three-dimensional numerical simulation of fluid flow with phase change heat transfer in an...

ned

0 NaEm

Porous media

geumeonteldRayne,

eat traicationndustr4]. In mporouhase c

phase change, it is thus helpful in engineering design and predic-tion. However, the difculties encountered in the numerical studyof uid owwith phase change heat transfer in porousmedia is dueto the strongly nonlinear and coupled nature of the governing

ed Kirchhoff method was better in handling the rapid change inthe diffusion coefcient.

Generally, in the literature [12e19], numerical simulations arebased only on a two-dimensional (2D) setting which cannotcapture the three-dimensional (3D) effects which may be impor-tant under certain circumstances. Furthermore, large discrepanciesbetween numerical prediction and the experimental data may arisefrom a 2-D simulation. Although the assumption of a 2D setting inthe numerical simulation is applicable in some very specic cases,

* Corresponding author. Tel.: 65 6790 5596; fax: 65 6792 4062.

Contents lists availab

International Journal

w.e

International Journal of Thermal Sciences 49 (2010) 2363e2375E-mail address: [email protected] (K.C. Leong).such systems is of fundamental interest. Therefore, extensivestudies have been carried out by many investigators since the late1970s [5e11].

Numerous studies have been performed to investigate single-component uid ow with phase change heat transfer in porousmedia both experimentally and numerically [12e19]. Comprehen-sive reviews of these studies are well documented in the books ofNield and Bejan [20], Kaviany [21], Ingham and Pop [22], Vafai [23]and Vadasz [24]. As numerical simulation can provide usefulinformation and preliminary results in complex systems with

mixture model to study the transient behavior of uid ow withphase change heat transfer in porous media. In their work, theeffects of heat ux locations on the ow and heat transfer eldswere investigated. The results showed that the liquid and vaporow elds, as well as the temperature and liquid saturation eldsexhibit distinctly different features for different heating locations.In addition, the discontinuous diffusion coefcient in the energyequation was handled with the modied Kirchhoff method [26]in their work. The performance of this method and the harmonicmean method [27] were compared. It was found that the modi-1. Introduction

Fluid ow with phase change hoccurs in many engineering applprocesses in the food [1] and paper i[3] and chemical catalytic reactors [two-phase owwith phase change inUnderstanding the uid ow with p1290-0729/$ e see front matter 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.07.018saturation increases with the increase of both the Peclet and Rayleigh numbers. 2010 Elsevier Masson SAS. All rights reserved.

nsfer in porous medias such as the dryingies [2], heat exchangersany of these systems,

smedia is encountered.hange heat transfer in

equations. The discontinuity in the thermal properties at the phasechange boundary requires careful consideration. In the numericalsimulation of the phase change problem, a front tracking approachwas needed to capture the evolving irregular interfaces as phasechange occurs. This may lead to the problem of singularity whenthe interface touches the bottom or the top boundary. The modeldeveloped by Wang [14], called the two-phase mixture modeladdresses this difculty. Recently, Li et al. [25] used the two-phaseTwo-phase mixture modelPhase changewhile an increase in the Rayleigh number helps to spread the heat to a larger region of the domain. Thedistribution of the liquid saturation on the heated section of the wall indicates that the minimum liquidThree-dimensionalliquid zone and the two-phase zone. An increase in the Peclet number decreases the two-phase regionThree-dimensional numerical simulatioheat transfer in an asymmetrically heat

H.Y. Li a, K.C. Leong a,*, L.W. Jin a, J.C. Chai b

a School of Mechanical and Aerospace Engineering, Nanyang Technological University, 5bMechanical Engineering Department, The Petroleum Institute, Abu Dhabi, United Arab

a r t i c l e i n f o

Article history:Received 13 February 2010Received in revised form17 June 2010Accepted 27 July 2010

Keywords:

a b s t r a c t

Fluid ow with phase chanheating from one side is ndeal with the spatial discvelocity and temperaturewith different Peclet andbypasses the two-phase zo

journal homepage: wwson SAS. All rights reserved.of uid ow with phase changeporous channel

nyang Avenue, Singapore 639798, Republic of Singaporeirates

heat transfer in a three-dimensional porous channel with asymmetricallyrically studied in this paper. The modied Kirchhoff method is used toinuity in the thermal diffusion coefcient in the energy equation. Thes, as well as the liquid saturation eld on the heated section of the wallleigh numbers are investigated. The results show that the liquid owwhile the vapor ows primarily to the interface between the sub-cooled

le at ScienceDirect

of Thermal Sciences

lsevier .com/locate/ i j ts

j diffusive mass ux (kg/m s)

s solid

Thea 3D numerical simulation is more desirable since it is more real-istic for comparison with actual experimental data. It is especially

