Model equilibrium dispersive

In the equilibrium dispersive model it is assumed - like in the ideal model - that the mobile and the stationary phases are permanently in the equilibrium state. In addition to adsorption and convection all band-broadening transport effects are considered. Thereby, all kinetic effects like axial dispersion, film and pore diffusion are lumped together in the apparent dispersion coefficient Djp. As the basis of the model the following partial differential equation can be written  

The concentration in the stationary phase is calculated as a function of the chosen isotherm model from the concentration in the mobile phase at any time. 

The apparent diffusion coefficient D p is estimated from the apparent number of theoretical stages. This estimation is correct as long as the number of theoretical stages is at least 100. 

In nonlinear chromatography Dap is not independent of the concentration as it is considered in this model. However, this approximation is suitable for the description of many chromatographic separations [45]. The performance of the equilibrium dispersive model is confirmed in many cases in the hterature. A detailed description is given by Seidel-Morgenstern [46]. 

The application of the model is limited if the elution profile is strongly influenced by mass-transfer effects and an equilibrium state between the two phases cannot be assumed. In that case more detailed models have to be apphed and for the description of the time-based change of the concentration in the stationary phase an additional balance equation has to be solved. 

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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 ... 

Thomas Model Equilibrium Dispersive Model Equilibrium Transport Model... 

The reaction is reversible and therefore the products should be removed from the reaction zone to improve conversion. The process was catalyzed by a commercially available poly(styrene-divinyl benzene) support, which played the dual role of catalyst and selective sorbent. The affinity of this resin was the highest for water, followed by ethanol, acetic acid, and finally ethyl acetate. The mathematical analysis was based on an equilibrium dispersive model where mass transfer resistances were neglected. Although many experiments were performed at different fed compositions, we will focus here on the one exhibiting the most complex behavior see Fig. 5. 

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. 

The underlying equilibrium-dispersion model, for which the mass balance for solute / in a A component mixture and a volume element is given in equation (21-2), has been very often successfully applied to quantify chromatographic processes under overloaded conditions. 

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. 

Equilibrium dispersive model Adsorption chromatography for products with low molecular weights High accuracy only if N > 100... 

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

In other words, the simplification of using the number of stages for process optimization is best applied if either mass transfer or dispersion dominates the peak broadening. Therefore, the optimization strategies discussed later in this chapter apply a validated transport dispersive model, which can flexibly consider mass transfer and/ or dispersion effect. Here, the number of stages is used as independent variable for the optimization criteria like productivity or eluent consumption. Another possible approach would be the use of simplified simulation model like equilibrium dispersive model (Seidel-Morgenstern, 1995). 

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. 

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]. 

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. 

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