Shock layer thickness

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]. 

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. 

Properties of the Shock Layer Thickness in Frontal Analysis.662... 

If 6 = 0.05, for example, the shock layer will be the region where 90% of the total concentration variation takes place. The shock layer thickness is the distance between the two corresponding points on the breaktlu-ough curve... 

In this section, we discuss the consequences of Eq. 14.33a and the experimental results reported by Zhu et al. [19,20] regarding the dependence of the shock layer thickness (SET) on the main experimental parameters, the mobile phase velocity and the height of the concentration step. 

It is possible also to derive from D an apparent plate number, N = Lu/ 2Da). These data are shown in Figure 14.8. From these results as well as from those of Zhu et al. [19,20] discussed above, we conclude that there is a good agreement between the predictions of the shock layer theory and the experimental measurements of the shock layer thickness, but that it seems that the less valid one of the several assumptions made in the derivation of the shock layer theory is the assumption that kf is independent of the concentration. A significant variation of kf with the component concentration would explain most of the deviations reported. These results are in agreement with the conclusions of the comparisons... 

Because it is possible to calculate the shock layer thickness with a lumped kinetic model and with the equilibrium-dispersive model, a comparison of these two expressions provides an attractive method of investigation of the range of validity of the latter model. In the equilibrium-dispersive model, the apparent dispersion coefficient is assumed to be given by the equation... 

If the shock layer theory is applied within the framework of the equilibrium-dispersive model (Eqs. 14.44 and 14.45), the shock layer thickness becomes [17,19]... 

Jaulmes and Vidal-Madjar [51] studied the influence of the mass transfer kinetics on band profiles, using a Langmuir second-order kinetics, and a constant axial dispersion coefficient, D. They derived numerical solutions using a finite difference algorithm. The influence of the rate constant on the band profile at various sample sizes is illustrated in Figure 14.18. As the mass transfer kinetics slows down, the band broadens and the shock layer becomes thicker. When the sample size increases, however, the influence of thermodynamics on the profile becomes more dominant, as shown by the change in shock layer thickness which decreases with increasing sample size. 

Shock Layer Thickness in Binary Frontal Analysis.740... 

In the case of two components, the shock layer thickness is defined as in the singlecomponent case (Figure 14.3), with the help of threshold concentrations CJ and Cy, j = 1, 2), and the auxiliary parameter 6 (Eq. 14.21). hi displacement chromatography, there is a shock layer on each side of the isotachic bands, and we need to distinguish C (band rear) and cj (band front), and c. In the case... 

Shock Layer Thickness Controlled by Axial Dispersion... 

For equal Peclet numbers and infinitely fast mass transfers, the profQe of the shock layer thickness is given by the following equation [3] ... 

Assuming that the contributions of axial dispersion and mass transfer resistances are additive, the breakthrough profile of the first component is given by the same equation as in the single-component case, but with a higher plateau concentration, Cf (Eq. 8.12a). Accordingly, the shock layer thickness for this... 

Figure 16.5 shows the dependence of the SLT on the separation factor oc = of the two components [11]. The SLT increases dramatically as oc decreases toward unity. It tends slowly toward 0 with increasing value of oc. Thus, it is no more possible to separate a binary mixture with a very small value of oc (e.g., components differing by an isotopic substitution) by displacement than by overloaded elution. The shock layer thickness in the isotachic train would be so large that the shock layer would encompass the whole bands, and the time required to achieve isotachic conditions would result in a very low throughput. The isotachic train... 

Efficiency, N The column efficiency characterizes the combined effects of the sources of band broadening due to axial dispersion and mass transfer resistance. It is derived from the width of the elution peak observed as the response to the injection of a small, narrow pulse of a dilute solution of a compoimd. It is difficult to correct for the contribution of the extracolumn sources of band broadening which have to be kept small. In preparative and nonlinear chromatography, there is a correlation between the colmnn efficiency and both the steepness of the shock layer and the duration of the band beyond the retention time However, the column efficiency is essentially a concept of linear chromatography, and it is difficult to extend to and use in nonlinear chromatography, except through the shock layer thickness concept. 

Shock Layer Thickness, SLT Distance (in time, space or dimensionless coordinates) between the two points of the constant-pattern breakthrough profile where the concentration is equal to an arbitrary fraction of the concentration step. 

---