Solid film linear driving force model

An iterative procedure using the solid film linear driving force model has been used with a steric mass action isotherm to model displacement chromatography on ion exchange materials and the procedure applied to the separation of horse and bovine cytochrome c using neomycin sulfate as the displacer.4 The solid film linear driving force model is a set of two differential equations imposing mass transfer limitations. 

In this report, a kinetic model based on the solid film linear driving force assumption is used. Unlike the equilibrium-dispersive model, which lumps all transfer and kinetic effects into an effective dispersion term, the kinetic model is effective when the column efficiency is low and the effects of column kinetics are significant. 

In this second lumped kinetic model and in contrast to the first one, we assume that the kinetics of adsorption-desorption is infinitely fast but that the mass transfer kinetics is not. More specifically, the mass transfer kinetics of the solute to the surface of the adsorbent is given by either the liquid film linear driving force model or the solid film linear driving force model. In the former case, instead of Eq. 6.41, we have for the kinetic equation ... 

If we instead use the solid film linear driving force model of mass transfer kinetics, we have... 

The results obtained with the solid film linear driving force model, the pore diffusion model, and the micropore diffusion model were compared by Ruthven [14]. In contrast to linear chromatography, numerical solutions obtained with different models are different, especially in the initial time region. For moderate loadings i.e., for Req > 0-5), the differences remain small. As the loading increases, however, and Req becomes lower than 0.5, the differences between the numerical solutions derived from the various models studied increase. Accordingly, differences observed between experimental results and the profiles predicted by a kinetic model are most often due to the selection of a somewhat inappropriate model. 

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

Equation 14.45 applies in linear chromatography. The correct HETP equation in frontal analysis imder constant pattern behavior, and with the same solid film linear driving force model is Eq. 14.36b. Comparison of these two equations shows that an error is made when the latter is used to replace the former, in the equilibrium-dispersive model. We should replace in Eq. 14.44 and 14.45 Atq by k — FAq)/AC. In the case of the Langmuir isotherm, this would give k = fc )/(l + bCo) = X. 

The transport-dispersive model assumes infinitely fast kinetics of adsorption-desorption but a finite rate of mass transfer, following the solid film linear driving force equation... 

The overloaded band profiles of the two enantiomers were recorded and compared to those calculated using the ED model. This simple model did not produced profiles in good agreement with the experimental ones. A much better agreement was observed between the experimental chromatograms and those calculated with the TD model, using the solid film linear driving force model to account for a slow mass transfer kinetics on both types of adsorption sites. In this case, the kinetic model is summarized as... 

This result means that even if the kinetics follows the solid film linear driving force model the breakthrough curve can be fitted successfully to the Thomas model, provided that the apparent desorption rate corrstant, k, given by Eq. 14.82b is used. 

In Figure 14.15a, we compare two series of band profiles. The first series was calculated as numerical solutions of the transport model (no axial dispersion, solid film linear driving force kinetics), with a number of transfer units, Nm —... 

Experimental and Computational Study of Preparative Gradient Elution of Peptides Using the Solid Film Linear Driving Force Model.719... 

A more comprehensive analysis of constant pattern behavior for a binary system has been given by Rhee and Amundson [3]. In this work, these authors have extended to binary systems the analysis of the combined effects of mass transfer resistance and axial dispersion that they had previously made in the case of single-component bands [4]. Rhee and Amundson [3] assumed the solid film linear driving force model, finite axial dispersion, and no particular isotherm model. The system of equations becomes... 

In these kinetic models of chromatography, all the sources of mass transfer resistance are lumped into a single equation. In the case of the solid film linear driving force model, we have for each component i... 

The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example. Figure 16.10 shows the displacement chromatogram calculated with kpi = kfg = = 50 s . The bands in the isotachic train are clearly formed... 

Solid Film Linear Driving Force Model Simple model of mass transfer in chromatography, where it is assumed that the rate of variation of the stationary phase concentration is proportional to the difference between the local concentration of the component in the stationary phase and the stationary phase concentration which would be in equilibrium with the local liquid phase concentration (Eq. 2.42). 

Mass transfer kinetics are accurately accounted for either by the solid film or by the liquid film linear driving force model. 

The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation q = qsbC)/ l bC)) in the partial differential equation of tire solid fihn linear driving force model (Eq. 14.3)... 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

In linear chromatography, the solid film driving force model (Eq. 6.43), and the liquid film driving force model (Eq. 6.42) are special cases of the first-order linear kinetics (Eq. 6.41), which can be rewritten as... 

Thus, in linear chromatography the liquid film driving force model (Eq. 6.42) is equivalent to the solid film driving force model (Eq. 6.43) with k m = akm and is a special case of the linear kinetic model, with kd = k m/a and ka = k m- The solution of the kinetic model derived by Lapidus and Amundson [3], which was obtained for a first-order linear kinetic model, is directly applicable to both the solid film and the liquid film driving force models. We now discuss these equivalent solutions. 

Some of the solutions which have been obtained in this way are summarized in Table 8.6. For a nonlinear system the linearized rate expression written in terms of a fluid phase concentration as the driving force is not exactly equivalent to the solid film driving force model and the expression obtained for the breakthrough curves for these two cases are therefore different. 

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