International Journal of Heat and Mass Transfer 67 (2013) 164–172

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effect of endothermic reaction mechanisms on the coupled heat andmass transfers in a porous packed bed with Soret and Dufour effects

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.004

⇑ Corresponding author. Tel.: +86 13604187687.E-mail address: liming-799@163.com (M. Li).

Mingchun Li ⇑, Yusheng Wu, Zhongliang ZhaoSchool of Material Science and Engineering, Shenyang University of Technology, Shenyang 110023, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 November 2011Received in revised form 3 April 2013Accepted 3 August 2013Available online 4 September 2013

Keywords:Porous packed bedHeat and mass transferReaction mechanismsCross-diffusion effectsNon-thermal equilibrium

Taking the influences of chemical reaction mechanisms into account, a mathematical model has beendeveloped to simulate the coupled heat and mass transfer as well as endothermic reactions in a non-ther-mal equilibrium packed bed with Soret and Dufour effects, which was numerically solved and validatedby comparing with experimental data. It was found that the chemical reaction mechanisms have a greatinfluence on the calculated solutions, and the cross-diffusion effects on the reactive characteristics of thepacked beds are diverse under different chemical reaction mechanism. For the heat transfer controlledmechanism, the maximum reductions of the concentration of gas product and the solid fractional conver-sion induced by the Soret and Dufour effects are 25.1% and 14.4% respectively at superficial velocity53.3 cm s�1 and the Nusselt number 4.1. However for the mass transfer controlled model, the calculatedsolid fractional conversions are higher than that obtained by neglecting the cross-diffusion effects and theincrease being less than 16% at superficial velocity 11.9 cm s�1 and the Sherwood number 47.1. The dif-ferences induced by the Soret and Dufour effects are demonstrated numerically to increase graduallywith the Nusselt number or the Sherwood number.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Heat and mass transfer with chemical reactions in porouspacked beds is of considerable importance in chemical and metal-lurgical industries, such as hydrodesulfurization multiphasereactors, flue gas denitrification reactors, enzymatic acidolysisreactors, catalytic conversion reactors, ore calcinations furnaces,etc. The reaction conversion processes are mainly influenced bydiffusion and convection, and in turn the heat and mass transfercan be altered tremendously by chemical reactions. Dependingon the chemical reaction occurs whether at a phase interface oras a single phase volume reaction, which can be classified as heter-ogeneous and homogeneous processes. In general, most researchesrelated the heat and mass transfer in porous media with chemicalreactions focus on the transport processes within catalytic porousmedia or the first-order reaction systems [1,2]. Chao et al. [3]investigated the heat and mass transfer of chemically-reactivesteady stagnation flows in catalytic porous beds according toArrhenius kinetics and Singular perturbation analysis. Ulson DeSouza et al. [4] used a two scale model to study the mass transferin a packed bed with a heterogeneous chemical reaction by takinginto account dispersion in the main fluid phase, internal diffusionof the reactant in the pores of the catalyst, and surface reaction

inside the catalyst. They employed volume averaging and Darcy’slaw for a spatially periodic porous medium. Beg et al. [5] examinedthe steady double-diffusive free convective heat and mass transferof a chemically-reacting micropolar fluid flowing through a Dar-cian porous regime adjacent to a vertical stretching plane. A sim-plified first order homogenous reaction model was used tosimulate the chemical reaction in the flow. In their analysis, Chem-ical reaction was shown to decelerate the flow and also micro-rota-tion values, in particular near the wall. Alam et al. [6] investigatedthe effects of thermophoresis and the homogeneous first orderchemical reactions on magneto-hydrodynamic mixed convectiveflow past a heated inclined permeable flat plate in the presenceof heat generation or absorption. Zueco et al. [7] investigated thefree convection boundary layer flow and heat and mass transferacross an isothermal cylinder embedded in an isotropic, homoge-nous, saturated porous regime also with a first-order chemicalreaction in the diffusing species. An exponent-function kineticmodel was constructed in their work for the simulation of the cat-alytic coupling reaction. The effect of first order chemical reactionand thermal radiation on hydromagnetic free convection heat andmass transfer flow of a micropolar fluid via a porous mediumbounded by a semi-infinite porous plate with constant heat sourcein a rotating frame of reference was studied by Das et al. [8].

Most of the above studies only considered the rate of thechemical reaction itself based on the micro-kinetics using simplereaction model. However, it is known that the actual chemical

Nomenclature

a, b, c stoichiometric coefficientsc1, c2 mass fraction of species C and inertial gas in the main

flowing gasC0, C0 concentration of species C at the surface of solid pellets

and the equilibrium concentration (kg m�3)Cp,1, Cp,d specific heat of species C and inertial gas (kJ mol�1 K�1)d particle diameter (m)D, Dse effective diffusivities of gas product in packed bed and

solid product layer (m2 s�1)D0, D00 thermal diffusion coefficient and the Dufour coefficient

