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Soret and Dufour effects in strongly endothermic chemical reaction system of porous media
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Transactions of Nonferrous Metals Society of China Trans. Nonferrous Met. SOC.China 16(2006) 1200-1204 www.csu.edu.cnJysxb/
Soret and Dufour effects in strongly endothermic chemical reaction system of porous media LI Ming-chun(+fl%)’”,
TIAN Yan-wen(Hg*)l, ZHAI Yu-chun(@3%)’
1. School of Materials and Metallurgy, Northeastern University, Shenyang 110004, China; 2. School of Materials Science and Engineering, Shenyang University of Technology, Shenyang 110023, China Received 3 1 October 2005; accepted 20 March 2006 Abstract: Taking account of the thermal-diffusion (Soret) and the diffusion-themo (Dufour) effects, the properties of the heat and mass transfers in a strongly endothermic chemical reaction system for a porous medium are numerically studied. Through the theory of the thermodynamics of irreversible processes, a coupled mathematical model describing the heat and mass transfers in a porous system for the calcination of limestone is formulated. The governing partial differential equations are numerically solved by the implicitly finite volume method through decomposing the equations to a. set of coupled differential equations. The results indicate that when the convectional velocity is lower or when the initial temperature of the feeding gas is higher, Soret and Dufour effects can’t be ignored. The distribution figures for the temperature field of the gas in the system, the concentration field of the product gas and the solid conversion ratio are provided. Key words: porous media; heat transfer; mass transfer; Soret phenomenon; Dufour phenomenon; decomposition reaction
1 Introduction The heat and mass transfers in a porous system have been stressed by theoretical and experimental studies owing to their wide applications, such as geothermal systems, energy-storage units, heat insulation, heat exchangers for the packed bed, drying technology, catalytic reactors, nuclear waste repository. The analysis of the heat and mass transfers in porous media becomes complicated when a strongly endothermic chemical reaction takes place within the porous solid matrix. In addition, the heat and mass transfers simultaneously affecting each other will also cause the cross-diffusion effect. The mass transfer caused by the temperature gradient is called Soret effect, while the heat transfer caused by the concentration gradient is called Dufour effect. ECKERT and DRAKE[l] presented several cases of Dufour effect. WEAVER et a1[2] have pointed out that when the differences of the temperature and the concentration are large or when the difference of the molecular mass of the two elements in a binary mixture is great, the coupled interaction is significant. A primary discussion on the effect of the cross-coupled diffusion in a system with horizontal temperature and concentration
gradients was made by MALASHETTY et a1[3]. BENANO-MELLY et a1[4] analyzed the heat diffusion of a binary fluid mixture in a porous medium with a horizontal thermal gradient. Up to now, most investigations on the heat and mass transfers in porous media were focused on the situation of an inertly porous matrix[5-9]. However, the heat exchange in a porous medium with chemical reactions is little reported[lO, 111. The calcination of limestone is an important gas-solid reaction existing in metallurgy and chemical industry[l2]. The thermal decomposition of calcium carbonate is a strongly endothermic chemical reaction, forming C 0 2 gas. At a higher rate, the reaction must absorb a plenty of heat from the envirmmeut a& causes a macroscopic volume flux of the product gas from the reaction interface to the main stream. Based on the characteristics mentioned above, the cross-coupled effect among the chemical reaction, the heat transfer and the mass transfer must be considered during the calcination of limestone when the temperature gradient and the concentration gradient exist simultaneously. In the present study, the heat and mass transfers in a strongly endothermic chemical reaction of a porous medium are studied when the Soret and Dufour effects exist simultaneously. In accordance with the thermo-
Foundation item: Project(50174015) supported by the National Natural Science Foundation of China Corresponding author: TIAN Yan-wen;Tel: +86-24-83677737; Fax: +86-24-83687737; E-mail: [email protected]
LI Ming-chun, et al/Trans. Nonferrous Met. SOC.China 16(2006)
dynamic laws of irreversible processes, a coupled mathematical model describing the heat, mass transfers and the chemical reaction in a porous system for the calcination of limestone is formulated. And the model is solved numerically by the implicitly finite volume method. The influences of the Soret and Dufour effects on the heat transfer, mass transfer, and the chemical reaction in the porous medium are discussed. The reaction features of the packed bed of pellets are analyzed under different conditions by varying the key parameters.
