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Prediction of ternary excess enthalpies at high pressures using an equation of state
Thermochimica Acta, 235 (1994) 1- 10 Elsevier Science B.V., Amsterdam SSDI 0040-6031(93)01539-F
Prediction of ternary excess enthalpies using an equation of state Tatsuhiko
at high pressures
Ohta
Department of Chemistry and Chemical Engineering, Division of Physical Sciences, Kanazawa University, Kodatsuno 2-40-20, Kanazawa, Ishikawa 920 (Japan) (Received
26 July 1993; accepted
3 August
1993)
Abstract H E for the carbon dioxide( l)-n-hexPredictions of the ternary excess enthalpies ane( 2)-toluene( 3) system in the vicinity of the critical region and of the methane( 1)carbon dioxide( 2)-hydrogen sulfide( 3) system in the gaseous region were performed by use of the PRSV equation of state with the NRTL mixing rule. Good predictions are obtained without adding any ternary parameters. LIST OF SYMBOLS
a a,
B b ‘,J,
DzJ
C G:
HE AH’ P R T V X Z
parameter of the equation of state binary interaction parameter coefficient defined by eqn. (4) parameter of the equation of state coefficients of eqn. (11) numerical constant excess Gibbs energy at infinite pressure excess enthalpy enthalpy departure function total pressure gas constant absolute temperature molar volume mole fraction compressibility factor
Greek letters 4
P
a,
r 0 ICI
amounts of liquid and vapor mixtures in equilibrium non-randomness parameter binary parameter defined by eqn. (10) acentric factor pure component parameter
0040-6031/94/$07.00
0
1994 - Elsevier
Science
B.V. All rights reserved
2
T. OhtalThermochim.
Acta 235 (1994) I-10
Subscripts C
i,.i m 19g co
critical property components actual composition of mixture liquid and vapor phases at infinite pressure
Superscript
E
excess property
INTRODUCTION
The excess enthalpies HE of mixtures at high pressures can be measured efficiently with a flow calorimeter. The HE data provide a useful test for a variety of equation of state models. In a previous work [I], it was shown that the Stryjek-Vera modification of the Peng-Robinson (PRSV) equation of state [2,3] coupled with the NRTL mixing rule [4] gives a good capability of representing various kinds of binary HE data at high pressures, including supercritical conditions. For ternary systems at high pressures, Casielles et al. [5] and Barry et al. [6] have discussed the representation of ternary HE data using equations of state. The present work shows the abilities of the PRSV equation of state with the NRTL mixing rule in predicting ternary HE data from constituent binary data alone. The ternary systems carbon dioxide( I)-n-hexane(2)toluene(3) in the vicinity of the critical region and methane( 1)carbon dioxide(2))hydrogen sulfide(3) in the gaseous region are studied in this work. CALCULATION
OF EXCESS ENTHALPY
The excess enthalpy HE is given by the equation ~~=drHl-~x,
AH:
(1)
The enthalpy departure functions for pure components and mixtures, AH: and AH’, can be obtained from an equation of state using the thermodynamic relation [ 71 AH’ =
” [T(dP/dT), sx
- P] du + RT(z
- 1)
(2)
T. OhtalThermochim.
Acta 235 (1994) l-10
3
where for the PRSV equation of state chosen here AH’ = (l/2,/%&
- T(&/8T)]
In Z+B(L-*)+RT(z-l) 2+&l +J$
(3)
with
Pv
’ =RT The derivative of parameter a with respect to T, da/aT, is obtained analytically from its temperature dependence form and the mixing rule. For mixtures, the following NRTL mixing rule proposed by Huron and Vidal [4] was used for the calculation of the parameters in the PRSV equation of state
(6) b =
1
x,b,
