- Email: [email protected]
A semi-theoretical viscosity model for non-polar liquids
The Chemical
Engineering
Journal,
163
47 (1991) 163-167
A semi-theoretical viscosity model for non-polar liquids L. G. Du and T. M. Guo* Universityof Petroleum, P.O. Box 902, Beiiing
loo083 (China)
(Received November 23, 1990; in final form May 28, 1991)
Abstract An empiricalradialdistributionfunctionwas developedbased on the experimentalP-V-T and viscosity data. The theoreticalviscosity equationdevelopedby Davis et al. (1961) for a model fluid with molecules, interact according to a square-wellpotential, when combined with the proposedradialdistributionfunctionand the generalizedcorrelationsfor square-wellparameters, has been successfullyextendedto the calculationof the viscositiesof non-polarliquids,including heavy hydrocarbons.
to the following pair potential:
1. Introduction
Davis et al. [l] presented a kinetic theory
V(r,) = 0
rij > a2
for the transport properties of a model liquid with molecules interacting according to a square-well potential. The theory results in
V(r,> = - E a, < rij I a2
explicit equations for the transport properties, which can be calculated from the square-well potential parameters and radial distribution function of the liquid. Davis and Luks [2] showed that the square-well equations can describe the viscosity and thermal conductivity of liquid argon over the entire liquid range. Based on the radial distribution function developed by Lowry et al. [ 31using a perturbation technique, Luks et al. [4] successfully applied the square-well equations to predict the transport properties of noble liquids. The major objective of this work is to extend the application of the above-mentioned theoretical viscosity equation to non-polar hydrocarbon liquids. For this purpose, an empirical distribution function was established on the basis of experimental P-V-T and viscosity data, and generalized correlations for evaluating square-well parameters were developed.
where rij is the distance of separation of molecules i and j, E is the well depth, and al and a2 are the inner and outer boundaries of the square-well potential respectively. Based on the argument that only binary collisions are important in a square-well fluid, Davis et al. [ 1] derived a modified Maxwell-Boltzmann integral differential equation for this fluid, in which only binary collisions (occurring at rij = aI or rij = ~2) contribute to the dissipative process. The integral differential equation, when solved, eventually leads to the following formulae for the equation of state and shear viscosity.
2. Viscosity of a square-well
77=77*
fluid
A square-well fluid is defined as one composed of N molecules which interact according
(1)
V(r,> = + C0 rij < CT1
Pressure p=
pkT{l + Wg(uI) +J%(u2>(1- exp(Jkr))]) (2)
Shear viscosity
t
1+ Wb(Ul) +R3g(u2>@12 g(uJ +fh(uz) W+ HdQ2 1 (3)
*Author to whom correspondence should be addressed.
Elsevier Sequoia, Laussnne
164
with
TABLE 1. Square-well potential parameters for 13 substances*
(4)
$q3
b=
(5)
Substance
01 @I
R
dk o(>
Ar Xe
3.162 3.760 3.917 3.299 3.400 3.535 4.418 4.812 6.397 3.347 4.316 4.103 5.422
1.85 1.85 1.83 1.87 1.85 1.652 1.464 1.476 1.314 1.677 1.460 1.480 1.450
69.4 127.7 119.0 53.7 88.8 244.0 347.0 387.0 629.0 222.0 339.0 191.1 382.6
co2
R= u2/q
(6)
and g(aJ and g(a2) refer to the radial distribution functions at the distance (TVand a2 respectively. The functions H and Q, are detined as
s m
eT
H=exp(r/kZJ-
-2
X2
0
X(x2 + JkT)ln
exp( -x2)
“&uoted from Reed and Gubbins 151.
dx
s
(7)
m
I+
5
exp(JW
(8)
(elkT)‘/2
X
exp( -x2)x2
NZ CK W, C,H&l n-WI0 n-C&, CA W% CF, WW,
1
dx
parameter values determined depend on the experimental data chosen. Table 1 lists the parameter values obtained from the second viral coefficient for 13 substances (taken from Reed and Gubbins [ 5 I). Based on these values the following generalized correlations for evaluating square-well parameters were established by the authors a, =(1.94131+
1.20678~)[P,/(101.3T,)]-l/3
Since it is cumbersome to evaluate II and
Qi from eqns. (7) and (S), the following simplified correlations were developed by authors to approximate the functions
(11) R= 1.2+exp[
-(0.387740+5.11067zu)]
(12) dk=T,
- [1.05885+0.289331(ElkT)] @=
1 - exp(dkT)
+ gT
(9)
[(1.93245
+0.442521)(e/kT)“a+0.179515(e/kT)] (10) In the range of (JkT) from 0.1 to 2.5, the calculated values of H and @ from eqns. (9) and (10) match closely to those calculated from eqns. (7) and (8); the deviations are within &-0.5%. 3. Square-well
exp[ -(0.101465+0.0015382)/(&w)]
parameters
Square-well parameters (vi, R and E) can be evaluated from second viral coefficient and transport properties of gases. However, the
(13) where P, is the critical pressure in kPa, T, the critical temperature in K, 2, the critical compressibility, and w is the acentric factor. The absolute average deviations of the calculated parameters from above correlations are: 3.9% for a,, 4.6% for R and 1.7% for e/k respectively.
