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A PY-type pseudocubic perturbed hard-sphere equation of state
43
Fluid Phase Equilibria, 63 (1991) 43-48 Elsevier Science Publishers B.V., Amsterdam
A PY-TYPE EQUATION
PSEUDOCUBIC OF STATE
MASAHIRO KATO, MASATO and KUNIHIKO SAN0 Department of Industrial Fukushima 963, Japan (Received
PERTURBED
YAMAGUCHI,
Chemistry,
March 19, 1990; accepted
Faculty
HARD-SPHERE
KAZUNORI of Engineering,
in final form October
AIZAWA Nihon
University,
Koriyama,
pseudocubic
perturbed
4, 1990)
ABSTRACT Kato, M., Yamaguchi, M., Aizawa, K. and Sano, K., 1991. A PY-type hard-sphere equation of state. Fluid Phase Equilibria, 63: 43~ 48.
A pseudocubic perturbed hard-sphere equation of state is proposed combining the hard-sphere expression of Percus and Yevick and the empirical cubic form for apparent volume. The present equation of state is successfully applied to the saturated properties of pure substances, including polar components.
INTRODUCTION
A pseudocubic perturbed hard-sphere equation of state has been recently proposed by the authors (Kato et al., 1989), combining the repulsion term of Carnahan and Starling (1969) and the cubic form for apparent volume. In the present study, a pseudocubic perturbed hard-sphere equation of state is newly proposed combining the hard-sphere expression of Percus and Yevick (1958) and the empirical cubic form for apparent volume. This equation of state is simpler than the previous pseudocubic perturbed hardsphere equation of state. The present equation of state gives satisfactory results for saturated properties. EQUATION
OF STATE
The proposed p=
IL
-
037%3812/91/$03.50
equation
of state is given as follows:
(1)
(V*+“l;V*-c,)
0 1991 - Elsevier Science Publishers
B.V.
44
V* -=4y+ V
(1 -Y)’
* 7 Y=
l+Y+Y
The parameters,
b TV
a, b, cl and c2 can be expressed
(2) as follows:
(3) (4) The dimensionless parameters L? can consequently be evaluated using the following equations which can be derived from the critical point requirements and the assumption that (V/V,) p= ~ = l/4. Q” = (1 - a)’
(5)
Q2, = Z,
(6)
ii,
(1
a,
= --_(~Cz+1)-(a+Z,) (1-a)
(7)
(1-a) -y-(\/i--4u-1)+(a+zc)
= (1
(8)
where cx = (9,‘28)
Z,
(9)
in which, 2, denotes the experimental critical compressibility factor. Given the temperature and pressure, the present equation of state has a maximum of three real roots for the volume, as do the conventional cubic equations of state. The temperature dependence of the parameter a was empirically introduced as follows: a=K
* . a,
00)
in which K, = 1 - S(1 - T2’3) S=
1 - Kcb
- 0.002.
(II) pr2. AT,
I - F3 AT,=
T,-
Tr”b
pr ’ = $/[9.8694
(12) (13)
x W5(
R . r,)‘/P,l
(14)
The superscript ‘nb’ means the normal boiling point. In eqn. (14) the units of I*, R, T, and PC are debye, kPa m3 kmol-’ K-i, K, and kPa,
45 TABLE
I
Accuracy
of saturated
Substance
properties
T, range
N a
of pure substances, This work s
Acetylene
0.62-1.0
Ammonia
0.60-l
Benzene
0.55-1.0
.O
including
polar substances IML
IAP,’
lAI’“/
lAI”/
IAP/
IAV”/
PI
V”(
V’I
PI
V’l
IAV’,’
V’l
av(%)
av(%)
av(%)
av(W)
av(%)
av(%) 1.99
23
0.642
2.04
4.43
2.41
2.08
2.48
30
0.689
2.09
5.12
3.73
1.82
2.73
1.99
48
0.632
0.38
3.12
5.40
1.04
3.42
2.81
i-Butane
0.63-1.0
30
0.638
0.43
2.13
4.92
2.05
4.20
n-Butane
0.64-1.0 0.65-l .o
29 27
0.647
0.33
2.22
2.28
1.44
3.18
2.61 2.10
0.675
0.64
3.00
2.96
1.09
4.14
4.95