where c is a numerical constant equal to 2,,@ln[(2 + ,,/$/(2 - $)] for the equation of state. The NRTL equation [8] was utilized to represent the excess Gibbs free energy at infinite pressure GE, C r/rG,,x1 GE,/RT=xx,
I
‘c G x kl
(8) k
k
with G, = exp( - aI,rl,) =,, = a,,lT
(9) (10)
where ai, ( =ocj,) is a non-randomness constant. The binary energy parameters a,, were assumed to be expressed by a linear function of temperature ai, = Cij + D,(T - 273.15)
(11)
The simplex method of Nelder and Mead [9] was used to determine the adjustable parameters, C,, and D,, by minimizing the sum of squares of deviations in experimental and calculated HE values. The excess enthalpies of mixing in the two-phase region from an equation of state have been derived by Lewis et al. [lo] and Casielles et al. [5]. A mixture is the sum of amounts c( of liquid mixture and /? of vapor mixture. The vapor and liquid equilibrium compositions are xg and xl respectively. When the actual composition of the mixture is given by x,, the following equations must be satisfied
573.15
470.15
413.15
358.15
28 25 27 25 28 28 26 24 34 38
dioxide( 1) -toulene(2) 7.60 12.67 7.60 12.67 7.60 12.67 7.60 12.67 7.60 12.67
573.15
470.15
413.15
358.15
Carbon 308.15
No. of data points
dioxide( 1) -n -hexane( 2) 7.50 29 12.50 22 7.50 29 12.50 31 7.50 30 12.50 28 7.50 31 12.50 31 7.50 30 12.50 39
Pressure in MPa
Temperature in K
arithmetic
Carbon 308.15
and absolute
1
Binary parameters
TABLE
534.09 985.74 655.32 432.55 - 726.33 11.66 73.25 472.85 1761.59 302.88
468.43 1191.36 1260.02 925.63 - 328.21 1244.89 433.85 1438.16 1502.38 2299.92
C,, in K
Parameters
mean deviations
-2.0456 0.6314 6.7235 0.8824 26.2765 8.1646 7.4236 3.7539 - 5.6029 8.6600
- 3.9192 0.4626 16.0620 5.5348 22.6738 4.5452 16.2852 2.3811 -4.8174 - 1.7638
D,Z
for binary
- 19.95 - 162.62 - 705.58 46.27 ~ 574.94 -3.17 - 165.85 - 114.69 - 325.39 - 242.76
4041.90 - 275.74 3141.02 -75.13 -420.85 - 107.95 - 140.96 - 2.05 - 1442.33 - 942.23
C,, in K
systems
-0.6597 - 2.7844 -4.6255 - 1.2334 - 1.1498 -0.8272 - 0.4097 - 1.2095 0.9440 -0.2399
22.6331 -4.1402 14.6799 - 2.5678 - 1.9436 -2.8971 1.1117 - 1.8546 7.2179 5.5824
D,,
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
El2
76.4 2.3 17.4 9.3 25.0 10.4 39.3 18.8 191.5 55.3
76.8 10.0 76.7 19.0 48.6 22.5 38.7 23.6 37.0 41.1
Abs. arith. mean dev. in J mol-’
14 14 14 14 14 14 15 15 15 15
12 12 12 12 12 12 13 13 13 13
Ref.
P
313.15
305.15
Carbon 293.15
313.15
305.15
Methane( 293.15
313.15
305.15
4 5 4 4 4 4
29 25 24 38 24 35 30 30 27 32
0.507 1.520
1.520
dioxide( 1) -hydrogen 0.507 1.419 0.507
sulfide(2) 4 4 4 5 4 4
1) -hydrogen sulfide( 2) 0.507 6 1.520 5 0.507 4 1.520 4 0.507 4 1.520 4
dioxide( 2) 0.507 1.520 0.507 1.520 0.507 1.520
I) -carbon
Methane( 293.15
573.15
470. I5
413.15
358.15
1) -toulene(2) 7.50 12.50 1.50 12.50 7.50 12.50 7.50 12.50 7.50 12.50
n-Hexane( 308.15
504.26 695.71 525.77 661.93 902.89 579.31
3301.83 897.54 675.17 423.97 125.47 286.43
318.74 438.66 320.46 392.44 270.20 545.14
- 304.68 115.93 2.50 158.96 228.92 224.03 -71.54 -118.36 - 96.75 - 137.70
9.5846 1.7630 - 0.2547 - 0.6269 11.7091 2.3300
-78.5112 -4.6170 -5.1730 -4.8512 -4.6880 - 2.3245
-0.3309 0.2565 4.8846 1.8104
1.6604
- 3.8720
- 1.3312 2.5188 -0.5171 0.3835 2.4691 2.3154 1.3138 - 0.0363 0.8702 0.7073
- 140.25 470.80 55.39 295.04 - 191.87 376.25
117.55 340.23 227.00 553.92 155.63 366.79
312.76 988.35 1260.24 468.74 420.64 639.09
27.44 11.64 120.58 266.15 38.62 70.82 264.02 223.88 231.09 274.35
- 1.6408 1.8359
1.0760
- 5.7675 1.8936 0.2115
- 19.1459 -8.3753 -2.9712 - 4.8003 - 2.4777 -4.8754
8.5942 3.5505 1.0029 22.1505 2.2021
1.4800
- 0.9656 - 2.4660 -0.0561 0.4875 - 1.0749 - 1.0078 -0.6811 -0.9618 - 1.3101 -0.9339
