- Email: [email protected]
A new generalized equation for predicting volumes of compressed liquids
Fluid Phase Equilibria, 58 (1990) 231-238 EIsevier Science Publishers B.V., Amsterdam -
231 Printed in The Netherlands
A NEW GENERALIZED EQUATION FOR PREDICTING VOLUMES OF COMPRESSED LIQUIDS CHIEN-HOU CHANG and XINGMIN ZHAO Department of Chemical Engineering, Tianjin University, Tianjin 3OGO72(China) (Received August 11,1989)
ABSTRACT Chang, C.-H. and Zhao, X.-M., 1990. A new generalized equation for predicting volumes of compressed liquids. Fluid Phase Equilibria, 58: 231-238. On the basis of reduced PVT data of liquids in the literature, a new generalized PVT equation for predicting volumes or densities of compressed liquids is proposed: A+ C’D-“‘B(P, - P,,,) V=l/p=v,. A + C(P, - P,.,) where A = C:woaiT: and B = E:_,,b#. AI1 the parameter values are determined by fitting with experimental data. In comparing the predicted volumes with experimental values, the overall average absolute percent deviation of 1126 data points is 0.96 for the equation proposed and 2.83 for the Yen-Woods correlation; and that of 1065 data points is 0.84 for the equation proposed and 1.44 for the COSTALD correlation.
INTRODUCTION
Various equations have been suggested for estimating volumes or densities of liquids. These equations can be divided into two groups: those for saturated liquids, and those for compressed liquids. Among the equations of the first group, the Gunn-Yamada equation (Gunn and Yamada, 1971, 1973) and the modified Rackett equation (Rackett, 1970; Spencer and Danner, 1972) are noted for their higher accuracy of calculation (Spencer and Danner, 1972; Spencer and Adler, 1978; Veeranna and Rihani, 1980; Reid et al., 1977), the former giving slightly better results. Recently, the modified Rackett equation has been further improved by Campbell and Thodos (1985). Representative of the correlations for compressed liquids are the Tait equation (Thomson et al., 1982), the Lu correlation (Lu, 1959), the Yen-Woods equation (Yen and Woods, 1966), the Chueh-Prausnitz equation (Chueh and Prausnitz, 1967, 1969) and the COSTALD correlation 0378-3812/90/$03.50 0 1990 - Else&r Science Publishers B.V.
232
(Thomson et al., 1982). The Yen-Woods and Chueh-Prausnitz equations have been extensively tested and shown to have comparable accuracy, but both are very complex, with numerous coefficients. The COSTALD correlation has the higher accuracy. The present work proposes a new generalized equation to improve the prediction of volumes or densities of compressed liquids. DEVELOPMENT OF THE CORRELATION EQUATION
The Yen-Woods equation is the analytical form based on the tabulated corresponding-states correlation developed by Lyderson et al. (Yen and Woods, 1966) for estimating pure-liquid densities at any pressure and at temperatures below T, = 1 in terms of T,, P, and the critical-compressibility factor 2,. Based on the same data, a generalized diagrammatic correlation of plotting pr versus P, at different constant T, has been presented (Smith and Van Ness, 1975). A new diagrammatic correlation with saturation condition as the point of reference can be obtained from the same data by changing the coordinates from p, versus P, to p,/p,, versus (P, - P,,,), as shown in Fig. 1. The curves in Fig. 1 all start from the point (0, l.O), are all concave downward, and all tend to approach asymptotic values as (P, - P,,,) increases to high values. These characteristics suggest that the family of curves may be represented by an equation of the form
where A, E and I; are coefficients.
0
I
2
3
4
5
6
7
6
9
IO
Pr-Ps,r
Fig. 1. Generalized diagrammatic correlation for liquids with saturation condition as points of reference.
