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Vapour—liquid equilibria in the argon—ethane and argon—methane—ethane systems at 15·5 K
Vapour-liquid equilibrium data were measured for the binary system argon-ethane and the ternary system argon-methane-ethane at 115.5 K. A forced recirculation apparatus was used for the determinations. The total pressure method previously proposed was modified and extended to evaluate the liquid phase activity coefficients and vapour compositions for the ternary system.
Vapour-liquid equilibria in the argon-ethane and argon-methane-ethane systems at 15.5 K I. M. Elshayal and B. C-Y. Lu
Vapour-liquid equilibrium data for the binary system argon-ethane and the ternary system a r g o n - m e t h a n e ethane were determined at 115-5 K as part of a continuing programme to investigate the non-ideal behaviour of mixtures of simple molecules. Phase equilibria of these systems are important for design purposes and for testing solution theories. Equilibrium data for the binary system argonethane were previously reported by Eckert and Prausnitz.1 However, there were only six experimental points, each measured at a different temperature. In this investigation, 38 experimental points were determined for the binary system, and 23 experimental points for the ternary system.
NOMENCLATURE B,C,D K P e o R T X X f
Y F
Redlich-Kister constants vapour phase correction term total vapour pressure of the system vapour pressure of pure component gas constant system temperature, K mole fraction in the liquid phase mole fraction of component 1 in the initial binary liquid mixture along a constant x l / x 2 secant mole fraction in the vapour phase molar volume of pure component second virial coefficient activity coefficient in the liquid phase
Experimental details
7
The equilibrium cell, the cryostat, the equipment assembly, and the experimental technique employed in this investigation were essentially the same as those previously reported, z The equilibrium cell was made of a 100 ml Jerguson transparent gauge. A dewar flask of 181 capacity was employed as the cryostat into which the equilibrium cell was submerged, with isopentane used as the bath liquid. In addition, two variable speed stirrers, a heating element, a refrigeration coil, and a resistant type temperature sensing probe were installed inside the cryostat. The temperature of the cryostat was regulated by controlling the evaporation rate of liquid nitrogen in the refrigeration coil, and was controlled to + 0-01 K. The equilibrium temperature was measured by means of two thermocouples of the protective type, one for the liquid phase and one for the vapour phase, in conjunction with a Leeds and Northrup K-3 potentiometer and a Tinsely SRI galvanometer. The thermocouples were calibrated against the vapour pressure of argon. 6 The total pressure of the system was measured by means of
Subscripts 12 hypothetical binary of components 1 and 2 3 third component of a ternary mixture i, ii, L Jl pure component j k hypothetical binary of components j and k
The authors are with the Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada. Received 15 March 1971. CRYOGENICS . AUGUST 1971
three pre-tested 12 in (30 cm) Heise gauges ( 0 - 6 0 , 0 - 2 5 0 , and 0 - 5 0 0 lb in-2 [ 1 lb in-2 = 6-9 kN m - 2 ] . The accuracy of these gauges is 0-1% of the full scale. A mercury manometer was used at low pressures. In addition to the cryostat, the equipment assembly 2 consisted of a feed measuring and charging device, a closed recirculation loop, a volume regulator, and sampling facilities. An electromagnetic pump was employed for recirculating the vapour through the liquid in the equilibrium cell. Research grade gases, supplied by Matheson of Canada Ltd, were used without further purification. The specified minimum purities of these gases were as follows: Argon 99"998 mole % Methane 99"99 mole % Ethane 99"9 mole % 285
Analyses of the liquid and the vapour samples were made with a Fisher-Hamilton gas partitioner, model 29V, using silica gel ( 2 8 - 3 0 mesh) as the column packing material, and helium as the carrier gas. Peak heights were employed in the calibration. The average deviation obtained in the calibration was less ~ a n + 0-001 mole fraction. The cell temperature of the gas partitioner was maintained at 70 C, and the flow rate of helium was maintained at 100 ml per minute at 35 lb in -2 . A retention time of 6 minutes was required for separating a three-component mixture.