(m2 s�1 K�1)fs solid fractional conversionGA moles of reactant A (mol)hd interfacial mass transfer coefficient (m s�1)hf interfacial heat transfer coefficient (W m�2 K�1)DH enthalpy of dissociation (kJ mol�1)kf fluid conductivity (W m�1 K�1)L reactor length (m)L1q, Lq1 cross-diffusion coefficients (kg K m�1 s�1)M1, M2, Ms molecular mass of species C, inertial gas and solid

reactant (g mol�1)Nu the Nusselt numberNc diffusion rate of species C across solid product layer and

gas film (kg s�1)Pe the Peclet numberPCO2 equilibrium dissociation pressure (Mpa)qc heat transfer rate (W)�qw dimensionless constant heat flux at the wallr coordinate variable in radial direction (m)rc un-reacted core radius (m)rin initial pellet radius (m)rs radius of limestone pellets (m)Pe dimensionless radial coordinate, r/R

R cylindrical reactor diameter (m)RA overall reaction rate of a single particle (mol s�1)Rgas the gas constant (J K�1 mol�1)

RðiÞREV overall reaction rate on the scale of REV (mol s�1)i superscript indicates the reaction mechanism, i = 1 ,2SREV surface area of REV (m2)Sh the Sherwood numbert time (s)�t dimensionless time scale, �t ¼ tD=L2

T temperature of environment (K)Tb, Tc, Ts temperatures of the flowing gas, the unburnt core and

the surface of solid pellets (K)Tin initial thermal gas temperature (K)Th the Thiele numberT; Tc; Ts dimensionless temperatures of the flowing gas, the un-

burnt core and the surface of solid pelletsv effective velocity of flowing gas in porous media (m s�1)vb superficial velocity of fluid (m s�1)VREV volume of REV (m3)x longitudinal coordinate (m)�x dimensionless axial length scale, x/L

Greek symbolse porosityep porosity of limestone pelletske effective thermal conductivity of the solid product

(W m�1 K�1)l dynamic viscosity (kg s�1 m�1)q density of the mixture gas (kg m�3)qs density of the solid reactant (kg m�3)qp density of limestone pellets (kg m�3)

M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172 165

reactions in packed beds are accompanied by the heat and masstransfer processes through gas film and solid product layers inaddition to pure chemical reaction [9]. To predict accurately theflow behavior and reaction rate, it is necessary to take macro-kinetics reaction mechanisms into account. The analysis of the heatand mass transfer in porous media becomes complicated whenmultiphase chemical reactions related to heat and (or) massexchanges between the flowing fluid and the porous solid matrix.Maikov and Director [10] studied the mass transfer processes inpolydisperse porous media and investigated the kinetic regime ofmethane pyrolysis in a porous carbon skeleton consideringexternal and internal diffusion resistances for different initialdistributions of particles forming the porous medium. Valipour[11] presented a mathematical model to simulate the multipleheterogeneous reactions with complex set of physicochemicaland thermal phenomena in a moving bed of porous pellets. Daggu-pati [12] examined a solid conversion process during hydrolysisand decomposition of cupric chloride in a fluidized bed reactor,in which the reaction rate was determined by shrinking-coremodel. Sahir [13] presented the kinetics of copper oxidation inthe air reactor of achemical looping combustion system using thelaw of additive reaction times, which took into account the porediffusion of gaseous species in the interior of the porous oxygencarrier but only appropriate when pore diffusion is controlling.Although these studies take the multiphase chemical reactionsand the transfer processes together into account, but no consider-ation about the cross-diffusion effects and no discussions on theinfluences of chemical reaction mechanisms were given in thesestudies.

The roasting, reduction and decomposition of ores are theimportant kinds of endothermic gas–solid reactions employedextensively in metallurgy and chemical industries [14]. These ther-mal decomposition reactions with gaseous and solid products arequite complex reactions influenced by many factors, includingthe chemical reaction itself, diffusion transport on different scalesin the porous solid, external mass transfer, heat transfer, etc. So thecross-coupled effects among the heat and mass transfer as well asthe endothermic reactions must be considered as the temperaturegradient and the concentration gradient exist simultaneously.

The energy flux caused by a composition gradient is termed theDufour or diffusion-thermo effect. On the other hand, mass flux canalso be created by temperature gradients and this embodies theSoret or thermal-diffusion effect. Ecket and Drake [15] presentedseveral cases that the Dufour effect can’t be neglected. A primarydiscussion on the effect of the cross-coupled diffusion in a systemwith horizontal temperature and concentration gradients wasmade by Malashetty et al. [16]. Benano-Melly et al. [17] analyzedthe heat diffusion of a binary fluid mixture in a porous mediumwith a horizontal thermal gradient. Adrian Postelnicu [18] ana-lyzed numerically the heat and mass transfer characteristics of nat-ural convection about a vertical surface embedded in a saturatedporous medium subjected to a chemical reaction, by taking into ac-count the Dufour and Soret effects. Mansour et al. [19] investigatedthe effects of chemical reaction, thermal stratification, Soret num-ber and Dufour number on MHD free convective heat and masstransfer of a viscous, incompressible and electrically conductingfluid on a vertical stretching surface embedded in a saturated por-ous medium. Ibrahim et al. [20] discussed the effect of chemical

166 M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172

reaction on free convection heat and mass transfer for a non-New-tonian power law fluid over a vertical flat plate embedded in afluid-saturated porous medium in the presence of the yield stressand the Soret effect.

