Quench propagation in high Tc superconductors

Quench propagation in high Tc superconductors R.H. Bellis* and Y. Iwasa Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 25 April 1993

The quenching process in composite high Tc and low Tc superconductors has been studied in detail. Specifically, thermal stability and quench protection issues were addressed to gain a better understanding of how high T¢ superconductors may be best used in superconducting magnet applications. A series of experiments has been performed to measure quench propagation velocities - a parameter that quantifies the quenching process - along short samples (~12cm) of both low Tc (niobium tin) and high Tc [BiPbSrCaCuO(2223)] composite superconductors as a function of operating temperature, transport current and background magnetic induction. A computer simulation has also been developed to predict the behaviour of these conductors during a quench. The simulation includes the temperature and magnetic field dependence of material properties and generates temperature profiles that are marched out in time. Simulated voltage traces are also generated for comparison with the experimentally recorded voltage traces. The assumption that quench propagation velocities are constant, generally used in quench analysis for low Tc conductors, is also reasonable for high Tc conductors and thus, velocities may still be experimentally extracted from high T¢conductors. In general, quench propagation velocities are two or three orders of magnitude slower in high Tc conductors than in low Tc conductors. This indicates that magnets wound with these conductors are unlikely to be self-protecting. For the same reasons that they are difficult to protect, they are likely to be much more stable in response to thermal disturbances than their low Tc counterparts.

Keywords: 7"=superconductors; quench propagation velocity, computer simulation

The advent of high Tc superconductors has greatly intensified the study of applied superconductivity. In recent years, much effort has been expended to successfully transform these promising raw superconducting materials into practical magnet conductors. While much progress is being made in this area, however, few experimental and/or analytical studies have been reported regarding the thermal behaviour of these materials during a quench. Without sufficient understanding of this behaviour, magnet designers will not be able to determine the appropriate quench protection scheme to employ. This paper investigates the issue of quench propagation in high T¢ superconductors, both experimentally and numerically by computer simulation, and compares the quenching process in these conductors with that in low Tc superconductors.

*Present address: Division of Naval Reactors, US Department of Energy, Washington, DC 20585, USA 0011-2275/941020129-16 © 1994 Butterworth-Heinemann Ltd

Quenching process in composite conductors Basic governing power density equation

The general one-dimensional differential equation that governs the quenching process in a composite conductor (superconductor and matrix) metal) is given by Ca,g(T) - - = at

ka,s(T

"l-gdist(l )

+pm(T)JcdJm(T)

h(T)P (T- Too) (I) A -

-

where Tis temperature, t is time and x is the spatial coordinate. The left-hand side of Equation (1) represents the time rate of change of internal energy in the section of superconductor. C,,,.(T) is the volumetrically averaged heat capacity. ~he first term on the righthand side represents thermal conduction along the superconductor, kavs (T) is the volumetrically averaged thermal conductivity, pm(T)J,~J,,,(T) represents the

Cryogenics 1994 Volume 34 Number 2 129

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa FuRySuper~ Current

Joule heat generation. When the conductor is superconducting, this term is zero. Pm (T) is the electrical resistivity of the matrix and Jcd is the overall conductor current density, defined as the transport current It, divided by the cross-sectional area of the conductor A. Jm(T) is the matrix current density and is determined by the critical properties of the superconductor which are, in turn, a function of temperature. The next term represents external heat sources such as conductor motion or epoxy cracking. The last term represents convective cooling; h(T) is the heat transfer coefficient, P is the perimeter of the superconductor exposed to the coolant and T® is the ambient temperature. The heat capacity and thermal conductivity are volumetrically averaged as follows because of the composite nature of all practical conductors Ca,,g (T) =

fCm(T )+ (1 --f) Csc(T )

kavg(r) = fkm (r) + (1 - f ) k~c(r)

where the subscripts avg, m and sc stand for average, matrix and superconductor, respectively, f i s defined as Am~A, where Am is the cross-sectional area of the matrix. Note that in general f is a volume ratio. In this case, however, it may be written as an area ratio since it is assumed that the matrix/superconductor ratio is independent of length. Also, in general, conduction occurs transverse to the conductor as well as along it. The above averaged value for kavg is applicable only to the one-dimensional case. In practice, kavg may be approximated as kavg(T)~fkm(T) because km(T)~' ksc(T) for most composite conductors.

Current sharing and composite generation An important complication introduced into the quenching process because of the composite nature of magnet conductors is the existence of the current sharing state between the fully superconducting and fully normal states. Since the conductor transport current cannot shift abruptly from the superconductor to the matrix during a quench, there exists a finite length of conductor where current is flowing in both regions, as shown in Figure 1. This region is referred to as the current sharing region. The length of the region is dependent on two factors: 1, the temperature gradient along the conductor, and 2, the absolute temperature range over which current sharing is possible for that particular superconductor. This second factor is in turn dependent on the slope of the critical current versus temperature curve for the given superconductor. These factors will be discussed in more detail below. For now it is sufficient to acknowledge that, since the slope of the lc versus T curve is always negative, regardless of the particular superconductor, all composite conductors undergoing a quench will experience current sharing over at least a small length of conductor. As can be seen in Figure 1, as long as the conductor temperature is below T~, the current sharing temperature, the conductor will be fully superconducting and there will be no generation since none of the transport current It is flowing in the matrix. At Tcs, It is exactly equal to the conductor critical current corresponding to Tcs, lc(T~). Above Tcs, the superconductor carries

130 Cryogenics 1994 Volume 34 Number 2

Nomml

&

/, i !

A. ,,

/

i ,/ 0

Top

(2)

(3)

Fully

To,

Tc

T

Figure 1 Upper plot shows a critical current versus temperature curve at a given magnetic induction, approximated by a straight line. Lower plots show I=c(T) ( ) and Ira(T) ( - - - ) curves as conductor temperature is increased from Top to a temperature above Tc It(T), which is less than It and decreases with temperature. The excess current flows through the matrix. Thus as T is increased above Tcs, the superconductor current I~:(T) must decrease and the matrix current Ira(T) must increase. At the critical temperature Tc and beyond, for cases when Am is not negligible compared with A, virtually all of the transport current flows in the matrix because the matrix resistivity is much smaller than the superconductor resistivity: pm(T>~Tc)'¢ p,c(T~ Tc)- Thus, for temperatures above T~, Ira= It and l,c ~ 0. The effect of current sharing is most prominent on the generation term in Equation (1). In non-composite conductors, as soon as the temperature rises past the transition temperature, Tt where the I~ versus T curve for the given superconductor is crossed, generation G,~ will jump from zero to a constant value, Gc = pJ2cd. That is 0

Gnc(T)

= G¢ =pJ2cd

T < Tt (It)

T~ T,(I,)

(4)

Gc depends only on the transport current It, which is assumed to be fixed. This is shown by the solid line in the lower plot in Figure2. The generation term is not nearly as simple in the composite case because of current sharing. A better understanding may be obtained by looking at the origin of the generation term. O n the simplest level, the generation power density G(T) is simply the transport current density Jcd = It/A multiplied by the electric field E

G(T)= JedE= (~) dt,(T) dx

(5)

where d~,(T) is the temperature dependent voltage across the differential length dx. This voltage may be computed by multiplying Ira(T) by the matrix resistance dRm (T)

dr(T) =

Ira(T)dRm(T) = I~(T) ×

pro(T)dx Am

(6)

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa

~(T)

only for very low temperatures (below ~-25 K) where the matrix resistivity Pm is independent of temperature. Since low Tc superconductors operate in this temperature range, this is a valid assumption. However, it is not a good assumption when working with high T¢ superconductors that could easily be operated at temperatures above 25 K. Indeed, special care must be taken when analysing high T¢ superconductors, as discussed below.

