IEEE TRANSACTIONS ON MAG"lCS. VOL. 28, NO.1, JANUARY 1992 LIMITING CURRENTS IN SUPERCONDUCTING

VEKEILIN, V.R.ROMANOVSKI1

I. K h r c h a t o v Institute of Atomic Energy, USSR 123182, Moscow

1I.THE

77 1

COMPOSITES

PROBLEM STATEMENT

Abstract-The results of numerical and analytical calculations of the process of current charging into a round superconducting composite with properties homoge- nized over cross-section are presented.In the numerical solution taken was into account a common proceeding of the thermal and electromagnetic processes. A wire with real volt-ampere characteristics approximated by expo - nential dependence was considered. The calculations carried out at various rates of current charging, volt- ampere characteristics, matrix materials, heat transfer coefficients and other parameters showed: - the existence of characteristic limiting value of current below which the wire remains in a superconduc - ting s ta te if the current charging ceases and above which - changes into a normal state; this current is somewhat less than a quench current; - the existence of finite value for limiting current a t any low heat t ransfer from a surface.

The analytical solution of the problem is given. I t permitted t o write the stability criterion from which the dependence of limiting currents on initial parame - ters follows. The wire nonisothermality, i t s heat capa- city, thermal and electric conductivities are taken in- to account additionally, as compared to results pub - lished earlier.

I. INTRODUCTION

The current-carrying capacity of superconducting wire is one of its main characteristics necessary to design superconducting magnets. Therefore significant atten - tion is bein given to a study of this problem [1:6]. As it was siown in [l], the limiting current Im which can be input into a composite without breaking down its superconducting state decreases monotonically from a cntical value to zero at a increase of parameter

A=p0I1)! S/(4~hp% ) (1)

from zero to infinity. Here I - the current charging rate, Jc - the critical current density, 7) - the frac - tion of wire cross-section filled over a superconductor S - the wire cross-section area, h - the heat transfer coefficient, p - the cooled perimeter, T6 - the tempe - rature parameter of smoothness (T6= (TcB Top; 6 - the smoothness parameter of volt-am ere characteristics [2-51, T, B-the superconductor criticay temperature, To- the coolant temperature). The refined numerical calcu - lation of the current charging process [4] has been showed that at high rates of changes in current the wi- re heat capacity is essential too,

Conditions of the wire transition into a normal sta- te at an input with a constant rate were formulated in this paper by means of the numerical experiment. Limi - ting currents were calculated which do not lead to ir - reversible heating. Using the numerical results,an app- roximate equation was written making it possible to simplify calculations for limiting currents.

Manuscript received June 25, 1991

Let us determine distributions of temperature T, electric field E and current density J in a cooled round untwisted wire in self field while injecting in it a current with a preset rate. The continuous medium model and following equation system are used for calcu- lations in a simple case when T,EJ changes only along the radius

C a T - h a aT (2) at - i a h ) +

aJ = 1 a ( 8 5 , (3) Po at i a r r E

- T C B E = J,p,exp( Js + 7 ) = Jnpn, (3-1) s6 6

J = qJs + (1 - v&, 6 = +/< = T6/(TCB - To)

with initial and boundary conditions

T(r,O) =To, E(r,O) 4,

aT (0,t) = 0, E (0,t) = 0, (4)

h Tr + h(T - To) = 0, Tr = sio a at r = r d

where C,A are average values of specific heat and heat conductivity, r is the radius, t is the time, p is the resistivity.

The iven formulation allows the heat flow and electric f ield diffusion to be calculated inside the round wire assuming that the superconductor is uniform- ly distributed over the cross-section. Boundary condi - tions take into account a convective heat exchange and change in an electric field strength on the composite surface at varying the current charging rate. The volt- ampere characteristics was described by the exponential dependence [ 1-71. The physical and methodical problems of this form of a volt-current characteristics are dis- cussed in [5,7]According these papers critical current corresponds to zero argument in exponent in expression (3-1) at T=To, and 6 determinds a quantitatively a smoothness of the transition to the normal state. It should be noticed that so far obtained experimenal data are not sufficient to be sure of perfect validity of (3-1), . but now it is not more better model for a full domain of temperature and current.