4. Determination function
of radial
distribution
The radial distribution function g(r) can be measured experimentally by X-ray or neutron diffraction techniques. A typical experimental radial distribution curve for argon reported by Mikolaj and Pings [S] is shown in Pig. 1: the state conditions are T= - 125 “C and p = 0.982 g cmW3. The distribution curve indicates that,
165
2
4
6
8 ~1
10
where a and r,, are the two parameters to be determined from experimental data. Based on P-V-T and viscosity data, values of a and rb were determined from eqns. (2) and (3) by using the Newton-Raphson iteration technique. It was found that rb is almost a constant for a specific substance in its whole liquid region. Parameter a varies with both temperature and density. The plot of ?-b against ~~(1 +R)a,/2 is shown in Pig. 2, which can be approximately represented by the following formula
12
0 A
Fig. 1. Experimental radial distribution function for argon, - 125 “C, 0.982 g cmd3 161.
at large separations, g(r) approaches unity, which means that the long-range structure is random. We assume that the distribution function can be approximated by the following expression: g(T, p, r)=a
r,=
-0.902012+0.51557(1
u=exp{7.1981+(2.7788-0.32772a,)a, - (26.278 - 24.688~)
- (6.8468 + 20.366w)w]
14
12.
IO-
6.
4
6
8 a,(1
Fig. 2. The plot of r,, VS. ~,(l +R)/2.
10
+R)/2
(15)
The values of the parameter a determined from 25 substances were correlated by the following generalized function:
- [(1.7032+0.11952a,)ff,
exp[-(r-rJ2]+exp(-l/r) (14)
2
+R)q
12
14
In(p)
166 TABLE 2. Test results on liquid viscosity calculations
Substance
TR range (R)
NP
P range
Ref.
AAD (96)
(bar) This work
Jossi et al.
Przezdziecki et al.
26.84 9.02 24.01 16.03
14.12 32.64 17.43 28.01
17.40 28.44 5.00 4.39
Methane i-Butane n-Butane n-Pentane
110 60 7 (:;‘)*
0.52-0.94 0.76-0.93 0.50-0.64 0.33-0.65
1.0-500.0 6.9-551.6 1.013 1.013
n-Heptane i-Octane n-Octane
25 17 (&
0.54-0.99 0.69-0.98 0.48-0.97
0.05-24.7 1.04-22.3 0.004-20.0
3.29 3.30 3.54
27.72 54.23 24.89
10.97 17.67 12.92
n-Nonsne
(::)
0.44-0.71
1 .OO-22.3
3.30
10.17
29.52
la1
4.73
26.11
40.65
[Cl
13.8-551.6
136
0.50-0.83
n-Decane n-Dodecane
(104) 5 11
0.50-0.83 0.41-0.57
1.013 1.013
4.65 9.87
30.40
40.65 41.34
Ial [al
n-Tridecane
(3) 16
0.40-0.70
1.013
6.28
33.37
45.27
Ial
n-Tetradecane
(7) 10
0.41-0.54
1.013
12.53
33.12
48.51
Ial
n-Pentadecane n-Hexadecane
(1) 9 16
0.41-0.53 0.4 l-O.72
1.013 1.013
8.57 6.36
35.25
54.03 54.01
n-Heptadecane
(6) 18
0.4 l-O.78
1.013
7.32
32.63
57.08
n-Octadecane n-Eichosane
(7) 8 13
0.41-0.50 0.4 l-O.64
1.013 1.013
6.91 9.07
29.74
63.42 62.93
2,2-Dimethyl-propane Cyclopentane Methyl-cyclopentane Cyclohexane Benzene Toluene
(2) 77 5 7 7 42 10
0.72-0.95 0.5SO.61 0.49-0.59 0.53-0.64 0.62-0.98 0.46-0.63
5.9-551.6 1.013 1.013 1.013 0.09-42.2 1.013
5.71 5.35 10.62 24.37 4.29 29.60
56.14 37.17 35.24 57.55 33.41 5.88
