Density dependence and composition dependence of the viscosity of neon-helium and neon-argon mixtures

Physica 106A (1981) 415-442 North-Holland

Publishing Co.

DENSITY DEPENDENCE AND COMPOSITION DEPENDENCE OF THE VISCOSITY OF NEON-HELIUM AND NEON-ARGON MIXTURES J. KESTIN and G. KORFALI* Division of Engineering, Brown Unioersity, Providence, RI 02912, USA and J.V. SENGERS

and B. KAMGAR-PARS1

Institute for Physical Science and Technology, Unioersity of Maryland, College Park, MD 20742, USA

Received 2 October 1980

In this paper we present a detailed experimental study of the density dependence of the viscosity of He-Ne and Ne-Ar mixtures at various compositions. The data are analyzed in terms of a density expansion including a logarithmic contribution whose presence is predicted by the kinetic theory of gases. The composition dependence of the viscosity of the dense gas mixtures is compared with estimated values predicted on the basis of the Enskog-Thorne theory for dense mixtures of gases of hard spheres.

1. Introduction In two previous publications”‘) we have reported a detailed experimental study of the density dependence of the viscosity of one-component gases such as He, Ar, N2 and CO*. Our interest was motivated by the earlier theoretical discovery’) that the transport properties of gases cannot be expanded in pure powers of the density p, but that in addition the presence of a term proportional to p* In p is to be expected. Our work, as well as that of Van den Berg and Trappeniers4% has indicated that the experimental data are consistent with the possible presence of a logarithmic density dependence, but they do not by themselves establish it definitely. In this paper we report a detailed experimental study of the density dependence of the viscosity of mixtures of gases. We thought it to be of interest to investigate whether the nature of the density dependence of gases consisting of disparate molecules would or would not be similar to that of * Present address: Director, Research and Development Cad., no. 50, Zincirlikuyu, Istanbul, Turkey. 415

Center, Alardo Holding, Kore Sehitleri

416

J. K E S T I N et al.

gases consisting of identical molecules. For this purpose we measured the viscosity of N e - H e and Ne-Ar mixtures. In order to analyze the density dependence with the methods of statistical analysis one needs accurate data at a large number of closely spaced densities. Our earlier measurements 6) of the viscosity of these mixtures were, therefore, not quite adequate for the purpose. Our results indicate that the density dependence of the viscosity of these mixtures is, in fact, quite similar to that of the viscosity of the pure gases. In addition, the data to be reported allow us to evaluate the composition dependence of the viscosity of these mixtures at elevated densities. A method for computing the composition dependence of the viscosity of dense gas mixtures was earlier proposed by Di Pippo et al.7). This method is based on the Enskog theory as applied to mixtures. In this paper we reconsider the application of this method incorporating some suggestions proposed subsequently by Sandler and FiszdonS).

2. Experimental method The viscosity, r/, was measured by the same method employed in our previous studies of the density dependence of the viscosity of one-component fluids ~'2) and described in detail by Kestin and Leidenfrostg). As oscillating disk with radius r and thickness d suspended from a stress-relieved elastic wire, is located between two fixed horizontal plates separated by equal vertical gaps b~ =/~2 =/~ from the upper and lower surface of the disk. In the experiment one measures the period T and the logarithmic decrement A of the damped harmonic oscillation performed by the system, T0 and A0 being the values of these quantities when the system is oscillating in vacuo. These experimental quantities satisfy a working equation that contains the instrumental constants listed in table I and a calibration function C(6) which is a unique function of the boundary layer thickness 8 = (~q]'0/2~rp). The function C(6) is determined by calibrating the instrument with fluids of known viscosity~°'~). In order to obtain optimum accuracy it is desirable that the logarithmic decrement should not exceed values of the order of about 0.05 which corresponds to about 6 observable oscillations. To obtain this condition for the present experiment, the gap distance /~ was increased to 0.127cm as compared to the distance 0.09 cm used in our previous experiments for carbon dioxide2). Using a longer suspension wire, we increased To and, consequently, the boundary layer thickness 3, enabling us to maintain values of 6//~ of order unity, to be preferred for relative measurements. The values of the instrumental constants after these modifications are presented in table I. The

VISCOSITY OF N E O N - H E L I U M A N D N E O N - A R G O N M I X T U R E S

417

TABLE I Instrumental constants Disk radius Disk thickness Moment of inertia Gaps Oscillation period in vacuo Damping constant in vacuo

r = (3.4906 -+ 0.001) cm d = (0.10431 - 0.00012) cm I = (53.6032 +_0.0006) g cm: /~ =/~1 =/~2 = (0.1269 -+ 0.0001) cm T0 = (31.264 _+0.002) s at 25°C Ao = (40 "4"4) x 10-6

1.12

i

i

i

i

1.10 C

1.08 1.06 1.04 °N 2 1.02 1.00 0

i

i

i

t

0.2

0.4

0.6

0.8

1.0

~, cm

Fig. 1. The calibration factor C as a function of the boundary layer thickness &

viscometer was recalibrated at 25°C with respect to nitrogen in the pressure range of 1 to 68 atm. The calibration procedure was the same as that described in previous publications~'2). The new calibration function is plotted in fig. 1. Within a standard deviation of - 1.7 × 10 -4, it can be represented by C = 1.1179473

0.0058258 0.0004313 0.0000183 +0.07 (~ +0.07)2+ (~ +0.07)3,

(1)

where the boundary layer thickness a is expressed in cm. He-Ne and Ne-Ar mixtures were prepared in a series of cylinders by weighing the masses of the components with a high-precision, high-capacity balance. This gravimetric method allowed us to determine the composition in mole fractions to about four significant figures. Each mixture was kept in its container for at least one week after preparation in order to secure complete interdiffusion. The purity of the gases, as specified by the suppliers, was 99.995% for He, 99.997% for Ne and 99.998% for Ar. In our previous studies ~'2) of the density dependence of the viscosity of gases, we measured the viscosity as a function of pressure using a dead-

418

J. K E S T I N et al.

weight gauge with an accuracy of about 0.0003 MPa at pressures up to 3 MPa and 0.003 MPa at pressures beyond 3 MPa. The densities at the experimental pressures and temperatures were then calculated from the equation of state. We continued with the same procedure for the N e - H e mixtures and for some of the Ne- Ar mixtures. However, the equations of state of these mixtures are less well known than for the pure components. Therefore, a provision was made to measure the densities of the mixtures. This provision enabled us to determine the viscosity for the remaining mixtures directly as a function of density and provided us with a check of the accuracy of the equations of state employed in calculating the density corresponding to the data points earlier measured as a function of pressure. The general experimental arrangement is shown in fig. 2. The environment of the entire experimental arrangement was kept at a temperature close to 25°C. After evacuating the viscometer, the viscometer was connected to the mixture cylinder. After allowing enough time for the system to reach equilibrium, the mixture cylinder would be disconnected and its weight would be determined. The volumes of the two mixture cylinders used in the measurements for the Ne - A r mixtures were determined prior to the experiments. For this purpose we filled each mixture cylinder with pure argon and pure neon at a given pressure and temperature, determined the weight of the charges and calculated the density from the known equation of state data for neon ~2) and argon13). This method enabled us to determine the volume of the mixture

viscometer[ ~

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flexible tubing__

~

H~

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cGlinder ~..

