Chromatography axial dispersion model

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

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... 

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. 

This model of nonlinear chromatography, the simplest model, was formulated and studied first by Wicke [3], Wilson [4], and DeVault [5]. It assumes that the column efficiency is infinite. There is no axial dispersion and the two phases are constantly at equilibrium. 

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. 

For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other nvass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In aU other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. 

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. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

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... 

Wade et al. [46] have compared the experimental band profiles of p-nitrophenyl-ff-D-mannopyranoside on silica-bonded Concanavalin A, obtained in affinity chromatography, and the best fit parameters to their model. This model i.e., Thomas model) uses a Langmuir kinetic and neglects the axial dispersion. The best values of the parameters are calculated using a Simplex program to minimize the sum of the residuals of the predided and experimental band profiles. Figure 14.10 illustrates the results obtained and shows excellent agreement. 

Tingyue et al. [43] presented a general rate model for the study of gradient elution in multicomponent nonlinear chromatography. The model considers axial dispersion, film mass transfer, intraparticle diffusion, and a second-order kinetic of adsorption-desorption. This model can simulate various gradient operations including linear, nonlinear, and stepwise gradients. 

In the case of a step input, the numerical solution of the system of Eqs. 16.30 and 16.31 has been discussed in the literature for multicomponent mixtures [16]. The numerical solution of Eqs. 16.30 and 16.31 without an axial dispersion term i.e., with Di = 0) has been described by Wang and Tien [17] and by Moon and Lee [18], in the case of a step input. These authors used a finite difference method. A solution of Eq. 16.31 with D, = 0, combined with a liquid film linear driving force model, has also been described for a step input [19,20]. The numerical solution of the same kinetic model (Eqs. 16.30 and 16.31) has been discussed by Phillips et al. [21] in the case of displacement chromatography, using a finite difference method, and by Golshan-Sliirazi et al. [22,23] in the case of overloaded elution and displacement, also using finite difference methods. 

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. 

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