Stability considerations for design of a high temperature superconductor

Stability considerations for design of a high temperature superconductor A. Abeln, E. Klemt and H. Reiss Asea Brown Boveri, Corporate Research Heidelberg, D-6900 Heidelberg, Germany

Received 16 July 199 I; revised 25 September 1991 Predictions based on the well known cryogenic stability criterion for the permissible ratio

J/Jc of transport to critical current density of high Tc superconductors are compared with a transient analysis. The results demonstrate that calculation of the transient 'temperature history' of the superconductor after a thermal disturbance, as well as knowledge of the magnitude of this disturbance, is necessary when designing a conductor for a particular application. While in other published work on stability analysis, constant heat transfer coefficients were applied, an analysis like the one presented here allows one to calculate more accurately heat transfer by using the transient temperature difference between superconductor and coolant.

Keywords: superconductor stability; high Tc superconductors; heat transfer

field) that can be tolerated if flux jumps and quenching are to be avoided. The quantities Jc, To, density and In terms of classical superconductor terminology, heat capacity of the superconductor have to be known as stability analysis covers the following items (arranged in 'input' values to the adiabatic stability formula. In addiincreasing order of complexity): cryogenic stability (the tion, dynamic stability requires knowledge of ct(AT) and well known Stekly criterion, which is a simple heat of the magnetic diffusivity of the superconductor. Both balance), adiabatic stability (predictions based on the criteria allow inclusion of the transport current density. critical state model, under the influence of a thermal Figure 1 describes schematically the present stability disturbance) and dynamic stability (same basis as for scenario and the 'input' values for the different stability adiabatic conditions but including thermal and magnetic models (and where the analysis to be presented here is diffusion and heat transfer to the coolant). A thermal located in this scheme). These criteria, rather than disturbance can be initiated by, for example, a conductor cryogenic stability, have to be observed when a supermovement, a fluctuation of magnetic field or heat leaks. conductor is to be developed for a particular technical Stekly's criterion assumes that heat produced in the application (additional considerations have to be normal conducting state of a superconductor is removed directed towards an estimate of a.c. losses). With regard by heat transfer to the coolant. It allows an estimate of to thin film preparation, prescription of, for example, the required ratio P/A of perimeter P and cross-section conductor thickness has a direct impact on materials A of a superconductor provided the normal state development. resitivity On, critical current density Jc, critical The above three classical tools, however, have been temperature Tc, temperature of coolant Tcool,atand heat derived using some simplifying assumptions: transfer coefficient cKAT) to the coolant are known (AT denotes the temperature difference between superconductor and coolant). In some derivations of this 1 A serious drawback of the classical stability criteria is that they do not, or at least do not explicitly, concriterion, we find also the ratio X of the cross-sections sider transient behaviour (temperature fields, current of superconductor to a stabilizing metallic conductor distribution) of a superconductor. As will be (see, for example, Reference 1). discussed below, this may have serious consequences Adiabatic and dynamic stability criteria are more for estimates of heat transfer, which always establish explicitly directed towards an estimate of conductor an important part of a stability analysis. A later secgeometry. The literature mostly considers the case of an tion of this paper deals with a comparison of results external magnetic field orientated parallel to the surface obtained using the conventional cryogenic stability of a superconducting slab (again see Reference 1). The criterion and a transient analysis of temperatures in adiabatic stability criterion, under the simplifying a high Tc superconductor performed with computer assumption of homogeneous transport properties, then simulations. yields an estimate of the slab thickness (or of penetration . . . . . 0011 - 2275/92/03026910 © 1992 Butterworth - Heinemann

Ltd

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Design of a high temperature superconductor: A. Abeln et al. C l a a a l c a l Stabfllty. Crlterla

thermal disturbance, in the presence of a transport current and with heat transfer to liquid nitrogen (LN2). It was also the aim of these investigations to allow estimates of the maximum transport current density that can be tolerated in a thin film, band-like superconductor. The predictions from the computer simulations will be compared with predictions based on the simple cryogenic stability criterion. An analysis that also takes into account anisotropy of transport properties will be presented in a subsequent publication.

c=~roge=~s t e b ~ MInimum Input: Pn. a(AT). J. Tc(J) or Jc. Tc. P/A, ~.

