AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2015/825683 825683 Research Article Optimal Homotopy Asymptotic Solution for Exothermic Reactions Model with Constant Heat Source in a Porous Medium Mabood Fazle 1 http://orcid.org/0000-0002-0945-7344 Pochai Nopparat 2, 3 Clayton John D. 1 Department of Mathematics, University of Peshawar, Peshawar Pakistan upesh.edu.pk 2 Department of Mathematics, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520 Thailand kmitl.ac.th 3 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400 Thailand 2015 192015 2015 27 05 2015 07 06 2015 192015 2015 Copyright © 2015 Fazle Mabood and Nopparat Pochai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The heat flow patterns profiles are required for heat transfer simulation in each type of the thermal insulation. The exothermic reaction models in porous medium can prescribe the problems in the form of nonlinear ordinary differential equations. In this research, the driving force model due to the temperature gradients is considered. A governing equation of the model is restricted into an energy balance equation that provides the temperature profile in conduction state with constant heat source on the steady state. The proposed optimal homotopy asymptotic method (OHAM) is used to compute the solutions of the exothermic reactions equation.

1. Introduction

In physical systems, energy is obtained from chemical bonds. If bonds are broken, energy is needed. If bonds are formed, energy is released. Each type of bond has specific bond energy. It can be predicted whether a chemical reaction would release or need heat by using bond energies. If there is more energy used to form the bonds than to break the bonds, heat is given off. This is well known as an exothermic reaction. On the other hand, if a reaction needs an input of energy, it is said to be an endothermic reaction. The ability to break bonds is activated energy.

Convection has obtained growth uses in many areas such as solar energy conversion, underground coal gasification, geothermal energy extraction, ground water contaminant transport, and oil reservoir simulation. The exothermic reaction model is focused on the system in which the driving force was due to the applied temperature gradients at the boundary of the system. In , they proposed the investigation of Rayleigh-Bernard-type convection. They also study the convective instabilities that arise due to exothermic reactions model in a porous medium. The exothermic reactions release the heat, create density differences within the fluid, and induce natural convection that turn out the rate of reaction affects . The nonuniform flow of convective motion that is generated by heat sources is investigated by . In , they propose the two- and three-dimensional models of natural convection among different types of porous medium.

In this research, the optimal homotopy asymptotic method for conduction solutions is proposed. The model equation is a steady-state energy balance equation of the temperature profile in conduction state with constant heat source.

The optimal homotopy asymptotic method is an approximate analytical tool that is simple and straightforward and does not require the existence of any small or large parameter as does traditional perturbation method. As observed by Herişanu and Marinca , the most significant feature OHAM is the optimal control of the convergence of solutions via a particular convergence-control function H and this ensures a very fast convergence when its components (known as convergence-control parameters) are optimally determined. In the recent paper of Herişanu et al.  where the authors focused on nonlinear dynamical model of a permanent magnet synchronous generator, in their study a different way of construction of homotopy is developed to ensure the fast convergence of the OHAM solutions to the exact one. Optimal Homotopy Asymptotic Method (OHAM) has been successfully been applied to linear and nonlinear problems [16, 17]. This paper is organized as follows. First, in Section 2, exothermic reaction model is presented. In Section 3, we described the basic principles of the optimal homotopy asymptotic method. The optimal homotopy asymptotic method solution of the problem is given in Section 4. Section 5 is devoted for the concluding remarks.

2. Exothermic Reactions Model

In this section, we introduce a pseudohomogeneous model to express convective driven by an exothermic reaction. The case of a porous medium wall thickness (0<z<L) is focused. The normal assumption in the continuity and momentum equations in the steady-state energy balance presents a nondimensional form of a BVP for the temperature profile [5, 13]: (1)d2θ0dz2+Bϕ21-θ0Bexpγθ0γ+θ0=0.Here, θ0 is the temperature, the parameter B is the maximum feasible temperature in the absence of natural convection, ϕ2 is the ratio of the characteristic time for diffusion of heat generator, and γ is the dimensionless activation energy. In the case of the constant heat source, (1) can be written as(2)d2θ0dz2+Bϕ21-θ0B=0,subject to boundary condition(3)dθ0dz=0,at  z=0,θ0=0,at  z=1.

3. Basic Principles of Optimal Homotopy Asymptotic Method

We review the basic principles of the optimal homotopy asymptotic method as follows.

(i) Consider the following differential equation:(4)Aux+ax=0,xΩ,where Ω is problem domain, A(u)=L(u)+N(u), where L, N are linear and nonlinear operators, u(x) is an unknown function, and a(x) is a known function.

(ii) Construct an optimal homotopy equation as(5)1-pLϕx;p+ax-HpAϕx;p+ax=0,where 0p1 is an embedding parameter and Hp=i=1mpiKi is auxiliary function on which the convergence of the solution greatly dependent. Here, Kj are the convergence-control parameters. The auxiliary function Hp also adjusts the convergence domain and controls the convergence region.

