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Vapour-liquid and liquid-liquid phase equilibria of binary mixtures containing helium: comparison of experiment with predictions using equations of state
Rglg PILilb"E EIIUIUBRIA ELSEVIER
FluidPhaseEquilibria 122(1996) 1-15
Vapour-liquid and liquid-liquid phase equilibria of binary mixtures containing helium: comparison of experiment with predictions using equations of state Ya S o n g Wei, Richard J. Sadus * Computer Simulation and Physical Applications Group, School of Computer Science and Software Engineering, Swinburne University of Technology, PO Box 218, Hawthorn, Vic. 3122, Australia
Received 19 December 1995; accepted 12 February 1996
Abstract
The unlike interaction parameters for binary mixtures of helium + non-polar gases are obtained by comparing experimental critical data with calculations using the Guggenheim and Heilig-Franck equations of state. These interaction parameters and equations of state are used to predict a priori both the vapour-liquid and liquid-liquid equilibria for helium + non-polar gas mixtures. The predicted phase diagrams are compared with experimental data for a wide range of pressure, temperature and composition. Good agreement between theory and experiment is obtained for the isobaric temperature-composition phase behaviour and isothermal pressurecomposition phenomena can be predicted qualitatively. Keywords: Theory; Equations of state; Vapour-liquid equilibria; Liquid-liquid equilibria; Binary mixtures; Helium
1. I n t r o d u c t i o n
Considerable progress [ 1] has been achieved in the application of theory to the calculation of binary mixture phase equilibria. The importance of vapour-liquid equilibria and liquid-liquid equilibria to many industrial processes provides a considerable incentive to improve the quality of phase equilibria prediction. Historically, the prediction of phase transitions has relied on empirical methods, but more realistic equations of state are being developed that attempt to account for the various aspects of intermolecular interactions. Some of these equations have been applied successfully to the prediction of phase transitions. For example, the Christoforakos-Franck [2] equation of state has been used to
* Corresponding author. 0378-3812/96/$15.00 Copyright© 1996ElsevierScienceB.V. All rightsreserved. PH S0378-3812(96)03045-2
2
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996) 1-15
calculate critical phenomena and binodal equilibria in aqueous mixtures [2], and the simplified perturbed hard-chain equation of state [3] has been applied extensively to both pure components [4,5] and mixtures [6]. There has also been progress in the development of theoretically based equations of state for macromolecules [7-9]. Traditionally, the accuracy of equations of state has been refined by comparison with experimental data. There are few "real" systems that can be used to evaluate unambiguously the accuracy of equation of state calculations. This situation has been alleviated partly by modem molecular simulation techniques [10] that provide exact data to test the accuracy of theory. However, simulations are limited to relatively simple intermolecular potentials. Consequently, the refinement of equations of state continues to rely on inputs from both simulation and direct comparison with experiment. Most successful equations of state use a "hard sphere + attractive" term approximation. Mixtures of the noble gases can be used legitimately to evaluate the accuracy of the "hard sphere + attractive" term approximation for vapour-liquid equilibria. However, liquid-liquid phase coexistence in binary mixtures almost invariably requires a dipolar component. Binary mixtures containing helium are noteworthy exceptions. Mixtures of helium + a non-polar component are rare examples of non-polar systems exhibiting both vapour-liquid and liquid-liquid phase transitions. They are also relatively free from theoretical complications, with the possible exception of quantum effects at low temperatures. The aim of this work is to establish whether or not theoretically based equations of state can be used to predict adequately the phase diagrams of binary mixtures containing helium. The unlike intermolecular parameters required for the Guggenheim [11 ] and Heilig-Franck [12] equations of state are obtained by fitting theory with experiment for the critical curves of eight helium + gas mixtures. These interaction parameters and equations of state are used subsequently to predict both vapour-liquid and liquid-liquid equilibria in helium-containing mixtures. The results are compared with experimental data over an extensive range of temperature, pressure and composition.
2. Theory Details of the calculations are given elsewhere [1,13] and only a brief outline is given below. The binodal curves at any temperature (T) and pressure ( p ) are obtained by solving the conditions for phase coexistence between the two phases (denoted by I and II) T I = T II
(1)
pt = pn
(2)
I' - -
j .v
A + pV
=
-- x 2
(3)
- -
3x2
r,v
]'[ =
A + pV--
x2
/
~
r.v
1I'
(4)
Y.S. Wei, R.I. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
3
where A, V and x denote the Helmholtz function, volume and mole fraction, respectively. The critical properties are obtained by satisfying the critical conditions for a binary mixture
W=
( a2A -IT~)T
a2A -
T
Ox21
X=
y=
r,v
=0
/ -0--VT
(o a/ (o a/
(5)
=0
r
(6)
>0
(7)
x,vlT lOg--TIT,, The Helmholtz function is obtained from conformal solution theory using the one-fluid model [1 4]
a =f~sA o (V/hes,T/f~s) - RTln h~s + R T ~ . , x i l n x i
(8)
i
where R is the universal gas constant, A o is the configurational contribution to the Helmholtz function and f~s and h~s are the characteristic conformal parameters of the equivalent substance. The conformal parameters can be deduced from the critical temperatures and critical volumes. It is customary to choose one of the components of the mixture as the reference substance (denoted by the subscript 0) and obtain the conformal properties of the pure substance relative to it, i.e. fLl = T~I/T~o, hll = V~/V~o, etc. The pure component data were obtained from a compilation by Ambrose [15]. The conformal parameters for the equivalent substance are obtained from the van der Waals prescriptions feshes = x2fl,h,l + 2x(1 - x)f12h12 + (1 -- x)2f22h22
hes=x2hll + 2x(1 - x)hl2 + (1 -- x)2h22
(9) (10)
where the contribution of unlike interactions is given by /.jl/3"~ 3 h~2 = 0-125~(h',( 3+,.22 j
(11)
f,2 = ~(f,,f22)o5
(12)
where ~ and ~ are adjustable parameters. The configurational Helmholtz function ( A o ) can be evaluated by direct integration of an equation of state with respect to volume. We have chosen two equations of state which differ in their treatment
4
Y.S. Wei, R,I. Sadus/Fluid Phase Equilibria 122 (1996) 1-15
of both attractive and repulsive contributions to intermolecular interaction. The Guggenheim [11] equation of state incorporates an improved hard-sphere repulsion term in conjunction with the simple van der Waals description of attractive interactions
RT P=
v(,
b
4
a V2
(13)
The covolume ( b = 0.18284RTC/p c) and attractive ( a = 0.49002 R 2TC2/pC) equation of state parameters are related to the critical properties. The Guggenheim equation has been used [16] successfully to calculate critical behaviour in a wide variety of binary and ternary mixtures. Some calculations using the Guggenheim equation have been attempted for vapour-liquid phase coexistence [17], and liquid-liquid phase equilibria at high pressures [18]. There is some evidence [12,13] that the Heilig-Franck equation of state is suitable for the calculation of the high pressure phase behaviour of binary mixtures. The Heilig-Franck equation of state uses the Carnahan-Starling [19] representation of repulsive forces between hard spheres, and a square-well representation for attractive forces.
RT(1 + y
P=
+ y2 _
V(1 _y)3
y3)
RTBa"
+ V2_(Catt/Batt) V
(14)
Heilig and Franck [12] used the following temperature-dependence [2] for the y term y = "~---~
(15)
where b c is the covolume at the critical point. However, in this work, y was made independent of temperature by using z = 0. The Batt and Qtt terms in Eq. (14) represent the attractive contributions from the second and third virial coefficients, respectively, of a hard-sphere fluid interacting via a square-well potential. The square-well is characterised by three parameters related to the hard-sphere collision diameter (~), the depth of the well (e) and the relative width of the well (k). The second and third virial coefficients for the square-well potential can be solved exactly [20] for both pure fluid and binary mixtures. Details of the calculation of the pure component parameters are reported elsewhere [ 18]. The unlike interaction contributions for these parameters are obtained from 0"12 = ~(0"11
+ 0"22) 2
'~12 = ~1'¢22
(16) (17)
It is also assumed that h = h~l = X22 ~- ~k12.
