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Larhyss Journal, ISSN 1112-3680, n°43, Sept 2020, pp. 13-22
© 2020 All rights reserved, Legal Deposit 1266-2002
© 2020 Achour B. and Amara L.; This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and
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NEW FORMULATION OF THE DARCY-WEISBACH
FRICTION FACTOR
NOUVELLE FORMULATION DU COEFFICIENT DE FROTTEMENT
DE DARCY-WEISBACH
ACHOUR B.1, AMARA L.2
1 Professor, Research laboratory in subterranean and surface hydraulics, LARHYSS,
University of Biskra, 07000, Biskra, Algeria
1,2 Associate Professor, Department of Civil Engineering and Hydraulics, Faculty of
Science and Technology, University of Jijel, Ouled Aissa, 18000 Jijel, Algeria
bachir.achour@larhyss.net
Research Article – Available at http://larhyss.net/ojs/index.php/larhyss/index
Received May 8, 2020, Received in revised form September 12, 2020, Accepted September 15, 2020
ABSTRACT
The proper assessment of the friction factor f is of a great importance in the sound
resolve of turbulent flow problems. The current rational formulation of f is that
developed by Colebrook stating that f depends on the relative roughness
/h
D
and the
Reynolds number R, through an implicit equation. The new formulation developed
herein presents f as a function not of the usual Reynolds number R but of a
dimensionless parameter, denoted
*
R
, representing the ratio of the friction forces to the
viscous forces. Acting as a Reynolds number, it is shown that
*
R
is governed by an
implicit equation of
/h
R
and R. The calculation of the friction factor value using the
new formulation gives a maximum deviation of 0.25% in comparison with the exact
value of f derived from Colebrook equation. At the end of an additional calculation step,
the deviation drops down to a maximum of 0.04% only. This calculation step is
recommended for solving problems requiring high accuracy. All the formulas developed
herein can be classified in the category of short equations, easily memorized, handy, and
of good accuracy.
Keywords: Friction factor, Darcy-Weisbach, Reynolds number, Pipe-flow.
Achour B. & Amara L. / Larhyss Journal, 43 (2020), 13-22
14
RESUME
L'évaluation appropriée du coefficient de frottement f est d'une grande importance dans
la bonne résolution des problèmes d'écoulement turbulent. La formulation rationnelle
actuelle de f est celle développée par Colebrook en montrant que f dépend de la rugosité
relative
/h
D
et du nombre de Reynolds R, à travers une équation implicite. La
nouvelle formulation développée ici présente f en fonction non pas du nombre de
Reynolds habituel R mais d'un paramètre sans dimension, noté
*
R
, représentant le
rapport des forces de frottement aux forces visqueuses. Agissant comme un nombre de
Reynolds, il est démontré que
*
R
est régi par une équation implicite de
/h
R
et de R.
Le calcul de la valeur du coefficient de frottement à l'aide de la nouvelle formulation
donne un écart maximal de 0,25% par rapport à la valeur de f dérivée de l’équation de
Colebrook. À la fin d'une étape de calcul supplémentaire, l'écart tombe à un maximum
de 0,04% seulement. Cette étape de calcul est recommandée pour résoudre les
problèmes nécessitant une grande précision. Toutes les formules développées ici
peuvent être classées dans la catégorie des équations courtes, facilement mémorisables,
maniables et de bonne précision.
Mots clés : Coefficient de frottement, Darcy-Weisbach, nombre de Reynolds, Conduite.