Table 1. Experimental and calculated results for the binary system argon-ethane at 115"5 K P lb in -2 (1 Ib in-2 = 6"9 k N m -2)
24"6 25"2 25"2 29"7 29"7 30"8 33"8 34"1 36"0 41 "2 43"4 48"4 51 "0 57"8 63"9 70"7 80" 1 81"8 87"2 87"4 90" 9 94"3 98"6 100"9 105"5 108"1 109"8 110"8 113"0 115"2 116"2 116"3 118"5 123"1 124"5 129"1 130"5 133" 1
XA
0"0690 0"0710 0"0712 0"0926 0"0928 0"0980 0"1119 0"1135 0"1222 0" 1450 0"1546 0"1760 0"1873 0"2176 0"2470 0"2835 0"3422 0"3540 0"3946 0"3969 0" 4259 0"4577 0"5046 0"5355 0"6120 0"6638 0"6978 0"7304 0"7581 0"8004 0"8210 0"8230 0"8619 0"9151 0"9265 0"9570 '0"9654 0"9813
(YA)calc 0"9990 0"9990 0"9990 0"9992 0"9992 0"9993 0"9993 0"9993 0"9994 0"9995 0'9995 0"9995 0"9995 0"9996 0"9996 0"9997 0" 9997 0"9997 0"9997 0"9997 0" 9997 0"9997 0'9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9999 0"9999 0"9999 0"9999 0"9999 1 "0000
7A
2"7434 2"7343 2"7334 2"6368 2"6362 2"6127 2"5550 2"5480 2"5131 2"4247 2"3891 2"3129 2"2749 2" 1770 2"0886 1"9867 1 "8387 1 "8096 1 "7175 1 "7125 1"6509 1 "5866 1 "4980 1 "4432 1 "3200 1"2468 1 "2033 1"1649 1"1351 1"0946 1 "0772 1 "0757 1 "0474 1 "0187 1"0142 1"0050 1 "0032 1 "0009
7C2 1 "0421 1 "0419 1 "0424 1 "0469 1 '0476 1 "0475 1"0513 1"0509 1 "0529 1 "0589 1"0619 1 "0691 1 "0735 1 "0859 1 "1006 1 "1216 1" 1635 1 "1775 1 "2109 1 "2131 1"2436 1 '2853 1 "3587 1 "4110 1 "5923 1 "7624 1 "9054 2"0600 2"2326 2"5291 2"7114 2"7267 3"1422 3"9233 4"1604 4"8089 4"9761 5"4854
Experimental results
The experimental equilibrium temperature, pressure, and liquid composition data determined in this investigation for the binary system argon-ethane are listed in Table 1, and shown in Figure I. The concentration of ethane in the vapour phase was found to be extremely low, and the error in its determination was rather high. For this reason, it was decided to evaluate the ethane composition of the vapour from the total vapour pressure-liquid composition measurements. At 115.5 K the system pressure is relatively low, and the total pressure method previously proposed 4 was employed for this purpose. The calculated values of liquid phase activity coefficient, 7, and the vapour phase composition, y, are also listed in Table 1. In Figure 2, In 7 values were plotted against liquid composition to illustrate the non-ideal behaviour of the binary system. The experimental points for the ternary system argonmethane-ethane were determined along constant Xcl/Xc2 and constant xA/xc2 secants. The arrangement is shown in Figure 3 and was planned for the purpose of reducing the ternary system to a hypothetical binary so that the total pressure method proposed for binary systems 4 could be extended and applied. Evaluation of ternary vapour phase compositions
When the components of a ternary solution vary along a constant Xl/X2 secant, the mole fractions of components 1
t40
Argon- ethane 115.5K
120
IOO
80 o 7 .c.
60
40
20
O
0.2
0'4
0-6
0'8
I.O
XA Figure 1. Total vapour pressure--liquidcompositiondiagram for the binary system argon--ethane at 115-5 K ( 1 Ib in -2 = 6.9 kN m -2)
286
CRYOGENICS . AUGUST 1971
ln3'3 =(1 --X3) 2 [B+C(4x3 -- 1) 1.8
+ D (2Xa - 1) (6x3 -- 1)]
Argon - ethone 115.5K
In 3't2 = x 2 [B + C (4x3 - 3) + D (2x3 - 1) (6x3 - 5)]
t.6
. . . (3b) Equations 2 and 3 indicate that the total pressure is a function of the Redlich-Kister constants and the liquid composition x3. Hence at constant temperature,
Id-
P=P(B, C,D, x3)
,.2
...(4)
Assuming a set of initial values for the constants B, C, and D, then
I.O
aP 63P aP 6 P = - ~ S B + -ff--~S C + "~ 6D
>o
_5
. . . (3a)
...
(5)
0.8
where 6 P = P e x p t -- P e ~
0.6
. .. (6)
and 63P_ 63P 63'73 + . 63P . 03'12 . . 63B 633'3 63B 633'12 aB
0.4
Similar equations may be written for 63P/63Cand 63P/63D. Furthermore from (3a) and (3b)
02
O
(7)
-
2
4
xA
6
8
073 63B =73 (1 - x 3 ) =
tO
63"/12 Figure 2. Liquid phase activity coefficient-composition curves for the binary system argon--ethane at 1 15.5 K
63B
-'r~2 x ]
...(8a)
...