Fig. 2. Experimental a r r a n g e m e n t for the m e a s u r e m e n t of the viscosity of dense gas mixtures.

VISCOSITY OF NEON-HELIUM AND NEON-ARGON MIXTURES

419

cylinders as a function of pressure. It is estimated that the densities of the Ne-Ar mixtures were determined with an accuracy of about 0.1%.

3. Equation of state In the pressure range of our experiments the compressibility factor Z = P V / R T can be represented by a virial expansion truncated after the third termS4), PV R T = 1 + Bp + Cp 2,

(2)

with B = x~Bjl + 2x.ix2B12

+

(3)

x~B22,

(4)

C = x~Clll + 3x~x2Cl12 + 3x~xjCl:2 + x~C222.

Here x~ is the mole fraction of component i in the mixture and B, and C,~ are the second and third virial coefficients, respectively, of pure component i. The virial coefficients B, and C,~ were obtained by fitting (2) to the compressibility isotherms of Michels and coworkers for helium15), neon 12) and argon ~3) at 25°C in the pressure range 0 < P < 160atm corresponding to the range of our experiments. These virial coefficients are included in table II. They are in satisfactory agreement with the virial coefficients quoted by Brewer and Vaughn 16) and by Levelt Sengers et al.~7), but refitting (2) to the original data enables us to retain enough significant figures in the coefficients so that (2) can

TABLE IIA Virial coefficients of N e - A r mixtures at 2~C

TABLE l i b Virial coefficients of H e - N e mixtures at 25°C

Xar

B (cm3/mol)

C (cm3/mol) 2

xNe

B (cm3/mol)

C (cm3/mol) z

0.0000 0.2727 0.5467 0.7722 1.0000

11.43 9.16 3.00 -4.98 - 15.73

228 388 611 849 1145

0.0000 0.1979 0.4035 0.7828 1.0000

11.75 11.93 11.99 11.76 11.43

80 101 127 187 228

420

J. K E S T I N et al.

be used to reproduce the original densities. The coefficients BI2 in (3) and Ct12, C122 in (4) are interaction coefficients which are represented by the combination rules ~6) B12

=

C112 =

(5)

E + (Bll + B22)/2,

(C211C222) I/3,

Ci22 =

(Cl11C~22) 1/3.

(6)

For the mixtures considered here the excess second virial coefficient E was determined experimentally by Brewer and Vaughn16). However, their experiments were restricted to a pressure range of about 2atm and the question arises whether the above equations can be used to calculate the densities at pressures up to 160atm encountered in our experiments. To investigate this question we determined the density as a function of pressure for two Ne-Ar mixtures up to 160atm. The densities were determined as described in the preceding section and the experimental results are presented in table III. The experimental values of the compressibility factor Z = PV/RT were fitted to the above equations with the excess second virial coefficient E as the only adjustable parameter, yielding ENe-Ar= 12.95 cm3/mol at 25°C. The differences between experimental and calculated values for Z are also included in table III. Our result ENe-Ar= 12.95 cm3/mol is in excellent agreement with the value ENe-Ar= 13.00 cm3/mol quoted by Brewer and Vaughn16), validating the procedure. In the case of the He-Ne mixtures, where the third virial yields smaller contributions, we therefore adopted also the above equations with the value EHe-Ne= 0.76 cm3/mo! at 25°C reported by Brewer and Vaughn16).

4. Experimental viscosity data In a previous publication ~) we have reported an extensive set of precise viscosity data for helium and argon, but not for neon. An experimental study of the density dependence of pure neon was, therefore, included in the present project. As a check we also remeasured the viscosity of helium at two different densities. The viscosity of helium at 25°C as a function of density is shown in fig. 3. Our new data points agree within 0.1% with those of Kestin, Payko9 and Sengers~), thus confirming the consistency between our present and previous viscosity data. The experimental viscosity data obtained for Ne are presented in table IV,

V I S C O S I T Y OF N E O N - H E L I U M

AND NEON-ARGON

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II

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MIXTURES

421

422

J. K E S T I N et al. 20.2r I

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Fig. 3. Viscosity of helium at 25°C as a function of density.

TABLE IV Experimental viscosity data for neon P (MPa)

t (°C)

p × 10 3 (mol/cm 3)

r I at 25.00°C ( 10 6 Pa s)

P (MPa)

t (°C)

p × 103 (mol/cm s)

"q at 25.00°C ( I0 6 Pa s)

0.2133 0.3128 0.3576 0.4583 0.5148 0.5463 0.6076 0.7042 0.7470 0.8532 0.9228 0.9904 1.1823 1.4318 1.8561 2.2156 2.9665 3.543

25.08 25.05 25.04 24.86 24.84 24.88 24.87 24.85 24.82 24.78 24.76 24.75 24.95 24.93 24.86 24.86 24.88 24.92

0.08591 0.1259 0.1440 0.1845 0.2072 0.2199 0.2445 0.2832 0.3004 0.3430 0.3709 0.3979 0.4743 0.5738 0.7425 0.8849 1.181 1.406

31.804 31.80~ 31.799 31.800 31.800 31.81; 31.803 31.80~ 31.808 31.80~ 31.815 31.809 31.82~ 31.836 31.83~ 31.864 31.88~ 31.916

4.028 4.497 5.035 5.655 6.375 7.041 7.641 8.354 9.075 9.558 10.113 10.643 11.119 11.606 12.137 12.768 13.369 14.113

24.87 24.84 24.82 24.94 25.00 25.05 25.16 25.04 24.97 24.96 24.96 24.98 24.88 24.88 24.90 24.82 25.00 24.88

1.595 1.777 1.985 2.222 2.497 2.749 2.973 3.242 3.510 3.689 3.893 4.087 4.262 4.438 4.629 4.857 5.069 5.334

31.949 31.96; 32.00~ 32.031 32.086 32.116 32.157 32.216 32.277 32.32~ 32.337 32.404 32.429 32.476 32.524 32.577 32.598 32.68,,

for H e - N e mixtures in table V and for N e - A r mixtures in table VI. Throughout this paper the c o m p o s i t i o n of the mixtures is specified in terms of mole fractions. In the case of N e and the H e - N e mixtures, the viscosity was measured as a function of pressure; the densities were calculated from the experimental pressures and temperatures as described in section 3. In the case

