Protection of superconducting coils by induction coupling

A model that predicts temperature rise in perfectly coupled circuits is presented. The use of frozen cryogen in a secondary protection circuit coupled to a superconducting coil is discussed in this paper. Formulas for the required amount of the frozen cryogen and conducting metal in the protection circuit are derived as a function of the desired final temperature in the protected coil.

Protection of superconducting coils by induction coupling Y.M. Eyssa Key words: superconducting device, cryogen, superconducting coil The use of a shorted secondary protection circuit that is inductively coupled to the primary superconducting coil circuit has been recently suggested as a method of quench protection for large superconducting coils) '2,a Green 1 has reported that coils which use long time constant shorted circuits will dump most of their energy into the secondary circuit when the magnet quenches. Green also has looked into the mechanism by which the primary coil is turned normal by heating back from the secondary circuit (Quench back) either by thermal conduction or ac loss heating in the superconductor due to magnetic field decay. Satti 2 has experimented on a low current superconducting coil wound with insulated strand cable. A superconducting secondary loop was made within a cable of a 6.2 Henry dipole coil. When quenching occurred current was induced in the secondary strand above the critical value. The normal strand quenched the whole coil due to good thermal contact. Eyssa a has suggested using the massive cold aluminium structure that is used to carry the axial forces in a single layer energy storage magnet to share the energy dump during a quench. This paper describes an analytical model that relates the temperature in the primary circuit to the temperature in the secondary circuits during the quench. The model also predicts the final temperature in each circuit and quench time constant. The use of solid cryogens such as nitrogen to provide more heat capacity to limit the final temperature in the conductor is analysed and discussed.

normal faster than it would through normal zone propagation! Assuming perfect coupling and no thermal contact, equations that govern the temperature rise for each circuit are

J'~(t) pj(t)dt and

JxPl

=

J2P2 =

- . . . . . =JjPj

Zj(Tj) = f rj Cj(T) pj(T) dT = constant

(3)

4.2 Equation (3) relates the temperature, Tj, in each circuit to each other. The total energy dissipated is

TFc E = Vc =

Model

The author is from the University of Petroleum and Minerals, Dhahran, Saudi Arabia. Paper received 1 April 1982.

(2)

where Jj is the electric current density in the jth circuit, p is the electric resistivity, and C is the heat capacity per unit volume. Equation (2) results from the fact that the electric field in each circuit is the same because of the perfect coupling. If the circuits are not closely coupled, (2) is not valid. Numerical calculations should be used to find out the relation between currents in each circuit. Using (1) and (2), it is easy to show that the function Zj(T) is

Cc(T) dT +

~4.2

The model assumes a primary circuit which is the magnet winding and secondary circuits which are well coupled with the primary. The secondary circuits can be the coil structure, the coil bobbin or a special protection circuit that is wound closely with the primary superconducting winding. In the latter case the coupling is perfect while in the other cases the coupling is good only for large coils. The assumption of perfect coupling in this case makes it easy to treat the problem analytically. The model also assumes no thermal contact between the coupled circuits during the short time of quench. The assumption does not recommend poor thermal contact between the coupled circuit. In fact the presence of good thermal contact makes the shorted secondary circuit cause the magnet to become

(1)

= Cj(T)dT

i~

gi

'

~!TFi Ci(T) d T .2

(4)

where Vc and Vi are the volume of conductor in the primary circuit and the secondary circuits respectively, Tvc and Tvi are the final temperature in each circuit. Using (3), (4) may be written as

TFc E = Vc f

.2

Co(T)

~i.~7_,~

~-c

aT

(s)

Equation (5) shows an increase in heat capacity due to the presence of the secondary protection circuits. For coupling coefficient less than unity, the increase in heat capacity shown by (5) would be reduced. Under the assumptions made previously, the currents divide between the circuits according to their resistances, (2). In that case the current density in the primary circuit is

0 0 1 1 - 2 2 7 5 / 8 2 / 0 0 9 4 6 9 - 0 4 $03.00 © 1982 Butterworth & Co (Publishers) Ltd. C R Y O G E N I C S . SEPTEMBER 1982

469

jc(t ) =

I(t) [A¢+ ~ , ~Pc(T) Ai]

(6)

where I(t) is the total current = the sum of the currents in all the circuits, A c, and A i are the cross-sectional areas of the primary circuit and the ith secondary circuit. Using (1), (5), and (6) it is possible to relate the total current I to the temperature in the primary circuit as

/a(T)_

~

. Pi(T.OVc

1+

d~-

(7)

Pi(Ti)Vc

dT

1

- --~--

Cc(T) 1+ o

i

(8)

Taking into consideration the heat capacity of the metal in the secondary circuit and that of the cryogen, (3) may be written as f4 TF c .2

o

A2c C(T)

l°2- T

CscPs dT +

"4.2

f TFs

PsCsdT (11)

4.2

,,., Zc(TFc) -- Zs(TFs) -- P(TFs) [h(TF s) - h(4.2)1

(12)

where h is the enthalpy of the cryogen per unit volume and Tvs in this case is the boiling temperature of the cryogen.