(Sb)
and
63P x3P3 633'3 K~ _
. . . (9a)
63P _ ( 1 -xa)e~2 and 2 depend on that of component 3 as follows
a3q2
Kt2
. . . (9b)
xt = x ' ( 1 - x a ) x2 = (1 - x'X1 - x3)
...(1)
where x' represents the mole fraction of component 1 in the initial binary solution 1 - 2 . The mole fraction of the third component along the secant, is taken as the independent variable, and the ternary mixture may therefore be regarded as a binary one. Assuming that the vapour phase is an ideal solution, but not an ideal gas, the total pressure of the system may be represented by the following equation
o
p _ X 3 P _ 3'3 Ka
+ (1 -
x3)Pl°2 "rl2
Argon-methane-ethane ~
/
~
. . . (2)
Kt2
where
K3 =exp [ (f133 - v3)(PKa2 =exp [ ([3t2- v t 2 ) ( P In this investigation liquid phase activity coefficients 7, were correlated by means of the Redlich-Kister equation as follows CRYOGENICS . AUGUST 1971
C2
Cf
Figure 3. Liquid compositions investigated for the ternary system argon--methane--ethane at 115-5 K 287
By substituting (8) and (9) into (7), the following expression is obtained
and
i=l
a__P_x~ (I -x~)" 3`~P; aB
+
xs2 (1 - x s ) 3`12 el~
Ks
...(10)
K12
,=,
Similar expressions for OP/~Cand aP/aD may be obtained in the same manner. Substituting aP/OB, ~P/aC and aP/aD into (5) for all the experimental points gives the following set of normal equations
...(13) These corrections for B, C, and D are employed to calculate the actual values of B, C, and D which in turn are used to evaluate the liquid phase activity coefficients, 3'3 and 3'12. Using the liquid phase activity coefficients, (14) may be applied to evaluate the vapour phase compositions.
Yi =
8B ~ (OPt 2
"
+ 5D
p
:,
.
= ~, 6P
i=1
[=1
...(11)
,=,
(xiPf3`i)/Ki (x,e;' 3`iJ/Ki + (x
. . . (14)
3`j )/K jk
The modified total pressure method was used to predict the vapour phase composition of the ternary system argon-methane -ethane. The numerical routine for determining the vapour phase composition may be summarized as follows:
OP
+ $D
i=1
~aC]
=
i=1
...(12)
1. Initial values of B, C and D were assumed. 2. Liquid phase activity coefficients were calculated from (3). 3. Total pressure of the system was calculated at each experimental point, using the pure components properties
Table 2. Experimental and calculated results for the ternary system argon-methane-ethane at 115.5 K
P Ib in -2 (1 Ib in-2 = 6.9 kN m-2) 59"8 71 '1 80"0 89" 5 94'4 98"5 51"0 64"6 70"7 79"1 84"4 90"4 99"5 107"5 51 "8 59"8 69"5 79"5 89"2 98"0 78"3 81 "0 84"0
YA XA 0"2572 0"3241 0"4177 0" 5148 0"5889 0"6553 0"1977 0"3046 0"3547 0"4521 0"4925 0"5571 0"6625 0"7564 0"2810 0"3398 0"4198 0"5083 0"6135 0"6924 0"4431 0"4606 0"4854
Xcl
Expt
Calc
Expt
YCl Calc
YC2 Calc
0"1859 0"1743 0"1521 0" 1271 0"1079 0"0980 0"3981 0"3066 0"2737 0"2351 0"2187 0"1887 0"1481 0'1071 0"5164 0"4746 0"4143 0"3551 0"2803 0"2044 0"2317 0"2096 0"1675
0"9164 0"9332 0"9525 0" 9646 0"9761 0"9762 0"8400 0"8852 0"9056 0"9300 0"9377 " 0"9489 0"9617 0"9730 0"8018 0"8359 0"8715 0"8984 0"9242 0"9469 0"9309 0"9380 0"4510
0"9158 0"9338 0"9518 0' 9645 0"9717 0"9751 0"8395 0"8839 0"9059 0"9302 0"9378 0"9492 0"9620 0"9726 0"8026 0"8363 0"8716 0"8984 0"9244 0"9465 0"9304 0"9383 0"9519
0"0836 0"0668 0"0475 0" 0354 0"0279 0"0238 0"1600 0"1148 0"0944 0"0700 0"0623 0"0511 0"0383 0"0270 0"1982 0"1641 0"1285 0"1016 0"0758 0"0531 0"0691 0"0620 0"0490