VISCOSITY OF NEON-HELIUM AND NEON-ARGON MIXTURES

423

TABLE V Experimental viscosity data for He-Ne mixtures P (MPa)

t (°C)

p x 103 (mol/cm 3)

-q at 25.00°C (10 _6 Pa s)

0.08613 0.1275 0.1702 0.2057 0.2406 0.2785 0.3514 0.4014 0.4909 0.5733 0.6996 0.8081 1.016 1.159 1.372 1.612 1.793 1.958 2.169 2.365 2.551 2.764 2.960 3.142 3.332 3.511 3.675 3.906 4.073 4.269 4.414 4.646

24.213 24.224 24.226 24.212 24.216 24.227 24.211 24.215 24.225 24.212 24.21o 24.21j 24.21o 24.199 24.195 24.2(h 24.186 24.207 24.186 24.204 24.194 24.204 24.213 24.211 24.215 24.21s 24.22~ 24.220 24.21s 24.230 24.237 24.240

0.09463 0.09482 0.1239 0.1588 0.2023 0.2386 0.2859 0.3226 0.3530 0.3974 0.4604

27.317 27.332 27.327 27.333 27.333 27.329 27.327 27.31 j 27.326 27.314 27.331

P (MPa)

t (°C)

p x 103 (mol/cm3)

~ at 25.00°C (10 -6 Pa s)

25.02 25.01 25.01 25.00 24.99 24.99 24.82 24.80 24.93 24.92 24.90 24.84 24.92 24.91 24.90 24.90 24.90 24.87 24.87 24.92 24.90 24.86 24.83

0.5497 0.6860 0.8396 1.044 1.229 1.400 1.601 1.795 2.010 2.204 2.358 2.566 2.764 2.979 2.980 3.162 3.340 3.531 3.702 3.900 4.092 4.259 4.432

27.317 27.311 27.334 27.327 27.323 27.325 27.343 27.339 27.342 27.345 27.346 27.366 27.355 27.376 27.364 27.370 27.384 27.392 27.404 27.412 27.420 27.433 27.452

xNe = 0.7828 0.2463 25.06 0.2856 25.07 0.3801 2 5 . 1 1 0.4353 25.12 0.5322 24.89 0.6059 2 4 . 9 3 0.6741 2 4 . 9 2 0.7569 2 4 . 9 3 0.8208 25.06 0.8969 2 5 . 1 3 1.0018 25.06 1.2996 2 5 . 0 8 1.5524 25.06 1.9845 25.06 2.5388 25.06 3.063 25.05 3.595 25.04 4.029 25.04 4.557 25.05 5.036 25.05

0.09920 0.1150 0.1530 0.1751 0.2142 0.2437 0.2711 0.3043 0.3297 0.3600 0.4021 0.5208 0.6214 0.7928 1.012 1.218 1.426 1.594 1.798 1.983

30.685 30.686 30.687 30.687 30.68s 30.691 30.690 30.689 30.696 30.695 30.70j 30.707 30.696 30.7h 30.730 30.740 30.769 30.773 30.790 30.813

XNe= 0.1979 0.2136 2 4 . 7 6 0.3163 24.76 0.4225 2 4 . 7 6 0.5107 2 4 . 7 5 0.5976 2 4 . 7 5 0.6920 2 4 . 7 5 0.8740 2 4 . 7 5 0.9992 2 4 . 8 1 1.2235 2 4 . 8 6 1.4304 2 4 . 8 7 1.7482 2 4 . 8 7 2.0215 2 4 . 8 2 2.5483 24.76 2.9099 24.74 3.454 24.73 4.070 24.72 4.539 24.83 4.967 24.86 5.514 24.83 6.028 24.84 6.517 24.84 7.083 25.03 7.607 25.12 8.089 25.02 8.599 25.02 9.078 24.94 9.523 25.03 10.150 25.05 10.609 25.12 11.147 25.14 11.547 25.20 12.187 25.17 xN~ = 0.4035 0.2350 0.2353 0.3076 0.3946 0.5028 0.5932 0.7111 0.8030 0.8789 0.9899 1.1479

25.13 24.89 24.97 24.98 24.95 24.95 24.95 25.04 25.04 25.02 25.02

1.3718 1.7149 2.1028 2.6206 3.091 3.528 4.045 4.545 5.104 5.610 6.014 6.559 7.085 7.654 7.657 8.143 8.620 9.133 9.595 10.138 10.661 1I. 116 11.592

424

J. K E S T I N et al.

TABLE V (cont.) P (MPa)

t (°C)

p×103 (mol/cm 3)

r/ at25.00°C (10 6 Pa s)

P (MPa)

t (°C)

pxl0 ~ (mol/cm ~)

r t at 2 _5 . 0 0° ( (10 6 Pa s)

5.614 5.973 6.463 7.091 7.496 8.086

25.04 25.04 25.06 25.06 24.93 24.98

2.½05 2.342 2.528 2.766 2.920 3.140

30.836 30.848 30.864 30.895 30.926 30.943

8.630 9.098 9.562 10.076 10.482

25.08 25.14 25.05 24.92 24.94

3.342 3.515 3.687 3.878 4.026

30.95~ 30.982 31.016 31.044 31.05~

TABLE Vl Experimental viscosity data for N e - A r mixtures t (°C)

p × l03 (mol/cm 3)

~ at 25.00°C (10 6 Pa s)

t (°C)

0 x 10~ (mol/cm ~)

"O at 25.00°C (10 6 Pa s)

0.06100 0.09061 0. 1292 0. 1749 0.2128 0.2515 0.2915 0.3310 0.3826 0.4182 0.4950 0.5871 0.6549 0.7155 0.7978 0.8681 1.066 1. 255 1.459 1.645 1.817 2.004 2. 184 2.378 2.588 2.773 2.965 3. 158 3.344 3.536 3.739

28.75S 28.774 28.801 28.788 28.78a 28.792 28.82O 28.8 IO 28.823 28.833 28.854 28.882 28.871 28.874 28.89~ 28.914 28.96O 28.984 29.046 29.09O 29. 133 29. 194 29.23t 29.31 t 29.363 29.417 29.458 29.544 29.597 29.67~ 29.769

25.01 25.00 24.99 24.98 25.04 25.03 25.03 25.02 25.03 25.04 25. 11 25. 16 25.23

3.905 4. 108 4.293 4.478 4.671 4.831 5.031 5. 194 5.429 5.625 5.822 5.986 6. 168

29.8h 29.92t 29.961 30.086 30.08O 30. 173 30.250 30.335 30.417 30.514 30.609 30.65,, 30.747

0.09076 0. 1080 0. 1344 0. 1546 0. 1734 0. 1993 0.2244 0.2402 0.2571 0.2807 0.3013 0.3185 0.3553 0.3781 0.4177 0.5241

