Graetz problem with non-linear viscoelastic fluids in non-circular tubes

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International Journal of Thermal Sciences 61 (2012) 50e60

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International Journal of Thermal Sciences

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Graetz problem with non-linear viscoelastic fluids in non-circular tubes

A. Filali a, L. Khezzar a,*, D. Siginer b, Z. Nemouchi c

aDepartment of Mechanical Engineering, Petroleum Institute, Abu Dhabi, United Arab EmiratesbCentro de Investigación en Creatividad y Educación Superior, Universidad de Santiago de Chile, Santiago, ChilecDepartment of Mechanical Engineering, University of Mentouri Constantine, Algeria

a r t i c l e i n f o

Article history:Received 5 July 2011Received in revised form13 February 2012Accepted 14 June 2012Available online 21 July 2012

Keywords:ViscoelasticGraetz problemPhan-ThieneTanner modelHeat transfer enhancementNusselt number

* Corresponding author.E-mail address: [email protected] (L. Khezzar).

1290-0729/$ e see front matter � 2012 Elsevier Mashttp://dx.doi.org/10.1016/j.ijthermalsci.2012.06.011

a b s t r a c t

A numerical investigation of the Graetz problem in straight pipes of circular and non-circular cross-sections is carried out to analyze the influence of the rheological parameters on the heat transferenhancement with negligible axial heat conduction and viscous dissipation for a class of non-linearviscoelastic fluids constitutively represented by the simplified Phan-ThieneTanner (SPPT) model. Theanalysis considers both constant wall heat flux and constant wall temperature thermal boundaryconditions and concludes that the combined elastic and shear-thinning effects represented by theparameter 3We2 lead to heat transfer enhancement for low values of the parameter of order O(1) whereasvalues of the parameter 3We2 > O(10) lead to a decrease in the heat transfer rate in the case of constantwall heat flux. Nusselt number distributions in the entrance region of tubes of equilateral triangular,square and rectangular cross-sections as well as Nusselt numbers Nu ¼ f( 3We2) for the fully developedflow in these non-circular tubes are reported for the first time for non-linear viscoelastic fluids of theSPPT type. It is concluded that for small values of elasticity (We), the computations based on the methodsincluded in the Polyflow software are in full agreement with analytical results when available and thatdiscrepancies exist for high values of We. Such limitations may not exist with pseudo-spectral methods.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

This paper investigates the Graetz problem for laminar flow ofNewtonian and viscoelastic fluids in ducts of circular and three non-circular cross sections with symmetry equilateral triangular, squareand rectangular. Heat transfer in the entry region of non-circularducts is of particular interest in the design of compact heatexchangers in chemical, pharmaceutical, petrochemical, and foodindustries. Most common heat exchangers in industry use circularand rectangular ducts, which are easy tomanufacture,maintain andclean. As most fluids used in chemical, pharmaceutical, petro-chemical, and food industry processes are non-Newtonian it isimportant to determine the characteristics of forced convective heattransfer in steady laminar non-Newtonian flow in ducts with non-circular cross section to exercise proper control over the perfor-mance of the heat exchanger and to optimize the process.

The classic comprehensive work compiled by Shah and London[1] addresses convective heat transfer for circular and non-circularducts for Newtonian fluids, and an updated review has been re-ported by Shah and Bhatti [2]. Entrance problems with Newtonian

son SAS. All rights reserved.

fluids have been given adequate attention in the literature usingtheoretical, numerical or experimental approaches. However withnon-Newtonian fluids the literature is much thinner. Graetzproblem involving non-Newtonian power law fluids in circularpipes was considered by Lyche and Bird [3] and a semi-analyticsolution was found. Their work was extended by Toor [4] whosolved the field equations analytically taking into account internalheat generation. A closed-form solution for laminar heat transfer topower law fluids with different power-law indices in circular ductswithout viscous dissipation was obtained for constant walltemperature by using confluent hypergeometric functions andLaplace transform [5].Most analytical solutions follow thepattern ofreducing the governing equations to a SturmeLiouville problem andnumerically calculating the eigenvalues and eigenvectors [6]. Parikhand Mahalingam [7] developed analytical solutions for the walltemperature profile of a power law fluid in laminar flow in a circulartube for constant wall temperature and arbitrarily varying heat fluxat thewall. Flores et al. [8] obtained asymptotic behaviour for powerlaw fluids for both axisymmetric and plane geometries withconstant wall temperature. The early work of McKillop [9] reportson numerical solution for power law pseudoplastic fluids in circulartubes. Chandrupatla and Sastri [10] used the finite differencemethod for laminar flow forced convection heat transfer ofNewtonian and pseudoplastic fluids in the thermal entrance region

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e60 51

of a square duct for different thermal boundary conditions. Thenumerical solution for Newtonian fluids gives values of the limitingNusselt numbers in excellent agreement with those available in theliterature. In general shear thinning fluids constitutively repre-sented by power lawgenerate a higher heat transfer coefficient thanNewtonian fluids.

