Self-sharpening

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.)... 

The switch from feed to displacer changes M roots, generating M boundaries. All of these boundaries are self-sharpening, except the boundary associated with the transition from to Oti that can be self-sharpening or diffuse depending on the relative value of these... 

The adjusted propagation velocity of each self-sharpening boundary is ... 

The switch from carrier to feed generates self-sharpening boundaries, since the h s (upstream) are larger than the h s (downstream). The switch from feed to displacer generates two self-sharpening... 

FIG. 16 36 Dimensionless time-distance plot for the displacement chromatography of a binary mixture. The darker lines indicate self-sharpening boundaries and the thinner lines diffuse boundaries. Circled numerals indicate the root number. Concentration profiles are shown at intermediate dimensionless column lengths = 0.43 and = 0.765. The profiles remain unchanged for longer column lengths. 

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. 

Narahari, C.R., Strong, J.C., and Frey, D.D., Displacement chromatography of proteins using a self-sharpening pH front formed by adsorbed buffering species as the displacer, /. Chromatogr. A, 825, 115, 1998. 

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. 

This work prompted a flurry of activity in the mid- to late 1980s to find the type IM isotherm. A number of inventions can be found in which alumina, or silica gel are blended with zeolites type X or Y to mimic the shape of the isotherm that Collier defined. Mol Sieve type DDZ-70(g) is in fact one of only a few true type IM isotherms. This product and Engelhard s type ETS-10 both have the required isotherm shape for water and deliver the benefits expected, to wit excellent capacity for water, self-sharpening mass transfer zone and low energy investment required to regenerate. Mol Sieve type DDZ-70(g) is used commercially in rotors... 

Liquid-solid chromatography is representative of this theory because nonlinearity effects are usually appreciable. Mass transfer is fast and longitudinal diffusion effects may be ignored in describing the system. The net result is that the bands (zones) develop self-sharpening fronts and diffuse rear boundaries. It is because of this tailing that this technique is to be regarded... 

It is evident from the above expressions that the appropriate diffusion coefficient must also be measured in order that molecular or particle masses may be determined from sedimentation velocity data. In this respect, a separate experiment is required, since the diffusion coefficient cannot be determined accurately in situ, because there is a certain self-sharpening of the peak due to the sedimentation coefficient increasing with decreasing concentration. 

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