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International Journal of Heat and Fluid Flow 28 (2007) 1144–1156

Numerical simulation and theoretical analysis of thermal boundarycharacteristics of convection heat transfer in porous media

Pei-Xue Jiang *, Xiao-Chen Lu

Key Laboratory for Thermal Science and Power Engineering, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China

Received 24 May 2005; received in revised form 6 July 2006; accepted 9 November 2006Available online 6 February 2007

Abstract

The thermal boundary characteristics at the contact interface between a porous media and an impermeable wall subject to a constantheat flux on its upper surface with and without consideration of the thermal contact resistance were investigated using a particle-levelnumerical simulation of single phase fluid flow and convection heat transfer in porous media. The numerical simulations assumed anideal packed bed (simple cubic structure) formed by uniform diameter particles with small contact areas and a zero- or finite-thicknesswall subject to a constant heat flux at the surface which mirrors the experimental setup. The numerical simulations showed that the tem-perature distribution at the contact interface is non-uniform for the porous media with a zero-thickness impermeable wall, and in theporous media with a finite-thickness impermeable wall with a thermal contact resistance between the particles and the plate (such as witha non-sintered porous media) with a constant heat flux on the outer surface, while the heat flux distribution at the contact interface isquite uniform for these cases. However, in the porous media with a finite-thickness impermeable wall without a thermal contact resis-tance between the particles and the plate (such as with a sintered porous media) with a constant heat flux on the outer surface, the heatflux distribution at the contact interface is very non-uniform, while the temperature distribution at the contact interface is quite uniform.Numerical simulations of the thermal boundary characteristics of the convection heat transfer in the porous media were used to inves-tigate the applicability of various boundary conditions for the energy equations with and without a thermal contact resistance betweenthe particles and the plate wall.� 2006 Elsevier Inc. All rights reserved.

Keywords: Thermal boundary characteristics; Heat transfer; Porous media; Particles; Thermal contact resistance

1. Introduction

Forced convection heat transfer in porous media hasmany important applications, such as geothermal energyextraction, catalytic and chemical particle beds, petroleumprocessing, transpiration cooling, solid matrix heat exchang-ers, packed-bed regenerators, and heat transfer enhance-ment. Therefore, fluid flow and convective heat transfer inporous media have received much attention for many years.

There have been numerous theoretical and numericalinvestigations of convection heat transfer in fluid-saturated

0142-727X/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.ijheatfluidflow.2006.11.009

* Corresponding author. Tel.: +86 10 62772661; fax: +86 10 62770209.E-mail address: jiangpx@tsinghua.edu.cn (P.-X. Jiang).

porous media. The numerical investigations of convectionheat transfer in porous media have used two different mod-els for the energy equation: the local thermal equilibriummodel and the local thermal non-equilibrium model. Thelocal thermal equilibrium model assumes that the solid-phase temperature is equal to the fluid temperature, i.e.,local thermal equilibrium between the fluid and solidphases at every location in the porous media. This modelsimplifies the theoretical and numerical investigations,but the assumption of local thermal equilibrium betweenthe fluid and solid phases is inadequate for many situa-tions. In recent years the local thermal non-equilibriummodel has been used more frequently in the energy equa-tion in theoretical and numerical research on convectionheat transfer in porous media (Amiri and Vafai, 1994;

Nomenclature

a central distance between two adjacent particles(m)

cp specific heat (J/kg K)dp particle diameter (m)F inertia coefficienth convection heat transfer coefficient (W/m2 K)K porous medium permeability (m2)p pressure (Pa)q heat transfer (W/m2)T temperature (K)u, v, w velocity (m/s)u0 inlet fluid velocity (m/s)x, y, z coordinates (m)

Greek symbols

d wall thickness (mm)e porosity

k thermal conductivity (W/m K)l absolute viscosity (N s/m2)q fluid density (kg/m3)

Subscripts

0 inletC correlation resultsf fluidm meanN numerical resultso outp particles solidw wall

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1145

Jiang and Ren, 2001; Jiang et al., 1996, 2002, 2004a; Hsiehand Lu, 2000; Alazmi and Vafai, 2002) to more accuratelymodel the convection heat transfer processes in porousmedia.

With the thermal non-equilibrium model, the treatmentof the boundary conditions for the energy equation signif-icantly affects the simulation results. A number of papershave analyzed this problem, e.g. Amiri et al. (1995), Quin-tard (1998), Peterson and Chang (1998), Martin et al.(1998), Lee and Vafai (1999), Jiang and Ren (2001), Jianget al., 2002, 2004a,b and Alazmi and Vafai (2002). For theconstant heat flux boundary condition, there are severaldifferent methods to treat the energy equation boundaryconditions. The main differences among them are whetherthe fluid temperature is equal to the solid phase tempera-ture at the boundary surface and how to determine the heatflux transferred through the solid phase and the fluid. Ala-zmi and Vafai (2002) analyzed 8 different forms of the con-stant wall heat flux boundary conditions in the absence oflocal thermal equilibrium conditions in porous media.They found that different boundary conditions may leadto substantially different results. At the same time, theypointed out that selecting one model over the others isnot an easy issue since previous studies validated each ofthe two primary models. In addition to that, the mechanicsof splitting the heat flux between the two phases is not yetresolved.