3. Results and discussion It is well documented [1] that the accurate prediction of binary mixtures requires accurate information concerning the strength of the unlike interaction parameters. For relatively simple cases
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
5
30
i i i T I
400
25--
350
i
,
i
',
i #
300 20--
\
I I
/
29.91 K C
/
c,
250
"1 m
Q. 13.
200
!
15~
,'
#
32.89
K
I
Q_
,"
150
10
100
~,
/ <, ~_, ¢ , y'
,'
,35.9 K
is
"' ' .
.
//
1 ,-~
"
I
ci ,
4~b
J
.
G
50
j_
I,
c'c
I'
. 41.9 K
,2,
38.88 K
/
......
,:
i
':'
,5~ ~
:3 ~" /
L
0 0
50
100
150
200
250
300
350
400
450
T/K Fig. 1.
Comparison of the experimental critical curves
012
0.4
0.6
0.8
1.0
X (He)
(o)
of binary helium + gas mixtures with calculations using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. Also shown are critical properties estimated from
experimental data ( × ).
0.0
Fig. 2. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + neon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Critical points estimated from experimental data are illustrated ( X ) .
[21,22], reliable a priori estimates of the ~ and ~ parameters are possible. However, the interaction parameters for liquid-liquid equilibria are generally obtained from experimental data. In this work, we have obtained the values of h, ~ and ~ by optimizing the agreement between theory and experiment for the p-T projection of the critical curve. The optimal agreement was obtained by trial and error adjustment of the interaction parameters and repeated calculation of the critical curve as described elsewhere [1]. Experimental high pressure and critical data are available for the helium + argon [23-25], xenon [26], hydrogen [27,28], nitrogen [25,29,30], ethylene [31] and propane [32] mixtures. The critical data are compared in Fig. 1 with calculations of the critical curve using both the Guggenheim and Heilig-Franck equations of state. The interaction parameters obtained from the analysis are included in Table 1. The critical curves and the interaction parameters for the helium + neon [33] and helium + methane [34,35] systems were estimated by assuming that the experimental critical pressure coincided with the maximum pressure of the phase envelope. It should be stressed that, in reality, the critical pressure can occur below the pressure maximum. The comparison of theory with experiment indicates good agreement for the p-T projection of the critical curve. It is almost impossible to
6
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996)1-15
Table 1 Combining rule parameters for binary helium+gas mixtures obtained by optimizing agreement between theory and experiment for the p - T projection of the critical locus Gas
Guggenheim
Neon Argon Xenon Hydrogen Nitrogen Methane Ethylene Propane
1.55 1.50 1.40 1.07 1.37 1.30 1.25 1.23
Heilig-Franck
0.70 1.02 1.07 1.06 1.03 1.05 1.06 1.06
1.10 1.35 1.27 1.50 1.55 1.65 1.60 1.50
1.62 1.55 1.59 1.08 1.38 1.30 1.28 1.50
0.80 1.02 1.08 1.07 1.03 1.05 1.06 1.09
distinguish between the critical curve calculated using either the Guggenheim or Heilig-Franck equations of state. It is apparent from Fig. 1, that the critical loci of the helium + gas systems all commence at the critical point of the gas component and thereafter, rise rapidly to very high pressures. The mixtures exhibit Type III behaviour in accordance with the phase behaviour classification scheme of van Konynenburg and Scott [36]. However, there is a subtle and interesting distinction between the different critical curves. The critical curves of helium + the gases methane, ethylene, propane and xenon, occur at temperatures and pressures above the critical point of the gas. This phenomenon is characteristic of so-called "gas-gas" immiscibility [37] of the first kind (Type IIIa behaviour). In contrast, the critical curve for the remaining mixtures extends initially to temperatures below that of the gas. For the helium + nitrogen mixture, the experimental data indicate a region of "gas-gas" immiscibility of the second kind (Type IIIb behaviour) at very high pressures. There are insufficient experimental data to assign Type IIIb behaviour to the other mixtures and the calculated critical properties always occur at temperatures below the critical temperature of the gas. The phenomenon of "gas-gas" immiscibility of the first kind is rare and it is confined almost exclusively to mixtures containing helium. The neon + water mixture [38] is the only other reported Type IIIa mixture. Temkin [39] proposed two simple criteria to predict "gas-gas" immiscibility of the first kind that are based on the relative magnitude of the van der Waals equation of state parameters. In terms of the critical volume (V c) and temperature (T c) of the components, the criteria that must be satisfied to observe Type IIIa behaviour are V•, >__0.42V2C2
(18)
T~,V?, <
(19)
0.052T~2V;2
where component 2 is the component with the higher critical temperature. These simple criteria predict successfully Type IIIa behaviour for several mixtures [13,38]. However, the above criteria have mixed success in predicting "gas-gas" immiscibility of the first kind for the helium + gas mixtures. Type IIIa behaviour is not predicted for the helium + propane mixture (Eq. (18) is not satisfied) whereas the phenomenon is incorrectly predicted for both the helium + argon and helium + nitrogen mixtures. However, Type IIIa behaviour is correctly predicted for mixtures of helium + xenon, methane and ethylene.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
7
The helium + noble gas mixtures are the simplest possible binary mixtures and as such they represent a suitable test for "hard sphere + attractive term" equations of state. The p - x vapour-liquid phase envelopes for the helium + neon mixture have been measured [33] at several different temperatures. These experimental data are compared in Fig. 2 with the predictions of both the Guggenheim and Heilig-Franck equations of state. Reasonably good agreement between theory and experiment is obtained at all temperatures. At pressures > 5 MPa, there is a noticeable difference in the position of the critical p - x locus predicted by the two equations of state. At low temperatures, the Heilig-Franck equation of state is slightly more accurate than the Guggenheim equation. Phase separation at high pressures is also predicted at unrealistically low temperatures (Fig. 1). This may be caused by errors in the unlike combining rule parameters obtained by comparing theory with the estimated low pressure vapour-liquid critical points. Considerable experimental high pressure data are available for the helium + argon mixture [23-25]. The equations of state predictions for vapour-liquid equilibria of the helium + argon mixture
'
o
~
,' 60
t
i ,
~
' i
J© "
,
I, ~ L
i o I
! J
'
,
,
i '~ o
c"
,,
,:~
t ,'
'
I
i
,i ~
looo 900
,, '
o
'
'I
o
t~
I~'x ,
/
"
!,
o
,'!
o
~.
/ [,
]
t~
400 ./
o_ii
~
'o,,,, , { - / / I ,/, /' "/'1 '/E
lo-i~
o
..../ / / ' "
2~
./
o
_~- o I
0.2
j
,,
,
-
'/
/"
/ o//
0 0.0
j
o/
',"o fx '/,,-" o / /
,
'
0.4
!o
J
"
/,i, °
!
0.6
i 0.8
aoo
200 Ioo
'
0 1.0
x (He)
Fig. 3. Comparison of experimental vapour-liquid equilibrium data (o) for the helium + argon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the estimated experimental critical points are identified ( X ).
0.0
0.2
0.4
0.6
0.8
1.0
x(He) Fig. 4. Comparison of the experimental isothermal pressure-composition behaviour (o) of the helium + argon
mixture with predictions using the Guggenheim (--) and Heilig-Franck (. . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data (x) are also identified.
