Ind. Eng. Chem. Res. 2008, 47, 10041–10052
10041
Addition of the Hydrogen Sulfide Group to the PPR78 Model (Predictive 1978, Peng–Robinson Equation of State with Temperature Dependent kij Calculated through a Group Contribution Method) Romain Privat, Fabrice Mutelet, and Jean-Noe¨l Jaubert* Laboratoire de Thermodynamique des Milieux Polyphase´s, Nancy-UniVersite´, 1 rue GrandVille, B.P. 20451, F-54001 Nancy Cedex, France
In 2004, we started to develop the PPR78 model which is a group contribution method aimed at estimating the temperature dependent binary interaction parameters (kij(T)) for the widely used Peng–Robinson equation of state. In our previous papers, 13 groups were defined: CH3, CH2, CH, C, CH4 (methane), C2H6 (ethane), CHaro, Caro, Cfused_aromatic_rings, CH2,cyclic, CHcyclic or Ccyclic, CO2, and N2. It was thus possible to estimate the kij for any mixture containing alkanes, aromatics, naphthenes, carbon dioxide, and nitrogen whatever the temperature. In this study, the PPR78 model is extended to systems containing hydrogen sulfide. To do so, the group H2S was added. From a general overview on the results obtained from the whole constituted experimental data bank, one can see that the PPR78 model is able to quite accurately predict the behavior of the systems containing H2S. Introduction During the last five years, Jaubert and co-workers1–11 developed a group contribution method12–14 allowing the estimation of the temperature dependent binary interaction parameters (kij(T)) for the widely used Peng–Robinson equation of state (EOS). Jaubert et al. called their model PPR78 (predictive 1978, Peng–Robinson EOS) because it relies on the Peng–Robinson equation of state as published by Peng and Robinson15 in 1978 and because the addition of a group contribution method to estimate the kij value makes it predictive. In their previous papers,1–5 Jaubert et al. defined 13 groups: CH3, CH2, CH, C, CH4 (methane), C2H6 (ethane), CHaro, Caro, Cfused_aromatic_rings, CH2,cyclic, CHcyclic or Ccyclic, CO2, and N2. To predict the phase behavior of petroleum fluids (crude oils, condensate gases, natural gases, etc.), it was obviously missing the H2S group. This is the reason why, in this study, the use of the PPR78 model is extended to systems containing hydrogen sulfide. The interactions between this new group and the 13 ones previously defined (a total of 26 parameters) are determined. Today, it is thus possible to estimate, at any temperature, the kij between two components in any mixture containing paraffins, naphthenes, aromatics, CO2, N2, and H2S, i.e., in any petroleum fluid. The PPR78 Model 15
Equation of State. In 1978, Peng and Robinson published an improved version of their well-known equation of state, noted PR78 in this paper. For a pure component, the PR78 EOS is P)
ai(T) RT – V – bi V(V + bi) + bi(V – bi)
(1)
with * Author to whom the correspondence should be addressed. E-mail:
[email protected]. Fax number: +33 3 83 17 51 52.
