The Equilibrium-Dispersive Model

Here /x is the mobile phase velocity, L the column length and N the number of theoretical plates. The equilibrium-dispersive (ED) model [13, 109] was used to calculate the overloaded band profiles in papers III-VI. 

The classical retention factor is related to the numerical coefficients of the Langmuir isotherm by the following equation under linear conditions (i.e., at infinite dilution)  

Here F is the phase ratio (Vs/Vm) and a is the equilibrium constant at infinite dilution which coincides with the initial slope of the isotherm. For a heterogeneous surface with two types of adsorption sites the retention factor k is the sum of two contributions, originating from type-I and type-II sites, and a general expression of the retention factors of the two enantiomers under linear conditions can be expressed as  

---

To simulate the empirical concentration profiles, an appropriate mass-transfer model has to be used. One of the simplest models is the model based on the equilibrium-dispersive model, frequently used in column chromatography [1]. It can be given by the following equation ... 

According to the equilibrium dispersive model and adsorption isotherm models the equilibrium data and isotherm model parameters can be calculated and compared with experimental data. It was found that frontal analysis is an effective technique for the study of multicomponent adsorption equilibria [92], As has been previously mentioned, pure pigments and dyes are generally not necessary, therefore, frontal analysis and preparative RP-HPLC techniques have not been frequently applied in their analysis. 

Using this methodology via measurement of adsorption isotherms, Guiochon and coworkers investigated site-selectively the thermodynamics of TFAE [51] and 3CPP [54] on a tBuCQD-CSP under NP conditions using the pulse method [51], the inverse method with the equilibrium-dispersive model [51, 54], and frontal analysis [54]. 

The ideal model and the equilibrium-dispersive model are the two important subclasses of the equilibrium model. The ideal model completely ignores the contribution of kinetics and mobile phase processes to the band broadening. It assumes that thermodynamics is the only factor that influences the evolution of the peak shape. We obtain the mass balance equation of the ideal model if we write > =0 in Equation 10.8, i.e., we assume that the number of theoretical plates is infinity. The ideal model has the advantage of supplying the thermodynamical limit of minimum band broadening under overloaded conditions. 

The equilibrium-dispersive model is defined by Equation 10.8 and as with the ideal model, an isotherm equation should be used to relate the mobile phase and stationary phase concentrations. 

The degree of agreement between the ideal and the equilibrium-dispersive models depends on the value of the effective loading factor, a dimensionless number which is also known as the Shirazi number [1] ... 

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 order to evaluate the performance of the controller, various scenarios have been simulated on a virtual platform. This means that instead of a real plant a SMB model based on the equilibrium dispersive model is used [7], In the following an example is given to show the flexibility and performance of the controller. 

Adsorption isotherms of the bi-Langmuir-type were measured (see Tab. 2). An initial set of parameters was obtained by the perturbation method [6], Subsequently, a peak fitting approach based on the equilibrium-dispersive model was used for refinement (for details, see [4]). Fig. 4 shows a good agreement between models and experiments. 

Fig. 6 shows performance predictions obtained with the equilibrium-dispersive model for such single-column recycling with and without ideal solvent removal (TSR). The same requirements were used as in section 3. The process is basically infeasible without ISR. Also shown is the steady state performance of an SMB-based process (6 columns, ISR cf Fig. 3a). As is often found, the SMB process achieves a lower productivity, but at the same time allows for significantly lower solvent consumption.

Basically, models using only one effect to describe band spreading lump all effects in one model parameter, which is straightforward for linear isotherms (Section 6.5.3.1) but is also commonly applied in the nonlinear range. Of these models, listed in the second level from the bottom in Fig. 6.2, the equilibrium dispersive model plays a prominent role. 

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. 

Although Eq. 6.61 is different from the exact analytical solution of the equilibrium dispersive model (Pallaske, 1984, Levenspiel and Bischoff, 1963, Guiochon et al., 1994b, Guiochon and Lin, 2003), the resulting moments derived from this analytical solution are equal to Eqs. 6.63 and 6.64. 

The basic material balance of the mobile phase for all lumped rate models is based on Eqs. 6.3, 6.4 and 6.13-6.17 and can be derived in the same manner as the equilibrium dispersive model (Eq. 6.58) ... 

Notably, concerning the overall peak shape, as with the equilibrium dispersive model (Section 6.2.4.1), the analytical solution of the transport dispersive model is always an asymmetric peak, and the asymmetry is enhanced by increasing Dax as well as decreasing keff (Lapidus and Amundson 1952). 

