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An apparatus to measure the specific thermal conductivity of gases at low temperatures
Physica X, no 7
Juli 1943
T H E SPECIFIC THERMAL CONDUCTIVITY OF G A S E S A T L O W TEMPERATURES
AN A P P A R A T U S
TO M E A S U R E
J. B. U B B I N K and W. J. DE HAAS Communication No. 266c from the l~amerlingh Onnes Laboratorium, Leiden
1. Introduction. The specific thermal conductivity ;~ of gases under normal conditions is measured in general either by the hot wire method 1) 2), or by the plate method 3) 4) 5). In the former method the heat flows from a metal wire stretched in a cylindrical vessel, filled with the gas to be examined, to the walls, the metal wire serving at the same time as a source of heat and as a thermometer. In the latter method the temperature difference and the heat flow between two horizontal plates are measured. In both cases the heat conductivity can be calculated from the heat flow through the gas and the temperature difference. The two main sources of error characteristic of the wire method are the convection currents and the temperature jump between the wire and the walls. The latter is caused by the fact that the energy exchange between the wire and the colliding gas molecules is only partial 6). As this temperature jump is proportional to the heat gradient it becomes of special importance near the wires, where large gradients occur. From the geometrical arrangement in this method follows that convection currents must disturb the measurements ~) s). When using this method the influence of theseerrors is in general eliminated by measuring the k in a large pressure region 7) s) 9) 10) 11). At low pressures the apparent thermal conductivity k,, varies with pressure as a result of the dependence of the temperature jump on the free path of the molecules, whereas at high pressures km changes with pressure, as a result of the increasing convection. At the intermediate pressures, where Xmis constant, k,,, should be identical with - -
451
- -
452
J . B . UBBINK AND W. J. DE HAAS
the real thermal conductivity of the gas k. The wire method can not be used at low temperatures for two reasons: Firstly, the occurrence of convection becomes more probable as a result of the relatively high densities at low temperatures. Secondly if a real pressure dependence of X, due to deviations from M a x w e 1 l's theory, should exist, this would be entirely obscured b y the pressure dependence due to the temperature jump and to convection. In the plate method the convection is suppressed b y mounting the hotter plate above the other one. The temperature jump is small and can be eliminated in principle b y varying the distance between the plates, as then the change in resistance must be entirely due to the change in thickness of the gas layer between the plates. For this reason the plate method was here chosen to measure the heat conductivity of gases. The theory of the apparatus is developed in par. 2, a description follows in par. 3, and finally in par. 4 a few results on the conductivity of hydrogen and air are given.
J2. The method. The apparatus consists of two parallel plates each containing a thermometer. The lower plate forms the bottom of a copper tube containing the apparatus and is in good heat contact with a bath of constant temperature. The upper plate contains a heating wire and can be placed at a variable distance from the lower plate. A guard-plate containing a heating wire and a thermometer is placed at a fixed distance above the upper plate in such a way that the heat contact with the upper plate is as poor as possible. By varying the heat development in the upper plate it is possible to equalise the temperatures T~ and T~ of the guard plate and the upper plate, while the heat development in the guard plate is being kept constant. Consequently, in that case, most of the heat developed in the upper plate will reach the lower plate b y gas conduction, whereas a small fraction will be transferred by conduction to the walls of the tube and b y radiation both to the walls and to the lower plate. The correction terms due to these heat losses can be accounted for b y varying the distance between the plates. This same method also eliminates the errors due to transition resistances between the thermometer and the plate and to the temperature jump at the plates, mentioned in the introduction.
AN APPARATUS TO M E A S U R E T H E R M A L C O N D U C T I V I T Y SPECIFIC
453
The effective area of the upper plate was determined (par. 3) by measuring the capacity of a model 4) 5). It was found that this effective area could be represented by S~11= S + bx, in which S represents the geometrical area, x the plate distance and b a constant. The form factor for the apparatus thus becomes S,tt/x = Six + b. Hence the form factor contains a term independent of the distance x. If I1 and 12 represent the current of heat from the guard-plate to the upper plate and the walls respectively, the total heat development of the guard plate 18 can be written als follows, 18 = 11 + Is
(1)
If the temperature of the lower plate and the walls is given by To and if W1 and W2 are respectively, the heat resistances between the guard-plate and the upper plate, and between the guard-plateand the walls, it is evident that T 8 - - T,, = I l W l
(2)
Tg-
(3)
T O -~. I 2 W 2
If, finally, I,, is the heat development in the upper plate, and W is the total heat resistance between the upper plate and the lower plate and the walls, then T , , - To = (Ii + I,,)W
(4)
Eliminating I1 and 15 from (1), (2) and (3) we find I~ =
T8 - - T,, + T ~ -
Wl
To
W2
(5)
which may be written as follows, - - WI ( T , , W I + W2
T8-T,,-
To) -~
WIW2 1~ W I + W2 "
(6)
Eliminating I1 from (2) and (4), we get T~ - - To = T g - T~ + I,, W W1
(7)
while (6) and (7) give WW2 W(W1 + W2) I , + Ig (8) T , , - - To = I ¥ 1 + W2 + W WI + W2 + W "
454
J. B. U B B I N K AN'D W. J. DE HAAS
These linear relationships can now be used to determine the value of W b y means of the following procedure. For a fixed distance of the plates and a fixed value of I v the heat development in the upper plate Iu is varied, and from a plot of T ~ - T,, against T , , - To according to (6), the value of T , , - To can be read, for which T 8 = T,,. Then
(T,, - - To) 7"~~ 2",, = I~W2 The corresponding value of I,, can then be found by plotting Tu - - To against I,, according to (8), and reading at (T,, - - To)zg=r,,. From (8) follows that
