- Email: [email protected]
n-octacosane
MgFHASE EOuIUgRIA ELSEVIER
Fluid Phase Equilibria 128 (1997) 229-240
Effect of pressure on solid-liquid equilibrium for the system carbon dioxide/n-decane/n-octacosane H y o - G u k L e e , P h i l i p A. S c h e n e w e r k , J o a n n e W o l c o t t , F r a n k R. G r o v e s Jr. * Chemical Engineering Department, Petroleum Engineering Department, Louisiana State University, Baton Rouge, LA 70803, USA
Received 9 December 1995; accepted 17 July 1996
Abstract Solid-liquid saturation lines (pressure vs temperature for a saturated solution of given composition) were determined for n-octacosane in mixtures of n-decane and carbon dioxide. The carbon dioxide content of the solutions ranged from 0 to 90 mol %, at pressures up to 25 MPa. The data were correlated by means of the perturbed hard sphere chain equation of state. The equation of state requires three pure component constants for each component plus three binary interaction parameters. Two of the interaction parameters were obtained from binary data. The third was adjusted to fit the ternary data. The correlation was satisfactory for carbon dioxide contents up to 80 mol %. For higher carbon dioxide levels the octacosane solubility predicted by the correlation was consistently low. Keywords: Experimental; Data SLE; Methods of calculation; Equations of state
1. I n t r o d u c t i o n There is relatively little data in the literature on the effect o f pressure on solid-liquid equilibria. The subject is o f interest in the petroleum production industry where deposition o f wax or asphaltenes is sometimes associated with changes in pressure, temperature, or composition during oil production. A particular area o f interest is the enhanced oil recovery process that involves injection o f carbon dioxide. Solid deposition is observed for some crude oils. This paper reports experimental equilibrium data for a simple well-defined carbon d i o x i d e / h y d r o c a r b o n system. The system may serve as a first approximation to the more complex mixtures involved in oil production.
* Corresponding author. 0378-3812/97/$17.00 Copyright© 1997 ElsevierScienceB.V. All fights reserved. PH S0378-3812(96)03166-4
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
230
The objectives of this research were: 1. to obtain experimental data on the solubility of n-octacosane in various mixtures of carbon dioxide and n-decane. The data were to cover pressures ranging from 2 to 25 MPa and carbon dioxide concentrations from 0 to 90 mol %. 2. to correlate the resulting data over the whole range of pressure and composition with the aid of an appropriate equation of state.
2. Related work A number of papers have appeared describing binary carbon dioxide-hydrocarbon systems. McHugh et al. (1984), Yau and Tsai (1993) and Reverchon et al. (1993) determined the solubility of solid n-octacosane in supercritical carbon dioxide. These results are useful in evaluating the carbon dioxide-octacosane interaction parameter for an equation of state. Luks and coworkers measured solid-liquid-vapor equilibria for various binary systems containing carbon dioxide. Hydrocarbon components were: n-eicosane, Huie et al. (1973); n-dotriacontane and n-docosane, Fall and Luks (1984); n-nonadecane and n-heneicosane, Fall et al. (1985). While these papers provide useful results, they do not provide quantitative correlations of the data by equation of state or other methods. Some more recent papers on solid-liquid-vapor equilibrium for ternary systems include: Hong and Luks (1991) on carbon dioxide/n-hexane/n-hexatriacontane; Hong and Luks (1992) on carbon dioxide/toluene/naphthalene; Hong et al. (1993) on methane with n-hexane/n-hexatriacontane, toluene/naphthalene, and n-hexane/naphthalene. In the latter paper the authors correlate the data using the Soave-Redlich-Kwong and Peng-Robinson equations of state and report values of the carbon dioxide-hydrocarbon and methane-hydrocarbon interaction parameters. In the correlations, the work focuses on the vapor-liquid equilibrium rather than on the solubility of the solid hydrocarbon. 3. Theory To develop an equation for octacosane solubility we visualize pure solid octacosane in equilibrium with a liquid solution containing carbon dioxide, n-decane and dissolved n-octacosane. The basic condition for equilibrium is the equality of the fugacity of the pure solid octacosane and the fugacity of the octacosane dissolved in the liquid phase. fff = f/L The fugacity of the pure solid octacosane is given by:
(vs
f/s = Pi, ex p _.~(p
_ Pi* )
)
(1)
(2)
The fugacity of the dissolved octacosane can be obtained from: (3)
f i L ~-- X i q o i P
Equating these fugacities and solving for the solubility gives:
Vi*exp(vS/RT( P - ei* )) Xi =
~i P
(4)
231
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
The fugacity coefficient, q~/, of octacosane in the liquid phase can be calculated with the aid of an appropriate equation of state. Because the right hand side is a function of octacosane solubility, x i, Eq. (4) must be solved iteratively. The pure solid molar volume of n-octacosane was extrapolated from the data of Templin (1956) giving the equation vs = 0.11828T + 381.62
(5)
To obtain the vapor pressure of the pure solid, Pi* we first extrapolated liquid vapor pressure data to the subcooled liquid region. Then a simple thermodynamic cycle was used to evaluate the ratio of solid vapor pressure to subcooled liquid vapor pressure. The liquid vapor pressure data of Chirico et al. (1989) was extrapolated below the octacosane triple point using the Cox correlation given in the Chirico paper, ln( P ~ ]
Pref/
= (1.O-705.0/T)exp(a
l+b,T+c,T
2)
(6)
where P r e f = 1.01325 × 105 Pa, a~ = 3.41304, b I = - 1.8894 × 10 -3, and c I = 1.04575 × 10 -6. A simple thermodynamic cycle (Prausnitz et al., 1986), involving alternate paths between solid and liquid, then gives l n t Pi ]
RT m T -
1
RE
-~-
1
(7)
Thermodynamic properties used in this and other calculations are given in Table 1. The fugacity coefficient, q~i, for octacosane in the liquid phase can be obtained from an equation of state via the basic thermodynamic relation
RT In q~i = [,..
