Analytical Solution of Non-Isothermal Diffusion-Reaction Processes and Effectiveness Factors

e mathematical modeling of nonlinear boundary value problems in catalytically chemical reactor is discussed. In this paper, we obtain the approximate analytical solution and the effectiveness factors for the evolution of single-step transformations under non-isothermal conditions using homotopy perturbation method. We have applied it to many reaction models and obtained very simple analytical expressions for the shape of the corresponding transformation rate peaks. ese analytical solutions represent a signi�cant simpli�cation of the system�s description allowing easy curve �tting to experiment. e accuracy achieved with our method is checked against several reaction models and numerical results. A satisfactory agreement is noted.


Introduction
Non-isothermal systems, where reaction and diffusion take place, are typical in the chemical process industry [1] and also in biological systems [2][3][4].e chemical reaction is always central in these systems, because the rate of the reaction oen will determine how fast chemicals can be produced.A high rate can be realized when the reaction is far from equilibrium.But an operation far from equilibrium is also an operation in which the energy dissipation is large.With the present interest to save valuable resources, chemical reactors should be studied also from the perspective of obtaining a more energy-efficient operation, in addition to maintaining the production of chemicals.In biological systems, one may expect that energy efficiency is an issue of survival, especially under harsh conditions [5].In such cases and probably many others, a thermodynamic description will be important to understand the transport phenomena involved [4,6].Studies of minimum energy dissipation start with an expression for the entropy production [7][8][9].
Chemical reactions are inherently non-linear processes, and are most successfully described in the �eld of reaction kinetics by the law of mass action [10,11].e reaction rate is not commonly expressed as a function of the reaction Gibbs energy.is is not surprising, because classical nonequilibrium thermodynamics [12,13] assumes a linear relation between these two variables, and experimental evidence indicates that this is only correct very close to chemical equilibrium.e �rst to address this problem successfully was Kramers [14] who described the reaction as a diffusion process along a reaction coordinate.
e extension in the context of non-equilibrium thermodynamics was �rst proposed by Prigogine and Mazur [15][16][17].By integrating over these variables to obtain the thermodynamic level, one can describe several phenomena, which are non-linear on the macroscopic level, and which retain a linear force-�ux relation on the mesocopic level.is applies not only to chemical reactions [18], but for instance also to adsorption [19], nucleation [20], electrode over potentials [21] and active transport in biology [4].e number of cases studied is now growing fast.e coupling of chemical reactions to other processes is then important [8,18,[22][23][24][25].Non-equilibrium thermodynamics is not only a theory for transport processes, it is also a theory for �uctuations.It has been demonstrated that the �uctuating contributions to the thermodynamic �uxes in a non-equilibrium system satisfy the �uctuation-dissipation theorem just like they do in equilibrium [26].
e theory of non-equilibrium thermodynamics is based on the assumption of local thermodynamic equilibrium.e validity of this assumption has been established by molecular dynamics simulations in several cases [27][28][29].Fluctuations and the resulting correlation functions away from equilibrium were then not considered.One of the major �ndings has been that although local equilibrium is valid for the description of the mean values of thermodynamic �elds, it is no longer valid for the description of the �uctuations around their average non-equilibrium values [26].Recently Vergara et al. [30] developed the multicomponent diffusion system including cross-term diffusion coefficients relating to �ux of the component  to concentration gradients of component .But in our problem the cross-diffusion is neglected.Ikeda et al. [31] analyzed this problem for a reaction-diffusion problem with a temperature gradient using a linear approximation for the description of the reaction.For the reaction-diffusion problem the assumption of local equilibrium has to be extended to be valid also along the reaction coordinate.However, to the best of our knowledge, till date no general analytical expressions of mass concentrations and effectiveness factors have been reported.e purpose of this communication is to derive the approximate analytical expression of mass concentration for planar particles by solving the non-linear differential equations using He's homotopy perturbation method [32][33][34][35].

Mathematical Formulation of the Problem and Analysis
e mathematical description of a catalytic chemical reactor (        is given by [1]  2   2 −  2  ( = 0 for 0 <  < where  is the spatial coordinate, ( is the reaction rate function (which is non-linear), and  is the half thickness of porous slab (.e mass concentration  is de�ned as the following function: where   is the volumetric molar concentration of the key component ,  AS is the surface value of the key component , and  is the corresponding iele modulus which is de�ned as follows: Here   ,   , and  BS are the speci�c kinetic constant, effective diffusivity coefficient, and the dimensionless concentration of the component , respectively.Also the temperature  and the mass concentration  are no longer independent, which satis�es the following relation: where  is the thermicity of the reaction and is de�ned as follows: Here −Δ is the reaction heat;  is the effective thermal conductivity inside the porous slab; and   is the dimensional temperature at the external pellet surface.Δ  represents the entropy change "Δ" of a system under this process.Entropy increases in all spontaneous processes.Hence entropy may be regarded as a measure of disorder or randomness of the molecules of the system.For isothermal process, the entropy changes of the universe during a reversible process are zero.e entropy of the universe increases in an irreversible process.e parameter  represents the deviation from isothermal conditions, being  < 0 and   0 for endothermic and exothermic reactions, respectively.Now the dimensionless reaction rate function ( including (1) is given by where the parameter  is de�ned as follows: where  is the Arrhenius group and is de�ned as Here  denotes the activation energy;  is the universal gas constant.Hence the corresponding non-linear boundary value problem is given by Using the following dimensionless variables: Now (9) becomes in dimensionless form as follows: e internal effectiveness factor ( is a measure of the relative importance of diffusion to reaction limitations.at is,

