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A comparison of many different mixing rules in a cubic equation of state for 1-alkanol + n-alkane mixtures
Fluid Phase Equilibria, 88 (1993) 63-78 Elsevier Science Publishers B.V., Amsterdam
63
A comparison of many different mixing rules in a cubic equation of state for 1-alkanol + n-alkane mixtures Petr Vo&a’,
Pave1 Dittrich’
and Jorgen
‘Department of Physical Chemistry, 166 28 Prague 6, Czechoslovakia
Lovland*
Prague Inst. of Chemical
*Department of Chemical NTH, Norway
Engineering,
University
Keywords: cubic equation alkanol + n-alkane
of state, mixing
Technology,
of Trondheim,
7034 Trondheim
rules, false phase splitting,
ABSTRACT Different density dependent and density independent mixing rules have been tested for a modified Redlich-Kwong EOS. The attractive parameter of the pure component is considered to be temperature dependent, and adjustable parameters in the attractive term are fitted to achieve a good agreement between experimental and calculated vapour pressure data for the pure compounds. The general form of the considered mixing rule has eight adjustable parameters, and it is possible to choose an arbitrary subset of the adjustable parameters in fitting experimental VLE data. Mixing rules with one, two, three or four parameters were tested for a number of I-alkanol + n-alkane systems. For such systems false liquid-liquid splits occur. They may be avoided by including liquidliquid critical points into the fit or by a constrained parameter fit.
INTRODUCTION The topic of the present communication is a critical evaluation of the possibility of describing 1-alkanol + n-alkane VLE data using different mixing rules in a cubic EOS. Mixtures of 1-alkanol + n-alkane arc important from an industrial point of view and they are known to have difficult behaviour including liquid phase splitting(Englezos et al., 1989, Margerum and Lu, 1990). Above the actual liquidliquid region the y,-x, curve for a binary system has a combination of very steep and very flat portions. Trying to describe such a behaviour typically gives an S-shape behaviour of the calculated y,-x, curve, i.e. false liquid-liquid splits are predicted. The calculated upper liquid-liquid critical point (for considered pressure and given set of adjustable parameters) then lies inside or above the experimental VLE region. We
0378-3812/93/$06.00
01993 Elsevier Science Publishers B.V. All rights reserved
64
have tested most ordinary mixing rules in combination with a cubic equation of state on a large number of binary systems of methanol, ethanol, and 1-propanol in combination with n-alkanes. The mixing rules include density independent ones (Anderko, 1990) and density dependent ones (Mollerup, 1981, 1983, 1985, Whiting and Prausnitz, 1982). Density independent mixing rules with cross term asymmetric in composition (Adachi and Sugie, 1985, 1986, Margerum and Lu, 1990, Huron and Vidal, 1979) are not considered because of their incompatibility with the low density limit. An improved constrained parameter fit is proposed and two methods for avoiding false liquid-liquid splits (i.e. including liquid-liquid critical points and constrained parameter fit) are discussed.
PURE COMPOUND DESCRIPTION A modified Redlich-Kwong RT P =--v-b
equation
a v(v+b)
(1)
was used with the following rules for parameters a and b a = a, exp(E j=l
CI, (TT - 1))
b = b,
(2)
where values of a, and b, are determined from the conditions at the critical point. Wagner’s set (Wagner, 1973) of exponents {m} = [ 1, 1.5, 3, 6} was used, because this set of exponents provides good description of vapour pressure data (Vonka et al., 1989). Adjustable parameters were determined to achieve sufficient agreement between experimental and calculated vapour pressure data. Usually four or three parameters were necessary for less than 0.2 % deviation. For components with low value of critical temperature (methane, ethane and propane) PVT data or second virial coefficient data for higher temperatures were included in the fit to guarantee a good description of the state behaviour at the temperature of the binary system. For NP = 3 or NP = 4 the calculated values of a, or a, were always negative. This fact guarantees sufficiently small, even though not zero, value of the second virial coefficient B (B = b - aIRT) for T + 00 .
65
MIXING RULES The classical quadratic mixing rules
b = b,,xf + 2b,,x,x, + b&
W
a = allxf + 2al,x,xz + a&
(W
were used. The following general forms of cross terms were considered
b 11 42
=
a12
= a;[1
+ 42
2
W
(1 - k, - 2)
- k, -
4
r
+
(k3 +
k,b
7’
,I - fUw,M.,&~
(4b)
and (4c)
apz=&Z
where H is a non-random term originally proposed by Mollerup (1981, 1983, 1985) on the basis of the local composition concept. It has the following form 1
v12c12
WZlC21
s 12
s 21
*=l(-+-
@a)
>
where
Cl2
=
0 aI2 -42 r 41
C21
I-
= @I2
+ x2
S2,
42
&‘-c21k”“” +$)I
&(-c,,R,W +$)I
4, = exp[
Sl2
-41
=a,,0
E21= ev[
= 3
(5b)
+ E21x2
External surface areas q, are calculated by the method of Bondi(1968). Originally the non-random term had only one constant (k., = k,) and only molecules of equal sizes were considered(q, = qj = 1). For such one-constant symmetrical non-random term cIZ = cZ1and El2 = E,, . Extension to mixtures of molecules of unequal sizes was done by Whiting and Prausnitz (1982). A two-constant non-random term (k,# k5 ) is used
66
in this work. In such case the calculated results don’t depend on the values of q1 and q2 because these values can be included into the values of the adjustable parameters k4 and k5 . An arbitrary subset of eight parameters in eqn. (4a,4b) can be chosen, so there exist 2* = 256 possible different mixing rules. We have tested about 20 of them chosen to include all the common ones in the literature.
METHOD OF CALCULATION Let “k” be the chosen set of adjustable parameters in the mixing rule. Values of adjustable parameters are determined in one or two steps. Since many experimental data are only total pressure measurements where the vapour phase composition is not determined, the following objective function 0, is considered in the first step
and the least squares method is used for searching the minimum of the objective function (6). The percentage mean square weighted deviation SP(%) 6P(%)
=
loo J
3 a
serves as a measure of goodness of fit. The normalization factor a doesn’t depend on the mixing rules, and in the case of constant relative error of experimental values of pressure (i.e. Hopi= const. * Pi for all i), 6P(%) is equal to the mean square percentage deviation. Thus, values of 6P(%) enable both comparison between several mixing rules and give some quantitative idea about agreement between experimental and calculated values of pressure. When the gas composition was available, a comparison between the experimental and calculated values were calculated using an analogous definition
In the case where the standard deviations oy, have a constant value, Sy is the mean square absolute deviation. Since the values of Ayi are not included in the objective function, the value of 6y has a predictive character. Because of the risk of getting false liquid-liquid splits in the fit, we calculated Gil (the second derivative of the molar reduced Gibbs energy GIRT with respect of composition) from the fitted model in each experimental point. The percentage of the
67
points with a negative GII (i.e. local instability) gives information about the extent of the false liquid-liquid splits. There exist two basic ways to avoid false liquid-liquid splits. The first of them is a physical approach which is based on including liquid-liquid critical points in the fit. It is necessary to include at least two critical points to describe well the dependence of the critical temperature on the critical pressure. In this case the objective function has the following form
(9) where AT, or AX, are differences between experimental and calculated critical temperatures or composition for a given value of the critical pressure. The second term on the right side is multiplied by an empirical constant $, since it is necessary to put sufficient weight on each critical point to achieve at least such agreement between each experimental and calculated critical point that the calculated critical point is forced out of the experimental VLE region. If this is the case, then will be no false liquid-liquid split. The second way - a mathematical approach - is based on a constrained parameter fit. Englezos et al. (1989) proposed the following two step procedure. The unconstrained problem (6) is solved in the first step. After that the point “z”, z = (x,T,P), for which GII has the most negative value is determined. In the second step the constrained problem mi0s
;
e3 = $+ + A(GIl(z) - e)
(10)
is solved, where h is a Lagrange multiplier and E:is a small positive number. We have tested this approach, and in our experience the method is successful (i.e. GII > 0 for all points, not only for point “z”) at the most for one parameter mixing rules. For more complicated mixing rules it is necessary to require not only Gil (z) = E but also GIlI = 0, i.e. we require a minimum value of the function GII with respect to composition in the point “2”. Thus we then use the objective function t#~~= $1 + A(Gll(z)
-
e) +EGlll(z)
(11)
where h and 5 are Lagrange multipliers.
