Enhanced cooling of cryogenic electrical leads

Enhanced cooling of cryogenic electrical leads D. G. Walmsley,

G . R. C l a r k e , a n d H . L o n d o n

When large currents areintroduced to a liquid helium bath the design of the leads is subject to two conflicting requirements: to keep thermal conduction losses to the bath small the leads should be long and thin but if Joule heating is to be minimized short thick leads are preferable. Heat exchange between the leads and the effluent helium gas reduces the amount of heat reaching the bath and thus the boil-off. These factors have been considered by a number of authors 1 - 5 and optimal design criteria have been suggested. The analyses have usually been developed for situations in which a steady large current is fed into the cryogenic environment. In some research work the maximum current is flowing for only a small fraction of the total experimental time. In this situation it can be shown that by inserting a heater in the liquid and arranging it so as to provide an enhanced evaporation dependent upon the current flowing, an overall reduction in the consumption of helium is achieved. The argument is developed as follows. First consider a non-current-carrying lead in thermal contact with effluent helium gas. The heat transferred to the gas per unit length of conductor, q, is given by

=nCp-

(i)

where k is the thermal conductivity, A the cross-sectional area, and dT/dx the thermal gradient of the lead; n is the mass flow rate of the helium gas and Cp its specific heat. Integration yields dT

kA--= nCpT+c dx

(2)

The constant of integration, c, may be neglected without serious error for the particular case of a helium bath at 4.2 K. This follows because the heat conducted by the lead into the bath produces a boil-off

leads to the temperature distribution along the length of the lead:

~

l X

nCp

- dT =

x

T2 T

(3)

A

The electrical resistance of such a lead, length L, and with its cold end at 4.2 K and its hot end at Th, can be written

R(Th)= f

fh

L p(x) 1 -dx= 0 A

A

d~ p(T) - dT

4.2

(4)

dT

Substitution for dx/dT from (2) allows us to put 1

Th

p

(5)

J4.2 WT dT

R(Th)- ,,Cp

where W (= 1/~) is the thermal resistivity of the leads. Combination of (3) and (5) yields an expression for the resistance in terms only of the lead geometry and the electrical and thermal resistivities of the lead material:

f

x

R(Th)= A

h

P d

4.2 WT

/

/jhl --

4.2 WT

dT

(6)

An equivalent analysis of the behaviour of a lead carrying a large current proceeds in a similar way. Joule heat is removed by the effluent gas and, for sufficiently large currents, longitudinal thermal conduction is of negligible importance. Equating the Joule heat arising from a current, I, to the heat removed by the gas:

120 - -

dx

= n

Cp dT

(7)

A n,~-

dx where L is the latent heat of evaporation of liquid helium. Then

and integrating, gives the temperature distribution along the lead

~

2 dT

T2 p c = ~kA - dx

12 - - -

(8)

1-

which is small since at 4.2 K, Cn = 1.25 cal g-1 K-1 (1 cal g-1 = 4.187 kJ kg"1) and L = 5 ~al g-1. A second integration

The electrical resistance now is

p(x) "R(Th) = 0

DGW is with the School of Physical Sciences, The New University of Ulster, Coleraine, N Ireland. GRC is with the Public Relations Branch, UKAEA, 11 Charles II Street, London. This paper is based on research carried out by DGW, GRC, and the late Dr London at AERE, Harwell, UK. Received 1 July 1971.

54

x

n CpA

A

1 ~h dx dx = p(T) - - dT A

4.2

(9)

dT

and substitution from (7) gives

.% R(Th) = 12

(Th - 4.2)

CRYOGENICS.

(10)

FEBRUARY

1972

I.o[-

Combining (8) and (10) we obtain a result depending on geometry and electrical resistivity:

~(Th)=x(Th_4.2/( j ~h

(ll)

The temperature distributions described by (3) and (8) are remarkably similar; they would be identical were it not for deviations from the Wiedemann-Franz-Lorenz law. This is of central importance to the present discussion. Let us now leave these two idealized situations where either Joule heating or longitudinal thermal conduction is neglected and return to the original problem. Suppose that initially there is no current in the lead and a current is built up. Joule heat will cause an increase in temperature in the lead. One possible consequence is that the lead temperature will rise to its melting point and suffer irreversible damage. A less drastic outcome is for the temperature increase in the lead to result in a larger conducted heat leak to the bath which in turn produces a temperaturestablizing increase in the boil-off. In the latter situation the Joule heating may be considerably greater than the minimum necessary for temperature stability. But if instead of relying on thermal conduction to provide the increased helium evaporation an additional resistor is placed in the circuit, below the liquid level, it will produce a quadratically increasing boil-off with current increase. Then there can be a balance between the additional heat produced in the leads and the additional cooling. The temperature distribution in the leads will now lie somewhere between the forms expressed in equations 3 and 8; at very low currents it will be close to that of equation 3 and at very high currents it will resemble the form of equation 8. For materials which do not deviate greatly from the Wiedemann-Franz-Lorenz law these forms, as has already been indicated, are closely similar. Hence the heat leak to the bath due to thermal conduction is almost independent of current. The value, r, of the required resistor is found by setting the helium mass flow rate arising from evaporation equal to that required for Joule heat removal: i2r

i2~'(

L

Th )

Cp(T h - 4.2)

and hence r-

R(rh) L

(12)

Cp(T h - 4.2) For leads to a room temperature of 290 K, this gives r--~ ~ / 7 2 and if the Wiedemann-Franz-Lorenz law is approximately obeyed r ~ R/72 also. R and ~' are defined in (6) and (11). The temperature distributions of (3) and (8) and the resistances R (T h = 290 K) and ff[(T h = 290 K) have been computed for copper of different purities. Procedures prescribed by White 6 for the calculation of electrical and thermal resistivities as functions of temperature have been followed. The Bloch-Gruneisen form involving the J5 transport integral was used for the intrinsic electrical resistivity, Pi, and the J3 integral for the intrinsic thermal

CRYOGENICS. FEBRUARY 1972

k o.

