Proportionate pattern behavior

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. 

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... 

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). 

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. ... 

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. 

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... 

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

In the examples selected for detailed discussion the profiles have all had the form of either simple proportionate-pattern waves or simple shocks. When there is an inflection in the characteristic a combined wave, consisting partly of a shock and partly of a simple wave, may occur (see Section 8.4, Figure 8.6). However such details do not affect the broad pattern of the behavior of the system. 

A pure thermal wave is always a shock and from Eq. (9.36) it is clear that its velocity is given by o/(l + [(1 - f)/i.]CjC ) while the velocity of the mass transfer front is given by o/ l +[(1 - e)/e]d(j dc. For the thermal wave to precede the concentration front (adsorption) therefore requires dq /dc o > Cs / Cf while for a pure thermal wave lagging the concentration front to be formed during desorption (dq /dc Q < CjCy. At the watershed point Tn), dq /dc o = CJC and the velocity of the limiting proportionate-pattern wave is the same as that of the thermal shock. A more detailed discussion of the conditions of thermal wave formation and the practical importance of this type of behavior has been given by Basmadjian. ... 

Morphogens are form-generating substances, whose configuration within a tissue prefigures the pattern (Meinhardt, 1983). These substances set up an extracellular concentration gradient, which in turns orchestrates a coherent set of cellular behaviors that will eventually result in the proportionate growth of an or-... 

LeVan MD (1989) Asymptotic fixed bed behavior proportionate and constant patterns. In Adsorption Science and Technology , vol 158, NATO-ASI Series. Kluwer, Amsterdam, pp 149-168... 

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