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On the density expansion for viscosity in gases
Physica
54 (1971) 1-19 0 North-Holland
ON THE DENSITY
Publishing
EXPANSION
Co.
FOR VISCOSITY
IN GASES
J. KESTIN Brown
University,
Providence, R. I., USA
E. PAYKOC Brown
University, Providence,
R. I.,
USA *
and J. V. SENGERS Institute
for Molecular
Physics,
University
of Maryland,
College Park, Maryland and National Bureau
of Standards,
Received
Washington,
D. C., USA
11 January 1971
Synopsis The paper presents the results of new, precise measurements of the viscosity of nitrogen, argon, and helium at 25°C. The measurements were performed over a nominal range of pressures l-100 atm and at very closely spaced density intervals. The data were subjected to a stringent statistical analysis in order to determine whether the density expansion consists of a pure polynomial or whether a term of the form pe In p must be included in it. The existence of such a term was discovered theoretically by several investigators. The analysis indicates that if such a term exists, its factor must be very small. Moreover, the statistically significant interval of values which this factor can assume includes zero in it. This result is interpreted as indicating that correlations which extend over distances of the order of a mean free path are negligible when compared with correlations which extend over distances of the order of the range of molecular interactions.
1. Introduction. The viscosity of a gas in the low-density limit can be described by the Chapman-Enskog theory which is based on Boltzmann’s equationr). This theory is akin to the perfect-gas law in that it assumes that the molecules are randomly distributed in configuration space. In addition, the theory supposes that the velocities of two molecules which are about to collide are uncorrelated (assumption of molecular chaos). A first attempt to extend the theory to higher densities was made by Enskog for the case of a gas of hard spheress). In this theory an estimate was made to account for the effect of correlations in configuration space; these were assumed to be the same as those for a gas in equilibrium. The as* Now at Middle East Tech. University, Ankara, Turkey.
2
J. KESTIN,
sumption
of molecular
E. PAYKOC
chaos was retained
velocity space. The theory viscosity of the form 11= 70 +
TlP
+
AND
q2p2
+
implies
93p3
+
J. V. SENGERS
for the probability
the existence
... .
Such a series is analogous to the virial ties of a gas. The coefficients of the successively higher-order correlations of these correlations is closely related action.
distribution
in
of a power series for the
(1)
expansion for the equilibrium properpower series represent the effect of in the position variables. The range to the range of the molecular inter-
A proper description of the density dependence of the transport properties also requires the inclusion of an estimate of the deviations from molecular chaos in the velocity distribution. Such deviations are brought about by sequences of correlated collisions between the molecules. In the past it was naturally assumed that these effects could also be represented by a power series in the density. However, the range of such velocity correlations is not restricted to the range of the molecular interaction, but extends to distances of the order of the mean free path. Since the mean free path is itself a function of the density, it is not evident that a power series which is formed with density-independent coefficients does exist. In fact, a systematic generalization of the Boltzmann equation indicates that the coefficient of the quadratic term contains a contribution which is proportional to the logarithm of density, so that we should writes) 7 =
~0 +
171~ +
qSp21np
+
q2p2
+
....
(4
as a replacement for eq. (1). Nevertheless, in the past experimental results on transport properties were habitually represented by a power series like that in eq. (1). As a result of the theoretical developments alluded to before, the adequacy of a power series cannot be taken for granted and a critical re-examination of the experimentally-established density dependence of these properties is called for. Hanley, McCarty and Sengers4) undertook an investigation to determine whether existing experimental data would favour the theoretically derived representation (2) in preference to the power series (1). Data on viscosity which were available at the time did not enable them to perform such a discrimination owing to inadequate precision. Kao and Kobayashi5) and Gracki, Flynn and ROSS~) reached a similar conclusion. On the other hand, the more extensive data for thermal conductivity7y s) seemed to favour the inclusion of the logarithmic term. This conclusion was confirmed by Le Neindre and coworkers 8). In spite of this agreement, we hesitate unequivocally to accept the conclusion that the presence of the logarithmic term has been verified experimentally. First, in one case, the precision of the measurements was only
ON THE DENSITY one of f 1%. Secondly,
EXPANSION
3
FOR VISCOSITY IN GASES
the density intervals of both sets of measurements
were quite large, and this necessitated
the inclusion
of data pertaining
to
excessively high densities (800 amagat units) in order to extract the required, statistically significant, information. The latter point is particularly important owing to the circumstance that the omission of higherorder terms beyond those quoted in eqs. (1) and (2) may not be justified in such an extended range. The preceding reasons induced us to undertake new measurements of a transport property and to arrange them so that they would be conducive to a more stringent statistical analysis. We chose to measure the viscosity of nitrogen, argon and helium in an oscillating-disk instruments) because we could obtain in it a precision of a few parts in ten thousand. Furthermore, we were able to make measurements at very closely spaced intervals of pressure and thus to provide a large population of data in a moderate density range. 2. Experimental. Measurements were performed on nitrogen, argon, and helium at a nominal temperature of 25°C and over a range of pressures, as follows : Ns up to P =
105atm
and
p = 0.12 g/ems,
Ar
100atm
and
p = 0.17 g/ems,
100atm
and
p = 0.02 g/cm3.
up to P =
He upto
P=
They were evaluated on a relative basis 1%ii). The temperature in the instrument was maintained by thermostating the room in which the instrument was accommodated. The characteristics of the suspension system are given in table I. TABLE I
Characteristics of suspension system Suspension wire
0.002in. diam. 92%
Pt-8%
W.
stress relieved
Wire damping Total separation between plates Upper and lower separations Disk radius Disk thickness Moment
of inertia of suspension system
Natural periods of oscillation
A0 = 0.000040f 0.000004 D = (0.28443 f 0.00005) cm b = bl = bz = (0.09006 & 0.00005) cm R = (3.4906 f 0.001) cm d =
10.10431 f.
0.00012j cm (53.6032 f 0.0006) g cm2 To = (29.244 f 0.002)sat 25%
I =
(first calibration) TO =
(29.216 f 0.002)sat (second calibration1
25°C
J. KESTIN,
4
E. PAYKOC
AND
J. V. SENGERS
The working equation 11) xpR4d 21 -
-zzz
contains the calibration layer thicknessli)
0,
factor C which is a unique function of the boundary-
6 = (rlTo/“p)k.
(3a)
Here, R denotes the radius of the disk, d is its thickness, I its moment of inertia, bl and b2 are the upper and lower spacings, respectively, both equal (bl = b2 = b), To is the period of oscillation in zluc~o, 19= T/To is the ratio of the actual period to that in a vacuum, and A is the logarithmic decrement of the damped harmonic oscillation performed by the system, AO denoting its value in ZIUCMO. Finally xo = a/R = (1 /R) (VT0/2xp)4,
W)
computed numerically from eq. (3), determines the viscosity, ye. In order to perform the three series of measurements, it turned out to be necessary to calibrate the instrument twice; owing to a mishap, the suspension wire was damaged before the measurements on argon were completed, and the oscillating system had to be re-assembled. The points obtained with the aid of the second calibration are marked with asterisks in the tables of results. The calibration curve C(6) was determined with respect to nitrogen in the pressure range l-25 atm and at 25°C. The calibration values were taken directly from table XV of ref. 9. Since the temperature in the instrument was not exactly equal to 25”C, differing from it, however, by at most 0.3”C, a small correction was applied. We based this correction on the circumstance that the excess viscosity is a function of density, V(P~ T)
-
~jlo(O, T)
=
in both representationsla), p = const, we find that V(P> T*)
=
rl(p> T)
-
(4)
f(p),
eqs.
{rlo(Q
T)
(1) and
-
vo(O,
(2).
T*))>
Hence,
along
an isochore
@a)
where T* = 25°C. In this manner the only correction that needs to be applied is deduced from the zero-density values, ~0, a linear estimate of 0.47 ppoise/“C being adequate for the purpose. The corrected value now refers to the actual density maintained in the instrument but to a slightly different temperature. To account for this fact, the pressure recorded during the measurement must be corrected accordingly.
ON THE DENSITY
The density,
EXPANSIOK
p, was calculated
PM/pRT = 1 +
FOR VISCOSITY
5
IN GASES
from the equation
B1P+BzP2+B3P3,
(5)
with
R =
82.082
atm crns/g
K, (on carbon
M = 28.014 g/gmol
TABLE
12 scale). II
Virial coefficients for nitrogen, argon, and helium Gas
Temperature “C
Rl
B2
B3
(atm-l)
(atm-2)
(atm-3)
N2t
20 30
-2.42 -1.47
x lo-4 x 10-4
2.13 x 10-s 2.04 x lo@
Artt
20 30
-7.5194 -6.2381
x IO-4 x 10-4
2.389 x 10-s 2.1046 x lo-”
He ttt
20 30
3.84 x 10-g 2.64 x 10-s -
4.633 x 10-4 4.471 x LO-4
t From ref. 13. tt From ref. 14. ttt From ref. 15.
The virial factors Bl,Bz,B3 lijtcd in table II wzre interpolated from NBS Circular 564ls), and a further linear interpolation routine between 20°C and 30°C was programmed in conjunction with the working equation (3). In the latter, the values of C were evaluated from known values of x0, the reverse procedure bei. g used for the determination of the viscosity, 7, in measurements proper. The two calibration lated as follows:
curves
Cr = 1.0890645-
(standard
(standard
in fig. 1; they
0.002 5248
0.000 7367
6 + 0.07
(6 + 0.07)s
deviation
Cs = 1.089 5202 -
are seen plotted
0.000 0334 (6 + 0.07)3 ’
(W
1.5 x 1O-4),
0.003 1036 -___ 6 + 0.07
deviation
--
can be corre-
I .8 x
-
0.000 53 15 (6 + 0.07)s
0.000 02 17 -
(6 + 0.07)3
’
(w
10-4).
