Column chromatography mass-transfer model

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 ... 

In order to estimate resolution among peaks eluted from a chromatography column, those factors that affect N must first be elucidated. By definition, a low value of Hs will result in a large number of theoretical plates for a given column length. As discussed in Chapter 11, Equation 11.20 obtained by the rate model shows the effects of axial mixing of the mobile phase fluid and mass transfer of solutes on Hs. 

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

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. 

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. 

The classical shift-invariant convolution permits a simple calculation of the combined effects of multiple sources of band broadening when the column efficiency is not very low. This approach gives correct results in linear chromatography but is incorrect in nonlinear chromatography [1]. The simplest model that takes axial dispersion and mass transfer kinetics into account is the equilibrium-dispersive model. This model permits, with a good approximation, the accurate prediction of the importance of the self-sharpening and dispersive phenomena due to thermodynamics and kinetics of phase equilibria. This, in turn, results in correct prediction of the band profiles and the achievement of often excellent... 

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. 

The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example. Figure 16.10 shows the displacement chromatogram calculated with kpi = kfg = = 50 s . The bands in the isotachic train are clearly formed... 

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. 

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. 

As explained in Sec. 4.4.4, the movement of components through a chromatography column can be modelled by a two-phase rate model, which is able to handle multicomponents with nonlinear equilibria. In Fig. 1 the column with segment n is shown, and in Fig. 2 the structure of the model is depicted. This involves the writing of separate liquid and solid phase component balance equations, for each segment n of the column. The movement of the solute components through the column occurs by both convective flow and axial dispersion within the liquid phase and by solute mass transfer from the liquid phase to the solid. 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

Similarly, a decision must be made whether or not to take into account the influence on band profiles of such phenomena as axial dispersion (dispersion in the direction of the concentration gradient in the column) and resistance to mass transfer (i.e.. the fact that equilibration between mobile and stationary phases is not instantaneous). These phenomena are responsible for the finite efficiency of actual columns. Neglecting them and assuming the column to have infinite efficiency leads to a model of ideal chromatography. Taking them into account results in one of the models of nonideal chromatography. 

In species exchange processes (e.g., ion-exchange chromatography column), conventional adsorption rate models describe mass transfer (or exchange) between phases, assuming the existence of a counterpart species. In contrast, the adsorption models may not be useful in an inert environment (or inactive zone) where adsorption/desorption caimot take place because of lack of counterpart species. 

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