{
R ) 8.314 472 J·mol–1·K–1 RTc,i bi ) 0.077 796 073 9 Pc,i
ai ) 0.457 235 529
[ ( )]
R2Tc,i2 1 + mi 1 – Pc,i
T Tc,i
if ωi e 0.491 mi ) 0.374 64 +
2
(2)
1.542 26ωi – 0.269 92ωi2 if ωi > 0.491 mi ) 0.379 642 + 1.485 03ωi – 0.164 423ωi2 + 0.016 666ωi3
where P is the pressure, R is the ideal gas constant, T is the temperature, V is the molar volume, Tc is the critical temperature, Pc is the critical pressure, and ω is the acentric factor. In this paper, the PR78 EOS is used. To apply such an EOS to mixtures, mixing rules are used to calculate the values of a and b of the mixtures. Classical mixing rules are used in this study
{
N
a)
N
∑ ∑ z z √a a (1 – k (T)) i j
i)1 j)1 N
b)
∑
i j
ij
(3)
zibi
i)1
where zk represents the mole fraction of component “k” in a mixture and N represents the number of components in the mixture. The term kij(T) is the so-called binary interaction parameter characterizing molecular interactions between molecules “i” and “j”. In this paper, in order to obtain a predictive model and to define the PPR78 model (predictive, 1978 PR EOS), kij, which depends on temperature, is calculated by a group contribution method through the following expression:
10.1021/ie800799z CCC: $40.75 2008 American Chemical Society Published on Web 11/19/2008
a
A23 ) 51.47 B23 ) 79.61 A24 ) 88.53 B24 ) 315.0 A25 ) 36.72 B25 ) 108.4 A26 ) 31.23 B26 ) 84.76 A27 ) 29.78 B27 ) 58.17 A28 ) 3.775 B28 ) 144.8 A29 ) 3.775 B29 ) 144.8 A2–10 ) 12.78 B2–10 ) 28.37 A2–11 ) –54.90 B2–11 ) –319.5 A2–12 ) 136.9 B2–12 ) 254.6 A2–13 ) 82.28 B2–13 ) 202.8 A2–14 ) 134.6 B2–14 ) 138.3
0
CH2 (group 2)
A34 ) –305.7 B34 ) –250.8 A35 ) 145.2 B35 ) 301.6 A36 ) 174.3 B36 ) 352.1 A37 ) 103.3 B37 ) 191.8 A38 ) 6.177 B38 ) –33.97 A39 ) 6.177 B39 ) –33.97 A3–10 ) 101.9 B3–10 ) –90.93 A3–11 ) –226.5 B3–11 ) –51.47 A3–12 ) 184.3 B3–12 ) 762.1 A3–13 ) 365.4 B3–13 ) 521.9 A3–14 ) 193.9 B3–14 ) 307.8
0
CH (group 3)
A45 ) 263.9 B45 ) 531.5 A46 ) 333.2 B46 ) 203.8 A47 ) 158.9 B47 ) 613.2 A48 ) 79.61 B48 ) –326.0 A49 ) 79.61 B49 ) –326.0 A4–10 ) 177.1 B4–10 ) 601.9 A4–11 ) 17.84 B4–11 ) –109.5 A4–12 ) 287.9 B4–12 ) 346.2 A4–13 ) 263.9 B4–13 ) 772.6 A4–14 ) 305.1 B4–14 ) –143.1
0
C (group 4)
A67 ) 41.18 B67 ) 50.79 A68 ) –3.088 B68 ) 13.04 A69 ) –3.088 B69 ) 13.04 A6–10 ) 8.579 B6–10 ) 76.86 A6–11 ) 10.29 B6–11 ) –52.84 A6–12 ) 135.5 B6–12 ) 239.5 A6–13 ) 61.59 B6–13 ) 84.92 A6–14 ) 157.2 B6–14 ) 217.1
0
A56 ) 13.04 B56 ) 6.863 A57 ) 67.26 B57 ) 167.5 A58 ) 139.3 B58 ) 464.3 A59 ) 139.3 B59 ) 464.3 A5–10 ) 36.37 B5–10 ) 26.42 A5–11 ) 40.15 B5–11 ) 255.3 A5–12 ) 137.3 B5–12 ) 194.2 A5–13 ) 37.90 B5–13 ) 37.20 A5–14 ) 181.2 B5–14 ) 288.9
0
C2H6 (group 6)
CH4 (group 5)
A78 ) –13.38 B78 ) 20.25 A79 ) –13.38 B79 ) 20.25 A7–10 ) 29.17 B7–10 ) 69.32 A7–11 ) –26.42 B7–11 ) –789.2 A7–12 ) 102.6 B7–12 ) 161.3 A7–13 ) 185.2 B7–13 ) 490.6 A7–14 ) 21.96 B7–14 ) 13.04
0
CHaro (group 7)
A89 ) 0.0 B89 ) 0.0 A8–10 ) 34.31 B8–10 ) 95.39 A8–11 ) –105.7 B8–11 ) –286.5 A8–12 ) 110.1 B8–12 ) 637.6 A8–13 ) 284.0 B8–13 ) 1892 A8–14 ) 1.029 B8–14 ) –8.579
0
Caro (group 8)
A9–10 B9–10 A9–11 B9–11 A9–12 B9–12 A9–13 B9–13 A9–14 B9–14
0 ) ) ) ) ) ) ) ) ) ) 34.31 95.39 –105.7 –286.5 267.3 444.4 718.1 1892 1.029 –8.579
Cfused_aromatic_rings (group 9)
A10–11 B10–11 A10–12 B10–12 A10–13 B10–13 A10–14 B10–14
0 ) ) ) ) ) ) ) ) –50.10 –891.1 130.1 225.8 179.5 546.6 120.8 163.0
CH2,cyclic (group 10)