The experimental results are given in tables III-V. The density of nitrogen was evaIuated in the same way as during a calibration, that of argon and helium was evaluated in like manner, except for the virial coefficients which are also listed in table II.
Point
0.85037 0.64939
1.000
1.716
1.000
1.714
1
2
20.068
24.638
20.051
24.644
16
17
0.13663 0.12966
39.958
44.663
49.840
55.106
60.602
65.419
39.919
44.682
49.853
55.126
60.603
65.434
20
21
22
23
24
25
0.10862
0.11241
0.11750
0.12310
0 14609
34 807
19
0.15753
29.806
29.838
34 816
18
0.17274
0.19095
0.22165
14.864
0.27082
14.880
0.28460
8.964
8.954
12
15
0.011378
0.30053
8.036
8.029
0.24056
0.31503
7.312
7.307
10 11
9.922 12.579
0.33116
6.616
6.613
9
9.931
0.36034
5.583
5.579
8
12.567
0.38142
4.981
4.979
7
13
0.010279
0.40438
4.344
4.341
6
14
0.0042597
0.44124
3.718
3.715
5
0.075 132
0.069635
0.063344
0.057304
0.051357
0.045946
0 040016
0.034258
0.028307
0.023045
0.017057
0.014433
0.0092126
0.0083814
0.0075827
0.0063979
0.0057072
0.0049765
0.0034470
0.49049
0.0026365
0.56057
2.302
3.009
2.300
3.009
4
24.727
60.083
55.688
50.657
45.826
41.070
25.070
25.003
25.113
25.077
25.131
24.712
25.084 36.743
25.318
27.396 32.001
25.068
24.749
18.429 22.637
25.334
24.702
25.280
24.65 1
24.744
24.804
24.874
24.784
24.890
24.819
24.722
13.641
11.542
9.0991
8.2202
7.3674
6.7027
6.0639
5.1164
4.5641
3.9797
3.4065
24.959
2.1084 2.7566
24.864
24.920
(2)
ture
1.5710
0.91590
0.0011453 0.0019645
P (amagat)
P k/cm3)
3
(cm)
(at4
(atm)
d
tempera-
thickness
pressure
mental
Experi-
layer
pressure
Density&)
Boundary
Corrected
mental
P
n
of nitrogen
Density
The viscosity
Experi-
p,
number
;
TABLE III
177.94
190.44
189.05
187.89
186.58
185.52
184.2g
183.49
182.64
181.48
180.54
180.04
179.45
179.30
178.88
178.7s
178.72
178.67
178.4g
178.39
178.32
178.1g
178.18
178.00
178.00
190.41
189.05
187.84
186.54
185.46
184.4g
183.45
182.49
180.66 181.45
179.88
179.59
179.17
179.04
178.90
178.81
178.73
178.59
178.44
178.32 178.41
178.19
178.13
178.0~
177.98
(eoise)
rl
(at 25°C)
viscosity
Standard
m
84.009
90.643
94.555 97.821
105.237
29
30
31
33
1.ooo
* 1.497
2.041
* 2.480
3.034
* 3.500
4.048
* 4.514
5.110
* 5.487 6.035
* 6.501
7.158
* 7.488
* 8.498
8.586
1
3
4
5
6
7
8
9
10 11
12
13
14
15
16
8.585
8.581
7.484
7.155
6.496
5.484 6.032
5.113
4.509
4.050
3.496
3.034
2.483
2.041
1.498
1.ooo
P
(at4
p,
pressure
Corrected
g/ems.
105.251
97.783
94.530
90.624
83.970
80.375
73.560
70.42 1
(at@
2
n
mental
pressure
Experi-
Point
1 amagat
= 0.00125046
80.403
28
number
“)
73.599
27
32
70.40 1
26
0.27425
0.27564
0.29354
0.3004 1
0.31508
0.34287 0.32714
0.35518
0.37817
0.39893
0.42949
0.46106
0.50924
0.56212
0.65559
0.80283
fcm)
6
thickness
layer
Boundary
0.08852
0.09120
0.09248
0.09414
0.09727
0.09911
0.10305
0.10504
of argon
0.014095
0.013943
0.012279
0.011737
0.010651
0.0089853 0.0098873
0.0083756
0.0073832
0.0066301
0.0057211
0.0049633
0.0040593
0.0033360
0.0024472
0.0016340
P k/cm9
Density
The viscosity
TABLE IV
0.11979
0.11156
0.10795
0.10360
0.096140
0.092094
0.084386
0.080825
b)
7.9012
7.8160
6.8832
6.5794
5.9706
5.0369 5.5425
4.695 1
4.1388
3.7166
3.2071
2.7823
2.2755
1.8701
1.3718
0.91597
P (amagat)
Density
95.797
89.215
86.328
82.850
76.884
73.648
67.484
64.636
25.036
25.177
25.150
25.124
25.212
25.155 25.141
24.826
25.3 15
24.837
25.302
24.982
24.687
25.040
24.872
24.911
(2,
temperature
mental
Experi-
24.957
25.079 25.115
25.062
25.139
25.102
25.162
24.916
226.74 ~226.8~ 226.86
226.95 226.70 227.0s
227.24 227.4~ 227.42 227.69 227.75
227.39 227.57 227.5s 227.82 227.7s
227.25
226.70
226.69
227.35
226.61
226.39
227.1 s
226.45
226.4s
227.06
226.29
226.20
227.17
226.34
226.2s
227.01
(ppoise)
r!
(at 25°C)
viscosity
Standard
201.67
199.31
198.30
197.20
195.37
194.32
192.47
191.65
&poise)
rle
Viscosity
201.65
199.36
198.34
197.23
195.44
194.37
192.55
191.61
pressure
mental
0.020417 0.025193 0.033506 0.037476 0.040727
0.067227 0.075656 0.064481 0.088920
0.25375 0.22826 0.20580 0.17890 0.16939 0.16255 0.14735 0.13619 0.12774 0.12077 0.11467 0.11200
12.405
15.279
20.258
22.626
24.562
29.996
35.198
40.193
45.120
50.256
52.832
10.029
12.403
15.288
20.266
22.637
24.542
29.985
35.156
40.191
45.090
50.261
52.869
18
19
20
21
22
23 24
25
26
27
28
29
P
25.067
25.111
18.782
24.896
27.969 32.914
0.049894 0.058715
25.083 25.172
61.792 67.874
0.11023 0.12108
0.10146 0.09729
65.144
71.380
65.162
71.421
33
34
g/cm3
using calibration
96.037
0.17132
0.08365
100.084
100.032
40
performed
91.289 0.16285
0.08548
95.253
95.303
39
= 0.0017839
25.156
66.395
0.15412
0.08751
90.274
90.303
38
b) 1 amagat
25.095
81.232
0.14491
0.08986
85.020
85.029
37
* Calculations
24.842
76.277
0.13607
0.09233
79.968
24.845
25.03 1
25.012
71.702
0.12791
0.0949 1
75.297
75.300
79.926
35
36
257.55
255.67
253.57
251.42
249.2s
247.5s
246.2s
243.84
242.06
25.161 56.909
0.10152
0.10535
60.122
60.154
241.36
32
239.9s
24.971
0.097371
238.7s
25.280
51.608 54.583
0.092064
0.11014 0.10741
54.655
57.725
54.650
25.211
49.846
57.779
25.028
47.357
30
239.66
24.801
42.410
237.10
24.986
235.70
234.01
232.75
231.19
231.05
230.41
229.26
228.56
228.07
227.9s
(Ccpoise)
r1e
Viscosity
37.685
24.644
24.763
22.830
25.145
25.183
14.122 21.008
24.956
24.924
11.446
9.2415
a.7382
(amagat)
ture
31
2
0.015588 0.016486
0.26073
9.489
10.032
* 9.491
17
P
k/cm3)
(4
(at4
(atm)
6
P
n
Experitempera-
b)
thickness
Density mental
Density
layer
Boundary
p,
pressure
Corrected
Experi-
Point
number
TABLP: IV (continued)
257.66
255.56
253.50
249.36 251.4a
247.5s
246.1 i
241.95 243.7s
238.76 237.24 239.51 239.9s 241.16
235.71
231.36 232.83 234.26
230.94
s
3
K
2
b 4
g
2
F
m
“Z
fi
+
228.59 229.13 230.33
228.12
227.85
Lupoise)
?1
(at 25°C)
viscosity
Standard
co
55.186
55.160
60.134
64.855
69.857
75.061
79.960
85.165
90.064
94.589
100.049
22
23
24
25
26
27
28
29
30
= 0.0001785
49.990
g/ems.
