Quench energies of composite superconductors

Quench energies of composite superconductors A.VI. Gurevich, R.G. M i n t s and A.A. Pukhov The Institute for High Temperatures, Moscow, 12741 2, USSR Received 3 June 1988

The quench energy of composite superconductors is treated theoretically. Universal analytical formulae for quench energies are obtained for the cases of one-, two- and three-dimensional heat propagation. Quench energy is shown to be both higher and lower than stationary enthalpy of the minimum propagating zone.

Keywords: superconductors; composites; critical energies

The minimum energy of a local heat pulse Qo, initiating normal zone propagation is an important characteristic of the stability of a superconducting composite. There are many papers concerning, both experimentally and theoretically (usually numerically), the value of Qc for composite superconductors (see, e.g., References 1-3 and references therein). In particular, simple analytical formulae for quench energy of a local heat pulse were obtained by Pasztor and Schmidt 4 and Dresner 5. In these papers ~'5 the model of anisotropic effective continuum was treated, the dependence of specific heat and thermal conductivity of the superconductor on temperature and the heat transfer to the coolant being neglected. The latter assumption restricted these results either by the case of thermally insulated composites or by currents I which are substantially greater than the minimum propagating current I w In the present paper the formulae for Qc have been obtained for the cases of one-, two- and threedimensional heat propagation, which take into account heat transfer to the coolant. We consider the model of anisotropic effective continuum disregarding the current sharing, which allows us to obtain the universal formulae for Qo(I) in the whole region Ip < I < I¢, where Ic is the critical current of the superconductor. The heat balance equation describing the temperature distribution in the superconducting composite is

C OT = k x OZT 02T 02T Ot ~x 2 + ky ~ + k= ~z 2 + pj2q( T - Tr) - h(T - To) + Q6(t)6(x)6(y)6(z)

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00 -&- = A0 - ~0 + q(0 - 1) + qr(z)6(rOr(r2)6(r3)

(1)

(2)

where: A = 02/Or~ + O2/Or2 + 02/0r~ is the Laplace operator; r t, r2 and r 3 are dimensionless coordinates; z is dimensionless time; and r 1 = x ~/(T, - To)kx' rE = y x / t T

~.( r3 = z

To)k/

pjZ

(3)

T,- To)k~

T - To

tpj 2

Q

0 = T~ -~---~o' z = (T~ - To)C' q - Od

¢=

[ T~(j)- To]h

pj2

Qd = Qd =

where: C is the specific heat per unit volume; kx,y,= is the thermal conductivity along x-, y-, z-axes, respectively, the superconducting filaments direct along the x-axis; h is the effective heat transfer coefficient; To is the coolant temperature; p is the resistivity of the composite in the normal state; j is the current density averaged over the cross-section; q(x) = 1 at x ~>0; ~/(x) = 0 at x < 0; T,(j) is the temperature of the resistive transition (I = I¢(T~)); Q is the energy of a local heat pulse; and 6(x) is the delta-function. 0011-2275/89/030188-03 $03.00 © 1989 8utterworth & Co (Publishers) Ltd

Let us consider the case when the dependences of C, k, h and p on T may be neglected. Then it is convenient to rewrite Equation (1) in the following dimensionless form

Qd

(4) (5)

AlTo(j)- To]a/2Cx/~, d = 1

(6)

lI-T~q) - To]2C k"/k~k-~k~,d =

(7)

j~

pj2

2

[T~(j)- To]s/2Ck~---~kyk~, d = 3

p3/2j3

="

(8)

where: d is space dimension; A is the cross-section area of the superconducting wire (d = 1); and b is the thickness of the superconducting winds layer (d = 2). The parameter ~(I) varies from zero to 0.5 with the current diminishing from I¢ to lp. If, for example, the dependence I¢(T) is linear, then Or = 1 - i and 1-i =

cti2

where ~ is the Stekly parameter, i = I/I~.