dV - RT In Z
(8)
T,V ,n
The experimental data cover a wide range of mixture compositions, from zero carbon dioxide (octacosane in decane) to 90 mol % carbon dioxide. The solvent in the latter mixture might be
Table 1 Thermodynamic properties of n-octacosane Property Unit T~
Pc To T,, A H,,
Tt AH,
K MPa K K J mol- l K Jmol i
a Gasem and Robinson (1990). b Dreisbach (1959). c Schaerer et al. (1955).
Value
Source
827.4 0.661 704.75 334.35 64643 331.15 35438
a a b c c c c
232
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
Table 2 Parameters for equation of state Pammeter Unit CO 2 n-Clo n-C28
r none 4.275 7.083 19.4625
~ m 2.228x10 -~° 3.40x10 - t ° 3.30x10 10
e/k K 121.7 202.7 198.8
characterized as supercritical carbon dioxide with n-decane as cosolvent. The correlation of such a wide range of data presents a considerable challenge for the equation of state. Attempts to correlate the data using a simple cubic equation of state ( S o a v e - R e d l i c h - K w o n g equation) failed, often yielding qualitatively incorrect results as carbon dioxide content increased. The challenge was met by choosing the perturbed hard sphere chain equation of state of Song et al. (1994a,b). The equation uses a hard sphere chain equation of state as the reference system and adds a Van der Waals attractive term as the perturbation. The reference part of the equation for mixtures is derived theoretically and involves no mixing rules. It requires three pure component parameters: r, o-, and E. The parameter r represents the effective number of hard spheres in the molecule. The parameters E and o- come from a pair potential for hard sphere interactions: E is the depth of the potential minimum, and o- is the distance between hard sphere centers at this minimum. The values used for these pure component parameters are summarized in Table 2. For carbon dioxide and n-decane all three parameters were obtained from the paper of Song et al. (1994a). For n-octacosane the parameters were fitted to vapor pressure and saturated liquid density data between 300 and 705 K. The data of Chirico et al. (1989), extrapolated to 300 K were used for the vapor pressure. For liquid density the data of Templin (1956) were extrapolated to 705 K using the correlation of Gunn and Yamada (1971). In addition to the three parameters, two universal functions, F a (kTs - l ) and F h ( k T s - l ) , are required. They were taken from the paper of Song et al. (1994a). The perturbation part of the equation of state has the form
(P) -~
am pe. = V RT
(9)
The mixture parameter, am, must be obtained form the pure component parameters,
a i = 2/37rNAFatr3Ei
(10)
with the help of an appropriate mixing rule. In this work two mixing rules were studied. The first, referred to hereafter as the empirical mixing rule, is the conventional Van der Waals geometric mean formula, corrected by a binary interaction parameter. m
am =
Exixjrirjaij(1 i,j
-- k i j )
(11)
with
aij= a~iai
(12)
H.-G. Lee et al. / Fluid Phase Equilibria 128 (1997) 229-240
233
The other mixing rule, suggested by Song et al. (1994b), is more in harmony with the perturbed hard sphere chain theory. It uses
aij = 2/3"rt'NAO'i~EijFij
(13)
and retains the geometric mean definition for EuFij, but uses the arithmetic mean for o'i), o]j=
tr, + o) 2
(14)
in accordance with the role of or as the distance between hard sphere centers at minimum potential. It is referred to hereafter as the theoretical mixing rule. To use the equation of state, it is necessary to fit the binary interaction coefficients, k u, to experimental data. In this work the coefficient kz3 for decane-octacosane was determined from data on the solubility of octacosane in pure decane. Data on solubility of octacosane in pure supercritical carbon dioxide, McHugh et al. (1984), was used to evaluate k13. Finally the carbon dioxide-decane coefficient, k12, was fitted to the ternary data. A computer program was written to solve Eq. (4) iteratively for octacosane solubility. Newton's method, with the derivative evaluated numerically, was used to converge the iterations.
4. Experimental The experiments were done in a Ruska pressure-volume-temperature (PVT) apparatus shown schematically in Fig. 1. The PVT apparatus has two 191 mL equilibrium cells in a temperature
c t
D
A. RUSKA PUMP
G, CO 2 RESERVOIR
B. C. D. E.
H. FLASH SEPARATOR I . WET TEST METER TC,TEMPERATURE CONTROLLER T I . TEMPERATURE INDICATOR
PRESSURE GAUGE P V T CELL AIR BATH CATHETOMETER
F. MERCURY RESERVOIR
Fig. 1. Experimental apparatus.
