Equilibrium-dispersive model numerical solution

The equilibrium dispersive model is widely applied in chromatography owing to the equivalence with standard dispersion models in chemical engineering in the case of linear isotherms (Levenspiel and Bischoff, 1963, Danckwerts, 1953) as well as its simplified numerical solution. 

This system of equations is an extension of the classical equilibrium-dispersive model to problems with two spadal dimensions, e.g., to the cases of a column having a cylindrical symmetry. It has no analytical solution but it is possible to write simple computational schemes for the calculation of its numerical solutions, using finite difference algorithms (see Chapter 10) [60]. 

In Figure 6.19, we compare two band profiles corresponding to the injection of the same amount of solute and calculated as numerical solutions of the equilibrium-dispersive model for a nonlinear isotherm, as described in Chapter 10. These profiles were calculated using a one-site (solid line) and a two-site isotherm model (dotted line), respectively. For both sites, we chose Langmuir isotherms. For the single-site model, this isotherm is q = 24C/(1 - - 6C). For the two-site model, the... 

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 mathematical origin of these concentration discontinuities or weak solutions of Eq. 7.1 has been explained by Courant and Friedrichs [24] and by Lax [25]. The mathematical backgroimd has been reviewed in cormection with the discussion of the numerical solution of the equilibrium-dispersive model given by Rou-chon et al. [26]. In the traditional theory of partial differential equations, a solution should be continuous. Lax [27] generalized the concept of solution to include weak solutions, which are not continuously differentiable. A solution of Eq. 7.1 that includes a continuous part, or diffuse bormdary, and a concentration shock is a weak solution of this equation [1,26-28]. A serious problem then arises, since there is no unique weak solution of Eq. 7.1. It is necessary to define the weak solution that is acceptable for the physical problem in order to achieve the determination of the band profile. This solution must make physical sense and prevent the crossing of the characteristics. Oleinik has suggested a selection rule that can... 

Figure 10.2 Comparison between the band profiles derived from the Houghton equation, the Haarhoff and Van der Linde equation and the numerical solution of the equilibrium-dispersive model using the Rouchon procedure, (a) Influence of the isotherm model used.

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... 

With the advent of fast computers, more exact numerical solutions have appeared. An original numerical solution of the equilibrium-dispersive model was developed [46], applied first to the prediction of band profiles in gas chromatography and later adapted to liquid chromatography [47,48]. It is used to predict the band profiles of large size samples. It requires prior determination of the column HETP imder linear conditions at the selected flow velocity, the column void volume, the extra-column volume, and the adsorption isotherm [48]. Other similar algorithms are available, and we now give a general presentation. 

Equation 10.79 permits the calculation of the concentration at the new space position, n + 1, knowing the concentration at the previous space position, n (Godunov scheme). This method calculates band profiles along the coliunn at successive time intervals. The elution profile is the history of concentrations at z = I. The forward-backward difference scheme was first used to calculate solutions of the equilibrium-dispersive model of chromatography by Rouchon et al. [46,47]. Since then, it has been widely used. It is particularly attractive because of its fast execution by modern computers [50]. Czok [50] and Felinger [54] have shown how the CPU time required can be further shortened by eliminating the needless computation of concentrations below a certain threshold. The dramatic increase over the last fifteen years of the speed of the computers available for the numerical calculations of band profiles has considerably reduced this advantage of the Forward-Backward scheme over the other possible ones. 

Figure 10.9 Comparison between the band profiles predicted by the ideal model and the numerical solution of the equilibrium-dispersive model for a Langmuir isotherm. Constant column efficiency, 2000 theoretical plates, (a) Classical C vs. f profile. Sample size given as loading factor, (b) Reduced profiles, plots of bC vs. (t — fo)/(fR,o — to)- Sample size given as apparent loading factor, m = [Icq/(1 + J q)] NLj. Similar chromatograms, corresponding to intermediate loading factors, are given in Figure 10.8.

Figure 10.12 Comparison of the numerical solutions of the equations of the equilibrium-dispersive model calculated with the forward-backward algorithm (dashed line) and the method of orthogonal collocation on finite elements (dotted line). 250 iL injection of a 15 g/L solution of (+)-Tr6ger s base. Column length, 25 cm column efficiency, Np = 146 plates F = 0.515 u = 0.076 cm/s. Isotherms in Figure 3.25...

Figure 11.8 Illustration of the hodograph transform applied to actual chromatograms. Numerical solutions of the equilibrium-dispersive model for a 310 (Top left) and a 120 (Top center) theoretical plate column, a = 1.8, = 0.7. (Bottom left and center) Hodograph...

Numerical solutions of the former [50,51] and latter [52] models have been calculated with finite difference methods. Any of the schemes discussed in Chapter 10 can be used for the calculation of solutions of these kinetic models. However, the generation of the error due to the numerical dispersion is more complex and it is more difficult to control, because two differential equations are now involved for each component, instead of one with the equilibrium-dispersive model. 

Bellot et al. [59] combined the use of the equilibrium-dispersive model (see Chapter 10) and the SDM isotherm (Eq. 15.57) to calculate numerical solutions of the elution profiles of binary mixtures of proteins. A linear gradient profile was used in these calculations. The counter-ion is small and the gradient started immediately after the feed injection. 

The series of figures 16.6 to 16.9 illustrates the influence on their band profiles of the number of mass transfer units (k kf/k) of the column for the two components of a 1 9 binary mixture [22]. In Figure 16.6, this number is large (fcy i = kf 2 = 50 s ). The profiles obtained for the two components are the same as those obtained by numerical solution of the equilibrium-dispersive model (Chapter 11). Figure 16.7 shows the profiles obtained imder the same conditions, except for the value of the mass transfer coefficient kf i — kf 2 = 0.05 s ), resulting in 1,000 times fewer mass transfer units for the column. The change in band profiles from... 

Another possible way to improve the separation performance of SMB units is to change the feed concentration during each successive period. This operation mode, called the Modicon process, was suggested by Schramm et al. [72]. These authors optimized the process using numerical solutions of the equilibrium-dispersive model and compared the performance of the Modicon process with that of conventional SMB. They concluded that a cyclic modulation of the feed concentration allows a significant improvement of the separation performance The productivity and the product concentration can be increased and the specific solvent consumption decreased compared to those achieved with conventional SMB. 

Mihlbachler et al. [26,59] studied the separation by SMB of the enantiomers of TrSger s base on Chiralpak AD, a cellulose carbamate CSP (see isotherm data in Figure 4.28). They compared their experimental results with the numerical solutions of an equivalent TMB process that was written within the framework of the equilibrium-dispersive model. The mass balance equation for each component i in each column k of each section j of the SMB is given by... 

When there are no yield constraints, the peld calculated at the maximum production rate is usually aroimd 60% [21,23]. Figure 18.7 shows a plot of the production rate of the second component versus the sample size. In this figure, the solid lines are given by Eq. 18.53a, while the symbols are derived from the calculation of numerical solutions of the equilibrium-dispersive model [21]. Equation 18.51a predicts correctly the optimum sample size for maximum production rate of the second component, although it has been derived from the ideal model [2]. 

Figure 18.7 Influence of the sample size on the production rate of a preparative chromatographic column. Plot of the production rate per unit mobile phase flow rate, Pr/F , versus the loading factor, Lj2- Solid lines derived from Eq. 18.53a. Symbols from numerical solution of the equilibrium-dispersive model. Experimental conditions Column length 25 cm particle size 10 im efficiency 5000 theoretical plates. Isotherm coefficients ...

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