Proportionate pattern

Fig. 12. (a) Development of the physically unreasonable overbanging concentration profile and the corresponding shock profile for adsorption with a favorable isotherm and (b) development of the dispersive (proportionate pattern) concentration profile for adsorption with an unfavorable isotherm (or for... 

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.

Favorable and unfavorable equihbrium isotherms are normally defined, as in Figure 11, with respect to an increase in sorbate concentration. This is, of course, appropriate for an adsorption process, but if one is considering regeneration of a saturated column (desorption), the situation is reversed. An isotherm which is favorable for adsorption is unfavorable for desorption and vice versa. In most adsorption processes the adsorbent is selected to provide a favorable adsorption isotherm, so the adsorption step shows constant pattern behavior and proportionate pattern behavior is encountered in the desorption step. 

The solution gives all of the expected asymptotic behaviors for large N—the proportionate pattern spreading of the simple wave if R > 1, the constant pattern if R < 1, and square root spreading for R = 1. 

As shown in Table III, mean fecal calcium losses tended to be higher when the higher fat diet was fed in comparison to results when the lower fat diet was fed. Therefore, apparent calcium absorption was higher when the low fat diet was fed. These differences were significant at only the P< 0.075 level hence, only a trend was illustrated. In this study no attempt was made to equalize fatty acid proportionality patterns or cholesterol intake. These or other dietary or non-dietary factors may have influenced the observed apparent trends. Other studies with human adults have not demonstrated any apparent influence on level of dietary fat on calcium absorption. 

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... 

In the case of an unfavorable isotherm (or equally for desorption with a favorable isotherm) a different type of behavior is observed. The concentration front or mass transfer zone, as it is sometimes called, broadens continuously as it progresses through the column, and in a sufficiently long column the spread of the profile becomes directly proportional to column length (proportionate pattern behavior). The difference between these two limiting types of behavior can be understood in terms of the relative positions of the gas, solid, and equilibrium profiles for favorable and unfavorable isotherms (Fig. 7). 

There is no published data on desorption of water vapour from a column by purging the column with a water free gas. However, such a desorption process produces a proportionate pattern MTZ within the column for a type I adsorption isotherm. The zones are generally governed by local adsorption equilibria within the column [18,19]. 

Analytical equations to describe isothermal proportionate pattern water desorption... 

B. Limiting Cases of Equilibrium Behavior 1. Proportionate-Pattern Case (Unfavorable Equilibrium)... 

The proportionate-pattern case is a classical one in the theory of chromatography, and was treated by DeVault (D2), Walter (Wl), Wilson (W7), and Weiss (W3). It is assumed that equilibrium is maintained everywhere in the column, that is, that N approaches infinity, due to high mass-transfer rates or to long residence times. 

The limits of validity are x = 0 at Z = 1/r x = 1 at Z = r. This is the desired concentration-history equation. It provides a proportionate pattern, because x depends upon Z only and not upon N or v. In this case the relative sharpness of the breakthrough curve cannot be increased by lengthening the column. 

It is apparent that Eq. (121) contains the J function as a limiting case, at r = 1. This equation also has been shown to reduce to the constant-pattern result (Eq. 96) with r mathematical viewpoint, and has presented limiting forms which give an accurate approximation in certain regions his variables u, s, and y correspond respectively to the present x, N, and ZN. 

The attention of many workers has been given to the equilibrium-limited case (proportionate-pattern) of multiple adsorption (D2, Wl, W3, W7, among others). In the constant-pattern case, Fujita s work has already been discussed (Section III, B, 2b). Also, using the theoretical-plate approximation to a packed column, plate-by-plate calculations can be carried out in the constant-pattern case exactly as for continuous (countercurrent) distillation this treatment is suggested from work on chromatographic and displacement problems by Mair (M2), Spedding (S6), and Glueckauf (G3). Moreover the linear-equilibrium result can be extended, in a nearly trivial fashion, to any number of components. 

FIGURE 83 (a) Development of physically unrealistic overhanging concentration profile and corresponding shock profile for a favorable isotherm fi< l.O). (6) Development of dispersive (proportionate-pattern) concentration profile for desorption of a uniformly saturated bed when the isotherm is favorable for adsorption, unfavorable for desorption < 1.0 > 1.0) (after... 

In most adsorption systems the isotherm is favorable for adsorption and therefore unfavorable for desorption. In desorption the mass transfer zone is therefore dispersive, leading to a continuously spreading concentration prOfife (proportionate-pattern behavior) while in adsorptioii the mass transfer zone is compressive, leading to constant-pattern behavior. For example, for a system which obeys the Langmuir isotherm (Eq. (8.6)] I... 

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. ... 

---