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

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.     

SYMBOLIC OPERATOR SOLUTIONS OF LAPLACE'S AND STOKES’ EQUATIONS PART I LAPLACE'S EQUATION

Symbolic operator techniques are employed to derive a general solution of Laplace's equation in the infinite space external to a sphere. This is done for the case where the function vanishes on the sphere surface and arbitrary continuous asymptotic boundary data are imposed at infinity, such data being prescribed in the form of a solution of Laplace's equation that is analytic at the origin. In contrast with other standard methods for solving Laplace's equation, e.g., Green's functions, eigenfunction expansions, etc., the novelty of the proposed method lies in the fact that the solution can be expressed in a completely explicit form, directly in terms of (radial derivatives of)the given “undisturbed” field at infinity

A reciprocal theorem is derived and used to demonstrate that certain integral properties of the field can be obtained directly from the prescribed data at infinity, without recourse to a detailed solution of the relevant boundary-value problem. This global symbolic operator technique is illustrated for ellipsoidal as well as spherical particles

The elementary scalar harmonic analysis of the present paper serves as an entré to a companion paper (Part II), concerned with the application of similar symbolic techniques to the solution of more difficult vector biharmonic boundary-value problems, relevant to hydrodynamic Stokes flows in the infinite region external to a particle.     

A SIMPLIFIED DESCRIPTION OF CHAR COMBUSTION

A simplified analysis of carbonaceous particle combustion is presented that includes the effects of pore diffusion and growth as well as gas-phase heat and mass transfer. The combustion dynamics are described by time-dependent equations for particle temperature, radius and a number of intraparticle conversion variables. These are coupled to pseudosteady equations for gas-phase transport and internal reaction and diffusion. The differential equations for gas-phase transport are reduced by quadrature to a nonlinear boundary condition to the intraparticle boundary value problem. Numerical calculations are performed for conditions relevant to pulverized coal combustion. An analytical solution of the intraparticle problem, pertinent to the regime of strong diffusional limitations, reduces the intraparticle solution into a set of two quadratures which drastically simplifies the numerical calculations. The simplified intraparticle solution is in excellent agreement with the full solution at 1800 K free stream temperature and fair agreement at 1500 K.     

Boundary value problems in transport with mixed and oblique derivative boundary conditions-II: Reduction to first order systems

Mixed second derivative or oblique derivative boundary conditions for the steady state heat conduction equation with heat generation in the two-dimensional plane are of importance in applications[1]. In general, they lead to non-selfadjoint boundary value problems or to singular integral equations.In this paper, the second order partial differential equations have been decomposed into first order systems, which under suitable circumstances can be conveniently adapted to satisfy the mixed or oblique derivative boundary conditions. Furthermore, the differential operator with respect to one of the variables is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors. The solution to the boundary value problems is obtained by expansion in terms of these eigenvectors.The method works in infinite regions with separable coordinates, where it gives the same solution as that by infinite Fourier transform. It also works on finite regions, when periodic boundary conditions are used such as in right circular cylinders. Solutions are presented for an infinite slab and concentric annulus for both mixed and oblique derivative boundary conditions.     

Boundary value problems in transport with mixed or oblique derivative boundary conditions—I: Formulation of equivalent integral equations

When an internally heated body is cooled along its boundary by a peripherally flowing fluid that is continually replenished from an external source, a differential energy balance on the boundary leads to unfamiliar boundary conditions. Such boundary conditions involve mixed second derivatives with respect to spatial variables, which under additional assumptions (such as an infinite heat transfer coefficient) lead to oblique derivative boundary conditions; i.e. at the boundary an oblique derivative of the temperature is specified. Classical attempts at solution of elliptic partial differential equations with oblique derivative boundary conditions have been through the establishment of equivalent singular integral equations, using complex analytic continuation. The theory of singular integral equations is complicated, however.Using appropriate Green's functions, the boundary value problems of interest have been reduced to equivalent integral equations in this work. While oblique derivative boundary value problems are shown to lead to singular integral equations, the mixed derivative boundary value problem is shown to yield Fredholm integral equations directly. This surprising finding is mathematically significant, because Fredholm integral equations are solved more easily, and physically significant because the mixed derivative boundary condition is the more realistic condition in the present context. A method of solution of Fredholm integral equations is discussed.More complicated boundary conditions in which axial conduction in the coolant fluid is important have also been shown to lead to Fredholm integral equations. Finally a transient problem has been formulated.     