233
It can be seen from eqn. (1) that, for any temperature, Q’V= 1 when P, = P,.,, and
3 _ F&-P,,,) F V - E( P, - P,,,) = 3
(2)
if P, increases to values high enough that coefficient A becomes relatively negligible. F/E is the asymptotic value and, as shown in the figure, a function of temperature. In order to express the asymptotic value F/E as a function of temperature, a reference reduced temperature which may be only hypothetical is taken at T, = d, at which the asymptotic value F/E can be approximated as equal to unity, and an exponential function of T, may be assumed to express F/E approximately as follows: F/E
= C@
(3)
where AT, = T, - d and C is an appropriate constant to satisfy the functional relation. Although F/E as a ratio is a function of q, either F or E may for simplicity be kept constant, with the other varying with temperature. So F is here chosen to be a constant and, to reduce the number of parameters, arbitrarily set to equal C, which also thus determines the magnitude of E. Then we obtain C/E = CoTc
(4)
and E at any temperature is determined by the following form, rearranged from eqn. (4): E = C” -AT,)= c(D-r,)
(5)
where D = 1 + d. For the purpose of generalizing and improving the functional relation, an exponent is introduced into eqn. (5), as follows: E = C(D-T,JB
(6)
where B is a function of the acentric factor of each substance and has the polynomial form B = cbjco’
(7)
To express coefficient A as a function of temperature, a polynomial is again adopted:
For prediction of V, eqn. (1) can be rearranged with F substituted by C and E by eqn. (6):
where C and D are constants, A is a function of temperature as expressed by eqn. (8), and B is a function of the acentric factor as shown by eqn. (7). DETERMINATION OF PARAMETER VALUES
The parameters of eqn. (9) are a,, bj, C and D. For lack of data of the asymptotic values represented by eqn. (2) the values of C and d or D
TABLE 1 Temperature and pressure ranges of the data used No.
1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18
Substance
No. of data points
Methane 79 n-Pentane 59 n-Heptane 103 n-Decane 91 Nitrogen 93 Oxygen 87 Carbon dioxide 36 Ammonia 119 F-12 32 Methyl acetate 32 Ethylene 10 Argon 31 Benzene 57 Toluene 55 p-Xylene 55 Cycle hexanone 55 Water 100 F-22 32
T range (K)
T, range
100 -190 310.9-444.3 277.6-510.9 310.9-410.9 75 -120 80 -150
0.525-0.997 0.661-0.945 0.514-0.946 0.503-0.825 0.59440.950 0.517-0.969
P range (0.1 MPa)
1 13.813.813.85 ~ 1 -
500 690 690 690 500 500
P, range
Data source
0.024-10.77 0.004-20.41 0.504-25.20 0.650-32.48 0.417-14.72 0.020- 9.82
a b ’ d a a
273.2-300 0.898-0.987 310 -390 0.764-0.961 253.2-313.2 0.568-0.813
40 - 600 0.5422 8.13 a 20 -1000 0.177-16.98 a 287 -1608 6.947-38.95 ’
253.2-313.2 0.490-0.606 273.1 0.966 90 -145 0.596-0.961 280 -550 0.498-0.978 303.2-323.2 0.512-0.631 303.2-323.2 0.492-0.524
197 -1567 50 - 700 5 - 200 1 - 500 2.0- 104 2.0- 104
303.2-323.2 0.482-0.514 273.2-423.2 0.422-0.948 253.2-313.2 0.686-0.848
2.0- 104 0.0533 2.74 ’ 1 -1000 0.005-11.64 ’ 287 -1596 5.760-32.07 ’
a Vargaftik (1975). b Sage and Lacey (1942). ’ Nichols (1955). d Reamer et al. (1942). ’ Kumagai and Iwasaki (1978). ’Experimentally determined by the present authors. g Bain (1964).
4.190-33.41 0.988-13.83 O.lOO- 4.00 0.020-10.16 0.050- 2.56 0.059- 3.00
e a a a
’ f
235
TABLE 2 Values of parameters a0
01
a2
99.42 6.502 -78.68
a3
a4
a5
bo
4
-75.18 41.49 7.257 0.38144 -0.30144
b2
C
D
-0.08457 2.810 1.1
cannot be determined using eqn. (3). In order to determine the values of the parameters of eqn. (9), a series of values for D were tried: namely, 1.0, 1.05, 1.1,1X and 1.2. For each of these values assigned to D, all other parameter values were determined by fitting eqn. (9) with experimental PVT data for the first ten liquid substances shown in Table 1, including elementary substances, and organic and inorganic, polar and non-polar compounds. The final D value, together with all other corresponding parameter values, as shown in Table 2, was determined by the best fit. Application of eqn. (9) requires either an experimental or a predicted value of Vs. For the present study, values of V, were estimated for all the substances by the modified Rackett equation (Spencer and Danner, 1972) v, = l/p, =
( ATJP,) Zz(* - T,)2/7
with the parameter Z, being determined by the method suggested by Campbell and Thodos (1985), or by fitting with experimental data (Table 3). TEST OF THE CORRELATION
Equation (9) with the parameter values shown in Table 2 was tested by comparing the volumes predicted with experimental values. Besides the ten substances mentioned above with respect to data fitting, the test was extended to eight other substances (Tables 1 and 3). For the purpose of comparison, similar calculations were also performed for the Yen-Woods correlation with T, < 1; and for the COSTALD correlation, which is limited to T, < 0.95. The COSTALD method was selected for comparison because of its high accuracy of prediction, while the Yen-Woods method was selected because of its relationship to eqn. (9) through the correlation of Lyderson et al. The calculation and comparison results are shown in Table 3. It can be seen that the overall average absolute percent deviation of 1126 data points is 0.96 for the present correlation (eqn. (9)) and 2.83 for the Yen-Woods correlation; and that of 1065 data points is 0.84 for eqn. (9) and 1.44 for the COSTALD correlation. For eqn. (9) and T, < 1, the overall average absolute percent deviation is 0.91 for the last eight substances, as compared with 0.99 for the first ten substances.