J(s) capillary pressure functionkrl relative permeability of liquidkrv relative permeability of vaporkeff effective thermal conductivity (W/mK)K permeability of the porous medium (m2)L length of the simulation domain (m)p pressure (Pa)Pe Peclet number, Pe uinH/aq00 heat ux (W/m2)Ra Rayleigh number, Ra KHgrlCpl/ylkeffs liquid saturationS energy source term (W/m3)t time (s)T temperature (C)u velocity vector (m/s)Nomenclature

Cp specic heat of uid (J/kg K)Ds capillary diffusion coefcient (m2/s)f(s) hindrance functiong gravity vector (m/s2)hfg latent heat of phase change (J/kg)h enthalpy (J/m3)H height of the simulation domain (m)

2

H.Y. Li et al. / International Journal of2364important for understanding and predicting the development ofcomplex ow structures and the two-phase zone for cases whichcannot be simplied to a 2D setting such as the case shown inFig. 1a. However, to the best of the knowledge of the authors, thereis no 3D numerical study of phase change heat transfer in porousmedia based on the two-phase mixture model.

In the present study, a 3D numerical code is developed byadopting the two-phase mixture model. The primary objective ofthe current article is to study the uid owwith phase change heattransfer in porousmedia in a 3D domain based on the developed 3Dcode. It is a continuation of the authors previous study on phasechange heat transfer in porous media in a 2D setting [25]. Thevelocity and temperature elds under different Rayleigh and Pecletnumbers in a 3D domainwith asymmetrically heated from one sideare analyzed. In this article, the discontinuity in the thermaldiffusion coefcient at the phase change boundary is also treatedwith the modied Kirchhoff method [26]. This 3D code can beapplied to simulate the uid ow and heat transfer for a large classof phase change ow problems.

2. Mathematical formulations

2.1. Problem description

The schematic diagram of the problem is shown in Fig. 1. A 3Dchannel with dimensions LWH is lled with a porous medium.A nite heat source with constant heat ux is applied on one sidewall as shown in Fig. 1a. The rest of the channel wall is perfectlyinsulated. An external pressure difference drives the sub-cooledwater with low temperature Tin through the channel. The owingliquid is heated as it ows past the heated section of thewall. Whenthe heat ux is increased sufciently, boiling occurs at the heatedsection and thus a two-phase zone is formed. Further increase inthe heat uxwill lead to a superheated vapor zone. This could cause

sat saturationv vaporW width of the simulation domain (m)

Greek symbolsa thermal diffusivity of porous media (m2/s)b thermal expansion coefcient (1/K)g advection correction coefcientG diffusion coefcientGh diffusion coefcient in enthalpy equationDr rl rv (kg/m3)3 porosityl relative mobilitym dynamic viscosity (kg/m s)y kinematic viscosity (m2/s)r density (kg/m3)s surface tension (N/m)U effective heat capacitance ratio

Subscriptsl liquidin inleto initial state

rmal Sciences 49 (2010) 2363e2375dry-out in which no liquid exists on the heated section. The vaporlayer blankets the heated section of thewall and greatly reduces theheat transfer between the heated section and the uid. Conse-quently, the local temperature at the dry-out location increasessharply. This is unfavorable to the cooling system. Hence, thepresent work considers only the case without any occurrence ofa superheated vapor zone.

2.2. Governing equations

The present model considers an isotropic and homogeneousporous medium. The local thermal equilibrium model is used here.This model assumes that the solid and uid phases are at the sametemperature. The governing equations obtained by adopting thetwo-phase mixture model [14] are:

Conservation of mass:

3vr

vt V$ru 0 (1)

Conservation of momentum:

u KmVp rk rog (2)

Conservation of energy:

Uvhvt

V$ghuh V$GhVh Vf sKDrhfg

nvg

(3)

The mixture variables and the properties in Eqs. (1)e(3) are listedin Table 1. The temperature and liquid saturation s can be calculatedfrom the enthalpy as:

h rC T 2h

(11)

Fig. 1. Schematic of the problem and the coordinate systems (a) the simulationdomain; (b) selected planes.

Table 1Variables in the two-phase mixture model.

Variables Expressions

Density r rls rv1 sVelocity ru rlul rvuvEnthalpy rh rlshl rv1 shvKinetic density rk rl1 blT To lls rv1 bvT Tsat lvs

Viscosity m rls rv1 skrl=nl krv=nvAdvection correction

coefcientgh

rv=rl1 s shvsat1 ll hlsatll 2hvsat hlsats rvhvsat=rl1 s

Effective heatcapacitance ratio

U 3 rsCps1 3dTdH

Effective diffusioncoefcient

Gh 1

1 1 rv=rlhvsat=hfgD keff dTdH

Capillary diffusioncoefcient

Ds 3K0:5s

ml

krlkrvnv=nlkrl krv

J0s

Relative motilities lls krl=nl

krl=nl krv=nv; lvs krv=nvkrl=nl krv=nv

Hindrance function f s krvkrl=nlkrl=nl krv=nv

Relative permeabilities krl s3; krv 1 s3

Capillary pressurefunction

Js 1:4171 s 2:1201 s2 1:2631 s3

H.Y. Li et al. / International Journal of Theu v w 0 (18)l pl in vsat

u uin; v 0; w 0 (12)At the outlet x L and t> 0,

vhvx

0 (13)

vuvx

vvvx

vwvx

0 (14)

