General Rate Models

With this model, most chromatographic separations can be described. If particles with a particle size 100 pm are used the model is not applicable because then the diffusion inside the particles cannot be neglected [46]. The consideration of this effect leads to the general rate model, which is described in the following 

The general rate model represents the most complex case of the mathematical modeling. From this model, two sets of partial balance equations result one for the region of the free fluid at the particle surface and one for the region inside the pores. The balance equation of the mobile phase will remain unchanged. 

For the mass-transfer parameter now the concentration of the pore fluid at the edge of the particle Cp.i.r p is used. 

The first two terms stand for the accumulation in the pores and on the solid surface. The term on the right side describes the diffusion inside the pores from Pick s 2nd law. Thereby, a parabolic concentration profile is assumed. The concentration of the substances adsorbed on the solid surface is balanced with the concentration in the particle pore Cp and this equilibrium balance is given by the adsorption isotherm. 

The general rate model was used as a basis for the development of a computer program for the simulation of chromatographic processes by Gu [53]. The solution of the partial differential equations in the nonlinear range of the adsorption isotherms can be obtained by application of numerical methods. One drawback for the modeling of real chromatographic separations with this model is the multitude of physical parameters, which cannot be determined experimentally and have to be estimated by approximations. In practice, these parameters are often only inaccurately fittable, so that a reasonable calculation is impossible. This model is rather applicable for theoretical studies [54]. 

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The most detailed model is the general rate model [2]. This model has been studied by several authors [3]. The moments calculated from the general rate model allow the derivation of a most detailed plate height equation for both particulate and monolith columns [4]. 

The general rate model of chromatography is the most complex of all the models used in this field. In this model, it is assumed that the mobile phase percolates through the interstitial volume between stationary phase particles, diffusion takes place from this stream into the particles and inside the pores of the stationary phase particles, where the mobile phase is stagnant, and adsorption-desorption takes place between the stagnant mobile phase within the pores and the adsorbent surface. 

The lumped pore model (often referred to as the POR model) was derived from the general rate model by ignoring two details of this model [5]. The first assumption made is that the adsorption-desorption process is very fast. The second assumption is that diffusion in the stagnant mobile phase is also very fast. This latter assumption leads to the consequence that there is no radial concentration gradient within a particle. Instead of the actual radial concentration profile across the porous particle, the model considers simply its average value. 

The connection of the overall mass transfer coefficient of the lumped kinetic and the parameters of the general rate model is... 

In this form, mass transfer by pore and surface diffusion instead of an overall mass transfer resistance is assumed inside the particle (internal mass transfer). Equation 6.31 represents a boundary condition of the general rate model, which is further discussed in Section 6.2.6. 

Table 6.1 M ass balance equations (general rate model).

General rate models (GRM) are the most detailed models. In addition to axial dispersion they are characterized by a minimum of two other parameters describing mass transport effects. These two parameters may combine mass transfer in the liquid... 

Gu et al. (1990a and 1990b) proposed an even more reduced general rate model, which only considers pore diffusion inside the particles (pore diffusion model). Thus, Eq. 6.84 is replaced by Eq. 6.85 ... 

As mentioned in Section 6.2.2, with general rate models, boundary conditions for the adsorbent phase are necessary in addition to the conditions at the column inlet and outlet (Section 6.2.7). The choice of appropriate boundary conditions is mathematically subtle and often a cause for discussion in the literature. The following is restricted to the form of the boundary condition derived by Ma et al. (1996) for a complete general rate model. 

The decision for a certain model has also to include considerations of methods to measure or estimate the model parameters. For the general rate model (Eq. 6.81 and Eq. 6.85) it is not possible to derive independently different transport parameters such as Dpore and krim for a given column from a chromatogram. Therefore, one parameter has to be calculated (e.g. by correlations in Section 6.5.8) and used to determine the other. 

General rate model Chromatographic separation of solutes with complex mass transfer and adsorption behavior (e.g. bio separations or ion exchange chromatography)... 

Additional parameters arise according to the model selected and hence the physical effects that are taken into account. Parameters appropriate for the general rate model are given by Berninger et al. (1991) and Ma et al. (1996). 

Equation 6.138 defines a formal connection between the effective mass transport and the film transport, the pore diffusion and the adsorption rate coefficient. It illustrates that keff is a lumped parameter", composed of several transport effects connected in series. This also gives reasons for the use lumped rate models as it proves that the impact of the lumped parameters on the most important peak characteristics, retention time and peak width, is identical to the effect described by general rate model parameters in linearized chromatography. 

Fig. 6.47 Enlarged simulated and measured (sampled) concentration profile in the SMB for fructose-glucose-sucrose (other data see Fig. 6.46).Table 6.1 Mass balance equations (general rate model).

As discussed above, the task of the controller is to optimize the performance of the process over a certain horizon in the future, the prediction horizon. Specifications of product purities, equipment limitations and the dynamic process model (a full hybrid model of the process, including the switching of the ports and a general rate model of all columns) appear as constraints. The control algorithm solves the following nonlinear optimization problem online ... 

The prediction horizon is discretized in cycles, where a cycle is a switching time tshift multiplied by the total number of columns. Equation 9.1 constitutes a dynamic optimization problem with the transient behavior of the process as a constraint f describes the continuous dynamics of the columns based on the general rate model (GRM) as well as the discrete switching from period to period. To solve the PDE models of columns, a Galerkin method on finite elements is used for the liquid... 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

The detailed study of the mass transfer kinetics is necessary in certain problems of chromatography in which the column efficiency is low or moderate. Complex models are then useful. The most important ones are the General Rate Model [52,62] and the FOR model (see next Section) [63]. To study the mass transfer kinetics, these models need to consider separately the mass balance of the feed components in the two different fractions of the mobile phase the one that percolates through the bed of the solid phase (column packed with fine particles or monolithic column) and the one that is stagnant inside the pores of the packing material. 

In dimensionless form, the general rate model consists of the set of equations described in the rest of this section. Each of these equations must be written for each component of the system. These equations are then completed with an appropriate set of kinetics equations. The corresponding models are discussed in the next section. Before discussing these equations, however, we must address an issue that must be carefully clarified to avoid confusions and errors. 

This set of equations (Eqs. 2.25 to 2.35) constitutes the general rate model of chromatography. 

The preparative separations of certain polar (e.g., strongly basic) compounds and of many large molecular compotmds e.g., peptides and proteins) usually involve a complex mass transfer mechanism that is often slower than the mass transfer kinetics of small molecules. This slow kinetics influences strongly the band profiles and its mechanism must be accovmted for quantitatively. The accurate prediction of band profiles for optimization purposes requires a correct mathematical model of the various mass transfer processes involved. The piupose of the general rate model (GRM) is to accormt for the contributions of all the sources of mass transfer resistances to the band profiles [52,62,94,95]. The mass transfer of molecules from the bulk of the mobile phase percolating through the bed to the surface of an adsorbent or the mass of a permeable resin particle involves several steps that must be identified. 

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