Jiang (2001) numerically analyzed the heat transferboundary characteristics between the surface of a thin plateand a solid particle in contact with each other with convec-tion heat transfer due to water or air in the voids betweenthe solids and a constant heat flux on the outer plate wall.The analysis studied the influence of the thermal contactresistance between the plate wall and the solid particleand the wall thickness on the surface thermal characteris-

tics. The results showed that if the thermal conductivitiesof the fluid and the solid particle were similar, the temper-ature and heat flux distributions on the contact surfacebetween the thin plate and the particle and on the convec-tion heat transfer surface of the thin plate were quite uni-form, which means that local thermal equilibrium existsfor such conditions. If the thermal contact resistance onthe wall surface between the solid particles and the platewall is considered, the solid particle temperature on thecontact surface and the fluid temperature on the convectionheat transfer surface of the thin plate near the particle aresignificantly different, while the heat fluxes transferredthrough the solid particles and the fluid adjacent to the sur-face are similar. If the thermal contact resistance on thewall surface between the solid particles and the wall isnot considered and the plate is relatively thick (e.g. 1–2 mm), the heat fluxes through the solid particles and thefluid near the wall are significantly different, while the tem-peratures of the solid particles and the fluid near the wallare fairly close, that is Tfw � Tsw.

Jiang and Ren (2001) and Jiang et al. (2002, 2004a)investigated the various treatments of the constant heatflux boundary conditions by comparing numerical resultswith experimental data for forced convection heat transferof water and air in plate channels filled with non-sinteredmetallic or non-metallic particles (packed beds) or sinteredporous media using a non-thermal equilibrium model. Theresults showed that for numerical simulations of convec-tion heat transfer in porous media, the boundary conditionmodel significantly influences the numerical results and theproper model must be chosen for the numerical results toagree well with the experimental data. Simulations of con-vection heat transfer of air or water in non-sintered packedbeds using a local thermal non-equilibrium model with theassumption that the heat fluxes transferred by the solid

1146 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

phase and the fluid from the heat transfer surface were thesame with a variable porosity model agreed well with theexperimental data. For numerical simulations of convec-tion heat transfer in sintered porous media, the particletemperatures and the fluid temperatures on the wall surfacewere assumed to be equal. Simulations of convection heattransfer of air or water in sintered bronze porous mediausing the local thermal non-equilibrium model and the var-iable porosity model with the wall effect corresponded wellto experimental data. The convection heat transfer coeffi-cient in sintered porous media is much higher than thatin non-sintered packed beds due to the reduced thermalcontact resistance and reduced porosity near the wall insintered porous media.

Kim and Kim (2001) investigated the heat transfer phe-nomenon at the interface between a porous medium and animpermeable wall subject to a constant heat flux at the sur-face. The focus of their paper was to determine which ofthe thermal boundary conditions is more appropriate foraccurately predicting the heat transfer characteristics in aporous channel. They numerically examined the heat trans-fer at the interface between a microchannel heat sink(assumed to be an ideally organized porous medium) anda finite-thickness substrate. The results clarified that theheat flux distribution at the interface was not uniform foran impermeable wall with finite thickness. Thus, a non-uni-form heat flux distribution and a uniform temperature dis-tribution at the interface were determined to be the mostphysically realistic.

Jiang and Lu (2006) presented particle-level simulationsof convection heat transfer in a simple cubic structureformed by particles with the same diameters with smallcontact areas and a finite-thickness wall subject to a con-stant heat flux at the surface. The permeability and inertiacoefficient were calculated numerically according to themodified Darcy’s model. The numerical results were inagreement with well-known correlation results with the cal-culated local heat transfer coefficients on the plate channelsurface agreeing well with the experimental data. The con-vection heat transfer coefficients between the solid particlesand the fluid and the volumetric heat transfer coefficients inthe porous media from the simulation increase withincreasing mass flow rate and decrease with increasing par-ticle diameter. The particle-level direct simulations of thefluid flow and convection heat transfer in porous mediaestablish a good basis for investigations of the boundarycharacteristics and internal phenomena controlling the heattransfer in porous media.

The main purpose of the present study is to analyze thethermal boundary characteristics for convection heat trans-fer in porous media to better understand the heat transferphenomena and boundary conditions at the interface onthe basis of the previous works in Jiang (2001) and Jiangand Lu (2006). Numerical simulations were performedfor forced convection heat transfer in a simple cubic (SC)structure formed by uniform diameter particles with smallcontact areas and a zero- or finite-thickness substrate wall

subject to a constant heat flux at the surface. The temper-ature differences between the fluid and solid phases in theporous media were examined along with the influence ofthe heated wall with and without a thermal contact resis-tance between the particles and the wall on the convectionheat transfer in the porous media. The proper models forthe boundary conditions for the energy equations wereidentified for flows with and without a thermal contactresistance between the particles and the wall.

