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Determining velocity and friction factor for turbulent
ow in smoothtubes
Dawid Taler
Cracow University of Technology, Faculty of Environmental Engineering, ul. Warszawska 24, 31-145 Cracow, Poland
a r t i c l e i n f o
Article history:
Received 5 October 2015Received in revised form11 January 2016Accepted 16 February 2016
Keywords:
Turbulent tube
owSmooth-wall tubeFriction factorMomentum conservation equationVelocity pro
le
a b s t r a c t
The most popular explicit correlations for the friction factor in smooth tubes are reviewed in this paper.The friction factor for the turbulent
ow in smooth tubes is required in some correlations when calcu-lating the Nusselt number. To calculate the friction factor, the velocity pro
le in a turbulent smooth wall-tube must be estimated at
rst. The radial velocity distribution was determined using either universalvelocity pro
le found experimentally by Reichardt or by integration the momentum equation using theeddy diffusivity model of Reichardt. The friction factor obtained by using the universal velocity pro
legives better results than that obtained from the momentum equation when compared with the Prandtl
e
von K
arm
an
e
Nikuradse equation. Based on the velocity pro
les proposed by Reichardt the frictionfactor was calculated as a function of the Reynolds number and subsequently two formulas for thefriction were proposed. They have satisfactory accuracy when comparing with the implicit Prandtl
e
vonK
arm
an
e
Nikuradse equation. Thus, it was concluded that the universal velocity pro
le proposed byReichardt will provide good results when it is taken into account while integrating the energy conser-vation equation. There is also a considerable number of experimental correlations for the friction factorin smooth tubes. All these relationships were compared with the experimental data and with the implicitPrandtl
e
von K
arm
an
e
Nikuradse equation that is considered as a standard to test the explicit approxi-mations. The Colebrookand Filonienko explicit correlations arewidelyused when calculating the Nusseltnumber for the turbulent
ow but they have noticeable errors for small Reynolds number ranged from3000 to 7000 for the Colebrook relation and from 3000 to 30,000 for the Filonienko relation. For thisreason, a new simple and accurate correlation for the friction factor for Reynolds numbers between 3000and 10
7
is proposed in the paper.
©
2016 Elsevier Masson SAS. All rights reserved.
1. Introduction
The determination of a friction factor in turbulent tube
ow isessential not only to pressure drop calculations in pipelines andheat exchangers [1] but also is needed for calculating the Nusseltnumber in turbulent tube
ow [2,3]. The correlations for the fric-tion factor
x
can be found experimentally based on the measuredpressure drop in a tube over a given distance or on the measuredradial velocity pro
le. The latter way of the friction factor deter-mining is also important in driving the heat transfer correlationbecause it allows to choose the most appropriate velocity pro
leindirectly. Solving the energyconservation equation using accurateuniversal velocity pro
le will yield the Nusselt numbers as func-tions of the Reynolds and Prandtl numbers that are consistent withthe experiment.Blasius was the
rst who proposed an explicit correlation forturbulent tube
ow that is valid for Reynolds numbers between3000 and 10
5
. The Blasius correlation is still in use [4]. Sheikho-leslami et al. [5] studied turbulent
ow and heat transfer in the airtowaterdouble-pipe heatexchanger. TheBlasius formulawas usedto calculate the friction factorneeded for theestimation of the heattransfer by the Gnielinski correlation. Only about 20 years later, animplicit relationship for determining the friction factor wasdevelopedbyPrandtl,vonK
arm
an,andNikuradse(PKN)[6
e
8].ThePKN equation for the friction factor for the turbulent
ow in asmooth tube is widely accepted and has become a model equationfor explicit approximations. The PKN correlation is implicit in
x
because the friction factor
x
appears on both equation sides. Inotherwords,itisanonlinearalgebraicequationthatmustbesolvedeither iteratively or graphically. This inconvenience can be cir-cumvented using a numerous explicit approximation to the PKN
E-mail address:
dtaler@pk.edu.pl.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences
journal homepage: www.elsevier.com/locate/ijts
http://dx.doi.org/10.1016/j.ijthermalsci.2016.02.0111290-0729/
©
2016 Elsevier Masson SAS. All rights reserved.
International Journal of Thermal Sciences 105 (2016) 109
e
122
equation[1,9
e
11].Evenalargernumberofexplicitcorrelationswasproposed for implicit Colebrook
e
White equation used to deter-mine the coef
cient of friction in the rough pipes. Many compari-sons of explicit approximations to the Colebrook
e
White equationwere conducted over the past two decades. Examples of suchcomparative reviews can be publications [1,9
e
11]. Unfortunately,the explicit equations for rough pipes cannot be used for smoothpipes, since they were derived for the relative surface roughnessgreater than zero.Petukhov and Kirillov [2,12] proposed in 1958 a formula for theNusselt number and suggested to use the explicit correlation of Filonienko [13] for calculating the friction factor. Gnielinski [14]
extended the application of the Petukhov
e
Kirillov equation tolower Reynolds numbers and continued to calculate the frictionfactor using the Filonienko correlation. Since that time, the Filo-nienko correlation is widely utilized in the calculation of the Nus-selt number for the transitional and turbulent
ow in tubes[15
e
24]. Mirth and Ramadhyani [15] applied the Gnielinski [14]
correlation in conjunction with the Filonienko friction factor tocalculate thewater-side heat transfercoef
cient inside the tubes of a
nned-tube chilled-water cooler. In all the experiments, highwater mass
ow rates were maintained to provide a turbulent
owof water. Fernando et al. tested a mini channel aluminum tube heatexchanger for water-to-water operation [16]. They found that theNusselt numbers obtained in the experiment agree with thosepredicted by the Gnielinski correlation [14] within an accuracy of
±
5% in the transition Reynolds number range of 2300
e
6000. Thefriction factor was calculated using the Filonienko approximation.A lotof papers is devoted tothe intensi
cation of heat exchangein tubes [17
e
22].Li et al. [17] measured the turbulent tube
ow in a micro-
ntube using water and oil. The friction factor and Nusselt numbersin a smooth tube were
rst estimated experimentally andcompared with the Filonienko and Gnielinski correlations,respectively,tovalidatetheexperimentalset-upanddatareductionprocedure. The Reynolds numbers varied from 2500 to 90,000 forwaterandfrom2500to12,000foroil.Theresultsofmeasurementsand calculations agree very well, even for small Reynolds numbersnear
Re
¼
2500. The maximum relative differences between themeasured friction factor and the empirical correlation by Filo-nienko does not exceed 10%. Similar experiments were carried outby Li et al. with rough tubes [18]. As in the previous study [17], the
measurements were conducted for the turbulent
ow of water in asmooth tube. The Reynolds number varied from 7000 to 90,000.Differencesbetween measured andcalculated friction factors usingthe Filonienko formula are small. Li et al. [19] used the Filonienkocorrelationtoshowtheincreaseinthefrictionfactorof thediscretedouble inclined ribbed tubes in relation to the smooth pipes. In thepaper of Ji et al. [20] developed turbulent heat transfer in internalhelically ribbed tubes is studied experimentally. To test the reli-ability of the test facility, the experimental results of the frictionfactor were
rstly compared with the Filonienko correlation. TherelativediscrepancybetweentheexperimentaldataandFilonienkopredictions was within
±
5% for the Reynolds number ranged from8000 to 90,000.Flow heat transfer and pressure drop measurements in doublyenhanced tubes were conducted with water and ethylene glycol inthe laminar-transition turbulent
ow regime by Raj et al. [21]. Theaim of the study was to investigate the usefulness of doublyenhanced tubes for lower duty heat exchangers in the laminar-transition-turbulent
