Francisco J. Valdés-Parada

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

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.

Upscaling Diffusion and Nonlinear Reactive Mass Transport in Homogeneous Porous Media

In this work, we revisit the upscaling process of diffusive mass transfer of a solute undergoing a homogeneous reaction in porous media using the method of volume averaging. For linear reaction rate kinetics, the upscaled model exhibits a vis-à-vis correspondence with the mass transfer governing equation at the microscale. When nonlinear reactions are present, other methods must be adopted to upscale the nonlinear term. In this work, we explore a linearization approach for the purpose of solving the associated closure problem. For large rates of nonlinear reaction relative to diffusion, the effective diffusion tensor is shown to be a function of the reaction rate, and this dependence is illustrated by both numerical and analytical means. This approach leads to a macroscale model that also has a similar structure as the microscale counterpart. The necessary conditions for the vis-à-vis correspondence are clearly identified. The validation of the macroscale model is carried out by comparison with pore-scale simulations of the microscale transport process. The predictions of both concentration profiles and effectiveness factors were found to be in acceptable agreement. In an appendix, we also briefly discuss an integral formulation of the nonlinear problem that may be useful in developing more accurate results for the upscaled transport and reaction equations; this approach requires computing the Green function corresponding to the linear transport problem.

Letter to the editor: A Green's function formulation of nonlocal finite-difference schemes for reaction-diffusion equations

Reaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Greens function formulation of reaction-diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h^2) accuracy and nonlocal approximations in the whole domain (including boundary nodes). By doing so, the approach departs from a Greens function formulation of the boundary-value problem to pose an approximation problem based on a domain decomposition. Within each subdomain, the corresponding integral equation is forced to have zero residual at given grid points. Different FD schemes are obtained depending on the numerical scheme used for computing the Greens integral over each subdomain. Dirichlet and Neumann boundary conditions are considered, showing that the FD scheme based on the Greens function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation.

An Approximate Solution for a Transient Two‐Phase Stirred Tank Bioreactor with Nonlinear Kinetics

The derivation of an approximate solution method for models of a continuous stirred tank bioreactor where the reaction takes place in pellets suspended in a well‐mixed fluid is presented. It is assumed that the reaction follows a Michaelis‐Menten‐type kinetics. Analytic solution of the differential equations is obtained by expanding the reaction rate expression at pellet surface concentration using Taylor series. The concept of a pelletapos;s dead zone is incorporated; improving the predictions and avoiding negative values of the reagent concentration. The results include the concentration expressions obtained for (a) the steady state, (b) the transient case, imposing the quasi‐steady‐state assumption for the pellet equation, and (c) the complete solution of the approximate transient problem. The convenience of the approximate method is assessed by comparison of the predictions with the ones obtained from the numerical solution of the original problem. The differences are in general quite acceptable.

An Integral Equation Formulation for Solving Reaction-Diffusion-Convection Boundary-Value Problems

Many interesting problems that include convective transport arise in chemical reactor engineering (for example, tubular reactors). To solve these boundary-value problems, finite-difference schemes with a type of discretization of the convection term have been traditionally used. Some controversy about the discretization form of the convection term has arisen because of the different possibilities, including backward, forward and central discretizations. To overcome this problem, the usage of Greens function formulations for the numerical solution of typical chemical engineering problems with both linear and nonlinear kinetics, diffusion and convection phenomena, is presented. A distinctive feature of the proposed scheme is that boundary conditions are exactly incorporated. The results show that the integral formulation is, in general, superior in accuracy to the different finite-differences schemes. That is, more accurate calculations of the performance factor are obtained in terms of less mesh points and computer time.

