Ind. Eng. Chem. Fundam. 1984, 23,8-13
8
The computer solution involves selecting a dimensionless force F and stepping down the spinline in small increments of Ax*. determining V*(x*)at each Doint bv" a trial-and-error procedure so as to satisfy eq k-2. Literature Cited v
~,
Bogue, D. C.; White, J. L. "Engineering Analysis of Non-Newtonian Fluids", NATO AQardoQraDh144, Nat. Tech. Info Serv., Springfield, VA (Doc. No. AD-710-324), -1970. Chang, J. C.; Denn, M. M. J. Non-Newtonian Fluid Mech. 1878, 5 , 369. Chang, J. C.; Denn, M. M.; Geyiing, F, T. Ind. Eng. Chem. Fundam. 1881, -20- , 147 . .. . Chen, I.J.;Bogue, D. C. Trans. SOC.Rheol. 1872, 16, 59. Denn, M. M.; Marrucci, G. J. Non-Newtonian FluidMech. 1877, 2. 159. Denn, M. M.; Petrie, C. J. S.; Avenas, P. AIChE J. 1975, 21, 791. Dietz, W.; Bogue, D. C. Rheol. Acta 1978, 17, 595. Fisher, R. J.; Denn, M. M. Chem. Eng. Scl. 1975, 30, 1129. Fisher, R. J.; Denn, M. M. AIChE J. 1978, 22, 236. Fisher, R. J.; Denn, M. M. AIChE J. 1877, 23,23. Han, C. D.; Kim, Y. W. J. Appl. folym. Sci. 1978, 20, 1555. Ide, Y.; White, J. L. J. Appl. folym. Sci. 1878, 22, 1061. Kase, S.; Matsuo, T.; Yoshimoto, Y. Seni Klkai Gakkaishi 1988, 19, T63. Keunings, R.; Crochet, M. J.; Denn, M. M. Ind. Eng. Chem. Fundam. 1983, 22,347.
Matsui. M.; Bogue, D. C. folym. Eng. Scl. 1976. 76, 735. Matsumoto*T.: WW D. C. W ' m . EW. SCl. 19789 7 8 3 564. Minoshima, W.; White, J. L.; Spruieii, J. E. J. Appl. folym. Sci. 1880a, 25. 287. Minoshima, W.; White, J. L.; Spruieii, J. E. folym. Eng. Sci. 1880b, 20, 1166. Nam, S. Ph.D. Dissertation, University of Tennessee, Knowville, TN, 1982. Pearson, J. R. A.; Matovich, M. A. Ind. Eng. Chem. Fundam. 1989, 8, 605. Petrie, C. J. S. J. Non-NewtonIan Fluid Mech. 1978, 4, 137. Petrie, C. J. S.;Denn, M. M. AIChE J. 1878, 22, 209 Racin, R.; Bogue, D. C. J. Rheol. 1879, 23, 263. Shah, Y. T.; Pearson, J. R. A. Ind. Eng. Chem. Fundam. 1972, 1 1 , 145. Tanner, R. I. J. folym. Sci. A - 2 1970, 8, 2067. Tsou, J. D. Ph.D. Dlssertatlon, University of Tennessee, Knoxville, TN, 1984. White, J. L. folym. Eng. Rev. 1981, 7 , 297. White, J. L.; Ide, Y. J. Appl. folym. Sci. 1878, 22, 3057. White, J. L.;Roman, J. F. J. Appl. folym. Scl. 1978, 20, 1005. Wust, C. J., Jr. M.S. Thesis, University of Tennessee, Knoxville, TN, 1978. Yamane, H. M.S. Thesis, University of Tennessee, Knoxville, TN, 1982. Zeichner, G. R. M.S. Thesis, University of Deiaware, Newark, DE, 1972. Zeichner, G. R.; Patel. P. D., paper presented at 2nd World Congress on Chemical Engineering, Montreal, Oct 1981.
