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Measurement of thermal conductivity at low temperatures by a non-stationary method
M e a s u r e m e n t of T h e r m a l C o n d u c t i v i t y a t Low T e m p e r a t u r e s by a N o n - s t a t i o n a r y M e t h o d E. Donth C. Gl(td'ltn
Technische Universitat, JDreaden z)eut,che Akademie der Wissenschaften "zu Berlin, Dresden
Received 12 February 1962
THERMAL conductivity is often measured in such a manner 1,2 that one point of the sample to be tested is maintained at a constant temperature, followed either by setting up a stationary temperature gradient and measuring the temporarily constant differences of temperature, or by generating a temporarily variable temperature gradient by means of temporary variation of heat supply and calculating thermal conductivity from the temperature versus time curves. In both cases, during adjustment of the temperature gradient variation of the constant temperature must be small compared with the differences of temperature to be measured. The method of measuring thermal conductivity which is described in this paper, and which is similar to that in reference 3, avoids this point of constant temperature. Thus, fine adjustment of temperature is not necessary. The heat conductivity can be measured continuously as a function of temperature over a wide temperature range. Because of the non-stationary character of the method, corrections must be made, but these will, in general, be small. Principle of the measuring method
The rod (or disk) whose thermal conductivity is to be measured (S in Figure I) is attached to a heat sink K with a high heat capacity. This is suspended and provides good thermal insulation. The heat generated in the heating
element H flows through the rod, which has two thermometers attached a distance l apart, and slowly warms the whole arrangement. The temperature of one thermometer (e.g. To) and the temperature difference between the two, AT = T o - T t , are measured simultaneously as a function of time t. If the arrangement is such that the temperature is constant within every section of the rod, the thermal conductivity will- be defined by the following equation dT
dA[dT~ 2
. d 2T
7-dT"adL~~ ~\~xx] = o with the boundary condition
a(To)~
...(2)
where y is specific heat/volume, F is rod section, O(0) is heat flow through the rod at x = 0 (approximately the total heat supply). By integration we obtain in first approximation A(0) = jO(O), c-~{t
12
[ "0"dT°
[AT~2dA(0)]
2a(6SaTL~'t ) d / - - i T ) d~-J+'"j ...(3) Corrections will be small provided that
2ATA
;z
a(0) F
o =
12y(dTo/dt) "K
...(1)
ylF - 2-C '~ I
...(4)
C ~- ~K ~)K K: Heat sink H : Heating element
S : Rod-shaped sample
Tt, To : Thermometers
(where v K is the volume of the storing body K), and
Figure 1. Pt@ciple of the measuring arrangement
o--
:To , ~H
C R Y O G E N I C S - J U N E 1962
The first term of the correction results from the decrease of heat flow within the measuring section due to the rise 223
in temperature of the rod. The second term arises from the Taylor expansion for . A T dA •~(T,/2) = A(0)+-~-~--~+...
The time of measurement or the required heat supply will then be given by
(dTo/dt) ~- Q/c and the temperature difference by ~ QI
AT_
If Q/F, I,and C are conveniently chosen, it will always be possible to keep the corrections small and the times of measurement reasonably short.
"-
Description of the apparatus
In order to test the method, the thermal conductivity of aluminium (99.5 per cent) was determined. Figure 2 shows schematically the test arrangement. The heating element, made of manganine wire (approximately 45 f2 at 300 ° K), was wound on to the end of the rod. Outside the cryostat Pump
a resistor of approximately the same magnitude was connected with the heating circuit so that the total heat supply was kept nearly constant; the variation of only a few per cent was corrected. The temperatures were measured by lead resistance thermometers (To, Tt) of approximately 110f~ at 300 ° K, the supports of which were soldered on the rod. The measuring current of the thermometers was chosen in such a manner that the average rise of temperature of the thermometers was equal to that of the rod. It was not necessary to correct for the transfer of heat from the rod to the thermometers or for the heat capacity of the thermometers. The temperature difference was measured continuously in a bridge circuit using the deflection method. Simultaneously, the temperature of the lower thermometer was also continuously measured with a second galvanometer. The rod was soldered into a lead cylinder K weighing about 140 g. A radiation shield R was attached to this cylinder. Gas flowing through a german silver spiral C (2 x 0.2 m m diameter, 600 m m long) which had been melted into the lead, served as coolant. It was pre-cooled to the temperature of the bath liquid in a cooling spiral P. The connections (spiral and wires) had dimensions such that in spite of increasing temperature difference between lead sink and cooling bath no disturbing heat transfer occurred, allowing the measurement of a wide range of temperatures. The vacuum chamber V was flushed with carbon dioxide and evacuated to 10 -2 m m Hg. When subsequently cooling below 100°K, a sufficiently high vacuum is obtained. The total cooling period--pre-cooling with liquid nitrogen to 78 ° K, then with liquid hydrogen--was about 1 hr, the cooling gas (hydrogen or helium) circulating with varying velocity (20-1,000 l./hr). It is intended to provide a small pressure vessel in the lead cylinder to obtain still lower temperatures by the expansion of helium.
