Forced convection heat transfer in tube banks in cross flow

Chemical Engineering Science57(2002)379–

391

https://www.360docs.net/doc/fb15511101.html,/locate/ces

Forcedconvection heat transfer in tube banks in cross ow

V.K.Mandhani a,R.P.Chhabra a;?,V.Eswaran b

a Department of Chemical Engineering,Indian Institute of Technology,Kanpur208016,India

b Department of Mechanical Engineering,Indian Institute of Technology,Kanpur208016,India

Received11April2001;receivedin revisedform27August2001;accepted18September2001

Abstract

The forcedconvection heat transfer characteristics for an incompressible,stead y andNewtonian uid ow over a bund le of circular cylinders has been investigated numerically.The inter-cylinder hydrodynamic interactions have been approximated by employing a simple cell mod el.The momentum andenergy equations have been solvedby using a?nite d i erence basednumerical solution proced ure for a range of physical and kinematic conditions.Furthermore,the role of the type of thermal boundary condition,namely,a constant temperature or a constant heat ux,imposed on the surface of the cylinder has also been elucidated.Extensive results on the temperature?elds,and on the variation of the Nusselt number on the surface of a typical cylinder in the assemblage have been obtained for two values of the Prandtl number(correspond ing to air andwater).The Reynold s number of ow was variedin the range1–500andthe void age of the assemblage rangedfrom0.4to0.99thereby covering the entire range of interest as encounteredin tubular heat exchangers andin?brous bed s.The paper is conclud edby presenting extensive comparisons with the limitedanalytical=numerical and=or experimental results available in the literature for the case of a single cylinder as well as that for tube bundles.?2002Elsevier Science Ltd.All rights reserved.

Keywords:Nusselt number;Tube bundles;Heat transfer;Cell model

1.Introduction

The ow of uids and heat transfer in tube banks represents an idealization of many industrially important processes.Typical examples include?ltration, ow in bi-ological systems,tubular heat exchangers, ow andheat transfer in?brous media as encountered in polymer process-ing and in insulation materials,etc.Additional examples where this ow con?guration is of relevance include ow in the uid izedbedd rying of?brous materials(such as coconut shell,rice husk)andpaper pulp suspensions andin foodprocessing applications(Kiljanski&Dziubinski,1996; Mauret&Renaud,1997).Notwithstanding the importance of the d etailedkinematics of the ow andtemperature?eld s, it is readily agreed that the variables of central interest in all these applications are the frictional pressure gradient and the convective heat transfer coe cient(or the rate of heat transfer)as functions of the pertinent system variables.Con-sequently,over the years,considerable research e ort has

?Corresponding author.Department of Engineering,Indian Insti-tute of Technology,Kanpur208016,India.Tel.:+91-512-597393;fax: +91-512-590007.

E-mail address:rpc@iitk.ernet.in(R.P.Chhabra).been expended in developing satisfactory methods for the prediction of pressure drop for the ow of incompressible Newtonian uids in cross- ow con?guration over a collec-tion of circular cylinders,as can be gauged from the number of books andsurvey articles in this?eld(e.g.,Zukauskas, 1987a,b,C hap.6;Drummond&Tahir,1984;Nishimura, 1986;Satheesh,Chhabra,&Eswaran,1999;Shibu,Chhabra, &Eswaran,2001).Perhaps it is fair to point out here that a bulk of the literature relates to momentum transfer or the es-timation of frictional pressure drop incurred during the cross ow of uids over tube banks in the low Reynolds number region.This work addresses the question of the prediction of the Nusselt number as a function of the Reynolds and Prandtl numbers for a range of voidages of tube banks.It is,however,instructive anduseful to brie y summarize the pertinent studies available in the literature.

2.Previous work

Over the years,considerable research e ort has been expend edin investigating the forcedconvection even from a single long cylinder,and an exhaustive compilation of such studies has been given by Ahmad(1996),though it

0009-2509/02/$-see front matter?2002Elsevier Science Ltd.All rights reserved. PII:S0009-2509(01)00390-6