The ability of viscoelastic fluids to enhance heat transfer inlaminar flow in tubes of non-circular cross section was firstdemonstrated by Hartnett and Kostic [11]. Exact analytical andsemi-analytical solutions with fluids obeying the simplified affinePhan-Thien Tanner (SPTT) model [12] were obtained by Coelhoet al. [13] for two steady problems of laminar forced convection inpipes and plane two dimensional channels under fully developedconditions with constant wall temperature. Exact analytical resultspoint at an increasing normalized heat transfer coefficient, anincrease thatmay reach up to 14%, with increasing combined elasticand elongational effects asmeasured by 3

1/2We due to the increasedlevel of shear-thinning governed by 3 in the SPTT model. Thecircular pipe problem has an exact analytical solution under theassumption of equilibrium between viscous dissipation and radialheat conduction with negligible axial convection and a constantwall temperature whereas the Nusselt number for the planeproblem was obtained numerically by successive approximations.

Analytical solutions for the Poiseuille flow of non-linear visco-elastic fluids represented by the non-affine PTTmodel were derivedindependently by Letelier and Siginer [14,15] and Oliveira andPinho [16]. Coelho et al. [17] carried out a theoretical study of theGraetz problem in both plane channels and axisymmetric tubesusing the velocity field derived in Ref. [16], and discussed thevariation of the Nusselt number for different Brinkman numbers interms of the combined elastic and elongational effects representedby the Weissenberg number and the constitutive parameter 3.Oliveira et al. [18] obtained a semi-analytical solution for the Graetzproblem of highly viscoelastic liquids of the FENE-P constitutivetype in pipes and channels including viscous dissipation underprescribed constant wall temperature and wall heat flux thermalboundary conditions. They determined that the difference betweenthe behaviour of the PTT and FENE-P fluids in this shear flow lies onthe effects of 3 and the extensibility parameter L2, respectively.Increases in both 3 and We lead to an increase of the Nusseltnumber as in the work of Coelho et al. [17]. L2 and 3

1/2We haveopposite effects on heat transfer enhancement. Increases in theparameter 3

1/2We lead to enhanced Nusselt numbers and highervalues of L2 lead to decreased shear-thinning and heat transfer.

Viscoelastic Graetz problem in ducts of arbitrary shape has notbeen addressed in the literature. Analytical solution of the Graetzproblem for PTT fluids is very difficult to obtain in tubes of arbitrarycross-sectional shape and in general numerical techniques must beused. The closely related problem of heat transfer with non-linearviscoelastic liquids in fully developed flow in tubes of non-circular shape with realistic constitutive models has not beenaddressed in the literature until recently primarily due to thedifficulties involved both analytically and numerically. Howeversome progress has been made in the last decade [13,19]. The lastauthors analytically derived for the first time the variation of theNusselt number with elasticity in arbitrary cross-sectional tubesunder constant boundary heat flux and fluids that abide by the non-affine PTT constitutive model. The objective of this paper is toextend the work of Coelho et al. [17] and investigate numericallythe Graetz problem in tubes of selected non-circular cross-sectionusing the simplified PTT constitutive equation (SPTT) with theexponential form of the embedded function f (thereafter calledexponential SPTT) which incorporates the constitutive parameter 3

governing shear-thinning in steady laminar flow with no viscousdissipation. The simulations ultimately yield the fully developed

Nusselt number as well for the selected ducts of circular, square,rectangular and triangular cross sections. Numerical results for theaverage wall Nusselt number are presented and discussed in termsof the combined effects of elasticity represented by the Weissen-berg number WehluR and shear-thinning effects governed by theconstitutive parameter 3. The numerical techniquemakes use of theelasticeviscous split-stress (EVSS) finite element method [20].Computational results are validated against previous analytical andexperimental results when possible.

2. Mathematical equations and Nusselt number

The problem considered is that of the entrance flow in threedimensional ducts with symmetry of circular, square, rectangularand triangular cross section. A number of simulations were alsoperformed using for two-dimensional axisymmetric geometries forthe purpose of validating the modelling approach. The fluid isincompressible and obeys the differential exponential SPTTconstitutive model. Flow is steady and laminar. The field equationsread as,

vuivxi

¼ 0; (1)

v�rujui

�vxj

¼ �vpvxi

þ vsijvxj

; (2)

v�rcpujT

�vxj

¼ v

vxj

kvTvxj

!; (3)

ui, p, r, k, cp, T and sij represent the velocity vector, the pressure, thefluid density, the thermal conductivity, the specific heat, thetemperature field and the total extra-stress tensor, respectively. Thetotal extra-stress tensor sij is decomposed into a viscoelasticcomponent s1 and a purely viscous Newtonian component s2 in theelasticeviscous split-stress (EVSS) method,

s ¼ s1 þ s2; (4)

s2 ¼ 2m2D; (5)

s1 is computed differently for each type of viscoelastic model. In theSPTT constitutive model chosen for this work slip between themolecular network and the surrounding continuum is neglectedand the constitutive equation [12] for s1 reads as,

f s1 þ ls1V ¼ 2m1D: (6)

The upper convected derivative of the viscoelastic extra-stress is,

s1V ¼ Ds1

Dt� s1$Vui � VuTi $s1; (7)

and the rate of deformation tensor D and the exponential form ofthe function f [21] are defined as

Dij ¼12

vuivxj

þ vujvxi

!; (8)

f ¼ exp�

3l

m1trðs1Þ

�: (9)

The constitutive parameter 3 governs the elongational and theshear-thinning behaviour. l, m1, m2 stand for the relaxation time, andthe polymeric and solvent viscosities, respectively. The zero-shear

Table 1Mesh distribution for tested axisymmetric cases.

