Dispersion Displacement chromatography

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... 

In the case of a step input, the numerical solution of the system of Eqs. 16.30 and 16.31 has been discussed in the literature for multicomponent mixtures [16]. The numerical solution of Eqs. 16.30 and 16.31 without an axial dispersion term i.e., with Di = 0) has been described by Wang and Tien [17] and by Moon and Lee [18], in the case of a step input. These authors used a finite difference method. A solution of Eq. 16.31 with D, = 0, combined with a liquid film linear driving force model, has also been described for a step input [19,20]. The numerical solution of the same kinetic model (Eqs. 16.30 and 16.31) has been discussed by Phillips et al. [21] in the case of displacement chromatography, using a finite difference method, and by Golshan-Sliirazi et al. [22,23] in the case of overloaded elution and displacement, also using finite difference methods. 

The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example. Figure 16.10 shows the displacement chromatogram calculated with kpi = kfg = = 50 s . The bands in the isotachic train are clearly formed... 

Dipole interact ions,. tee Electrostatic forces Dispersion forces (energies), 44-47 on alumina, 245 in gas-solid adsorption, 243-245 Displacement chromatography, 34-36 Distribution coefficient A, lOi-ll calculation (examples), 385-396 correlation between different adsorbent batches, 148-149... 

Isolation procedures for many biochemicals are based on chromatography. Practically any substance can be selected from a crude mixture and eluted at relatively high purity from a chromatographic column with the right combination of adsorbent, conditions, and eluant. For bench scale or for a small pilot plant, such chromatography has rendered alternate procedures such as electrophoresis nearly obsolete. Unfortunately, as size increases, dispersion in the column ruins resolution. To produce small amounts or up to tens of kilograms per year, chromatography is an excellent choice. When the scale-up problem is solved, these procedures should displace some of the conventional steps in the chemical process industries. 

Displacement and Dispersion of Particles of Finite Size in Flow Channels with Lateral Forces. Field-Flow Fractionation and Hydrodynamic Chromatography, J. C. Giddings, Sep. Sci. Technol., 13, 241 (1978). 

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). 

Thus, we can conclude that, as long as the mass transfer kinetics is reasonably fast, the equilibrium-dispersive model can be used as a first approximation to predict shock layer profiles. As a consequence, the results of calculations of band profiles, breakthrough curves, or displacement chromatograms made with this model can be expected to agree well vsdth experimental results. Conclusions based on the s) stematic use of such calculations have good predictive value in preparative chromatography. 

There is a minimum of the SLT for an intermediate value of the mobile phase velocity, as in linear chromatography [11]. At low velocities, axial dispersion is large due to the long migration time during which axial diffusion proceeds constantly to relax the concentration gradients, while at high velocities, the finite rate of the mass transfer kinetics causes the SLT to increase in proportion to the velocity. If we assume as above the Van Deemter equation for the axial dispersion term (Eq. 14.3Q2), we obtain for the optimum velocity for minimum SLT in displacement (imder isotachic conditions)... 

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