IgW2 (I") ~=~"
--
W
Hence, the heat resistance W can be calculated from the value of ( T , , - - To)~=~,, and (I,,)~g=7.:
W=(T'-T°) I,,
"
(9)
Tg ~ T u
Ifi~o.rder to find the specific thermal conductivity X of the gas from this value of the total resistance W, we must consider the total conductivity I / W als being composed of two fractions. a. The layer fraction is ascribed to the thermal conduction of the gas, with a form factor (S + bx)/x. In consequence of the zero correction of the distance measurement of the plates, the measured distance X : x - - d differs b y a quantity d from the real distance x. The total thermal conductivity of the gas is therefore given b y
), S + b(X + d) 1 1 x + d -- X + ~ + - V xS bk
(IO)
The two terms can be interpreted as the heat resistance X + d / ~ S of the cylindrical gas layer of surface S and the heat resistance 1/bk of the rim regions placed parallel. This expression must still be corrected because of the presence of a transition resistance between the thermometer and the plate mass in each of the plates and the temperature jump at the surface of the plates which, too, may be looked upon as being replaced by a heat resistance. These resistances are placed in series with each of the resistances of (10), so that the heat resistances
AN APPARATUS TO MEASURE THERMAL CONDUCTIVITY SPECIFIC
455
corresponding to these two terms becom~
X +d ?~S -~ ra and
~_~
+ r2
respectively. b. Secondly there is the conductivity C' ascribed to radiation. Addition of these three contributions leads to 1
--=C' W
+
1
X+d XS
1
+ - 1 + rt - ~ --~ r2
Defining the terms, introduced by the application of the corrections as new constants" I C = C ' + - , 1
X--b -~ r2 d R =-~-ff + r t 1/W can be written as 1 l W = C + X XS + R
(1 ~)
The quantity C is assumed to be independent of X, since the shifts of the upper plates are small as compared with the other dimensions of the apparatus. Moreover, C is always small. The value of ~ is found by determination of W for a series of distances X of the two plates. Then a value of C is chosen in such a way that [ I / W - - C] -1, plotted against X, gives a straight line, as must be the case according to 1 1
X = Z--S + R.
(12)
----C
W
Now X may be directly calculated from the slope of this line.
3. Construction o/the apparatus. In fig. 1 the internal construction of the apparatus is given to a scale of I • 0.75. It is made chiefly of copper. The apparatus mainly consists of the lower plate A, the upper plate B, and the guard-plate C, while the gas layer whose conductivity is to be measured lies between A and B. Both B and C are
456
j.B.
U B B I N K AND W. J. DE HAAS
provided with a thermometer Th and a heating element St, whereas A carries a thermometer only.
St
Th St Th
Th
F i g . 1.
On B, a plate of ebonite is placed, having three small legs *) supporting C. The plate B is drawn tightly against C b y means of a phosphorbronze spring connected to B by a thin thread of D.M.C. yarn and a glass cylinder. In constructing the apparatus in this way a great heat resistance between B and C is obtained. The plate C is slightly larger than B, and has a hole in its centre, through which run the leads to the upper plate and the phosphorbronze spring. The ebonite plate of C is fixed to a brass cylindrical body D (fig. 2) b y three screws *). This body can slide up and down in the tube H, a large portion of its area has been cut away in order to keep the heat capacity as small as possible. In mounting the apparatus we took care that the three screws were adjusted, so that the (optically) plainly-ground surfaces of the plates A and B were parallel. The tube H was screwed, on one side, to the lower plate, and *) For the sake' of clearness only two are drawn in fig. 1.
AN A P P A R A T U S TO M E A S U R E T H E R M A L C O N D U C T I V I T Y S P E C I F I C
457
on the o t h e r side to the inside of the cover. The cylindrical brass jacket fits r o u n d H and A. W h e n the a p p a r a t u s was m o u n t e d , it was soldered v a c u u m tight with W o o d's metal, thus forming a closed box. The cover W of this box was c o n n e c t e d to a G e r m a n silver v a c u u m tube Zb, t h r o u g h which the box can be e v a c u a t e d and filled with the gas to be e x a m i n e d .
®
-[
.... f
~Zb
®
~H
~D
®
®
F i g . 2.
All leads were stuck on the tube H before leaving the a p p a r a t u s b y the v a c u u m tube, and were thus brought to the t e m p e r a t u r e of the bath. In this way no energy from the w a r m e r surroundings could flow to the plates t h r o u g h the wires.
458
•
J . B . U B B I N K AND W. J. DE HAAS
The sliding body D was soldered to a German silver tube (fig. 2) .which is movable inside the vacuum tube Zb and was coupled to a micrometer screw, permitting only a vertical displacement of B and C together. The apparatus was placed in a cryostat containing the refrigerating liquid, the temperature of which can be regulated. The temperature di]/erences between the plates were measured by means of the three resistance thermometers Th, each consisting of a platinum wire in series with a phosphorbronze wire. The resistances, which at 0°C amounted to a 30 ~, were determined by the compensation method of D i e s s e 1 h o r s t. Between 273°K and 14°K the platinum wire is used as thermometer; in the region below 4.2°K the phosphorbronze wire is used, the resistance coefficient of platinum being too small. The thermometers not being quite free from tension, t h e y were calibrated in each series of measurements. The guard-plate as well as the upper plate could be heated by the constantan wires St the resistance of which was about 400 ~. As the energy was developed electrically, it could be easily determined. Only in the region below 4.2°K it was necessary to make a small .correction for the energy, developed by the measuring current running through the thermometers. When the middle plate B touched the lower plate A, an electrical circuit was closed, consisting of an extra wire connected to B, the box and an ampere-meter. In this way the zero-position could be determined with a maximum error of 0.01 ram. By turning the micrometer screw, every plate distance could be adjusted to within error 0.001 mm. The distance between A and B varied from 0.1 to 1 mm. Hence the relative error in x is about 1%. An error in the zero-position as well as any obliquity of the plates relatively to each other, would result in an additional resistance, for which corrections were made.