At the position of thewall that is heatedwith constant heat ux andt> 0,

Ghr

vhvy

q00 (15)

u v w 0 (16)At the non-heated portions of the side walls and t> 0,

Ghr

vhvy

0 (17)T

8>>>>>>>:

h2rlhvsatrlCpl

hrl2hvsat hlsatTsat rl2hvsathlsat < hrvhvsat

Tsath rvhvsatrvCpv

rvhvsat < h(4)

s

8>>>>>:

1 hrl2hvsathlsat h rvhvsatrlhfgrl rvhvsat

rl2hvsathlsat < hrvhvsat0 rvhvsat < h

(5)

Subscripts l and v in Eqs. (4) and (5) refer to the liquid and vapor,respectively. The individual velocities of the liquid and vapor can berecovered from

rlul llru j (6)

rvuv lvru j (7)where j is the total mass ux which is expressed as

j rlDsVs f sKDrnv

g (8)

2.3. Boundary conditions

The initial and boundary conditions for the present problem aregiven as follows:

At t 0,

h rlCplTin 2hvsat

(9)

u v w 0 (10)At the inlet x 0 and t> 0,

rmal Sciences 49 (2010) 2363e2375 2365At the upper and lower walls and t> 0,

and height H of the channel are prescribed as 20 mm. The length ofthe heated section on the side wall is 20 mm. The lengths of theunheated sections of the side wall namely, l1/H and l2/H are set to 1.These lengths are chosen because the inlet and exit boundaryconditions have no effects on the solution.Water enters the domainwith a uniform velocity and at a constant temperature of 22 C. Aconstant heat ux q00 200 kW/m2 is imposed at the heated sectionof the side wall. The porous medium considered in the currentsimulation is the graphite foam developed at Oak Ridge NationalLaboratory (ORNL), USA in 1997 [29]. This material consists ofpredominantly spherical pores with small openings between theligaments. The porosity of this graphite foam is 0.75 and its thermalproperties arewell documented in the report of Klett et al. [30]. Theeffective thermal conductivity for this graphite foam is calculatedbased on the model proposed by Tee et al. [31].

A grid independency study was carried out and the results showthat a mesh of 60 20 20 control volumes with Dt 1.0 s

Fig. 2. Comparison of 3D results under symmetric boundary conditions with 2Dresults (a) x 1/3L; (b) x 1/2L.

The3. Results and discussionGhr

vhvz

f sKDrhfgvv

g 0 (19)

u v w 0 (20)

2.4. Numerical procedure

The momentum equation [Eq. (2)] is different from the tradi-tional NaviereStokes equation. In the present problem, themomentum equation [Eq. (2)] is rst substituted into the continuityequation [Eq. (1)] to obtain an equation for the pressure. Theresulting pressure equation is then solved by a line-by-line tri-diagonal matrix algorithm. Upon obtaining the pressure, themixture velocity eld can be calculated using the momentumequation [Eq. (2)]. The individual velocities of the liquid and vaporcan be obtained from the mixture velocity using Eqs. (6) and (7).These velocities are stored at the interfaces of the control volumes.The energy equation [Eq. (3)] can be written as a general con-vectiveediffusive equation of the form

vrfvt

vrujf

vxj

vvxj

Gvf

vxj

! S (21)

where f, G and S are dened as the dependent variables, diffusioncoefcient and source term. These equations are solved using thenite-volume method [27]. The power-law is used to treat thecombined convection-diffusion term. Unlike thework ofWang [14],the discontinuous diffusion coefcient in this problem is treatedwith the modied Kirchhoff method [26]. With this, a smootherinterface between the liquid and two-phase regimes can becaptured. The performance of the modied Kirchhoff method hasbeen demonstrated in the authors previous work [25]. Thedescriptions of this method are well documented in the work ofVoller and Swaminathan [26]. The current work will adopt thesame terminology of modied Kirchhoff method for easy refer-ence. However, unlike the work of Voller and Swaminathan [26]where the integral variable is temperature, the present studyadopts enthalpy h as the integral variable given the form of theenergy equation expressed in terms of enthalpy. The performanceof this method is compared with the harmonic mean method [27].It shows that the former approach produces smoother isothermallines. These simulationswere carried out on a desktopwith Intel (R)Core (TM) 2 Quad CPU Q9450 @ 2.66GHz and 3.25GB RAM. Arelative error of less than 106 is required for both the velocity andtemperature elds between successive iterations to achieveconvergence.