Only the last line of this table, relative to H2S, was determined in this study. The first 13 lines of this table were determined in our previous papers.1–5
0 A12 ) 74.81 B12 ) 165.7 CH (group 3) A13 ) 261.5 B13 ) 388.8 C (group 4) A14 ) 396.7 B14 ) 804.3 CH4 (group 5) A15 ) 32.94 B15 ) –35.00 C2H6 (group 6) A16 ) 8.579 B16 ) –29.51 CHaro (group 7) A17 ) 90.25 B17 ) 146.1 Caro (group 8) A18 ) 62.80 B18 ) 41.86 Cfused aromatic rings A19 ) 62.80 (group 9) B19 ) 41.86 CH2,cyclic (group 10) A1–10 ) 40.38 B1–10 ) 95.90 CHcyclic or Ccyclic A1–11 ) 98.48 B1–11 ) 231.6 (group 11) CO2 (group 12) A1–12 ) 164.0 B1–12 ) 269.0 N2 (group 13) A1–13 ) 52.74 B1–13 ) 87.19 H2S (group 14) A1–14 ) 158.4 B1–14 ) 241.2
CH3 (group 1) CH2 (group 2)
CH3 (group 1)
Table 1. Group Interaction Parameters: (Akl ) Alk)/MPa and (Bkl ) Blk)/MPaa
A11–12 B11–12 A11–13 B11–13 A11–14 B11–14
0 ) ) ) ) ) )
91.28 82.01 100.9 249.8 –16.13 –147.6
A12–13 B12–13 A12–14 B12–14
0 ) ) ) )
0
N2 H2 S (group 13) (group 14)
98.42 0 221.4 134.9 A13–14 ) 319.5 201.4 B13–14 ) 550.1
CO2 CHcyclic or Ccyclic (group 11) (group 12)
10042 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 2. List of the 33 Pure Components Used in This Study component short name
hydrogen sulfide nitrogen carbon dioxide methane ethane propane 2-methyl propane n-butane 2,2-dimethyl propane 2-methyl butane n-pentane n-hexane n-heptane 2,2,4-trimethyl pentane n-octane n-nonane n-decane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-eicosane benzene methyl benzene (toluene) 1,3-dimethyl benzene (meta-xylene) n-propyl benzene 1,3,5-trimethyl benzene cyclohexane methyl cyclohexane ethyl cyclohexane n-propyl cyclohexane isopropyl cyclohexane H2S N2 CO2 1 2 3 2m3 4 22m3 2m4 5 6 7 224m5 8 9 10 12 13 14 15 16 20 B mB 13mB prB 135mB C6 mC6 eC6 prC6 iprC6
– 1 2
[ ∑ ∑ (R Ng Ng
ik – Rjk)(Ril – Rjl)Akl
k)1 l)1
(
kij(T) ) 2
√ai(T)aj(T)
(
298.15 T
√ai(T) – √aj(T) bi
)( ) –
Akl
Bkl
–1
bj
] ) 2
bibj (4)
More information on eq 4 may be found in our first paper.1 The PPR78 model is thus defined by eqs 1–4. In eq 4, T is the temperature. ai and bi are simply calculated by eq 2. Ng is the number of different groups defined by the method (for the time being, 14 groups are defined and Ng ) 14). The term Rik is the fraction of molecule i occupied by group k (occurrence of group k in molecule i divided by the total number of groups present in molecule i). Akl ) Alk and Bkl ) Blk (where k and l are two different groups) are constant parameters determined either in this study or in our previous papers1–5 (Akk ) Bkk ) 0). For the new group added in this paper (group 14 ) H2S), we have to estimate the interactions between this new group and the 14 ones previously defined. We thus need to estimate 26 parameters (13 Akl and 13 Bkl values). These parameters have been determined in order to minimize the deviations between calculated and experimental VLE data from an extended database. The corresponding Akl and Bkl values (expressed in megapascals) are summarized in Table 1. An example of kij calculation may be found in our first article.1
Database and Reduction Procedure
Table 2 presents the list of pure components used in this study. Their physical properties (Tc, Pc, and ω) originate from
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 10043