In order to determine how the e]/ective area of the upper plate varies as a function of the plate distance, a brass model of the apparatus on an enlarged scale was built. Of this model the capacity between the upper and the lower plate was measured as a function of the distance. The effective area S,jI can be represented by S~II----4~CX, (C = capacity, X = plate distance). As the rim corrections are by no means negligible, S~II is not a constant. It was found that S,I! = S + bX (S the geometrical area, b a constant). From direct measurements of the upper plate a value of S ~ 7.035 -¢- 0.015 cm 2
AN APPARATUS TO M E A S U R E THERMAL C O N D U C T I V I T Y S P E C I F I C
459
for the a p p a r a t u s was found, in agreemetlt with the results of the d e t e r m i n a t i o n of the c a p a c i t y of the m o d e l which g a v e a v a l u e of 7.07 -4- 0.04 c m 2 w h e n reduced to the dimensions of the a p p a r a t u s . At each m e a s u r e m e n t S was c o r r e c t e d for t h e r m a l expansion. 4. Control m e a s u r e m e n t s . Control m e a s u r e m e n t s on h y d r o g e n at 19°K a n d on h y d r o g e n a n d air at 0°C showed t h a t the t h e o r y of the a p p a r a t u s given in par. 2 agrees w i t h the e x p e r i m e n t a l results. According to the p r e v i o u s discussion each d e t e r m i n a t i o n of X at one t e m p e r a t u r e consists of a n u m b e r of series of m e a s u r e m e n t s , all being carried out w i t h the s a m e h e a t d e v e l o p m e n t Ig in the g u a r d - p l a t e . I n e a c h series, with a definite p l a t e distance, the t e m p e r a t u r e - d i f f e r e n ces T , - - To a n d T,, - - To were d e t e r m i n e d as a function of the h e a t d e v e l o p m e n t I,, in the u p p e r plate. ~wcJ I.,
~o~ °
\
I 0 ~
Tu- T O
O~
0
2 n de(j C
-0.5
Fig. 3. Hydrogen. T = 19.2°1(. O = a : I , = 39.0 × 10-4 cal/sec p -- 44 cm Hg [] = b : Ig = 60.9 x 10-4cal/sec p = 4 4 c m H g A = c : Ig = 39.0 × 10-4cal/sec p = 2 c r n H g . F o r h y d r o g en at the v a p o u r pressure (44 cm Hg) at the m e a n t e m p e r a t u r e 19°K in fig. 3 all values of Tg - - T,,, corresponding to a given value of Ig ---- 39.0 × 10 -+ cal/sec, b u t to different values of I,, and X, h a v e been p l o t t e d as a function of T , , - - To. T h e points fit a s t r a i g h t line (a) in accordance with (6). The intersection of this line
j.B.
460
UBBINK AND W. J. DE HAAS
with the horizontal axis directly gives the value of (T, - - TO)Tg=Tu , where T, -----T,,. For the purpose of a further justification of the equation (6) the measurements have been repeated for another value of the heat development in the guardplate I = 60.9 × 10-4 cal/sec (line b) which according to (6) should give a parallel shifting of the line. This is confirmed b y the approximate equality of the values of W1 and W2 at 44 cm pressure given in table I. To investigate the possible influence of convection, the pressure has also been varied to 2 cm Hg, keeping the value of I~ at 39.0 × × 10~ cal/sec. As might be expected, this changes the value of W2 and, therefore, of both coefficients in eq. (6), whereas W1 remains constant within experimental accuracy. The experimental points fit the line c in fig. 3. 2.0deg~
/
t5
/
I•0
• {xs
o -50
Fig. 4. Hydrogen T = 19.2°K. = 39.0 × 10-4cal/sec p = 4 4 c m H g [-:-] = b : I g -----60.9 x 10-4cal/sec p = 44cmHg.
(~) =
a : Ig
The corresponding values of (I,,)rg:r,, are found from a plot of T , , - To against I,, as given in fig. 4, reading I,, at the value of ( T , - To) for which T, = T,,. For one value of Ig(a) the points corresponding to one series at a definite plate distance lie in a straight line. Variation of the distance must effect the slope of the line according to (8), as the coefficients of (8) contain W. The point of inter-
AN APPARATUS TO MEASURE THERMAL CONDUCTIVITY
SPECIFIC
461
section with the horizontal axis is the same for all distances, being given b y I , = - - W2/WI + W2 Zg. For a higher value of Ig (b) all lines are shifted parallel to a larger value of I,,. From the corresponding values of ( T , , - T0)r~r,, and (I,,)Tg=r,, the values of W can be found from (9). In fig. 5, W has been plotted against X, together with the values of (W - 1 - C) -1, where C has 400ca~tsQc ~ 4
,~/ / ( ~ / ;
//"
200
f
// .0.02
~
X
J 0
0.04.
