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The low temperature thermal conductivity of 4He
Physica 103B (1981) 212-225 © North-Holland Publishing Company
THE LOW TEMPERATURE THERMAL CONDUCTIVITY OF 4He II. MEASUREMENTS IN THE CRITICAL REGION A. ACTON* and K. KELLNER**
PhysicsDepartment, University of Southampton, Southampton, UK Received 28 August 1979 Revised 1 July 1980
Measurements of the thermal conductivity of helium-4 along five isotherms between 5.1 and 5.6 K and covering a range of densities from 30 to 120 kg m-3 are reported. These results were obtained with a guarded flat plate apparatus using temperature differences from 0.025 to 0.1 K and fluid layer thicknesses of 0.4 and 0.2 mm. Experimental tests showed that convection probably contributes less than 5% to the measured thermal conductivity in all but a few cases. The experimental data are compared at the critical density with recent theoretical predictions. Although the agreement between theory and experiment is not excellent, it is good enough to demonstrate that the theory can be applied to helium-4 and to show that sufficiently close to the critical temperature helium-4 appears to behave like a classical fluid.
1. Introduction In part I of this paper [1], measurements of the thermal conductivity were reported which covered the dilute gas, the dense gas and the normal liquid in the temperature region between 3.3 and 20 K and at pressures up to 2.5 MPa. In this part, additional measurements are discussed which were carried out along five isotherms in the critical region. All five isotherms exhibit an anomalous behaviour of the thermal conductivity as the critical density is approached either from above or from below. The experimental arrangement was exactly the same as that for the measurements of part I, indeed the measurements were all made at the same time, using the cell described by Acton and Kellner [2]. The division of the paper into two parts is therefore not due to a difference in experimental technique but rather due to a difference in the physical state of the fluid in the critical and non-critical regions. The large value of the de Boer parameter [3] for helium-4 A* = h/(me)~o of 2.68, where h is Planck's * Now with Engineering Sciences Data Unit Ltd, 251-9 Regent Street, London W1R 7AD, UK. ** Now at Institute of Cryogenics, University of Southampton, Southampton, U.K. 212
constant, m the atomic mass, e the depth of the attractive potential well and o the (hard core) atomic diameter, is a measure for the considerable quantum mechanical influence on the physical properties. This has already been referred to in part I where it was shown that the thermal conductivity o f the liquid has a positive temperature coefficient in contrast to the negative temperature coefficient of simple classical liquids. Further quantum influence on the thermal conductivity may well be expected in the liquid-gas critical region although, in their static properties, the critical behaviour of quantum fluids does n o t differ greatly from that of classical fluids [4, 5].
2. Thermal conductivity in the critical region The anomalous behaviour in the fluid critical region of some physical properties is well established by now. Divergencies in the specific heats, Cp and co, and the isothermal compressibility, K T, have been known for a long time. The thermal conductivity has been investigated only relatively recently but divergence of this property in the critical region has been shown to occur for a number of fluids [6, 7].
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
The enhancement of the thermal conductivity can quite easily be confused with an increase in convection in the critical region and tests have to be carried out in order to establish that convection is either absent or, at least, reduced to negligible proportions. In the case of a direct determination of the conductivity these tests depend on the type of apparatus used for the measurements. For example in the case of a concentric cylinder cell the onset of convection can be varied by turning the cylinders from a vertical to a horizontal position or vice versa [8]. In the case of a parallel plate cell a similar test can be made by tilting the plates away from their normal horizontal position. It is also possible with this type of cell to establish the absence of convection by changing the gap between the plates. Of course, in the first place, the cell has to be designed in such a way so as to reduce convection as much as possible.
3. Measurements
As in part I, pure helium gas was used and all temperatures were measured on the 1965 NBS Provisional Temperature Scale (based on the acoustic thermometer). Further information on the gas and on the accuracy of temperature measurement are given in part I. Values of the critical temperature, Tc, and the critical density, Pc, are taken from Kierstead's work [9] and are 5.198 K and 69.6 kg m -3, respectively. The experimental technique was similar to that adopted for the measurements in the non-critical region reported in part I, and similar corrections were applied to the measured heat flows and temperature differences. The thermal conductivity was also calculated in a similar way except that in the critical region the cell constant, A/s, could be taken to be invariant for each isotherm and individual determinations of its value for each experimental point were unnecessary. This permitted the direct determination of the fluid density from the measured cell capacitance, see section 5. The thermal conductivity measurements made for five isotherms close to T c are listed in tables I to V. Each measurement has been allotted an identification code using the same system as in part I. In the tables, T a is the average temperature of the fluid layer and P is its pressure; the temperature difference across the
213
layer and the heat flow through it are AT and W, respectively. The cell capacitance is C, p is the fluid density and ?~is the thermal conductivity. The variation of thermal conductivity with temperature and density is illustrated in fig. 1. Full lines in this figure have been faired through the data on each isotherm. Experimental data for which convection was purposely introduced and experimental data from run L (indicated by the prefix L to the code in table II and for which the test gap, s ~ 0.2 mm) have been omitted from this figure; these data are discussed in sections 4.1 and 4.2, respectively. The curve for 5.248 K has not been extended through the three data points nearest to the peak in ?, because these data are believed to be affected by convection (see the discussion in section 4.2). Apart from those results affected by convection, which are marked with an asterisk in tables II and III, the measured thermal conductivities are expected to be accurate to -+5%with AT = 0.025 K and more accurate than this with the larger temperature differences. Complete discussion of the expected accuracy is given in part I. The results in the critical region were, however, nearly all obtained during one experimental run (run K, s ~. 0.4 mm) during which the cell was maintained at low temperature. Most of the error in stems from the irreproducibility of the germanium thermometers on thermal cycling between room and low temperatures, and so the results of run K illustrated in fig. 1 would be expected to exhibit a scatter of rather less than 5%. This is indeed the case, the precision of k being about +1%. Another possible source of error, boundary resistance, was undetectable for the results in the non-critical region reported in part I, and there is, therefore, good reason to suppose that no significant boundary resistance occurs in the critical region. Enhancement of thermal conductivity as the critical density is approached along an isotherm from either above or below is dearly apparent from fig. 1. Also, as T a approaches T c from above, the enhancement at any given value of p increases. The behaviour is similar to the effects observed with CO 2 and CH4 [7] ; A [7, 8] ; 3He [I0] ; and in another study with 4He [11].
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
214
Table I Thermal conductivity of 4He at 5.1 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (mW)
C (pF)
p (kg m -3)
X (W m -1 K -1)
K239 K247 K246 K245 K244 K243 K242 K241 K240
5.097 5.098 5.097 5.097 5.098 5.097 5.097 5.098 5.097
0.1870 0.1954 0.2027 0.2078 0.2147 0.2204 0.2302 0.2768 0.382
0.1048 0.0548 0.0549 0.0254 0.0254 0.0550 0.1047 0.1050 0.1049
6.545 3.572 3.779 1.923 2.371 5.221 9.985 10.34 11.03
45.52 45.57 45.64 45.71 46.70 46.75 46.80 46.95 47.12
30.2 33.3 37.4 41.5 96.9 99.7 102.9 111.1 120.4
1.23E-02 1.28E-02 1.35E-02 1.49E-02 1.84E-02 1.87E-02 1.88E-02 1.94E-02 2.07E-02
Table II Thermal conductivity of 4He at 5.25 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (mW)
C (pF)
p (kg m -a)
X (W m -1 K -1)
K248 K260 K249 K250 K251 K252 K263 K253 K261" K262" K254" K264 K265 K255 K266 K256 K257 K258 K267 K259
5.248 5.248 5.248 5.247 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248
0.1979 0.2152 0.2196 0.2271 0.2309 0.2330 0.2342 0.2349 0.2352 0.2367 0.2369 0.2376 0.2392 0.2410 0.2412 0.2452 0.2531 0.2752 0.333 0.445
0.1053 0.1051 0.0546 0.0549 0.0249 0.0249 0.0248 0.0249 0.0250 0.0252 0.0248 0.0247 0.0251 0.0247 0.0251 0.0252 0.0551 0.0551 0.1047 0.1046
6.720 7.236 3.929 4.266 2.172 2.275 2.553 2.818 2.954 3.164 3.101 2.911 2.519 2.373 2.408 2.411 5.305 5.395 10.62 1t.26
45.52 45.62 45.67 45.77 45.85 45.92 46.01 46.08 46.10 46.25 46.28 46.34 46.45 46.52 46.52 46.60 46.68 46.80 46.95 47.11
30.2 36.4 39.2 44.6 49.0 53.0 58.0 62.3 63.6 71.8 73.4 76.8 83.1 87.1 87.1 91.5 96.0 102.9 111.0 120.1
1.26E-02 1.36E-02 1.42E-02 1.53E-02 1.72E-02 1.80E-02 2.03E-02 2.23E-02 2.33E-02 2.47E-02 2.46E-02 2.32E-02 1.97E-02 1.89E-02 1.89E-02 1.88E-02 1.89E-02 1.93E-02 2.00E-02 2.12E-02
Cell constant = 10.310 m L57 L58 L59 L60 L61 L62 L63
5.248 5.248 5.248 5.248 5.248 5.248 5.248
0.1986 0.2310 0.2350 0.2364 0.2385 0.2468 0.445
0.0249 0.0250 0.0252 0.0251 0.0251 0.0251 0.0250
* Indicates result significantly affected by convection.
