Page 68 - Separation process principles 2

2.2 Phase Equilibria 33

Table 2.3 Useful Expressions for Estimating K-Values for Vapor-Liquid Equilibria (Ki - yi/xi)
Equation Recommended Application

Hydrocarbon and light gas mixtures from cryogenic
temperatures to the critical region

(2) Activity coefficient All mixtures from ambient to near-critical temperature

Approximate forms:
Pi"
K. - - Ideal solutions at near-ambient pressure
(3) Raoult's law (ideal)
'- P
?ILP,S
(4) Modified Raoult's law K, = - Nonideal liquid solutions at near-ambient pressure
P
(5) poynting correction Kl = nibv (z) (A ( dP) Nonideal liquid solutions at moderate pressure and
exp
V,L
below the critical temperature
(6) Hemy's law Low-to-moderate pressures for species at supercritical
temperature

Since 1960, (2-27) has received considerable attention with required (i.e., vapor pressure, activity coefficient, and fugac-
applications to important industrial systems presented by ity coefficients). For practical applications, the choice of K-
Chao and Seader (C-S) [9], with a modification by Grayson value formulation is a compromise among considerations of
and Streed [lo]. accuracy, complexity, convenience, and past experience.
Table 2.3 is a summary of useful formulations for estimat- For liquid-liquid equilibria, (2-9) becomes
ing K-values for vapor-liquid equilibrium. Included are the
two rigorous expressions given by (2-26) and (2-27), from
which the other approximate formulations are derived. The where superscripts (1) and (2) refer to the two immiscible
so-called Raoult's law or ideal K-value is obtained from liquid phases. A rigorous formulation for the distribution
(2-27) by substituting from Table 2.2, for an ideal gas and coefficient is obtained by combining (2-23) with (2-20) to
ideal gas and liquid solutions, yiL = 1.0, hL = Pf/P, and obtain an expression involving only activity coefficients:
$i = 1 .O. The modified Raoult's law relaxes the assumption
of an ideal liquid solution by including the liquid-phase ac-
tivity coefficient. The Poynting-correction form for moderate
pressures is obtained by approximating the pure-component
liquid fugacity coefficient in (2-27) by the expression
For vapor-solid equilibria, a useful formulation can be
derived if the solid phase consists of just one of the compo-
nents of the vapor phase. In that case, the combination of
(2-9) and (2-25) gives
where the exponential term is the Poynling factor or cor-
rection. If the liquid molar volume is reasonably constant
over the pressure range, the integral in (2-28) becomes
U,L(P - Pf). For a light gas species, whose critical tempera- At low pressure, $iv = 1.0 and the solid fugacity can be ap-
ture is less than the system temperature, the Henry's law form proximated by the vapor pressure of the solid to give for the
for the K-value is convenient provided that a value of Hi, the vapor-phase mole fraction of the component forming -the
empirical Henry's law coefficient, is available. This constant solid phase:
for a particular species, i, depends on liquid-phase composi-
tion, temperature, and pressure. As pointed out in other chap-
ters, other forms of Henry's law are used besides the one in
Table 2.3. Included in Table 2.3 are recommendations for the For liquid-solid equilibria, a similar useful formulation
application of each of the vapor-liquid K-value expressions. can be derived if again the solid phase is a pure component.
Regardless of which thermodynamic formulation is used Then the combination of (2-9) and (2-23) gives
for estimating K-values, the accuracy depends on the partic-
ular correlations used for the thermodynamic properties