Received for review February 25, 1983 Accepted August 4, 1983
Applications of Kinetic Gas Theories and Multiparameter Correlation for Prediction of Dilute Gas Viscosity and Thermal Conductivity Tlng-Horng Chung,' Lloyd L. Lee, and Kenneth E. Starling School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 730 19
Kinetic gas theories have been applied for the development of a correlation of gas viscosity and thermal conductivity. Employing the acentric factor (w), the dipole moment (p), and the association parameter (K) to characterize the effects of molecular shape and anisotropic intermolecular forces, the resultant multiparameter correlations are self-consistent for viscosity and thermal conductivity and generalized for polar and nonpolar gases. The results for pure gases are outstanding, not only in accuracy but also in applicability for such wide classes of fluids which include polar and hydrogen-bonding compounds.
Introduction The pace of progress in the study of transport properties is much slower than that of thermodynamic properties. The difficulties faced in the study of the transport properties come from the complexity of the mathematical treatment for rigorous theories and the uncertainty of the experimental data, especially the data of thermal conductivity and self-diffusivity. In the theoretical development, it has been over one century since the fundamental equation fo? the gas kinetic theory was first established by Maxwell and Boltzmann. The solution of the Boltzmann equation for monatomic gases was given independently by Chapman and Enskog in 1916-1917 (Hirschfelder et al., 1954). After about 80 years, the kinetic theories for polyatomic gases were successfully derived by Wang Chang and Uhlenbeck (1951),and Taxman (1958) using different approaches. However, direct application of these rigorous theories to calculate the transport properties for polyatomic gases is still too complicated even for modern high-speed computing machines. Because of the difficulty in describing the dynamics of collision for anisotropic molecules, which involve energy transfer between translation and rotation, empirical correlations or equations from simplified theories for transport properties have been the principal methods used for practical calculations. The differences in the transport properties of monatomic and polyatomic fluids are significant, especially for the
thermal conductivity, even in the dilute gas region. In this paper, we present a method of utilizing the acentric factor (w), the dipole moment ( p ) , and the association parameter ( K ) to characterize the effects of molecular shape and anisotropic intermolecular forces. This work will concentrate on the study of each of these effects. Equations for the viscosity and the thermal conductivity are developed based on kinetic gas theories and correlated with experimental data. The resultant multiparameter correlations are self-consistent for viscosity and thermal conductivity and generalized for polar and nonpolar gases. The average absolute deviations (AAD) from experimental data of predicted pure dilute gas properties is 1.5% for viscosity and 2.0% for thermal conductivity. Application of these equations for gas mixtures is very easy and accurate with the conformal solution model (Mo and Gubbins, 1976) and other available models (Reid et al., 1977). Viscosity of Dilute Gases The Chapman-Enskog (CE) theory treats only molecules which do not possess any internal degrees of freedom. The CE theory for the dilute gas viscosity is 5 (rmkT)1/2 VCE =
E *02Q*(2,2)(P)
(1)
where m is the mass of a molecule, k is the Boltzmann constant, Tis the absolute temperature, the reduced tem-
0196-4313/84/1023-0008$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984 9 2.01
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Table I. Association Parameters
I
compound
K
methanol ethanol l-propanol 2-propanol l-butanol 2-methyl-l-propanol l-pentanol l-hexanol l-heptanol acetic acid water
0.21 51 75 0.174823 0.143453 0.143453 0.1 31671 0.131671 0.121555 0.114230 0.108674 0.091549 0.07 5908
the relative orientation is fixed, to reduce the WCU theory to a mathematically treatable form T’
Figure 1. Comparison of the Chapman-Enskog Theory (CE)with the experimental dilute gas viscosity of polyatomic gases.
