I

2109 IEEE TRANSACTIONS ON MAGNETICS, VOL. 25, NO. 2, MARCH 1989

INITIATION OF FLUX JUMP IN SC COMPOSITE BY HEAT PULSE E.Yu. Klimenko and N.N. Martovetsky

Kurchatov Institute of Atomic Energy, 123182 Moscow, USSR

Abstract -__--

Nonisothermal diffusion of magnetic flux after heat pulse shot on the surface of SC composite has been studied numerically taking into account smoothed transition characteristic of the superconductor. It is shown that for SC composite with poor stabilizatioh the current and heat redistribution changes significantly the estimations of stability based on steady state functions of heat generation and heat transfer. The critical pulsed energy strongly depends on initial current dist- ribution over the conductor cross section and the energy may be much less for a conductor with growing current than for a conductor with the same current in steady state. It has been found that undercritical heat pulses only slightly affect current density profde but stability increases as time delay increases between the current input halt and the pulse shot.

It has been found that for a SC composite with poor stabilization the transve- rsal thermal conductivity is more important than electrical resistivity of the mat- rix from the stability standpoint. The critical energy decreases when the thick- ness of SC filaments increases and this may explain unstable behaviour of the wires with thick fdaments.

Introduction ____-___ The theory of develo d on the basis of modelofcritical state3

(MCS) was modified in early 8@s4~'using more realistic description of smoothed transition characteristic of superconductor6 to the normal state. This small from the first sight correction resulted in principally new understanding concerning rea- sons for stability of non steady state stabilized SC composites.

the electrical field of taking off and level of permissible disturbances in a wire', relation between losses and stability in varying felds4, existence of critical rates of changing of external field or current chargin$, the superconductor is stable under these critical rates if there are no other disturbances. Some of these prin- cipal predictions were verified experimentally7>*.

Since far modified theory, taking into account real transition characteristics (RTC-theory) dealt with either small disturbances or with disturbances at uniform current distribution over the wire cross section. Yet most practically interesting are the cases where initial distributions are not uniform because these cases cor- respondto real conditions in a magnet.

Correct analysis requires dealing with thermal and magnetic diffusion because additional heat release caused by the processes may be higher than initial pulsed energy.

ssible only with RTC description of superconductor because for simplified MCS approach the current states with high current are unstable even against i n f ~ t e l y small disturbances4.

In our paper we studied numerically thermal and electromagnetic processes in a multifiiament wire at heat pulses imposed on its surface.

To understand the problem we are starting from MCS concept. There is no ot- her approach for spread disturbance along the wire but analysis of steady state functions of heat generation G and heat transfer Q. The functions are shown in Fig.1 for different stabilizer resistivity valuesThe current is assumed to be distri- buted uniformly.

G Q

Series of qualitatively new results were obtained such as the connection between

The formulation itself this problem for steady state unstabilized SC wire is po-

I / / I

Tb ' ci

Fig.1. Heat generation and heat transfer functions vs temperature

If cooling is good enough to be close to steady state stability (ai2-1, where a is Stekly pammeterj=I/I> then permissible overheating of the wire strongly de- pends on stabilizer resistivity. If cooling is poor, then matrix resistivity is insign- ificant and Tci (temperature of the begining of sharing) is upper limit of heating.

Manuscript received August 22, 1988

Meanwhile the practice of the SC magnets use shows that matrix material has a strong effect on magnet stability even if it operates in poor stabilization conditi- ons. For instance, attainable current in a magnet wound by SC composite with copper-nickel matrix was much less than in similar magnet with copper matrix composite9. Experiments "showed strong dependence of critical energy on mat- rix material. The results were tried to explain by longitudinal thermal conducti- vity and minimal propagation zone model2. But estimations of MPZ lenghts ap- peared to be near or less than wire diameter, therefore disturbances should be considered as long ones and longitudinal heat transfer might be neglected.

been developed. It makes diffcult to estimate quantatively the advantages o f . this or that composite design and to optimize it. For instance, what are quanta- tive advantages of the wire design with high resistive matrix and copper radial paths for heat removal in comparison with the similar composite without paths?