3.249 4.328 5.572 5.856 5.390 4.976 5.543
92.39 93.07 93.55 93.81 94.21 94.61 95.57
31.0 50.0 63.5 70.9 82.0 93.1 119.4
1.26E-02 1.68E-02 2.15E-02 2.26E-02 2.08E-02 1.92E-02 2.15E-02
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region Table III Thermal conductivity of 4He at 5.3 K
215
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (roW)
C (pF)
p (kg m -3)
h (W m -1 K -1)
K194 K268 K195 K196 K197 K269 K198 K270 K199 K200 K201 K202 K203 K279 K204 K271 K272" K273" K274" K275" K276 K277 K278
5,297 5.297 5.297 5,297 5.297 5,298 5,298 5.298 5,298 5.297 5.297 5,298 5,297 5.298 5.297 5.298 5.297 5,297 5.298 5.298 5.297 5.298 5.298
0.2022 0.2201 0.2264 0.2362 0.2418 0.2436 0.2448 0.2458 0.2468 0.2497 0.2516 0.2610 0.2869 0,349 0,463 0,2460 0.2459 0,2459 0.2459 0.2459 0.2460 0.2459 0.2461
0.1047 0.1046 0.0551 0.0552 0.0249 0.0252 0.0250 0.0246 0.0249' 0.0250 0.0249 0.0549 0.0546 0.1046 0.1046 0.0255 0.0246 0.0246 0.0245 0.0249 0.0249 0.0247 0.0249
6,738 7,274 3.969 4.473 2.399 2.636 2,764 2.737 2.712 2.487 2.405 5.300 5.337 10.69 11.36 2,796 3.925 3.648 3,328 3,006 2,720 2,596 2,701
45.52 45.63 45.69 45.83 46.00 46.11 46.21 46.27 46.34 46.46 46.51 46.64 46.79 46.94 47.11 46.27 46.27 46.27 46.27 46.27 46.28 46.28 46.29
30.5 36.6 40.2 48.0 57.8 64.0 69.4 73.1 76.7 83.8 86.5 93.5 102.2 110.8 120.1 73.2 73,2 73.2 73.2 73.2 73.5 73.3 73.8
1.27E-02 1.37E-02 1.42E-02 1.59E-02 1.90E-02 2.06E-02 2.17E-02 2.19E-02 2.14E-02 1.96E-02 1.90E-02 1.90E-02 1.92E-02 2.01E-02 2.14E-02 2.16E-02 3.14E-02 2.91E-02 2.67E-02 2.38E-02 2.15E-02 2.07E-02 2.14E-02
* Indicates result significantly affected by convection.
Table IV Thermal conductivity of 4He at 5.4 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
I¢ (mW)
C (pF)
p (kg m -a)
h (W m -1 K -1)
K205 K206 K207 K208 K209 K210 K211 K212 K213 K214 K215 K216
5,397 5.397 5.398 5.397 5.397 5.397 5.397 5,397 5.397 5,397 5.397 5,397
0.2094 0.2356 0.2492 0.2572 0.2603 0.2627 0.2681 0.2734 0.2838 0.317 0.380 0.500
0.1050 0.0548 0.0552 0.0253 0.0250 0.0249 0.0248 0.0247 0.0550 0.0550 0.1048 0.1048
6.838 3,964 4.435 2.305 2.429 2.455 2.421 2.363 5.331 5.457 10.83 11.51
45.52 45.68 45.84 46.02 46.13 46.23 46.40 46.50 46.61 46.79 46.94 47.10
30.4 39.8 48.7 58.9 64.9 70.4 80.0 85.9 92.1 101.9 110.6 119.6
1.28E-02 1.42E-02 1.58E-02 1.80E-02 1.91E-02 1.94E-02 1.92E-02 1.89E-02 1.91E-02 1.95E-02 2.03E-02 2.16E-02
216
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
Table V Thermal conductivity of 4He at 5.6 K
Cell constant = 5.080 m
Code
Ta (K)
P (MPa)
AT (K)
W (roW)
C (pF)
p (kg m -3)
h (W m -1 K -1)
K217 K218 K219 K220 K221 K222 K223 K224 K225 K226 K227 K228
5.597 5.598 5.597 5.598 5.597 5.598 5.598 5.598 5.598 5.598 5.598 5.598
0.2230 0.2538 0.2726 0.2866 0.2925 0.2976 0.306 0.317 0.339 0.374 0.435 0.546
0.1054 0.1051 0.0551 0.0550 0.0254 0.0250 0.0250 0.0551 0.0549 0.1050 0.1049 0.1046
7.038 7.652 4.425 4.834 2.311 2.396 2.400 5.366 5.420 10.61 11.07 11.68
45.51 45.67 45.82 46.00 46.11 46.20 46.33 46.47 46.63 46.77 46.91 47.06
30.3 39.5 48.2 58.3 64.3 69.6 76.9 84.7 93.6 101.4 109.3 117.8
1.31E-02 1.43E-02 1.58E-02 1.73E-02 1.79E-02 1.89E-02 1.89E-02 1.92E-02 1.94E-02 1.99E-02 2.08E-02 2.20E-02
2"5 o o
Tc = 5-198K o
Ta/K
25
5598 5-597 5 297
21
~" 5 - 2 4 8 "~ 5 . 0 9 7
T - ~ 1.9
~E
o 1.7 ,.-<
1"5
1.5
1~20
50
410
I
510
!'i °
8'0
p /kg m-3
910
'
I00
IllO
'
120
'
150
Fig. 1. Isotherms of the thermal conductivity of 4He in the critical region.
4. E s t i m a t e o f c o n v e c t i o n It was s h o w n in p a r t I t h a t the m e a s u r e m e n t s in the n o n - c r i t i c a l region were n o t a f f e c t e d b y convection. As far as t h e critical region is c o n c e r n e d , convect i o n c a n be d i s c o u n t e d e x c e p t for a few m e a s u r e m e n t s
very close to t h e critical p o i n t w h e r e its i n f l u e n c e is e s t i m a t e d t o give an a p p a r e n t t h e r m a l c o n d u c t i v i t y t h a t is n o t m o r e t h a n 10% t o o high. This is discussed below. A d e t a i l e d analysis o f free ( n a t u r a l ) c o n v e c t i o n in a near-critical fluid c o n t a i n e d in a c o n d u c t i v i t y cell,
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region similar to the one used in this work, was given by Michels and Sengers [12], and is discussed in part I. This analysis shows that convection can occur if the test gap is not quite horizontal. Apart from those measurements discussed in section 4.1 for which it was deliberately tilted away from the horizontal, the top plate of the Dewar vessel containing the thermal conductivity cell was levelled to within +4.4 mrad (0.25 ° ) for the measurements in the critical region. This should have ensured that the test gap was horizontal to within the same range of angle. However, the experiment discussed in section 4.1 showed that the test gap could, in fact, have been out of the horizontal by up to 12 mrad (0.69°). From Michels and Sengers' analysis, if Xa is the apparent thermal conductivity with convection present and X is the true thermal conductivity, Xa -- )k X
/S'~ Ra sin
= qR1 ~A)
360
(1)
In eq. (1), R 1 is the radius of the hot plate (emitter), A/s (the cell constant) is the effective heat transfer area of the hot and cold plates divided by their separation and ~ is the angle that the normal to the plates makes with the horizontal. The Rayleigh number is Ra = p2CpZgS3AT/~X, where z = -(1/p)(ap/aT)p is the volume expansion coefficient of the fluid, g is the acceleration due to gravity, 77is the coefficient of shear viscosity and the other symbols have their usual meanings. Because the emitter is surrounded by a guard ring, only a fraction of the convected heat flow across the test gap actually comes from the emitter; this fraction is represented by q. Values of the Rayleigh number were estimated for the whole range of conditions under which the experimental results were obtained using the tables of Thermophysical Properties of 4He by McCarty [13] with the exception of X, for which experimental values from this work were used. The highest Rayleigh number for all of the experimental data in the critical region (K262 in table II; T a = 5.25 K, AT = 0.025 K, p = 71.8 kg m -3 and s = 0.4 mm) is estimated to be 4 X 106. Taking ¢ = 12 mrad (the upper limit) and setting q equal to its maximum value of unity, eq. (1) predicts that Xa is in error by 64% due to convection! Therefore, detailed investigation of convection in the critical region is warranted. It can be seen from eq. (1)
217
that the error in k a due to convection depends on the value of Ra, the cell dimensions R 1, s and A and on the angle 4). For the cell used in this work R 1 (and consequently A) is a fixed quantity, but by changing AT, ¢ or s, it is possible to test for convection experimentally. For the measurements in the non-critical region, reported in part I, it was possible to show that convection was insignificant by varying the heat flow in the test gap and thus varying AT whilst maintaining the average fluid temperature, Ta, constant. However, in the critical region, k is a strong function of temperature and large values of AT cannot generally be expected to yield accurate values of k at the average fluid temperature even in cases where convection is insignificant. Thus, the variation of AT as a test for convection is not reliable unless AT is always small in comparison with T a - Tc, a condition that could not be achieved for these experiments. Consequently, tests for convection were restricted to varying ¢ and s.