ow regime. To check out the experimentalset-up and the data processing methodology, the tube-side heattransfer and friction factor were
rst determined in a 2590 mmlong copper smooth tube with an inner diameter of 15.88 mm.Turbulent
ow friction factors determined experimentally for de-ionized water compared to within
±
5% of the friction factors pre-dicted by the Filonienko equation.The paper by Zhang et al. [22] reported the thermo-hydraulic
Nomenclature
c
1
,
c
2
constants
d
w
inner diameter of a circular tube,
d
w
¼
2
r
w
, m
e
i
relative difference
i
node number
k
turbulence kinetic energy, N/(s m
2
)
L
distance between pressure taps, m
n
number of nodes in the
nite difference grid
p
pressure, PaPKN Prandtl
e
von K
arm
an
e
Nikuradse
r
radial coordinate, m
r
w
inner radius of the tube, m
r
2
coef
cient of determination
r
þ
dimensionless radius,
r
þ
¼
ru
t
=
v
R
dimensionless radius,
R
¼
r
=
r
w
Re
Reynolds number,
Re
¼
w
m
d
w
=
v
u
t
friction velocity,
u
t
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t
w
=
r
p
, m/s
w
m
mean velocity
w
x
velocity component in the
x
direction
w
r
,
w
x
time averaged velocity component in the
x
and
r
direction, respectively, m/s
x
a spatial coordinate in Cartesian or cylindricalcoordinate systems, m
y
a spatial coordinate in a Cartesian system or distancefrom distance from the wall surface, m
y
þ
dimensionless distance from the tube wall,
y
þ
¼
yu
t
=
v
Greek symbols
D
p
pressure drop, Pa
D
y
þ
dimensionless spatial step
3
turbulence dissipation rate, N/(s m
2
)
3
t
eddy diffusivity for momentum transfer (turbulentkinematic viscosity), m
2
/s
m
dynamic viscosity, kg/(m s)
k
the von K
arm
an constant
n
kinematic viscosity,
v
¼
m
=
r
, m
2
/s
x
Darcy
e
Weisbach friction factor
r
uid density, kg/m
3
t
shear stress, Pa
t
w
shear stress at wall surface, Pa
Subscripts
m mean
i
node numberw wall surface
Superscripts
time averaged
þ
dimensionless
D. Taler / International Journal of Thermal Sciences 105 (2016) 109
e
122
110
evaluation of the heat transfer enhancement in the smooth tubes
tted with rotor-assembled strands of various diameters. The re-sults of preliminary experiments carried out on a plane tube werecompared with the empirical correlation of Gnielinski for theNusselt number and with the Filonienko correlation for the frictionfactor. The friction factors found experimentally agree within 15%with the Filonienko equation [22].Hojjat et al. [23] investigated convective heat transfer of non-Newton nano
uids through a uniformly heated circular tube.TheycalculatedtheNusseltnumbersforde-ionizedwaterusingtheexperimental data and compared with the Nusselt numbers pre-dicted by the Gnielinski equation [14] in which the Filonienkoempirical correlation was applied to calculate the friction factor.The very satisfactory agreement was obtained between the exper-imental results and those obtained by the Gnielinski or the Dit-tus
e
Boelter correlations in the range of
Re
from 3000 to 20,000.Heat transfer of nano
uids in a turbulent pipe
ow was studiedtheoretically by Corcione et al. [24] who demonstrated that nano-
uids behave like single-phase
uids. They recommended usingthe Gnielinski correlation [14] for the heat transfer. The frictionfactor may be calculated using either the Filonienko correlation ortraditional power-type Blasius [4] or Moody [25] empirical
correlations.One of the
rst explicit approximation to the PKN equationproposed Colebrook [26,27]. The almost identical correlation wassuggestedonceagainsixyearslaterbyKonakov[28].Becauseofthebetter accuracy of the Konakov explicit representation of the fric-tion factor in comparison with the Filonienko correlation, Gnie-linski recommended it to use it in his correlation for the Nusseltnumber [3,29,30].A continuous explicit relationship for the friction factor for fullydeveloped laminar, transitional and turbulent
ows in smooth andrough pipes developed Churchill [31]. Churchill's formula wasmodi
ed by Schroeder [32], and Rennels and Hudson [33] to get
better compatibility with the Hagen
e
Poiseuille law and Cole-brook
e