Solution of Boundary Value Problem Using HPM
Recently, many authors have applied the homotopy perturbation method (HPM) to solve the non-linear problems in physics and engineering sciences [36][37][38][39].Recently this method is also used to solve some of the non-linear problem in physical sciences [32][33][34].is method is a combination of homotopy in topology and classic perturbation techniques.He used to solve the Lighthill equation [32], the Diffusion equation [33], and the Blasius equation [34,35].e HPM is unique in its applicability, accuracy and efficiency.e HPM uses the imbedding parameter  as a small parameter, and only a few iterations are needed to search for an asymptotic solution.Using this method, we can obtain the following solution to (11) and (12) for the following three cases (see Appendices A-C): Case 1.When the reaction orders    and   , the analytical solution of ( 11) to (12) using homotopy perturbation method [35,[40][41][42][43][44] is where provided   .Using ( 14), the effectiveness factor  is given by Case 2. When the reaction orders    and   1, we can obtain the analytical solution of ( 11) to ( 12) using homotopy perturbation method as follows: where provided   1.Using ( 14), the effectiveness factor  is given by Case 3. When the reaction orders    and   , the analytical solution of ( 11) to ( 12) using the homotopy perturbation method is given by e above expression is valid only if   2. Using ( 14), the effectiveness factor  is given by

Numerical Simulation
e non-linear equations ( 11) to (12) for the �ve cases are solved by numerical methods.e function pdex4, in Matlab soware, is used to solve two-point boundary value problems (BVPs) for ordinary differential equations which are given in Appendices D-H.In Tables 1, 2, 3, 4, and 5, the numerical results are also compared with the obtained analytical expressions (see (15), (20), and ( 25)) and Villa et al. [1� results for some �xed value of   .5.

Results and Discussions
Tables 1-5 represent the dimensionless mass concentration ) versus the dimensionless spatial coordinate  for the following different values of the dimensionless parameters  and : (i) when   ,   ,   .5,   , and   .
(ii) when   ,   ,   .5,   , and   .From these tables it is evident that the values of the dimensionless mass concentration  decrease, when dimensionless parameters  and  decrease.In Tables 1-5, our analytical results for the mass concentrations  are compared with the numerical results and Villa et al. results [1].Villa et al. [1] obtained the analytical solution of this problem only for taking the parametric restrictions.
In Tables 1-4, our analytical results are compared with the numerical results and Villa et al. [1] results.A good agreement between them is noted.In Table 5  our analytical results and Villa et al. [1] results are compared with the numerical results.Our analytical result gives good agreement with the numerical results.In Table 6, the effectiveness factors for the Cases 1 and 2, a satisfactory agreement between our results and Villa et al. [1] results is noted.

Conclusion
e steady state non-linear reaction-diffusion equation has been solved analytically and numerically.A simple and approximate dimensionless mass concentrations  are derived by using the HPM for all values of dimensionless parameters , , and .e HPM is an extremely simple method and it is also a promising method to solve other nonlinear equations.is method can be easily extended to �nd the solution of all other non-linear equations.e proposed formulas are used to �nd the thiele module range, in which multiple values of the effectiveness factor should be searched.e present method is quick and efficient and is able to reduce T 6: Comparison of our effectiveness factors (( 19), (24), and ( 29)), and Villa et al. [1] when   .(20) in this paper is derived.�o �nd the solution of ( 11) and ( 12), when  = 0 and  = 1.When ( + 1) small, then (11)    Effectiveness factor for the heterogeneous chemical reaction.
[1] the Case 3, T 3: Comparison of our analytical e�pression of concentration pro�les  with numerical results and Villa et al.[1]results corresponding to Case 2, for      , and   , when    and   .Comparison of our analytical expression of concentration pro�les  with numerical results and Villa et al.[1]results corresponding to Case 3, for      , and   , when    and   .
[1] 4: Comparison of our analytical e�pression of concentration pro�les  with numerical results and Villa et al.[1]results corresponding to Case 2, for      , and   , when    and   .