Physica
1OOA(1980) 335-348 @ North-Holland
Publishing Co.
DENSITY EXPANSION OF THE VISCOSITY OF CARBON DIOXIDE NEAR THE CRITICAL TEMPERATURE J. KESTIN h’vision of Engineering,
and 6. KORFALI?
Brown University, Providence,
RI 02912, USA
J. V. SENGERS Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
Received 10 September
1979
We present a detailed experimental study of the viscosity of carbon dioxide as a function of density at a temperature near the critical temperature. Kinetic theory predicts the presence of a logarithmic term in the density expansion for the viscosity of gases. While it is difficult to prove the existence of such a term experimentally, the data are consistent with a possible logarithmic contribution of an order of magnitude estimated theoretically by Kan.
1. Introduction Modern kinetic theory of gases predicts that the transport coefficients of gases unlike the compressibility factor cannot be expanded in terms of power series in the density p. Instead, a transport coefficient such as the shear viscosity n of a moderately dense gas should be represented by an expansion of the form’“) rl=770+77,~+7)~p’Inp+772p*+...,
(1)
where the coefficients no, ul, q$ and u2 are functions of the temperature. The theoretical issues related to this prediction were recently reviewed by Kan et aL4T5). Several attempts have been made to investigate the possibility of experimentally detecting a logarithmic density dependence of the transport coefficients of moderately dense gases&l2). For this purpose one needs a large number of accurate experimental data in a range. of densities sufficiently restricted so that one may neglect contributions from higher order terms ’ Present address: Director, Research and Development Center, Alardo Holding, Kore Sehitleri Cad., no. 50, Zincirlikuyu, Istanbul, Turkey. 335
J. KESTIN
336
AND ii. KORFALI
beyond those quoted in (1). In a previous paper’) we reported the results of a study of the density expansion of the viscosity of nitrogen and argon at 25°C. It was concluded that the data could actually be represented by a polynomial in p implying relatively narrow bounds for the values of the coefficient 9; of the logarithmic terms. Recently, Van den Berg and Trappeniers”) reported a detailed experimental study of the density dependence of the viscosity of krypton at 25°C and reached similar conclusions. The studies mentioned above were conducted at temperatures well above the critical temperature. On the other hand, Hanley and Haynes’), from an analysis of viscosity data for fluorine, claimed to have found evidence for the presence of a logarithmic term at lower reduced temperatures. Since the coefficients in (1) are expected to depend on temperature, an appreciably larger value of 77; at lower reduced temperatures cannot be excluded a priori. For this reason we investigated the density expansion of the viscosity of a gas at a temperature near the critical temperature. The instrument at Brown University most suitable for the purpose is restricted to operation slightly above room temperature. For this study we therefore selected carbon dioxide which has a critical temperature of approximately 31°C.
2. Experimental
method
The viscosity was determined using an oscillating-disk instrument. An oscillating disk with radius R and thickness d, suspended from a 92% Pt-8% W stress-relieved elastic wire, is located between two fixed horizontal plates separated by equal vertical gaps from the upper and lower surface of the disk. To reduce the volume of the gas cylindrical aluminum blocks were placed in the space above and below the fixed plates. Apart from this detail the instrumental arrangement was identical to that described by Kestin and Leidenfrost”). The characteristics of the suspension system used in the experiment reported here are given in table I. In the experiment one measures the period T and the logarithmic decrement A of the damped harmonic TABLEI
Characteristics Disk radius Disk thickness Moment of inertia Gaps Oscillation period in vacua Damping constant in vacua
of suspension
system
R = (3.4906 + O.OOl)cm d = (0.10431 ~0.00012)cm
