Lumped kinetic models, chromatography

In linear chromatography, these last two linear kinetic models are particular cases of the model used by Lapidus and Amundson [85] (Eq. 2.22). By contrast, the different lumped kinetic models give different solutions in nonlinear chromatography. Investigations of the properties of these models and especially of the relationship between the band profiles and the value of the kinetic constant have been carried out for many single-component problems. Numerous studies of the influence of the mass transfer kinetics on the separation of binary mixture have been published in the last ten years. These results are discussed in Chapters 14 and 16, respectively. 

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. 

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. 

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. 

These models recognize that equilibrium between the mobile and stationary phases can never actually be achieved, and use a kinetic equation to relate bq/ht, the partial differential of q with respect to time, and the local concentrations, q and C. Several different kinetic models are possible depending on which step is assumed to be rate controlling. All are called lumped kinetic models because, for the sake of simplicity, the contributions of all other steps are lumped together with the one considered to be most important. These models are discussed and compared in detail in [24]. All such models are equivalent in linear chromatography, but not when the equilibrium isotherm is nonlinear [16]. The kinetic equation can be written as ... 

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... 

Since Eq. 2.2 contains two functions, C, and another equation or relationship between them is necessary for its solution. Depending on the model of chromatography used, Eq. 2.2 will be accompanied by a mass balance in the stationary phase and a kinetic equation, by a lumped mass transfer kinetic equation, or by an adsorption isotherm (Section 2.1.3). 

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