TABLE
3
0.324 0.086 0
32
n-Pentane n-Heptane
n-Decane Nitrogen
oxygen Carbon dioxide
Ammonia F-12 Methyl
acetate Ethylene
Argon Benzene Toluene
p-Xylene Cyclo-
2 3
4 5
6 7
8 9 10
11
12 13 14
15 16
-
1.26 2.13 0.48
0.2775
1.85 0.56 0.55 0.81 0.28
b
0.2699
0.2490
0.2680
b
0.2624
b
0.96
1.58 0.16
0.40 1.65
b
0.2553
b
b
b
b
b
0.60 0.92
0.53
0.2517
b
0.52 1.52 0.79
Eqn. (9)
2.83
4.38 2.48
6.20
2.91 1.81
1.66
1.55 1.39 1.19
1.89 1.00 3.74
4.62 7.89
1.31 1.16 1.20 1.74
Yen-Woods
Av. % deviation a
b
ZRA
T,) (Campbell and Tbodos, 1985). c Using true w values, because no COSTALD values available. d Total number of points.
b Calculated
0.443 0.348 0.215
0.210 0.257 0.324
- VexpI/V,,.
1126 d
100 32
55
55 55
57
10 31
0.351 0.490
103 91 93 87 0.040 0.021
0.007 0.251
79 59
a Percent deviation = 100 1V,, by Z RA = a + fl(1
17 Water 18 F-22 Overall av. % deviation
hexanone
0.225 0.250 0.176
36 119 32
Methane
w
1
T, -c 1
points
No. of
results
Substance
and comparison
No.
Calculation
1065 d
32
55 100
55
53 55
28
32 0
19 109 32
80
103 91 85
59
77
T. i 0.95
No. of points
0.84
0.16
0.28 1.58
0.81
0.45 0.48 0.55
0.40
2.05 0.48
1.07
0.46 0.80
0.79 0.53
1.52
0.44
Eqn. (9)
1.44
3.26 0.61
2.09
2.25
0.61 2.58
0.39
c
c
1.65 ’
1.40
0.69 2.25
1.00 0.71
1.08
0.97 0.70
0.57
COSTALD
Av. % deviation e
231
Equation (9) would also be applicable to liquid mixtures, given appropriate selection or formulation of the necessary mixing rules.
CONCLUSION
A new generalized Pm equation for predicting volumes or densities of compressed liquids has been proposed:
v= l/p=
v,*
A +
P-q
P, - P, ,)
A+C(P,-P,,,)’
where A =~~=oaiTr’and B= C&objwi, the parameter values being determined by fitting with experimental data (Table 2). As it is applicable to a wide range of substances with T, < 1, the equation proposed gives better prediction results than the Yen-Woods and COSTALD correlations with which it has been compared. It is an extremely simple method as compared with the high complexity of the Yen-Woods method; and a truly generalized method in the sense that it uses the true acentric factor, as opposed to the COSTALD method which requires the curve-fitted acentric factor.