0.08 e m
Fig. 5. Hydrogen. T = 19.2°I(. p = 44cmHg
[]
--C
[] ......
p =
2cmHg
W
A [-~ - I- C -1 L
A
J
w
been chosen in such a w a y that the plot shows a straight line. F r o m the slope of this line, the value of the heat conductivity can directly be determined. From the figure in table I it appears that, althougl ! as a result of the convection C probably varies from 2.55 × 10 -a cal sec -1 to 0.55 × 10 -a cal sec -1 when the pressure diminishes, the value of x remains practically constant in the case of hydrogen.
462
j.B.
UBBINK
AND W. J. DE HAAS
TABLE I Pressure in cm H g Mean T (°K) I s, (10-* eal see - t ) WI, (10' ca1-1 sec deg) W~, (10 s c a l -1 sec deg) C, (lO - s c a l sec -1) X, (I0 - s e a l cm -1 sec -1 deg -1)
44 cm (b)
44 cm (a)
2 cm (c)
19.33 60.9 9.0 2.43 2.55 3.75
19.08 39.0 9.3 2.51 2.55 3.67
19.31 39.0 8.7 3.84 0.55 3.75
From these measurements it follows that, at low temperatures, the apparatus gives satisfactory results. As a result of the small heat capacities, equilibrium was always reached within 1 rain.
/
/
J
J
"U
/ o ---y--
Fig.
6.
10
Hydrogen. /k
-- C
Air.
--C
[]
15
20.10-2cm
R . 101 ca1-1 deg sec --R.
102cal - l d e g s e c
The measurements of dry air, free from C02, and hydrogen at 0°C als0 gave satisfactory results, although the apparatus is less suited for these temperatures, as it takes too long a time (about 1 h) to reach equilibrium. No pressure dependency of either ;~ or C is found when the pressure is varied from l atm to a few cm Hg. In fig. 6 the
AN APPARATUS
TO MEASURE
THERMAL
CONDUCTIVITY
SPECIFIC
463
values of (W 1 1 - C) -1 ---R have been plotted against X, and in table n the values of W1, W2, C and 9, are tabulated. T A B L E II
M e a n t (°C) W , , (caW* see deg) W , , (eal - t see deg) C, ( 1 0 - ' cal sec -1 d e g -1) ),, (10 - 5 ca] c m -1 sec - l deg-*)
air
hydrogen
0.6 542 228 1.68 5.85
0.5 153 58 7.60 42.1
Table III gives a comparison of these values with those found by other authors. TABLE III A. Plate method
investigator
year 1909 1919 1935
T o d d Hercules Hereus
and Laby and Sutherland
ref.
I
air
14 4 5
4.94 5.40 5.724-0,04
air
H2
B. H o t w i r e m e t h o d
year
investigator
ref.
1917 1926 1926 1927
S. W e b e r Schneider Gregory and Archer T h e s a m e , c o r r e c t e d for t e m p . j u m p by Hereus and Laby S. W e b e r
7 I6 8
1927 1933 1934 1934 1936 1937
Kannuluik Milverton Gregory Archer Nothdurft
and M a r t i n
15 12 13 17 9 18 19
5.680 5.83 4- 0.02
5.85 5.740 5.76 4- 0.03 5.81
5.777 4- 0 . 0 1 2
S~ 41.65 41.80 40.43
40.6 41.3 42.0 41.8 42.45
Finally our sincere thanks are due to Mr. N. F. I. S c h w a r z, nat. phil. cand., for his valuable assistance in performing this investigation. Received april
24, 1943,
464
A N A P P A R A T U S TO M E A S U R E T H E R M A L C O N D U C T I V I T Y S P E C I F I C
REFERENCES I) 2) 3) 4) 5) 6) 7) 8) 9) 10) II) 12) 13) 14) 15) 16) 17) 18) 19)
A. S c h l e i e r m a c h e r , Wied. Ann. 36,346, 1889. R. G o l d s c h m i d t , Phys. Z. i2,418, 19ll. C. C h r i s t i a n s e n, Wied. Ann. 14, 23, 1881. E.O. H e r c u s ' a n d T . H. L a b y , Proc. roy. Soc.,LondonA95, 190, 1919. E.O. H e r c u s andD. M. S u t h e r l a n d , Proc. roy,.Soc.,LondonA i45, 599, 1935. M. K n u d s e n , Ann. Phys. (4) 46,641, 1915. S. W e b e r , Ann. Phys. (4) 54,325,437 and481, [917. H. G r e g o r y andC. T. A.rcher, Proc. roy. Soc.,LondonAt10,91,1926. H. G r e g o r y, Proc. roy. Soc., Loudon A 140, 35, 1934. H. G r e g o r y andC. T. A r c h e r , PhiL Mag. (7) 1,593,1926. S.W. M i l v e r t o n , Proc. roy. Soe., London A150, 287, 1934. S. W e b e r , Ann. Phys. (4) 82, 479, 1927. W.G. Kannuluik andL. H. M a r t i n , Proc. roy. Soc., London A144, 436, 1933. G . W . T o d d, Proc. roy. Soc., London A 83, 19, 1909. E.O. H e r c u s andT. H. L a b y , Phil. Mag.]7)3, I061, 1927. E. S c h n e i d e r , Ann. Phys. (4) 80,215, 1927. S . W . M i l v e r ton, Phil. Mag.(7) 17,397, 1934. C.T. A r c h e r , Nature London138,286, 1936. W. N o t h d u r f t , Ann. Phys.(5) 28, 137, 1937.