G(T)

Comparison of low Tc and high Tc superconductors

G~

0

Top

Tt ,T~,

T¢

T

Rgure 2 For same I.(T) curve given in Figure 1, power generation G(T) Ilower plots) behaves differently for noncomposite ( ) and composite ( - - - ) superconductors. Note that for the non-composite there is no Tea and the transition occurs at Tt. G= is the critical generation; Gc is constant for temperatures below ==25 K where the matrix resistivity pm is independent of temperature

where Am(=fm) is the matrix cross-section area. Combining Equation (6) with Equation (5) gives

G(T)=pm(,T,) rIt]rIm(r)~ ~ j [ - - ~ j=Pm(T)J~Jm(T)

(7)

Equation (7) is the most general form of the generation term for composite conductors. Equation (7) may be simplified somewhat by introducing a linear approximation of the lc versus T curve, as done in Figure 1. Thus

Io l¢(r) = - -- r + lo

where Io is the critical current at zero temperature. This approximation may then be used to obtain an expression for the matrix current lm

f O

(9)

G(T)

T>~ Tc

The value of T~ is dependent on transport current It. Incorporating this approximation into Equation (7), the generation term may be expressed as

T<-Tcs

O

I

h

T<~T~

/t X T-T-

It

p~(T)J~d

G(T) =

~(T)

(8)

r,

Ira(T) =

Because the thermal behaviour of superconductors is controlled by the heat generation term, which is a strong function of temperature, the behaviours of low T¢ and high T¢ superconductors during a quench are quite different, chiefly because relevant material properties are greatly temperature dependent over the temperature range of interest, 2 ~ 100 K. Between 4.2 and 77 K, the specific heat of a typical matrix metal increases by three orders of magnitude. Thus, much more energy is required to raise the temperature by a given amount for a superconductor operating at 77 K than that required for a superconductor operating at 4.2K. It may be inferred that, if high T¢ superconductors are operated at such elevated temperatures, they will be inherently much more stable than their low Tc counterparts 1. A second notable difference occurs in matrix resistivity Pm because of its characteristic dependence on temperature over this temperature range. Below~-25 K, pm is practically constant; above ---25K it varies roughly linearly with temperature. This variation affects the generation term, as shown in Figure 3. Generation for a low Tc composite, operating below

f

p,.(r)]~ f

x

r- r.

T~- T~

Tcs~< T~< Tc

(I0)

T>~Tc

This linear approximation of the G(T) curve is shown by the dashed line in the lower plot in Figure 2. It should be noted that the plots in the figure are valid

I.,o~-Te

/

Top T~oT¢

~/~/High-T~

T~o

T~

T

Rgure 3 Upper figure shows critical current versus temperature plots for low T© conductor and high Tc conductor. The corresponding Joule heat generation versus temperature plots for the two conductors ere shown in the lower figure. Note that the heat generation curve for the high Tc composite takes into account the fact that the matrix resistivity increases with temperature above - 1 5 K (silver)

Cryogenics 1994 Volume 34 Number

2

131

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa 25K, will be linear with temperature while in the current sharing regime. Above Tc, it will be constant up to ~--25K. The generation in a high T¢ superconductor operating above 25 K will vary as T 2 in the current sharing regime because both p,, and Jm vary linearly with temperature in this range. Above To, generation will vary approximately linearly with temperature. A final significant difference between low T¢ and high Tc superconductors can be seen in the slopes of the respective I~ versus T curves, as shown in Figure3. The slope of the curve for high T¢ superconductors is much, much smaller in magnitude than that of low T¢ superconductors. This results in a significant difference in the size of the temperature range, T¢-T~, over which current sharing occurs. Thus because Tcs and T~ are separated generally by only a few kelvin in low Tc conductors, the spatial span of the current sharing region is negligible and can effectively be ignored. The conductor essentially jumps from the fully superconducting state to the fully normal state. The current sharing range cannot be ignored in high Tc conductors, however, because of the large value of Tc-T,~. Indeed, the current sharing region can extend over much of the length of the conductor. As discussed below, current sharing has a significant effect on the quenching process and its analysis. Further, since current sharing occurs over such a large temperature range in high Tc conductors, their behaviour during a quench is markedly different from that of low T¢ conductors. The implications of these observations on superconducting magnet design are discussed in the next section.

Quench propagation velocity The quench propagation velocity is a useful parameter that magnet designers use to quantify the natural ability of a superconductor to protect itself from quenchinduced damage. It is the speed with which the normal zone grows in size, as driven by thermal conduction. More specifically, it is the speed with which the normal/ superconducting boundary propagates further into the superconducting region, after all transient effects have disappeared. Thus it is a measure of the speed with which a travelling temperature wave carries the Joule heat generation forward along the conductor. In general, a high velocity implies that the quench volume grows rapidly which enables the stored electromagnetic energy to be dissipated over a larger volume and thus reduces the risk of quench-induced damage. The general approach for obtaining the quench propagation velocity is to assume that it has reached a steady state value and to use a translating coordinate system2-5. By doing so the partial differential equation [Equation (1)] becomes an ordinary differential equation having a one-dimensional travelling waveform solution. To do so, the transformation z = x - U t t must be applied, where Ue is the longitudinal quench propagation velocity and z is the new translating spatial co-ordinate; z = 0 corresponds to the normal/ superconducting boundary. The resulting ordinary differential equation can be solved to give an explicit expression for the quench propagation velocity Ue if adiabatic conditions are

132

Cryogenics 1994 Volume 34 Number 2

assumed (h = 0 and gdist (t) = 0) 5

Ue=Jca p(T,)lcn(T,) C"(TO k,(T,) dT T=T,

The subscripts s and n correspond, respectively, to the superconducting state and normal state. For constant material properties, Equation (11) may be simplified to

~. Ue=J~

pk, C,,Cs(Tt-Top)

(12)

This expression is valid only for unclad superconductors in an adiabatic environment and thus is quite limited in its usefulness. In composite conductors, however, the normal/ superconducting boundary is ill-defined because of current sharing. There is no distinct transition temperature Tt that can be used to define the z = 0 co-ordinate. To properly account for current sharing, we must have a third equation governing the heat balance in this region in addition to the equations for the superconducting and normal regions. The resulting three equations are coupled and may be reduced to two simultaneous equations for the propagation velocity and the length of the current sharing region by applying heat flow continuity across both the superconducting/ current sharing interface and the current sharing/ normal interface. Unfortunately, because of the nonlinearity of the current sharing term, no general solution may be obtained. Nevertheless, by using an approximation for the heat generation term, Equation (11) may be applied to composite conductors with reasonably good accuracy 6. If the composite generation term is approximated as a linear function of temperature in the current sharing region, then the average generation t~ in this range may be written as G = Gc/2. This same generation can be obtained if half of the current sharing region is lumped with the normal region and the other half is lumped with the superconducting region. An imaginary superconducting/normal boundary can thus be defined by 1 Tt = Tcs + ~ (To - Tcs)

(13)

The generation term may thus be approximated as a step function that is zero for T < Tt and equal to Gc for T > T~. This is shown in Figure 4. It has been verified experimentally that, by replacing Tt in Equation (11) with Equation (13), reasonable predictions of quench propagation velocities in low Tc composite conductors can be obtained 5-a. This approximation is reliable for low T¢ superconductors because the length of the current sharing region is small. This is so for three reasons: 1, the temperature range T,-Tcs over which current sharing occurs is small; 2, the matrix resistivity at low operating temperatures is independent of temperature; and 3, propagation velocities are relatively fast. Since these conditions cannot be met in composite

Quench propagation in high Tc superconductors: R.H. Be//is and Y. /wasa I

=

It(T)

X

Normal

Ctmmt

Supetcondu~ag

Sbarmg I

.