INTRODUCTION
In a turbulent flow regime, the friction factor, denoted f, plays a very important role. It
is a dimensionless parameter that relates the head loss in a pipe to its length/diameter
ratio and dynamic pressure (Jaeger, 1956). It is governed by the well-known implicit
Colebrook formula (1939) which states that f depends on the relative roughness
/h
D
and the Reynolds number R. Colebrook formula is one of the few relationships that has
aroused so much interest probably due to its importance in solving a number of main
problems such as pressure drop calculation in pipe-flow. Various approaches are
available to solve the Colebrook equation and find the appropriate value of f. The best
known of the time was the graphical solution using the Rouse and Moody diagrams,
established respectively in the years 1943 and 1944. Due to the low accuracy, the value
of f provided by the reading of theses charts should be as mere guidance value or
approximate. With the advent of modern laptops, the Colebrook equation can be solved
iteratively using an Excel spreadsheet or a programming solver, but this approach
requires more computational time. Recently (Brkić, 2011), Lambert W function was
used for the calculation of f and this approach seems to avoid iterative calculation and
reduces the relative errors. This probably has a purely theoretical and mathematical
interest, but has no practical impact in the daily work of an engineer who needs a fast
and reliable calculation of the friction factor. It is also the major concern of students
faced with their exam problems. There is also the trial-and-error method which is today
New formulation of the Darcy-Weisbach friction factor
15
completely obsolete. One of the research workers’ most noticed and preferred
approaches is to solve the Colebrook equation by an explicit approximate equation. In
this regard, one may counts up in the literature no less than forty approximate formulas
solving for f. These differ in both form and accuracy (Zeghadnia et al., 2019). The craze
to find an easier-to-compute formula result still remains today. There are short and
simple formulas but less accurate than the long and more developed formulas. The most
effective will be the one which will admit the best trade-off between time of
computation and precision of results. It is better to use short formulas that are less
accurate than long formulas that are more accurate for at least three reasons. Long
formula requires a longer computation time, they are not easy to memorize and present
the risk of omitting terms when key stroking. The short formulas available in the
literature generate a deviation in f of the order of 2% to 3% in the whole range of
turbulent flow corresponding to Reynolds number
2300R
. Considering, for instance,
the three main problems encountered in turbulent pipe-flow, 2% deviation in f causes
the same deviation in the slope of the energy grade line computation, 1% in the
discharge calculation, and only 0.4% in the determination of the pipe diameter. It is
therefore not a “disaster”.
In this technical note, a new formulation of the friction factor f is presented not
dependent on the usual Reynolds number R but on a dimensionless number denoted
*
R
.
This acts as a Reynolds number taking into account the effect of the friction forces. It
can be, therefore, considered within a constant as the shear Reynolds number. It is
shown that
*
R
depends on both the Reynolds number R and the relative roughness
/h
D
, through an implicit equation. A good approximate relationship has been found
which allows calculating f with an acceptable accuracy when compared to the f value
given by the Colebrook equation. If the problem under considerations requires a more
accurate value of f, an additional step calculation is then necessary. This causes a
significant drop down in the deviation in f. Thanks to the introduction of the
dimensionless parameter
*
R
, it was possible to derive an explicit approximate
relationship of the shear velocity
*
u
.
NEW FORMULATION OF THE FRICTION FACTOR
Using the Rough Model Method (Achour and Bedjaoui, 2006) or eliminating the
friction factor f between the Darcy-Weisbach (1854) and Colebrook (1939) equations,
the following dimensionally consistent uniform flow relationship can be worked out as:
*
10.04
4 2 log 14.8
hh
Q gA R S RR
(1)
Achour B. & Amara L. / Larhyss Journal, 43 (2020), 13-22
16
where Q is the discharge, S is the slope of the energy grade line,
is the absolute
roughness, g is the acceleration due to gravity, A is the wetted area,
h
R
is the hydraulic
radius and
is the kinematic viscosity. The parameter
*
R
is a dimensionless number,
acting as a Reynolds number, expressed as:
3
*32 2 h
gR S
R
(2)
Eq.(1) is valid for any channel and pipe shape. It is also applicable in the whole domain
of turbulent flow, including smooth, transitional, and rough flow regimes. The Darcy-
Weisbach and Colebrook equations can be written as follows:
2
8h
V
Sf
gR
(3)
1 2.51
2log 14.8 h
R
f R f
(4)
where V is the mean velocity of the flow.
The quantity
h
gR S
in Eq.(2) corresponds to the shear velocity u* also called friction
velocity (Schlishting, 1979) having dimension of velocity. Thus, Eq.(2) can be re-
written in the following form:
*
*32 2 h
uR
R
(5)
With regard to the form of the Eq.(5), the dimensionless number
*
R
would give a
measure of the ratio of friction forces to viscous forces and consequently the relative
importance of these kind of forces. On the other hand, Eq.(3) allows writing that:
2
*2 8
V
uf
(6)
or:
*8
f
uV
(7)
Multiplying both sides of Eq.(7) by
32 2 /
h
R
and knowing that
4/
hh
DR
, one
may derived the following result:
New formulation of the Darcy-Weisbach friction factor
17
*/
32 2 32 2 48
hh
u R VD f
(8)
which is reduced to :
*4R R f
(9)
When the proper values of
R
and f are known, Eq.(9) gives the exact value of
*
R
. It is
worth noting that the equality
*
RR
is obtained for
1/ 16 0.0625f
. This
corresponds to the relative roughness
0.037/h
D
according to Eq.(4) for
R
.
The flow is in the fully rough domain.