0.71-1.0
33
0.513
0.67
2.28
1.86
0.28
2.61
2.22
3.93
4.22
1.17
2.68
1.91
12.79
0.77
2.54
1.29 9.72
l-Butene Carbon dioxide Carbon
0.51ll.O
monoxide
0.60-l
Ethane
.O
25
0.706
28
0.662
0.80 0.49
Ethanol
0.72-1.0
16
0.441
0.33
1.63
1.51
0.81
1.51
2.88
Ethylene
0.60-1.0
22
0.738
0.76
3.16
5.10
1.64
2.44
7.04
Heavy water n-Hexane
0.43-1.0
39
0.674
1.09
2.61
4.46
3.14
3.44
2.75
0.54-1.0
44
0.67
2.41
3.12
2.24
0.59-1.0
30
0.38
3.28
3.43
1.59 1.73
2.19
Methane
0.603 0.745
4.18
2.31
Methanol
0.62-1.0
21
0.562
1.20
3.90
2.34
0.92
4.16
3.28
Nitrogen
0.61-1.0
19
0.718
0.43
25
0.679
1.02
1.74 1.25
2.53
0.58-1-O
3.15 14.87
5.42
Oxygen n-Pentane
3.50 3.29
3.29
8.92
0.6661.0
31
0.624
0.20
1.96
1.54
0.52
2.53
2.45
Propane
0.62-l
32
0.671
0.18
2.88
2.37
0.70
3.70
2.51
Propylene
0.62-1.0
27
0.647
0.38
1.16
7.62
0.67
1.57
4.52
.O
Rll
0.4551.0
30
0.641
1.13
3.12
4.64
1.64
3.31
2.97
R12
0.47-l
31
0.647
1.03
2.53
5.12
2.71
3.60
2.32
R13
0.46-1.0
31
0.654
0.72
2.70
3.07
0.84
2.46
2.20
R13Bl
0.47- 1 .o
31
0.628
0.56
1.54
9.14
1.17
1.26
5.41
R21
0.4991.0
31
0.646
0.54
2.17
3.72
3.02
4.10
2.17
R114
0.48-1.0
31
0.605
0.95
2.39
3.55
1.10
2.75
1.76
36 44
0.617
0.41 1.69
2.17
4.12
3.45
7.60
2.20 1.07
3.72 1.93
2.85 6.54
0.77
2.79
4.70
1.44
3.06
3.48
.o
Sulfur dioxide
0.59-l
Water
0.42-1.0
.o
Average ’
N = Number
h Equation
0.666
of data points.
of state proposed
by Iwai et al. (1988).
respectively. Critical properties and normal boiling points were obtained from Canjar and Manning (1967) and JSME (1983), and the normal boiling point of ammonia from Reid et al. (1977). The dipole moments were obtained from Reid et al. (1977). The values of K, were first calculated to
Reduced
Density
Fig. 1. Density vs. pressure diagram at saturation for water: (0) experimental (JSME, 1983); ) this work; (- - -) SRK (Soave, 1972); (. . . . .) CCOR (Guo et al.. 1985). (-
satisfy the vapor pressures over the whole temperature range for many substances. The calculated K, values were then empirically correlated with eqns. (ll)-(14). The present equation of state strictly satisfies the critical point requirements.
RESULTS
Table 1 represents the accuracy obtained for vapor pressures, saturated vapor volumes and saturated liquid volumes for pure substances, including polar substances, compared with the IML equation of state proposed by Iwai et al. (1988). In the calculations, the saturated data of pure substances were obtained from Canjar and Manning (1967) and JSME (1983). Table 1 shows that satisfactory results were obtained except for the saturated liquid volumes of ethane and oxygen. In the present study, no attempt was made to fit the saturated liquid volumes in the correlation of K,. The critical point requirements were strictly satisfied by the present equation of state, but failed to improve on the average accuracies obtained using the IML equation of state (Iwai et al., 1988). Figures 1 and 2 show the saturated volumetric behavior for water and ammonia, showing excellent calculation results using the present equation of state. The computation time required was reduced to 30% of that for the previous equation of state (Kato et al., 1989) by changing the hard-sphere
47
Fig. 2. Density vs. pressure diagram at saturation for ammonia: (0) experimental (Canjar and this work; (- - -) SRK (Soave, 1972); (. . . . . .) CCOR (Guo et Manning, 1967); ( -) al., 1985).
