Soret and Dufour effects in strongly endothermic chemical reaction system of porous media

LI Ming-chun(李明春)^{1, 2}, TIAN Yan-wen(田彦文)¹, ZHAI Yu-chun(翟玉春)¹

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 31 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[1] presented several cases of Dufour effect. WEAVER et al[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 al[3]. BENANO-MELLY et al[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[10, 11].

The calcination of limestone is an important gas-solid reaction existing in metallurgy and chemical industry[12]. The thermal decomposition of calcium carbonate is a strongly endothermic chemical reaction, forming CO₂ gas. At a higher rate, the reaction must absorb a plenty of heat from the environment and 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- 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.

2 Modeling and formulation

2.1 Problem description

There is a rectangular or cylindrical 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.

Fig.1 Schematic diagram of reactor filled with porous media

2.2 Mathematical models

The decomposition of limestone can be expressed as follows

CaCO₃=CaO+CO₂                            (1)

When the flow in the porous media is incompressible, i.e., the bulk density (ρ) of the gas is constant, then on the scale of a representative elementary volume, the mass balance of the product gas CO₂ is given by

ερ?c₁/?t=-εdivJ₁-εdiv(ρ₁v)+γ₁J                  (2)

where  ε is the porosity, c₁ is the mass fraction of species CO₂ in the gas, J₁ is the diffusion flux of CO₂, ρ₁ is the density of CO₂, the coefficient γ₁ divided by the molecular mass M₁ of component CO₂ is proportional to the stoichiometric coefficient of CO₂ 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, the value of v is determined by the superficial velocity v_b and the porosity ε, v=v_b/ε. According to the Darcy law, v_b is determined from the following equations:

v_b=-(k/η) ?P/?x=-(k/η)(P_L-P₀)/L

where  η is the dynamic viscosity, L is the length of the reactor, k is the penetration ratio calculated by the Ergun correlation[13]:  where d_p is the diameter of pellet.

The balance equation of energy for the gas flows in the porous media is given by

ερ?u/?t=-εdiv(ρuv+J_q) -γ₁J?H                  (3)

where  J_q is the heat flow, ?H is the enthalpy of reaction. According to the thermodynamics of irreversible processes[14], the mass flow J₁ and the heat flow J_q can be written as follows:

J₁=-ρc₁c₂D′gradT-ρDgradc₁                   (4)

J_q=-λgradT-ρ₁TD″gradc₁                  (5)

where  λ=L_qq/T², represents the heat conductivity; D″= L_q1/(ρc₁c₂T²), denotes the Dufour coefficient; D′= L_1q(ρc₁c₂T²), is the thermal diffusion coefficient; D= (L₁₁)/(ρc₂T), is the diffusion coefficient. = (RTM₁)/{c₁[M₁-c₁(M₁-M₂)]}. The order of magnitude of D″ and D′ of the gas is 10^-4-10^-6 cm²×s^-1×K^-1. c₂ is the mass fraction of the inert gas, M₂ is the relative molecular mass of the inert gas. The coefficients L_qq and L₁₁ are related to the heat conductivity and the diffusion coefficients. L_1q and L_q1 are the cross-diffusion coefficient, with the Onsager relations L_q1=L_1q. Substitution of Eqns.(4) and (5) into Eqns.(2) and (3) yields

(6)

(7)

where  c_p=(?u/?T)_p is the specific heat at constant pressure.  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

chemical reaction rate is  is the phenomenological coefficient,  is the chemical affinity of the chemical  reaction, ,   is

true stoichiometric coefficient in the chemical reaction, is the chemical potential per mole of species i. The formulation of  in reaction(1) can be derived as

is the equilibrium partial pressure of CO₂, is the partial pressure of CO₂ at any time, is the standard atmosphere pressure. Then, we can obtain the following relation:

  (8)

where  N₁ is the molar concentration of species CO₂, which can be presented by equation: N₁=c₁M₂/[M₁+ c₁(M₂-M₁)], when the mass density of the bulk stream is constant. K is the equilibrium constant. Thus, Eqn.(8) can be expressed as

  (9)

The rate of the reaction can be given with the substance quantity of the pellet of calcium carbonate:

 (10)

where  r_c is radius of the unreacted core, r_s is radius of the initial pellet, ρ_A is the density of the solid pellet, -dG_A/dt denotes the exhaustive amount of calcium carbonate per unit volume and unit time, and  Accordingly, the solid conversion degree can be expressed as follows:

  (11)

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, transfer and diffusion can be analyzed in detail[15]. 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.[15] and [17]. The initial and boundary conditions for the problem are: c₁(0, t)=0, [?c₁(x, t)/ ?x|_x=L]=0, T(0, t)=T_in, [T(x, t)/ ?x]|_x=L=0, c₁(x, 0)=0, T(x, 0)=T_c, r_c(x, 0)=r_s, T_c=1 163 K.

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 calculated at each time increment. This calculation is operated iteratively until the results suffice to the convergence criterion. In this work, the convergence criterion is satisfied when the absolute difference between two consecutive iterations is less than 10^-6. Uniform grid arrangements are employed in the present study (the time step ?t=6 s, and the spatial step ?x= 0.000 3 m). The results are independent of the grid system.