234
H.-G. Lee et al./Fluid Phase Equilibria 128 (1997) 229-240
controlled air bath. The cells have windows through which their contents can be observed by means of a cathetometer telescope. A mechanism for inverting and rocking the cells for mixing is provided. A temperature controller (Omega) maintains a constant temperature in the air bath. The temperature of the cells is measured by platinum resistance thermometers connected to an Omega temperature indicator. Two Ruska high pressure syringe type pumps are used for metering fluids into the equilibrium cells. The maximum pressure for the PVT apparatus is 27.6 MPa (4000 psia). In a typical experiment, weighed amounts of decane and melted octacosane were poured through a port into an equilibrium cell at atmospheric pressure. The temperature of the cell was then raised so that the hydrocarbon mixture (of known composition) remained liquid. Mercury was pumped into the cell with a Ruska pump to displace air and raise the pressure. A known volume of carbon dioxide was now displaced from a reservoir into the cell by metering mercury into the reservoir with the second Ruska pump. The pressure of the cell was kept constant during addition of carbon dioxide by removing mercury from the cell with the other Ruska pump. At this point the cell contained a liquid mixture of carbon dioxide, n-decane, and n-octacosane of known composition at a known temperature and pressure. Solubility data were now obtained by determining the pressures at which the mixture was just saturated at various temperatures. The temperature was first set at a desired value such that the
Table 3 Saturation conditions for n-octacosane in n-decane T (K)
P (MPa)
Calculated Xc2s
x¢28 = 0.06013 308.0 309.0 310.4 311.9 312.5
4.51 8.72 14.34 20.51 22.44
0.05907 0.06083 0.06410 0.06684 0.06832
Average absolute error:6.9%,empiricalrule;8.9%,theoreticalrule. Xc2s = 0.08198 310.3 311.0 312.2 313.4 314.4 314.7
1.792 4.58 9.68 15.47 19.54 21.23
0.08031 0.08191 0.08268 0.08294 0.08467 0.08461
Average absolute error: 1.8%, empirical rule; 0.85%, theoretical rule. Xc28 = 0.1074 312.7 313.5 314.4 315.3 316.3 317.4
1.103 4.51 8.93 12.58 16.92 21.61
Average absolute error: 2.8%, empirical rule; 6.3%, theoretical rule.
0.1031 0.1034 0.1028 0.1043 0.1058 0.1081
H.-G. Lee et d/Fluid
Phase Equilibria
128 (1997) 229-240
235
Data of McHugh et al. (1984) A
307.7 K
0
318.4 K
0
.0004__
0
.0003_. g ‘S 0 e Lu $ w V h
.0002_-
.OOOl__
0, 0
I 10
I
P, MPa
Fig. 2. Solubility of n-octacosane
20 in supercritical
30 CO,.
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
236
o . o. o . o. o. o .
:F.
m. o LL
<~-
0
0
0
0
0
O
t~
E 0
c-~ ~
o
~
~
~
~
?
"--
~.
o
0
~q b~ 0
0
0
0
0
0
- - c/~----:~-o-~ -m~mZmm<~
0
"7,
II
. ~ . ~"
e-
64 .&
0 0
0
0
~
0
0
0
0
"E
O ¢.,
.~
e~
E O
~3
~.
e-,
m. o
o
~.
o
" d
e-
q q q q q
e'-'
0
0
0
0
0
0
o ~
u
d II ~o
~e
o
~d NNN~N
<~
<~
o
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
237
e~
~eq
0 0
,:~
° .
,2.~~
0
~. o
"2.
0
0
0
°.
e~
t~
e~
oO
,.2
0 e-
U
ll
~,e4
t~
t~
~5 II o
~8 II
N
~8
H.-G. Lee et al./Fluid Phase Equilibria 128 (1997) 229-240
238
mixture was all liquid. The pressure was then slowly raised, while observing the solution through the viewing window, until solid particles appeared. The pressure was then gradually reduced until the particles went back into solution. This procedure was repeated, slowly raising and lowering the pressure, to determine the pressure at which the solution was just saturated. During these observations the cell was manually rocked to hasten the attainment of equilibrium. For some experiments at high carbon dioxide levels, the system was very sensitive to pressure changes. Under these conditions the pressure was increased until solid appeared. Saturation conditions were then determined by slowly raising the temperature until the solid just disappeared. By this procedure, saturation lines--pressure vs temperature at saturation for the known composit i o n - w e r e obtained. The procedure was repeated with various compositions to obtain extensive solid-liquid equilibrium data for the ternary system. The temperature measurement is believed accurate to within 0.2 K. Pressure could be read within 0.03 MPa on a large Bourdon gage. The reproducibility of the pressure measurements was +0.3 MPa. The maximum error in pressure is estimated to be 0.7 MPa. The mass of the sample was measured within 0.01 g, and the maximum error in composition is estimated to be less than 1.0%. The materials used in this research were n-decane, Aldrich, 99 + %; n-octacosane, Aldrich, 99 + %, and carbon dioxide, Liquid Carbonic, 99 + %. They were used without further purification.
5. Results Table 3 shows saturation conditions for octacosane in pure decane. These data were used to fit the decane-octacosane interaction coefficient, k23, of the perturbed hard sphere chain equation of state. Calculated solubilities using the best fit are also shown in the table. The calculated solubilties were obtained using the empirical Van der Waals mixing rule with geometric mean o-,.~.The goodness of fit is indicated in the table by the average absolute percentage error of the calculated solubilities at each solubility level. The fit was also done using the theoretically more sound mixing rule with arithmetic mean o-ij. The average absolute errors were similar with this approach. Fig. 2 shows the data of McHugh et al. (1984) on the solubility of octacosane in supercritical carbon dioxide at 307.7 and 318.4 K. These data were used to fit the carbon dioxide-octacosane interaction coefficient, kl3. The calculated lines based on the best fit using the empirical mixing rule are also shown. The theoretical mixing rule was also used to fit the data. The goodness of fit was approximately the same using both mixing rules and was only semi-quantitative. The best fit interaction coefficients were as follows: k23
=
0 . 0 0 0 4 6 T - 0.13869,T < 311
k23
=
0.00017333T- 0.049537,T > 311
k13
=
0.000607477T- 0.0825206
(15)
Table 4 shows the saturation conditions for octacosane in carbon dioxide-decane mixtures. Values of octacosane solubility, calculated using the equation of state for a solvent of the same composition at the same temperature and pressure, are given for comparison. The calculated solubilities are based on the empirical mixing rule.