4.1. Variation o f ¢ A test for convection was carried out by tilting the fluid layer with respect to its normal horizontal position. This was done for the 5.3 K isotherm very close to the thermal conductivity peak. The results are given in table III (coded K271 to K278 inclusive) and are illustrated in fig. 2. Fig. 2 shows that the apparent thermal conductivity is a minimum for an angle of tilt of 9 mrad instead of the hoped-for angle of zero. This is probably due to uncertainty in the angle of tilt of the test gap. Since, during an experiment, the angle of tilt of the test gap could only be obtained from the measured angle of tilt of the Dewar top plate, this experiment indicates that the tilt of the test gap does not quite follow the tilt of the top plate. Furthermore, this tilting test should have been repeated in a plane perpendicular to that of the first test in order to fmd the position of minimum Xa; unfortunately this could not be done without causing damage to the apparatus and time did not permit a more thorough investigation. The difference in Xa between zero tilt (K278) and the minimum value of k a is about 3%. Assuming that the levelling error in the perpendicular plane is similar to that in the tested plane, +9 mrad, the maximum magnitude that ¢ can attain in any plane is 12 mrad. Assuming further that the minimum k a in fig. 2
218
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region 34
52 J 2 7 2
50
T
2"8
T
~" _o 2-6 x
2.4
2.2
2<
K271
K277
20
50
40
Measured angle of tilt/mrad
50
60
70
(10 mrad=0.57 o)
Fig. 2. Effect of tilting the Dewar vessel on the apparent thermal conductivity at 5.3 K. corresponds to negligible convection, the figure shows that a value of ¢ = 12 mrad corresponds to ~'a being in error by about 4%. Putting ~b= 12 mrad into the righthand side of eq. (1), taking q = 1 and other parameters appropriate to the conditions of experimental points K271 to K278, Xa is predicted to be in error by 28% due to convection. Therefore, eq. (1) with q = 1 is conservative for this apparatus. 4.2. Variation o r s
A few measurements were made along the 5.25 K isotherm during run L, with s ~ 0.2 ram, in order to provide a second test for convection. Fig. 3 shows a comparison of these results with the 5.25 K results of run K, with s ~ 0.4 mm. The figure shows that the results of the two runs are in good agreement except for a region close to the critical density, where the run L values are up to 10% lower than those of run K. This behaviour strongly suggests that several of the run K results are influenced by convection. For the result from run L closest to Pc, (L60 in table II; T a = 5.25 K, A T = 0.025 K, p = 70.9 kg m - 3 , A / s = 10.31 m and
taking ¢ = 12 mrad) eq. (1) predicts that Xa is in error by 6.7% due to convection. This represents the worst case for the run L results and, as eq. (1) was shown to be conservative in section 4.1, it is clear that the results L57 to L63 are almost certainly free of significant convective influence. We may conclude from this that the three results of run K near the thermal conductivity maximum, K261, K262 and K254 are most probably significantly affected by convection, but to an extent not exceeding 10% of the values given in table II. The results that are significantly affected by convection are marked with asterisks in tables II and III. For results not marked with asterisks, it is believed that convection contributes less than 5% to the measured value of the thermal conductivity.
5. Determination of the fluid density in the critical region The thermal conductivity in the critical region is best studied from a conductivity-density graph and, therefore, reasonably accurate values of the density
219
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
must be obtained. In this work, the average density of the fluid sample in the conductivity cell was deduced directly from measurements of the cell capacitance using the Clausius-Mossotti relation. This method was adopted rather than using McCarty's tables [13] because, in the latter case, the evaluation of the density from the measured values of pressure and temperature could be in considerable error in the critical region. The Clausius-Mossotti relation, which connects the relative permittivity, er, of a fluid with its density, p, may be written as er - 1
=pp,
(2)
er+ 2 where p is the specific polarization of the fluid. For helium-4, an empirical relation for p deduced from the results of Kerr and Sherman [14] is p =E 1
-
(3)
E2P ,
where E 1 = 1.29095 X 10 - 4 m 3 kg -1 a n d E 2 = 1.5 X 10 - 9 m 6 kg-2; hence
e r -- 1 er+2
Since E 2 / E 1 is small, the right-hand side of eq. (4) can be expanded and, neglecting powers o f E 2 / E 1 higher than the first, one obtains e r -- 1 P ~ ' E l ( e r + 2) --
(E2/E1)(e
r
(5)
-- 1)"
All measuretnents made outside the critical region show that the variation of cell constant with pressure is negative, approximately linear with pressure and small (0.2%/MPa for a 0.4 m m gap). For the critical region the pressure changes were limited to a range of some 0.3 MPa; consequently, the variation of the cell constant was expected to be very small indeed. In order to check this, and also to obtain appropriate values of the cell constant, eq. (4) was used to find the relative permittivity of the fluid at the lower and upper extremes of each conductivity isotherm..These extremes were arranged to be well removed from the critical region and so McCarty's tables [13] provided
I
2.5
(4)
- p ( E 1 - E 2 P ).
Run
5% °
~
o + [
s/mm K
04
L
0-2
2.3 +
21 T 19
15
1.3
1120
,
J
J
I
Th ~
30
40
50
60
70 p/kg
I 80 m-3
l
I
J
j
90
I00
I10
120
Fig. 3. Effect of changing the test gap on the apparent thermal conductivity at 5.25 K.
220
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
values of # for use in eq. (4) which were accurate to within about 0.5%, yielding values of er accurate to within about 0.02% at worst. The cell constants, A/s, could then be evaluated from the expression for the capacitance, C, of a parallel plate condenser A - =
s
C 8.8542 X 10-12e r '
(6)
(see part I, p. 193) for the two density, and hence pressure, extremes. These calculations showed that the variation of cell constant with pressure was almost negligible for the range of pressures covered for the measurements in the critical region. It was considered sufficient, therefore, to average the two values of the cell constant which were obtained for each conductivity isotherm. The average cell constants were then used to calculate values of p from the measured capacitance over the whole of each isotherm using eqs. (6) and (5). It is these values of p that are given in tables I to V. The approximation involved in using eq. (5) can be seen to be quite justified since, in no case, did the second term of the denominator exceed 0.2% of the first term. The accuracy of p determined in this way was governed mainly by the precision of the determination of C. Assuming that the only error here was due to the capacitance bridge, the measurements of which were precise to within 0.02%, p was accurate to within 2% at 30 kg m -3 ranging to 0.5% at 120 kg m -3. It is reassuring to note that the cell constants in tables I - V (except that for the nominal 0.2 mm gap), and which were separately determined over a period of eleven days, lie all within 0.02% of one another. Except for a few points above 0.3 MPa, where bourdon gauges were used, measurements of cell pressure were made with a mercury manometer, suitably corrected for temperature, latitude, altitude and hydrostatic effects. The accuracy of pressure measurement was better than +-0.03% with the mercury manometer, and better than +1% with the bourdon gauges. 6. E q u a t i o n o f state data
The pressures and densities measured in this study can be compared with equation of state data recently obtained by other workers. This comparison is iUus-
trated in fig. 4 where the full lines represent our five isotherms in the critical region. Also shown are ten points taken from independent P---O- T measurements by Roach [15] (corrected to the NBS Provisional Temperature Scale used in this work). Roach also used a capacitive method to determine p and consequently assumed the validity of the Clausius-Mossotti relation in the critical region. As can be seen from fig. 4, the agreement between the two sets of data is very good which is not really surprising since, in both studies, the experimental methods were based on the same principles. More recent measurements by Kierstead [9] using a volumetric method, show also very good agreemelat with Roach's results [5]. This demonstrates that the Clausius-Mossotti relation can be used with confidence in the critical region.
7. Dilute gas, excess and ideal c o n t r i b u t i o n s t o the thermal conductivity It is customary to consider the thermal conductivity of a gas to be due to the sum of various contributions [7], illustrated schematically for two temperatures (T 2 > T 1 > Tc) in figs. 5a and 5b. Taking density and temperature as the independent variables, the thermal conductivity for all p, T can be represented by
?t(p, 73 = ?,0(73 + Ax(;, 73,
(7)
where ?t0(T) is the dilute gas thermal conductivity (the low-density limit that depends on temperature alone) and Ak(,o, T) is the excess thermal conductivity. In the critical region there is an anomalous increase in k which can be accounted for by considering the excess thermal conductivity as being made up of two contributions A?t(p, T) = A?tid(,O, T) + A?tc(p, T),
(8)
where AXia is the ideal, or background, excess thermal conductivity and A?tc is the critical excess thermal conductivity. Values of ?t are obtained experimentally but, in order to understand the critical anomaly, values of A?tc are required. The dilute gas thermal conductivity, )t0, is often known as a function of temperature. Normally (for classical gases), A?tid is a very weak
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
221
o 32 03i
Rooch,/~
0 30
5 ~
0 29 0 28[ 027
5"397 0 26I 0 25 5 248 o 24~ 023 5097 0 22 02
plkg
m -5
Fig. 4. Equation of state data in the critical region.
a
b
/ /.f___/~.-" "~?'TkT_, ,__~,°,~.,~)
),(&,T 2)
),(p= ,TI) ....
-t ....
.~-_ __t ......
4- ....
I
P, pc
~- .....
'. ....
J .....
ko(T2)
ko(Ti )
P
I
p,p~
Fig. 5. Contributions to the thermal conductivity. (a) At T 1 ; (b) at T 2.
p
222
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
function of temperature. Consequently, it is only necessary to determine the variation of A)kid with [ at a single temperature in order to obtain 2x)kc at all p and T from eqs. (7) and (8). This is not so for helium (and hydrogen) for which A)kid depends on both temperature and density. In this case it is appropriate to define the ideal thermal conductivity, )kid, which is the thermal conductivity the fluid would have if A)kc did not exist
all, least-squares polynomial curve fits of X as a function of p along the experimental isotherms in the noncritical region were made. These polynomials were then used to generate values of X along isochores, which were subsequently cross-plotted to give a graph of )kid versus p for the isotherms used in the critical region. Fig. 6 is a small-scale version of this graph. Values of )kid taken from the original graph have been subtracted from the experimental values of X in the critical region to give the experimental points illustrated in fig. 7. Only data at temperatures above T c have been treated in this way as insufficient data for analysis are available at temperatures below T c. Data that were influenced by convection and data from run L have been omitted from fig. 7 ; extra data from part I that show thermal conductivity enhancement have been included. These extra data are at temperatures of 6, 7 and 8 K and are all from run K with AT 0.2 K. Full lines are curves faired through the experimental points. The reduced relative temperature and the reduced relative density defined as follows are useful,
(9)
)kid(O, T) = )k0(T) + A)kid(o , 7).