100.152
90.080 94.712
85.207
79.963
75.109
69.853
64.889
60.148
45.097
39.918
34.921
29.968
24.885
50.002
24.882
15
19.627
20 21
19.622
14
17.230
45.090
17.227
13
14.867
19
14.866
12
39.919
12.397
11
9.965 12.404
18
9.954
10
9.057
29.965
9.049
9
8.036
34.918
8.028
8
6.993
16
6.987
7
6.015
5.007
4.026
3.003
17
5.001
6.008
5
4.021
4
6
3.000
3
2.016
0.24271
0.24934
0.25542
0.26233
0.27052
0.27876
0.28877
0.29920
0.31062
0.32381
0.33990
0.35747
0.37948
0.40543
0.43710
0.47920
0.53884
0.57487
0.61852
0.67682
0.75460
0.79142
0.83996
0.90039
0.97070
1.0639
1.1859
1.3729
1.6757
2.1798
1.191
1.190
2.014
6 (cm)
P
(at4
P,
1
a) 1 amagat
layer
(atm)
2
n
pressure
Boundary thickness
mental
number
Corrected
pressure
Experi-
Point
0.015667
0.014851
0.014153
0.013416
0.012619
0.011878
0.011073
0.010308
0.0095753
0.0088047
0.0079940
0.0072274
0.0064122
0.0056220
0.0048353
0.0040244
0.0031816
0.0027962
0.0024153
0.0020173
0.0016225
0.0014752
0.0013096
0.0011401
0.00098111
0.00081704
0.00065719
0.00049045
0.00032937
0.00019465
k/cm3)
P
Density
of helium
TABLE V The viscosity
87.770
83.199
79.289
75.160
70.695
66.543
62.034
57.748
53.643
49.326
40.490 44.784
35.923
31.496
27.089
22.546
17.824
15.665
13.531
11.301
9.0896
8.2644
7.3367
6.3871
5.4964
4.5773
3.6817
2.7476
1.8452
1 0905
198.29
198.37
24.613 24.692
198.3s
198.35
198.41
198.32
198.37
198.27
198.50
198.35
198.44
198.44
198.3s
198.55
198.49
198.55
198.4s
198.54
198.5s
198.54
198.5,~
198.5s
198.51
198.59
198.63
198.55
198.59
198.61
198.7s
198.71
(ppoise)
7e
Viscosity
24.947
24.854
24.987
24.811
25.017
24.845
24.932
24.862
25.073
24.955
25.007
24.977
24.975
24.966
24.930
24.943
24.973
24.839
24.658
24.744
24.693
24.734
24.644
24.647
24.662
24.718
24.751
24.807
T,
CT)
temperature
mental
Experi-
P
“)
(amagat)
Density
198.44
198.55
198.40
198.4s
198.41
198.41
198.37
198.34
198.5s
198.41
198.40
198.46
198.3s
198.56
198.50
198.57
198.51
198.56
198.54
198.62
198.6s
198.64
198.66
198.71
198.79
198.72
198.74
198.74
198.85
198.80
(ypoise)
rl
(at 25°C)
viscosity
Standard
J. KESTIN,
10
E. PAYKOC
AND J. V. SENGERS
1.09
1.09
=2 1.07
1.06
1.09
1.05
1.08
I .04
-
1.07
I .03
-
1.06
1.02
.
1.05
I.01
.
1.04
1.00
-
1.03
1.02
_l.oo~ 1.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I.0
S,cm
Fig. 1. Calibration curves with respect to nitrogen at 25°C nominal and in the pressure range l-25 atm.
The data for argon were taken from the measurements performed by Whalley, Lupien and Schneideria), those for helium were taken from the work of Schneider and Duffiei5) ; it proved to be sufficient to retain terms up to 0(P2) only. In all cases a small correction for temperature was applied to reduce the data to 25°C as mentioned earlier in connection with the calibration curves. The actual temperature corrections were deduced from runs at slightly different temperatures, and turned out to have the following values: Ar
0.7 1 ppoise/“C,
He
0.46 ppoise/“C.
ON THE
DENSITY
In this manner, III-V,
the standard
represents
that prevailed corresponding,
EXPANSION
the viscosity
FOR VISCOSITY
viscosity,
IN GASES
listed in the last column
I1
of tables
of the gas at 25°C and at the actual density
during the measurement. For this reason, we evaluated the corrected pressure and listed it in the second column in the
tables, as already mentioned. Large graphs of standard viscosity
versus density show that the data are
smooth to a very satisfactory degree. We estimate that the accuracy of the measurements is of the order of f0.2%. The precision and internal consistencyis, of course, much better, as will emerge from the analysis. In assessing the reliability of the preceding experimental values, it is necessary to realize, as pointed out in our earlier experimental work, that eq. (3) is quite insensitive to the assumed value of density. This is due to the fact that the damping in an oscillating-disk viscometer which is equipped with closely spaced fixed plates is very nearly independent of density. For example, if the density in eqs. (3) and (3a) were changed by as much as l%, the viscosity evaluated from eq. (3b) would change by less than 0.1%. Thus, an accurate knowledge of the density is inessential for the evaluation of viscosity. By contrast, the values of density assigned to the corrected pressures at 25°C must be computed with great care when an analysis of the density expansion of viscosity is undertaken. Nevertheless, the analysis is not sensitive to uncertainties in the actual values used for the virial coefficients Bk in eq. (5). A choice of other values for these virial coefficients would represent a coordinate transformation p’ = f(p) where f(p) is a power series in p. Such a coordinate transformation does not affect the presence or absence of a logarithmic term in eq. (2). 3. Method of analysis. The verification of a proposed mathematical form for the density dependence of the viscosity is hampered by the fact that a series expansion contains an infinite number of terms. Moreover, no rigorous expressions are available for the numerical evaluation of the coefficients in such an expansion beyond the Chapman-Enskog zero-density value of 70. Thus there exists no a priori information regarding the number of terms of the expansion that would be adequate in representing the viscosity in a specified density interval. Even the dilute-gas value, ~0, cannot be established independently with sufficient precision, and its most probable value must also be extracted from the experimental data. Nevertheless, it is possible to formulate heuristic criteria that ought to be satisfied, if a postulated equation is to be consistent with a population of experimental data. In a previous paper (with Hanley and McCarty4)) such a set of criteria was formulated in conjunction with a systematic algorithm for the determination of successively higher-order terms in an equation such as (1) or (2), but truncated at some point. In order to understand the algorithm, we consider the succession of
12
J. KESTIN,
E. PAYKOC
AND J. V. SENGERS
I
n=40
I
n=40 mT=
0.06
0
Fig. 2. Deviation plots for argon. (a) Eq. (7~); (b) eq. (7d).
polynomials yI = TO +
TlP,
q =
q1p
^170+
r = ^Ilo+
(74 +
?IlP +
(7b)
rlZP2> rlZP2 +
173P3,
(74
each containing one more term than its predecessor. We now fit the first equation to the first rt = 5 points in the table of results for a particular gas, and record the least-square values of 70 and ~1 together with their standard deviations and the variance, c~, for the set. This procedure is continued with n increased by unity in each successive fit. It is clear that at first the variance, (TV,decreases with n, but after a certain value PL= n’ (or p = p’) has been reached, the variance, having passed through a stationary value, begins to increase, because the quadratic term comes into play. The same occurs with respect to the standard deviations of the coefficients. The calculation is then stopped, and the same procedure is followed with eqs. (7b) and (7~) except that progressively larger values of n and n’ are
ON THE DENSITY
used.
In this
manner
EXPANSION
the algorithm
FOR VISCOSITY
allows
IN GASES
us to determine
13
an interval
0 < p < p’ which provides the best fit of the data to a particular equation. As a last step, we repeat the same calculation in relation to the alternative form q
=
~0
+
wp
+
$2~~
In
P +
172~~.
(74
In addition, detailed deviation plots are prepared to ensure that the distribution of the former is random, as expected. We say that an equation is consistent with the experimental data if the following, obviously necessary, conditions are satisfied : (i) The deviations of the experimental data from those calculated with a fitted equation should be random, so that the standard deviation can be interpreted as a true measure of the experimental precision. (ii) When the values returned for the coefficients of the equation are studied as functions of n (or p), they should be independent of n within the limits prescribed by their respective standard deviations. (iii) Within their standard deviation the coefficients should not change when the next term is added to the equation and the density interval is increased correspondingly. It should be noted that the algorithm ensures that criteria (i) and (ii) are satisfied. The final decision as to whether a power series or the series with the logarithmic term included yields a superior empirical representation is made with reference to criterion (iii). 4. Numerical experiment. Before applying the procedure described in the preceding section to a set of experimental results, it was thought advisable to perform a preliminary test on a set of artificially generated numbers and so to ascertain whether the algorithm is capable of producing a clear discriminant generated
for the presence of the logarithmic term. data from the following three equations
we
11 = 226.09
+ 0.199~ +
1.5 x 10-3~3 -
q = 226.09
+ 0.199~ -
0.2 x
10-3~2 In p + 2 x 10-3~2,
(8b)
7 = 226.09
+ 0.199~ -
2.0
10-3~2 In p + 2 x 10-3~~.
(8~)
x
1.8 x
For this purpose
10-6~3,
(84
The coefficients in these equations have been chosen to correspond closely to the case of argon. Numerical data were generated at the densities (in amagat units) listed in table IV for argon with a random error added on to them; the latter was taken from a normal distribution with a variance of d = 0.07t. t The authors are indebted to Dr. D. T. Gillespie for providing them with suitable sets of random numbers.
eq.(8~)
eq.(8b)
eq.(8a)
Data generated with
6 52 96 96
12 30 40 40
226.10& 0.04 226.00f 0.04 226.03+ 0.04 226.09& 0.03 (226.09)
(exact)
15 8 24 28 31 55 40 96 (exact)
7 72 96 96
226.08& 0.05 226.07f 0.03 226.09* 0.03 226.10f 0.03 (226.09)
226.08h 0.05 226.06& 0.02 226.07-+ 0.03 226.09& 0.03 (226.09)
13 35 40 40
(exact)
p’
n %,
VI
0.187f 0.009 0.242f 0.008 0.230& 0.008 0.198+ 0.006 (0.199)
0.204f 0.012 0.207f 0.002 0.204& 0.004 0.198f 0.006 (0.199)
%I,) x
lo3
(2)
-6.1 & 0.3 -4.8 i 0.4 2.1f 0.4
(2)
1.06f 0.03 1.2 & 0.1 2.1 f 0.4
(1.5)
1.37+ 0.07 1.5 f 0.1 2.5 f 0.4
(72 f
ofthe algorithm
0.202+ 0.014 0.202f 0.003 0.198& 0.004 0.192f 0.006 (0.199)
71 f
Test
TABLE
%,I
x
-33*5 (0)
-2.0 f 0.7 (0)
-
(-1.8)
i
-
%*d
x
lo3
-2.03 f 0.08 (-2)
-0.23 f 0.08 (-0.2)
-
-0.23 f 0.09 (0)
l@:(r12'
-2.1 * 0.7
(r3 f
0.07 0.08 0.07 0.07 (0.07)
0.07 0.07 0.07 0.07 (0.07)
0.08 0.07 0.07 0.08 (0.07)
ON THE
Eqs.
