Unfavorable Isotherm

Levenspicl and K. B. Bischoff, Advances in Chemical Engineering, Vol. 4, 95. Academic Press, New York, 1963. 

DYNAMICS OF ADSORPTION COLUMNS Multiple-Transition Systems 

In the preceding chapter we restricted our consideration of adsorption column dynamics to isothermal or near isothermal systems containing no more than two components. Indeed, most of the discussion was further restricted to systems containing only one adsorbable species in an inert carrier or solvent. The distinguishing feature of such systems is that the concentration profile shows only a single transition or mass transfer zone. In many adsorption systems of practical interest the situation is more complicated because the column is run adiabatically rather than isothermally and there are commonly more than one adsorbable components present in the feed. In such systems the concentration profile comprises more than one mass transfer zone. 

At higher sorbate concentrations, and for nonisothermal systems this simple pattern of behavior no longer applies since, outside the Henry s law region, 

As in Section 8.1 we consider an element of the bed through which a stream containing concentration c,(z,r) of adsorbabie species / is flowing. Assuming that the flow pattern can be described by the axially dispersed plug flow model, the differential fluid phase mass balance equation for each component is 

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Fig. 11. (a) Equilibrium isotherm and (b) dimensionless equilibrium diagram showiag favorable, linear, and unfavorable isotherms. 

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.

Historically, isotherms have been classified as favorable (concave downward) or unfavorable (concave upward). These terms refer to the spreading tendencies of transitions in fixed beds. A favorable isotherm gives a compact transition, whereas an unfavorable isotherm leads to a broad one. 

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.]... 

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. 

The Brunauer type I is the characteristic shape that arises from uniform micro-porous sorbents such as zeolite molecular sieves. It must be admitted though that there are indeed some deviations from pure Brunauer type I behavior in zeoHtes. From this we derive the concept of the favorable versus an unfavorable isotherm for adsorption. The computation of mass transfer coefficients can be accompHshed through the construction of a multiple mass transfer resistance model. Resistance modehng utilizes the analogy between electrical current flow and transport of molecular species. In electrical current flow voltage difference represents the driving force and current flow represents the transport In mass transport the driving force is typically concentration difference and the flux of the species into the sorbent is resisted by various mechanisms. 

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). 

Reduced (dimensionless) parameters for some of the IE systems subjected to numerical calculations are shown in Table 1. Variants I through III (high selectivity, Kp /Kps > 1, of invading B ion or favorable isotherm) and variants I.e through Ill.e (low selectivity, Kp /Kpa < 1, of B ion or unfavorable isotherm) are listed in this Table. 

With the unfavorable isotherm and at the the relation > Dg, rapid migration of A ions out of the bead into the solution favors the sharpening of profiles (Fig. 3, Vg, curve Ill.e). However, for variant Ill.e the profiles are spread due to the effect of the selectivity factor (Kg gg > 1). 

In the favorable isotherm region for the B ion (where Kra rb 1) the slowest process is that for which the relation Dg = O.ID is applicable (Fig. 5, curve III, solid line). For an unfavorable isotherm the slowest... 

Figure 5 Calculated dependencies of exchange rate (Tq 5) versus seleaivity factor (KRj /Kgg) at various diffusion foctors (Dj /Dg) including variants I through III and I.e through Ill.e K, ylCRB>l—favorable isotherm (solid lines) KRy /KRB

The different effect of diffusivity ratio on kinetic rate for favorable and unfavorable isotherms can be visualized as follows the low mobility of displaced A ions leads to its accumulation in the resin bead and a lower ratio of free-ion concentrations (Cg/C ). The relative variation of concentration as the ratio varies is considerably smaller in the case of the favorable isotherm than the unfavorable isotherm and in the long run may result in variation of the kinetic rate as shown in Fig. 5. 

Equations for Highly Favorable or Highly Unfavorable Isotherms... 155... 

The slope of a favorable isotherm is thus a decreasing function, while the slope of an unfavorable isotherm is an increasing function, of c. It will be shown later that the equilibrium constants describing the isotherms are numerically larger in favorable than in unfavorable cases. In fixed-bed adsorption, favorable equilibria lead to relatively sharp concentration gradients in the direction of flow, while unfavorable equilibria lead to more diffuse boundaries. 

Generally speaking, favorable isotherms are promising for uptake. In contrast, adsorbents that exhibit unfavorable isotherms would not perform well in most adsorption applications. 

An isotherm that is concave upward is called unfavorable because relatively low solid loadings are obtained and because it leads to quite long mass-transfer zones in the bed. Isotherms of this shape are rare, but they are worth studying to help understand the regeneration process. If the adsorption isotherm is favorable, mass transfer from the solid back to the fluid phase has characteristics similar to those for adsorption with an unfavorable isotherm,... 

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