2. Physical model, governing equations and numerical

method

Particle-level direct numerical simulations of the fluidflow and convection heat transfer in porous media as inJiang and Lu (2006) were used to numerically investigatethe thermal boundary characteristics for convection heattransfer in porous media. A fully numerical solution ofthe Navier–Stokes equations for the convection heat trans-fer in a porous media with all the particles is very difficultdue to the huge amount of calculation time. Therefore, alimited number of identical particles was used to modelthe entire porous media with two layers of 5 · 5 particlesas the porous media. The physical model and the coordi-nate system are shown in Fig. 1. The calculational regionincluded an entrance region with a length of 11 dp and anexit region with a length of 6 dp. The packed bed was madeof homogeneous, uniform-sized solid particles. The smallchanges of porosities were obtained by changing the parti-cle sizes and contact areas between the particles. The topsurface of the porous media was touching a heated wallwhose top surface received a constant heat flux, qwo, whilethe bottom surface of the channel was thermally insulated.Fluid entered the channel with a uniform velocity, u0, andconstant temperature, Tf0. The fluid flow was steady, sin-gle-phase and three-dimensional. The heated wall wasassumed to have either zero thickness or a finite thicknessof 1 mm. For the porous media with the finite thicknessheated wall, cases with and without a thermal contact resis-tance between the particles and the wall were considered.

Table 1 presents the dimensions and porosities of theporous media used in the calculations. ew is the wall poros-ity which is equal to one minus the ratio of the contact areaof the particles with the plate to the plate surface area. Thecontact areas between the particles and the wall surface andthe contact areas between particles were assumed to beconstant based on the desired porosity. The centerline dis-tance between adjacent particles, a, for actual sintered por-ous media is a little less than the particle diameter due tothe sintering technology.

With non-sintered porous media (packed beds), a ther-mal contact resistance is known to exist between the parti-cles and between the particles and the wall. These finitecontact resistances are due principally to surface roughnesseffects (Incropera and DeWitt, 1996). The contact resis-tance may then be viewed as two parallel resistances dueto the contact spots and due to the gaps. The contact area

Table 2Thermophysical properties

Material q (kg/m3) k (W/m K) cp (J/kg K)

Wall Copper 8954 398 384

Particle Bronze 8666 54 343Glass 2500 0.76 840

Fluid Water 1000 0.6 4183

x

y

qwo

u0

Fig. 1. Schematic diagram of the physical system.

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1147

is typically small, and especially for rough surfaces, themajor contribution to the resistance is due to the gaps.The surface roughness for metal plates is usually 1–30 lm(Incropera and DeWitt, 1996; Cengel and Turner, 2001).According to Incropera and DeWitt (1996), the thermalcontact resistance of an aluminum interface with interfacialair is 2.75 · 10�4 m2 K/W. However, it is very difficult toaccurately determine the thermal contact resistancebetween particles and the wall in a porous media. Fornon-sintered porous media, the joint pressure between theparticles is usually not large, so the gap between the parti-cles is reasonably assumed to be larger than between two

Table 1Dimensions and porosities of the porous media

dp = 1.7 mm dp = 1.2 mm

a (mm) em ew a (mm) em

1.68 0.458 0.973 1.18 0.1.64 0.420 0.934 1.16 0.1.75 0.520 1.0 1.25 0.

plates at high pressure. Jiang (2001) assumed a 50 lmgap between two adjacent particles for non-sintered porousmedia with the resulting thermal contact resistance being2.08 · 10�3 m2K/W which is 7.6 times that between plates.The purpose of the present research is to investigate thethermal boundary characteristics in sintered and non-sin-tered porous media. The qualitative results are moreimportant, and the value of the thermal contact resistancedoes not change the qualitative results and conclusions.Therefore, the present paper for non-sintered porous mediaalso used a 50 lm gap between adjacent particles with theresults having a thermal contact resistance at the particleinterfaces with water of 8.33 · 10�5 m2K/W which is rea-sonable for thermal contact resistances. As shown in Table1, the centerline distance between two adjacent particles, a,is larger than the particle diameter for non-sintered porousmedia.

The calculations assume that the wall is made of copperand the heat flux at the top of the wall is 1 · 105 W/m2.Water flows through the porous media and removes heatfrom the porous media and the impermeable wall. Thewater inlet velocity is 1 mm/s. In the calculations all thephysical properties are assumed to be constant. The valuesof the thermophysical properties used in the calculation arelisted in Table 2.

The three-dimensional conjugate heat transfer problemwas solved using the CFD software FLUENT 6.1. TheReynolds numbers based on the particle diameters andthe inlet velocity for the three geometries listed in Table 1were 1.6, 1.2 and 0.6. Laminar flow was assumed in the cal-culations. The SIMPLE algorithm was used to couple thepressure and the velocities with the second order upwindmethod used for the advection terms in the momentumequations and for the energy equation.