26. 157 26. 174 26. 1%. 26. 178 26. 178 26.200 26.206 26.2~ 26. 1% 26.224 26.21 26. 234 26.233 26.25, 26.26~ 26.30~

XAT= 0.2727 25.14 24.96 24.96 24.98 24.97 24.97 24.96 24.96 24.85 24.79 24.77 24.93 24.98 24.93 25.08 25.06 25.06 25.02 25.03 25.02 25.04 25.04 25.04 25.04 25.04 25.05 25.05 25.05 25.07 25.01 25.00

XAT= 0.5467 25.01 25.05 25.07 25.09 25.08 25.09 25.04 25.03 25.04 25.04 25.04 25.04 25.08 25.08 25.08 25.09

VISCOSITY OF N E O N - H E L I U M AND NEON-ARGON MIXTURES TABLE VI (cont.) t (°C)

p X 103 (mol/cm 3)

7"/ at 25.00°C (10-6 Pa s)

25.10 25.10 25.06 25.12 25.14 25.03 25.06 25.03 25.04 25.02 25.08 25.09 25.08 25.03 25.05 25.04 25.08 25.07 25.03 25.03 25.03 25.07 24.96 24.94 25.08 25.04 25. I 1 25.10 25.00 24.94 25.06 25.08 25.05 25.04 25.04 25.01 24.91 25.05 XAr= 0.7722

0.6207 0.7338 0.8511 0.9890 1.184 1.401 1.593 1.727 1.854 2.005 2.207 2.350 2.582 2.720 2.830 3.013 3.244 3.424 3.611 3.805 3.955 4.064 4.200 4.328 4.470 4.574 4.719 4.861 4.988 5.191 5.269 5.365 5.721 5.847 6.234 6.429 6.619 6.78 !

26.329 26.373 26.417 26.464 26.517 26.583 26.674 26.722 26.762 26.826 26.926 26.994 27.102 27.189 27.233 27.332 27.444 27.539 27.657 27.789 27.845 27.909 28.027 28.07o 28.163 28.249 28.336 28.396 28.555 28.624 28.705 28.744 28.976 29.143 29.42s 29.54o 29.698 29.787

25.09 25.05 25.04 25.04 25.12 25.13 25.04 25.06 25.06

0.06151 0.08905 0.1196 0.1447 0.1721 0.2021 0.2230 0.2482 0.2836

24.272 24.285 24.30t 24.306 24.326 24.341 24.347 24.356 24.364

t (°C) 24.99 24.87 25.06 25.06 25.07 25.07 25.09 25.10 25. l I 25.11 25.03 25.06 25.06 25.05 25.12 25.16 25.20 25.21 24.93 25.03 25.02 25.05 25.07 25.03 25.03 25.06 25.01 25.02 25.01 25.02 25.10 25.09 25.08 25.03 25.10 25.06 25.02 25.06 25.06 25.07 25.08 25.03 25.05 25.07 25.08 25.11 25.08 25.06

p × 103 (mol/cm 3) 0.3082 0.3343 0.3580 0.4085 0.4651 0.5528 0.6389 0.7555 0.8545 0.9432 1.042 1.249 1.332 1.415 1.498 1.644 1.788 1.964 2.056 2.122 2.250 2.444 2.797 2.964 3.103 3.271 3.378 3.470 3.540 3.629 3.708 3.787 3.836 3.894 3.998 4.100 4.197 4.267 4.360 4.443 4.540 4.604 4.724 4.872 5.047 5.185 5.356 5.533

-r/ at 25.00°C (10 -6 Pa s) 24.378 24.387 24.40j 24.414 24.438 24.466 24.5 l j 24.574 24.600 24.642 24.693 24.794 24.843 24.873 24.921 24.991 25.081 25.189 25.236 25.287 25.354 25.47o 25.684 25.821 25.902 26.04s 26.121 26.187 26.232 26.305 26.383 26.440 26.48o 26.554 26.625 26.708 26.776 26.846 26.909 26.95~ 27.06j 27.087 27.297 27.332 27.462 27.616 27.774 27.910

425

426

J. KESTIN et al.

of the N e - A r mixtures the viscosity was measured directly as a function of density except for the first 20 data points at xA, = 0.5467 for which the densities again were calculated f r o m the experimental pressures and temperatures. In all cases a small t e m p e r a t u r e correction was applied to the data so that all quoted viscosities c o r r e s p o n d to the t e m p e r a t u r e t = 25.00°C at the experimental density. In the case of neon we used the t e m p e r a t u r e coefficient d'o/dt = 0 . 0 7 2 x 10 6 p a s / K as previously determined by Kestin, Ro and Wakekam18). In the case of the N e - H e mixtures we used the values drl/dt = 0.0583 x 10 6 Pa s/K (XNe= 0.1979), d~/dt = 0.0665 x 10 6 Pa s/K (XNe = 0.4035), drl/dt = 0 . 0 7 0 8 x 10 6 p a s / K (XNe=0.7828) and in the case of the N e - A r mixtures d'o/dt = 0.0735 x 10 -6 Pa s/K (XAr = 0.2727), d'q/dt = 0.0690 X 10 6 Pa s/K (XA, = 0.5467), drl/dt = 0.0678 × 10 --6 P a s/K (xA, = 0.7722), as previously determined by Kestin and Nagashima6). Our data have a precision of about -+0.05% and an estimated a c c u r a c y of about -+0.2%. As shown in fig. 4, our data are slightly larger than those obtained by Flynn et at. 2") and by Trappeniers and c o w o r k e r s 2') with the use of a capillary flow viscometer. The same trend can be seen for our previous m e a s u r e m e n t s of the viscosity of argon, shown in fig. 5, where the difference between our data and those of the investigators at the Van der Waais laboratory 2:) reaches a m a x i m u m of about 0.6%. This difference was noted earlier by Kestin and Wang 23) and Kestin and Leidenfrostg). The agreement of our data with those of Flynn et al. :°) for helium is good, as shown in

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o

o o o °rs

°x c This work u Kestin, White4aw x Flynn el ~Jl

315

v

31.0

I

•

0

I

3

Trappeniers et ai.

1

1

4

5

p x l O s, m o l e / c m s

Fig. 4. Viscosity of neon at 25°C as a function of density.