We have already shown" that the taking into account of the transversal dif- fusion of heat and magnetic field essentially changes understanding of the SC composite stability against heat disturbances. Using developed in paper" methods we consider some aspects of stability problem for a SC composite.

So, .nethods for analyses of influence of mairix resistivity on stability have not

Nonlinear diffusion of magnetic flux . . . . . . . . . . . . . . . . . . . . . . We shall deal with round SC composite neglecting influence of changing fded

on current distribution over the wire cross section (supposing the saturated zone due to changing field to be thin). Then we have equations:

(2) ar E) ar =p ai ~7 (1) C ??=Xi b(rgx)+ Ej+W a7 I ar ar

Where W - external heating, relations between electrical field for matrix material E=jstpst and for superconductor E=jqnexp(-Tc(B)/To+T/To+jJjo) where jo and

Current and temperature distributions previous to heat pulse has been calculated by numerical solution of equations (1-2) with boundary conditions:+l then ?=O and E' +, r=ro then X?=-h(T-Tb) andE'=jli/2rrro, where I i s current charging rate. Disturbance is described as an additional heating of a layer near the surface with a depth of 1 per cent of a radius. Time duration was choosen rather short which permits to have only total energy as a parameter of disturb- mce. Analysis is carried out in a framework of continuum model; current which flows through element of cross section of composite dS is a sum of current thro- ugh the superconductor qj dS and through the matrix (lq)js dS where q is a f&g factor. Effective trgmersal thermal conductivity X=Asdl-q)/(l~); heat capacity C = T $ ~ ~ I - ~ ) C ~ ~ After charging halt heat pulse is shooting immedeat- ly and boundary condition for electrical field on the surface is E'+.

I e"+e'/r=( 1 -q)aj/ a7 ( 1 a);

To are parameters of the transition smoothness. We neglected self field effect which is not essential for not very sharp twisting.

The problem was treated in dimensionless form:

(1 /( l-qH"+u'/r + ~ 4 1 +qXej+g)/( 1 -# (2a j

Scale for dimensionless current jc was choosen the one which makes zero an argument of exponential function in description of superconductor state at Tb. It should be kept in mind that maximal attainable current (current of taking off ItJis always some less than critical value in such determination5

Current charging rate was choosen rather high but less than critical one7, i.e. at this rate instabilities do not grow up to the taking off current in steady state.

Unfortunately in paper' dimensionless rate DIDT was rinted incorrectly. Right def i t ion is DIDT=pI/mpstjc and values in paper d e f ~ t i o n .

Analyses of stability at infinitely small disturbances in linear RTC-theory4 shows that differential resistivity of the superconductor p=aE/aj is very small up to electrical field of taking off if overheating is not considered. So, in linear ap- proximation instability

But we have shown' 'that if magnetic diffusion develops at nonisothermal conditions with growing temperature, then diffusion may take place with very high rate, because in this condition differential resistivity of the composite may be locally negative due to rapid temperature rise and this pecularity results in more uniform current distribution and higher stability than it is expected from 'frozen flux " approximation.

From the other side, boundary of the saturated zone where SC current flows can not move sharply inside at once because differential resistivity on the bound- ary is very low. These considerations show that simple model of frozen flux is

correspond to this

ows at unchanged current profile.

0018-9464/89/0300-2109$01 .WO 1989 IEEE

-1-

21 10

> w

C

-1 2 0.05

8 0.1

zi 3

not applicable for analysis of composite stability at strong disturbances and nu- merical analysis of this nonlinear process is needed.

Calculations which were carried out c o n f m s our qualitative expectations. Af- ter pulse heating we noticed intensive redistribution of current density in satu- rated region but if pulsed energy was smaller than critical one, even very close to it(95 per cent), dimension of saturated area was almost unchanged. Influence of matrix material on SC composite stability

q=3.5; S=(102; pn=6'1010hm'm; T,(B)= 6.5 K; C=1.5 J/m3K; Tb=4.2 K h=800 W/rnzK; pst changed from 210"o to 2*lOq O h m and thermal con- ductivity was determined by law ?~=2.45*10* T. Here S=jJjc =TJ(Tc(B)-Tb) is dimensionless parameter of transition smoothing.