(9)

Quench energies of superconductors." A. V/. Gurevich et al. Quench energy q, is the minimum value of q, beyond which the normal zone initiated by the local heat pulse will extend infinitely due to the Joule heating. It follows from Equation (2) that q~ is a universal function qd(~) depending on the space dimension d and one dimensionless parameter ~(j) only, i.e., Q~ = qd(~)Qd

(10)

In particular, at ~ << 1 (thermally insulated composite h --, 0 or the case I -~ I¢) the energy qo is equal to the value qd(0) depending on the space dimension d only. In this case the dependence of Qo on the current and other parameters is defined by Equations (6}-(8) which ought to multiply by the numerical coefficients q~(0), q2(0) and q3(0), respectively. Thus, Equations (6)-(8) defining the value of Qo at h - - 0 may be obtained by the dimensional analysis 4"5. We found the universal functions qa(O by the numerical integration of Equation (2) at d = 1, 2, 3 with the following initial and boundary conditions 0(~, 0) = 0 and 0(0% t) = 0, respectively. The results of these calculations are shown in Figure 1. The functions qd(~) calculated with accuracy not less than 3% may be approximated by the following formulae: q~(~) = 2.3(1 -- 2~) -'/2, d = 1, tr = 1.7%

(11)

q2(¢) = 1

_6"12~expL_{0"837~1/3'~ ;, d = 2, tr = 5.3%

q~(¢)=(1

---~-f)3/213"2exp ( 1"66~1/3]S 2~ f"~' d=3,

(12)

a=3.6% (13)

Here a is the mean square-root dispersion of Equations (11)-(13) with respect to the numerical data distributing homogeneously in the regions 0.025 < ¢ < 0.475 (d = 1), qO

0.025 < ~ < 0.435 (d = 2) and 0.025 < ~ < 0.375 (d = 3). Note that formula (11) was obtained previously 6.3. Equations (1 I)--(13) determine quench energy Q~ in the case of a local heat pulse and h 4=0. With decreasing of I the function qd(~) increases from the value qd(0) at / = I c (~ = 0) to infinity at I = lp (¢ = 0.5). Thus, the quench energy Qo increases with the decrease of the current, and in the region I ~ It we have Q¢ oc ( I ¢ - 1) I +d[2 whereas in the region I ~ lp we have Q¢ oc(l - lp)-~/2 (d = 1), Q~ oc exp[aalp/(I -- lp)] (d = 2, 3), where a a are numbers of the order of unity. Let us compare the obtained quench energy with the enthalpy Q~Pz calculated in accordance with the minimum propagating zone (MPZ) theory 7's. The dimensionless enthalpy q~Pz is given by: (14)

qMPZ = f d3r0(~)

where qMPz= QMPz/Qd, r = ( r l , r2, r3) and 0(~) is the stationary solution of Equation (2) with the boundary condition 0(oo)= 0. The enthalpy qMPz is the minimum value of q, demanded for minimum propagating zone creating (see Figure 2). The MPZ profile O(r) depends on r = (r~ + r~ + r~) 1/2 due to O(r) spherical symmetry. To obtain the expression for q~Pz, let us substitute the formula 0 = [A0 + rl(O - 1)]/~, which follows from Equation (2) at aO/dt = q = 0, in Equation (14). The contribution of the first term in the qMPz is equal to zero due to the fact that d0/&(0) = 30/&(oo) = 0. Thus 2R qMPz =-~-, d = 1

(15)

rcR2

quPz = - - ~ , d = 2

(16)

4rcR3 qMpz = - - , d= 3 33

(17)

Here 2R is the dimensionless length of the normal zone O(r) > 1 (see Figure 2). The relation between R a n d ~ can be found by solving the stationary Equation (2). This yields:

1

30

1

R = f---~ In _1- - - ~ , d = I

(18)

= I,(R ~/~)Ko(R V/'~) + Io(R V/-~)K,tR x/~)' d = 2 (19) -o

~"

20

~3/2 = R x/~ - tanh R ~ d= 3 g(1 + tanh R x/~) '

(20)

e (r) d=2 l0

!

I

I

I

0.1

0,2

0.3

0.4

Figure 1 Universal functions qd(~) for d = 1, 2, 3

I

O.