6

VISCOSITY OF N E O N - H E L I U M A N D N E O N - A R G O N MIXTURES 26.0

--

,

1

i

427

i

ARGON

o

25°0

o

25.5

'7 u

o o

25.0

o~ o

7

o

24.5 O_

~d

o

o u 7

o

~- 24.0

J OiE%

23.5 0

7

~c o Kesfin, Peyko~, S e n g e r s • Kesfin, Le]derffrost

2&O

o K e s l i n , Whitelow

o~

× F l y n n et al. -~ M i c h e l s , B o l z e n , S c h u u r m a n

22s8-

i

I

4 p x IOs, mole/cm 3

Fig. 5. Viscosity of argon at 25~C as a function of density.

fig. 3. Flynn and coworkers 2°) claimed an accuracy of 0.1%; we now believe that the accuracy of their data is about 0.4%. Trappeniers and coworkers 2') claimed an accuracy of 0.3%; again, a new analysis of the working equation of the instrumen¢) indicates the possibility of larger systematic deviations. Haynes 24) has measured the viscosity of argon as a function of density with a torsionally oscillating crystal with an estimated accuracy of 2%; his data agree with our data within the claimed accuracy. In figs. 6 and 7 we show the viscosity of He-Ne and of Ne-Ar, respectively, as a function of density. For comparison we have also included in these figures the viscosity of the pure components. In figs. 8 and 9 we show the viscosity rl0 in the limit of low densities as a function of the composition of the mixtures in mole fractions. In these figures we have also included viscosity data for these mixtures previously reported in the literature. Our data as a function of composition are in good agreement with the viscosity data for these mixtures .previously determined by Kestin and Nagashima 6) and extrapolate smoothly to the viscosity data obtained by Kestin, Payko9 and Sengers') for helium and argon. The experimental data of

428

J. K E S T I N et al.

34

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

T

. . . . .

~

He ~ Ne 25°0 52

........ , . :::z: :

NEON , ,..~, .....

• ....

. . . . ......

,

x. ~0.~828

50

£

28

. . . . . . . . . . . . . . . . . . . . . . .

XNe=0.4055

2e x

24

"~J:'

'

'

i

,

,

....

....

XNe=O.1979

22 /

HELIUM

[ 0

I

2

3 4 p x l O 3, m o l e / c m 3

5

6

Fig. 6. Viscosity of H e - N e mixtures at 25°C as a function of density.

54

................................................. Ne-Ar 25°0

NEON

52

50 co EL

28

->.( 2 6

£RGON

24

~,~,- 0 . 0 0 0 0 ~,~=0 2 7 2 7

22

~"

×~, - 0 . 5 4 6 7

I

×A - 0 . 7 7 2 2 , x~:~l.O000

2O ............................................................. 0 I 2 .~

I j

4

5

6

p x l ~(m3 , ' , m o l e / c, rT/~

Fig. 7. Viscosity of N e - A r mixtures at 25°C as a function of density.

7

VISCOSITY OF N E O N - H E L I U M AND N E O N - A R G O N MIXTURES 34 He-Ne 25°C

32

50

28

~0- 26 xo F24 • This work 22 t r

,~ Kestn, Neg0shimo • De Troyer et oL o Traulz, Binkele v Troutz, Kipphon

20

18

_

I

0.2

0

J

01.4

0.6

018

XNe

Fig. 8. Dilute gas viscosity ~10 of H e - N e mixtures as a function of composition.

54 Ne-Ar 25°C

• This work a Kestin, Poyko~, Sengers ,~ Kestin, Nagashimo • Rietveld, van Itterbeek o Troutz, Binkele

52 ~ 5O

o~ 28 xo ~.- 26

24

22

20 0

012

0!4

016

o'.8

xA r

Fig. 9. Dilute gas viscosity ~10 of N e - A r mixtures as a function of composition.

429

430

J. K E S T I N et al.

van Itterbeek and coworkers 25'2~) have an internal consistency of about 2%. The old data of Trautz and coworkers 27'28) were determined relative to the viscosity of air; as noted by Van den Berg and Trappeniers 5) use of more recent values for the viscosity of air would increase the data by about 0.6%. In view of these uncertainties it is possible to assert that the older data are in acceptable agreement with the current values.

5. Density expansion of viscosity Modern kinetic theory of gases predicts that the density dependence of the viscosity of a moderately dense gas should be represented by an expansion of the form 3'29) "q = 7/+ "q~P + ~/~p2 In p + ~2p 2 +. • ..

(7)

Assuming this expansion to be valid, we try to deduce estimates for the possible values of the coefficient -t/~ of the logarithmic contribution consistent with the experimental data. For this purpose we follow the same procedure used in previous publications~'2). We first fit the experimental data to the linear equation = 7/o + "q~p

(8)

in a density range where the higher order terms in (7) are not yet statistically significant. Treating ~ as a parameter, we adopt a possible value for 7/; and fit the experimental values of T / - ~ p : in p to a quadratic equation • / - */'P2 In P = ~o+ *hP + ~12P2.

(9)

The adopted value for ~ is said to be consistent with the data, if the fits to the eqs. (8) and (9) yield, within two standard deviations, the same values for the coefficients ~0 and ~ , the condition for ~j being the more sensitive one. In implementing this procedure we expressed previously the density in terms of the dimensionless amagat unit"2). H o w e v e r , we find this unit less convenient here, since it would require a determination of the density of each mixture at 0°C and 1 atm. We therefore continued to express the density in terms of mol/cm 3. Since the terms T/~p2 in p and ~2p 2 are both present in (9), the results will be independent of the unit chosen for p. As an example we consider the data for the N e - A r mixture with XAr = 0.2727. In fig. 10a we have plotted the values obtained for "q~_+2tr,, as a function of the number, u, of data points when the data are fit to the linear equation (8). The values of ~1 increase systematically for u > 18, so that the linear range is restricted to u -< 18 which corresponds to p -< 1.3 x 10 3 mol/cm 3

VISCOSITY OF NEON-HELIUM

quadratic

quadratic fit

hnear fit -6 E

AND NEON-ARGON MIXTURES

5

~ : + 001 Pa.s cm6/mole 2

~9~2:-0003

431

quadratic fll

fit Pa.s cm6/mole 2

~2: + 0 0 2 3 Pas cm6/mole 2

E co

g2 x

b:l

c~

gI 10

I 15

I

I

I

I

I

20

20

25

30

55

I 20

z/

(o)

(b)

I 25

I 50

3~5

21O

215

v

z/

(c)

(d)

I 50

J 55

Fig. 10. The coefficient r h - 2trn[ as a function of the n u m b e r , v, of data points for the N e - A r mixture with xA, = 0.2727.

and we obtain for v = 18 rio - 2trn 0 = (28.755 --- 0.009) x 10 -6 P a s, "01 - 2(r,, = (1.84 --- 0.15) x 10 -4 P a s cm3/mol,

(10)

with (r, = 0 . 0 3 % . W e s u b s e q u e n t l y fit the data to (9), v a r y i n g ri~ f r o m - 0 . 0 0 3 P a s(cm3/mol) 2 to + 0.023 P a s(cma/mol)2; the results o b t a i n e d for r i t 2(rn, are s h o w n in figs. 10b-10d. T h e e q u a t i o n r e p r e s e n t s the data up to v = 34 w h i c h c o r r e s p o n d s to p <- 4.3 × 10 -3 m o l / c m 3. F o r ri~ = 0.01 P a s(cm3/mol) 2 we obtain ri0 - 2try0 = (28.754 - 0.008) x 10 -6 P a s, ril - 2o-~, = (1.81 - 0.12) × 10 -4 P a s cm3/mol,