Literature review shows that no study on the influences of ki-netic reaction mechanisms in a non-thermal equilibrium porouspacked bed with endothermic reaction, forced convection and thecross-diffusion effects were reported so far. In the present paper,the coupled heat and mass transfer in a non-thermal equilibriumpacked bed considering Soret and Dufour effects are studied underdifferent macro-kinetics reaction mechanisms, in accordance withthe dictates of thermodynamics of irreversible processes, reactionengineering principles and the local thermal non-equilibriummodel.

2. Modeling and formulation

2.1. Problem description

A cylindrical reactor filled with spherical specimens subjects toa constant heat flux boundary condition as shown in Fig. 1(a). Thediameter of the packed bed is 2R, and the length of which is L.The solid sphere particles are of uniform shape and undeformable.The high temperature feeding gas flows through the bed by forcedconvective and exchanges heat with the solid matrix. The endo-thermic decomposition reactions take place as the particle temper-ature rises. The gas product diffuses from the reactive interfaceinto the main stream under the co-acting of the pressure differenceand the temperature difference. Here, the natural convection andthe radiation heat transfer are neglected, and the flowing gas is as-sumed to be incompressible.

2.2. Reaction kinetics

The present model describes the kinetic and thermal behaviorof solid pellets in packed bed, in which a heterogeneous endother-mic reaction of the type given by Eq. (1) takes place.

aAðsolidÞ �heating

bBðsolidÞ þ cCðgasÞ ð1Þ

The overall processes contain essentially heat and mass transportsteps and chemical reaction at interface as is schematically shownin Fig. 1(b), which can be summarized as the following several stages:(1) Heat transfer from environment to the surface of solid particle. (2)Heat transfer from the particle’s external surface to the reactioninterface, the process of the heat conductivity through the solidproduct layer is assumed a pseudo-steady state. (3) Endothermicdecomposition reaction begins at the reaction interface, and the tem-perature of the unreacted core remained constant. (4) Gas product C

Fig. 1. The packed bed with endothermic reaction and the

diffuses through the porous solid layer, which extends from the reac-tion interface to the particle’s exterior. (5) Diffusion of gas product Cfrom particle’s exterior surface to the environment. Appearance of anoverall decomposition process substantially changes with differentcontributions of each elemental step.

2.2.1. Heat transfer controlled schemeConsider a spherical sample of radius rin and at an initial

temperature Tc, be suspended in a large furnace maintained at atemperature Tb. Based on the additive theory of the transferresistances, the amount of heat that reaches the reactive interfacefrom environment of the pellet is given by [21]

qc ¼ 4pr2inðTb � TcÞhf=½1þ ðhf rin=keÞðrin=rc � 1Þ� ð2Þ

Defining the modified Nusselt number Nu ¼ ðhf rin=keÞ.Assuming the entire amount of heat transferred from the envi-

ronment to the sample was used up for the endothermic reac-tion(qc = RADH), and the initial temperature of gas product is thesame to the interface temperature. Then the overall reaction rateof a single pellet RA in heat transfer controlled regime (i = 1) canbe given as follows

RA ¼ 4pr2inðTb � TcÞhf= ½1þ ðhf rin=keÞðrin=rc � 1Þ�DHf g ð3Þ

In a packed bed, the minimal unit is the representative elemen-tary volume (REV). The surface area of REV can be written asfollows

SREV ¼ VREV3ð1� eÞ=rin ð4Þ

According to Eqs. (3) and (4), the overall reaction rate of REV in heattransfer controlled regime (i = 1) can be derived as:

Rð1ÞREV ¼ VREV½3ð1� eÞ=rin�ðTb

� TcÞhf= ½1þ ðhf rin=keÞðrin=rc � 1Þ�DHf g ð5Þ

The quantity Rð1ÞREV denotes the production mole of the gas product Cper unit time in REV, which equals to the exhaustive mole of reac-tant A per unit time in REV – (c/a) dGA/dt, that is

VREV½3ð1� eÞ=rin�ðTb � TcÞhf

½1þ ðhf rin=keÞðrin=rc � 1Þ�DH¼ � c

aVREVð1� eÞ

4pr3in=3

ð4pr2cqsÞ

Ms

drc

dtð6Þ

Separating the variables in Eq. (6) and integrating gives the positionof the calcination front rc as a function of time and the surroundingtemperature.

½ahf Ms=ðcrinDHqsÞ�Z t

0ðTb � TcÞdt

¼ ½1� ðhf rin=keÞ�½1� ðrc=rinÞ3�=3þ ðhf rin=keÞ½1� ðrc=rinÞ2�=2 ð7Þ

decomposition scheme for a single spherical specimen.