0.30 0.30 0.30 0.30 0.30 0.30
0.40 0.40 0.40 0.40 0.40 0.40
0.40 0.40 0.40 0.40 0.40 0.40
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
0.2 0.4 0.8 2.2 0.6 2.0
0.2 0.3 0.2 3.3 2.1 2.9
0.4 0.8 0.4 0.1 0.4
0.1
7.1 3.5 6.0 6.4 3.1 6.5 2.4 4.9 8.6 5.3
19 19 19 19 19 19
18 18 18 18 18 18
17 17 17 17
16 16 16
z
I
S
% S
6
x,a
T. OhtalThermochim.
Acta 23.5 (1994) l-10
+xgp=x,
(1 -x,)cc +(l
(12) -X&I
= 1 --CC,
(13)
where x,, xl and xg are mole fractions of the same component. When we solve for CIand p, the enthalpy departure of the mixture is given by AH’ = a( AH:,), + j?(AH:,),
(14)
where (AH:,), and (AH&), are the enthalpy departure functions of a liquid mixture of compositions x1 and a vapor mixture of composition 5. RESULTS
The two ternary systems carbon dioxide( l)-n-hexane(2)-toluene(3) and methane( l)-carbon dioxide( 2)-hydrogen sulfide( 3) were investigated at high pressures. To use the PRSV equation of state, it is necessary to obtain pure component parameters for each component, T,, PC, co, and rcI, which are available in the literature [3, 111. The values of Cli,for the related binary systems were taken from a previous paper [ 11, except for the two systems containing hydrogen sulfide. In this work, the values of CI,,for these systems were empirically assigned as follows: methane-hydrogen sulfide, 0.40; carbon dioxide-hydrogen sulfide, 0.30. Table 1 presents the binary parameters for the related binary systems and absolute arithmetic mean deviations. Agreement with the experimental data seems to be acceptable. The carbon dioxide(l)-n-hexane(2)of the critical region
toluene(3) system in the vicinity
The excess enthalpy data for this ternary system were studied at five temperatures between 308.15 and 573.15 K and at pressures of 7.50 and 12.50 MPa. The ternary HE data were determined by adding carbon dioxide to an n-hexane-toluene mixture of known composition. Three mixtures with n-hexane mole fractions of 0.25, 0.50 and 0.75 were measured [20-241. Two-phase regions appear at 308.15, 358.15, 413.15, 470.15 and 573.15 K and 7.50 MPa, and at 358.15, 413.15 and 470.15 K and 12.50 MPa. Large changes in sign and magnitude of the ternary HE values with the variations in temperature, pressure and composition are observed. The predicted results for the system are listed in Table 2, together with the maximum and minimum values of each set of ternary HE data. Figures 1 and 2, respectively, indicate the isotherms with x*/x3 = 3 at 7.50 and 12.50 MPa. Similar results are obtained for the other two series of ternary HE determinations with x2/x3 = l/3 and 1. Isotherms at some temperatures show a minimum and a maximum connected by a linear section which
T. OhtalThermochim. TABLE
1
Acta 235 (1994) l-10
2
Predicted
results for the carbon
Temperature K
308.15 358.15 413.15 470.15 573.15
in
dioxide( 1) -n-hexane(
2) -toluene(
3) system
Pressure in MPa
No. of data points
H&, in J mol-’
H&, in J mol-’
Abs. arith. mean dev. in J molV’
Ref.
7.50 12.50 1.50 12.50 7.50 12.50 7.50 12.50 7.50 12.50
83 76 92 84 84 15 82 74 82 80
-4650.0 -45.0 -2130.0 -2100.0 - 860.0 - 966.0 146.0 9 1.O 297.0 244.0
137.0 831.0 576.0 242.0 2200.0 984.0 4660.0 2400.0 7120.0 4300.0
173.3 76.5 69.6 139.3 123.9 91.2 139.8 34.3 258.8 83.2
20 20 21 21 22 22 23 23 24 24
I
I
I
I
2
-7 8 c, x
1 0
1
%-1 -2 -3 -4 -5 0.0
0.2 Mole
0.4 fraction
0.6 of
0.8
1.0
CO2
Fig. 1. Comparison of experimental and predicted excess enthalpies for the carbon dioxtde( I)-n-hexane( 2)-toluene( 3) system at 7.50 MPa and x2/x3 = 3. Experimental: 0, at 308.15 K; A, at 358.15 K; n , at 413.15 K; +, at 470.15 K; V, at 573.15 K. Predicted: --.
T. OhtalThermochim.
5
I
I
I
Acta 235 (1994) I-10
I
4
3 ‘i’ “E c, x 11
2
*z h 0
-1
-2 0.0
0.2
0.4 Mole fraction
0.6 of
0.8
1.0
CO2
Fig. 2. Comparison of experimental and predicted excess enthalpies for the carbon dioxide( I)-n-hexane(2)-toluene(3) system at 12.50 MPa and x,/x, = 3. Experimental: 0, at 308.15 K; A, at 358.15 K; n , at 413.15 K; +, at 470.15 K; 7, at 573.15 K. Predicted: -.