1+

• 2Vc

i T Fs

where a is the volume ratio of the cryogen to the metal in the secondary circuit, Ps is the resistivity of the metal wires imbedded in the cryogen, Csc is the heat capacity of the cryogen per unit volume which includes heat of fusion and vaporization, Cs is the heat capacity of the metal in the protection circuit per unit volume, TFc and TFs are the desired final temperature in the primary and secondary circuit respectively. Since most of the heat capacity of the cryogen is heat of vaporization, (1 I) may be reduced and arranged as:

where Io is the total initial current and L is the circuit inductance. The time constant r is Z(r) f r =

PcCcdT = a

T

fo C(T)

[

Pi(Ti)Ve E Pc(T)Vi ] dT 1 + ,' ~ V c J dT

To find out the volume of conductor required in the secondary circuit as a function of TFc and TFs, (4) may be rearranged as ( . TF c E/Vc - | CcdT Fs _ "4.2 ~ Vc fry c PcCc 4.2 ~ dT

(9) where T(r) is the temperature in the primary circuit at t = r. T(r) is calculated using (5) for the value of E equal to E = [1 - e -z] Eo

(13)

(10)

where E o is the initial energy stored and e = 2.718. As shown by (9) the time constant r can be increased by an order of magnitude if Pi(TO is kept small compared to pc(T). The idea of using frozen cryogen to keep the temperature low in the secondary circuit will be discussed in the next section.

Equation (13) can be further reduced for Ts ~ TF s and approximated as: .TF c

Frozen cryogen protection circuits

where 13is the ratio of conductor volume in the secondary circuit to the conductor in the primary circuit Vs/Vc.

The idea of using frozen cryogens to share the energy deposited in the winding of large energy storage coils during a quench has been considered recently.4,s It was suggested that frozen cryogens such as ammonia and nitrogen can be incorporated with the conductor to provide a large amount of heat capacity via heats of fusion and vaporization. The poor thermal conductivity of the solid cryogens requires a large contact area between the conductor and the cryogen in order to diffuse the heat dissipated quickly to the cryogen. Such a requirement complicates the design of the conductor and limits the area in contact with helium required for superconductor stability. A better way to use the large heat capacity of the solid cryogen is to incorporate it in a secondary winding that is inductively coupled to the superconducting winding. The suggested secondary protective winding may be small aluminum wires in a matrix of solid cryogen. During the quench resistance of the secondary is kept low due to the presence of the solid cryogen in good thermal contact with the aluminum wires. As the primary winding gets warmer than the secondary, more current will be inductively diverted to the cryogen matrix.

470

¢l ~ Ps(TFs)

4.2

Zc(Tr~)

(14)

7 6

,~

....

Aluminium~ 50o/0 R~R= 5, 500/0 RIRR-- 2 5 0 1 Copp¢% RRR = 150 ~ / / I

5

E

-> 2 I

0 80

I 0 0 120

140 160 180 2 0 0 220 240 260 280 Temperature, K T Fig. 1 Function Z ( T ) = f p c d T vs T for copper conductor 4.2 ( R R R = 150) and aluminum conductor (50% higher strength aluminum R R R = 5 and 50% high purity aluminum R R R = 250)

CRYOGENICS. SEPTEMBER 1982

Equations (12) and (14) gives the required amount of conductor and cryogen in the protection secondary circuit as a function of the final temperature in the conductor, TFc , and the cryogen, TFs. Fig. 1 is a plot of the function Z(T) for aluminium and copper conductors. Fig. 2 is a plot offCcdT for the same conductors. Table 1 lists thermodynamic properties of different possible cryogens used in protective secondary circuits. 6

°t

!08

/

//

~--"

