The Approximate Analytical Solution of Non-Linear Equation for Simultaneous Internal Mass and Heat Diffusion Effects

For the first time a mathematical modelling of porous catalyst particles subject to both internal mass concentration gradients as well as temperature gradients, in endothermic or exothermic reactions has been reported. This model contains a non-linear mass balance equation which is related to rate expression. This paper presents an approximate analytical method (Modified Adomian decomposition method) to solve the non-linear differential equations for chemical kinetics with diffusion effects. A simple and closed form of expressions pertaining to substrate concentration and utilization factor is presented for all value of diffusion parameters. These analytical results are compared with numerical results and found to be in good agreement.


Introduction
In many engineering and industrial applications, catalytic processes in chemical reactors are often considered to be very useful.This induces particular attention to the study of catalytic reactions at the single-particle level [1].Moreover, the reaction behavior of porous catalyst particles had been studied over nearly a quarter of a century [2]- [4].Majority of chemical reactions are accompanied by heat transfer effects; they either release or absorb heat.This can lead to appreciable increase (or decrease) of temperature toward the particle centre [5]- [7].Since chemical reaction rates vary rapidly increase with temperature, this effect could radically change the behavior of the catalyst particles.Analysis of chemical kinetics with diffusion effects usually leads to solving strongly nonlinear differential equations.Detailed reviews of mathematical models describing reactions in porous catalyst particle can be found in [8].Assuming a flat geometry for the particle and that conductive heat transfer is negligible compared to convective heat transfer.The approximate behavior of the functional forms is sufficiently similar for various geometric forms [9] [10] so that the spherical particle is a approximation [11] [12] for most cases encountered, such as cylindrical pellets, or irregular granules.When the chemical reaction is accompanied by a heat effect, not only a mass concentration gradient, but also appreciable temperature gradients can exist within the particle.Weisz and Hicks [13] solved the non-linear mass balance equation using numerical method.
However, to the best of our knowledge, there was no rigorous analytical solution for the concentration of reactant of catalyst having been reported.The purpose of this communication is to derive simple analytical expression for concentration and utilization factor for all possible values of reaction/diffusion parameters using the modified Adomian decomposition method.

Mathematical Formulation of the Problem
The dimensionless mass transport equation of porous catalyst particle is [13] ( ) where ( ) where y is the dimensionless concentration, x is the dimensionless radius of the spherical catalyst pellet, c is the dimensionless concentration of reactant, K is thermal conductivity, H is molar heat of reaction.The parameter γ expresses the sensitivity of the reaction rate to temperature; β is the maximum temperature variation ( ) max T ∆ which could exist within the particle relative to the boundary temperature.The boundary conditions are The utilization factor ( )

Analytical Expression of the Concentration Using Modified Adomian Decomposition Method (MADM)
In the recent years, much attention is devoted to the application of the Adomian decomposition method to the solution of various scientific models [14].The MADM yields, without linearization, perturbation, transformation or discretisation, an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms.The decomposition method is simple and easy to use and produces reliable results with few iterations.The rate of convergence of modified Adomian decomposition method is higher than standard Adomian decomposition method [15]- [17].Using this method (see Appendix A), we can obtain the analytical expression of concentration (see Appendix B), of the substrate as follows: Using Equations ( 5) and ( 6), we can obtain the effectiveness factor ( ) The Equation ( 6) and ( 7) represent the new and simple analytical expression of concentration of substrate and effectiveness factor provided

Numerical Simulation
The diffusion Equation (1) for the boundary condition (Equation ( 4)) is also solved numerically.We have used the function pdex1 in MATLAB software to solve numerically the initial-boundary value problem for the nonlinear differential equation.This numerical solution is compared with our analytical results in Figure 1 and

Discussion
The nonlinear system for coupled heat and mass transfer in a spherical non-isothermal catalyst pellet is solved analytically.The concentration of substrate depends on the following three factors , γ β and 0 φ .γ is the ac- tivation energy parameter and β is the heat of reaction parameter which represents the ratio of the characteris- tic time of the enzymatic reaction to that of substrate diffusion.the concentration of ( ) 1 y x = when 1 x = and also for all values of 0 φ , γ and β .From these figure, it should be noted that the value of the concentration of substrate decreases for all values of γ .From this Figures, it is apparent that the value of the concentration of substrate increases when the values of β increases.
The variation in effectiveness factor for various values of , γ β and 0 φ using Equation ( 7) is shown in

Conclusion
In this work, we have discussed the mathematical model of catalyst particle in a porous medium through which reactants diffuses.We have obtained the approximate analytical expression for the steady state concentration of substrate for all values of γ and β in a packed bed reactor using the modified Adomian decomposition me- thod.A satisfactory agreement with the numerical result is noted.Moreover, we have also presented a closed form expression for the utilization factor.The proposed model can be used to solve the nonlinear convective mass and heat diffusion problems.

Fig- ure 2 .
Upon comparison, it gives a satisfactory agreement for all values of the dimensionless parameters γ , β and φ .The Matlab program is also given in Appendix C.

Figure 1 (Figure 1 .
Figure 1.Plot of dimensionless concentration y versus dimensionless pellet radius x.The concentrations were computed for various values of the dimensionless parameter φ when (a)

Figure 2 .
Figure 2. The normalized dimensionless concentration y versus dimensionless pellet radius x and dimensionless parameters 0 φ , γ and β calculated using Equation (6).The plot was constructed for the values of (a)

Fig- ure 3 and Figure 4 .Figure 3 ,
From it is evident that the effectiveness factor increases with the increasing value of the dimensionless parameter γ .From Figure4, it is also observed that the effectiveness factor increases with the increasing value of the dimensionless parameter β .

Figure 3 .
Figure 3. Plot of the utilization factor η versus dimensionless parameter 0 φ for various values of (a) β and (b) γ .

Figure 4 .
Figure 4. Plot of the utilization factor η versus dimensionless parameter 0 φ for various values of γβ .