T,

:=::--[.Tit

...................l ....... \

Step Function ~ Approximation

G'On x'Nk ~ ]~

Actual Composite GenerationCurve ,

0 T~

0

To~ Tc, T~, + A T

z

Tel ................ i .............................

high T¢ superconductors, however, the length of conductor over which current sharing occurs is quite large. Thus there is no approximation that can accurately account for the current sharing effect in high Tc superconductors because there is simply no adequate way to define the normal/superconducting boundary z = 0. This effect, combined with a temperature dependent matrix resistivity, makes the high T¢ generation term (shown in Figure 3) too complicated to be approximated as a simple on/off function as can be done for composite low Tc conductors. These observations indicate that it is impossible to satisfactorily write an approximate, explicit expression for the propagation velocity in high T¢ superconductors. Thus the only analytical prediction tool remaining in the high T¢ case is to numerically solve the governing partial differential equation (PDE) by computer. (It is also possible to numerically solve the simultaneous ordinary differential equations obtained using the transformation above but, by solving the PDE, one gets much more information for roughly the same effort since the PDE solution produces temperature profiles that contain much more than just velocities.) Although no adequate expression for the quench propagation yelocity may be obtained for high Tc superconductors, an observation may be made regarding quench propagation in such conductors. From Equation (12) one can see that the quench propagation velocity is inversely proportional to specific heat and directly proportional to the square root of heat generation. 1

C(T)

O~/-O (T)

T

•

Rgum 4 Schematic showing configuration used to determine transition temperature Tt in low Tc composite conductors for use in computing the quench propagation velocity. The actual composite generation curve is represented by a linear approximation s

U< oc ~

.

Figure S Plot showing comparison of amount of current flowing in matrix in low Tc and high Tc conductors in response to a given increase in temperature ATabove T=. Note that It for each conductor is selected not only to be identical but also to make T= for each conductor identical. Im for each conductor represents current that flows through the matrix. Clearly Im for low Tc conductors is much greater than Im for high Tc conductors

gatttr¢

i

Top_

i\L,-.--- .r<(~ (I~-T<)

(14)

One therefore expects normal zone propagation to be much slower in high Tc conductors than in low T~

conductors1'9A°. This is so for two reasons. First, the specific heat is much greater because of the higher operating temperature. A higher specific heat means that more energy must be absorbed before the temperature may be raised in a section of superconductor. Thus, less energy will be available to drive the propagation. Second, for a given increase in temperature AT of a section of composite superconductor, low Tc conductors will have significantly more current flowing in the matrix, which determines heat generation, than will high Tc conductors because of their steeper Ic versus T curves. This effect can be clearly seen in Figure 5 where Im for low Tc conductors is much larger than that for high T~ conductors. Thus for a unit increase in temperature AT much less heat will be generated in high Tc conductors. The effect on generation is shown clearly in Figure 3. The inherent slowness of propagation velocities in high Tc conductors implies that it is not possible to rely on normal zone propagation to spread out the normal region to achieve nearly uniform dissipation within the magnet winding. The dissipation will instead occur over a small fraction of the winding volume, possibly subjecting the coil to extreme temperatures and/or voltages. One may therefore observe that magnets wound with high Tc conductors are unlikely to be selfprotecting and will thus require protection schemes. Experimental measurement propagation velocities

of quench

In this section, the experimental method used to study the quenching process in both low Tc and high Tc superconductors is described. In particular, normal zone propagation was studied in short lengths of conductor (--10 cm) as a function of operating temperature, transport current and background magnetic induction. The two materials studied were niobium tin (Nb3Sn) and BiPbSrCaCuO(2223), henceforth referred to as BPSCCO. The parameter of interest was the quench propagation velocity Ue. Quench propagation velocity measurement was first performed on Nb3Sn tapes. The test rig built for these

Cryogenics 1994 V o l u m e 34 N u m b e r 2

133

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa Voltage t a p s

BPSCCO Suoerconductor

Current [

G-10 Support Blocks Figure 6 Schematic of BPSCCO sample holder used for quench propagation velocity measurement. /)tot,l, /)1, /)2 and 1)3 are voltages recorded in the experiment

measurements is described elsewhere ]1. It uses a circular tape configuration that allows experiments to be performed in small bore, high field magnets. For BPSCCO tapes, a test rig accommodating ---10cm lengths of straight, fiat tape conductor was employed; it is described below.

Experimental set-up A schematic of the BPSCCO sample holder is shown in Figure 6. The sample holder consists of a BPSCCO tape --10era long, a dummy copper tape, two current leads with copper blocks at their ends, an additional small copper block and a three-piece G-10 support. The two tapes are sandwiched between the G-10 pieces and are soldered to the copper blocks. A heater is mounted at the far end of the BPSCCO tape and four voltage taps are equally spaced along the length of the BPSCCO tape. The heater is a strain gauge epoxied to the tape to provide good thermal contact. To prevent strain damage in the tape at the voltage tap and heater locations due to uneven solder connections, a thin sheet of G-10 with holes at these locations is placed between the sections of the G-10 support (this sheet is not shown in Figure 6). The holder is mounted between two 6mm thick, 12.5cm diameter copper plates and placed inside a cryostat. While the holder remains in gas, the copper plates are thermally linked by copper pipes to a bath of liquid helium located below the bottom plate. Two carbon-glass resistor temperature sensors monitor the temperatures of the copper plates. One sensor is mounted to each plate. The initial temperature of the superconductor tape is inferred from the temperature of the plates since the sample is thermally anchored to the plates via the current leads. The average temperature of the two plates is assigned to the sample.

Experimental procedure The basic procedure for velocity measurement of both Nb3Sn and BPSCCO tapes is as follows. A transport current It is supplied to the tape through the current leads from the external power supply. A voltage pulse of known amplitude and duration is then applied to the heater at the far end of the tape. If the

134

Cryogenics 1994 V o l u m e 34 N u m b e r 2

heater pulse is large enough to overpower any cooling effects, a quench will ensue. As the quench progresses, the normal zone will grow in size and increase in temperature. Its rate of growth, and thus the normal zone propagation velocity, may be measured by monitoring the voltages across the voltage taps. As soon as the normal zone grows past one of the taps, voltage will appear across the set. This voltage will continue growing until the normal zone has propagated past the second tap in the pair. This process occurs for each of the three sets of taps. The quench propagation velocity may be obtained by dividing the known tap separation by the rise time required for the voltage across a set of taps to rise from zero to a steady state value. Unfortunately, this velocity measurement technique did not work as well for BPSCCO tapes as it did for NbaSn tapes. There are several reasons for this. First, the heat generation term is not a simple step function of temperature in high T¢ conductors; rather it is nonlinear and varies continuously with temperature, as noted above. Second, the velocities were so slow that the temperature of the normal zone could not be modelled as constant. This would not be a problem except for the fact that the BPSCCO experiments were often started at temperatures where the matrix resistivity was a function of temperature, i.e. above ~15 K for the high purity silver matrix. Thus the voltage across a given set of taps would rise without bound, rather than levelling off. (Even if experiments were run at temperatures where resistivity was constant, the voltage traces would still rise without bound because of the non-linear generation term). Finally, the extremely slow velocities made is impossible to ignore the end effects of the copper block at each end of the tape. Although only a limited set of voltage traces could be recorded with BPSCCO tapes, much meaningful data were obtained regarding the quenching process in high T¢ superconductors. Table 1 presents pertinent parameters of the tapes used in this experiment. Computer

simulation

As discussed previously, the quenching process in all superconductors is governed by a non-linear partial Table 1 Parameters of short (==12cm) sample Nb3Sn and BPSCCO superconducting tapes