380V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391

only relates to the ow of air,i.e.,the Prandtl number of about0.7;subsequently,more accurate results for this ow con?guration have been reportedby Lange,Durst,and Breuer(1998)for the ow of air over a heatedcylind er. In contrast,the analogous problem of forcedconvection from tube bundles in cross- ow con?guration has been studied much less extensively,both experimentally as well as analytically=numerically.From a theoretical standpoint, a mathematical description of the cylinder–cylinder interac-tions is needed in addition to the governing equations of con-tinuity,momentum andenergy.C urrently,there are two d is-tinct approaches available to incorporate this feature into an analysis.In the?rst scheme,the momentum andenergy equations are solvedanalytically and=or numerically for periodic arrays with di erent geometrical arrangements. For instance,Laund er andMassey(1978)reportedlim-itedresults on pressure d rop andheat transfer in ow over a staggeredarrangement of circular cylind ers with equal transverse andlongitud inal pitch andreportedsatisfactory agreement with the experimental results of Bergelin,Brown, andDoberstein(1952).However,the main thrust of their study was on the numerical aspects of the solution proce-dure for a Prandtl number of20and the Reynolds number in the range of50–75.In a two-part paper,Wung andC hen (1989)andC hen andWung(1989)have re-investigated this problem by considering the steady and incompressible ow over staggeredandin-line arrays of circular cylind ers for the ratio of the longitudinal to transverse pitches of two. They solvedthe Navier–Stokes andthe energy equations for the Reynold s numbers of40,120,400and800andfor the Prandtl number values of0.1,1and10.The motivation of including the rather small value of Prandtl number of0.1 is not immediately obvious.While the major thrust of their study was on the elucidation of the evolution of the ow ?eld(stream lines,second ary ow,etc.)with the increasing Reynolds number of ow,they(Chen&Wung,1989)did present limitedresults on the mean Nusselt number as a function of the Reynolds number and Prandtl number.The average Nusselt number was shown to scale as Re Pr?where rangedfrom0.40to0.45and?varieda little in the range 0.37–0.38for the in-line andstaggeredarrays,respectively. Conversely,while the actual geometrical arrangement of an array might exert a strong in uence on the detailed kinemat-ics of ow,the gross heat transfer characteristics seemed to be little e ectedby such d etails of an array.More re-cently,Martin,Saltiel,andShyy(1998)have numerically stud iedthe frictional losses andconvective heat transfer in sparse periodic(square and triangular)arrays of cylinders in cross- ow(voidage?0:8).They have also reportedthe surface averagedvalues of Nusselt number for the ow of air andfor the values of Reynold s number ranging from3 to160.Furthermore,they also examinedthe e ect of the type of the thermal boundary condition,i.e.,a constant tem-perature or constant heat ux imposedon the surface of the cylinder.As expected,the Nusselt number for the constant heat ux condition tends to be somewhat higher than that under the constant temperature boundary condition,though the di erence between the two values is strongly dependent on the porosity of the array andthe Reynold s number.Fur-thermore,this study also shows that the dependence of the Nusselt number on the Reynolds number progressively be-comes weaker with increasing porosity,i.e.,as the distance between the cylinders increases.Likewise,Wilson and Bassiouny(2000)have numerically studied heat transfer between air andin-line as well as staggeredarrays com-prising two rows of cylinders in fully turbulent regime (Reynolds number upto50,000).They employed the con-stant temperature condition on the surface of the cylinders. In addition to these numerical studies,Sangani and Acrivos (1982)presentedan analytical=numerical solution for Nus-selt number which is applicable in the low Reynolds number and Prandtl number region.Such studies tend to be not only highly computational intensive andmust be repeatedfor a range of values of the geometrical details(pitch and type of arrangement)of the arrays to embrace the wide range of porosities encounteredin process applications but are also of little practical utility owing to the limitation of the low Reynolds number.

The second approach to the modeling of the cylinder–cylinder interactions relies on the use of the so-called cell models.In this approach,the viscous interactions between the uidandthe tube bank are mimickedby enclosing each cylinder into a hypothetical concentric cylindrical enve-lope of the uidof the size such that the void age of each cell is equal to the overall mean voidage of the assem-blage.Amongst such models,two cell models,namely,the free surface andthe zero vorticity cell mod els(Happel& Brenner,1965)seem to have gainedwid e acceptance in predicting macroscopic transport phenomena in concen-tratedsystems.Thus,for instance,both the sphere-in-sphere and the concentric cylinders versions of such cell models have been shown to yieldsatisfactory pred ictions of fric-tional losses for the ow of Newtonian andnon-Newtonian uids in beds of spherical and?brous beds as well as in a collection of circular cylinders up to moderate values of the Reynolds number(Kawase&Ulbrecht,1981a,b;Tri-pathi&Chhabra,1992,1996;Chhabra,1993a;Satheesh et al.,1999;Vijaysri,Chhabra,&Eswaran,1999;Dhotkar, Chhabra,&Eswaran,2000;Chhabra,Dhotkar,Eswaran, Satheesh,&Vijaysri,2000;Chhabra,Comiti,&Machac, 2001;Shibu et al.,2001).Similarly,this approach has also been shown to yieldreasonable estimates of the free rise velocity of bubble swarms in Newtonian andpower-law type non-Newtonian uids in the creeping and the high Reynolds number ow regimes(Gummalam&Chhabra, 1987;Chhabra,1998;Zhu,2001).The range of applica-tions shows the extremely versatile nature of this simple approach.The two cell models are exactly identical to each other except for one boundary condition at the surface of the hypothetical outer cylinder.While the condition of the zero shear stress(frictionless)is prescribedin the free surface cell model(Happel,1959),Kuwabara(1959)suggested

