Numerical Solutions of the Equilibrium-Dispersive Model

As we have shown clearly in the previous section, there are no closed-form analytical solutions of the equilibrium-dispersive model. There exists only an approximate solution in the case when the loading factor is low, and bCmax is smaller than 0.05 to 0.1 [15], with b, the second coefficient of the Langmuir isotherm and CMax 

Seshadri and Deming [44] have used the Craig model to calculate band profiles in chromatographic systems. However, they selected an unrealistic isotherm, (Ji = ( j + biCj)Ci, i.e., an isotherm which, for each component i, is linear in respect to its concentration, but with a retention factor that is a linear function of the other component concentration. There is little physical basis for this model, and this prevented them from deriving any useful conclusions. More recently, Eble et al. [45] have used the Craig model to calculate band profiles in isocratic elution and to develop general correlations between the sample size on the one hand and the apparent retention factor and the column efficiency on the other. Experimental data confirm the approximate validity of the relationships obtained [24,25] (Eig-ure 10.3). The use of such empirical relationships allowed an estimation of the band shape on a personal computer for column efficiencies not exceeding a few hxmdred theoretical plates. 

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. 

The equilibrium-dispersive model correctly takes into accoimt the influence of the column efficiency on the profile of the elution bands. The mathematical basis of all the numerical methods is a modification of Eq. 10.10 

Single-Component Profiles with the Equilibrium Dispersive Model 

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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.

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 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...

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. 

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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