Optimal boundary control of a diffusion–convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process

In this paper the optimal boundary control problem for diffusion–convection-reaction processes modeled by partial differential equations (PDEs) defined on time-dependent spatial domains is considered. The model of the transport system with time-varying domain arises in the context of high energy consuming Czochralski crystal growth process in which the crystal temperature regulation must successfully account for the change in the crystal spatial domain due to the crystal growth process realized by the pulling crystal out of melt. Starting from the first principles of continuum mechanics and transport theorem the time-varying parabolic PDE describing temperature evolution is derived and represented as a nonautonomous parabolic evolution system on an appropriately defined function space which is exactly transformed in the infinite-dimensional boundary control problem for which a boundary linear quadratic regulator is proposed. Properties of the solution of the time-varying parabolic PDEs given by the two-parameter evolutionary system are utilized in the synthesis of the optimal boundary regulator, and the control law is applied to the model given by a two-dimensional partial differential equation in the cylindrical coordinates representing the Czochralski crystal growth process with one-dimensional growth direction. Finally, numerical results demonstrate optimal stabilization of the two-dimensional temperature distribution in the crystal.     

Simultaneous fluid-solid reactions in porous solids—II: Reactions between one fluid and two solid reactants

A general model describing simultaneous independent reactions between one fluid reactant and two solid reactants is presented. Such reaction systems are attracting much attention in the areas of oil shale retorting, reduction of mixed metal oxides with a reducing gas, leaching of mixed minerals, and roasting of mixed sulfides.The model has been formulated in general terms so as to allow the incorporation of specific details of an actual system. The law of additive reaction times previously developed for single fluid-solid reactions and found useful for the analysis of the reaction between one solid and two fluid reactants has been applied to the reaction system under consideration. It has yielded a useful approximate solution, which obviates the necessity for repeated numerical solution of a second-order differential equation with split boundary conditions and removes the possibility of encountering the numerical stiffness problem.     

A data-driven optimization algorithm for differential algebraic equations with numerical infeasibilities

Support vector machines (SVMs) based optimization framework is presented for the data-driven optimization of numerically infeasible differential algebraic equations (DAEs) without the full discretization of the underlying first-principles model. By formulating the stability constraint of the numerical integration of a DAE system as a supervised classification problem, we are able to demonstrate that SVMs can accurately map the boundary of numerical infeasibility. The necessity of this data-driven approach is demonstrated on a two-dimensional motivating example, where highly accurate SVM models are trained, validated, and tested using the data collected from the numerical integration of DAEs. Furthermore, this methodology is extended and tested for a multidimensional case study from reaction engineering (i.e., thermal cracking of natural gas liquids). The data-driven optimization of this complex case study is explored through integrating the SVM models with a constrained global grey-box optimization algorithm, namely the ARGONAUT framework.     

On the accuracy of the calculated stress field around a craze

Several recent calculations of the stress field around crazes in glassy polymers have been based on experimentally obtained craze opening displacements. Although various techniques for performing these calculations have been presented (i.e., Fourier transform, boundary integral, and finite element methods), all are based on the solution of the same ill-conditioned boundary value problem. Specific boundary conditions required to calculate the stresses around a craze lead to a large change in the stress field with small changes in the craze profile which are within the reported experimental tolerance.     

Solution of moving boundary problems for gas—solid noncatalytic reactions by orthogonal collocation

An integral transformation, a coordinate transformation for immobilization of the moving boundary, and orthogonal collocation are used to reduce a nonlinear initial-boundary value problem in time and space to a set of ordinary differential equations in time with given initial conditions. The method is developed for solution of models for gas—solid noncatalytic reactions and is especially useful for moving boundary, two-stage reaction problems. The method represents an advantage and an alternative to the available finite difference techniques. Results of various gas—solid reaction models are analyzed.     

