Tsonopoulos
Tsonopoulos
Tsonopoulos
261
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ABSTRACT
Tsonopoulos, C. and Heidman, J.L., 1990. From the virial to the cubic equation of state.
Fluid Phase Equilibria, 51: 261-276.
Second virial coefficients are used extensively in low-pressure vapor-liquid equilibrium
calculations to predict the vapor-phase fugacity coefficient. Tsonopouloss 1974 B (second
virial coefficient) correlation is reviewed, with particular emphasis on the dependence of the
polar parameter a on the reduced dipole moment. New results for the B of water are
analyzed and the 1974 recommendation is discarded in favor of a = -0.0109. New data also
make it possible to correlate the characteristic constant kij for water/n-alkane binaries with
the critical volume of the hydrocarbon. For calculations at reduced densities significantly
higher than 0.25, B is combined with the Redlich-Kwong equation to form the Virial-RK
equation of state. This equation predicts reasonably good estimates for the third virial
coefficient of non-polar gases, such as methane, but the predictions for water are very poor.
However, the fugacity coefficients predicted by virial-RK for the water/methane binary are
reliable up to very high pressures, except in the vicinity of the critical point.
INTRODUCTION
(1)
That is, the fugacity of component
* Paper presented at a special symposium to celebrate the 60th birthday of Professor John M.
Prausnitz at the AIChE Annual Meeting, Washington, DC, November 30, 1988.
037%3812/90/$03.50
262
(2)
xi
(3)
(py
Equation (3) provides the simplest and most direct method for VLE calculations-it
requires no standard states or special procedures as the critical
point is approached-provided
the equation of state applies to both phases.
Some equations do well, if we exclude strongly polar compounds and
electrolytes.
For mixtures containing strongly polar compounds or electrolytes, VLE
calculations are commonly carried out with hybrid models; that is, models
which use activity coefficients for the liquid and fugacity coefficients for the
vapor, which are calculated with an equation of state. In that case, eqn. (2) is
replaced by
&YiP = YiXif,O
(4
Z=jp=l+F
temperature,
(7)
263
(8)
yiBii - ln 2,
+y* = -242 u2
ln
(9)
22
B CORRELATION
00)
where
f( T,) = 0.1445 - 0.330/T, - 0.1385/q2
- 0.0121/T,3 - 0.000607/T,*
- 0.008/T,*
- 0.423/q3
(12)
03)
264
04
(15)
However, for alkyl halides, mercaptans, sulfides, and disulfides, the bottom
curve (2) in Fig. 1 should be used.
(16)
-0.09
-0 08
0.07
-0
06
-0.05
a
-0
Ketones,
Aldehydes,
Alkyl Nitriles.
Ethers,
Corboxylic
Acid Esters
04
50
150
100
REDUCED
DIPOLE
MOMENT,
200
pr
250
265
Equations (15) and (16) present a satisfactory correlation for most nondimerizing polar organic compounds. These two equations approach closely
at high k, suggesting that a unique relationship between a and pr is
possible only for strongly polar, but non-dimerizing, organic compounds.
Second virial coefficient of water
Figure 2 presents all the B data for water reported by Cholinski et al.
(1986) along with the recent measurements of Eubank et al. (1988) who
reported two sets of data. Also shown in Fig. 2 are the calculations with the
1974 correlation; first, with f (2) = 0 (i.e. without a polar term), then with the
1974 recommendation
f (2) = 0.0279/q6
- 0.0229/T8
07)
08)
Eubank(l988)
Eubank(l988)
Vukolovich
Collins
Kell (1968)
1,
(1975)
McCullough
Noppe
LeFevre (1975)
-_12=0
---1974
----Revised
250
(1967)
q Bohlonder
-1750
: SetII
: Set III
(1952)
(1976)
Correlation
Correlation
I,
350
I
450
550
650
750
850
950
1050
1150
1250
f (0
Fig. 2. Second virial coefficient of water. All references are in Cholinski et al. (1986), except
for Eubank et al. (1988).
266
50
25
Fb
0
V
is
E
3
0 -25
B
;
&
m -50
w
V
V
eo
Eubcmk(1966)
Vukolovich
Collins
Kell (1965)
Kell (1968)
Bohlander
III
(197:)
)c McCullough
-75
: Set
(1967)
(1952)
-100
250
350
450
550
650
15
850
LeFevre
950
(1975)
1050
,150
,230
T (K),
Fig. 3. Second virial coefficient of water. Deviation from revised correlation: fc2) =
-0.0109/T,6. All references are in Cholinski et al. (1986), except for Eubank et al, (1988).
far too negative at low temperatures. This is because the 1974 recommendation was based, at low temperatures, on the data of Kell et al. (1968). These
data have been shown, most recently and conclusively by Eubank et al.