expression of Carnahan Yevick (1958).
and Starling
(1969)
to the simpler one of Percus and
CONCLUSIONS
A pseudocubic perturbed hard-sphere equation of state was proposed successfully applied for vapor pressures, saturated vapor volumes, saturated liquid volumes for non-polar and polar components. NOMENCLATURE
a,
b,
cl,
c2
Ka P R S T V .Y Z
parameters in eqn. (1) temperature parameter given by eqn. (11) pressure gas constant parameter given by eqn. (12) temperature volume function defined by eqn. (2) compressibility factor
Greek letters
dipole moment dimensionless parameters
given by eqns. (Z+-(8)
and and
48
Subscripts C
r
critical state property reduced property
Superscripts nb *
normal boiling point apparent value
REFERENCES Canjar, L.N. and Manning, F.S., 1967. Thermodynamic Properties and Reduced Correlations for Gases. Gulf Publishing, Houston, TX. Carnahan, N.F. and Starling, K.E., 1969. Equation of state for nonattracting rigid spheres. J. Chem. Phys., 51: 635-636. Guo, T.M., Kim, H., Lin, H.M. and Chao, K.C., 1985. Cubic chain-of-rotators equation of state. 2. Polar substances. Ind. Eng. Chem. Process Des. Dev., 24: 764-767. Iwai, Y., Margerum, M.R. and Lu, B.C.-Y., 1988. A new three-parameter cubic equation of state for polar fluids and fluid mixtures. Fluid Phase Equilibria, 42: 21-41. JSME, 1983. Thermophysical Properties of Fluids. Japan Sot. of Mech. Engrs.. Tokyo. Kato, M., Yamaguchi, M. and Kiuchi, T., 1989. A new pseudocubic perturbed hard-sphere equation of state. Fluid Phase Equilibria. 47: 171-187. Percus, J.K. and Yevick, G.J., 1958. Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev., 110: l-13. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd Edn. McGraw-Hill, New York. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197-1203.
Fluid Phase Equilibria, 63 (1991) 43-48 Elsevier Science Publishers B.V., Amsterdam
A PY-TYPE EQUATION
PSEUDOCUBIC OF STATE
MASAHIRO KATO, MASATO and KUNIHIKO SAN0 Department of Industrial Fukushima 963, Japan (Received
PERTURBED
YAMAGUCHI,
Chemistry,
March 19, 1990; accepted
Faculty
HARD-SPHERE
KAZUNORI of Engineering,
in final form October
AIZAWA Nihon
University,
Koriyama,
pseudocubic
perturbed
4, 1990)
ABSTRACT Kato, M., Yamaguchi, M., Aizawa, K. and Sano, K., 1991. A PY-type hard-sphere equation of state. Fluid Phase Equilibria, 63: 43~ 48.
A pseudocubic perturbed hard-sphere equation of state is proposed combining the hard-sphere expression of Percus and Yevick and the empirical cubic form for apparent volume. The present equation of state is successfully applied to the saturated properties of pure substances, including polar components.
INTRODUCTION
A pseudocubic perturbed hard-sphere equation of state has been recently proposed by the authors (Kato et al., 1989), combining the repulsion term of Carnahan and Starling (1969) and the cubic form for apparent volume. In the present study, a pseudocubic perturbed hard-sphere equation of state is newly proposed combining the hard-sphere expression of Percus and Yevick (1958) and the empirical cubic form for apparent volume. This equation of state is simpler than the previous pseudocubic perturbed hardsphere equation of state. The present equation of state gives satisfactory results for saturated properties. EQUATION
OF STATE
The proposed p=
IL
-
037%3812/91/$03.50
equation
of state is given as follows:
(1)
(V*+“l;V*-c,)
0 1991 - Elsevier Science Publishers
B.V.