Energy equation for the upper copper plate:

o2T w

ox2þ o2T w

oy2þ o2T w

oz2¼ 0 ð1Þ

dp = 0.6 mm

ew a (mm) em ew

450 0.965 0.59 0.450 0.965423 0.937 0.58 0.423 0.937537 1.0 0.65 0.588 1.0

1148 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

Energy equation for the solid particles:

o2T s

ox2þ o2T s

oy2þ o2T s

oz2¼ 0 ð2Þ

Mass, momentum and energy equations for the fluid:

qouoxþ ov

oyþ ow

oz

� �¼ 0 ð3Þ

q uouoxþ v

ouoyþ w

ouoz

� �¼ � op

oxþ l

o2uox2þ o2u

oy2þ o2u

oz2

� �ð4Þ

q uovoxþ v

ovoyþ w

ovoz

� �¼ � op

oyþ l

o2vox2þ o2v

oy2þ o2v

oz2

� �ð5Þ

q uowoxþ v

owoyþ w

owoz

� �¼ � op

ozþ l

o2wox2þ o2w

oy2þ o2w

oz2

� �ð6Þ

qcp uoToxþ v

oToyþ w

oToz

� �¼ k

o2Tox2þ o2T

oy2þ o2T

oz2

� �ð7Þ

Fig. 2 shows the grid distribution in the porous media.The unstructured mesh used about 400,000 elements. Eachparticle had 400–800 nodes with 2000–4000 elements. Thespaces between adjacent particles had about 500 nodes.Calculations with various numbers of elements showedthat the results were grid independent. The convergencecriteria required a decrease of at least 6 orders of magni-tude for the residuals with no observable change in the sur-face temperatures for an additional 200 iterations.

The reliability of the calculations was verified in Jiangand Lu (2006) where the Reynolds numbers based on the

Fig. 2. Grid distribution in the po

particle diameters (dp = 1.7, 1.2 and 0.6 mm) and the inletvelocity were 4.57–231, so the fluid flow in the porousmedia was in the laminar, transitional and turbulent flowregions. The k–x turbulent model was used in the numeri-cal calculations for the turbulent flow. The calculations fora porous medium of glass and water predicted an effectivethermal conductivity for the porous media of 0.668 W/(m K) for em = 0.354 which agreed well with the value fromthe well-known equations proposed by Zehener (1970) ofkm = 0.699 W/(m K) for the same porosity of em = 0.354.Comparisons of the predicted values of K and F from thenumerical results and those calculated using well-knowncorrelations (Ergun, 1952; Vafai, 1984, 1986) for the threeparticle diameters showed that the numerical results agreewithin 15% of the predictions except for a relatively largedeviation in F for the 0.6 mm diameter particles. Jiangand Lu (2006) also compared the calculated local heattransfer coefficients for convection heat transfer of waterin the sintered bronze porous plate channels with the exper-imental data of Jiang et al. (2004b) for 1.7, 1.2 and 0.6 mmdiameter particles with three different velocities for eachparticle size. The numerical results agreed well with theexperimental data, indicating that the particle-level directnumerical simulations can model the fluid flow and heattransfer in the complete porous media. The small devia-tions of the numerical results from the correlations andthe experimental data may have been caused by differencesbetween the mean porosities in the numerical model and inthe experimental porous plate channel and by the non-uni-form particle diameters, non-uniform porosities and ran-dom particle distributions in the experimental geometry.

rous media for dp = 1.7 mm.

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1149

3. Numerical results and theoretical analysis

3.1. Temperature distributions in the porous media

The two-dimensional temperature distributions in thex–y-plane in the porous media with the zero and finite-thickness walls with and without thermal contactresistances are shown in Fig. 3. As mentioned above, thethermal contact resistance between particles and betweenthe particles and the wall in non-sintered porous mediawas modeled using a 50 lm gap filled with water betweenthe solid surfaces. The porous media temperature increaseswith increasing x and decreases with increasing distancefrom the contact interface. With the zero-thickness wallwithout thermal contact resistance between the particles,the solid particle temperature differs greatly from the adja-cent fluid temperatures near the wall (Fig. 3a). Close to thecontact surface between the wall and the upper row of solidparticles, the solid particle temperatures are lower than thenearby fluid temperature. With the finite-thickness wallwithout thermal contact resistance, the wall temperatureis quite uniform due to its high thermal conductivity(Fig. 3b). With the finite-thickness wall with thermal con-tact resistance (Fig. 3c), the wall temperature varies withthe temperatures at the contact points with the solid parti-cles being much lower than that of the areas in contact withthe fluid. For this case, the heat is not easily transferredinto the channel from the top surface, so the fluid temper-atures in the channel are much lower than the particletemperatures.

Fig. 4 shows the two-dimensional temperature distribu-tion at the wall–particle interface between the solid parti-cles and the wall for the three sets of conditions. For theporous media with zero-thickness wall without thermalcontact resistances, the solid temperature is much lowerthan the nearby fluid temperatures at the contact interfacedue to the large thermal conductivity differences betweenthe fluid and the solid particles (Fig. 4a). The temperatureincreases along the fluid flow direction as the fluid temper-ature increases. For the porous media with the finite-thick-ness wall without thermal contact resistances, thetemperature distribution is much more uniform althoughthe fluid temperatures are still lower than the solid temper-atures at adjacent locations (Fig. 4b). For the porous mediawith the finite-thickness heated wall with thermal contactresistances, the solid temperatures are also much lowerthan the nearby fluid temperatures at the contact interface,but the difference between the solid and fluid temperaturesis less than for the zero-thickness wall due to heat conduc-tion in the wall (Fig. 4c).