Parameter

Nb3Sn

BPSCCO

Overall tape thickness (cm) Overall tape width (cm) Overall tape cross-sectional area, A (cm 2) Superconductor area (cm z) Matrix (copper or silver) area, Am (cm 2)

0.0119 0.6350 0.0075

0.0173 0.3980 0.0063

0.0005 0.0032"

0.0019 0.0044

f = AmlA

0.421 2 698 18

0.695 1 or 2.5 228 106

Voltage tap separation (cm) Io = Ic(4.2 K, 0T) b (A) Nominal Tc(0A, 0T) (K)

"For Nb3Sn, the matrix consists of copper, stainless steel, unreacted niobium end solder. In this case the matrix area is assumed to be the copper cross-sectional area only bDefined at an electric field of ==0.1/~Vcm-1

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa differential equation, or PDE [Equation (1)]. Typically, analysis of the quenching process is simplified by assuming that, at steady state, a quench propagates at constant velocity2-5. This constant velocity assumption with a co-ordinate transformation results in an explicit expression for the propagation velocity. Unfortunately, such a simplification, while valid, does not help in the analysis of high Tc conductors because of the large effect of current sharing, as discussed previously. Therefore, to gain any insight into the quenching process in high T, conductors, one must analyse the full non-linear PDE. Such a complicated equation cannot be solved analytically and must be discretized and solved numerically by computer. A computer program written for this purpose is described below.

Basic algorithm The basic idea behind the simulation code is to solve a discretized version of Equation (1) for one-dimensional temperature profiles at discrete instants in time ti: T(x, t~). The numerical method employed is adapted from the methods presented by Patankar 12. To begin, the tape is discretized into approximately 1000 differential elements dx. Within each element, the temperature is assumed to be a function of time only. Thus at any given instant in time, the temperature of a given element is constant. This means that the temperature dependent material properties are also constant. Within each element, Equation (1) therefore reduces to a linear ordinary differential equation. By simultaneously solving for the temperature of each element, a temperature profile for that given instant in time may be obtained, T(x, ti). The basic algorithm is as follows. First, an initial constant temperature profile is assumed. Using this profile, the material properties are evaluated within each discrete element along the tape. These properties include thermal conductivity, specific heat and resistivity. In addition, the amount of current in the matrix, and thus the amount of heat generation, at each location is temperature dependent and is therefore evaluated in the same manner. Second, based upon these material property 'profiles', the interconnected discrete equations for each location along the tape are solved simultaneously using the tri-diagonal matrix algorithm (Thomas algorithm), resulting in a new temperature profile for the next time step. This new temperature profile is then used to update the material properties in each element. The loop then repeats itself, marching out the temperature profile in time. The program outputs data at selected multiples of the time increment into two files - one for the temperature profile T(x, t~) and the other for the voltage traces ~,;(t). These'files are then loaded into Matlab .(a matrix operation/graphics software utility) to generate the plots shown here.

Modelling assumptions Piecewise-linear approximations to material property versus temperature curves are built into the program. Thermal conductivity and specific heat values were volumetrically averaged, as discussed previously. To

simulate runs in a background magnetic field, the matrix electrical resistivity and thermal conductivity curves had to be modified to account for. magnetic field effects. Magnetoresistive effects were accounted for by incorporating Kohler's rule which relates the change in resistivity Ap to the magnetic induction Bo. The mathematical form of the rule is

Ap(T, Bo) = f u n c t i o n [ Bpo~T)] (15) po(T) where po(T) is the zero-field resistivity at temperature T and Ap(T, Bo)/po(T) monotonically increases with Bo/Po(T) (reference 13). Kohler plots for copper and silver used in the present study are based on measured magnetoresistivity data 14'15. To obtain a first order approximation of the effects of magnetic fields on thermal conductivity, it was assumed that the Wiedemann-Franz-Lorenz law for metals is valid in magnetic fields and at all temperatures of interest. (The validity of the law in a magnetic field is disputed in the literature; see, for example, work by Arenz 16 and Fevrierl7.) The Wiedemann-Franz-Lorenz law is given by A=

kp

(16) T where A is the Lorenz number, 2.443 x 10-sV2K -2 (reference 13). Therefore, the thermal conductivity in a field may be given by -

-

k(T, Bo) = [1 + 0.05Bo]

po(r) ko(r) p(T, Bo)

(17)

where ko (T) is the zero-field thermal conductivity at T. The factor 1 + 0.05 Bo accounts for a slight dependence of A on Bo (in tesla), as discussed by Arenz 16 (the dependence is roughly 5% per tesla). The simulation uses piecewise-linear approximations of the Nb3Sn critical current versus temperature curves for different background magnetic inductions Bo. The data for these curves were obtained from quench propagation velocity measurements by Bellis 11. For the BPSCCO conductor, simple linear It(T, Bo) curves were employed based on data obtained by Sato et al. ~s (see Figure 7). Using these curves and Equation (9), the current in the matrix lm could be determined as a

250 i

i

!

~0r

]

....91/~..................................................................... . . :.......... . . .~....~ : ~........ ~ .79..6............

i~il., T o

0

20

i

:............... ...............

i

40

60

80 .

.

.

.

.

.

.

.

.

.

.

.

.

.

100

r (K) Rllure 7 conductor measured

C r i t i c a l c u r r e n t properties of h y p o t h e t i c a l B P S C C O used in s i m u l a t i o n . Properties adapted from data of S a t o e t al. TM

Cryogenics 1994 Volume 34 Number 2 135

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa function of T. This, combined with the resistivity curve, is then used to calculate heat generation as a function of temperature using Equation (10). The boundary conditions at the ends of the tape, where the tape was soldered to the copper blocks (see Figure 6), are determined by a heat balance updated in each time step. Specifically, the endpoint temperature is set equal to the temperature of the corresponding copper block at that time step. The copper block temperature is determined by a heat balance unique for each end of the tape and for each particular test rig. Perhaps the greatest uncertainty in the simulation is modelling of cooling effects. Although both experiments were designed to minimize the influence of direct convective cooling on the tape, they are not strictly adiabatic. To account for cooling in the real experiments with the simplest possible model, the simulation incorporates a temperature dependent cooling term qcoot that lumps all cooling effects into one constant heat transfer coefficient he. Thus

q¢ool = h¢[T(x, ti)

-

-

Tinf]

and discussion

Quench analysis of low Tc (niobium tin) tapes The first case studied was that of the niobium tin tape in zero background field. Figure 8 shows an osciliogram of

t l

j

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I

m

//1 it,.,t

18 16 14 120

2

4

6

8

~(cm)