8
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996) 1-15 310-
310 -
p ~ 90.3975 MPa
p = 50.6225 MP a
300
p29O
290-
0 280
. 0.0
320 140-
140-
130-
190 -
'
i ' i ' ' 0.4, 0.6 018 110 X (He)
012
2800.0
320-
p = 81.06 MPa
310-
0.2
0.4 0.6 X (He) p = 101.325 MPa
0.8
"i
1.0
310,'
0
300110"
110-
100-
100-
290
90
'
0.0
I
0.2
190-
'
I
'
t
'
I
'
~
0.4 0.6 0.8 1.0 X (He) p = 200 MPa
,7o:
90
~
0.0
0.2
190 7
I
1.o: ,5o-
'901 /o/° 190-1
o
280
t
0.4 0.6 0.8 1.0 x (He) p = 4 0 0 MPa
,7o- 1
290-
\ \ °o
i
0.0 330
0.2
04 0.6 X (He)
0.8
1.0
,
~
310 -
,'
1.0
0.0
0.2
0.4 0.6 x (He)
0.8
Fig. 5. Comparison of experimental isobaric temperaturecomposition behaviour (o) of the helium + argon mixture with predictions using the Guggenheim ( - - ) and HeiligFranck ( . . . . ) equations of state. The experimental ( • ) and calculated ( × ) critical points are identified.
1.0
i 1.0
0.8
O
,,
©
/',"',t,l,I ,"
310 -
0
29011-
290 0.8
p = 162.12 M ~ , , , .
C)
290
0.4 0.6 X (He)
0.4 0.6 X (He)
O
300
0.2
i 0.2
350-
/'
0.0
' 0.0
330-
p = 131.7225 MPa
320 -
290
, 0.0
0.2
0.4 0.6 X (He)
0.8
j 1.0
280 k 0.0
.
' 0.2
0.4 0.6 X (He)
0.8
t 1.0
Fig. 6. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + xenon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( - ) and calculated ( × ) critical points are identified.
are compared with experiment in Fig. 3. At pressures < 70 MPa (Fig. 3), there is satisfactory agreement of theory with experimental vapour-liquid equilibria data in the p - x projection. The predictions of both equations of state are of similar accuracy. However, the transition between vapour-liquid and liquid-liquid equilibria is not predicted accurately. The very high pressure liquid-liquid phase coexistence of helium + argon at different temperatures is examined in Fig. 4. The agreement of theory with experiment is only satisfactory at moderately high pressures ( < 400 MPa). At very high pressures there is a substantial discrepancy between the predicted and experimental critical composition. Consequently, the predicted coexistence curves are shifted to lower compositions. The isobaric T - x behaviour for liquid-liquid equilibria is examined in Fig. 5. In contrast to the predicted isothermal behaviour, the equation of state calculations at isobaric conditions are in exceptionally good agreement with experiment for the entire composition range.
Y.S. Wei, R J. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
9
De Swaan Arons and Diepen [26] measured liquid-liquid equilibria for the helium + xenon mixture. Their experimental data at several different temperatures are compared with equation of state predictions in Fig. 6. The predicted liquid-liquid equilibria are in reasonably good agreement with experiment. However, a deterioration in the quality of prediction occurs at helium-rich compositions. This can be attributed partly to the inability of theory to adequately account for the considerable size difference between the component molecules.
35-
p = 3.447 MPa
30v
p_ 25-
20-
,!
i
15 ~ , 0.0 0.2
J
o
I
'
'
'
'
I
0.4 0.6 0.8 1.0 X (He) p = 20.0 MPa
30-
o
Oo ~. o
c~
# 25-
A
i
s
i
i
/
13_
20- /
//o i
l/
/
0.0
I
oq
0 /0 f
0
t°I
ol I ol ol
80 ~
0.2
0.4
0.6
0.8
X (He)
Fig. 7. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data ( X ) are also identified.
0.4 0.6 0.8 1.0 x (He) p = 500,0 MPa .X.
," ,"" " "
" " " "~''',,,
#
,/ 0.0
0.2
lOO-
60-
1.0
40 0.0
0.2
0.4 0.6 x (He)
0.8
1.0
Fig. 8. Comparison of experimental isobaric temperaturecomposition behaviour (o) of the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( . ) and calculated ( X ) critical points are identified.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
10
140- p = 82.783 MPa 130 -
//(5"~
120-.
100 11o-_ 90-
140
/
120
O
p = 137.903 MPa
1
O
\°o
11ol
i i ~
80-
70, . . . . 0.0 0.2 0.4 0.6 X (He) 140- p = 206.804 MPa
0.8
0.2
0.4 0.6 X (He)
0.8
1.0
O~,,.,.:k~,... ii•
130 -
120 - /
"
J
110-
,
120
~~(3
90 80-
70
.
. . . . 0.2 0.4 0.6 X 140~ p =344.91 M P ~ L I I O 0.0
,301 12oI
'/9
(~)
0
0
100 -
IZ El.
0.0
1.0
(He)
0.8
i 1.0
1130 90 80 700.0 . 0.2. 0.4. . 0.6. 140 7
p=413.71
X (He) M ~ ( ~
0.8
1.0
0 0
\°o
/o
\o
\o 110 100 00
0.0
0.2
0.4
0.6
0.8
1.0
x (He)
Fig. 9. Comparison of experimental isothermal pressurecomposition behaviour (o) of the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the experimental critical points are identified ( ' ) .
0.0
0.2
0.4 0~6 X (He)
0.8
1.0
0.0
0.2
0.4 0.6 X (He)
0.0
1.0
Fig. 10. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + nitrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( ' ) and calculated ( × ) critical points are identified.
The phase behaviour of helium + simple diatomic mixtures can be represented by systems containing either hydrogen or nitrogen. The isothermal p - x vapour-liquid coexistence curves at several different temperatures have been reported [27] for the helium + hydrogen mixture. These data are compared with equation of state predictions in Fig. 7. It is apparent from Fig. 7 that either the Guggenheim or Heilig-Franck equations of state can predict accurately the vapour-liquid coexistence of this mixture. Very impressive agreement with experiment [27] is also obtained for the isobaric temperature-composition phase coexistence at low pressures (Fig. 8). The helium + hydrogen mixture also exhibits liquid-liquid separation at very high pressures [28]. The experimental isothermal liquid-liquid coexistence data are compared with equation of state calculations in Fig. 9. At pressures > 50 MPa there is a significant discrepancy between the predicted
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
11
16
,
:
ii
s~
,/
12
'
,/
!
ItI
/
,// ;/
7; i
/
5• i
q
/
;
/
/ o
,
,,' ~
,
0.2
0.4
0.6
0.8
1.0
X (He) Fig. 11. Comparison of experimental isothermal pressure-
composition behaviour ( o ) of the helium + nitrogen mixture w i t h predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data ( × ) are also identified.
f
j_.
0 0.0
i
,/ ,•/
0.0
0.2
i 0.4
0.6
0.8
1.0
x (He) Fig. 12. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + methane mixture with
predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold.
and experimental critical composition. Nonetheless, there is generally satisfactory agreement between theory and experiment at temperatures < 80 K and pressures < 600 Mpa. The experimental [29] isobaric T - x behaviour of liquid-liquid coexistence for the helium + nitrogen mixture is compared with equation of state predictions in Fig. 10. Irrespective of the pressure, there is good agreement between theory and experiment for the entire composition range. For this mixture, several isothermal measurements at very high pressures have been reported [25]. These data are compared with equation of state predictions in Fig. 11. There is generally only qualitative agreement between theory and experiment at all temperatures. In particular, theory fails to account for the coexistence isotherms at very high pressures. Experimental measurements for helium + a non-polar polyatomic molecule are limited to mixtures containing a hydrocarbon gas. The helium + methane mixture represents the simplest possible helium + hydrocarbon combination. Sinor et al. [34] measured the isothermal p - x vapour-liquid coexistence of the helium + methane mixture at several temperatures. It is apparent from Fig. 12 that reasonably accurate predictions for these experimental data are obtainable with either the Guggenheim or Heilig-Franck equations of state. Streett et al. [35] reported liquid-liquid separation for this system
12
Y.S. Wei, R J . Sadus / Fluid Phase Equilibria 122 (1996) 1-15 340
p
=196.132 MPa
320
. -
•
O %
v
# 300 -
280 0.2 380 -
360 -
340 b---
0.4
0.6 X (He)
I 1.0
0.8
p = 392.26 MPa
~ O ( D
-
320
n Q.
