N IIf P q
R
T v, 2 2
a:
= = = =
= = = = = =
0
=
E
= =
X p p
c
s2
= =
= =
molar flux, gram moles/sq.cm.-sec. molecular weight pressure, atm. tortuosity factor, dimensionless radius of porous disk, em. temperature, K. average pore velocity, cm./sec. mole fraction. distance, cm. constant, defined by Equation 7 inhomogeneity factor, dimensionless porosity, dimensionless thermal expansion coefficient, cm./cm.-’ K. viscosity, gram/sec.-sq.cm. density, gram/cc. molecular diameter, Angstrom units collision integral, dinleiisionless
SUBSCRIPTS -4,B = molecular species o
= zero net flux
Literature Cited
Boyd, C. A., Stein, N., Steingrimsson, V., Rumpel, W. F., J . Chem. Phys. 19, 548 (1951). Ember, G., Ferron, J. R., Wohl, L., J . Chem. Phys. 37, 891 (1962). Evans, R. B., Truitt, J., Watson, G. M., J . Chem. Eng. Data 6, 522 (1961). Evans, R. B., Watson, G. M., Mason, E. A., J . Chem. Phys. 36, 1894 (1962). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Wiley, New York, 1954. Klibanov, T. M., Pomerantsev, V. V., Frank-Kamenentskii, D. A., J . Tech. Phys. U S S R 12, 14 (1942). Mason, E. A., Weissmari, S., Wendt, R. P., Phys. Fluids 7, 174 (1964). Pakurar, T. A.,Ferron, J. R., IND.ESG.CHEY.FUXDAMESTALS 6 , 553 (1966). Pall, D. P., Ind. Eng. Chem. 46, 1197 (1953). Perkins, T. K., Johnston, 0. C., Soc. Petrol. Engrs. J . 3, No. 3, 70 . (1963) -- _ ~ _-,
Rumpel, W. F., University of Wisconsin, Kava1 Research Laboratory, Rept. CM-861 (1955). Scott, D. S., Cox, K. E., Can. J . Chern. Eng. 38, 201 (1960). Scott, D. S..Dullien. F. A. L.. A.I.Ch.E.J. 8. 113 (1962). Walker, IL‘E., Westenberg, A. A., J. Chem. Phys. 29, 1139 (1958). Walker, R. E., U’estenberg, A. A., J . Chem. Phys. 29, 1147 (1958b). Westenberg, A. A., Advan. Heat Transfer 3, 253 (1966). RECEIVED for review June 26, 1968 ACCEPTED May 15, 1969
Boardman, I,. E., WildJ X. E., Proc. Roy. SOC.(London) A162, 511 (1937).
CORRESPONDENCE P R I N C I P L E FOR TRANSPORT PROPERTIES OF DENSE FLUIDS P u r e Monatomic Fluids M. J . THAM’ AND K . E. GUBBINS Department of Chemical Engineering, University of Florida, Gainesville, Fla. 52601 The two-parameter law of corresponding states is obeyed closely for viscosity, thermal conductivity, and self-diffusivity of pure monatomic fluids over the entire range of temperature and density conditions for which data are available. Potential parameters determined from gas-phase viscosities are used in the reduction, and correlate the data for both gas and liquid phases.
HE principle of corresponding states has proved of great Tvalue to chemical engineers in the correlation and prediction of thermodynamic properties of dense gases and liquids (Hirschfelder et al., 1954; Hougen et al., 1959); the principle was first derived in a rigorous may from statistical mechanics by Pitzer (1939). While the simple corresponding states law works well for transport properties of dilute gases (Hirschfelder et al., 1954), comparatively little attention has been given to its application to transport properties for dense gases and liquids. Thermodynamic properties of inert gas fluids are known to obey the same reduced relationships closely (Crivelli and Danon, 1967; Danon and Pitzer, 1962; Keeler et al., 1965), the correspondence having been confirmed for argon and xenon by shock experiments to pressures of 200,000 atm. The principle has been found to apply for the viscosity of inert gases in the dilute and dense gas phase at temperatures above the critical (Trappeniers et al., 1965a, b ) , and has been tested ‘Present address, Shell Development Co., Houston, Tex. 77001.
for saturated liquid viscosity for quantum fluids (Rogers and Brickwedde, 1966) and for inert gases and simple polyatomic liquids (Boon e t al., 1967; Cini-Castagnoli et al., 1959). The latter tests were carried out using potential parameters determined from gas-phase property data, and correlations were rather poor. Better results have been obtained by Preston e t al. (1967). However, in their method it is necessary to use the liquid-phase transport properties themselves t o determine potential parameters. The correspondence principle has also been tested for the self-diffusion coefficient of dense gases and liquids using critical parameters in the reduction (Tee et al., 1966b). No systematic and consistent test of the principle for all transport properties over a wide range of temperatures and densities seems to have been made. The transport properties of fluids at high densities (and thus high pressures) is of particular interest, as it is in this region that deviations from correspondence may occur due to nonadditivity of pair potentials and differences in the potential energy curves. Reduced relationships are here presented for four transport VOL.
