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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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