Chromatography ideal model

SAMs are generating attention for numerous potential uses ranging from chromatography [SO] to substrates for liquid crystal alignment [SI]. Most attention has been focused on future application as nonlinear optical devices [49] however, their use to control electron transfer at electrochemical surfaces has already been realized [S2], In addition, they provide ideal model surfaces for studies of protein adsorption [S3]. 

Zhong G., Guioehon G. (1996) Analytieal Solution for the Linear Ideal Model of Simulated Moving Bed Chromatography, Chem. Eng. Sci. 51 4307-4319. 

Golshan-Shirazi, S. and Guiochon, G., Analytical solution for the ideal model of chromatography in the case of a Langmuir isotherm, Anal. Chem., 60 , 2364, 1988. 

Brus et al. prepared isolated silicon particles by high temperature pyrolysis of disilane with a subsequent passivation of the surface by oxidation [33]. The particles of various size are then processed by high-pressure, liquid-phase, size exclusion chromatography to separate sizes and obtain various fractions of monosize particles. Such particles represent an almost ideal model of silicon quantum dots. 

I. Hagglund and J. Stahlberg, Ideal model of chromatography applied to charged solutes in reversed-phase liquid chromatography, J. Chromatogr. A 761 (1997), 3 7. 

Zhong, G., Guiochon, G. Analytical solution for the linear ideal model of simulated moving bed chromatography, Chem. Eng. Sci., 1996, 51(18), 4307 1-319. 

In the last part of this book, we apply the different models discussed earlier, particularly the ideal model and the equilibrium-dispersive model, to the investigation of the properties of simulated moving bed chromatography (Chapter 17) and we discuss the optimization of the batch processes used in preparative chromatography (Chapter 18). Of central importance is the optimization of the column operating and design parameters for maximum production rate, minimum solvent use, or minimum production cost. Also critical is the comparison between the performance of the different modes of chromatography. 

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 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. 

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. 

Furthermore, the theoretical analysis of the single-component problem in the ideal model provides some of the fimdamental concepts in nonlinear chromatography, such as the notions of the velocity associated with a concentration, of concentration shocks, and of diffuse bormdaries [1,2]. It also provides an understanding of the relationship between the thermod5mamics of phase equilibria, the shape of the isotherm (i.e., convex upward, linear, convex downward, or S-shaped) and the band profiles. Finally, it provides an explanation of the relative importance of the influences of the thermodynamics and the kinetics on the band profile. These concepts will provide a most useful framework for imderstanding the phenomena that occur in preparative chromatography. 

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 system of Eqs. 8.1a and 8.1b is the classical system of reducible, quasihnear, first-order partial differential equations of the ideal model of chromatography [1, 2,4r-6,9-17]. The properties of these equations have been studied in detail [4,9,10, 18-24], We discuss here those properties that are important for the xmderstanding of the solutions of the ideal model in the case of elution or displacement of a binary mixture. They are the existence of characteristic fines, called characteristics, the coherence condition, and the properties of the hodograph transform. 

Band Profiles in Displacement Chromatography with the Ideal Model... 

Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the... 

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