Geometry Mesh Nx Nr Number ofelements

Number ofnodes

2D Axisymmetricpipe

M1 75 10 750 836M2 120 15 1800 1936M3 300 20 6000 6416M4 1025 50 51,250 52,326

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e6052

viscosity is given by m0 ¼ m1 þ m2, where the polymeric viscosity m1was set to 0.00445 (Pa s). The ratio of Newtonian viscosity m2 to thetotal viscosity b which is defined as m2/m0 equals 1/9 which is theoften used value [22,23] and recommended by Polyflow 12.1 [24].The reasons behind this restriction lie with the outstanding unre-solved problem of High Weissenberg Number Limit (HWNL) facedin computational studies of viscoelastic fluid flow using any avail-able numerical method regardless the mesh size and type and theconstitutive equation used, differential or integral, and the type offlow simulated. Depending on the type of flow and the rheologicalequation of state used at a Weissenberg number sufficiently highbut different for each case numerical calculations brake down. Thisloss of convergence of the numerical algorithm at relatively lowmeasures of the fluid elasticity to be of any use in industrialapplications came to be known as the high Weissenberg numberproblem (HWNL). As early as 1985 Van Schaftingen and Crochet[25] showed analytically that for a solvent (Newtonian) to poly-meric viscosity ratio of m2/m1 � 1/8 and We > 0.3497 it is possiblethat infinite stresses and pressure gradients may manifest them-selves in points of the flow field inside the tube, expected to causeproblems for numerical simulations. The dilemma of HWNLremains unsolved although incremental progress continues to bemade [26] thus the requirement that b w 1/9 in numericalcomputations has taken into account the Newtonian contributionm2 to the denominator and given that m1 [ m2.

The peripherally averaged, but axially local, heat transfer coef-ficient hx is defined by

qw ¼ hxðTb � TwÞ; (10)

qw is the constant wall heat flux and Tw is the mean wall temper-ature. The fluid bulk temperature Tb at an arbitrary cross-section isdefined as [1]

Tb ¼ 1Acum

ZAc

u$T dAc; (11)

where um is the fluid mean axial velocity and Ac is the flow area.Hence, the peripheral average local Nusselt number is defined as,

Nux ¼ hxDk

¼ qwDkðTb � TwÞ

: (12)

3. Numerical technique and modelling strategy

3.1. Numerical technique

The computations reported in this work were performed usingPOLYFLOW 12.1 software. The elasticeviscous split-stress (EVSS)method splits the viscoelastic extra-stress-tensor sij into purelyviscous and elastic components, which are solved for separately.Combining both viscous and elastic components recovers the actual

Tinlet = constant

Fully developed velocity profile

u = 0, v = 0

r

x

Fig. 1. Geometry and mesh considered with bou

viscoelastic stress tensor. In 1994 the EVSS finite element methodincorporating the non-consistent streamline-upwind/Petrov-Galerkin technique known as the EVSS/SU was proposed by Debaeet al. [27] and found to be accurate and stable for viscoelastic flowproblems with non-smooth boundaries and is therefore used here.The EVSS combined with streamline up-winding (SU) technique isused to reach numerical convergence [28]. An appropriatenumerical strategy must be used to reach convergence in compu-tational non-linear viscoelastic fluid flow problems. In the presentstudy, an iterative evolution technique is used. For differential typeviscoelastic fluids the appropriate evolution is usually based oneither the relaxation time (l) or the flow rate Q providing gradualincreases in the values assigned to the Weissenberg numberWe. Inthis work the evolution technique is applied to the relaxation timesuch that li ¼ f(Si) � lnom, where lnom is the nominal relaxationtime and f(Si) ¼ Si ¼ (Si�1 þ DSi�1) where initial and final values ofthe S are S0 ¼ 0 and Sfinal ¼ 1 respectively and the initial value of DSis DS0 ¼ 0.01, with S representing the evolution variable. Thesolution of the non-linear problem is obtained using Newton’siterative scheme with the converged solution of the previous stepused as the initial guess when available. For example, for i ¼ 1, wehave l1 ¼ f(S1)� lnom, if the solution converges DS1 ¼ DS0 � 1.5 andS2 ¼ S1 þ DS1, then for i ¼ 2, l2 ¼ f(S2) � lnom. But if the solutiondiverges, DS10 ¼ DS1/2 and S2 ¼ S1 þ DS10 and the iteration is re-done until DSi is less than a minimum DSmin ¼ 1 � 10�4 and thesimulation stops. For Newtonian fluids considered in this paper allsimulations converge without recourse to the evolution function.But for non-linear viscoelastic fluids an evolution function wasnecessary to reach convergence except for very low values ofcombined elastic and elongational effects ( 3We2 ¼ 0.001). Thisparameter appears naturally in the analytical solution for fullydeveloped channel and pipe flow of SPTT fluids, see equations(13)e(15). The criterion of convergence based on the relative errorwas taken to be equal to 10�6. For each type of field, the modifi-cation at every node between two successive iterations is comparedto the maximum value of the field at the current iteration. Theglobal test is based on the highest relative variation for most fields.The number of iterations needed for convergence was 10. Thecomputations were performed on a workstation, using Intel (R)Xeon (R) CPU X5550 @ 2.67 GHz with 12 GB of RAM. The memorycapacity has been used to the maximum possible extent for the 3Dsimulations.

qw = constant or T = constant

Axis of symmetry

Outflow

ndary conditions for 2D axisymmetric case.