0"0840 0"0660 0"0480 0" 0353 0"0280 0"0245 0'1603 0"1158 0"0938 0"0695 0"0619 0"0505 0"0376 0"0269 0"1970 0"1632 0"1279 0"1010 0"0751 0"0580 0"0693 0"0614 0"0478
0"0002 0"0002 0"0002 0" 0002 0"0003 0"0003 0"0003 0'0003 0"0003 0"0003 0"0003 0"0003 0"0004 0"0005 0"0005 0"0005 0"0005 0"0005 0"0005 0"0005 0"0003 0"0003 0"0002
7A
7C 1
3`C2
3"5694 2"5350 1 "5119 3"0153 2"1964 1 "7420 2"4247 1 '8479 2"2033 1"9559 1"5827 3"0046 1 '6738 1 "4331 4"0395 1"4469 1 "3198 5"7975 3"0720 2"1149 1"7569 2"2713 1 " 6 4 3 9 2"4917 2"1390 1"5893 2"6935 1"7973 1 "4205 3"5983 1"6778 1"3654 4"1301 1"5214 1 "3017 5"1739 1 "2942 1 "2233 8"4816 1"1556 1 " 2 1 1 3 4"0292 1"5470 1"1582 5"9968 1 "4628 1"1386 6"8501 1 " 3 6 7 3 1 " 1 2 4 7 8"1797 1"2604 1"1136 10"7773 1 " 1 6 2 5 1 " 1 2 6 4 15"2237 1" 1288 1" 1597 17" 1085 1°1855 1 "4547 3"3821 1"8630 1 "4730 3"3345 1 " 9 2 3 7 1 " 5 3 5 4 3"1154
Root mean square deviation in vapour phase compositions of argon and methane = 0-002
288
CRYOGENICS . AUGUST 1971
Table 3. Physical properties of pure components
Pc, arm *
Vc, c m 3
m o l e -1
Tc, K P~ (at 115" 5 ° K), atm * Liquid molal v o l u m e ,
Argon
Methane
48"0 75'2 150"72 - 0"002
45"8 98"72 191"06 0'0104
9"26 33"1
1 "3
Ethane
Reference
48"5 141 "72 305"5 0"105
6
0"00145
38"35
48'25
- 290"08
- 2 1 6 1 "82
6
6 6 6
cm 3 m o l e -~ (at 115"5 K)
[3ii , cm 3 m o l e -1 (at 115.5 K)
- 141 "19
* 1 atm = 101 k N m -2
reported in Table 3. Second virial coefficients values for pure components and the hypothetical binaries were evaluated using the method of O'Connell and Prausnitz. s 4.8B, 5C, 5D were evaluated from (11)-(13). 5. New values of B, C, and D were obtained. 6. The process was repeated until the tolerance of pressure was satisfied. This procedure was first applied to the secants of constant xCl/XC2 for the evaluation of YA and then applied to the secants of constant x A / x ¢ 2 , which were fixed in such a manner that they intercept the secants of constant xCl/XC2 at each of the experimental points, for the evaluation of Y c l " The vapour phase composition of the least volatile component, Yc2 was determined by (15). Yc2 = 1 - (YA + YCl )
. . . (15)
An IBM 360 computer programme was written to perform the above calculations. The required binary equilibrium data of the system methane-ethane were interpolated from those of Chang and Lu 2,3 and Canfield. 7 The calculated results for the ternary system are given in Table 2. The root mean square deviation, which is defined by =
A.
Ycalc)2
exp
.
.
(16) /
CRYOGENICS . AUGUST 1971
The authors are indebted to the National Research Council of Canada for financial support.
REFERENCES
1. ECKERT, C. A., and PRAUSNITZ, J. M. AIChE J ll, 886 (1965) 2. CHANG, S-D., and LU, B. C-Y. Chem Eng Progr Symposium Series No 81 63, 18 (1967) 3. LU, B. C-Y., CHANG, S-D., ELSHAYAL, 1. M., Yu, P., GRAVELLE, D., and POON, D. P. L. Proceedings of First International Conference of Calorimetry and Thermodynamics, Warsaw, 1969, p 755 4. HO, J. C. K., BOSHKO, O., and LU, B. C-Y. Can J Chem Eng 39,205 (1961) 5. O'CONNELL, J. P., and PRAUSNITZ, J. M. Ind Eng Chem Process Design and Develop 6,245 (1967) 6. DIN, F. Thermodynamic Functions of Gases, Vols 2 and 3 (Butterworths, 1956)
0 °5
.
is only 0"002 in the vapour phase compositions of argon and methane, indicating the good agreement obtained between the experimental and calculated results, and the validity of the proposed calculation procedure.