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Minimum Input: de, J , To, heat capacity, dimension of conductor, a(Kl3, thermal conductivity tensor; a complete model would require the Dyrmmlc Stability Input plus thermal conductivity tensor, heat resistance to substrate ./ .1

/ /

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Figure 1 Classical stability criteria (cryogenic, adiabatic and dynamic stability), i.e. the present stability scenario, and the relative position ( • ) of computer simulations (this work) and of the intrinsic stability model (References 4 and 5; see text for details). The symbols denote the 'minimum input' required for application of the different models for stability calculations

2 The three classical tools assume isotropic transport properties of current, heat and magnetic diffusion for the superconductor. Since these conditions are not fulfilled with high temperature superconductors, new stability models have to be established, or the existing models must be modified thoroughly. In view of the expected strong anisotropies, for example of J~ or of thermal conductivity (considered in the ab plane or in the c direction), it is not permissible simply to apply the above formulations to ceramic superconductors. Estimates based on methods that still follow the conventional models (for example in References 2 and 3) can give only orders of magnitude. A step in the right direction has been made recently by the introduction of an intrinsic stability criterion 4'5 which considers anisotropy of heat transport in the conductor, but not, as yet, anisotropic critical current density or magnetic field. 3 The classical models, and also recently published work on intrinsic stability 4"5, assume a constant and uniform heat transfer coefficient or(AT). In reality, this quantity depends strongly on the temperature difference AT between cooled surface and coolant. In this paper, we will focus our attention primarily on the third problem. We describe computer simulations for investigating transient temperature fields in a high T~ superconductor under the influence of a

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A computer program has been written for calculating transient temperature fields in a cooled, thin film high Tc superconductor (a 'band-like' conductor) in the presence of a transport current and a thermal disturbance. Thus the program is to some extent a combination of the conventional cryogenic and adiabatic stability criteria mentioned above. However, it explicitly yields transient temperature fields in the superconductor, which establish a sensitive means of stability analysis. All calculations are performed with temperature dependent parameters. Like the conventional models, the program does not take into account anisotropy of current and heat transport properties, because the models required for this purpose do not yet exist. Also, the calculations have been performed without explicitly accounting for a magnetic field (the possible use of Jc(B) as an input parameter is the only response of the present form of this program to a magnetic field B). Under these conditions, the question the program can answer is: if all relevant materials parameters of a high temperature superconductor are given, if the heat transfer coefficients to the coolant are known, if its transport properties are assumed isotropic and if a thermal disturbance occurs, what is the maximum allowable transport current for the case of a particular film thickness? Figure 2a gives an overview of the conductor configuration. All calculations were made in one dimension only. The band-like conductor (width 10 mm, with variable thickness) is deposited onto a substrate and cooled by boiling nitrogen (or forced convection; see below). The superconductor is connected at both ends to a copper current feed-through (no optimization is made with respect to the feed-throughs). A finite differences scheme is applied to define heat balances for each element, 1 _
Design of a high temperature superconductor: A. Abeln Input data

a: Finite Differences Scheme LN 2

.

LN=

//Superconductor KI Connections '\ 8ubstrate

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LN= QCond ~---

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Substrate LN=

Oe..=b= Figure 2 Finite differences scheme for computer simulations. (a) Overall structure of the conductor; (b) details of heat flow components and source function for superconductor element N and its surroundings

sample. For calculation of U we need, in principle, a three-dimensional diagram showing the electrical resistivity p of the superconductor in all its thermodynamical states, i.e. as a function of T and J, as shown schematically in Figure 3 (assuming B = 0). The function p(T,J) exceeds zero only in the shaded region. A voltage U and thus heat generation occurs not only if T > Tc (ohmic resistance) but also if J _ J~ (T < To), which relies on flux flow resistance.