(11)

ri2 - 2o'~2 = (7.8 - 0.3) x 10 -2 P a s(cm3/mol) ~, with trn = 0.04%. On c o m p a r i n g (10) and (11), we c o n c l u d e that the values r e t u r n e d for rl0 and riz are in a g r e e m e n t . If we a d o p t ri~ = - 0 . 0 0 3 P a s ( c m 3 / mol) 2, we obtain a value ri1 - 2trn, = (1.4 - 0.1) × 10 -4 P a s cm3/mol in d i s a g r e e m e n t with the result quoted in (10) f r o m the linear fit, while a d o p t i n g r i i = + 0 . 0 2 3 P a s ( c m 3 / m o l ) 2 tends to yield values of ri1 larger than that o b t a i n e d f r o m the linear fit. A s s u m i n g that we c a n neglect higher o r d e r t e r m s b e y o n d those retained in (9), we c o n c l u d e that a n y possible values f o r the coefficient ri~ are restricted to ri~ = (0.010 - 0.013) P a s(cm3/mol) 2. T h e d e n s i t y d e p e n d e n c e of the e x p e r i m e n t a l v i s c o s i t y data of the m i x t u r e s at all c o n c e n t r a t i o n s w a s a n a l y z e d b y this p r o c e d u r e . T h e results thus

432

J. K E S T I N

et al.

obtained for the coefficients ?7o, 37j and 37~ of the theoretically predicted density expansion (7) are presented in table VII. The coefficient 371 is plotted as a function of concentration in fig. 11. For the H e - N e mixtures the first density correction 371 varies from a small negative value for helium to a small positive value for neon; as a result the viscosity of these mixtures changes little with density and it becomes difficult to deduce narrow bounds for the coefficient 37~. For the H e - N e mixtures and for the N e - A r mixtures the possible range for 37.~ includes zero so that we cannot prove the positive existence of a logarithmic density contribution from the data. The logarithmic contribution to the density dependence of the transport properties has been investigated theoretically for a Lorentz gas 3°-32) and for a gas of hard spheres33-36). In the case of the Lorentz gas one considers the diffusion coefficient of a light particle in a medium of heavy particles as a simplified model for the mutual diffusion coefficient of a binary gas mixture. Theoretical estimates of the logarithmic contribution for the viscosity are only available in the case of a one-component fluid of hard spheres. For a gas of hard spheres it is convenient to rewrite the density expansion (7) in the form & = 1 + 37~'n~r3 + 37~'(mr3) 2 In mr 3 + 37~(no'3)2 + . • -, 370

(12)

.... 5 / m k T \ j/2 370-- l.OlOl-~--~ k ~ ) •

(13)

with

Here n is the number density, ~r the molecular diameter, m the molecular mass and k Boltzmann's constant. The coefficient 37~' of the contribution

t?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ne -L,r

He N,~

25°C

~-

4

E c)

q

J -

-

2

0

02

,04

06

£,,.R

,0

C'

02

(}4

:} 6

x~t

XNe

(a)

{~;

~-

Fig. 1 1. The coefficient *q~ ± 2 ~ , for H e - N e and N e - A r mixtures as a function of composition.

VISCOSITY OF NEON-HELIUM

AND NEON-ARGON

MIXTURES

433

+1 ~.

+1

~

"H

+1

+1

~

+1

~.

I

:z

o

+1 ~.

~ ÷ ' ~ ~¢ 2~~_ -

+1

÷1 ÷1

0

~

"H

°

.~.

oo +1 -H ~

~

[...., .~

¢:, ÷1

[..., .N

o 0

+1

~.

•

"~-8 0

r,,,)

xxx-~

x x x ~

---~~ , ~ . ~ . , b b b~7~

+1 +1 +1 ~

x

¢'-11'~11P, I

+1 ÷1 +1

~

,'~ X

.~,.- .~,,.,

434

J. KEST1N et al.

linear in no "3 is 37) "OT =0.403 -+0.002. For the coefficient rl~' two theoretical estimates have been reported in the literature. Kan 36) reported rl~'= 0.641, while Gervois et al. 33'34) found -0,~' = 21 or r/~' = 42 as reinterpreted by Kan36). In order to compare these predictions with our experimental results, we attributed to each mixture an effective molecular diameter ~ren by identifying the observed value of ~/0 with that of a gas of hard spheres given by (13). The results then deduced from our experiments for "qT and r/~' are included in table VII. Because of the different effects of the kinetic and potential energy contributions, a gas of hard spheres is not a realistic model for the first density correction "0T for a gas of molecules with attractive forces2'38). However, the coefficient r/~' has a purely kinetic origin arising from the same sequences of successive binary collisions as those for a gas of hard spheres. Thus, unless special cancellations are present, one may expect the coefficient "O~' of real gases to be of the same order of magnitude as for a gas of hard spheres. From the information provided in table VII we find that Kan's estimate 36) r/~'= 0.641 is within the range of possible values consistent with the experimental data while the estimate of Gervois et al. 33'34) is not. The same conclusion was reached earlier from the experimentally observed density dependence of the viscosity of one-component fluids2'4). For practical purposes, such as interpolation, the viscosity data at each composition can be represented empirically by a cubic polynomial (14)

1"I = 1~0+ l~lp + ~2p 2-1- ~3p 3,

with the values of the effective coefficients fi~ listed in table VIII appropriate for the density range Ap of our experimental data.

6. Composition dependence of the viscosity at low densities The viscosity "q0 of gas mixtures in the limit of low densities is given by the Chapman-Enskog solution of the linearized Boltzmann equation. Retaining one Sonine polynomial in the perturbation solution, one obtains 39'4°)

[ x~ with

x~

xlx2

H°2 =

2xlx2H°2"~[ 1_ (H°2) 2 ~ ' ,

m,2 .( s_2__ )

"q~12(ml + m2)k3A'~2

1 ,

x?± xlx2 m,2 ( 5 . m2"~ H°' = ~ ~--~2 ( m ~ m2jk-3A--~2± m,]' x~ ± x,x2 H2°2 = ~

_m,2_ .( 5___~ m,) (ml + m2)k3A'~2 +m22 "

(15)

(16) (17a) (17b)

VISCOSITY O F N E O N - H E L I U M A N D N E O N - A R G O N M I X T U R E S

435

TABLE VIIIA

Coefficients -~ in the empirical density expansion rl = E~-~ ~p~ for N e - A r mixtures XAr

"00× 106 "~1× 104 (Pa s) (Pa s cm3/mol)

"02× 102 • (Pa s(cm3/mol) 2)