RESULTS We have performed calculations for 12 binary methanol + n-alkane systems, 10 binary ethanol + n-alkane systems and 6 binary propanol + n-alkane systems. Information about the experimental data and the results for the methanol, ethanol and propanol series are presented in the Tables l-6. Odd numbered tables contain information about the number and range of data points, where C, everywhere means the carbon number of the alkane. Even numbered tables contain cumulative results
68 TABLE 1 Information
about the experimental
data of methanol + n-alkane systems.
PMW
VW G
min
max
min 1.4
max
Ref.
N
1
273.15
300.0
41.4
43
a
2
260.0
373.15
.45
6.0
33
bed
3
313.1
371.1
.35
3.9
25
e
4 5
273.15 303.00
373.15 422.6
.006 .lOl
1.72 2.53
34 38
fg h i
6 7
322.6 332.0
492.6 531.5
.lOl .101
6.0 6.8
36 26
j k 1 k
8
335.9
539.2
.lOl
6.9
48
1 k
10
422.4
526.4
1.3
7.3
30
k
12
421.3
522.0
1.3
8.0
42
k
14
422.2
524.5
1.3
8.5
32
k
16
298.15
318.15
15
m
.03
.057
“Hong et al., 1987, bZeck and Knapp, 1986, ‘Oghaki et al., 1976, dMa and Kohn, 1964, %alivelSolastiouk et al., 1986, ‘Kretschmer and Wiebe, 1952, %eu at al., 1989, ‘Budantseva et al., 1975c, ‘Wilsak et al., 1987, ‘Budantseva et al., 1975a, tie Loos et al., 1988,‘Budantseva et al., 1975b, “Tucker et al., 1969.
for each considered mixing rule. In the tables, abbreviation 1+2+7 means mixing rule with adjustable parameters k,, k, and k, (other parameters are equal zero) etc. and abbreviations 4 or 1+4 or 1+2+4 mean one- or -two or three-parameter mixing rules containing the sy_mmetrical Goon-random term (Sa) and (Sb) where k5 = kb The cumulative quantities P(%) or Sy are defined by the relation (12)
iiP(96)
(12)
where P,(%) is quantity (7) for the j-th alkane member of each series (i.e. PI(%) is quantity (7) for the system methanol + methane in the methanol series etc.). M is the total number of alkanes in each series (12 or 10 or 6 for the methanol or ethanol or propanol series). Cumulative averages may contain great deviations. This is the reason why the maximum value of quantity (7) is also presented @P_(%) or Sy,,) with the corresponding carbon number of the n-alkane for which the maximum value of 6P(%) or Sy occurs. The quantity U(%) is the cumulative percentage amount of unstable points, and is defined by the relation
69
TABLE 2 Results of unconstrained mix.rule
SP(%)
fit for methanol + n-alkane systems ~Plnaxx(%)
c,
ZY
6Y,,
C,
UC%)
no par. 35.0 166 1 .085 .169 7 2 ____________________~~~_~_~~~~~~~~~~~__-_~~-~~~~~~__---~~~~~~~~~__-----~~~~~~___---~~~~~~ 1 16.2 23.2 1 .088 .205 6 11 2
14.1
21.3
1
.093
.225
6
17
3
16.2
23.3
8
.093
.230
6
9
4 16.2 50.3 16 .093 .211 6 20 ____-_-----___-_-___-----~~~~~~~~---~---~~~-~~~~-~---~~~~~~~~~~~_---~~-~~~~~~----~~~~~~~~ 1+2 8.9 20.6 1 .071 .176 7 14 2+7
8.2
18.2
5
.075
.184
7
7
2+3
6.5
15.0
5
.065
.138
7
9
3+8
11.5
18.5
5
.072
.152
7
10
1+4
8.7
24.1
16
,058
.123
6
6
4+5 5.2 10.9 2 .060 .146 6 16 ______________-________--____________--_-__-______---___________--____-_____~____________ 1+2+7 4.3 10.0 5 .084 .224 6 11 1+2+4
4.7
18.8
1
.035
.082
6
9
1+2+3
3.2
7.2
9
.061
.133
7
12
2+3+7
4.2
14.6
5
.071
.I62
6
10
2+3+8
4.3
14.5
5
.070
.161
6
10
1+4+5 2.3 5.2 3 .029 .067 5 5 _____--_______-_____-~~~-~~~~~~~~~--~~-~~~~-~~~~-~~~-~~~~~~~~~-~~-~~~~-~----~~~~~~~~~~~-1+2+6+7 3.9 9.8 5 .071 .236 6 10 1+2+3+6
2.8
6.6
8
.068
.181
6
10
1+4+5+6
1.5
4.2
1
.026
.066
5
2
(13) where Wj or Nj are, respectively, the number of points with an unstable liquid phase (i.e. Gil < 0) and the total number of points for the j-th alkane in the series. Table 7 contains a comparison between our results (“VDL”) for mixing rule 1+4+5+6 and the hole quasichemical group contribution EOS proposed by Victorov and Fredeslund (1990), (“VF”). Table 7 enables a comparison between two quite different approaches in the description of VLE data. A comparison of unconstrained and constrained fits is presented in Fig. 1. Results from the different approaches are demonstrated for the isobaric (101.3 kPa) system methanol-heptane (7 K above liquid-liquid critical temperature). Two critical points (P(MPa), T(K), x) = (0.1,324.5,0.588),(25, 332.5,0.580),0, = 0.5 K and 6, = 0.01 measured by Ott et al. (1986) were included in the objective
70
TABLE 3 Information about exnerimental data ethanol + n-alkane systems P[MPa]
TlKl C,
min
max
min
max
Ref.
N
1
298.15
498.15
1.410
31.53
36
2
313.4
498.15
1.307
11.31
32
ab ab
3
273.15
500.00
.041
6.205
91
cd
4
298.45
345.65
.086
.881
65
e
5
283.15
460.93
.016
5.516
57
6
298.15
353.15
.020
.212
145
fg hij
7
303.27
343.15
.Oll
.096
82
k
8
318.15
348.15
.OlO
.095
73
1
10
315.48
433.15
.lOl
.lOl
27
mn
16
325.25
325.25
.004
.032
29
0
et al., 1990, bBrunner and Hultenschmidt, 1990, ‘Kretschmer and Wiebe, 1951, dGomezNieto and Thodos, 1978, “Holderbaum et al., 1991, ‘Ishii, 1935, KMcCracken et al., 1960, ‘Wolff and Goetz, 1976, ‘Iguchi, 1978, JSugi and Katayama, 1978, kBerro et al., 1982, ‘Boublfkova and Lu, 1969, “Koshelnikov et al., 1974, “Ellis and Spurr, 1961, “French et al., 1979. ‘Suzuki
function (9). For each approach the adjustable parameters of the mixing rules 1+4+5 were determined from the whole set of experimental methanol-heptane VLE data (see Table l.), i.e. not only from the isobaric data presented in Fig.1. Curve (1) in Fig. 1 corresponds to the experimental data and curve (2) to the unconstrained fit using objective function (6). The values of W(%) = 0.72 and Sy = 0.035 were achieved for this fit (for the whole set of data). Curve (3) corresponds to the mathematical approach (constrained problem (1 l)), giving 6P(%) = 1.0 and Sy = 0.026. Finally, curve (4) corresponds to the physical approach (objective function (9)) with q= 1000. For such a choice of q we get an almost perfect agreement between calculated and experimental critical points (ATi < 0.3 K i = 1,2), but SP(%) = 14 and Sy = 0.12. If we choose q = 50 then the calculated temperature of the low pressure critical point corresponds with the lowest temperature of VLE data and we obtain approximately the same results as for the mathematical approach.
DISCUSSION It follows from Tables 2,4 and 6 that the best unconstrained fits were achieved with mixing rules containing the two parameter non-random term, i.e. 4+5, 1+4+.5 or 1+4+5+6. The first row of Tables 2, 4 and 6 contain information about the results for no parameters. Even though the decrease of the value of the objective function using only one parameter mixing rules is very significant, these one-
71
TABLE 4 Results of unconstrained fit for ethanol + n-alkane systems mixrule
z?P(%)
~P.n#X(%)
C”
SY
8Y,.X(%)
c.