perature T* = kT/e, u and t are the potential distance and energy parameter, and Q*(2,2) is the reduced collision integral which is related with the potential model. Equation 1 is accurate for nonpolar monatomic gases with the collision integral evaluated using the LennardJones (12-6) potential model, which has been correlated as a function of reduced temperature and given by Neufeld et al. (1972) as
G P B sin ( S P W- P) (2) where A = 1.16145, B = 0.14874, C = 0.52487,D = 0.77320, E = 2.16178,F = 2.43787, G = -6.435 X lo4, W = -0.76830, P = 7.27371, and S = 18.0323. Equation 1 can be written in the reduced form as
The reduced viscosity, q*, is defined here by the relation q* = i26.693 X
lo* (M e/k)’/2u’-2)-1q
(4)
where q has the unit of poise and M is molecular weight. In this work, the two parameters, u and E , are evaluated from the critical density (pc) and critical temperature (T,) for all substances using the empirical relations
2 = O.3189/pc; e / k = TJ1.2593
(5)
With u and t / k determined by eq 5, the experimental viscosity data can be reduced via eq 4 and plotted vs. the reduced temperature. Figure 1 presents this plot, which compares the CE theory with the experimental results for some substances. It can be seen that the data points for the more complicated molecules do not correspond to the CE theory. Therefore, the direct application of the CE theory for anisotropic gases is not adequate. The difference between the CE theory and the theories for polyatomic gas viscosity (WCU theory or Taxman theory) is in the collision integral. For structural molecules, the interaction potential between molecules is angle-dependent and energy transfer between rotation and translation is possible on energy collision. These problems cause difficulty in the analysis of the dynamics of collision between two molecules and thus retard the application of the WCU (or Taxman) theory. Later, Monchick and Mason (1961) made two assumptions: (1) the inelastic collisions have little effect on the trajectories; (2) during collision,
where the angular average collision integral, (Q*),can be calculated for the angle-dependent potential. Calculations for some potential models have been made (e.g., Monchick and Mason, 1961; Smith and Munn, 1967). The recent study from computer simulation for the collision dynamics given by Evans and Watts (1976) shows that the assumption of fixed orientation during molecular collision is incorrect. Besides, the application of eq 6 for real fluids is still impractical because of the complexity in the potential model for the real fluid. In the kinetic theory for rough-spheres (Chapman and Cowling, 1970), the resulting equation for viscosity treats the rotational effect as a correction factor for the elastic smooth-sphere (S.S.) viscosity, i.e. qRS. = %.S. i1 - f(a)\ (7) where the parameter a is related with the momnt of inertia of the rough spherical particle. Based on the same concept, a correction factor, f(W,p*,K), which accounts for the anisotropic effects was introduced to make the CE theory suitable for prediction of the viscosity for polyatomic, polar, and hydrogen-bondinggases. Thus the equation for dilute gas viscosity is given as qO* = VC!E*(l - f ( O & * , K ) J (8) The correction factor, f(u,p*,K), can be correlated with Pitzer’s acentric factor (w) for polyatomic gases, the reduced dipole moment ( p * ) for polar gases, and the association parameter (K) for hydrogen-bonding compounds as f(W,p*,K) = 0.27560 - 0.059035p*4- K (9) The reduced dipole moment is p* = p / ( ~ u ~ ) where ’ / ~ , p is the dipole moment. The association parameter K was determined individually from the viscosity data for fluids having the hydrogen-bonding effect. The values for K for some compounds are given in Table I. For the group of alcohols, the values of K change linearly with the weight ratio of the OH group in the molecule (as shown in Figure 2) and can be correlated as 17 X no. of OH groups K = 0.0682 0.276659 X molwt
+
(
The values of K show that the hydrogen-bonding effect decreases as the share of the OH groups in the molecular weight decreases. A comparison of the predicted results from this work and previously recommended methods (Reid et al., 1977) for some nonpolar, polar, and hydrogen-bonding gases is given in Table 11. It can be seen that eq 8 is superior to the other methods not only in accuracy but also in its
10
Ind. Eng. Chem. Fundam., Vol. 23,No. 1, 1984
Table 11. Comparison between Calculated and Experimental Values of Low-Pressure Gas Viscosity deviation, %
data point temp range, K
compound methane ethane propane 1-butane 1-pentane isobu tane ethylene propylene 1-butene acetylene cyclohexane benzene toluene carbon dioxide carbon disulfide carbon tetrachloride chlorine
5 4 5 3 4 3 4 4 3 3 5
293.2-773,2 293.2-523.,2 293.2-548.2 293.2-393.2 398.2-573,2 293..2-393.2 273,2-523.2 29 3..2-523.2 293,2-393.2 303,2-4.73.2 3 08,2- 57 3.2 301.2-47 3.2 333.2-523.2 303.2-473.2 303.2-4 73.2 398.2-573.2 293.2-473.2
3
3 3 3 3 3
this work exptla q , pp
Thodos et al.