With
/V Q A
, Eq.(3) expresses the discharge Q as:
8h
Q A gR S
f
(10)
Comparing Eqs.(1) and (10) results in:
*
1 10.04
2log 14.8 h
RR
f
(11)
The friction factor f is therefore presented as a function of both the relative roughness
/h
R
and the dimensionless number
*
R
.
Extracting
1/ f
from Eq.(9) and inserting it in Eq.(11) gives:
**
1 10.04
log
2 14.8 h
RR RR
(12)
Eq.(12) shows that
*
R
is governed by an implicit function of both the relative roughness
/h
R
and the Reynolds number R. To avoid the implicit calculation induced by the
Eq.(12), one may use the following explicit approximate relationship :
1
*0.9
/5.45
2 log 3.7 h
D
RR R
(13)
Achour B. & Amara L. / Larhyss Journal, 43 (2020), 13-22
18
As shown in Fig.1, the maximum deviation between Eqs.(12) and (13) is about 1% only.
The deviation depends strongly on the relative roughness and the Reynolds number
values.
Figure 1: Deviation in
*
R
between Eqs.(12) and (13)
When the relative roughness
/h
D
and the Reynolds number R are given, Eq.(11)
along with Eq.(13) allows computing the friction factor f with a maximum deviation of
less than 0.25% as shown in Fig.2.
Figure 2: Deviation between Eqs.(4) and (11)
SHEAR VELOCITY
The shear velocity
*
u
can be expressed when combining Eqs.(5) and (13). Hence:
1
*
0.9
/5.45
32 2 2 log 3.7
hh
u R D
RR
(14)
Rearranging Eq.(14) results finally in :
-2
-1
0
1
2
0
0.00001
0.0001
0.001
0.01
0.05
/h
D
3
10
4
10
5
10
6
10
7
10
8
10
R
deviation in R* (%)
0.25
0.25
0
3
10
4
10
5
10
6
10
7
10
8
10
/h
D
0.01
0.05
0.001
0.0001
0.00001
0
R
deviation in (%)f
New formulation of the Darcy-Weisbach friction factor
19
1
1
*0.9
/5.45
log 14.8
16 2
hh
RR
uR R
(15)
Knowing that
4/
h
R VR
where V is the mean velocity, Eq.(15) can be rewritten as :
1
*
0.9
/
1 5.45
log 14.8
42 h
R
u
VR
(16)
IMPROVEMENT OF THE FRICTION FACTOR CALCULATION ACCURACY
To improve the friction factor f calculation accuracy, one may try to find a substitute
relation to the equation (13), but it is not self-obvious. The simplest way is to consider
an additional step in the calculation of f using Eqs.(9), (11), and (13). The calculation
procedure can be described by the following steps:
1. Let’s denote
*
0
R
the value of
*
R
given by Eq.(13) for the known values of both the
Reynolds number R and the relative roughness
/h
D
. The corresponding value of f is
*
10
()Rf
, worked out from Eq.(11). We have seen earlier that the maximum deviation
caused by this first step calculation on the f value, when compared to Colebrook Eq.(4),
is less than 0.25%.
2. Compute in this additional step
*
11
()Rf
using Eq.(9), whence:
*
11
4R R f
(17)
3. Introduction
*
11
()Rf
so calculated into Eq.(11) gives
2
f
as the second value of f.
It was observed that the maximum deviation between
2
f
and f given by the Colebrook
Eq.(4) is less than 0.04% as it is reported in Fig.3. This result was obtained for
2300R
and
0 0.05/h
D
, thus encompassing the whole domain of turbulent
flow. Thus, at the end of the second calculation step, the deviation on f experienced a
significant drop down from 0.25% to 0.04%. In fact,
*
0
R
served as an appropriate initial
guess value rapidly converging the only two-steps iterative process. The computation
can stop at this step because the relative error in f is largely sufficient to solve accurately
practical problems.
Achour B. & Amara L. / Larhyss Journal, 43 (2020), 13-22
20
Figure 3: Deviation between Eqs.(4) and (11) at the end of an additional
calculation step
NUMERICAL EXAMPLE
For the following data, compute the Darcy-Weisbach friction factor f using Eq.(11)
along with Eq.(13), after the first calculation step and then after the second calculation
step. What should be the deviation in comparison with the value of f given by
Colebrook equation?
0.00001/h
D
,
2,000,000R
Let’s assume the following definitions:
e
f
= the “exact” value of f computed using Colebrook Eq.(4).
1,a
f
= the approximate value of f after the first step of calculation.
2,a
f
= the approximate value of f after the second step of calculation.
*
0,a
R
= initial guess value of
*
R
computed using the approximate Eq.(13) and giving
1,a
f
by the use of Eq.(11).