UC%)
no par. 25.9 35.1 5 .OSl .162 8 2 --________--__-_____~~~~-~~~~~~~~~~~~~-~~~~-~~~~~~~~-~~~~~~~~~~~~-~~~~-~~~~~~~~~~~~~~~~~~ 1 18.2 31.1 16 .066 .141 7 21 2
17.8
32.1
16
.066
.133
7
20
3
18.9
32.1
16
.066
.137
7
20
4 18.6 32.1 16 .059 .130 8 16 ______________-_____~~~~-~~~~~~~~~~~~~-~~~~-~~~~~~~~-~~~~~~~~~~~~-~~~~-~~~~~~~~~~~~~~~~~~ 1+2 10.4 21.8 1 .056 .094 8 18 2+7
13.7
24.6
5
.063
.119
8
18
2+3
12.0
3+8
15.1
18.9
5
.056
.114
8
16
32.1
16
.070
.118
8
1+4
15.9
29.7
18
16
.068
.164
7
7
___4’5__________814_______26._q_______-_~~_-~~~~______,o55_________1_o__-_~________________ 1+2+7 8.0 14.3 4 .054 .086 2 18 1+2+4
6.9
12.7
1
.042
.lOO
8
10
1+2+3
7.4
14.1
1
.046
.090
8
16
2+3+7
10.2
18.4
4
.065
.106
2
14
2+3+8
10.2
18.4
4
.058
.105
8
14
1+4+5
6.3
13.7
10
.021
.040
10
6
1+2+6+7
8.0
14.3
4
.054
.086
2
18
1+2+3+6
7.4
14.1
1
.046
.090
8
16
1+4+5+6
6.3
13.7
10
.021
.040
10
6
parameter mixing rules are not able to describe 1-alkanol + n-alkane mixture with less than 10 % average deviation in pressure. Note also the similarity of the oneparameter results. For two- and three-parameter mixing rules too, the cumulative results do not differ much between the alternatives without the two-parameter non-random term. For instance, the combination 1+2 does not give worse overall results than mixing rules with temperature dependence (parameter 6 or 7 or S), with density dependence (parameter 3) or both (3+8). For three parameters, density dependence seems to be slightly better than temperature dependence (1+2+3 vs. 1+2+7), while both together (2+3+7 or 2+3+8) look slightly worse. Mollerup and Clark (1988) and Mollerup (1983,1985) tested mixing rules 1+2,1+4 and 1+2+4 on several types of systems. They obtained from a more limited set of data the following conclusion for methanol - hydrocarbon mixtures : (a) the mixing rule 1+4 gives significantly better results than the mixing rule 1+2, (b) introduction of a I?, (i.e. mixing rule 1+2+4) does not improve the correlation. Our results do not support that conclusion.
72
TABLE 5 Information C,
about the experimental
data of propanol + n-alkane systems.
WI
P[MPa] N
Ref.
min
max
min
max
1
313.4
333.4
1.41
10.197
10
a
2
283.15
333.4
.l
6.742
16
ab
5
313.1
317.1
.05
.134
16
C
6
298.15
361.75
.013
.l
19
de
7 10
333.15 347.75
523.15 398.95
.l .017
62 24
fgh i
4.319 .091
‘Suzuki et al., 1990, bBen-Naim and Yaacobi, 1974, ‘Rice et al., 1990, dIguchi, 1978, “Rhim and Kwak, 1981, ‘Zawisza and Vejrosta, 1982, *Pena and Cheda, 1970, hSabarathiham and Andiappan, 1985, ‘Ellis et al., 1960.
TABLE 6 Results of unconstrained mixrule
fit for propanol + n-alkane systems
ZP(%)
8PlnJ%)
C”
SY
8Y,,(%)
C”
U%)
7 0 .059 .122 35.6 5 no par. 24.4 _______________________~~~~~~~~~~~~~~___~__-~~-~~~~~~~~~__~______-~~~~~~~~~~~~___________ 7 7 .034 .089 16.7 5 1 9.7 2 3
8.9
12.8
5
.052
.130
5
6
9.2
13.6
5
.034
.085
7
6
7 6 .035 .094 12.9 10 4 7.8 ______________-__________~___________---__--________-________---______-________--________ 7 11 .028 .058 9.5 1 1+2 5.9 2+7
7.1
12.8
5
.036
.081
7
5
2+3
5.2
8.8
7
.026
.070
7
3
3+8
7.2
13.6
5
.030
.082
7
4
1+4
6.8
11.2
7
.036
.089
7
3
5 8 .019 .027 4.5 1 4+5 3.1 ~~__________~~~~~______~_~__~_________-_----________-_____----_____----_-________________ 1+2+7 4.1 7.3 .019 5 .058 7 8 1+2+4
4.0
9.4
1
.013
.035
7
8
1+2+3
2.7
4.8
10
.013
.033
7
6
2+3+7
4.6
8.8
7
.021
.070
7
2
2+3+8
4.5
8.8
7
.020
.070
7
2
1+4+5 1.9 ------________-_________-_1+2+6+7 3.9
2.7 10 .013 10 4 .019 ________________--__~~~_______~~~~~________-_~__________--_--_ 7.2 .018 5 .056 7 10
1+2+3+6
2.2
4.8
10
.013
.023
7
4
1+4+5+6
1.7
217
10
.013
.019
10
0
73
TABLE 7 Comparison
between results of this work (VDL) and results of Victorov and Fredeslund SP[%]
@F)
100 Sy
N
PAMpal
VDL
VF
VDL
VF
11
41.4
250.03
7.4
6.5
.4
.08
12
41.4
273.16
4.5
5.4
.7
.15
12
41.4
290.0
3.9
5.9
.8
.16
11 41.4 330.0 9.6 5.4 1.2 __---------_______-_________--_---_-______-__-----_--____-____________~~~~~~~~~~----~-methanol + 11 2.3 273.16 1.3 1.4 -
.33
System methanol methane
ethane
+
5
4.1
‘UK1
298.16
1.9
1.9
-
-
5 6.0 373.15 3.2 1.7 5.7 4.4 ____________________-__________________-----________---_--_-_____-_____-___~___~~~~~~~~ methanol + 8 1.4 313.1 3.3 1.3 1.0 .6 propane
8
2.5
343.1
2.4
1.6
2.1
8 4.3 373.1 3.3 1.2 2.0 ____________________-____________---_-------_________---------__--_____-_________~____~ methanol + 11 .l 303-338 1.2 4.6 6.2 n-pentane
9
.83
372.69
1.0
4.2
.9
9 2.5 422.59 .3 3.5 1.3 __________________-________------------___-_______-----__-_______-______~_~~~~~~~~~~~~~ methanol + 7 .03 298.16 3.1 2.0 9.9 n-hexane
1.5 2.5 4.9 2.5 2.7 2.3
23
.05
308.16
1.0
1.6
-
-
23
.25
348.16
1.7
2.5
-
-
473.02
5.0
.4
-
-
1
4.6
The improvement in going from three to four parameters is slight, and in addition four parameters gave numerical problems unless both temperature and pressure region were large. Density dependent mixing rules makes it possible to satisfy not only the low density limit (i.e. quadratic dependence of the second virial coefficient on composition) but also a reasonable high density limit. If we consider the nonrandom term (5a) and (5b) for a binary mixture of equisized molecules and of components having equal molar volumes, then the high density limit is equal to the three-parameter Wilson (1964) equation (Whithing and Prausnitz (1982), Won (1983)), where the parameters of the Wilson equation are determined from the values of k4 (mixing rule 4) or k4 and k5 (mixing rule 4+5). Two independent values of the adjustable parameters k4 and k, makes it possible to achieve an arbitrary asymmetrical dependence of the excess Gibbs energy calculated from Wilson equation on composition. This is, in our opinion, the reason why mixing rules containing the two parameter non-random term give better result than other mixing rules having the same number of parameters. The actual values of the cumulative quantities in Tables 2,4,6 ( &‘(%) and 8~) are not so important, because they are strongly infhtenced by the experimental
74
0.76
-
0.72
-
0.64
-
Yl
0.60
’
I
I
I
I
I
0.0
0.2
0.4
0.6
0.8
1.0
xl
Fig. 1 Experimental(l) and calculated y1 - x1 curves of methanol-heptane (7 K above lJCS7J and x, E (.05;.95) (2) . .. unconstrained fit (3) . .. constrained fit eqn. (11) (4) . .. physical approach eqn. (9) with q = lo3
system for
101.32 kPa
data. As can also be seen from Fig 1, data near the liquid-liquid critical points will be harder to describe, and will lead to larger deviations. However, using the same experimental data for all the alternatives, the various mixing rules can be compared. The comparison with the complex group-contribution method in Table 7 shows that our results are very similar to theirs, and they also report false liquidliquid splits. The most noticable result in Tables 2,4 and 6 is that all but two of the cumulative results show a number of points with an unstable liquid phase, i.e. the equation of state will give a false liquid-liquid phase split irrespective of the of the number of parameters and mixing rules. The cumulative results mask any trend with alkane carbon number. For each alcohol, the data for the alcohol - methane binaries were fit without any unstable points for all mixing rules. Likewise, the mixtures with ethane and propane had relatively few or no unstable points. This is related to the change in upper critical solution point with carbon number. The experimental data are all at ambient temperature or above, while the UCSP for methanol methane (from extrapolation) is below 200 K, and for methanol - hexane 308 K at 1 bar. Since the amount and the range of the experimental data are different for each system, it is impossible to correlate the results with the UCSP’s for each system.