AAD MAX AAD MAX
I. Nonpolar Gases 109-227 0.28 0.7 90.1-153 1.50 -2.0 80.6-142 1.48 -2.5 73.9-99.8 1.22 1.5 91.7-130 0.59 0.7 74.4-99.5 1.10 1.3 94.5-168 0.31 0.5 84.3-147 1.54 -1.8 76.1-99.8 1.63 -2.3 102-155 0.60 0.6 72.3-129 1.46 2.3 73.2-117 0.56 -1.0 78.9-123 4.10 5.0 151-219 1.48 -2.9 94.6-151 4.62 -5.7 133-190 1.59 -2.4 133-209 3.32 -4.5
average error
0.38 1.92 2.28 0.53 0.60 1.90 1.15 1.52 2.90 0.63 1.54 1.96 3.43 2.96 9.76 3.13 2.00
1..6
0.7 2.4 4.1 0.6 0.7 2.0 -1.4 2.1 4.3 -0.8 3.5 3.3 -4:O 4.8 11.0 4.2 2.9
other methodsa Golubev Reichenberg AAD
MAX
AAD MAX
___
--9.12 7.62 2.26 3.07 3.93 9.50 5.75 5.00 5.83 4.08 4.66 1.06 9.80 12.66 5.23 4.40
2.2
23.0 16.0 3.0 4.8 4.3 26.0 11.0 6.4 12.0 5.4 6.9 -1.8 10.0 15.0 6.1 5.8
8.0
2.17 0.82 1.80 1.22 1.73 0.25 1.60 3.16 -1.26 3.20 2.66 2.53
-2.4 -1.7 -2.8 -1.6 2.1 -0.4 2.4 4.6 2.4 -4.2 4.4 -3.1
2.03
-2.6
-----
-----
___
_._
1.9
11. Polar Gases
acetone methyl ether ethyl ether ethyl acetate sulfur dioxide ammonia chloroform methyl chloride methylene chloriide methanol ethanol 1-propanol 2-propanol
4 3 4 4 7 3 5 4 4 6 4 3 4
37 3.2-598.2 293,2-393.2 298.2-573.2 298.2-598.2 .283.2-1173.2 273.2-673.2 293.2-623.2 293.2-403,2 293.2-573.2 308.2-593.2 383.2-573.2 298.2-548.2 393.2-573.2
93.3-153 .90.9- 123 99.1-141 101-153 120-432 90-251 100-208 106-147 98.5-193 101-195 111-165 104-144 103- 150
average error a
1.22 0.26 0.82 0.50 1..39 5.78 1.87 4.28 0.25 0.28 0.29 0.81 0.20
1.8 -0.4 1.0 1.0 -3.0 11.3 3.4 5.0 0.5 0.7 0.5 1.1 0.3
4.77 8.66 3.55 2.80 1.60 2.0 1.10 1.12 3.20 2.46 1.50 2.63 6.22
1.38
7.1 -9.2 4.2 4.0 -3.1 3.7 1.8 2.1 3.5 -5.2 3.0 3.6 8.6
4.87 0.86 2.22 1.55 24.68 10.26 7.44 1.60 5.57 22.16 10.75 6.26 5.90
3.1
-6.2 -1.1 4.2 -2.2 93.0 -13.0 8.0 -2.6 7.0 -22 -11 -7.1 -6.7
7.5
1.55 1.13 0.37 2.57
2.5 2.2 0.7 -4.1
0.70 2.72 6.37 1.86 0.25 1.00 3.02
-1.4 4.1 7.8 2.8 0.7 -1.6 4.2
-----
2.0
Given by Reid et al. (1977), Table 9-4.
I 0.25
1
!
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5 4
A'
3 2
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0.1 0.2 0.3 0.4 0.5 0.6 OH
M.W. Figure 2. Association parameter for alcohol group.
applicability for such wide classes of substances. It is more general than the method of Thodos et al. (Reid et al., 1977) which correlates different classes of fluids with different equations. An extensive test of eq 8 for 40 substances (Chung, 1980) yields the overall result of about 1.5% AAD. Thermal Conductivity For thermal conductivity, the CE theory gives the well-known Eucken relation ACE* = TCE* (11) The thermal conductivity is reduced by A* = (19.891 X ( C / ( M ~ ) ) ~ / ~ U - ~ ) -(12) 'X
I W
0
1
2
3
4
5
6
9' Figure 3. Eucken relation for monatomic and diatomic gases.