280
' 0.2
400
I 0.4
'
~ ' 0.6 X (He)
'
I 1.0
p = 588.397 MPa
360 380 -
#
I 0.8
Q ~ (~0 i
340 320 300 280
o.o
0.2
0.4
0.6
0.8
1.o
' 0.2
,
x (He)
Fig. 13. Comparison of experimental isothermal pressurecomposition behaviour ( o ) of the helium + methane mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the experimental critical points are identified ( • ).
, 0.4
, 0.6 X (He)
, 0.8
1.0
Fig. 14. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + ethylene mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( - ) and calculated critical points ( × ) are illustrated.
under isothermal conditions. Their data are compared with equation of state predictions in Fig. 13. In common with the other helium mixtures, the agreement between theory and experiment deteriorates progressively at very high pressures. Experimental liquid-liquid coexistence data are available for the helium + ethylene [31] and helium + propane [32] mixtures. The predicted liquid-liquid coexistence of helium + ethylene is compared with experiment in Fig. 14. The agreement with theory at different pressures is satisfactory at most compositions. In contrast, the isobaric T - x liquid-liquid coexistence predicted for the Y.S. Wei, R.1. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
410-
410-
p = 98.066MPa
400
p = 196.132MPa
400-
~C
390-
"
•
0
0" 380 -
~
0
0.60
x (He) 430-t p = 294.198MPa 420 4104oo-
390-
',
,"
'
=
0.60
X (He)
;'
=
0.80
0
;'
.
0
, ~ 0.60 0.80 x(He)
I
1.00
-
©
'
,,"
'~
400390 -
o '
0
410-
'".0
0 0
0
,"
0 0
370 0.40
420
,"
©
430-
ioo,/,"
370 / 0.40
1.00
0.80
,," 0
0 3~o-
380-
0
370 ,' 0.40
~.
13
o°/
380-" 0 I
1.00
370 0.40
i°o
:'
"© 0.60
x (He)
0.80
1.00
Fig. 15. Comparison of experimentalliquid-liquid equilibrium data (o) for the helium+ propane mixture with predictions using the Guggenheim(--) and Heilig-Franck (. . . . ) equations of state. The experimental( . ) and calculated (×) critical points are identified.
helium + propane mixture (Fig. 15) occurs over a narrower range of composition than the experimental data. At high pressure, the Heilig-Franck equation is marginally more accurate than the Guggenheim equation of state. T h e deterioration in the quality of prediction for this mixture can be attributed partly to the significantly non-spherical geometry of the propane molecule. An interesting aspect of the comparison of experiment with theory, is the consistently good results obtained using the relatively simple Guggenheim equation. In most cases, it is almost impossible to distinguish between the results obtained for either the Guggenheim or Heilig-Franck equations. We may anticipate that the temperature-dependence of the Heilig-Franck equation may improve the prediction of phase coexistence in dipolar mixtures. However, a recent comparison [ 18] of the binodal curves of aqueous mixtures also found very little quantitative difference between the accuracy of the Guggenheim and Heilig-Franck equations of state. The accurate evaluation of the unlike interaction parameters appears to be the crucial influence which determines the quality of equation of state predictions. For Type III systems, the analysis of the p-T behaviour of the critical curve appears to yield combining rule parameters which are satisfactory for both vapour-liquid and liquid-liquid equilibria. In contrast to other mixtures, e.g. Type II systems, the high-temperature critical curves of Type III mixtures are characterized by a continuous transition between vapour-liquid and liquid-liquid equilibria which may partly explain the similar accuracy of the unlike interaction parameters for both phenomena.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
14
4. Conclusions
The comparison of theory with experiment demonstrates that the "hard-sphere -I- attractive term" formulation of either the Guggenheim or Heilig-Franck equations of state is adequate to predict accurately the isobaric temperature-composition behaviour of binary mixtures containing helium. In contrast, the predicted isothermal pressure-composition behaviour is generally only in qualitative agreement with experimental data. It is also demonstrated that both vapour-liquid and liquid-liquid equilibria for helium + gas mixtures can be calculated from the same set of combining rule parameters. The predictions using either the Guggenheim or Heilig-Franck equations of state are of similar accuracy.
5. List of symbols
A b B C f h P R T V W x
X Y Y
Helmholtz function Molecular covolume parameter Second virial coefficient of a square-well fluid Third virial coefficient of a square-well fluid Conformal solution parameter Conformal solution parameter Pressure Universal gas constant Temperature Volume Determinant defined by Eq. (5) Composition Determinant defined by Eq. (6) Packing fraction defined by Eq. (15) Determinant defined by Eq. (7)
Greek letters 8
h o"
Depth of square-well Width of square-well Hard-core intermolecular separation Combining rule parameter (Eq. (12)) Combining rule parameter (Eq. (11))
Subscripts and superscripts c 1,2 att
Critical property Property of component 1, component 2 Attractive property
Y.S. Wei, R.I. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
15
Acknowledgements Y . S . W . thanks the Australian G o v e r n m e n t f o r an Australian P o s t g r a d u a t e A w a r d . T h e calculations w e r e p e r f o r m e d on the S w i n b u r n e C r a y Y M P - E L c o m p u t e r .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [ 15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
R.J. Sadus, High Pressure Phase Behaviour of Multicomponent Fluid Mixtures, Elsevier, Amsterdam, 1992. M. Christoforakos and E.U. Franck, Ber. Bunsenges. Phys. Chem., 90 (1986) 780-789. C.-H. Kim, P. Vimalchand, M.D. Donohue and S.I. Sandler, AIChE J., 32 (1986) 1726-1734. A. van Pelt, C.J. Peters and J. de Swaan Arons, Fluid Phase Equilibria, 74 (1992) 67-83. D. Pla6kov, D.E. Mainwaring and R.J. Sadus, Huid Phase Equilibria, 109 (1995) 171-182. A. van Pelt, C.J. Peters and J.de Swaan Arons, J. Chem. Phys., 95 (1991) 7569. Y. Song, S.M. Lambert and J.M. Prausnitz, Macromolecules, 27 (1994) 441-448. J. Chang and S.I. Sandier, J. Chem. Phys., 103 (1995) 3196-3211. R.J. Sadus, J. Phys. Chem., 99 (1995) 12363-12366. A.Z. Panagiotopoulos, Mol. Phys., 61 (1987) 813-826. E.A. Guggenheim, Mol. Phys., 9 (1965) 199-200. M. Heilig and E.U. Franck, Ber. Bunsenges. Phys. Chem., 93 (1989) 898-905. N.G. Sretenskaja, R.J. Sadus and E.U. Franck, J. Phys. Chem., 99 (1995) 4273-4277. C.P. Hicks and C.L. Young, Chem. Rev., 75 (1975) 119-175. D. Ambrose, Vapour-Liquid Critical Properties of Pure Substances,National Physical Laboratory Report, Teddington, UK, 1980. R.J. Sadus, AIChE J., 40 (1994) 1376-1403. C.P. Hicks and C.L. Young, Aust. J. Chem., 30 (1977) 2583-2590. Y.S. Wei, R.J. Sadus and E.U. Franck, Fluid Phase Equilibria, (1996) in press. N.F. Carnahan and K.E. Starling, AIChE J., 18 (1972) 1184--1189. J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York, 1954, p. 158-162. R.J. Sadus, J. Phys. Chem., 93 (1989) 3787-3790. R.J. Sadus, Ber. Bunsenges. Phys. Chem., 96 (1992) 1454--1459. W.B. Streett, Trans. Faraday Soc., 65 (1969) 696-702. W.B. Streett, Trans. Faraday Soc., 67 (1971) 622-630. W.B. Streett and A.L. Erickson, Phys. Earth Planet. Interiors, 5 (1972) 357-366. J. de Swaan Arons and G.A.M. Diepen, J. Chem. Phys., 44 (1966) 2322-2330. W.B. Streett, R.E. Sonntag and G.J. van Wylen, J. Chem. Phys., 40 (1964) 1390-1395. W.B. Streett, Astrophys. J., 186 (1973) 1107-1125. W.B. Streett and J.L.E. Hill, J. Chem. Phys., 52 (1976) 1402-1406. W.B. Streett, Chem. Eng. Prog. Symp. Sen, 63 (1967) 37-42. D.S. Tsiklis, Dokl. Acad. Nauk. 91 (1953) 1361. D.S. Tsiklis, Dokl. Acad. Nauk., 101 (1955) 129-130. C.K. Heck and P.L. Barrick, Adv. Cryogenic Eng., 12 (1967) 714-718. J.E. Sinor, D.L. Schindler and F. Kurata, AIChE. J., 12 (1966) 353-357. W.B. Streett, A.L. Erickson and J.L.E. Hill, Phys. Earth Planet. Interiors, 6 (1972) 69-77. P.H. van Konynenburg and R.L. Scott, Philos. Trans. Roy. Soc. London A, 298 (1980) 495-540. G.M. Schneider, in M.L. McGlashan (Ed.), Chemical Thermodynamics, Vol. 2, A Specialist Periodical Report, The Chemical Society, London, 1978. A.E. Matber, R.J. Sadus and E.U. Franck, J. Chem. Thermodyn., 25 (1993) 771-779. M.I. Temkin, Russ. J. Phys. Chem., 33 (1969) 275.