Both of the methods suggested will remove the false phase split, but at the expense of a poorer fit of the VLE data. The constrained fit can only assure correct phases in the region of the VLE data, but with minimum loss in the VLE data fit. Including upper critical solution points in the fit may in principle make the region of false phase splits almost disappear, but at a corresponding much worse fit of the VLE data. Both methods will be treated in more detail in subsequent publications.
CONCLUSION Cubic equations of state with ordinary mixing rules will predict false liquidliquid splits for 1-alkanol + n-alkane binaries when the binary parameters are determined from VLE data, unless the data are limited to temperatures much above the upper critical solution points. Disregarding the false phase splits, the best fits are obtained using density dependent mixing rules containing two parameter non-random term, preferably in combination with an additional parameter in the repulsive term. False phase splits may be avoided by either including UCSPs in the fit with an appropriate weight, or by a constrained fit to force a correct curvature of the Gibbs energy function. Such additional requirement will necessarily give a poorer fit to the VLE data.
LIST OF SYMBOLS
a, ai, b, b, Gll (Glll) H 4 L M N NP P i T T, u UCSP V
attractive parameter in EOS repulsive parameter in EOS the second (third) derivative of the molar reduced Gibbs energy with respect to liquid composition at constant temperature and pressure non-random term parameter in a mixing rule liquid number of alkanes number of points number of adjustable parameters in Eqn.(2) pressure external surface area gas constant temperature reduced temperature percentage amount of the unstable points upper critical solution point molar volume
number of unstable points liquid mole fraction of the first component value of ‘Ix” for the i-th point gas mole fraction of the first component adjustable parameter in Eqn.(2) difference between experimental and calculated value mean deviation standard deviation of an experimental value objective function Langrange multipliers empirical constant
ACKNOWLEDGEMENT This research was financed by the Norwegian research council NTNF, and mainly by its SPUNG program.
LIST OF REFERENCES Adachi, Y. and Sugie, H., 1985. Effects of mixing rules on phase equilibrium calculations. Fluid Phase Equilibria, 28: 353-362 Adachi, Y. and Sugie, H., 1986. A new mixing rule - modified conventional mixing rule. Fluid Phase Equilibria, 24: 103-l 18 Anderko, A., 1990. Equation-of-state methods for the modelling of phase equilibria. Fluid Phase Equilibria, 61: 145-225 Ben-Naim, A. and Yaacobi, M., 1974. Solvophobic interaction. J.Phys.Chem., 78: 175-178 Berro, C., Rogalski, M. and Peneloux, A., 1982. A new ebulliometric technique. Vapor-liquid equilibriums in the binary systems ethanol - n-heptane and ethanol - n-nonane. Fluid Phase Equilibria, 8: 55-73 Bondi, A., 1968. Physical Properties of Molecular Crystals, Liquids and Glasses. Wiley, New York Boublfkova, L. and Lu, B.C.I., 1969. Isothermal vapor-liquid equilibria for the ethanol - n-octane system. J.Appl.Chem., 19: 89-92 Brunner, E. and Hultenschmidt, W., 1990. Fluid mixtures at high pressures. VIII. Isothermal phase equilibria in the binary mixtures : (ethanol + hydrogen or methane or ethane). J.Chem.Thermodynamics, 22: 73-84 Budantseva, L.S., Lesteva, T.M. and Nemtsov, MS., 1975a. Liquid-vapor equilibriums in methanol - C, hydrocarbons of different classes. Zh.Fiz.Khim., 49: 260-261 Budantseva, L.S., Lesteva, T.M. and Nemtsov, MS., 1975b. Liquid-vapor equlibrium in systems methanol - C,., hydrocarbons of different classes. Zh.Fiz.Khim., 49: 1844-1846 Budantseva, L.S., Lesteva, T.M. and Nemtsov, M.S., 1975~. Liquid-vapor equilibriums in systems methanol - C, hydrocarbons of different classes. Zh.Fiz.Khim., 49: 1848-1850 De Loos, TbW., Poot, W. and de Swaan Arons, J., 1988. Vapor-liquid equilibria and critical phenomena in methanol + n-alkane systems. Fluid Phase Equilibria, 42: 209-227 Ellis, S.R.M., McDermott, C. and Williams, J.C.L., 1960. Proc. of the Intern. Symp. on Dist., Inst.Chem.Engrs., London Ellis, S.R.M. and Spurr, J.M., 1961. Isobaric vapor-liquid equilibriums. Brit.Chem.Eng.,6: 92
77
Englezos, P., Kalogerakis, N. and Bishnoi, P.R., 1989. Estimation of binary interaction parameters for equations of state subject to liquid phase stability requirements. Fluid Phase Equilibria, 53: 81-88 French, H.T., Richards, A. and Stokes, R.H., 1979. Thermodynamics of the partially miscible system ethanol + hexadecane. J.Chem.Thermodynamics, 11: 671-686 Galivel-Solastiouk, F., Laugier, S. and Richon, D., 1986. Vapor-liquid equilibrium data for the propane - methanol and propane - methanol - carbon dioxide system. Fluid Phase Equilibria, 28: 73-85 Gomez-Nieto, M. and Thodos, G., 1978. Vapor-liquid equilibrium measurements for the propane ethanol system at elevated pressures. AIChE J., 24: 672-678 Holderbaum, T., Utzig, A. and Gmehling, J., 1991. Vapor-liqiud equilibria for the system butane/ethanol at 25.3, 50.6 and 72.5 ‘C. Fluid Phase Equilibria, 63: 219-226 Hong, J.H., Malone, P., Jett, M.D. and Kobayashi, R., 1987. The measurement and interpretation of the fluid phase equlibria of a normal fluid in a hydrogen bonding solvent : the methane methanol system. Fluid Phase Equilibria, 38: 83-96 Huron, M.J. and Vidal, .I., 1979. New mixing rules in simple equations of state for representing vapor-liquid equilibriums of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271 Iguchi, A., 1978. Vapor-liquid equlibriums at 25 ‘C for binary systems between alcohols and hydrocarbons. Kagaku Sochi, 20: 66-8 Ishii, N., 1935. Volatility of fuels contg. EtOH (V.) vapor pressures of mixts. of EtOH and methylcyclohexane, (VI.) vapor pressures of mixts. of EtOH and n-hexane. J.Soc.Chem.Ind.Jap., 38: 659 Koshelnikov, V.A., Pavlenko, T.G. and Timofeev, V.S, 1974. Phase equilibriums in the wateraliphatic alcohols (C,-G) - n-paraffinic hydrocarbons (C,-C,,) systems. Tr.Altaisk.Politekh.Inst., 41: 74 Kretschmer, C.B. and Wiebe, R., 1951. The solubility of propane and the butanes in ethanol. J.Am.Chem.Soc., 73: 3778-3781 Kretschmer, C.B. and Wiebe, R., 1952. Solubility of gaseous paraffins in methanol and isoprppyl alcohols. J.Am.Chem.Soc., 74: 1276-1277 Leu, A.D., Chen, C.J. and Robinson, D.B., 1989. Vapor-liquid equilibrium in selected binary systems. AICHE Symp.Series, 271/85: 11-16 Ma, Y.H. and Kohn, J.P., 1964. Multiphase and volumetric equilibria of the ethane - methanol system at temperatures between -40°C and 100°C. J.Chem.Eng.Data, 9: 3-5 Margerum, M.R. and Lu, B.C., 1990. VLE correlation for binary 1-alkanol + n-alkane mixtures. Fluid Phase Equilibria, 56: 105-118 McCracken, P.G., Storvick, T.S.and Smith, J.M., 1960. Phase behavior from enthalpy measurements. Benzene - ethyl alcohol and n-pentane - ethyl alcohol systems. J.Chem.Eng.Data, 5: 130-132 Mollerup, J., 1981. A note on excess Gibbs energy models, equations of state and the local , composition concept. Fluid Phase Equilibria, 7: 121-138 Mollerup, J., 1983. Correlation of thermodynamic properties of mixtures using a random-mixture reference state. Fluid Phase Equilibria, 15: 189-207 Mollerup, J., 1985. Correlation of gas solubilities in water and methanol at high pressure. Fluid Phase Equilibria, 22: 139-154 Mollerup, J. and Clark, W.M., 1988. Correlation of solubilities of gases and hydrocarbons in water. AICHE spring national meeting, New Orleans, USA Oghaki, K., Sano, F. and Katayama, T., 1976. Isothermal vapor-liquid equilibrium data for binary systems containing ethane at high pressures. J.Chem.Eng.Data, 21: 55-58 Ott, J.B., Hoeltscher, I.F. and Schneider, G.M., 1986. (Liquid-liquid) phase equilibria in (methanol + heptane) and (methanol + octane) at pressures from 0.1 to 150 MPa. J.Chem.Thermodynamics, 18: 815-826
