General rate model numerical solution

All the solutions discussed above are given in the form of infinite integrals. In the case of a pulse injection, a similar analytical solution has not been derived yet, except for Carta s solution of Rosen s model. However, the numerical evaluation of the inverse Laplace transform is possible. It has been calculated in the case of the general rate model i.e., Eqs. 6.58 to 6.64a) by Lenhoff [38]. The numerical integral derived by Lenhoff is given by ... 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

The derivation of numerical solutions of the general rate model (GR) for singlecomponent problems is a special case of that of nmnerical solutions of this model for multi-component systems, which will be discussed in Chapter 16 and that there is no need to reproduce here. However, since several authors have used this model to compare the experimental overloaded profiles of piue-component bands with those calculated with this and other models, we discuss briefly the numerical solutions of the general rate model for single-components. 

Numerical Solution of The General Rate Model of Chromatography. 754... 

Compared to our discussion of the general rate model in Chapter 6 for linear chromatography, the only important changes are the need to use nonlinear competitive isotherms and to consider the possible dependence of diffusion and the mass transfer rate coefficients on the local concentrations of the feed components. These changes increase dramatically the mathematical complexity of the problem and only numerical solutions are possible. 

Many authors have described procedures for the calculation of numerical solutions of the general rate model of chromatography with a variety of initial and boundary conditions corresponding to practically all the modes of chromatography (with the notable exception of system peaks). Orthogonal collocation on finite elements seems to be the most popular approach for these calculations. 

Numerical Solution of a Simplified General Rate Model, the Lumped Pore Diffusion Model (FOR)... 

The general rate model was used as a basis for the development of a computer program for the simulation of chromatographic processes by Gu [53]. The solution of the partial differential equations in the nonlinear range of the adsorption isotherms can be obtained by application of numerical methods. One drawback for the modeling of real chromatographic separations with this model is the multitude of physical parameters, which cannot be determined experimentally and have to be estimated by approximations. In practice, these parameters are often only inaccurately fittable, so that a reasonable calculation is impossible. This model is rather applicable for theoretical studies [54]. 

Full rate modeling Accurate description of transitions Appropriate for shallow beds, with incomplete wave development General numerical solutions by finite difference or collocation methods Various to few... 

Fu and Zhang (1991) developed a general method for calculating burning rate of coal and char particles, and a chart was made through numerical solution for the model equations, as shown in Fig. 5. The dimensionless... 

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