M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172 167

Noting that fs ¼ 1� r3c=r3

in, where fs is the solid fractional conversionof REV at time t, Eq. (7) may be restated as follows

½ahf Ms=ðcrinDHqsÞ�Z t

0ðTb � TcÞdt

¼ ½1� ðhf rin=keÞ�fs=3þ ðhf rin=keÞ½1� ð1� fsÞ2=3�=2 ð8Þ

2.2.2. Mass transfer controlled schemeOn the basis of pseudo-steady state assumption, the rate of dif-

fusion of the gas product C away from the decomposition interfacemay be given by [21]

NC ¼ 4pDseðC0 � C0Þ½rinrc=ðrin � rcÞ� ð9Þ

The diffusion rate of gas product C from particle’s exterior surface toenvironment is given by

NC ¼ 4pr2inhdðC0 � qc1Þ ð10Þ

Eqs. (9) and (10) may be solved for C0

C0 ¼ ½DseC 0rinrc=ðrin � rcÞ þ r2inhdqc1�=½r2

inhd þ Dserinrc=ðrin � rcÞ�ð11Þ

Substituting the value of C0 in Eq. (10) and rearranging, the overallreaction rate of a single pellet RA in mass transfer controlled regimecan be given as follows

RA ¼ 4pr2inhdðC0 � qc1Þ=½1þ ðhdrin=DseÞðrin=rc � 1Þ� ð12Þ

Defining the modified Sherwood number Sh = (hdrin/Dse). Hence theoverall reaction rate of REV in mass transfer controlled regime(i = 2) can be given by

Rð2ÞREV ¼ VREV½3ð1� eÞ=rin�hdðC 0 � qc1Þ=½1þ ðhdrin=DseÞðrin=rc � 1Þ�ð13Þ

Similar to the derivation of Eq. (7), the position of the calcinationfront rc in mass transfer controlled regime as a function of timeand the surrounding concentration are obtained as follows

½aMs=ðcqsÞ�Z t

0ðC 0 �qc1Þdt¼ rin½1�ðrc=rinÞ3�=3hdþðr2

in=DseÞf½1

�ðrc=rinÞ2�=2�½1�ðrc=rinÞ3�=3g ð14Þ

Fig. 2. Comparison of the calculated results under different chemical reactionmechanisms with the experimental data in literature [14]. (Tc = 1073 K; ep = 0.09;qp = 2.47 � 103 kg m�3; vb = 1.32 cm s�1; e = 0.47) —calculated results, D0 = D00 = 0,i = 1; ——calculated results, D0 = D00 = 0, i = 2; s calculated results,D0 = D00 = 0.0008 cm2 s�1 K�1, i = 1; r calculated results, D0 = D00 = 0.0008 cm2 s�1 -K�1, i = 2.

2.3. Conservation equations

The model is two-dimensional and transient. The variables andparameters are therefore functions of the axial position in thepacked bed x, the radial position in the packed bed r and time t.Throughout what follows, the variables v(r), c1(x, r, t), Tb(x, r, t),Ts(x, r, t), rc(x, r, t) and fs(x, r, t) will be termed principal variables,and are calculated respectively from the above reaction kineticsmodels and the conservation equations given below.

The mass balance for gas product C considering the cross diffu-sion effects on the scale of REV is given by

eq@c1

@t¼ eq

1r@

@rrD@c1

@r

� �þ e

@

@xðqc1c2D0gradT þ qDgradc1Þ

� eqm@c1

@xþ RðiÞREV=VREV ð15Þ

The energy conservation equation of the main stream consider-ing the cross diffusion effects is given by

eq@½ð1� c1ÞCp;d þ c1Cp;1�Tb

@t¼ e

1r@

@rrkf

@Tb

@r

� �þ e

@

@xðkf gradTb

þ qc1lc11TbD00gradc1Þ

� emq@½ð1� c1ÞCp;d þ c1Cp;1�Tb

@x� Svhf ðTb � TsÞ� ðRðiÞREV=VREVÞCP;1ðTb � TcÞ; ð16Þ

RðiÞREV=VREV ¼½3ð1�eÞ=rin�ðTb�TcÞhf=f½1þðhf rin=keÞðrin=rc�1Þ�DHg; i¼1½3ð1�eÞ=rin�hdðC0 �qc1Þ=½1þðhdrin=DseÞðrin=rc�1Þ�; i¼2

�ð17Þ

The effective velocity v may be determined from the superficialvelocity vb and the porosity of packed beds v = vb/e, and the Ergun–Forchheimer–Brinkman equation [22] is used for analyzing thefluid flow in the porous packed bed,

�dPdx¼ 150

lmb

d2

ð1� eÞ2

e3 þ 1:75qfm2

b

dð1� eÞ

e3 � le

1r

ddr

rdmb

dr

� �ð18Þ

The surface temperature of the solid pellets Ts as a function ofsurrounding temperature Tb and the calcination front rc is givenby the following relationship which takes into account the heattransfer resistances of both the external film and the solid productlayer [22].

Ts ¼ ½ðhf rin=keÞTb þ rcTc=ðrin � rcÞ�=½hf rin=ke þ rc=ðrin � rcÞ� ð19Þ

The function of the calcination front rc included in Eqs. (15),(16), (17), and (19) depend on the reaction kinetics model chosen.The expression of Eq. (7) is used in the heat transfer controlledmodel, while that of Eq. (14) is used in the mass transfer controlledmodel.

The following dimensionless variables are introduced for nor-malizing the governing equations

�t¼ tD=L2; �x¼ x=L; �r¼ r=R; T ¼ Tb=T in; Tc ¼ Tc=T in; Ts ¼ Ts=T in

ð20Þ

Utilizing Eq. (20), the dimensionless governing balance equation ofEqs. (15)–(17) can be rewritten as:

@c1

@�t¼ L2

R2�r

@

@�r�r@c1

@�r

� �þ @

2c1

@�x2 þD0T in

D@

@�xc1c2

@T@�x

!

