International Communications in Heat and Mass Transfer 35 (2008) 1262–1266

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International Communications in Heat and Mass Transfer

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Computation of turbulent heat transfer in a moving porous bed using a macroscopictwo-energy equation model☆

Marcelo J.S. de Lemos ⁎, Marcelo B. SaitoDepartamento de Energia - IEME, Instituto Tecnológico de Aeronáutica - ITA, 12228-900 - São José dos Campos - SP - Brazil

☆ Communicated by W.J. Minkowycz.⁎ Corresponding author.

E-mail address: delemos@ita.br (M.J.S. de Lemos).

0735-1933/$ – see front matter © 2008 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2008.08.014

a b s t r a c t

a r t i c l e i n f o

Available online 7 October 2008

Heat transfer between phas

Keywords:TurbulencePorous bedMoving bedModeling

es in a moving porous bed is analyzed. This work proposes a set of transportequations for solving problems involving turbulent flow and heat transfer in a moving bed equipment. Thedevice is modeled as a saturated porous matrix in which the solid phase moves with a steady imposedvelocity. Additional drag terms appearing the momentum equation, as well as interfacial heat transferbetween phases, are assumed to be a function of the relative velocity between the fluid and solid phases.Turbulence transport equations are here also dependent on the speed of the solid material. Results indicatethat, as the phases attain velocities of equal order, turbulence in damped and heat transfer between solid andfluid occurs mainly by conduction mechanism.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The use of biomass in modern combustion system has called theattention worldwide for its potential substitution of non-renewablefossil fuels. Biomass pelletization and preparation for energy produc-tion may consider systems having a moving porous bed [1–6]. Theability to realistic model such systems is of great advantage to anumber of materials, food and energy production processes.

Accordingly, a turbulencemodel for flow in a fixed and rigid porousmedia has been proposed [7–9], which has been extended to non-buoyant heat transfer under local thermal equilibrium [10,11], buoyantflows [12–18], mass transfer [19] and double diffusion [20], includingapplications to channels with porous inserts [21] and baffles [22].

In addition, in an accompanying paper [23], movement of the solidphase in a porous bedwas considered. However, in [23] only isothermalflow was treated. The purpose of this contribution is to extend theprevious work on moving porous media [23], accounting now for theenergy transfer between the fluid and the moving solid matrix.

2. Macroscopic model for fixed bed

Amacroscopic formof the governing equations is obtained by takingthe volumetric average of the entire equation set. In the developmentnext, theporousmediumis considered to be rigid,fixedand saturatedbythe incompressible fluid. Asmentioned, derivation of this equation set isalready available in the literature [7–9] so that details need not to be

l rights reserved.

repeated here. Nevertheless, for the sake of completeness, the final formof the modeled equations is here presented:

j � uD ¼ 0 ð1Þ

ρ j � uDuD

/

� �� �¼ −j /hpii

� �þ μj2uD þj � −ρ/huVuVii

� �

−μ/K

uD þ cF/ρjuDjuDffiffiffiffiK

p� �

;

ð2Þ

ρj � uDhkii� �

¼ j � μ þμ t/

σk

� �j /hkii� �� �

þ Pi þ Gi−ρ/heii ð3Þ

ρj: uDheii� �

¼ j � μ þμt/

σ e

� �j /heii� �� �

þ c1Pi heiihkii

þ c2heiihkii

Gi−ρ/heii� �

ð4Þ

where the c's are constants, Pi ¼ −ρhuVuVii : juD is the productionrate of ⟨k⟩i due to gradients of uD and Gi ¼ ckρ/hkiijuDj=

ffiffiffiffiK

pis the

generation rate of the intrinsic average of k due to the action of theporous matrix.

For a fixed bed, temperatures for the fluid and solid phase aregoverned by,

ρcp

fj � uDhTf ii� �

¼ j � Keff ;f �jhT f iin o

þ hiai hTs ii−hTf ii� �

ð5Þ

0 ¼ j � Keff ;s �jhTs iin o

−hiai hTs ii−hTf ii� �

ð6Þ

Nomenclature

cF Forchheimer coefficientc's Constants in Eqs. (3) and (4)cp Specific heatD Particle diameter, size of square rod.D Deformation rate tensor, D=[∇u+(∇u)T]/2Gi Production rate of ⟨k⟩i due to the porous matrixH Distance between channel wallsk Turbulent kinetic energy per unit mass,

thermal conductivity⟨k⟩v Volume (fluid+solid) average of k⟨k⟩i Intrinsic (fluid) average of kK PermeabilityL Channel lengthp Thermodynamic pressure⟨p⟩i Intrinsic (fluid) average of pressure pPi Production rate of k due to mean gradients of uD,Re Reynolds number⟨T f⟩ Averaged fluid temperature.⟨T s⟩ Averaged fluid temperature.hi Interfacial heat transfer coefficientu Microscopic time-averaged velocity vector⟨u⟩i Intrinsic (fluid) average of uuD Darcy velocity vector, uD=ϕ⟨u⟩i

urel Relative velocity based on total volume.