FluidPhaseEquilibria 122(1996) 1-15
Vapour-liquid and liquid-liquid phase equilibria of binary mixtures containing helium: comparison of experiment with predictions using equations of state Ya S o n g Wei, Richard J. Sadus * Computer Simulation and Physical Applications Group, School of Computer Science and Software Engineering, Swinburne University of Technology, PO Box 218, Hawthorn, Vic. 3122, Australia
Received 19 December 1995; accepted 12 February 1996
Abstract
The unlike interaction parameters for binary mixtures of helium + non-polar gases are obtained by comparing experimental critical data with calculations using the Guggenheim and Heilig-Franck equations of state. These interaction parameters and equations of state are used to predict a priori both the vapour-liquid and liquid-liquid equilibria for helium + non-polar gas mixtures. The predicted phase diagrams are compared with experimental data for a wide range of pressure, temperature and composition. Good agreement between theory and experiment is obtained for the isobaric temperature-composition phase behaviour and isothermal pressurecomposition phenomena can be predicted qualitatively. Keywords: Theory; Equations of state; Vapour-liquid equilibria; Liquid-liquid equilibria; Binary mixtures; Helium
1. I n t r o d u c t i o n
Considerable progress [ 1] has been achieved in the application of theory to the calculation of binary mixture phase equilibria. The importance of vapour-liquid equilibria and liquid-liquid equilibria to many industrial processes provides a considerable incentive to improve the quality of phase equilibria prediction. Historically, the prediction of phase transitions has relied on empirical methods, but more realistic equations of state are being developed that attempt to account for the various aspects of intermolecular interactions. Some of these equations have been applied successfully to the prediction of phase transitions. For example, the Christoforakos-Franck [2] equation of state has been used to
* Corresponding author. 0378-3812/96/$15.00 Copyright© 1996ElsevierScienceB.V. All rightsreserved. PH S0378-3812(96)03045-2
2
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996) 1-15
calculate critical phenomena and binodal equilibria in aqueous mixtures [2], and the simplified perturbed hard-chain equation of state [3] has been applied extensively to both pure components [4,5] and mixtures [6]. There has also been progress in the development of theoretically based equations of state for macromolecules [7-9]. Traditionally, the accuracy of equations of state has been refined by comparison with experimental data. There are few "real" systems that can be used to evaluate unambiguously the accuracy of equation of state calculations. This situation has been alleviated partly by modem molecular simulation techniques [10] that provide exact data to test the accuracy of theory. However, simulations are limited to relatively simple intermolecular potentials. Consequently, the refinement of equations of state continues to rely on inputs from both simulation and direct comparison with experiment. Most successful equations of state use a "hard sphere + attractive" term approximation. Mixtures of the noble gases can be used legitimately to evaluate the accuracy of the "hard sphere + attractive" term approximation for vapour-liquid equilibria. However, liquid-liquid phase coexistence in binary mixtures almost invariably requires a dipolar component. Binary mixtures containing helium are noteworthy exceptions. Mixtures of helium + a non-polar component are rare examples of non-polar systems exhibiting both vapour-liquid and liquid-liquid phase transitions. They are also relatively free from theoretical complications, with the possible exception of quantum effects at low temperatures. The aim of this work is to establish whether or not theoretically based equations of state can be used to predict adequately the phase diagrams of binary mixtures containing helium. The unlike intermolecular parameters required for the Guggenheim [11 ] and Heilig-Franck [12] equations of state are obtained by fitting theory with experiment for the critical curves of eight helium + gas mixtures. These interaction parameters and equations of state are used subsequently to predict both vapour-liquid and liquid-liquid equilibria in helium-containing mixtures. The results are compared with experimental data over an extensive range of temperature, pressure and composition.
2. Theory Details of the calculations are given elsewhere [1,13] and only a brief outline is given below. The binodal curves at any temperature (T) and pressure ( p ) are obtained by solving the conditions for phase coexistence between the two phases (denoted by I and II) T I = T II
(1)
pt = pn
(2)
I' - -
j .v
A + pV
=
-- x 2
(3)
- -
3x2
r,v
]'[ =
A + pV--
x2
/
~
r.v
1I'
(4)
Y.S. Wei, R.I. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
3
where A, V and x denote the Helmholtz function, volume and mole fraction, respectively. The critical properties are obtained by satisfying the critical conditions for a binary mixture
W=
( a2A -IT~)T
a2A -
T
Ox21
X=
y=
r,v
=0
/ -0--VT
(o a/ (o a/
(5)
=0
r
(6)
>0
(7)
x,vlT lOg--TIT,, The Helmholtz function is obtained from conformal solution theory using the one-fluid model [1 4]
a =f~sA o (V/hes,T/f~s) - RTln h~s + R T ~ . , x i l n x i
(8)
i
where R is the universal gas constant, A o is the configurational contribution to the Helmholtz function and f~s and h~s are the characteristic conformal parameters of the equivalent substance. The conformal parameters can be deduced from the critical temperatures and critical volumes. It is customary to choose one of the components of the mixture as the reference substance (denoted by the subscript 0) and obtain the conformal properties of the pure substance relative to it, i.e. fLl = T~I/T~o, hll = V~/V~o, etc. The pure component data were obtained from a compilation by Ambrose [15]. The conformal parameters for the equivalent substance are obtained from the van der Waals prescriptions feshes = x2fl,h,l + 2x(1 - x)f12h12 + (1 -- x)2f22h22
hes=x2hll + 2x(1 - x)hl2 + (1 -- x)2h22
(9) (10)
where the contribution of unlike interactions is given by /.jl/3"~ 3 h~2 = 0-125~(h',( 3+,.22 j
(11)
f,2 = ~(f,,f22)o5
(12)
where ~ and ~ are adjustable parameters. The configurational Helmholtz function ( A o ) can be evaluated by direct integration of an equation of state with respect to volume. We have chosen two equations of state which differ in their treatment
4
Y.S. Wei, R,I. Sadus/Fluid Phase Equilibria 122 (1996) 1-15
of both attractive and repulsive contributions to intermolecular interaction. The Guggenheim [11] equation of state incorporates an improved hard-sphere repulsion term in conjunction with the simple van der Waals description of attractive interactions
RT P=
v(,
b
4
a V2
(13)
The covolume ( b = 0.18284RTC/p c) and attractive ( a = 0.49002 R 2TC2/pC) equation of state parameters are related to the critical properties. The Guggenheim equation has been used [16] successfully to calculate critical behaviour in a wide variety of binary and ternary mixtures. Some calculations using the Guggenheim equation have been attempted for vapour-liquid phase coexistence [17], and liquid-liquid phase equilibria at high pressures [18]. There is some evidence [12,13] that the Heilig-Franck equation of state is suitable for the calculation of the high pressure phase behaviour of binary mixtures. The Heilig-Franck equation of state uses the Carnahan-Starling [19] representation of repulsive forces between hard spheres, and a square-well representation for attractive forces.
