Shock layer theory

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]. 

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. 

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

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... 

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]... 

The solution of this system of equations is a mathematical problem similar to the one encoimtered in the study of the propagation of shock layers in compressible fluids. The shock layer theory developed by von Mises [5] and by Gilbarg [6] can be applied. The treatment is similar to the one previously discussed in Chapter 14, in the case of a single component. The concentration profiles in the shock are given by the system of two nonlinear differential equations (i = 1,2)... 

Figure 18.25 Comparison of experimental and theoretical results. Optimization of the dis-placer concentration for maximum production rate xmder isotachic train conditions, (a) Experimental results. Plot of the maximum production rate versus the normalized breakthrough time of the displacer, (b) Optimum calculated with the shock layer theory. F = 0.416 Dl = 0.000023 cm /s fcy = 0.33 s. Langmuir isotherm coefficients (and k values) k y = 1.5, = 2.5, k 2 = 4.0, = 6.0 k = 9.0 = 0.04 bz = 0.07 bg = 0.12 bi = 0.18 b =...

CONSTANT PATTERN, SELF-SHARPENING EFFECT, SHOCK-LAYER THEORY... 

An equivalent analysis of the combined effects of axial dispersion and mass transfer resistance has been presented by Rhee and Amundson, based on shock layer theory. From the mass balance over the shock layer it may be shown that the propagation velocity [Eq. (8.13)] is not affected by mass transfer resistance or axial dispersion. For an equilibrium system with axial dispersion the differential mass balance [Eq. (8.1)] becomes, under constant pattern conditions. 

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