Dependences of dimensionless critical pulsed energy ( scaled by Cnc(Tc-Tb) on current are shown in Fig.2 for different matrixes.

Dimensionless parameters used for numerical analyses correspond to ro=lmm;

1 1 I

-

p=2 w 9 ; X=5.14

-

2.10-*-

0 0.2 0 4 0.6 0.8

DIMENSIONLESS CURRENT, ]/Ito ( lt0=0.67 1, Fig.2. Critical energy vs current

Current charging rate corresponding to initial nonuniform current distribution was 'about 500 4/s

Critical energies in Fig.? may be compared with so called adiabatic energy for a composite with pst of 2.1040hm.m (shown in Fig.2) which is calculated by temperature of crossing heat generation and heat transfer functions in steady state. From this comparison it is seen that critical energies for a composite with nonuniform current distribution is essentially less than adiabatic energy where current distribution is assumed to be uniform". This energy difference, although not so great was seea in experiment^'^?^^ and was explained by additional ener- gy release whde redistribution occured. Source and value of the additional heat was determined by energy stored in ma etic profde. Such explaination seemed to be consistent with our calculationsl'where we saw that maximum stored energy in magnetic field profde coinsides with m e u m difference between pulsed critical energy for nonuniform distribution and adiabatic energy as cur- rent was increasel. However detailed study shows that noticeable changes of ma- gnetic field profde begins after heating the conductor above the equilibrium tem- perature which corresponds to the transport current, so main heat release is not a reason but mainly is a consequence of the transition.

Since both the fast diffusion of magnetic flux within the saturated zone with high electrical field and slow diffusion on the boundaly of the saturated zone depend on mainly properties of superconductor we expected small effect of el- ectrical resistivity of the matrix on the stability of the composite. As it is seen from Fig.? our expectations is valid in a region of pa of 210"o - 21O4 O h m ( @ a.676-67.6) but stability sharply decreases at pst=2.10* and fur- ther. Meanwhile magnetic diffusion at high matrix resistivities was qualitatively the same.

It is not seen another reasons for stability worsening but thermal conductivity. Rough appmximation of the process may be represented in frozen flux model, Le. we assume constant current distribution during heat process evolution.Hence, internal volume of the composite i s only a heat capacitor. In Fig.3 functions of heat generation for uniform current distribution are shown schematically. One can see that for nonuniform distribution overheating which result in irreversible transition to the normal state may be much less than for uniform one.

If heat conductivity of the composite is low enough and overheating of satu- rated zone exceeds equilibrium temperature while heat has not yet reached the internal zone, stability decreases sharply. Estimations made according such a picture show that there is critical value of heat conductivity and for case under consideration it is about 10 W/m'K which agrees with numerical result.

To verify that thermal conductivity is more responsible for stability than re- sistivity of the matrix wecalculated critical energy for a composite with electrical resistivity which corresponds to curve 2 in Fig.2 @=2'lO+Ohm.m) and thermal conductivity which corresponds to curve 3 in Fig.2 ( x 3 . 1 4 W/m.K).

Proximity of the result to curve 3 (see Fig.4) c o n f m s our qualitative conside-

ration.

Fig.3. Scetch of heat generation functions for uniform (1) and nonuniform (2) current distributions

So, for better performance of SC composite with high resistive matrix it is desirable to ensure radial channels with lllgh conductive material, usually cop- p r 1 4 and it promises to rise stability against heat pulses several times as much.

0.15 I 1

L I 0 0.2 0.4 0 6 0.8 l.o

DIMENSIONLESS CURRENT, I/Ito Fig.4. Critical energy vs current

Influence of initial current distribution on composite stability

I t is desirable to obtain quantative relations between level of nonuniformity of the initial current distribution in a SC composite and its stability. Nonuni- form distribution results from nonzero current charging rate or (and) changing external field. After the current charging halt, diffusion of magnetic flux takes place and distribution is getting more uniform. This circumstance was used for studying dependence of stability on initial current distribution.