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properties of inert gas fluids over the entire range of temperature and density conditions for which data are available, using readily available potential parameters determined from gas-phase data. Reduced relationships for transport properties may be derived by dimensional analysis or from statistical mechanics, The former procedure (Hirschfelder et al., 1954) fails to clarify the assumptions involved or the domains of validity of the resulting equations. I n this paper particular emphasis is placed on the molecular basis of the principle and the validity of the necessary assumptions. Fluids composed of monatomic molecules should fulfill the assumptions most closely, and are therefore most suited to a test of the reduced equations. I n a subsequent paper the principle will be extended to polyatoniic fluids. Molecular Basis
The transport properties of a pure fluid mav be formally related to equilibrium time correlation functions (Mazo, 196i; Zwanzig, 1965). The equations for the self-diffusion coefficient, D , the shear and bulk viscosities, q and t i , and the thermal conductivity, X, are
D = L r n ( u L( 0 )( t ~ )dt ~
1
(1 1
rm
=
K*
ti*(T*, v*,h*) = ti62 (me)1/2
A*
=
X*(T*, Ti", h,
where
T* = LT/E, V* =
V/a1r0~3,
h* = ~ / u ( v L E ) ~ ' ~
The variable h* accounts for quantum effects, and may be omitted for classical fluids. The correspondence principle may be derived under less restrictive conditions than those used above. Donth (1966) has shomn that the derivation may be performed without pairwise additivity of intermolecular potentials (assumption 2 ) , provided that the total potential energy of the system may be expressed in the form
where @ is a universal function of reduced position coordinates. The extent to which multibody potential terms obey Equation 10 is uncertain, however. Thus the Axilrod-Teller formula for three-body dispersion interaction would introduce an additional reduced variable a* = ./a3 on the right side of Equation 10, and therefore in Equations 6 to 9 also. Even if such nonadditive contributions are of a significant magnitude, the effect on corresponding states correlations may be small because a* is relatively constant from one molecule to another (Reed and Gubbins, 1970).
I
0-9T-
where the functions contained in brackets ( ) are appropriate time correlation functions, the enclosed quant,ities being averaged over a canonical ensemble. These play a somewhat analogous role in the statistical mechanical treatment of transport properties to that of the partition function in statistical thermodynamics. -4lthough these equations are very difficult to solve for dense fluids, they may be used as the basis for a rigorous derivation of the corresponding states principle for transport properties. Such a derivation has been performed by Helfand and Rice (l960), based on the following assumpt'ions :
1. The intramolecular degrees of freedom (rotation, vibration) are independent of translational degrees of freedom and of densit,y. 2 . Intermolecular potentials are pairwise additive. 3. The potential energy for a pair of molecules has the form u = €+(?./a) ( 51 where E and u are energy and distance parameters of the potential function, and 4 is a universal function of reduced distance. The resulting reduced relationships are
i
2
1
co
T
0'
I&EC
FUNDAMENTALS
0.5
Figure 1. Smoothing Lennard-Jones (6,12) energy parameter
0
Ar
d
Xe
Kr
-0 N e
M
N:!
Q
02
*
c02
%
CH4
+
CCI4
r
792
0.4
W
0 co
D" = D*(T*, V", it*) =
03
02
F2
Hakala (1967) has derived the correspondence principle for more general pair potential functions than t h a t of Equation 5, using dimensional analysis. The result is to introduce additional dimensionless variables (shape and electrostatic parameters) into Relationships 6 to 9. Although the principle is then of more general application, it loses the attractive simplicity of the two-parameter law. Test for lnerf Gases
The essential simplifying factor for monatomic (as opposed to polyatomic) fluids is that assumption 1 is fully satisfied.
