Dispersive model of chromatography

Figure 6.2 Comparison of the chromatogram given by the equilibrium-dispersive model of chromatography with a Gaussian profile. Dimensionless plot of = Ctn/Ap versus frf = f/tj . Solid line Gaussian profile with N theoretical plates. Dotted line equilibrium-dispersive model with an "open-open" boundary condition and (2D )-...

Accordingly, the equilibrium-dispersive model of chromatography for a single component is represented by one single partial differential equation, the mass balance equation... 

In the equilibrium-dispersive model of chromatography, however, we assume that Eq. 10.4 remains valid. Thus, we use Eq. 10.10 as the mass balance equation of the component, and we assume that the apparent dispersion coefficient Da in Eq. 10.10 is given by Eq. 10.11. We further assume that the HETP is independent of the solute concentration and that it remains the same in overloaded elution as the one meastued in linear chromatography. As shown by the previous discussion this assxunption is an approximation. However, as we have shown recently [6], Eq. 10.4 is an excellent approximation as long as the column efficiency is greater than a few hundred theoretical plates. Thus, the equilibriiun-dispersive model should and does account well for band profiles under almost all the experimental conditions used in preparative chromatography. In the cases in which the model breaks down because the mass transfer kinetics is too slow, and the column efficiency is too low, a kinetic model or, better, the general rate model (Chapter 14) should be used. 

Equation 10.79 permits the calculation of the concentration at the new space position, n + 1, knowing the concentration at the previous space position, n (Godunov scheme). This method calculates band profiles along the coliunn at successive time intervals. The elution profile is the history of concentrations at z = I. The forward-backward difference scheme was first used to calculate solutions of the equilibrium-dispersive model of chromatography by Rouchon et al. [46,47]. Since then, it has been widely used. It is particularly attractive because of its fast execution by modern computers [50]. Czok [50] and Felinger [54] have shown how the CPU time required can be further shortened by eliminating the needless computation of concentrations below a certain threshold. The dramatic increase over the last fifteen years of the speed of the computers available for the numerical calculations of band profiles has considerably reduced this advantage of the Forward-Backward scheme over the other possible ones. 

In this case, the system of equations of a model of chromatography (Chapter 2) must be solved numerically. These models are very general. They apply to all modes of chromatography, independently of the model of competitive isotherms selected. Most theoretical studies used the equUibrium-dispersive model of chromatography because the mass transfer kinetics are fast xmder the experimental conditions employed in the study of system peaks. These theoretical studies also used the Langmuir competitive model because it is both general and convenient. 

Apparent dispersion coefficient, Dapi The apparent dispersion coefficient lumps all the contributions to axial dispersion arising from axial molecular diffusion, tortuosity, eddy diffusion, and from a finite rate of mass transfer, adsorption-desorption, or other phenomena, such as reactions, in which the eluites may be involved. It is used in the equilibrium-dispersive model of chromatography to ac-coimt for the finite efficiency of the column (Eq. 2.53 and 10.11). See equilibrium-dispersive model. 

Batch Decomposition of Acetylated Castor Oil 243 Dispersion Model for Chromatography Columns 483 Stagewise Model for Chromatography Columns 486 Complex Reaction 237 Continuous Flow Tank 406... 

Frontal chromatography can be used in combination with chromatographic models to study mass-transfer and dispersion processes (e.g., the equilibrium dispersive or the transport model of chromatography [7]). 

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. 

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. 

In the modeling of chromatography, the contributions of aU the phenomena that contribute to axial mixing are lumped into a single axial dispersion coefficient. Two main mechanisms contribute to axial dispersion molecular diffusion in the interparticle pores and eddy diffusion. In a first approximation, their contributions are additive, and the axial dispersion coefficient, Di, is given by... 

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. 

Finally, Kvaalen et al have shown that the system of equations of the ideal model for a multicomponent system (see later, Eqs. 8.1a and 8.1b) is strictly h5q3er-bolic [13]. As a consequence, the solution includes two individual band profiles which are both eluted in a finite time, beyond the column dead time, to = L/u. The finite time that is required for complete elution of the sample in the ideal model is a consequence of the assumption that there is no axial dispersion. It contrasts with the infinitely long time required for complete elution in the linear model. This difference illustrates the disparity between the hyperbolic properties of the system of equations of the ideal model of chromatography and the parabolic properties of the diffusion equation. 

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. 

Equilibrium-dispersive model Model of chromatography assuming near equilibrium between the stationary and the mobile phases. Specifically, it assumes that the concentrations in these two phases are related by Ae isotherm equation, and that the effect of the finite rate of mass transfer can be lumped together with the axial dispersion coefficient. This model is valid when the column efficiency is larger than a few hundred plates. 

Ideal model of chromatography A model of chromatography assuming no axial dispersion and no mass transfer resistance, i.e., that the column efficiency is infinite (fi = 0). This model is accurate for high-efficiency, strongly overloaded columns. It permits an easy study of the influence of the thermodynamics of phase equilibrium (i.e., of the isotherm) on the band profiles and the separation. See Chapters 7 to 9. 

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. 

The answers to all these questions are now well known. The ideal model of chromatography responds to the first and second questions [68]. The equilibrium-dispersive model speaks to the first three questions, at least so long as the column efficiency under linear conditions exceeds 30-50 plates [22]. The last question is outside the scope of this review, but has been abundantly discussed in the literature [23]. 

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 ... 

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