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the value of v is determined by the superficial velocity vb and the porosity E, v=vd&. According to the Darcy law, vb is determined from the following equations:
vb=-(kitl) a~iaF-(kitl)(pL-p,,)/~ where q is the dynamic viscosity, L is the length of the reactor, k is the penetration ratio calculated by the Ergun correlation[131: k = ~~d;/150(1-E * ) , where dp is the diameter of pellet. The balance equation of energy for the gas flows in the porous media is given by &paU/at=-&diV@UVCJq) -Y~JAH
2 Modeling and formulation 2.1 Problem description There is a rectangular or cylindncal reactor whose length is L, filled with spherical carbonate pellets, as shown in Fig. 1. The solid sphere particles are of uniform shape and undeformable. It is supposed that the reactor has the same properties within the same cross section. Thus one can only study the changes of the physical and chemical quantity fields along the axial direction of the reactor. The high temperature gas flows through the bed by forced convective and exchanges heat with the solid matrix. Here, the natural convection and the radiation heat transfer are neglected.
Fulid flow L Tin Fig.1 Schematic diagram of reactor filled with porous media
(3)
where Jq is the heat flow, AH is the enthalpy of reaction. According to the thermodynamics of irreversible processes[l4], the mass flow J1and the heat flow Jq can be written as follows:
J1=-pc1c2DrgradT-pDgradcl
(4)
Jq=-AgradT-pl pt,TDffgradcl
(5)
where A=Lqq/p,represents the heat conductivity; D"= Lq1/@clc$), denotes the Dufour coefficient; D'= LI,@clcZp), is the thermal diffusion coefficient; D= (L11 pfl )l@czT), is the diffkion coefficient. = (RrM~)/(c,[Ml-cl(Ml-M2)1). The order of magnitude of D" and D' of the gas is 10-4-1&-6 crn2.s-'.R1. cz is the mass fraction of the inert gas, M2 is the relative molecular mass of the inert gas. The coefficients L, and LI1are related to the heat conductivity and the diffusion coefficients. L1, and Lql are the cross-diffusion coefficient, with the Onsager relations Lql=Llq. Substitution of Eqns.(4) and ( 5 ) into Eqns.(2) and (3) yields
2.2 Mathematical models The decomposition of limestone can be expressed as follows CaC03=CaO+C02
(1)
When the flow in the porous media is incompresGble, i.e., the bulk density @) of the gas is constant, then on the scale of a representative elementary volume, the mass balance of the product gas COz is given by &paCllat=-&diVJ1-&diV@lv)+ylJ
(2)
where E is the porosity, cl is the mass fraction of species COz in the gas, J1is the difhsion flux of COz,p1 is the density of COz, the coefficient y1 divided by the molecular mass M I of component COz is proportional to the stoichiometric coefficient of COz in reaction(1). The quantity J is called the chemical reaction rate of reaction(1). v is the velocity of the gas flows in the pore,
where c,=(auldT), is the specific heat at constant pressure. J = 7 is the rate of the chemical reaction described with molar concentrations and true stoichiometric coefficients, a is the normalization constant. The linear phenomenological equation of the
a,
chemical reaction rate is
J=-TAlT,
phenomenological coefficient, affinity of the chemical reaction,
2
7
is the
is the chemical
2=
n i=l
ripi, ./i
is
LI Ming-chun, et al/Trans. Nonferrous Met. Soc.China lq2006)
1202
The discrete equation sets of the concentration and the temperature, through coupling each other, can’t be solved independently. Therefore, Gauss-Seidel iteration method was used to compute the model in the present study. To assure the iterations convergent in the program, the difference between two calculation results should be pcoz,eis the equilibrium partial pressure of C02, calculated at each time increment. This calculation is pco, is the partid pressure of COZ at any time, peis operated iteratively until the results suffice to the the standard atmosphere pressure. Then, we can obtain convergence criterion. In this work, the convergence the followingrelation: criterion is satisfied when the absolute difference between two consecutive iterations is less than lo6. Uniform grid arrangements are employed in the present = -ZR(ln Nl - In K) study (the time step M s, and the spatial step Ax= 0.000 3 m). The results are independent of the grid where Nl is the molar concentration of species COZY system. which can be presented by equation: N ~ = C ~ M ~ I [ M I + cl(M2-M1)], when the mass density of the bulk stream is 4 Results and discussion constant. K is the equilibrium constant. Thus, Eqn.(8) can be expressed as To study the coupled heat and mass transfers and the effects of the heat and mass transfers on the thermal J = -FR{ln(clM2) - ln[Ml + c1( M 2 -Ml)] - InK} . (9) decomposition reaction in a porous medium with a strongly endothermic chemical reaction, the temperature The rate of the reaction can be given with the distribution of the gas, the concentration of the product substance quantity of the pellet of calcium carbonate: gas and the conversion degree of the solid pellets are calculated under different conditions by the established model. The results are shown in Figs.2-7. The conditions are L=0.15 m, Lw=161.91 kJ.mol-’, ~ 0 . 3 2p=0.336 , 8X 10-~g ~ m - ~ . where r, is radius of the unreacted core, r, is radius of At different convection rates, the influences of Soret the initial pellet, p A is the density of the solid pellet, and Dufour effects on the transfer and reaction -dGA/dt denotes the exhaustive amount of calcium characteristics in a porous system are compared, as carbonate per unit volume and unit time, and shown in Figs.24. It can be seen from Figs.24 that the dGA/ dt = = -J. Accordingly, the solid conversion larger the convection rate is, the less the effects of Soret degree can be expressed as follows: and Dufour coefficients are. From the comparison of curve 3 and 6 in Fig.2 we find that when the superficial rate of gas vb increases to 50 c d s , the ignored effects of Soret and Dufour can’t cause much error. However, true stoichiometric coefficient in the chemical reaction, ,iii is the chemical potential per mole of species i. The formulation of 2 in reaction(1) can be derived as
rAJ
Eqns.(6), (7), (9) and (11) are the coupled model.
