Self-sharpening waves

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be self-sharpening. (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded oed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.)... 

Equation 17.75 is important as it illustrates, for the equilibrium case, a principle that applies also to the non-equilibrium cases more commonly encountered. The principle concerns the way in which the shape of the adsorption wave changes as it moves along the bed. If an isotherm is concave to the fluid concentration axis it is termed favourable, and points of high concentration in the adsorption wave move more rapidly than points of low concentration. Since it is physically impossible for points of high concentration to overtake points of low concentration, the effect is for the adsorption zone to become narrower as it moves along the bed. It is, therefore, termed self-sharpening. 

As is the case with adsorption isotherms, those curves in Figure 18.3 which are concave to the concentration axis for the mobile phase are termed favourable and lead to self-sharpening ion exchange waves. 

Constant pattern or self-sharpening front—usually a sharp, syimnetric transition observed during uptake under favorable (Type I) equilibrium. The physical limit (without mass transfer effects) is called a shock wave. 

F. Generalize. The single shock wave that results when two steps are input occurs because of the self-sharpening behavior of shock waves. Diffuse waves can also be sharpened if they are followed with a shock wave. This phenomenon is used in commercial cycles. The regeneration cycle will require much less purge if an adsorbate tail (called a heel) is left in the column (see Figure 18-16B1. This diffuse wave is then sharpened when the next feed step forms a shock wave. 

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