(1988), to be too negative owing to significant adsorption effects that had
not been accounted for.
The problem with the data of Kell et al. (1968) is clearly shown in Fig. 3,
where the difference between experimental and calculated (revised recommendation, eqn. (18)) B values is plotted vs. temperature. The data of Kell
et al. (1968), along with the 353 K value from Vukalovich et al. (1967) (see
Fig. 2), suggest a much steeper dependence on temperature, which is not
supported by the recent results of Eubank et al. (1988), as well as of others.
Equation (18), the revised recommendation for water, fits all 118 data
points with an average deviation of 11.0 cm3 mol-. As shown in Fig. 3, the
deviations exceed 25 cm3 mol- only below 450 K.
CROSS-COEFFICIENTS
FOR MIXTURES
267
(lo)-(13) are Pcij, Tcij, wij, U,j, and bij. The following mixing rules make it
possible to relate these characteristic constants to pure-component parameters.
Tcij =
p
(TciTcj)l(l
4Lj(
09)
kij)
pcici/G
pcjvc.j/T,)
(20)
ClJ
(vy3
ldij
lg3
(21)
os( q + Wj)
(22)
b,, = 0
(23)
For polar/polar
assuming that
to Bij is calculated
by
(24)
(25)
The most sensitive mixing rule is eqn. (19). Tcij can be assumed to be the
geometric mean of Tci and Tcj ( kij = 0) only when i and j are very similar
in size and chemical nature. Otherwise, in the absence of any specific
chemical interaction between i and j, kij should be positive and thus Tcij
would be less than the geometric mean.
CROSS-COEFFICIENTS
FOR WATER/n-ALKANE
BINARIES
268
0.55
kii
0.35
0.30
0.25
-kii
= 0.6114 - 2.7135/~c
05
0.20
0
100
200
CRITICAL
300
VOLUME.vc
400
(cd/mot)
binaries.
These early results have been confirmed by analyzing B,, values derived
from the excess enthalpy measurements of Wormald and co-workers. The
results for C&s n-alkanes are given in Table 1 and plotted in Fig. 4, which
TABLE 1
Optimum kij values for water/n-alkane
binaries
Alkane
kc;
Methane
0.34
Ethane
0.37
Propane
0.418
Butane
0.45Cj
Pentane
Hexane
0.46g
Heptane
0.473
Octane
0.492
0.463
269
(26)
for all water/
VIFUALEQUATIONOF STATE
The virial equation
then one quarter the
subcritical conditions).
third virial coefficient,
The density series
z=1+:+
c+ ...
(27)
V2
...
(28)
where B = B/RT and C = (C - B)/( RT)2; for example, see Prausnitz et
al. (1986). Equation (27) can be used with confidence up to one half the
critical density. Lee et al. (1978, 1979) have presented graphically the
reduced pressure and temperature ranges that can be described by eqns. (27)
or (28) with B, B and C, or even higher virial coefficients. However, even an
infinite series cannot describe accurately the critical region. What is more
important is that so little is known about C, especially for polar systems,
that in general the virial equation is truncated after the B (or B) term.
When only the second virial coefficient is used, it is important to
differentiate between T < T, and T > T,. At supercritical temperatures, the
truncated pressure series is superior to the truncated density series. Indeed,
for non-polar gases, C = 0 for T, > 1.4 (Chueh and Prausnitz, 1967), and
therefore the truncated pressure series is equivalent to eqn. (28).
What is not so well known is that the truncated density series is the
superior form at subcritical temperatures. This is clearly illustrated in Fig. 5.
As shown, the truncated density series predicts reliably the compressibility
factor of saturated steam up to T, = 0.9, but then it breaks down, predicting
complex values at T, > 0.925. In contrast, the truncated pressure series is
reasonably good only up to T, = 0.8.
In order to go to T, > 0.9, we must use either C or a closed-form equation
of state, such as a cubic equation. However, most cubic equations give poor
results for polar gases. This limitation can be drastically diminished if we
270
0.7i!
0.60.6- ----2 = 1 + B/v
-.-._z = 1 + BPlRl
0.40.5
0.6
0.6
0.7
0.9
1.0
1,
Fig. 5. Compressibility
EQUATION
This
OF STATE
%k(T)
d + brk)
(29
where
ark = 0.42748 ...