3 Numerical resolutions The non-linear governing equations are rendered discrete using the implicitly finite volume method, by which the properties of conservation, transfa and diffusion can be analyzed in detail[151. In order to avoid the fluctuation of the solution and to ensure that the discrete equations simultaneously reflect the effects of diffusion and convection, a mixed formulation suggested by SPALDING[16] is used. The validity of the mixed format has been testified in Refs.[l5] and [17]. The initial and boundary conditions for the problem are: q(0, o=o, [ W x , 0 1 ~ X L L l = O , T(0, t w , [ W , 0 1 ~lLL=o, cl(x, O)=O, T(x, O)=T‘, rc(x, O)=r,, T,=l 163 K.
-0 S, Tj,=l274 K 1 -Df=LY‘=OcrnZ/(s*K),fi=lOcm/s 2 - Df=o”=o cm2/(s*K),q,=30cm/s 3 - Df=IY’=ocrnV(s-K),vb=50 cm/s 4 - D’=D’L-0.0008 crnZ/(s-K),%=lo cm/s 5 - D’=D”=O.OOO8 crnZ/(s-K), %=3Ocm/s 6 -D’=D’=O.OOO8 cm2/(s-K), %=SO cm/s 0
2
4
6
8 x/cm
1 0 1 2 1 4
Fig.2 Influence of Soret and Dufour effects on concentration distribution of product gas for different convection velocities
1
LI Ming-chun, et aliTrans. Nonferrous Met. SOC.China 16(2006)
1 400
1 300
1203
F60 s, vb’lo cm/s
- D’=D”=O cm2/(s.K), Tln=l 173 K 2 - D’=D”=O cm2/(s* K), Tin=1 274 K 3 - D’=LY’=O cm2/(s*K),Ti,=l 383 K I
- 0‘=0”=0.0008 cm2/(s*K),Tk=l 173K 5 - 0‘=0‘‘=0.000 8 cm2/(s.K), Tin=1274K 6 - D’=D”=O.OOO 8 cm2/(s-K), Tin=l 383 K
4
E l 200
1 100
i
1ooob
i ;; i
I0
12
I4
’
.ricru Fig3 Influence of Soret and Dufour effects on temperature fields of bulk stream for different convection velocities
1 000I 2
0
4
8
6
1 0 1 2 1 4
.rlcm
Fig.6 Influence of Soret and Dufour effects on temperature fields of bulk stream for different initial temperatures t=60 s, vL,=10 cm/s
- D’=LY‘=O cmT/(s *K), T,n=l 173 K 2 - D’=n”O cmV(s.K), Tln=l383 K 1
D’=D”O.O008cni~/(s~K), Tln=l 173K 4- 0’=0”=0.000 8 cni’/(s* K), T,,=l 383 K
0.25. 30 h
0
2
4
6
8 .x/cm
1 0 1 2 1 4
Fig.4 Influence of Soret and Dufour effects on conversion degree of solid matrix for different convection velocities
0.20 0.16 C T 0.12
Dr=D”=Ocm2/(s*K), Tln=l 173K
0.08
0.04 t
- D‘=D”=O cml/(s.K), Tln=l274K 3 - D’=D=O cml/(s.K), T,n=l383 K
2
-
O’=D’’=O.OOO 8 cmZ/(s.K). Tin=l 173 K 5 - o’=D’=O.OOO 8 cmV(s-K), Tln=l 27433 6 7D’=y=O.OOO 8 ~m2/(s.K),Tln=l 383; K 2 4 6 8 10 12 14
4
1,cm
Fig.5 Influence of Soret and Dufour effects on concentration
distribution of product gas for different initial temperatures when the superficial rate is much smaller (%=lo c d s ) , Soret and Dufour effects have much larger influence (curve 1 and 4 in Fig.2). As seen fiom Figs.2-4, the concentration distribution for the product gas in the reactor and the temperature distribution for the flowing
0
2.
