Correlation of the Phase Behavior in the Systems Hydrogen Sulfide-Water and Carbon Dioxide-Water Katherine A. Evelein, R. Gordon Moore,* and Robert A. Heldemann Department of Chemical Engineering, The University of Calgary, Calgary,Alberta, Canada T2N IN4
The Redlich-Kwong equation has been used to correlate carbon dioxide-water and hydrogen sulfide-water phase behavior. Soave’s correlation for coefficients a and b, with a slight modification for water, has been used. Empirical “interaction parameters” have been employed in the mixing rules for both a and b. The three-phase pressures which are predicted in these two systems by the proposed correlations agree with the experimental data within a fraction of an atmosphere. Good accuracy is obtained in the predicted phase behavior of the hydrogen sulfide-water system to pressures of 70 atm at 31 1 K and to 200 atm at 444 K. The water content of saturated carbon dioxide is predicted with reasonable accuracy up to 500 atm at 298 K and up to 1000 atm at 523 K. The corresponding calculated solubility of carbon dioxide in water is of variable accuracy, being most accurate near 348 K. A procedure is proposed for limiting computational difficulties near the three-phase line, where spurious equilibria may be computed.
Introduction Natural gas systems containing water, carbon dioxide, and hydrogen sulfide are commonly encountered. Predictive methods for the phase behavior of such gas mixtures could play a role in the design of processing equipment such as field separators for removal of an aqueous liquid stream and would also assist in anticipating disposal problems in waste waters contaminated with dissolved acid gases and hydrocarbons. Most presently used phase equilibria calculation routines do not predict the water content of natural gases and, particularly in the case of “sour” gas streams, the designer has to resort to somewhat crude empirical techniques (Campbell, 1970; Maddox, 1974). Recently, computed schemes for vapor-liquid equilibria in which both phases are described by some modification of the Redlich-Kwong equation of state have come into widespread use (Wilson, 1969; West and Erbar, 1973; Peng et al., 1975,1976). I t is the purpose of this paper to demonstrate that the applicability of such schemes can be extended to mixtures involving water and the acid gases. Attention here is restricted to the C02-H20 and H2S-HzO binary systems and we have used the Redlich-Kwong equation with Soave’s (1972) correlation for coefficients a and b. The results show that a satisfactory description of the phase behavior of these nonhydrocarbon binaries is obtained and that the method is easily extended to natural gas systems. Experimental Phase Behavior The principal experimental works on the system COpH20 to which we have had reference in this paper are those of Wiebe (1941), Wiebe and Gaddy (1939,1940,1941),Todheide and Franck (1963), and Kuenen and Robson (1899). Wiebe and Gaddy studied the phase boundaries over the ranges 298-353 K and to pressures of 700 atm. Todheide and Franck extended these ranges to 623 K and 3500 bars. The early work of Kuenen and Robson established the existence and the location of a three-phase L1-Lz-G line in the system. Among the other experimental works of interest are the measurements of the solubility of water in compressed C o n by Coan and King (1971) which, in the main, confirmed the data of Wiebe and Gaddy and provided some additional information. Takenouchi and Kennedy (1964) have reported high-temperature and high-pressure data in ranges which virtually duplicate the conditions studied by Todheide and Franck (1963). Unfortunately, the quantitative agreement
between these two sets of high-pressure high-temperature data is poor over the whole range of conditions studied but Takenouchi and Kennedy do confirm the general picture of the phase behavior presented by Todheide and Franck. The C02-HpO system is of type I11 in the scheme of classification proposed by Scott and van Konynenburg (1970), which is based on the P-T projection of the critical line. One critical line extends from the critical point of pure COz (C1) to an upper critical end point (UCEP) which is the upper termination of the three-phase line Ll-LrG. A second critical line extends from the critical point of water and would apparently terminate on the melting curve a t some very high pressure outside the range of the existing experimental data. This behavior is presented schematically in Figure 1. Figure 2 presents the pressure-composition diagrams corresponding to the four temperatures indicated in Figure 1. Figure 2A shows the phase behavior corresponding to a temperature T I below the critical temperature of pure CO2 (Cl). The qualitative behavior for the C02-H20 system a t a temperature T2 between the COz critical (C,) and the upper critical end point temperature (UCEP) is shown in Figure 2B. The vapor-liquid lense for the two CO2 rich phases is seen t o terminate at a critical point. So-called “gas-gas equilibria of the second type” (Schneider, 1972) are illustrated in Figures 2C and 2D. Figure 2C represents the behavior of the COpH20 binary a t a temperature T3 above the upper critical end point temperature (UCEP) but less than the critical temperature of pure water. Figure 2D shows gas-gas phase envelopes corresponding to a temperature T4 which is below the critical temperature of H20 (C,) but above the minimum upper critical solution temperature. The system H20-HzS has not been studied as extensively as the HzO-CO2 system. We have used the data of Selleck et al. (1952) in this paper. These data extend t o 444 K in temperature and to 340 atm in pressure. The location of a liquid-liquid-gas three-phase line and the nature of the critical curve extending from the H2S critical point to an upper critical end point (as in the COZ-H~Osystem) were established by Selleck et al. We are not aware of any measurements that would establish the location of the critical line which extends from the critical point of water in this system.