Z = (53.6032 + 0.0006)gcm2 b = (0.09006? 0.00005)cm T0 = (29.250? 0.002)s at 31°C A0=(40?4)~10-6
DENSITY EXPANSION OF THE VISCOSITY OF CO2
337
oscillation performed by the system, To and A0 being the values of these quantities when the system is oscillating in vacua. The working equation is14)
where I is the moment of inertia of the suspension system, 8 = T/T,, the ratio of the actual period to that in vacua and C a calibration factor which is a unique function of the boundary layer thickness S = (nTo/27rp)“‘. Finally, the quantity x0 = 6/R = R-‘(~To/2rp)“2, computed numerically from (2), yields the viscosity rj. The function C(6) is determined by calibrating the instrument with fluids of known viscosity. The calibration data were taken using argon in a pressure range of 1 to 27 atm and nitrogen in a pressure range of 1 to 68 atm. In order to reproduce a situation similar to that encountered in the subsequent measurements of the viscosity of carbon dioxide, the calibration was performed at 30°C. At this temperature the viscosity of argon and nitrogen may be represented as 71= no+ G(P),
(3)
with no = 22.906pPa.s for Ar and no = 18.002 pPa.s for N2 as determined previously”). The function Q(P), often referred to as the excess viscosity, was determined by Kestin and Leidenfrost13). It may be represented empirically by 77=(P)= 171P+
7?2P29
(4)
with 771= 9.310 X IO-’ kPa.s/(kg/m’), 772= 75.% x 10V6pPa.s/(kg/m3)2 for Ar and 71 = 10.052 X low3kPa.s/(kg/m3), n2 = 91.89 x 10” pPa.s/(kg/m3)2 for N2. The calibration equation reproduces the viscosity of the calibrating fluids with a standard deviation of approximately 0.03%. In the actual experiments the data are determined as a function of pressure. For argon and nitrogen the experimental pressures were converted into densities using the procedure described in a previous publication’). The resulting calibration function C(6) is plotted in fig. 1. It is represented by C = 1.0920754 -
0.0030381 0.0005768 0.0000237 s + 0.07 - (6 + 0.07)2 + (S + O-07)3’
(5)
where the boundary layer thickness 6 is expressed in cm. This equation reproduces the experimental data for C(S) with a standard deviation of 3.6x 10-4.
338
J. KESTIN
AND
ij. KORFALI
1.08 I.06
0
0.2
0.4
0.6
/ 0.8
I.0
6, cm Fig. 1. The calibration
3. Experimental
factor C as a function
of the boundary
layer thickness
6.
results
The experimental results obtained for carbon dioxide are presented in table II. The viscosity was measured as a function of pressure and temperature. The pressure was measured with a dead-weight gauge of an accuracy of about 0.0003 MPa at pressures up to 3 MPa and 0.003 Mpa at pressures beyond 3 MPa. The temperature of the instrument was measured with an accuracy of about O.Ol”C using five chromel-alumel thermocouples. The densities were deduced from the experimental pressures and temperatures using the equation of state data of Michels and Michels16). In calculating these densities we applied an estimated correction of 0.04”C to account for the difference between Michels’s temperature scale and the current international temperature scalei7J8). Because of the appearance of a logarithmic term in (l), we prefer to express the density in dimensionless units. For this we have chosen the practical amagat unit. The density in amagat units is the ratio of the actual density to the density of the fluid at a standard temperature (0’0 and pressure (1 atm). Where necessary, a small temperature correction was applied to the viscosity so that the values quoted in table II all refer to the same temperature, t = 31.63”C, and the experimental densities. The viscosity of fluids is known to exhibit a critical enhancement in the vicinity of the critical point. At the critical temperature the anomalous behavior extends over a density range of about &25% of the critical density pc which for COz corresponds to 175 am < p < 300 am’S*o). In order to make sure that the analysis of the density dependence of the viscosity at low and moderately low densities was not affected by the critical anomaly, we concentrated on determining the viscosity at densities not exceeding 155 amagat.