A Green's function approach for the numerical solution of a class of fractional reaction-diffusion equations

Reaction-diffusion equations with spatial fractional derivatives are increasingly used in various science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and anomalous diffusive transport mechanisms. Most numerical schemes to solve fractional reaction-diffusion equations use finite difference schemes based on the Grunwald-Letnikov formula. This work introduces a new systematic approach based on Greens function formulations to obtain numerical schemes for fractional reaction-diffusion equations. The idea is to pose an integral formulation of the equation in terms of the underlying Greens function of the fractional operator to subsequently use numerical quadrature to obtain a set of ordinary differential equations. To illustrate the numerical accuracy of the method, dynamic and steady-state situations are considered and compared with analytical and numerical solutions via Grunwald finite differences schemes. Numerical simulations show that the scheme proposed improves the performance and convergence of traditional finite differences schemes based on Grunwald formula. Numerical solutions of fractional RD systems are given using Greens function formulations.The scheme proposed exhibit global approximation orders of O ( h α ) .Proposed scheme exhibit better numerical approximation than traditional schemes.

A volume averaging approach for asymmetric diffusion in porous media

Asymmetric diffusion has been observed in different contexts, from transport in stratified and fractured porous media to diffusion of ions and macromolecular solutes through channels in biological membranes. Experimental and numerical observations have shown that diffusion is facilitated in the direction of positive void fraction (i.e., porosity) gradients. This work uses the method of volume averaging in order to obtain effective medium equations for systems with void fraction gradients for passive and diffusive mass transport processes. The effective diffusivity is computed from the solution of an associated closure problem in representative unit cells that allow considering porosity gradients. In this way, the results in this work corroborate previous findings showing that the effective diffusivity exhibits important directional asymmetries for geometries with void fraction gradients. Numerical examples for simple geometries (a section with an obstacle and a channel with varying cross section) show that the diffusion asymmetry depends strongly on the system configuration. The magnitude of this dependence can be quantified from the results in this work.

An Integral Formulation Approach for Numerical Solution of Tubular Reactors Models

In this paper, we derive an integral formulation approach based on Green’s function for the numerical solution of tubular reactor models described by reaction-diffusion-convection (RDC) equations with Danckwerts-type boundary conditions. The integral formulation approach allows the direct incorporation of boundary conditions and leads to a stable and accurate numerical integration with smooth round-off error. Numerical simulations of two of tubular reactors models are presented in order to illustrate the numerical accuracy of the method. The results are compared with those resulting from using standard finite difference method. Our results show that the integral formulation approach improves the performance of classical FD schemes.

A Priori Parameter Estimation for the Thermodynamically Constrained Averaging Theory: Species Transport in a Saturated Porous Medium

The thermodynamically constrained averaging theory (TCAT) has been used to develop a simplified entropy inequality (SEI) for several major classes of macroscale porous medium models in previous works. These expressions can be used to formulate hierarchies of models of varying sophistication and fidelity. A limitation of the TCAT approach is that the determination of model parameters has not been addressed other than the guidance that an inverse problem must be solved. In this work we show how a previously derived SEI for single-fluid-phase flow and transport in a porous medium system can be reduced for the specific instance of diffusion in a dilute system to guide model closure. We further show how the parameter in this closure relation can be reliably predicted, adapting a Green’s function approach used in the method of volume averaging. Parameters are estimated for a variety of both isotropic and anisotropic media based upon a specified microscale structure. The direct parameter evaluation method is verified by comparing to direct numerical simulation over a unit cell at the microscale. This extension of TCAT constitutes a useful advancement for certain classes of problems amenable to this estimation approach.

A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory

The thermodynamically constrained averaging theory (TCAT) is an evolving approach for formulating macroscale models that are consistent with both microscale physics and thermodynamics. This consistency requires some mathematical complexity, which can be an impediment to understanding and efficient application of this model-building approach for the non-specialist. To aid understanding of the TCAT approach, a simplified model formulation approach is developed and used to show a more compact, but less general, formulation compared to the standard TCAT approach. This new simplified model formulation approach is applied to the case of binary species diffusion in a single-fluid-phase porous medium system, clearly showing a TCAT approach that is applicable to many other systems as well. Recent extensions to the TCAT approach that enable a priori parameter estimation, and approaches to leverage available TCAT modeling building results are also discussed.