Ideal model of chromatography

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. 

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. 

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. 

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. 

A general mathematical treatment of system peaks and of the closely related method of vacancy chromatography was given by Helfferich and Klein [8]. This work includes a detailed analysis of the phenomena that take place upon injection of a sample into a chromatographic column. It is based on the use of the solution of the ideal model of chromatography for multicomponent systems, with competitive Langmuir isotherms (see Chapters 8 and 9), and of the ft-transform. 

There are simple algebraic solutions for the linear ideal model of chromatography for the two main coimter-current continuous separation processes. Simulated Moving Bed (SMB) and True Moving Bed (TMB) chromatography. Exphcit algebraic expressions are obtained for the concentration profiles of the raffinate and the extract in the columns and for their concentration histories in the two system effluents. The transition of the SMB process toward steady state can be studied in detail with these equations. A constant concentration pattern can be reached very early for both components in colimm III. In contrast, a periodic steady state can be reached only in an asymptotic sense in colunms II and IV and in the effluents. The algebraic solution allows the exact calculation of these limits. This result can be used to estimate a measure of the distance from steady state rmder nonideal conditions. 

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. 

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 ... 

The answers to all these questions are now well known. The ideal model of chromatography responds to the first and second questions [68]. The equilibrium-dispersive model speaks to the first three questions, at least so long as the column efficiency under linear conditions exceeds 30-50 plates [22]. The last question is outside the scope of this review, but has been abundantly discussed in the literature [23]. 

Guiochon et al. [3] also arrived at equations for retention and efficiency, based upon the solution of differential mass balance equations for chromatography using the Ideal Model of chromatography. This makes the major assumption that the column efficiency is infinite, under which conditions it is possible to reach an analytical solution of the equations. Their equation for capacity factor converges with that of Snyder et al. at high values of efficiency and has the virtue of simplicity ... 

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