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. 

Constant Separation-Factor Treatment If the valences of all species are equal, the separation factor Oti applies, where... 

For the constant separation factor case, the c contours in a plot hke Fig. 16-8 are hnear. 

Commonly used forms of this rate equation are given in Table 16-12. For adsorption bed calculations with constant separation factor systems, somewhat improved predictions are obtained using correction factors f, and fp defined in Table 16-12 is the partition ratio... 

Alternate driving force approximations, item 2B in Table 16-12, for solid diffusion, and item 3B in Table 16-12, for pore diffusion, provide somewhat more accurate results in constant pattern packed-bed calculations with pore or solid diffusion controlling for constant separation factor systems. 

FIG. 16-14 Constant separation factor batch adsorption curves for external mass-transfer control with an infinite fluid volume and n j = 0. 

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. 

For constant-separation factor systems, the li-transformation of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AIChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Freuz and Hoi-vath [AIChE J., 31, 400 (1985)] and are based on the h-transformatiou. They refer to the separation of a mixture of M — 1... 

In general, for a constant separation factor system, complete separation is obtained when ... 

Table 10 Values of pAa s, extraction constants, separation factors, and calculated point charges in...

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... 

Series, No. 107, Nijhoff, Dordrecht, 1986, pp. 119-1451. Constant Separation-Factor Treatment If the valences of all species are equal, the separation factor atJ applies, where atJ = KtJ= Z (16-46) cjn/ FIG. 16-8 Ideal mass-action equilibrium for three-component ion exchange with unequal valences. K AC = 8.06 K B C = 3.87. Duolite C-20 polystyrenesul-fonate resin, with Ca as A, Mg as B, and Na as C. [Klein et al., Ind. Eng. Chem. Fund., 6, 339 (1967) reprinted with permission.]... 

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... 

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. 

The use of a constant separation factor is a reasonable approximation if the actual factor does not vary significantly. Film thickness for spherical particles is related to the Sh number via the relation... 

Calculations Permeability Constants, Separation Factors and Scaling of Results for System Design... 

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