186
Ind. Eng. Chem. Fundam. 1980, 19, 186-188
The Viscosity and Thermal Conductivity of Simple Dense Gases Yoram Cohen and Stanley I. Sandler' Department of Chemical Engineering, University of Delaware, Newark, Delaware 197 I 1
One generally expects physical properties correlations based on theory to be more satisfactory than strictly empirical correlations. For this reason, we have explored the use of the Enskog dense gas theory as a basis for correlating the density dependence of the viscosity and thermal conductivity of simple gases (O,,N, Ar, and CH4). It is shown that the correlations developed here from the Enskog theory, together with equation of state data, lead to accurate predictions for the dense gas transport properties.
Here Atr and Aint are the translational and internal contributions to the total thermal conductivity and k is the Boltzmann constant. (It should be noted that eq 4 and 5 contain the approximation that the rate of rotational relaxation is sufficiently rapid that the internal degrees of freedom are in equilibrium with the translational degrees of freedom, so that there is a diffusional transfer of internal energy. This approximation fails for molecular fluids with large rotational relaxation numbers, such as hydrogen. Also, the form of eq 4 and 5 ensures that the low density thermal conductivity A. is exact and that the density dependence is correct.) Combining eq 3, 4, and 5 gives 1.2- 0.755Y - A- - -1+ - + Aobp Y A A
Introduction Since there is no satisfactory theory of transport in real dense gases, engineers generally use transport coefficient correlations which are either empirical or based on some theoretical foundation. A priori, one would expect a theoretically based correlation to be more satisfactory in that better accuracy may be obtained and fewer difficulties might be encountered with extrapolations outside the range of the correlation. For this reason, we have explored the use of generalized correlations for the viscosity and thermal conductivity of pure dense gases of simple molecules based on the Enskog theory of dense gas transport. The results of this work are described below. Theory Enskog (Chapman and Cowling, 1970; Hirschfelder e t al., 1954) developed an approximate theory for the transport coefficients of a dense rigid sphere gas. His analysis led to the following expressions for viscosity (7) and thermal conductivity (A) 7
1
7obP
y
- = - + 0.8 + 0 . 7 6 1 ~
(1)
- A-
(2)
where A = 4 Ao/15kqois equal to unity for a monatomic gas. Next, one must account for the fact that the interaction between real molecules is not of a hard-sphere nature; at present this cannot be done in a theoretically exact way. Rather than attempting to develop a new kinetic theory, we take a more pragmatic approach. Instead of using the Enskog theory in which b and y are chosen from dB b=B+T(7) dT and
and AobP
- -1 + 1.2 + 0 . 7 5 5 ~ y
where qo and A. are the dilute gas viscosity and thermal conductivity, p is the molecular density, b = 2ru3/3 where u is the molecular diameter, y = bpx is the Enskog modulus, and x , which is a function of density, is the radial distribution function a t contact for hard spheres. Several generalizations of these expressions are needed t o make them applicable to real fluids. First, the nonspherical nature and internal degrees of freedom of the molecules must be taken into account. For relatively simple molecules (for example, nearly spherical molecules such as the diatomic gases or methane, neopentane, etc.), the only fluids of interest to us here, eq 1 with appropriately chosen parameters may still be used for viscosity predictions, but, for the thermal conductivity, eq 2 is replaced with (Mason et al., 1962, 1978; Hanley et al., 1972) A = Atr + Aint (3) where
where B is the second virial coefficient, as suggested by Hanley et al. (1972, 1976) and DiPippo et al. (1977), we have developed correlations for these quantities (in fact, two sets of correlations, one for viscosity and another for thermal conductivity), which lead to reasonably accurate predictions for the density dependence of both q and A. Before proceeding to the correlations, we note that from experimental data a t fixed temperature, q/qop and A/&p have minima as a function of density. From eq 1 and 6 these minima occur a t 9
= 2.5447b7 qop lmin
and Aint
=
Y
respectively, where we have used the terms 6, and bh to allow for the fact that for real fluids the b parameters determined from eq 9 and 10 will not be the same. Consequently, eq 9 and 10 together with experimental data can
(5)
0196-4313/80/1019-0186$01.00/0
(9)
0
1980 American Chemical Society