DENSITY
EXPANSION
OF THE VISCOSITY
OF CO*
TABLE II Experimental viscosity data for carbon dioxide Pressure 0IPa)
Temperature (“0
Density (amagat)
Viscosity at 31.6PC (W Pas)
0.2131 0.2566 0.3055 0.3515 0.3919 0.4441 0.4999 0.5476 0.6021 0.6510 0.7085 0.8700 0.9139 1.0173 1.3434 1.6245 2.0430 2.5718 3.0477 3.525 4.034 4.586 5.075 5.578 5.894 6.035 6.269 6.437 6.537 6.695 6.773 6.775 6.786 6.861 6.904 6.951 6.%6 7.012 7.014 7.053 7.093 7.119 7.122 7.170 7.220 7.221 7.239 7.261 7.330
31.65 31.66 31.67 31.71 31.78 31.81 31.81 31.61 31.64 31.65 31.69 31.82 31.57 31.57 31.59 31.69 31.69 31.57 31.48 31.52 31.56 31.58 31.50 31.57 31.48 31.47 31.49 31.52 31.63 31.67 31.56 31.56 31.50 31.61 31.66 31.51 31.66 31.62 31.52 31.57 31.62 31.63 31.63 31.68 31.72 31.65 31.64 31.60 31.63
1.89 2.28 2.72 3.14 3.51 3.98 4.50 4.94 5.44 5.90 6.44 7.97 8.40 9.40 12.62 15.50 19.97 25.99 31.82 38.10 45.43 54.28 63.27 73.91 81.83 85.71 92.78 98.42 101.9 108.3 112.2 112.3 113.0 116.4 118.5 121.9 122.1 125.2 125.8 128.2 130.8 132.7 133.0 136.6 140.8 140.8 143.6 146.4 155.3
15.24 15.24 15.24 15.25 15.25 15.25 15.26 15.26 15.27 15.28 15.28 15.30 15.29 15.30 15.35 15.38 15.47 15.62 15.76 15.92 16.15 16.52 16.88 17.43 17.84 17.99 18.32 18.68 19.10 19.36 19.84 19.86 19.64 20.16 20.23 20.44 20.48 20.73 20.79 21.01 21.02 21.42 21.42 21.63 22.00 22.00 22.19 22.49 23.12
a 1 amagat = 1.9764 kg/m).
339
340
J. KESTIN
AND
ii.
KORFALI
A large number of data were taken to make the results amenable to statistical analysis. The precision of the experimental data, as determined by the standard deviation when the data are fitted to a variety of functional forms, varies from 0.05% at atmospheric pressure to 0.3% at the highest pressures. This is in contrast to our preceding measurements for argon and nitrogen where the random error was 0.03% at all pressures’); the difference must be attributed to the more difficult conditions in the present experiment close to the critical temperature. In particular in our experiment the temperature of the instrument is maintained by thermostating the room at the experimental temperature of (31.6?0.2)“C, a temperature appreciably higher than in our preceding experiments with the same instrument. The experimental viscosity obtained for CO* is plotted as a function of density in figs. 2 and 3. In these figures we have also included data for CO* at the same temperature from some other source?‘-**). As noted earlie?), the data of Michels et al. are larger, the difference increasing with increasing density. This effect is due to a breakdown of the assumptions in the working equation for viscosity applied by Michels et al. to the capillary flow measurements in the vicinity of the critical point”). Our values are slightly larger than the values determined earlier by Kestin, Whitelaw and Zien”), the maximum difference being 1% at the higher densities. This result is acceptable in view of the appreciably larger gap (b =0.9cm) in the previous experiment. After our experiment was completed, we received an extensive set of viscosity data for CO2 of comparable accuracy from Iwasaki and Takahashi*‘)
1
I oThls work x Iwasaki, Tokahoshl ~Michels, Botzen, Schuurman
v
-
0
0
x 0
x
co* t=31.63”C
15.0 L 0
5
I IO
I 15 Density,
Fig. 2. Viscosity
I 20
I 25
I 30
Amagot
of CO* at 31.63”C at densities
up to 35 amagat.