4
8
6 \
I0
12
14
Clll
Fig.7 Influence of Soret and Dufour effects on conversion
degree of solid matrix for different initial temperatures gas in the pore decrease, and the conversion degree of the solid pellet also decreases due to the Soret and Dufour effects. Because the Soret effect stands for the mass transfer caused by the heterogeneity of the temperature field, and the Dufour effect denotes the heat transfer caused by the heterogeneity of the concentration field. Thus, as the effects of Soret and Dufour coefficients are taken into account, the velocity of the mass and heat transfers increases. Under a condition of superficial rate of gas being 10 c d s , when the initial temperature of the feeding gas is changed, the influences of the Soret and Dufour effects on the concentration field of the product gas, temperature field of the bulk flow and the solid fractional conversion in a porous system are exhibited in Figs.5-7. Fig.5 illustrates that a higher initial temperature will result in a larger concentration difference between the two groups of the product gas concentration distribution lines, which are calculated separately for D’=D”=O and Df=D”= 0.000 8 cm2/(sK). As the initial gas temperature increases, the heterogeneity of the temperature field of
LI Ming-chun, et al/Tm. Nonferrow Met. SOC. China 16(2006) [3] MALASHEW M S, GAIKAD S N. Effect of cross diffusion on the bulk stream in porous media increases, causing 1204
increment of the gas temperature gradient, as shown in Fig.6. As a result, the mass and heat transfers caused by the Soret and Dufour effects increase. Variation of the solid fractional conversion caused by the effects of the Soret and Dufour coefficients at various Th is shown in Fig.7.
[5]
5 Conclusions
[6]
Through the thermodynamics of irreversible processes, a mathematical model describing the coupling among the multi-irreversible processes in a porous system with a strongly endothermic chemical reaction was established and solved by the implicitly finite volume method. The calculated results show that when the convectional velocity is lower or when the initial temperature of the feeding gas is higher, the Soret and Dufour effects can't be ignored. The quickened rate of outward transfer of the product gas and the heat caused by the Soret and Dufour effects make the concentration distribution of the product g a s decrease, the temperature of bulk stream decrease, the reaction rate and the conversion degree of the solid pellet decrease. Both the temperature field of the bulk flow and the solid hctional conversion increase with an increase in the convection velocity or in the initial temperature of the feeding gas. While the concentration distribution of the product gas decreases with an increase in the convection velocity and increases with an increase in the initial temperature of the feeding gas.
[4]
[7]
[8]
[9]
[lo]
[ll]
[12]
[ 131
[14] [15] [16]
References [17] [l] [2]
ECKERT E R G DRAKE R M. Analysis of Heat and Mass Transfer [MI. New York:McGraw-Hill, 1972. WEAVER J A, VISKANTA R. Natural convection due to horizontal temperature and concentration gradients(2): species interdiffusion, Soret and Dufour effects [Jl. Int J Heat and Mass Transfer, 1991, 34(12): 3121-3133.