78
Pena, M.D. and Cheda, D.R., 1970. Liquid-vapor equlibrium. III. Systems of n-propanol - nhexane at 50°C and n-propanol - n-heptane at 6O’C. An.Quim., 66: 747 Rhim, J.N. and Kwak, C., 1981. Determination of vapor liquid equilibrium from boiling point curve. Hwahak Konghak, 19: 199-206 Rice, P., El-Nikheli, A. and Teja, A.S., 1990. Isothermal vapor liquid equlibrium data for the system n-pentanefn-propanol. Fluid Phase Equilibria, 56: 303-312 Sabarathiham, P.L. and Andiappan, A.N., 1985. Isobaric vapor-liquid equlibria for binary systems : n-propanol, isopropanol, isobutanol and tert-butanol with n-heptane. Indian.J.Technol., 23: 101 Sugi, H. and Katayama, T., 1978. Ternary liquid-liquid and miscible binary vapor-liquid equilibrium data for the two systems n-hexane - ethanol - acetonitrile and water - acetonitrile ethyl acetate. J.Chem.Eng.Japan, 11: 167-172 Suzuki, K. et al., 1990. Isothermal vapor-liquid equilibrium data for binary systems at high pressures : carbon dioxide - methanol, carbon dioxide - ethanol, carbon dioxide - 1-propanol, methane - ethanol, methane - I-propanol, ethane - ethanol. and ethane - 1-propanol systems. J.Chem.Eng.Data, 35: 63-66 Tucker, E.E., Famham, S.B. and Christian, S.D., 1969. Association of methanol in vapor and in n-hexadecane. A model for the association of alcohols. J.Phys.Chem., 73: 3820-3829 Victorov, A.I. and Fredeslund, Aa., 1991. Application of the hole quasichemical group contribution equation of state for phase equilibria calculation in systems with association. Fluid Phase Equilibria, 66: 77-101 Vonka, P., Novak, J.P. and Dittrich, P., 1989. Determination of parameters of EOS from vapour pressure data. Collect.Czech.Chem.Commun., 54: 1446-1463 Wagner, W., 1973. New vapor pressure measurements for argon and nitrogen and a new method for establishing rational vapor pressure equation. Cryogenics, 13: 470 Whithing, W.B. and Prausnitz, J.M., 1982. Equations of state for strongly nonideal fluid mixtures : application of local compositions towards density-dependent mixing rules. Fluid Phase Equilibria, 9: 119-147 Wilsak, R.D., Campbell, S.W. and Thodos, G., 1987. Vapor-liquid equilibrium measurements for n-pentane - methanol system at 372.7, 397.7 and 422.6 K. Fluid Phase Equilibria, 33: 157-171 Wilson, G.M., 1964. Vapor-liquid equilibrium. XI. A new expression for the excess free energy of mixing. J.Am.Chem.Soc., 86: 127-130 Wolff, H. and Goetz, R., 1976. Die Assoziation verschieden deuterierter Athanole in n-Hexan (nach Dampfdruckmessungen). Z.Phys.Chem.(Frankfurt am Main), 100: 25-36 Won, K.W., 1983. Thermodynamic calculation of supercritical-fluid equilibriums : new mixing rules for equations of state. Fluid Phase Equilibria, 10: 191-210 Zawisza, A. and Vejrosta, J., 1982. High-pressure liquid-vapour equilibria, critical state, and p(V,T,x) up to 573.15 K and 5.066 MPa for (heptane + propan-l-01). J.Chem.Thexmodynamics, 14: 239-249 Zeck, S. and Knapp, H., 1986. Vapor-liquid and vapor-liquid-liquid phase equilibria for binary and ternary systems of nitrogen, ethane and methanol : experiment and data reduction. Fluid Phase Equilibria, 25: 303-322
63
A comparison of many different mixing rules in a cubic equation of state for 1-alkanol + n-alkane mixtures Petr Vo&a’,
Pave1 Dittrich’
and Jorgen
‘Department of Physical Chemistry, 166 28 Prague 6, Czechoslovakia
Lovland*
Prague Inst. of Chemical
*Department of Chemical NTH, Norway
Engineering,
University
Keywords: cubic equation alkanol + n-alkane
of state, mixing
Technology,
of Trondheim,
7034 Trondheim
rules, false phase splitting,
ABSTRACT Different density dependent and density independent mixing rules have been tested for a modified Redlich-Kwong EOS. The attractive parameter of the pure component is considered to be temperature dependent, and adjustable parameters in the attractive term are fitted to achieve a good agreement between experimental and calculated vapour pressure data for the pure compounds. The general form of the considered mixing rule has eight adjustable parameters, and it is possible to choose an arbitrary subset of the adjustable parameters in fitting experimental VLE data. Mixing rules with one, two, three or four parameters were tested for a number of I-alkanol + n-alkane systems. For such systems false liquid-liquid splits occur. They may be avoided by including liquidliquid critical points into the fit or by a constrained parameter fit.
INTRODUCTION The topic of the present communication is a critical evaluation of the possibility of describing 1-alkanol + n-alkane VLE data using different mixing rules in a cubic EOS. Mixtures of 1-alkanol + n-alkane arc important from an industrial point of view and they are known to have difficult behaviour including liquid phase splitting(Englezos et al., 1989, Margerum and Lu, 1990). Above the actual liquidliquid region the y,-x, curve for a binary system has a combination of very steep and very flat portions. Trying to describe such a behaviour typically gives an S-shape behaviour of the calculated y,-x, curve, i.e. false liquid-liquid splits are predicted. The calculated upper liquid-liquid critical point (for considered pressure and given set of adjustable parameters) then lies inside or above the experimental VLE region. We
0378-3812/93/$06.00
01993 Elsevier Science Publishers B.V. All rights reserved
64
have tested most ordinary mixing rules in combination with a cubic equation of state on a large number of binary systems of methanol, ethanol, and 1-propanol in combination with n-alkanes. The mixing rules include density independent ones (Anderko, 1990) and density dependent ones (Mollerup, 1981, 1983, 1985, Whiting and Prausnitz, 1982). Density independent mixing rules with cross term asymmetric in composition (Adachi and Sugie, 1985, 1986, Margerum and Lu, 1990, Huron and Vidal, 1979) are not considered because of their incompatibility with the low density limit. An improved constrained parameter fit is proposed and two methods for avoiding false liquid-liquid splits (i.e. including liquid-liquid critical points and constrained parameter fit) are discussed.
