Equilibrium isotherm unfavorable

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

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.)... 

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. 

Since the method depends on constant-pattern behavior it should never be applied to a system in which equilibrium isotherm is unfavorable and it clearly does not provide any information concerning the required regeneration conditions. 

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. 

In the special case of ion exchange and unfavorable equilibrium, i.e. aA B < 1, with A originally in the solution, under the condition of sufficiently long bed, Walter s solution could be used. Walter s equation is a special case of the Thomas model for arbitrary isotherm and the kinetic law equivalent to a reversible second-order chemical reaction (Helfferich, 1962) ... 

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). 

Not all adsorption beds will develop stable MTZs. One requirement for stability (i.e., the MTZ reaches a limiting size) is that the equilibrium line must be favorable. In the case of a single adsorbate isothermally removed from a non-adsorbable component, the curve of loading as a function of composition must be concave downward in the region of loading below the stoichiometric point to be favorable. This effect is described in more detail in Section 7.9. In non-isothermal adsorption it is possible for the temperature effects to cause a favorable isotherm to become an unfavorable equilibrium line. This was discussed previously in the context of the crossover ratio R. 

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. 

Isotherms having an inflection point correspond to ranges of both favorable and unfavorable equilibrium. Since separation operations only involve the part of the isotherm which lies below the feed-concentration, the occurrence of an inflection point corresponding to a higher concentration has no effect. If the inflection point occurs at a lower concentration, fixed-bed separation calculations must be carried out by approximate methods to be described below. 

Algebraic relations for equilibria that are partly favorable and partly unfavorable are included in Section II, A, 2. Such relations can sometimes be used for separation calculations where only the highly favorable part of the isotherm, or only the highly unfavorable part, is involved. In all other such cases the equilibrium data can be utilized numerically, by dividing the concentration range into several sections and determining an average r for each interval. The use of non-constant r values will be described in Section III, D, 4. 

These deaggregated particles are in the state of partial equilibrium their aggregation is thermodynamically unfavorable, while isothermal mass transfer and coalescence, leading to a decrease in degree of dispersion, are possible. At the same time, if isothermal mass transfer does not occur within realistic observation time periods, the equilibrium from partial turns into complete the... 

All regression fits have nearly the same correlation coefficient so that cannot be used to determine which is the best fit. However, the Langmuir isotherm gives a negative value for K. If K is to represent an equilibrium constant, which must be positive, the Langmuir description must be rejected. The standard deviation of the slope of the Freundlich isotherm is twice as large as the slope itself. This would seem to be unfavorable. Thus, the linear description seems to be the best. but not excellent choice. However, the Freundlich isotherm is usually preferred for this kind of system, even though that choice is not supported by the data in this case. 

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. 

When the 2.5 N sodium chloride solution is fed to the column, the copper equivalent fraction, and concentration in the feed drop to zero. Since the column is at 2.5 N when the sodium wave reaches any part of the column, we use that equilibrium curve in Figure 18-19. This is an unfavorable isotherm for copper thus, with a drop in copper concentration in the feed a shock wave results. Then use Eq.( 18-461 to calculate the shockwave velocity. 

Since an increase in the temperature from i,a to i.co occurs, this causes a decrease in the solvent absorption capacity of the soluted substance according to Henry s law. The absorption efficiency reduces due to an unfavorable equilibrium position at the higher temperature, (see Chapter 1.4.3.3). Consequently, isothermic absorption is aimed for in practice. The heat flux to be removed under isothermal absorption at (t / a = di u) follows from Eq. (3-7). 

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