DENSITY
(7) were fitted
EXPANSION
FOR
VISCOSITY
to the data thus generated
IN GASES
by the method
15
of least
squares. We used the Omnitab computer program for statistical analysis which uses a Gram-Schmidt orthogonalization precedurel6). In a comparative study by Wampleri7) this method was shown to be reliable for the purpose at hand. The results obtained for the fitted equations were studied as a function of p’ to determine the range 0 < p ==cp’, if any, in which criteria (i) and (ii) were satisfied. The most probable values for the coefficients of eqs. (7) are presented in table VI. In this and the subsequent table PL refers to the number of data points. It is seen that the data generated by the cubic equation (8a) produce polymonials (7a), (7b) and (7~) which do satisfy criterion (iii). More precisely, the quadratic equation reproduces the values of qo and ~1 obtained with the linear equation and the cubic equation reproduces the values of 70, 71 and 7s derived from the quadratic equation. Thus we conclude that the cubic polynomial is consistent with the data set. The value for ~1 obtained from the logarithmic equation (7~) is somewhat lower than the corresponding term in the linear equation. Nevertheless, the two values agree within their standard deviation so that our criteria do not permit us to rule out a logarithmic term when it is multiplied with a coefficient of 2 x 10-4. This conclusion is confirmed by the results obtained from the data generated according to equation (8b). This equation contains a logarithmic term with a coefficient of this magnitude. According to our criteria, both the cubic equation (7~) and the logarithmic equation (7d) are consistent with this data set. Finally, we considered the data generated by eq. (8~) where the coefficient of p2 In p is ten times larger than that in the previous equation. Our procedure now shows very clearly that the polynomials are not consistent with the data set. In other words, the quadratic and cubic equations do not reproduce the values of ~0 and 71 of the linear equation, nor does the cubic equation reproduce the value of q2 of the quadratic equation. On the other hand, the logarithmic equation (7d) does reproduce the factors ~0 and 71 within their combined standard deviations and is thus consistent with the data. We conclude that our algorithm can, indeed, distinguish between a polynomial such as eq. (7~) or a logarithmic equation (7d). Of course, for data with a finite precision, the distinction can only be made with a finite resolution. With our current relative precision of 3 x 10-A we can no longer detect a logarithmic contribution if its coefficient is as small as 2 x 10-4. 5. Results of analysis. The experimental viscosity data were subjected to the analysis described in the previous two sections. The most probable values thus deduced for the coefficients of eqs. (7a), (7b), (7~) and (7d)
16
J. KESTIN.
E. PAYKOC
AND
J. V. SENGERS
ON THE
DENSITY
are listed in table VII. to the individual same results
EXPANSION
In fitting
data points.
are obtained
FOR VISCOSITY
the equations
However,
IN GASES
17
equal weights were assigned
it was verified
when a weight proportional
that essentially
the
to the value of vis-
cosity is assigned to the data. We note that the analysis indicates that the relative precision of the experimental data is better than 3 x 10e4. Examples of two deviation plots for the cubic equation (7~) and the logarithmic equation (7d) are shown in fig. 2; they are virtually indistinguishable. Evidently the two equations satisfy criterion (i) equally well, as they do also for nitrogen. If we apply in addition criterion (iii), we see from table VII that the power series is consistent with the experimental data for nitrogen and argon to a high degree of satisfaction. This means that the linear, quadratic and cubic equations lead to the same coefficients. The logarithmic equation (7d) does not reproduce the value of 71 of the linear equation nearly as well. Thus we conclude that from viscosity data with a relative precision of 3 x 10-d a p2 In p term cannot be detected in the case of nitrogen or argon. It turns out that in the case of helium we cannot obtain significant information for all coefficients, but the results for the linear and quadratic equation are again in satisfactory agreement. As demonstrated in the previous section our procedure would have revealed the existence of a logarithmic term if its coefficient were significantly larger than 2 x lo-4 ppoise/am 2. Therefore, we conclude that such a coefficient is not larger than the above value, if it exists at all. In order to make this statement more precise, we rewrite eq. (7d) as f(q) = r -
G2
lnp = TO + VIP + q2p2.
(9)
In this new equation 7; is not considered as a coefficient to be determined, but as a parameter. Each value for the parameter $ defines a new set of data points f(q). For the “correct” value of 7; the data f(q) should satisfy our criteria when fitted to a polynomial. Even when we relax our criteria to allow variations within twice the standard deviations, we conclude that values for $ such that I$] > 6 x 10-J ppoise/ams cannot possibly be reconciled with our experimental data for nitrogen or argon. 6. Conclusions. It has been demonstrated that the experimentally determined density dependence of the viscosity of argon and nitrogen at 25°C can be consistently represented by a power series in the de’nsity. Implicit in this conclusion is the assumption that the first four terms of the density expansion (1) and (2) are adequate to describe the data in the density range under consideration. The existence of a logarithmic term of the form ps In p cannot be detected on the basis of viscosity data whose relative precision is 3 X 10M4. If such a term exists, it is concluded that the absolute value of its factor, vi, must be appreciably smaller than 6 x 10-J ppoise/am2 for
18
J. KESTIN,
E. PAYKOC
AND
J. V. SENGERS
argon as well as nitrogen at 25°C. From the practical point of view, there is no need to include such a term in describing the density dependence of the viscosity within the best precision currently obtainable. In view of the discussion in the introduction, we suggest that these results indicate that correlations that extend in the gas over distances of the order of a mean free path are quantitatively insignificant compared to correlations that extend over distances of the order of the range of molecular interactions. APPENDIX
In addition to the principal conclusions of the present paper regarding the form of the density expansion, the experimental data can be utilized in a practical way to provide us with very precise data on the zero-density value of the viscosity of the three gases and on the linear correction term. Referring to table VI1 we can summarize them as follows: Nitrogen 70 = (177.85 & 0.02) ppoise
at 25”C,
71 = (0.132 & 0.007) ppoise/amagat
at 25°C.
Argon 70 = (226.09 & 0.04) ppoise
at 25”C, at 25°C.
71 = (0.198 * 0.009) ppoise/amagat Helium 70 = (198.825 & 0.02) ppoise ~1 = -(0.019
f
at 25”C,
0.003) ppoise/amagat
Evidently,
the preceding
existence
of unaccounted-for
optimum
at 25°C.
values disregard
systematic
the possibility
of the
errors.
A c k n o w 1e d g e m e n t s. The experimental results described in this paper were obtained with funds provided by the National Science Foundation under Grant GK 2133X to Brown Univeristy. Mr. R. Paul was very helpful in maintaining the instrument in a first-class working condition. The analysis was performed with funds provided by the Arnold Engineering Development Center, Tennessee, under Contract F40600-69-CO002 with the University of Maryland and Contract (40-600) 66-494 with the National Bureau of Standards. The authors thank Dr. H. J. M. Hanley for some stimulating discussions.
ON THE DENSITY
EXPANSION
FOR VISCOSITY
IN GASES
19
REFERENCES 1) Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press (London, 3rd ed., 1970). 2) See Chapter 16 of ref. 1. 3) Dorfman, J. R. and Cohen, E. G. D., J. math. Phys. 8 (1967) 282. 4) Hanley, H. J. M., McCarty, R. D. and Sengers, J. V., J. them. Phys. 50 (1969) 857. 5) Kao, J. T. F., Viscosity of He-N2 mixtures at low temperatures and high pressures, Thesis, Rice Univ., Houston, Texas, 1966. 6) Gracki, J. A., Flynn, G. P. and Ross, J., J. them. Phys. 51 f 1969) 3856. 7) Sengers, J. V., Balk, W. T. and Stigter, C. J., Physica 30 (1964) 1018. 8) Le Neindre, B., Tufeu, R., Bury, P., Johannin, P. and Vodar, B., The thermal conductivity coefficients of some noble gases, Proc. 8th Thermal Conductivity Conference, C. Y. Ho and R. E. Taylor, eds., Plenum Press (New York, 1968) p. 75. 9) Kestin, J. and Leidenfrost, W., Physica 25 (1959) 1033. 10) Kestin, J. and Wang, H., J. appl. Mech. 24 (1957) 197. 11) Kestin, J., Leidenfrost, W. and Liu, C. Y., 2. angew. math. Phys. 10 (1959) 558. 12) Sengers, J. V., Transport properties of compressed gases, Recent Advances in Engineering Science, Vol. 3, A. C. Eringen, ed., Gordon and Breach (New York, 1968) p. 153. 13) Hilsemath, J. et al., Tables of Thermodynamic and Transport Properties, NBS Circular 564 (U. S. Government Printing Office, Washington, D. C., 1955). 14) Whalley, E., Lupien, Y. and Schneider, W. G., Canad. J. Chem. 31 (1953) 722. 15) Schneider, W. G. and Duffie, J. A. H., J. them. Phys. 17 (1949) 751. 16) Hilsenrath, J., Ziegler, G. C., Messina, C. G., Walsh, P. J. and Herbold, R. J,, “Omnitab”, NBS Handbook 101 (U. S. Government Printing Office, Washington, D. C., 2nd ed., 1968). 17) Wampler, R. H., J. Res. NBS, 73B (1969) 59.