where the units of X are cal/(cm s K). The Eucken relationship has been checked with experimental data for monatomic and diatomic gases in Figure 3. The diagonal line represents the Eucken relation. For monatomic gases the ratio of the experimental thermal conductivity to the viscosity is described by the Eucken relation (within 2% deviation). For diatomic gases, good results cannot be expected. For molecules with internal degrees of freedom, such as polyatomics, energy can
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
11
Table III. Calculated Results for Thermal Conductivity of Pure Gas at About 1 atm dev, %
a
substance
data points
temp range, K
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane isobutane isopentane ethylene propylene acetylene carbon monoxide carbon dioxide nitrogen oxygen benzene ammonia acetone ethyl acetate ethyl ether sulfur dioxide water methanol ethanol 1-propanol
23 18 23 11 9 7 11 10 8 12 13 9 4 10 38 33 31 10 10 10 6 5 10 25 13 13 13
193-673 233-513 253-823 273-413 313-473 313-573 373-613 433-633 210-413 273-457 300-500 298-623 198-313 373-1273 186-1400 80-1400 100-1400 325-450 300-500 273-113 3 19-673 273-573 213-1173 313-933 450-570 450-570 450-570
Vargaftik (1975).
z-
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data ref
4.995-23.71 3.298-15.75 3.059-25.43 3.131-9.178 3.774-8.481 4.897-11.01 4.491-11.58 5.638-11.47 3.123-7.152 -2.912-7.520 4.891-12.06 3.968-14.63 2.810-7.110 7.199~19.26 ,2.073-20.12 1.869-20.79 2,211-2 2.06 2.987-6.071 5.903-12.55 2.220-16.11 2.880-12.50 3.100-11.95 2.007-12.40 5.860-21.83 7.553-11.18 7.553-11.28 7.075-10.95
a a a a a a a a a a a a
Reid et al. (1917),Table 10-4. I
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b a
qb,c a a a, b a a, b 0, b qb
a a a a a
MAX 1.88 2.33 5.23 -3.77 3.86 -2.15 -2.70 2.83 2.38 -8.04 2.56 1.21 -0.75 -1.09 -6.,31 9.82 7.95 -3.26 -3.69 7.36 -9.62 6.76 7.57 9.00 0.75 0.87 0.80
AAD 1.29 1.77 1.47 0.99 1.34 0.15 0.95 1.48 0.74 2.43 1.62 0.42 0.45 0.56 -1.37 1.40 2.19 2.20 2.31 3.14 6.35 3.69 4.33 5.17 0.20 0.49 0.21
pD/q
1.28 1.37 1.41 1.43 1.45 1.45 1.43 1.41 1.43 1.44 1.36 1.40 1.29 1.33 1.33 1.2Q 1.17 1.58 1.08 1.42 1.44 1.48 1.16 0.78 1.31 1.38 1.43
Johnston and Grilly (1946). I 1
bc
of two contributions: the translation and the internalenergy contributions, i.e.
x = At,
+ xint
(13) The WCU theory gives the expressions for A,, and Aint as
elr
$-
gal (D-
b-
(15)
el-
0
1
2
3
4
5
6
7
8
9
101112
T'
where Cht is the internal heat capacity at constant volume. The integrals X and Y can be expressed in terms of the viscosity q and the relaxation time 7
Figure 4. Plot of reduced thermal conductivity for dilute polyatomic fluids.
be absorbed and expended through rotational energy transfer and vibrational energy transfer during molecular collisions. Therefore, the dependence of the thermal conductivity on molecular structure is very much more pronounced than in the case of the viscosity as can be seen from the comparison of Figure 1 and Figure 4. Any corresponding states theory cannot correctly take into account the effect of internal degrees of freedom on the thermal conductivity. In this work, a method was developed to study the contribution of the internal degrees of freedom to the property of thermal conductivity. The derivation is based on the WCU theory and the approximations given by Mason and Monchick (1962). For polyatomic dilute gases, the kinetic theory has been successfully derived (WCU theory of Taxman theory). Several forms of approximation have been developed from these theories. Almost all the formulas for the thermal conductivity of polyatomic gases are written as the sum
Mason and Monchick (1962) approximate the 2 integral as
and the correction factor ( q / p T ) is given as -v= PT
4 4 , U
where ZmUis defined as the number of collisions required to interchange a quantum of internal energy with translational energy. Letting @ = q / p D and combining (16), (17), (18), and (19) into (14), (15), and (13), we can get
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Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
hO* _ - 1 + C*h, no*
t
x
0.26665 + (0.215 - 1.06l/3)/ZCou+ 0.28288C*ht/Zcoll 0 + 0.6366/ZcO11+ 1.061~C*~nt/Zco~ (20)
where qo* is reduced dilute gas viscosity given by eq 8. C*ht (= C,/R) is the reduced internal heat capacity at constant volume, i.e., C*i,t = C*, - C*tr and C*,, = 3/2, where C*, can be obtained from any available correlation for C*p.