Fig. 5 shows the contact interface temperature distribu-tions for the three conditions. Here, Tw1 is the temperatureon the wall–particle contact surface along a line parallel tothe x-axis in the flow direction across the fluid and solidphase contact points at the interface (line 1 in Fig. 4), whileTw2 is the wall surface temperature along a line betweenadjacent particles (line 2 in Fig. 4). Tf is the bulk fluid tem-

perature. For the porous media with zero-thickness wallwithout thermal contact resistances, the fluid and solidtemperatures at the contact interface differ greatly for thisporous media with a large thermal conductivity differencebetween the fluid and the solid particles. For the porousmedia with finite-thickness heated wall without thermalcontact resistances, the temperature distribution is muchmore uniform, while for the porous media with finite-thick-ness wall with thermal contact resistances, the solid temper-ature is again much less than the nearby fluid temperaturesat the contact interface. The difference between the solidand fluid temperatures is less than the case with the zero-thickness wall due to heat conduction in the heated wall.Comparing the temperature distributions at the contactsurface in the porous media for zero and finite-thicknesswalls without thermal contact resistances shows that theconvection heat transfer coefficient with the finite-thicknesswall is larger than with the zero-thickness wall. Therefore,the heated wall increases the fin effect in the porous mediawhich intensifies the overall convection heat transfer. How-ever, the thermal contact resistance between the particlesand the wall reduces the overall convection heat transfer.

The temperature distributions show that the heat trans-fer coefficients in the porous media without thermal con-tact resistances are much larger than those in the porousmedia with thermal contact resistances which is consistentwith the experimental results in Jiang et al. (2004b).

3.2. Heat flux distributions at the contact interface in the

porous media

Fig. 6 shows the heat flux distributions at the wall–par-ticle interface in the porous media with a finite wall thick-ness without thermal contact resistances for variousporosities. The solid line is the heat flux distribution onthe wall–particle contact surface along a line parallel tothe x-axis across the fluid and solid phases at the interface(line 1 in Fig. 4), while the dotted line is the heat flux dis-tribution on the wall surface along a line between adjacentparticles (line 2 in Fig. 4). With a constant wall heat fluxboundary condition on the top wall surface, the heat fluxestransferred into the fluid and the solid phases at adjacentlocations on the contact interface are very different. Thefluid and solid temperatures at adjacent locations are alsonot the same as shown in Fig. 5, but the temperature differ-ences are much less than the heat flux differences at the con-tact interface shown in Fig. 6. The contact surface areabetween the particles and the wall is bigger in Fig. 6b thanin Fig. 6a due to smaller porosity. As shown in Fig. 6a andb, the heat flux to the solid phase is much more than that tothe fluid phase because of the large difference in the thermalconductivities. In Fig. 6a, the heat flux at the top wall sur-face is 1 · 105 W/m2, while the maximum heat flux to thesolid phase reaches 6.6 · 106 W/m2 with the heat flux tothe fluid phase of only about 1.6 · 104 W/m2. Similarly,in Fig. 6b, the maximum heat flux to the solid phasereaches 2.2 · 106 W/m2, while the heat flux to the fluid

Fig. 3. Temperature distributions in the porous media with zero and finite-thickness heated walls with and without thermal contact resistances(u0 = 1 mm/s, qwo = 1 · 105 W/m2, dp = 1.7 mm, Tf0 = 300 K, em = 0.458). (a) Zero-thickness wall without thermal contact resistances, (b) finite-thicknesswall without thermal contact resistances, (c) finite-thickness wall with thermal contact resistances.

1150 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

Fig. 4. Temperature distributions at the wall–particles interface with zero and finite-thickness heated walls with and without thermal contact resistances(u0 = 1 mm/s, qwo = 1 · 105 W/m2, dp = 1.7 mm, Tf0 = 300 K, em = 0.458). (a) Zero-thickness wall without thermal contact resistances, (b) finite-thicknesswall without thermal contact resistances, (c) finite-thickness wall with thermal contact resistances.

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1151

phase is only about 1.3 · 104 W/m2. Therefore, the maxi-mum heat flux to the solid phase increases with increasingporosity; although, the total heat transferred through thesolid phase increases with decreasing porosity due to thelarger contact surface area.

Fig. 7 shows the heat flux distributions at the wall–par-ticle interface with a finite wall thickness with thermal con-tact resistances for the constant wall heat flux boundaryconditions on the top wall surface. The heat fluxes trans-ferred by the fluid and the solid at adjacent locations on

the contact interface are very similar for this case withthe heat flux differences being much less than for the casewithout thermal contact resistances.