V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391381

the vorticity to vanish at the hypothetical uidcylind er in the zero vorticity cell model.Though extensive discussions on the relative merits and demerits of both these models are available in the literature(Happel&Brenner,1965; Chhabra,1993a),detailed comparisons between theoretical predictions and experimental results reveal the experiments to lie somewhere in between the predictions of these two models(Chhabra,1993b).In contrast to the?rst approach of periodic arrays,the cell approach,though perhaps less rigorous,allows reasonable predictions of macroscopic transport parameters not only with much less computational requirements but also over the entire range of porosity,from the case of sparse to highly dense arrays.The concentric sphere-in-sphere free surface cell model has been shown to give acceptable predictions of mass transfer in the region of low Reynolds and high Peclet numbers(Pfe er,1964;Pf-e er&Happel,1964).As far as known to us,only Le Clair andHamielec(1968)have employedthe zero vorticity cell model to study forced convection mass transfer between air andan assemblage of cylind ers.They invokedthe thin concentration boundary layer approximation togetherwith some other simpli?cations(such as neglecting the angular di usion term,and assuming(@2T=@r2) (1=r)(@T=@r), etc.)andobtainedapproximate numerical results basedon a linear velocity pro?le.They reportednumerical values of the point andmean values of the Nusselt number for mass transfer andcorrelatedtheir results in the form of a j-factor for mass=heat transfer.They suggestedthese results to be applicable for heat transfer by noting the equivalence of the Prandtl and Schmidt numbers.While there is no justi?cation for this analogy to apply over wide ranges of conditions,these appear to work satisfactorily at low rates of heat andmass transfer when the change in surface area due to mass transfer is negligible.As far as known to us, there has been no other study on heat transfer using the cell model approach.In contrast to the aforementioned limited analytical=numerical studies,the literature is inundated with experimental results andempirical expressions covering a wide variety of tube con?gurations and ow conditions (Reynolds and Prandtl numbers).Most of these have been summarizedby Ad ams andBell(1968),Nishimura(1986), Zukauskas(1987a,b,Chap.6),Nishimura,Itoh,Ohya, &Mirashita(1991)andNishimura,Itoh,&Miyashita (1993).

From the foregoing description,it is thus abundantly clear that virtually no theoretical=numerical predictions are available of forcedconvection heat transfer between a uid andan assemblage of heatedcircular cylind ers within the framework of the cell models.In this paper,new extensive results basedon the free surface cell mod el are presentedfor the Nusselt number as a function of the Reynolds number of ow(1–500),Prandtl number(0.71and7.7,correspond-ing to air andwater,respectively)andover wid e ranges of voidage(0.4–0.999).Furthermore,in some cases the e ect of the type of the thermal boundary condition(constant temperature or constant heat ux)imposedon the surface of the cylinder is also elucidated.The paper is concluded by presenting detailed comparisons with the previous the-oretical andsuitable experimental results available in the literature.

3.Problem statement,idealisation and governing equations

Consider the steady incompressible ow of a Newtonian uidwith constant thermo-physical properties(d ensity , viscosity ,thermal conductivity k,heat capacity C p)over a collection of circular cylinders which are oriented normal to the direction of ow(Fig.1a).The free surface cell model envisions each cylinder of radius(r o)to be surrounded by a hypothetical concentric cylindrical envelope of radius r∞, as shown schematically in Fig.1(b).A cylindrical coordi-nate system?xedat the center of the cylind er is convenient, and?is measuredfrom the front stagnation point as shown in Fig.1(b).The uidis approaching the tube bund le with a constant anduniform velocity U andat a uniform temper-ature T o.Owing to the two-dimensional nature of the ow, the ow variables are functions of r and?only.

The governing equations thus comprise the continuity equation,r-and?-components of the Navier–Stokes equa-tions andthe energy equation(in the absence of viscous dissipation)which in their dimensionless forms are written as follows:

Continuity equation

1

r

@

@r

(rV r)+

1

r

@V?

@?

=0;(1) r-Component

@V r

@t

+

1

r

@

@r

(rV2r)+

1

r

@

@?

(V r V?)?

V2?

r

=?

1

2

@p

@r

+

2

Re

@

@r

1

r

@

@r

(rV r)

+

1

r2

@2V r

@?2

?

2

r2

@V?

@?

;(2a)?-Component

@V?

@t

+

1

r

@

@?

(V2?)+

1

r

@

@?

(V r V?)+

V r V?

r

=

1

2r

@p

@?

+

2

Re

@

@r

1

r

@

@r

(rV?)

+

1

r2

@2V?

@?2

+

2

r2

@V r

@?