O
"
1
0.61
A comparison of the correspondence behavior for monatomic and polyatomic fluids may therefore throw light on the validity of this assumption. Lennard-Jones (6, 12) potential parameters are used in testing the principle. If assumption 3 is accurately obeyed, it is immaterial whether potential or critical reducing parameters are used (Prigogine et al., 1957). However, in practice the group of molecules do not accurately obey the same potential function, and potential parameters are force-fitted to some semiempirical equation-e.g., the Lennard-Jones (Tee et al., 1966a). The latter parameters are therefore preferable, and have been found in practice to yield better results than critical constants. Because of substantial differences in values of E and u reported by different workers, potential parameters determined from gas-phase viscosity data (Hirschfelder et al., 1954; Tee et al., 1966a) were smoothed by the method proposed by Tee, Gotoh, and Stewart (1966a). Figures 1 and 2 show the plots of reduced parameters against Piteer factor for 20 molecules; where several values for E and u were reported for a given molecule, the maximum deviation is indicated. The straight lines in these figures were fitted by 0.3,
O"r
0.3l 0.0
I
0.1
I
I
0.3
0.2
I
J
0.4
0.5
0.1:
W
Figure 2. Smoothing distance parameter
0"
0.05-
Lennard-Jones (6,12)
0.02 -
Symbols as in Figure 1
-
-1
2
I
'
'
1
I
'
1
I
3
Figure 5. Reduced self-diffusion coefficient for saturated liquid inert gases
1
?*
'
0.01
I
50
0
I
-.-Experiment
4.0
3 .O K"
01
0 3
05
0.9
0.7
11
1.3
1.5
2.0
l/T*
Figure 3.
Reduced viscosity of inert gases 1.0
1.0
19
1.2
1.1
1.3
1.4
1.5
llT*
Figure 6. argon
Reduced bulk viscosity of saturated liquid Naugle ef a!., 1964
Parameters for Monatomic Molecules
Table 1.
Potential Parameters u, A .
Neon
Figure 4.
Reduced thermal conductivity of inert gases
Argon Krypton Xenon
2.79 3.45 3.70 4.03 V O l .
8
e/k,
O
K,
35.3
119.8 166.1
229.8
NO. 4
T,,
K.
P o , Atm.
w
26.9 48.0 54.3 58.0
0 0 0 0
44.5 151.0 209.4
289.8 NOVEMBER
1969
793
Table II. Average Errors between Experiment and Equation 13 for Viscosity
Reduced Temperature Range 1.4-10 0.7-1.1 0.7-1.3 0.8-10
Reduced Pressure, P"
0 .O (gas phase) 0 .O (liquid) Saturated liquid 1.o
h-0. of
Points 29 23 39 22
Av. 76 Error
1.O
2.0 3.4 2.3
2.0
1.4-10
17
1.1
3 .O 4 .O
1.8-10 1.8-10
10 10
6.4 6.4
Table 111.
Average Errors between Experiment and Equation 14 for Thermal Conductivity
Reduced Temperature P" Range No. of Points 0.8-10.0 27 0 .O (gas phase) 0.7-1.2 24 0 . 0 (liquid phase) 0.6-1.3 37 Saturated liquid Reduced Pressure,
1.o
0.7-10 2.5-10 2.5-10 2.5-10
2 .o 3.0 4 .O
Viscosity References Flynn et al., 1963; Michels et al., 1954; Trappeniers et al., 1964, 1965a, b Cook, 1961; Johnson, 1960; Lowry et al., 1964 Cook, 1961; Johnson, 1960 Johnson, 1960; Michels e t al., 1954; Reynes and Thodos, 1964; Trappeniers et al., 1964, 1965a, b Johnson, 1960; hlichels et al., 1954; Reynes and Thodos, 1964; Trappeniers et al., 1964, 1965a, b Michels et al., 1954; Trappeniers et al., 1964, 1965a, b Michels et al., 1954; Trappeniers et al., 1964, 1965a, b
23 6 6 6
AV. %
Error
3.6 1.8 3.5 1.6 0.8
0.9 1.5
Thermal Conductivity References Ikenberry and Rice, 1963; Johnson, 1960; Michels et al., 1963; Uhlir, 1952 Cook, 1961; Ikenberry and Rice, 1963; Johnson, 1960 Cook, 1961; Horrocks and McLaughlin, . 1960; Ikenberry and Rice, 1963; Johnson, 1960 Ikenberrv and Rice, 1963: Michels et al.. 1963: Senners - et al., 1964 Michels et al., 1963; Sengers et al., 1964 ' Michels et al., 1963; Sengers et al., 1964 Michels et al., 1963; Sengers et al., 1964
least squares and have the equations
( P , / k T , ) ' ' 2 ~= 0.4583
Table IV. Average Errors between Experiment and Equation 15 for Self-Diffusion Coefficient
+ 0.1213~
+ alT*+ + U ~ T *+- ~a3T*-3 + a4T*-4 In A* = bo + blT"l + b2T*--2+ b3T*-3 + b4TW
(12)
794
l&EC
..