Greekμ Fluid dynamic viscosityμt Turbulent viscosityμtϕ Macroscopic turbulent viscosityε Dissipation rate of k, e ¼ μjuV: juVð ÞT =ρ⟨ε⟩i Intrinsic (fluid) average of ερ Densityϕ Porosityγ Phase identifier

Subscripts,f s=solid, f=fluid

1263M.J.S. de Lemos, M.B. Saito / International Communications in Heat and Mass Transfer 35 (2008) 1262–1266

where ⟨T f⟩, ⟨T s⟩i, K's, hi and ai are the fluid and solid temperatures, the

conductivity tensors, the interfacial heat transfer coefficient and theinterfacial area per unit volume, respectively. The effective conductivetensors for the fluid and solid phases, Keff,f and Keff,s, respectively,account for all mechanisms contributing to the energy budget. Acomplete review of Eqs. (5) and (6) is beyond the scope of the presenttext and details of their derivation can be found in references [24–26].

3. Interfacial heat transfer coefficient

In Eqs. (5) and (6) the heat transferred between the two phaseswas modeled by means of a film coefficient, hi, such that:

hiai hTs ii−hTf ii� �

¼ 1ΔV

ZAi

ni � kfjTf dA ¼ 1ΔV

ZAi

ni � ksjTsdA: ð7Þ

where Ai is the interfacial area between the two phases and ai, asmentioned above, is the interfacial area per unit volume or ai=Ai/ΔV.In foam-like or cellular media, the high values of ai make themattractive for transferring thermal energy via conduction through thesolid followed by convection to a fluid stream.

For numerically determining hi, Kuwahara et al. (2001) [27]modeled a porous medium by considering it as an infinite numberof solid square rods of size D, arranged in a regular triangular pattern.They numerically solved the governing equations in the void region,exploiting to advantage the fact that for an infinite and geometricallyordered medium a repetitive cell can be identified. Periodic boundaryconditions were then applied for obtaining the temperature distribu-tion under fully developed flow conditions. A numerical correlationfor the interfacial convective heat transfer coefficient was thenproposed by Kuwahara et al. (2001) [27] for laminar flow as:

hiDkf

¼ 1þ 4 1−/ð Þ/

� �þ 12

1−/ð Þ1=2ReDPr1=3; valid for 0:2b/b0:9; ð8Þ

Results in Eq. (8) depend on porosity and are valid for packed beds ofparticle diameter D. Saito and de Lemos (2005) [24] also obtained theinterfacial heat transfer coefficient for laminar flows though an infinitesquare rod array using the samemethodology as Kuwahara et al. (2001)[27]. For turbulent flow, Saito and de Lemos (2006) [25] extended thework in [24] for ReD up to 107. Both Low Reynolds and High Reynoldsturbulence models were applied in [25]. The following expression wasreviewed in de Lemos and Saito (2008) [26]:

hiDkf

¼ 0:08ReD/

� �0:8

Pr1=3; for 1:0x104bReD/

b2:0x107; valid for 0:2b/b0:9;

ð9ÞOther correlations for determining interfacial heat transfer in fixedbeds have been published (see for example Zhukauskas [28]).

4. Macroscopic model for moving bed

Here, only cases where the solid phase velocity is kept constantwill be considered. A moving bed crosses a fixed control volume inaddition to a flowing fluid, which is not necessarily moving with avelocity aligned with the solid phase velocity. The steps below showfirst some basic definitions prior to presenting a proposal for a set oftransport equations for analyzing such systems.

A general form for a volume-average of any property φ, distributedwithin a phase γ that occupy volume ΔVγ, can be written as [29,30],

huiγ ¼ 1ΔVγ

ZΔVγ

udVγ ð10Þ

In the general case, the volume ratio occupied by phase γ will beϕγ=ΔVγ/ΔV.

If there are two phases, a solid (γ=s) and a fluid phase (γ= f),volume average can be established on both regions. Also,

/s ¼ ΔVs=ΔV ¼ 1−ΔVf =ΔV ¼ 1−/f ð11Þ

and for simplicity of notation one can drop the superscript “f “ to getϕs=1−ϕ.