The critical energy values were determined as a dependence on time interval between charging halt and begining of a pulse shot.

Result of the calculation is presented in Fig.5 for a composite with $=67.6 @=2.10' 0hm. r ) and at i=I/Ic (I/40@.5).

The dimensionless current charging rate was the same, as for previous study DIDT=10" . Dimensionless duration of a pulse was 0.15 which corresponds to lo4 s and it is much less than magnetic diffusion time.

In the inserts in Fig.5 the evolution of current distribution is presested as a function of time. I t should be noted that diffusion occurs much faster than in MCS model where diffusion through the ends of the composite is possible

/

only. Rather mall characteristic time of the diffusion makes it necessary to take

this circumstance into consideration at experimental study of stability on short samples. This behaviour allows to explain naturally well known fact of increasing magnet critical current when charging rate decreases.

One would expect12 that undercritical thermal disturbances should result in more uniform current distribution. But our calculations does not c o n f m it.

Even disturbances with 95 per cent of critical value does not result in appea- rent broadening saturated zone, though electrical fields have increased by few orders of magnitude simultaneously with heat pulse. We could see evolution of current profde which occured only in saturated zone where weak wave of cur- rent density ran from surface to inside and then relaxed near the boundary.

2111

0.4 a

0.3 's 8 3 Q

3 F4

0.2

0.1 I

i 4 .52 It0

/ 0 I

io5 io' lo9

DIMENSIONLESS TIME DELAY

Fig.5. Wire stability dependence on initial current distribution

Meanwhile current density near surface increased and full process looked like reflection of the wave of current density against boundary of saturated zone.

This qualitative picture is valid of course only if time of the existence of high electrical field is short. Long enough heating will result in uniform current distribution without any

pecularities. As transport current increases diffusion time decreases. For instance, at i=

0.55 which is 82 per cent of maximal taking off current, critical energy does not depend on time after 7 = 2.104 which corresponds to 12 s. After such a small delay stability is getting five time as much. So, current charging halt sometimes may be favorable for prevention of flux jumps initiated by small disturbances in magnets. In paper ''experimental investigation of small electri- cal fields originated previously to flux jumps is described, moreover, current charging halt permited to avoid development of flux jump.

energy does not reach adiabatic limit to about 10 per cent. Appearently the difference is due to limited transversal heat conductivity of the composite.

Influence of smoothness of the transition characteristic on SC composite

I t should be noticed that even at initial uniform current distribution critical

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stability _____

Analyses of SC composite stability which is not steady state stabilized against small disturbances shows that smoothness of transition characteristics is the main stabilizing factor which makes possible to use superconductors for large currents at poor cooling. The question arises - whether the optimal smoothness of the transition characteristic exists for given operation conditions?

We consider only one aspect of the problem, namely: if SC composite inten. ded to use at given charging rate and given amplitude (as in accelerator dipoles) is it reasonable to increase smoothness of transition characteristic for stability enhancement but sacrifiiing the current carring capacity of SC composite to some extent? This question is of practical importance, because the main trend in increasing

of current carring capacity is in enhancing of purity of Nb-Ti rods, in enhancing of geometrical perfection of fiaments etc., that actually increases current carry- ing capacity and simultaneously decreases the smoothness of the transition characteristic.

is appearent that the increasing of current carring capacity enhances stability even if smoothness does not grow, because it increases permissible overheating (the current sharing temperature increases).

production increment as a response to pulsed heating of SC composite with sharper transition may be much higher and this composite would be less stable in spite of its higher potential current carrying capasity in comparison with smoother transition composite.

To illustrate it we calculated stability against pulsed heating for two compo- sites with different smoothing parameters 6 and rest parameters and conditions ( jc/2; h; b S t ) were the same.

