Constant pattern behavior approach

Fig. 13. Schematic diagram showing (a) approach to constant pattern behavior for a system with a favorable isotherm and (b) approach to proportionate pattern behavior for a system with an unfavorable isotherm, jy axis cj qlj q,----------------------- < q,-- From ref. 7.

Proportionate Pattern Behavior. If the isotherm is unfavorable (as in Fig. 1,111), the stable dynamic situation leading to constant pattern behavior can never be achieved. The equilibrium adsorbed-phase concentration then lies above rather than below the actual adsorbed-phase profile. As the mass transfer zone progresses through the column it broadens, but the limiting situation, which is approached in a long column, is simply local equilibrium at all points (c = c ) and the profile therefore continues to... 

FIGURE 7 Schematic diagram showing (a) approach to constant-pattern behavior for a system with favorable equilibrium and (b) approach to proportionate-pattern limit for a system... 

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]. 

By comparing the concentration profiles derived from the full solution to the differential model equations with the asymptotic profiles calculated from the constant-pattern condition it is possible to determine the dimensionless distance required to approach constant-pattern behavior. The results of such calculations are summarized in Figure 8.21. 

The asymptotic behavior of transitions under the influence of mass-transfer resistances in long, deep beds is important. The three basic asymptotic forms are shown in Fig. 16-2. With an unfavorable isotherm, the breadth of the transition becomes proportional to the depth of bed it has passed through. For the linear isotherm, the breadth becomes proportional to the square root of the depth. For the favorable isotherm, the transition approaches a constant breadth called a constant pattern. 

At sufficiently large values of X the saturation curves approach a constant pattern form, and thereafter the concentration front progress through the columii at a steady velocity, governed by the capacity of the adsorbent and the feed concentration, with no further change in the shape of the curve. Such behavior is characteristic of systems with a favorable equilibrium isotherm (12). The constant pattern limit is reached when the dimensionless concentration profile in fluid phase and adsorbed phase become practically coincident, and the asymptotic form of the break-... 

Representative adsorption and desorption curves are shown in Figure 8.17. The curves for the various models are qualitatively similar and show the same general trends. When the isotherm approaches linearity ()8-> 1.0) the adsorption and desorption curves become mirror images and coincide with the theoretical curve calculated from Rosen s analysis. The adsorption curves for the nonlinear system show the expected approach to the constant-pattern form. As the isotherm becomes increasingly favorable (/8->0) the distance required to approach the constant-pattern limit decreases and the form of the constant-pattern breakthrough curve approaches eventualljr the form calculated for an irreversible isotherm (Table 8.3). Meanwhile the desorption curves approach proportionate-pattern behavior so a pronounced asymmetry develops between adsorption and desorption curves. ... 

The variation of the temperatures of the midpoints of the peaks related to water (bound, interphasal, and free) as a function of water content followed the same pattern a gradual increase of the temperature to a less negative value and then a more or less constant temperature. Such behavior was observed for systems A [10], B [45], D [42], and E [42] for the binary system water-Ci2(EO)8 [45] and for aqueous solutions of polymers (here as a function of the sorbed water content) such as poly(4-hydroxystyrene) [40] and mucopolysaccharides [83], For example, system A has an endothermic peak at about -10°C, which is ascribed to the melting of interphasal water (and dodecane). In fact, it begins at about - 12°C, increases in height with increasing water content, and levels off at about - 10°C, when (total) water content approaches 30 wt%. For the binary system water + Ci2(EO)s, interphasal water melts between -3 and -4°C [45,84],... 

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