I 35
DENSITY EXPANSION 24
oThis 22 -
I
I
I
I
341
OF THE VISCOSITY OF CO2
I
I
1
x0
work
a
x Iwosoki, Takahashi OK&in, Whitelaw, Zien vMichels,
B&en,
Schuurman
I 25
I 50
I 75
v o” .+ 00
20_
12
0
I 100
Density,
I 125
I 150
175
Amagat
Fig. 3. Viscosity of CQ at 31.63”C at densities up to 155 amagat. TABLE III Viscosity of carbon dioxide at 31.63”C as determined by Iwasaki and TakahashiB) Density (amagat)
Viscosity (10e6 Pas)
Density (amagat)
Viscosity (lo+ Pa.s)
0.88 1.45 2.45 4.37 6.20 9.01 11.94 16.74 24.31
15.22 15.24 15.25 15.28 15.29 15.33 15.35 15.42 15.57
38.76 52.84 69.39 86.07 104.7 122.5 138.6 152.6
15.97 16.48 17.17 18.03 19.13 20.33 21.71 22.95
obtained with the same oscillating disk viscometer as was used earlier for ethylenes). The primary purpose of the work of Iwasaki and Takahashi was to determine the anomalous behavior of the viscosity close to the critical point, but a substantial number of data points at lower densities were taken as well. The viscosity data in the density range of our experiment obtained by Iwasaki and Takahashi are listed in table III. These data are also included in figs. 2 and 3 and they agree with our data within the accuracy of our experiment.
342
J. KESTIN AND 6. KORFALI
4. Analysis of results
We want to investigate whether the experimental data are consistent with the presence of a logarithmic term in the density expansion (1) for the viscosity. However, when the data are fitted to (1) retaining the first four terms of the density expansion, the coefficient 7; does not become fully significant. When the data are fitted to a cubic polynomial, the coefficient of the cubic term does not become significant. On the other hand, when the data are fitted to a quadratic polynomial, the coefficients appear to vary with the density range. As an alternate procedure, we assume the validity of the nature of the theoretically predicted density expansion (1) and inquire which range of values for the coefficient 75 is compatible with the experimental results’). For this purpose we rewrite (1) as f(7) = 7) - 75~’ ln P =
710 +
TIP
+
772pz,
(6)
where the coefficient 71 is treated as a parameter. Each value for the parameter 71;defines a new data set associated with the experimental viscosities and we investigate whether the data f(n) satisfy the conditions for a power series in the density. The procedure has been described in detail in previous publications6*‘). Briefly, we investigate the range where f(n) can be represented by a linear equation and a larger range where f(v) can be represented by a quadratic polynomial and then require that the coefficients v. and v1 from the quadratic fit agree within error with those returned from the linear fit. In this paper we shall estimate the error in the coefficient qi as twice the standard deviation a,,. This procedure assumes that we can find a finite experimentally accessible density range where the quadratic term in (6) is significant, while higher-order terms can still be neglected. The validity of this assumption is far from obvious, as can be seen from the theoretically known equation for the density dependence of the diffusion coefficient of a two-dimensional Lorentz However, in the absence of any theoretical information concerning gas 5*26*27). the higher-order terms in (6) for three dimensional fluids, there is little else we can do. As recently emphasized by Codastefano et al.‘*), the procedure also assumes the existence of a finite density range where the contribution from the linear term is important while those from the logarithmic and the quadratic term are still small compared to the experimental precision. Neglecting the logarithmic term we would expect the linear and quadratic terms to be of an order of magnitude similar to those predicted by the theory of Enskog and the assumption would be justified”). From a practical point of view, evidence in support for the validity of the assumption is provided a posteriori by the results of the analysis.
DENSITY EXPANSION
343
OF THE VISCOSITY OF CO2
For our final analysis presented here we have combined our data in table II with the data of Iwasaki and Takahashi in table III, increasing the population of data for statistical analysis and reducing spurious effects due to possible systematic errors in either experiment. The principal results of the analysis, using the values vi= 0 and n$ = 0.4 x lo-” Pa.s/am*, are presented in table IV. We focus our attention in particular upon the coefficient 71. In fig. 4 we have plotted n1 + 2a,,, as a function of the density range Ap when the data are fitted to the eq. (6) assuming 71:= 0. The linear equation yields n1 _+2a,, = (1.00 + 0.17) x lo-* Pa.s/am when fitted up to Ap = 10 am and n1 + 2u,,, = (1.05 2 0.10) x lo-’ Pas/am when fitted up to Ap = 15 am. When the data are fitted to the quadratic equation, the coefficient n1 is constant in a range up to Ap = 75 amagat, but the value nl f 2u,,, = (0.67 f 0.08) x lo-* Pa.s/am does not agree with the value returned from the linear fit. In fig. 5 we have plotted the same coefficient n1 but now assuming 75 = 0.4 x lo-” Pa.s/am*. As can be TABLEIV
Results of analysis of CO2 data Range Ap (amagat)
(T70‘t2um)X 106 (Pa.s)
(Slk 20,,) x 108 (Pa.s/am)
(72 +2&J x 1O’O (Pas/am*)
7: x 1O’O (Pas/am*)
2)
10 75
15.218 f 0.009 15.224 ? 0.007
1.00~0.17 0.67 f 0.08
3.1 kO.1
0 (fixed) 0 (fixed)
0.05 0.08
10 75
15.221~0.009 15.217 f 0.008
0.9lkO.17 0.88 + 0.08
1.1*0.1
0.4 (fixed) 0.4 (fixed)
0.06 0.09
1614 _ 12 _ IO-
!