corresponds to a two-phase region for the ternary mixture. The accuracy of the predictions by the present equation of state can be considered satisfactory. In this work, the predictions of the HE data in the two-phase region were made on the assumption that the boundaries of the two-phase region in the ternary mixture can be approximately expressed by the straight lines connecting the limits of those in the binary systems carbon dioxide-n-hexane and carbon dioxide-toluene, because the vapor-liquid equilibrium data for the ternary system are not reported and the shape of the two-phase region is unknown. Details of the calculational procedure for this estimation have been described by Casielles et al. [5]. In their work, the Peng-Robinson equation of state was employed to predict ternary HE data for the same system from binary data alone. On the whole, the predictive ability of the present approach seems to be slightly better than that of Casielles et al. Further improvements of the ternary HE predictions in the two-phase
T. OhtalThermochim. TABLE Predicted
3 results for the methane(
Temperature K
in
293.15
1) -carbon
No. of data points
Pressure in MPa
0.507 1.520 0.507 1.520 0.507 1.520
305.15 313.15
region
9
Acta 235 (1994) I-10
might
equilibrium
be
obtained
by
dioxide(2)
-hydrogen
sulfide( 3) system
HE,, in J mol-’
Hz,, in J mol-’
Abs. arith. mean dev. in J mol-’
Ref.
29.9 73.7 27.3 57.2 25.9 38.1
50.6 129.9 41.2 99.1 32.3 90.8
3.6 7.9 11.3 4.7 2.9 4.7
6 6 6 6 6 6
using
more
accurate
ternary
vapor-liquid
data.
The methane(l)-carbon gaseous region
dioxide(2)-hydrogen
suljide(3) system in the
Predictions of ternary H” data for the gaseous mixtures were carried out at 293.15, 305.15 and 313.15 K and at 0.507 and 1.520 MPa. The experimental values of HE under these conditions are small and endothermic [6]. The predicted results for the system are shown in Table 3. The present equation of state provides good results for the ternary system. The mean percentage deviation between the experimental and predicted values of the ternary excess enthalpies is, in most cases, below lo%, and it is comparable with that of Barry et al. [6] who utilized the Redlich-Kwong and the Benedict-Webb-Rubin equations of state to represent the ternary HE data using the binary interaction parameters obtained from the ternary data. CONCLUSIONS
The PRSV equation of state coupled with the NRTL mixing rule was utilized for the prediction of ternary excess enthalpy data at high pressures. For the carbon dioxide( 1) -n-hexane(2) -toluene( 3) system in the vicinity of the critical region and for the methane( 1) -carbon dioxide( 2) -hydrogen sulfide( 3) system in the gaseous region, the equation of state provides good predictions of ternary HE data using only binary data. REFERENCES 1 T. Ohta, Thermochim. Acta, 230 (1993) 1. 2 R. Stryjek and J.H. Vera, Equations of State Ser., (1986) 560.
Theories
and Applications,
ACS Symp.
10
T. OhtalThermochim.
Acta 235 (1994) l-10
3 R. Stryjek and J.H. Vera, Can. J. Chem. Eng., 64 (1986) 323. 4 M.J. Huron and J. Vidal, Fluid Phase Equilibria, 3 (1979) 255. 5 A.G. Casielles, C. Pando, J.A.R. Renuncio, J.J. Christensen and R.M. Izatt, Thermochim. Acta, 154 (1989) 57. 6 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 28 (1983) 375. 7 R.C. Reid, J.M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, 4th edn., McGraw-Hill, New York, 1987. 8 H. Renon and J.M. Prausnitz, AIChE J., 14 (1968) 135. 9 J.A. Nelder and R. Mead, Comput. J., 7 (1965) 308. 10 K.L. Lewis, SE. Mosedale and C.J. Wormald, J. Chem. Thermodyn., 9 (1977) 121. 11 P. Proust and J.H. Vera, Can. J. Chem. Eng., 67 ( 1989) 170. 12 J.J. Christensen, T.A.C. Walker, R.S. Schofield, P.W. Faux, P.R. Harding and R.M. Izatt, J. Chem. Thermodyn., 16 (1984) 445. 13 J.J. Christensen, D.M. Zebolsky and R.M. Izatt, J. Chem. Thermodyn., 17 (1985) 183. 14 C. Pando, J.A.R. Renuncio, R.S. Schofield, R.M. lzatt and J.J. Christensen, J. Chem. Thermodyn., 15 (1983) 747. 15 J.J. Christensen, D.M. Zebolsky and R.M. Izatt, J. Chem. Thermodyn., 17 (1985) 1. 16 P.W. Faux, J.J. Christensen, and R.M. Izatt, J. Chem. Thermodyn., 19 (1987) 757. 17 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 27 (1982) 258. 18 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 27 (1982) 436. 19 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, Can. J. Chem. Eng., 61 (1983) 241. 20 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 20 (1988) 897. 21 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 20 (1988) 559. 22 P.W. Faux, J.J. Christensen, R.M. Izatt, C. Pando and J.A.R. Renuncio, J. Chem. Thermodyn., 20 (1988) 503. 23 P.W. Faux, J.J. Christensen, R.M. Izatt, C. Pando and J.A.R. Renuncio, J. Chem. Thermodyn., 20 (1988) 1297. 24 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 21 (1989) 41.