Fig. 2. Grid sensitivity test for 2D circular axisymmetric case: Nusselt number vs.normalized axial distance x0 for 3We2 ¼ 1.

Table 2Mesh distribution for 3D circular pipe geometry.

Geometry Mesh Nx Nr Nq Number ofelements

Number ofnodes

3D circular pipe M4 100 10 15 18,000 19,481M5 120 15 15 27,000 29,161M6 154 15 15 34,650 37,355

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e60 53

3.2. Geometries considered

The ducts considered in this paper for the thermally developingregion have circular and non-circular cross-sections. 2Dmeshes areused in axially symmetric cross-sections for validation andcomparison of the 3D simulations. For each selected cross-sectioncalculations are performed for both Newtonian and non-linearviscoelastic fluids of the SPTT type. The cross-sections used forboth 2D axisymmetric and 3D ducts considered are shown in Figs. 1and 3. Constant wall temperature and constant flux boundary caseshave been investigated.

The present results for Newtonian and SPTT fluids pertain tox0 > 10�5, where x0 ¼ x/D$Re$Pr is the normalized axial distancefrom the entrance cross-section. For most viscoelastic fluids thePrandtl number assumes values in the neighbourhood of Prw 50 orhigher. And because increasing Reynolds and/or Prandtl numberslead to an increase of the entrance length of the pipe a relativelylow value of the Prandtl number Pr ¼ 30 is used to optimize thecomputations. All the computations reported were conducted fora Reynolds number of 50. It has been numerically verified that theNusselt number Nu does not depend on Re with fluids obeying theSPTT model by computing Nu for different Re with all otherconditions kept the same thus confirming the findings of Siginerand Letelier [19,33] for the simplified or original version of the PTTmodel.

Grid sensitivity tests were conducted with different number ofcells to calculate the Nusselt number with two values of theparameter 3We2 of 0.001 and 1 for 2D axisymmetric cases. The

Walls

Planes of symmetry

Fig. 3. Computational domains for 3D simulations, x-axis is perpendicular to thefigure.

difference between the values of 3We2 of 0.001 and 1 is insignifi-cant. Grid sensitivity tests performed using non-uniform meshesare presented in Table 1 and the variation of the Nusselt numberwith the normalized distance x0 is plotted for different static gridmeshes in Fig. 2 for 3We2 ¼ 1. As the difference in the values of thefully developed Nusselt number between M2, M3 and M4 isinsignificant for both values of 3We2, mesh M2 is used in allsimulations to optimize the computation time.

Fig. 3 shows the static grid meshes used for all computationsand the boundary conditions for the cross-sections considered,circular, rectangular b/a ¼ 1/2 (n ¼ 2), triangular (n ¼ 3) and squareb/a ¼ 1 (n ¼ 4) with n representing the number of planes ofsymmetry. The size mesh for the different channel considered wasrefined and decreased gradually in the vicinity of channel wallsand, most importantly near the corners for the case of the trian-gular and rectangular channel cross-section. The hydrodynamicand thermal boundary conditions are illustrated in Fig. 1 with theregion between two planes of symmetry used for each cross-section allowing a much reduced computational domain andleading to improved grid resolution. For 3D geometries, again gridsensitivity tests were conducted for two values of the parameter3We2 of 0.001 and 1 on the variation of the fully developed Nusseltnumber with the normalized axial distance x0 are summarized inTable 2. Nx, Nr and Nq represent the number of cells in the axial,radial and tangential directions respectively. Grid sensitivity testswith non-uniform meshes for circular tubes 3We2 ¼ 1 are shown inFig. 4. The difference between meshes M5 and M6 is insignificantfor both values of 3We2. The comparison between the 3D and the 2Daxisymmetric cases shows some disagreement for 10�5 < x0 < 10�4

that is very near the duct entrance. Beyond a small neighbourhoodof the tube entrance the values are in excellent agreement all theway up to the fully developed region of the tube. Table 3 shows thedetails of the meshes used for selected non-circular ducts (Trian-gular, square b/a ¼ 1 and rectangular b/a ¼ 1/2). The solution of the3D problems investigated requires large memory space around 12Gbytes of RAM to store the stiffness matrix. The meshes used in thepresent study were the largest allowed by the computer memoryavailable.

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

Nu

x`

10

100

1000

1.E-05 1.E-04

Nu

x`

Zoom

M4 M5 3D pipeM62D Axisymmetric

Fig. 4. Grid sensitivity test for 3D circular pipe: Nusselt number vs. normalized axialdistance x0 for 3We2 ¼ 1 and comparison with 2D axisymmetric solution.