c*, = c*P - l Equation 20 contains two parts; the first part is the translational contribution; the second part is due to the internal degrees of freedom. For monatomic molecules, C*ht is zero and eq 20 reduces to the CE theory expression, eq 11. If the term in brackets is a constant (0.26665) then eq 20 reduces to the Eucken correlation formula
M
15 + (C, - -R
_.-
?
-
IR)
4
At high temperatures, the value of Zmuis very large (of the order of lo3 to lo'); then eq 20 reduces to 15 -AM- - -R + ?
4
(C,
-
:R)
Bl m
rl
!7
(22)
?
Equation 22 is fairly good in predicting the thermal conductivity of gases at high temperatures. For most simple linear molecules, the value of p D / q is about 1.32, which simplifies eq 22 to the so-called Modified Eucken Correlation
M
- = 1.32CV+ 3.52 17
(23)
If the rational term in the bracket of eq 20 is expanded in series to the first order of (l/Zcou)and p D / q = 1.32, we get the approximation 0.886Cbt Ah4 - 1.32CV+ 3.52 (24) tl
3
Zcoll
The last term, which contains the rotational collision number,,,Z , has been correlated by Bromley (Reid et al., 1977). So, eq 20 is a generalized expression for the individual contribution from the translation and the internal energy. Based on this expression, we made a detailed investigation of the behavior of the thermal conductivity. Figure 4 is a plot of the smoothed experimental results of thermal conductivity vs. temperature in reduced quantities. The data include the dilute gases of diatomic and more complex molecules. It shows the effect of molecular structure, i.e.: (1)the more complex the molecular structure the higher the value of the reduced thermal conductivity; (2) the value of the reduced thermal conductivity for a straight-chain molecule gas is greater than that of the branch-chain isomer, which has more centered molecular mass; (3) molecular size also affects the value of the dilute gas thermal conductivity (e.g., O2> CO > NJ. The separation of each line in the plot is due to the second part of eq 20. In other words, energy transfer due to the internal degrees of freedom is significant for the thermal conductivity, which is not the case for viscosity. In eq 20 there are two unknown quantities, /3 and Zc0n. The value of Zcou varies widely from one order of magnitude or less for rotational relaxation to the order of lo7for vibrational relaxation (Mason and Monchick, 1962). From
$1 w
01
MI Q
3
CI
e
z
i
d l l Q,
$ h
r-
YI
0
*a a
E
u0 0 U
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984 03 t 9
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scribed in this work and expressed by eq 20. The results show that the developed correlations are outstanding, not only in accuracy but also in applicability.
Nomenclature CE = Chapman-Enskog theory Ci,, = internal heat capacity at constant volume per mole, cal/(mol K) C*int = Ci,,/R = reduced internal heat capacity );C , ideal gas heat capacity at constant pressure, cal/(mol
0
1
2
3
4
5
T'
6
7
8
9
1
0
Figure 5. Plot of (X*o/q*o - l)/C*v,btvs. reduced temperature for dilute polyatomic gases.