54 (1971) 1-19 0 North-Holland
ON THE DENSITY
Publishing
EXPANSION
Co.
FOR VISCOSITY
IN GASES
J. KESTIN Brown
University,
Providence, R. I., USA
E. PAYKOC Brown
University, Providence,
R. I.,
USA *
and J. V. SENGERS Institute
for Molecular
Physics,
University
of Maryland,
College Park, Maryland and National Bureau
of Standards,
Received
Washington,
D. C., USA
11 January 1971
Synopsis The paper presents the results of new, precise measurements of the viscosity of nitrogen, argon, and helium at 25°C. The measurements were performed over a nominal range of pressures l-100 atm and at very closely spaced density intervals. The data were subjected to a stringent statistical analysis in order to determine whether the density expansion consists of a pure polynomial or whether a term of the form pe In p must be included in it. The existence of such a term was discovered theoretically by several investigators. The analysis indicates that if such a term exists, its factor must be very small. Moreover, the statistically significant interval of values which this factor can assume includes zero in it. This result is interpreted as indicating that correlations which extend over distances of the order of a mean free path are negligible when compared with correlations which extend over distances of the order of the range of molecular interactions.
1. Introduction. The viscosity of a gas in the low-density limit can be described by the Chapman-Enskog theory which is based on Boltzmann’s equationr). This theory is akin to the perfect-gas law in that it assumes that the molecules are randomly distributed in configuration space. In addition, the theory supposes that the velocities of two molecules which are about to collide are uncorrelated (assumption of molecular chaos). A first attempt to extend the theory to higher densities was made by Enskog for the case of a gas of hard spheress). In this theory an estimate was made to account for the effect of correlations in configuration space; these were assumed to be the same as those for a gas in equilibrium. The as* Now at Middle East Tech. University, Ankara, Turkey.
2
J. KESTIN,
sumption
of molecular
E. PAYKOC
chaos was retained
velocity space. The theory viscosity of the form 11= 70 +
TlP
+
AND
q2p2
+
implies
93p3
+
J. V. SENGERS
for the probability
the existence
... .
Such a series is analogous to the virial ties of a gas. The coefficients of the successively higher-order correlations of these correlations is closely related action.
distribution
in
of a power series for the
(1)
expansion for the equilibrium properpower series represent the effect of in the position variables. The range to the range of the molecular inter-
A proper description of the density dependence of the transport properties also requires the inclusion of an estimate of the deviations from molecular chaos in the velocity distribution. Such deviations are brought about by sequences of correlated collisions between the molecules. In the past it was naturally assumed that these effects could also be represented by a power series in the density. However, the range of such velocity correlations is not restricted to the range of the molecular interaction, but extends to distances of the order of the mean free path. Since the mean free path is itself a function of the density, it is not evident that a power series which is formed with density-independent coefficients does exist. In fact, a systematic generalization of the Boltzmann equation indicates that the coefficient of the quadratic term contains a contribution which is proportional to the logarithm of density, so that we should writes) 7 =
~0 +
171~ +
qSp21np
+
q2p2
+
....
(4
as a replacement for eq. (1). Nevertheless, in the past experimental results on transport properties were habitually represented by a power series like that in eq. (1). As a result of the theoretical developments alluded to before, the adequacy of a power series cannot be taken for granted and a critical re-examination of the experimentally-established density dependence of these properties is called for. Hanley, McCarty and Sengers4) undertook an investigation to determine whether existing experimental data would favour the theoretically derived representation (2) in preference to the power series (1). Data on viscosity which were available at the time did not enable them to perform such a discrimination owing to inadequate precision. Kao and Kobayashi5) and Gracki, Flynn and ROSS~) reached a similar conclusion. On the other hand, the more extensive data for thermal conductivity7y s) seemed to favour the inclusion of the logarithmic term. This conclusion was confirmed by Le Neindre and coworkers 8). In spite of this agreement, we hesitate unequivocally to accept the conclusion that the presence of the logarithmic term has been verified experimentally. First, in one case, the precision of the measurements was only
ON THE DENSITY one of f 1%. Secondly,
EXPANSION
3
FOR VISCOSITY IN GASES
the density intervals of both sets of measurements
were quite large, and this necessitated
the inclusion
of data pertaining
to
excessively high densities (800 amagat units) in order to extract the required, statistically significant, information. The latter point is particularly important owing to the circumstance that the omission of higherorder terms beyond those quoted in eqs. (1) and (2) may not be justified in such an extended range. The preceding reasons induced us to undertake new measurements of a transport property and to arrange them so that they would be conducive to a more stringent statistical analysis. We chose to measure the viscosity of nitrogen, argon and helium in an oscillating-disk instruments) because we could obtain in it a precision of a few parts in ten thousand. Furthermore, we were able to make measurements at very closely spaced intervals of pressure and thus to provide a large population of data in a moderate density range. 2. Experimental. Measurements were performed on nitrogen, argon, and helium at a nominal temperature of 25°C and over a range of pressures, as follows : Ns up to P =
105atm
and
p = 0.12 g/ems,
Ar
100atm
and
p = 0.17 g/ems,
100atm
and
p = 0.02 g/cm3.
up to P =
He upto
P=
They were evaluated on a relative basis 1%ii). The temperature in the instrument was maintained by thermostating the room in which the instrument was accommodated. The characteristics of the suspension system are given in table I. TABLE I
Characteristics of suspension system Suspension wire
0.002in. diam. 92%
Pt-8%
W.
stress relieved
Wire damping Total separation between plates Upper and lower separations Disk radius Disk thickness Moment
of inertia of suspension system
Natural periods of oscillation
A0 = 0.000040f 0.000004 D = (0.28443 f 0.00005) cm b = bl = bz = (0.09006 & 0.00005) cm R = (3.4906 f 0.001) cm d =
10.10431 f.
0.00012j cm (53.6032 f 0.0006) g cm2 To = (29.244 f 0.002)sat 25%
I =
(first calibration) TO =
(29.216 f 0.002)sat (second calibration1
25°C
J. KESTIN,
4
E. PAYKOC
AND
J. V. SENGERS
The working equation 11) xpR4d 21 -
-zzz
contains the calibration layer thicknessli)
0,
factor C which is a unique function of the boundary-
6 = (rlTo/“p)k.
(3a)
Here, R denotes the radius of the disk, d is its thickness, I its moment of inertia, bl and b2 are the upper and lower spacings, respectively, both equal (bl = b2 = b), To is the period of oscillation in zluc~o, 19= T/To is the ratio of the actual period to that in a vacuum, and A is the logarithmic decrement of the damped harmonic oscillation performed by the system, AO denoting its value in ZIUCMO. Finally xo = a/R = (1 /R) (VT0/2xp)4,
W)
computed numerically from eq. (3), determines the viscosity, ye. In order to perform the three series of measurements, it turned out to be necessary to calibrate the instrument twice; owing to a mishap, the suspension wire was damaged before the measurements on argon were completed, and the oscillating system had to be re-assembled. The points obtained with the aid of the second calibration are marked with asterisks in the tables of results. The calibration curve C(6) was determined with respect to nitrogen in the pressure range l-25 atm and at 25°C. The calibration values were taken directly from table XV of ref. 9. Since the temperature in the instrument was not exactly equal to 25”C, differing from it, however, by at most 0.3”C, a small correction was applied. We based this correction on the circumstance that the excess viscosity is a function of density, V(P~ T)
-
~jlo(O, T)
=
in both representationsla), p = const, we find that V(P> T*)
=
rl(p> T)
-
(4)
f(p),
eqs.
{rlo(Q
T)
(1) and
-
vo(O,
(2).
T*))>
Hence,
along
an isochore
@a)
where T* = 25°C. In this manner the only correction that needs to be applied is deduced from the zero-density values, ~0, a linear estimate of 0.47 ppoise/“C being adequate for the purpose. The corrected value now refers to the actual density maintained in the instrument but to a slightly different temperature. To account for this fact, the pressure recorded during the measurement must be corrected accordingly.
ON THE DENSITY
The density,
EXPANSIOK
p, was calculated
PM/pRT = 1 +
FOR VISCOSITY
5
IN GASES
from the equation
B1P+BzP2+B3P3,
(5)
with
R =
82.082
atm crns/g
K, (on carbon
M = 28.014 g/gmol
TABLE
12 scale). II
Virial coefficients for nitrogen, argon, and helium Gas
Temperature “C
Rl
B2
B3
(atm-l)
(atm-2)
(atm-3)
N2t
20 30
-2.42 -1.47
x lo-4 x 10-4
2.13 x 10-s 2.04 x lo@
Artt
20 30
-7.5194 -6.2381
x IO-4 x 10-4
2.389 x 10-s 2.1046 x lo-”
He ttt
20 30
3.84 x 10-g 2.64 x 10-s -
4.633 x 10-4 4.471 x LO-4
t From ref. 13. tt From ref. 14. ttt From ref. 15.
The virial factors Bl,Bz,B3 lijtcd in table II wzre interpolated from NBS Circular 564ls), and a further linear interpolation routine between 20°C and 30°C was programmed in conjunction with the working equation (3). In the latter, the values of C were evaluated from known values of x0, the reverse procedure bei. g used for the determination of the viscosity, 7, in measurements proper. The two calibration lated as follows:
curves
Cr = 1.0890645-
(standard
(standard
in fig. 1; they
0.002 5248
0.000 7367
6 + 0.07
(6 + 0.07)s
deviation
Cs = 1.089 5202 -
are seen plotted
0.000 0334 (6 + 0.07)3 ’
(W
1.5 x 1O-4),
0.003 1036 -___ 6 + 0.07
deviation
--
can be corre-
I .8 x
-
0.000 53 15 (6 + 0.07)s
0.000 02 17 -
(6 + 0.07)3
’
(w
10-4).