The equilibrium dispersive model offers an acceptable accuracy only for N > 100, which may be sufficient for many practical cases. However, sometimes the number of stages is considerably lower and more sophisticated models have to be applied. 

The resulting equation for the equilibrium dispersive model is given by Eq. 6.65. 

In the last part of this book, we apply the different models discussed earlier, particularly the ideal model and the equilibrium-dispersive model, to the investigation of the properties of simulated moving bed chromatography (Chapter 17) and we discuss the optimization of the batch processes used in preparative chromatography (Chapter 18). Of central importance is the optimization of the column operating and design parameters for maximum production rate, minimum solvent use, or minimum production cost. Also critical is the comparison between the performance of the different modes of chromatography. 

If it is assumed that the equilibrium between the two phases is instantaneous and, at the same time, that axial dispersion is negligible, the column efficiency is infinite. This set of assumptions defines the ideal model of chromatography, which was first described by Wicke [3] and Wilson [4], then abundantly studied [5-7,32-39] and solved in a number of cases [7,33,36,40,41]. In Section 2.2.2, which deals with the equilibrium-dispersive model, it is shown how small deviations from equilibrium can be handled while retaining the simplicity of Eq. 2.4 and of the ideal model. 

Table 2.1 System of Equations of the Equilibrium-Dispersive Model. Binary Mixture and Pure (or weakly adsorbed) Mobile Phase...

These equations form a system of partial differential equations of the second order. Examples of two complete systems are given in Table 2.1 (a binary mixture and a pure mobile phase or a mobile phase containing only weakly adsorbed additives, a two-component system) and Table 2.2 (a binary mixture and a binary mobile phase with a strongly adsorbed additive, a three-component system). For the sake of simplicity, the equilibrium-dispersive model (see Section 2.2.2) has been used in both cases. The problem of the choice of the isotherm model will be discussed in the next two chapters. 

Chapters 10 to 13 review the solutions of the equilibrium-dispersive model for a single component (Chapter 10), and multicomponent mixtures in elution (Chapter 11) and in displacement (Chapter 12) chromatography and discuss the problems of system peaks (Chapter 13). These solutions are of great practical importance because they provide realistic models of band profiles in practically all the applications of preparative chromatography. Mass transfer across the packing materials currently available (which are made of very fine particles) is fast. The contribution of mass transfer resistance to band broadening and smoothing is small compared to the effect of thermodynamics and can be properly accounted for by the use of an apparent dispersion coefficient independent of concentration (Chapter 10). 

In these models, the mass balance equation (Eq. 2.2) is combined vHth a kinetic equation (Eq. 2.5), relating the rate of variation of the concentration of each component in the stationary phase to its concentrations in both phases and to the equilibrium concentration in the stationary phase [80-93]. Although in principle kinetic models are more exact than the equilibrium-dispersive model, the difference between the individual band profiles calculated using the equilibrium-dispersive model or the linear driving force model, for example, is negligible when the rate constants are not very small i.e., when the column efficiency exceeds a few him-dred theoretical plates), as shown in Chapter 14 (Section 14.2). 

For mass balance reasons, Cbfl = Cafi. The problem is to find a relationship between the isotherm parameters and the parameters and of the f-th harmonic. In the calculations made to relate the equilibrium isotherm and the response of the system (Eq. 3.102), the equilibrium-dispersive model is used (Chapter 2, Section 2.2.2) and the mass balance equation is integrated with the Danck-werts boundary conditions (Chapter 2, Section 2.1.4.3) and with the initial conditions C = Ca,o, q = qiCa,o). 

Figure 4.29 Perturbation signals recorded on a four-compound plateau. Top, experimental results. Bottom signal calculated with the equilibrium-dispersive model and the best coefficients of a competitive quaternary bi-Langmuir isotherm. Compoxmds enantiomers of methyl- and ethyl-mandelate on a chiral phase. Perturbation as in Eq. 4.104 Reproduced with permission from J. Lindholm, P. Porssen, T. Fomstedt, Anal. Chem., 76 (2004) 5472 (Fig. 3). 2004, American Chemical Society.

From the Lumped Kinetic Model back to the Equilibrium-Dispersive Model. . 300... 

In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by... 

In Figures 6.2a and 6.2b, we compare the profile of a Gaussian distribution and the exact solutions of the equilibrium-dispersive model with the two different bormdary conditions, for columns having 100 (Figure 6.2a) and 1000 (Figure 6.2b) theoretical plates, respectively. In each figure, the dotted lines are the band profiles... 

---