;(2b)

382V.K.Mandhani et al./Chemical Engineering Science57(2002)

379–391

Fig.1.Schematics andmod el representation of ow. Energy Equation

@T @t +

1

r

@

@r

(rV r T)+

1

r

@

@?

(TV?)

=

2

RePr

1

r

@

@r

r

@T

@r

+

1

r2

@2T

@?2

:(3)

In Eqs.(1)–(3),the velocity components have been scaled using the free stream velocity U,the radial distance using r o andthe pressure term as(p?p )=( U2=2).The temperature T has been scaledeither as{(T?T o)=(T s?T o)}when the surface of the solidcylind er(r=1)is maintainedat a uni-form temperature of T s(referredto hereafter as C ase I)or as (T?T o)=(q w r o=k)when a constant heat ux,q w,is imposed at r=1(referredto hereafter as C ase II).The two d imen-sionless multipliers,namely,the Reynolds number and the Prandtl number are de?ned as follows:

Re= Ud

(4)

and

Pr=C p

k

;(5)

where d=2r o is the diameter of the solid cylinder.

The physically realistic boundary conditions in the frame-work of the free surface cell model are those of no-slip at r=1andthe frictionless surface at r=r∞,i.e.,

At r=1

V r=0;V?=0(6a) T=1(Case I)(6b)

or @T

@r

=?1(Case II)(6c)

At r=r∞

V r=?cos?; r?=0(6d)

T=0for C ase I andC ase II(6e) In addition,the conditions of symmetry of ow and temper-ature were also imposedat?=0and?= planes.

As mentioned previously,the radius of the outer cylindri-cal cell boundary is related to the size of the cylinder(r o) andthe porosity(e)of the assemblage through the simple relation as

r∞=(1?e)?1=2:(7) Thus,by simply varying the value of r∞,assemblages of

di erent porosities can be simulated.The two limiting cases of e=1and e=0,respectively,correspondto the ow over a single cylinder and no ow situation(zero ow area). Thus,Eqs.(1)–(3)together with the boundary conditions outlinedin Eq.(6)provid e the theoretical framework for mapping the entire ow domain(16r6r∞;06?6 ) in terms of the velocity andtemperature?eld s as V r(r;?);V?(r;?)and T(r;?).These in turn can be processed further to deduce the values of global quantities such as pressure andfrictional components of the uidd ynamic drag on the tube bundle and the Nusselt number as functions of the Reynolds and Prandtl numbers,the voidage and the type of the thermal boundary condition used,i.e.,Case I and Case II.From a knowledge of the temperature?eld,it can readily be shown that the local value of the Nusselt number at r=1for the two cases is simply obtainedas follows: Case I:Nu?=

hd

k

=?2

@T

@r

r=1

;(8) Case II:Nu?=

2

T

r=1

(9)

andthe surface averagedvalue of the Nusselt number is obtainedsimply as

Nu=

1

Nu?d?:(10) While extensive results on the velocity andpressure?eld s, and drag coe cient for wide ranges of the Reynolds number (16Re6500)andvoid age(0:46e60:99)have been reportedelsewhere(Satheesh et al.,1999;C hhabra et al., 2000;Shibu et al.,2001),the results for the temperature?eld as well as of the local andmean Nusselt numbers are reported in this paper.In essence,the velocity?eldobtainedby Shibu et al.(2001)was usedas the input,except for a few test cases to cross check the accuracy of the solution procedure employedby Shibu et al.(2001).The salient features of the numerical solution proced ure andthe choice of gridsize, etc.for the energy equation are described brie y in the next section.

4.Numerical solution procedure

The governing equations solvedin this work includ e the equations of continuity,momentum andenergy,i.e., Eqs.(1)–(3)subject to the boundary conditions outlined in Eq.(6).The uid ow equations,Eqs.(1)and(2)have been solvedusing a SMAC-type?nite d i erence algorithm that was made implicit for the viscous terms.This leads to improvednumerical stability at low Reynold s numbers, as detailed elsewhere(Shibu et al.,2001).The velocity ?eld s are obtainedon a staggeredgridusing a two-step

V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391383

predictor=corrector methodthat(a)uses a“guessed”pres-sure?eld to obtain“predicted”values of the velocity?eld from the Navier–Stokes equations,andthen(b)corrects the velocity andpressure?eld s to ensure their compliance with the continuity equation.The viscous terms are approximated by using the center di erence scheme while the convec-tive terms are d iscretisedby the upwindscheme.The“pres-sure”usedis actually a pseud o-pressure which just enforces the continuity.The use of the pseudo-pressure with simple boundary conditions(i.e.,homogeneous Neumann boundary condition at solid boundaries)is easier than that of the true pressure with its complicated boundary conditions.How-ever,once the fully converged steady-state velocity?elds are obtained,the true pressure?eldcan be computedby solv-ing a Poisson equation with the correct boundary conditions consistent with the Navier–Stokes equations.