7 SHANA'A, M. Y., and CANFIELD, F. B. Trans Faraday Soc 64, 2281 (1968)
289
Vapour-liquid equilibria in the argon-ethane and argon-methane-ethane systems at 15.5 K I. M. Elshayal and B. C-Y. Lu
Vapour-liquid equilibrium data for the binary system argon-ethane and the ternary system a r g o n - m e t h a n e ethane were determined at 115-5 K as part of a continuing programme to investigate the non-ideal behaviour of mixtures of simple molecules. Phase equilibria of these systems are important for design purposes and for testing solution theories. Equilibrium data for the binary system argonethane were previously reported by Eckert and Prausnitz.1 However, there were only six experimental points, each measured at a different temperature. In this investigation, 38 experimental points were determined for the binary system, and 23 experimental points for the ternary system.
NOMENCLATURE B,C,D K P e o R T X X f
Y F
Redlich-Kister constants vapour phase correction term total vapour pressure of the system vapour pressure of pure component gas constant system temperature, K mole fraction in the liquid phase mole fraction of component 1 in the initial binary liquid mixture along a constant x l / x 2 secant mole fraction in the vapour phase molar volume of pure component second virial coefficient activity coefficient in the liquid phase
Experimental details
7
The equilibrium cell, the cryostat, the equipment assembly, and the experimental technique employed in this investigation were essentially the same as those previously reported, z The equilibrium cell was made of a 100 ml Jerguson transparent gauge. A dewar flask of 181 capacity was employed as the cryostat into which the equilibrium cell was submerged, with isopentane used as the bath liquid. In addition, two variable speed stirrers, a heating element, a refrigeration coil, and a resistant type temperature sensing probe were installed inside the cryostat. The temperature of the cryostat was regulated by controlling the evaporation rate of liquid nitrogen in the refrigeration coil, and was controlled to + 0-01 K. The equilibrium temperature was measured by means of two thermocouples of the protective type, one for the liquid phase and one for the vapour phase, in conjunction with a Leeds and Northrup K-3 potentiometer and a Tinsely SRI galvanometer. The thermocouples were calibrated against the vapour pressure of argon. 6 The total pressure of the system was measured by means of
Subscripts 12 hypothetical binary of components 1 and 2 3 third component of a ternary mixture i, ii, L Jl pure component j k hypothetical binary of components j and k
The authors are with the Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada. Received 15 March 1971. CRYOGENICS . AUGUST 1971
three pre-tested 12 in (30 cm) Heise gauges ( 0 - 6 0 , 0 - 2 5 0 , and 0 - 5 0 0 lb in-2 [ 1 lb in-2 = 6-9 kN m - 2 ] . The accuracy of these gauges is 0-1% of the full scale. A mercury manometer was used at low pressures. In addition to the cryostat, the equipment assembly 2 consisted of a feed measuring and charging device, a closed recirculation loop, a volume regulator, and sampling facilities. An electromagnetic pump was employed for recirculating the vapour through the liquid in the equilibrium cell. Research grade gases, supplied by Matheson of Canada Ltd, were used without further purification. The specified minimum purities of these gases were as follows: Argon 99"998 mole % Methane 99"99 mole % Ethane 99"9 mole % 285
Analyses of the liquid and the vapour samples were made with a Fisher-Hamilton gas partitioner, model 29V, using silica gel ( 2 8 - 3 0 mesh) as the column packing material, and helium as the carrier gas. Peak heights were employed in the calibration. The average deviation obtained in the calibration was less ~ a n + 0-001 mole fraction. The cell temperature of the gas partitioner was maintained at 70 C, and the flow rate of helium was maintained at 100 ml per minute at 35 lb in -2 . A retention time of 6 minutes was required for separating a three-component mixture.
