Page 79 - Separation process principles 2

44 Chapter 2 Thermodynamics of Separation Operations

From (2-49), Values of kii, back-calculated from experimental data, have
+ been published for both the S-R-K and P-R equations.
a = y;ap + 2ypy~(apa~)O.~ Y&B
Knapp et al. [22] present an extensive tabulation. Generally,
= (0.3949)'(836.7) + 2(0.3949)(0.6051)[(836.7)(2,072)]~.~
kd is taken as zero for hydrocarbons paired with hydrogen or
+ (0.6051)'(2,072) = 1,518 kPa-m6/kmo12
other hydrocarbons.
From (2-47) and (2-48) using SI units, Although the S-R-K and P-R equations were not in-
tended to be applied to mixtures containing polar organic
compounds, they are finding increasing use in such applica-
tions by employing large values of kij, in the vicinity of 0.5,
as back-calculated from experimental data. However, a
preferred procedure for mixtures containing polar organic
From (2-46), we obtain the cubic Z form of the R-K equation:
compounds is to use a more theoretically based mixing rule
such as that of Wong and Sandler, which is discussed in de-
tail in Chapter 11 and which bridges the gap between a cubic
Solving this equation gives one real root and a conjugate pair of
complex roots: equation of state and an activity-coefficient equation.
Another theoretical basis for polar and nonpolar sub-
Z = 0.7314, 0.1314 + 0.042431', 0.1314 - 0.042431'
stances is the virial equation of state due to Thiesen [23] and
The one real root is assumed to be that for the vapor phase. Onnes [24]. A common representation of the virial equation,
which can be derived from the statistical mechanics of the
From (2) of Table 2.5, the molar volume is
forces between the molecules, is a power series in l/v for Z:
ZRT (0.7314)(8.314)(477.59)
v=-- - = 1.027 m3/kmol
P 2,829
The average molecular weight of the mixture is computed to
An empirical modification of the virial equation is the
64.68 kghol. The specific volume is
Starling form [5] of the Benedict-Webb-Rubin (B-W-R)
equation of state for hydrocarbons and light gases in both the
gas and liquid phases. Walas [25] presents an extensive dis-
Glanville et al. report experimental values of Z= 0.7128 and
cussion of B-W-R-type equations, which because of the
VIM = 0.2478 ft3/lb, which are within 3% of the above estimated
large number of terms and species constants (at least 8), is not
values.
widely used except for pure substances at cryogenic tempera-
tures. A more useful modification of the B-W-R equation is a
Following the success of earlier work by Wilson [21],
generalized corresponding-states form developed by Lee and
Soave [6] added a third parameter, the acentric factor, w, de- Kesler [26] with an important extension to mixtures by
fined by (2-45), to the R-K equation. The resulting, so-called
Plocker et al. [8]. All of the constants in the L-K-P equation
Soave-Redlich-Kwong (S-R-K) or Redlich-Kwong-Soave
are given in terms of the acentric factor and reduced tempera-
(R-K-S) equation of state, given as (4) in Table 2.5, was
ture and pressure, as developed from P-V-T data for three
immediately accepted for application to mixtures containing simple fluids (w = O), methane, argon, and krypton, and a ref-
hydrocarbons and/or light gases because of its simplicity and
erence fluid (o = 0.398), n-octane. The equations, constants,
accuracy. The main improvement was to make the parameter
and mixing rules in terms of pseudo-critical properties are
a a function of the acentric factor and temperature so as to given by Walas [25]. The Lee-Kesler-Plocker (L-K-P)
achieve a good fit to vapor pressure data of hydrocarbons and equation of state describes vapor and liquid mixtures of
thereby greatly improve the ability of the equation to predict
hydrocarbons and/or light gases over wide ranges of temper-
properties of the liquid phase.
ature and pressure.
Four years after the introduction of the S-R-K equation,
Peng and Robinson [7] presented a further modification of
Derived Thermodynamic Properties
the R-K and S-R-K equations in an attempt to achieve im- from P-v-T Models
proved agreement with experimental data in the critical re-
gion and for liquid molar volume. The Peng-Robinson In the previous subsection, several useful P-V-T equations
(P-R) equation of state is listed as (5) in Table 2.5. The of state for the estimation of the molar volume (or density)
S-R-K and P-R equations of state are widely applied in or pure substances and mixtures in either the vapor or liquid
process calculations, particularly for saturated vapors and phase were presented. If a temperature-dependent, ideal-gas
liquids. When applied to mixtures of hydrocarbons and/or heat capacity or enthalpy equation, such as (2-35) or (2-36),
light gases, the mixing n~les are given by (2-49) and (2-50), is also available, all other vapor- and liquid-phase properties
except that (2-49) is often modified to include a binary inter- can be derived in a consistent manner by applying the classi-
action coefficient, kd: cal integral equations of thermodynamics given in Table 2.6.
c TC 1 These equations, in the form of departure (from the ideal
gas) equations of Table 2.4, and often referred to as residu-
als, are applicable to vapor or liquid phases.