RT(1 + y
P=
+ y2 _
V(1 _y)3
y3)
RTBa"
+ V2_(Catt/Batt) V
(14)
Heilig and Franck [12] used the following temperature-dependence [2] for the y term y = "~---~
(15)
where b c is the covolume at the critical point. However, in this work, y was made independent of temperature by using z = 0. The Batt and Qtt terms in Eq. (14) represent the attractive contributions from the second and third virial coefficients, respectively, of a hard-sphere fluid interacting via a square-well potential. The square-well is characterised by three parameters related to the hard-sphere collision diameter (~), the depth of the well (e) and the relative width of the well (k). The second and third virial coefficients for the square-well potential can be solved exactly [20] for both pure fluid and binary mixtures. Details of the calculation of the pure component parameters are reported elsewhere [ 18]. The unlike interaction contributions for these parameters are obtained from 0"12 = ~(0"11
+ 0"22) 2
'~12 = ~1'¢22
(16) (17)
It is also assumed that h = h~l = X22 ~- ~k12.
3. Results and discussion It is well documented [1] that the accurate prediction of binary mixtures requires accurate information concerning the strength of the unlike interaction parameters. For relatively simple cases
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
5
30
i i i T I
400
25--
350
i
,
i
',
i #
300 20--
\
I I
/
29.91 K C
/
c,
250
"1 m
Q. 13.
200
!
15~
,'
#
32.89
K
I
Q_
,"
150
10
100
~,
/ <, ~_, ¢ , y'
,'
,35.9 K
is
"' ' .
.
//
1 ,-~
"
I
ci ,
4~b
J
.
G
50
j_
I,
c'c
I'
. 41.9 K
,2,
38.88 K
/
......
,:
i
':'
,5~ ~
:3 ~" /
L
0 0
50
100
150
200
250
300
350
400
450
T/K Fig. 1.
Comparison of the experimental critical curves
012
0.4
0.6
0.8
1.0
X (He)
(o)
of binary helium + gas mixtures with calculations using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. Also shown are critical properties estimated from
experimental data ( × ).
0.0
Fig. 2. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + neon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Critical points estimated from experimental data are illustrated ( X ) .
[21,22], reliable a priori estimates of the ~ and ~ parameters are possible. However, the interaction parameters for liquid-liquid equilibria are generally obtained from experimental data. In this work, we have obtained the values of h, ~ and ~ by optimizing the agreement between theory and experiment for the p-T projection of the critical curve. The optimal agreement was obtained by trial and error adjustment of the interaction parameters and repeated calculation of the critical curve as described elsewhere [1]. Experimental high pressure and critical data are available for the helium + argon [23-25], xenon [26], hydrogen [27,28], nitrogen [25,29,30], ethylene [31] and propane [32] mixtures. The critical data are compared in Fig. 1 with calculations of the critical curve using both the Guggenheim and Heilig-Franck equations of state. The interaction parameters obtained from the analysis are included in Table 1. The critical curves and the interaction parameters for the helium + neon [33] and helium + methane [34,35] systems were estimated by assuming that the experimental critical pressure coincided with the maximum pressure of the phase envelope. It should be stressed that, in reality, the critical pressure can occur below the pressure maximum. The comparison of theory with experiment indicates good agreement for the p-T projection of the critical curve. It is almost impossible to
6
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996)1-15
Table 1 Combining rule parameters for binary helium+gas mixtures obtained by optimizing agreement between theory and experiment for the p - T projection of the critical locus Gas
Guggenheim
Neon Argon Xenon Hydrogen Nitrogen Methane Ethylene Propane
1.55 1.50 1.40 1.07 1.37 1.30 1.25 1.23
Heilig-Franck
0.70 1.02 1.07 1.06 1.03 1.05 1.06 1.06
1.10 1.35 1.27 1.50 1.55 1.65 1.60 1.50
1.62 1.55 1.59 1.08 1.38 1.30 1.28 1.50
0.80 1.02 1.08 1.07 1.03 1.05 1.06 1.09
distinguish between the critical curve calculated using either the Guggenheim or Heilig-Franck equations of state. It is apparent from Fig. 1, that the critical loci of the helium + gas systems all commence at the critical point of the gas component and thereafter, rise rapidly to very high pressures. The mixtures exhibit Type III behaviour in accordance with the phase behaviour classification scheme of van Konynenburg and Scott [36]. However, there is a subtle and interesting distinction between the different critical curves. The critical curves of helium + the gases methane, ethylene, propane and xenon, occur at temperatures and pressures above the critical point of the gas. This phenomenon is characteristic of so-called "gas-gas" immiscibility [37] of the first kind (Type IIIa behaviour). In contrast, the critical curve for the remaining mixtures extends initially to temperatures below that of the gas. For the helium + nitrogen mixture, the experimental data indicate a region of "gas-gas" immiscibility of the second kind (Type IIIb behaviour) at very high pressures. There are insufficient experimental data to assign Type IIIb behaviour to the other mixtures and the calculated critical properties always occur at temperatures below the critical temperature of the gas. The phenomenon of "gas-gas" immiscibility of the first kind is rare and it is confined almost exclusively to mixtures containing helium. The neon + water mixture [38] is the only other reported Type IIIa mixture. Temkin [39] proposed two simple criteria to predict "gas-gas" immiscibility of the first kind that are based on the relative magnitude of the van der Waals equation of state parameters. In terms of the critical volume (V c) and temperature (T c) of the components, the criteria that must be satisfied to observe Type IIIa behaviour are V•, >__0.42V2C2
(18)
T~,V?, <
(19)
0.052T~2V;2
where component 2 is the component with the higher critical temperature. These simple criteria predict successfully Type IIIa behaviour for several mixtures [13,38]. However, the above criteria have mixed success in predicting "gas-gas" immiscibility of the first kind for the helium + gas mixtures. Type IIIa behaviour is not predicted for the helium + propane mixture (Eq. (18) is not satisfied) whereas the phenomenon is incorrectly predicted for both the helium + argon and helium + nitrogen mixtures. However, Type IIIa behaviour is correctly predicted for mixtures of helium + xenon, methane and ethylene.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
7
The helium + noble gas mixtures are the simplest possible binary mixtures and as such they represent a suitable test for "hard sphere + attractive term" equations of state. The p - x vapour-liquid phase envelopes for the helium + neon mixture have been measured [33] at several different temperatures. These experimental data are compared in Fig. 2 with the predictions of both the Guggenheim and Heilig-Franck equations of state. Reasonably good agreement between theory and experiment is obtained at all temperatures. At pressures > 5 MPa, there is a noticeable difference in the position of the critical p - x locus predicted by the two equations of state. At low temperatures, the Heilig-Franck equation of state is slightly more accurate than the Guggenheim equation. Phase separation at high pressures is also predicted at unrealistically low temperatures (Fig. 1). This may be caused by errors in the unlike combining rule parameters obtained by comparing theory with the estimated low pressure vapour-liquid critical points. Considerable experimental high pressure data are available for the helium + argon mixture [23-25]. The equations of state predictions for vapour-liquid equilibria of the helium + argon mixture
'
o
~
,' 60
t
i ,
~
' i
J© "
,
I, ~ L
i o I
! J
'
,
,
i '~ o
c"
,,
,:~
t ,'
'
I
i
,i ~
looo 900
,, '
o
'
'I
o
t~
I~'x ,
/
"
!,
o
,'!
o
~.