H . - G . L e e et a l . / F l u i d P h a s e E q u i l i b r i a 128 (1997) 2 2 9 - 2 4 0
239
6. Discussion of results Correlation of data for such a wide range of carbon dioxide contents, ranging from 0 to 90 mol %, is a considerable challenge for the equation of state. The correlation was done with both mixing rules. The carbon dioxide-decane interaction coefficient, k~2, was adjusted to give the best fit. Average percentage errors for the two rules are shown in Table 4 at each solubility level. Overall, the empirical mixing rule gives somewhat better results. It represents the data well for carbon dioxide contents up to 80 mol % using a constant carbon dioxide-decane interaction coefficient, k~2 = 0.05. The average absolute per cent error for all the data in this concentration range is 5.03%. The maximum absolute per cent eror is 15.45%. For carbon dioxide contents of 85 mol % and higher the correlation is only semi-quantitative. At 85 tool % carbon dioxide the calculated solubilities were uniformly low by about 25%. For 90 mol % carbon dioxide they were low by about 50%. For the empirical mixing rule there did not seem to be any consistent trend of k12 with temperature. Rather, it was possible to improve the correlation at high carbon dioxide levels by adjusting k~2 as a function of carbon dioxide content. This suggests that a modification of the equation of state to make the perturbation term density dependent might succeed in covering the whole range of carbon dioxide content.
7. Conclusions Solubility data for octacosane in mixtures of carbon dioxide and decane were obtained covering a wide range of carbon dioxide contents. The data were correlated by means of a perturbed hard sphere chain equation of state. The correlation was good for carbon dioxide contents of 80 mol % or less but gave low results for higher carbon dioxide percentages.
8. Nomenclature a i a ?~l ~-.
L= Fo, 6Hm= AH,
=
kij
=
n i =
P
=
Pi*
=
Pref" ri
R T=
=
equation of state parameter for component i, perturbation mixture parameter for perturbation term fugacity of component i, Pa universal functions for equation of state, dimensionless enthalpy change for melting, J m o l - l enthalpy change for transition, J m o l - t equation of state interaction parameter for i , j pair number of moles of component i pressure, Pa vapor pressure of component i, Pa reference pressure, Pa effective number of hard spheres per molecule universal gas constant, J K - i mol absolute temperature, K
240
m =
r,= Ui
V= X i -~-
Z= gOi = E i ~-
o-i= O =
H . - G . L e e et al. / F l u i d P h a s e Equilibria 1 2 8 ( 1 9 9 7 ) 2 2 9 - 2 4 0
melting point temperature, K transition point temperature, K molar volume of component i, m3/mole volume, nl 3 mole fraction of component i compressibility factor, d i m e n s i o n l e s s G r e e k symbols fugacity coefficient for component i, dimensionless potential well depth parameter, J m o l - l pair potential distance parameter, m density, mol m 3
Acknowledgements
This work was supported by member companies of the Louisiana State University CO 2 Flooding Consortium including ARCO Oil and Gas, Chevron Oilfield Research, Greenhill Petroleum, and Texaco.
References Chirico, R.D., Nguyen, A., Steele, W. and Strube, M.M., 1989. J. Chem. Eng. Data, 34: 149. Dreisbach, R.R., 1959. Adv. Chem. Set., 22: 178. Fall, D.J., Fall, J.L. and Luks, K.D., 1985. J. Chem. Eng. Data, 30: 82. Fall, D.J. and Luks, K.D., 1984. J. Chem. Eng. Data, 29: 413. Gasem, K.A.M. and Robinson, R.L. Jr., 1990. Fluid Phase Equilibria, 58: 13. Gunn, R.D. and Yamada, T., 1971. AIChE J., 17: 1341. Hong, S.P., Green, K.A. and Luks, K.D., 1993. Fluid Phase Equilibria, 87: 255. Hong, S.P. and Luks, K.D., 1991. J. Supercritical Fluids, 4: 227. Hong, S.P. and Luks, K.D., 1992. Fluid Phase Equilibria, 74: 133. Huie, N.C., Luks, K.D. and Kohn, J.P., 1973. J. Chem. Eng. Data, 18:311. McHugh, M.A., Seckner, A.J. and Yogan, T.J., 1984. Ind. Eng. Chem. Fund,, 23: 493. Prausnitz, J.M., Lichtenthaler, R.N. and Azevedo, E.G. de, 1986. Molecular Theory of Fluid-Phase Equilibria, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ, p. 418. Reverchon, E., Russo. P. and Stassi, A., 1993. J. Chem. Eng. Data, 38: 458. Schaerer, A.A., Busso, C.J., Smith, A.E. and Skinner, L.B., 1955. J. Am. Chem. Soc., 77: 2017. Song, Y., Lambert, S.M. and Prausnitz, J.M., 1994a. Ind. Eng. Chem. Res., 33: 1047. Song, Y., Lambert, S.M. and Prausnitz, J.M., 1994b. Ind. Eng. Chem. Res., 49: 2765. Templin, P.R., 1956. Ind. Eng. Chem., 48: 154. Yau, J.-S. and Tsai, F.-N., 1993. J. Chem. Eng. Data, 38: 171.