2.5. Code validation

A 2D code has been developed in a previous work by the authors[25]. This 2D code was validated with the experimental results ofEasterday et al. [28].With symmetric boundary conditions enforcedalong the width direction in the 3D domain (uid ows alonglength (x) direction and heating along height (z) direction), theresults from the present 3D code should reduce to its 2D counter-part with these boundary conditions. This comparison is given inFig. 2. The temperature distributions obtained from the 2D and 3Dcodes were found to overlap at different locations. The validity ofthe 3D code is therefore demonstrated.

H.Y. Li et al. / International Journal of2366The present study focuses on uid ow with phase change heattransfer in a 3D porous channel. For all cases studied, the width Wrmal Sciences 49 (2010) 2363e2375produces a grid-independent solution. All subsequent computa-tions were performed using this time-step and mesh size.

The effects of the Rayleigh and Peclet numbers are discussed inthe present article. There are two Rayleigh numbers based ondifferent ow regimes [13,32]. One is for the liquid regime and isbased on the temperature difference. The second Rayleigh number isfor the two-phase regime and involves the density differencebetween the liquid and vapor phases. Detailed information can befound in the works of Ramesh and Torrance [13,32]. As the primaryinterest in the currentwork is to study theuidowandheat transferbehavior in the two-phase regime in the porous media, the authorshave chosen the form of the Rayleigh number based on the two-phase regime. The mathematical expression of this Ra number is:

RahKgrlCplHnlkeff

(22)

The Peclet number is dened as:

PehuinHa

(23)

3.1. Effects of Peclet number on uid ow and heat transfer

In the current work, 3D uid ow with phase change heattransfer in a porous channel (Fig. 1a) with asymmetrically heatingfrom one side is studied. In order to describe the characteristics ofthe uid ow clearly, results for a series of planes are shown. Fig. 1bshows the selected planes. Although transient uid owwith phasechange heat transfer was carried out as part of the simulation, onlythe steady-state results are discussed in the following sections.

For the ow elds in the present article, the interface between thesub-cooled liquid and the two-phase zones, viz. the condensationfront is shown as the solid curve.

Fig. 3 shows the ow elds for Ra 226 and Pe 0.06. As theliquid ows into the channel with a uniform velocity prole, itabsorbs heat when it ows past the heated section of the wall. Itstemperature increases, leading to a lower density. Acted upon bythe buoyancy force, the heated liquid ows in a slightly upwardmanner. The liquid upstream of the heated section of the wall,which is cooler and therefore denser, tends to ow downward toreplace the upward ow of the heated liquid. As the buoyancy forcebecomes important, this combination of downward ow of thecooler liquid from the upstream of the heated section and upwardow of the hotter liquid downstream results in a circulatory ow.This circulation ow is located at the leading edge of the two-phasezone (Fig. 3a). The recirculation hinders the liquid upstream frompenetrating the circulation cell. Therefore, the liquid in the sub-cooled liquid zone from the inlet ows decisively downward tobypass the recirculation (Fig. 3a). When it approaches thecondensation front, it shows a slightly upward motion. This slightlyupwardmotion of the liquid is due to a decrease in the permeabilitywhich causes the sub-cooled liquid to ow through the two-phasezone. The two-phase zone is occupied by both the liquid and vapor.The presence of the generated vapor in the void space reduces thepermeability for the sub-cooled liquid to ow through the two-phase zone. The liquid therefore has difculty in penetrating thevoid space in the porous media which also contains vapor. Whenthe incoming ow is weak, the liquid in the sub-cooled liquid zone

H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e2375 2367Fig. 3. Liquid velocity in different planes for Pe 0.06 and Ra 226 (a) the xz plane with(e) x/H 0.5.y/H 0.5; (b) the xy plane with z/H 0.5; the yz plane at (c) x/H 0.1; (d) x/H 0.3;

is blocked as it hits the condensation front. The blocked liquid thusreverses its ow direction to the inlet as the two-phase zonepresents a region of high ow resistance. This can be seen in the xyplane with z/H 0.5 (Fig. 3b). Further away from the heated sectionof the wall, less vapor is generated. The ow resistance caused bythe presence of the vapor to the owing liquid is therefore smaller.Hence, the liquid away from the heated section in the sub-cooledliquid zone ows across the condensation front into the two-phasezone (Fig. 3b). The volume fraction occupied by the liquid in thetwo-phase zone is called the liquid saturation [21], denoted as s.The vapor volume fraction is given by 1-s. In the locations near theheated section of the wall, more liquid vaporizes, leading toa smaller liquid saturation. In contrast, at locations away from theheated section, the liquid saturation is higher. This difference in theliquid saturation causes a liquid saturation gradient in the two-phase zone. Such a liquid saturation gradient gives rise to thecapillary-induced force. The liquid is driven by the capillary force toow from the locations with high volume fraction of the liquid tothe locations with low volume fraction of the liquid; viz. thecapillary force drives the liquid in the two-phase zone towards theheated section of the wall. This can be seen from the motion ofthe liquid in the vicinity of the heated section of the wall shown inthe xy planewith constant z/H 0.5 (Fig. 3b). In the regionwith lowliquid saturation gradient, the liquid ows approximately parallelto the outlet. Under the combined effect of the buoyancy andcapillary forces, the liquid from the inlet divides into two streams.While one stream ows backwards toward the inlet in a circulatory