F U N D A M E N T A L S
Ar Kr
5 5 6
Xe
Av.
D
Error
References
4.9 7.4 2.3
Naghizadeh and Rice, 1962 Xaghizadeh and Rice, 1962 Naghizadeh and Rice, 1962
coefficients were found to fit the equation In D* = do
+ dll'*-'
The coefficients of the above three equations were evaluated by a least-squares fit to experimental data using the computer, and Tables I1 to IV show average errors between the resulting corresponding states curves and experimental data. The average errors seem to be within the limits of experimental accuracy in most cases. Conclusions
(13)
The correspondence principle is closely obeyed by inert gas fluids over the entire range of temperatures and densities for which data are available. Potential parameters determined from gas-phase viscosity data and smoothed in the way described by Tee et al. (1966a) correlate the data for both gas and liquid phases for thermal conductivity, viscosity, and self-diffusion coefficient. The good agreement a t the highest densities indicates that any errors introduced by assuming additive pair potentials are within the limits of accuracy of the experimental data. This may indicate that nonadditive potential terms can be approximately accounted for by a function of the type shown in Equation 10, as suggested by Donth (1966). Apparently assumption 3 concerning the form of the pair potential function does not lead to significant departures from correspondence, in spite of known differences in the potential curves for inert gases (Smith, 1966).
(14)
Acknowledgment
a t constant reduced pressure or along the saturation curve, where UO,al, . , bo, bl, . are constants. Self-diffusion
..
of Points ?io.
Figures 3, 4, and 5 are reduced plots for viscosity, thermal conductivity, and self-diffusion coefficient of neon, argon, krypton, and xenon, using potential parameters from Equations 11 and 12. These fluids may be expected to behave classically over most of the temperature range. Reduced temperature and pressure are chosen as more convenient independent variables than T * and V*. Data plotted in Figures 3 and 4 cover densities ranging from dilute gas to dense liquid, and pressures from 0 to 2000 atm. Selfdiffusion coefficients for liquid inert gases reported by different workers show substantial deviations, in some cases as large as 100% (Xaghizadeh and Rice, 1962; Yen and Norberg, 1963). The values shown in Figure 5 are those of Naghizadeh and Rice. Good agreement is obtained for reduced viscosity over the entire range of temperatures and pressures. For thermal conductivity poor agreement is observed for neon in the liquid phase a t P* = 0, although agreement is good for this fluid a t higher temperatures. This discrepancy may be due to quantum effects a t low temperature (Rogers and Brickwedde, 1966). The only bulk viscosity measurements available seem to be those for liquid argon (Naugle, 1966; Naugle et al., 1966; Swyt et al., 1967), and are shown in reduced form in Figure 6. Severe experimental difficulties are involved in bulk viscosity measurements, and even among the data of the same worker deviations may be as large as 15%. Reduced viscosities and thermal conductivities were found to fit equations of the form In v* = a.
Molecule
Computations were carried out on a digital computer a t the University of Florida Computing Center.
Nomenclature
. . . a4 . . . ba
= coefficients in Equation 13 = coefficients in Equation 14 = self-diffusion coefficient, sq. cm./sec. D* = reduced self-diffusion coefficient = coefficients in Equation 15 do, di h = Planck constant h* = reduced Planck constant J“” (O), J“” ( t ) = diagonal elements of viscosity dyadic at times 0 and t Jzw (0), Jzu ( t ) = nondiagonal elements of viscosity dyadic
ao, a1
bo, bl D
a t times 0 and t Boltzmann constant molecular mass, g. N number of molecules in system ATo Avogadro number P pressure, atm. P* Pu3/e = reduced pressure Pc critical pressure, atm. r = distance separating two molecules, cm. S“(O), P ( t ) = z-component of heat current vector a t times 0 and t T = temperature, K. T” = reduced temperature Tc = critical temperature, E(. t = time, sec. = total potential energy of system, ergs u U = pair potential energy, ergs V = molal volume, cc. = molecular velocity in 2-direction at times u,(O), u,(t) 0 and t
k m
= = = = = = =
GREEKLETTERS a E
17 K
x U
LJ
= polarizability of molecule, cc. = energy parameter in pair potential function,
ergs = shear viscosity, g./cm. sec. = bulk viscosity, g./cm. sec. = thermal conductivity, cal./sec. cm. K. = distance parameter in pair potential function, cm. = universal function of reduced distances = universal function of reduced intermolecular distance = Pitzer’s acentric factor
literature Cited
Boon, J. P., Legros, J. C., Thomaes, G., Physica 33, 547 (1967). Cini-Castagnoli, G., Pizeella, G., Ricci, F. P., Nuovo Cimento 11, 466 (1959).
