Can Pipes Be Actually Really Smooth
Can Pipes Be Actually Really Smooth
Can Pipes Be Actually Really Smooth
w w w . i i fi i r . o r g
Short Communication
Dejan Brkic*
Petroleum and Natural Gas Engineer, Strumicka 88, 11050 Beograd, Serbia
Article history: In some recent papers a few approximations to the implicit NikuradseePrandtleKarman
Received 16 May 2011 equation were shown. The NikuradseePrandtleKarman equation for calculation of the
Received in revised form hydraulic friction factor is valid for the hydraulically smooth regime of turbulence. Accu-
17 September 2011 racy of these approximations for the friction factor in so called smooth pipes is checked
Accepted 24 September 2011 and related problems from the hydraulics are analyzed in the spotlight of the recently
Available online 2 October 2011 developed equations. It can be concluded that pipes can be treated as smooth below certain
value of the Reynolds number but after that even new polished pipes with a minor
Keywords: roughness follow the transitional and subsequently the rough law of flow at a higher
Accuracy values of the Reynolds number.
Comparison ª 2011 Elsevier Ltd and IIR. All rights reserved.
Flow rate
Friction
Hydraulics
Piping
Perfectly smooth surfaces do not exist (Taylor et al., 2006). In their recent paper Li et al. (2011) analyze the flow friction
Hydraulically smooth regime does not occur only in absence factor with the special attention to so called “smooth” pipes.
of the roughness (i.e. only when ε/D ¼ 0). This means that They note that the implicit equation developed by Colebrook
smooth regime can occur even if the relative roughness exists (1939) is valid for rough pipes which should imply that its
(if it is minor, i.e. if ε/D/0). This problem is shown in the accuracy for “smooth” pipes can be disputed. The Colebrook
spotlight of some recent new formulas. equation is valid for the entire turbulent regime which
includes the turbulent regime in the hydraulically smooth recommendation by AGA (American Gas Association) and
pipes, the transient (partially) turbulent regime and the fully American Bureau of Mines.
turbulent regime in the hydraulically rough pipes. This is All equations in this short report are presented using the
obvious from the title of the paper of Colebrook “Turbulent Darcy friction factor where the DarcyeWeisbach or the Moody
flow in pipes with particular reference to the transition region friction factors are synonyms (here noted as l). In the other
between the smooth and rough pipe laws”. The Colebrook hand, some researchers use the Fanning friction factor ( f ). This
equation is not valid for the laminar regime which occurs for is also correct where the connection between these two factors
approximately Re < 2320. It is valid for 2320 < Re < 108 (the is (l ¼ 4f ). The Fanning friction factor is Cf in Li et al. (2011).
turbulent regime). It has to be noted that for the laminar
regime, there are no smooth and rough pipes (Fig. 1).
Furthermore, in the laminar regime, all pipes are hydraulically 3. Determination of hydraulic regime
smooth. If the pipe roughness (protrusions of inner pipe
surface) is completely covered by the laminar sub-layer, the As noted before, the turbulent regime can be divided into the
surface is smooth from the hydraulic point of view. In the three sub-regimes; i.e. in the “smooth” (introductory) turbu-
laminar flow there is no laminar sub-layer, or better to say the lent regime, the partially (transient) turbulent regime and the
main and only layer of flow is laminar, hence, the prefix ‘sub’ rough (fully) turbulent regime.
is sufficient (there is no turbulent layer). In other words, in the The turbulent regime usually occurs when Re > 2300 or
laminar regime, all pipes are “smooth” as mentioned before. slightly above. The Reynolds number was introduced by
With further increasing of the Reynolds number, thickness of Reynolds (1883a,b) first in the “Proceedings of the Royal
the laminar sub-layer decreases baring the protrusions and Society” followed by a longer paper in the “Philosophical
fluid flow through a pipe becomes consequently hydraulically Transactions of the Royal Society”. Special issue of the “Phil-
smooth, and then gradually roughs, both from the hydraulic osophical Transactions of the Royal Society” dedicated to
point of view (Fig. 1). Hence the introductory turbulent flow these papers was published in 2008 with title “Turbulence
through the rough pipes (because the perfectly smooth pipes transition in pipe flow: 125th anniversary of the publication of
do not exist) can be noted as the hydraulically smooth. In the Reynolds’ paper”. Further about the history of the Reynolds
turbulent regime a rough pipe can be treated as smooth or number can be seen in Rott (1990).
rough which depends on the circumstances (Fig. 1). As shown in Brkic (2011a), the smooth regime of turbu-
Accuracy of the Colebrook equation can perhaps be lence occurs only if x < 16 and if Re > 2320 while the rough
disputed, but up to date it has been an accepted standard for turbulent regime occurs if x > 200. Between is the transient
the calculation of the friction factor in the turbulent flow both (partial) turbulent regime (16<x < 200). The Reynolds number
in, “smooth” and rough pipes. The well known Rouse and (Re) is a well known parameter while a parameter x is defined
Moody diagrams (or better to say, their turbulent part) had by (2):
been constructed using Colebrook’s formula (1):
ε pffiffiffi
x ¼ $Re$ l (2)
D
1 2:51 ε
pffiffiffi ¼ 2$log10 pffiffiffi þ (1) The value of parameter x as defined is valid for the Darcy
l Re$ l 3:71$D
friction factor (regarding the value of x for the Fanning friction
For flow of natural gas or other gaseous fluids the coefficient factor readers can consult Abodolahi et al. (2007) where the
2.51 should be replaced with 2.825 according to the hydraulically smooth regime occurs if xf < 8 and if Re > 2320).