Table 1. Experimental and calculated results for the binary system argon-ethane at 115"5 K P lb in -2 (1 Ib in-2 = 6"9 k N m -2)
24"6 25"2 25"2 29"7 29"7 30"8 33"8 34"1 36"0 41 "2 43"4 48"4 51 "0 57"8 63"9 70"7 80" 1 81"8 87"2 87"4 90" 9 94"3 98"6 100"9 105"5 108"1 109"8 110"8 113"0 115"2 116"2 116"3 118"5 123"1 124"5 129"1 130"5 133" 1
XA
0"0690 0"0710 0"0712 0"0926 0"0928 0"0980 0"1119 0"1135 0"1222 0" 1450 0"1546 0"1760 0"1873 0"2176 0"2470 0"2835 0"3422 0"3540 0"3946 0"3969 0" 4259 0"4577 0"5046 0"5355 0"6120 0"6638 0"6978 0"7304 0"7581 0"8004 0"8210 0"8230 0"8619 0"9151 0"9265 0"9570 '0"9654 0"9813
(YA)calc 0"9990 0"9990 0"9990 0"9992 0"9992 0"9993 0"9993 0"9993 0"9994 0"9995 0'9995 0"9995 0"9995 0"9996 0"9996 0"9997 0" 9997 0"9997 0"9997 0"9997 0" 9997 0"9997 0'9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9998 0"9999 0"9999 0"9999 0"9999 0"9999 1 "0000
7A
2"7434 2"7343 2"7334 2"6368 2"6362 2"6127 2"5550 2"5480 2"5131 2"4247 2"3891 2"3129 2"2749 2" 1770 2"0886 1"9867 1 "8387 1 "8096 1 "7175 1 "7125 1"6509 1 "5866 1 "4980 1 "4432 1 "3200 1"2468 1 "2033 1"1649 1"1351 1"0946 1 "0772 1 "0757 1 "0474 1 "0187 1"0142 1"0050 1 "0032 1 "0009
7C2 1 "0421 1 "0419 1 "0424 1 "0469 1 '0476 1 "0475 1"0513 1"0509 1 "0529 1 "0589 1"0619 1 "0691 1 "0735 1 "0859 1 "1006 1 "1216 1" 1635 1 "1775 1 "2109 1 "2131 1"2436 1 '2853 1 "3587 1 "4110 1 "5923 1 "7624 1 "9054 2"0600 2"2326 2"5291 2"7114 2"7267 3"1422 3"9233 4"1604 4"8089 4"9761 5"4854
Experimental results
The experimental equilibrium temperature, pressure, and liquid composition data determined in this investigation for the binary system argon-ethane are listed in Table 1, and shown in Figure I. The concentration of ethane in the vapour phase was found to be extremely low, and the error in its determination was rather high. For this reason, it was decided to evaluate the ethane composition of the vapour from the total vapour pressure-liquid composition measurements. At 115.5 K the system pressure is relatively low, and the total pressure method previously proposed 4 was employed for this purpose. The calculated values of liquid phase activity coefficient, 7, and the vapour phase composition, y, are also listed in Table 1. In Figure 2, In 7 values were plotted against liquid composition to illustrate the non-ideal behaviour of the binary system. The experimental points for the ternary system argonmethane-ethane were determined along constant Xcl/Xc2 and constant xA/xc2 secants. The arrangement is shown in Figure 3 and was planned for the purpose of reducing the ternary system to a hypothetical binary so that the total pressure method proposed for binary systems 4 could be extended and applied. Evaluation of ternary vapour phase compositions
When the components of a ternary solution vary along a constant Xl/X2 secant, the mole fractions of components 1
t40
Argon- ethane 115.5K
120
IOO
80 o 7 .c.
60
40
20
O
0.2
0'4
0-6
0'8
I.O
XA Figure 1. Total vapour pressure--liquidcompositiondiagram for the binary system argon--ethane at 115-5 K ( 1 Ib in -2 = 6.9 kN m -2)
286
CRYOGENICS . AUGUST 1971
ln3'3 =(1 --X3) 2 [B+C(4x3 -- 1) 1.8
+ D (2Xa - 1) (6x3 -- 1)]
Argon - ethone 115.5K
In 3't2 = x 2 [B + C (4x3 - 3) + D (2x3 - 1) (6x3 - 5)]
t.6
. . . (3b) Equations 2 and 3 indicate that the total pressure is a function of the Redlich-Kister constants and the liquid composition x3. Hence at constant temperature,
Id-
P=P(B, C,D, x3)
,.2
...(4)
Assuming a set of initial values for the constants B, C, and D, then
I.O
aP 63P aP 6 P = - ~ S B + -ff--~S C + "~ 6D
>o
_5
. . . (3a)
...
(5)
0.8
where 6 P = P e x p t -- P e ~
0.6
. .. (6)
and 63P_ 63P 63'73 + . 63P . 03'12 . . 63B 633'3 63B 633'12 aB
0.4
Similar equations may be written for 63P/63Cand 63P/63D. Furthermore from (3a) and (3b)
02
O
(7)
-
2
4
xA
6
8
073 63B =73 (1 - x 3 ) =
tO
63"/12 Figure 2. Liquid phase activity coefficient-composition curves for the binary system argon--ethane at 1 15.5 K
63B
-'r~2 x ]
...(8a)
...