Ind. Eng. Ghem. Fundam., Vol. 19, No. 2, 1980
187
T R : T / T ~ ( f o r b,)
v
v 0 0
t t
Oxygen Nitrogen Methane
c
A
7-0
I
0
4
- 2
-
v
TR:T/Tc
5
'0
be used to determine the b parameters. This was first suggested by Enskog (see Chapman and Cowling, 1970; Hirschfelder et al., 1954) and used by Michels et al. (1954) and others. The Correlation Equations 1and 6 are the basis of the present correlation, where b and y have been chosen to give good agreement with experimental data. In brief, b, and bh were found for argon, nitrogen, and oxygen using eq 9 and 10, the viscosity and thermal conductivity data of Hanley et al. (1974), and the equation of state contained therein. The b, and bh parameters obtained in this way are presented in Figure 1;they are similar to those obtained earlier by Gubbins (1968). These data were fitted with polynomials in reduced temperature (TR = TIT,) yielding - = 0.15423
+ 1.7561Tf1
-
2.27436T~-' +
v c
1.32177Tf3 (11) and bh _
- -0.09421 + 3.9322TR-1 - 7.1700T~-'+ 4.9401TR-3
vc
(12) These correlations, which are shown in Figure 1, lead to average errors in b, and bh of 3.56 and 4.1570,respectively, over the range of reduced temperature of 1.15 I:TR I:4.0. Near the critical temperature, the b parameters (and indeed the transport properties) are very sensitive to small changes in temperature. With the b parameters found from eq 9 and 10, the y functions were obtained using experimental transport property and equation of state data, eq 1 and 6, and the procedure developed by Fiszdon and Sandler (1979). The functions y, and yh so obtained are presented as a function of the reduced density bp in Figures 2 and 3, respectively. These functions were correlated as follows
In
():
= 0.01134 + 0.1764~3+ 0.4673 ( P * ) + ~ 0.08106(p*)3 (13)
and In
($)
= 4.26 X
02
03 0 4 0 5 06 07 0 8 09
IO
II
12
13
b7P
Figure 1. The parameters b/ V , for viscosity and thermal conductivity. Solid lines are the correlations obtained using the argon, oxygen, and nitrogen data. The methane points (m and 0)were not used in developing the correlations.
b,
01
( f o r b,)
+ 0.14543 - 1.1630p2+ 2.4105F3 - 0.97793" (14)
Figure 2. The function y, as a function b,p obtained using the viscosity and equation of state data for argon, nitrogen, and oxygen. The solid line is the correlation given in the text.
A Argon
AAA A
I__.,
0 0
01
A 0 2 03
.
,
,
0 4 0 5 06
,
,
,
07 08 0 9
,
,
IO
I1
0
Nitrogen
0
,Oxygen -, , I2
13
14
1, I5
bkP
Figure 3. The function y, as a function of b,p as in Figure 2.
where p* = bop and p = b,p/A'I2. The y, correlation has an average error of 3.2% and is valid over the density range 0.003 < p* C 1.31, while the yx correlation has an average error of 3.75% over the density range of 0.003 Ip I1.67. (Note that there is significant scatter in the y functions obtained from experiments above p* and 3 equal to unity, resulting in average errors of 13 and 18% in y, and y h but only 4.7 and 5.3% in the viscosity and thermal conductivity, respectively.) T o test our correlation we have used the viscosity, thermal conductivity, and equation of state data for methane, the only other gas for which all these properties have been critically evaluated by a single investigator (Hanley et al., 1977). Over the temperature range studied (250 to 500 K), the value of the A parameter for methane ranges from 1.5 to 1.8, compared to a value of unity for argon and a range of 1.27 to 1.32 for nitrogen and oxygen, and thus represents an extrapolation of our correlation. The results are shown in Figures 4 and 5. In all cases the agreement between experiment and prediction is quite good, especially when one takes into account that the experimental uncertainty of the transport property data a t high densities, and that errors in the density (Le., equation of state) are also reflected in transport property predictions. In fact, for the data in Figures 4 and 5 , the average error in the predicted viscosity is only 2.36% and that in the thermal conductivity is 1.9270,over a very large density range and 250 K range in temperature. Note also that to use our correlations only the dilute gas viscosity and thermal conductivity, and the equation of state are needed. Further, if dilute gas transport properties are not available,
188
Ind. Eng. Chem. Fundam., Vol. 19, No. 2, 1980
PIP,
Figure 4. The viscosity of methane as a function of reduced density at 250 K (0,l),300 K (V,2), 400 K (0,3)and 500 K (A,4). Points are the experimental data, and solid lines are the predictions. 35
30
25 X/k0
20
P/PC
Figure 5. The thermal conductivity of methane as a function of reduced density. Legend as in Figure 4.
