On Green’s function methods to solve nonlinear reaction–diffusion systems

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Abstract

Recent studies have shown that the usage of classical discretization techniques (e.g., orthogonal collocation, finite-differences, etc.) for reaction–diffusion models cannot be stable in a wide range of parameter values as required, for instance, in model parameter estimation. Oriented to reduce the adverse effects of numerical differentiation, integral equation formulations based on Green’s function methods have been considered, in the chemical engineering fields. In this paper, a further exploration of this approach for nonlinear reaction–diffusion transport is carried out. To this end, the Green’s function problem is presented and solved for three geometries (i.e., rectangular, cylindrical and spherical), and three representative examples are worked out to illustrate the ability of the method to describe accurately the phenomena with respect to analytical and numerical solutions via finite-differences. Our results show that: (i) by avoiding numerical differentiation, the round-off error propagation is significantly reduced, (ii) boundary conditions are exactly incorporated without approximation order reduction and (iii) more accurate calculations are performed making use of less mesh points and computer time.

Introduction

Modern computer-based process design, optimization and control methodologies can require massive on-line solution of detailed, commonly distributed-parameter models. For instance, optimization of chemical reactors with selectivity criteria requires the solution of the reaction-transport (diffusion and convection) model for catalytic pellet and/or reactor scales. At each step of the optimization cycle the underlying reaction-transport model must be solved numerically by means of stable and robust schemes. Finite-differences and finite-elements schemes are widely used given the existence of both theoretical results on stability and computational techniques for implementation. In this way, it is apparent that the development of stable procedures for massive solution of distributed-parameter models seems to be a solved issue. Recent studies have shown that the usage of classical discretization techniques (e.g., orthogonal collocation, finite-differences, etc.) for reaction–diffusion models cannot be stable in a wide range of parameter values as required, for instance, in model parameter estimation Agrawal et al., 2006, Asteasuain et al., 2001. The main source of instability is the lower order approximations for boundary conditions compared to the higher-order used in the domain. Hence, the development of stable and robust numerical procedures for distributed parameter processes is still of prime importance within advanced process design and optimization methodologies.

Inaccuracies in the numerical solution of distributed parameter models are induced by inaccurate approximations for spatial derivatives of any order. From signal processing practice it is known that differentiators are, in fact, highly sensitive to round-off errors. Differentiation schemes are very likely to magnify the propagation of approximation errors and, hence, to reduce the accuracy of numerical solutions. Generally, this drawback is compensated by the usage of refined meshes. Since differential operators are analytically inverted within a Green’s function formulation for distributed parameter processes, integral equation formulations for distributed parameter processes become a serious alternative to avoid the usage of approximate differentiators. In this approach, the differential equation is converted into an integral, Fredholm-type, equation where boundary conditions are incorporated exactly. As is known in signal processing and process control theories, integrators are welcome because of their ability to wash-out and smooth round-off errors. In this way, integral equation formulations offer the advantage that approximations for differentiators have no longer to be considered, and potential numerical schemes could depend on numerical quadratures, that are unconditionally stable.

The application of Green’s functions for solving reaction–diffusion processes in chemical engineering can be traced back to Amundson and Schilson (1961), who obtained the Green’s function for isothermal linear reaction in a sphere, and solved the resulting linear Fredholm integral equation via a successive approximation technique. Kesten (1969) applied Green’s function analysis to obtain concentration profiles for ammonia decomposition in a spherical catalytic pellet. Dixit and Tavlarides (1982) were the first to use Newton iteration schemes to solve nonlinear Fredholm equations arising from reaction in a sphere, and applied their results to the Fischer-Tropsch synthesis. Subsequently, Mukkavilli, et al., 1987a, Mukkavilli et al., 1987b presented and solved numerically an integral equation formulation for reaction in a finite cylinder with Dirichlet and Robin-type boundary conditions. They solved the underlying Green’s function differential equation by means of eigenfunction expansions. Recently, Onyejekwe, 1995, Onyejekwe, 1996, Onyejekwe, 2002 used integral equation formulation to propose a Green element solution for nonlinear reaction–diffusion equations. In that work, the main idea was to invert the diffusion operator, via Green’s functions, to subsequently obtain a set of nonlinear (algebraic) equations from a suitable spatial discretization. Extensive numerical simulations showed the stability and accuracy of the proposed method compared to standard finite-difference schemes.

Maybe due to the boom of finite-differences and finite-element methods, the application of Green’s function theory for solving nonlinear reaction-transport process has been rarely explored. However, as mentioned earlier, integral equation formulations offer interesting implementation advantages, including exact incorporation of boundary conditions and enhanced stability in the face of round-off errors. In this paper, a further exploration of integral equation formulation for nonlinear reaction–diffusion transport is carried out. To this end, the Green’s function problem is posed and solved for three geometries (i.e., rectangular, cylindrical and spherical), and three representative examples are worked out to illustrate the ability of the method to describe accurately the phenomenae with respect to analytical and numerical solutions via finite-differences.

Section snippets

Reaction–diffusion model

Following the classical formulation described by Aris (1975), which assumes a homogeneous porous pellet and using effective transport coefficients, the steady state concentration and temperature profiles for a single chemical reaction are given bym2y=ϕ2R(y,θ)andm2θ=βϕ2R(y,θ)where the one-dimensional operator m2 is given bym2=1xmddxxmddxx=x/lm is the dimensionless spatial coordinate, y and θ are dimensionless concentration and temperature, respectively. In addition, ϕ is the Thiele

Integral equation formulation

The reaction–diffusion model described in the above section is represented as a differential equation. In this section, following a Green’s function formalism (Greenberg, 1971), we shall provide an integral equation formulation of the phenomena, which will be advantageous to compute approximate solutions for the nonlinear case.

Solution through nonlinear iterative process

In general, the integral equation formulation for the reaction–diffusion process can be expressed asy(x)=ϕ(x)+N(y(z),x)where N() is the nonlinear integral operator 01z2G(z,x)Q()dz, and ϕ(x)=dG(1,x)/dz. In order to solve (30) one should consider a finite-dimensional approximation, say yaRn, for the infinite-dimensional function y(x). For instance, an easy way for ya is to take a n-dimensional vector corresponding to a regular mesh in the normalized domain [0,1]. That is, ya=[y1,y2,,yn]T,

Numerical results

The aim of this section is to explore the ability of the integral equation formulation to produce accurate solutions for nonlinear reaction–diffusion equations. To this end, we consider three examples; namely, one linear having analytical solution, and other two nonlinear to compare with solutions obtained with a finite-differences scheme.

Example 1

Consider the isothermal first-order kinetics in a catalytic pellet. As a comparison parameter, consider the effectiveness factor, which is the overall

Conclusions

In this paper, the ability of integral equation formulations to provide accurate numerical solutions for reaction–diffusion models was explored. Integral equations formulations are obtained from Green’s function theory, and have the following advantage relative to more traditional discretization schemes, like finite-differences and finite-elements, (i) by avoiding numerical differentiation, the round-off error propagation is significantly reduced, and (ii) boundary conditions are exactly

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