Once the steady-state velocity and pressure?elds are available,the energy equation is solvedon a staggeredgrid to obtain the temperature?eldby employing a false-transient time stepping algorithm with a fully implicit center di er-ence scheme for both convective andd i usion terms.The system of equations is solvedusing a Gauss–Sied el method with successive over-relaxation.The boundary conditions in all cases are implementedusing?ctitious cells at the bound-aries.

5.Results and discussion

In this study,the thermal energy equation has been solved numerically in the range of conditions as:16Re6500; 0:46e60:99andfor two values of the Prand tl number (Pr=0:71and7.7)correspond ing to the ow of air andof water,respectively,for the two types of thermal boundary conditions.In particular,results are obtained to elucidate the role of porosity,Reynolds and Prandtl numbers on the local andsurface averagedvalues of the Nusselt number andon the temperature?eld.It is,however,desirable?rst to estab-lish the accuracy andreliability of the numerical solution procedure used here which in turn facilitates the delineation of the accuracy of the new results presentedherein.

5.1.Validation of results

The numerical solution procedure employed herein has been validated by comparing the present predictions with the previous analytical and=or numerical results available in the literature.While detailed discussions on the accuracy of the solution proced ure employedfor the momentum andcon-tinuity equations are available elsewhere(Satheesh et al., 1999;Chhabra et al.,2000;Shibu et al.,2001),the relia-bility andaccuracy for the solution of the energy equation is establishedhere by comparing the present values with the literature values for a single cylinder and for a collec-tion of cylinders within the framework of the zero vortic-ity cell model.In this work,the case of the ow over a single cylinder was simulated by setting e=0:999which makes the radius of the outer hypothetical uid envelope30 cylinder radii.In a recent paper,Lange et al.(1998)have studied the convective heat transfer to air(Pr=0:715)from a heatedcylind er maintainedat a constant temperature.In the range Re6200,they correlatedtheir numerical results as follows.

Nu=0:082Re0:5+0:734Re x;(11a) where

x=0:05+0:226Re0:085:(11b) The present values of the average Nusselt number of1.86 and4.96for Re=10and Re=100for e=0:999compare very well with the corresponding values of1.82and5.1, respectively,predicted using Eq.(11).The discrepancies of this order between various predictions are indeed not un-common in the literature.Thus,for instance,Ahmad(1996) cites the values of the mean Nusselt number in the range 4.816–5.166for Re=100for the ow of air over a heated cylinder whose surface is maintained at a constant temper-ature.Thus,the present value of Nu=4:96lies well within this range.Furthermore,small di erences can also be as-cribed to the slightly di erent values of Prandtl number used by di erent investigators.

Similarly,Le C lair andHamielec(1968)have stud ied convective mass transfer between a tube bundle and air (Pr=0:74)using the zero vorticity cell model and these were assumedto be applicable for heat transfer also.With this proviso andbearing in mindthe approximations and simpli?cations inherent in their analysis(as described pre-viously),they correlatedtheir results in the form of the j factor as

j H e0:833=1:05Re?0:582(12) where

j H=

Nu

RePr1=3

(13)

A few results have been obtainedhere with the zero vor-ticity boundary condition for Pr=0:74anda comparison between the present andtheir values(Le C lair andHami-elec,1968)is shown in Table1for type I thermal boundary condition at r=1.An examination of these results reveals qualitative agreement between the present values andthose of Le C lair andHamielec(1968).The d iscrepancy between the two values is seen to somewhat vary with the voidage andthe Reynold s number,albeit the two values are gen-erally within20–25%of each other.The correspondence between the two improves somewhat with the increasing val-ues of e andRe.In assessing the comparison shown in Table 1,it must be borne in mindthat the results of Le C lair and Hamielec(1968)are basedon the approximate andsimpli-?edform of the energy(continuity)equation andsecond ly, the errors associatedwith Eq.(12)are also not known.On the other hand,the present results have been checked for

384V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391

Table1

C omparison between present results andthose of Le C lair andHamielec (1968)for Pr=0:74for zero vorticity cell model

e Re Nu

Present Eq.(12) 0.9950 4.69 4.91 0.40107.99 5.33

509.3310.45

10011.7813.97

0.5010 5.94 4.42

507.648.68

10010.0511.50

0.6010 4.56 3.81

50 6.517.46

1008.639.96

gridind epend ence andalso are basedon the solution of the complete energy equation.In view of the above-noted factors,the present results are believedto be more accurate than those of Le C lair andHamielec(1968).

Basedon the aforementionedcomparisons andon previ-ous experience(Satheesh et al.,1999;Chhabra et al.,2000; Shibu et al.,2001),the new results reportedherein are believedto entail an uncertainty of not more than1%or so.