PURE COMPOUND DESCRIPTION A modified Redlich-Kwong RT P =--v-b
equation
a v(v+b)
(1)
was used with the following rules for parameters a and b a = a, exp(E j=l
CI, (TT - 1))
b = b,
(2)
where values of a, and b, are determined from the conditions at the critical point. Wagner’s set (Wagner, 1973) of exponents {m} = [ 1, 1.5, 3, 6} was used, because this set of exponents provides good description of vapour pressure data (Vonka et al., 1989). Adjustable parameters were determined to achieve sufficient agreement between experimental and calculated vapour pressure data. Usually four or three parameters were necessary for less than 0.2 % deviation. For components with low value of critical temperature (methane, ethane and propane) PVT data or second virial coefficient data for higher temperatures were included in the fit to guarantee a good description of the state behaviour at the temperature of the binary system. For NP = 3 or NP = 4 the calculated values of a, or a, were always negative. This fact guarantees sufficiently small, even though not zero, value of the second virial coefficient B (B = b - aIRT) for T + 00 .
65
MIXING RULES The classical quadratic mixing rules
b = b,,xf + 2b,,x,x, + b&
W
a = allxf + 2al,x,xz + a&
(W
were used. The following general forms of cross terms were considered
b 11 42
=
a12
= a;[1
+ 42
2
W
(1 - k, - 2)
- k, -
4
r
+
(k3 +
k,b
7’
,I - fUw,M.,&~
(4b)
and (4c)
apz=&Z
where H is a non-random term originally proposed by Mollerup (1981, 1983, 1985) on the basis of the local composition concept. It has the following form 1
v12c12
WZlC21
s 12
s 21
*=l(-+-
@a)
>
where
Cl2
=
0 aI2 -42 r 41
C21
I-
= @I2
+ x2
S2,
42
&‘-c21k”“” +$)I
&(-c,,R,W +$)I
4, = exp[
Sl2
-41
=a,,0
E21= ev[
= 3
(5b)
+ E21x2
External surface areas q, are calculated by the method of Bondi(1968). Originally the non-random term had only one constant (k., = k,) and only molecules of equal sizes were considered(q, = qj = 1). For such one-constant symmetrical non-random term cIZ = cZ1and El2 = E,, . Extension to mixtures of molecules of unequal sizes was done by Whiting and Prausnitz (1982). A two-constant non-random term (k,# k5 ) is used
66
in this work. In such case the calculated results don’t depend on the values of q1 and q2 because these values can be included into the values of the adjustable parameters k4 and k5 . An arbitrary subset of eight parameters in eqn. (4a,4b) can be chosen, so there exist 2* = 256 possible different mixing rules. We have tested about 20 of them chosen to include all the common ones in the literature.
METHOD OF CALCULATION Let “k” be the chosen set of adjustable parameters in the mixing rule. Values of adjustable parameters are determined in one or two steps. Since many experimental data are only total pressure measurements where the vapour phase composition is not determined, the following objective function 0, is considered in the first step
and the least squares method is used for searching the minimum of the objective function (6). The percentage mean square weighted deviation SP(%) 6P(%)
=
loo J
3 a
serves as a measure of goodness of fit. The normalization factor a doesn’t depend on the mixing rules, and in the case of constant relative error of experimental values of pressure (i.e. Hopi= const. * Pi for all i), 6P(%) is equal to the mean square percentage deviation. Thus, values of 6P(%) enable both comparison between several mixing rules and give some quantitative idea about agreement between experimental and calculated values of pressure. When the gas composition was available, a comparison between the experimental and calculated values were calculated using an analogous definition
In the case where the standard deviations oy, have a constant value, Sy is the mean square absolute deviation. Since the values of Ayi are not included in the objective function, the value of 6y has a predictive character. Because of the risk of getting false liquid-liquid splits in the fit, we calculated Gil (the second derivative of the molar reduced Gibbs energy GIRT with respect of composition) from the fitted model in each experimental point. The percentage of the
67
points with a negative GII (i.e. local instability) gives information about the extent of the false liquid-liquid splits. There exist two basic ways to avoid false liquid-liquid splits. The first of them is a physical approach which is based on including liquid-liquid critical points in the fit. It is necessary to include at least two critical points to describe well the dependence of the critical temperature on the critical pressure. In this case the objective function has the following form
(9) where AT, or AX, are differences between experimental and calculated critical temperatures or composition for a given value of the critical pressure. The second term on the right side is multiplied by an empirical constant $, since it is necessary to put sufficient weight on each critical point to achieve at least such agreement between each experimental and calculated critical point that the calculated critical point is forced out of the experimental VLE region. If this is the case, then will be no false liquid-liquid split. The second way - a mathematical approach - is based on a constrained parameter fit. Englezos et al. (1989) proposed the following two step procedure. The unconstrained problem (6) is solved in the first step. After that the point “z”, z = (x,T,P), for which GII has the most negative value is determined. In the second step the constrained problem mi0s
;
e3 = $+ + A(GIl(z) - e)
(10)
is solved, where h is a Lagrange multiplier and E:is a small positive number. We have tested this approach, and in our experience the method is successful (i.e. GII > 0 for all points, not only for point “z”) at the most for one parameter mixing rules. For more complicated mixing rules it is necessary to require not only Gil (z) = E but also GIlI = 0, i.e. we require a minimum value of the function GII with respect to composition in the point “2”. Thus we then use the objective function t#~~= $1 + A(Gll(z)
-
e) +EGlll(z)
(11)
where h and 5 are Lagrange multipliers.
RESULTS We have performed calculations for 12 binary methanol + n-alkane systems, 10 binary ethanol + n-alkane systems and 6 binary propanol + n-alkane systems. Information about the experimental data and the results for the methanol, ethanol and propanol series are presented in the Tables l-6. Odd numbered tables contain information about the number and range of data points, where C, everywhere means the carbon number of the alkane. Even numbered tables contain cumulative results
68 TABLE 1 Information
about the experimental
data of methanol + n-alkane systems.
PMW
VW G
min
max
min 1.4
max
Ref.
N
1
273.15
300.0
41.4
43
a
2
260.0
373.15
.45
6.0
33
bed
3
313.1
371.1
.35
3.9
25
e
4 5
273.15 303.00
373.15 422.6
.006 .lOl
1.72 2.53
34 38
fg h i
6 7
322.6 332.0
492.6 531.5
.lOl .101
6.0 6.8
36 26
j k 1 k
8
335.9
539.2
.lOl
6.9
48
1 k
10
422.4
526.4
1.3
7.3
30
k
12
421.3
522.0
1.3
8.0
42
k
14
422.2
524.5
1.3
8.5
32
k
16
298.15
318.15
15
m
.03
.057
“Hong et al., 1987, bZeck and Knapp, 1986, ‘Oghaki et al., 1976, dMa and Kohn, 1964, %alivelSolastiouk et al., 1986, ‘Kretschmer and Wiebe, 1952, %eu at al., 1989, ‘Budantseva et al., 1975c, ‘Wilsak et al., 1987, ‘Budantseva et al., 1975a, tie Loos et al., 1988,‘Budantseva et al., 1975b, “Tucker et al., 1969.
