Numerical Solution of the Model System

Having set up a model to describe the dynamics of the system, a very important first step is to compare the numerical solution of the model with any experimental results or observations. In the first stages, this comparison might be simply a check on the qualitative behaviour of a reactor model as compared to experiment. Such questions might be answered as Does the model confirm the experimentally found observations that product selectivity increases with temperature and that increasing flow rate decreases the reaction conversion ... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

This chapter presents an analysis of the development of dynamic models for packed bed reactors, with particular emphasis on models that can be used in control system design. Our method of attack will be first to formulate a comprehensive, relatively detailed packed bed reactor model next to consider the techniques available for numerical solution of the model then, utilizing... 

In spite of the eleven assumptions we have made, numerical solution of the mathematical system comprising the governing equations discussed in the previous sections is still time-consuming. Various simplifications can be made to obtain simpler mathematical systems or models. Next, five models are presented that were found to allow adequate description of the BSR in a limited range of conditions (that will be indicated for each model). To illustrate the different models, their distinguishing features are explained graphically in Fig. 13. 

The simultaneous numerical solution of the ODE system was performed using Matlab (The MathWorks). The kinetic and model parameters are summarized in Table 1,... 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

The chemical reactions are commonly described by global reactions and rate formulations of the Langmuir-Hinshelwood type. Such a description requires less computational effort compared to detailed elementary step mechanisms. However, kinetic parameters must be fitted to measurement data in advance. More details on the model equations and the numerical solution of the resulhng system can be found elsewhere [15, 16, 20, 22-25]. 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The configuration of a system in which percolation may occur is classically treated as one in which the ingredients do not move. Considerable work has been devoted to these static models, leading to numerical solutions of the critical concentrations and cluster sizes associated with a percolation threshold. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

Results for two types of model systems are shown here, each at the two different inverse temperatures of P = 1 and P = 8. For each model system, the approximate correlation functions were compared with an exact quantum correlation function obtained by numerical solution of the Schrodinger equation on a grid and with classical MD. As noted earlier, testing the CMD method against exact results for simple one-dimensional non-dissipative systems is problematical, but the results are still useful to help us to better imderstand the limitations of the method imder certain circumstances. 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

At first glance the appearing equations seem to be very complex. But the numerical solution of the equations is a process which can be done with a computer program. The analytical model offers several advantages compared to simulations. Since such a theoretical ansatz needs only a small amount of computing time, more complex systems can be studied. Moreover our models are not restricted to small lattices which are inavoidably used in computer... 

Fig. 7. Numerical simulation of equation (29) for the Brusselator model on a ring. At t = 0 a clockwise wave propagates along the ring at t = 4.49 the effect of the counterclockwise external field deforms this wave appreciably after a while the sense of rotation is reversed and for t > 22.49 one obtains a stable wave solution in the counterclockwise direction. The period of both the external field and of the linearized solution of the unperturbed system is 4.28 time units.

---