Modeling of particle deposition in a vertical turbulent pipe flow at a reduced probability of particle sticking to the wall

Particle deposition on the wall in a dilute turbulent vertical pipe flow is modeled. The different mechanisms of particle transport to the wall are considered, i.e., Brownian motion, turbulent diffusion and turbophoresis. The Saffman lift force, the electrostatic force, the virtual mass effect and wall surface roughness are taken into account in the model developed. A boundary condition that accounts for the probability of particle sticking to the wall is suggested. An analytical solution for deposition of small Brownian particles is obtained. A particle relaxation time range, where the model developed is reliably applicable, is evaluated. Computational results obtained at different particle-wall sticking probabilities in the wide particle relaxation time range are presented and discussed.     

CONJUGATE ANALYSIS OF NATURAL CONVECTIVE DRYING OF BIOLOGICAL MATERIALS

This paper presents a numerical analysis of heat and mass transport during natural convective drying of an extruded com meal plate. The conjugate problem of drying and natural convection boundary layer Is modeled. The finite volume technique was used to discretize and solve the highly nonlinear system of coupled differential equations governing the transport inside the plate. The boundary layer solution was obtained by means of a finite difference software package that utilizes Runge-Kutta's 5th order method to solve the inherent transport equations. A methodology for evaluating the heat and mass transfer coefficients during the numerical simulation was developed and successfully implemented. The results showed that there is no analogy between heat and mass transfer coefficients for this type of problem.     

Mass transport effects during measurements of gas-solid reaction kinetics in a fluidised bed

Design, modelling and scaling of many gas-solid reaction processes in a fluidised bed (FB) require reliable intrinsic kinetic data obtained under conditions similar to those in industrial-scale units. The determination of such data in a laboratory-scale FB seems to be the best route. Despite this, owing to practical difficulties, few experimental reactivity data are available from FB. One of the drawbacks of FB reactors is the plausible interference of physical processes on kinetic measurements. In this work a simple methodology is developed for the assessment of fluid-dynamic and mass transport effects during kinetic experiments in FB (FBKE). A modelling approach is proposed, which combines a kinetic particle model with a simple two-phase flow model. The parameters resulting from the model are expressed in terms of three observable quantities, making it possible to evaluate the transport effects in a straightforward way from gas concentration measurements. Charts for direct evaluation of transport effects in FBKE are included. The analysis presented aims at facilitating the selection of optimum operating conditions for FB tests to determine gas-solid kinetics. Moreover, it may support dimensioning of new rigs designed for this purpose. This study primarily aims at the assessment of transport effects in batch-operated laboratory FBKE, where isothermal conditions are assumed and only one reaction occurs: the reaction between CO₂ and char is taken as an example. The methodology developed can be applied to other isothermal systems to estimate the influence of diffusional effects on the observed reaction rate, i.e., char oxidation and other catalytic or non-catalytic systems.     

A Fixed-Point Iteration Approach for Solving a BVP Arising in Chemical Reactor Theory

A numerical scheme aimed at solving a broad class of second-order nonlinear boundary value problems (BVPs) is presented, described, and then applied to a problem that appears in chemical reactor theory. In particular, the steady-state solution of the equation governing the model of the adiabatic tubular chemical reactor is investigated. The proposed method is based on expressing the particular solution of the governing equation in terms of an integral involving Green’s function. Then, the Krasnoselskii–Mann’s fixed-point scheme is implemented to an amended version of the resulting integral operator. The convergence of the iterative scheme is proved and its efficiency and applicability are demonstrated by solving the equation for selected values of the parameters that appear in the model. Residual error computation is adopted to confirm the accuracy of the results. In addition, the numerical outcomes of the proposed method are compared with those obtained by other existing numerical approaches.     

The general theory of polarographic kinetic currents

A unified mathematical formulation of the transport problem to the dropping electrode, accounting for diffusion, chemical reactions in solution and electrode reactions, is given. The method for the reduction of the problem to a diffusion problem where the effect of chemical reactions is restricted to boundary conditions is described. Conditions are defined for appearance of a kinetic current and, on the contrary, of a diffusion current dependent on equilibrium constants of chemical reactions. The general results are exemplified by the case of a single chemical and a single electrode reaction.     