for each considered mixing rule. In the tables, abbreviation 1+2+7 means mixing rule with adjustable parameters k,, k, and k, (other parameters are equal zero) etc. and abbreviations 4 or 1+4 or 1+2+4 mean one- or -two or three-parameter mixing rules containing the sy_mmetrical Goon-random term (Sa) and (Sb) where k5 = kb The cumulative quantities P(%) or Sy are defined by the relation (12)
iiP(96)
(12)
where P,(%) is quantity (7) for the j-th alkane member of each series (i.e. PI(%) is quantity (7) for the system methanol + methane in the methanol series etc.). M is the total number of alkanes in each series (12 or 10 or 6 for the methanol or ethanol or propanol series). Cumulative averages may contain great deviations. This is the reason why the maximum value of quantity (7) is also presented @P_(%) or Sy,,) with the corresponding carbon number of the n-alkane for which the maximum value of 6P(%) or Sy occurs. The quantity U(%) is the cumulative percentage amount of unstable points, and is defined by the relation
69
TABLE 2 Results of unconstrained mix.rule
SP(%)
fit for methanol + n-alkane systems ~Plnaxx(%)
c,
ZY
6Y,,
C,
UC%)
no par. 35.0 166 1 .085 .169 7 2 ____________________~~~_~_~~~~~~~~~~~__-_~~-~~~~~~__---~~~~~~~~~__-----~~~~~~___---~~~~~~ 1 16.2 23.2 1 .088 .205 6 11 2
14.1
21.3
1
.093
.225
6
17
3
16.2
23.3
8
.093
.230
6
9
4 16.2 50.3 16 .093 .211 6 20 ____-_-----___-_-___-----~~~~~~~~---~---~~~-~~~~-~---~~~~~~~~~~~_---~~-~~~~~~----~~~~~~~~ 1+2 8.9 20.6 1 .071 .176 7 14 2+7
8.2
18.2
5
.075
.184
7
7
2+3
6.5
15.0
5
.065
.138
7
9
3+8
11.5
18.5
5
.072
.152
7
10
1+4
8.7
24.1
16
,058
.123
6
6
4+5 5.2 10.9 2 .060 .146 6 16 ______________-________--____________--_-__-______---___________--____-_____~____________ 1+2+7 4.3 10.0 5 .084 .224 6 11 1+2+4
4.7
18.8
1
.035
.082
6
9
1+2+3
3.2
7.2
9
.061
.133
7
12
2+3+7
4.2
14.6
5
.071
.I62
6
10
2+3+8
4.3
14.5
5
.070
.161
6
10
1+4+5 2.3 5.2 3 .029 .067 5 5 _____--_______-_____-~~~-~~~~~~~~~--~~-~~~~-~~~~-~~~-~~~~~~~~~-~~-~~~~-~----~~~~~~~~~~~-1+2+6+7 3.9 9.8 5 .071 .236 6 10 1+2+3+6
2.8
6.6
8
.068
.181
6
10
1+4+5+6
1.5
4.2
1
.026
.066
5
2
(13) where Wj or Nj are, respectively, the number of points with an unstable liquid phase (i.e. Gil < 0) and the total number of points for the j-th alkane in the series. Table 7 contains a comparison between our results (“VDL”) for mixing rule 1+4+5+6 and the hole quasichemical group contribution EOS proposed by Victorov and Fredeslund (1990), (“VF”). Table 7 enables a comparison between two quite different approaches in the description of VLE data. A comparison of unconstrained and constrained fits is presented in Fig. 1. Results from the different approaches are demonstrated for the isobaric (101.3 kPa) system methanol-heptane (7 K above liquid-liquid critical temperature). Two critical points (P(MPa), T(K), x) = (0.1,324.5,0.588),(25, 332.5,0.580),0, = 0.5 K and 6, = 0.01 measured by Ott et al. (1986) were included in the objective
70
TABLE 3 Information about exnerimental data ethanol + n-alkane systems P[MPa]
TlKl C,
min
max
min
max
Ref.
N
1
298.15
498.15
1.410
31.53
36
2
313.4
498.15
1.307
11.31
32
ab ab
3
273.15
500.00
.041
6.205
91
cd
4
298.45
345.65
.086
.881
65
e
5
283.15
460.93
.016
5.516
57
6
298.15
353.15
.020
.212
145
fg hij
7
303.27
343.15
.Oll
.096
82
k
8
318.15
348.15
.OlO
.095
73
1
10
315.48
433.15
.lOl
.lOl
27
mn
16
325.25
325.25
.004
.032
29
0
et al., 1990, bBrunner and Hultenschmidt, 1990, ‘Kretschmer and Wiebe, 1951, dGomezNieto and Thodos, 1978, “Holderbaum et al., 1991, ‘Ishii, 1935, KMcCracken et al., 1960, ‘Wolff and Goetz, 1976, ‘Iguchi, 1978, JSugi and Katayama, 1978, kBerro et al., 1982, ‘Boublfkova and Lu, 1969, “Koshelnikov et al., 1974, “Ellis and Spurr, 1961, “French et al., 1979. ‘Suzuki
function (9). For each approach the adjustable parameters of the mixing rules 1+4+5 were determined from the whole set of experimental methanol-heptane VLE data (see Table l.), i.e. not only from the isobaric data presented in Fig.1. Curve (1) in Fig. 1 corresponds to the experimental data and curve (2) to the unconstrained fit using objective function (6). The values of W(%) = 0.72 and Sy = 0.035 were achieved for this fit (for the whole set of data). Curve (3) corresponds to the mathematical approach (constrained problem (1 l)), giving 6P(%) = 1.0 and Sy = 0.026. Finally, curve (4) corresponds to the physical approach (objective function (9)) with q= 1000. For such a choice of q we get an almost perfect agreement between calculated and experimental critical points (ATi < 0.3 K i = 1,2), but SP(%) = 14 and Sy = 0.12. If we choose q = 50 then the calculated temperature of the low pressure critical point corresponds with the lowest temperature of VLE data and we obtain approximately the same results as for the mathematical approach.
DISCUSSION It follows from Tables 2,4 and 6 that the best unconstrained fits were achieved with mixing rules containing the two parameter non-random term, i.e. 4+5, 1+4+.5 or 1+4+5+6. The first row of Tables 2, 4 and 6 contain information about the results for no parameters. Even though the decrease of the value of the objective function using only one parameter mixing rules is very significant, these one-
71
TABLE 4 Results of unconstrained fit for ethanol + n-alkane systems mixrule
z?P(%)
~P.n#X(%)
C”
SY
8Y,.X(%)
c.
UC%)
no par. 25.9 35.1 5 .OSl .162 8 2 --________--__-_____~~~~-~~~~~~~~~~~~~-~~~~-~~~~~~~~-~~~~~~~~~~~~-~~~~-~~~~~~~~~~~~~~~~~~ 1 18.2 31.1 16 .066 .141 7 21 2
17.8
32.1
16
.066
.133
7
20
3
18.9
32.1
16
.066
.137
7
20
4 18.6 32.1 16 .059 .130 8 16 ______________-_____~~~~-~~~~~~~~~~~~~-~~~~-~~~~~~~~-~~~~~~~~~~~~-~~~~-~~~~~~~~~~~~~~~~~~ 1+2 10.4 21.8 1 .056 .094 8 18 2+7
13.7
24.6
5
.063
.119
8
18
2+3
12.0
3+8
15.1
18.9
5
.056
.114
8
16
32.1
16
.070
.118
8
1+4
15.9
29.7
18
16
.068
.164
7
7
___4’5__________814_______26._q_______-_~~_-~~~~______,o55_________1_o__-_~________________ 1+2+7 8.0 14.3 4 .054 .086 2 18 1+2+4
6.9
12.7
1
.042
.lOO
8
10
1+2+3
7.4
14.1
1
.046
.090
8
16
2+3+7
10.2
18.4
4
.065
.106
2
14
2+3+8
10.2
18.4
4
.058
.105
8
14
1+4+5
6.3
13.7
10
.021
.040
10
6
1+2+6+7
8.0
14.3
4
.054
.086
2
18
1+2+3+6
7.4
14.1
1
.046
.090
8
16
1+4+5+6
6.3
13.7
10
.021
.040
10
6
parameter mixing rules are not able to describe 1-alkanol + n-alkane mixture with less than 10 % average deviation in pressure. Note also the similarity of the oneparameter results. For two- and three-parameter mixing rules too, the cumulative results do not differ much between the alternatives without the two-parameter non-random term. For instance, the combination 1+2 does not give worse overall results than mixing rules with temperature dependence (parameter 6 or 7 or S), with density dependence (parameter 3) or both (3+8). For three parameters, density dependence seems to be slightly better than temperature dependence (1+2+3 vs. 1+2+7), while both together (2+3+7 or 2+3+8) look slightly worse. Mollerup and Clark (1988) and Mollerup (1983,1985) tested mixing rules 1+2,1+4 and 1+2+4 on several types of systems. They obtained from a more limited set of data the following conclusion for methanol - hydrocarbon mixtures : (a) the mixing rule 1+4 gives significantly better results than the mixing rule 1+2, (b) introduction of a I?, (i.e. mixing rule 1+2+4) does not improve the correlation. Our results do not support that conclusion.
72
TABLE 5 Information C,
about the experimental
data of propanol + n-alkane systems.