The subscripts “1”, “2”, “0”, “
a
”, and “e” denote respectively “first step calculation”
“second step calculation”, “initial value”, “approximate value”, and “e” “exact value”.
1. The iterative process applied to the implicit Eq.(4) of Colebrook gives
e
f
value as:
0.0107206
e
f
2. The exact value of
*
R
is given by the implicit Eq.(12). The calculation shows that:
3
10
4
10
5
10
6
10
7
10
8
10
0.04
0.02
0
0.02
0.04
0 / 0.05
h
D
R
(%)
f
f
New formulation of the Darcy-Weisbach friction factor
21
*828322.642
e
R
Note that the exact value of
*
R
can also be computed using Eq.(9) in which
e
ff
.
3. The intial approximate value of
*
R
is easily worked out from Eq.(13). The final
result is:
*
0, 825804.52
a
R
4. The deviation between the exact value of
*
R
given in step 2, i.e.
*
e
R
, and the
approximate value computed in step 3, i.e.
*
0,a
R
, is then :
**
0,
*
825804.52 828322.642
100 100 0.304%
828322.642
ae
e
RR
R
5. The approximate value of f, i.e.
1,a
f
, is given by Eq.(11) in which the value of
*
R
is
that of
*
0,a
R
computed in step 3. Hence:
1, 0.01072536
a
f
6. The deviation between
e
f
and
1,a
f
is then:
1, 0.01072536 0.0107206
100 100 0.0443%
0.0107206
ae
e
ff
f
7. This step aims to compute the deviation between
e
f
and
2,a
f
after an additional
calculation step following the procedure described above. Eq.(9) gives:
*
1 1,
4a
R R f
The final result is:
*
1828506.369R
8. Introducing
*
1
R
in Eq.(11) gives a second approximate value of f , i.e.
2,a
f
, as:
2, 0.0107202
a
f
Achour B. & Amara L. / Larhyss Journal, 43 (2020), 13-22
22
9. The deviation between
e
f
and
2,a
f
is:
CONCLUSIONS
A new formulation of the friction factor
*
/( , )
h
RRf
was presented [Eq(11)], where
*
R
is a dimensionless parameter, acting as a Reynolds number taking into account the
effect of the friction forces. It was shown that
*
R
depends on both the Reynolds number
R and the relative roughness
/h
R
through an implicit equation [Eq.(12)]. An
approximate relationship has been derived giving
*
R
-value with a maximum deviation
of 1% [Fig.(1)]. The friction factor
*
/( , )
h
RRf
was then calculated with a maximum
deviation of 0.25% [Fig.(2)] when compared to the friction factor given by the
Colebrook equation [Eq.(4)]. If the problem under considerations requires a more
accurate friction factor value, an additional step calculation has been suggested to
improve the accuracy causing the deviation on f to drop down significantly from
0.25% to 0.04% [Fig.(3)].
REFERENCES
ACHOUR B., BEDJAOUI A. (2006). Discussion to “Exact solution for normal depth
problem, by SWAMME P.K. and RATHIE P.N., Journal of Hydraulic Research,
Vol.44, Issue 5, pp.715-717.
BRKIĆ D. (2011). W solutions of the CW equation for flow friction. Applied
Mathematics Letters, Vol.24, Issue 8, pp.1379-1383.
COLEBROOK C.F. (1939). Turbulent Flow in Pipes with Particular Reference to the
Transition Region Between Smooth and Rough Pipe Laws, Journal of the
Institution of Civil Engineers, Vol.11, pp.133-156.
DARCY H. (1854). Sur les recherches expérimentales relatives au mouvement des eaux
dans les tuyaux, Comptes rendus des séances de l’Académie des Sciences, n°. 38,
pp. 1109-1121.
JAEGER C. (1956) Engineering Fluid Mechanics, Blackie & Son Ltd., Glasgow.
MOODY L.F. (1944). Friction factors for pipe flow, Transactions of the American
Society of Mechanical Engineers, ASME, Vol.66, Issue 8, pp.671-684.
2, 0.0107202 0.0107206
100 100 0.0037%
0.0107206
ae
e
ff
f
New formulation of the Darcy-Weisbach friction factor
23
ROUSE H. (1943). Evaluation of boundary roughness, Proceeding of the 2nd
Hydraulics Conference, New-York, Vol.27, pp.105-116.
SCHLICHTING H. (1979). Boundary layer theory, New York, McGraw-Hill.
ZEGHADNIA L., ROBERT, J. L., ACHOUR, B. (2019). Explicit Solutions For
Turbulent Flow Friction Factor: A Review, Assessment and Approaches
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