Prediction of ternary excess enthalpies using an equation of state Tatsuhiko
at high pressures
Ohta
Department of Chemistry and Chemical Engineering, Division of Physical Sciences, Kanazawa University, Kodatsuno 2-40-20, Kanazawa, Ishikawa 920 (Japan) (Received
26 July 1993; accepted
3 August
1993)
Abstract H E for the carbon dioxide( l)-n-hexPredictions of the ternary excess enthalpies ane( 2)-toluene( 3) system in the vicinity of the critical region and of the methane( 1)carbon dioxide( 2)-hydrogen sulfide( 3) system in the gaseous region were performed by use of the PRSV equation of state with the NRTL mixing rule. Good predictions are obtained without adding any ternary parameters. LIST OF SYMBOLS
a a,
B b ‘,J,
DzJ
C G:
HE AH’ P R T V X Z
parameter of the equation of state binary interaction parameter coefficient defined by eqn. (4) parameter of the equation of state coefficients of eqn. (11) numerical constant excess Gibbs energy at infinite pressure excess enthalpy enthalpy departure function total pressure gas constant absolute temperature molar volume mole fraction compressibility factor
Greek letters 4
P
a,
r 0 ICI
amounts of liquid and vapor mixtures in equilibrium non-randomness parameter binary parameter defined by eqn. (10) acentric factor pure component parameter
0040-6031/94/$07.00
0
1994 - Elsevier
Science
B.V. All rights reserved
2
T. OhtalThermochim.
Acta 235 (1994) I-10
Subscripts C
i,.i m 19g co
critical property components actual composition of mixture liquid and vapor phases at infinite pressure
Superscript
E
excess property
INTRODUCTION
The excess enthalpies HE of mixtures at high pressures can be measured efficiently with a flow calorimeter. The HE data provide a useful test for a variety of equation of state models. In a previous work [I], it was shown that the Stryjek-Vera modification of the Peng-Robinson (PRSV) equation of state [2,3] coupled with the NRTL mixing rule [4] gives a good capability of representing various kinds of binary HE data at high pressures, including supercritical conditions. For ternary systems at high pressures, Casielles et al. [5] and Barry et al. [6] have discussed the representation of ternary HE data using equations of state. The present work shows the abilities of the PRSV equation of state with the NRTL mixing rule in predicting ternary HE data from constituent binary data alone. The ternary systems carbon dioxide( I)-n-hexane(2)toluene(3) in the vicinity of the critical region and methane( 1)carbon dioxide(2))hydrogen sulfide(3) in the gaseous region are studied in this work. CALCULATION
OF EXCESS ENTHALPY
The excess enthalpy HE is given by the equation ~~=drHl-~x,
AH:
(1)
The enthalpy departure functions for pure components and mixtures, AH: and AH’, can be obtained from an equation of state using the thermodynamic relation [ 71 AH’ =
” [T(dP/dT), sx
- P] du + RT(z
- 1)
(2)
T. OhtalThermochim.
Acta 235 (1994) l-10
3
where for the PRSV equation of state chosen here AH’ = (l/2,/%&
- T(&/8T)]
In Z+B(L-*)+RT(z-l) 2+&l +J$
(3)
with
Pv
’ =RT The derivative of parameter a with respect to T, da/aT, is obtained analytically from its temperature dependence form and the mixing rule. For mixtures, the following NRTL mixing rule proposed by Huron and Vidal [4] was used for the calculation of the parameters in the PRSV equation of state
(6) b =
1
x,b,
where c is a numerical constant equal to 2,,@ln[(2 + ,,/$/(2 - $)] for the equation of state. The NRTL equation [8] was utilized to represent the excess Gibbs free energy at infinite pressure GE, C r/rG,,x1 GE,/RT=xx,