� mLD@c1

@�xþ RðiÞREV=VREV

� �ð21Þ

168 M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172

@½1þ ðCp;1=Cp;d � 1Þc1�T@�t

¼ L2

R2

1qCp;dD

1�r@

@�r�rkf

@T@�r

!þ kf

qCp;dD@2T@�x2

þ D00

DCp;d

@

@�xðc1lc

11T@c1

@�xÞ

� LmD@½1þ ðCp;1=Cp;d � 1Þc1�T

@�x

� L2

DeqCp;dSvhf ðT � TsÞ

� ðRðiÞREV=VREVÞCP;1ðT � TcÞ

Cp;dð22Þ

RðiÞREV=VREV ¼½3ð1� eÞ=rin�L2hf TinðT � TcÞ=fDeq½1þ ðhf rin=keÞðrin=rc � 1Þ�DHg; i ¼ 1

�½3ð1� eÞ=rin�L2hdðqc1 � C0Þ=fDeq½1þ ðhdrin=DseÞðrin=rc � 1Þ�g; i ¼ 2

(ð23Þ

Let Pe = vL/D, Th=R(i)REVL2/(Deq), Pe is the modified Peclet number

which represents the relative convection velocity and diffusionvelocity, Th is the Thiele modulus that reflects the relative endo-thermic chemical reaction rate and diffusion velocity.

The initial and boundary conditions for the problem are:

c1ð�x;�r;0Þ ¼ 0; Tð�x;�r;0Þ ¼ Tc; rcð�x;�r; 0Þ ¼ rin; c1ð0;�r;�tÞ ¼ 0;

Tð0;�r;�tÞ ¼ 1; @c1ð�x;0;�tÞ=@�r ¼ 0; @Tð�x; 0;�tÞ=@�r ¼ 0;

@c1ð1;�r;�tÞ=@�x ¼ 0; @Tð1;�r;�tÞ=@�x ¼ 0; @c1ð�x;1;�tÞ=@�r ¼ 0;

@Tð�x;1;�tÞ=@�r ¼ �qw

2.4. Expression of parameters

The specific heats, density and the enthalpy of reaction areknown for a given reaction and constituents. This work takes thelimestone decomposition as an example for computation. Theother parameters are determined as follows.

The heat transfer coefficient hf and mass transfer coefficient hd

through the stagnant gas film are calculated from the followingequations [23],

hf ¼ kf ½2:0þ 1:1ðCp;1l=kf Þ1=3ðqmd=lÞ0:6�=d;

hd ¼ D½2:0þ 0:6ðlq=DÞ1=3ðqmd=lÞ1=2�=d ð24Þ

The Soret and Dufour coefficients are expressed by [24]

D0 ¼ L1q=ðqc1c2T2Þ; D00 ¼ Lq1=ðqc1c2T2Þ ð25Þ

Fig. 3. Influences of the superficial velocity on the concentration gradient and the D

the Dufour coefficient is equal to the Soret coefficient in value forthe same research system by the Onsager’s reciprocity relationL1q = Lq1, and the value of which is 0.0008 cm2 s�1 K�1 in this workcorresponding to the conditions of the flowing gas in the fixed bedof limestone calcinations at 1373 K.

lc11 ¼ ðRgasTM1Þ=fc1½M1 � c1ðM1 �M2Þ�g ð26Þ

In the present work, the equilibrium concentration C0is calcu-lated from the equilibrium dissociation pressure of gaseous prod-uct (carbon dioxide) over the solid pellets as follows [25]

log pCO2¼ �8792:3=T þ 10:4; C 0 ¼ pCO2

=RgasT ð27Þ

3. Results and discussion

The ADI (alternate dimension implicit) method [26,27] wasused to solve numerically the two-dimensional mathematicalmodel established in previous sections. Computations have beencarried out for a half of the cylindrical packed bed (L � R). The re-sults are independent of the time step size Dt = 0.01 s and thenumber of grids 50 � 40. L = 0.18 m, R = 0.045 m, rin = 0.004 m.

3.1. Model validation

A comparison is made among the results of the experimentaldata in literature [14] and the two groups of numerical solutionsobtained by the established mathematical model under differentchemical reaction mechanisms as shown in Fig. 2. It can be seenin Fig. 2 that the chemical reaction mechanisms have a great influ-ence on the simulated solutions. The results calculated by the heattransfer controlled regime are in good agreement with the experi-mental data, especially at the initial and middle stages of thedecomposition, the maximum relative error being less than13.5%. But as the solid fractional conversion exceeds 80%, the max-imum relative error between the simulation results and experi-mental data increases to 23.6%. In contrast, the simulation resultsunder mass transfer control mechanism are much higher thanthe experimental results at the initial and middle stages of thethermal decomposition and the maximum relative error reaches43.4%, while in good agreement with the experimental results atthe latter stage (fs > 80%) with the maximum relative error beingless than 12.1%. This shows that with the continuously thickening

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

D'=D''=0

---- D'=0, D''=0.0008 cm2.s-1.K-1f s

x/L

1- vb=18.7 cm.s-1

2- vb=37.9 cm.s-1

3- vb=53.3 cm.s-1

3

21

(b). Solid fractional conversion along axial direction

ufour effect. (i = 1; Tc = 1170 K; Tin = 1373 K; e = 0.42; t = 120 s; Nu = 1.9; �r ¼ 0).