double diffusive in the presence of horizontal grdents [Jl. Int J of Engin-g Science, 2002,40(7): 773-787. BENANO-MELLY L B, CALTAGIRONE J P, FAISSAT B, M O m L F, COSTESEQUE P. Modeling Soret coefficient measurement experhen@in porous media considering thermal and solutal convection [JJ. Int J Heat Mass Transfer, 2001, 44(7): 1285-1297. VAFAI K,SOZEN M. Analysis of energy and momentum transport for liquid flow through a porous bed [Jl. J Heat Transfer Traas ASME, 1990, 112(4): 690-699. QUINTM, WHITAKER S. Two-phase flow in heterogeneous porous media (I): The influence of large spatial and temp& @eats [J'J Transport in Porous Media, 1990,5(4): 341-379. LEE D Y, VAFAI K. Comparative analysis of jet impingement and mimhaunel cooling for high heat flux applications [Jl. Int J Heat Mass Transfer, 1999,42(9): 1555-1568. ALAZMI B, VAFAI K. Constant wall heat flux boundary conditions in porous media under local thermal non-equIlibrium conditions [J]. Int JHeat Mass Transfer, 2002,45(15): 3071-3087. XU Zmg-he, WANG Jm-xia Analytical solution for coupled diffusion of three-components liquid in porous pellet [J]. Trans N O I I ~ I IM Uet S SOCC h i 2005,15(1): 176-183. HARRIS KENDALL T, HAJI-SHEIKH A, A G W "ANNA A G Phase-change phenomena in porous manon-local thermal equilibrium model [Jl. Int J Heat Mass Transfer, 2001, 44(8): 1619-1625. MARAFIE A, VAFAI K Analysis of nondarcian effects on temperature differentids in porous media [J]. Int J Heat Mass Transf, 2001,44(23): 401-4411. TURKDOGAN E T, OLSSON R G,WRIEDT H A, DARKEN L S. Calcination of limestone [J]. Trans Soc Min Eng AJME, 1973, 254 9-2 1. ERGUN S. Fluid flow through packed columns [Jl. Chem Eng Prog, 1952,48(2): 89-94. DEGROOT S R, MAZUR P. Non-Equilibrium Thermodynamics [MI. Amsterdam: North-Holland Publisher, 1962.235-284. PATANKAR S V. Numerical Heat Transfer and Fluid Flow [MI. New York McGraw-Hill, 1980.142-145, 165-168. SPALDING D B. Novel fmite difference formulation for differential expressions involving both first and second derivativea [Jl. International Joumal for Numerical Methods in Engineerin& 1972, 4(4): 551-559. TAO W Q, SPARROW E M. Transportive property and convective numerical stability of the steady-state convection-diffiion finite difference equation [J]. Numerical Heat Transfer, 1987, 11(4): 49 1 4 9 7 . mated by YANG Blng)
Science Press
Transactions of Nonferrous Metals Society of China Trans. Nonferrous Met. SOC.China 16(2006) 1200-1204 www.csu.edu.cnJysxb/
Soret and Dufour effects in strongly endothermic chemical reaction system of porous media LI Ming-chun(+fl%)’”,
TIAN Yan-wen(Hg*)l, ZHAI Yu-chun(@3%)’
1. School of Materials and Metallurgy, Northeastern University, Shenyang 110004, China; 2. School of Materials Science and Engineering, Shenyang University of Technology, Shenyang 110023, China Received 3 1 October 2005; accepted 20 March 2006 Abstract: Taking account of the thermal-diffusion (Soret) and the diffusion-themo (Dufour) effects, the properties of the heat and mass transfers in a strongly endothermic chemical reaction system for a porous medium are numerically studied. Through the theory of the thermodynamics of irreversible processes, a coupled mathematical model describing the heat and mass transfers in a porous system for the calcination of limestone is formulated. The governing partial differential equations are numerically solved by the implicitly finite volume method through decomposing the equations to a. set of coupled differential equations. The results indicate that when the convectional velocity is lower or when the initial temperature of the feeding gas is higher, Soret and Dufour effects can’t be ignored. The distribution figures for the temperature field of the gas in the system, the concentration field of the product gas and the solid conversion ratio are provided. Key words: porous media; heat transfer; mass transfer; Soret phenomenon; Dufour phenomenon; decomposition reaction