Equation of State Although the observed behavior in these systems is complex it has been demonstrated that equations of state can predict, Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
423
Table I. Correction to Soave’s Correlation for Water
Temp, K
kii
298.0 304.0 311.1 344.4 348.0 377.8
0.0242 0.0230 0.0216 0.0153 0.0145 0.0098 0.0048 0.0005
411.1
444.4
(1972) has been adapted for COZand H2S. That is, for the ith component Figure 1. Schematic representation of phase behavior for C02-H20 binary.
where
mi = 0.480 + 1.574wi - 0.176wi2
(4)
Soave’s correlation was obtained by fitting vapor pressure data. We found that it predicts the vapor pressures of carbon dioxide and hydrogen sulfide very accurately but that the predictions for water are not quite so good. We have found values of ai for water that do predict accurate vapor pressures. These values can be represented by the equation O
MOLE F R A C T I O N n20
Io
(5) where ai is the value calculated from Soave’s correlation. Parameter kii, which is tabulated in Table I a t a number of temperatures, is never very large. The mixing rules employed here are
u O
MOLE FRACTION
n20
Io
and
IL O
MOLE F R A C T I O N n2o
Io
Figure 2. Schematic pressure-composition diagrams for C02-H20 binary. Temperaturescorrespond to isotherms shown in Figure 1: (A), T = T I ;(B) T = Tp; (‘2) T = 7’3; (D) T = Tq.
a t least qualitatively, all the experimental phenomqna, including three-phase lines and gas-gas equilibria. Scott (1972) and van Konynenburg and Scott (1970) have shown that the van der Waals equation can predict almost all the known types of binary phase behavior. Furthermore, van Konynenburg (1968) has computed binary phase diagrams corresponding to type 111systems such as COpH2O. Peter and Wenzel(l972) have applied the Redlich-Kwong equation to obtain semiquantitative agreement with experimental data in gas-gas equilibria of both the first and the second type. Heidemann * (1974) has used Wilson’s modification of the Redlich-Kwong equation to calculate the water content of hydrocarbon rich phases in three-phase equilibria, in some cases with quantitative accuracy. The Redlich-Kwong equation has been used also in this paper. The equation is
where coefficient b is evaluated in the usual way; that is, for the ith component
bi = 0.08664RT~i/P~i
(2)
For a ( T ) , the temperature dependence proposed by Soave 424
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 3, 1976
(7) where x , and x, are mole fractions and k,, and c,, are empirical “interaction” parameters. In general, k,, is zero for all i. I t has been found convenient to permit k,, for water to take on the values in Table I and to take a, for all three components, including water, to be the value given by Soave’s correlation. This procedure permits the small deviations from Soave’s correlation to be dealt with in the mixing rule. The mixing rule of eq 6 does not differentiate between polar-polar and polar-nonpolar interactions and it does not deal with the possibility that water and carbon dioxide associate in the gas phase. de Santis et al. (1974) have correlated a and b in the Redlich-Kwong equation for COz and HzO and their mixtures, and have separated a for water into polar and nonpolar terms. The particular values they have found for a and b are based on gas phase data only and could not be expected to be suitable for phase equilibrium computations with the same Redlich-Kwong equation being used for all phases a t equilibrium. (Elsewhere, de Santis et al. (1975) have suggested that the Krichevski-Kasarnovski equation could be used to correlate liquid phase compositions in mixtures of supercritical components with water when the gas phase is described by an equation of state.) In this paper we simply regard k,, for the binary pairs as a parameter which may be used to correlate data. Further, k,, is restricted to be temperature independent. The mixing rule for b , eq 7, becomes the usual linear rule if cy = 0. There are precedents for using other than the arithmetic-mean in “volume” terms for binary pairs. Winnick