Fig. 4. Influences of the Soret and Dufour effects on the temperature fields for both the flowing gas and the solid matrix, the concentration distribution of the product gas andthe solid fractional conversion at different Nusselt number. (i = 1; Tc = 1170 K; Tin = 1373 K; e = 0.42; t = 120 s; vb = 53.3 cm s�1; �r ¼ 0).

Fig. 5. Influence of the Nusselt number on the ratio of Th/Pe in the packed bed.(i = 1; Tc = 1170 K; Tin = 1373 K; e = 0.42; t = 120 s; vb = 53.3 cm s�1; �r ¼ 0).

M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172 169

of the solid product layer, the controlling step of reaction rate isalso continually changing. For the heat transfer controlled model(i = 1), the solid fractional conversions calculated considering thecross-diffusion effects are lower than those obtained by neglectingthe cross-diffusion effects and the decrease being less than 10%.However for the mass transfer controlled model (i = 2), the solidfractional conversions calculated considering the cross-diffusioneffects are higher than that obtained by neglecting the cross-diffu-sion effects and the increase being less than 12.6%. Hence, it is nec-essary to discuss and identify the specific chemical reactionmechanism for an endothermic or exothermic reaction systemwith gas product during the researches of the cross-diffusioneffects.

3.2. Discussion with heat transfer controlled scheme

The temperature of the flowing gas in the pores of packed bedsis a key factor affecting the endothermic reaction rate under heattransfer control mechanism (i = 1). The Dufour coefficient D00 de-notes the heat transfer resulting from concentration gradient,which affects chiefly the temperature distribution and chemicalreaction rate as i = 1. Fig. 3 shows the variation of the concentrationof gas product andthe solid fractional conversion with the superfi-cial velocity vb. As seen from Fig. 3, for heat transfer control mech-anism, the concentration gradient of the gas product increaseswith the superficial velocity and the decrease in the solid fractionalconversion caused by the Dufour effect intense accordingly. Themaximum relative error of the solid fractional conversion increasesfrom 4.7% to 11.1% as the superficial velocity increases from 18.7 to53.3 cm s�1.

Fig. 4 shows the variation of the temperature fields of the flow-ing gas and the solid matrix, the concentration distribution of thegas product and the conversion ratio of the solid pellets with theNusselt number and the cross-diffusion coefficients. FromFig. 4(a)–(c) we can see that, as the Nusselt number increases,

the temperature field of the flowing gas decreases, while the con-centration distribution of gas product and the surface temperaturefield of the solid pellets increase. This is expected since a higherNusselt number can improve the quantity of heat exchange be-tween the flowing gas and the solid matrix, which results in the de-crease in the temperature of the flowing gas and the increase in thedecomposition rate near the entrance correspondingly. It was alsofound that the solid conversion degree increases near the entranceas the Nusslet number increases. But with the region where thedecomposition reaction takes place deepens down to the exit, anincrease in the Nusslet number results in an opposite effect forthe solid conversion degree (see Fig. 4(d)). Fig. 5 portrays the vari-ations of the ratio of Th/Pe with spatial coordinate under differentNusselt number, which means that relative to the ability of heat

170 M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172

and mass transfer, the ability of endothermic chemical reaction in-creases in the upstream of the packed bed, whereas decreases inthe downstream as the Nusselt number increases. The overall reac-tion rate is determined by the product of the effective heat transfercoefficient and the temperature difference between the twophases. Hence, the influences of the Nusselt number on the endo-thermic reaction rate are diverse at different spatial coordinate.

The effect of the Nusslet number on the cross-diffusion effects isalso shown in Fig. 4. A higher Nusselt number results in a largerdifference between the two groups of results calculated separately

Fig. 6. Influences of the superficial velocity on the temperature gradient and the So

Fig. 7. Influences of the Soret and Dufour effects on the temperature fields of flowing gasdifferent Sherwood number.(i = 2; Tc = 1170 K; Tin = 1373 K; e = 0.42; t = 120 s; vb = 11.9

at D0 = D00 = 0 and D0 = D00 = 0.0008 cm2 s�1 K�1. As the Nusselt num-ber increases from 0.93 to 4.1, the temperature gradient of the bulkstream and the concentration gradient of the gas product in thepacked bed both increase, which improve the Soret and Dufour ef-fects. As a result, the maximum relative error of the concentrationof the gas product increases from 3.2% to 25.1% and that of the so-lid fractional conversion increases from 2.6% to 14.4%. Because thepositive thermal diffusion coefficient implies the mass transferresulting from higher temperature to lower temperature, theexisting temperature gradient (see Fig. 4(a)) in the endothermic

ret effect. (i = 2; Tc = 1170 K; Tin = 1373 K; e = 0.42; t = 120 s; Sh = 23.5; �r ¼ 0).