~3 (Pa s(cm3/mol) 3)

O',~ (%)

Ap× 103 (mol/cm 3)

0.0000 0.2727 0.5467 0.7722 1.0000

31.793 28.765 26.144 24.265 22.608

3.308 4.190 4.815 7.313 7.714

-1.934 -1.885 -0.890 -2.473 - 2.114

0.02 0.06 0.06 0.05 0.03

5.3 6.2 6.8 5.5 4.3

0.445 1.343 2.561 3.326 4.445

TABLE VIIIB

Coefficients fl~ in the empirical density expansion -q = E~=o f/~p~ for H e - N e mixtures XNe

0.0000 0.1979 0.4035 0.7828 1.0000

~/0 × 106 (Pa s)

hi × 104 (Pa s cm3/mol)

"02 × 102 (Pa s(cm3/mol) 2)

0% (%)

(mol/cm 3)

Ap × 103

19.879 24.224 27.327 30.679 31.784

- 0:304 - 0.252 - 0.104 + 0.400 +0.735

0.587 0.630 0.833 1.364 1.816

0.03 0.03 0.03 0.02 0.03

3.9 4.7 4.4 4.0 5.3

Here mi and xi are the molecular weight and the mole fraction of component i in the mixture, rote = 2mlmE/(m~ + mE), ~/0 is the dilute gas viscosity ~0 of pure component i and A~: a dimensionless ratio of collision integrals which, to a good approximation may be taken to be 4~) AI'2 = 1.10. The quantity ~°2, commonly referred to as the interaction viscosity, is the viscosity of a hypothetical one-component gas with molecules of mass m12 with an interaction potential of that between molecules of components 1 and 2. For noble gas mixtures this interaction viscosity can be predicted from the extended law of corresponding states formulated by Kestin and coworkers4~). Conversely, this interaction viscosity can be deduced by fitting (15) to our experimental values for ~0 yielding ~°e_Ne=21.95X 10-6pas,

"q°e_A~=26.38X 10-6pas.

(18)

The values thus calculated from (15) are represented by the curves in figs. 8 and 9 and reproduce our experimental data for the viscosity ~/o of the mixtures within 0.05%.

436

J. KESTIN et al.

7. Composition dependence of the viscosity at elevated densities; comparison with the theory of Enskog and Thorne An attempt to formulate a method for estimating the viscosity of dense gas mixtures, given the viscosity of the pure components at elevated densities and that of the mixtures at low densities, was proposed by Di Pippo et al.7). This method uses as a model the composition dependence of the viscosity predicted by the theory of Enskog and Thorne for dense gas mixtures of hard spheres45). H o w e v e r , in view of some subsequent developmentsS), we find it desirable to reconsider the method in some detail. According to the theory of Enskog, the viscosity ~i of a dense onec o m p o n e n t gas of hard spheres with molecular mass rn~ and diameter o-~ can be represented by 45'46) r/i = r/°(1 + 3biPxi) + Xi

(bip)2xi('rrmikr) j/2,

(19)

while the equation of state is given by PV = 1 + bipxi. RT

(20)

Here bi is the covolume associated with the molecules of c o m p o n e n t i such that bip = ~rrno'~. In this paper we continue to express the density in terms of mol/cm 3 so that p = n/N, where N is Avogadro's number. Thus

bi = 3z"n'No'~.

(21)

The quantity Xi is the value of the radial distribution of c o m p o n e n t i at separation o'i; it has a density expansion of the form X~ = 1 + Xll)p + . •.,

(22)

XI~I= 8~b~.

(23)

with

Since the mixtures formulas, earlier introduced in the previous section, refer to the first Sonine approximation, we consider, for consistency, also the equation for the viscosity of the dense gas in the first Sonine approximation. In this approximation "0° = (5/16~o'~)(1rmikT) I/2, and eq. (19) can be written as

r/= r/°[(1 +~ bipxi)2+ 48 Xi

257r

(bip)2xi

]

"

(24)

The theory of Enskog was extended to a dense binary gas mixture by Thorne45). Introducing the combination rule ~J2 = (trl + ~r2)/2, we can write the

VISCOSITY OF N E O N - H E L I U M AND N E O N - A R G O N MIXTURES

437

theoretical formula for the viscosity of a dense binary gas mixture as

48 r 2 0 + 2-~tx ,rl ~(blp) 2XH + 2XlXzrl°2(bl2p)zx,2 + x~'o°(b2p)2x22],

(25)

with 2 3 = l ( b I/3 + bl/3)3, b12 = ~rNtr12

Yl = xt[1

2 +~(x,bNpxH + ml2 m---'~x2bl2PXl2)],

2 Hn

(26)

m12

x,x2x,2( m,2 rl°2 \rnl + m2/\3A*2

i

X2XIIIXIX2XI2( m12

H,,=~-

)(

m12

~{

(28)

l 5

no2 \mr+m2~ ~

n22=xZx22+xix2x12( no

(27a)

+m2)

(29a)

m,/' 5

+if/l)

rl°2 \ml + m2/\-fA-~2~2 ~ } '

(29b)

Here '}'102 = (5/167rtrE2)(TrmlzkT) 1/2 is the interaction viscosity at low densities introduced in the previous section. The equation of state of the binary mixture is given by PV

RT

= 1 + (x2blpXH

+ 2XlXEblEOX12+ x2622pX22).

(30)

The quantities X~J represent the values of the radial distribution of the mixtures between molecules i and i at the point of contact. They have a density expansion of the form X~J= 1 +Xu(1),~ e +'",

(31)

Xt~ = ~{Xlbl[8 - 3(~r,/cr,2)] + x2b218 -- 3(Or2/Or12)]},

(32)

with

Xt~) = ~bl{5xl + x2[1 +

16(O'122/Orl) 3 -- 12(Or12/Orl)2]},

(33a)

X~ = ~b2{5x2 + Xl[1 + 16((r22hr2) 3 - 12(o'12/(r2)2]}.

(33b)

On comparing (32) and (33) with (23), we note that the coefficients X!]) for the mixture can be related to the coefficients X! 1) of the pure components by 3 ~'~')" - Xt '~'' X/~ = XlXl 1) -t- X2X ~1)1t- 3 (XIXlI) -- X2X~I)) X~I)11'"t- Xt I)11''

(34)

438

J. KESTIN et al. 1

(1)

r

+ 2(X~''/3 + Xt'") 3 - 3Xt'"~(Xt'"~ + Xt""~):],

(35a)

1 r ~1) Xt~ ~ = x 2 x t " + ~ x l t x ~ + 2(X~ ''~3 + Xt')"') 3 - 3Xt"/3(Xt ''13 + Xt"/3)21.