White equation. A piecewise friction factor formula forlaminar, transition and turbulent
ow in smoothpipes was derivedby Joseph and Yang [33]. The data in the transition region is pro-cessed by
tting
ve points with a logistic dose algorithm.Friction factors and Nusselt numbers for all
ow regimes inround tubes and parallel-plate channels were successfully deter-mined using CFD (Computational Fluid Dynamics) modeling[34
e
38]. The RANS (Reynolds-Averaged Navier
e
Stokes) equationsof the mass and momentum conservation, and SST (Shear StressTransport) turbulence model were used to study the transitionfrom laminar to intermittent and turbulent
ow or internal
owswhich transit from turbulent through intermittent to laminar. Thefriction factors obtained from numerical experiments by Abrahamet al. [34,37,38] were approximated by polynomials of various de-grees to get explicit formulas suitable for engineering applications.A review of published papers shows that for the calculation of the friction factor for turbulent tube
ow several empirical corre-lations of various accuracy are used. There are various ranges of Reynolds numbers, in which these correlations are valid. There is,therefore, a need to compare the currently used correlations withthe standard PKN equation and other recent experimental studies.The aim of this study is to
nd a suitable universal velocitypro
le for turbulent
uid
ow in a tube to determine the frictionfactor in smooth tubes. An accurate pro
le of the universal velocityis also essential in calculating the Nusselt number as a function of Prandtl and Reynolds numbers by integrating the time-averagedequation of energy conservation.The velocity pro
le will
rst be determined using the universalvelocity distribution found by Reichardt [39,40], and then themomentumconservationequationwillbeappliedtodeterminethevelocity pro
le. After determining the average velocity of
uidusing the previously determined velocity pro
les, the friction fac-tors will be calculated. Based on a comparison of the determinedfriction factors with the friction factors derived from the referencePKN and experimental correlations, more accurate method fordetermining the velocity pro
le will be selected. A more accuratevelocity pro
le can be used to determine the distribution of tem-perature and heat
ux in the tube cross-section, which will alsoprovidegreateraccuracyindeterminingtheNusseltnumberonthetube surface.Another important goal of this paper is a comparative study of existing explicit correlations for the smooth tube friction factor,which are used in calculating the Nusselt number using Petukhovor Gnielinski correlations or other heat transfercorrelations, whichhave smooth tube friction factors.New simple but accurate correlations for the estimation of thefriction factor in smooth tubes will also be offered.
2. Mathematical formulation of the problem
The Darcy
e
Weisbach friction factor
x
for circular tubes is esti-mated experimentally using the following data reduction equation(Fig.1)
x
¼
4
D
pr
w
r
Lw
2m
(1)
The friction factor
x
can also be determined based on the radialvelocity pro
le obtained experimentally or determined by inte-grating the momentum conservation equation.At
rst, the velocity distribution and the friction factor will bedetermined based on the solution of the momentum conservationequationforturbulent
owwhen
Re
>
3000.Theturbulentvelocitypro
le
w
x
is obtained by solving the time-averaged momentumconservation equation [41
e
43].
1
r
dd
r
r
r
ð
v
þ
3
t
Þ
d
w
x
d
r
¼
d
p
d
x
(2)
Equation (2) was obtained from a momentum conservation equa-tion written in cylindrical coordinates with the followingassumptions
w
r
¼
0
;
v
w
r
v
r
¼
0
;
v
w
x
v
x
¼
0
;
vv
x
r
w
0
x
w
0
r
¼
0
where
w
0
x
and
w
0
r
are the
uctuating components of the longitu-dinal and radial velocity, respectively.The momentum eddy diffusivity
3
t
is de
ned as
r
w
0
x
w
0
r
¼
r
3
t
v
w
x
v
r
Equation (2) is subject to the following boundary conditions
w
x
j
r
¼
r
w
¼
0 (3)d
w
x
d
r
r
¼
0
¼
0 (4)
The shear stress
t
is de
ned as [41
e
43].