CONJUGATE ANALYSIS OF NATURAL CONVECTIVE DRYING OF BIOLOGICAL MATERIALS

ABSTRACT

This paper presents a numerical analysis of heat and mass transport during natural convective drying of an extruded com meal plate. The conjugate problem of drying and natural convection boundary layer Is modeled. The finite volume technique was used to discretize and solve the highly nonlinear system of coupled differential equations governing the transport inside the plate. The boundary layer solution was obtained by means of a finite difference software package that utilizes Runge-Kutta's 5th order method to solve the inherent transport equations. A methodology for evaluating the heat and mass transfer coefficients during the numerical simulation was developed and successfully implemented. The results showed that there is no analogy between heat and mass transfer coefficients for this type of problem.     

A COMPREHENSIVE MATHEMATICAL MODEL FOR A MULTIZONE TUBULAR HIGH PRESSURE LDPE REACTOR

In this study a comprehensive mathematical model of high pressure tubular ethylene polymerization reactors is presented. A fairly general reaction mechanism is employed to describe the complex kinetics of ethylene polymerization. To determine the variation of molecular properties along the reactor length the method of moments is applied to the infinite set of species balance equations to transform it into a low order system of differential equations in terms of the leading moments of the number chain length distribution. Detailed algebraic equations are given describing the variation of kinetic rate constants, thermodynamic and transport properties of the reaction mixture with temperature, pressure and composition. A new correlation is derived to describe the change of reaction viscosity with reactor operating conditions. The model permits a realistic calculation of temperature and pressure profiles, monomer and initiator concentrations, molecular properties of LDPE (i.e. M_n, M_m, LCB and SCB) as well as the variation of inside film heat transfer coefficient with respect to the reactor length. Simulation results are presented illustrating the effects of initiator concentration, inlet pressure, chain transfer concentration and wall fouling on the polymer quality and reactor operation. The present model predictions are in good agreement with experimental observations in industrial high pressure tubular LDPE reactors.     

BOUNDARY CONDITIONS FOR A TUBULAR NONISOTHERMAL NONADIABATIC PACKED BED REACTOR WITH WALL HEAT TRANSFER IN THE FORE AND AFTER SECTIONS

An analysis of the boundary conditions for a nonadiabatic steady-state flow reactor with axial dispersion is presented. Conclusions regarding reactor behavior are reached as a result of solution of six differential equations for the reaction section, fore and after sections. Axial dispersion and thermal conductivity as well as wall heat transfer occur in the fore and after sections. It is shown that the new boundary conditions exert only a weak effect on the shape of temperature and concentration profiles and that the classical Danckwerts boundary conditions represent a good approximation.     

BOUNDARY CONDITIONS FOR A TUBULAR NONISOTHERMAL NONADIABATIC PACKED BED REACTOR WITH WALL HEAT TRANSFER IN THE FORE AND AFTER SECTIONS

An analysis of the boundary conditions for a nonadiabatic steady-state flow reactor with axial dispersion is presented. Conclusions regarding reactor behavior are reached as a result of solution of six differential equations for the reaction section, fore and after sections. Axial dispersion and thermal conductivity as well as wall heat transfer occur in the fore and after sections. It is shown that the new boundary conditions exert only a weak effect on the shape of temperature and concentration profiles and that the classical Danckwerts boundary conditions represent a good approximation.     

Liquid distribution and redistribution in packed columns-I Theoretical

When liquid is uniformly distributed at the top of a packed column it is found that there is a preferential flow to the wall, and early theoretical work has suggested that the observed radial velocity is proportional to the radial gradient of axial velocity. A set of consistent boundary conditions has not been found. In this paper the experimental observation of preferential liquid flow is interpreted as a difference in permeability between the wall and bulk region of packing, and the existence of a potential for liquid redistribution is inferred from an examination of experimental work on two-phase flow in porous media, and of the internal consistency of the early relation between radial velocity and the radial gradient of axial velocity. The existence of a potential for liquid redistribution, and a difference in permeability between the wall and bulk regions are shown to lead to a differential equation describing liquid distribution in the packing, defined by consistent and well-posed boundary conditions that are determined from the physical analysis.The solution to the partial differential equation describing redistribution from an initial axisymmetric distribution is given.