Table 1 e Equations for hydraulically smooth regime developed for the total absence of roughness.
Equation Name
0:25
l ¼ 0:316=Re Blasius (for Re < 2$104)
l ¼ 0:184=Re0:2 Blasius (for Re > 2$104)
l ¼ ð0:79$lnðReÞ 1:64Þ2 Filonenko
1
pffiffiffi ¼ 0:8686$lnðRe$=1:964$lnðReÞ 3:8215Þ Techo et al.
l !
1 1 1:73718$Co $lnðCo Þ 2:62122$Co $½lnðCo Þ2 3:03568$Co $½lnðCo Þ3
pffiffiffi ¼ $ Co þ þ Co ¼ 4$log10 ðReÞ 0:4 Danish et al. (2011)
l 2 1:73718 þ Co ð1:73718 þ Co Þ3 ð1:73718 þ Co Þ4
" !#2
150:39 152:66
l ¼ 0:25$ log10 Fang et al. (2011)
Re0:98865 Re
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1 0:0015702 0:3942031 2:5341533
pffiffiffi ¼ 0:8685$ln Re$ þ þ 0:198 Li et al. (2011) a
l lnðReÞ ln2 ðReÞ ln3 ðReÞ
a Also note that ln(Re)2 and ln(Re)3 are actually ln2(Re) and ln3(Re), respectively, i.e. (ln(Re))2 and (ln(Re))3. According to Eq. (7) in Li et al. (2011),
one can assume that using logarithmic rule, ln(Re)2 and ln(Re)3 can be rearranged to 2$ln(Re) and 3$ln(Re), respectively which produce error
(because 2 and 3 are power of whole logarithm and not only of Re).
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 2 0 9 e2 1 5 213
1 pffiffiffi turbulent and even the laminar flow of the Newtonian fluid
pffiffiffi ¼ 1:93$log10 Re$ l 0:537 (5)
l (including the zone between them). According to the recent
paper by Brkic (2011c), the approximations of the Colebrook
The NPK equation with the coefficients adjusted by McKeon
equation by Romeo et al. (2002), Buzzelli (2008), Serghides (1984),
et al. (2004) is also implicit in flow friction factor, hence the
Zigrang and Sylvester (1982) and Vatankhah and Kouchakzadeh
methodology proposed by Li et al. (2011) and Danish et al.
(2008) are among the five most accurate up to date. Their rela-
(2011) can be applied to it. The Princeton and the Oregon
tive error is no more than 0.15% compared to the iterative
equations for the fully developed pipe flow can be seen in
solution of the implicit Colebrook equation (for the whole
McKeon et al. (2005). These equations for the fully developed
turbulent regime). The other three approximations mentioned
pipe flow can be used as an improved substitution for the
in the paper of Li et al. (2011) are not among the most accurate.
Colebrook equation valid for the whole regime of turbulence.
These mentioned approximations are by Haaland (1983) with
the relative error of no more than 1.5%, Swamee and Jain (1976)
with the relative error of no more than 2.5% and Avci and
6. Comparison of different formulas Karagoz (2009) with the relative error up to 5%.
Measuring the CPU time is a good approach for a compar-
From Figs. 3 and 4, it can be seen that different equations ison of the formulas in hydraulics (Giustolisi et al., 2011;
produce the different results. But from Fig. 2, it also can be Danish et al., 2011; Li et al., 2011). Also, one has to be aware
seen that these differences have a minor or none influence on that computational speed does not depend only on the
the calculation. From Fig. 2, it can be clearly seen that only the problem size but also on the computing environment (the type
effect of roughness can make an influence on the final results of CPU or other hardware components). Giustolisi et al. (2011)
(to increase the accuracy of the final results). observe that the computation of logarithm in the computer
Same as the implicit Colebrook relation, its explicit approx- languages is based on the series of expansions that require
imations are valid for the whole turbulent regime. Up to date, several powers of the argument to be computed and added to
only the approximation made by Churchill (1977) is valid for the each other. Note that the explicit equations proposed by
Fig. 3 e Maximal relative error of presented equations for hydraulically smooth regime developed for the total absence of
roughness.
214 i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 2 0 9 e2 1 5
Fig. 4 e Comparison of the most accurate explicit approximations of the NPK equation.
Danish et al. (2011) and Li et al. (2011) contain many loga- Avci, A., Karagoz, I., 2009. A novel explicit equation for friction
rithmic expressions. Approximations to the Colebrook equa- factor in smooth and rough pipes. J. Fluid Eng. ASME 131 (6)
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