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This term is evaluated in the same manner as all of the material properties. One should note, however, that the simulation is applicable to one-dimensional situations only. It cannot be used to predict quench behaviour in magnet windings, whether low T¢ or high Tc. In higher dimensions other factors must be taken into account. First, heat conduction, and thus quench propagation, may occur transverse to the conductor into the next layer of the winding. Second, the thermal behaviour of the coil, expressed as a 2-D or 3-D equivalent of Equation (1), must be coupled to the electrical circuit characteristics of the coil. Several simulation studies have been developed to model these higher-dimensional phenomena 6-8"19'2°. However, these codes are based on the quench propagation velocity model and, as such, are probably not applicable to high Tc magnets. Results

21 20

,,

~'m

~A="

I ,.,,m

v

Figure 8 Typical oscillogram showing voltage traces for NbaSn tape. Experimental conditions: Top= 12K, I t = 2 2 5 A , B o = 0 T . Vertical scale: 1 mV/div; horizontal scale: 19.82ms/div. From top-left to bottom-right, the traces are vtotat, P~, v2 and ~3. Tap separation is 2cm, beginning 0.6cm from edge of heater. Heater pulse applied was 7.5W for 5 ms +~

136 Cryogenics 1994 Volume 34 Number 2

2 1 0

o

20

4o

6o so t (ms)

,

,

,

too

12o

14o

Figure 9 Simulated version of event shown in Figure 8, presenting temperature profiles and voltage traces. The lower horizontal line on the temperature profile is To== 13.7 K; the top line is Tc= 16.5K. Temperature profiles are 25ms apart, hc= 24,5 mW cm -2 K-1. The voltage traces, from top-left to bottomright, are Vtot.~, P1, v2 and ~3. Tap separation is 2cm, beginning at x = 1.5cm. Heater pulse was 3 W for 2ms

voltage traces (Vtotal, vt, v2 and v3 of Figure 6) recorded during quenching at Top= 12K with It = 225 A. Note that each voltage levels off after a certain period of time. This indicates two things. First, the copper matrix resistivity is constant with temperature, implying that the peak temperature remained below ~25K. Second, there is very little current sharing because the amount of current in the matrix must be constant if the voltage is constant. In turn, this indicates that the generation in the normal zone is independent of temperature. Figure 9 shows typical simulated temperature profiles and the associated simulated voltage versus time traces for the event of Figure 8. Each temperature profile is at a distinct instant in time and the profiles are 25 ms apart. Note that the lower-most profile corresponds to time t = 0, the instant in time at which the heater is turned on. To obtain the simulated voltages, the computed generation in each element is multiplied by the volume of the element and divided by the transport current It to give the voltage v across the element. The voltage between a given set of taps at any instant in time is thus simply the sum of the voltages across all of the elements that are located between the taps. The voltage traces are in excellent agreement in shape, time scale and final steady state value. The fact that they match in steady state value indicates that the

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa Pm versus T curve used in the simulation is accurate. Further, all other material properties, including the critical current versus temperature curve, are of the correct magnitude. The simulated voltage traces are generated from the temperature profiles numerically computed by the code discussed in the previous section. Note also that the peak temperature does remain below =25K, as inferred from the experimental voltage traces. It should be noted that the transverse cooling coefficient hc in the simulation was chosen such that experiment and simulation matched. In order to use the simulation as a predictive tool, a cooling correlation had to be developed. Since reliable cooling values could not be obtained with any simple correlation, cooling was instead based on the experimental results themselves. Specifically, cooling coefficients were inferred from the recovery current which, for a given temperature and field, is defined as the current at which heat generation (a function of current) exactly balances cooling 21. Therefore, any normal zone created will grow if the transport current is greater than the recovery current and shrink if it is smaller. In this situation, the voltage across a given set of taps rises, levels off and then drops back to zero. Thus, the appropriate cooling coefficient he could be extracted by adjusting its value until the simulated voltage traces matched the experimental traces when simulating the exact operating conditions at which the niobium tin tape recovered experimentally. To simplify the simulation, hc was assumed to be constant in both space and time. To give the cooling term temperature dependence, the actual cooling parameter qcoo~ used in the simulation was given by Equation (18), as mentioned in the previous section. Unfortunately, although the he obtained by this recovery current method is of the right magnitude (10-100mWcm-2K-l), it did not remain constant as operating conditions were varied. We present one more set of experimental and simulated NbaSn results with the tape exposed to an ambient magnetic induction of 8 T. (The study covered inductions of 0, 4, 8 and 12T.) Figure 10 presents an oscillogram recorded with the tape carrying It = 150 A at Top = 5.5 K; Figure 11 shows simulated temperature and voltage traces for the same event. Again, agreement between experiment and simulation is excellent. Several observations may be made about the niobium tin simulations in a field. First, note that the voltage levels in the simulation very nearly match those

-

/

f

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Rgure 10 Experimental voltage traces for NbaSn tape at No = 8T, Top= 5.5K and /t--150A. Vertical scale: 2.5mWdiv; horizontal scale: 19.82 ms/div. Note that ~)tOtalsaturates at ==11 mV

,=

,

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.

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.

.

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.

.

.

.

.

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.

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140

Rgure 11 Simulated version of event s h o w n in Figure 10, presenting temperature profiles and voltage traces. The lower horizontal line on the temperature plot is Tc, : 7 K; the upper line is Tc = 11 K. Temperature profiles are 2 5 m s apart, h e = 54 mW cm -2 K-1. Tap separation is 2 m, beginning at x = 1.5 cm (or 0.6cm from edge of heater)

in the experiment. This indicates that the magnetic field dependence of electrical resistivity was correctly accounted for. Since this parameter is independent of the cooling, one may conclude that this characteristic of the simulation is quite reliable. Second, note that the voltage traces level off, indicating that the temperature remains below ~25 K where resistivity is constant and that the effect of current sharing is small. This also agrees with the simulated temperature profiles. Unfortunately, the speed of propagation, that is how fast the simulated voltage traces obtained their steady state values, was strongly dependent on the value of the cooling coefficient hc and thus the simulations cannot be used to reliably predict this parameter unless cooling can be better modelled. One will note that the experimental and simulated voltages rise smoothly and at a steady rate. Further, the temperature profiles are nearly equally separated in time and space. These two observations indicate that propagation does indeed occur at a constant velocity, as assumed by Whetstone and Roos2; and others 3-5. Thus the use of an explicit, though approximate, expression for the quench propagation velocity (Equation 1) is indeed validated, at least for these high field, low Te examples. Indeed, looking at the zero-field case (Figures 8 and 9), it seems that the voltage traces and temperature profiles are not quite equally separated, indicating that velocity is not exactly constant. These anomalous results, however, may be due to the

Cryogenics 1994 Volume 34 Number 2 137

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa influence of end effects, which are apparently more prevalent at higher temperatures; it is difficult to make any conclusion in this case.

