Isotherms constant separation factor

FIG. 16-7 Constant separation factor isotherm as a function of the separation factor r (or interchangeably R). Each isotherm is symmetric about the perpendicular hne connecting (0,1) and (1,0). Isotherms for r and 1/r are symmetric about the 45 hne. 

The treatment here is restricted to the Langmuir or constant separation factor isotherm, single-component adsorption, dilute systems, isothermal behavior, and mass-transfer resistances acting alone. References to extensions are given below. Different isotherms have been considered, and the theory is well understood for general isotherms. 

TABLE 16-13 Constant Pattern Solutions for Constant Separation Factor Isotherm (R < 1)... 

Constant pattern solutions for the individual mechanisms and constant separation factor isotherm are given in Table 16-13. The solutions all nave the expected dependence on R—the more favorable the isotherm, the sharper the profile. 

When the adsorption equihbrium 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. Wmen R > 1 (unfavorable), the opposite is true. 

FIG. 16-35 EluHon curves under trace conditions with a constant separation factor isotherm for different feed loadings and N = 80. Solid lines, rate model dashed line, local equilihrium theory for Xp= 0.4. 

This term is analogous to relative volatility or its reciprocal (or to an equilibrium selectivity). Similarly, the assumption of a constant separation factor is a useful assumption in many sorptive operations. [It is constant for the Langmuir isotherm, as described below, and for mass-action equilibrium with za = zh in Eq. (16-24).] This gives the constant separation factor isotherm... 

The Langmuir isotherm, Eq. (16-13), corresponds to the constant separation factor isotherm with... 

Example 10 Transition Types For the constant separation-factor isotherm given by Eq. (16-31), determine breakthrough curves for r = 2 andr = 0.5 for transitions from cf=0 to cf = 1. 

In general, full time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been obtained for nonlinear isotherms. Only for reaction kinetics with the constant separation factor isotherm has a full solution been found [Thomas, J. Amer. Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phys. Colloid Chem., 54, 812 (1950) Hiester and Vermeulen, Chem. Eng. Progress, 48,505 (1952) Gilliland and Baddour, Ind. Eng. Chem., 45, 330 (1953) Vermeulen, Adv. in Chem. Eng., 2,147 (1958)]. The solution to Eq. (16-130) for item 4C in Table 16-12 for the same initial and boundary conditions as Eq. (16-146) is... 

For constant-separation factor systems, the /(-I rails formal ion of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AlChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Frenz and Horvath [AlChE ]., 31, 400 (1985)] and are based on the h I ransiomialion. They refer to the separation of a mixture of M — 1 components with a displacer (component 1) that is more strongly adsorbed than any of the feed solutes. The multicomponent Langmuir isotherm [Eq. (16-39)] is assumed valid with equal monolayer capacities, and components are ranked numerically in order of decreasing affinity for the stationary phase (i.e., Ki > K2 > Km). 

That is why La is also called the separation factor. Furthermore, it is clear that the constant separation factor aA B in an ion-exchange system means that in practice, this system obeys a Langmuiran equilibrium isotherm, in which La is of course constant. 

Let us first focus on a nonreactive system with constant separation factors. Typical examples are distillation processes with constant relative volatilities or adsorption processes described by competitive Langmuir isotherms. For nonreactive systems with constant separation factors, the constant pattern waves and spreading waves are... 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlhng. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve very slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant The driving force is written for a constant separation factor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system Iw its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

---