/ [,
]
t~
400 ./
o_ii
~
'o,,,, , { - / / I ,/, /' "/'1 '/E
lo-i~
o
..../ / / ' "
2~
./
o
_~- o I
0.2
j
,,
,
-
'/
/"
/ o//
0 0.0
j
o/
',"o fx '/,,-" o / /
,
'
0.4
!o
J
"
/,i, °
!
0.6
i 0.8
aoo
200 Ioo
'
0 1.0
x (He)
Fig. 3. Comparison of experimental vapour-liquid equilibrium data (o) for the helium + argon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the estimated experimental critical points are identified ( X ).
0.0
0.2
0.4
0.6
0.8
1.0
x(He) Fig. 4. Comparison of the experimental isothermal pressure-composition behaviour (o) of the helium + argon
mixture with predictions using the Guggenheim (--) and Heilig-Franck (. . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data (x) are also identified.
8
Y.S. Wei, RJ. Sadus/Fluid Phase Equilibria 122 (1996) 1-15 310-
310 -
p ~ 90.3975 MPa
p = 50.6225 MP a
300
p29O
290-
0 280
. 0.0
320 140-
140-
130-
190 -
'
i ' i ' ' 0.4, 0.6 018 110 X (He)
012
2800.0
320-
p = 81.06 MPa
310-
0.2
0.4 0.6 X (He) p = 101.325 MPa
0.8
"i
1.0
310,'
0
300110"
110-
100-
100-
290
90
'
0.0
I
0.2
190-
'
I
'
t
'
I
'
~
0.4 0.6 0.8 1.0 X (He) p = 200 MPa
,7o:
90
~
0.0
0.2
190 7
I
1.o: ,5o-
'901 /o/° 190-1
o
280
t
0.4 0.6 0.8 1.0 x (He) p = 4 0 0 MPa
,7o- 1
290-
\ \ °o
i
0.0 330
0.2
04 0.6 X (He)
0.8
1.0
,
~
310 -
,'
1.0
0.0
0.2
0.4 0.6 x (He)
0.8
Fig. 5. Comparison of experimental isobaric temperaturecomposition behaviour (o) of the helium + argon mixture with predictions using the Guggenheim ( - - ) and HeiligFranck ( . . . . ) equations of state. The experimental ( • ) and calculated ( × ) critical points are identified.
1.0
i 1.0
0.8
O
,,
©
/',"',t,l,I ,"
310 -
0
29011-
290 0.8
p = 162.12 M ~ , , , .
C)
290
0.4 0.6 X (He)
0.4 0.6 X (He)
O
300
0.2
i 0.2
350-
/'
0.0
' 0.0
330-
p = 131.7225 MPa
320 -
290
, 0.0
0.2
0.4 0.6 X (He)
0.8
j 1.0
280 k 0.0
.
' 0.2
0.4 0.6 X (He)
0.8
t 1.0
Fig. 6. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + xenon mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( - ) and calculated ( × ) critical points are identified.
are compared with experiment in Fig. 3. At pressures < 70 MPa (Fig. 3), there is satisfactory agreement of theory with experimental vapour-liquid equilibria data in the p - x projection. The predictions of both equations of state are of similar accuracy. However, the transition between vapour-liquid and liquid-liquid equilibria is not predicted accurately. The very high pressure liquid-liquid phase coexistence of helium + argon at different temperatures is examined in Fig. 4. The agreement of theory with experiment is only satisfactory at moderately high pressures ( < 400 MPa). At very high pressures there is a substantial discrepancy between the predicted and experimental critical composition. Consequently, the predicted coexistence curves are shifted to lower compositions. The isobaric T - x behaviour for liquid-liquid equilibria is examined in Fig. 5. In contrast to the predicted isothermal behaviour, the equation of state calculations at isobaric conditions are in exceptionally good agreement with experiment for the entire composition range.
Y.S. Wei, R J. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
9
De Swaan Arons and Diepen [26] measured liquid-liquid equilibria for the helium + xenon mixture. Their experimental data at several different temperatures are compared with equation of state predictions in Fig. 6. The predicted liquid-liquid equilibria are in reasonably good agreement with experiment. However, a deterioration in the quality of prediction occurs at helium-rich compositions. This can be attributed partly to the inability of theory to adequately account for the considerable size difference between the component molecules.
35-
p = 3.447 MPa
30v
p_ 25-
20-
,!
i
15 ~ , 0.0 0.2
J
o
I
'
'
'
'
I
0.4 0.6 0.8 1.0 X (He) p = 20.0 MPa
30-
o
Oo ~. o
c~
# 25-
A
i
s
i
i
/
13_
20- /
//o i
l/
/
0.0
I
oq
0 /0 f
0
t°I
ol I ol ol
80 ~
0.2
0.4
0.6
0.8
X (He)
Fig. 7. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data ( X ) are also identified.
0.4 0.6 0.8 1.0 x (He) p = 500,0 MPa .X.
," ,"" " "
" " " "~''',,,
#
,/ 0.0
0.2
lOO-
60-
1.0
40 0.0
0.2
0.4 0.6 x (He)
0.8
1.0
Fig. 8. Comparison of experimental isobaric temperaturecomposition behaviour (o) of the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( . ) and calculated ( X ) critical points are identified.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
10
140- p = 82.783 MPa 130 -
//(5"~
120-.
100 11o-_ 90-
140
/
120
O
p = 137.903 MPa
1
O
\°o
11ol
i i ~
80-
70, . . . . 0.0 0.2 0.4 0.6 X (He) 140- p = 206.804 MPa
0.8
0.2
0.4 0.6 X (He)
0.8
1.0
O~,,.,.:k~,... ii•
130 -
120 - /
"
J
110-
,
120
~~(3
90 80-
70
.
. . . . 0.2 0.4 0.6 X 140~ p =344.91 M P ~ L I I O 0.0
,301 12oI
'/9
(~)
0
0
100 -
IZ El.
0.0
1.0
(He)
0.8
i 1.0
1130 90 80 700.0 . 0.2. 0.4. . 0.6. 140 7
p=413.71
X (He) M ~ ( ~
0.8
1.0
0 0
\°o
/o
\o
\o 110 100 00
0.0
0.2
0.4
0.6
0.8
1.0
x (He)
Fig. 9. Comparison of experimental isothermal pressurecomposition behaviour (o) of the helium + hydrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the experimental critical points are identified ( ' ) .
0.0
0.2
0.4 0~6 X (He)
0.8
1.0
0.0
0.2
0.4 0.6 X (He)
0.0
1.0
Fig. 10. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + nitrogen mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( ' ) and calculated ( × ) critical points are identified.
The phase behaviour of helium + simple diatomic mixtures can be represented by systems containing either hydrogen or nitrogen. The isothermal p - x vapour-liquid coexistence curves at several different temperatures have been reported [27] for the helium + hydrogen mixture. These data are compared with equation of state predictions in Fig. 7. It is apparent from Fig. 7 that either the Guggenheim or Heilig-Franck equations of state can predict accurately the vapour-liquid coexistence of this mixture. Very impressive agreement with experiment [27] is also obtained for the isobaric temperature-composition phase coexistence at low pressures (Fig. 8). The helium + hydrogen mixture also exhibits liquid-liquid separation at very high pressures [28]. The experimental isothermal liquid-liquid coexistence data are compared with equation of state calculations in Fig. 9. At pressures > 50 MPa there is a significant discrepancy between the predicted
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
11
16
,
:
ii
s~
,/
12
'
,/
!