(Sb)
and
63P x3P3 633'3 K~ _
. . . (9a)
63P _ ( 1 -xa)e~2 and 2 depend on that of component 3 as follows
a3q2
Kt2
. . . (9b)
xt = x ' ( 1 - x a ) x2 = (1 - x'X1 - x3)
...(1)
where x' represents the mole fraction of component 1 in the initial binary solution 1 - 2 . The mole fraction of the third component along the secant, is taken as the independent variable, and the ternary mixture may therefore be regarded as a binary one. Assuming that the vapour phase is an ideal solution, but not an ideal gas, the total pressure of the system may be represented by the following equation
o
p _ X 3 P _ 3'3 Ka
+ (1 -
x3)Pl°2 "rl2
Argon-methane-ethane ~
/
~
. . . (2)
Kt2
where
K3 =exp [ (f133 - v3)(PKa2 =exp [ ([3t2- v t 2 ) ( P In this investigation liquid phase activity coefficients 7, were correlated by means of the Redlich-Kister equation as follows CRYOGENICS . AUGUST 1971
C2
Cf
Figure 3. Liquid compositions investigated for the ternary system argon--methane--ethane at 115-5 K 287
By substituting (8) and (9) into (7), the following expression is obtained
and
i=l
a__P_x~ (I -x~)" 3`~P; aB
+
xs2 (1 - x s ) 3`12 el~
Ks
...(10)
K12
,=,
Similar expressions for OP/~Cand aP/aD may be obtained in the same manner. Substituting aP/OB, ~P/aC and aP/aD into (5) for all the experimental points gives the following set of normal equations
...(13) These corrections for B, C, and D are employed to calculate the actual values of B, C, and D which in turn are used to evaluate the liquid phase activity coefficients, 3'3 and 3'12. Using the liquid phase activity coefficients, (14) may be applied to evaluate the vapour phase compositions.
Yi =
8B ~ (OPt 2
"
+ 5D
p
:,
.
= ~, 6P
i=1
[=1
...(11)
,=,
(xiPf3`i)/Ki (x,e;' 3`iJ/Ki + (x
. . . (14)
3`j )/K jk
The modified total pressure method was used to predict the vapour phase composition of the ternary system argon-methane -ethane. The numerical routine for determining the vapour phase composition may be summarized as follows:
OP
+ $D
i=1
~aC]
=
i=1
...(12)
1. Initial values of B, C and D were assumed. 2. Liquid phase activity coefficients were calculated from (3). 3. Total pressure of the system was calculated at each experimental point, using the pure components properties
Table 2. Experimental and calculated results for the ternary system argon-methane-ethane at 115.5 K
P Ib in -2 (1 Ib in-2 = 6.9 kN m-2) 59"8 71 '1 80"0 89" 5 94'4 98"5 51"0 64"6 70"7 79"1 84"4 90"4 99"5 107"5 51 "8 59"8 69"5 79"5 89"2 98"0 78"3 81 "0 84"0
YA XA 0"2572 0"3241 0"4177 0" 5148 0"5889 0"6553 0"1977 0"3046 0"3547 0"4521 0"4925 0"5571 0"6625 0"7564 0"2810 0"3398 0"4198 0"5083 0"6135 0"6924 0"4431 0"4606 0"4854
Xcl
Expt
Calc
Expt
YCl Calc
YC2 Calc
0"1859 0"1743 0"1521 0" 1271 0"1079 0"0980 0"3981 0"3066 0"2737 0"2351 0"2187 0"1887 0"1481 0'1071 0"5164 0"4746 0"4143 0"3551 0"2803 0"2044 0"2317 0"2096 0"1675
0"9164 0"9332 0"9525 0" 9646 0"9761 0"9762 0"8400 0"8852 0"9056 0"9300 0"9377 " 0"9489 0"9617 0"9730 0"8018 0"8359 0"8715 0"8984 0"9242 0"9469 0"9309 0"9380 0"4510
0"9158 0"9338 0"9518 0' 9645 0"9717 0"9751 0"8395 0"8839 0"9059 0"9302 0"9378 0"9492 0"9620 0"9726 0"8026 0"8363 0"8716 0"8984 0"9244 0"9465 0"9304 0"9383 0"9519
0"0836 0"0668 0"0475 0" 0354 0"0279 0"0238 0"1600 0"1148 0"0944 0"0700 0"0623 0"0511 0"0383 0"0270 0"1982 0"1641 0"1285 0"1016 0"0758 0"0531 0"0691 0"0620 0"0490
0"0840 0"0660 0"0480 0" 0353 0"0280 0"0245 0'1603 0"1158 0"0938 0"0695 0"0619 0"0505 0"0376 0"0269 0"1970 0"1632 0"1279 0"1010 0"0751 0"0580 0"0693 0"0614 0"0478