21.78 31.79 42.41 24.91 22.20 18.37
o-Xylene
(8) (Z)
0.43-0.66
1.013
5.82
1.68
2.58
oxygen Nitrogen
140 107
0.49-0.91 0.52-0.87
1.0-300.0 1.0-300.0
5.42 8.21
29.15 33.30
28.85 49.08
9.27
30.22
31.12
n-Decane
Overall AAD (%)
b-4
Ial [al
*Numbers in parentheses indicate the data points tested for Jossi et al., correlation within its application range. Data sources: [a] N. B. Vargaftik, Tables on the ZYmrnwphysical Properties of Liquids and Gases, Hemisphere, Washington, DC, 1975. [b] M. H. Gonzalez and A. L. Lee, J. Churn. Eng. Data, 11, 3 (1966) 357. [c] A. L. Lee and R. T. Elhngton, J. Chem. Eng. Data, IO, 4 (1965) 346. [d] M. H. Gonzalez and A. L. Lee, J. C&em. Eng. Data, 13, 1 (1968) 66.
6. Results
-[(0.76340-O.l9473a,)oj
+(0.70038+11.6512u)20]p + 0.80203~ + 1.554 In@)}(E/kT,
(16)
The proposed semi-theoretical viscosity model has been tested on the liquidviscosity data of 25 non-polar substances (a total of 931 data points); the test results are listed in
167 3 B. A. Lowry, H. T. Davis and S. A. Rice, Phys. Fluids,
Table 2. The critical properties were taken from Reid et al. [7], except that the critical compressibility factors for heavy hydrocarbons (with w>O.65) were evaluated from the following correlation 2, = 0.291 - 0.08w
7 (1964) 402. 4 K. D. Luks, M. A. Miller and H. T. Davis, AIChE J., 12, 6 (1966) 1079-1086. 5 T. M. Reed and K. E. Gubbins, Applied Statistical Mechanics, McGraw-Hill, New York, 1973. 6 P. G. Mikolqj and C. J. Pings, J. Chin. Phys., 46 (1967) 1401. 7 R. C. Reid, J. M. Prausnitz and B. E. Poling, The Properties of Gases and Liquids, 4th ed., McGrawHill, New York, 1987. 8 A. J. Jo&, L. I. Stiel and G. Thodos, AIChE J., 8, 1 (1962) 59-63. 9 J. W. Przezdziecki and T. Sridhar, AIChE J., 31 (1985) 333.
(17)
The overall average deviation of the calculated viscosities for 25 non-polar liquids (including heavy hydrocarbons up to C,,) is 9.27%. For comparison, the calculation results obtamed by using the correlations of Jossi et al. [ 81 and Przezdziecki and Sridhar [9] are also given in Table 2. The superiority of the new viscosity model for the substances tested is obvious.
Appendix
a b
6. Conclusion
m
H k
The theoretical viscosity equation derived by Davis et al. [l] for a model fluid with a square-well potential, when coupled with the empirical radial distribution function and generalized correlations for square-well parameters developed in this work, has been successfully extended to the calculation of viscosities of non-polar liquids (including heavy hydrocarbons up to C,,).
Acknowledgment
N”
PP PC R ?-i.f ?+b
T TC V(qjl W
Financial support from the China National Petroleum and Natural Gas Corporation for this work is gratefully acknowledged.
References 1 H. T. Davis, S. A. Rice and J. V. Sengers, J. Cha. Phys., 35 (1961) 2210-2233. 2 H. T. Davis and K. D. Luks, J. Phys. Chem., 69 (1965) 869-880.