At uniform current distribution and high heat conductivity of the matrix it

But if current distribution is nonuniform due to f ~ t e charging rate, then heat

In Fig.6 results of the calculation are shown for two matrix materials with pst =2*10-'O Ohm'm and pst= 2'10- Ohm'm.

DIMENSIONLESS CURRENT, I/Ic

Fig.6. Critical e n e m dependence vs cment in a wire for different

2) 6 = 0.01, pSt =2.10-1° Ohmx~, DIDT=lO"; 3) 6= 0.02, pSt=2*1Oq sc composites: 1) 6= 0.02, pst=2.10-100hm'm, DIDT=~O-~ ;

Ohm, DIDT 4) 6=0.01, pst=2-10-7~hm.m, DIDT = io".

Different dimensionless current charging rates for different matrix resistivities

Voltage current characteristics in steady state for compared composites are shown in inserts of Fig.6.

I t is seen that in wide current interval the composite with smoother transi- tion is essentially more stable though its potential current carring capacity is lower.

sharp transition given current charging rate turned out to be higher than critical one and current attained at this charging rate for this composite is lower than could be attained in steady state (about 0.7 instead of 0.82 ).

enhancing of of homogenity of the SC composite results in enhancing of its performance.

Parameters used for curve 1 inFig6. correspond to multiGlamentary wire with copper matrix and ones for curve 4 may be considered as correspondent to monofdament wire with reduced current carrying capacity

Comparing the curves 1 and 4 we speculate that success in applied super- conductivity reached by transition from monofdament to mul t i f i i en t wires can be explained not by obtaining of internal stability in general i.e. not by avoiding spontaneous flux jumps without external disturbances, because it may be avoided by decreasing of charging rate or rate of external field change7 but mainly by great difference between critical energies for flux jump initiation for monofdament and multifiiament conductors.

Noticeably, the lower the smoothness the less critical energy initiating flux jump is.

It should be mentioned, that monofdament wires have even smaller parameters of smoothness than we used in our calculations (6 =jo/ jC/, = 0.01).

For instance, Nb-Zr monofdament wire7 has 6= 0.001 - 0.005 at B =I - 3 Teslas.

So, the use of multifiiament composites in practice of applied superconduc- tivity resulted in advance not only due to increasing of heat conductivity and electrical conductivity (in the normal state) but due to enhancing of smoothness of transition characteristic as well.

Numerical investigation of nonlinear diffusion of magnetic flux into the wire permited to obtain reasonable explaination of some experimental facts

correspond to the same actual current charging rates of about 500 A/s.

One can see as well that for the composite with high resistive matrix and

Conclusion which may be extracted from the Fig.6 is that not always the

2112

which could not be explained on MCS basis and even on linear RTC-theory.

our paper were always less than adiabatic energy, while experimentsl0'l2 gave values much more than adiabatic one even in quasiadiabatic conditions where transient heat transfer could not be involved.

solving the contradiction.

But we can not ignore the fact that critical energy of disturbances obtained in

I t seems doubtful that two dimensional approach may result in

We guess that in the experiments two sourses of errors may appeared : depen- dable calibration of pulsed heat absorbed by composite and wrong determination of current charing tem rature where smoothness must be taken into account.

One may that longitudinal heat removal is the main reason. For long length heaters the longitudinal heat transfer should become unim-

portant and accordance between calculated and measured values could have been reached but it is not seen in experiment". Moreover detailed taking into account of longitudinal heat transfer did not result in agreement yet. So verification of the results of our study is needed.

Conclusion

Numerical analysis of nonlinear Mus ion of magnetic flux into non stabilized steady state SC composite allowed to establish that: 1. SC composite stability at pulsed disturbances is ensured by smoothness of

transition characteristic. 2. Electrical resistivity of the matrix is not of great importance for stability. 3. Good thermal conductivity is essential but there is a limit exceeding of

which does not result in better stability. 4. Stability of the wire while current charging is lower than that after some

time delay at the same current. 5. Short undercritical pulses not always result in more uniform current distri-

bution over cross section. 6. Better homogenity of the wire and higher current carrying capacity does not

always mean better performance.

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