I
‘c G x kl
(8) k
k
with G, = exp( - aI,rl,) =,, = a,,lT
(9) (10)
where ai, ( =ocj,) is a non-randomness constant. The binary energy parameters a,, were assumed to be expressed by a linear function of temperature ai, = Cij + D,(T - 273.15)
(11)
The simplex method of Nelder and Mead [9] was used to determine the adjustable parameters, C,, and D,, by minimizing the sum of squares of deviations in experimental and calculated HE values. The excess enthalpies of mixing in the two-phase region from an equation of state have been derived by Lewis et al. [lo] and Casielles et al. [5]. A mixture is the sum of amounts c( of liquid mixture and /? of vapor mixture. The vapor and liquid equilibrium compositions are xg and xl respectively. When the actual composition of the mixture is given by x,, the following equations must be satisfied
573.15
470.15
413.15
358.15
28 25 27 25 28 28 26 24 34 38
dioxide( 1) -toulene(2) 7.60 12.67 7.60 12.67 7.60 12.67 7.60 12.67 7.60 12.67
573.15
470.15
413.15
358.15
Carbon 308.15
No. of data points
dioxide( 1) -n -hexane( 2) 7.50 29 12.50 22 7.50 29 12.50 31 7.50 30 12.50 28 7.50 31 12.50 31 7.50 30 12.50 39
Pressure in MPa
Temperature in K
arithmetic
Carbon 308.15
and absolute
1
Binary parameters
TABLE
534.09 985.74 655.32 432.55 - 726.33 11.66 73.25 472.85 1761.59 302.88
468.43 1191.36 1260.02 925.63 - 328.21 1244.89 433.85 1438.16 1502.38 2299.92
C,, in K
Parameters
mean deviations
-2.0456 0.6314 6.7235 0.8824 26.2765 8.1646 7.4236 3.7539 - 5.6029 8.6600
- 3.9192 0.4626 16.0620 5.5348 22.6738 4.5452 16.2852 2.3811 -4.8174 - 1.7638
D,Z
for binary
- 19.95 - 162.62 - 705.58 46.27 ~ 574.94 -3.17 - 165.85 - 114.69 - 325.39 - 242.76
4041.90 - 275.74 3141.02 -75.13 -420.85 - 107.95 - 140.96 - 2.05 - 1442.33 - 942.23
C,, in K
systems
-0.6597 - 2.7844 -4.6255 - 1.2334 - 1.1498 -0.8272 - 0.4097 - 1.2095 0.9440 -0.2399
22.6331 -4.1402 14.6799 - 2.5678 - 1.9436 -2.8971 1.1117 - 1.8546 7.2179 5.5824
D,,
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
El2
76.4 2.3 17.4 9.3 25.0 10.4 39.3 18.8 191.5 55.3
76.8 10.0 76.7 19.0 48.6 22.5 38.7 23.6 37.0 41.1
Abs. arith. mean dev. in J mol-’
14 14 14 14 14 14 15 15 15 15
12 12 12 12 12 12 13 13 13 13
Ref.
P
313.15
305.15
Carbon 293.15
313.15
305.15
Methane( 293.15
313.15
305.15
4 5 4 4 4 4
29 25 24 38 24 35 30 30 27 32
0.507 1.520
1.520
dioxide( 1) -hydrogen 0.507 1.419 0.507
sulfide(2) 4 4 4 5 4 4
1) -hydrogen sulfide( 2) 0.507 6 1.520 5 0.507 4 1.520 4 0.507 4 1.520 4
dioxide( 2) 0.507 1.520 0.507 1.520 0.507 1.520
I) -carbon
Methane( 293.15
573.15
470. I5
413.15
358.15
1) -toulene(2) 7.50 12.50 1.50 12.50 7.50 12.50 7.50 12.50 7.50 12.50
n-Hexane( 308.15
504.26 695.71 525.77 661.93 902.89 579.31
3301.83 897.54 675.17 423.97 125.47 286.43
318.74 438.66 320.46 392.44 270.20 545.14
- 304.68 115.93 2.50 158.96 228.92 224.03 -71.54 -118.36 - 96.75 - 137.70
9.5846 1.7630 - 0.2547 - 0.6269 11.7091 2.3300
-78.5112 -4.6170 -5.1730 -4.8512 -4.6880 - 2.3245
-0.3309 0.2565 4.8846 1.8104
1.6604
- 3.8720
- 1.3312 2.5188 -0.5171 0.3835 2.4691 2.3154 1.3138 - 0.0363 0.8702 0.7073
- 140.25 470.80 55.39 295.04 - 191.87 376.25
117.55 340.23 227.00 553.92 155.63 366.79
312.76 988.35 1260.24 468.74 420.64 639.09
27.44 11.64 120.58 266.15 38.62 70.82 264.02 223.88 231.09 274.35
- 1.6408 1.8359
1.0760
- 5.7675 1.8936 0.2115
- 19.1459 -8.3753 -2.9712 - 4.8003 - 2.4777 -4.8754
8.5942 3.5505 1.0029 22.1505 2.2021
1.4800
- 0.9656 - 2.4660 -0.0561 0.4875 - 1.0749 - 1.0078 -0.6811 -0.9618 - 1.3101 -0.9339