Figure 1. Prediction of isothermal and isobaric dew and bubble curves for the system methane(1) + hydrogen sulfide(2) using the PPR78 model: (solid lines) predicted curves with the PPR78 model. (a) Prediction at two different temperatures: T1 ) 223.15 K (kij ) 0.0899), T2 ) 252.00 K (kij ) 0.0828). Experimental dew and bubble points at T1 ) 223.15 K (+), and T2 ) 252.00 K (×). (b) Prediction at three different temperatures: T1 ) 277.60 K (kij ) 0.0776), T2 ) 310.90 K (kij ) 0.0719), T3 ) 344.30 K (kij ) 0.0674). Experimental dew and bubble points at T1 ) 277.60 K (+), T2 ) 310.90 K (×), and T3 ) 344.30 K (/). Experimental binary mixture critical points (O). (c) Prediction at pressure P ) 27.58 bar. experimental bubble points (+), experimental dew points (/). (d) Prediction at pressure P ) 41.37 bar. Experimental bubble points (+), experimental dew points (/). (e) Prediction at two different pressures: P ) 55.16 and 82.74 bar. Experimental bubble points at P ) 55.16 bar (+), experimental dew points at P ) 55.16 bar (/). Experimental bubble points at P ) 82.74 bar (×), experimental dew points at P ) 82.74 bar (]). (f) Prediction at pressure P ) 110.32 bar. Experimental bubble points (+), experimental dew points (/).
Poling et al.16 Table 3 details the sources of the binary experimental data used in our evaluations17–60 along with the temperature, pressure, and composition range for each binary system. Most of the data available in the open literature (2299 bubble points + 1561 dew points + 63 mixture critical points) have been collected. Our database includes VLE data on 32
binary systems. The 26 parameters (13 Akl and 13 Bkl) determined in this study (see Table 1), are those which minimize the following objective function: Fobj )
Fobj,bubble + Fobj,dew + Fobj,crit.comp + Fobj,crit.pressure (5) nbubble + ndew + ncrit + ncrit
10044 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 3. Binary Systems Database binary system (1st compound–2nd compound) N2–H2S CO2–H2S 1–H2S 2–H2S H2S–3 H2S–2m3 H2S–4 H2S–22m3 H2S–2m4 H2S–5 H2S–6 H2S–7 H2S–224m5 H2S–8 H2S–9 H2S–10 H2S–12 H2S–13 H2S–14 H2S–15 H2S–16 H2S–20 H2S–B H2S–mB H2S–13mB H2S–prB H2S–135mB H2S–C6 H2S–mC6 H2S–eC6 H2S–prC6 H2S–iprC6
{
temperature range (K)
pressure range (bar)
x1 range (1st compound liquid mole fraction)
y1 range (1st compound gas mole fraction)
0.0004–0.1610 0.0090–0.9670 0.0007–0.6000 0.0085–0.9741 0.0090–0.9960 0.0048–0.9988 0.0099–0.9916 0.0100–0.9980 0.0090–0.9930 0.0062–0.9733 0.0230–0.9140 0.0177–0.9270 0.0246–0.2598 0.0393–0.0508 0.0419–0.2088 0.0181–0.9940 0.0267–0.9020 0.0187–0.3047 0.0284–0.0615 0.1670–0.7610 0.0205–0.8780 0.0669–0.9080 0.0339–0.9360 0.0310–0.9540 0.0206–0.9888 0.0930–0.9210 0.0280–0.9953 0.0020–0.8900 0.0290–0.9700 0.0163–0.9841 0.0418–0.9832 0.0389–0.1937
0.1080–0.9873 0.0350–0.9690 0.0117–0.9690 0.0888–0.9703 0.0400–0.9740 0.0093–0.9989 0.0195–0.9820 0.0340–0.9990 0.0710–0.9960 0.0559–0.9930 0.1840–0.9874 0.1040–0.9970
200.15–344.26 3.34–207.05 224.82–366.65 6.89–89.97 188.70–366.48 0.48–145.30 199.93–360.39 0.65–83.93 182.33–367.05 0.17–81.60 277.65–398.15 2.07–88.87 182.33–418.15 0.06–78.74 323.15–413.15 5.58–74.53 323.15–413.15 3.10–83.77 223.17–444.26 0.17–89.63 288.15–422.65 1.01–75.45 310.90–477.59 1.60–95.49 323.15–473.15 2.39–16.58 288.15–303.15 1.01–1.01 310.93–477.59 1.37–27.65 277.59–523.15 1.01–133.42 288.15–434.40 1.01–56.75 323.15–523.15 2.02–11.23 288.15–343.15 1.01–1.01 422.60–422.60 11.30–91.00 298.15–523.15 1.01–74.14 323.10–423.30 4.04–76.74 304.30–422.65 1.02–98.00 310.93–477.59 2.03–115.76 310.90–477.60 1.46–131.20 313.20–473.50 3.95–129.70 310.90–477.60 1.31–137.50 283.16–422.65 0.20–94.95 310.90–477.60 2.50–94.53 310.93–477.59 1.70–125.00 310.93–477.59 1.77–130.52 310.93–477.59 1.12–29.65