As such, calling the instantaneous local velocities for the solid andfluid phases, us and u, respectively, one can obtain the average for thesolid velocity, within the solid phase, as follows,

huis ¼ 1ΔVs

ZΔVs

usdVs ð12Þ

which, in turn, can be related to an average velocity referent to theentire REV as,

uS ¼ ΔVs

ΔV

z}|{1−/ð Þ

1ΔVs

ZΔVs

usdVs

|{z}huis

: ð13Þ

Fig. 1. Porous bed reactor with a moving solid matrix.

1264 M.J.S. de Lemos, M.B. Saito / International Communications in Heat and Mass Transfer 35 (2008) 1262–1266

A further approximation herein is that the porous bed is rigid andmoves with a steady average velocity uS. Note that the condition ofsteadiness for the solid phase gives uS= uS=const where the overbardenotes, as usual in the literature, time-averaging.

For the fluid phase, the intrinsic (fluid) volume average gives, afterusing the subscript “i” also for consistency with the literature,

huii ¼ 1ΔVf

ZΔVf

udVf : ð14Þ

Both velocities can then be written as,

uD ¼ /huii; uS ¼ 1−/ð Þhuis ¼ const: ð15ÞA total-volume based relative velocity is defined as,

urel ¼ uD−uS: ð16Þ

Incorporating now in Eq. (2) a model for the Macroscopic ReynoldsStresses −ρ/huVuVii (see Ref. [7–26] for details), and assuming that arelative movement between the two phases is described by Eq. (16),the momentum equation reads,

ρ j � uDuD

/

� �� �−j � μ þ μ t/

� �juD þ juDð ÞTh in o

¼ −j /hpii� �

−μ/K

urel þcF/ρjureljurelffiffiffiffi

Kp :

ð17Þ

A corresponding transport equation for ⟨k⟩i can be written as,

ρ j � uDhkii� �h i

¼ j � μ þμt/

σk

� �j /hkii� �� �

−ρhuVuVii : juD þ ckρ/hkiijureljffiffiffiffi

Kp|{z}Gi

−ρ/heii

ð18Þ

where the generation rate due to the porous substrate, Gi, which wasincluded in Eq. (3), now depends on |urel| and reads,

Gi ¼ ckρ/hkiijurelj=ffiffiffiffiffiK:

pð19Þ

For analyzing moving beds, the energy equation for the fluid (Eq. (5)remains the same,whereas convective transport is added to the solid-phaseenergy balance Eq. (6). A modeled form for then for a moving bed reads,

ρcp

fj � uDhTf ii� �

¼ j � Keff ; f �jhT f iin o

þ hiai hTs ii−hTf ii� �

ð20Þ

ρcp

sj � uShTs ii� �

¼ j � Keff ;s �jhTs iin o

−hiai hTs ii−hTf ii� �

: ð21Þ

The interstitial heat transfer coefficient hi is also calculated bycorrelations (8) and (9) for laminar and turbulent flow, respectively.However, since the relative movement between phases is seen as thepromoter of convective heat transport from the fluid to the solid, orvice-versa, a relative Reynolds number defined as,

ReD ¼ ρjureljDμ

ð22Þ

is used in the correlations (8) and (9) instead of a Reynolds numberbased on the absolute velocity of the fluid phase. Accordingly, whenthe solid phase velocity approaches the fluid velocity, the onlymechanism for transferring heat between phases is conduction.

5. Results and discussion

In order to apply the mathematical model proposed above, anumerical example is shown next. The flow under consideration isschematically presented in Fig. 1, where a channel is completely filledwith a moving layer of a porous material. The channel shown in thefigure has length and height given by L and H, respectively. The porous

matrix moves with constant velocity uS and its temperature at theentrance in kept constant and equal to Ti,in. A fluid flows longitudinallyfrom left to right permeating through such moving porous structurewith an inlet temperature Tf,in. The channel is kept insulated at northand bottom plates. Further, results at the channel center (y=H/2) are arepresentative of uniform one-dimensional fully developed flow aftera certain developing length. The numerical method applied was theControl Volume technique [31].

Temperature are plotted in terms of their non-dimensional value,defined as,

Θs;f ¼Ts;f −Tmin

Tmax−Tminð23Þ

where the subscripts s,f refers to the solid and fluid temperature,respectively. In addition, the maximum and minimum temperatureare set as Ts,in and Tf,in, respectively.