(30)
brk = 0.08664 - * *
(31)
p=RT_
" - brk
(brk-B)RT
+'+
where
B = brk- urk( T)/RT
brk)
(33)
(34)
271
Prediction of C
Equation (33) now reproduces the B as given by eqn. (lo), but can also
predict C (and higher virial coefficients):
c = b,k + b&.z,~(T)/RT
Wa)
or
C = brk(2& - B)
(35b)
15000
Predicted
---Methane
---Hz0
5000
--
&g
rD
--
-A
a-!i
AA -
E
25
O3
0
0
-,oo*o
ExDerimental
0
H20,Eubonk
Methane.
et al. (1968)
Clourlin
et al. (i964
-1ooooc
300
400
500
T(K)
Fig. 6. Virial-RK prediction of third virial coefficients.
600
272
interestingly, is very similar to that of the Stockmayer potential (Hirschfelder et al., 1964). In spite of the poor results for the C of water, the next
section shows that virial-RK can be used successfully to predict the fugacity
coefficient of water in methane at high pressures.
Mixing rules
The mixing rules in the cubic equations of state provide formulas for
predicting the ark and b& of mixtures. The most common or classical
mixing rules are the one-fluid van der Waals mixing rules:
a rk,m
brk,m
i
i
Cyiyjark,ij
(36)
Lhyjbrk,ij
j
brk,ij
0a5(
brk,i
brk, j)
(37)
In the case of the virial-RK equation, we have two choices for ark,ij. If we
treat it as a virial equation, then
a rk,ij=
cbrk,ij-
(38)
Bij)RT
and the value of a&ii depends on Bij, which in turn depends on the value
of the characteristic binary constant kij. In this approach, therefore, we can
use the kij values determined by regressing Bij data; for example, the values
in Table 1 or Fig. 4.
Joffe (1978) made extensive calculations with eqn. (38), especially on
water/alkane
systems. He confirmed that the ki js from Bij analysis gave
good results, although in some cases it was noted that slightly lower kijs
gave better results at high pressures. For example, for water/ethane,
kij =
0.37 was best at low pressures, but at P > 200 bar the optimum value was
kij = 0.33. With that value, the fugacity coefficient of water in ethane
predicted with eqn. (33) was only 4% too high at 414 bar, while the original
Redlich-Kwong equation was 67% below the experimental value (Reamer et
al., 1943). Joffe also found eqn. (33) to be a significant improvement over
the equations of state of de Santis et al. (1974) and Nakamura et al. (1976).
The second choice for ark, is to treat virial-RK as a cubic equation of
state. Then we introduce the binary constant Cii to correct for the deviation
of ark,ii from the geometric mean.
j
a rk,ij
(ark,ia,,j)05(1
where
a rk,i
cbrk,i
Bi)
RT
ij)
(3%
273
1
0.8
0.6
0.4
Experimental
q
n
---
0.2
0
0
10
30
20
40
50
P (MPa)
The two approaches for a,,ij produce similar results, although the values
of kjj and Cij differ. For example, the optimum binary constants for
water/methane
are kij = 0.34 and Cij = 0.57.
Figure 7 presents calculations and experimental data for the fugacity
coefficient of water in methane. At 410.93 K, the virial-RK matches closely
the data of Joffrion and Eubank (1988) up to 40 MPa, where the truncated
virial equation is about 30% too low. However, at 344.26 K the virial-RK is
much less satisfactory (25% too low at 40 MPa). As shown, the truncated
virial equation is considerably worse and breaks down above 20 MPa. Not
shown are the results with the original Redlich-Kwong equation, which is in
error by more than 30%.
In spite of the discrepancy at 344.26 K (which may be due to experimental uncertainties related to the very low water levels), Fig. 7 supports the
conclusion that virial-RK can be used to predict vapor-phase fugacity
coefficients up to much higher pressures than with the truncated virial
equation-and
for systems that are more polar than with the original
Redlich-Kwong equation of state.
274
CONCLUDING REMARKS
a, b
a rk,
brk
275
c
cij
fi
f(O)
kij
f(l)
f(2)
Ki
P
R
T
u
xi
Yi
Greek letters
a
Yi
gi
w
equation of state
Subscripts
C
i, j
ij
m
r
rk
critical property
property of component i, j
property of i-j interaction
mixture property
reduced property
Redlich-Kwong equation of state parameters
Superscripts
L
V
0
I
cc
liquid-phase property
vapor-phase property
reference property
virial coefficients in pressure series (eqn. (28))
infinite-dilution property
276
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