0.30 0.30 0.30 0.30 0.30 0.30
0.40 0.40 0.40 0.40 0.40 0.40
0.40 0.40 0.40 0.40 0.40 0.40
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
0.2 0.4 0.8 2.2 0.6 2.0
0.2 0.3 0.2 3.3 2.1 2.9
0.4 0.8 0.4 0.1 0.4
0.1
7.1 3.5 6.0 6.4 3.1 6.5 2.4 4.9 8.6 5.3
19 19 19 19 19 19
18 18 18 18 18 18
17 17 17 17
16 16 16
z
I
S
% S
6
x,a
T. OhtalThermochim.
Acta 23.5 (1994) l-10
+xgp=x,
(1 -x,)cc +(l
(12) -X&I
= 1 --CC,
(13)
where x,, xl and xg are mole fractions of the same component. When we solve for CIand p, the enthalpy departure of the mixture is given by AH’ = a( AH:,), + j?(AH:,),
(14)
where (AH:,), and (AH&), are the enthalpy departure functions of a liquid mixture of compositions x1 and a vapor mixture of composition 5. RESULTS
The two ternary systems carbon dioxide( l)-n-hexane(2)-toluene(3) and methane( l)-carbon dioxide( 2)-hydrogen sulfide( 3) were investigated at high pressures. To use the PRSV equation of state, it is necessary to obtain pure component parameters for each component, T,, PC, co, and rcI, which are available in the literature [3, 111. The values of Cli,for the related binary systems were taken from a previous paper [ 11, except for the two systems containing hydrogen sulfide. In this work, the values of CI,,for these systems were empirically assigned as follows: methane-hydrogen sulfide, 0.40; carbon dioxide-hydrogen sulfide, 0.30. Table 1 presents the binary parameters for the related binary systems and absolute arithmetic mean deviations. Agreement with the experimental data seems to be acceptable. The carbon dioxide(l)-n-hexane(2)of the critical region
toluene(3) system in the vicinity
The excess enthalpy data for this ternary system were studied at five temperatures between 308.15 and 573.15 K and at pressures of 7.50 and 12.50 MPa. The ternary HE data were determined by adding carbon dioxide to an n-hexane-toluene mixture of known composition. Three mixtures with n-hexane mole fractions of 0.25, 0.50 and 0.75 were measured [20-241. Two-phase regions appear at 308.15, 358.15, 413.15, 470.15 and 573.15 K and 7.50 MPa, and at 358.15, 413.15 and 470.15 K and 12.50 MPa. Large changes in sign and magnitude of the ternary HE values with the variations in temperature, pressure and composition are observed. The predicted results for the system are listed in Table 2, together with the maximum and minimum values of each set of ternary HE data. Figures 1 and 2, respectively, indicate the isotherms with x*/x3 = 3 at 7.50 and 12.50 MPa. Similar results are obtained for the other two series of ternary HE determinations with x2/x3 = l/3 and 1. Isotherms at some temperatures show a minimum and a maximum connected by a linear section which
T. OhtalThermochim. TABLE
1
Acta 235 (1994) l-10
2
Predicted
results for the carbon
Temperature K
308.15 358.15 413.15 470.15 573.15
in
dioxide( 1) -n-hexane(
2) -toluene(
3) system
Pressure in MPa
No. of data points
H&, in J mol-’
H&, in J mol-’
Abs. arith. mean dev. in J molV’
Ref.
7.50 12.50 1.50 12.50 7.50 12.50 7.50 12.50 7.50 12.50
83 76 92 84 84 15 82 74 82 80
-4650.0 -45.0 -2130.0 -2100.0 - 860.0 - 966.0 146.0 9 1.O 297.0 244.0
137.0 831.0 576.0 242.0 2200.0 984.0 4660.0 2400.0 7120.0 4300.0
173.3 76.5 69.6 139.3 123.9 91.2 139.8 34.3 258.8 83.2
20 20 21 21 22 22 23 23 24 24
I
I
I
I
2
-7 8 c, x
1 0
1
%-1 -2 -3 -4 -5 0.0
0.2 Mole
0.4 fraction
0.6 of
0.8
1.0
CO2
Fig. 1. Comparison of experimental and predicted excess enthalpies for the carbon dioxtde( I)-n-hexane( 2)-toluene( 3) system at 7.50 MPa and x2/x3 = 3. Experimental: 0, at 308.15 K; A, at 358.15 K; n , at 413.15 K; +, at 470.15 K; V, at 573.15 K. Predicted: --.
T. OhtalThermochim.
5
I
I
I
Acta 235 (1994) I-10
I
4
3 ‘i’ “E c, x 11
2
*z h 0
-1
-2 0.0
0.2
0.4 Mole fraction
0.6 of
0.8
1.0
CO2
Fig. 2. Comparison of experimental and predicted excess enthalpies for the carbon dioxide( I)-n-hexane(2)-toluene(3) system at 12.50 MPa and x,/x, = 3. Experimental: 0, at 308.15 K; A, at 358.15 K; n , at 413.15 K; +, at 470.15 K; 7, at 573.15 K. Predicted: -.
corresponds to a two-phase region for the ternary mixture. The accuracy of the predictions by the present equation of state can be considered satisfactory. In this work, the predictions of the HE data in the two-phase region were made on the assumption that the boundaries of the two-phase region in the ternary mixture can be approximately expressed by the straight lines connecting the limits of those in the binary systems carbon dioxide-n-hexane and carbon dioxide-toluene, because the vapor-liquid equilibrium data for the ternary system are not reported and the shape of the two-phase region is unknown. Details of the calculational procedure for this estimation have been described by Casielles et al. [5]. In their work, the Peng-Robinson equation of state was employed to predict ternary HE data for the same system from binary data alone. On the whole, the predictive ability of the present approach seems to be slightly better than that of Casielles et al. Further improvements of the ternary HE predictions in the two-phase
T. OhtalThermochim. TABLE Predicted
3 results for the methane(
Temperature K
in
293.15
1) -carbon
No. of data points
Pressure in MPa
0.507 1.520 0.507 1.520 0.507 1.520
305.15 313.15
region
9
Acta 235 (1994) I-10
might
equilibrium
be
obtained
by
dioxide(2)
-hydrogen
sulfide( 3) system
HE,, in J mol-’
Hz,, in J mol-’
Abs. arith. mean dev. in J mol-’
Ref.