LIST OF SYMBOLS
parameter in eqn. (9), function of temperature i = l-5 parameters in eqn. (8) parameter in eqn. (9), function of w j = O-2 parameters in eqn. (7) parameters in eqn. (9) D parameters in eqn. (1) F pressure, in units consistent with PC gas constant absolute temperature, in K volume of liquid, in units consistent with V, compressibility factor constant in Rackett equation GA
A
a,, B b,, C, E, P R T V Z
Greek fetters
a, B P
0
empirical constants for estimating 2, density, in units consistent with p, acentric factor
238
Subscripts C
cal exp r s
critical state calculated experimental reduced property saturation
REFERENCES Bain, R.W., 1964. Steam Tables. HMSO, London. Campbell, S.W. and Thodos, G., 1985. J. Chem. Eng. Data, 30: 102. Chueh, P.L. and Pratt&z, J.M., 1967. AIChE J., 13: 1099. Chueh, P.L. and Prausnitz, J.M., 1969. AIChE J., 15: 471. Gunn, R.D. and Yamada, T., 1971. AIChE J., 17: 1341. Gunn, RD. and Yamada, T., 1973. J. Chcm. Eng. Data, 18: 234. Kumagai, A. and Iwasaki, H., 1978. J. Chem. Eng. Data, 23: 193. Lu, B.C.-Y., 1959. Chem. Eng., 66: 137. Nichols, W.B., 1955. Ind. Eng. Chem., 47: 2219. Rackett, H.G., 1970. J. Chem. Eng. Data, 15: 514. Reamer, H.H., Olds, R.H., Sage, B.H. and Lacey, W.N., 1942. Ind. Eng. Chem., 34: 1526. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York. Sage, B.H. and Lacey, W.N., 1942. Ind. Eng. Chem., 34: 730. Smith, J.M. and Van Ness, H.C., 1975. Introduction to Chemical Engineering Tbcrmodynamics, 3rd edn. McGraw-Hill, New York. Spencer, C.F. and Adler, S.B., 1978. J. Chem. Eng. Data, 23: 82. Spencer, CF. and Danner, R.P., 1972. J. Chem. Eng. Data, 17: 236. Thomson, G.H., Brobst, K.R. and Hankinson, R.W., 1982. AIChE J., 28: 671. Vargaftik, N.B., 1975. Tables on the Thermophysical Properties of Liquids and Gases, 2nd edn. Hemisphere Publishing Corporation, Washington, DC. Veeranna, D. and Rihani, D.N., 1980. J. Chem. Eng. Data, 25: 267. Yen, L.C. and Woods, S.S., 1966. AIChE J., 12: 95.
231 Printed in The Netherlands
A NEW GENERALIZED EQUATION FOR PREDICTING VOLUMES OF COMPRESSED LIQUIDS CHIEN-HOU CHANG and XINGMIN ZHAO Department of Chemical Engineering, Tianjin University, Tianjin 3OGO72(China) (Received August 11,1989)
ABSTRACT Chang, C.-H. and Zhao, X.-M., 1990. A new generalized equation for predicting volumes of compressed liquids. Fluid Phase Equilibria, 58: 231-238. On the basis of reduced PVT data of liquids in the literature, a new generalized PVT equation for predicting volumes or densities of compressed liquids is proposed: A+ C’D-“‘B(P, - P,,,) V=l/p=v,. A + C(P, - P,.,) where A = C:woaiT: and B = E:_,,b#. AI1 the parameter values are determined by fitting with experimental data. In comparing the predicted volumes with experimental values, the overall average absolute percent deviation of 1126 data points is 0.96 for the equation proposed and 2.83 for the Yen-Woods correlation; and that of 1065 data points is 0.84 for the equation proposed and 1.44 for the COSTALD correlation.
INTRODUCTION
Various equations have been suggested for estimating volumes or densities of liquids. These equations can be divided into two groups: those for saturated liquids, and those for compressed liquids. Among the equations of the first group, the Gunn-Yamada equation (Gunn and Yamada, 1971, 1973) and the modified Rackett equation (Rackett, 1970; Spencer and Danner, 1972) are noted for their higher accuracy of calculation (Spencer and Danner, 1972; Spencer and Adler, 1978; Veeranna and Rihani, 1980; Reid et al., 1977), the former giving slightly better results. Recently, the modified Rackett equation has been further improved by Campbell and Thodos (1985). Representative of the correlations for compressed liquids are the Tait equation (Thomson et al., 1982), the Lu correlation (Lu, 1959), the Yen-Woods equation (Yen and Woods, 1966), the Chueh-Prausnitz equation (Chueh and Prausnitz, 1967, 1969) and the COSTALD correlation 0378-3812/90/$03.50 0 1990 - Else&r Science Publishers B.V.
