Page 78 - Separation process principles 2

2.5 Nonideal Thermodynamic Property Models 43

engineers because its range of application is too narrow. corresponding to the vapor phase-that is, Zv-and the
However, its development did suggest that all species might smallest Z (smallest v) corresponding to the liquid phase-
have equal reduced molar volumes, v, = vlv,, at the same that is, ZL. The intermediate value of Z is of no practical use.
reduced temperature, x. = TIT,, and reduced pressure, To apply the R-K equation to mixtures, mixing rules are
p, = PIP,. This finding, referred to as the law (principle or used to average the constants a and b for each component in
theorem) of corresponding states, was utilized to develop the the mixture. The recommended rules for vapor mixtures of C
equation of state given as (2) in Table 2.5. That components are
equation defines the coinpressibility factol; Z, which is a
function of P,, T,, and the critical compressibility factor, Z,,
or the acentric factol; w, which is determined from experi-
mental P-V-T data. The acentric factor, introduced by Pitzer
et a]. [17], accounts for differences in molecular shape and is
determined from the vapor pressure curve:

EXAMPLE 2.5
Glanville, Sage, and Lacey [20] measured specific volumes of
This definition results in a value for w of zero for symmetric
vapor and liquid mixtures of propane and benzene over wide
Some typical values of w are 0.264, 0.490, and ranges of temperature and pressure. Use the R-K equation to esti-
0.649 for toluene, n-decane, and ethyl alcohol, respectively, mate specific volume of a vapor mixture containing 26.92 wt%
as taken from the extensive tabulation of Poling et al. [ll]. propane at 400°F (477.6 K) and a saturation pressure of 410.3 psia
In 1949, Redlich and Kwong [I81 published an equation (2,829 kPa). Compare the estimated and experimental values.
of state that, like the van der Waals equation, contains only
two constants, both of which can be determined from T, and
SOLUTION
PC, by applying the critical conditions
Let propane be denoted by P and benzene by B. The mole fractions
($), are
0
(g)Tc and = o
=
However, the R-K equation, given as (3) in Table 2.5, is a
considerable improvement over the van der Waals equation.
A study by Shah and Thodos [19] showed that the simple
R-K equation, when applied to nonpolar compounds, has an
The critical constants for propane and benzene are given by Poling
accuracy that compares quite favorably with equations con- et al. [ll]:
taining many more constants. Furthermore, the R-K equa-
tion can approximate the liquid-phase region.
If the R-K equation is expanded to obtain a common Propane Benzene
denominator, a cubic equation in v results. Alternatively,
(2) and (3) in Table 2.5 can be combined to eliminate v to
give the compressibility factor, Z, form of the R-K equation:

From the equations for the constants b and a in Table 2.5 for the
R-K equation, using SI units,
where

Equation (2-46), which is cubic in Z, can be solved ana-
Similarly,
lytically for three roots (e.g., see Perry's Handbook, 7th ed.,
bB = 0.08263 m3/kmol
P. 4-20). In general, at supercritical temperatures, where
only one phase can exist, one real root and a complex conju- a~ = 2,072 kPa-m6/kmo12
gate pair of roots are obtained. Below the critical tempera-
From (2-50),
ture, where vapor and/or liquid phases can exist, three real
roots are obtained, with the largest value of Z (largest v)