X 2,
A: Nomenclature
parameter in eqn. (14) parameter defined in eqn. (5) radial distribution function function defied by eqn. (7) Boltzmann constant (k = 1.3805 X 1O-23 J K-r) mass of particle (kg) number of experimental data points pressure (kPa) critical pressure &Pa) defined by eqn. (6) distance of separation between molecules i and j parameter in eqn. (14) temperature (K) critical temperature (K) intermolecular potential acentric factor =rr2/aI critical compressibility depth of square-well potential shear viscosity (pP> de6ned by eqn. (4) number density (km01 rne3) inner and outer boundaries of squarewell potential (A, m) function de8ned by eqn. (8)
Engineering
Journal,
163
47 (1991) 163-167
A semi-theoretical viscosity model for non-polar liquids L. G. Du and T. M. Guo* Universityof Petroleum, P.O. Box 902, Beiiing
loo083 (China)
(Received November 23, 1990; in final form May 28, 1991)
Abstract An empiricalradialdistributionfunctionwas developedbased on the experimentalP-V-T and viscosity data. The theoreticalviscosity equationdevelopedby Davis et al. (1961) for a model fluid with molecules, interact according to a square-wellpotential, when combined with the proposedradialdistributionfunctionand the generalizedcorrelationsfor square-wellparameters, has been successfullyextendedto the calculationof the viscositiesof non-polarliquids,including heavy hydrocarbons.
to the following pair potential:
1. Introduction
Davis et al. [l] presented a kinetic theory
V(r,) = 0
rij > a2
for the transport properties of a model liquid with molecules interacting according to a square-well potential. The theory results in
V(r,> = - E a, < rij I a2
explicit equations for the transport properties, which can be calculated from the square-well potential parameters and radial distribution function of the liquid. Davis and Luks [2] showed that the square-well equations can describe the viscosity and thermal conductivity of liquid argon over the entire liquid range. Based on the radial distribution function developed by Lowry et al. [ 31using a perturbation technique, Luks et al. [4] successfully applied the square-well equations to predict the transport properties of noble liquids. The major objective of this work is to extend the application of the above-mentioned theoretical viscosity equation to non-polar hydrocarbon liquids. For this purpose, an empirical distribution function was established on the basis of experimental P-V-T and viscosity data, and generalized correlations for evaluating square-well parameters were developed.
where rij is the distance of separation of molecules i and j, E is the well depth, and al and a2 are the inner and outer boundaries of the square-well potential respectively. Based on the argument that only binary collisions are important in a square-well fluid, Davis et al. [ 1] derived a modified Maxwell-Boltzmann integral differential equation for this fluid, in which only binary collisions (occurring at rij = aI or rij = ~2) contribute to the dissipative process. The integral differential equation, when solved, eventually leads to the following formulae for the equation of state and shear viscosity.
2. Viscosity of a square-well
77=77*
fluid
A square-well fluid is defined as one composed of N molecules which interact according
(1)
V(r,> = + C0 rij < CT1
Pressure p=
pkT{l + Wg(uI) +J%(u2>(1- exp(Jkr))]) (2)
Shear viscosity
t
1+ Wb(Ul) +R3g(u2>@12 g(uJ +fh(uz) W+ HdQ2 1 (3)
*Author to whom correspondence should be addressed.
Elsevier Sequoia, Laussnne
164
with
TABLE 1. Square-well potential parameters for 13 substances*
(4)
$q3
b=
(5)
Substance
01 @I
R
dk o(>
Ar Xe
3.162 3.760 3.917 3.299 3.400 3.535 4.418 4.812 6.397 3.347 4.316 4.103 5.422
1.85 1.85 1.83 1.87 1.85 1.652 1.464 1.476 1.314 1.677 1.460 1.480 1.450
69.4 127.7 119.0 53.7 88.8 244.0 347.0 387.0 629.0 222.0 339.0 191.1 382.6
co2
R= u2/q
(6)
and g(aJ and g(a2) refer to the radial distribution functions at the distance (TVand a2 respectively. The functions H and Q, are detined as
s m
eT
H=exp(r/kZJ-
-2
X2
0
X(x2 + JkT)ln
exp( -x2)
“&uoted from Reed and Gubbins 151.
dx
s
(7)
m
I+
5
exp(JW
(8)
(elkT)‘/2
X
exp( -x2)x2
NZ CK W, C,H&l n-WI0 n-C&, CA W% CF, WW,
1
dx
parameter values determined depend on the experimental data chosen. Table 1 lists the parameter values obtained from the second viral coefficient for 13 substances (taken from Reed and Gubbins [ 5 I). Based on these values the following generalized correlations for evaluating square-well parameters were established by the authors a, =(1.94131+
1.20678~)[P,/(101.3T,)]-l/3
Since it is cumbersome to evaluate II and
Qi from eqns. (7) and (S), the following simplified correlations were developed by authors to approximate the functions
(11) R= 1.2+exp[
-(0.387740+5.11067zu)]
(12) dk=T,
- [1.05885+0.289331(ElkT)] @=
1 - exp(dkT)
+ gT
(9)
[(1.93245
+0.442521)(e/kT)“a+0.179515(e/kT)] (10) In the range of (JkT) from 0.1 to 2.5, the calculated values of H and @ from eqns. (9) and (10) match closely to those calculated from eqns. (7) and (8); the deviations are within &-0.5%. 3. Square-well
exp[ -(0.101465+0.0015382)/(&w)]
parameters
Square-well parameters (vi, R and E) can be evaluated from second viral coefficient and transport properties of gases. However, the
(13) where P, is the critical pressure in kPa, T, the critical temperature in K, 2, the critical compressibility, and w is the acentric factor. The absolute average deviations of the calculated parameters from above correlations are: 3.9% for a,, 4.6% for R and 1.7% for e/k respectively.