Juli 1943
T H E SPECIFIC THERMAL CONDUCTIVITY OF G A S E S A T L O W TEMPERATURES
AN A P P A R A T U S
TO M E A S U R E
J. B. U B B I N K and W. J. DE HAAS Communication No. 266c from the l~amerlingh Onnes Laboratorium, Leiden
1. Introduction. The specific thermal conductivity ;~ of gases under normal conditions is measured in general either by the hot wire method 1) 2), or by the plate method 3) 4) 5). In the former method the heat flows from a metal wire stretched in a cylindrical vessel, filled with the gas to be examined, to the walls, the metal wire serving at the same time as a source of heat and as a thermometer. In the latter method the temperature difference and the heat flow between two horizontal plates are measured. In both cases the heat conductivity can be calculated from the heat flow through the gas and the temperature difference. The two main sources of error characteristic of the wire method are the convection currents and the temperature jump between the wire and the walls. The latter is caused by the fact that the energy exchange between the wire and the colliding gas molecules is only partial 6). As this temperature jump is proportional to the heat gradient it becomes of special importance near the wires, where large gradients occur. From the geometrical arrangement in this method follows that convection currents must disturb the measurements ~) s). When using this method the influence of theseerrors is in general eliminated by measuring the k in a large pressure region 7) s) 9) 10) 11). At low pressures the apparent thermal conductivity k,, varies with pressure as a result of the dependence of the temperature jump on the free path of the molecules, whereas at high pressures km changes with pressure, as a result of the increasing convection. At the intermediate pressures, where Xmis constant, k,,, should be identical with - -
451
- -
452
J . B . UBBINK AND W. J. DE HAAS
the real thermal conductivity of the gas k. The wire method can not be used at low temperatures for two reasons: Firstly, the occurrence of convection becomes more probable as a result of the relatively high densities at low temperatures. Secondly if a real pressure dependence of X, due to deviations from M a x w e 1 l's theory, should exist, this would be entirely obscured b y the pressure dependence due to the temperature jump and to convection. In the plate method the convection is suppressed b y mounting the hotter plate above the other one. The temperature jump is small and can be eliminated in principle b y varying the distance between the plates, as then the change in resistance must be entirely due to the change in thickness of the gas layer between the plates. For this reason the plate method was here chosen to measure the heat conductivity of gases. The theory of the apparatus is developed in par. 2, a description follows in par. 3, and finally in par. 4 a few results on the conductivity of hydrogen and air are given.
J2. The method. The apparatus consists of two parallel plates each containing a thermometer. The lower plate forms the bottom of a copper tube containing the apparatus and is in good heat contact with a bath of constant temperature. The upper plate contains a heating wire and can be placed at a variable distance from the lower plate. A guard-plate containing a heating wire and a thermometer is placed at a fixed distance above the upper plate in such a way that the heat contact with the upper plate is as poor as possible. By varying the heat development in the upper plate it is possible to equalise the temperatures T~ and T~ of the guard plate and the upper plate, while the heat development in the guard plate is being kept constant. Consequently, in that case, most of the heat developed in the upper plate will reach the lower plate b y gas conduction, whereas a small fraction will be transferred by conduction to the walls of the tube and b y radiation both to the walls and to the lower plate. The correction terms due to these heat losses can be accounted for b y varying the distance between the plates. This same method also eliminates the errors due to transition resistances between the thermometer and the plate and to the temperature jump at the plates, mentioned in the introduction.
AN APPARATUS TO M E A S U R E T H E R M A L C O N D U C T I V I T Y SPECIFIC
453
The effective area of the upper plate was determined (par. 3) by measuring the capacity of a model 4) 5). It was found that this effective area could be represented by S~11= S + bx, in which S represents the geometrical area, x the plate distance and b a constant. The form factor for the apparatus thus becomes S,tt/x = Six + b. Hence the form factor contains a term independent of the distance x. If I1 and 12 represent the current of heat from the guard-plate to the upper plate and the walls respectively, the total heat development of the guard plate 18 can be written als follows, 18 = 11 + Is
(1)
If the temperature of the lower plate and the walls is given by To and if W1 and W2 are respectively, the heat resistances between the guard-plate and the upper plate, and between the guard-plateand the walls, it is evident that T 8 - - T,, = I l W l
(2)
Tg-
(3)
T O -~. I 2 W 2
If, finally, I,, is the heat development in the upper plate, and W is the total heat resistance between the upper plate and the lower plate and the walls, then T , , - To = (Ii + I,,)W
(4)
Eliminating I1 and 15 from (1), (2) and (3) we find I~ =
T8 - - T,, + T ~ -
Wl
To
W2
(5)
which may be written as follows, - - WI ( T , , W I + W2
T8-T,,-
To) -~
WIW2 1~ W I + W2 "
(6)
Eliminating I1 from (2) and (4), we get T~ - - To = T g - T~ + I,, W W1
(7)
while (6) and (7) give WW2 W(W1 + W2) I , + Ig (8) T , , - - To = I ¥ 1 + W2 + W WI + W2 + W "
454
J. B. U B B I N K AN'D W. J. DE HAAS
These linear relationships can now be used to determine the value of W b y means of the following procedure. For a fixed distance of the plates and a fixed value of I v the heat development in the upper plate Iu is varied, and from a plot of T ~ - T,, against T , , - To according to (6), the value of T , , - To can be read, for which T 8 = T,,. Then
(T,, - - To) 7"~~ 2",, = I~W2 The corresponding value of I,, can then be found by plotting Tu - - To against I,, according to (8), and reading at (T,, - - To)zg=r,,. From (8) follows that
IgW2 (I") ~=~"
--
W
Hence, the heat resistance W can be calculated from the value of ( T , , - - To)~=~,, and (I,,)~g=7.:
W=(T'-T°) I,,
"
(9)
Tg ~ T u
Ifi~o.rder to find the specific thermal conductivity X of the gas from this value of the total resistance W, we must consider the total conductivity I / W als being composed of two fractions. a. The layer fraction is ascribed to the thermal conduction of the gas, with a form factor (S + bx)/x. In consequence of the zero correction of the distance measurement of the plates, the measured distance X : x - - d differs b y a quantity d from the real distance x. The total thermal conductivity of the gas is therefore given b y
), S + b(X + d) 1 1 x + d -- X + ~ + - V xS bk
(IO)
The two terms can be interpreted as the heat resistance X + d / ~ S of the cylindrical gas layer of surface S and the heat resistance 1/bk of the rim regions placed parallel. This expression must still be corrected because of the presence of a transition resistance between the thermometer and the plate mass in each of the plates and the temperature jump at the surface of the plates which, too, may be looked upon as being replaced by a heat resistance. These resistances are placed in series with each of the resistances of (10), so that the heat resistances
AN APPARATUS TO MEASURE THERMAL CONDUCTIVITY SPECIFIC
455
corresponding to these two terms becom~
X +d ?~S -~ ra and
~_~
+ r2
respectively. b. Secondly there is the conductivity C' ascribed to radiation. Addition of these three contributions leads to 1
--=C' W
+
1
X+d XS
1
+ - 1 + rt - ~ --~ r2
Defining the terms, introduced by the application of the corrections as new constants" I C = C ' + - , 1
X--b -~ r2 d R =-~-ff + r t 1/W can be written as 1 l W = C + X XS + R
(1 ~)
The quantity C is assumed to be independent of X, since the shifts of the upper plates are small as compared with the other dimensions of the apparatus. Moreover, C is always small. The value of ~ is found by determination of W for a series of distances X of the two plates. Then a value of C is chosen in such a way that [ I / W - - C] -1, plotted against X, gives a straight line, as must be the case according to 1 1
X = Z--S + R.
(12)
----C
W
Now X may be directly calculated from the slope of this line.
3. Construction o/the apparatus. In fig. 1 the internal construction of the apparatus is given to a scale of I • 0.75. It is made chiefly of copper. The apparatus mainly consists of the lower plate A, the upper plate B, and the guard-plate C, while the gas layer whose conductivity is to be measured lies between A and B. Both B and C are
456
j.B.
U B B I N K AND W. J. DE HAAS
provided with a thermometer Th and a heating element St, whereas A carries a thermometer only.
St
Th St Th
Th
F i g . 1.
On B, a plate of ebonite is placed, having three small legs *) supporting C. The plate B is drawn tightly against C b y means of a phosphorbronze spring connected to B by a thin thread of D.M.C. yarn and a glass cylinder. In constructing the apparatus in this way a great heat resistance between B and C is obtained. The plate C is slightly larger than B, and has a hole in its centre, through which run the leads to the upper plate and the phosphorbronze spring. The ebonite plate of C is fixed to a brass cylindrical body D (fig. 2) b y three screws *). This body can slide up and down in the tube H, a large portion of its area has been cut away in order to keep the heat capacity as small as possible. In mounting the apparatus we took care that the three screws were adjusted, so that the (optically) plainly-ground surfaces of the plates A and B were parallel. The tube H was screwed, on one side, to the lower plate, and *) For the sake' of clearness only two are drawn in fig. 1.
AN A P P A R A T U S TO M E A S U R E T H E R M A L C O N D U C T I V I T Y S P E C I F I C
457
on the o t h e r side to the inside of the cover. The cylindrical brass jacket fits r o u n d H and A. W h e n the a p p a r a t u s was m o u n t e d , it was soldered v a c u u m tight with W o o d's metal, thus forming a closed box. The cover W of this box was c o n n e c t e d to a G e r m a n silver v a c u u m tube Zb, t h r o u g h which the box can be e v a c u a t e d and filled with the gas to be e x a m i n e d .
®
-[
.... f
~Zb
®
~H
~D
®
®
F i g . 2.
All leads were stuck on the tube H before leaving the a p p a r a t u s b y the v a c u u m tube, and were thus brought to the t e m p e r a t u r e of the bath. In this way no energy from the w a r m e r surroundings could flow to the plates t h r o u g h the wires.
458
•
J . B . U B B I N K AND W. J. DE HAAS
The sliding body D was soldered to a German silver tube (fig. 2) .which is movable inside the vacuum tube Zb and was coupled to a micrometer screw, permitting only a vertical displacement of B and C together. The apparatus was placed in a cryostat containing the refrigerating liquid, the temperature of which can be regulated. The temperature di]/erences between the plates were measured by means of the three resistance thermometers Th, each consisting of a platinum wire in series with a phosphorbronze wire. The resistances, which at 0°C amounted to a 30 ~, were determined by the compensation method of D i e s s e 1 h o r s t. Between 273°K and 14°K the platinum wire is used as thermometer; in the region below 4.2°K the phosphorbronze wire is used, the resistance coefficient of platinum being too small. The thermometers not being quite free from tension, t h e y were calibrated in each series of measurements. The guard-plate as well as the upper plate could be heated by the constantan wires St the resistance of which was about 400 ~. As the energy was developed electrically, it could be easily determined. Only in the region below 4.2°K it was necessary to make a small .correction for the energy, developed by the measuring current running through the thermometers. When the middle plate B touched the lower plate A, an electrical circuit was closed, consisting of an extra wire connected to B, the box and an ampere-meter. In this way the zero-position could be determined with a maximum error of 0.01 ram. By turning the micrometer screw, every plate distance could be adjusted to within error 0.001 mm. The distance between A and B varied from 0.1 to 1 mm. Hence the relative error in x is about 1%. An error in the zero-position as well as any obliquity of the plates relatively to each other, would result in an additional resistance, for which corrections were made.