5.2.Variation of Nusselt number

Figs.2and3show representative results on the e ect of porosity,Reynolds number and the type of thermal boundary condition on the variation of the local Nusselt number over the surface of the solidcylind er.From an examination of these?gures,the following overall trends can be observed.

(a)The extent of variation of the Nusselt number on the

surface of the cylinder r=1is determined by a com-plex interplay between the kinematic(Re;Pr;Pe)and the physical characteristics(e)of the system.At small values of Peclet number,one wouldintuitively expect the value of Nu?to show virtually no or small varia-tion as the surface of the cylinder is traversed from the front(?=0)to the rear(?= )stagnation points.This is so due to the fact that at low values of Re and=or Pe, heat transfer occurs primarily by conduction with a very small contribution from convection.Irrespective of the type of thermal boundary condition satis?ed at r=1, this behavior was observedup to about Pe～1.For in-stance,while for Pe=0:7,the value of Nu?at r=1varied by less than2–3%in the interval0:406e60:99,the corresponding variation for Pe=7was seen to range from～15%at e=0:4to～50%at e=0:99for the isothermal cylinder case.Qualitatively,similar trends are present for the case of the constant heat ux thermal boundary condition at r=1.(b)As the value of Peclet number is progressively

increasedby increasing either the value of the Reynold s number or the Prandtl number or both,the contribution of convection gradually rises.Under these conditions, the Nusselt number is seen to be maximum at the front stagnation point andit progressively d ecreases with displaying a distinct minimum in concentrated systems (small values of e,e.g.,see Figs.2a and3a)whereas the Nusselt number is seen to decrease monotonically in sparse systems(high values of e).In dense sys-tems(e～0:4–0.5),the minimum value of the Nusselt number is seen to occur in the rear of the cylinder,?～140–150?.At very high values of Peclet num-ber,the local Nusselt number exhibits another local peak(as seen clearly in Figs.2b,c and3b,c)in the rear region of the cylinder,following which its value again decreases.Qualitatively similar trends are seen to be present when the constant heat ux condition is appliedon the cylind er surface insteadof the constant temperature condition.Undoubtedly,the range of pos-sibilities seen here re ect a complex interplay between the pertinent variables.

(c)The value of the local Nusselt number is seen to de-

crease with increasing porosity andwith d ecreasing value of the Prandtl number.(see Figs.2a–c and3a–c) This is understandable since the prevailing velocity andtemperature grad ients are strongly in uencedby porosity,being smaller in sparse systems andsteeper in dense systems.The observed e ect of Prandtl number is explainedby the d ecreasing ratio of the thickness of the thermal boundary layer to that of the momentum boundary layer.As the value of the Reynolds number increases,the wake region also grows in size thereby enhancing heat transfer under these conditions.

Representative temperature pro?les are shown in Figs.4 and5where isotherms are shown for a range of combi-nations of porosity,Reynolds number and Prandtl number. In these?gures,the isotherms are plottedat an interval of 0.05.The fore andaft symmetry of these plots at low Peclet numbers(showing the dominance of conduction over con-vection)was seen to be destroyed progressively as the value of the Reynolds or Prandtl numbers or both is gradually in-creased.However,smaller is the value of porosity,higher is the value of Peclet number upto which such a symme-try persists.This observation is qualitatively consistent with our previous?ndings regarding the limiting value of the Reynolds number marking the cessation of creeping ow (Jaiswal,Sundararajan,&Chhabra,1993;Satheesh et al., 1999).Furthermore,as the value of the Reynolds number or of Prandtl number or of both increases,more complicated patterns appear in the re-circulation zone in the rear endof the cylinder,whereas the isotherms are seen to be little in u-encedin the front region.The size of the re-circulating zone itself grows rapidly with the increasing value of voidage (e),e.g.,see Figs.4a,b and5a,b.Also,much sharper tem-

V.K.Mandhani et al./Chemical Engineering Science 57(2002)379–391

385

Fig.2.Variation of local Nusselt number on the surface of a representative cylinder in the bundle for Case I:(a)Re =20;Pr =0:7;(b)Re =100;Pr =0:7;(c)Re =500;Pr =0:7.

perature gradients are seen to appear closer to the cylinders as the Reynolds number increases.As inspection of Figs.4and5clearly shows a markedd epend ency upon the type of thermal boundary condition imposed at r =1.Indeed,

these complicatedtemperature pro?les are d irectly respon-sible for the increasing complexity of the surface variation

386V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391

Fig.3.Variation of local Nusselt number on the surface of a representative cylinder in the bundle for Case II:(a)Re=20;Pr=0:7;(b)Re=100;Pr=0:7;

(c)Re=500;Pr=0:7.