To solve the problem formulated in (2-4) the finite difference method was used. A control of calculation accuracy was run under the condition of current conser- vation

r dI

The parameters of the finite-difference model were cho- sen so that the difference of the numerical integral value in the left side of equality (5) did not exceed 1% of the current's true value. The following parame - ters were taken in carrying-out the calculations

0018-9464/92$03.00 0 1992 IEEE

\nz C=103 J/(m3. K), ~ = 0 5 , J, 4 . 1 8 A/m2, ro =5.1C4m, ps =5.1(r7 n-m, T, 4 2 K, x B = 9 K, pn=2.10-'o Q-m, A=171 W/(m-K) - for copper matrix denoted as HC, pn=10-7Q*m, A=0343 W/(m*K) - for matrix of copper - nickel type, designated as LC.

(It should be noted that it was not necessary to con - sider non-linear heat conduction e uations which take into account the temperature depenlence of the thermo - physical y m e t e r s . The results obtained in this way can not c ange the main physical regulanties).

1II.THE NUMERICAL ANALYSIS RESULTS

Fig.1 depicts the time dependence of. composite surface tem rature. The calculation was carried out for "good congcting matrix HC at h = l W/m2/K, dI/dt=1000 A/s and various values for the smoothness parameter. It should be noted that during the entire current chargng pro - cess a temperature distribution inside the wire with " ood" conducting matrix depends rather weakly on ra - i u s . For composites with a "poorly" conducting matrix LC an insignificant non-uniformity of temperature field occurs directly before an irreversible transition into a normal state. Therefore the wire surface temperature chan e characterizes enouFh accurately its thermal sta- te. 6 e two kind calculations has been done:at a conti- nuous input and injection with stop. In the last case when currents Io indicated in the figure were reached in the wire it was assumed in the calculated model that dI dt=O. k e calculation results given in Fig.1 make it possib-

le to formulate a determination for a limiting current. The limiting current is stipulated by a limited possi - bility of heat removal released at a change in current. lThis results in the existence of finite value for a permitted heat release density, which depends on a rate of input, heat transfer coefficient and other thermal and electrophysical arameters. If it is exceeded an irreversible growth o! temperature and electric field takes lace despite a stop of input (the quench pro - cess). herefore the current value intermediate between the maximum stop current at which the transition does not yet occur and the minimum stop current at which the transition takes place, can be envlsaged as a limiting current at a given rate of input, For condiditions shown in Fig.1 the limiting current is preset between 350 A and 351 A for 6 =1% and between 301 A and 302 A for 6=2%. Fig2 illustrate calculation results of limiting cur -

rents normalized to the composite critical current (im=Im/r)/Jc/S) depending on a rate of input at various values for heat transfer coefficient and smoothness rameter for a wire with 'tell" conducting matrix. numerical calculation shows the presence of three typi- cal regions on curves im. First region: limiting cur - rent are maximum. It changes weak1 along with the rate of input. This is the region d full realization of current-carrying capacity. Its existence was previously observed in [MI. Its characteristic is the full fil - ling of a cross-section with current and the low over- heating with respect to the coolant temperature. As follows from the calculations, the limiting current for

ven parameters is described by the value suggested in E1

5-1,=302 A - 4

6-1,=301 A -

4 / I I I , 1 I I , 1 , I , , , I I I I I , , , , , , , , , , , , , , , , , , I , , , , , , , , , , , ,

Time,( s) 0.0 0.1 0.2 0.3 0.4 0.5

Fig.1. The change in the composite surface temperature in time at various conditions of input: 1,4 - the

uninterrupted input; 2$,5,6-the input with a stop.

10 100 1000 0 00 Current charging rate,( A/sg

Fig2 The limiting current de ndence on the rate of input at varioui heat t r a n s g coefficient ( 1-h = 3O00 W/m2/K, 2-h=100 W/m2/K, 3-h=0.1W/m2 K) and the smoothness parameter& 6=1%, - - - 6=2%), calculated numerical1 and with the aid of the approximated model 6)-(8)(- - g=l%).

The second characteristic region of im is the region of minimum current-carryin ca acity also de nds weak- ly on the rate of input. fn tiis case the &tine cur- rent different from zero is determined by an adiabatic heating of the wire and is characterized by a high overheating and a small depth of current penetration.. Let us note for the given region the difference from the results known earlierAccording to [l] the limiting current at dI/dt- is equal to zero (not taking into account heat capacity)A priori assumption of TcB as an upper boundary for permissible heating has led the authors of [4] to a nonmonotonic dependence of im which has not been observed a rigorous solution of the prob - lem.

In the transient re ion with the increase in the rate of input the depth of current penetration decreases and the permissible overheating increases.