Table 3Mesh distribution for the 3D non-circular geometries.

3D geometries Nx Ny Nz Number ofelements

Number ofnodes

Triangular 120 15 15 21,120 24,200Square b/a ¼ 1 120 15 15 21,120 24,200Rectangular b/a ¼ 1/2 120 15 20 36,000 40,656

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e6054

4. Results and discussion

First the results of a series of validation calculations are pre-sented. These consider Newtonian and viscoelastic fluids inaxisymmetric and 3D circular geometries with constant heat fluxthermal boundary condition. Numerical results for non-circularducts are discussed next for both constant heat flux and constantwall temperature in terms of the variation of the average Nusseltnumber with the normalized axial distance and its fully developedvalue.

Fig. 5. Comparison of the fully developed velocity profiles for the axisymmetric case with thby three orders of magnitude.

4.1. Circular pipe geometries: velocity results

To validate the numerical computations and the globalapproach, numerical results for steady isothermal fully developedpipe flow of SPTT fluids are compared with the solution obtainedwith the semi analytical solution reported by Oliveira and Pinho[16] for the exponential version of the SPTT model. The numericalresults are obtained using 2D axisymmetric and 3Dmeshes. The 3Dmesh cell distributions are used to validate their further use forother non-circular cross sections investigated. The computationsare performed for values of the parameter 3We2 ranging from 0.001to 100. These correspond to equivalent values of We ranging from0.0632 to a maximum of 20 setting 3at a fixed value of 0.25. Theparticular choice of the parameter 3We2 to illustrate the effect ofelasticity follows the approach used by Refs. [16,17] and stems fromthe simple fact that it appears naturally in the analytic expressionsfor the normalized velocity profile and governs the flow in the caseof the fully developed flow in circular tubes derived by Refs. [15,16].In all the computations reported in this paper 3is set at 0.25, which

ose of Letelier and Siginer [15] and Oliveira and Pinho [16] for values of 3We2 that differ

Fig. 6. Fully developed dimensionless velocity profiles of the exponential SPTT modelas a function of the dimensionless group 3We2.

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e60 55

is typical for concentrated polymer solutions or polymer melts[23,28]. Typical values of We for viscoelastic fluids of concentratedsolutions in entry flows may reach values of the order of 16.84 [29]and the value of 4 [30] giving typical maximum values of 3We2 ofw100.

Normalized fully developed velocity profiles obtained from thepresent simulations and from Ref. [16] for the exponential versionof the SPTTmodel and fromRefs. [14,15] for the linear version of theSPTT model are shown in Fig. 5. The analytical non-dimensionalaxial velocity for the linear version of the SPTT model is given byEqs. (13) and (14) drawn from the work of Letelier and Siginer [15]and Oliveira and Pinho [16], respectively. u and uN represent theaverage velocity of the SPTT fluid and the average velocity of theNewtonian or the upper convected Maxwell fluids, respectively,

uðrÞu

¼ 63þ8 3We2

�1�ðr=RÞ2

��1þ2 3We2

�1þðr=RÞ2

��; (13)

uðrÞu

¼ 2�uN=u

��1�ðr=RÞ2

��1þ16 3We2

�uN=u

�2�1þðr=RÞ2��

:

(14)

The value of uN=u is calculated from the following equation

uNu

¼ð432Þ1=6

�d2=3 � 22=3

�6b1=2d1=3

;

bhð64=3Þ 3We2;

d ¼ a1=2 þ b; a ¼ 33bþ 4; b ¼ 33=2b1=2:

(14.a)

The semi-analytical non-dimensional axial velocity for the expo-nential version of the SPTT model is given by [16]

uðrÞu

¼ 2uNu

exp�b�uN=u

�2�b�uN=u

�2�1�exp

��b�uN=u

�2�1�ðr=RÞ2���

;

(15)

where uN=u is calculated from a non-linear equation solved bya straightforward but robust bisection method,

1 ¼ 2uNu

exp�b�uN=u

�2�b�uN=u

�2 1�

1� exp�� b�uN=u

�2�b�uN=u

�2!;

bh32 3We2:

(15.a)

An excellent agreement is shown for 3We2 varying from 0.001 to 0.1especially in the near wall region. A maximum difference in thecentreline region of 4.7% is observed between the present resultsand those of Oliveira and Pinho [16]. For 3We2 ¼ 1 the numericalresults appear to be closer to the analytical predictions of the linearSPTT model in the central core region. With growing 3We2 thedifference between numerical and analytical predictions growswith the maximum difference in each case on the centreline.Numerical simulations with the exponential SPTT model for values3We2 [ 1 differ from the analytical predictions for the exponentialSPTT model [16] by about 30% on the axis for 3We2 ¼ 10 and byabout 40% for 3We2 ¼ 100, numerical simulations always predictinga higher value for the velocity on the axis. The analytical velocitypredictions with the exponential SPTTmodel derived in [16] lead toa flatter velocity profile with growing 3We2. The predictions of theSPTT model with the linear form of the function f incorporating theconstitutive parameter 3consistently fall between the numericalsimulations and analytical predictions with the exponential form ofthe function f for 3We2 [ 1. There is no clear explanation of the

increasingly larger gap between the analytical and numericalpredictions with the exponential form of the embedded function f.Given that analytical results are correct, which no doubt they are,and that the ability of the mesh used to deliver accurate results hasbeen verified as well the only open avenue of speculation is theability of the POLYFLOW to deliver accurate results at highWe [ O(10). However we did not investigate this possibilityfurther.