Fluid Phase Equilibria 128 (1997) 229-240
Effect of pressure on solid-liquid equilibrium for the system carbon dioxide/n-decane/n-octacosane H y o - G u k L e e , P h i l i p A. S c h e n e w e r k , J o a n n e W o l c o t t , F r a n k R. G r o v e s Jr. * Chemical Engineering Department, Petroleum Engineering Department, Louisiana State University, Baton Rouge, LA 70803, USA
Received 9 December 1995; accepted 17 July 1996
Abstract Solid-liquid saturation lines (pressure vs temperature for a saturated solution of given composition) were determined for n-octacosane in mixtures of n-decane and carbon dioxide. The carbon dioxide content of the solutions ranged from 0 to 90 mol %, at pressures up to 25 MPa. The data were correlated by means of the perturbed hard sphere chain equation of state. The equation of state requires three pure component constants for each component plus three binary interaction parameters. Two of the interaction parameters were obtained from binary data. The third was adjusted to fit the ternary data. The correlation was satisfactory for carbon dioxide contents up to 80 mol %. For higher carbon dioxide levels the octacosane solubility predicted by the correlation was consistently low. Keywords: Experimental; Data SLE; Methods of calculation; Equations of state
1. I n t r o d u c t i o n There is relatively little data in the literature on the effect o f pressure on solid-liquid equilibria. The subject is o f interest in the petroleum production industry where deposition o f wax or asphaltenes is sometimes associated with changes in pressure, temperature, or composition during oil production. A particular area o f interest is the enhanced oil recovery process that involves injection o f carbon dioxide. Solid deposition is observed for some crude oils. This paper reports experimental equilibrium data for a simple well-defined carbon d i o x i d e / h y d r o c a r b o n system. The system may serve as a first approximation to the more complex mixtures involved in oil production.
* Corresponding author. 0378-3812/97/$17.00 Copyright© 1997 ElsevierScienceB.V. All fights reserved. PH S0378-3812(96)03166-4
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
230
The objectives of this research were: 1. to obtain experimental data on the solubility of n-octacosane in various mixtures of carbon dioxide and n-decane. The data were to cover pressures ranging from 2 to 25 MPa and carbon dioxide concentrations from 0 to 90 mol %. 2. to correlate the resulting data over the whole range of pressure and composition with the aid of an appropriate equation of state.
2. Related work A number of papers have appeared describing binary carbon dioxide-hydrocarbon systems. McHugh et al. (1984), Yau and Tsai (1993) and Reverchon et al. (1993) determined the solubility of solid n-octacosane in supercritical carbon dioxide. These results are useful in evaluating the carbon dioxide-octacosane interaction parameter for an equation of state. Luks and coworkers measured solid-liquid-vapor equilibria for various binary systems containing carbon dioxide. Hydrocarbon components were: n-eicosane, Huie et al. (1973); n-dotriacontane and n-docosane, Fall and Luks (1984); n-nonadecane and n-heneicosane, Fall et al. (1985). While these papers provide useful results, they do not provide quantitative correlations of the data by equation of state or other methods. Some more recent papers on solid-liquid-vapor equilibrium for ternary systems include: Hong and Luks (1991) on carbon dioxide/n-hexane/n-hexatriacontane; Hong and Luks (1992) on carbon dioxide/toluene/naphthalene; Hong et al. (1993) on methane with n-hexane/n-hexatriacontane, toluene/naphthalene, and n-hexane/naphthalene. In the latter paper the authors correlate the data using the Soave-Redlich-Kwong and Peng-Robinson equations of state and report values of the carbon dioxide-hydrocarbon and methane-hydrocarbon interaction parameters. In the correlations, the work focuses on the vapor-liquid equilibrium rather than on the solubility of the solid hydrocarbon. 3. Theory To develop an equation for octacosane solubility we visualize pure solid octacosane in equilibrium with a liquid solution containing carbon dioxide, n-decane and dissolved n-octacosane. The basic condition for equilibrium is the equality of the fugacity of the pure solid octacosane and the fugacity of the octacosane dissolved in the liquid phase. fff = f/L The fugacity of the pure solid octacosane is given by:
(vs
f/s = Pi, ex p _.~(p
_ Pi* )
)
(1)
(2)
The fugacity of the dissolved octacosane can be obtained from: (3)
f i L ~-- X i q o i P
Equating these fugacities and solving for the solubility gives:
Vi*exp(vS/RT( P - ei* )) Xi =
~i P
(4)
231
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
The fugacity coefficient, q~/, of octacosane in the liquid phase can be calculated with the aid of an appropriate equation of state. Because the right hand side is a function of octacosane solubility, x i, Eq. (4) must be solved iteratively. The pure solid molar volume of n-octacosane was extrapolated from the data of Templin (1956) giving the equation vs = 0.11828T + 381.62
(5)
To obtain the vapor pressure of the pure solid, Pi* we first extrapolated liquid vapor pressure data to the subcooled liquid region. Then a simple thermodynamic cycle was used to evaluate the ratio of solid vapor pressure to subcooled liquid vapor pressure. The liquid vapor pressure data of Chirico et al. (1989) was extrapolated below the octacosane triple point using the Cox correlation given in the Chirico paper, ln( P ~ ]
Pref/
= (1.O-705.0/T)exp(a
l+b,T+c,T
2)
(6)
where P r e f = 1.01325 × 105 Pa, a~ = 3.41304, b I = - 1.8894 × 10 -3, and c I = 1.04575 × 10 -6. A simple thermodynamic cycle (Prausnitz et al., 1986), involving alternate paths between solid and liquid, then gives l n t Pi ]
RT m T -
1
RE
-~-
1
(7)
Thermodynamic properties used in this and other calculations are given in Table 1. The fugacity coefficient, q~i, for octacosane in the liquid phase can be obtained from an equation of state via the basic thermodynamic relation
RT In q~i = [,..
dV - RT In Z
(8)
T,V ,n
The experimental data cover a wide range of mixture compositions, from zero carbon dioxide (octacosane in decane) to 90 mol % carbon dioxide. The solvent in the latter mixture might be
Table 1 Thermodynamic properties of n-octacosane Property Unit T~
Pc To T,, A H,,
Tt AH,
K MPa K K J mol- l K Jmol i
a Gasem and Robinson (1990). b Dreisbach (1959). c Schaerer et al. (1955).
