The Lumped Kinetics Model

The lumped kinetic model can be obtained with further simplifications from the lumped pore model. We now ignore the presence of the intraparticle pores in which the mobile phase is stagnant. Thus, p = 0 and the external porosity becomes identical to the total bed porosity e. The mobile phase velocity in this model is the linear mobile phase velocity rather than the interstitial velocity u = L/Iq. There is now a single mass balance equation that is written in the same form as Equation 10.8. 

In the lumped kinetic model, various kinetic equations may describe the relationship between the mobile phase and stationary phase concentrations. The transport-dispersive model, for instance, is a linear film driving force model in which a first-order kinetics is assumed in the following form ... 

Note that this kinetic equation is rather similar to Equation 10.15. The major difference between Equations 10.15 and 10.19 is that the general rate and the lumped pore models assume that adsorption takes place from the stagnant mobile phase within the pores, while the lumped kinetic model assumes that the mobile phase concentration is the same in the pores and between the particles. 

The GRM is the most comprehensive model of chromatography. In principle, it is the most realistic model since it takes into account all the phenomena that may have any influence on the band profiles. However, it is the most complicated model and its use is warranted only when the mass transfer kinetics is slow. Its application requires the independent determination of many parameters that are often not accessible by independent methods. Deriving them by parameter identification may be acceptable in practical cases but is not easy since it requires the acquisition of accurate band profile data in a wide range of experimental conditions. This explains why the GRM is not as popular as the equilibrium-dispersive or the lumped kinetic models. 

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

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

The study of the lumped kinetic models shows that, as long as the equilibration kinetics is not very slow and the column efficiency exceeds 25 theoretical plates (a condition that is satisfied in all the cases of practical importance), the band profile is a Gaussian distribution. We can thus identify all independent sources of band broadening, calculate their individual contributions to the variance of the Gaussian distribution, and relate the column HTTP to the sum of these variances. The method is simple and efficient. It has been used successfully for over 40 years [29]. We may want a more rigorous approach. 

Note that the rate coefficient kf used in Eq. 6.82 was defined in Eq. 5.70 and has dimensions LT. By contrast, in the lumped kinetic models, the rate coefficient km in Eq. 6.43 or fcy in Eq. 14.3) has dimensions T . The third, fourth, and fifth moments are given by more complicated expressions and can be formd in the literature [30,31], In practice, only the first and second moments of a band are determined, the first to characterize its retention and calculate the equilibrium constant, the second to characterize and study the band spreading, hence the mass transfer kinetics. 

The above equation can be compared with Eq. 6.56, which was obtained with the lumped kinetic model. The two expressions are identical only if n = Nm, i-e. the average number of adsorption-desorption events in the microscopic model and the number of mass transfer units in the macroscopic kinetic model are analogous terms. Felinger et al. showed that not only the first and the second moments but also the whole band profiles obtained as the solutions of the microscopic model and the macroscopic lumped kinetic model are completely identical [9]. 

It would be very attractive to derive analytical expressions for the optimum experimental conditions from the solution of a realistic model of chromatography, i.e., the equiUbriiun-dispersive model, or one of the lumped kinetic models. Approaches using analytical solutions have the major advantage of providing general conclusions. Accordingly, the use of such solutions requires a minimum number of experimental investigations, first to validate them, then to acquire the data needed for their application to the solution of practical problems. Unfortunately, as we have shown in the previous chapters, these models have no analytical solutions. The systematic use of these numerical solutions in the optimization of preparative separations will be discussed in the next section. 

Models of chromatography Besides linear chromatography (Chapter 6), which assumes a linear equilibrium isotherm, there are four main models, differing in their treatment of the mass transfer kinetics. In the ideal model (Chapters 7 to 9), the column is assumed to have an infinite efficiency there is no axial dispersion and the mass transfer kinetics is infinitely fast. In the equilibrium-dispersive model (Chapters 10 to 13), the rate of mass transfer is assumed to be very fast and is treated as a contribution to axial dispersion, independent of the concentration. In the lumped kinetic models (Chapters 14 to 16), the rate of mass transfer is still high, but their dependence on the concentration is accounted for. The general rate model (Chapters 14 and 16) takes into account all the possible sources of deviation from eqtulibrium. 

Neither of the above models explicitly include the original claim that the resources are a part of the biomass and their availability may thus be constrained. This feature is readily built into either model. For example, we may denote the resources by R, their concentration by r, and postulate that their formation occurs by growth, and consumption by their allocation for enzyme synthesis. Admittedly this is somewhat of a narrow view of how resources may be used in metabolic processes but this simplification is virtually at the same level as that Inherent in the lumped kinetic models of the past. The differential equation for r may be written as... 

These models involve writing and integrating a differential mass-balance equation. The basic assumptions associated with writing such an equation are discussed in the literature [16], There are two important models of this type, the equilibrium-dispersive model [22]. and the lumped kinetic models, in addition to the general rate model. The differential mass balance equation is... 

Table 2 The Lumped Kinetics Models for Fixed-Bed FTS over 1 ron-Based Catalyst (Wang et al., 1999) ... 

The lumped kinetic models developed via the top-down route have limited extrapolative power. The rate constants generally depend on feedstock, catalyst, and reactor hardware configuration. As such, the models must be continually retuned for new situations. Moreover, the lack of sufficient molecular information makes it difficult to predict subtle changes in product qualities and properties. The bottom-up approach attempts to rectify this state of affairs, as discussed next. 

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