0.9975–0.9987
0.4370–0.9966 0.3520–0.9990 0.5592–0.9997 0.7870–0.9999 0.6247–0.9999 0.4770–0.9895 0.4110–0.9990 0.3444–0.9998 0.6356–0.9998 0.5779–0.9970 total number of points
∑ 0.5( x|∆x| + x|∆x| ) 1,exp
i)1
2,exp i
(
∑
|∆y| |∆y| Fobj,dew ) 100 0.5 + y y 1,exp 2,exp i)1
)
(
|∆xc| |∆xc| Fobj,crit.comp ) 100 0.5 + x x c1,exp c2,exp i)1
∑
Fobj,crit.pressure ) 100
∑ i)1
(
|Pcm,exp – Pcm,cal| Pcm,exp
refs 17–19 20–22 23–30 31–33 30, 34–39 40–42 30, 34, 40 43 43 30, 44 45, 46 47, 48 49 46 50 46, 49, 51 46, 52 49 46 45 46, 49, 53 54 45, 55 47 56 57 52, 56 45, 58 59 60 60 50
)
)
For all the 3923 data points included in our database, the objective function defined by eq 5 is Fobj ) 9.2%.
nbubble
) 0.025
∆x1% + ∆x2% Fobj,bubble ) ) 10.3% 2 nbubble
The average overall deviation on the gas phase composition is
i
Results and Discussion
i)1
and
i
nbubble, ndew, and ncrit are the number of bubble points, dew points, and mixture critical points respectively. x1 is the mole fraction in the liquid phase of the most volatile component, and x2 is the mole fraction of the heaviest component (it is obvious that x2 ) 1 – x1). Similarly, y1 is the mole fraction in the gas phase of the most volatile component, and y2 is the mole fraction of the heaviest component (it is obvious that y2 ) 1 – y1). xc1 is the critical mole fraction of the most volatile component, and xc2 is the critical mole fraction of the heaviest component. Pcm is the binary critical pressure.
∑ (|x
1,exp – x1,cal|)i
∆x% )
with |∆xc| ) |xc1,exp – xc1,cal|)|xc2,exp – xc2,cal| ncrit
{ {
0 10 9 6 20 6 5 2 2 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63
nbubble
∆x1 ) ∆x2 )
i
with |∆y| ) |y1,exp – y1,cal|)|y2,exp – y2,cal| ncrit
76 176 237 124 254 83 56 28 33 53 25 69 0 0 15 44 0 0 0 5 0 0 24 27 29 25 44 24 29 28 35 18 1561
The average overall deviation on the liquid phase composition is
with |∆x| ) |x1,exp – x1,cal|)|x2,exp – x2,cal| ndew
71 177 215 128 369 83 276 28 33 202 29 66 19 4 15 98 39 25 6 5 54 28 46 27 29 25 44 48 29 28 35 18 2299
0.5449–0.9975 0.9000–0.9990
nbubble
Fobj,bubble ) 100
number of number of number of bubble points dew points binary critical (T, P, x) (T, P, y) points (Tcm, Pcm, xc)
ndew
1,exp – y1,cal|)i
∆y1 ) ∆y2 )
i)1
ndew
) 0.015
and
∆y% )
{
∑ (|y
∆y1% + ∆y2% Fobj,dew ) ) 7.7% 2 ndew
The average overall deviation on the critical composition is ncrit
∑ (|x
c1,exp – xc1,cal|)i
∆xc1 ) ∆xc2 )
i)1
ncrit
) 0.029
and
∆xc% )
∆xc1% + ∆xc2% Fobj,crit.comp ) ) 9.67% 2 ncrit
The average overall deviation on the binary critical pressure is
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 10045
Figure 2. Prediction of the global phase equilibrium diagram for the system methane(1) + hydrogen sulfide(2) using the PPR78 model (a) in the (P,T) plane and (b) in the (T,x1) plane: (solid lines) predicted critical loci and 3-phase lines (liquid-liquid-vapor); (dashed lines) vapor pressure curves of pure components; (dashed lines) 3-phase curves (liquid–liquid–vapor); (+) critical points of pure components; (2) predicted upper critical end point; (O) experimental binary mixture critical points.
