Shock layer

More recently, there have been attempts to study band patterns as they are affected by shock layers in nonlinear chromatography.42 Shock layers are steep boundaries that develop when the boundary front of an elution band becomes very steep and self-sharpening at high concentrations. While comparison of predicted and experimental data was promising, this study, like the others mentioned above, was done with single-component samples and awaits further analysis with the kinds of multi-component feeds more frequently encountered in process purifications. 

Constant Pattern, Self-Sharpening Effect, Shock-Layer Theory... 

A very detailed study of the combined effects of axial dispersion and mass-transfer resistance under a constant pattern behavior has been conducted by Rhee and Amundson [10]. They used the shock-layer theory. The shock layer is defined as a zone of a breakthrough curve where a specific concentration change occurs (i.e., a concentration change from 10% to 90%). The study of the shock-layer thickness is a new approach to the study of column performance in nonlinear chromatography. The optimum velocity for minimum shock-layer thickness (SLT) can be quite different from the optimum velocity for the height equivalent to a theoretical plate (HETP) [9]. 

Frontal analysis can easily be extended to binary mixtures. The shape of the breakthrough profiles and the effect of axial dispersion on these shapes have been studied theoretically [93,94] and experimentally [14,73,95-99]. These profiles are characterized by the successive elution of two steep fronts (shock layers) for a binary mixture. The use of these profiles for the determination of the competitive isotherms of two components has been developed by Jacobson et al. [14]. 

Figure 4.21 [14] shows the breakthrough ciuves obtained in two-component frontal analysis with competitive Langmuir isotherms, with four successive concentration steps, and with a column efficiency of 5000 theoretical plates. The thin and thick solid lines correspond to the first and the second components, respectively. The first step gives a different profile from all the following steps because the column is initially equilibrated with the pure mobile phase only [initial condition, Q(x, f = 0) = 0]. For this first step, two shock layers signal the successive exit of the lesser and the more retained components. The first component subplateau is more concentrated than the feed there is no subplateau for the second... 

We have shown in Chapters 7, 8, and 9 that the ideal model gives a good first approximation of the band profiles observed under conditions of strongly nonlinear behavior of the isotherm. This approximation is better at high column efficiency. Even then, however, the profiles predicted by the ideal model are angular and have no rormd comers like those observed experimentally. Indeed, experimental profiles do not show concentration discontinuities but shock layers of various thicknesses (see Chapter 14), while the other features of the solutions are smoothed out, eroded by the influence of axial dispersion and of the resistances to mass transfer in the column. A more accurate model is needed to account for the actual band profiles and to predict them when needed. 

The ideal model (Chapter 7) assiunes an infinite column efficiency. This makes the band profiles that it predicts unrealistically sharp, especially at low concentrations. This sharpness is explained by the fact that the ideal model propagates concentration discontinuities or shocks. For a hnear isotherm, the elution profile would be identical to the input profile, clearly an unacceptable conclusion. The effects of a nonideal column are significant in three parts of the band profile. The shock is replaced by a steep boimdary, the shock layer, whose thickness is related to the coefficients of the column HETP (axial dispersion and mass transfer resistance see Chapter 14). The top of the band profile is roimd, instead of being... 

However, in practice, the deviation between the solutions of the ideal and the equilibrium-dispersive models is small along most of the profiles. The deviations of the plot of the experimental values of C2 versus C from the hodograph transform of the ideal model solution are also small in most parts of the graph, see Figure 11.8, top and bottom left [5,14,15]. The only exceptions are the parts of this plot that correspond to the beginning of the second shock layer and to the... 

As demonstrated in Chapter 8, the retention time of the end of the second component tail is constant and equal to 2 the retention time of the second component under linear conditions. The retention time of the band front decreases with increasing sample size. Therefore, as the sample size increases, the band spreads, and its front moves forward, driving the displacement of the first component. Consider the production of the second component. As the sample size increases, the production rate for the second component increases. With increasing sample size, the displacement effect creates a first component tail which penetrates into the front shock layer of the second component. At this point, the peld begins to decrease, while the production rate continues to increase and a mixed band is generated. When the tail penetrates too deeply into the second component band, the production rate begins to drop. 

Significant or even large differences between band profiles calculated with the two finite difference methods arise for aU mixture compositions at low column efficiencies. Such differences also arise at high efficiencies, when the relative concentration of the second component is low, i.e., when the tag-along effect is important [6-8]. In this last case, the only significant differences between the profiles calculated with the different methods are in the steepness of the shock layers in the mixed zone and in the retention time of the second component front [6-9,28]. The numerical problems have been discussed above, in Sections 11.1.3 and 11.1.4, and examples shown in Figures 11.5 and 11.6. 

We compare in Figure 12.1a the isotachic trains calculated with the ideal and the equilibrium-dispersive models under the experimental conditions given in Chapter 9. The major difference observed concerns the boundary between adjacent zones. These boundaries are no longer vertical fronts or shocks they have become shock layers (Chapter 16, Section 16.1.4). The shock layer thickness is quite significant compared to the natural width of these zones, and it increases... 

Although the general phenomena and the qualitative results described in this section remain valid for any isotherm model, provided they are convex upward and do not intersect, the quantitative results of the shock layer theory presented in Chaptersl4 and 16 are valid only when the adsorption behavior of the mixture components is properly described by the competitive Langmuir isotherm model. The theory shows conclusively that, when the separation factor decreases, the shock layer thickness, hence the width of the mixed zone in the isotachic train, increases in proportion to oc + l)/ a — 1) (Eqs. 16.27a and 16.27b). At the same time, the column length required to reach isotachic conditions increases also indefinitely, as predicted by the ideal model. 

As a demonstration of the validity of this theoretical result, we show in Figure 12.7 a fully developed isotachic train of inosine, deoxyinosine, adenosine, and deoxyadenosine [11]. This figiue illustrates how strongly the thickness of the shock layer between bands in the isotachic train depends on the relative retention, and how widely the size of the mixed zones can vary with a — 1, as predicted by Eq. 16.27. The main features of this displacement chromatogram are (i) the very close plateau concentrations of the first two and of the last two com-... 

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