5.1. Effect of Reynolds number, ReD

Fig. 2 shows temperature along the centerline of the channel of Fig.1for different ReynoldsnumberReD based on uD and for a slip ratio (us/uD)=0.5. As the inlet fluid velocity decreases, the fluid gets hotter and thetemperature Θf rises faster along the axial direction. Consequently, alower drop in the solid temperature is obtained. On the other hand, thecooling effect of the hot porousmaterial iswell seen for a higher value ofReD, when a greater reductionon the solid temperature occurs,mostly inthebeginningof the channel. Final equilibrium temperaturewill then bereduced with increase of the mass flow rate through the bed.

5.2. Effect of porosity, ϕ

The effect of porosity on longitudinal temperature distribution isshown in Fig. 3. The slip ratio uS/uD and the Reynolds number are keptconstant for all curves. Thefigure indicates that for a high porosity case,a smaller heat transfer area per unit volume, given by ai=4×(1−ϕ)/Dfor a porousmedium formed by square rods of sizeD (see Ref. [24–26]),decreases the total heat transfer rate from the solid material to thefluid. In suchhigh porositymedia, a longer developing length is neededfor both materials to reach thermal equilibrium. Also, for a fixedReynolds number based on uD=ϕ⟨u⟩i, an increase inϕ corresponds to areduction on the fluid velocity ⟨u⟩i, which further reduces the coolingeffect by reducing the heat transfer coefficient hi between phases.Consequently, the product hiai in Eqs. (20) and (21)will be decreased asϕ increases, which, ultimately, indicates damping of convectivetransfer through the interfacial area.

5.3. Effect of slip ratio, uS/uD

Fig. 4 shows values for the non-dimensional turbulent kineticenergy, ⟨k⟩v/|uD|2, and for the non-dimensional temperatures, Θf andΘs, along the channel mid-height. As the relative phase velocity

Fig. 2. Non-dimensional temperatures as a function of ReD for uS/uD=0.5, ρs/ρf=10,(ρcp)s/(ρcp)f=25 and ks/kf=25.

1265M.J.S. de Lemos, M.B. Saito / International Communications in Heat and Mass Transfer 35 (2008) 1262–1266

decreases, the amount of fluid disturbance past the solid obstacles isreduced, implying then in a reduction of the final level of ⟨k⟩i, ac-cording to Gi in Eq. (19) (Fig. 4a) [32]. Also, with an increase in uS/uD,the solid carries more thermal energy by convection towards theinside of the reactor, raising the solid temperature for higher inletsolid velocity (Fig. 4b). The figure indicates that when the solidvelocity approaches the fluid velocity, transfer of heat between phasestends to occur by a pure conduction mechanism, which requires alonger entry length for the two temperatures to achieve an equilib-rium value. Finally, in Fig. 4c the raise of Θf along the channel ispresented. The figure seems to indicates that there is an optimumvelocity ratio, above which the fluid temperature at the channelexit decreases. For the example here plotted, this optimal value isabout uS/uD≈0.75. This behavior could be explained by noting that theheat transfer between phases is proportional to hiaiΔT, as seen inEqs. (20) and (21). Increasing uS/uD raises Θs, Fig. 4b, while reduces theproduct hiai due to the use of a relative velocity to calculate ReD, asshown in Eq. (22). Then, these two opposing trends will compete witheach other as uS/uD is varied, indicating the two-fold behavior of thecurves in Fig. 4c.

Fig. 3. Non-dimensional temperatures as a function of porosityϕ for uS/ uD=0.5, ρs/ρf=10,(ρcp)s/(ρcp)f=25 and ks/kf=25.

Fig. 4. Effect of uS/uD for ϕ=0.6: a), ⟨k⟩v/|uD|2, ReD=5×104; b) Solid temperature, Θs,ReD=104; c) Fluid temperature, Θf, ReD=104.

1266 M.J.S. de Lemos, M.B. Saito / International Communications in Heat and Mass Transfer 35 (2008) 1262–1266

6. Conclusions

Numerical solutions for turbulent flow in a moving porous bedwere obtained for different ratios uS/uD. Governing equations werediscretized and numerically solved. Increasing the solid speed reducesthe interfacial drag forces as well as the transfer of energy betweenphases, ultimately indicating that energy transport between phasesmainly occurs due to conduction. Results herein may contribute to thedesign and analysis of engineering equipment where a moving porousbody can be identified.

Acknowledgments

The authors are thankful to CNPq and FAPESP, Brazil, for theirinvaluable support during the course of this research.

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