29.9 73.7 27.3 57.2 25.9 38.1
50.6 129.9 41.2 99.1 32.3 90.8
3.6 7.9 11.3 4.7 2.9 4.7
6 6 6 6 6 6
using
more
accurate
ternary
vapor-liquid
data.
The methane(l)-carbon gaseous region
dioxide(2)-hydrogen
suljide(3) system in the
Predictions of ternary H” data for the gaseous mixtures were carried out at 293.15, 305.15 and 313.15 K and at 0.507 and 1.520 MPa. The experimental values of HE under these conditions are small and endothermic [6]. The predicted results for the system are shown in Table 3. The present equation of state provides good results for the ternary system. The mean percentage deviation between the experimental and predicted values of the ternary excess enthalpies is, in most cases, below lo%, and it is comparable with that of Barry et al. [6] who utilized the Redlich-Kwong and the Benedict-Webb-Rubin equations of state to represent the ternary HE data using the binary interaction parameters obtained from the ternary data. CONCLUSIONS
The PRSV equation of state coupled with the NRTL mixing rule was utilized for the prediction of ternary excess enthalpy data at high pressures. For the carbon dioxide( 1) -n-hexane(2) -toluene( 3) system in the vicinity of the critical region and for the methane( 1) -carbon dioxide( 2) -hydrogen sulfide( 3) system in the gaseous region, the equation of state provides good predictions of ternary HE data using only binary data. REFERENCES 1 T. Ohta, Thermochim. Acta, 230 (1993) 1. 2 R. Stryjek and J.H. Vera, Equations of State Ser., (1986) 560.
Theories
and Applications,
ACS Symp.
10
T. OhtalThermochim.
Acta 235 (1994) l-10
3 R. Stryjek and J.H. Vera, Can. J. Chem. Eng., 64 (1986) 323. 4 M.J. Huron and J. Vidal, Fluid Phase Equilibria, 3 (1979) 255. 5 A.G. Casielles, C. Pando, J.A.R. Renuncio, J.J. Christensen and R.M. Izatt, Thermochim. Acta, 154 (1989) 57. 6 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 28 (1983) 375. 7 R.C. Reid, J.M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, 4th edn., McGraw-Hill, New York, 1987. 8 H. Renon and J.M. Prausnitz, AIChE J., 14 (1968) 135. 9 J.A. Nelder and R. Mead, Comput. J., 7 (1965) 308. 10 K.L. Lewis, SE. Mosedale and C.J. Wormald, J. Chem. Thermodyn., 9 (1977) 121. 11 P. Proust and J.H. Vera, Can. J. Chem. Eng., 67 ( 1989) 170. 12 J.J. Christensen, T.A.C. Walker, R.S. Schofield, P.W. Faux, P.R. Harding and R.M. Izatt, J. Chem. Thermodyn., 16 (1984) 445. 13 J.J. Christensen, D.M. Zebolsky and R.M. Izatt, J. Chem. Thermodyn., 17 (1985) 183. 14 C. Pando, J.A.R. Renuncio, R.S. Schofield, R.M. lzatt and J.J. Christensen, J. Chem. Thermodyn., 15 (1983) 747. 15 J.J. Christensen, D.M. Zebolsky and R.M. Izatt, J. Chem. Thermodyn., 17 (1985) 1. 16 P.W. Faux, J.J. Christensen, and R.M. Izatt, J. Chem. Thermodyn., 19 (1987) 757. 17 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 27 (1982) 258. 18 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, J. Chem. Eng. Data, 27 (1982) 436. 19 A.O. Barry, S.C. Kaliaguine and R.S. Ramalho, Can. J. Chem. Eng., 61 (1983) 241. 20 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 20 (1988) 897. 21 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 20 (1988) 559. 22 P.W. Faux, J.J. Christensen, R.M. Izatt, C. Pando and J.A.R. Renuncio, J. Chem. Thermodyn., 20 (1988) 503. 23 P.W. Faux, J.J. Christensen, R.M. Izatt, C. Pando and J.A.R. Renuncio, J. Chem. Thermodyn., 20 (1988) 1297. 24 C. Pando, J.A.R. Renuncio, P.W. Faux, J.J. Christensen and R.M. Izatt, J. Chem. Thermodyn., 21 (1989) 41.