to the heated section. This is different to the case of single phasenatural convection in a cavity with heating from one side. As theuid ows into themiddle of the domainwith the heated section atx/H 0.3, an extensive region of the plane is covered by the two-phase zone. The sub-cooled liquid zone is pushed to a small arealocated in the lower right corner of the plane (Fig. 3d). This two-phase zone grows to become almost symmetrical to the diagonal ofthe plane. When the liquid leaves the heated section at x/H 0.5(Fig. 3e), the two-phase zone expands. As most of the uid owsparallel to the outlet of the domain (Fig. 3a and b), the liquid ow inthe yz plane becomes insignicant.

Fig. 4 shows the characteristics of the vapor ow in differentplanes. The vapor is rst generated on the heated section of thewalland then it moves away from the heated source in differentdirections. A portion of the vapor is carried downstream in the formof a vaporeliquid mixture while the other portion ows upwarddue to buoyancy force (Fig. 4a and b). As a result, the upper portionof the channel is progressively richer in vapor along the owdirection. On the other hand, the lower portion of the channel isrelatively rich in liquid. This phase separation is caused by the largedensity difference between the liquid and vapor. The quantity ofthe vapor increases along the ow direction since more liquidvaporizes (Fig. 4d). It is noted that the vapor ows toward thecondensation front where it is condensed.

Figs. 3e8 demonstrate the effects of Peclet number, Pe on thevelocity elds for a xed Rayleigh number of 226. The Pecletnumber is directly related to the inlet velocity. At low Pe (or low

H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e23752368manner, the other stream ows to the outlet. In the yz plane withconstant x/H 0.1 (Fig. 3c), the two-phase zone occupies a smallportion of the plane in the upper-left corner. Given the buoyancyforce and the weak ow from inlet, the vapor rises and spreads tothe upstream of the heated section. At certain elevation, thebuoyancy force assists the liquid to ow upward while the liquidbelow such elevation ow directly downward (Fig. 3c). Driven bythe capillary force, this downward ow was dragged to ll theempty space caused by the upward ow in a slightly lateral mannerFig. 4. Vapor velocity in different planes for Pe 0.06 and Ra 226 (a) the xz plane with(e) x/H 0.5.inlet velocity), the buoyancy force is dominant. The circulatory owcaused by the buoyancy force is therefore strengthened. The eye ofthe recirculatory ow is shifted upstream. Generally, a smaller Pesuggests a weaker convection effect, i.e. a longer residence time ofthe uidwithin the channel. The uid absorbsmore heat, leading tothe generation of more vapor. Therefore, a larger two-phase zone isdeveloped in the channel. The interface shape between the sub-cooled liquid zone and the two-phase zone is a direct result of thestrong upward vapor ow. At Pe 0.12, the increase in the relativey/H 0.5; (b) the xy plane with z/H 0.5; the yz plane at (c) x/H 0.1; (d) x/H 0.3;

Fig. 5. Liquid velocity in different planes for Pe 0.12 and Ra 226 (a) the xz plane with y/H 0.5; (b) the xy plane with z/H 0.5; the yz plane at (c) x/H 0.1; (d) x/H 0.3;(e) x/H 0.5.

H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e2375 2369strength of the inlet ow reduces the circulatory ow (Fig. 5a). Theinertial of the liquid increases so that it gradually changes thedirection for the back ow of the liquid near the condensation frontin the xy plane (Fig. 5b). When Pe is sufciently high (Pe 0.24), thestrong incoming liquid pushes the liquid near the condensationfront in the xy plane decisively downstream (Fig. 7b). With the

increase of the inlet velocity, the heat transfer from the heated

Fig. 6. Vapor velocity in different planes for Pe 0.12 and Ra 226 (a) the xz plane with(e) x/H 0.5.section of the wall to the uid is increased. A small amount of vaporis generated and the two-phase zone shrinks substantially. It isobserved that the two-phase zone recedes appreciably away fromthe inlet (Fig. 7b). The vapor in the two-phase zone ows partiallyto the condensation front and partially to the outlet.

