Ind. Eng. Chem. Res. 1995,34, 314-323
314
New Procedures for Application of the Wong-Sander Mixing Rules to the Prediction of Vapor-Liquid Equilibria? Philip T. Eubank,* Guor-ShiarnShyu, and Nishawn S. M. Hanif Texas A&M University, Department of Chemical Engineering, College Station, Texas 77843-3122.
New procedures are developed for use with the recent Wong-Sandler mixing rules for the prediction of mixture phase equlibria from cubic equations of state. Improvements in the apriorz evaluation of both the liquid-phase parameter A! and the gas-phase parameter BY:’ (or k12) are made. An important new procedure is developed where the latter parameter is found from experimental measurements of the cross second virial coefficient BfT in the single-phase vapor mixture at low pressures. Apriori prediction of phase eqilibna to high pressures and temperatures then follows when the liquid-phase parameter A! is estimated from a groupcontribution method, such as UNIFAC. An order-of-magnitude analysis shows the resultant phase equlibria to be more sensitive to the gas-phase parameter BY:’. These new procedures are most promising in the prediction (not correlation) of the phase behavior of complex systems, such as COz/H20/0il in petroleum reservoirs undergoing COZ and/or steam flooding.
Introduction Recently, Wong and Sandler (1992) and Wong et al. (1992)published two important articles presenting first a new set of mixture-combiningrules (MCR)and second their application to correlation of vapor-liquid (VLE) and vapor-liquid-liquid equilibrium (VLLE) over wide ranges of temperatures and pressures for binary and ternary systems containing polar components. As applied to common cubic equations of state (EOS) containing two parameters, a, and b,, the new combining rules are composed of one equation representing the liquid state and a second equation representing the gaseous state. The liquid-side equation is
(a,/b,)
=
C (xiai/bi)- A!(X)/O,
(1)
1
where (1)u is a constant dependent on the EOS used, (2) ai, bi are the pure component parameters in the cubic EOS, (3) a, b, are the corresponding mixture parameters and (4)A: is the excess Helmholtz energy for a fixed liquid composition, xi, and temperature, T, in the infinite pressure limit. The gas-side equation is based upon the well-known quadratic rule from statistical mechanics for the second virial coefficient:
The WS/MCR contain more basic physics of how liquid and vapor mixtures should behave than does the original one-fluid MCR of van der Waals (VdW/MCR). The use of A: in eq 1, as opposed to the excess Gibbs energy, GE,of the Huron-Vidal type MCR developed by Huron et al. (1979), is advantageous as is discussed below. Further, the WS/MCR are not difficult to use nor is excessive computer time required-two separate graduate classes (a total of 34 students) taught recently by the senior author have written VLE and critical loci computer programs using the WS/MCR with all of the different common cubic EOS, including the original Redlich-Kwong equation where
B = b - (a/RT3I2).
(6)
Implementation and New Procedures for the WS/MCR Liquid Phase. Any cubic EOS using the VdW repulsion term, (7) yields, following the procedure of Sandler (1977), an expression for the excess internal energy:
where yi is the vapor composition. For the common cubic EOS of van der Waals (VdW), Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR),
B = b - (dRT)
(3)
uE= Z X i ( U i - Tdi)hi- (a, - Ta”)h, + 1
2RT{C[Xib’,l(Vi - bill
-
[b’,/(V,
- b,)]}
+
1
ZTa,b’,(ah,lab,),m
causing eq 2 to provide the second MCR as
- ~ ~ i a i b ’ ~ ( a h i / a b(8) ~),~l, 1
where a’
(da/dT), b’
(db/dT) and
Combination of eqs 1 and 4 gives b, as t Presented at the Annual Meeting AIChE, St. Louis, MO., November 1993. * Author to which correspondence should be addressed.