08_
I II I’
06_ 04_
-it
,I);=0 02_ 01
2
/
4
/
6
I
8
Ap,
I
IO amogat
I
I2
I
14
1
16
oi 20
quadratic
30
40
50
60
fit
70
80
Ap, amagat
Fig. 4. The coefficient r), + 2rv,, as a function of the density range Ap assuming q: = 0.
344
J. KESTIN
AND ij. KORFALI
16_
‘6, 14
14 _ t 12 -
ot 2
4
quadratic
fit
1
I
I
/
1
8
IO
12
14
Ap,
0
77;=0_4 x IO’O Pa.shr?
lineor
6
Fig. 5. The coefficient 0.4 X lo-” Pa.s/am*.
-0.21
77;=04Xld’oPas/am2
20
30
40
I
I
I
I
I
I
4
6
8
IO
12
p, amagat
50
60
70
80
Ap, amagat
as a function
2
(01
o/
I 16
amagat
7, ?20,,,
fit
of the density
I-o.21 14
16
0
range
I
I
I
IO
20
30
Ap assuming
1):’
I
I
I
I
I
40
50
69
70
80
p, omogoi
(b)
Fig. 6. Plot of 15.221 X 10m6Pa.s,
deviations (qell)- 77,.1~)/~& (a) qlcalc= q. + q1 + TIW In P with Ilo = 7: = 0.4 X 10-‘” Pa.s/am*. gl = 0.895 X lo-’ Pa.s/am, (b) l)C.lC= with qo = 15.217 x 10e6Pa.s, q1 = 0.877 X lo-* Pa.s/am, 770 + VIP + 4p* In P + q2pz s:= 0.4 X lo-” Pa.s/am2, q2 = 1.14 X lO-‘OPa.s/am*. The circles indicate our data and the crosses those of Iwasaki and Takahashi”).
seen, the values returned for vl are now in acceptable agreement. A further increase of 7; makes it difficult to isolate a quadratic range where ~2 is independent of the range. From the information provided in table IV we conclude that the experiinental data for CO2 are consistent with the following
DENSITY
EXPANSION
OF THE VI$COSITY
values of the first three terms in the density expansion
OF CO2
345
(1) for the viscosity:
q. = (15.22 + 0.01) X 10” Pa.s ~1 = (0.9 * 0.2) X 1Om8 Pa.s/am
)
at 3 1.63”C.
(7)
-qS= (0.4 2 0.4) X 10-r” Pa.s/am2 1 Deviation plots for the equations with these coefficients
5. Comparison
are presented
in fig. 6.
with theory
Theoretical studies of the coefficients in the density expansion (1) of the viscosity have been made for a gas of hard spheres. In this case it is convenient to write the density expansion in the form n = 70[ I+ 77Tfzo3+ ~P’(na~)~ In no3 + ~T(na~)~ + - * -1,
(8)
with?
