Denis Constales

Multi-zone TAP-reactors theory and application : I. The global transfer matrix equation

Abstract A general theory of single-pulse state-defining experiments for a multi-zone TAP (temporal analysis of products) reactor, is developed using the Laplace transform formalism; the theory gives explicit expressions for the moments of the outlet flux, series expansions for the transient values of this flux, and offers an efficient means to compute the actual profiles of gas concentration in the reactor and the values of the outlet flux numerically, using e.g. Fast Fourier Transform. The central concept of the theory is the global transfer matrix equation, which determines completely the dynamic behavior of the reactor. Using efficient computer algebra methods, the theory generates previous theoretical results reported in the literature for all the known TAP–reactor configurations, and yields new results related to the reversible adsorption/reaction–diffusion case and the thin-zone case. It can be used for further theoretical studies in the area of diffusion/reaction dynamics.

Szegö and Polymonogenic Bergman Kernels for Half-Space and Strip Domains, and Single-Periodic Functions in Clifford Analysis

In this paper, we consider half-space domains (semi-infinite in one of the dimensions) and strip domains (finite in one of the dimensions) in real Euclidean spaces of dimension at least 2. The Szegö reproducing kernel for the space of monogenic and square integrable functions on a strip domain is obtained in closed form as a monogenic single-periodic function, viz a monogenic cosecant. The relationship between the Szegö and Bergman kernel for monogenic functions in a strip domain is explicitated in the transversally Fourier transformed setting. This relationship is then generalised to the polymonogenic Bergman case. Finally, the half-space case is considered specifically and the simplifications are pointed out.

Hilbert spaces of solutions to polynomial Dirac equations, Fourier transforms and reproducing Kernel functions for cylindrical domains

In this paper, we consider L2 spaces of functions that satisfy polynomial Dirac equations. Fourier transformation methods and methods from harmonic analysis are then applied to treat Hilbert spaces of Clifford algebra valued functions that are either square-integrable over a cylinder or square-integrable over its boundary, and which satisfy in its interior the generalized Cauchy-Riemann system. In particular, explicit representation formulas for the Bergman and Szegö reproducing kernel of several types of cylindrical domains are developed.

On the order of basic series representing Clifford valued functions

In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point.

Implementation of Homotopy Perturbation Method to Solve a Population Balance Model in Fluidized Bed

Abstract: A particle population balance formulation for a circulating fluidized bed, involving aggregation and breakage of particles, is solved using the homotopy perturbation method (HPM). The homotopy method deforms a difficult problem into a simple problem, which then can be easily solved. The HPM solution is compared with the solution obtained using a standard finite difference method. Using homotopy, a good approximation of the finite difference solution is obtained within a few iteration steps. The results reveal that the homotopy method is an effective and simple tool to solve nonlinear partial integro-differential equations and has a wide scope and applicability to solve complex engineering problems. The reliability of the algorithm is tested using three different feed inlet particle size (diameter) distributions to indicate the robustness of this method.

Multi-zone TAP-reactors theory and application: II. The three-dimensional theory

The rigorous three-dimensional theory for a TAP (Temporal Analysis of Products) Knudsen pulse response experiment is developed for the combined diffusion and reaction cases for multi-zone packing, in order to determine the domain of validity of the commonly used one-dimensional model. The analysis is based on a specific modification of the transfer matrix formalism previously introduced for one-dimensional TAP-reactor models. The outlet flux can be written as a sum of three independent terms: one corresponding to the one-dimensional solution, a term accounting for axially symmetric radial nonuniformity, and a term needed for a fully three-dimensional model. The theory provides a method for estimating the accuracy of the one-dimensional model and for finding the domain of its validity. The theory is illustrated by the diffusion-only case for a one-zone reactor. It is shown that the one-dimensional model is valid for aspect ratios L/R>3.5.

Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications

In this paper, we discuss several essentially different formulas for the general derivatives q(n)(z) of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis, upon,which - among other important applications - the theory of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition methods, we obtain a compact integral representation formula over a half-space, which also lends itself to establish upper bounds on the values parallel toq(n)(z)parallel to. A second formula that we discuss is a recurrence formula involving permutational products of hypercomplex variables by which these estimates can be obtained immediately. We further prove several formulas for q(n)(z) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in terms of permutational products but using different and fewer hypercomplex variables than used in the recurrence relations. Summing up a fixed q(n). over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for q(n)(z) to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.

Parameter identification by a single injection?extraction well

In this paper, we present numerical modelling techniques supporting the determination of parameters for the contaminant transport by underground water flow. The parameter identification is based on measurements obtained by a single injection–extraction well. The underground water flow is modelled using a Dupuit–Forchheimer approximation for the unsaturated–saturated aquifer.

Determination of Soil Parameters via the Solution of Inverse Problems in Infiltration

In this paper, we propose an efficient method for the identification of soil parameters in unsaturated porous media, using measurements from infiltration experiments. The infiltration is governed by Richards nonlinear equation expressed in terms of effective saturation. The soil retention and hydraulic permeability functions are expressed using the Van Genuchten-Mualem ansatz in terms of the soil parameters. The mathematical algorithm is based on a transformation of Richards equation to a system of ordinary differential equations completed by the governing equation for the movement of the wetness front. This system can be efficiently solved by specialized packages for the solution of stiff systems of ODE. The unknown parameters are determined using the optimization approach of minimizing a cost functional for the discrepancy between the model output and the measurements. The gradient and Hessian of the solution with respect to soil parameter vector are determined using automatic differentiation. Several numerical experiments are included.

Intersections and coincidences in chemical kinetics: Linear two-step reversible–irreversible reaction mechanism

Abstract We investigate the qualitative behavior of chemical reaction mixtures in their time evolution, specifically when starting from initial conditions that are extremal in the sense that they cannot be extrapolated back in time without nonphysical, negative concentration values arising. Several extremal initial conditions will exist and the maxima and mutual intersections of the corresponding concentration time evolutions are of surprising complexity as to their ordering in time and in value, and as to the coincidences that may occur among them for special values of the rate constants. These extremal time evolutions also exhibit properties of mutual reciprocity that can be expressed by invariants involving their joint kinetics. Such invariants arise naturally, since they curtail the complexity of the coincidence analysis, e.g., by excluding certain intersections and by identifying the times at which certain maxima are reached. In this paper, we analyze the reversible–irreversible three-step linear sequential mechanism A ↔ B → C and provide charts of parametric subdomains and their boundary curves. Generally, such properties of simple linear and non-linear systems in fact reflect their unexpected complexity. We propose to use a special term for defining this phenomenon: simplexity.