^{1}

^{2}

^{3}

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.

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 [

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.

The dimensionless mass transport equation of porous catalyst particle is [

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

The utilization factor

In the recent years, much attention is devoted to the application of the Adomian decomposition method to the solution of various scientific models [

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

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

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

The normalized numerical simulation of three dimensional substrate concentration y versus dimensionless pellet radius x is shown in Figures 2(a)-(c). The time independent concentration y is represented in Figures 2(a)-(c). For fixed value of

the concentration of

The variation in effectiveness factor for various values of

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

The authors express their gratitude to the reviewers for their valuable comments to improve the quality of the manuscript. This work was supported by the Department of Science and Technology (DST) (No. SB/SI/PC- 50/2012), New Delhi, India. The authors are thankful to the Head of the Department of Mathematics, Principal and Chairman of Sethu Institute of Technology, Kariapatti for their encouragement.

Mayathevar Renugadevi,Saminathan Sevukaperumal,Lakshmanan Rajendran, (2016) The Approximate Analytical Solution of Non-Linear Equation for Simultaneous Internal Mass and Heat Diffusion Effects. Natural Science,08,284-294. doi: 10.4236/ns.2016.86033

Consider the nonlinear differential equation in the form

with initial condition

where

So, the problem (A.1) can be written as,

The inverse operator

Applying

By operating

The Adomian decomposition method introduce the solution

where the components

By substituting (A.8) and (A.9) into (A.7),

Through using the Adomian decomposition method, the components

which gives

From (A.9) and (A.12), we can determine the components

In this appendix, we derive the general solution of nonlinear Equation (1) by using the Adomian decomposition method. We write the Equation (1) in the operator form,

where

where A and B are the constants of integration. We let,

and

where

In view of Equations (B. 3 - B. 5), Equation (B. 2) gives

We identify the zeroth component as

Using the boundary condition (4) we get,

and the remaining components can be obtained using the recurrence relation

where

The remaining polynomials can be generated easily, and so,

Adding (B. 8), (B. 12) and (B. 13) we get the Equation (6) in the text.

function pdex1

m = 2;

x = linspace(0,1);

t = linspace(0,100);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

u = sol(:,:,1);

surf(x,t,u)

title('Numerical solution computed with 20 mesh points.')

xlabel('Distance x')

ylabel('Time t')

figure

plot(x,u(end,:))

title('Solution at t = 2')

xlabel('Distance x')

ylabel('u(x,2)')

% --------------------------------------------------------------

function [c,f,s] = pdex1pde(x,t,u,DuDx)

c = 1;

f = DuDx;

Q=1;

B=1.5;

r=1;

s =-(Q^2)*u*exp(r*B*(1-u)/(1+B*(1-u)));

% --------------------------------------------------------------

function u0 = pdex1ic(x)

u0 = 1;

% --------------------------------------------------------------

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)

pl = 0;

ql = 1;

pr = ur-1;

qr = 0;

C_{A} Concentration of reactant A inside the catalyst pellet (mole/cm^{3})

C_{A,s} Concentration of reactant A at the surface of catalyst pellet (mole/cm^{3})

^{2}∙s^{−1})

E Activation energy (kJ∙mol^{−1})

r_{A} Arrhenius reaction rate (s^{−1})

R_{g} Universal gas constant (8.3145 J∙k^{−1}∙mol^{−1})

T Temperature inside the catalyst pellet (K)

T_{ref} Reference temperature (K)

T_{s} Temperature at the surface of catalyst pellet (K)

x Dimensionless radius of the spherical catalyst pellet (none)

y Dimensionless concentration along radial direction of catalyst pellet (none)

Greek Symbols

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