ItI
/
,// ;/
7; i
/
5• i
q
/
;
/
/ o
,
,,' ~
,
0.2
0.4
0.6
0.8
1.0
X (He) Fig. 11. Comparison of experimental isothermal pressure-
composition behaviour ( o ) of the helium + nitrogen mixture w i t h predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold. Direct experimental critical point measurements ( • ) or critical points estimated from experimental data ( × ) are also identified.
f
j_.
0 0.0
i
,/ ,•/
0.0
0.2
i 0.4
0.6
0.8
1.0
x (He) Fig. 12. Comparison of experimental vapour-liquid equilibrium data ( o ) for the helium + methane mixture with
predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold.
and experimental critical composition. Nonetheless, there is generally satisfactory agreement between theory and experiment at temperatures < 80 K and pressures < 600 Mpa. The experimental [29] isobaric T - x behaviour of liquid-liquid coexistence for the helium + nitrogen mixture is compared with equation of state predictions in Fig. 10. Irrespective of the pressure, there is good agreement between theory and experiment for the entire composition range. For this mixture, several isothermal measurements at very high pressures have been reported [25]. These data are compared with equation of state predictions in Fig. 11. There is generally only qualitative agreement between theory and experiment at all temperatures. In particular, theory fails to account for the coexistence isotherms at very high pressures. Experimental measurements for helium + a non-polar polyatomic molecule are limited to mixtures containing a hydrocarbon gas. The helium + methane mixture represents the simplest possible helium + hydrocarbon combination. Sinor et al. [34] measured the isothermal p - x vapour-liquid coexistence of the helium + methane mixture at several temperatures. It is apparent from Fig. 12 that reasonably accurate predictions for these experimental data are obtainable with either the Guggenheim or Heilig-Franck equations of state. Streett et al. [35] reported liquid-liquid separation for this system
12
Y.S. Wei, R J . Sadus / Fluid Phase Equilibria 122 (1996) 1-15 340
p
=196.132 MPa
320
. -
•
O %
v
# 300 -
280 0.2 380 -
360 -
340 b---
0.4
0.6 X (He)
I 1.0
0.8
p = 392.26 MPa
~ O ( D
-
320
n Q.
280
' 0.2
400
I 0.4
'
~ ' 0.6 X (He)
'
I 1.0
p = 588.397 MPa
360 380 -
#
I 0.8
Q ~ (~0 i
340 320 300 280
o.o
0.2
0.4
0.6
0.8
1.o
' 0.2
,
x (He)
Fig. 13. Comparison of experimental isothermal pressurecomposition behaviour ( o ) of the helium + methane mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The calculated critical curve is illustrated in bold and the experimental critical points are identified ( • ).
, 0.4
, 0.6 X (He)
, 0.8
1.0
Fig. 14. Comparison of experimental liquid-liquid equilibrium data (o) for the helium + ethylene mixture with predictions using the Guggenheim ( - - ) and Heilig-Franck ( . . . . ) equations of state. The experimental ( - ) and calculated critical points ( × ) are illustrated.
under isothermal conditions. Their data are compared with equation of state predictions in Fig. 13. In common with the other helium mixtures, the agreement between theory and experiment deteriorates progressively at very high pressures. Experimental liquid-liquid coexistence data are available for the helium + ethylene [31] and helium + propane [32] mixtures. The predicted liquid-liquid coexistence of helium + ethylene is compared with experiment in Fig. 14. The agreement with theory at different pressures is satisfactory at most compositions. In contrast, the isobaric T - x liquid-liquid coexistence predicted for the Y.S. Wei, R.1. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
410-
410-
p = 98.066MPa
400
p = 196.132MPa
400-
~C
390-
"
•
0
0" 380 -
~
0
0.60
x (He) 430-t p = 294.198MPa 420 4104oo-
390-
',
,"
'
=
0.60
X (He)
;'
=
0.80
0
;'
.
0
, ~ 0.60 0.80 x(He)
I
1.00
-
©
'
,,"
'~
400390 -
o '
0
410-
'".0
0 0
0
,"
0 0
370 0.40
420
,"
©
430-
ioo,/,"
370 / 0.40
1.00
0.80
,," 0
0 3~o-
380-
0
370 ,' 0.40
~.
13
o°/
380-" 0 I
1.00
370 0.40
i°o
:'
"© 0.60
x (He)
0.80
1.00
Fig. 15. Comparison of experimentalliquid-liquid equilibrium data (o) for the helium+ propane mixture with predictions using the Guggenheim(--) and Heilig-Franck (. . . . ) equations of state. The experimental( . ) and calculated (×) critical points are identified.
helium + propane mixture (Fig. 15) occurs over a narrower range of composition than the experimental data. At high pressure, the Heilig-Franck equation is marginally more accurate than the Guggenheim equation of state. T h e deterioration in the quality of prediction for this mixture can be attributed partly to the significantly non-spherical geometry of the propane molecule. An interesting aspect of the comparison of experiment with theory, is the consistently good results obtained using the relatively simple Guggenheim equation. In most cases, it is almost impossible to distinguish between the results obtained for either the Guggenheim or Heilig-Franck equations. We may anticipate that the temperature-dependence of the Heilig-Franck equation may improve the prediction of phase coexistence in dipolar mixtures. However, a recent comparison [ 18] of the binodal curves of aqueous mixtures also found very little quantitative difference between the accuracy of the Guggenheim and Heilig-Franck equations of state. The accurate evaluation of the unlike interaction parameters appears to be the crucial influence which determines the quality of equation of state predictions. For Type III systems, the analysis of the p-T behaviour of the critical curve appears to yield combining rule parameters which are satisfactory for both vapour-liquid and liquid-liquid equilibria. In contrast to other mixtures, e.g. Type II systems, the high-temperature critical curves of Type III mixtures are characterized by a continuous transition between vapour-liquid and liquid-liquid equilibria which may partly explain the similar accuracy of the unlike interaction parameters for both phenomena.
Y.S. Wei, RJ. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
14
4. Conclusions
The comparison of theory with experiment demonstrates that the "hard-sphere -I- attractive term" formulation of either the Guggenheim or Heilig-Franck equations of state is adequate to predict accurately the isobaric temperature-composition behaviour of binary mixtures containing helium. In contrast, the predicted isothermal pressure-composition behaviour is generally only in qualitative agreement with experimental data. It is also demonstrated that both vapour-liquid and liquid-liquid equilibria for helium + gas mixtures can be calculated from the same set of combining rule parameters. The predictions using either the Guggenheim or Heilig-Franck equations of state are of similar accuracy.
5. List of symbols
A b B C f h P R T V W x
X Y Y
Helmholtz function Molecular covolume parameter Second virial coefficient of a square-well fluid Third virial coefficient of a square-well fluid Conformal solution parameter Conformal solution parameter Pressure Universal gas constant Temperature Volume Determinant defined by Eq. (5) Composition Determinant defined by Eq. (6) Packing fraction defined by Eq. (15) Determinant defined by Eq. (7)
Greek letters 8
h o"
Depth of square-well Width of square-well Hard-core intermolecular separation Combining rule parameter (Eq. (12)) Combining rule parameter (Eq. (11))
Subscripts and superscripts c 1,2 att
Critical property Property of component 1, component 2 Attractive property
Y.S. Wei, R.I. Sadus / Fluid Phase Equilibria 122 (1996) 1-15
15
Acknowledgements Y . S . W . thanks the Australian G o v e r n m e n t f o r an Australian P o s t g r a d u a t e A w a r d . T h e calculations w e r e p e r f o r m e d on the S w i n b u r n e C r a y Y M P - E L c o m p u t e r .
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