Fig. 9 shows the isotherms within the porous medium. The

temperature difference between successive isotherms is 15 C. The

y/H 0.5; (b) the xy plane with z/H 0.5; the yz plane at (c) x/H 0.1; (d) x/H 0.3;


H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e23752370isotherm of 100 C is not smooth. This is caused by the interpola-tion error in plotting the data, a limitation of the plotting software.The 100 C isotherm is the interface between the sub-cooled liquidand two-phase zone. Within the two-phase zone, the liquid-vapormixture remains at the saturation point of 100 C. The results from3D simulation are more realistic compared with those from a 2D

simulation since the heat transfer eld can be clearly seen in the

Fig. 8. Vapor velocity in different planes for Pe 0.24 and Ra 226 (a) the xz plane with(e) x/H 0.5.whole domain. For different Peclet numbers, the temperaturecontours are all inclined towards the outlet due to the incomingsub-cooled liquid ow. The presence of the circulatory ow in theupper left-hand corner (Figs. 3a, 5a and 7a) of the plane actuallyhinders cooler liquid from reaching the heated section. This resultsin poor heat transfer from the heated section to the surrounding

liquid. Therefore, denser temperature contours are observed in the


f TheH.Y. Li et al. / International Journal oregion with circulatory ow. This indicates a large temperaturegradient. For large Pe, the isotherms are prominently inclinedtowards the outlet due to the strong incoming ow. The increase ofPe increases the convection heat transfer between the heatedsection of the wall and the uid. The region covered by the two-phase zone is therefore reduced greatly.

The variation of the liquid saturation s on the wall with heatedsection under different Peclet numbers is shown in Fig. 10. Theliquid saturation s is kept at a constant value of 1 at the entrance fordifferent inlet velocities, indicating a sub-cooled liquid at thisregion. However, it begins to drop as the liquid approaches theheated section of the wall, reaches a minimum somewhere withinthe heated section and then increases at the point in which theliquid leaves the heated section as a liquidevapor mixture.Generally, s reduces along the direction of the height (z direction) asthe vapor ows upward; indicating that the upper wall is relatively

Fig. 9. Temperature distributions in the 3D domain for Ra 226 and (a) Pe 0.06;(b) Pe 0.12; (c) Pe 0.24.rmal Sciences 49 (2010) 2363e2375 2371rich in vapor. Special attention should be paid to such problems inengineering applications. An implication of this result could be thatin thermal systems with heat sources located at the side walls,a dry-out zone could appear near the upper wall within the heatedsection. The temperatures at these locations would increasesignicantly and local hot spots may be induced. This could bepotentially disastrous for sensitive electronic devices and clearlydemonstrates the need for detailed 3D studies to accurately capturethe locations of minimum liquid saturation. The increase of Pe

Fig. 10. Liquid saturation on the wall with heated section for Ra 226 and(a) Pe 0.06; (b) Pe 0.12; (c) Pe 0.24.

Fig. 11. Liquid velocity in different planes for Pe 0.06 and Ra 113 (a) the xz plane with y/H 0.5; (b) the xy plane with z/H 0.5; the yz plane at (c) x/H 0.1; (d) x/H 0.3;

H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e23752372pushes the two-phase zone downstream. As a result, the locationwhere s drops from 1 recedes away from the inlet. The minimumliquid saturation also increases with the increase of the inletvelocity.

3.2. Effects of Rayleigh number on uid ow and heat transfer

(e) x/H 0.5.Attention is now focused on the effects of Rayleigh number onthe velocity, temperature and liquid saturation elds. The Rayleigh

Fig. 12. Vapor velocity in different planes for Pe 0.06 and Ra 113 (a) the xz plane with(e) x/H 0.5.number, Ra is associated with buoyancy driven ow. According tothe denition of Ra adopted in the present work, a change in Rawhile keeping the other parameters constant implies a variation inthe permeability of porous media. In the current study, threedifferent Rayleigh numbers, viz. three different permeabilities ofthe porous media, are studied. For all cases, the Peclet number is setto 0.24.

The ow elds for both liquid and vapor under different Ray-

leigh numbers are shown in Figs. 11e14. Generally, a large Ra



H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e2375 2373suggests signicant buoyancy effect. In this case, buoyancy worksagainst convection. At high Rayleigh number, Ra 226, as shown inFig. 3, the buoyancy force produces a strong circulatory ow in thexz plane. The recirculation deects the liquid from inlet to owdownwards. Such a downward ow is accelerated in a narrow

region adjacent to the lower wall. The circulatory ow reduces

Fig. 14. Vapor velocity in different planes for Pe 0.06 and Ra 56 (a) the xz plane with(e) x/H 0.5.convection heat transfer near the upper wall while the acceleratedliquid increases convection near the lower wall. Therefore, the two-phase zone blankets the upper wall in a larger region comparedwith that on the lower wall, leading to a more inclined condensa-tion front. This is different fromwhat was observed at low Rayleigh