In order to determine how the e]/ective area of the upper plate varies as a function of the plate distance, a brass model of the apparatus on an enlarged scale was built. Of this model the capacity between the upper and the lower plate was measured as a function of the distance. The effective area S,jI can be represented by S~II----4~CX, (C = capacity, X = plate distance). As the rim corrections are by no means negligible, S~II is not a constant. It was found that S,I! = S + bX (S the geometrical area, b a constant). From direct measurements of the upper plate a value of S ~ 7.035 -¢- 0.015 cm 2
AN APPARATUS TO M E A S U R E THERMAL C O N D U C T I V I T Y S P E C I F I C
459
for the a p p a r a t u s was found, in agreemetlt with the results of the d e t e r m i n a t i o n of the c a p a c i t y of the m o d e l which g a v e a v a l u e of 7.07 -4- 0.04 c m 2 w h e n reduced to the dimensions of the a p p a r a t u s . At each m e a s u r e m e n t S was c o r r e c t e d for t h e r m a l expansion. 4. Control m e a s u r e m e n t s . Control m e a s u r e m e n t s on h y d r o g e n at 19°K a n d on h y d r o g e n a n d air at 0°C showed t h a t the t h e o r y of the a p p a r a t u s given in par. 2 agrees w i t h the e x p e r i m e n t a l results. According to the p r e v i o u s discussion each d e t e r m i n a t i o n of X at one t e m p e r a t u r e consists of a n u m b e r of series of m e a s u r e m e n t s , all being carried out w i t h the s a m e h e a t d e v e l o p m e n t Ig in the g u a r d - p l a t e . I n e a c h series, with a definite p l a t e distance, the t e m p e r a t u r e - d i f f e r e n ces T , - - To a n d T,, - - To were d e t e r m i n e d as a function of the h e a t d e v e l o p m e n t I,, in the u p p e r plate. ~wcJ I.,
~o~ °
\
I 0 ~
Tu- T O
O~
0
2 n de(j C
-0.5
Fig. 3. Hydrogen. T = 19.2°1(. O = a : I , = 39.0 × 10-4 cal/sec p -- 44 cm Hg [] = b : Ig = 60.9 x 10-4cal/sec p = 4 4 c m H g A = c : Ig = 39.0 × 10-4cal/sec p = 2 c r n H g . F o r h y d r o g en at the v a p o u r pressure (44 cm Hg) at the m e a n t e m p e r a t u r e 19°K in fig. 3 all values of Tg - - T,,, corresponding to a given value of Ig ---- 39.0 × 10 -+ cal/sec, b u t to different values of I,, and X, h a v e been p l o t t e d as a function of T , , - - To. T h e points fit a s t r a i g h t line (a) in accordance with (6). The intersection of this line
j.B.
460
UBBINK AND W. J. DE HAAS
with the horizontal axis directly gives the value of (T, - - TO)Tg=Tu , where T, -----T,,. For the purpose of a further justification of the equation (6) the measurements have been repeated for another value of the heat development in the guardplate I = 60.9 × 10-4 cal/sec (line b) which according to (6) should give a parallel shifting of the line. This is confirmed b y the approximate equality of the values of W1 and W2 at 44 cm pressure given in table I. To investigate the possible influence of convection, the pressure has also been varied to 2 cm Hg, keeping the value of I~ at 39.0 × × 10~ cal/sec. As might be expected, this changes the value of W2 and, therefore, of both coefficients in eq. (6), whereas W1 remains constant within experimental accuracy. The experimental points fit the line c in fig. 3. 2.0deg~
/
t5
/
I•0
• {xs
o -50
Fig. 4. Hydrogen T = 19.2°K. = 39.0 × 10-4cal/sec p = 4 4 c m H g [-:-] = b : I g -----60.9 x 10-4cal/sec p = 44cmHg.
(~) =
a : Ig
The corresponding values of (I,,)rg:r,, are found from a plot of T , , - To against I,, as given in fig. 4, reading I,, at the value of ( T , - To) for which T, = T,,. For one value of Ig(a) the points corresponding to one series at a definite plate distance lie in a straight line. Variation of the distance must effect the slope of the line according to (8), as the coefficients of (8) contain W. The point of inter-
AN APPARATUS TO MEASURE THERMAL CONDUCTIVITY
SPECIFIC
461
section with the horizontal axis is the same for all distances, being given b y I , = - - W2/WI + W2 Zg. For a higher value of Ig (b) all lines are shifted parallel to a larger value of I,,. From the corresponding values of ( T , , - T0)r~r,, and (I,,)Tg=r,, the values of W can be found from (9). In fig. 5, W has been plotted against X, together with the values of (W - 1 - C) -1, where C has 400ca~tsQc ~ 4
,~/ / ( ~ / ;
//"
200
f
// .0.02
~
X
J 0
0.04.