Figure 3. Prediction of isothermal and isobaric dew and bubble curves for the system ethane(1) + H2S(2) using the PPR78 model: (solid lines) predicted bubble, dew, and critical curves with the PPR78 model; (dashed lines) predicted azeotropic locus with the PPR78 model; (O) experimental binary mixture critical points; (+) critical points of pure components. (a) Prediction at four different temperatures: T1 ) 199.93 K (kij ) 0.0889), T2 ) 227.93 K (kij ) 0.0874), T3 ) 255.32 K (kij ) 0.0865), T4 ) 283.15 K (kij ) 0.0860). Experimental dew and bubble points at T1 ) 199.93 K (+), T2 ) 227.93 K (×), T3 ) 255.32 K (/), and T4 ) 283.15 K (]). (b) Prediction at six different temperatures: T1 ) 306.62 K (kij ) 0.0859), T2 ) 309.54 K (kij ) 0.0859), T3 ) 313.90 K (kij ) 0.0859), T4 ) 323.77 K (kij ) 0.0860), T5 ) 341.48 K (kij ) 0.0862), T6 ) 360.39 K (kij ) 0.0865). (c) Prediction at seven different pressures: P1 ) 27.58 bar, P2 ) 37.92 bar, P3 ) 48.26 bar, P4 ) 51.71 bar, P5 ) 58.61 bar, P6 ) 65.50 bar, P7 ) 72.40 bar. Experimental dew and bubble points at P1 ) 27.58 bar (+), P2 ) 37.92 bar (×), P3 ) 48.26 bar (/), P4 ) 51.71 bar (]), P5 ) 58.61 bar (0), P6 ) 65.50 bar (b), P7 ) 72.40 bar ([).
10046 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008
Figure 4. Prediction of the global phase equilibrium diagram using the PPR78 model for two systems; (a) system ethane(1) + hydrogen sulfide(2), (b) system hydrogen sulfide(1) + propane(2); (solid lines) predicted critical locus and azeotropic locus, (dashed lines) vapor pressure curves of pure components, (dashed lines) azeotropic locus; (+) critical points of pure components, (O) experimental binary mixture critical points, (0) critical azeotropic end point (AEP), ([) temperature minimum critical point (Cmin).
{
ncrit
∆Pc )
∑
(|Pcm,exp – Pcm,cal|)i
i)1
ncrit
) 2.67 bar
and
∆Pc% )
Fobj,crit.pressure ) 3.02% ncrit
The deviations observed in this study are higher than those observed with hydrocarbons,1–3 with CO2 4 and even with N2.5 Indeed, for the 403 binary systems studied in our previous papers,1–5 i.e., for more than 60 000 data points, the objective function is Fobj ) 6.2%. The main reason why, in this paper, the objective function is higher than in the previous ones is very simple: many of the binary H2S + hydrocarbon vapor–liquid equilibrium data reported in the literature are generally not internally consistent and are mutually conflicting, i.e., there is a great deal of scatter among the experimental points. This scatter inevitably increases the objective function. In order to illustrate the accuracy and the limitations of the proposed model, it was decided to define several families of binary systems. It is indeed impossible to show the results for all the studied systems.