, the concentration distribution of gas product and the solid fractional conversion atcm s�1; �r ¼ 0).

M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172 171

reaction packed bed improves the transmission speed of the gasproduct to downstream which results in the decrease of the con-centration fields (see Fig. 4(b)). The positive Dufour coefficient D00

means the heat transfer resulting from higher concentration tolower concentration. As the Dufour effect is taken into account,the heat transmission rate is reduced, which leads to the lowertemperatures for both the flowing gas in the pore and the solid ma-trix as well as a lower endothermic reaction rate (see Fig. 4(d)).

3.3. Results with mass transfer controlled scheme

For the mass transfer controlled scheme (i = 2), the concentra-tion of gas product in the bulk flow is a key factor affecting theendothermic reaction rate. The Soret coefficient D0 denotes themass transfer resulting from temperature gradient, which chieflyaffects the concentration distribution and the reaction rate. Fig. 6shows the variation of the temperature gradient and the solid frac-tional conversion with the superficial velocity vb. As seen fromFig. 6, the temperature gradient of the flowing gas decreases withthe superficial velocity and the increase in the solid fractional con-version induced by the Soret effect decreases accordingly. Themaximum relative error of the solid fractional conversion increasesfrom 2.7% to 13.7% as the superficial velocity decreases from 47.4to 11.9 cm s�1.

The influences of Soret and Dufour effects on the reaction char-acteristics, the heat and mass transfer in the packed bed at differ-ent Sherwood number are compared in Fig. 7. As seen from Fig. 7(a) and (b), the concentration of gas product and the temperatureof the flowing gas both decrease due to the additional mass andheat transfer caused by cross-diffusion effects. Hence, the solidconversion fraction calculated considering the cross-diffusion ef-fects are higher than that obtained by neglecting the cross-diffu-sion effects as i = 2 (see Fig. 7 (c)). From the comparison of thesolid curves (D0 = D00 = 0) and the dashed lines (D0 = D00 =0.0008 cm2 s�1 K�1) at different Sherwood number in Fig. 7, theauthors find that the higher Sherwood number is, the larger effectsof the Soret and Dufour coefficients are. Under the mass transfercontrolled mechanism, a high Sherwood number results in a highdecomposition rate meaning the increases in both the thermal gra-dient of flowing gas and the concentration gradient of gas product.As a result, the Soret and Dufour effects are intensified. The maxi-mum relative error of the concentration of gas product increasesfrom 7.2% to 17.7% and that of the solid fractional conversion in-creases from 6.5% to 15.8%, as the Sherwood number increasesfrom 11.8 to 47.1.

4. Conclusion

Through the chemical reaction kinetics, the thermodynamics ofirreversible processes and the assumption of local thermal non-equilibrium, a two dimensional mathematical model describingthe coupling among the heat transfer, the mass transfer and theendothermic reaction in a non-thermal equilibrium packed bedwas established and solved. The influences of the Soret and Dufoureffects under different chemical reaction mechanisms werediscussed.

The influence of the cross-diffusion effects on the reactive char-acteristics of packed beds is diverse at different chemical reactionmechanisms. The solid fractional conversion calculated consider-ing the cross-diffusion effects are higher than that obtained byneglecting the cross-diffusion effects under mass transfer con-trolled mechanism. While the cross-diffusion effects results in anopposite effect for the solid fractional conversion under heat trans-fer controlled mechanism. The weakened heat transfer and theintensified mass transfer caused by the Soret and Dufour effects

lead to a decrease in the temperature of the flowing gas and areduction in the concentration of gas product. The differences be-tween the results including the temperature fields, the concentra-tion of gas product and the solid fractional conversion calculatedseparately at D0 = D00 = 0, D0 = D00 = 0.0008 cm2 s�1 K�1, are all inten-sified by the increase either in the Nusselt number or in the Sher-wood number. Whatever the chemical reaction mechanism, theignored Soret and Dufour effects willcause much error for theendothermic reaction packed bed with high reaction rate.

Acknowledgments

This work was financially supported by the National NaturalScience Foundation of China (51004071), the Special Project forHigh-end CNC machine tools and basic manufacturing equipmentof China (2012ZX04007-021) and Liaoning Province College Excel-lent Young Talents Fund Project (LJQ2013012).

References

[1] J.W. Veldsink, R.M.J. van Damme, G.F. Versteeg, W.P.M. van Swaaij, The use ofthe dusty-gas model for the description of mass transport with chemicalreaction in porous media, Chem. Eng. J. 57 (1995) 115–125.

[2] D.N. Jaguste, S.K. Bhatia, Simulation of reaction and transport in catalystparticles with partial external and internal wetting, Int. J. Heat Mass Transfer38 (1995) 1443–1455.

[3] B.H. Chao, H. Wang, P. Cheng, Stagnation point flow of a chemically reactivefluid in a catalytic porous bed, Int. J. Heat Mass Transfer 39 (1996) 3003–3019.

[4] S.M.A.G. Ulson De Souza, S. Whitaker, Mass transfer in porous media withheterogeneous chemical reaction, Braz. J. Chem. Eng. 20 (2003) 191–199.

