Equilibrium-dispersive model single components

We have attempted to present here, in a rather condensed form, a vievc of the present status of the fxmdamentals of preparative and nonlinear chromatography. The fundamental problems and the various models used to model chromatography are discussed first (Chapter 2). As the thermodynamics of phase equilibrium is central to the separation process, whatever model is used, we devote two chapters to the discussion of equilibrium isotherms, for single components (Chapter 3) and mixtures (Chapter 4). A chapter on the problems of dispersion, mass transfer and flow rate in chromatography (Chapter 5) completes the fundamental bases needed for the thorough discussion of preparative chromatography. 

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). 

Single-Component Profiles with the Equilibrium Dispersive Model... 

Accordingly, the equilibrium-dispersive model of chromatography for a single component is represented by one single partial differential equation, the mass balance equation... 

All the results presented here combine to demonstrate the practical usefulness of the equilibrium-dispersive model, at least in the case of single-component band profiles. Even when the column efficiency is poor e.g., Troger s base on cel-... 

Figure 12.10 Enrichment of jS-naphthylamine traces in methanol solutions by displacement with diethyl phthalate. Experimental (symbols) and calculated (solid lines) chromatograms. Calculations made with the equilibrium-dispersive model, a competitive Langmuir isotherm using the coefficients of the single-component isotherms, N = 3600 plates, and fluorescence quenching, (a) Displacement of 0.45 jig (6-naphthylamine by di-ethylphthalate (200 mg/mL) in methanol-water (70 30). (b) Same experiment with 275 mg/mL diethylphthalate. Reproduced with permission from R. Ramsey, A.M. Katti and G. Guio-chon, Anal. Chem., 62 (1990) 2557 (Figs. 7b and 8a). (5)1990, American Chemical Society.

Figure 12.29 Comparison of theoretical and experimental displacement separations of resorcinol and catechol by phenol. Calculations using the equilibrium-dispersive model, the LeVan- Vermeulen isotherm model, and single-component adsorption data. Experimental results on a 4.6x250 CIS Nucleosil 5 fim column, F = 0.4 carrier, water, Fj, = 0.2 mL/min, T = 20°C 1 1 mixture, = 0.5 mL displacer, 80 g/L phenol in water = 30%, Lf = 16.5%. (a) Calculation with LeVan-Vermeulen isotherm, (b) Calculation with quadratic isotherm, three floating parameters, (c) Calculation with competitive Langmuir isotherm, single-component isotherm parameters, (d) Calculation with Langmuir isotherm, best adjusted parameters. Reproduced with permission from. C. Bellot and J.S. Condoret, J. Chromatogr., 657 (1994) (Figs. 3c, 4c, 6c, 8c) 305.

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. 

The basic assumptions implied in the homogeneous model, which is most frequently applied to single-component two-phase flow at high velocities (with annular and mist flow-patterns) are that (a) the velocities of the two phases are equal (b) if vaporization or condensation occurs, physical equilibrium is approached at all points and (c) a single-phase friction factor can be applied to the mixture if the Reynolds number is properly defined. The first assumption is true only if the bulk of the liquid is present as a dispersed spray. The second assumption (which is also implied in the Lockhart-Martinelli and Chenoweth-Martin models) seems to be reasonably justified from the very limited evidence available. 

Figure 2.6 Relationship between the equilibrium isotherms and the band profiles for single components. Top row isotherm second row, isotherm differential third row, same as second row, but symmetry aroimd the first bisector fourth row, band profiles with the ideal model bottom row, band profiles with axial dispersion.

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