Fig. 6 shows the velocity profiles obtained with the 2Daxisymmetric and 3D mesh for several values of the parameter3We2. To provide further validation and increase the confidence in3D simulations, the 3D calculationswere carried out and contrastedwith the 2D results. Clearly the fully developed velocity fields ob-tained with both 2D and 3D meshes are identical and therefore itcan be assumed that the 3D simulations with the mesh sizes usedprovide acceptable accuracy.

Fig. 7 illustrates the behaviour of the shear and normal stressesacross the circular pipe for the axisymmetric mesh [Fig. 7a(1) andb(1)] and for the 3D mesh [Fig. 7a(2) and b(2)] for a series of Wevalues at constant 3¼ 0.25 for the exponential SPTT model.

Fig. 7a(1) and b(1) shows a comparison with the analyticalresults of Oliveira and Pinho [16]. The numerical results for bothaxisymmetric and 3Dmeshes are exactly the same. The comparisonof the numerical values of the shear and normal stresses with theanalytical solution of [16] is in general quite good. For values of3We2 less than 1 the agreement is very good. A discrepancy existsfor larger values of the parameter and can reach a value of less than10% and 20% at the wall for the shear and normal stresses respec-tively. As plotted in Fig. 7b(1) and b(2) at constant 3 and low

a

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e6056

Weissenberg numbers normal stresses sxx increase, but for highelasticity that is for high We normal stresses decrease. Theanalytical solution predicts that for high We normal stressesdepend on We alone and sxx is proportional to We�1/3 hence con-firming the decrease of sxx observed in Fig. 7b(1) and b(2).

But if normal and shear stresses are scaled with the value of theshear stress at the wall the analytical solution given by equations(16) and (17) predicts that the non-dimensional ratio increasesmonotonically with We as shown in Fig. 8a.

b

Fig. 8. Normal stress sxx/(sxy)w and shear stress sxy/(sxy)w normalized with wall shearstress (sxy)w for 2D axisymmetric case of (a) Oliveira and Pinho [16] and (b) presentnumerical results.

a

b

Fig. 7. a(1). Normalized shear stress component, variation with 3We2 for 2D axisym-metric mesh. a(2). Normalized shear stress components, variation with 3We2 for 3Dcircular mesh. b(1). Normalized normal stress component, variation with 3We2 for 2Daxisymmetric mesh. b(2). Normalized normal stress components, variation with 3We2

for 3D circular mesh.

sxx�sxy�W

¼ 4kWe�uNu

�yH

�2; (16)

�sxy��

sxy�W

¼ yH: (17)

Fig. 9. Variation of the normalized friction coefficient Cf/CfN vs. the parameter 3We2 fordifferent geometries considered.

Fig. 12. Nusselt number vs. axial distance x0 ¼ x/D$Re$Pr under constant wall heat fluxqw for equilateral triangular cross-sections and 0.001 < 3We2 < 100. The final Nu valuesreported are those related to the fully developed region.

Fig. 10. Nusselt number vs. axial distance x0 ¼ x/D$Re$Pr under imposed positive heatflux qw for axisymmetric geometry and 0.001 < 3We2 < 100. The final Nu values re-ported are those related to the fully developed region.

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e60 57

The non-dimensional normal stress profiles obtained from thenumerical computations are shown to increase monotonically aspredicted by equations (16) and (17) for values of 3We2 less than 1and to decrease for higher values 3We2 ¼ 10 and 100 as shown inFig. 8b. These trends are expected from the behaviour of thevelocity profiles which become fuller with growing 3We2 when3We2 > O(1).

The skin friction coefficient Cf is defined as [1]

Cf ¼ sw12rðumÞ2

; (18)

where sw represents the mean wall shear stress. The theoreticalwall skin friction coefficient for the 2D axisymmetric case forexponential PTT model [16] is given by:

sxy4m1u=H

¼ ��uNu

�yH

�: (19)

The variation of the normalized skin friction coefficient Cf/CfN vs.the parameter 3We2 is presented in Fig. 9 for different geometriesconsidered where CfN represents the skin friction coefficient for theNewtonian fluid. Results indicate that the skin friction coefficientfor viscoelastic fluids is small compared with the Newtonian limit

Fig. 11. Nusselt number vs. axial distance x0 ¼ x/D$Re$Pr under imposed positive heatflux qw for 3D geometry and 0.001 < 3We2 < 100. The final Nu values reported arethose related to the fully developed region.

case and decreases monotonically with increasing 3We2, henceconfirming the decrease of the shear stresses presented in Fig. 7a(1)and a(2) induced by the shear-thinning behaviour. The comparisonof the numerical results with theoretical results of Oliveira andPinho [16] show a good agreement for very low values of theparameter 3We2 ranging from 0 to 0.1. For 3We2 � 1, the numericalresults differ from the theoretical predictions as expected from theshear stress profiles presented in Fig. 7a(1).