they can be separately predicted (Reid et al., 1977), though this will introduce an additional error. We have compared our methane viscosity predictions with those based on the correlations of Jossi et al. (1962) and Reichenberg (1973,1975),and our methane thermal conductivity predictions with those based on the correlations of Stiel and Thodos (1964) and the recommended correlations of Reid et al. (1977). The overall performance of all these correlations were comparable. We find this encouraging since the empirical correlations of Thodos et al. and Reichenberg were developed using available methane data, while our correlations did not. Finally, it is perhaps worthwhile to compare the basis for the correlation proposed with other correlations of dense gas transport properties. The viscosity correlation of Jossi et al. (1962) and the thermal conductivity correlation of Stiel and Thodos (1964) are strictly empirical, using the experimental observation that 7 - qo and X - Xo are mainly functions of density. The viscosity correlation of Reichenberg (1973, 1975) is also completely empirical, but it does have the advantage of using temperature and pressure as variables, so that an equation of state is not needed. The corresponding states correlation of Tham and Gubbins (1969, 1970) involves a clever choice of dimen-
sionless groups based on the molecular theory of time correlation functions. Unfortunately, the dimensionless groups involve, among other quantities, the parameters in the molecular potential function and, once these groups have been chosen, the correlation is completely empirical with no further recourse to theory. Only Damasius and Thodos (1963) have even considered the use of the Enskog theory as a basis for correlating dense gas properties. However, they obtained values for the size parameter b and the Enskog modulus y solely from an equation of state without reference to transport property data, and they implicitly assume that such values should also be satisfactory for transport properties predictions. The work here, and that of Fiszdon and Sandler (1979), shows that this is not so. Nomenclature A = 4Xo/15k7 B = second virial coefficient b = 2xa3f 3 b,, bk = molecular size parameter for viscosity and thermal conductivity correlations, respectively k = Boltzmann constant P = pressure R = gas constant T = temperature T , = critical temperature TR = reduced temperature V = volume V , = critical volume x = radial distribution function for hard spheres at contact y = bpx y,, y x = y functions for viscosity and thermal conductivity correlations, respectively Greek Letters p = density ~ ( 7 =~ viscosity ) (at low density) X(X,) = thermal conductivity (at low density) a = molecular diameter Literature Cited Chapman, S., Cowling, T. G., "The Mathematical Theory of Non-Unlform Gases", 3rd ed, Chapter 16, Cambridge University Press, London, 1970. Damasius, G., Thodos, G.. Ind. Eng. Chem. Fundam., 2 , 73 (1963). DiPippo, R., Dorfman, J. R., Kestln, J., Khalifa, H. E., Mason, E. A,, Physica, 88A, 205 (1977). Fiszdon, J. K., Sandler, S. I., Physica, 95A, 602 (1979). Gubbins, K. E., J . Chem. Phys., 48. 1405 (1968). Hanley, H. J. M.. McCarty. R . D., Cohen, E. G. D., Physica, 80, 322 (1972). Hanley, H. J. M., McCarty, R. D., Haynes, W. M., J . Phys. Chem. Ref. Data, 3, 979 (1974). Hanley, H. J. M., Cohen, E. G. D., Physica, 83A, 215 (1976). Hanley, H. J. M., Haynes, W. M., McCarty, R. D., J . Phys. Chem. Ref. Data, 8, 597 (1977). Hirschfelder, J. O., Curtiss. C. F., Bird, R. B., "Molecular Theory of Gases and Liquids", p 634 ff, Wiley, New York, N.Y., 1954. Jossi. J. A,, Stiel, L. A,, Thodos, G., AIChE J., 8, 59 (1962). Mason, E. A., Monchick, L., J . Chem. Phys., 38, 1622 (1962). Mason, E. A., Khalifa, H. E.. Kestin, J., DiPippo, R., Dorfman, J. R., Physica, 91A, 377 (1978). Michels, A., Botzen, A., Schuurnan, W., Physica 20, 1141-1148 (1954). Reichenberg, D., AIChE J., 19, 654 (1973). Reichenberg, D., AIChE J., 21, 181 (1975). Reid, R. C., Prausnitz, J. M.. Sherwood, T. K., "The Propertles of Gases and Liquids", 3rd ed, Chapters 9 and 10, McGraw-Hiil, New York, N.Y., 1977. Stiel, L. A,, Thodos, G., AIChE J., IO, 26 (1964). Tham, M. J., Gubbins, K. E., Ind. Eng. Chem. Fundam., 8, 791 (1969). Tham, M. J., Gubbins. K. E., Ind. Eng. Chem. Fundam., 9, 63 (1970).
Received for review June 25, 1979 Accepted January 16, 1980
This research was supported in part by the National Science Foundation under Grant ENG 76-82102 t o the University of Delaware.