Quench analysis of high Tc (BPSCCO) tapes After gaining experience studying the quenching process in niobium tin, a study of high Tc (BPSCCO) quenching was underaken. Figures 12 and 13 show,

respectively, an oscillogram (v,, z'2, v3) and simulated results of a quenching event for the BPSCCO tape carrying It = 100 A at 40 K in zero magnetic field. The simulated voltage traces do emulate the experiment reasonably well in both shape and magnitude. This indicates that the temperature dependence of matrix resistivity was correctly accounted for in the simulation. Note that the peak temperature is nearly 200 K and that the temperature gradient along the tape is =20 K cm-', an extraordinarily large gradient but still consistent

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Rgure 12 Typical oscillogram showing voltage traces for BPSCCO tape. Experimental conditions: Top = 40 K, /t = 100 A, Bo = 0T. Vertical scale: 25 mV/div; horizontal scale: 9.91 s/div. Voltage tap separation is 2.5 cm, beginning 0.5 m from edge of heater. From left to right, the traces are v,, ~2 and v3. Top trace is the heater pulse, 17W for ==15s

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1~

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IL, 001111111

IRgure 14 Experimental BPSCCO oscillogram showing two heating events. In the first event, the conductor recovered, while in the second, when a heat pulse was applied ==8s after recovery, the conductor quenched. Experimental conditions: Top= 14K, I,= 125A, Bo = 0T. Vertical scale: 10mV/div; horizontal scale: 4.95s/div. Voltage tap separation is 1 cm, beginning 1 cm from edge of heater. Heater pulse was ==10W for ~5s

250

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t (s) Figure 13 Simulated version of event shown in Figure 12, presenting temperature profiles and voltage traces. The lower horizontal line on the temperature plot is To== 54.1 K; the top line is Tc = 93 K. Temperature profiles are 10 s apart. The voltage traces, from left to right, are ~ , i/2 and #3 and are 2.5cm apart, beginning at x = 1.135 cm. hc = 22.9 mW cm -2 K-'. Heater pulse was 1.5W for 15s

138

Cryogenics 1994 Volume 34 Number 2

0

0

4

8

12

14

t (s) Rgure 15 Simulated version of recovery event shown in Figure 14, presenting temperature profiles and voltage traces. The lower horizontal line on the temperature plot is Tcs = 44.4 K; the upper line is T©= 93 K. Temperature profiles are 2.5 s apart. The voltage taps ere l cm apart, beginning at x = 1.635cm. hc = 24.5 m W c m -2 K-'. The heater pulse was 3W for 5

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa 300

g

200

100 50 2

4

6

8

=

40, 3O 2o

10

O~

4

8

12 t

16 18

(.)

Figure 16 Simulated version of quench event shown in Figure 14, presenting temperature profiles and voltage traces. The lower horizontal line on the temperature plot is T= = 44.4 K; the top line is Tc=93K. Temperature profiles are 3s apart. The voltage taps are 1 cm apart, beginning at x = 1.635cm. he= 24.5mWcm-2K -1. The heater pulse was 3.2W for 5s

with the voltage levels (the experiment or simulation). As additional proof of these high temperatures, it should be noted here that during some of the first experiments the tape actually burned out, severing the tape and charting the G-10 supports over a --2cm section, indicating that extremely high temperatures are indeed possible over a localized section. Figures 14, 15 and 16 illustrate two additional heating events, experimental and simulated. Figure 14 shows an oscillogram (v,, P2, v3) for two heating events, both with It = 125 A at Top = 14 K in zero magnetic field. In the first event, applied heating was insufficient and the tape recovered, while in the second event, initiated --8 s after recovery, the tape was driven to quench. Figure 15 presents simulated temperature and voltage traces corresponding to the recovery event, while Figure 16 presents those corresponding to the quench event. Note that in both events the hot spot temperature, in simulation, exceeded 200 K. As with the niobium tin simulation, the BPSCCO simulation ran into difficulties when modelling cooling. A similar cooling parameter qo~ =h¢[T(x, ti)-7"=] was used. Determining h¢ was again the problem. In this case, however, a physical justification for the cooling coefficients needed to match simulation to experiment (hc=10-50mWcm-2K -I) was easier to arnve at. Since the BPSCCO tape is sandwiched between two thick, plain slabs of G-10, a cooling

coefficient based on conduction into the G-10 is justified. Analysis of thermal conduction in the G-10 indicates that an equivalent cooling coefficient of the order of --25 mW cm-2 K-1 is not unreasonable. Such h~ values produced simulated results that matched the experimental results quite well, though the matches were not as good as those achieved in the niobium tin simulations. Perhaps the most important observation to make regarding the high Tc voltage traces is the time scale over which the quench occurs. From the plots it is clear that the time scale of the quench for the high T~ tape is more than a factor of --100 longer than that of the niobium tin tape. As predicted previously, this has important implications for the design of high Tc superconducting (HTS) magnets. Also note that in both the experimental and simulated voltage traces there is a sharp slope change in the first trace. This occurs at the instant when the heater is shut off. Because cooling is strong, the peak temperature drops for a period of time before it begins to rise again, as can be seen in the temperature profiles. Other than this slope change due to the heater, however, there are no equally spaced, clean break points in the traces where they level off as there was in niobium tin. Such break points would indicate the passage of the normal/superconducting boundary, which is ill-defined in the high Tc case because of the large current sharing regime, as discussed above. Without these break points, it was impossible to experimentally measure quench propagation velocities using the given experimental apparatus, as discussed previousl.y. Nevertheless, as will be discussed below, velocities can be experimentally extracted under the right conditions.

Simulated adiabatic high Tc quenching Since thc complicated nature of highT.~ quenching limited our experimental data base, several important cases were considered in simulation. To simplify matters, the tape was assumed adiabatic, thus eliminating the need for any cooling correlation (he =0). Further, the length of the tape was increased to reduce the influence of end effects on the quenching process. The endpoint temperatures could thus be simply fixed at 7"= for the duration of the simulation since they were no longer a factor. The heater was also moved to the centre of the tape and the voltage taps were moved further away from the heater. Finally, the heater pulse used was boosted in magnitude and shortened in width to minimize the effects of the heater (and any other initial conditions) on propagation. All of the above measures were necessary because of the extremely slow propagation velocities inherent in high Tc conductors. To make the analysis more manageable, the number of variables was restricted. Specifically, only three sets of temperature and field combinations were studied: 1, 20T at 4.2K; 2, 5T at 25 K; and 3, 0T at 65 K. The first two of these combinations are likely modes of operation for high T¢ magnets in the near future (see references 9 and 10); the third approaches the limit of practical operating conditions attainable with present high Tc materials in the future. The main thrust of these simulations was to see

Cryogenics 1994 Volume 34 Number 2

139

Quench propagation in high Tc superconductors: R.H. Bellis and Y. Iwasa if high Tc conductors obeyed the constant quench propagation velocity assumption used in most analyses of quenchingF-s. A second motivation was to study how high Tc conductors behaved in the presence of a background field since this was not done experimentally. A third reason was to see if it was at all possible to extract quench propagation velocity data experimentally from voltage measurements. Finally, by simulating a very long tape, some insight on the basic nature of normal zone propagation in high Tc superconductors could be gained.