0.08 e m
Fig. 5. Hydrogen. T = 19.2°I(. p = 44cmHg
[]
--C
[] ......
p =
2cmHg
W
A [-~ - I- C -1 L
A
J
w
been chosen in such a w a y that the plot shows a straight line. F r o m the slope of this line, the value of the heat conductivity can directly be determined. From the figure in table I it appears that, althougl ! as a result of the convection C probably varies from 2.55 × 10 -a cal sec -1 to 0.55 × 10 -a cal sec -1 when the pressure diminishes, the value of x remains practically constant in the case of hydrogen.
462
j.B.
UBBINK
AND W. J. DE HAAS
TABLE I Pressure in cm H g Mean T (°K) I s, (10-* eal see - t ) WI, (10' ca1-1 sec deg) W~, (10 s c a l -1 sec deg) C, (lO - s c a l sec -1) X, (I0 - s e a l cm -1 sec -1 deg -1)
44 cm (b)
44 cm (a)
2 cm (c)
19.33 60.9 9.0 2.43 2.55 3.75
19.08 39.0 9.3 2.51 2.55 3.67
19.31 39.0 8.7 3.84 0.55 3.75
From these measurements it follows that, at low temperatures, the apparatus gives satisfactory results. As a result of the small heat capacities, equilibrium was always reached within 1 rain.
/
/
J
J
"U
/ o ---y--
Fig.
6.
10
Hydrogen. /k
-- C
Air.
--C
[]
15
20.10-2cm
R . 101 ca1-1 deg sec --R.
102cal - l d e g s e c
The measurements of dry air, free from C02, and hydrogen at 0°C als0 gave satisfactory results, although the apparatus is less suited for these temperatures, as it takes too long a time (about 1 h) to reach equilibrium. No pressure dependency of either ;~ or C is found when the pressure is varied from l atm to a few cm Hg. In fig. 6 the
AN APPARATUS
TO MEASURE
THERMAL
CONDUCTIVITY
SPECIFIC
463
values of (W 1 1 - C) -1 ---R have been plotted against X, and in table n the values of W1, W2, C and 9, are tabulated. T A B L E II
M e a n t (°C) W , , (caW* see deg) W , , (eal - t see deg) C, ( 1 0 - ' cal sec -1 d e g -1) ),, (10 - 5 ca] c m -1 sec - l deg-*)
air
hydrogen
0.6 542 228 1.68 5.85
0.5 153 58 7.60 42.1
Table III gives a comparison of these values with those found by other authors. TABLE III A. Plate method
investigator
year 1909 1919 1935
T o d d Hercules Hereus
and Laby and Sutherland
ref.
I
air
14 4 5
4.94 5.40 5.724-0,04
air
H2
B. H o t w i r e m e t h o d
year
investigator
ref.
1917 1926 1926 1927
S. W e b e r Schneider Gregory and Archer T h e s a m e , c o r r e c t e d for t e m p . j u m p by Hereus and Laby S. W e b e r
7 I6 8
1927 1933 1934 1934 1936 1937
Kannuluik Milverton Gregory Archer Nothdurft
and M a r t i n
15 12 13 17 9 18 19
5.680 5.83 4- 0.02
5.85 5.740 5.76 4- 0.03 5.81
5.777 4- 0 . 0 1 2
S~ 41.65 41.80 40.43
40.6 41.3 42.0 41.8 42.45
Finally our sincere thanks are due to Mr. N. F. I. S c h w a r z, nat. phil. cand., for his valuable assistance in performing this investigation. Received april
24, 1943,
464
A N A P P A R A T U S TO M E A S U R E T H E R M A L C O N D U C T I V I T Y S P E C I F I C
REFERENCES I) 2) 3) 4) 5) 6) 7) 8) 9) 10) II) 12) 13) 14) 15) 16) 17) 18) 19)
A. S c h l e i e r m a c h e r , Wied. Ann. 36,346, 1889. R. G o l d s c h m i d t , Phys. Z. i2,418, 19ll. C. C h r i s t i a n s e n, Wied. Ann. 14, 23, 1881. E.O. H e r c u s ' a n d T . H. L a b y , Proc. roy. Soc.,LondonA95, 190, 1919. E.O. H e r c u s andD. M. S u t h e r l a n d , Proc. roy,.Soc.,LondonA i45, 599, 1935. M. K n u d s e n , Ann. Phys. (4) 46,641, 1915. S. W e b e r , Ann. Phys. (4) 54,325,437 and481, [917. H. G r e g o r y andC. T. A.rcher, Proc. roy. Soc.,LondonAt10,91,1926. H. G r e g o r y, Proc. roy. Soc., Loudon A 140, 35, 1934. H. G r e g o r y andC. T. A r c h e r , PhiL Mag. (7) 1,593,1926. S.W. M i l v e r t o n , Proc. roy. Soe., London A150, 287, 1934. S. W e b e r , Ann. Phys. (4) 82, 479, 1927. W.G. Kannuluik andL. H. M a r t i n , Proc. roy. Soc., London A144, 436, 1933. G . W . T o d d, Proc. roy. Soc., London A 83, 19, 1909. E.O. H e r c u s andT. H. L a b y , Phil. Mag.]7)3, I061, 1927. E. S c h n e i d e r , Ann. Phys. (4) 80,215, 1927. S . W . M i l v e r ton, Phil. Mag.(7) 17,397, 1934. C.T. A r c h e r , Nature London138,286, 1936. W. N o t h d u r f t , Ann. Phys.(5) 28, 137, 1937.