V.K.Mandhani et al./Chemical Engineering Science 57(2002)379–391

387

Fig.4.Representative temperature pro?les (Pr =0:7)for Case I:(a)Re =500;e =0:4;(b)Re =500;e =0:

9.

Fig.5.Representative isotherms for Pr =0:7under constant heat ux condition:(a)Re =500;e =0:4;(b)Re =500;e =0:9.

of the Nusselt number seen in Figs.2and3.Qualitatively,similar temperature pro?les in staggeredarrays have also been documented in the literature (e.g.,see Chen &Wung,1989).

5.3.Average Nusselt number

Notwithstanding the importance of the detailed temper-ature ?eld,it is readily agreed that the mean value of the Nusselt number (averagedover the surface)represents an important design parameter for this problem.The scaling of the ?eldequations andthe bound ary cond itions reveals the mean Nusselt number to be a function of Re;Pr and e for a given type of thermal boundary condition.It is also custom-ary to introduce the so-called j H factor of heat transfer which is de?ned by Eq.(13).Fig.6shows the variation of the j H factor with the Reynolds number and porosity for the ow of air (Pr =0:715)for the two types of thermal boundary condition imposed on the surface of the cylinder.Qualita-tively,this type of functional dependence is consistent with the previously reportedtheoretical andexperimental results available in the literature.Similar type of dependence is

388V.K.Mandhani et al./Chemical Engineering Science 57(2002)

379–391

Fig.6.Dependence of average Nusselt number on Reynolds number for Pr =0:7(Fig.6a)and7.7(Fig.6b)for a range of porosity values.Solidlines for C ase I andbroken lines refer to C ase II.

obtainedwhen the present results for Pr =7:7and =or for the constant heat ux condition are plotted.As remarked ear-lier,the j H factor for Case I is always lower than that for Case II,especially at high Reynolds numbers.

In summary,the value of the mean Nusselt number in-creases with decreasing porosity and increasing Reynolds number and =or Prandtl number.Also,the imposition of the constant heat ux condition at r =1(cylinder surface)results in marginally higher values of the Nusselt number than those obtainedwith the constant temperature cond ition;however,the di erence between the two values progressively dimin-ishes as the value of the Reynolds number decreases and =or the porosity decreases.

https://www.360docs.net/doc/fb15511101.html,parison with literature studies

While numerous analytical studies based on the periodic arrays are available in the literature,only one approximate study based on the zero vorticity cell model is due to Le C lair andHamielec (1969).Some of these relate to low Peclet number andlow Reynold s number e.g.,see Sangani andAcrivos (1982)while in many other cases d ue to the in-herently di erent geometrical con?gurations employed and due to the lack of su cient details,it is not possible to make direct comparisons between these and the present re-

sults (https://www.360docs.net/doc/fb15511101.html,under &Massey,1978;Wilson &Bassiouny,2001).However,it is possible to contrast the present pre-d ictions with the results of C hen andWung (1989)andof Martin et al.(1998).C hen andWung (1989)have presented the mean values of Nusselt number for Case I for one value of porosity,namely,e =0:8;Re =40,120,400and800and for Pr =0:1,1and10.They correlatedtheir results using the following empirical expressions:?For in-line arrays Nu =0:8Re 0:4Pr 0:37(14a)

?For staggeredarrays Nu =0:78Re 0:45Pr 0:38:

(14b)

It is readily conceded that while the geometry of the array has a profounde ect on the d etailedkinematics of ow,the gross parameters like the mean Nusselt number are seen to be little in uenced,for the values of the empirical con-stants in Eqs.(14a)and(14b)are virtually id entical for the two types of arrays.Table 2shows a comparison between the present results andthe pred ictions of Eq.(14)over the common ranges of conditions.An inspection of these results shows that while the present values are consistently higher than the predictions of Eq.(14a)generally by 30–45%,good correspondence is seen to exist with the predictions of

V.K.Mandhani et al./Chemical Engineering Science 57(2002)379–391

389

Table 2

C omparison between the present results andthat of C hen andWung (1989)for e =0:8Re

Pr

Value of Nu Present

Chen &Wung In-line

Staggered 500.715 4.67 3.35 3.967.710.868.149.851000.715 6.31 4.42 5.417.715.7010.7413.462000.7158.90 5.847.397.722.9114.1718.38500

0.71514.398.4211.167.7

37.39

20.45

27.76

https://www.360docs.net/doc/fb15511101.html,parison between the present predictions and those of Martin et al.(1998).Open symbols for type I boundary condition.

Eq.(14b);the divergence being of the order of 15–25%.The discrepancy is somewhat smaller at Pr =7:7than that at Pr =0:715.