4. Determination function
of radial
distribution
The radial distribution function g(r) can be measured experimentally by X-ray or neutron diffraction techniques. A typical experimental radial distribution curve for argon reported by Mikolaj and Pings [S] is shown in Pig. 1: the state conditions are T= - 125 “C and p = 0.982 g cmW3. The distribution curve indicates that,
165
2
4
6
8 ~1
10
where a and r,, are the two parameters to be determined from experimental data. Based on P-V-T and viscosity data, values of a and rb were determined from eqns. (2) and (3) by using the Newton-Raphson iteration technique. It was found that rb is almost a constant for a specific substance in its whole liquid region. Parameter a varies with both temperature and density. The plot of ?-b against ~~(1 +R)a,/2 is shown in Pig. 2, which can be approximately represented by the following formula
12
0 A
Fig. 1. Experimental radial distribution function for argon, - 125 “C, 0.982 g cmd3 161.
at large separations, g(r) approaches unity, which means that the long-range structure is random. We assume that the distribution function can be approximated by the following expression: g(T, p, r)=a
r,=
-0.902012+0.51557(1
u=exp{7.1981+(2.7788-0.32772a,)a, - (26.278 - 24.688~)
- (6.8468 + 20.366w)w]
14
12.
IO-
6.
4
6
8 a,(1
Fig. 2. The plot of r,, VS. ~,(l +R)/2.
10
+R)/2
(15)
The values of the parameter a determined from 25 substances were correlated by the following generalized function:
- [(1.7032+0.11952a,)ff,
exp[-(r-rJ2]+exp(-l/r) (14)
2
+R)q
12
14
In(p)
166 TABLE 2. Test results on liquid viscosity calculations
Substance
TR range (R)
NP
P range
Ref.
AAD (96)
(bar) This work
Jossi et al.
Przezdziecki et al.
26.84 9.02 24.01 16.03
14.12 32.64 17.43 28.01
17.40 28.44 5.00 4.39
Methane i-Butane n-Butane n-Pentane
110 60 7 (:;‘)*
0.52-0.94 0.76-0.93 0.50-0.64 0.33-0.65
1.0-500.0 6.9-551.6 1.013 1.013
n-Heptane i-Octane n-Octane
25 17 (&
0.54-0.99 0.69-0.98 0.48-0.97
0.05-24.7 1.04-22.3 0.004-20.0
3.29 3.30 3.54
27.72 54.23 24.89
10.97 17.67 12.92
n-Nonsne
(::)
0.44-0.71
1 .OO-22.3
3.30
10.17
29.52
la1
4.73
26.11
40.65
[Cl
13.8-551.6
136
0.50-0.83
n-Decane n-Dodecane
(104) 5 11
0.50-0.83 0.41-0.57
1.013 1.013
4.65 9.87
30.40
40.65 41.34
Ial [al
n-Tridecane
(3) 16
0.40-0.70
1.013
6.28
33.37
45.27
Ial
n-Tetradecane
(7) 10
0.41-0.54
1.013
12.53
33.12
48.51
Ial
n-Pentadecane n-Hexadecane
(1) 9 16
0.41-0.53 0.4 l-O.72
1.013 1.013
8.57 6.36
35.25
54.03 54.01
n-Heptadecane
(6) 18
0.4 l-O.78
1.013
7.32
32.63
57.08
n-Octadecane n-Eichosane
(7) 8 13
0.41-0.50 0.4 l-O.64
1.013 1.013
6.91 9.07
29.74
63.42 62.93
2,2-Dimethyl-propane Cyclopentane Methyl-cyclopentane Cyclohexane Benzene Toluene
(2) 77 5 7 7 42 10
0.72-0.95 0.5SO.61 0.49-0.59 0.53-0.64 0.62-0.98 0.46-0.63
5.9-551.6 1.013 1.013 1.013 0.09-42.2 1.013
5.71 5.35 10.62 24.37 4.29 29.60
56.14 37.17 35.24 57.55 33.41 5.88
21.78 31.79 42.41 24.91 22.20 18.37
o-Xylene
(8) (Z)
0.43-0.66
1.013
5.82
1.68
2.58
oxygen Nitrogen
140 107
0.49-0.91 0.52-0.87
1.0-300.0 1.0-300.0
5.42 8.21
29.15 33.30
28.85 49.08
9.27
30.22
31.12
n-Decane
Overall AAD (%)
b-4
Ial [al
*Numbers in parentheses indicate the data points tested for Jossi et al., correlation within its application range. Data sources: [a] N. B. Vargaftik, Tables on the ZYmrnwphysical Properties of Liquids and Gases, Hemisphere, Washington, DC, 1975. [b] M. H. Gonzalez and A. L. Lee, J. Churn. Eng. Data, 11, 3 (1966) 357. [c] A. L. Lee and R. T. Elhngton, J. Chem. Eng. Data, IO, 4 (1965) 346. [d] M. H. Gonzalez and A. L. Lee, J. C&em. Eng. Data, 13, 1 (1968) 66.