232
(Thomson et al., 1982). The Yen-Woods and Chueh-Prausnitz equations have been extensively tested and shown to have comparable accuracy, but both are very complex, with numerous coefficients. The COSTALD correlation has the higher accuracy. The present work proposes a new generalized equation to improve the prediction of volumes or densities of compressed liquids. DEVELOPMENT OF THE CORRELATION EQUATION
The Yen-Woods equation is the analytical form based on the tabulated corresponding-states correlation developed by Lyderson et al. (Yen and Woods, 1966) for estimating pure-liquid densities at any pressure and at temperatures below T, = 1 in terms of T,, P, and the critical-compressibility factor 2,. Based on the same data, a generalized diagrammatic correlation of plotting pr versus P, at different constant T, has been presented (Smith and Van Ness, 1975). A new diagrammatic correlation with saturation condition as the point of reference can be obtained from the same data by changing the coordinates from p, versus P, to p,/p,, versus (P, - P,,,), as shown in Fig. 1. The curves in Fig. 1 all start from the point (0, l.O), are all concave downward, and all tend to approach asymptotic values as (P, - P,,,) increases to high values. These characteristics suggest that the family of curves may be represented by an equation of the form
where A, E and I; are coefficients.
0
I
2
3
4
5
6
7
6
9
IO
Pr-Ps,r
Fig. 1. Generalized diagrammatic correlation for liquids with saturation condition as points of reference.
233
It can be seen from eqn. (1) that, for any temperature, Q’V= 1 when P, = P,.,, and
3 _ F&-P,,,) F V - E( P, - P,,,) = 3
(2)
if P, increases to values high enough that coefficient A becomes relatively negligible. F/E is the asymptotic value and, as shown in the figure, a function of temperature. In order to express the asymptotic value F/E as a function of temperature, a reference reduced temperature which may be only hypothetical is taken at T, = d, at which the asymptotic value F/E can be approximated as equal to unity, and an exponential function of T, may be assumed to express F/E approximately as follows: F/E
= C@
(3)
where AT, = T, - d and C is an appropriate constant to satisfy the functional relation. Although F/E as a ratio is a function of q, either F or E may for simplicity be kept constant, with the other varying with temperature. So F is here chosen to be a constant and, to reduce the number of parameters, arbitrarily set to equal C, which also thus determines the magnitude of E. Then we obtain C/E = CoTc
(4)
and E at any temperature is determined by the following form, rearranged from eqn. (4): E = C” -AT,)= c(D-r,)
(5)
where D = 1 + d. For the purpose of generalizing and improving the functional relation, an exponent is introduced into eqn. (5), as follows: E = C(D-T,JB
(6)
where B is a function of the acentric factor of each substance and has the polynomial form B = cbjco’
(7)
To express coefficient A as a function of temperature, a polynomial is again adopted:
For prediction of V, eqn. (1) can be rearranged with F substituted by C and E by eqn. (6):
where C and D are constants, A is a function of temperature as expressed by eqn. (8), and B is a function of the acentric factor as shown by eqn. (7). DETERMINATION OF PARAMETER VALUES
The parameters of eqn. (9) are a,, bj, C and D. For lack of data of the asymptotic values represented by eqn. (2) the values of C and d or D
TABLE 1 Temperature and pressure ranges of the data used No.
1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18
Substance
No. of data points
Methane 79 n-Pentane 59 n-Heptane 103 n-Decane 91 Nitrogen 93 Oxygen 87 Carbon dioxide 36 Ammonia 119 F-12 32 Methyl acetate 32 Ethylene 10 Argon 31 Benzene 57 Toluene 55 p-Xylene 55 Cycle hexanone 55 Water 100 F-22 32
T range (K)
T, range
100 -190 310.9-444.3 277.6-510.9 310.9-410.9 75 -120 80 -150
0.525-0.997 0.661-0.945 0.514-0.946 0.503-0.825 0.59440.950 0.517-0.969
P range (0.1 MPa)
1 13.813.813.85 ~ 1 -
500 690 690 690 500 500
P, range
Data source
0.024-10.77 0.004-20.41 0.504-25.20 0.650-32.48 0.417-14.72 0.020- 9.82
a b ’ d a a
273.2-300 0.898-0.987 310 -390 0.764-0.961 253.2-313.2 0.568-0.813
40 - 600 0.5422 8.13 a 20 -1000 0.177-16.98 a 287 -1608 6.947-38.95 ’
253.2-313.2 0.490-0.606 273.1 0.966 90 -145 0.596-0.961 280 -550 0.498-0.978 303.2-323.2 0.512-0.631 303.2-323.2 0.492-0.524
197 -1567 50 - 700 5 - 200 1 - 500 2.0- 104 2.0- 104
303.2-323.2 0.482-0.514 273.2-423.2 0.422-0.948 253.2-313.2 0.686-0.848
2.0- 104 0.0533 2.74 ’ 1 -1000 0.005-11.64 ’ 287 -1596 5.760-32.07 ’
a Vargaftik (1975). b Sage and Lacey (1942). ’ Nichols (1955). d Reamer et al. (1942). ’ Kumagai and Iwasaki (1978). ’Experimentally determined by the present authors. g Bain (1964).
