Applications of the Equilibrium-Dispersive Model

There is an abimdant literature on the comparison between experimental and calculated band profiles for binary mixtures. The most popular methods used have been the forward-backward finite difference scheme and the OCFE method. The former lends itself readily to numerical calculations in many cases representative of the present preoccupations in preparative chromatography. We present first a comparison between the band profiles obtained with the ideal and the equilibrium-dispersive model to illustrate the dispersive influence of the column efficiency. Related to the comparison between these two models is the issue of the use of the hodograph transform of experimental results discussed in Section 11.2.2. Computer experiments are easy to carry out and most instructive because it is possible to show e effects of the change of a single parameter at a time. Some 

1 Comparison of Solutions of the Ideal and the Equilibrium-Dispersive Models 

When a wide rectangular injection pulse is injected in a column and the width is such that the plateau is not completely eroded when it is eluted, the solution of the system of equations of the ideal model (Eqs. 8.1a and 8.1b) includes a constant state, followed by a simple wave, as shown by the theory of partial differential equations [12,13]. The importance of this result is due to the existence of a relationship between the concentrations of the two components of the binary mixture in the simple wave region. This relationship is independent of the position of the band along the column. We have discussed the properties of the hodograph transform in the case of the ideal model (Oiapter 8, Sections 8.1.2 and 8.8). In the case of the equilibrium-dispersive model (Eqs. 11.1 and 11.2), this result is no longer valid. However, the plots of Ci versus C2 are often close to the simple wave solu- 

As demonstrated in Chapter 8, the retention time of the end of the second component tail is constant and equal to 2 the retention time of the second component under linear conditions. The retention time of the band front decreases with increasing sample size. Therefore, as the sample size increases, the band spreads, and its front moves forward, driving the displacement of the first component. Consider the production of the second component. As the sample size increases, the production rate for the second component increases. With increasing sample size, the displacement effect creates a first component tail which penetrates into the front shock layer of the second component. At this point, the peld begins to decrease, while the production rate continues to increase and a mixed band is generated. When the tail penetrates too deeply into the second component band, the production rate begins to drop. 

the separation factor can be increased by changing the experimental conditions of the separation, such as by decreasing the organic modifier concentration, the buffer concentration, the type of buffer used, or the pH of the mobile phase. These methods must be used to optimize the separation factor. Provided the coliunn saturation capacity does not decrease when cc increases, which may happen with macromolecules, this means maximizing a. 

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Chapters 10 to 13 review the solutions of the equilibrium-dispersive model for a single component (Chapter 10), and multicomponent mixtures in elution (Chapter 11) and in displacement (Chapter 12) chromatography and discuss the problems of system peaks (Chapter 13). These solutions are of great practical importance because they provide realistic models of band profiles in practically all the applications of preparative chromatography. Mass transfer across the packing materials currently available (which are made of very fine particles) is fast. The contribution of mass transfer resistance to band broadening and smoothing is small compared to the effect of thermodynamics and can be properly accounted for by the use of an apparent dispersion coefficient independent of concentration (Chapter 10). 

The equilibrium-dispersive model had been discussed and studied in the literature long before the formulation of the ideal model. Bohart and Adams [2] derived the equation of the model as early as 1920, but it does not seem that they attempted any calculations based on this model. Wicke [3,4] derived the mass balance equation of the model in 1939 and discussed its application to gas chromatography on activated charcoal. In this chapter, we describe the equilibrium-dispersive model, its historical development, the inherent assumptions, the input parameters required, the methods used for the calculation of solutions, and their characteristic features. In addition, some approximate analytical solutions of the equilibrium-dispersive model are presented. 

This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses. 

Prominent models for estimating peak profiles carry out a differentiation of the equilibrium isotherm with approximations for the mass transfer contribution. The equilibrium-dispersive model, above, assumes that all contributions due to nonequilibrium can be lumped into an apparent axial dispersion term. It further assumes that the apparent dispersion coefficient of the solutes remain constant, independent of the concentration of the sample components. For small particles, these approximations are reasonable for many applications. The ideal model assumes that the column efficiency is infinite. There is no axial diffusion, and the two phases are constantly at equilibrium. The band profiles obtained as solutions are in good agreement with experimental chromatograms for columns with N > 1000 having high loading factors. On the other... 

For soils without appreciable clay aggregation, the experimental results and theoretical analysis described here indicate that when diffusion is rate-limiting, the ratio of the hydrodynamic dispersion coefficient to the estimated soil self-diffusion coefficient may be a useful indicator of the applicability of the local equilibrium assumption. For reacting solutes in laboratory columns such as those used in this study, systems with ratios near unity can be modeled using equilibrium chemistry. 

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 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... 

Figure 7.36a and b proves the applicability of shortcut calculations based on the ideal equilibrium model for the estimation of process conditions. The results of rigorous process simulation based on the transport-dispersive model are in very good agreement with the shortcut calculation for isocratic (a) as well as nonisocratic (b) SMB processes. Expectedly safety margins have to be taken into account when the process conditions of an SMB process are estimated by shortcut calculation. The scattering of the numerical data results from an increased grid size for the numerical calculations that has been chosen in order to reduce computer time. The model parameters coincide with the data for the protein separation presented in Section 6.6.2.2.3 the separation quality of the SMB process was set to 99.9% purity. 

E.g. in the so-called "pseudo-equilibrium model, developed by Sylvester [53-56], the same design procedure is used as in a single phase catalytic gas phase reaction, where the mass transfer resistance is replaced by a suitable overall term. Bulk flow and dispersion of the liquid phase are neglected and the whole transport mechanisms are lumped into the equilibrium of the reactant concentrations between gas-, liquid- and particle phase. It is an application of the same principle used successfully in fluid/fluid reactions [57], But the necessary precondition is that the rate of reaction is slow compared to the transfer rate across the phase boundaries, so that equilibrium can really by assured. This might be justified in some of the hydrotreating processes, but certainly not in case of an aqueous liquid phase, existing in waste water treating. Earlier models used in petroleum industry have taken in-... 

We then proceed to review results on the applicability of the homogeneous equilibrium assumptions (Kukkonen et al., 1993, 1994 Nikmo et al., 1994). We have compared thermodynamic model predictions with those of the heavier-than-air cloud dispersion model DRIIT (Webber et al., 1992). These investigations have shown that the simpler model does indeed provide a good description for many release situations and guidance is given on where the homogeneous equilibrium model is not likely to be adequate. 

By contrast, when the mass transfer resistances and/or axial dispersion are considered, there is no analytical solution for an SMB operated under nonlinear isotherm conditions. A numerical solution of the applicable mathematical model must be used instead to calculate the performance of the SMB, to simulate the influence of the various design and operating parameters, and to search for the optimum flow rates and switching time that give the desired results. In this quest, the selection as a starting point of the optimum set of flow rates and switching time derived from the equilibrium theory permits a considerable reduction of the number of calculations. As discussed earlier by Ruthven and Ching [27], four... 

Fortunately, many applications can be predicted accurately via the simplest models. They ignore transport phenomena, mainly account for mass conservation, and are called equilibrium models. Though the concept is simple, such a model can account for a wide variety of effects, including flow, pressure, and composition variations, of course, as well as nonlinear isotherms and dispersive effects due to dead space, say, at the entrance or exit of the adsorbent bed. 

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