the analysis of the experimental data (Figure 5 ) , it was found that the values in the bracket of eq 20 become nearly constant when the reduced temperature is about 4 or greater. Since a t higher temperatures, the contribution due to internal vibration becomes more significant, the value of Zd becomes very large and all the terms involving ZCou approach zero. All the terms involving Zcouhave significance only in low temperature cases. Accordingly, Zcouwas correlated as a function of temperature only. Zcou= 2.0
+ 6.6P'
(25)
The parameter @ is defined as the ratio of the viscosity of density and internal self-diffusivity (pD)at dilute gas conditions. However, we could not obtain the value of @ through this definition, because of the lack of experimental data for the internal self-diffusivity. Here, we treat @ as a parameter. From Figure 5, it can be seen that the value of p can be determined from the thermal conductivity data of high temperature. The obtained @ values for some compounds are listed in Table 111. For nonpolar hydrocarbons, /3 was correlated with Pitzer's acentric factor w (7) and the product
0 = 0.786231 - 0.710907~+ 1.31683~'
(26)
The calculated results for the thermal conductivity of dilute gases are summarized in Table 111. The substances include nonpolar, polar, and hydrogen-bonding compounds. The predicted results are generally within experimental errors (about 1.6% AAD). When compared with other recommended methods (Table IV),the results of eq 20 are more accurate and simpler to apply for a wide range of fluids. Conclusions This work provides multiparameter corresponding states correlations for the viscosity and thermal conductivity of nonpolar, polar, and associating pure gases. Polyatomic and polar effects are accounted for using defined, measurable quantities, the acentric factor, and the dipole moment. For description of the effects of association, an association parameter which must be determined from the transport data has been introduced. These five parameters can be used consistently for dilute gas and also for high density fluid; the latter application will be presented subsequently. The contribution of the internal degrees of freedom to the thermal conductivity, which is much more pronounced than in the case of the viscosity, has been adequately de-
C,* = C,/R = reduced ideal gas heat capacity C, = ideal gas heat capacity at constant volume, cal/(mol K) C"* = C,/R = reduced ideal gas heat capacity D = internal self-diffusion coefficient, cmz/s int = internal energy k = Boltzmann constant m = mass molecule M = molecular weight n = number of molecules in system P = pressure, dyn/cm2 R = universal gas constant R.S.= rough-sphere S.S. = smooth-sphere T = absolute temperature, K T, = critical temperature, K T* = kT/c = reduced temperature tr = translational energy ,Z , = collision number Greek Letters a = rotational coefficient 6 = q / p D = diffusion term t = energy potential parameter q = shear viscosity, g/(cm s) q* = reduced shear viscosity qo = dilute gas shear viscosity, g/(cm s) qo* = reduced dilute gas shear viscosity K = association parameter X = thermal conductivity, cal/(s cm K) A* = reduced thermal conductivity Xo* = reduced dilute gas thermal conductivity = dipole moment p = density u = distance potential parameter 7 = relaxation time o = Pitzer's acentric factor Q* = reduced collision integral Literature Cited Chapman, S.; Cowling, T. 0. "The Mathematical Theory of Nonuniform Gases": Cambridge Universlty Press: New York, 1970. Chung, T. H. Ph.D. Dissertation, University of Oklahoma, Norman, OK, 1980. Evans D. J.; Watts, R. 0. Mol. Phys. 1878, 32, 995. Hirschfeider. J. 0.; Curtlss, C. F.; Bird, R. B. "Molecular Theory of Gases and Liquids"; Wiley: New Ywk, 1954. Johnston, H. L.; Grllly, E. R. J . Chem. Phys. 1948, 74, 233. Mason, E. A.; Monchick, L. J. Chem. Phys. 1982, 36, 1622. Mason, E. A,; Saxena, S. C. Phys. Flukfs 1958, 1, 361. Mo, K. C.; Gubbins, K. E. Mol. Phys. 1978, 31, 835. Monchick, L.; Mason, E. A. J . Chem. Phys. 1981, 35, 1679. Neufeld, P. D.: Janzen, A. R.; Aziz, R. A. J . Chem. Phys. 1872, 57, 1100. Reid, R. C.; Prausnk, J. M.; Sherwood, T. K, "The Properties of Gases and Liquids"; McGraw-Hili: New York, 1977. Smith, F. J.: Munn, R. J. J. Chem. Phys. 1987, 46, 317. Taxman, N. Phys. Rev. 1958. 170, 1235. Vargaftlk, N. B. "Tables on the Thermophysicai Properties of LiquMs and Gases"; Wiley: New York, 1975. Wang Chang, C. S.; Uhbnbeck, G. E. "Transport Phenomena in Poiyatomic Molecules": Michigan University Engineering Research Institute Report CM-681, 1951.
Received for review June 21, 1982 Revised manuscript received July 1, 1983 Accepted August 3, 1983 We are grateful to the National Science Foundation for the support of this research under grant ENG77-21551.