Experimental results
ici 2
5
Figure 2. Schematic diagram o f the apparatus 224
The results are shown in Figure 3. The thermal conductivity A was calculated from the T(t) and AT(t) curves in accordance with equation (3) at different heat supplies (0-05, 0.12, 0-22, 0-88 W). The maximum corrections amounted to 10 per cent; at lower heat supplies they were in all cases less than 3 per cent. Measuring the curve took approximately 40 min at Q = 0.05 W between 14 and 30 ° K, about 120 min between 20 and 70 ° K at Q = 0.12 W, and about 15 min at Q = 0.88 W. In the obviously most favourable case, Q = 0-22 W, the measurement took about 70 min. The deviation of the curves shows no systematic correlation with the heat supply and amounts to a maximum of 3 per cent. The results agree with values obtained by other methods. An estimate of the errors occurring in our tests resulted in an error of 3 per cent for the absolute values and of about 0.5 per cent for the relative values. C R Y O G E N I C S . J U N E 1962
Conclusions
6.0
The results obtained by this method appear to make it worth developing into a precision method for the measurement of heat conductivity by taking into account corrections of the second order (for the heat capacity and heat supply of the thermometers as well) and by improving the measurin'g devices. The method allows the determination, continuously and quickly (15-120 min), of the heat conductivity over a wide temperature range. The consumption of cooling agents (I 1. liquid hydrogen in the supply vessel) is low. We think that this method will be especially suited to those applications where a large number of specimens is to be tested, as for example in analytical processes.
E
~" 5.~ 5"0
l
4"S 4"0 I !
3"S
l
I
3-0
0
I i
~'%.
20
30
40
50 T
60 '
70
80
90 (°K)
1 : Q" = 0.22 w (21 November 1961) 2: Q = 0-88 W (29 November 1961) 3: Q = 0-12 w (5 December 1961) 4: Q = 0.056 W (12 December 1961) 5 : 4 from 28 ° K, continued with Q = 0.12 W
REFERENCES 1. WroTE, G. K. Experimental Techniques in Low-temperature Physics (Oxford University Press, London, 1959) 2. EDER, F. X., Moderne Messmethoden der Physik. Teil 2, p. 320 (Deutscher Verlag der Wissenschaften, Berlin, 1956) 3. W~nn, F. J., WILKINSOn, K. R., and WILKS, J. Proc. roy. Soc. .4,214, 546 (1952)
Figure 3. Heat conductivity of aluminium (99"S per cent) as a function of temperature
CRYOGEN]CS.~U~
1962
225
Technische Universitat, JDreaden z)eut,che Akademie der Wissenschaften "zu Berlin, Dresden
Received 12 February 1962
THERMAL conductivity is often measured in such a manner 1,2 that one point of the sample to be tested is maintained at a constant temperature, followed either by setting up a stationary temperature gradient and measuring the temporarily constant differences of temperature, or by generating a temporarily variable temperature gradient by means of temporary variation of heat supply and calculating thermal conductivity from the temperature versus time curves. In both cases, during adjustment of the temperature gradient variation of the constant temperature must be small compared with the differences of temperature to be measured. The method of measuring thermal conductivity which is described in this paper, and which is similar to that in reference 3, avoids this point of constant temperature. Thus, fine adjustment of temperature is not necessary. The heat conductivity can be measured continuously as a function of temperature over a wide temperature range. Because of the non-stationary character of the method, corrections must be made, but these will, in general, be small. Principle of the measuring method
The rod (or disk) whose thermal conductivity is to be measured (S in Figure I) is attached to a heat sink K with a high heat capacity. This is suspended and provides good thermal insulation. The heat generated in the heating
element H flows through the rod, which has two thermometers attached a distance l apart, and slowly warms the whole arrangement. The temperature of one thermometer (e.g. To) and the temperature difference between the two, AT = T o - T t , are measured simultaneously as a function of time t. If the arrangement is such that the temperature is constant within every section of the rod, the thermal conductivity will- be defined by the following equation dT
dA[dT~ 2
. d 2T
7-dT"adL~~ ~\~xx] = o with the boundary condition
a(To)~
...(2)
where y is specific heat/volume, F is rod section, O(0) is heat flow through the rod at x = 0 (approximately the total heat supply). By integration we obtain in first approximation A(0) = jO(O), c-~{t
12
[ "0"dT°
[AT~2dA(0)]
2a(6SaTL~'t ) d / - - i T ) d~-J+'"j ...(3) Corrections will be small provided that
2ATA
;z
a(0) F
o =
12y(dTo/dt) "K
...(1)
ylF - 2-C '~ I
...(4)
C ~- ~K ~)K K: Heat sink H : Heating element
S : Rod-shaped sample
Tt, To : Thermometers
(where v K is the volume of the storing body K), and
Figure 1. Pt@ciple of the measuring arrangement
o--
:To , ~H
C R Y O G E N I C S - J U N E 1962
The first term of the correction results from the decrease of heat flow within the measuring section due to the rise 223
in temperature of the rod. The second term arises from the Taylor expansion for . A T dA •~(T,/2) = A(0)+-~-~--~+...
