Pattern Behavior for Favorable Isotherms

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.

FIG. 16-2 Limiting fixed-bed behavior simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodtigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dotdtecht, The Nethedands, 1989 reptinted withpeimission.]... 

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... 

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 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. 

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). 

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. ... 

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