The critical excess thermal conductivity is then given by
(1o)
AXc(O, T) = )k(p, T) -- hid@, T).
The ideal thermal conductivity, kid , can be estimated by extrapolation into the critical region of data sufficiently far removed from the critical point and for which AXc is negligibly small. This has been done in the following way using the data in part I. First of 29
To/K 8
27
25
6
23
5.6 5-4 'i2
I
o
~-19
1.7
1.5
13
I t:~0
I
30
40
50
I
60
I
I
70 80 p/kg m-3
I
90
Fig. 6. Ideal thermal conductivity.
I
I00
I
I10
J
120
A. Acton and K. Kellner/Thermal conductivity o f 4He in the critical region
223
1.0
O 0.9 Exp. Points
0.8
[]
0.7
0-6 T T E
[]
Temp. Ta/K
Cole. Points v =0.543 v =0.63 ~¢o = 0 - 3 6 0 ~¢o= 0 . 2 2 Ref. 20 Ref. 21
5 "248 5.297 5.397 5.598 5.997
® ® ® ® ®
[] [] [] @ []
6.998 7.997
(~)
Similar to
®
®.®
0"5 x
0.3
®
® 0.1
30
I
I
I
40
50
60
I
70
80
90
,oo
,Yo
,2o
p/kg m-5 Fig. 7. Isotherms of the critical excess thermal conductivity of 4He.
A7~ - Ta - Tc
Tc
and
A/3 =
P - Pc Pc
An estimated value of the thermal conductivity at 6 K and at the critical density was given in fig. 3 of part I to fill a gap in the experimental data. This estimate was obtained from a least-squares linear fit of the values of log (A~c) at Pc versus log (AT~) for temperatures of 5.3, 5.4, 5.6, 7 and 8 K, giving A~c(Pc , 6) = 1.9 X 10 -3 W m -1 K -1. Adding the ideal thermal conductivity gave k(Pc, 6) = 1.80 X 10 -2 W m -1 K -1.
As discussed in section 8, A~ c is only given approximately by this method at temperatures above about 5,4 K. However, a reasonable estimate of the (total) thermal conductivity at 6 K is obtained because A3.c contributes only about 10% to the total in this case.
8. Comparison with theory It is theoretically predicted [16] that, along the critical isochore and within a certain range of temperature above the critical temperature, A?~c should be
224
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
proportional to (AT) -a where a is a universal (fluidindependent) constant called the critical exponent of the thermal conductivity. Furthermore, theory predicts that the constant a should satisfy the equality involving two other comparably-defined critical exponents,
critical point is approached and vanishing far away from the critical point. From an analysis of the data for carbon dioxide and, to a lesser extent, for methane and argon it is proposed that F(AT, A~3)= exp (-[18.66(A7~) 2 + 4.25(A~) 4] }. (13)
a = 7 - v.
(11)
In this equality 3' is the critical exponent of the isothermal compressibility, K T, and u is the critical exponent of the correlation length, ~, The correlation length can be visualised as the mean radius of molecular clusters. Although the equality (11) is generally accepted there exists a fairly wide range of experimentally determined values of the exponents, particularly for v. Normally, 3' is about 1.19 although in the case of helium-4 a value of 1.1743 --- 0.0005 has been determined for a particular equation of state [5, 9]. At present, the generally accepted value is v = 0.63 -+0.02 [5, 17] although in the case of helium-4 values as low as 0.543 -+ 0.046 and as high as 0.67 -+ 0.02 have been reported [18]. Measurements of the thermal conductivity of carbon dioxide [19] have given a = 0.60 ---0.05 and measurements of the thermal conductance of a cell filled with helium-4 [11] have given a = 0.67 +- 0.31, values that are consistent with the above values of 3' and v and the equality (11). Whilst it would have been interesting to obtain a value of the exponent a from our results for comparison with the above, there are insufficient data to do this. Only a very limited number of isotherms were studied and, according to Hanley et al. [7] the simple proportionality between AXc and (A]~)-a ceases to be valid for A7~ > 0.03 (Ta > 5.4 K) approximately. The peak of the 5.25 K isotherm was not resolved in our measurements and so only two points are available, at 5.3 and 5.4 K. Therefore, an alternative comparison between theory and experiment is made. Hanley et al. [7] in a development of Kadanoff and Swift's theory [16] show that the critical excess thermal conductivity of classical fluids can be estimated from the equation K T F(AT, At5).
(12)
The function F(AT, Ats) is empirically determined to fit the experimental data, approaching unity as the
In order to compare the predictions of eq. (12) with our experimental results for helium-4 the parameters in the equation have been evaluated as follows. Boltzmann's constant, k, and the absolute temperature, T(= Ta), are known. The dynamic viscosity, ~/, and the isochoric pressure derivative, (~P/aT)o, are relatively slow-moving parameters in the critical region and are therefore obtained by interpolation within McCarty's tables [13]. The correlation length, ~, is evaluated along the critical isochore using the equation :~0(AT~ -v.
(14)
Two values of v and their corresponding two values of G0 are selected from those listed by Roe and Meyer [18] : Ohbayashi and Ikushima's [20] values, v = 0.543 -+ 0.046 and G0 = 0.360 -+ 0.078 nm, because these values fit our experimental data best and Tominaga's [21] values, v = 0.63 + 0.1 and G0 = 0.22 _+0.06 nm, because the exponent closely reproduces the value expected from other experimental and theoretical studies [17]. No experimental determinations of ~ are known for p 4: Pc" Whilst in principle it is possible to calculate for all/9 using methods outlined by Hanley et al. [7] this is not a trivial task and has not been attempted here. Therefore, the comparison between theory and experiment is restricted to the critical isochore only. The isothermal compressibility, KT, is estimated using the equation F
- ) _ .~ r r = ~ ( ATs8
(15)
Eq. (15), valid along the critical isochore only, is taken from the paper by Levelt-Sengers et al. [5], with constants from table 17 of their paper. The critical pressure (Pc)is 0.2275 MPa, I~ = 0.1611 and 3' = 1.174. The parameter AT58 is in terms of the (extrapolated) 1958 Helium-4 Vapour Pressure Temperature Scale.
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region The necessary corrections to our temperatures, measured on the 1965 NBS Provisional Temperature Scale are estimated using the data in ref. [22]. Values of A?~c at Pc calculated using eqs. ( 1 2 ) - ( 1 5 ) are compared with the experimental results in fig. 7. The agreement between experiment and theory is quite good at temperatures between 5.25 and 5.6 K and is best when the values of v and 40 from ref. 20 are used rather than those from ref. 21. In this temperature region, the function F ( A 7~, AtS) is not significantly different from unity and helium-4 behaves very much like the classical fluids carbon dioxide, methane and argon. At higher temperatures, however, the calculated values of A?~c are lower than the experimental results indicating that the term in A T in eq. (13) is incorrect for helium-4 (the coefficient 18.66 is too big). The thermal conductivity measurements at 6, 7 and 8 K are accurate to about +--1%,see part I. It is believed that the estimation o f hid was carried through with similar precision and so it is unlikely that the disagreement between experiment and theory at these higher temperatures is spurious. However, more data at temperatures between 5.6 and 10 K are needed to establish firmly the temperature dependence o f F(A7~, At3) for helium-4.
9. Conclusions The thermal conductivity of helium-4 shows enhancement in the critical region. The experimental data above the critical temperature show that theoretical prediction methods recently developed for classical fluids are successful for helium-4 at the critical density and for reduced relative temperature, A7~, less than about 0.08. It is shown that at larger values of A7~ the enhancement falls off less rapidly with increasing temperature with helium-4 than it does with the classical fluids carbon dioxide, methane and argon. Further data for the correlation length and the thermal conductivity are needed to extend the theoretical predictions to the whole range of densities and temperatures for which enhancement has been demonstrated.
225
Acknowledgement We are grateful to the Science Research Council for providing financial support for this work.
References [1] A. Acton and K. Kellner, Physica 90B (1977) 192. [2] A. Acton and K. Kellner, J. Phys. (E): Sci. Instrum. 9 (1976) 1144. [3] J. de Boer, Physica 14 (1948) 139. [4] L. P. Kadanoff et al., Rev. Mod. Phys. 39 (1967) 395. [5 ] J. M. H. Levelt-Sengers, W. L. Greet and J. V. Sengers, J. Phys. Chem. Ref. Data 5 (1976) 1. [6] J. V. Sengers, Critical Phenomena, Proc. Intern. School of Physics Enrico Fermi, Course LI, M. S. Green, ed. (Academic Press, New York, 1971) p. 445. [7] H. J. M. Hanley, J. V. Sengers and J. F. Ely, Proc. 14th Intern. Conf. on Thermal Conductivity, P. G. Klemens and T. K. Chu, eds. (Plenum, New York, 1976) p. 383. [8] B. J. Bailey and K. Kellner, Brit. J. Appl. Phys. 18 (1967) 1645. [9] H. A. Kierstead, Phys. Rev. A7 (1973) 242. [10] J. F. Kerrisk and W. E. Keller, Phys. Rev. 177 (1969) 341. [11] L. E. Weaver, Ph.D. Thesis, University of Illinois (1970). [12] A. Michels and J. V. Sengers, Physica 28 (1962) 1238. [13] R. D. McCarty, NBS Technical Note 631 (1972). [14] E. C. Kerr and R. H. Sherman, J. Low Temp. Phys. 3 (1970) 451. [15] P. R. Roach, Phys. Rev. 170 (1968) 213. [16] L. P. Kadanoff and J. Swift, Phys. Rev. 166 (1968) 89. [17] J. V. Sengers and J. M. H. Levelt-Sengers, Progress in Liquid Physics, C. A. Croxton ed. (Wiley, New York, 1978) p. 164. [18] D. B. Roe and H. Meyer, J. Low Temp. Phys. 30 (1978) 91. [19] J. V. Sengers and P. H. Keyes, Phys. Rev. Lett. 26 (1971) 70. [20] K. Ohbayashi and A. Ikushima, J. Low Temp. Phys. 15 (1974) 33. [21] A. Tominaga, J. Low Temp. Phys. 16 (1974) 571. [22] The 1976 Provisional 0.5 K to 30 K Temperature Scale, Translation of the official French text (Bureau International des Poids et Mesures, 1978).