4.190-33.41 0.988-13.83 O.lOO- 4.00 0.020-10.16 0.050- 2.56 0.059- 3.00
e a a a
’ f
235
TABLE 2 Values of parameters a0
01
a2
99.42 6.502 -78.68
a3
a4
a5
bo
4
-75.18 41.49 7.257 0.38144 -0.30144
b2
C
D
-0.08457 2.810 1.1
cannot be determined using eqn. (3). In order to determine the values of the parameters of eqn. (9), a series of values for D were tried: namely, 1.0, 1.05, 1.1,1X and 1.2. For each of these values assigned to D, all other parameter values were determined by fitting eqn. (9) with experimental PVT data for the first ten liquid substances shown in Table 1, including elementary substances, and organic and inorganic, polar and non-polar compounds. The final D value, together with all other corresponding parameter values, as shown in Table 2, was determined by the best fit. Application of eqn. (9) requires either an experimental or a predicted value of Vs. For the present study, values of V, were estimated for all the substances by the modified Rackett equation (Spencer and Danner, 1972) v, = l/p, =
( ATJP,) Zz(* - T,)2/7
with the parameter Z, being determined by the method suggested by Campbell and Thodos (1985), or by fitting with experimental data (Table 3). TEST OF THE CORRELATION
Equation (9) with the parameter values shown in Table 2 was tested by comparing the volumes predicted with experimental values. Besides the ten substances mentioned above with respect to data fitting, the test was extended to eight other substances (Tables 1 and 3). For the purpose of comparison, similar calculations were also performed for the Yen-Woods correlation with T, < 1; and for the COSTALD correlation, which is limited to T, < 0.95. The COSTALD method was selected for comparison because of its high accuracy of prediction, while the Yen-Woods method was selected because of its relationship to eqn. (9) through the correlation of Lyderson et al. The calculation and comparison results are shown in Table 3. It can be seen that the overall average absolute percent deviation of 1126 data points is 0.96 for the present correlation (eqn. (9)) and 2.83 for the Yen-Woods correlation; and that of 1065 data points is 0.84 for eqn. (9) and 1.44 for the COSTALD correlation. For eqn. (9) and T, < 1, the overall average absolute percent deviation is 0.91 for the last eight substances, as compared with 0.99 for the first ten substances.
TABLE
3
0.324 0.086 0
32
n-Pentane n-Heptane
n-Decane Nitrogen
oxygen Carbon dioxide
Ammonia F-12 Methyl
acetate Ethylene
Argon Benzene Toluene
p-Xylene Cyclo-
2 3
4 5
6 7
8 9 10
11
12 13 14
15 16
-
1.26 2.13 0.48
0.2775
1.85 0.56 0.55 0.81 0.28
b
0.2699
0.2490
0.2680
b
0.2624
b
0.96
1.58 0.16
0.40 1.65
b
0.2553
b
b
b
b
b
0.60 0.92
0.53
0.2517
b
0.52 1.52 0.79
Eqn. (9)
2.83
4.38 2.48
6.20
2.91 1.81
1.66
1.55 1.39 1.19
1.89 1.00 3.74
4.62 7.89
1.31 1.16 1.20 1.74
Yen-Woods
Av. % deviation a
b
ZRA
T,) (Campbell and Tbodos, 1985). c Using true w values, because no COSTALD values available. d Total number of points.
b Calculated
0.443 0.348 0.215
0.210 0.257 0.324
- VexpI/V,,.
