Philippe Saucez

Modeling and simulation of a SMB chromatographic process designed for enantioseparation

Abstract This paper focuses on modeling of a simulated moving bed process (SMB) dedicated to the separation of racemic mixtures. In the first approach, a true moving bed model is derived, which assumes an equivalent counter-current movement of the solid phase. The good agreement between the model and the real system is demonstrated with experimental results. Then, a more rigorous approach is developed, which considers the system as an arrangement of static chromatographic columns and takes into account periodic switching. Attention is focused on model formulation and numerical solution techniques in order to develop efficient dynamic simulation programs.

Upwinding in the method of lines

The method of lines (MOL) is a procedure for the numerical integration of partial differential equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences (FDs). If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations (ODEs) results through the algebraic approximation of the spatial derivatives.

An adaptive method of lines solution of the Korteweg-de Vries equation

Abstract Following a method of lines formulation, the Korteweg-de Vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle, which allows the number of nodes to be significantly reduced as compared to a fixed-grid solution. Several finite difference schemes, including direct and stagewise procedures, are compared and the results of a large number of computational experiments are presented, which demonstrate that the selection of a spatial approximation scheme for the third-order derivative term is the primary determinant of solution accuracy.

Modeling and numerical simulation of secondary settlers: a Method of Lines strategy.

In this paper, attention is focused on a parabolic partial differential equation (PDE) modeling sedimentation in a secondary settler and the proper formulation of the problem boundary conditions (i.e., the conditions prevailing at the feed, clear water and sludge outlets). The presence of a diffusion term in the equation not only allows the reproduction of experimental observations, as reported in a number of works, but also makes the numerical solution of the initial-boundary value problem significantly easier than the original conservation law (which is a nonlinear hyperbolic PDE problem requiring advanced numerical techniques). A Method of Lines (MOL) solution strategy is then proposed, based on the use of finite differences or spectral methods, and on readily available time integrators. The efficiency and flexibility of the general procedure are demonstrated with various numerical simulation results.

Matlab implementation of a moving grid method based on the equidistribution principle

The objective of this paper is to report on the development of a method of lines (MOL) toolbox within MATLAB, and especially, on the implementation and test of a moving grid algorithm based on the equidistribution principle. This new implementation includes various spatial approximation schemes based on finite differences and slope limiters, the choice between several monitor functions, automatic grid adaptation to the initial condition, and provides a relatively easy tuning for the non-expert user. Several issues, including the sensitivity of the numerical results to the tuning parameters, are discussed. A few test problems characterized by solutions with steep moving fronts, including the Buckley-Leverett equation and an extended Fisher-Kolmogorov equation, are investigated so as to demonstrate the algorithm and software performance.

Simulation of ODE/PDE Models with MATLAB®, OCTAVE and SCILAB

Simulation of ODE/PDE Models with MATLAB, OCTAVE and SCILAB shows the reader how to exploit a fuller array of numerical methods for the analysis of complex scientific and engineering systems than is conventionally employed. The book is dedicated to numerical simulation of distributed parameter systems described by mixed systems of algebraic equations, ordinary differential equations (ODEs) and partial differential equations (PDEs). Special attention is paid to the numerical method of lines (MOL), a popular approach to the solution of time-dependent PDEs, which proceeds in two basic steps: spatial discretization and time integration. Besides conventional finite-difference and element techniques, more advanced spatial-approximation methods are examined in some detail, including nonoscillatory schemes and adaptive-grid approaches. A MOL toolbox has been developed within MATLAB/OCTAVE/SCILAB. In addition to a set of spatial approximations and time integrators, this toolbox includes a collection of application examples, in specific areas, which can serve as templates for developing new programs. Simulation of ODE/PDE Models with MATLAB, OCTAVE and SCILAB provides a practical introduction to some advanced computational techniques for dynamic system simulation, supported by many worked examples in the text, and a collection of codes available for download from the books page at www.springer.com. This text is suitable for self-study by practicing scientists and engineers and as a final-year undergraduate course or at the graduate level.

The laryngeal channel as a variable venturi-meter

A model of the larynx, based on the analogy between the laryngeal channel and a constriction in a pipe, was developed by integrating the actual movements of the vocal cords during the quiet respiratory cycle. The model allows one to simulate and explain (1) the shape (tilted 8 with counterclockwise sign and cycle aperture) of the upper airway pressure-flow diagram, (2) the asymmetry between inspiratory and expiratory phases and (3) the variation of upper airway resistance with gas mixture.

A non-linear multialveolar model of the mechanical behavior of the human lung

A mathematical multialveolar model has been developed which describes the mechanical behavior of the human ventilatory system in healthy subjects, under plethysmograph test conditions. It is based on the morphometric scheme of Weibel (1989) and uses the equations described by Pedley (1970) for modeling the airflow resistance. The 23 airways generations are grouped in four different levels on the basis of similar dimensions and mechanical properties. The dynamic effects of the changes in lung volume and transmural pressure on airway dimensions have been incorporated. In this first approach, the upper airways model is simplified and based on Rohrers equation. The simulation results obtained from this model allow the authors to explain and retrieve global parameters available in the usual medical literature and obtained in healthy subjects during clinical tests.<<ETX>>

Two Dimensional and Time Varying Spatial Domains

Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.

Finite Elements and Spectral Methods

In this chapter, the weighted residual methods are introduced. These methods represent the solution as a series of basis functions whose coefficients are determined to make the PDE (and BC) residuals as small as possible (in an average sense). In particular, attention is focused on the Galerkin and collocation methods, and the use of global and local basis functions, the latter leading to the famous Finite Element Method (FEM). Together with finite difference methods, FEM is one of the most popular methods for solving IBVP. Various basis functions can be used in conjunction with the FEM, especially Lagrange or Hermite polynomials. The construction of the FEM matrices is presented in detail for these two types of polynomials, when using a Galerkin method, and for Hermite polynomials, when using an orthogonal collocation approach. It is also possible to compute, using experimental or simulation data, optimal basis functions, i.e., the ones that are able to capture most of the solution features with a limited number of functions. This is the essence of the proper orthogonal decomposition (POD). Several examples are studied including the heat equation and the Brusselator.