9(T,J)

The computer simulations were performed using experimental data of p of a superconducting thin film of YBaCuO produced by chemical vapour deposition (CVD) at the ABB Heidelberg Research Center. Details of the deposition process are given elsewhere 6. A complete experimental p(T,J) diagram, as indicated in Figure 3, was not yet available for this film. Instead, we first measured p(J) at constant temperature (77 K). Figure 4a shows the measured voltage U versus the current I of a typical CVD sample of 1.1 /zm thickness, 27/zm width and 120/zm length, that was prepared on a SrTiO3 substrate. The critical current of this sample corresponds to Jc = 1.4 × 105 A cm -2. The U versus I curve shown in Figure 4a has been measured by a pulse method. A triangular voltage profile from a function generator is fed (after proper amplification) on to a circuit including the sample immersed in liquid nitrogen, and an additional ohmic resistance. The current increases, by virtue of the voltage profile, from zero to its maximum value within some milliseconds. The voltage over the sample and over the ohmic resistance is measured in parallel and amplified by two differential amplifiers and a storage oscilloscope. Since sample voltage did not show any hysteresis, we concluded that the sample temperature had not risen during the short current pulse. For a derivation of the required temperature dependence of p(T, J), we consider first the case T T~ in ohmic resistive states (see below and

Figure 4b). Since the temperature dependence of Jc needed for this procedure had not yet been measured using our sample, we applied a frequently used, linear relation instead

T

L(T) = Jc(77 K) × (T~ - T)/(Tc - 77 K)

J Figure 3 Schematic diagram showing specific electrical resistance p(T, J) (J = current density, T = temperature). The shaded region denotes resistive states of the superconductor and the dashed line corresponds to the transport current density J. See text for definitions of curves C(T') and C(77 K)

(1)

The source function aHeat for the conductor element N is then calculated using U = p [ l - l c ( 7 7 K)] × (L/A) x [ I - lc(T)], where L and A are the conductor length and cross-section, respectively, of this element. Next, we consider the case T >_ Tc. Production of Joule heat is calculated again using experimental values of electrical resistivity p,(T) obtained with a CVD film. These measurements were made using a sample mounted on a cold head. The temperature of the sample was measured using a resistance thermometer. Voltage

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Design of a high temperature superconductor: A. Abeln et al.

u (77K) [pv]

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Cryogenics 1992 Vol 32, No 3

and thus specific resistance p. were detected by applying a small current (of several microamperes) to the sample during cooling from 300 K to temperatures below To. The source function is then calculated using U = p,(T) × (L/A) × I, with linear approximations in the two sections a and b to the measured values of p,(T), as shown schematically in Figure 4b. A value p* = p,(98 K) = 3.57 × 10 -7 ~ m was detected at the intercept of the two linear approximations (filled circle in Figure 4b). It is understood that the data given in Figure 4a, the value p,* and the slope of the linear fits to p, in the sections a and b are valid only with reference to a single thin film sample, and they might not be representative for a large collection of other samples. A plot of 0,(300 K) versus Jc using data taken from the literature for a variety of high T~ samples is shown in Figure 4c. Compared with the (solid line) fit curve, the Pn values applied in the computer calculations are smaller, but they are within the scatter of data observed in this figure, Heat transfer by conduction Qco,a from element N (Figure 2b) to its neighbours is calculated using the temperature dependent (isotropic) thermal conductivity of the bulk material. Measurements of anisotropic thermal conductivity of thin films are not yet available. We plan to measure these quantities using non-stationary techniques, such as laser flash (an experimental programme has recently been started). The temperature dependent heat capacity is also taken from the bulk material. The data for thermal conductivity and heat capacity used here are from Reference 7. No experimental data for heat resistances between thin film superconductors and substrates (or buffer layers or stabilizing metal) are currently available (so we will measure this quantity also). Accordingly, the heat transfer OCo.t.ct between a conductor element and a substrate is accounted for by assuming a conduction contact heat transfer coefficient of 1000 W m -2 K -]. This value is a worst case assumption in that it presumably underestimates the thermal coupling of the superconductor to part of its surroundings, according to experiences with heat resistances of thin metal films with A12038. In other words, the heat resistance assumed here is possibly too large. However, additional phonon scattering may arise in thin disordered layers in either the superconductor or substrate close to the interface. A high thermal resistance between the superconductor and (low heat conducting) substrate would establish a 'thermal mirror' to the superconductor. This would reduce the cooled surface of the conductor to effectively about half its value and, as a consequence, would have the same impact as an uncertainty factor of 2 with regard to heat transfer from the superconductor to a coolant! Figure 4 (a) Voltage U v e r s u s current I measured with a pulse technique applied to a CVD thin film YBaCuO superconductor sample prepared on SiTi03 (Reference 6; for details o f the measurement see text). The sample was held under liquid nitrogen. The solid curve is a second order polynomial least squares fit: U = U ( I - Ic(77 K)) to the data. (b) Schematic graph of specific resistance Pn v e r s u s temperature T with linear approximations used in regions a and b. Experimental value P n * = 3.57 x 10 -7 ~ m (taken at T = 98 K). (c) Ohmic resistance pn(300 K) v e r s u s critical current density J c for a variety of high Tc superconductors. Data were taken from measurements from this w o r k and from the literature. The solid curve is a least squares fit to the data