0888-5885/95/2634-0314$09.00/0
where h-
h(b). Values of g(V,b), h(V,b),h-, u, (ah/
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 315 Table 1. Terms and Constants in Various Cubic E09 term VdW RK alfi
a(T)
a
g(v)
v2
SRK a(T)
V(V + b )
V(V
h“
(lib)
(In 2)lb
(In 2)lb
u = bh”
1
0.69315
0.69315
(ahlab),
0
( ahlab;)
0
-
m[Z(xibti)- b’,]
+ !&,b’,(ah,/ab,)~m
(2&)-11nr
V + b(l b ( l - &I] &) +
+
( 2 h b ) - l ln(-) 2 + & 2-45 0.62323
AE = Cr,aihi - a m h m
- RT
1 1 (vm
In
- bm)
(15)
n ( V i - biF
i
L i
u: = s i c u i - Ta’i)h; - (a, - Tu’,)h; + i
V2+2Vb-b2
- b-’ l nV (+7b )
ab), and (ahlab);are given in Table 1for the VdW, RK, SRK, and PR/EOS. At the infinite pressure limit (V, b, and Vi bi) so that
-
a(T)
+ b)
b-‘ l nV (+7b )
b-‘ l nV(+7b )
[b(b + VI-’
PR
J
At the infinite pressure limit, the last term of eq 15 is zero from L’Hospital’s rule so that
-
i
Because cannot be divergent at infinite pressure, an important constraint for any MCR used with a cubic EOS employing VdW repulsion is that
b’i
0 and b’,
=0
(11)
There is no problem with conventional, one fluid VdW/ MCR of the form
b, = s i b i
which is eq 1. In terms of excess entropy,
sE= (uE- A ~ ) / T or
SE= a’,h,
+
(16)
I 1,
- B i d i h i R In (vm - bm) i n ( V i - biY L i
(12)
(17)
J
where eq 8 has been used with the constraint of eq 11. Then,
i
and
( S ~ O=)(dm/bm)- CJci(di/bi),
(18)
i
where k;z is the binary interaction constant of the VdWMCR. As long as bi is independent of temperature, 0 SO eq 10 becomes b’m
( U ~ G=)&(u~ - Tdi)/bi - (u, - Ta’,)/b,
(14)
1
However, with the WSNCR there is a natural tendency for bm to vary slightly with temperature, through eq 5, even when b’i 0. This variation should be suppressed by the constraint of eq 11 through variation of A! with temperature (constant x ) . While the above problem can be viewed as an artifact of the failure of the VdW repulsion to model the liquid at high pressures, use of a more realistic repulsion term of the Carnahan-Starling type alters the simple cubic nature of the EOS and will not be pursued here. Because no temperature derivatives of the pressure are involved in the thermodynamic identities for the Helmholtz energy, we find, as in Wong and Sandler (19921, the simpler expression:
so that Sf is only the first term (RHS) when ai is independent of temperature as for the original derivation of Wong and Sandler (1992) using the VdW/EOS. With conventional VdWMCR, Sf is zero when ai and k;2 are independent of temperature. In the second article, Wong et al. (1992) showed that AE(T,P,x)is not sensitive to pressure at low reduced temperatures where the vapor pressure is low. This liquid-phase result is due to
dAE = -SE(dZ‘) - P ( d p ) (constant x)
(19)
where VE at lower pressures is nearly the same as v“, = bE 3 (b, - C xibi). Further, at low temperatures and pressures,
G~
+ PP
(20)
Then precise low-pressure VLE measurements (e.g., via the total pressure method), conventionally performed to provide activity coefficients, yi, and hence, @, a t some low temperature, TI, yield
316 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
AE(T+)
= GE(T,, low P,z)
(21)
The first equality of eq 20 shows GE to be highly sensitive to pressure causing Wong et al. (1992)to argue correctly the superiority of their correlation over those using GE. Revised Liquid-Phase Procedures. The only remaining question is how does A: vary with temperature? Equations 8, 15, and 17 are consistent with the thermodynamic identities (Kreglewski, 1984)
[
]
a(AE/RT) (aAE/anV, = -sE; aT
=-(UE/RT). (22) V,
Here the constant volume, V, means that both V, and Vi are fixed which also implies constancy of VE. As P m, however, P = bE and bE cannot vary with temperature under eq 11. Then
-
-e
ture dependency of the constants in the @ (or activity coefficient) model. Wong et al. (1992)used the PWEOS with the athermal assumption (H: = 0) resulting in constancy of the van Laar constants over wide ranges of temperature. Our results are more consistent with the assumptions behind eq 1because of our use of the regular solution assumption, corrected numerically by eq 23. Sample calculations for methanolhenzene show that the van Laar constants diminish with increasing temperature more than that given by regular solutions when eq 23 is used. However, none of these improvements in how A! is evaluated at higher temperatures and pressures create major numerical changes in VLE forecasts from the WS/MCR. A later order-of-magnitude analysis shows the dominance of the gas-phase interaction term of eq 25, below, over At. Gas Phase. For the gas phase, Wong and Sandler (1992) and Wong et al. (1992) used eq 4 with
(aAf/aT), = -sE = a~'&,w~/b,) - ( ~ y b ~ (23) ) i 1
(25)
where equality of the first and last terms is consistent with eq 1. Often, @ measurements are fit to an empirical model (e.g., van Laar or Margules):
+ px21(van Laar)
(24a)
(GEIxlx&T) = A l p l + A,,x,, (Margules)
(24b)
(GE/x,x&T) = @/[axl
where a, /3, A12, and A21 are all functions of temperature only. Wong et al. (1992) have assumed a and ,8 to be independent of temperature, the athermal solution assumption, working with the PR/EOS and ai(T). Thus, A: is proportional to temperature (fixed x), allowing for eq 21. While thermodynamic consistency can be maintained with the athermal assumption, we use here tke\regular solution assumption of SE = 0 or A t independent of temperature causing (alT1= a2T21, (PITI = /32T2) and likewise for the Margules constants. There are several reasons for this preference. First, Abbott et al. (1989) found that roughly 80% of all nonideal liquid solutions are more nearly regular as compared to athermal in the sense that (HE/RT)> (SE/R). Second, when ai is independent of temperature, eq 1 provides either (amIRT)a linear function of inverse temperature or a, independent of temperature for athermal and regular solution assumptions, respectively. A detailed numerical study by Shyu (1993) with the methanol(l)/ benzene(2) system has shown that variations of ai(T) with temperature are strong enough with the PR/EOS to provide reasonable variation of a , with temperature under either assumption. To predict or correlate fluid phase equilibria involving a vapor phase, it is necessary to tune the attraction constants a,(T), to provide the experimental vapor The liquid-phase eq 1 is unchanged, pressures, as shown in the previous section. ai(T) can be found numerically from (1)tuning onto the experimental vapor pressures, (2) the original corresponding states formulas associated with the P m O S and SRWEOS or (3) the more recent PRSV formulation of Stryjek and Vera (1986). Use of the WS/MCR with either the P m O S or the SRWEOS over wide ranges of pressure and temperature should be improved by using the regular solution assumption to find numerically the tempera-
c(T).