3.3. Boundary conditions for the energy equations

in porous media

These results have shown that the thermal contact resis-tance at the wall between the solid particles and the wallsignificantly affect the heat fluxes through the solid parti-

0.00 1.68 3.36 5.04 6.72 8.40300

320

340

360

380

400 Tw1

Tw2

Tf

T(K

)

x (mm)

0.00 1.68 3.36 5.04 6.72 8.40300

310

320

330

340

350

360

370

Tw1

Tw2

Tf

T (

K)

x (mm)

0.00 1.75 3.50 5.25 7.00 8.75300

320

340

360

380

400

420

440

Tw1

Tw2

Tf

T(K

)

x (mm)

Fig. 5. Temperature distributions at the wall–particle interface with zeroand finite-thickness heated walls with and without thermal contactresistances (u0 = 1 mm/s, qwo = 1 · 105 W/m2, dp = 1.7 mm, Tf0 =300 K, em = 0.458). (a) Zero-thickness wall without thermal contactresistances, (b) finite-thickness wall without thermal contact resistances,(c) finite-thickness wall with thermal contact resistances.

0.00 1.68 3.36 5.04 6.72 8.400.1

1

10

100

q w/q

wo

x (mm)

x (mm)

qw1

qw2

0.00 1.64 3.28 4.92 6.56 8.200.1

1

10

100

q w/q

wo

qw1

qw2

Fig. 6. Heat flux distributions at the wall–particle interface in the porousmedia with a finite-thickness wall without thermal contact resistances forvarious porosities (u0 = 1 mm/s, qwo = 1 · 105 W/m2, dp = 1.7 mm, Tf0 =300 K). (a) em = 0.458, (b) em = 0.420.

0.00 1.68 3.36 5.04 6.72 8.400.1

1

10

100

qw/q

wo

x (mm)

qw1

qw2

Fig. 7. Heat flux distribution at the wall–particle interface with a finite-thickness wall with thermal contact resistances (u0 = 1 mm/s, qwo = 1 ·105 W/m2, dp = 1.7 mm, Tf0 = 300 K, em = 0.458).

1152 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

cles and the fluid near the wall. Without the thermalcontact resistance, the temperatures of the solid particlesand the fluid near the wall are quite similar, while the heatfluxes are quite different. However, when the thermal con-

tact resistance at the wall between the solid particles andthe wall is considered, the particle temperatures at thecontact surface are quite different from the fluid tempera-tures in the pores near the wall, while the heat fluxes

Table 3Comparison of the thermal boundary characteristics for different cases in a porous media of bronze and water: ks/kf = 54/0.6 = 90

dp (mm) em ew �qs

qwo

� �N

�qf

qwo

� �N

�qs

qwo

� �C

�qf

qwo

� �C

�kmoToy

� �f ;w;N

qwo

qw;N

qwo

1.7 0.458 0.973 34.3 0.14 26.4 0.294 1.03 1.061.7 0.420 0.934 11.3 0.11 13.1 0.145 0.892 0.8491.2 0.450 0.965 25.8 0.13 21.9 0.243 0.982 1.031.2 0.423 0.937 12.8 0.12 13.6 0.151 0.965 0.9190.6 0.450 0.965 30.9 0.14 21.9 0.243 1.07 1.220.6 0.423 0.937 15.3 0.13 13.6 0.151 1.05 1.091.7C 0.520 0.973 1.40 0.98

Note: superscript ‘‘C’’ indicates the case with thermal contact resistance.

qw;N ¼ �ewkfoT f

oy

� �w;N

� ð1� ewÞksoT s

oy

� �w;N

:

Table 4Comparison of the thermal boundary characteristics for different cases in a porous media of glass and water: ks/kf = 0.76/0.6 = 1.27

dp (mm) em ew �qs

qwo

� �N

�qf

qwo

� �N

�qs

qwo

� �C

�qf

qwo

� �C

�kmoToy

� �f ;w;N

qwo

qw;N

qwo

1.7 0.458 0.973 11.5 0.70 1.26 0.993 1.05 0.9921.7 0.420 0.934 4.71 0.62 1.24 0.983 0.945 0.8901.2 0.450 0.965 9.90 0.66 1.25 0.991 1.04 0.9831.2 0.423 0.937 5.51 0.60 1.25 0.983 0.958 0.9090.6 0.450 0.965 12.9 0.63 1.25 0.991 1.10 1.060.6 0.423 0.937 7.19 0.58 1.25 0.983 1.03 0.9961.7C 0.458 0.973 1.02 1.00

Note: superscript ‘‘C’’ indicates the case with thermal contact resistance.

qw;N ¼ �ewkf

oT f

oy

� �w;N

� ð1� ewÞks

oT s

oy

� �w;N

:

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1153

through the particles and through the fluid are quitesimilar. Tables 3 and 4 present the calculated heat fluxesat the wall–particle interface (bronze and glass).