Value
Source
827.4 0.661 704.75 334.35 64643 331.15 35438
a a b c c c c
232
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
Table 2 Parameters for equation of state Pammeter Unit CO 2 n-Clo n-C28
r none 4.275 7.083 19.4625
~ m 2.228x10 -~° 3.40x10 - t ° 3.30x10 10
e/k K 121.7 202.7 198.8
characterized as supercritical carbon dioxide with n-decane as cosolvent. The correlation of such a wide range of data presents a considerable challenge for the equation of state. Attempts to correlate the data using a simple cubic equation of state ( S o a v e - R e d l i c h - K w o n g equation) failed, often yielding qualitatively incorrect results as carbon dioxide content increased. The challenge was met by choosing the perturbed hard sphere chain equation of state of Song et al. (1994a,b). The equation uses a hard sphere chain equation of state as the reference system and adds a Van der Waals attractive term as the perturbation. The reference part of the equation for mixtures is derived theoretically and involves no mixing rules. It requires three pure component parameters: r, o-, and E. The parameter r represents the effective number of hard spheres in the molecule. The parameters E and o- come from a pair potential for hard sphere interactions: E is the depth of the potential minimum, and o- is the distance between hard sphere centers at this minimum. The values used for these pure component parameters are summarized in Table 2. For carbon dioxide and n-decane all three parameters were obtained from the paper of Song et al. (1994a). For n-octacosane the parameters were fitted to vapor pressure and saturated liquid density data between 300 and 705 K. The data of Chirico et al. (1989), extrapolated to 300 K were used for the vapor pressure. For liquid density the data of Templin (1956) were extrapolated to 705 K using the correlation of Gunn and Yamada (1971). In addition to the three parameters, two universal functions, F a (kTs - l ) and F h ( k T s - l ) , are required. They were taken from the paper of Song et al. (1994a). The perturbation part of the equation of state has the form
(P) -~
am pe. = V RT
(9)
The mixture parameter, am, must be obtained form the pure component parameters,
a i = 2/37rNAFatr3Ei
(10)
with the help of an appropriate mixing rule. In this work two mixing rules were studied. The first, referred to hereafter as the empirical mixing rule, is the conventional Van der Waals geometric mean formula, corrected by a binary interaction parameter. m
am =
Exixjrirjaij(1 i,j
-- k i j )
(11)
with
aij= a~iai
(12)
H.-G. Lee et al. / Fluid Phase Equilibria 128 (1997) 229-240
233
The other mixing rule, suggested by Song et al. (1994b), is more in harmony with the perturbed hard sphere chain theory. It uses
aij = 2/3"rt'NAO'i~EijFij
(13)
and retains the geometric mean definition for EuFij, but uses the arithmetic mean for o'i), o]j=
tr, + o) 2
(14)
in accordance with the role of or as the distance between hard sphere centers at minimum potential. It is referred to hereafter as the theoretical mixing rule. To use the equation of state, it is necessary to fit the binary interaction coefficients, k u, to experimental data. In this work the coefficient kz3 for decane-octacosane was determined from data on the solubility of octacosane in pure decane. Data on solubility of octacosane in pure supercritical carbon dioxide, McHugh et al. (1984), was used to evaluate k13. Finally the carbon dioxide-decane coefficient, k12, was fitted to the ternary data. A computer program was written to solve Eq. (4) iteratively for octacosane solubility. Newton's method, with the derivative evaluated numerically, was used to converge the iterations.
4. Experimental The experiments were done in a Ruska pressure-volume-temperature (PVT) apparatus shown schematically in Fig. 1. The PVT apparatus has two 191 mL equilibrium cells in a temperature
c t
D
A. RUSKA PUMP
G, CO 2 RESERVOIR
B. C. D. E.
H. FLASH SEPARATOR I . WET TEST METER TC,TEMPERATURE CONTROLLER T I . TEMPERATURE INDICATOR
PRESSURE GAUGE P V T CELL AIR BATH CATHETOMETER
F. MERCURY RESERVOIR
Fig. 1. Experimental apparatus.