4.2. Circular tube: heat transfer results

Analytical results for the mean Nusselt number for the Graetzproblem with viscoelastic fluids and constant heat flux at theboundary in an axisymmetric pipe are used for further validation.Figs. 10 and 11 show the comparison between the calculated valuesand those reported by Coelho et al. [17] and Shah and London [32]for both axisymmetric and 3D meshes for both Newtonian andviscoelastic cases for 0.001 < 3We2 < 100. Numerically computedNusselt number decreases with increasing viscoelastic and elon-gational (shear-thinning) effects for 10 < 3We2 < 100 for bothaxisymmetric and 3D circular pipes in contrast to the results ofCoelho et al. [17]. The discrepancy observed for the 3D mesh for10�5 < x0<10�4 is probably due to the high axial gradients existingin this region and is limited to a small region at the entrance.

The results thus indicate that there is an enhancement in heattransfer as 3We2 increases up to a value of 3We2 between 1 and 10.

Fig. 13. Nusselt number vs. axial distance x0 ¼ x/D$Re$Pr under constant wall heat fluxqw for square cross-sections and 0.001 < 3We2 < 100. The final Nu values reported arethose related to the fully developed region.

Fig. 16. Constant wall temperature with fluid heating; comparison of Newtonian andviscoelastic Nu values over dimensionless distance x0 with Coelho et al. [17] for 3Dcircular geometry.

Fig. 14. Nusselt number vs. axial distance x0 ¼ x/D$Re$Pr under constant wall heat fluxqw for rectangular (b/a ¼ 1/2) cross-sections and 0.001 < 3We2 < 100. The final Nuvalues reported are those related to the fully developed region.

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e6058

In the absence of viscous dissipation as 3/0 the Nusselt numberasymptotically tends to the well known Newtonian value of 4.364for fully developed flow for allWe [31]. The constitutive parameter 3

imparts shear-thinning to the viscoelastic fluid provided theparameter is not too small. The effect of 3is negligible when it is ofthe order of O(10�2). The value of the parameter 3used in ourcalculations is 3¼ 0.25. Some light is shed on the counter case ofvery small elasticity We and substantial elongational capability3s 0 in Figs. 8 and 9. 3We2 ¼ 0.001 corresponds to We ¼ 0.063representing a dilute solution when 3¼ 0.25. The deviation of Nufrom its Newtonian value for the fully developed region of the pipewhen 3We2 ¼ 0.001 is only 0.3% thus providing further support tothe statement that the coupling of shear-thinning to elasticity is thecrucial governing factor and not the shear-thinning capability byitself.

4.3. Nusselt number for arbitrary cross-sections with constant heatwall flux

Three non-circular ducts, triangular, square and rectangularwere considered in the present work for the Graetz problem withnon-linear viscoelastic fluids. In all cases the meshes are three-dimensional and use the symmetry features of the geometryinvestigated. Figs. 12e14 depict the variation of the average Nusseltnumber for the three geometries considered. There is an excellent

Fig. 15. Constant wall temperature with fluid heating; comparison of Newtonian andviscoelastic Nu values with Coelho et al. [17] for axisymmetric geometry.

agreement between the computed fully developed value of theNewtonian Nu and the results available in the literature, providingconfidence in the accuracy of the numerical results [32]. The fullydeveloped viscoelastic value of the Nusselt number increases in therange 0.001 < 3We2 < 1. However for 10 < 3We2 < 100 the Nusseltnumber decreases following the same trend inherent to computa-tional results for axisymmetric and 3D circular pipe. When3We2¼ 1, the Nusselt number for the 3D circular and square ducts islarger than the Newtonian value by about 7.8% and 6.1% respec-tively. But for triangular and rectangular (b/a ¼ 1/2) ducts, theincrease in the Nusselt number is relatively small compared withthe previous geometries, about 3.2% and 4% from the Newtonianvalue, respectively. The principal reason behind rather small devi-ations from the Newtonian values of the Nu for the same cross-section is the lack of secondary flow field in the cross-sectionpredicted by the SPTT constitutive structure. For both circular andnon-circular geometries, secondary flows appear only for non-affine motion (x s 0, x represents the slippage factor in the PTTmodel). The simplified PTTmodel, x¼ 0, does not predict secondaryflows [19,33]; in the latter case particle paths are straight andparallel along the duct for all duct cross-sections considered.

The fully developed Nu values reported in Figs. 12e14 asa function of the governing parameter 3We2 as well as the Nudistribution in the entrance region of the equilateral triangular,square and rectangular cross-sectional tubes are reported to ourknowledge for the first time in the literature.

Fig. 17. Constant wall temperature with fluid heating; Nusselt number vs. dimen-sionless distance x0 for equilateral triangular duct.

Fig. 18. Constant wall temperature with fluid heating; Nusselt number vs. dimen-sionless distance x0 for square duct.