THE LOW TEMPERATURE THERMAL CONDUCTIVITY OF 4He II. MEASUREMENTS IN THE CRITICAL REGION A. ACTON* and K. KELLNER**
PhysicsDepartment, University of Southampton, Southampton, UK Received 28 August 1979 Revised 1 July 1980
Measurements of the thermal conductivity of helium-4 along five isotherms between 5.1 and 5.6 K and covering a range of densities from 30 to 120 kg m-3 are reported. These results were obtained with a guarded flat plate apparatus using temperature differences from 0.025 to 0.1 K and fluid layer thicknesses of 0.4 and 0.2 mm. Experimental tests showed that convection probably contributes less than 5% to the measured thermal conductivity in all but a few cases. The experimental data are compared at the critical density with recent theoretical predictions. Although the agreement between theory and experiment is not excellent, it is good enough to demonstrate that the theory can be applied to helium-4 and to show that sufficiently close to the critical temperature helium-4 appears to behave like a classical fluid.
1. Introduction In part I of this paper [1], measurements of the thermal conductivity were reported which covered the dilute gas, the dense gas and the normal liquid in the temperature region between 3.3 and 20 K and at pressures up to 2.5 MPa. In this part, additional measurements are discussed which were carried out along five isotherms in the critical region. All five isotherms exhibit an anomalous behaviour of the thermal conductivity as the critical density is approached either from above or from below. The experimental arrangement was exactly the same as that for the measurements of part I, indeed the measurements were all made at the same time, using the cell described by Acton and Kellner [2]. The division of the paper into two parts is therefore not due to a difference in experimental technique but rather due to a difference in the physical state of the fluid in the critical and non-critical regions. The large value of the de Boer parameter [3] for helium-4 A* = h/(me)~o of 2.68, where h is Planck's * Now with Engineering Sciences Data Unit Ltd, 251-9 Regent Street, London W1R 7AD, UK. ** Now at Institute of Cryogenics, University of Southampton, Southampton, U.K. 212
constant, m the atomic mass, e the depth of the attractive potential well and o the (hard core) atomic diameter, is a measure for the considerable quantum mechanical influence on the physical properties. This has already been referred to in part I where it was shown that the thermal conductivity o f the liquid has a positive temperature coefficient in contrast to the negative temperature coefficient of simple classical liquids. Further quantum influence on the thermal conductivity may well be expected in the liquid-gas critical region although, in their static properties, the critical behaviour of quantum fluids does n o t differ greatly from that of classical fluids [4, 5].
2. Thermal conductivity in the critical region The anomalous behaviour in the fluid critical region of some physical properties is well established by now. Divergencies in the specific heats, Cp and co, and the isothermal compressibility, K T, have been known for a long time. The thermal conductivity has been investigated only relatively recently but divergence of this property in the critical region has been shown to occur for a number of fluids [6, 7].
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
The enhancement of the thermal conductivity can quite easily be confused with an increase in convection in the critical region and tests have to be carried out in order to establish that convection is either absent or, at least, reduced to negligible proportions. In the case of a direct determination of the conductivity these tests depend on the type of apparatus used for the measurements. For example in the case of a concentric cylinder cell the onset of convection can be varied by turning the cylinders from a vertical to a horizontal position or vice versa [8]. In the case of a parallel plate cell a similar test can be made by tilting the plates away from their normal horizontal position. It is also possible with this type of cell to establish the absence of convection by changing the gap between the plates. Of course, in the first place, the cell has to be designed in such a way so as to reduce convection as much as possible.
3. Measurements
As in part I, pure helium gas was used and all temperatures were measured on the 1965 NBS Provisional Temperature Scale (based on the acoustic thermometer). Further information on the gas and on the accuracy of temperature measurement are given in part I. Values of the critical temperature, Tc, and the critical density, Pc, are taken from Kierstead's work [9] and are 5.198 K and 69.6 kg m -3, respectively. The experimental technique was similar to that adopted for the measurements in the non-critical region reported in part I, and similar corrections were applied to the measured heat flows and temperature differences. The thermal conductivity was also calculated in a similar way except that in the critical region the cell constant, A/s, could be taken to be invariant for each isotherm and individual determinations of its value for each experimental point were unnecessary. This permitted the direct determination of the fluid density from the measured cell capacitance, see section 5. The thermal conductivity measurements made for five isotherms close to T c are listed in tables I to V. Each measurement has been allotted an identification code using the same system as in part I. In the tables, T a is the average temperature of the fluid layer and P is its pressure; the temperature difference across the
213
layer and the heat flow through it are AT and W, respectively. The cell capacitance is C, p is the fluid density and ?~is the thermal conductivity. The variation of thermal conductivity with temperature and density is illustrated in fig. 1. Full lines in this figure have been faired through the data on each isotherm. Experimental data for which convection was purposely introduced and experimental data from run L (indicated by the prefix L to the code in table II and for which the test gap, s ~ 0.2 mm) have been omitted from this figure; these data are discussed in sections 4.1 and 4.2, respectively. The curve for 5.248 K has not been extended through the three data points nearest to the peak in ?, because these data are believed to be affected by convection (see the discussion in section 4.2). Apart from those results affected by convection, which are marked with an asterisk in tables II and III, the measured thermal conductivities are expected to be accurate to -+5%with AT = 0.025 K and more accurate than this with the larger temperature differences. Complete discussion of the expected accuracy is given in part I. The results in the critical region were, however, nearly all obtained during one experimental run (run K, s ~. 0.4 mm) during which the cell was maintained at low temperature. Most of the error in stems from the irreproducibility of the germanium thermometers on thermal cycling between room and low temperatures, and so the results of run K illustrated in fig. 1 would be expected to exhibit a scatter of rather less than 5%. This is indeed the case, the precision of k being about +1%. Another possible source of error, boundary resistance, was undetectable for the results in the non-critical region reported in part I, and there is, therefore, good reason to suppose that no significant boundary resistance occurs in the critical region. Enhancement of thermal conductivity as the critical density is approached along an isotherm from either above or below is dearly apparent from fig. 1. Also, as T a approaches T c from above, the enhancement at any given value of p increases. The behaviour is similar to the effects observed with CO 2 and CH4 [7] ; A [7, 8] ; 3He [I0] ; and in another study with 4He [11].
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
214
Table I Thermal conductivity of 4He at 5.1 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (mW)
C (pF)
p (kg m -3)
X (W m -1 K -1)
K239 K247 K246 K245 K244 K243 K242 K241 K240
5.097 5.098 5.097 5.097 5.098 5.097 5.097 5.098 5.097
0.1870 0.1954 0.2027 0.2078 0.2147 0.2204 0.2302 0.2768 0.382
0.1048 0.0548 0.0549 0.0254 0.0254 0.0550 0.1047 0.1050 0.1049
6.545 3.572 3.779 1.923 2.371 5.221 9.985 10.34 11.03
45.52 45.57 45.64 45.71 46.70 46.75 46.80 46.95 47.12
30.2 33.3 37.4 41.5 96.9 99.7 102.9 111.1 120.4
1.23E-02 1.28E-02 1.35E-02 1.49E-02 1.84E-02 1.87E-02 1.88E-02 1.94E-02 2.07E-02
Table II Thermal conductivity of 4He at 5.25 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (mW)
C (pF)
p (kg m -a)
X (W m -1 K -1)
K248 K260 K249 K250 K251 K252 K263 K253 K261" K262" K254" K264 K265 K255 K266 K256 K257 K258 K267 K259
5.248 5.248 5.248 5.247 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248 5.248
0.1979 0.2152 0.2196 0.2271 0.2309 0.2330 0.2342 0.2349 0.2352 0.2367 0.2369 0.2376 0.2392 0.2410 0.2412 0.2452 0.2531 0.2752 0.333 0.445
0.1053 0.1051 0.0546 0.0549 0.0249 0.0249 0.0248 0.0249 0.0250 0.0252 0.0248 0.0247 0.0251 0.0247 0.0251 0.0252 0.0551 0.0551 0.1047 0.1046
6.720 7.236 3.929 4.266 2.172 2.275 2.553 2.818 2.954 3.164 3.101 2.911 2.519 2.373 2.408 2.411 5.305 5.395 10.62 1t.26
45.52 45.62 45.67 45.77 45.85 45.92 46.01 46.08 46.10 46.25 46.28 46.34 46.45 46.52 46.52 46.60 46.68 46.80 46.95 47.11
30.2 36.4 39.2 44.6 49.0 53.0 58.0 62.3 63.6 71.8 73.4 76.8 83.1 87.1 87.1 91.5 96.0 102.9 111.0 120.1
1.26E-02 1.36E-02 1.42E-02 1.53E-02 1.72E-02 1.80E-02 2.03E-02 2.23E-02 2.33E-02 2.47E-02 2.46E-02 2.32E-02 1.97E-02 1.89E-02 1.89E-02 1.88E-02 1.89E-02 1.93E-02 2.00E-02 2.12E-02
Cell constant = 10.310 m L57 L58 L59 L60 L61 L62 L63
5.248 5.248 5.248 5.248 5.248 5.248 5.248
0.1986 0.2310 0.2350 0.2364 0.2385 0.2468 0.445
0.0249 0.0250 0.0252 0.0251 0.0251 0.0251 0.0250
* Indicates result significantly affected by convection.
