Numerical Analysis of the Equilibrium-Dispersive Model

In the case of a multicomponent mixture, the individual band profiles are given by the equations of the equilibrium-dispersive model 

In order to solve the system of Eqs. 11.1, we need to specify the initial and the boimdary conditions of the problem. The general boundary conditions were discussed in Chapter 6, Section 6.2.1. Simpler boundary conditions were introduced and justified in Chapter 2, Section 2.1.4, and in the discussion of the single-component problem. In isocratic elution, the column contains no sample component but is filled with the pure mobile phase (when the mobile phase is a mixture and one of its components may compete with the sample components, we have a more complex problem, discussed in Chapter 13) 

The boundary condition expresses the continuity of the mass flux of solutes at the column inlet and outlet. The Danckwerts [1] conditions are written 

Finally, we need the isotherm equations that relate the concentration of each of the solutes in the stationary phase and the concentrations of all the solutes in the mobile phase. In general, these adsorption isotherms are competitive, meaning that the amount of component i adsorbed at equilibrium between phases from a solution with a constant concentration Q decreases with increasing concentrations of any one of the other components (Chapter 4). 

The methods of calculation of numerical solutions discussed in Chapter 10 for a single component can easily be extended to the case of multicomponent mixtures. In this case, we have n partial differential equations similar to Eqs. 10.78,10.80, or 10.82 (f = 1,2, , ) 

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In Chapters 3 and 4, we discussed the numerical analysis procedure suggested by James et al. [35] and applied by Felinger et al. [36] to calculate solutions of the inverse problem of ideal chromatography and, more specifically, to derive the best possible estimates of the numerical coefficients of an isotherm model together with a figure of merit for any isotherm model selected. The main drawback of this approach is that it is based on the use of the equilibrium-dispersive model since... 

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