44
V* -=4y+ V
(1 -Y)’
* 7 Y=
l+Y+Y
The parameters,
b TV
a, b, cl and c2 can be expressed
(2) as follows:
(3) (4) The dimensionless parameters L? can consequently be evaluated using the following equations which can be derived from the critical point requirements and the assumption that (V/V,) p= ~ = l/4. Q” = (1 - a)’
(5)
Q2, = Z,
(6)
ii,
(1
a,
= --_(~Cz+1)-(a+Z,) (1-a)
(7)
(1-a) -y-(\/i--4u-1)+(a+zc)
= (1
(8)
where cx = (9,‘28)
Z,
(9)
in which, 2, denotes the experimental critical compressibility factor. Given the temperature and pressure, the present equation of state has a maximum of three real roots for the volume, as do the conventional cubic equations of state. The temperature dependence of the parameter a was empirically introduced as follows: a=K
* . a,
00)
in which K, = 1 - S(1 - T2’3) S=
1 - Kcb
- 0.002.
(II) pr2. AT,
I - F3 AT,=
T,-
Tr”b
pr ’ = $/[9.8694
(12) (13)
x W5(
R . r,)‘/P,l
(14)
The superscript ‘nb’ means the normal boiling point. In eqn. (14) the units of I*, R, T, and PC are debye, kPa m3 kmol-’ K-i, K, and kPa,
45 TABLE
I
Accuracy
of saturated
Substance
properties
T, range
N a
of pure substances, This work s
Acetylene
0.62-1.0
Ammonia
0.60-l
Benzene
0.55-1.0
.O
including
polar substances IML
IAP,’
lAI’“/
lAI”/
IAP/
IAV”/
PI
V”(
V’I
PI
V’l
IAV’,’
V’l
av(%)
av(%)
av(%)
av(W)
av(%)
av(%) 1.99
23
0.642
2.04
4.43
2.41
2.08
2.48
30
0.689
2.09
5.12
3.73
1.82
2.73
1.99
48
0.632
0.38
3.12
5.40
1.04
3.42
2.81
i-Butane
0.63-1.0
30
0.638
0.43
2.13
4.92
2.05
4.20
n-Butane
0.64-1.0 0.65-l .o
29 27
0.647
0.33
2.22
2.28
1.44
3.18
2.61 2.10
0.675
0.64
3.00
2.96
1.09
4.14
4.95
0.71-1.0
33
0.513
0.67
2.28
1.86
0.28
2.61
2.22
3.93
4.22
1.17
2.68
1.91
12.79
0.77
2.54
1.29 9.72
l-Butene Carbon dioxide Carbon
0.51ll.O
monoxide
0.60-l
Ethane
.O
25
0.706
28
0.662
0.80 0.49
Ethanol
0.72-1.0
16
0.441
0.33
1.63
1.51
0.81
1.51
2.88
Ethylene
0.60-1.0
22
0.738
0.76
3.16
5.10
1.64
2.44
7.04
Heavy water n-Hexane
0.43-1.0
39
0.674
1.09
2.61
4.46
3.14
3.44
2.75
0.54-1.0
44
0.67
2.41
3.12
2.24
0.59-1.0
30
0.38
3.28
3.43
1.59 1.73
2.19
Methane
0.603 0.745
4.18
2.31
Methanol
0.62-1.0
21
0.562
1.20
3.90
2.34
0.92
4.16
3.28
Nitrogen
0.61-1.0
19
0.718
0.43
25
0.679
1.02
1.74 1.25
2.53
0.58-1-O
3.15 14.87
5.42
Oxygen n-Pentane
3.50 3.29
3.29
8.92
0.6661.0
31
0.624
0.20
1.96
1.54
0.52
2.53
2.45
Propane
0.62-l
32
0.671
0.18
2.88
2.37
0.70
3.70
2.51
Propylene
0.62-1.0
27
0.647
0.38
1.16
7.62
0.67
1.57
4.52
.O
Rll
0.4551.0
30
0.641
1.13
3.12
4.64
1.64
3.31
2.97
R12
0.47-l
31
0.647
1.03
2.53
5.12
2.71
3.60
2.32
R13
0.46-1.0
31
0.654
0.72
2.70
3.07
0.84
2.46
2.20
R13Bl
0.47- 1 .o
31
0.628
0.56
1.54
9.14
1.17
1.26
5.41
R21
0.4991.0
31
0.646
0.54
2.17
3.72
3.02
4.10
2.17
R114
0.48-1.0
31
0.605
0.95
2.39
3.55
1.10
2.75
1.76
36 44
0.617
0.41 1.69
2.17
4.12
3.45
7.60
2.20 1.07
3.72 1.93
2.85 6.54
0.77
2.79
4.70
1.44
3.06
3.48
.o
Sulfur dioxide
0.59-l
Water
0.42-1.0
.o
Average ’
N = Number
h Equation
0.666
of data points.