I

I

IOO

0

I

I 300

'

200 Temperature, K

Fig.1 Temperature distribution along copper leads under high current (curve A) and low current (curve B) conditions; electrical resistivity (curve C) and the product WT (curve D) are shown as functions of temperature

resistivity, Wi. Matthiessen's rule was assumed in adding in the temperature-independent components (/90, Wo) at all temperatures and the Debye Temperature, ®, was taken to be 310 K. Table 1 gives the results for the ratio of room temperature resistance, R290, to that when one end of the lead is at room temperature and the other at 4.2 K. The ratio is given for both the low current and the high current conditions and for a range Of Po/Pi(®) values. The value of r relevant to a particular electrical lead can easily be determined by interpolation of these results once the resistance ratio of the copper is known. Note that P i ( T = 290 K)/Pi(T= 310 K) = 0.928. An experimental system in our laboratory, originally designed to operate at an optimum current of 400 A, Table I.

po/Pi (®) R290/R(Th = 290 K)

0.0012

0.00234

0.004

0.008

0.012

42.6

29.3

21.5

14.4

11.4

63.5

39.3

27.0

16.8

12.8

R290f'~(T h = 290 K)

55

exhibited very large boil-offs at currents beyond 600 A. The resistance ratio of the copper, measured at 290 K and 4.2 K was 398. This corresponds to the second column of Table 1. The temperature distributions for the same purity copper calculated on the basis of equations 8 and 3 are displayed as curves A and B respectively in Fig.1. For reference, the electrical resistivity, p, and the product WT, are also shown (curves C and D) in reduced coordinates as functions of temperature. The value of the series resistor required to keep the temperature distribution between the two limiting curves was calculated from (12). When this was installed currents up to 1 200 A produced only moderate boil-offs. In conclusion, it would seem that the installation of a resistor below the liquid level of a helium bath has definite advantages if high currents are passed intermittently. It

also provides a simple method of uprating existing optimized lead assemblies and it may well offer a useful additional control in continuously operating high current cryogenic systems.

REFERENCES 1. McFEE, R. R e v S c i l n s t r u m 30, 98 (1959). 2. SOBOL, H., and McNICHOL, J. J. R e v S c i l n s t r u m 33,473 (1962). 3. WILLIAMS, J. E. C. Cryogenics 3,235 (1963). 4. DEINESS, S. Cryogenics 5,269 (1965). 5. KEILIN, V. E., and KLIMENKO, E. Yu. Cryogenics 6, 222 (1966). 6. WHITE, G. K. Experimental Techniques in Low Temperature Physics (OUP, 1959).

Automatic temperature control of a superconducting bolometer I. A. Khrebtov, N. M. Gopshtein, and G. A. Zaitsev The temperature of a superconducting bolometer has to be kept constant to an accuracy approaching 1.5 x 10-5 K for normal operation. 1 A system is described here for controlling the temperature of a bolometer and results of using it given. The design of the bolometer is discussed elsewhere. 2 A temperature is kept constant to an accuracy of 10-3 K by controlling the vapour pressure of boiling helium. An additional heater is used for accurate control, and the bolometer itself is used to react to temperature changes and connected in a bridge circuit supplied from an 800 Hz oscillator. The out-of-balance signal when the temperature changes is amplified, rectified by a phase sensitive detector, and after a constant current amplifier is fed to the heater. The heater is wound on the cylindrical brass bolometer base (Fig. I). The base is fixed via an insulating washer to the base of the helium reservoir. The washer and base act as a thermal filter for reducing the effect of fluctuations in the temperature of the liquid helium, produced by its boiling. The temperature controller is a system with negative feedback. The bolometer base is cooled by liquid helium

£o o

~'~

\ \'.6\

--,3s °

-40

-'k-.

-180[~j

= ,o

"'~ i ",400 w, r o d s-I

\

"-,\ k/cx~"~,

\

Fig.2 L o g a r i t h m i c amplitude and phase-frequency characteri-

stics of the system 1 and 6 - a m p l i t u d e and phase characteristics ~ ( m ) , 2 and 5 --

p ( t o ) / p o , 3 a n d 7 - ~P/Po" 4 -- ~p

at a temperature T c which changes by an amount A T c . The oscillations in bolometer temperature are

CAT AT =

J 4

I

I

Fig.1 Construction of the bolometer 1 -- b o l o m e t e r receiver element, 2 -- base, 3 -- heater: 4 -- washer

The authors are w i t h the Physics F a c u l t y , M o s c o w State U n i v e r s i t y , Moscow, USSR. Prib i Tekh ~ksper N o I, 247 (1971). Received 7 June 1971.

56

-

(1)

where ~"is the transfer function from cryostat to bolometer, ~, K/W is the transfer function from heater to bolometer and pW/K is the slope of the amplification stage of the system. In the stationary state ~"= ~'o --- 1. The system should not suppress temperature changes occurring at the modulation frequency of the illumination being measured. Parasitic temperature variations in the cryostat lie in the frequency range 0.01 to 0.1 Hz, while the modulation frequency is in units of 10 Hz. By using the thermal inertia of the heater, suppression of the optical signal can therefore be avoided. The frequency dependences of ~ and p are important for the operation of the system. For p it is determined CRYOGENICS. FEBRUARY 1972