1126 d
100 32
55
55 55
57
10 31
0.351 0.490
103 91 93 87 0.040 0.021
0.007 0.251
79 59
a Percent deviation = 100 1V,, by Z RA = a + fl(1
17 Water 18 F-22 Overall av. % deviation
hexanone
0.225 0.250 0.176
36 119 32
Methane
w
1
T, -c 1
points
No. of
results
Substance
and comparison
No.
Calculation
1065 d
32
55 100
55
53 55
28
32 0
19 109 32
80
103 91 85
59
77
T. i 0.95
No. of points
0.84
0.16
0.28 1.58
0.81
0.45 0.48 0.55
0.40
2.05 0.48
1.07
0.46 0.80
0.79 0.53
1.52
0.44
Eqn. (9)
1.44
3.26 0.61
2.09
2.25
0.61 2.58
0.39
c
c
1.65 ’
1.40
0.69 2.25
1.00 0.71
1.08
0.97 0.70
0.57
COSTALD
Av. % deviation e
231
Equation (9) would also be applicable to liquid mixtures, given appropriate selection or formulation of the necessary mixing rules.
CONCLUSION
A new generalized Pm equation for predicting volumes or densities of compressed liquids has been proposed:
v= l/p=
v,*
A +
P-q
P, - P, ,)
A+C(P,-P,,,)’
where A =~~=oaiTr’and B= C&objwi, the parameter values being determined by fitting with experimental data (Table 2). As it is applicable to a wide range of substances with T, < 1, the equation proposed gives better prediction results than the Yen-Woods and COSTALD correlations with which it has been compared. It is an extremely simple method as compared with the high complexity of the Yen-Woods method; and a truly generalized method in the sense that it uses the true acentric factor, as opposed to the COSTALD method which requires the curve-fitted acentric factor.
LIST OF SYMBOLS
parameter in eqn. (9), function of temperature i = l-5 parameters in eqn. (8) parameter in eqn. (9), function of w j = O-2 parameters in eqn. (7) parameters in eqn. (9) D parameters in eqn. (1) F pressure, in units consistent with PC gas constant absolute temperature, in K volume of liquid, in units consistent with V, compressibility factor constant in Rackett equation GA
A
a,, B b,, C, E, P R T V Z
Greek fetters
a, B P
0
empirical constants for estimating 2, density, in units consistent with p, acentric factor
238
Subscripts C
cal exp r s
critical state calculated experimental reduced property saturation
REFERENCES Bain, R.W., 1964. Steam Tables. HMSO, London. Campbell, S.W. and Thodos, G., 1985. J. Chem. Eng. Data, 30: 102. Chueh, P.L. and Pratt&z, J.M., 1967. AIChE J., 13: 1099. Chueh, P.L. and Prausnitz, J.M., 1969. AIChE J., 15: 471. Gunn, R.D. and Yamada, T., 1971. AIChE J., 17: 1341. Gunn, RD. and Yamada, T., 1973. J. Chcm. Eng. Data, 18: 234. Kumagai, A. and Iwasaki, H., 1978. J. Chem. Eng. Data, 23: 193. Lu, B.C.-Y., 1959. Chem. Eng., 66: 137. Nichols, W.B., 1955. Ind. Eng. Chem., 47: 2219. Rackett, H.G., 1970. J. Chem. Eng. Data, 15: 514. Reamer, H.H., Olds, R.H., Sage, B.H. and Lacey, W.N., 1942. Ind. Eng. Chem., 34: 1526. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York. Sage, B.H. and Lacey, W.N., 1942. Ind. Eng. Chem., 34: 730. Smith, J.M. and Van Ness, H.C., 1975. Introduction to Chemical Engineering Tbcrmodynamics, 3rd edn. McGraw-Hill, New York. Spencer, C.F. and Adler, S.B., 1978. J. Chem. Eng. Data, 23: 82. Spencer, CF. and Danner, R.P., 1972. J. Chem. Eng. Data, 17: 236. Thomson, G.H., Brobst, K.R. and Hankinson, R.W., 1982. AIChE J., 28: 671. Vargaftik, N.B., 1975. Tables on the Thermophysical Properties of Liquids and Gases, 2nd edn. Hemisphere Publishing Corporation, Washington, DC. Veeranna, D. and Rihani, D.N., 1980. J. Chem. Eng. Data, 25: 267. Yen, L.C. and Woods, S.S., 1966. AIChE J., 12: 95.