A Singular Perturbation Problem for Steady State Conversion of Methane Oxidation in a Reverse Flow Reactor

Abstract

The governing equations describing methane oxidation in a reverse flow reactor are given by a set of convective-diffusion equations with a nonlinear reaction term, where temperature and methane conversion are dependent variables. In this study, the process is assumed to be a one-dimensional pseudohomogeneous model and takes place with a certain reaction rate in which thewhole process ofthereactor is still workable. Thus, the reaction rate can proceed at a fixed temperature. Under these conditions, we can restrict ourselves to solving the equations for the conversion only. From the available data, it turns out that the ratio of the diffusion term to the reaction term is small. Hence, this ratio is considered as a small parameter in our model and this leads to a singular perturbation problem. Numerical difficulties will be found in the vicinity of a small parameter in front of a higher order term. Here, we present an analytical solutionby means of matched asymptotic expansions. The result shows that, up to and including the first order of approximation, the solution is in agreement with the exact and numerical solutions of the boundary value problem.

An Analytical Solution Of 1-D Pseudo Homoneneous Model For Oxidation Reaction Using Homotopy Perturbation Method

“…In previous research, Nuryaman et al (see [2]) constructed a singular perturbation problem using Equations ( 5)-( 8) at steady state and takes place with a certain reaction rate in which the whole process of the reactor still workable and with small temperature variation and no heat loss. This leads to an equation in terms of the conversion variable only.…”

“…But there are some researchers who carried out analytical approaches, using certain assumptions, to discuss the behavior of dependent variables. In [2], Nuryaman et al presented a singular perturbation problem for conversion variable in steady state conditions and solved using matched asymptotic methods. Furthermore, Nuryaman and Gunawan (see [3]) used the asymptotic expansion method to solve singular perturbation problems for temperature and concentration variables in steady state conditions.…”

Solution of the Reverse Flow Reactor Model Using Homotopy Analysis Method

“…Previous studies [5,6] used this model to investigate operating parameter sensitivities of RFR on the behavior of dependent variables for periodic feed gas by using a numerical approach. Whereas in [7,8], this model was used to construct singular perturbation problems by considering certain assumptions for steady state conditions and solved them using asymptotic methods. While for the unsteady state case, Nuryaman [9] reported an analytical solution for conversion equation that was derived from the 1-D pseudo homogeneous model which assumed that the reaction took place spontaneously at a certain reaction rate.…”

A Singular Perturbation Problem in Steady State of Methane Combustion Using Reverse Flow Reactor

“…The common features of combustion process in an RFR were described by convective-diffusion equations with the corresponding boundary and initial conditions [7]. Many previous studies [1][2][3][4][5][6] mostly determined the features of the RFR dynamic with numerical approach. For example, Gupta and Bhatia [1] proposed a Newton's technique to directly solve for the cyclic profiles.…”

“…As common convective-diffusion equation, some methods can be applied to this problem, for examples Green's function method (see [12]), classical integral transform techniques [13], generalized and extended classical integral transform method [14,3]. Nuryaman et al [2] presented an asymptotic solution of a singular perturbation problem for steady state conversion of methane oxidation in reverse flow reactor. They considered a one-dimensional pseudo-homogeneous model with a certain reaction rate in which the whole process of the reactor was still workable and in one direction only from the left to the right end.…”