The experimental results are given in tables III-V. The density of nitrogen was evaIuated in the same way as during a calibration, that of argon and helium was evaluated in like manner, except for the virial coefficients which are also listed in table II.
Point
0.85037 0.64939
1.000
1.716
1.000
1.714
1
2
20.068
24.638
20.051
24.644
16
17
0.13663 0.12966
39.958
44.663
49.840
55.106
60.602
65.419
39.919
44.682
49.853
55.126
60.603
65.434
20
21
22
23
24
25
0.10862
0.11241
0.11750
0.12310
0 14609
34 807
19
0.15753
29.806
29.838
34 816
18
0.17274
0.19095
0.22165
14.864
0.27082
14.880
0.28460
8.964
8.954
12
15
0.011378
0.30053
8.036
8.029
0.24056
0.31503
7.312
7.307
10 11
9.922 12.579
0.33116
6.616
6.613
9
9.931
0.36034
5.583
5.579
8
12.567
0.38142
4.981
4.979
7
13
0.010279
0.40438
4.344
4.341
6
14
0.0042597
0.44124
3.718
3.715
5
0.075 132
0.069635
0.063344
0.057304
0.051357
0.045946
0 040016
0.034258
0.028307
0.023045
0.017057
0.014433
0.0092126
0.0083814
0.0075827
0.0063979
0.0057072
0.0049765
0.0034470
0.49049
0.0026365
0.56057
2.302
3.009
2.300
3.009
4
24.727
60.083
55.688
50.657
45.826
41.070
25.070
25.003
25.113
25.077
25.131
24.712
25.084 36.743
25.318
27.396 32.001
25.068
24.749
18.429 22.637
25.334
24.702
25.280
24.65 1
24.744
24.804
24.874
24.784
24.890
24.819
24.722
13.641
11.542
9.0991
8.2202
7.3674
6.7027
6.0639
5.1164
4.5641
3.9797
3.4065
24.959
2.1084 2.7566
24.864
24.920
(2)
ture
1.5710
0.91590
0.0011453 0.0019645
P (amagat)
P k/cm3)
3
(cm)
(at4
(atm)
d
tempera-
thickness
pressure
mental
Experi-
layer
pressure
Density&)
Boundary
Corrected
mental
P
n
of nitrogen
Density
The viscosity
Experi-
p,
number
;
TABLE III
177.94
190.44
189.05
187.89
186.58
185.52
184.2g
183.49
182.64
181.48
180.54
180.04
179.45
179.30
178.88
178.7s
178.72
178.67
178.4g
178.39
178.32
178.1g
178.18
178.00
178.00
190.41
189.05
187.84
186.54
185.46
184.4g
183.45
182.49
180.66 181.45
179.88
179.59
179.17
179.04
178.90
178.81
178.73
178.59
178.44
178.32 178.41
178.19
178.13
178.0~
177.98
(eoise)
rl
(at 25°C)
viscosity
Standard
m
84.009
90.643
94.555 97.821
105.237
29
30
31
33
1.ooo
* 1.497
2.041
* 2.480
3.034
* 3.500
4.048
* 4.514
5.110
* 5.487 6.035
* 6.501
7.158
* 7.488
* 8.498
8.586
1
3
4
5
6
7
8
9
10 11
12
13
14
15
16
8.585
8.581
7.484
7.155
6.496
5.484 6.032
5.113
4.509
4.050
3.496
3.034
2.483
2.041
1.498
1.ooo
P
(at4
p,
pressure
Corrected
g/ems.
105.251
97.783
94.530
90.624
83.970
80.375
73.560
70.42 1
(at@
2
n
mental
pressure
Experi-
Point
1 amagat
= 0.00125046
80.403
28
number
“)
73.599
27
32
70.40 1
26
0.27425
0.27564
0.29354
0.3004 1
0.31508
0.34287 0.32714
0.35518
0.37817
0.39893
0.42949
0.46106
0.50924
0.56212
0.65559
0.80283
fcm)
6
thickness
layer
Boundary
0.08852
0.09120
0.09248
0.09414
0.09727
0.09911
0.10305
0.10504
of argon
0.014095
0.013943
0.012279
0.011737
0.010651
0.0089853 0.0098873
0.0083756
0.0073832
0.0066301
0.0057211
0.0049633
0.0040593
0.0033360
0.0024472
0.0016340
P k/cm9
Density
The viscosity
TABLE IV
0.11979
0.11156
0.10795
0.10360
0.096140
0.092094
0.084386
0.080825
b)
7.9012
7.8160
6.8832
6.5794
5.9706
5.0369 5.5425
4.695 1
4.1388
3.7166
3.2071
2.7823
2.2755
1.8701
1.3718
0.91597
P (amagat)
Density
95.797
89.215
86.328
82.850
76.884
73.648
67.484
64.636
25.036
25.177
25.150
25.124
25.212
25.155 25.141
24.826
25.3 15
24.837
25.302
24.982
24.687
25.040
24.872
24.911
(2,
temperature
mental
Experi-
24.957
25.079 25.115
25.062
25.139
25.102
25.162
24.916
226.74 ~226.8~ 226.86
226.95 226.70 227.0s
227.24 227.4~ 227.42 227.69 227.75
227.39 227.57 227.5s 227.82 227.7s
227.25
226.70
226.69
227.35
226.61
226.39
227.1 s
226.45
226.4s
227.06
226.29
226.20
227.17
226.34
226.2s
227.01
(ppoise)
r!
(at 25°C)
viscosity
Standard
201.67
199.31
198.30
197.20
195.37
194.32
192.47
191.65
&poise)
rle
Viscosity
201.65
199.36
198.34
197.23
195.44
194.37
192.55
191.61
pressure
mental
0.020417 0.025193 0.033506 0.037476 0.040727
0.067227 0.075656 0.064481 0.088920
0.25375 0.22826 0.20580 0.17890 0.16939 0.16255 0.14735 0.13619 0.12774 0.12077 0.11467 0.11200
12.405
15.279
20.258
22.626
24.562
29.996
35.198
40.193
45.120
50.256
52.832
10.029
12.403
15.288
20.266
22.637
24.542
29.985
35.156
40.191
45.090
50.261
52.869
18
19
20
21
22
23 24
25
26
27
28
29
P
25.067
25.111
18.782
24.896
27.969 32.914
0.049894 0.058715
25.083 25.172
61.792 67.874
0.11023 0.12108
0.10146 0.09729
65.144
71.380
65.162
71.421
33
34
g/cm3
using calibration
96.037
0.17132
0.08365
100.084
100.032
40
performed
91.289 0.16285
0.08548
95.253
95.303
39
= 0.0017839
25.156
66.395
0.15412
0.08751
90.274
90.303
38
b) 1 amagat
25.095
81.232
0.14491
0.08986
85.020
85.029
37
* Calculations
24.842
76.277
0.13607
0.09233
79.968
24.845
25.03 1
25.012
71.702
0.12791
0.0949 1
75.297
75.300
79.926
35
36
257.55
255.67
253.57
251.42
249.2s
247.5s
246.2s
243.84
242.06
25.161 56.909
0.10152
0.10535
60.122
60.154
241.36
32
239.9s
24.971
0.097371
238.7s
25.280
51.608 54.583
0.092064
0.11014 0.10741
54.655
57.725
54.650
25.211
49.846
57.779
25.028
47.357
30
239.66
24.801
42.410
237.10
24.986
235.70
234.01
232.75
231.19
231.05
230.41
229.26
228.56
228.07
227.9s
(Ccpoise)
r1e
Viscosity
37.685
24.644
24.763
22.830
25.145
25.183
14.122 21.008
24.956
24.924
11.446
9.2415
a.7382
(amagat)
ture
31
2
0.015588 0.016486
0.26073
9.489
10.032
* 9.491
17
P
k/cm3)
(4
(at4
(atm)
6
P
n
Experitempera-
b)
thickness
Density mental
Density
layer
Boundary
p,
pressure
Corrected
Experi-
Point
number
TABLP: IV (continued)
257.66
255.56
253.50
249.36 251.4a
247.5s
246.1 i
241.95 243.7s
238.76 237.24 239.51 239.9s 241.16
235.71
231.36 232.83 234.26
230.94
s
3
K
2
b 4
g
2
F
m
“Z
fi
+
228.59 229.13 230.33
228.12
227.85
Lupoise)
?1
(at 25°C)
viscosity
Standard
co
55.186
55.160
60.134
64.855
69.857
75.061
79.960
85.165
90.064
94.589
100.049
22
23
24
25
26
27
28
29
30
= 0.0001785
49.990
g/ems.