3.249 4.328 5.572 5.856 5.390 4.976 5.543
92.39 93.07 93.55 93.81 94.21 94.61 95.57
31.0 50.0 63.5 70.9 82.0 93.1 119.4
1.26E-02 1.68E-02 2.15E-02 2.26E-02 2.08E-02 1.92E-02 2.15E-02
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region Table III Thermal conductivity of 4He at 5.3 K
215
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
W (roW)
C (pF)
p (kg m -3)
h (W m -1 K -1)
K194 K268 K195 K196 K197 K269 K198 K270 K199 K200 K201 K202 K203 K279 K204 K271 K272" K273" K274" K275" K276 K277 K278
5,297 5.297 5.297 5,297 5.297 5,298 5,298 5.298 5,298 5.297 5.297 5,298 5,297 5.298 5.297 5.298 5.297 5,297 5.298 5.298 5.297 5.298 5.298
0.2022 0.2201 0.2264 0.2362 0.2418 0.2436 0.2448 0.2458 0.2468 0.2497 0.2516 0.2610 0.2869 0,349 0,463 0,2460 0.2459 0,2459 0.2459 0.2459 0.2460 0.2459 0.2461
0.1047 0.1046 0.0551 0.0552 0.0249 0.0252 0.0250 0.0246 0.0249' 0.0250 0.0249 0.0549 0.0546 0.1046 0.1046 0.0255 0.0246 0.0246 0.0245 0.0249 0.0249 0.0247 0.0249
6,738 7,274 3.969 4.473 2.399 2.636 2,764 2.737 2.712 2.487 2.405 5.300 5.337 10.69 11.36 2,796 3.925 3.648 3,328 3,006 2,720 2,596 2,701
45.52 45.63 45.69 45.83 46.00 46.11 46.21 46.27 46.34 46.46 46.51 46.64 46.79 46.94 47.11 46.27 46.27 46.27 46.27 46.27 46.28 46.28 46.29
30.5 36.6 40.2 48.0 57.8 64.0 69.4 73.1 76.7 83.8 86.5 93.5 102.2 110.8 120.1 73.2 73,2 73.2 73.2 73.2 73.5 73.3 73.8
1.27E-02 1.37E-02 1.42E-02 1.59E-02 1.90E-02 2.06E-02 2.17E-02 2.19E-02 2.14E-02 1.96E-02 1.90E-02 1.90E-02 1.92E-02 2.01E-02 2.14E-02 2.16E-02 3.14E-02 2.91E-02 2.67E-02 2.38E-02 2.15E-02 2.07E-02 2.14E-02
* Indicates result significantly affected by convection.
Table IV Thermal conductivity of 4He at 5.4 K
Cell constant = 5.081 m
Code
Ta (K)
P (MPa)
AT (K)
I¢ (mW)
C (pF)
p (kg m -a)
h (W m -1 K -1)
K205 K206 K207 K208 K209 K210 K211 K212 K213 K214 K215 K216
5,397 5.397 5.398 5.397 5.397 5.397 5.397 5,397 5.397 5,397 5.397 5,397
0.2094 0.2356 0.2492 0.2572 0.2603 0.2627 0.2681 0.2734 0.2838 0.317 0.380 0.500
0.1050 0.0548 0.0552 0.0253 0.0250 0.0249 0.0248 0.0247 0.0550 0.0550 0.1048 0.1048
6.838 3,964 4.435 2.305 2.429 2.455 2.421 2.363 5.331 5.457 10.83 11.51
45.52 45.68 45.84 46.02 46.13 46.23 46.40 46.50 46.61 46.79 46.94 47.10
30.4 39.8 48.7 58.9 64.9 70.4 80.0 85.9 92.1 101.9 110.6 119.6
1.28E-02 1.42E-02 1.58E-02 1.80E-02 1.91E-02 1.94E-02 1.92E-02 1.89E-02 1.91E-02 1.95E-02 2.03E-02 2.16E-02
216
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
Table V Thermal conductivity of 4He at 5.6 K
Cell constant = 5.080 m
Code
Ta (K)
P (MPa)
AT (K)
W (roW)
C (pF)
p (kg m -3)
h (W m -1 K -1)
K217 K218 K219 K220 K221 K222 K223 K224 K225 K226 K227 K228
5.597 5.598 5.597 5.598 5.597 5.598 5.598 5.598 5.598 5.598 5.598 5.598
0.2230 0.2538 0.2726 0.2866 0.2925 0.2976 0.306 0.317 0.339 0.374 0.435 0.546
0.1054 0.1051 0.0551 0.0550 0.0254 0.0250 0.0250 0.0551 0.0549 0.1050 0.1049 0.1046
7.038 7.652 4.425 4.834 2.311 2.396 2.400 5.366 5.420 10.61 11.07 11.68
45.51 45.67 45.82 46.00 46.11 46.20 46.33 46.47 46.63 46.77 46.91 47.06
30.3 39.5 48.2 58.3 64.3 69.6 76.9 84.7 93.6 101.4 109.3 117.8
1.31E-02 1.43E-02 1.58E-02 1.73E-02 1.79E-02 1.89E-02 1.89E-02 1.92E-02 1.94E-02 1.99E-02 2.08E-02 2.20E-02
2"5 o o
Tc = 5-198K o
Ta/K
25
5598 5-597 5 297
21
~" 5 - 2 4 8 "~ 5 . 0 9 7
T - ~ 1.9
~E
o 1.7 ,.-<
1"5
1.5
1~20
50
410
I
510
!'i °
8'0
p /kg m-3
910
'
I00
IllO
'
120
'
150
Fig. 1. Isotherms of the thermal conductivity of 4He in the critical region.
4. E s t i m a t e o f c o n v e c t i o n It was s h o w n in p a r t I t h a t the m e a s u r e m e n t s in the n o n - c r i t i c a l region were n o t a f f e c t e d b y convection. As far as t h e critical region is c o n c e r n e d , convect i o n c a n be d i s c o u n t e d e x c e p t for a few m e a s u r e m e n t s
very close to t h e critical p o i n t w h e r e its i n f l u e n c e is e s t i m a t e d t o give an a p p a r e n t t h e r m a l c o n d u c t i v i t y t h a t is n o t m o r e t h a n 10% t o o high. This is discussed below. A d e t a i l e d analysis o f free ( n a t u r a l ) c o n v e c t i o n in a near-critical fluid c o n t a i n e d in a c o n d u c t i v i t y cell,
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region similar to the one used in this work, was given by Michels and Sengers [12], and is discussed in part I. This analysis shows that convection can occur if the test gap is not quite horizontal. Apart from those measurements discussed in section 4.1 for which it was deliberately tilted away from the horizontal, the top plate of the Dewar vessel containing the thermal conductivity cell was levelled to within +4.4 mrad (0.25 ° ) for the measurements in the critical region. This should have ensured that the test gap was horizontal to within the same range of angle. However, the experiment discussed in section 4.1 showed that the test gap could, in fact, have been out of the horizontal by up to 12 mrad (0.69°). From Michels and Sengers' analysis, if Xa is the apparent thermal conductivity with convection present and X is the true thermal conductivity, Xa -- )k X
/S'~ Ra sin
= qR1 ~A)
360
(1)
In eq. (1), R 1 is the radius of the hot plate (emitter), A/s (the cell constant) is the effective heat transfer area of the hot and cold plates divided by their separation and ~ is the angle that the normal to the plates makes with the horizontal. The Rayleigh number is Ra = p2CpZgS3AT/~X, where z = -(1/p)(ap/aT)p is the volume expansion coefficient of the fluid, g is the acceleration due to gravity, 77is the coefficient of shear viscosity and the other symbols have their usual meanings. Because the emitter is surrounded by a guard ring, only a fraction of the convected heat flow across the test gap actually comes from the emitter; this fraction is represented by q. Values of the Rayleigh number were estimated for the whole range of conditions under which the experimental results were obtained using the tables of Thermophysical Properties of 4He by McCarty [13] with the exception of X, for which experimental values from this work were used. The highest Rayleigh number for all of the experimental data in the critical region (K262 in table II; T a = 5.25 K, AT = 0.025 K, p = 71.8 kg m -3 and s = 0.4 mm) is estimated to be 4 X 106. Taking ¢ = 12 mrad (the upper limit) and setting q equal to its maximum value of unity, eq. (1) predicts that Xa is in error by 64% due to convection! Therefore, detailed investigation of convection in the critical region is warranted. It can be seen from eq. (1)
217
that the error in k a due to convection depends on the value of Ra, the cell dimensions R 1, s and A and on the angle 4). For the cell used in this work R 1 (and consequently A) is a fixed quantity, but by changing AT, ¢ or s, it is possible to test for convection experimentally. For the measurements in the non-critical region, reported in part I, it was possible to show that convection was insignificant by varying the heat flow in the test gap and thus varying AT whilst maintaining the average fluid temperature, Ta, constant. However, in the critical region, k is a strong function of temperature and large values of AT cannot generally be expected to yield accurate values of k at the average fluid temperature even in cases where convection is insignificant. Thus, the variation of AT as a test for convection is not reliable unless AT is always small in comparison with T a - Tc, a condition that could not be achieved for these experiments. Consequently, tests for convection were restricted to varying ¢ and s.