234
H.-G. Lee et al./Fluid Phase Equilibria 128 (1997) 229-240
controlled air bath. The cells have windows through which their contents can be observed by means of a cathetometer telescope. A mechanism for inverting and rocking the cells for mixing is provided. A temperature controller (Omega) maintains a constant temperature in the air bath. The temperature of the cells is measured by platinum resistance thermometers connected to an Omega temperature indicator. Two Ruska high pressure syringe type pumps are used for metering fluids into the equilibrium cells. The maximum pressure for the PVT apparatus is 27.6 MPa (4000 psia). In a typical experiment, weighed amounts of decane and melted octacosane were poured through a port into an equilibrium cell at atmospheric pressure. The temperature of the cell was then raised so that the hydrocarbon mixture (of known composition) remained liquid. Mercury was pumped into the cell with a Ruska pump to displace air and raise the pressure. A known volume of carbon dioxide was now displaced from a reservoir into the cell by metering mercury into the reservoir with the second Ruska pump. The pressure of the cell was kept constant during addition of carbon dioxide by removing mercury from the cell with the other Ruska pump. At this point the cell contained a liquid mixture of carbon dioxide, n-decane, and n-octacosane of known composition at a known temperature and pressure. Solubility data were now obtained by determining the pressures at which the mixture was just saturated at various temperatures. The temperature was first set at a desired value such that the
Table 3 Saturation conditions for n-octacosane in n-decane T (K)
P (MPa)
Calculated Xc2s
x¢28 = 0.06013 308.0 309.0 310.4 311.9 312.5
4.51 8.72 14.34 20.51 22.44
0.05907 0.06083 0.06410 0.06684 0.06832
Average absolute error:6.9%,empiricalrule;8.9%,theoreticalrule. Xc2s = 0.08198 310.3 311.0 312.2 313.4 314.4 314.7
1.792 4.58 9.68 15.47 19.54 21.23
0.08031 0.08191 0.08268 0.08294 0.08467 0.08461
Average absolute error: 1.8%, empirical rule; 0.85%, theoretical rule. Xc28 = 0.1074 312.7 313.5 314.4 315.3 316.3 317.4
1.103 4.51 8.93 12.58 16.92 21.61
Average absolute error: 2.8%, empirical rule; 6.3%, theoretical rule.
0.1031 0.1034 0.1028 0.1043 0.1058 0.1081
H.-G. Lee et d/Fluid
Phase Equilibria
128 (1997) 229-240
235
Data of McHugh et al. (1984) A
307.7 K
0
318.4 K
0
.0004__
0
.0003_. g ‘S 0 e Lu $ w V h
.0002_-
.OOOl__
0, 0
I 10
I
P, MPa
Fig. 2. Solubility of n-octacosane
20 in supercritical
30 CO,.
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
236
o . o. o . o. o. o .
:F.
m. o LL
<~-
0
0
0
0
0
O
t~
E 0
c-~ ~
o
~
~
~
~
?
"--
~.
o
0
~q b~ 0
0
0
0
0
0
- - c/~----:~-o-~ -m~mZmm<~
0
"7,
II
. ~ . ~"
e-
64 .&
0 0
0
0
~
0
0
0
0
"E
O ¢.,
.~
e~
E O
~3
~.
e-,
m. o
o
~.
o
" d
e-
q q q q q
e'-'
0
0
0
0
0
0
o ~
u
d II ~o
~e
o
~d NNN~N
<~
<~
o
H.-G. Lee et a l . / Fluid Phase Equilibria 128 (1997) 229-240
237
e~
~eq
0 0
,:~
° .
,2.~~
0
~. o
"2.
0
0
0
°.
e~
t~
e~
oO
,.2
0 e-
U
ll
~,e4
t~
t~
~5 II o
~8 II
N
~8
H.-G. Lee et al./Fluid Phase Equilibria 128 (1997) 229-240
238
mixture was all liquid. The pressure was then slowly raised, while observing the solution through the viewing window, until solid particles appeared. The pressure was then gradually reduced until the particles went back into solution. This procedure was repeated, slowly raising and lowering the pressure, to determine the pressure at which the solution was just saturated. During these observations the cell was manually rocked to hasten the attainment of equilibrium. For some experiments at high carbon dioxide levels, the system was very sensitive to pressure changes. Under these conditions the pressure was increased until solid appeared. Saturation conditions were then determined by slowly raising the temperature until the solid just disappeared. By this procedure, saturation lines--pressure vs temperature at saturation for the known composit i o n - w e r e obtained. The procedure was repeated with various compositions to obtain extensive solid-liquid equilibrium data for the ternary system. The temperature measurement is believed accurate to within 0.2 K. Pressure could be read within 0.03 MPa on a large Bourdon gage. The reproducibility of the pressure measurements was +0.3 MPa. The maximum error in pressure is estimated to be 0.7 MPa. The mass of the sample was measured within 0.01 g, and the maximum error in composition is estimated to be less than 1.0%. The materials used in this research were n-decane, Aldrich, 99 + %; n-octacosane, Aldrich, 99 + %, and carbon dioxide, Liquid Carbonic, 99 + %. They were used without further purification.
5. Results Table 3 shows saturation conditions for octacosane in pure decane. These data were used to fit the decane-octacosane interaction coefficient, k23, of the perturbed hard sphere chain equation of state. Calculated solubilities using the best fit are also shown in the table. The calculated solubilties were obtained using the empirical Van der Waals mixing rule with geometric mean o-,.~.The goodness of fit is indicated in the table by the average absolute percentage error of the calculated solubilities at each solubility level. The fit was also done using the theoretically more sound mixing rule with arithmetic mean o-ij. The average absolute errors were similar with this approach. Fig. 2 shows the data of McHugh et al. (1984) on the solubility of octacosane in supercritical carbon dioxide at 307.7 and 318.4 K. These data were used to fit the carbon dioxide-octacosane interaction coefficient, kl3. The calculated lines based on the best fit using the empirical mixing rule are also shown. The theoretical mixing rule was also used to fit the data. The goodness of fit was approximately the same using both mixing rules and was only semi-quantitative. The best fit interaction coefficients were as follows: k23
=
0 . 0 0 0 4 6 T - 0.13869,T < 311
k23
=
0.00017333T- 0.049537,T > 311
k13
=
0.000607477T- 0.0825206
(15)
Table 4 shows the saturation conditions for octacosane in carbon dioxide-decane mixtures. Values of octacosane solubility, calculated using the equation of state for a solvent of the same composition at the same temperature and pressure, are given for comparison. The calculated solubilities are based on the empirical mixing rule.