2 0 T at 4.2K. The first case studied was at 20T and 4.2K. Two different levels of transport current were studied, It = 70 A and It = 91 A. The 91 A case represents operation of the conductor very close to the critical current versus temperature curve (i.e. where To,-Top is small), as can be seen in Figure 7. The current sharing temperature T~ in this case is ~-4.6 K and the 4.2K critical current is ~ 9 1 . 8 A . As the 20T critical temperature is 51 K, it is clear that the current sharing regime is very large. It was the 70A case, however, representing operation well below the critical current versus temperature curve, that presented particularly difficult problems for the simulation. The 70 A case is discussed first. To eliminate any initial or end effects, a tape length of 30 m was chosen. The results of this simulation are

shown in Figure 17. Looking at the voltage traces, one will notice that they are separated by a fixed interval of time that does not vary with voltage level. This indicates that the normal zone propagates at a constant velocity Ue = 10.8 cm s-I and that this quantity should be measurable experimentally. However, if one looks at the temperature profiles, they are not separated by equal distances. At low temperatures, they seem to indicate that the propagation is slowing down since the profiles get closer and closer as time progresses. At high temperatures, however, they seem to indicate that the propagation is accelerating, since the profiles get further and further apart. Perhaps this is not so surprising when one considers the tremendous temperature gradient that exists along the tape (~10 K cm -1) as well as the excessive peak temperature (>1000 K). Simulation was much easier with It = 91 A because, at 91 A, 4.2 K and 20T, the superconductor is very near its critical current versus temperature curve where T~-Top is small. Thus, the tape requires a much smaller heat pulse to initiate a quench than it does when running at 70 A. With a smaller heat pulse (0.1 W for 5 ms), the temperature profiles reached a constant shape much more quickly, and thus the simulation was only carried out to 0.6s (instead of 12 s) which, in turn, made it possible to use a tape only 10 m long to produce propagation that was not hindered by any end effects. Figure 18 shows this particular case. One will immedi3O

•

20

10 0 I~

15.0

15.4

15.8 ~: (m)

16.2 16.4

5 5

6

7

8

9

= (In)

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/.......

/ / 00

4

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t (s) Figure 17 Simulated temperature profiles and voltage traces for 30 m long BPSCCO tape at Top -- 4.2 K, It : 70 A and Bo : 20T. The lower horizontal line on the temperature plot is To= : 15.3 K; the top line is Tc : 51 K. In the temperature plot, profiles are 2 s apart. The voltage traces, from left to right, are vl, ~2 and us and voltage taps are 2 5 c m apart, beginning at x : 1550cm. Heater pulse was 2 5 W for 2 0 m s

140

Cryogenics 1994 Volume 34 Number 2

O0

0.2

/ 0.4

0.6

Figure 18 Simulated temperature profiles and voltage traces for 1 0 m long BPSCCO tape at Top=4.2K, I t = 9 1 A and Bo = 20T. The horizontal line on the temperature plot is To= = 4.6K. Temperature profiles are 0.1 s apart. The voltage traces, from left to right, are ul, v~ and us and are 5 0 c m apart, beginning at x = 600cm. Heater pulse was 0 . 1 W for 5 m s

Quench propagation in high Tc superconductors: R.H. Be//is and Y. Iwasa ately notice that in 0.6s the normal zone has already propagated more than 3 times further than in the 70 A case, yet the peak temperature does not even exceed 30 K. The propagation in this case, as obtained from the voltage traces, is --560 cm s-1, or more than 50 times faster than in the 70A case. In this case, where temperature gradients are not excessive, the velocity seems to be constant regardless of the temperature level.

5 T at 25 K. The next temperature and field combination studied was 5T and 25 K. Again two levels of transport current were used: 70 A, for comparison with the previous case, and 86.5 A, to simulate operation very near the critical current versus temperature curve where T=-Top is small (see Figure 11). At this higher temperature, propagation velocities are much slower, mainly because the heat capacity of the conductor is much larger, and thus the normal zone takes more time to reach the endpoint. Therefore, a 10 m long tape was sufficient to prevent end effects from influencing the simulation. Figure 19 shows the 70A simulation. Because specific heat is higher, a larger heat pulse (25 W, 20 ms) was also necessary and thus more time was required for the temperature profile to reach a constant shape. The simulation was therefore carried out to 12 s. Note that the voltage traces are separated

1600

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- -

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5.8

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z fro)

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Rgure 20 Simulated temperature profiles and voltage traces for 1 0 m long BPSCCO tape at Top= 25K, I t = 8 6 . 5 A and B o = 5T. The lower horizontal line on the temperature plot is To. = 25.4 K; the top line is Tc = 63 K. In the temperature plot, profiles are 2s apart. The voltage traces, from left to right, are vl, P2 and Y3 and voltage taps are 10cm apart, beginning at x = 530cm. Heating pulse was 25W for 2 0 m s

500

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= (m) 40

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t (s)

by fixed intervals in time and thus that quenching proceeds at a constant velocity (Ue--2.8cms-1). Note also that with the higher specific heat at 25 K, the propagation velocity is much slower than in the 4.2 K case. A consequence of this slope propagation velocity is a very high peak temperature (--700 K) and an even greater temperature gradient (--25 Kcm-I). As in the 4.2 K, 70 A case, the temperature profiles do not seem to indicate that the propagation velocity is constant. At 86.5 A, 25 K and 5 T, the propagation velocity is faster since the current is higher (Ue = 6.1 cm s-l). This case is shown in Figure 20. In this simulation, the heat pulse used had the same magnitude and width as in the 70A case despite being much closer to the critical current curve. This is again most likely due to the large specific heat at this temperature. The voltage traces indicate the velocity to be nearly constant but the temperature profiles seem to indicate that it is accelerating. As above, this seems to occur when excessive peak temperatures and temperature gradients are present.

Figure 19 Simulated temperature profiles and voltage traces for 10 m long BPSCCO tape at Top = 25 K, It = 70 A and Bo = 5 T. The lower horizontal line on the temperature plot is Tc, = 33.6 K; the top line is Tc = 63 K. In the temperature plot, profiles are 2s apart. The voltage traces, from left to right, are vl, v2 and va and voltage taps are 5 cm apart, beginning at x = 510 cm. Heater pulse was 25W for 2 0 m s

0 T at 65 K. The final combination of temperature and field studied is 0 T and 65 K. In this situation, specific heat is even greater than in the 25 K case and thus propagation velocity is even slower. However, since heat capacity does not increase as much between 25 and

Cryogenics 1994 Volume 34 Number 2

141

Quench propagation in high T~superconductors: R.H. Bellis and Y Iwasa 1200~

500

_

4OO s00i

200 100 0500

504

508

512

516

500

520

510

520

530

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x (c,-)

(cm) 65.8

74 2

65.6 N 65.4

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68 66 65500-'~ 510

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Figure 21 Simulated t e m p e r t u r e profiles f o r i O m long BPSCCO tape at T o p = 6 5 K , I t = 5 0 A and B o = 5 T . The l o w e r horizontal line on each plot is Ta, = 73.5 K; the t o p line is T¢ = 93 K. In both t e m p e r a t u r e plots, profiles are 2 s apart. Heating pulse was 2 5 W f o r 2 0 m s

65 K as it does between 4.2 K and 25 K, the decrease in velocity is not as sharp. Again, a 10m long tape eliminates any end effects. Two different levels of transport current were studied, 50 and 70A for comparison to the other cases. Coincidentally, 70A and 65 K puts the tape very close to its critical current versus temperature curve for 0 T (again, see Figure 11). The temperature profiles for the 50 A case are shown in Figure 21. The quench front appears to mov6 at a constant velocity lie of 1.5 cm s -1. Figure 22 shows the 70 A, 65 K, 0 T simulation; again only the temperature profiles are shown. The velocity in this case is faster than in the 50 A case because of the higher current level (Ue = 2.5 cm s-! versus Ue = 1.5 cm s-l). Also, one will notice that the peak temperature reached after 12 s is more than twice that in the 50A case. This is due to the higher current level as well. Observations. From the above six cases, several conclusions can be made. First, while propagation velocity in most cases is nearly constant, there is no fundamental reason that velocity should be constant; it is simply an experimental observation based on experience with low Tc conductors. At lower temperatures, it seems to be a fairly good approximation; at high temperatures, it is less so. Nevertheless, the quench propagation velocity concept is still an important tool for magnet design, even with high T¢ conductors. It is

