Power Series Solution for a Two-step Reactive-diffusive Equation with Variable Pre-exponential Factor 1

We discussed in this research a power series method for solution of a two-step reactive-diffusive equation with variable pre-exponential factor. A transformation was used to fit in the geometry of the model to allow an asymptotic series expansion which is invoked in a two compartmental regions with respect to the spatial variable. Adequate matching conditions were used at the interface between the two regions. The results were displayed graphically and compared with the previous work in the literature.


INTRODUCTION
Studies on combustion abound in the literature. A number of investigators have considered works in relation to chemical kinetics, thermodynamics, fluid mechanics and transport processes. In the industry rapid exothermic reaction processes which take place with the evolution of large amount of heat are very important. Energy released by a chemical explosion generally occurs from exothermic reactions.
Alao and Ayeni [3] in their research established the condition for the existence and uniqueness of solution for a transformed mathematical model describing a viscous reacting flow. They used the Arrhenius term step reaction and considered that the activation energy is small. Furthermore, Alao [2] working after their former work; generalized the viscous reacting flow. Similarly, solutions were investigated and the numerical solutions for the resulting differential equation were obtained. Adegbie and Ayeni [1] in their study discussed the existence and uniqueness of solution of a nonhomogenous branched chain reaction with slow radical recombination.
Okoya [10] brought to light a new analytical solution for the Frank-Kamenetskii parameter δ in the space in the special case of n = 2, Q 2 = 0 corresponding to the sensitized reaction. The analytical solution is in terms of Bernoulli's numbers, and it is thus in accordance with the numerical integration for which β (<< 1) is very small and for β→0. It reduces to the solution for the Frank-Kamenetskii case θ m is the dimensionless temperature at the centre of the slab. He also determined the transitional values δ, β and θ m numerically Olanrewaju et al. [11] investigated the effects of the geometry of vessel in the notation. They discovered that the maximum temperature in a non-uniform vessel occurs at the centre and that the maximum temperature for diverging or converging vessel is greater than the one of the diverging vessel

MATHEMATICAL FORMATION
The governing equation for the steady temperature, T, of a one-dimensional Slab with boundaries lying in the coordinate plane x = ±a may be written in term of physical variables, with the corresponding boundary conditions T 0 , in the initial temperature. All other variable or parameters are deferent in the Nomenclature.
Conversing that the consumption of the consultable in neglected, if d 2 = 0, it has been show experimentally that the model can predict the critical ignition temperature for variety of combustible material (Bowes [5] ).
Using the following non-dimensional variables.
on equation (2) and (3) we have after dropping the bars In (4) δ 1 and δ 2 are the Frank-Kamenetskii parameters which are the measures of the exothermicity of the reactions.
A number of attempts for the solution to determine the factors that control thermal ignition of combustion materials relating to the model in Eq. (4) and (5) have been done. Frank-Kamenetskii [8] obtained a closed form solution. Considering that Q 2 = 0, the limit of large activation energy (ε→0) Bowes [5] , Ward and Van De Velde [12] established the possibility of perturbation solution under physically reasonable assumption when Q 2 = 0 and n = 0. Numerical and asymptotic solutions were obtained for some Biot numbers and for some realistic conditions Burnell et al. [7] and Bilingham [4] . In these work we are using the spatial induced converging series method to obtain solution to the model in (4) and (5) Substituting (B), (C) and (D) in Eq. (2) and Taking the coefficients of y 0 , y 1 , y 2 Order y 0 : Order y (1) : ( a a a a ) a n a 1 a 2 − ν λ − λ β + + δ λ − λ β λ + ν + λ +

METHOD OF SOLUTION
The non-linear boundary valued problem in the Eq.
Similarly, substituting the series in the Eq. (11) into the Eq. (2) and collecting the coefficients of z 0 , z 1 , z 2 (where z = 1 -y). We have

DISCUSSION OF RESULTS AND CONCLUSION
The steady state solutions of temperature as shown in Fig 1-3 reveal the solution obtained by previous works in the literature and that the curve is continuous at the merging point. This is also an improvement over the previous work by other authors is that it caters for two-step activation energy rather than one which is simple to be comprehended.