1 Introduction The heat and mass transfers in a porous system have been stressed by theoretical and experimental studies owing to their wide applications, such as geothermal systems, energy-storage units, heat insulation, heat exchangers for the packed bed, drying technology, catalytic reactors, nuclear waste repository. The analysis of the heat and mass transfers in porous media becomes complicated when a strongly endothermic chemical reaction takes place within the porous solid matrix. In addition, the heat and mass transfers simultaneously affecting each other will also cause the cross-diffusion effect. The mass transfer caused by the temperature gradient is called Soret effect, while the heat transfer caused by the concentration gradient is called Dufour effect. ECKERT and DRAKE[l] presented several cases of Dufour effect. WEAVER et a1[2] have pointed out that when the differences of the temperature and the concentration are large or when the difference of the molecular mass of the two elements in a binary mixture is great, the coupled interaction is significant. A primary discussion on the effect of the cross-coupled diffusion in a system with horizontal temperature and concentration
gradients was made by MALASHETTY et a1[3]. BENANO-MELLY et a1[4] analyzed the heat diffusion of a binary fluid mixture in a porous medium with a horizontal thermal gradient. Up to now, most investigations on the heat and mass transfers in porous media were focused on the situation of an inertly porous matrix[5-9]. However, the heat exchange in a porous medium with chemical reactions is little reported[lO, 111. The calcination of limestone is an important gas-solid reaction existing in metallurgy and chemical industry[l2]. The thermal decomposition of calcium carbonate is a strongly endothermic chemical reaction, forming C 0 2 gas. At a higher rate, the reaction must absorb a plenty of heat from the envirmmeut a& causes a macroscopic volume flux of the product gas from the reaction interface to the main stream. Based on the characteristics mentioned above, the cross-coupled effect among the chemical reaction, the heat transfer and the mass transfer must be considered during the calcination of limestone when the temperature gradient and the concentration gradient exist simultaneously. In the present study, the heat and mass transfers in a strongly endothermic chemical reaction of a porous medium are studied when the Soret and Dufour effects exist simultaneously. In accordance with the thermo-
Foundation item: Project(50174015) supported by the National Natural Science Foundation of China Corresponding author: TIAN Yan-wen;Tel: +86-24-83677737; Fax: +86-24-83687737; E-mail: [email protected]
LI Ming-chun, et al/Trans. Nonferrous Met. SOC.China 16(2006)
dynamic laws of irreversible processes, a coupled mathematical model describing the heat, mass transfers and the chemical reaction in a porous system for the calcination of limestone is formulated. And the model is solved numerically by the implicitly finite volume method. The influences of the Soret and Dufour effects on the heat transfer, mass transfer, and the chemical reaction in the porous medium are discussed. The reaction features of the packed bed of pellets are analyzed under different conditions by varying the key parameters.