[5] O.A. Beg, R. Bhargava, S. Rawat, H.S. Takhar, T.A. Beg, A study of steadybuoyancy-driven dissipative micropolar free convection heat and masstransfer in a Darcian porous regime with chemical reaction, Nonlinear Anal.Model. Control 12 (2007) 157–180.

[6] M.S. Alam, M.M. Rahman, M.A. Sattar, Effects of chemical reaction andthermophoresis on magneto-hydrodynamic mixed convective heat and masstransfer flow along an inclined plate in the presence of heat generation and(or) absorption with viscous dissipation and Joule heating, Can. J. Phys. 86(2008) 1057–1066.

[7] J. Zueco, O.A. Bég, T.A. Bég, H.S. Takhar, Numerical study of chemically reactivebuoyancy-driven heat and mass transfer across a horizontal cylinder in a high-porosity non-Darcian regime, J. Porous Med. 12 (2009) 519–535.

[8] K. Das, Effect of chemical reaction and thermal radiation on heat and masstransfer flow of MHD micropolar fluid in a rotating frame of reference, Int. J.Heat Mass Transfer 54 (2011) 3505–3513.

[9] S. Homma, S. Ogata, J. Koga, S. Matsumoto, Gas–solid reaction model for ashrinking spherical particle with unreacted shrinking core, Chem. Eng. Sci. 60(2005) 4971–4980.

[10] I.L. Maikov, L.B. Director, Mathematical simulation of mass transfer processesunder heterogeneous reactions in polydisperse porous media, Chem. Eng. Sci.62 (2007) 1388–1394.

[11] M.S. Valipour, Y. Saboohi, Modeling of multiple noncatalytic gas–solidreactions in a moving bed of porous pellets based on finite volume method,Heat Mass Transfer 43 (2007) 881–894.

[12] V.N. Daggupati, G.F. Naterer, K.S. Gabriel, Diffusion of gaseous productsthrough a particle surface layer in a fluidized bed reactor, Int. J. Heat MassTransfer 53 (2010) 2449–2458.

[13] A.H. Sahir, J.S. Lighty, H.Y. Sohn, Kinetics of copper oxidation in the air reactorof a chemical looping combustion system using the law of additive reactiontimes, Ind. Eng. Chem. Res. 50 (2011) 13330–13339.

[14] E.T. Turkdogan, R.G. Olsson, H.A. Wriedt, L.S. Darken, Calcination of limestone,Trans. Soc. Min. Eng. AIME 254 (1973) 9–21.

[15] E.R.G. Eckert, R.M. Drake, Analysis of Heat and Mass transfer, McGraw HillKogakusha, New York, 1972.

[16] M.S. Malashetty, S.N. Gaikad, Effect of cross diffusion on double diffusive in thepresence of horizontal gradients, Int. J. Eng. Sci. 40 (2002) 773–787.

[17] L.B. Benano-Melly, J.P. Caltagirone, B. Faissat, F. Montel, P. Costeseque,Modeling Soret coefficient measurement experiments in porous mediaconsidering thermal and solutal convection, Int. J. Heat Mass Transfer 44(2001) 1285–1297.

[18] A. Postelnicu, Influence of chemical reaction on heat and mass transfer bynatural convection from vertical surfaces in porous media considering Soretand Dufour effects, Heat Mass Transfer 43 (2006) 595–602.

[19] M.A. Mansour, N.F. El-Anssary, A.M. Aly, Effects of chemical reaction andthermal stratification on MHD free convective heat and mass transfer over avertical stretching surface embedded in a porous media considering Soret andDufour numbers, Chem. Eng. J. 145 (2008) 340–345.

[20] F.S. Ibrahim, F.M. Hady, S.M. Abdel-Gaied, M.R. Eid, Influence of chemicalreaction on heat and mass transfer of non-Newtonian fluid with yield stress by

172 M. Li et al. / International Journal of Heat and Mass Transfer 67 (2013) 164–172

free convection from vertical surface in porous medium considering Soreteffect, Appl. Math. Mech. 31 (2010) 675–684.

[21] J. Szekely, J.W. Evans, H.Y. Sohn, Gas–Solid Reactions, Academic Press, NewYork, 1976.

[22] M.C. Li, Y.S. Wu, Y.W. Tian, Y.C. Zhai, Non-thermal equilibrium model of thecoupled heat and mass transfer in strong endothermic chemical reactionsystem of porous media, Int. J. Heat Mass Transfer 50 (2007) 2936–2943.

[23] F. Patisson, M.G. Francois, D. Ablitzer, A non-isothermal, non-equimolartransient kinetic model for gas–solid reactions, Chem. Eng. Sci. 53 (1998)697–708.

[24] S.R. DeGroot, P. Mazur, Non-Equilibrium Thermodynamics, North-HollandPub, Amsterdam, 1962.

[25] T. Rajeswara Rao, D.J. Gunn, J.H. Bowen, Kinetics of calcium carbonatedecomposition, Chem. Eng. Res. Des. 67 (1989) 38–47.

[26] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Co.,New York, 1980.

[27] W.Q. Tao, E.M. Sparrow, Transportive property and convective numericalstability of the steady-state convective-difference equation, Numer. HeatTransfer 11 (1987) 491–497.