WI
P[MPa] N
Ref.
min
max
min
max
1
313.4
333.4
1.41
10.197
10
a
2
283.15
333.4
.l
6.742
16
ab
5
313.1
317.1
.05
.134
16
C
6
298.15
361.75
.013
.l
19
de
7 10
333.15 347.75
523.15 398.95
.l .017
62 24
fgh i
4.319 .091
‘Suzuki et al., 1990, bBen-Naim and Yaacobi, 1974, ‘Rice et al., 1990, dIguchi, 1978, “Rhim and Kwak, 1981, ‘Zawisza and Vejrosta, 1982, *Pena and Cheda, 1970, hSabarathiham and Andiappan, 1985, ‘Ellis et al., 1960.
TABLE 6 Results of unconstrained mixrule
fit for propanol + n-alkane systems
ZP(%)
8PlnJ%)
C”
SY
8Y,,(%)
C”
U%)
7 0 .059 .122 35.6 5 no par. 24.4 _______________________~~~~~~~~~~~~~~___~__-~~-~~~~~~~~~__~______-~~~~~~~~~~~~___________ 7 7 .034 .089 16.7 5 1 9.7 2 3
8.9
12.8
5
.052
.130
5
6
9.2
13.6
5
.034
.085
7
6
7 6 .035 .094 12.9 10 4 7.8 ______________-__________~___________---__--________-________---______-________--________ 7 11 .028 .058 9.5 1 1+2 5.9 2+7
7.1
12.8
5
.036
.081
7
5
2+3
5.2
8.8
7
.026
.070
7
3
3+8
7.2
13.6
5
.030
.082
7
4
1+4
6.8
11.2
7
.036
.089
7
3
5 8 .019 .027 4.5 1 4+5 3.1 ~~__________~~~~~______~_~__~_________-_----________-_____----_____----_-________________ 1+2+7 4.1 7.3 .019 5 .058 7 8 1+2+4
4.0
9.4
1
.013
.035
7
8
1+2+3
2.7
4.8
10
.013
.033
7
6
2+3+7
4.6
8.8
7
.021
.070
7
2
2+3+8
4.5
8.8
7
.020
.070
7
2
1+4+5 1.9 ------________-_________-_1+2+6+7 3.9
2.7 10 .013 10 4 .019 ________________--__~~~_______~~~~~________-_~__________--_--_ 7.2 .018 5 .056 7 10
1+2+3+6
2.2
4.8
10
.013
.023
7
4
1+4+5+6
1.7
217
10
.013
.019
10
0
73
TABLE 7 Comparison
between results of this work (VDL) and results of Victorov and Fredeslund SP[%]
@F)
100 Sy
N
PAMpal
VDL
VF
VDL
VF
11
41.4
250.03
7.4
6.5
.4
.08
12
41.4
273.16
4.5
5.4
.7
.15
12
41.4
290.0
3.9
5.9
.8
.16
11 41.4 330.0 9.6 5.4 1.2 __---------_______-_________--_---_-______-__-----_--____-____________~~~~~~~~~~----~-methanol + 11 2.3 273.16 1.3 1.4 -
.33
System methanol methane
ethane
+
5
4.1
‘UK1
298.16
1.9
1.9
-
-
5 6.0 373.15 3.2 1.7 5.7 4.4 ____________________-__________________-----________---_--_-_____-_____-___~___~~~~~~~~ methanol + 8 1.4 313.1 3.3 1.3 1.0 .6 propane
8
2.5
343.1
2.4
1.6
2.1
8 4.3 373.1 3.3 1.2 2.0 ____________________-____________---_-------_________---------__--_____-_________~____~ methanol + 11 .l 303-338 1.2 4.6 6.2 n-pentane
9
.83
372.69
1.0
4.2
.9
9 2.5 422.59 .3 3.5 1.3 __________________-________------------___-_______-----__-_______-______~_~~~~~~~~~~~~~ methanol + 7 .03 298.16 3.1 2.0 9.9 n-hexane
1.5 2.5 4.9 2.5 2.7 2.3
23
.05
308.16
1.0
1.6
-
-
23
.25
348.16
1.7
2.5
-
-
473.02
5.0
.4
-
-
1
4.6
The improvement in going from three to four parameters is slight, and in addition four parameters gave numerical problems unless both temperature and pressure region were large. Density dependent mixing rules makes it possible to satisfy not only the low density limit (i.e. quadratic dependence of the second virial coefficient on composition) but also a reasonable high density limit. If we consider the nonrandom term (5a) and (5b) for a binary mixture of equisized molecules and of components having equal molar volumes, then the high density limit is equal to the three-parameter Wilson (1964) equation (Whithing and Prausnitz (1982), Won (1983)), where the parameters of the Wilson equation are determined from the values of k4 (mixing rule 4) or k4 and k5 (mixing rule 4+5). Two independent values of the adjustable parameters k4 and k, makes it possible to achieve an arbitrary asymmetrical dependence of the excess Gibbs energy calculated from Wilson equation on composition. This is, in our opinion, the reason why mixing rules containing the two parameter non-random term give better result than other mixing rules having the same number of parameters. The actual values of the cumulative quantities in Tables 2,4,6 ( &‘(%) and 8~) are not so important, because they are strongly infhtenced by the experimental
74
0.76
-
0.72
-
0.64
-
Yl
0.60
’
I
I
I
I
I
0.0
0.2
0.4
0.6
0.8
1.0
xl
Fig. 1 Experimental(l) and calculated y1 - x1 curves of methanol-heptane (7 K above lJCS7J and x, E (.05;.95) (2) . .. unconstrained fit (3) . .. constrained fit eqn. (11) (4) . .. physical approach eqn. (9) with q = lo3
system for
101.32 kPa
data. As can also be seen from Fig 1, data near the liquid-liquid critical points will be harder to describe, and will lead to larger deviations. However, using the same experimental data for all the alternatives, the various mixing rules can be compared. The comparison with the complex group-contribution method in Table 7 shows that our results are very similar to theirs, and they also report false liquidliquid splits. The most noticable result in Tables 2,4 and 6 is that all but two of the cumulative results show a number of points with an unstable liquid phase, i.e. the equation of state will give a false liquid-liquid phase split irrespective of the of the number of parameters and mixing rules. The cumulative results mask any trend with alkane carbon number. For each alcohol, the data for the alcohol - methane binaries were fit without any unstable points for all mixing rules. Likewise, the mixtures with ethane and propane had relatively few or no unstable points. This is related to the change in upper critical solution point with carbon number. The experimental data are all at ambient temperature or above, while the UCSP for methanol methane (from extrapolation) is below 200 K, and for methanol - hexane 308 K at 1 bar. Since the amount and the range of the experimental data are different for each system, it is impossible to correlate the results with the UCSP’s for each system.
Both of the methods suggested will remove the false phase split, but at the expense of a poorer fit of the VLE data. The constrained fit can only assure correct phases in the region of the VLE data, but with minimum loss in the VLE data fit. Including upper critical solution points in the fit may in principle make the region of false phase splits almost disappear, but at a corresponding much worse fit of the VLE data. Both methods will be treated in more detail in subsequent publications.
CONCLUSION Cubic equations of state with ordinary mixing rules will predict false liquidliquid splits for 1-alkanol + n-alkane binaries when the binary parameters are determined from VLE data, unless the data are limited to temperatures much above the upper critical solution points. Disregarding the false phase splits, the best fits are obtained using density dependent mixing rules containing two parameter non-random term, preferably in combination with an additional parameter in the repulsive term. False phase splits may be avoided by either including UCSPs in the fit with an appropriate weight, or by a constrained fit to force a correct curvature of the Gibbs energy function. Such additional requirement will necessarily give a poorer fit to the VLE data.
LIST OF SYMBOLS
a, ai, b, b, Gll (Glll) H 4 L M N NP P i T T, u UCSP V
attractive parameter in EOS repulsive parameter in EOS the second (third) derivative of the molar reduced Gibbs energy with respect to liquid composition at constant temperature and pressure non-random term parameter in a mixing rule liquid number of alkanes number of points number of adjustable parameters in Eqn.(2) pressure external surface area gas constant temperature reduced temperature percentage amount of the unstable points upper critical solution point molar volume
number of unstable points liquid mole fraction of the first component value of ‘Ix” for the i-th point gas mole fraction of the first component adjustable parameter in Eqn.(2) difference between experimental and calculated value mean deviation standard deviation of an experimental value objective function Langrange multipliers empirical constant
ACKNOWLEDGEMENT This research was financed by the Norwegian research council NTNF, and mainly by its SPUNG program.
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