Chromatography difference equation model

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 this table, we provide solubility parameters for some liquid solvents that can be used as modifiers in supercritical fluid extraction and chromatography. The solubility parameters (in MPa1/2) were obtained from reference 3, and those in cal1/2cm 3/2 were obtained by application of Equation 4.1 for consistency. It should be noted that other tabulations exist in which these values are slightly different, since they were calculated from different measured data or models. Therefore, the reader is cautioned that these numbers are for trend analysis and separation design only. For other applications of cohesive parameter calculations, it may be more advisable to consult a specific compilation. This table should be used along with the table on modifier decomposition, since many of these liquids show chemical instability, especially in contact with active surfaces. 

The outline of this chapter is as follows First, some basic wave phenomena for separation, as well as integrated reaction separation processes, are illustrated. Afterwards, a simple mathematical model is introduced, which applies to a large class of separation as well as integrated reaction separation processes. In the limit of reaction equilibrium the model represents a system of quasilinear first-order partial differential equations. For the prediction of wave solutions of such systems an almost complete theory exists [33, 34, 38], which is summarized in a second step. Subsequently, application of this theory to different integrated reaction separation processes is illustrated. The emphasis is placed on reactive distillation and reactive chromatography, but applications to other reaction separation processes are also... 

Equation 3.9 is a linear isotherm. The concentration of the solute in the stationary phase is proportional to that in the mobile phase. This isotherm is widely used in analytical chromatography, where it gives results that are most often satisfactory. When the concentration becomes high, however, deviations from linear behavior take place, competitive interactions between the different components of the feed appear, and a more complex model becomes necessary to accormt for these experimental results. As an example. Figure 3.2 [20] shows a comparison between a linear isotherm and two nonlinear models (see below). The difference is small at concentrations below 0.05 mM, but significant deviations take place at 0.2 mM. They are sufficient to cause an important decrease in the band retention and a marked asymmetry of its profile [20]. 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques. 

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. 

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. 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

For modeling and simulation of preparative chromatography experimentally determined adsorption equilibrium data have to be represented by suitable mathematical equations. From the literature a multitude of different isotherm equations is known. Many of these equations are derived from equations developed for gas phase adsorption. Detailed literature can be found, for example, in the textbooks of Guiochon et al. (2006), Everett (1984), and Ruthven (1984) or articles of Seidel-Morgenstern and Nicoud (1996) or Bellot and Condoret (1993). 

In nonlinear chromatography the equilibrium concentrations of a component in the stationary phase and mobile phase are not proportional, the adsorption isotherms are not linear. The loading of the stationary phase is dependent on the concentration in the mobile phase. The differential equations applied for modeling are interlinked and can be solved by application of different algorithms. 

When these equations are applied, one of molecules is the model phase and the target molecules are the analytes. The following equations can he used to study chromatographic retention in different types of chromatography. 

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