Chemical Engineering Journal 86 (2002) 369–374
Short communication
Improved explicit equations for estimation of the friction
factor in rough and smooth pipes
Eva Romeo, Carlos Royo, Antonio Monzón
∗
Department of Chemical and Environmental Engineering, Faculty of Science, University of Zaragoza, Zaragoza 50009, Spain
Received 26 March 2001; received in revised form 29 October 2001; accepted 29 October 2001
Abstract
The most common correlations for calculating the friction factor in rough and smooth pipes are reviewed in this paper. From these
correlations, a series of more general equations has been developed making possible a very accurate estimation of the friction factor without
carrying out iterative calculus. The calculation of the parameters of the new equations has been done through non-linear multivariable
regression. The better predictions are achieved with those equations obtained from two or three internal iterations of the Colebrook–White
equation. Of these, the best results are obtained with the following equation:
1
√
f
=-2.0 log
ε/D
3.7065
-
5.0272
Re
log
ε/D
3.827
-
4.567
Re
log
ε/D
7.7918
0.9924
+
5.3326
208.815 + Re
0.9345
.
© 2002 Elsevier Science B.V. All rights reserved.
Keywords: Friction factor; Colebrook–White equation; Darcy–Weisbach equation; Head loss in pipes; Non-linear regression
1. Introduction
The energy loss due to friction undergone by a Newtonian
liquid flowing in a pipe is usually calculated through the
Darcy–Weisbach equation [1]:
h
f
= f
L
D
u
2
2g
(1)
In this equation f is the so-called Moody or Darcy friction
factor (f
M
or f
D
, respectively) [1] which, from the above
equation, is calculated as follows:
f
M
= f
D
=
D
L
gh
f
1
2
u
2
=
D
L
P
1
2
ρu
2
(2)
In addition to the Moody factor, the Fanning friction factor
can also be used, which is defined as follows [2]:
f =
τ
w
1
2
ρu
2
=
1
4
D
L
P
1
2
ρu
2
(3)
∗
Corresponding author. Tel.: +34-976-76-11-57;
fax: +34-976-76-21-42.
E-mail address: amonzon@posta.unizar.es (A. Monz´ on).
From Eqs. (2) and (3) the relation between both friction
factors is deduced: f = f
M
= f
D
= 4f
F
.
The friction factor depends on the Reynolds number (Re),
and on the relative roughness of the pipe, ε/D. For laminar
flow (Re < 2100), the friction factor is calculated from the
Hagen–Poiseuille equation:
f =
64
Re
=
64µ
uDρ
(4)
For turbulent flow, the friction factor is estimated through
the equation developed by Colebrook and White [3,4]
1
√
f
=-2 log
ε/D
3.71
+
2.52
Re
√
f
(5)
The Colebrook–White equation is valid for Re ranging from
4000 to 10
8
, and values of relative roughness ranging from 0
to 0.05. This equation covers the limit cases of smooth pipes,
ε = 0, and fully developed turbulent flow [3,4]. For smooth
pipes, Eq. (5) turns into the Prandtl–von Karman [3,4]:
1
√
f
= 1.14 - 2 log
ε
D
=-2 log
ε/D
3.71
(6)
If the flow is fully developed, it is verified that Re(ε/D)
√
f>
200. In this case, the friction factor depends only on the
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