100.152
90.080 94.712
85.207
79.963
75.109
69.853
64.889
60.148
45.097
39.918
34.921
29.968
24.885
50.002
24.882
15
19.627
20 21
19.622
14
17.230
45.090
17.227
13
14.867
19
14.866
12
39.919
12.397
11
9.965 12.404
18
9.954
10
9.057
29.965
9.049
9
8.036
34.918
8.028
8
6.993
16
6.987
7
6.015
5.007
4.026
3.003
17
5.001
6.008
5
4.021
4
6
3.000
3
2.016
0.24271
0.24934
0.25542
0.26233
0.27052
0.27876
0.28877
0.29920
0.31062
0.32381
0.33990
0.35747
0.37948
0.40543
0.43710
0.47920
0.53884
0.57487
0.61852
0.67682
0.75460
0.79142
0.83996
0.90039
0.97070
1.0639
1.1859
1.3729
1.6757
2.1798
1.191
1.190
2.014
6 (cm)
P
(at4
P,
1
a) 1 amagat
layer
(atm)
2
n
pressure
Boundary thickness
mental
number
Corrected
pressure
Experi-
Point
0.015667
0.014851
0.014153
0.013416
0.012619
0.011878
0.011073
0.010308
0.0095753
0.0088047
0.0079940
0.0072274
0.0064122
0.0056220
0.0048353
0.0040244
0.0031816
0.0027962
0.0024153
0.0020173
0.0016225
0.0014752
0.0013096
0.0011401
0.00098111
0.00081704
0.00065719
0.00049045
0.00032937
0.00019465
k/cm3)
P
Density
of helium
TABLE V The viscosity
87.770
83.199
79.289
75.160
70.695
66.543
62.034
57.748
53.643
49.326
40.490 44.784
35.923
31.496
27.089
22.546
17.824
15.665
13.531
11.301
9.0896
8.2644
7.3367
6.3871
5.4964
4.5773
3.6817
2.7476
1.8452
1 0905
198.29
198.37
24.613 24.692
198.3s
198.35
198.41
198.32
198.37
198.27
198.50
198.35
198.44
198.44
198.3s
198.55
198.49
198.55
198.4s
198.54
198.5s
198.54
198.5,~
198.5s
198.51
198.59
198.63
198.55
198.59
198.61
198.7s
198.71
(ppoise)
7e
Viscosity
24.947
24.854
24.987
24.811
25.017
24.845
24.932
24.862
25.073
24.955
25.007
24.977
24.975
24.966
24.930
24.943
24.973
24.839
24.658
24.744
24.693
24.734
24.644
24.647
24.662
24.718
24.751
24.807
T,
CT)
temperature
mental
Experi-
P
“)
(amagat)
Density
198.44
198.55
198.40
198.4s
198.41
198.41
198.37
198.34
198.5s
198.41
198.40
198.46
198.3s
198.56
198.50
198.57
198.51
198.56
198.54
198.62
198.6s
198.64
198.66
198.71
198.79
198.72
198.74
198.74
198.85
198.80
(ypoise)
rl
(at 25°C)
viscosity
Standard
J. KESTIN,
10
E. PAYKOC
AND J. V. SENGERS
1.09
1.09
=2 1.07
1.06
1.09
1.05
1.08
I .04
-
1.07
I .03
-
1.06
1.02
.
1.05
I.01
.
1.04
1.00
-
1.03
1.02
_l.oo~ 1.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I.0
S,cm
Fig. 1. Calibration curves with respect to nitrogen at 25°C nominal and in the pressure range l-25 atm.
The data for argon were taken from the measurements performed by Whalley, Lupien and Schneideria), those for helium were taken from the work of Schneider and Duffiei5) ; it proved to be sufficient to retain terms up to 0(P2) only. In all cases a small correction for temperature was applied to reduce the data to 25°C as mentioned earlier in connection with the calibration curves. The actual temperature corrections were deduced from runs at slightly different temperatures, and turned out to have the following values: Ar
0.7 1 ppoise/“C,
He
0.46 ppoise/“C.
ON THE
DENSITY
In this manner, III-V,
the standard
represents
that prevailed corresponding,
EXPANSION
the viscosity
FOR VISCOSITY
viscosity,
IN GASES
listed in the last column
I1
of tables
of the gas at 25°C and at the actual density
during the measurement. For this reason, we evaluated the corrected pressure and listed it in the second column in the
tables, as already mentioned. Large graphs of standard viscosity
versus density show that the data are
smooth to a very satisfactory degree. We estimate that the accuracy of the measurements is of the order of f0.2%. The precision and internal consistencyis, of course, much better, as will emerge from the analysis. In assessing the reliability of the preceding experimental values, it is necessary to realize, as pointed out in our earlier experimental work, that eq. (3) is quite insensitive to the assumed value of density. This is due to the fact that the damping in an oscillating-disk viscometer which is equipped with closely spaced fixed plates is very nearly independent of density. For example, if the density in eqs. (3) and (3a) were changed by as much as l%, the viscosity evaluated from eq. (3b) would change by less than 0.1%. Thus, an accurate knowledge of the density is inessential for the evaluation of viscosity. By contrast, the values of density assigned to the corrected pressures at 25°C must be computed with great care when an analysis of the density expansion of viscosity is undertaken. Nevertheless, the analysis is not sensitive to uncertainties in the actual values used for the virial coefficients Bk in eq. (5). A choice of other values for these virial coefficients would represent a coordinate transformation p’ = f(p) where f(p) is a power series in p. Such a coordinate transformation does not affect the presence or absence of a logarithmic term in eq. (2). 3. Method of analysis. The verification of a proposed mathematical form for the density dependence of the viscosity is hampered by the fact that a series expansion contains an infinite number of terms. Moreover, no rigorous expressions are available for the numerical evaluation of the coefficients in such an expansion beyond the Chapman-Enskog zero-density value of 70. Thus there exists no a priori information regarding the number of terms of the expansion that would be adequate in representing the viscosity in a specified density interval. Even the dilute-gas value, ~0, cannot be established independently with sufficient precision, and its most probable value must also be extracted from the experimental data. Nevertheless, it is possible to formulate heuristic criteria that ought to be satisfied, if a postulated equation is to be consistent with a population of experimental data. In a previous paper (with Hanley and McCarty4)) such a set of criteria was formulated in conjunction with a systematic algorithm for the determination of successively higher-order terms in an equation such as (1) or (2), but truncated at some point. In order to understand the algorithm, we consider the succession of
12
J. KESTIN,
E. PAYKOC
AND J. V. SENGERS
I
n=40
I
n=40 mT=
0.06
0
Fig. 2. Deviation plots for argon. (a) Eq. (7~); (b) eq. (7d).
polynomials yI = TO +
TlP,
q =
q1p
^170+
r = ^Ilo+
(74 +
?IlP +
(7b)
rlZP2> rlZP2 +
173P3,
(74
each containing one more term than its predecessor. We now fit the first equation to the first rt = 5 points in the table of results for a particular gas, and record the least-square values of 70 and ~1 together with their standard deviations and the variance, c~, for the set. This procedure is continued with n increased by unity in each successive fit. It is clear that at first the variance, (TV,decreases with n, but after a certain value PL= n’ (or p = p’) has been reached, the variance, having passed through a stationary value, begins to increase, because the quadratic term comes into play. The same occurs with respect to the standard deviations of the coefficients. The calculation is then stopped, and the same procedure is followed with eqs. (7b) and (7~) except that progressively larger values of n and n’ are
ON THE DENSITY
used.
In this
manner
EXPANSION
the algorithm
FOR VISCOSITY
allows
IN GASES
us to determine
13
an interval
0 < p < p’ which provides the best fit of the data to a particular equation. As a last step, we repeat the same calculation in relation to the alternative form q
=
~0
+
wp
+
$2~~
In
P +
172~~.
(74
In addition, detailed deviation plots are prepared to ensure that the distribution of the former is random, as expected. We say that an equation is consistent with the experimental data if the following, obviously necessary, conditions are satisfied : (i) The deviations of the experimental data from those calculated with a fitted equation should be random, so that the standard deviation can be interpreted as a true measure of the experimental precision. (ii) When the values returned for the coefficients of the equation are studied as functions of n (or p), they should be independent of n within the limits prescribed by their respective standard deviations. (iii) Within their standard deviation the coefficients should not change when the next term is added to the equation and the density interval is increased correspondingly. It should be noted that the algorithm ensures that criteria (i) and (ii) are satisfied. The final decision as to whether a power series or the series with the logarithmic term included yields a superior empirical representation is made with reference to criterion (iii). 4. Numerical experiment. Before applying the procedure described in the preceding section to a set of experimental results, it was thought advisable to perform a preliminary test on a set of artificially generated numbers and so to ascertain whether the algorithm is capable of producing a clear discriminant generated
for the presence of the logarithmic term. data from the following three equations
we
11 = 226.09
+ 0.199~ +
1.5 x 10-3~3 -
q = 226.09
+ 0.199~ -
0.2 x
10-3~2 In p + 2 x 10-3~2,
(8b)
7 = 226.09
+ 0.199~ -
2.0
10-3~2 In p + 2 x 10-3~~.
(8~)
x
1.8 x
For this purpose
10-6~3,
(84
The coefficients in these equations have been chosen to correspond closely to the case of argon. Numerical data were generated at the densities (in amagat units) listed in table IV for argon with a random error added on to them; the latter was taken from a normal distribution with a variance of d = 0.07t. t The authors are indebted to Dr. D. T. Gillespie for providing them with suitable sets of random numbers.
eq.(8~)
eq.(8b)
eq.(8a)
Data generated with
6 52 96 96
12 30 40 40
226.10& 0.04 226.00f 0.04 226.03+ 0.04 226.09& 0.03 (226.09)
(exact)
15 8 24 28 31 55 40 96 (exact)
7 72 96 96
226.08& 0.05 226.07f 0.03 226.09* 0.03 226.10f 0.03 (226.09)
226.08h 0.05 226.06& 0.02 226.07-+ 0.03 226.09& 0.03 (226.09)
13 35 40 40
(exact)
p’
n %,
VI
0.187f 0.009 0.242f 0.008 0.230& 0.008 0.198+ 0.006 (0.199)
0.204f 0.012 0.207f 0.002 0.204& 0.004 0.198f 0.006 (0.199)
%I,) x
lo3
(2)
-6.1 & 0.3 -4.8 i 0.4 2.1f 0.4
(2)
1.06f 0.03 1.2 & 0.1 2.1 f 0.4
(1.5)
1.37+ 0.07 1.5 f 0.1 2.5 f 0.4
(72 f
ofthe algorithm
0.202+ 0.014 0.202f 0.003 0.198& 0.004 0.192f 0.006 (0.199)
71 f
Test
TABLE
%,I
x
-33*5 (0)
-2.0 f 0.7 (0)
-
(-1.8)
i
-
%*d
x
lo3
-2.03 f 0.08 (-2)
-0.23 f 0.08 (-0.2)
-
-0.23 f 0.09 (0)
l@:(r12'
-2.1 * 0.7
(r3 f
0.07 0.08 0.07 0.07 (0.07)
0.07 0.07 0.07 0.07 (0.07)
0.08 0.07 0.07 0.08 (0.07)
ON THE
Eqs.