/

o.I

0.01 [ - o

/ /

- Copper

200

300

400

Temperature, K T

4,2

200

2 300

Temperature, K Specific heat vs temperature for solid and liquid nitrogen

To illustrate the benefit of using frozen nitrogen in a secondary circuit we will consider examples of magnet systems storing energies higher than 150 J cm -3 of conductor. Using (12) and (14), the amount of nitrogen and aluminium conductor in the protection circuit are plotted vs the desired final average temperature for different values E/Vo Fig. 4. It is assumed that during the quench the primary circuit is short circuited. Without the frozen cryogen protection circuit the average final temperature for a system with E/V c = 7 × 10 s J m -3 would be 325 K. With the presence of the cryogen in the protection circuit, the final temperature in the conductor can be reduced significantly depending on the amount of conductor and the cryogen. The use of another protection scheme (external dump resistor 8) combined with the suggested frozen cryogen scheme may further reduce the final temperature in the primary circuit. The analysis of such hybrid schemes is not covered in the present study.

io 4.

f

IOO

From the economic and safety point of view it seems that nitrogen may be preferred over other cryogens. Its boiling temperature is lower than oxygen and it has a total heat capacity from 4.2 K to 77.4 of 320 J g-1. Fig. 3 is a plot of the specific heat vs temperature for solid and liquid nitrogen.

i0 5

Fig, 2

;

~

6

I00

/ = 5 x IO 8 Jm "3 . . . . / = _ 3×108 Jm "3 /z - 1.5x1©8 Jm "3

r

-o IO

0

t

/ £ ' / V - 7 x IO-8 j m -3

"

I.O_~

i/I

107

I

/

/

Fig. 3

~

,J

/

/

Equation (14) shows that in order to reduce the amount of metal in the protection circuit, Ps(Tvs) should be small which means low boiling point cryogens are preferred. Although ammonia has the highest heat capacity of any cryogen, it is not preferred because its boiling temperature is close to room temperature. The conclusion to draw here is that only the boiling temperature, Tvs , of the cryogen determines the amount of conductor in the protection circuit. However, the amount of heat capacity available from the cryogen determines what ratio of cryogen volume to conductor volume is required, (12), to achieve a final temperature TFc in the primary circuit.

109

I .s

cdT vs T for copper and aluminum

Table 1. Thermodynamicpropertiesof ammonia, oxygen, nitrogen and neon NH 3

02

N2

Ne

Melting p o i n t at 1 atm,

TM, K

195

54.9

63.4

24.47

Boiling p o i n t at 1 atm,

TB, K

239.8

90.1

77.4

27.2

Heat o f fusion

j g-1

352

14

25

17

h(TB ) _ h(TM )

j g-1

200

59

35

7

Heat o f v a p o r i z a t i o n

j g-1

1365

213

200

87

Total e n t h a l p y *

j g-1

1917

286

260

121

L i q u i d density

g cm -3

0.683

1.14

0.808

1.21

*Total enthalpy does not include the enthalpy of the solid cryogen.

CRYOGENICS. SEPTEMBER 1982

471

the primary circuit. This method of protection may be useful for large energy storage coils where other protection schemes could be difficult to use because of excessive voltage. The suggested secondary circuit must be designed so that during the magnet quench, the heat generated in that circuit will be conducted effectively to the cryogen. Contact surface area and diffusion of heat through the solid and liquid cryogen should be chosen so that the incremental temperature between the metal and the cryogen is small.

"t"

T ition

,3

Melting

Normal boiling pointj 77,4 K

0 r-

The work is supported by Wisconsin Electric Utilities. The author would like to thank R.W. Boom from the University of Wisconsin, Madison, Wisconsin and G.E. Mclntosh from Cryogenic Technical Services, Boulder, Colorado for their helpful discussions.

'U tO

References

I I0

O.

0

I 20

I 30

I 40

I 50

I 60

I 70

I 80

1 2 3 90

Tcmperatur¢~ K

Fig. 4 E/V

Values of fl and e vs temperature TFc for different values of

4 5

Conclusion

6

The analysis shown in this paper shows that the use of frozen cryogens in a well coupled secondary magnet protection circuit can significantly reduce the final temperature in

7

472

8

Green,M.A.IEEETransMagMAG-17 (1981) 1793 Satti,J.A.IEEETransMagMAG.17(1981)435 Eyssa, Y., MeIntosh, G., Hilal, M., Khalil, A., Nilsson, B. presented at the 9th Symposium on Engineering Problems of Fusion Research, Chicago (1981) Young,W.C. private communication Purcell,J. A protection method for SMES coils, presented at the US -Japan Workshop on Superconductive Energy Storage, (October 1981) University of Wisconsin, Madison, WI. Also General Atomic Report GA-16482, PO Box 81608, San Diego, CA 92138 Haselden,G.G. Cryogenic fundamentals, Academic Press, NY, (1971) Breehna,H. Superconducting magnet systems, SpringerVerlag, NY, (1973) 403 Iwasa,Y., Sinclair, M.W. Cryogenics 20 12 (1980) 711

C R Y O G E N I C S , S E P T E M B E R 1982