of state proposed
by Iwai et al. (1988).
respectively. Critical properties and normal boiling points were obtained from Canjar and Manning (1967) and JSME (1983), and the normal boiling point of ammonia from Reid et al. (1977). The dipole moments were obtained from Reid et al. (1977). The values of K, were first calculated to
Reduced
Density
Fig. 1. Density vs. pressure diagram at saturation for water: (0) experimental (JSME, 1983); ) this work; (- - -) SRK (Soave, 1972); (. . . . .) CCOR (Guo et al.. 1985). (-
satisfy the vapor pressures over the whole temperature range for many substances. The calculated K, values were then empirically correlated with eqns. (ll)-(14). The present equation of state strictly satisfies the critical point requirements.
RESULTS
Table 1 represents the accuracy obtained for vapor pressures, saturated vapor volumes and saturated liquid volumes for pure substances, including polar substances, compared with the IML equation of state proposed by Iwai et al. (1988). In the calculations, the saturated data of pure substances were obtained from Canjar and Manning (1967) and JSME (1983). Table 1 shows that satisfactory results were obtained except for the saturated liquid volumes of ethane and oxygen. In the present study, no attempt was made to fit the saturated liquid volumes in the correlation of K,. The critical point requirements were strictly satisfied by the present equation of state, but failed to improve on the average accuracies obtained using the IML equation of state (Iwai et al., 1988). Figures 1 and 2 show the saturated volumetric behavior for water and ammonia, showing excellent calculation results using the present equation of state. The computation time required was reduced to 30% of that for the previous equation of state (Kato et al., 1989) by changing the hard-sphere
47
Fig. 2. Density vs. pressure diagram at saturation for ammonia: (0) experimental (Canjar and this work; (- - -) SRK (Soave, 1972); (. . . . . .) CCOR (Guo et Manning, 1967); ( -) al., 1985).
expression of Carnahan Yevick (1958).
and Starling
(1969)
to the simpler one of Percus and
CONCLUSIONS
A pseudocubic perturbed hard-sphere equation of state was proposed successfully applied for vapor pressures, saturated vapor volumes, saturated liquid volumes for non-polar and polar components. NOMENCLATURE
a,
b,
cl,
c2
Ka P R S T V .Y Z
parameters in eqn. (1) temperature parameter given by eqn. (11) pressure gas constant parameter given by eqn. (12) temperature volume function defined by eqn. (2) compressibility factor
Greek letters
dipole moment dimensionless parameters
given by eqns. (Z+-(8)
and and
48
Subscripts C
r
critical state property reduced property
Superscripts nb *
normal boiling point apparent value
REFERENCES Canjar, L.N. and Manning, F.S., 1967. Thermodynamic Properties and Reduced Correlations for Gases. Gulf Publishing, Houston, TX. Carnahan, N.F. and Starling, K.E., 1969. Equation of state for nonattracting rigid spheres. J. Chem. Phys., 51: 635-636. Guo, T.M., Kim, H., Lin, H.M. and Chao, K.C., 1985. Cubic chain-of-rotators equation of state. 2. Polar substances. Ind. Eng. Chem. Process Des. Dev., 24: 764-767. Iwai, Y., Margerum, M.R. and Lu, B.C.-Y., 1988. A new three-parameter cubic equation of state for polar fluids and fluid mixtures. Fluid Phase Equilibria, 42: 21-41. JSME, 1983. Thermophysical Properties of Fluids. Japan Sot. of Mech. Engrs.. Tokyo. Kato, M., Yamaguchi, M. and Kiuchi, T., 1989. A new pseudocubic perturbed hard-sphere equation of state. Fluid Phase Equilibria. 47: 171-187. Percus, J.K. and Yevick, G.J., 1958. Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev., 110: l-13. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd Edn. McGraw-Hill, New York. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197-1203.