and Kong (1974) have introduced an empirical correction to the arithmetic mean unlike diameters in a partition function which was used to correlate excess volumes of mixtures containing polar liquids. Teja and Rowlinson (1973) have employed a similar parameter in the corresponding states prediction of critical and azeotropic states. Hissong and Kay (1970), in work more directly comparable with the present paper, have used an interaction parameter in the mixing rule for b in the Redlich-Kwong equation to obtain highly accurate predictions of the critical temperature and pressure of binary mixtures. Parameter c i j is regarded here as a second empirical parameter which is adjustable to match experimental data. I t too is independent of temperature. With the mixing rules of eq 6 and 7 the fugacity of the i t h component is given by
ii
Inx;P
-
where Z is the compressibility factor and A and E are defined by A = - aP
R2T2
(9)
B = -bP RT Equilibrium Computations The equation of state was solved for the compressibility roots by direct solution of the cubic equation. In case there were three real positive roots, the smallest was used if the phase in question was presumed to be liquid or the largest was used if the phase was a gas. The algorithm described by Starling and Han (1972) for flash calculations and a two-phase version of Heidemann’s (1974) flash calculation procedure were used in various parts of this work. For the H~S-HZOdata, a two-parameter search routine was used to determine k i j and cIJwhich minimized the function
Table 11. Binary Interaction Parameters
H2S-H20 C 02-H 2 0
0.163 0.280
Table 111. Critical Properties Component
T,,K
P,, atm
Wi
H20 H2S
647.4 373.7 304.2
218.3 88.94 72.88
0.100
con
N
”
2 y r In f r + L r=l
Three-phase Pressure The existence of three-phase equilibria in these two systems is predicted by the equation of state employed. This feature complicates the computations in the vicinity of the three phase line.
0.344 0.225
In Figure 3 is presented a section of the computed phase boundaries for the system H ~ S - H Z Oa t 444.3 K. At certain pressures, for instance a t 51 atm, it is possible to compute three different pairs of phase boundaries when only one pair of computed equilibrium phase compositions can have physical meaning. At all three pairs of compositions, however, fugacities of the two components are equal. The possibility of extraneous equilibrium solutions in three-phase binary systems has been discussed by Heidemann (1974). The three equilibrium solutions are (i) vapor-liquid equilibria with both phases rich in H2S, (ii) an HzO-rich liquid in equilibrium with an HzS rich vapor, and (iii) liquid-liquid equilibria. (Whether a phase is “vapor” or “liquid” depends on whether the largest or smallest positive compressibility root is used in computing fugacities.) Each of these three solutions can be viewed as an extension of phase boundaries which are physically meaningful in some pressure ranges. The situation should be clear from Figure 3. Further, the three-phase line may be located by the intersection of these phase boundaries. An alternative to the graphical procedure implied by Figure 3 was found for locating the three-phase line. This procedure is based on the minimization of Gibbs free energy and is easy to incorporate in a computer program. The assumption i5 that the mixture being flashed lies outside the region in which the equilibrium with two COz or Has-rich phases can be found. Then one phase is a liquid rich in water. The second phase is rich in the nonaqueous component and it remains only to decide whether the liquid or the vapor compressibility factor root (the smallest or the largest) should be used. Since the Gibbs free energy is a minimum a t equilibrium, then G’defined by
G‘ = V
The data of Selleck et al. (1952) a t 444.3 K were used to evaluate the binary interaction parameters. For the COz-HzO system, the criterion of fit was based on the data of Wiebe (1941) and Wiebe and Gaddy (1939,1940, 1941) a t 348 K and on the composition of the COSrich phase reported by Todheide and Franck (1963) to pressures of 1000 bars and temperatures up to 523 K. Values of the interaction parameters are given in Table 11. Table I11 lists the pure component critical properties used in this study.