Results for Mixtures of Hydrogen Sulfide + n-Alkanes. In this family, the binary system methane(1) + H2S(2) is the only one to exhibit a type III phase behavior in the classification scheme of Van Konynenburg and Scott.61 From our experience,1–7 it is very difficult to predict accurately the phase behavior of such systems with a cubic equation of state even with temperature dependent kij. This is particularly true at low temperature when the slope of the critical curve becomes very steep in the (P,T) plane. Because no high-pressure data are available in this domain, the phase behavior of the system methane(1) + H2S(2) is well reproduced by the PPR78 model (see Figures 1 and 2). For this system, the binary interaction parameter (kij) decreases with the temperature. Mixtures of H2S with ethane or with longer n-alkanes (at least up to eicosane) exhibit type I or II phase behavior. Such systems are generally the easiest to correlate. The objective function for these systems is however Fobj ) 11.9% which is higher than the average value. Figure 3 and 4 clearly show that for short n-alkanes, the PPR78 model is able to perfectly predict the phase behavior of such mixtures at high temperatures including the location of the azeotropes and the critical region. At lower temperatures, it is not scarce that different authors made the same measurements. A few examples are shown in Figure 5.
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 10047
Figure 5. Comparison of experimental points originating from different sources, measured through similar conditions: (solid lines) predicted bubble and dew curves with the PPR78 model. (a) System H2S(1) + propane(2) at T ) 243.2 K: (+) experimental dew and bubble points originating from ref 39; (/) experimental bubble points originating from ref 30; (]) experimental azeotropic point from ref 39. (b) System H2S(1) + propane(2) at T ) 273.1 K: (/) experimental dew and bubble points originating from ref 39; (+) experimental bubble points originating from ref 30; (]) experimental azeotropic point from ref 39. (c) System H2S(1) + propane(2) at T ) 288.5 K: (+) experimental dew and bubble points originating from ref 39; (/) experimental bubble points originating from ref 36; (]) experimental azeotropic point from ref 39. (d) System H2S(1) + n-butane(2) at T ) 263.1 K: (+ and /) experimental bubble points originating from ref 30. (e) System H2S(1) + n-butane(2) at T ) 283.2 K: (+ and /) experimental bubble points originating from ref 30. (f) System H2S(1) + n-butane(2) at T ) 293.4 K: (+ and /) experimental bubble points originating from ref 30.
From this figure, it is clear that there is a great deal of scatter among the experimental points which makes artificially increase
the objective function. One can however observe that the PPR78 model tries to find a compromise between the different authors.
10048 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008
Figure 6. Prediction of isothermal dew and bubble curves for two different systems using the PPR78 model and prediction of the critical locus for 6 systems: (solid lines) predicted bubble, dew, and critical curves with the PPR78 model. (a) System H2S(1) + n-pentane(2) at four different temperatures: T1 ) 223.2 K (kij ) 0.0364), T2 ) 243.1 K (kij ) 0.0383), T3 ) 263.1 K (kij ) 0.0403), T4 ) 277.6 K (kij ) 0.0417). Experimental dew and bubble points at T1 ) 223.2 K (+), T2 ) 243.1 K (×), T3 ) 263.1 K (/), and T4 ) 277.6 K (]). (b) System H2S(1) + n-pentane(2) at five different temperatures: T1 ) 310.9 K (kij ) 0.0449), T2 ) 344.3 K (kij ) 0.0482), T3 ) 377.6 K (kij ) 0.0516), T4 ) 410.9 K (kij ) 0.0551), T5 ) 444.3 K (kij ) 0.0588). Experimental dew and bubble points at T1 ) 310.9 K (+), T2 ) 344.3 K (×), T3 ) 377.6 K (/), T4 ) 410.9 K (]), and T5 ) 444.3 K (0). (c) System H2S(1) + n-decane(2) at four different temperatures: T1 ) 277.6 K (kij ) 0.0118), T2 ) 310.9 K (kij ) 0.0153), T3 ) 377.6 K (kij ) 0.0225), T4 ) 444.3 K (kij ) 0.0302). Experimental dew and bubble points at T1 ) 277.6 K (+), T2 ) 310.9 K (×), T3 ) 377.6 K (/), T4 ) 444.3 K (]). (O) Experimental critical points. (d) Prediction of several critical loci for six systems H2S(1) + n-alkane(2). (O) Experimental critical points.