4.1. Variation o f ¢ A test for convection was carried out by tilting the fluid layer with respect to its normal horizontal position. This was done for the 5.3 K isotherm very close to the thermal conductivity peak. The results are given in table III (coded K271 to K278 inclusive) and are illustrated in fig. 2. Fig. 2 shows that the apparent thermal conductivity is a minimum for an angle of tilt of 9 mrad instead of the hoped-for angle of zero. This is probably due to uncertainty in the angle of tilt of the test gap. Since, during an experiment, the angle of tilt of the test gap could only be obtained from the measured angle of tilt of the Dewar top plate, this experiment indicates that the tilt of the test gap does not quite follow the tilt of the top plate. Furthermore, this tilting test should have been repeated in a plane perpendicular to that of the first test in order to fmd the position of minimum Xa; unfortunately this could not be done without causing damage to the apparatus and time did not permit a more thorough investigation. The difference in Xa between zero tilt (K278) and the minimum value of k a is about 3%. Assuming that the levelling error in the perpendicular plane is similar to that in the tested plane, +9 mrad, the maximum magnitude that ¢ can attain in any plane is 12 mrad. Assuming further that the minimum k a in fig. 2
218
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region 34
52 J 2 7 2
50
T
2"8
T
~" _o 2-6 x
2.4
2.2
2<
K271
K277
20
50
40
Measured angle of tilt/mrad
50
60
70
(10 mrad=0.57 o)
Fig. 2. Effect of tilting the Dewar vessel on the apparent thermal conductivity at 5.3 K. corresponds to negligible convection, the figure shows that a value of ¢ = 12 mrad corresponds to ~'a being in error by about 4%. Putting ~b= 12 mrad into the righthand side of eq. (1), taking q = 1 and other parameters appropriate to the conditions of experimental points K271 to K278, Xa is predicted to be in error by 28% due to convection. Therefore, eq. (1) with q = 1 is conservative for this apparatus. 4.2. Variation o r s
A few measurements were made along the 5.25 K isotherm during run L, with s ~ 0.2 ram, in order to provide a second test for convection. Fig. 3 shows a comparison of these results with the 5.25 K results of run K, with s ~ 0.4 mm. The figure shows that the results of the two runs are in good agreement except for a region close to the critical density, where the run L values are up to 10% lower than those of run K. This behaviour strongly suggests that several of the run K results are influenced by convection. For the result from run L closest to Pc, (L60 in table II; T a = 5.25 K, A T = 0.025 K, p = 70.9 kg m - 3 , A / s = 10.31 m and
taking ¢ = 12 mrad) eq. (1) predicts that Xa is in error by 6.7% due to convection. This represents the worst case for the run L results and, as eq. (1) was shown to be conservative in section 4.1, it is clear that the results L57 to L63 are almost certainly free of significant convective influence. We may conclude from this that the three results of run K near the thermal conductivity maximum, K261, K262 and K254 are most probably significantly affected by convection, but to an extent not exceeding 10% of the values given in table II. The results that are significantly affected by convection are marked with asterisks in tables II and III. For results not marked with asterisks, it is believed that convection contributes less than 5% to the measured value of the thermal conductivity.
5. Determination of the fluid density in the critical region The thermal conductivity in the critical region is best studied from a conductivity-density graph and, therefore, reasonably accurate values of the density
219
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
must be obtained. In this work, the average density of the fluid sample in the conductivity cell was deduced directly from measurements of the cell capacitance using the Clausius-Mossotti relation. This method was adopted rather than using McCarty's tables [13] because, in the latter case, the evaluation of the density from the measured values of pressure and temperature could be in considerable error in the critical region. The Clausius-Mossotti relation, which connects the relative permittivity, er, of a fluid with its density, p, may be written as er - 1
=pp,
(2)
er+ 2 where p is the specific polarization of the fluid. For helium-4, an empirical relation for p deduced from the results of Kerr and Sherman [14] is p =E 1
-
(3)
E2P ,
where E 1 = 1.29095 X 10 - 4 m 3 kg -1 a n d E 2 = 1.5 X 10 - 9 m 6 kg-2; hence
e r -- 1 er+2
Since E 2 / E 1 is small, the right-hand side of eq. (4) can be expanded and, neglecting powers o f E 2 / E 1 higher than the first, one obtains e r -- 1 P ~ ' E l ( e r + 2) --
(E2/E1)(e
r
(5)
-- 1)"
All measuretnents made outside the critical region show that the variation of cell constant with pressure is negative, approximately linear with pressure and small (0.2%/MPa for a 0.4 m m gap). For the critical region the pressure changes were limited to a range of some 0.3 MPa; consequently, the variation of the cell constant was expected to be very small indeed. In order to check this, and also to obtain appropriate values of the cell constant, eq. (4) was used to find the relative permittivity of the fluid at the lower and upper extremes of each conductivity isotherm..These extremes were arranged to be well removed from the critical region and so McCarty's tables [13] provided
I
2.5
(4)
- p ( E 1 - E 2 P ).
Run
5% °
~
o + [
s/mm K
04
L
0-2
2.3 +
21 T 19
15
1.3
1120
,
J
J
I
Th ~
30
40
50
60
70 p/kg
I 80 m-3
l
I
J
j
90
I00
I10
120
Fig. 3. Effect of changing the test gap on the apparent thermal conductivity at 5.25 K.
220
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
values of # for use in eq. (4) which were accurate to within about 0.5%, yielding values of er accurate to within about 0.02% at worst. The cell constants, A/s, could then be evaluated from the expression for the capacitance, C, of a parallel plate condenser A - =
s
C 8.8542 X 10-12e r '
(6)
(see part I, p. 193) for the two density, and hence pressure, extremes. These calculations showed that the variation of cell constant with pressure was almost negligible for the range of pressures covered for the measurements in the critical region. It was considered sufficient, therefore, to average the two values of the cell constant which were obtained for each conductivity isotherm. The average cell constants were then used to calculate values of p from the measured capacitance over the whole of each isotherm using eqs. (6) and (5). It is these values of p that are given in tables I to V. The approximation involved in using eq. (5) can be seen to be quite justified since, in no case, did the second term of the denominator exceed 0.2% of the first term. The accuracy of p determined in this way was governed mainly by the precision of the determination of C. Assuming that the only error here was due to the capacitance bridge, the measurements of which were precise to within 0.02%, p was accurate to within 2% at 30 kg m -3 ranging to 0.5% at 120 kg m -3. It is reassuring to note that the cell constants in tables I - V (except that for the nominal 0.2 mm gap), and which were separately determined over a period of eleven days, lie all within 0.02% of one another. Except for a few points above 0.3 MPa, where bourdon gauges were used, measurements of cell pressure were made with a mercury manometer, suitably corrected for temperature, latitude, altitude and hydrostatic effects. The accuracy of pressure measurement was better than +-0.03% with the mercury manometer, and better than +1% with the bourdon gauges. 6. E q u a t i o n o f state data
The pressures and densities measured in this study can be compared with equation of state data recently obtained by other workers. This comparison is iUus-
trated in fig. 4 where the full lines represent our five isotherms in the critical region. Also shown are ten points taken from independent P---O- T measurements by Roach [15] (corrected to the NBS Provisional Temperature Scale used in this work). Roach also used a capacitive method to determine p and consequently assumed the validity of the Clausius-Mossotti relation in the critical region. As can be seen from fig. 4, the agreement between the two sets of data is very good which is not really surprising since, in both studies, the experimental methods were based on the same principles. More recent measurements by Kierstead [9] using a volumetric method, show also very good agreemelat with Roach's results [5]. This demonstrates that the Clausius-Mossotti relation can be used with confidence in the critical region.
7. Dilute gas, excess and ideal c o n t r i b u t i o n s t o the thermal conductivity It is customary to consider the thermal conductivity of a gas to be due to the sum of various contributions [7], illustrated schematically for two temperatures (T 2 > T 1 > Tc) in figs. 5a and 5b. Taking density and temperature as the independent variables, the thermal conductivity for all p, T can be represented by
?t(p, 73 = ?,0(73 + Ax(;, 73,
(7)
where ?t0(T) is the dilute gas thermal conductivity (the low-density limit that depends on temperature alone) and Ak(,o, T) is the excess thermal conductivity. In the critical region there is an anomalous increase in k which can be accounted for by considering the excess thermal conductivity as being made up of two contributions A?t(p, T) = A?tid(,O, T) + A?tc(p, T),
(8)
where AXia is the ideal, or background, excess thermal conductivity and A?tc is the critical excess thermal conductivity. Values of ?t are obtained experimentally but, in order to understand the critical anomaly, values of A?tc are required. The dilute gas thermal conductivity, )t0, is often known as a function of temperature. Normally (for classical gases), A?tid is a very weak
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
221
o 32 03i
Rooch,/~
0 30
5 ~
0 29 0 28[ 027
5"397 0 26I 0 25 5 248 o 24~ 023 5097 0 22 02
plkg
m -5
Fig. 4. Equation of state data in the critical region.
a
b
/ /.f___/~.-" "~?'TkT_, ,__~,°,~.,~)
),(&,T 2)
),(p= ,TI) ....