142 Cryogenics 1994 Volume 34 Number 2

'

530

540

(c,-) F i g u r e 22 Simulated t e m p e r a t u r e profiles f o r 1 0 m long BPSCCO tape at Ton = 65K, It = 7 0 A and Be = 5T. The l o w e r horizontal line on each plot is To, = 65.8 K; the top line is Tc = 93 K. In both t e m p e r a t u r e plots, profiles are 2 s apart. Heating pulse was 25 W f o r 20 ms

more difficult to extract experimentally, however, because velocities are inherently much slower in these conductors. This allows external factors such as end effects, transverse cooling and/or initial condition effects to have a more significant influence. Indeed, these effects can effectively mask the propagation, making this parameter difficult to measure accurately, particularly when conductor lengths are limited. In controlled situations, propagation velocities can be

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r. (K) Figure 23

BPSCCO p r o p a g a t i o n velocity as function o f operating temperature. In all cases, the f ul l - nor mal matrix current density Jm is 1 5 9 A m m - 2 . T h e BPSCCO data are f r o m simulation; the Nb3Sn point, also at Jr~ = 1 5 9 A m m -= and included f o r compar i son, is e x p e r i m e n t a l 1~

Quench propagation in high Tc superconductors: R.H. Be//is and Y. Iwasa Table 2

Summary of simulated high 7"= quench propagation velocities

B

Top

Tc

t,

jr.

To,

To,-Top

U~

Material

(T)

(K)

(K)

(A)

(A mm -2)

(K)

(K)

(cm s -~ )

BPSCCO

20 20 5 5 0 0

4.2 4.2 25 25 65 65

51 51 63 63 93 93

70 91 70 86.5 50 70

159 207 159 197 114 159

! 5.3 4.6 33.6 25.4 73.5 65.8

11.1 0.4 8.6 0.4 8.5 0.8

10.8 560 2.8 6.1 1.5 2.5

Nb3Sn

12

4.2

111.4

159

~5.6

~1.4

~60 °

7.5

aMeasured velocity11; simulated velocity, using recovery current technique, is ~-400 cm s-;

determined from voltage traces, although not in the same way as they can be in low Tc tapes; there is no clean break point as there is in low T~ conductors because of the very large current sharing regime. Figure 23 and Table 2 summarize the results of the above simulated quenches. As can be seen, the simulations seem to verify expected trends. Specifically, it is clear from Figure 23 that, for a given current density, velocity does indeed decrease with temperature. Also, from Table 2, it is clear that velocity increases with transport current for a given background magnetic field and operating temperature because this places the conductor closer to the Ic(T) curve where Tc~-Top is small. For comparison, a Nb3Sn conductor (carrying the appropriate transport current so that the current density in the matrix Jm is comparable with the three 70 A BPSCCO cases when fully normal) is also included in Figure 23 and Table 2. Based on this and other data it can be stated that, in general, quench propagation velocities in high T~ conductors are much slower than those in low T¢ conductors. However, under the right operating conditions - high current, low temperature and high field - fast propagation velocities (>1 m s-1) may be obtained in high T¢ conductors, at least in adiabatic, long-length configurations. These conditions are at low temperature, where specific heat is small, and at high currents and fields that place the conductor very near to its critical current versus temperature curve where T~-Top is small.

Conclusions Throughout this paper, an emphasis has been placed on issues relevant to magnet design using high T¢ conductors. It is clear that, although they are very promising materials, they present a unique set of challenges to the magnet designer. Perhaps the most important of these is that quench propagation is a very slow process in these conductors: high T¢ magnets are unlikely to be self-protecting and will require protection schemes. Since the normal zone barely propagates after initiation of a quench, the peak temperature in the hot spot can easily reach excessive values, as observed both experimentally and in simulation. If some sort of protection scheme is not employed, a magnet wound with these conductors is quite likely to burn out and thus be destroyed should a quench occur. Fortunately, for the same reason that propagation is very slow, high Tc conductors are exceptionally stable in the face of external disturbances. That is, it is

difficult, in general, to initiate a quench in high Tc conductors because of the shape of the Joule heat generation versus temperature curve. Temperature must be increased significantly before any major heat generation occurs. This is quite unlike the low Tc case where a small increase in temperature can produce a tremendous jump in heat generation. These generation curves have markedly different shapes because of the large difference in the temperature range over which current sharing occurs in each case. These two conclusions point to an interesting potential design philosophy for high Tc magnets: disposable coils. In the event of a quench, which is highly unlikely because of the inherent stability of high T¢ conductors, simply dispose of the magnet since protection from quench damage is a very difficult task. In terms of analysis, it can be concluded that the assumption that quench propagation velocity is constant is indeed reasonable, even for high T¢ conductors, under the right conditions. External factors can influence the propagation much more easily in this case because it occurs so slowly. The actual propagation may be masked by these external factors and thus care must be taken to avoid such contamination. On the experimental side, quench propagation velocities can indeed still be measured experimentally, albeit in a different manner than is typically used in low T¢ propagation velocity measurement. Fortunately, velocity information can still be extracted from a high T¢ superconductor by means of voltage taps. In the low T¢ case, voltage levels off after passing a set of taps because resistivity and matrix current were independent of temperature. In the high Tc case, this does not occur and velocity may be extracted only by measuring the time difference between voltage traces rather than the rise time for a given voltage trace. As an alternative, a more meaningful measurement technique may be to experimentally measure temperature profiles for comparison with simulated profiles. While this would provide more information, it is, however, much more difficult since it would require placing temperature sensors along the entire length of the tape. In terms of simulation, it seems that numerical solution of the governing partial differential equation is practically a necessity because of the temperature dependence of the generation term in high Tc conductors. The use of a transformation to convert the partial differential equation into an ordinary differential equation, while valid, does little to help simplify the analysis since it must be solved numerically as well. In

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Quench propagation in high Tc superconductors: R.H. Be//is and Y. Iwasa fact, since both must be solved numerically, for roughly the same effort one may obtain much more information regarding the quenching process by solving the governing partial differential equation. Solution of the PDE generates temperature profiles from which one may extract much information, including not only velocity, but also peak temperature, temperature gradient, etc. The problem with this is that when one goes to higher dimensions, the solution becomes much more difficult if not impossible. More sophisticated finite element and/or finite difference methods must be applied. In one dimension, temperature profiles were marched out in time; in two or three dimensions, temperature surfaces must be marched out in time. It is unclear whether the transformation to an ordinary differential equation will offer any help in the high Tc multidimensional case. Nevertheless, solution of the PDE is still the most fundamental method of analysis.

Acknowledgements The authors would like to thank K.-I. Sato of Sumitomo Electric for supplying the BPSCCO tapes. R.H.B. would also like to thank D. Johnson and J. Chadenoit for generously lending technical assistance throughout the development and construction of the experimental apparatus, and P. Michael and M. Yunus for expert advice and assistance in all phases of this project. The research described in this paper was sponsored by the Division of Conservation and Renewable Energy, US Department of Energy. It is based in part on a thesis submitted by R.H. Bellis in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering at the Massachusetts Institute of Technology, January 1993. The Francis Bitter National Magnet Laboratory is supported by the National Science Foundation.

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