A second reason explaining the high objective function value is the presence of azeotropic points. This is the case for the system H2S(1) + propane(2) for which many experimental VLE data are available. We know from our experience6 that azeotropes artificially increase the objective function. For longer n-alkanes, the results remain accurate (see Figure 6) even if large deviations may appear in the critical region (see Figure 6b–d). At this step, it was decided to deepen the situation and to understand why the PPR78 model was not able to accurately predict the phase behavior of the system H2S(1) + n-pentane(2) at T/K ) 444.3 (see the highest temperature in Figure 6b). Our model predicts k12 ) 0.0588. This value however strongly underestimates the critical pressure and the critical composition. To find the best kij value, we decided to build the isothermal P–xy phase diagram of the system H2S(1) + n-pentane(2) at T/K ) 444.3 giving to kij different values. Some results may be seen in Figure 7. From this figure, it is possible to conclude that two kij values make the Peng–Robinson EOS able to exactly reproduce the critical pressure (kij ) –0.2870 and kij ) 0.4850) and only one allows to restitute the critical composition
(kij ) –0.1550). Unfortunately, no kij value can both restitute the critical pressure and the critical composition. In other words, no kij value is able to correlate these data! In such a case, the PPR78 model is a compromise between the restitution of the critical composition and that of the critical pressure. All the previously explained phenomena increase the objective function up to 11.9%. However, from a general overview, one can claim that the PPR78 model remains an accurate predictive model leading to satisfying results for systems H2S + n-alkanes. Results for Mixtures of Hydrogen Sulfide + Branched Alkanes. Our data bank (see Table 3) contains VLE data for only four binary mixtures containing H2S + a branched alkane. All these systems exhibit a type I or II phase behavior. As shown in Figure 8, the PPR78 model is able to predict the phase behavior of these systems accurately. We can however imagine that if experimental data points were available for longer branched alkanes, the same kind of drawbacks as those previously observed with the n-alkane family would arise. In addition, it is often said through the literature that a temperature minimum on the critical curve involves an azeo-
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 10049
Figure 7. Prediction of isothermal dew and bubble curves for system H2S(1) + n-pentane (2) at T ) 444.3 K using the Peng–Robinson equation of state with classical mixing rules and a constant kij coefficient: (solid curve) predicted bubble and dew curves with PPR78 (kij ) 0.0588), (dashed curves) bubble and dew curves obtained with kij values restituting the experimental critical pressure (i.e., kij ) 0.4850 and kij ) –0.2870), (dotted curve) bubble and dew curves obtained with kij value restituting the experimental critical composition (i.e., kij ) –0.1550); (+) experimental bubble points, (/) experimental dew points.
tropic behavior. The system H2S(1) + methyl propane(2) is a perfect counterexample of this empirical rule. As shown in Figure 8, even though the system exhibits a minimum temperature critical point (called Cmin), it does absolutely not exhibit an azeotropic behavior. Results for Mixtures of Hydrogen Sulfide + Aromatic Compounds. VLE data are only known for five binary mixtures belonging to the H2S + aromatic compound family. Once again, all these systems exhibit type I or II phase behavior according to the classification scheme of Van Konynenburg and Scott. As shown in Figure 9a and b, VLE data, available at subcritical and supercritical temperatures, are accurately predicted by the PPR78 model. Let us note that among the aromatic compounds, none contains a fused aromatic carbon (see Table 2). As a
consequence, the values of the Akl and Bkl parameters of group 9 (fused aromatic carbon group) are the same as those of group 8 (aromatic carbon group). Results for Mixtures of Hydrogen Sulfide + Naphthenic Compounds (Also Called Naphthenes or Cycloparaffins). Once again, our data bank contains VLE data for only five systems belonging to the H2S + naphthenic compound family. Figure 9c shows predicted phase diagrams for the system H2S(1) + propyl cyclohexane(2). As a general characteristic of such systems, diagrams at subcritical temperatures seem to be perfectly reproduced. At higher supercritical temperatures, a drawback previously encountered arises. The binary mixture critical point is underestimated. Regarding the illustrated case (Figure 9c), at T ) 477.59 K, the absolute deviation in critical pressure is less than 20 bar. Results for Mixtures of Hydrogen Sulfide + Carbon Dioxide. Mixtures of CO2 + H2S have been measured extensively both at subcritical and supercritical temperatures. Eight predicted isobaric dew and bubble curves are shown in Figures 9d and e. As can be seen, the PPR78 model is able to predict accurately the behavior of this system. However, as shown by the critical curve displayed in Figure 9f, the PPR78 model tends to overestimate the critical pressures in spite of the absolute deviations remain quite small (