H . - G . L e e et a l . / F l u i d P h a s e E q u i l i b r i a 128 (1997) 2 2 9 - 2 4 0
239
6. Discussion of results Correlation of data for such a wide range of carbon dioxide contents, ranging from 0 to 90 mol %, is a considerable challenge for the equation of state. The correlation was done with both mixing rules. The carbon dioxide-decane interaction coefficient, k~2, was adjusted to give the best fit. Average percentage errors for the two rules are shown in Table 4 at each solubility level. Overall, the empirical mixing rule gives somewhat better results. It represents the data well for carbon dioxide contents up to 80 mol % using a constant carbon dioxide-decane interaction coefficient, k~2 = 0.05. The average absolute per cent error for all the data in this concentration range is 5.03%. The maximum absolute per cent eror is 15.45%. For carbon dioxide contents of 85 mol % and higher the correlation is only semi-quantitative. At 85 tool % carbon dioxide the calculated solubilities were uniformly low by about 25%. For 90 mol % carbon dioxide they were low by about 50%. For the empirical mixing rule there did not seem to be any consistent trend of k12 with temperature. Rather, it was possible to improve the correlation at high carbon dioxide levels by adjusting k~2 as a function of carbon dioxide content. This suggests that a modification of the equation of state to make the perturbation term density dependent might succeed in covering the whole range of carbon dioxide content.
7. Conclusions Solubility data for octacosane in mixtures of carbon dioxide and decane were obtained covering a wide range of carbon dioxide contents. The data were correlated by means of a perturbed hard sphere chain equation of state. The correlation was good for carbon dioxide contents of 80 mol % or less but gave low results for higher carbon dioxide percentages.
8. Nomenclature a i a ?~l ~-.
L= Fo, 6Hm= AH,
=
kij
=
n i =
P
=
Pi*
=
Pref" ri
R T=
=
equation of state parameter for component i, perturbation mixture parameter for perturbation term fugacity of component i, Pa universal functions for equation of state, dimensionless enthalpy change for melting, J m o l - l enthalpy change for transition, J m o l - t equation of state interaction parameter for i , j pair number of moles of component i pressure, Pa vapor pressure of component i, Pa reference pressure, Pa effective number of hard spheres per molecule universal gas constant, J K - i mol absolute temperature, K
240
m =
r,= Ui
V= X i -~-
Z= gOi = E i ~-
o-i= O =
H . - G . L e e et al. / F l u i d P h a s e Equilibria 1 2 8 ( 1 9 9 7 ) 2 2 9 - 2 4 0
melting point temperature, K transition point temperature, K molar volume of component i, m3/mole volume, nl 3 mole fraction of component i compressibility factor, d i m e n s i o n l e s s G r e e k symbols fugacity coefficient for component i, dimensionless potential well depth parameter, J m o l - l pair potential distance parameter, m density, mol m 3
Acknowledgements
This work was supported by member companies of the Louisiana State University CO 2 Flooding Consortium including ARCO Oil and Gas, Chevron Oilfield Research, Greenhill Petroleum, and Texaco.
References Chirico, R.D., Nguyen, A., Steele, W. and Strube, M.M., 1989. J. Chem. Eng. Data, 34: 149. Dreisbach, R.R., 1959. Adv. Chem. Set., 22: 178. Fall, D.J., Fall, J.L. and Luks, K.D., 1985. J. Chem. Eng. Data, 30: 82. Fall, D.J. and Luks, K.D., 1984. J. Chem. Eng. Data, 29: 413. Gasem, K.A.M. and Robinson, R.L. Jr., 1990. Fluid Phase Equilibria, 58: 13. Gunn, R.D. and Yamada, T., 1971. AIChE J., 17: 1341. Hong, S.P., Green, K.A. and Luks, K.D., 1993. Fluid Phase Equilibria, 87: 255. Hong, S.P. and Luks, K.D., 1991. J. Supercritical Fluids, 4: 227. Hong, S.P. and Luks, K.D., 1992. Fluid Phase Equilibria, 74: 133. Huie, N.C., Luks, K.D. and Kohn, J.P., 1973. J. Chem. Eng. Data, 18:311. McHugh, M.A., Seckner, A.J. and Yogan, T.J., 1984. Ind. Eng. Chem. Fund,, 23: 493. Prausnitz, J.M., Lichtenthaler, R.N. and Azevedo, E.G. de, 1986. Molecular Theory of Fluid-Phase Equilibria, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ, p. 418. Reverchon, E., Russo. P. and Stassi, A., 1993. J. Chem. Eng. Data, 38: 458. Schaerer, A.A., Busso, C.J., Smith, A.E. and Skinner, L.B., 1955. J. Am. Chem. Soc., 77: 2017. Song, Y., Lambert, S.M. and Prausnitz, J.M., 1994a. Ind. Eng. Chem. Res., 33: 1047. Song, Y., Lambert, S.M. and Prausnitz, J.M., 1994b. Ind. Eng. Chem. Res., 49: 2765. Templin, P.R., 1956. Ind. Eng. Chem., 48: 154. Yau, J.-S. and Tsai, F.-N., 1993. J. Chem. Eng. Data, 38: 171.