0.08 0.195
N r=l
x , In f I L
(12)
is also a minimum. In this expression x , and y r are mole fractions, L and V are the total number of moles in the water-rich phase and the second phase, respectively, and f l L is the fugacity of a component in the water-rich phase. The fugacity of a component in the second phase, f, can have one of two values depending on which compressibility factor is used. The correct compressibility factor is the one which gives the smaller value for G’. The logic described above can easily be programmed. I t is only necessary to check which of two values of G’ is smaller and to use the correct compressibility factor in the computations. An example of the use of minimum G’ to locate the three phase line is given in Figure 4, which is prepared for 0.5 mol of HzO and 0.5 mol of H2S a t 344.4 K. Two different equilibrium solutions are possible over the pressure range 50 to 61.2 atm. The values of G’ a t equilibrium using respectively the vapor and the liquid compressibility factors, intersect a t 51.4
”,
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
425
Table IV. Three-phase Pressures for Hydrogen Sulfide-Water Binary
400t
ASSUMING LARGEST C O M P R E S S I B I L I T Y ROOT ----ASSUMING S M A L L E S T C O M P R E S S I B I L I T Y ROOT
Pressure, atm ~
-
/
~~
Temp, K
Exptl (Selleck et al.)
Calcd
311.1 344.4 377.8
26.54 51.67 -
26.42 51.40 89.10
I/
T H RPIESSURE E E PHASE
1
I
3.92
46
50
48
I
52 54 56 PRESSURE, ATMOSPHERES
1
I
58
60
Table V. Three-phase Pressures for Carbon Dioxide- Water Binary 62
Pressure, atm
Figure 3. Effect of pressure and root selection on mixture free enegry function for HzS-HzO binary at 344.4 K.
vr
’;I
VAPOR’i
-?!+ _ -
\\, -ACTUAL \\ - - - -
00
I 0010
0 020
MOLE
,
0 030
Calcd
292.00 298 304.2
54.4
55.24 63.35 12.55
-
72.4
I
COMPOSITION EXTRANEOUS EXTRANEOUS COMPOSITION COMPOSITION
\
47
Exptl (Kuenen e t al.)
I
-- - -- A
\
49
Temp, K
0 040
0 050
FRACTION H 2 0
Figure 4. Pressure-compositiondiagram for H~S-HZO binary at 344.4 K showing existence of extraneous solutions in the H2S rich phase compositions. 0
atm. A t higher pressures, the correct equilibria is computed by using the liquid compressibility factor. This is the same three-phase condition as is inferred from the intersection of phase boundaries in Figure 3. Table IV presents comparisons of the calculated threephase pressures with the experimental values of Selleck et al. (1952). The agreement is excellent for the two temperatures at which experimental data are available. The predicted three-phase pressure at 377.8 K is not confirmed by experimental data since Selleck et al. report that the upper critical end point in this system is at 373.3 K, slightly below the critical temperature of H2S. Calculated three-phase pressures for the C02-HzO system are presented with the data of Kuenen and Robson (1899) in Table V. Excellent agreement was again obtained. Calculation of the vapor-liquid phase lense for the COz or HzS-rich phases is straightforward once the three-phase line is established. The lense is generated assuming the small compressibility root for the liquid phase and the large root for the vapor phase. Although the water mole fraction in both phases is very small, there are mixtures which, when flashed, separate into two C02 or HsS-rich phases. Calculated Phase Diagrams Calculated phase compositions are compared with experimental data in Figures 5-9. The calculated pressure-composition diagrams for H2S-H20 at 311.1 and 444.3 K are presented in Figures 5 and 6. In both diagrams there is good 426
Ind. Eng. Chem., Process
Des. Dev., Vol. 15, No. 3, 1976
I
1
I
005
010
015 MOLE
FRACTION
I
I
I
085 H20
090
095
Y
IOC
Figure 5. Pressure-composition diagram for H~S-HZObinary at 311.1 K.
0‘
01
02
03
I
I
04
05
06
07
08
09
I IO
MOLE F R A C T I O N H 2 0
Figure 6. Pressurexompositiondiagram for H~S-HZO binary at 444.4 K.
agreement between the calculated and experimental phase boundaries. Interaction parameters were calculated to fit the data of Figure 6 and the agreement there is excellent. In Figure 5 , the somewhat larger errors are tolerable.
- 4 - WlEBE
AND GADDY CALCULATED
COAN 8 KING
k 1 2 * 0 2 8 , C12*0195
N
0 0
z 0
0004
0008
a92
MOLE FRACTION n
N
098
p
Figure 7. Pressure-composition diagram for COpHzO binary a t 298
K.
I
z
00
0
L
L
a
(z U
W _I
0
z