Here n is the number density, u the molecular diameter, m the molecular mass, k Boltzmann’s constant and T the absolute temperature. The coefficient of the first density correction for a gas of hard spheres is”) T? = 0.403 f 0.002. For the coefficient 72’ of the logarithmic term Gervois et al.3’32)have reported a numerical estimate of 7)f’ = 21 or 42 as reinterpreted by Kan33). An independent attempt to evaluate this coefficient was made by Kan and Dorfman4,33). Evaluating some of the collision integrals using an approximation based on the BGK collision operator, Kan reported33) T#’ = 0.641. The reason for the appreciable difference between the two theoretical estimates is not known presently’). In comparing (8) with the viscosity of a real gas an ambiguity arises by virtue of attributing an effective hard sphere diameter u to the molecules; comparing with (8), one can only hope to obtain a similar order of magnitude for the coefficients of the density expansion. In this paper we have simply determined an effective diameter (T by identifying (9) with the coefficient v. determined experimentally. We thus obtain for CO2 from (7). u = 4.54 X 1O-'o m and 77T = 0.23 2 0.03,
@J’ = 0.4 2 0.4.
(10)
In discussing the first density correction it should be noted that the coefficient VT for hard spheres is the sum of a collisional transfer term +1.675 which is
346
.I. KESTIN AND ii. KORFALI
positive and a triple collision term -1.272 which is negative. For a real gas these separate contributions may be different functions of temperature and the coefficient 771is known to change sign as a function of temperature6). Moreover, at low temperature this coefficient will also be affected by the possible formation of bound molecular states”37) which are absent in the case of hard spheres. On the other hand, the logarithmic term arises only from sequences of successive binary collisions for real gases as well as for a gas of hard spheres. From (10) we conclude that the range of values for the coefficient r/2*‘,consistent with the experimental data, is of the same order of magnitude as the estimate reported by Kan, but substantially smaller than the estimate presented by Gervois et al. As emphasized by Gervois et al.32), one can always increase the deduced 77?’ by introducing a smaller effective diameter u, since the value of 77?’ deduced from the experimental data is inversely proportional to u6. Nevertheless, taking u = 3.68 X lo-” m, which corresponds to the range of the repulsive part of an effective m-6-8 potentia13*)for CO*, raises T# up to only 1.4. The same analysis can be repeated for our previous measurements’) of the viscosity of nitrogen and argon as was also done by Gervois et al.32). The results are summarized in table V. In this table we have also included the values recently deduced by Van den Berg and Trappeniers”) from very accurate capillary-flow measurements for krypton at 25°C when using the same criteria. It turns out that the possible bounds found by Van den Berg and Trappeniers for the coefficient 71 are of the same order of magnitude as those found in our experiment. From the information in table V we draw the following conclusions. (a) Within the present experimental resolution and for the simple fluids considered here the density expansion of the viscosity at reduced temperatures of TABLE V Coefficients of density expansion for viscosity Fluid
N2
Ar
Kr
co2
Investigators Reduced temperature T* 70 X 106,Pa.s 7, X loS,Pa.s/am 7: X lO’O,Pa.s/amz Density conversion factor, kglm’am Effective diameter (T, lo-‘0 rn tl;r sz*’
Kestin et al.‘) 2.36 17.785 1.3 0.0 2 0.6
Kestin et al.‘) 1.98 22.609 2.0 0.0 2 0.6
Van den Berg et al.“) 1.42 25.349 3.13 0.0 * 0.5
This work 1.00 15.22 0.9 0.4 + 0.4
1.2505 3.73 0.53 021.7
1.7839 3.62 0.69 O+ 1.6
3.7490 4.11 0.66 020.6
1.9764 4.54 0.23 0.420.4
DENSITY
EXPANSION
OF THE VISCOSITY
OF CO2
341
unity does not appear to differ fundamentally from that observed at higher reduced temperatures. (b) The experimental data are consistent with the possible existence of a logarithmic term with a coefficient of the order of magnitude estimated by Kan and Dorfman. (c) If we assume that an experimentally accessible density range exists where higher order terms in (1) can be neglected, we find it somewhat difficult to reconcile the experimental data with a coefficient of the logarithmic term of the magnitude estimated by Gervois et al.
Acknowledgments
We are indebted to H. Iwasaki and H.R. Van.den Berg and N.J. Trappeniers for providing us with their experimental data prior to publication. We also acknowledge valuable discussions with J.R. Dorfman. The research at Brown University was supported by NSF grant ENG 78-12380 and that at the University of Maryland by NSF grant DMR 79-10819. Computer time for this project was provided by the Computer Science Center at the University of Maryland.