The time of measurement or the required heat supply will then be given by
(dTo/dt) ~- Q/c and the temperature difference by ~ QI
AT_
If Q/F, I,and C are conveniently chosen, it will always be possible to keep the corrections small and the times of measurement reasonably short.
"-
Description of the apparatus
In order to test the method, the thermal conductivity of aluminium (99.5 per cent) was determined. Figure 2 shows schematically the test arrangement. The heating element, made of manganine wire (approximately 45 f2 at 300 ° K), was wound on to the end of the rod. Outside the cryostat Pump
a resistor of approximately the same magnitude was connected with the heating circuit so that the total heat supply was kept nearly constant; the variation of only a few per cent was corrected. The temperatures were measured by lead resistance thermometers (To, Tt) of approximately 110f~ at 300 ° K, the supports of which were soldered on the rod. The measuring current of the thermometers was chosen in such a manner that the average rise of temperature of the thermometers was equal to that of the rod. It was not necessary to correct for the transfer of heat from the rod to the thermometers or for the heat capacity of the thermometers. The temperature difference was measured continuously in a bridge circuit using the deflection method. Simultaneously, the temperature of the lower thermometer was also continuously measured with a second galvanometer. The rod was soldered into a lead cylinder K weighing about 140 g. A radiation shield R was attached to this cylinder. Gas flowing through a german silver spiral C (2 x 0.2 m m diameter, 600 m m long) which had been melted into the lead, served as coolant. It was pre-cooled to the temperature of the bath liquid in a cooling spiral P. The connections (spiral and wires) had dimensions such that in spite of increasing temperature difference between lead sink and cooling bath no disturbing heat transfer occurred, allowing the measurement of a wide range of temperatures. The vacuum chamber V was flushed with carbon dioxide and evacuated to 10 -2 m m Hg. When subsequently cooling below 100°K, a sufficiently high vacuum is obtained. The total cooling period--pre-cooling with liquid nitrogen to 78 ° K, then with liquid hydrogen--was about 1 hr, the cooling gas (hydrogen or helium) circulating with varying velocity (20-1,000 l./hr). It is intended to provide a small pressure vessel in the lead cylinder to obtain still lower temperatures by the expansion of helium.
Experimental results
ici 2
5
Figure 2. Schematic diagram o f the apparatus 224
The results are shown in Figure 3. The thermal conductivity A was calculated from the T(t) and AT(t) curves in accordance with equation (3) at different heat supplies (0-05, 0.12, 0-22, 0-88 W). The maximum corrections amounted to 10 per cent; at lower heat supplies they were in all cases less than 3 per cent. Measuring the curve took approximately 40 min at Q = 0.05 W between 14 and 30 ° K, about 120 min between 20 and 70 ° K at Q = 0.12 W, and about 15 min at Q = 0.88 W. In the obviously most favourable case, Q = 0-22 W, the measurement took about 70 min. The deviation of the curves shows no systematic correlation with the heat supply and amounts to a maximum of 3 per cent. The results agree with values obtained by other methods. An estimate of the errors occurring in our tests resulted in an error of 3 per cent for the absolute values and of about 0.5 per cent for the relative values. C R Y O G E N I C S . J U N E 1962
Conclusions
6.0
The results obtained by this method appear to make it worth developing into a precision method for the measurement of heat conductivity by taking into account corrections of the second order (for the heat capacity and heat supply of the thermometers as well) and by improving the measurin'g devices. The method allows the determination, continuously and quickly (15-120 min), of the heat conductivity over a wide temperature range. The consumption of cooling agents (I 1. liquid hydrogen in the supply vessel) is low. We think that this method will be especially suited to those applications where a large number of specimens is to be tested, as for example in analytical processes.
E
~" 5.~ 5"0
l
4"S 4"0 I !
3"S
l
I
3-0
0
I i
~'%.
20
30
40
50 T
60 '
70
80
90 (°K)
1 : Q" = 0.22 w (21 November 1961) 2: Q = 0-88 W (29 November 1961) 3: Q = 0-12 w (5 December 1961) 4: Q = 0.056 W (12 December 1961) 5 : 4 from 28 ° K, continued with Q = 0.12 W
REFERENCES 1. WroTE, G. K. Experimental Techniques in Low-temperature Physics (Oxford University Press, London, 1959) 2. EDER, F. X., Moderne Messmethoden der Physik. Teil 2, p. 320 (Deutscher Verlag der Wissenschaften, Berlin, 1956) 3. W~nn, F. J., WILKINSOn, K. R., and WILKS, J. Proc. roy. Soc. .4,214, 546 (1952)
Figure 3. Heat conductivity of aluminium (99"S per cent) as a function of temperature
CRYOGEN]CS.~U~
1962
225