Cook, G. A., ed., “Argon, Helium and the Rare Gases,” Chap. 9, Interscience, New York, 1961. Crivelli, I., Danon, F., J . Phys. Chem. 71, 2650 (1967). Danon, F., Pitxer, K. S., J . Am. Chenz. Soc. 66, 583 (1962). Donth. E.. Phvsica 32. 913 (1966’1. Flynn,’G.’P., Hanks, R . V.; Lemaire, N. A., Ross, J., J . Chem. P h i m 38. --,1,54 - _ - ilRfi3). ~-”--, b.
Hakala, R. W., J. Phys. Chem. 71, 1880 (1967). Helfand, E., Rice, S. A., J . Chenz. Phys. 32, 1642 (1960). Hirschfelder, J. O., Curtiss, C. F., Bird, It. B., “hlolecular Theory of Gases and Liquids,” Wiley, Xew York, 1954. Horrocks, J. K., LIcLaughlin, E., Trans. Faraday Soc. 66, 206 11960). Hougenj 0. A,, Watson, K. 31.) Ragatz, R. A , , “Chemical Process Principles,” Pt. 2, “Thermodynamics,” 2nd ed., M’iley, New York, 1959. Ikenberry, L. D., Rice, S. A., J . Chem. Phys. 39, 1561 (1963). Johnson, 5’. J., Wright-Patterson Air Force Base, Ohio WADD Tech. Rept. 60-66 (1960). Keeler, R. N., \-an Thiel, AI., Alder, B. J., Physica 31, 1437 (1965). Lowry, B. A,, Itice, S. A,, Gray, P., J . Chenz. Phys. 40, 3673 (1964). Mazo, It. AI., “Statistical Mechanical Theories of Transport Processes,” Chap. 10, Pergamon Press, Oxford, 1967. Physica 20, 1141 (1954). hlichels, A , , Botzen, A,, Schuurman, W., Michels, A., Sengers, J. V.,Tan de Klundert, L. J. AI., Physica 29. 149 119631. Naghizadeh, J., Rice, S. A., J . Chem. Phys. 36, 2710 (1962). Naugle, D. G., J . Chevz. Phys. 44, 741 (1966). Naugle, D. G., Lunsford, J. H., Singer, J. R., J . Chetn. Phys. 46, 4669 (1966). Pitzer, K . S., J . Chem. Phys. 7, 583 (1939). Preston, G. T.. ChaDman. T. W.,Prausnitz. J. 31..Crziooenics 7, 274 (1967). Prigogine, I., Bellemans, A., Mathot, J’.> “The Molecular Theory of Solutions,” p. 30, North-Holland Publishing Co , Amsterdam, 1957. Reed, T. &I.,Gubbins, K. E., “Applied Statistical Mechanics,” Chap. 12, IlcGraw-Hill, New York, in press, 1970. Reynes, E. G., Thodos, G., Physica 30, 1529 (1964). Rogers, J. D., Brickwedde, F. G., Physica 32, 1001 (1966). Sengers, J. V., Bolk, W. T., Stiger, C. J., Physica 30, 1018 (1964). Smith, E. B., Ann. Repts. Chem. Soe. (London) 63, 13 (1966). Swyt, D . S., Havlice, J. F., Carome, E. F., J . Chein. Phys. 47, llRR - - _ _IlFIA7). -.,. Tee, L. S., Gotoh, S., Stewart, IT. E., IND.EXG.CHEW FUNDAMENTALS 6, 356 (1966a). Tee, L. S., Kuether, G. F., Robinson, R. C., Stewart, R‘. E. 31st XIidvear Meetinn. American Petroleum Institute. Division of Refining, Houzon, Tex., 1966b. Tham, M. J., Ph.D. dissertation, University of Florida, 1968. Trappeniers, N. J., Botzen, A., Ten Seldam, C. A., Van den Berg, 13. R., Physica 31, 1681 (1965). Trappeniers, N. J., Botzen, A , , Tan den Berg, H. R., Van Oosten, J., Physica 30,985 (1964). Trappeniers, N. J., Botzen, A , , Van Oosten, J., Iran den Berg, H.It., Physica, 31, 945 (1965b). Uhlir, A., J . Chem. Phys. 20,463 (1952). Yen, W.M., Xorberg, R. E., Phys. Rw. 131,269 (1963). Zwanzig, R., Ann. Reo. Phys. Chem. 16,67 (1965). ”
Y
j - ”
RECEIVED for review August 2, 1968 ACCEPTEDJuly 22, 1969 Based on a Ph.D. dissertation (Tham, 1968)
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