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e60 59

4.4. Constant wall temperature condition for pipe flow

This section presents preliminary limited calculations for thecase of constant wall temperature and a comparison with previousresults available in the literature. The wall temperature is fixed andconsidered to be greater than the inlet flow temperature Tw > T0.The geometries considered are 2D axisymmetric pipe, 3D circularpipe, triangular, square (b/a ¼ 1) and rectangular (b/a ¼ 1/2). Thevalidation of the numerical results is considered first. The numer-ical results are compared with the solution for the axisymmetriccase reported by Coelho et al. [17] for Newtonian and the expo-nential version of the SPTT model. For thermally fully developedaxisymmetric flow and for 3We2 ¼ 1, Nusselt number increases by7.56% from the Newtonian value. Fig. 15 shows the comparison ofthe variation of the Nusselt number with the normalized distancex0. The agreement in both cases is quite good.

A good agreement is obtained for the value of the Nusseltnumber in the fully developed region for the 3D pipe geometry asshown in Fig. 16 with a difference of 2.98% and 2.24% for theNewtonian and viscoelastic cases, respectively, between thepresent results and those reported by Coelho et al. [16] with somediscrepancy in the region 10�5 < x0 < 10�3 with the axisymmetriccase most likely due to meshing in this region. For the non-circulargeometries investigated for 3We2 ¼ 1 Nusselt number increases by9.337%, 6.95% and 4.79% from the Newtonian value for the trian-gular, square and rectangular (b/a ¼ 1/2) ducts, respectively.Figs. 17e19 represent the variation of the normalized Nusselt

Fig. 19. Constant wall temperature with fluid heating; Nusselt number vs. dimen-sionless distance x0 for rectangular duct (b/a ¼ 1/2).

number for constant wall temperature boundary condition for the3D triangular, square and rectangular ducts respectively.

5. Conclusions

The Graetz problem for laminar flow of Newtonian and visco-elastic fluids described by the simplified affine version of the Phan-Thien Tanner model with the exponential form of the function frepresenting elongational (shear-thinning) effects in ducts ofcircular, triangular, square and rectangular cross sections is solvednumerically for constant heat flux and constant temperatureboundary conditions in the absence of viscous dissipation. Thesolution presented is compared with previous work available in theliterature and discussed in terms of the combined shear-thinningand elastic effects represented by the parameter 3We2 and thedimensionless axial distance x0.

For the geometries considered, the fully developed value of theNusselt number obtained for the Newtonian cases is in goodagreement with those available in the literature. The computationalviscoelastic resultswith SPTTmodel are also in goodagreementwiththose available in the literature up to a value of the governingparameter O(1) < 3We2 < O(10) for round tubes. However for3We2>O(10) the discrepancywith existing results for axisymmetrictubes grows with growing 3We2 reaching a maximum for3We2wO(102) in the range investigated. The velocity profiles for theexponential form of the function f for the SPTT model differ mark-edly from those predicted by the analytical solution. The latterbecomes gradually fuller with growing 3We2 with larger shear ratesnext to thewall. As a result the heat transfer rates obtained from thenumerical and analytical solutions in round tubes show opposingtrends decreasing and increasing, respectively, for 3We2 > O(10)although they do agree up to a value of the governing parameterO(1) < 3We2 < O(10). The mesh used in the computations has beenextensively studied and optimized by comparing 2D and 3D resultsand by comparison with existing results in the literature as well asdescribed above. Thediscrepancy in the velocity profiles aswell as inthe heat transfer rates is independent of themesh and lies primarilywith the difficulty to simulate flows with high elasticity We. Thesefindings confirm the present capability and limitation of traditionalmethods as those included in Polyflow. On the other hand, it is quitepossible to overcome such limitations at higher values of We withpseudo-spectral based techniques of integration [34,35].

Against this backdrop, the fully developed value of the Nusseltnumber for flow in circular, triangular, square and rectangular ductsis obtained fora large rangeof theparameter 0.001< 3We2<100. Theresults for the non-circular cross-sections investigated are obtainedfor the first time for polymeric liquids with SPTT type of constitutivestructure. In the light of the observations made above completeconfidence can be placed in the heat transfer rates for 3We2 < O(10).Howeveras in the caseof the round tubes the results for 3We2>O(10)show a decreasing trend under constant boundary flux. The resultsfor 3We2 > O(10) have thus to be treated with caution.

A limited set of results for the heat transfer rate when 3We2 ¼ 1are also presented for the constant wall temperature case in allnon-circular cross-sections investigated. As the value of the gov-erning parameter lies in the confidence range determined throughabove considerations and computations we think that the reportedheat transfer rates are trustworthy as well as a first with thesecross-sections.

Nomenclature

r fluid density (kg/m3)cp specific heat (J/kg K)k thermal conductivity (W/mK)

A. Filali et al. / International Journal of Thermal Sciences 61 (2012) 50e6060

Re Reynolds number ðruD=mÞNu Nusselt number (hD/k)T temperature (K)q heat flux (W/m2)r radial coordinate (m)R tube radius (m)D pipe diameter (m)x axial coordinate (m)x0 normalized axial coordinate (x/DRePr)u velocity (m/s)u average velocity (m/s)We Weissenberg number ðlu=RÞPr Prandtl number (mcp/k)Cf skin friction coefficientCfN skin friction coefficient of Newtonian fluid

Greek symbolsm fluid viscosity (Pa s)q non-dimensional temperature3 model parameterl relaxation time (s)s stress tensor

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