Let us particularly emphasize the effect of nonlinea- rity of the volt-ampere characteristics on limiting current values. Calculations of im for various values

of 6 show that it negatively influences on im. This re- sult occurs most essentially on initial regions of im curves. These values, as was mentioned above, can be estimated from (6). In Fig3 corresponding dependence is presented for the HC wire. Values for limiting cur - rents calculated even by means of such a simplified mo- del, confirm the conclusion on deterioration of cur - rentcarrying capacity with the increase of the smooth- ness arameter 8 . From (6) it is not difficult to ob - tainef l i m imax+l that proves convincingly regu-

larity to use the critical state model in similar in - vesbgations and permits also to draw the conclusion on full stability of superconducting state of a composite when current is slowly input into it (dI/dt+O) up to a critical one if a nonlinear section on the volt-ampere characteristic is absent.

The substantiation of these results follows from curves of composite's temperature changes presented in Fig.1. It is seen that the increase of temperature over the entire process of input takes place more intensive- ly with wire havinfi greater values of 6. The cause of it is simple: additional heating due to nonlinearity of the volt-ampere characteristic in the low range of electric field.Precisely the constant total increase of temperature is observed at the growth of 6 and as a consequence - the decrease in the permissible overheat- ing and earlier break - down. The im ortance of the gi- ven conclusion should be emphasizecf specifically, since in [l-31 the proportional increase in the permissible overheatin with the growth of 6 is postulated. So the authors fl-31 have come to the wrong conclusion on the positive effect of the nonlinearity of volt-ampere cha- racteristics on limiting currents.

Fi A shows limiting current dependence on heat transgr coefficient for HC and LC wires with a fixed parameter of smoothness (6=1%).First of all let us note rather weak effect of the matrix material on the limi - ting current.The highest difference is observed in tho- se cases when the current did not yet fill the cross - section and the development of the quench process in a "bad conducting composite is affected by heat release in the current region due to 'a low thermal conductivity coefficient. Besides, it is seen from Fig.4 that owing to the above mentioned reasons the wire is stable at a low but finite heat transfer coefficient. For wire with an abrupt volt-ampere characteristics ( 6 4 ) this state- ment seems to be valid also for uncooled wires.

1V.THE APPROXIMATE ANALYTICAL MODEL The above analysis points out the significant role

which the change in the composite temperature played by in the development of instability. The calculation car- ried out was based on the numerical solution of the systems consistin of thermal conductivity and Maxwell equations. Therefore the undertaken problem asked for, a development of a special program necessary for running computer calculations. The solution of initial problem by means of simple models not requiring clumsy calcula- tions is of obvious interest. Let us write an approxi - mate solution for the problem of the wire's supercon - ductin8 state stability while in utting a current ta - king into account the specific features mentioned abo - ve. Let us bear in mind the presence of maximum limi - ting current, the com osite heat capacity, the wire permissible temperatvre &pending on the depth of cur - rent penetration.

Let us integrate equation (2) in the plane of cross - section. Takin into account the boundary condition (4) let us write &e heat balance equation in the integral form

6+0

C l S F dS = -k - To)dp + EJdS . I,

773

.3 3

E 0.0

0.00 Smoothness parameter, 6

Fig3 The dependence of maximum limiting current on smoothness parameter at dI/dt+O.

c, 0.8

5 :: 0.6

0

w E

.r(

4 0.2

h

Heat transfer coefficient,(W/m2/K) Fig.4. The limiting current dependence on heat transfer

coefficient for composites with HC (- - -) and LC (-) matrix: 1-dI/dt =lOOA/s, 2-dI/dt=1000 A/s.

\ 1 -h=3000 W/m2/K

2 - h = 01 W/m2/K

- - - - _ 0.0 Lq , l l l i l l l , I 8 1 1 1 1 1 1 1 I ,,>:,,-, 1 1 1 1 1 1 # , T l l l l l l i , I

0.1 1 10 100 1000 10000 100000 D i m e n s i o n l e s s parameter, A

FigS. The comparison of estimated values for limiting currents, calculated by various models: - nume- merical calculation, --- * --- a proximate model (7)-(8), - - - calculation by the mofel, suggested in [l].

774 Based on it let us formulate the condition for the wire superconductivity conservation : heat released at an input of current causing a change in enthalpy and remo- ved to a cooling agent, should not result in heating up higher than the permissible level. For a current-carry- ing element having insignificant temperature field non- uniformity in the cross-section the boundary of stable states is described by the equation

where At, AT - are the permissible time of input and heating corresponding to it. To calculate Ohmic heating let us use the approximate formula

f EJdS N T ) Jc(l - Aj)J EdS S S

in which the electric field inside the round wire is determined similarly to [l]

Here A is some unknown value taking into account the decrease in the superconductor current-carrying capaci-

at the cost of the volt-ampere characteristics non - fnearity; d is the depth of , electric field penetration inside the composite, related in accordance with the critical state model, with the input current value by the equality

J

1

Im = 2nr) Jc(l - 4)rdr d

Assuming At = I m b and performing a necessary in - tegration, let us write the system

4rhpAT[ 1 + C i / (hpnJc L i ] +Un(d/ro)+ 1 -(d/rof = 0

C1,nJc S ( 1 - 4 ) i

l m ' = ( 1 - A, ) [ 1 - (d/ro)'] .