DENSITY
(7) were fitted
EXPANSION
FOR
VISCOSITY
to the data thus generated
IN GASES
by the method
15
of least
squares. We used the Omnitab computer program for statistical analysis which uses a Gram-Schmidt orthogonalization precedurel6). In a comparative study by Wampleri7) this method was shown to be reliable for the purpose at hand. The results obtained for the fitted equations were studied as a function of p’ to determine the range 0 < p ==cp’, if any, in which criteria (i) and (ii) were satisfied. The most probable values for the coefficients of eqs. (7) are presented in table VI. In this and the subsequent table PL refers to the number of data points. It is seen that the data generated by the cubic equation (8a) produce polymonials (7a), (7b) and (7~) which do satisfy criterion (iii). More precisely, the quadratic equation reproduces the values of qo and ~1 obtained with the linear equation and the cubic equation reproduces the values of 70, 71 and 7s derived from the quadratic equation. Thus we conclude that the cubic polynomial is consistent with the data set. The value for ~1 obtained from the logarithmic equation (7~) is somewhat lower than the corresponding term in the linear equation. Nevertheless, the two values agree within their standard deviation so that our criteria do not permit us to rule out a logarithmic term when it is multiplied with a coefficient of 2 x 10-4. This conclusion is confirmed by the results obtained from the data generated according to equation (8b). This equation contains a logarithmic term with a coefficient of this magnitude. According to our criteria, both the cubic equation (7~) and the logarithmic equation (7d) are consistent with this data set. Finally, we considered the data generated by eq. (8~) where the coefficient of p2 In p is ten times larger than that in the previous equation. Our procedure now shows very clearly that the polynomials are not consistent with the data set. In other words, the quadratic and cubic equations do not reproduce the values of ~0 and 71 of the linear equation, nor does the cubic equation reproduce the value of q2 of the quadratic equation. On the other hand, the logarithmic equation (7d) does reproduce the factors ~0 and 71 within their combined standard deviations and is thus consistent with the data. We conclude that our algorithm can, indeed, distinguish between a polynomial such as eq. (7~) or a logarithmic equation (7d). Of course, for data with a finite precision, the distinction can only be made with a finite resolution. With our current relative precision of 3 x 10-A we can no longer detect a logarithmic contribution if its coefficient is as small as 2 x 10-4. 5. Results of analysis. The experimental viscosity data were subjected to the analysis described in the previous two sections. The most probable values thus deduced for the coefficients of eqs. (7a), (7b), (7~) and (7d)
16
J. KESTIN.
E. PAYKOC
AND
J. V. SENGERS
ON THE
DENSITY
are listed in table VII. to the individual same results
EXPANSION
In fitting
data points.
are obtained
FOR VISCOSITY
the equations
However,
IN GASES
17
equal weights were assigned
it was verified
when a weight proportional
that essentially
the
to the value of vis-
cosity is assigned to the data. We note that the analysis indicates that the relative precision of the experimental data is better than 3 x 10e4. Examples of two deviation plots for the cubic equation (7~) and the logarithmic equation (7d) are shown in fig. 2; they are virtually indistinguishable. Evidently the two equations satisfy criterion (i) equally well, as they do also for nitrogen. If we apply in addition criterion (iii), we see from table VII that the power series is consistent with the experimental data for nitrogen and argon to a high degree of satisfaction. This means that the linear, quadratic and cubic equations lead to the same coefficients. The logarithmic equation (7d) does not reproduce the value of 71 of the linear equation nearly as well. Thus we conclude that from viscosity data with a relative precision of 3 x 10-d a p2 In p term cannot be detected in the case of nitrogen or argon. It turns out that in the case of helium we cannot obtain significant information for all coefficients, but the results for the linear and quadratic equation are again in satisfactory agreement. As demonstrated in the previous section our procedure would have revealed the existence of a logarithmic term if its coefficient were significantly larger than 2 x lo-4 ppoise/am 2. Therefore, we conclude that such a coefficient is not larger than the above value, if it exists at all. In order to make this statement more precise, we rewrite eq. (7d) as f(q) = r -
G2
lnp = TO + VIP + q2p2.
(9)
In this new equation 7; is not considered as a coefficient to be determined, but as a parameter. Each value for the parameter $ defines a new set of data points f(q). For the “correct” value of 7; the data f(q) should satisfy our criteria when fitted to a polynomial. Even when we relax our criteria to allow variations within twice the standard deviations, we conclude that values for $ such that I$] > 6 x 10-J ppoise/ams cannot possibly be reconciled with our experimental data for nitrogen or argon. 6. Conclusions. It has been demonstrated that the experimentally determined density dependence of the viscosity of argon and nitrogen at 25°C can be consistently represented by a power series in the de’nsity. Implicit in this conclusion is the assumption that the first four terms of the density expansion (1) and (2) are adequate to describe the data in the density range under consideration. The existence of a logarithmic term of the form ps In p cannot be detected on the basis of viscosity data whose relative precision is 3 X 10M4. If such a term exists, it is concluded that the absolute value of its factor, vi, must be appreciably smaller than 6 x 10-J ppoise/am2 for
18
J. KESTIN,
E. PAYKOC
AND
J. V. SENGERS
argon as well as nitrogen at 25°C. From the practical point of view, there is no need to include such a term in describing the density dependence of the viscosity within the best precision currently obtainable. In view of the discussion in the introduction, we suggest that these results indicate that correlations that extend in the gas over distances of the order of a mean free path are quantitatively insignificant compared to correlations that extend over distances of the order of the range of molecular interactions. APPENDIX
In addition to the principal conclusions of the present paper regarding the form of the density expansion, the experimental data can be utilized in a practical way to provide us with very precise data on the zero-density value of the viscosity of the three gases and on the linear correction term. Referring to table VI1 we can summarize them as follows: Nitrogen 70 = (177.85 & 0.02) ppoise
at 25”C,
71 = (0.132 & 0.007) ppoise/amagat
at 25°C.
Argon 70 = (226.09 & 0.04) ppoise
at 25”C, at 25°C.
71 = (0.198 * 0.009) ppoise/amagat Helium 70 = (198.825 & 0.02) ppoise ~1 = -(0.019
f
at 25”C,
0.003) ppoise/amagat
Evidently,
the preceding
existence
of unaccounted-for
optimum
at 25°C.
values disregard
systematic
the possibility
of the
errors.
A c k n o w 1e d g e m e n t s. The experimental results described in this paper were obtained with funds provided by the National Science Foundation under Grant GK 2133X to Brown Univeristy. Mr. R. Paul was very helpful in maintaining the instrument in a first-class working condition. The analysis was performed with funds provided by the Arnold Engineering Development Center, Tennessee, under Contract F40600-69-CO002 with the University of Maryland and Contract (40-600) 66-494 with the National Bureau of Standards. The authors thank Dr. H. J. M. Hanley for some stimulating discussions.
ON THE DENSITY
EXPANSION
FOR VISCOSITY
IN GASES
19
REFERENCES 1) Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press (London, 3rd ed., 1970). 2) See Chapter 16 of ref. 1. 3) Dorfman, J. R. and Cohen, E. G. D., J. math. Phys. 8 (1967) 282. 4) Hanley, H. J. M., McCarty, R. D. and Sengers, J. V., J. them. Phys. 50 (1969) 857. 5) Kao, J. T. F., Viscosity of He-N2 mixtures at low temperatures and high pressures, Thesis, Rice Univ., Houston, Texas, 1966. 6) Gracki, J. A., Flynn, G. P. and Ross, J., J. them. Phys. 51 f 1969) 3856. 7) Sengers, J. V., Balk, W. T. and Stigter, C. J., Physica 30 (1964) 1018. 8) Le Neindre, B., Tufeu, R., Bury, P., Johannin, P. and Vodar, B., The thermal conductivity coefficients of some noble gases, Proc. 8th Thermal Conductivity Conference, C. Y. Ho and R. E. Taylor, eds., Plenum Press (New York, 1968) p. 75. 9) Kestin, J. and Leidenfrost, W., Physica 25 (1959) 1033. 10) Kestin, J. and Wang, H., J. appl. Mech. 24 (1957) 197. 11) Kestin, J., Leidenfrost, W. and Liu, C. Y., 2. angew. math. Phys. 10 (1959) 558. 12) Sengers, J. V., Transport properties of compressed gases, Recent Advances in Engineering Science, Vol. 3, A. C. Eringen, ed., Gordon and Breach (New York, 1968) p. 153. 13) Hilsemath, J. et al., Tables of Thermodynamic and Transport Properties, NBS Circular 564 (U. S. Government Printing Office, Washington, D. C., 1955). 14) Whalley, E., Lupien, Y. and Schneider, W. G., Canad. J. Chem. 31 (1953) 722. 15) Schneider, W. G. and Duffie, J. A. H., J. them. Phys. 17 (1949) 751. 16) Hilsenrath, J., Ziegler, G. C., Messina, C. G., Walsh, P. J. and Herbold, R. J,, “Omnitab”, NBS Handbook 101 (U. S. Government Printing Office, Washington, D. C., 2nd ed., 1968). 17) Wampler, R. H., J. Res. NBS, 73B (1969) 59.