As mentioned in the Introduction, various methods totreat the energy equation boundary conditions can be usedfor the constant heat flux boundary condition on the outersurface of porous channels. The main differences arewhether the fluid temperature is equal to the solid phasetemperature at the boundary surface and how to determinethe heat flux transferred through the solid phase and thefluid. Alazmi and Vafai (2002) analyzed 8 different formsof constant wall heat flux boundary conditions. They foundthat different boundary conditions may lead to substantiallydifferent results and selecting one model over the others isnot an easy issue. Analysis of the different forms of the con-stant wall heat flux boundary conditions showed that thefollowing most accurately represents the actual conditions.

qw ¼ eqf þ ð1� eÞqs;qf

qs

¼ kf

ks

ð8Þ

These can be rearranged to give:

qs ¼ks

ekf þ ð1� eÞks

qw; qf ¼kf

ekf þ ð1� eÞks

qw ð9Þ

Tables 3 and 4 also compare the wall heat fluxes betweenthe solid particles and the wall calculated using Eq. (9). Ifthe thermal contact resistance is not considered (as insintered porous media) for the porous media of bronzeand water, the results calculated using Eq. (9) are similarto the numerical results, while for the porous media madeof glass and water, the results calculated using Eq. (9)are very different from the numerical results. Therefore,the model for the thermal boundary conditions, Eq. (9),is not universal. When thermal contact resistances areincluded (as in non-sintered porous media), the heat fluxestransferred by the fluid and the solid phases are very close.

Fig. 8 presents the average wall temperatures of theparts contacting the fluid (Twf) and the solid particles(Tws), the average wall temperatures at the wall–particleinterface (Tw), temperatures inside the wall (0.5 mm fromthe top surface, Twp, which corresponds with the thermo-couples in the experiments) and the fluid temperature (Tf)in the porous media for a finite wall thickness with andwithout thermal contact resistance. Without the thermalcontact resistances (Fig. 8a), the average wall temperaturesof locations in contact with the fluid (Twf) and the solidparticles (Tws), and the average wall temperature at the

0.00 1.68 3.36 5.04 6.72 8.40300

310

320

330

340

350

360

370

0.00 1.68 3.36 5.04 6.72 8.40300

310

320

330

340

350

360

370

Tws

T (

K)

x (mm)

T (

K)

x (mm)

Twf

Tw

Twp

Tf

0.00 1.75 3.50 5.25 7.00 8.75300

320

340

360

380

400

420

440

Tws

x (mm)

T (

K)

Twf

Tw

Twp

Tf

Fig. 8. Average wall temperatures on the wall surface and the fluidtemperature in the porous media with a finite wall thickness with andwithout thermal contact resistances (u0 = 1 mm/s, qwo = 1 · 105 W/m2,dp = 1.7 mm, Tf0 = 300 K, em = 0.458). (a) Finite-thickness wall withoutthermal contact resistances, (b) finite-thickness wall with thermal contactresistances.

0.00 1.68 3.36 5.04 6.72 8.400.1

1

10

100

0.00 1.68 3.36 5.04 6.72 8.400.1

1

10

100

qws

x (mm)

q w/q

wo

qwf

qw

0.00 1.75 3.50 5.25 7.00 8.750.1

1

10

100

0.00 1.75 3.50 5.25 7.00 8.750.1

1

10

100

qws

q w/q

wo

x (mm

q/q

wo

x (mm)

qwf

qw

Fig. 9. Average heat flux distributions on the wall surface in the porousmedia with a finite wall thickness with and without thermal contactresistances (u0 = 1 mm/s, qwo = 1 · 105 W/m2, dp = 1.7 mm, Tf0 = 300 K,em = 0.458). (a) Finite-thickness wall without thermal contact resistances,(b) finite-thickness wall with thermal contact resistances.

1154 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

contact interface (Tw) are very similar. However, with thethermal contact resistances between the particles and onthe wall (Fig. 8b), the average temperatures Twf, Tws andTw are quite different.

Fig. 9 shows the average heat flux distributions on thewall of locations in contact with the fluid (qwf) and the solidparticles (qws), and the overall average heat flux distribu-tion on the surface (qw) in the porous media with a finitewall thickness with and without thermal contact resis-tances. Without thermal contact resistances (Fig. 9a), theaverage heat fluxes through the solid phase (qws) and fluid(qwf) at the contact interface are very different, with a non-uniform overall average heat flux at the wall surface (qw).With thermal contact resistances between the particlesand on the wall surface (Fig. 9b), the average heat fluxesof the solid phase (qws) and fluid (qwf) at the contact inter-face are very similar. The general average heat fluxes of thesolid phase (qws) and the fluid (qwf) at the wall surface arefairly similar to the heat flux on the outer surface, and theoverall average heat flux at the contact interface (qw) isquite uniform.

As shown by Jiang (2001, 2004a), there are primarilyfour methods for treating the constant wall heat fluxboundary conditions for the energy equations:

Amiri et al. (1995) and Jiang et al. (1996, 1999):

qw ¼ �kfwðoT f=oyÞw ¼ �ksðoT f=oyÞw ð10Þ

Martin et al. (1998):

qw ¼ �ðekf þ ð1� eÞksÞwðoT=oyÞw; T ws ¼ T wf ð11Þ

Lee and Vafai (1999):

qw ¼ �ewkfwðoT f=oyÞw � ð1� ewÞksðoT s=oyÞw; T ws ¼ T wf

ð12Þ

Jiang et al. (2004a):

qw ¼ �kmðoT=oyÞw; T ws ¼ T wf ð13Þ

Where the effective thermal conductivity of the porousmedia can be calculated from the well-known equationsproposed by Zehener (1970):