1201
the value of v is determined by the superficial velocity vb and the porosity E, v=vd&. According to the Darcy law, vb is determined from the following equations:
vb=-(kitl) a~iaF-(kitl)(pL-p,,)/~ where q is the dynamic viscosity, L is the length of the reactor, k is the penetration ratio calculated by the Ergun correlation[131: k = ~~d;/150(1-E * ) , where dp is the diameter of pellet. The balance equation of energy for the gas flows in the porous media is given by &paU/at=-&diV@UVCJq) -Y~JAH
2 Modeling and formulation 2.1 Problem description There is a rectangular or cylindncal reactor whose length is L, filled with spherical carbonate pellets, as shown in Fig. 1. The solid sphere particles are of uniform shape and undeformable. It is supposed that the reactor has the same properties within the same cross section. Thus one can only study the changes of the physical and chemical quantity fields along the axial direction of the reactor. The high temperature gas flows through the bed by forced convective and exchanges heat with the solid matrix. Here, the natural convection and the radiation heat transfer are neglected.
Fulid flow L Tin Fig.1 Schematic diagram of reactor filled with porous media
(3)
where Jq is the heat flow, AH is the enthalpy of reaction. According to the thermodynamics of irreversible processes[l4], the mass flow J1and the heat flow Jq can be written as follows:
J1=-pc1c2DrgradT-pDgradcl
(4)
Jq=-AgradT-pl pt,TDffgradcl
(5)
where A=Lqq/p,represents the heat conductivity; D"= Lq1/@clc$), denotes the Dufour coefficient; D'= LI,@clcZp), is the thermal diffusion coefficient; D= (L11 pfl )l@czT), is the diffkion coefficient. = (RrM~)/(c,[Ml-cl(Ml-M2)1). The order of magnitude of D" and D' of the gas is 10-4-1&-6 crn2.s-'.R1. cz is the mass fraction of the inert gas, M2 is the relative molecular mass of the inert gas. The coefficients L, and LI1are related to the heat conductivity and the diffusion coefficients. L1, and Lql are the cross-diffusion coefficient, with the Onsager relations Lql=Llq. Substitution of Eqns.(4) and ( 5 ) into Eqns.(2) and (3) yields
2.2 Mathematical models The decomposition of limestone can be expressed as follows CaC03=CaO+C02
(1)
When the flow in the porous media is incompresGble, i.e., the bulk density @) of the gas is constant, then on the scale of a representative elementary volume, the mass balance of the product gas COz is given by &paCllat=-&diVJ1-&diV@lv)+ylJ
(2)
where E is the porosity, cl is the mass fraction of species COz in the gas, J1is the difhsion flux of COz,p1 is the density of COz, the coefficient y1 divided by the molecular mass M I of component COz is proportional to the stoichiometric coefficient of COz in reaction(1). The quantity J is called the chemical reaction rate of reaction(1). v is the velocity of the gas flows in the pore,
where c,=(auldT), is the specific heat at constant pressure. J = 7 is the rate of the chemical reaction described with molar concentrations and true stoichiometric coefficients, a is the normalization constant. The linear phenomenological equation of the
a,
chemical reaction rate is
J=-TAlT,
phenomenological coefficient, affinity of the chemical reaction,
2
7
is the
is the chemical
2=
n i=l
ripi, ./i
is
LI Ming-chun, et al/Trans. Nonferrous Met. Soc.China lq2006)
1202
The discrete equation sets of the concentration and the temperature, through coupling each other, can’t be solved independently. Therefore, Gauss-Seidel iteration method was used to compute the model in the present study. To assure the iterations convergent in the program, the difference between two calculation results should be pcoz,eis the equilibrium partial pressure of C02, calculated at each time increment. This calculation is pco, is the partid pressure of COZ at any time, peis operated iteratively until the results suffice to the the standard atmosphere pressure. Then, we can obtain convergence criterion. In this work, the convergence the followingrelation: criterion is satisfied when the absolute difference between two consecutive iterations is less than lo6. Uniform grid arrangements are employed in the present = -ZR(ln Nl - In K) study (the time step M s, and the spatial step Ax= 0.000 3 m). The results are independent of the grid where Nl is the molar concentration of species COZY system. which can be presented by equation: N ~ = C ~ M ~ I [ M I + cl(M2-M1)], when the mass density of the bulk stream is 4 Results and discussion constant. K is the equilibrium constant. Thus, Eqn.(8) can be expressed as To study the coupled heat and mass transfers and the effects of the heat and mass transfers on the thermal J = -FR{ln(clM2) - ln[Ml + c1( M 2 -Ml)] - InK} . (9) decomposition reaction in a porous medium with a strongly endothermic chemical reaction, the temperature The rate of the reaction can be given with the distribution of the gas, the concentration of the product substance quantity of the pellet of calcium carbonate: gas and the conversion degree of the solid pellets are calculated under different conditions by the established model. The results are shown in Figs.2-7. The conditions are L=0.15 m, Lw=161.91 kJ.mol-’, ~ 0 . 3 2p=0.336 , 8X 10-~g ~ m - ~ . where r, is radius of the unreacted core, r, is radius of At different convection rates, the influences of Soret the initial pellet, p A is the density of the solid pellet, and Dufour effects on the transfer and reaction -dGA/dt denotes the exhaustive amount of calcium characteristics in a porous system are compared, as carbonate per unit volume and unit time, and shown in Figs.24. It can be seen from Figs.24 that the dGA/ dt = = -J. Accordingly, the solid conversion larger the convection rate is, the less the effects of Soret degree can be expressed as follows: and Dufour coefficients are. From the comparison of curve 3 and 6 in Fig.2 we find that when the superficial rate of gas vb increases to 50 c d s , the ignored effects of Soret and Dufour can’t cause much error. However, true stoichiometric coefficient in the chemical reaction, ,iii is the chemical potential per mole of species i. The formulation of 2 in reaction(1) can be derived as
rAJ
Eqns.(6), (7), (9) and (11) are the coupled model.
3 Numerical resolutions The non-linear governing equations are rendered discrete using the implicitly finite volume method, by which the properties of conservation, transfa and diffusion can be analyzed in detail[151. In order to avoid the fluctuation of the solution and to ensure that the discrete equations simultaneously reflect the effects of diffusion and convection, a mixed formulation suggested by SPALDING[16] is used. The validity of the mixed format has been testified in Refs.[l5] and [17]. The initial and boundary conditions for the problem are: q(0, o=o, [ W x , 0 1 ~ X L L l = O , T(0, t w , [ W , 0 1 ~lLL=o, cl(x, O)=O, T(x, O)=T‘, rc(x, O)=r,, T,=l 163 K.