-R

Figure 2

r M P Z temperature profile

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189

Quench energies of superconductors: A. Vl. Gurevich et al. where l(x) and K(x) are modified Bessel functions. The enthalpy QMpz(I) increases with the decrease of I and in the region i ~ I v one has QuPz oc -- In(/-- Iv) (d = 1), Q M p z o c ( l - Ip) -d (d = 2, 3), whereas in the region I ~ 1~ one has Q M p z o c ( l c - l ) ( d = l ) , Q M p z w _ - l n - t ( l ¢ - l ) (d = 2), QMPZ OC (Ie -- 1)1/2 (d = 3). The relation between Qc and QMpz depends on I. As it is seen, the q~(~) increases more rapidly than q~u,z(~), so qMPz >>qc at ~ << 1 (I ~ I¢), whereas qMPz<< q¢ at ¢ ~ 0.5 (I ~ lp). Thus, Q¢ > QMPz at Ip < I < ld (~d < ~ <0.5) and Q¢ < QMPZat ld < I < I, (0 < ¢ < ~d). The value Cd can be calculated numerically from Equations (11)-(20). This yields: ~ = 0.46, d = 1

(21)

¢2 = 0.40, d = 2

(22)

~3 = 0.39, d = 3

(23)

Providing the linear dependence I~ on T (see Equation (9)) for id = I J l ¢ one finds: ia = ( 1 + 4~d2)~

2~1d

(24,

where C%=~d~. In the case ct>>l it follows from Equation (24) that ia=(ot~d) -t12 and so I~ = 1.04Ip ( d = l ) , 12=1.12Ip (d=2), I3=1.13Ip (d=3), where lp= (2/~)1/2I~. Hence, in the region I a < I < I¢ (Qc < QMPz) the most 'dangerous' disturbances are the local heat pulses destroying the superconducting state, whereas in the region Ip < I < I a these are the disturbances with the characteristic length of the order of R. To compare the computed values Qc with the experimental data it is necessary to take into account temperature dependences h(T), k(T), C(T) and p(T), and the non-stationary heat transfer effects, the geometry of the superconducting coil and so on, which are disregarded

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in the simple model considered above (see, also Reference 1). Nevertheless, this model allows us to obtain the universal formulae for Qc; in particular, at lp << I < lc one has Q= ~ Od. Note that the quench energies increase by 3-4 orders as the coolant temperature increases from helium to nitrogen temperatures. This fact may be important for high-To superconducting ceramics, for which the Stekly parameter ct forj~ ~ 105-106 A cm-2 is of the same order of magnitude as at To = 4.2 K. Thus, the high-T~ superconductors ought to be much more stable against the local heat pulses than the conventional ones, while the cryostatic stability of these superconductors does not change so drastically.

References 1 Wilson, M.N. Superconductor magnets Oxford University Press, Oxford, UK (1983)Ch 5, 79-84 2 Gurevieh,A.VL,Mints, R.G. and Rakhmanov, A.L The Physics of Composite Superconductors Moscow,USSR(1987)Ch 5, 186-197(in Russian) 3 Gurevich, A.VI. and Mints, R.G. Self-heatingin normal metals and superconductors Rev Mod Phys (1987)59 941-999 4 Pasztor, G. and Schmidt, C. Dynamic stress effects in technical superconductors and the 'training' problem of superconducting magnets J Appl Phys (1978)49 886-894 5 Dresner, L. Quench energies of potted magnets IEEE Trans Magn (1985) 21 392-395 6 Gurevich, A.VI., Kazantsev, N.A. and Parizh, M.B. The normal zone dynamics and stability of superconductorsagainst local heat pulses Zh Tech Phiz (1983)53 1678-1680(in Russian) 7 MartineUi, A.P. and Wipf, S.L Investigation of cryogenicstability Proc 1972 Appl Superconductivity Con,/" IEEE, New York, USA (1972) 325-330 8 Wilson, M.N. and Iwasa, Y. Stability of superconductors against localized disturbances of limited magnitude Cryogenics (1978) 18 17-25