(35b)

XIl~~ = x ~ x ~ ' + ~x2tx~

The theory of Enskog and Thorne for dense gas mixtures was revised by Van Beijeren and Ernst so as to ensure consistency with the Onsager reciprocal relations47). However, the resulting corrections do not affect the theoretical expressions for the viscosity of a mixture. Many investigators have tried to use the Enskog equation (24) to predict the viscosity of dense one-component gases. For a discussion of the results thus obtained for the viscosity of dense noble gases the reader is referred to some earlier reviews48-5°). Here we follow an opposite approach in that we try to determine effective values for the covolume b~ and the parameter Xi so that the Enskog equation (24) reproduces the known experimental viscosities of the dense one-component gas. For this purpose it is convenient to write (24) in the form 8) 7)

_

0

=__L_I +

biPXi

ot +/3bipxi,

(36)

.

(37)

with 4 a=5,

/3=

4(~_) 1+

It is well known that the kinematic viscosity ~/0, when plotted as a function of the density 0 reaches a minimum value (~/0)m~, at a certain density 0 = Omi~s~)• Eq. (37) predicts that this minimum value is given by (~-) min = (Og-~-2"~/-~)bi'f~ O.

(38)

If we adopt an effective value for the covolume bi, effective values of the parameter X~ can be obtained from the experimental viscosity data by solving (36)

biPXi,*_

0 -- a +--"X/(O - a ) 2 - 4 ~

=-

2/3

(39)

In the method, originally proposed by Di Pippo et al., an effective value of the covolume b~ was determined as b~ = B~i + T ( d B ~ J d T ) , where B~ is the second virial coefficient of the real gasT). This choice is obtained by identifying the thermal pressure T ( a P V / O T ) p = 1 + b~pxi of a gas of hard spheres with the thermal pressure of the real gas46). However, this procedure leads to a number of complications. First, it follows from (38) and (39) that (36) has a real solution at all densities only if b~ <- (rl/P)min/rlO(o~ + 2~v//3); this condition is

VISCOSITY OF NEON-HELIUM AND NEON-ARGON MIXTURES

439

not necessarily satisfied by the above mentioned procedure. Furthermore, for a gas of hard spheres X~ is a monotonically increasing function of p. Identifying X~ with the root X~,- so as to ensure the correct low density behavior, Di Pippo et al. found that at high densities the effective values obtained for X~ started to decrease with density reaching unphysical values smaller than unity. It was pointed out by Sandier and Fiszdon 8) that these complications are avoided if b~ is determined from (38) and X~ is identified with X~,- for p <- pm~,and with X~,+for p -> pmin- This procedure guarantees that the effective X~ is a real, continuous and monotonically increasing function of p. Actually, it appears that the difficulties, encountered by Di Pippo et al. do not occur in the density range of the experimental data presented in this paper. Nevertheless, we prefer to follow the method suggested by Sandier and Fiszdon, first because the method is theoretically more satisfactory and secondly because it allows one in principle to extend the method used here to higher densities. In order to determine (r//p)min for helium, neon and argon, we fitted the experimental viscosity to a polynomial in p. The data in this paper do not extend to high enough densities to determine this minimum. Therefore, we combined our data with those available elsewhere in the literature for helium2°'52'53), neon 2~'54) and argon22'24). The results obtained, together with the

54

. . . .

T

He-Ne

NEON

32~ ..............

xNd0.7828

30 28

xNe:0.4055

x g-

....................................................

xNe=0.1979

24 22 HELIUM

20 1

0

I

........

L. . . . .

2

i

I

3 4 pxlO 3, mole/cm 3

L

5

6

Fig. 12. Comparison of the viscosity of He-Ne mixtures as a function of density with the values estimated on the basis of the Enskog-Thorne theory for dense gas mixtures.

440

J, K E S T I N el

al.

corresponding values of bi, are presented in table IX. It should be noted that in the case of helium an extrapolation of the experimental data was necessary to estimate the value of (~/p)m,.. As a next step we determined for helium, neon and argon the coefficient )¢IT~ in the density expansion (22) of the effective ~ by substituting (22) into (36) and fitting the resulting equation to the experimental viscosity data. The values thus obtained for Xl ~ are also included in table IX. The viscosity values calculated from (36) with the adopted values of b~ and Xi are represented by the solid curves in figs. 12 and 13. The equation reproduces the experimental viscosity data of helium, neon and argon with a standard deviation o-~ of 0.04%, 0.07% and 0.12%, respectively. We thus conclude that terminating the density expansion of X~ after the term linear in p is adequate for the density range of our experimental data. We also note from (20) that this procedure is TABLE IX Quantities needed for a comparison with the theory of E n s k o g and Thorne

He Ne Ar

(/]/p)mm × 104 (Pa s cm3/mol)

b, (cm3/mol)

cri × 108 (cm)

Xc,n (cm3/mol)

5.304 14,13 27.87

10.44 17.39 48.22

2.023 2.398 3.369

9,60 10,77 15.87

25°o

5:0

N

..,'"

.

,"r'

....

o 78

~

...........

26 ~

.

.

.

.

.

.

.

.

....

r :.~ ~ ~

xA,: 0 0 0 0 0

~,~ ~

M/02727

22

xA: r) 7727 ×~,,:

~0 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I £ ~ ,4 :> X

!( I~,

mole/',

0000 C~

FT t

Fig. 13. Comparison of the viscosity of N e - A r mixtures as a function of density with the values estimated on the basis of the Enskog-Thorne theory for dense gas mixtures.

VISCOSITY OF N E O N - H E L I U M AND NEON-ARGON MIXTURES

441

consistent with a quadratic equation for the equation of state discussed in section 3. Having determined effective values of bi and Xi for the individual gases and knowing the dilute gas viscosities ri0 and rl °, we are now in a position to calculate the viscosities of the gas mixtures from the theoretical EnskogThorne equation (25). For this purpose we assume that the relationship between bij of the real gas mixtures and the bi of the individual components (see (26)), and the relationship between XiJ of the mixtures and the Xi of the individual components (see (34) and (35)) are the same as that for a mixture of gases of hard sphere molecules. The viscosity values thus deduced from the Enskog-Thorne equation as a function of density at the experimental compositions are represented by the dotted curves in figs. 12 and 13. Because of the approximate nature of the theoretical equation, one cannot expect agreement within the experimental accuracy. Nevertheless, from the information provided in figs. 12 and 13 we see that the method provides reasonable estimates for the viscosity of the dense gas mixtures. The difference between calculated and experimental data is generally within a few tenths of a percent with the maximum deviations never exceeding 0.4% in the density range of our experimental data.

Acknowledgements The authors acknowledge stimulating discussions with J.R. Dorfman and E.A. Mason. The research at the University of Maryland was supported by the National Science Foundation under grant DMR 79-10819. Computer time for this project was provided by the Computer Science Center at the University of Maryland. The research at Brown University was supported under several grants awarded by the National Science Foundation.

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