Similarly,Martin et al.(1998)have numerically solved the energy equation for the cross ow of air (Pr =0:71)past square andtriangular arrays of cylind ers for 0:86e 60:99and36Re 6160.Also,they have usedboth types of thermal boundary conditions.Fig.7shows a comparison between their andpresent values for e =0:8,0.9and0.99.While for e =0:8and0.9,the correspond ence between

the

Fig.8.Typical comparison between the present predictions (e =0:999;Pr =0:74)andthe experimental results d ue to Eckert and Soehngen (1952)for a single cylinder.

two sets of results is seen to be acceptable,the two results

are seen to diverge progressively with the increasing value of e ,especially beyond e =0:95irrespective of the type of the thermal boundary condition.Such high values of e ,however,are of little practical utility in the context of tubular heat exchangers.Once again,the correspondence between the two is seen to be about as goodas can be expectedin this kindof work.There are,however,no d is-cernable trends present either in Table 2or in Fig.7,except that the free surface cell model yields higher values of the Nusselt number than those basedon the period ic arrays.Some experimental data on heat transfer in such systems is available in the literature,albeit only some of these studies can be reduced to a form suitable for direct comparison with the present works.Eckert andSoehngen (1952)reportedthe values of the local Nusselt number for the ow of air over a heatedcylind er.A comparison between the present values for e =0:999andtheir values is shown in Fig.8,where a qualitative agreement is seen to exist.The discrepancy seen in Fig.8can be attributedin part at least to the ?nite values of L=d ratio andthe wall e ects encounteredin such

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379–391

Fig.9.Typical comparison between the present predictions,approximate results of Le C lair andHamielec (1968)andthe experimental results of Bergelin et al.(1952).

experimental studies and to the fact that the predictions relate to e =0:999andnot to e =1.

Similarly,Fig.9shows a comparison between the present predictions (free surface cell model),the approximate results of Le C lair andHamielec (1968)for the zero vorticity cell model and the experimental results of Bergelin et al.(1952).The modi?ed de?nitions of the j H factor andthe Reynold s number are,however,requiredto account for the fact that Bergelin et al.(1952)usedan interstitial Reynold s number basedon the minimum ow area.Thus,

j

H =j H (1?√1?e )(15a)and

Re =

Re

1?√1?e

:(15b)Once again a satisfactory correspondence is seen to exist between the two predictions and the experimental results in the ranges of conditions embraced in this work (Fig.9).In summary,the present predictions of heat transfer char-acteristics in tube bundles are consistent with the previous scant numerical predictions as well as the experimental re-sults available in the literature.However,the present results encompass much wider ranges of physical and kinematic conditions than those associated with previous studies.6.Concluding remarks

In this work,the ?eldequations have been solvednumer-ically to obtain detailed temperature ?elds and the distribu-tion of Nusselt number on the surface of a typical cylinder in a cylinder bundle for the steady incompressible ow of Newtonian uids.The role of the type of thermal boundary conditions,namely,constant temperature and constant heat ux,on overall heat transfer characteristics has been eluci-dated.Overall,the range of parameters encompassed herein include two values of Prandtl number (air and water)and 0:46e 60:99,while the Reynolds number of ow varied from 1to 500.The nature of variation of Nusselt number with the position on the surface of the cylinder is strongly dependent on the values of e;Pr and Re .The surface av-eragedvalue of Nusselt number increases with the d ecreas-ing value of porosity andincreasing values of Prand tl and Reynolds numbers.

The present results are in satisfactory agreement with the previous numerical andexperimental results for a sin-gle cylinder and for the arrays of cylinders in the overlap-ping ranges of conditions.The limited comparisons included herein seem to suggest that the overall heat transfer charac-teristics are primarily governedby the value of e andthe e ect of the actual geometrical arrangement of the array to be of secondary importance.However,more experimental results are needed to thoroughly validate the numerical pre-dictions reported in this study.Finally,this study clearly shows that the cell models o er a viable approach to the modelling of momentum,heat and mass transfer in such multi-cylinder systems.

Notation C p heat capacity,J =kg K d cylinder diameter,m

e porosity or voidage,dimensionless h heat transfer coe cient,W =m 2K j H heat transfer factor,dimensionless k thermal conductivity o

f uid,W =mK

Nu Nusselt number based on cylinder diameter,di-mensionless p pressure,Pa

Pr Prandtl number,dimensionless q w heat ux at r =1;W =m 2r radial coordinate,m

r o cylinder radius,dimensionless

r ∞radius of hypothetical cylindrical envelope,di-mensionless

Re Reynolds number,dimensionless

T dimensionless temperature,dimensionless T o temperature of the uidin the free stream,K T s temperature on the surface of the cylinder,K U free stream velocity,m =s

V r ;V ?

r -and ?-components of velocity,dimensionless

Greek letters ?cylindrical coordinate viscosity of uid,Pa =s

density of uid,kg =m 3

V.K.Mandhani et al./Chemical Engineering Science57(2002)379–391391

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