P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156 1155

km

kf

¼ ½1�ffiffiffiffiffiffiffiffiffiffiffi1� ep

�

þ 2ffiffiffiffiffiffiffiffiffiffiffi1� ep

1� rBð1� rÞBð1� rBÞ2

ln1

rB

� �� Bþ 1

2� B� 1

1� rB

" #

ð14Þ

where,

B ¼ 1:25ðð1� eÞ=eÞ10=9; r ¼ kf=ks

Previous analyses by Jiang and Ren (2001) and Jianget al. (2002, 2004a) showed that numerical simulations ofconvection heat transfer of air and water in non-sinteredpacked beds using a local thermal non-equilibrium modelwith the assumption that the heat fluxes into the solidand the fluid from the heat transfer surface were the same(as Eq. (10)) with a variable porosity model agreed wellwith the experimental data. For numerical simulations ofconvection heat transfer in sintered porous media, theboundary conditions on the wall should be that the particletemperatures are equal to the fluid temperatures (as in Eqs.(11)–(13)). Jiang et al. (2004a) showed that Eq. (13) gavesatisfactory results that compared well with experimentaldata for sintered porous media.

Tables 3 and 4 list the calculated results for �km(oT/oy)f,w,N/qw0 where km was calculated using Eq. (14) withthe mean porosity em. The values are quite close to 1 forboth types of the porous media with bronze and glass par-ticles. Tables 3 and 4 also list the calculated results for

�ewkfoT f

oy

� �w;N� ð1� ewÞks

oT s

oy

� �w;N

� ��qw0. From the

physical meaning, the value of �ewkfoT f

oy

� �w;N�

�1� ewð Þks

oT s

oy

� �w;N

��qw0 should be exactly equal to 1, so

the departure from 1 most likely indicates numerical errors.

Therefore, the results indicate that the distribution ofthe heat fluxes between the solid phase and the fluid atthe contact interface without thermal contact resistance isrelated to the ratio of the thermal conductivities of the solidand the fluid, the ratio of the thermal conductivities of theplate and the solid particles, the ratio of the wall thicknessto the particle diameter, the ratio of the wall porosity to themean porosity and the Biot number. Therefore, the propermodel for the constant wall heat flux boundary conditionsin porous media with a finite wall thickness without ther-mal contact resistance is

qw ¼ eqf þ 1� eð Þqs ð15Þqs

qf

¼ f ðks=kf ; kw=ks; ew=em; d=dp; hd=kwÞ ð16Þ

However, this is very complicated. From the abovemen-tioned results the following models should be a goodapproximation for the constant wall heat flux boundaryconditions in porous media with a finite wall thicknesswithout thermal contact resistance (as in sintered porousmedia):

qw ¼ �ewkfwðoT f=oyÞw � ð1� ewÞksðoT s=oyÞw; T ws ¼ T wf

ð17Þor

qw ¼ �kmðoT=oyÞw; T ws ¼ T wf ð18ÞFor convection heat transfer in porous media with a

finite wall thickness with thermal contact resistances (asin non-sintered porous media), the numerical simulationsin the present research and previous experimental andnumerical investigations in Jiang and Ren (2001) and Jianget al. (1996, 2002) suggest that the thermal boundary con-ditions for the constant wall heat flux boundary conditionon the outer surface should be

qw � qf � qs ð19Þ

4. Conclusions

(1) For porous media with a zero-thickness heated wall,the fluid and solid temperatures at the wall surfacedepend strongly on the thermal conductivity ratiobetween the fluid and solid particles. For porousmedia with a finite-thickness heated wall withoutthermal contact resistances, the temperature distribu-tion between the solid and fluid near the wall is quiteuniform, while for porous media with a finite-thick-ness heated wall with thermal contact resistances,the solid temperature is much lower than the nearbyfluid temperature near the wall surface.

(2) For the constant wall heat flux boundary conditionson the top wall surface without thermal contact resis-tance, the heat fluxes into the fluid and the solidphases at adjacent locations on the contact interfaceare quite different. The maximum heat flux to thesolid increases with increasing porosity, while thetotal heat transferred by the solid increases withdecreasing porosity due to the larger contact surfacearea with the smaller porosities. However, the heatflux distributions at the wall surface in the porousmedia with a finite wall thickness with thermal con-tact resistances are quite uniform.

(3) A good approximation for the constant wall heat fluxboundary condition for convection heat transfer inporous media with a finite wall thickness withoutthermal contact resistance (as in sintered porousmedia) is Tws � Twf, while the proper model for theconstant wall heat flux boundary condition for con-vection heat transfer in porous media with a finitewall thickness with thermal contact resistances (asin non-sintered porous media) is qw � qf � qs.

Acknowledgement

The project was supported by the National OutstandingYouth Fund from the National Natural Science Founda-tion of China (No. 50025617). We thank Prof. DavidChristopher for editing the English.

1156 P.-X. Jiang, X.-C. Lu / Int. J. Heat and Fluid Flow 28 (2007) 1144–1156

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