where kij is the Wong-Sandler binary interaction constant defined by eq 25 and differing from many other such interaction constants (see Reid et al., 1987). Both Sandler and ourselves take bi from the usual critical constraint formula, or 0.0778O(RTdPc)for the PWEOS. They did not tune [b - (a/RT)]i to pure second virial coefficient data because unfortunately this is not practical. One must adjust ai(T) t o provide the purecomponent, experimental vapor pressure for any accurate VLE predictiodcorrelation. It then makes little sense to adjust the repulsive, liquid parameter, bi(T), to provide Bii(T) in agreement with experiment. Without such tuning, however, cubic EOS are infamous for providing incorrect B(T) at subcritical temperatures. Near the critical temperature and above, they do much better with EOS constants from the well-known critical constraints. Indeed, the RWEOS provides an accurate estimate of the experimental second virial coefficient, B,, at the critical temperature, T,,
[P,$JRT,] = -0.34 f 0.01,
(26)
for all compounds. The PR/EOS provides a value of -0.38. The use of eq 25, however, reintroduces a binary interaction constant kb, related to the VdW/MCR k: as, k12[(a1+
a,)
- ( b i + b,)RTl = (a1 + a,) - [2&(l
- KS2)1. (27)
Wong and Sandler (1992) and Wong et al. (1992) then adjust k12 to ensure that @ ( X I = 0.5) computed from the EOS agrees with experimental VLE data at low temperatures and pressures. A more recent, third paper (Huang and Sandler, 1993)clarifies this point in arguing that k12 is not an additional parameter. Equally important from the VLE results of each of the first three Sandler papers is that k12 changes little with temperature, unlike most other binary interaction constants. That is, even when binary VLE data are available over wide ranges of temperature, independent tuning of k12 at each temperature to GEresults in little variation of k12. It may seem theoretically undesirable to adjust a gas-phase parameter k12 to match low-pressure @ data for the liquid phase under circumstances where vapor
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 317 60.00
70.00
-king Experimental B,,(k12-0.284)
O K !
60.004
50.00
?' 344.3 K 50.00 40.00
$
s 30.00
a
cn VI
E
20.00
10.00
0
0.00 250
0.0 300
350
400
450
500
Temperature (K)
Figure 1. P-T diagram for the propane(l)/n-heptane(2) system at z1 = 0.7154. Points are data of Kay (1971).
imperfections are usually ignored, especially when a liquid phase parameter A: already exists in the WS/ MCR. Nevertheless, the ability of Sandler's procedure in the prediction of high pressure VLE from like measurements of a single low-temperature (low-pressure) isotherm cannot be disputed. Accurate VLE measurements at the low temperature are required because both A: and k12 are based upon the measured
GE.
Revised Gas-PhaseProcedures. To reconnect the
. 0 0 0.2 0.4 0.6 0.8 Mixture Composition, xl, y,
0
1.o
Figure 2. P-x-y diagram for the carbon dioxide(l)/propane(2) system. Points are data of Reamer et al. (1951). Lines are predictions with Tsonopoulos Blz(K12 = 0.1) together with A: from GE data a t 277.6 K using NRTL Model.
5
m-
-
-100.00
w
3
..Y B -200.00-
........... :...::...: ..................................
................... ::.::............... .....................
Ls
/
z
-8.-n 5
-300.00-
_.-.-
._.-.-a-
_.-. _.--_.--
_.-.-e
-
,-.-