t
¼ ð
m
þ
r
3
t
Þ
d
w
x
d
r
¼
r
ð
v
þ
3
t
Þ
d
w
x
d
r
¼
m
1
þ
3
t
v
d
w
x
d
r
(5)
Writingthemomentumconservationequationforacontrolvolume
D. Taler / International Journal of Thermal Sciences 105 (2016) 109
e
122
111
of a diameter
d
w
¼
2
r
w
and a length
D
x
(Fig.1) yields
p
r
2w
p
ð
x
Þ ¼
p
r
2w
p
ð
x
þ
D
x
Þ þ
2
p
r
w
D
x
t
w
x
þ
D
x
2
(6)
t
w
x
þ
D
x
2
¼
r
w
2
p
ð
x
þ
D
x
Þ
p
ð
x
Þ
D
x
(7)
If
D
x
/
0 then Eq. (7) can be written as
t
w
¼
r
w
2d
p
d
x
(8)
Taking into account Eqs. (5) and (8), Eq. (2) can be rewritten in the
form
1
r
dd
r
ð
r
t
Þ
2
t
w
r
w
(9)
Integration of Eq. (9) with the boundary condition
t
j
r
¼
r
w
¼
t
w
(10)
gives
t
¼
t
w
r r
w
(11)
An analysis of expression (11) indicates, that the shear stress
t
is alinear function of the radius
r
(Fig. 1).Substitution of Eq. (11) into Eq. (5) leads to
d
w
x
d
r
¼
t
w
r r
w
1
m
1
þ
3
t
v
(12)
SolvingEq.(12)subjecttotheboundarycondition(3)givesthe
uidvelocity as a function of the radius. Based on the radial velocitypro
le
w
x
ð
r
Þ
the friction factor
x
can be determined. Consideringthat
d
p
d
x
¼
x
d
w
r
w
2m
2 (13)
and substituting Eq. (13) into Eq. (8) gives the expression for the
shear stress at the wall
t
w
¼
x
r
w
2m
8 (14)
where the mean velocity
w
m
is given by
w
m
¼
2
r
2w
Z
r
w
0
w
x
r
d
r
(15)
Introducing the so called friction velocity
u
t
given by Refs. [41
e
43]
u
t
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t
w
=
r
p
(16)
the dimensionless variables can be de
ned in the following way
y
þ
¼
yu
t
v
¼
y
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t
w
=
r
p
v
¼ ð
r
w
r
Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t
w
=
r
p
v
(17)
r
þ
¼
ru
t
v
¼
r
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t
w
=
r
p
v
;
r
þ
w
¼
r
w
u
t
v
¼
Re
2
ﬃﬃﬃ
x
8
r
(18)
R
¼
r r
w
¼
r
w
yr
w
¼
1
y
þ
r
þ
w
(19)
Rearranging Eq. (16) gives
t
w
¼
r
u
2
t
(20)
Inserting Eq. (20) for
t
w
into Eq. (14) yields
x
¼
8
u
2
t
w
2m
¼
8
u
þ
m
2
(21)
where the symbol
u
þ
m
denotes the dimensionless mean velocity
u
þ
m
¼
w
m
=
u
t
(22)
Equation (21) is used to calculate the friction factor for incom-pressible
ow over a
at surface or for fully developed
ow in atube [41
e
43]. The dimensionless velocity
u
þ
can be determined bysolving Eq. (12) with the boundary condition (3) or using the ve-
locity distribution
u
þ
determined experimentally. Next, the meanvelocity
u
þ
m
is calculated using Eqs. (15) and (22).
3. Determination of the dimensionless mean velocity andfriction factor
First,solvingequation(12)subjecttotheboundarycondition(3)
will be discussed in detail. Equation (12) rewritten in the dimen-sionless form
d
u
þ
d
y
þ
¼
1
r
þ
w
r
þ
w
y
þ
1
þ
3
t
=
v
(23)
is subject to the boundary condition (3) that takes the form
u
þ
y
þ
¼
0
¼
0 (24)
Using the Euler method to solve Eq. (23) gives
u
þ
j
þ
1
u
þ
i
D
y
þ
¼
12
r
þ
w
r
þ
w
y
þ
i
1
þ ð
3
t
=
v
Þj
y
þ
i
þ
r
þ
w
y
þ
i
þ
1
1
þ ð
3
t
=
v
Þj
y
þ
i
þ
1
#
;
i
¼
1
;
…
;
n
1 (25)
To increase the accuracy of the calculation, the right side of Eq. (25)is the arithmetic mean of the right-hand side of Eq. (2) that isevaluated at
y
þ
i
and
y
þ
i
þ
1
. The Euler method is simple and accurate
Fig.1.
Turbulent
uid
ow in the tube;
p
1
and
p
2
measured pressures at a distance of
L
for determining the friction factor
x
.
D. Taler / International Journal of Thermal Sciences 105 (2016) 109
e
122
112

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