(iii) Expand ϕx;p,Kj in Taylor’s series about p, one has an approximate solution: (6)ϕx;p,Kj=u0x+k=1ukx,Kjpk,j=1,2,3,.Many researchers have observed that the convergence of the series equation (6) depends upon Kj, j=1,2,,m; if it is convergent then, we obtain(7)v~=v0x+k=1mvkx;Kj.

(iv) Substituting (7) in (4), we have the following residual:(8)Rx;Kj=Lu~x;Kj+ax+Nu~x;Kj.If Rx;Kj=0, then v~ will be the exact solution. For nonlinear problems, generally, this will not be the case. For determining Kj, (j=1,2,,m), collocation method, Ritz method, or the method of least squares can be used.

(v) Finally, substituting the optimal values of the convergence-control parameters Kj in (7) one can get the approximate solution.

4. Application of OHAM to an Exothermic Reaction Model

Applying OHAM on (2), the zeroth, first, and second order problems are(9)1-pθ0-Hpθ+Bϕ21-θ0B=0.We consider θ0, Hp in the following manner:(10)θ=θ0,0+pθ0,1+p2θ0,2.H1p=pK1+p2K2,

4.1. Zeroth Order Problem

(11) θ 0,0 = 0 , with boundary conditions(12)θ0,01=0,θ0,00=0.The solution of (11) with boundary condition (12) is(13)θ0,0z=0.

4.2. First Order Problem

(14) θ 0,1 - K 1 ϕ 2 B = 0 , with boundary conditions(15)θ0,11=0,θ0,10=0.The solution of (14) with boundary condition (15) is(16)θ0,1z,K1=K1ϕ2B2z2-1.

4.3. Second Order Problem

(17) θ 0,2 z , K 1 , K 2 = K 1 ϕ 2 B + K 1 2 ϕ 2 B - 1 2 K 1 2 ϕ 4 B z 2 + 1 2 K 1 2 ϕ 4 B + 1 2 K 2 ϕ 2 B , with boundary conditions(18)θ0,21=0,θ0,20=0.The solution of (17) with boundary condition (18) is(19)θ0,2z,K1,K2=-124ϕ4K12Bz4+12ϕ2K1Bz2+12ϕ2K12Bz2+14ϕ4K12Bz2+12ϕ2K2Bz2-524ϕ4K12B-12ϕ2K1B-12ϕ2K12B-12ϕ2K2B.The final three terms solution via OHAM for p=1 is(20)θ~0z,K1,K2=θ0,0z+θ0,1z,K1+θ0,2z,K1,K2.The method of least squares is used to determine the convergence control parameters K1 and K2 in (20). In particular case for ϕ=1, B=10, the values of the convergence control parameters are K1=-0.8337205022 and K2=-0.02092667470.

By substituting the values of K1 and K2 in (20) and after simplification, we can obtain the second order approximate solution via OHAM. To check the accuracy of the OHAM solution, a comparison between the solutions determined by OHAM and numerical methods was made and is presented in Table 1. Graphical representation of the solution using finite difference technique , OHAM, and Runge-Kutta Fehlberg fourth fifth order method is shown in Figure 1; an excellent agreement can be observed. We can see that the OHAM gives a better accurate solution than the traditional finite difference technique of . On the other hand, the OHAM gives a continuity solution, but the traditional finite difference technique gives a discrete solution. It follows that the solutions of the OHAM is easier to implement than the finite difference solutions.

Comparison of θ0(z) via OHAM and RKF45 for ϕ=1,B=10.

Z FDM  RKF45 OHAM Percentage error
0.0 3.114344 3.518277 3.518285 0.000227
0.1 3.046176 3.485927 3.485969 0.001204
0.2 2.911251 3.388613 3.388675 0.001829
0.3 2.711819 3.225339 3.225359 0.000620
0.4 2.451166 2.994264 2.994284 0.000667
0.5 2.133897 2.693071 2.693037 0.001262
0.6 1.766284 2.318441 2.318432 0.000388
0.7 1.356680 1.866723 1.866701 0.001178
0.8 0.915960 1.333395 1.333311 0.006299
0.9 0.457980 0.713042 0.713046 0.000560
1.0 0.000000 0.000000 0.000000

Comparison of analytical and numerical solution.

In Figure 2, we exhibit the effect of different values of ϕ with fixed value of B on temperature profile.

5. Concluding Remarks

In this paper, one has described an optimal homotopy asymptotic technique for obtaining the temperature profiles in porous medium. We can see that the temperature reduces to the end. The OHAM scheme for obtaining the model is convenient to implement. The OHAM gives fourth order accurate solutions. It follows that the method has no instability problem. The model should be considered in the case of nonconstant heat source.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The authors greatly appreciate valuable comments received from Professor John D. Clayton and their reviewers.

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