-0 S, Tj,=l274 K 1 -Df=LY‘=OcrnZ/(s*K),fi=lOcm/s 2 - Df=o”=o cm2/(s*K),q,=30cm/s 3 - Df=IY’=ocrnV(s-K),vb=50 cm/s 4 - D’=D’L-0.0008 crnZ/(s-K),%=lo cm/s 5 - D’=D”=O.OOO8 crnZ/(s-K), %=3Ocm/s 6 -D’=D’=O.OOO8 cm2/(s-K), %=SO cm/s 0
2
4
6
8 x/cm
1 0 1 2 1 4
Fig.2 Influence of Soret and Dufour effects on concentration distribution of product gas for different convection velocities
1
LI Ming-chun, et aliTrans. Nonferrous Met. SOC.China 16(2006)
1 400
1 300
1203
F60 s, vb’lo cm/s
- D’=D”=O cm2/(s.K), Tln=l 173 K 2 - D’=D”=O cm2/(s* K), Tin=1 274 K 3 - D’=LY’=O cm2/(s*K),Ti,=l 383 K I
- 0‘=0”=0.0008 cm2/(s*K),Tk=l 173K 5 - 0‘=0‘‘=0.000 8 cm2/(s.K), Tin=1274K 6 - D’=D”=O.OOO 8 cm2/(s-K), Tin=l 383 K
4
E l 200
1 100
i
1ooob
i ;; i
I0
12
I4
’
.ricru Fig3 Influence of Soret and Dufour effects on temperature fields of bulk stream for different convection velocities
1 000I 2
0
4
8
6
1 0 1 2 1 4
.rlcm
Fig.6 Influence of Soret and Dufour effects on temperature fields of bulk stream for different initial temperatures t=60 s, vL,=10 cm/s
- D’=LY‘=O cmT/(s *K), T,n=l 173 K 2 - D’=n”O cmV(s.K), Tln=l383 K 1
D’=D”O.O008cni~/(s~K), Tln=l 173K 4- 0’=0”=0.000 8 cni’/(s* K), T,,=l 383 K
0.25. 30 h
0
2
4
6
8 .x/cm
1 0 1 2 1 4
Fig.4 Influence of Soret and Dufour effects on conversion degree of solid matrix for different convection velocities
0.20 0.16 C T 0.12
Dr=D”=Ocm2/(s*K), Tln=l 173K
0.08
0.04 t
- D‘=D”=O cml/(s.K), Tln=l274K 3 - D’=D=O cml/(s.K), T,n=l383 K
2
-
O’=D’’=O.OOO 8 cmZ/(s.K). Tin=l 173 K 5 - o’=D’=O.OOO 8 cmV(s-K), Tln=l 27433 6 7D’=y=O.OOO 8 ~m2/(s.K),Tln=l 383; K 2 4 6 8 10 12 14
4
1,cm
Fig.5 Influence of Soret and Dufour effects on concentration
distribution of product gas for different initial temperatures when the superficial rate is much smaller (%=lo c d s ) , Soret and Dufour effects have much larger influence (curve 1 and 4 in Fig.2). As seen fiom Figs.2-4, the concentration distribution for the product gas in the reactor and the temperature distribution for the flowing
0
2.
4
8
6 \
I0
12
14
Clll
Fig.7 Influence of Soret and Dufour effects on conversion
degree of solid matrix for different initial temperatures gas in the pore decrease, and the conversion degree of the solid pellet also decreases due to the Soret and Dufour effects. Because the Soret effect stands for the mass transfer caused by the heterogeneity of the temperature field, and the Dufour effect denotes the heat transfer caused by the heterogeneity of the concentration field. Thus, as the effects of Soret and Dufour coefficients are taken into account, the velocity of the mass and heat transfers increases. Under a condition of superficial rate of gas being 10 c d s , when the initial temperature of the feeding gas is changed, the influences of the Soret and Dufour effects on the concentration field of the product gas, temperature field of the bulk flow and the solid fractional conversion in a porous system are exhibited in Figs.5-7. Fig.5 illustrates that a higher initial temperature will result in a larger concentration difference between the two groups of the product gas concentration distribution lines, which are calculated separately for D’=D”=O and Df=D”= 0.000 8 cm2/(sK). As the initial gas temperature increases, the heterogeneity of the temperature field of
LI Ming-chun, et al/Tm. Nonferrow Met. SOC. China 16(2006) [3] MALASHEW M S, GAIKAD S N. Effect of cross diffusion on the bulk stream in porous media increases, causing 1204
increment of the gas temperature gradient, as shown in Fig.6. As a result, the mass and heat transfers caused by the Soret and Dufour effects increase. Variation of the solid fractional conversion caused by the effects of the Soret and Dufour coefficients at various Th is shown in Fig.7.
[5]
5 Conclusions
[6]
Through the thermodynamics of irreversible processes, a mathematical model describing the coupling among the multi-irreversible processes in a porous system with a strongly endothermic chemical reaction was established and solved by the implicitly finite volume method. The calculated results show that when the convectional velocity is lower or when the initial temperature of the feeding gas is higher, the Soret and Dufour effects can't be ignored. The quickened rate of outward transfer of the product gas and the heat caused by the Soret and Dufour effects make the concentration distribution of the product g a s decrease, the temperature of bulk stream decrease, the reaction rate and the conversion degree of the solid pellet decrease. Both the temperature field of the bulk flow and the solid hctional conversion increase with an increase in the convection velocity or in the initial temperature of the feeding gas. While the concentration distribution of the product gas decreases with an increase in the convection velocity and increases with an increase in the initial temperature of the feeding gas.
[4]
[7]
[8]
[9]
[lo]
[ll]
[12]
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[14] [15] [16]
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