number. With the decrease of Ra, the buoyancy force becomes


TheH.Y. Li et al. / International Journal of2374insignicant. This reduces the circulatory ow. At Ra 56 (Fig. 13),the recirculation near the upper wall is almost curtailed by theincoming ow. The leading edge of the two-phase zone on theupper wall is pushed further downstream. The condensation frontbecomes steeper (Fig. 13a). Unlike the case for Ra 226, no two-phase zone is observed in the yz plane with constant x/H 0.1 atRa 113 and Ra 56 (Figs.11c and 13c). The vapor ows in a similarfashion for different Ra, having a direction which is primarilynormal to the condensation front.

The temperature contours under Ra 113 and Ra 56 areshown in Fig. 15(a) and (b), respectively. Comparisons are madeagainst Ra 226 in Fig. 3. As indicated by the temperaturecontours, a large region of the domain is blanketed by the two-phase zone. The buoyancy force assists the vapor moving from theheated section of the wall to expand to a wide region. The heat isnot only expelled from the heated source downstream, but alsoswept upstream. An implication of this result in engineeringapplications could be that the region which is rich in vapor can besignicantly far away from the heat source. Hence, a detailed studyof two-phase heat transfer becomes necessary.

Fig.16 shows the distribution of the liquid saturation on thewallwith heated section for Ra 113 and Ra 56. A similar prole isobserved for different Rayleigh numbers. For ow with large Ra,a larger s is to be expected. For example, s reaches a minimum ofabout 0.2 for Ra 56, but 0.3 for Ra 113 and 0.4 for Ra 226. Ofparticular interest is the effect of the buoyancy force, although ithelps to spread the heat to a large region of the domain, it also, tosome extent, helps to promote mixing of the uid, reducing theliquid saturation in some local spots. Therefore, the minimumliquid saturation occurs at low Rayleigh number.

Fig. 15. Temperature distributions in the 3D domain for Pe 0.06 and (a) Ra 113; (b)Ra 56.rmal Sciences 49 (2010) 2363e23754. Conclusions

A 3D numerical study has been carried out for uid ow withphase change heat transfer in a porous channel with asymmetri-cally heating from one side. The velocity, temperature and liquidsaturation elds under different Peclet and Rayleigh numbers wereconsidered. The liquid ow in the xz plane shows a circulatory owat the leading edge of the two-phase zone. Such a recirculation isreducedwith the increase of Pe and decrease of Ra. The liquid in thexy plane bypasses the two-phase zone. In the two-phase zone, thecapillary force drives the liquid towards the heated section whilevapor ows primarily to the condensation front. An increase ofPeclet number decreases the size of the two-phase zone, while theincrease of Rayleigh number helps to spread the heat to a largerregion of the domain. The minimum liquid saturation s on the wallwith heated section increases with the increase of both Pe and Ra.

Acknowledgment

The authors gratefully acknowledge the nancial supportprovided under Defence Science and Technology Agency, SingaporeGrant no. DSTA-NTU-DIRP/2005/01 for the work described in thispaper.

References

[1] P.P. Tripathy, S. Kumar, Modeling of heat transfer and energy analysis ofpotato slices and cylinders during solar drying, Appl. Therm. Eng. 29 (2009)884e891.

Fig. 16. Liquid saturation on the wall with heated section for Pe 0.06 and (a)Ra 113; (b) Ra 56.

[2] A. Watzl, M. Rckert, Industrial through-air drying of nonwovens and paperbasic principles and applications, Drying Technol. 16 (1998) 1027e1045.

[3] G.F. Naterer, C.H. Lam, Transient response of two-phase heat exchanger withvarying convection coefcients, ASME J. Heat Transfer 128 (2006) 953e962.

[4] R.H. Nibbelke, J.H.B.J. Hoebink, G.B. Marin, Kinetically induced multiplicity ofsteady states in integral catalytic reactors, Chem. Eng. Sci. 53 (1998) 2195e2210.

[5] C.H. Sondergeld, D.L. Turcotte, An experimental study of two-phase convec-tion in a porous medium with applications to geological problems, J. Geophys.Res. 82 (1977) 2045e2053.

[6] H.H. Bau, K.E. Torrance, Boiling in low-permeability porous material, Int.J. Heat Mass Transfer 25 (1982) 45e55.

[7] H.H. Bau, K.E. Torrance, Thermal convection and boiling in a porous medium,Lett. Heat Mass Transfer 9 (1982) 431e441.

[8] H.H. Bau, K.E. Torrance, Low Rayleigh number thermal convection in a verticalcylinder lled with porous materials and heated from below, ASME J. HeatTransfer 104 (1982) 166e172.