0"0002 0"0002 0"0002 0" 0002 0"0003 0"0003 0"0003 0'0003 0"0003 0"0003 0"0003 0"0003 0"0004 0"0005 0"0005 0"0005 0"0005 0"0005 0"0005 0"0005 0"0003 0"0003 0"0002
7A
7C 1
3`C2
3"5694 2"5350 1 "5119 3"0153 2"1964 1 "7420 2"4247 1 '8479 2"2033 1"9559 1"5827 3"0046 1 '6738 1 "4331 4"0395 1"4469 1 "3198 5"7975 3"0720 2"1149 1"7569 2"2713 1 " 6 4 3 9 2"4917 2"1390 1"5893 2"6935 1"7973 1 "4205 3"5983 1"6778 1"3654 4"1301 1"5214 1 "3017 5"1739 1 "2942 1 "2233 8"4816 1"1556 1 " 2 1 1 3 4"0292 1"5470 1"1582 5"9968 1 "4628 1"1386 6"8501 1 " 3 6 7 3 1 " 1 2 4 7 8"1797 1"2604 1"1136 10"7773 1 " 1 6 2 5 1 " 1 2 6 4 15"2237 1" 1288 1" 1597 17" 1085 1°1855 1 "4547 3"3821 1"8630 1 "4730 3"3345 1 " 9 2 3 7 1 " 5 3 5 4 3"1154
Root mean square deviation in vapour phase compositions of argon and methane = 0-002
288
CRYOGENICS . AUGUST 1971
Table 3. Physical properties of pure components
Pc, arm *
Vc, c m 3
m o l e -1
Tc, K P~ (at 115" 5 ° K), atm * Liquid molal v o l u m e ,
Argon
Methane
48"0 75'2 150"72 - 0"002
45"8 98"72 191"06 0'0104
9"26 33"1
1 "3
Ethane
Reference
48"5 141 "72 305"5 0"105
6
0"00145
38"35
48'25
- 290"08
- 2 1 6 1 "82
6
6 6 6
cm 3 m o l e -~ (at 115"5 K)
[3ii , cm 3 m o l e -1 (at 115.5 K)
- 141 "19
* 1 atm = 101 k N m -2
reported in Table 3. Second virial coefficients values for pure components and the hypothetical binaries were evaluated using the method of O'Connell and Prausnitz. s 4.8B, 5C, 5D were evaluated from (11)-(13). 5. New values of B, C, and D were obtained. 6. The process was repeated until the tolerance of pressure was satisfied. This procedure was first applied to the secants of constant xCl/XC2 for the evaluation of YA and then applied to the secants of constant x A / x ¢ 2 , which were fixed in such a manner that they intercept the secants of constant xCl/XC2 at each of the experimental points, for the evaluation of Y c l " The vapour phase composition of the least volatile component, Yc2 was determined by (15). Yc2 = 1 - (YA + YCl )
. . . (15)
An IBM 360 computer programme was written to perform the above calculations. The required binary equilibrium data of the system methane-ethane were interpolated from those of Chang and Lu 2,3 and Canfield. 7 The calculated results for the ternary system are given in Table 2. The root mean square deviation, which is defined by =
A.
Ycalc)2
exp
.
.
(16) /
CRYOGENICS . AUGUST 1971
The authors are indebted to the National Research Council of Canada for financial support.
REFERENCES
1. ECKERT, C. A., and PRAUSNITZ, J. M. AIChE J ll, 886 (1965) 2. CHANG, S-D., and LU, B. C-Y. Chem Eng Progr Symposium Series No 81 63, 18 (1967) 3. LU, B. C-Y., CHANG, S-D., ELSHAYAL, 1. M., Yu, P., GRAVELLE, D., and POON, D. P. L. Proceedings of First International Conference of Calorimetry and Thermodynamics, Warsaw, 1969, p 755 4. HO, J. C. K., BOSHKO, O., and LU, B. C-Y. Can J Chem Eng 39,205 (1961) 5. O'CONNELL, J. P., and PRAUSNITZ, J. M. Ind Eng Chem Process Design and Develop 6,245 (1967) 6. DIN, F. Thermodynamic Functions of Gases, Vols 2 and 3 (Butterworths, 1956)
0 °5
.
is only 0"002 in the vapour phase compositions of argon and methane, indicating the good agreement obtained between the experimental and calculated results, and the validity of the proposed calculation procedure.
7 SHANA'A, M. Y., and CANFIELD, F. B. Trans Faraday Soc 64, 2281 (1968)
289