8003 1·10-5
1·104
Papaevangelou, Evangelides and
Tzimopoulos (36)
-0.8247 7.5·10-2
2·106
Zigrang and Sylvester (18) -1.0074 1·10-6
3·105
Haaland (20) +1.4083 2.5·10-4
9·104
Jain (11) -2.0437 1·10-2
1·104
Swame and Jain (10) -2.0404 1·10-2
1·104
Manadilli (26) -2.0651 1·10-2
1·104
Churchill (9) -2.1718 1·10-2
1·104
Churchill (12) -2.1914 1·10-2
1·104
Brkić (38) -2.2719 1·10-2
1·104
Brkić (37) +3.1560 1·10-6
1·104
a
Modified Colebrook (4) -3.2025 1·10-6
1·104
Avzi and Karagoz (35) -4.7857 7.5·10-2
1·108
Eck (8) +8.20 1·10-6
5·106
Round (16) +10.92 7.5·10-2
1·108
Moody (5) +21.49 7.5·10-2
1·104
Wood (6) +23.72 1·10-6
1·104
Rao and Kumar (30) +81.24 1·10-6
1·104
a
also in implicit form (constant 2.51 is replaced with 2.825, recommended by AGA-American
Gas Association)
Table 1 DB
Click here to download Table: Table 1 DB.doc

a
Complexity b
Complexity index
c
Eck (8) 27 1
Moody (5) 29 1.07
Churchil (9) 31 1.14
Haaland (20) 35 1.29
Jain (11) 35 1.29
Swame and Jain (10) 36 1.33
Round (16) 36 1.33
Manadilli (26) 44 1.62
Zigrang and Sylvester (18) 47 1.74
Avzi and Karagoz (35) 47 1.74
Rao and Kumar (30) 61 2.25
Brkić (38) 67 2.48
Sonnad and Goudar (28) 67 2.48
Papaevangelou, Evangelides and
Tzimopoulos (36)
67 2.48
Brkić (37) 69 2.55
Zigrang and Sylvester (19) 69 2.55
Vatankhah and Kouchakzadeh (28a) 77 2.85
Barr (17) 80 2.96
Chen (15) 91 3.37
Wood (6) 98 3.62
Buzzelli (32) 104 3.85
Churchil (12) 106 3.92
Serghides (22) 107 3.96
Romeo, Royo and Monzon (27) 125 4.62
Serghides (21) 144 5.33
a
number of estimated algebraic notation calculator key strokes required to solve the
approximation, i.e. to find value of friction factor λ (estimated, i.e. average number of strokes)
b
Complexity index is defined as quotient of key strokes required for an observed approximation
and the least complex one; here Eck (8)
c
the least complex approximation presented here
Table 2 DB
Click here to download Table: Table 2 DB.doc

MS Excel DB
Click here to download Background dataset for online publication only: el. appendix DB.xlsx

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