References 1) 2) 3) 4) 5)
J.R. Dorfman and E.G.D. Cohen, J. Math, Phys. 8 (1%7) 282. K. Kawasaki and I. Oppenheim, Phys. Rev. 139 (1965) 1763. S.G. Brush, Kinetic Theory, Vol. III (Pergamon Press, New York, 1972) pp. 125-129. Y. Kan and J.R. Dorfman, Phys. Rev. A 16 (1977) 2447. Y. Kan, J.R. Dorfman and J.V. Sengers, in Proceedings 7th Symposium on Thermophysical Properties, A. Cezairliyan, ed. (American Society of Mechanical Engineers, New York, 1977) p. 652. 6) H.J.M. Hanley, R.D. McCarty and J.V. Sengers, J. Chem. Phys. 50 (1969) 857. 7) J. Kestin, E. Paykoc and J.V. Sengers, Physica 54 (1971) 1. 8) J. Kestin, in Proceedings 4th International Conference on High Pressure (Kyoto, 1974) p. 518. 9) H.J.M. Hanley and W.M. Haynes, J. Chem. Phys. 63 (1975) 358. 10) A.A. Vasserman and B.A. Puttin, J. Eng. Phys. 29 (1975) 1384. 11) H.R. Van den Berg and N.J. Trappeniers, Chem. Phys. Lett. 58 (1978) 12. 12) P. Codastefano, D. Rocca and V. Zanza, Physica %A (1979) 454. 13) J. Kestin and W. Leidenfrost, Physica 25 (1959) 1033. 14) J. Kestin, W. Leidenfrost and C.Y. Liu, Z. angew. math. Phys. 10 (1959) 558. 15) J. Kestin, Y. Kobayashi and R.T. Wood, Physica 32 (1966) 1065. 16) A. Michels and C. Michels, Proc. Roy. Sot. (London) A160 (1937) 348. 17) J.M.H. Levelt Sengers and W.T. Chen, J. Chem. Phys. 56 (1972) 595. 18) J.M.H. Levelt Sengers, W.L. Greer and J.V. Sengers, J. Phys. Chem. Ref. Data 5 (1976) 1. 19) R.S. Basu and J.V. Sengers, J. Heat Transfer, Trans. ASME 101 (1979) 3. 20) J. Kestin, J.H. Whitelaw and T.H. Zien, Physica 30 (1964) 161. 21) H. Iwasaki and M. Takahashi, private communication.
348 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38)
J. KESTIN
AND 6. KORFALI
A. Michels, A. Botzen and W. Schuurman, Physica 23 (1957) 95. J. Kestin and J.H. Whitelaw, Physica 29 (1963) 335. H.R. Van den Berg, A. Botzen and N.J. Trappeniers, private communication. H. Iwasaki and M. Takahashi, in Proc. 4th Int. Conf. on High Pressure (Kyoto, 1974) p. 523. J.M.J. Van Leeuwen and A. Weijland, Physica 36 (1%7) 457; 38 (1968) 35. C. Bruin, Physica 72 (1974) 261. J.V. Sengers, Int. J. Heat Mass Transfer 8 (1%5) 1103. S. Chapman and T.G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge Univ. Press, 3rd ed., London, 1970). J.V. Sengers, D.T. Gillespie and J.J. Perez-Esandi, Physica 90A (1978) 365. A. Gervois and Y. Pomeau, Phys. Rev. A 9 (1974) 2196. A. Gervois, C. Normand-AlIC and Y. Pomeau, Phys. Rev. A 12 (1975) 1570. Y.H. Kan, Physica 93A (1978) 191. D.E. Stogryn and J.O. Hirschfelder, J. Chem. Phys. 31 (1959) 1545. S.K. Kim and J. Ross, J. Chem. Phys. 42 (1%5) 263. J.T. Lowry and R.F. Snider, J. Chem. Phys. 61 (1974) 2320. R.D. Olmsted and C.F. Curtiss, J. Chem. Phys. 63 (1975) 1%6. H.J.M. Hanley and M. Klein, J. Phys. Chem. 53 (1970) 4722.