Equilibrium isotherm favorable

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

The equilibrium isotherms were favorable type and the Langmuir equation represents our experimental data very well. 

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-... 

FIGURE 6 (a) Equilibrium isotherms and (b) dimensionless equilibrium diagram showing distinction between favorable, unfavorable, and linear systems. (Reprinted with permission from Ruthven, D. M. (1984). Principles of Adsorption and Adsorption Processes, copyright John Wiley Sons, New York.)... 

The distance required to approach the constant pattern limit decreases as the mass transfer resistance decreases and the nonlinearity of the equilibrium isotherm increases. However, when the isotherm, is liiglily favorable, as in many adsorption processes, this distance may be very small, a few centimeters to perhaps a meter. 

Favorable and unfavorable equilibrium 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 agreement that was observed between the experimental results and the prediction of a competitive Langmuir model based on the use of single-component Langmuir isotherms in the case of the adsorption of enantiomeric derivatives of amino acids on immobilized serum albumin [26] is imusual. It demonstrates the validity of the competitive Langmuir model based on the use of the parameters of the single-component Langmuir model. However, as explained before, the experimental conditions are exceptionally favorable since the column saturation capacities for the two enantiomers are equal. Nevertheless, Zhou et ah have shown that it is possible, in certain favorable cases, to derive the equilibrium isotherms of the pure enantiomers and to calculate isotherm equilibrium data for any mixture of... 

Such an expression contains four independent constants, so it will obviously provide a better fit to experimental data than the simple two-constant Langmuir expression. However, such a model makes physical sense for systems such as the adsorption of polar (or quadrupolar) molecules on a cationic zeolite, where the most favorable sites are those associated with the exchangeable cations and the less favorable sites correspond to adsorption elsewhere on the framework or simply within the micropores. For example, it has been shown that the analysis of equilibrium isotherms for CO2 on various different forms of zeolite A yields site densities that are consistent with structural information [4]. 

The process of adsorption takes place when the concentration of the adsorptive is greater than the equiUbrium value vahd for the given temperature however, desorption requires a fluid concentration of the adsorptive which is smaller than the equilibrium concentratiom An adsorption isotherm favorable for adsorption is unfavorable for desorption and vice versa. Condensation of gases or vapors and solidification or crystalhzation will start when the relative supersaturation becomes > 1. In the case of adsorbents with capillary or very narrow pores, capillary condensation is observed for relative saturations adsorption isotherm vahd for adsorption and desorption can sometimes be ejqrlained, see Fig. 2.4-2. Sohd materials exposed to drying (see Chap. 10) often show such hysteresis behavior which can sometimes be explained by the ciu-vature of the liqttid sttrface in capillaries The radius of this surface is greater in the case of adsorption in comparison to the radius valid for a desorption process, see Fig. 2.4-2. 

The Henry s law constant provides a direct measure of the interaction between an adsorbed molecule and the most favorable adsorption sites on the surface. Although the form of the equilibrium isotherm at higher sorbate concentrations is sensitive to a number of complicating effects arising from sorbate-sorbate interaction and energetic heterogeneity, one may expect that,... 

These expressions provide a simple and convenient model for the analysis of experimental uptake curves when the equilibrium isotherm is highly favorable and micropore diffusion is rapid. These conditions are amply fulfilled for the adsorption of water at or near ambient temperature in molecular sieve adsorbents. Experimental uptake curves for this system measured by Kyte are shown in Figure 6.9. The experimental conditions and the effective diffusivity calculated according to Eq. (6.27) are giveii in Tables 6.4 and 6.5. Under the experimental conditions the estimated value of the Knudsen diffusivity [from Eq. (5.17)] is much larger than the molecular diffusivity... 

In order to improve the accuracy with which the breakthrough time can be predicted for the more general case of a favorable equilibrium isotherm, a more detailed mathematical model has been developed as outlined below. ... 

Figure 7.1.2(b) illustrates a few more adsorption equilibrium isotherms. For the isotherm identified as favorable, it is obvious that if two concentrations Cizh and C,2 2 are significantly apart and C,2 2 > Q2I1. then... 

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... 

When the adsorption equilibrium is nonlinear, skewed peaks are obtained, even when N is large. For a constant separation-factor isotherm with R < 1 (favorable), the leading edge of the chromatographic peak is steeper than the trailing edge. When R > 1 (unfavorable), the opposite is true. 

The parameter La is also called the separation factor and provides a quantitative description of the equilibrium regions La = 0 for irreversible, La< 1 for favorable, La = 1 for lineal-, and La > 1 for unfavorable adsorption. The same holds for Fr in Freundlich s isotherm. 

Cooney defined qmgx as the solid concentration in equilibrium with C0. This number is dimensionless, and thus the concentration units are the same for the liquid and solid phase (e.g. mg/g). Cooney pointed out that for Bi < 0.5, complete dominance of the liquid-film resistance exists, while for Bi > 30, reasonably complete dominance of intraparticle resistance exists. The only restriction is that the isotherm be favorable. 

Figure 17.20. Control of temperature in multibed reactors so as to utilize the high rates of reaction at high temperatures and the more favorable equilibrium conversion at lower temperatures, (a) Adiabatic and isothermal reaction lines on the equilibrium diagram for ammonia synthesis, (b) Oxidation of SOz in a four-bed reactor at essentially atmospheric pressure, (c) Methanol synthesis in a four bed reactor by the ICI process at 50 atm not to scale 35% methanol at 250°C, 8.2% at 300°C, equilibrium concentrations.

Constant Pattern Behavior In a real system the finite resistance to mass transfer and axial mixing in the column lead to departures from the idealized response predicted by equilibrium theory. In the case of a favorable isotherm the shock wave solution is replaced by a constant pattern solution. The concentration profile spreads in the initial region until a stable situation is reached in which the mass transferrate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-phase profiles become coincident. This represents a stable situation and the profile propagates without further change in shape—hence the term constant pattern. 

The packed bed breakthrough method for investigation of mass transfer phenomena in sorbent systems can in many instances offer certain advantages not found in other experimental methods. The method is especially useful when the adsorption isotherms for the principal sorbate exhibit favorable curvature (convex toward loading axis). In such a case, there is the potential for a portion of the sorption front to approach a stable wave form (shape of the front invariant with time). Given the existence of a stable or "steady-state" mass transfer zone (MTZ) and a detailed knowledge of the equilibrium loading characteristics within that zone, one can extract local values of the effective mass transfer resistance at any concentration in the zone. 

When the equilibrium constant b is large (highly favorable adsorption) the Langmuir isotherm approaches irreversible or rectangular form,... 

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). 

Adsorption Isotherms. Equilibrium dialysis studies indicate around 10 repeat VP units (base moles) are required to form favorable complexes (89,90). This figure can rise to several hundred for methyl orange and other anions depending on structure (91,92). 

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