4 Results and discussion

To study the coupled heat and mass transfers and the effects of the heat and mass transfers on the thermal decomposition reaction in a porous medium with a strongly endothermic chemical reaction, the temperature distribution of the gas, the concentration of the product 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, ?H=161.91 kJ?mol^-1, ε=0.32, ρ=0.336 8×10^-3 g?cm^-3.

At different convection rates, the influences of Soret and Dufour effects on the transfer and reaction characteristics in a porous system are compared, as shown in Figs.2-4. It can be seen from Figs.2-4 that the larger the convection rate is, the less the effects of Soret and Dufour coefficients are. From the comparison of curve 3 and 6 in Fig.2 we find that when the superficial rate of gas v_b increases to 50 cm/s, the ignored effects of Soret and Dufour can’t cause much error. However, when the superficial rate is much smaller (v_b=10 cm/s), Soret and Dufour effects have much larger influence (curve 1 and 4 in Fig.2). As seen from Figs.2-4, the concentration distribution for the product gas in the reactor and the temperature distribution for the flowing 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.

Fig.2 Influence of Soret and Dufour effects on concentration distribution of product gas for different convection velocities

Fig.3 Influence of Soret and Dufour effects on temperature fields of bulk stream for different convection velocities

Fig.4 Influence of Soret and Dufour effects on conversion degree of solid matrix for different convection velocities

Under a condition of superficial rate of gas v_b being 10 cm/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″=0 and D′=D″=   0.000 8 cm²/(s?K). As the initial gas temperature increases, the heterogeneity of the temperature field of the bulk stream in porous media increases, causing 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 T_in is shown in Fig.7.

Fig.5 Influence of Soret and Dufour effects on concentration distribution of product gas for different initial temperatures

Fig.6 Influence of Soret and Dufour effects on temperature fields of bulk stream for different initial temperatures

Fig.7 Influence of Soret and Dufour effects on conversion degree of solid matrix for different initial temperatures

5 Conclusions

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 gas 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 fractional 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.

References

[1] ECKERT E R G, DRAKE R M. Analysis of Heat and Mass Transfer [M]. New York: McGraw-Hill, 1972.

[2] WEAVER J A, VISKANTA R. Natural convection due to horizontal temperature and concentration gradients(2): species inter-diffusion, Soret and Dufour effects [J]. Int J Heat and Mass Transfer, 1991, 34(12): 3121-3133.

[3] MALASHETTY M S, GAIKAD S N. Effect of cross diffusion on double diffusive in the presence of horizontal gradients [J]. Int J of Engineering Science, 2002, 40(7): 773-787.

[4] BENANO-MELLY L B, CALTAGIRONE J P, FAISSAT B, MONTEL F, COSTESEQUE P. Modeling Soret coefficient measurement experiments in porous media considering thermal and solutal convection [J]. Int J Heat Mass Transfer, 2001, 44(7): 1285-1297.

[5] VAFAI K, SOZEN M. Analysis of energy and momentum transport for liquid flow through a porous bed [J]. J Heat Transfer Trans ASME, 1990, 112(4): 690-699.

[6] QUINTARD M, WHITAKER S. Two-phase flow in heterogeneous porous media (I): The influence of large spatial and temporal gradients [J]. Transport in Porous Media, 1990, 5(4): 341-379.

[7] LEE D Y, VAFAI K. Comparative analysis of jet impingement and microchannel cooling for high heat flux applications [J]. Int J Heat Mass Transfer, 1999, 42(9): 1555-1568.

[8] ALAZMI B, VAFAI K. Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions [J]. Int J Heat Mass Transfer, 2002, 45(15): 3071-3087.

[9] XU Zeng-he, WANG Jin-xia. Analytical solution for coupled diffusion of three-components liquid in porous pellet [J]. Trans Nonferrous Met Soc China, 2005, 15(1): 176-183.

[10] HARRIS KENDALL T, HAJI-SHEIKH A, AGWU NNANNA A G. Phase-change phenomena in porous media—a non-local thermal equilibrium model [J]. Int J Heat Mass Transfer, 2001, 44(8): 1619-1625.

[11] MARAFIE A, VAFAI K. Analysis of non-darcian effects on temperature differentials in porous media [J]. Int J Heat Mass Transf, 2001, 44(23): 4401-4411.

[12] TURKDOGAN E T, OLSSON R G, WRIEDT H A, DARKEN L S. Calcination of limestone [J]. Trans Soc Min Eng AIME, 1973, 254: 9-21.

[13] ERGUN S. Fluid flow through packed columns [J]. Chem Eng Prog, 1952, 48(2): 89-94.

[14] DEGROOT S R, MAZUR P. Non-Equilibrium Thermodynamics [M]. Amsterdam: North-Holland Publisher, 1962. 235-284.

[15] PATANKAR S V. Numerical Heat Transfer and Fluid Flow [M]. New York: McGraw-Hill, 1980.142-145, 165-168.

[16] SPALDING D B. Novel finite difference formulation for differential expressions involving both first and second derivatives [J]. International Journal for Numerical Methods in Engineering, 1972, 4(4): 551-559.

[17] TAO W Q, SPARROW E M. Transportive property and convective numerical stability of the steady-state convection-diffusion finite difference equation [J]. Numerical Heat Transfer, 1987, 11(4): 491-497.

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: liming-799@163.com

(Edited by YANG Bing)