Adding to it the limiting transition l i m im-imax and d- 0 ~~

making some transformations let us reduce it to a transcendental equation with respect to the sought va - lue for limiting current

i C I im + i m h (1 - T~ ) + AT[ 1 + - 3 = 0, (7)

max A 5 hpT)J,i

where A is calculated from (1).

to the equation suggested in [l]. But as is seen from the numerical analysis results these assumption can significantly change im. Therefore their allowance has resulted in an appearance in (7) of additional di - mensionless parameters among which AT/TS according to the above-cited results, can differ markedly from 1. To estimate ATITS let us use the numerical results. At l%SSS5% we have

The deduced equation at imax-l, C-0, AT-T, changes

AT 3 5

T - = 1 + (imx/ im) - 32/(1 + im/ imx) . (8) 6

Chain-dotted lines in Fig22 show imvalues calculated with the aid of (7)-(8). Dimensionless limiting cur - rents as function of A depicted in Fig5 as solid lines for the indicated heat transfer coefficient values were numerically calculated for the taken-above thermal and electro hysical parameters in varying the rate of input and g=l%. Here dotted lines show im(A) calculated at imx=l, C=O, AT=T,. It is not difficult to see that the suggested model describes satisfactorily the dependence of limiting current on initial parameters. Fig5 de - monstrates clearly the necessity of accounting the abo- ve-mentioned additional dimensionless parameters since the real limiting current is not an one-valued function of dimensionless complex A monotonically tending to zero at its unlimited increase.

V.CONCLUSIONS

1. The problem of current charging into a superconduc - ting composite has been solved numerically and analyti- cally, which has made it possible to refine the results of previous works and define more exactly to reveal li- mitation of criteria, based on the studies of stabili - ties to low disturbances. 2. It has been shown that the limiting current is the current at an excess of which the composite changes to a normal state even after an interruption of the input. 3. It has been ascertained that there exists a direct connection between the depth of current flow, preceding the quench and the permissible overheat temperature. In this case a significant overheating of the wire ( over than by 1K) can take place without distraction of su- perconductivity. 4. It has been found that at a constant value for Jcthe smoothed transition to the normal state deteriorates workability conditions for superconductors, reducing the limiting currents. 5. There exists the region of parameters (a rapid input, p""' cooling) for which an adiabatic release of Ohmic osses results in minimum but finite values for the li- miting current. It monotonically decreases at the increase of the rate of input, approaching the minimum value. REFERENCE

V.VAndrianov, V.P.Baev, S.S.Ivanov, R.G.Mints, A.L.Rakhmanov,"Superconducting current stability in composite superconductors",Cryogenics, v01.22, n.2,

E.!u.Klimenko, N.N.Martovetsky, S.I.Novikov,"Stabi- lity of Sc wires with real transition characteris - tics". DAN SSSR(Sov. Phvs. Dokladv). ~01.261. n.6.

p 81-87, 1982.

<,*

050-1354, 1981. Ef$u.Klimenko, N.N.Martovetsky, S.I.Novikov,"On the maximal current in Sc wire". DAN SSSR. ~01.282. - , n.5, pp.1123-1127, 1985.

E.Yu.Klimenko, N.N.Martovetsky, "Stability of Sc composite at rapid current charging and against pulsed heating", IEEE Trans. on Mag., vo1.24, n.2, p 1167-1169, 1988.

N.k.Martovetskv. "Some asDects of modern theorv of ap lied superionductivity", fEEE Trans. on Mhg.., v o h , n.2, pp.1692-1697, 1989.

M.Polak, I.Hlasnik, L.Krempasky, "Voltage-current characteristics of NbTi and Nb3Sn superconductors in flux creep region", Cryogenics, vo1.13, 11.12,

G.L.Dorofejev, A.B.Imenitov, E.Yu.Klimenko, "Volta- pp.701-711, 1973.

ge-current characteristics of type I11 superconduc- tors", Cryogenics, v01.20, n.6, pp.307-312, 1980.