6. Results
-[(0.76340-O.l9473a,)oj
+(0.70038+11.6512u)20]p + 0.80203~ + 1.554 In@)}(E/kT,
(16)
The proposed semi-theoretical viscosity model has been tested on the liquidviscosity data of 25 non-polar substances (a total of 931 data points); the test results are listed in
167 3 B. A. Lowry, H. T. Davis and S. A. Rice, Phys. Fluids,
Table 2. The critical properties were taken from Reid et al. [7], except that the critical compressibility factors for heavy hydrocarbons (with w>O.65) were evaluated from the following correlation 2, = 0.291 - 0.08w
7 (1964) 402. 4 K. D. Luks, M. A. Miller and H. T. Davis, AIChE J., 12, 6 (1966) 1079-1086. 5 T. M. Reed and K. E. Gubbins, Applied Statistical Mechanics, McGraw-Hill, New York, 1973. 6 P. G. Mikolqj and C. J. Pings, J. Chin. Phys., 46 (1967) 1401. 7 R. C. Reid, J. M. Prausnitz and B. E. Poling, The Properties of Gases and Liquids, 4th ed., McGrawHill, New York, 1987. 8 A. J. Jo&, L. I. Stiel and G. Thodos, AIChE J., 8, 1 (1962) 59-63. 9 J. W. Przezdziecki and T. Sridhar, AIChE J., 31 (1985) 333.
(17)
The overall average deviation of the calculated viscosities for 25 non-polar liquids (including heavy hydrocarbons up to C,,) is 9.27%. For comparison, the calculation results obtamed by using the correlations of Jossi et al. [ 81 and Przezdziecki and Sridhar [9] are also given in Table 2. The superiority of the new viscosity model for the substances tested is obvious.
Appendix
a b
6. Conclusion
m
H k
The theoretical viscosity equation derived by Davis et al. [l] for a model fluid with a square-well potential, when coupled with the empirical radial distribution function and generalized correlations for square-well parameters developed in this work, has been successfully extended to the calculation of viscosities of non-polar liquids (including heavy hydrocarbons up to C,,).
Acknowledgment
N”
PP PC R ?-i.f ?+b
T TC V(qjl W
Financial support from the China National Petroleum and Natural Gas Corporation for this work is gratefully acknowledged.
References 1 H. T. Davis, S. A. Rice and J. V. Sengers, J. Cha. Phys., 35 (1961) 2210-2233. 2 H. T. Davis and K. D. Luks, J. Phys. Chem., 69 (1965) 869-880.
X 2,
A: Nomenclature
parameter in eqn. (14) parameter defined in eqn. (5) radial distribution function function defied by eqn. (7) Boltzmann constant (k = 1.3805 X 1O-23 J K-r) mass of particle (kg) number of experimental data points pressure (kPa) critical pressure &Pa) defined by eqn. (6) distance of separation between molecules i and j parameter in eqn. (14) temperature (K) critical temperature (K) intermolecular potential acentric factor =rr2/aI critical compressibility depth of square-well potential shear viscosity (pP> de6ned by eqn. (4) number density (km01 rne3) inner and outer boundaries of squarewell potential (A, m) function de8ned by eqn. (8)