-t ....
.~-_ __t ......
4- ....
I
P, pc
~- .....
'. ....
J .....
ko(T2)
ko(Ti )
P
I
p,p~
Fig. 5. Contributions to the thermal conductivity. (a) At T 1 ; (b) at T 2.
p
222
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
function of temperature. Consequently, it is only necessary to determine the variation of A)kid with [ at a single temperature in order to obtain 2x)kc at all p and T from eqs. (7) and (8). This is not so for helium (and hydrogen) for which A)kid depends on both temperature and density. In this case it is appropriate to define the ideal thermal conductivity, )kid, which is the thermal conductivity the fluid would have if A)kc did not exist
all, least-squares polynomial curve fits of X as a function of p along the experimental isotherms in the noncritical region were made. These polynomials were then used to generate values of X along isochores, which were subsequently cross-plotted to give a graph of )kid versus p for the isotherms used in the critical region. Fig. 6 is a small-scale version of this graph. Values of )kid taken from the original graph have been subtracted from the experimental values of X in the critical region to give the experimental points illustrated in fig. 7. Only data at temperatures above T c have been treated in this way as insufficient data for analysis are available at temperatures below T c. Data that were influenced by convection and data from run L have been omitted from fig. 7 ; extra data from part I that show thermal conductivity enhancement have been included. These extra data are at temperatures of 6, 7 and 8 K and are all from run K with AT 0.2 K. Full lines are curves faired through the experimental points. The reduced relative temperature and the reduced relative density defined as follows are useful,
(9)
)kid(O, T) = )k0(T) + A)kid(o , 7).
The critical excess thermal conductivity is then given by
(1o)
AXc(O, T) = )k(p, T) -- hid@, T).
The ideal thermal conductivity, kid , can be estimated by extrapolation into the critical region of data sufficiently far removed from the critical point and for which AXc is negligibly small. This has been done in the following way using the data in part I. First of 29
To/K 8
27
25
6
23
5.6 5-4 'i2
I
o
~-19
1.7
1.5
13
I t:~0
I
30
40
50
I
60
I
I
70 80 p/kg m-3
I
90
Fig. 6. Ideal thermal conductivity.
I
I00
I
I10
J
120
A. Acton and K. Kellner/Thermal conductivity o f 4He in the critical region
223
1.0
O 0.9 Exp. Points
0.8
[]
0.7
0-6 T T E
[]
Temp. Ta/K
Cole. Points v =0.543 v =0.63 ~¢o = 0 - 3 6 0 ~¢o= 0 . 2 2 Ref. 20 Ref. 21
5 "248 5.297 5.397 5.598 5.997
® ® ® ® ®
[] [] [] @ []
6.998 7.997
(~)
Similar to
®
®.®
0"5 x
0.3
®
® 0.1
30
I
I
I
40
50
60
I
70
80
90
,oo
,Yo
,2o
p/kg m-5 Fig. 7. Isotherms of the critical excess thermal conductivity of 4He.
A7~ - Ta - Tc
Tc
and
A/3 =
P - Pc Pc
An estimated value of the thermal conductivity at 6 K and at the critical density was given in fig. 3 of part I to fill a gap in the experimental data. This estimate was obtained from a least-squares linear fit of the values of log (A~c) at Pc versus log (AT~) for temperatures of 5.3, 5.4, 5.6, 7 and 8 K, giving A~c(Pc , 6) = 1.9 X 10 -3 W m -1 K -1. Adding the ideal thermal conductivity gave k(Pc, 6) = 1.80 X 10 -2 W m -1 K -1.
As discussed in section 8, A~ c is only given approximately by this method at temperatures above about 5,4 K. However, a reasonable estimate of the (total) thermal conductivity at 6 K is obtained because A3.c contributes only about 10% to the total in this case.
8. Comparison with theory It is theoretically predicted [16] that, along the critical isochore and within a certain range of temperature above the critical temperature, A?~c should be
224
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region
proportional to (AT) -a where a is a universal (fluidindependent) constant called the critical exponent of the thermal conductivity. Furthermore, theory predicts that the constant a should satisfy the equality involving two other comparably-defined critical exponents,
critical point is approached and vanishing far away from the critical point. From an analysis of the data for carbon dioxide and, to a lesser extent, for methane and argon it is proposed that F(AT, A~3)= exp (-[18.66(A7~) 2 + 4.25(A~) 4] }. (13)
a = 7 - v.
(11)
In this equality 3' is the critical exponent of the isothermal compressibility, K T, and u is the critical exponent of the correlation length, ~, The correlation length can be visualised as the mean radius of molecular clusters. Although the equality (11) is generally accepted there exists a fairly wide range of experimentally determined values of the exponents, particularly for v. Normally, 3' is about 1.19 although in the case of helium-4 a value of 1.1743 --- 0.0005 has been determined for a particular equation of state [5, 9]. At present, the generally accepted value is v = 0.63 -+0.02 [5, 17] although in the case of helium-4 values as low as 0.543 -+ 0.046 and as high as 0.67 -+ 0.02 have been reported [18]. Measurements of the thermal conductivity of carbon dioxide [19] have given a = 0.60 ---0.05 and measurements of the thermal conductance of a cell filled with helium-4 [11] have given a = 0.67 +- 0.31, values that are consistent with the above values of 3' and v and the equality (11). Whilst it would have been interesting to obtain a value of the exponent a from our results for comparison with the above, there are insufficient data to do this. Only a very limited number of isotherms were studied and, according to Hanley et al. [7] the simple proportionality between AXc and (A]~)-a ceases to be valid for A7~ > 0.03 (Ta > 5.4 K) approximately. The peak of the 5.25 K isotherm was not resolved in our measurements and so only two points are available, at 5.3 and 5.4 K. Therefore, an alternative comparison between theory and experiment is made. Hanley et al. [7] in a development of Kadanoff and Swift's theory [16] show that the critical excess thermal conductivity of classical fluids can be estimated from the equation K T F(AT, At5).
(12)
The function F(AT, Ats) is empirically determined to fit the experimental data, approaching unity as the
In order to compare the predictions of eq. (12) with our experimental results for helium-4 the parameters in the equation have been evaluated as follows. Boltzmann's constant, k, and the absolute temperature, T(= Ta), are known. The dynamic viscosity, ~/, and the isochoric pressure derivative, (~P/aT)o, are relatively slow-moving parameters in the critical region and are therefore obtained by interpolation within McCarty's tables [13]. The correlation length, ~, is evaluated along the critical isochore using the equation :~0(AT~ -v.
(14)
Two values of v and their corresponding two values of G0 are selected from those listed by Roe and Meyer [18] : Ohbayashi and Ikushima's [20] values, v = 0.543 -+ 0.046 and G0 = 0.360 -+ 0.078 nm, because these values fit our experimental data best and Tominaga's [21] values, v = 0.63 + 0.1 and G0 = 0.22 _+0.06 nm, because the exponent closely reproduces the value expected from other experimental and theoretical studies [17]. No experimental determinations of ~ are known for p 4: Pc" Whilst in principle it is possible to calculate for all/9 using methods outlined by Hanley et al. [7] this is not a trivial task and has not been attempted here. Therefore, the comparison between theory and experiment is restricted to the critical isochore only. The isothermal compressibility, KT, is estimated using the equation F
- ) _ .~ r r = ~ ( ATs8
(15)
Eq. (15), valid along the critical isochore only, is taken from the paper by Levelt-Sengers et al. [5], with constants from table 17 of their paper. The critical pressure (Pc)is 0.2275 MPa, I~ = 0.1611 and 3' = 1.174. The parameter AT58 is in terms of the (extrapolated) 1958 Helium-4 Vapour Pressure Temperature Scale.
A. Acton and K. Kellner/Thermal conductivity of 4He in the critical region The necessary corrections to our temperatures, measured on the 1965 NBS Provisional Temperature Scale are estimated using the data in ref. [22]. Values of A?~c at Pc calculated using eqs. ( 1 2 ) - ( 1 5 ) are compared with the experimental results in fig. 7. The agreement between experiment and theory is quite good at temperatures between 5.25 and 5.6 K and is best when the values of v and 40 from ref. 20 are used rather than those from ref. 21. In this temperature region, the function F ( A 7~, AtS) is not significantly different from unity and helium-4 behaves very much like the classical fluids carbon dioxide, methane and argon. At higher temperatures, however, the calculated values of A?~c are lower than the experimental results indicating that the term in A T in eq. (13) is incorrect for helium-4 (the coefficient 18.66 is too big). The thermal conductivity measurements at 6, 7 and 8 K are accurate to about +--1%,see part I. It is believed that the estimation o f hid was carried through with similar precision and so it is unlikely that the disagreement between experiment and theory at these higher temperatures is spurious. However, more data at temperatures between 5.6 and 10 K are needed to establish firmly the temperature dependence o f F(A7~, At3) for helium-4.
9. Conclusions The thermal conductivity of helium-4 shows enhancement in the critical region. The experimental data above the critical temperature show that theoretical prediction methods recently developed for classical fluids are successful for helium-4 at the critical density and for reduced relative temperature, A7~, less than about 0.08. It is shown that at larger values of A7~ the enhancement falls off less rapidly with increasing temperature with helium-4 than it does with the classical fluids carbon dioxide, methane and argon. Further data for the correlation length and the thermal conductivity are needed to extend the theoretical predictions to the whole range of densities and temperatures for which enhancement has been demonstrated.
225
Acknowledgement We are grateful to the Science Research Council for providing financial support for this work.
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