[9] C.Y. Wang, C. Beckermann, A two-phase mixture model of liquid-gas ow andheat transfer in capillary porous media, I. Formulation, Int. J. Heat MassTransfer 36 (1993) 2747e2758.

[10] Z.Q. Chen, P. Cheng, T.S. Zhao, An experimental study of two phase ow andboiling heat transfer in bi-dispersed porous channels, Int. Comm. Heat MassTransfer 22 (2000) 293e302.

[11] C.H. Sondergeld, D.L. Turcotte, Flow visualization studies of two-phase thermalconvection in a porous layer, Pure Appl. Geophys. 117 (1978) 321e330.

[12] P.S. Ramesh, K.E. Torrance, Stability of boiling in porous media, Int. J. HeatMass Transfer 33 (1990) 1895e1908.

[13] P.S. Ramesh, K.E. Torrance, Numerical algorithm for problems involvingboiling and natural convection in porous materials, Numer. Heat Transfer, PartB: Fundamentals 17 (1990) 1e24.

[14] C.Y. Wang, A xed-grid numerical algorithm for two-phase ow and heattransfer in porous media, Numer. Heat Transfer, Part B: Fundamentals 32(1997) 85e105.

[15] T.S. Zhao, P. Cheng, C.Y. Wang, Buoyancy-induced ows and phase-changeheat transfer in a vertical capillary structure with symmetric heating, Chem.Eng. Sci. 55 (2000) 2653e2661.

[16] T.S. Zhao, Q. Liao, Mixed convective boiling heat transfer in a vertical capillarystructure heated asymmetrically, J. Thermophys. Heat Transfer 13 (1999)302e307.

[17] M. Najjari, S.B. Nasrallah, Effects of latent heat storage on heat transfer ina forced ow in a porous layer, Int. J. Therm. Sci. 47 (2007) 825e833.

[18] M. Najjari, S.B. Nasrallah, Heat transfer between a porous layer and a forcedow: inuence of layer thickness, Drying Technol. 27 (2009) 336e343.

[19] K. Yuki, J. Abei, H. Hashizume, S. Toda, Numerical investigation of thermouidow characteristics with phase change against high heat ux in porous media,ASME J. Heat Transfer 130 (2008) 1e12.

[20] D.A. Nield, A. Bejan, Convection in Porous Media, third ed. Springer, New York,2006.

[21] M. Kaviany, Principles of Heat Transfer in Porous Media, second edition.Springer-Verlag, New York, 1995.

[22] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media III, Elsevier,Oxford, 2005.

[23] K. Vafai, Handbook of Porous Media, second ed. Taylor & Francis, New York,2005.

[24] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media.Springer, New York, 2008.

[25] H.Y. Li, K.C. Leong, L.W. Jin, J.C. Chai, Transient two-phase ow and heattransfer with localized heating in graphite foams, Int. J. Therm. Sci. 49 (2010)1115e1127.

[26] V.R. Voller, C.R. Swaminathan, Treatment of discontinuity thermal conduc-tivity in control-volume solutions of phase change problems, Numer. HeatTransfer, Part B: Fundamentals 24 (1993) 161e180.

[27] S.V. Patankar, Numerical Heat Transfer and Fluid Flow. Hemisphere PublishingCorporation, 1980.

[28] O.T. Easterday, C.Y. Wang, P. Zheng, A numerical and experimental study oftwo-phase and heat transfer in a porous formation with localized heatingfrom below, Proc. ASME Heat Transfer Fluid Eng. Division 321 (1995)723e732.

[29] J.W. Klett, Process for making carbon foam, US Patent 6033506 (2000).[30] J.W. Klett, R. Hardy, E. Romine, High thermal conductivity, mesophase-pitch-

derived carbon foam: effect of precursor on structure and properties, Carbon38 (7) (2000) 953e973.

[31] C.C. Tee, J.W. Klett, D.P. Stinton, N. Yu, Thermal conductivity of porous carbonfoam, Proc. of the 24 th Biennial Conference on Carbon, Charleston, USA, 1999.

[32] C.Y. Wang, C. Beckermann, C. Fan, Numerical study of boiling and naturalconvection in capillary porous media using the two-phase mixture model,Numer. Heat Transfer, Part A 26 (1994) 375e398.

H.Y. Li et al. / International Journal of Thermal Sciences 49 (2010) 2363e2375 2375

Three-dimensional numerical simulation of fluid flow with phase change heat transfer in an asymmetrically heated porous channelIntroductionMathematical formulationsProblem descriptionGoverning equationsBoundary conditionsNumerical procedureCode validation

Results and discussionEffects of Peclet number on fluid flow and heat transferEffects of Rayleigh number on fluid flow and heat transfer

ConclusionsAcknowledgmentReferences

three-dimensional numerical simulation of fluid flow with phase change heat transfer in an...

Documents