Design of a high temperature superconductor: A. Abeln et al. Heat transfer to the c o o l a n t OCoolingis described in the program with data for boiling nitrogen under atmospheric pressure 9,t°, or with the well known Dittus-Boelter equation for forced convection, assuming supercritical conditions (see below). Values of a(AT) for boiling heat transfer with nitrogen given in the older literature (e.g. Reference 1 l) should no longer be used (the magnitude of or(AT) has only slightly changed but there are significant corrections to AT: see References 9 and 10). Measurements of the heat transfer coefficient or(AT) between a heated surface to boiling nitrogen have been reported in the literature only for copper or stainless steel surfaces. Values of or(AT) using ceramics, especially ceramic superconducting surfaces, are not known. It may be expected that the or(AT) values of the high temperature superconductors would be about two or three times lower than values measured on copper. Surface roughness as well as degassing and thus wetting behaviour effects of LN 2 with ceramics have not been investigated. Another difficult problem with a(AT) is connected with transient heat transfer problems. It is not known, at least for liquid nitrogen, how long it takes to produce a N 2 bubble or to establish a vapour film on the heated surface; in other words, with what time-scale is it permissible to use or(AT), conventionally measured under stationary conditions, in a transient analysis? (The situation seems to be better with helium.) Taking into consideration for a moment the above mentioned stability criteria, e.g. cryogenic stability, it can be immediately recognized that the value of or(AT) may have equal importance in the estimation of conductor geometry as the squared critical current density. This influence of or(AT) must be shown explicitly in the computer simulations, which will be demonstrated by the results reported in a later section. Besides boiling heat transfer, forced convection heat transfer has also been considered (in sample calculations). In this case, supercritical nitrogen was considered as a coolant, and the heat transfer coefficient was calculated by application of the Dittus-Boelter equation. It should be noted that the critical pressure of nitrogen is = 3 3 . 5 a t m t. If it is not possible, under extreme operating conditions, to cool the conductor sufficiently simply by boiling heat transfer at atmospheric pressure, increased pressures up to 15 atm or even supercritical conditions cannot be excluded. Such situations could impose severe additional requirements with respect to mechanical strength of the cryostat and lifetime of the superconducting thin films. In the following it is assumed that a thermal disturbance (caused by conductor movement or by other heat generating processes) initiates quenching of the superconductor in a small volume at a time r'. This process will be active for a time interval AT. Heat generation resulting from the quench can be calculated, for example, by setting the electrical resistivity of element N (e.g. N = K/2) at the time r' to the value of p,* (the filled circle in Figure 4b) during the interval AT. In the simulations, as with the application of the simple cryogenic stability criterion, it was assumed that the full cross:1 a t m

=

101325

N m -2

section of the conductor element was subject to this transition to normal resistivity. Let us take Pn* = 3.57 X 10 -7 ~ m, Jc = 109 A m -2 and, for example, Ar = 0.1 ms. If the normal (transport) current equals the critical current, the heat generated amounts to = 104 W h m -3 or 1.6 W h kg -t. This amount is rather large (about 1000 times what can be expected in the case of a simple phase transition from the normal to the superconducting state; it is of the same order of magnitude as applied in Reference 4 for a thermal disturbance). The program calculates temperatures T(N, z) for all r > T'. As initial values, the program assumes T(N, r = 0) = 77 K for all elements, N, where 1 ___N _< K and for the substrate in the case of pool boiling under atmospheric pressure (for cases of supercritical operating conditions, all initial values of T were set to 67 K). The smallest time steps were 0.1 /zs, with element lengths of 10 or 100 mm (in view of the strong contributions from the source and cooling functions, the usual criteria for numerical stability that relate geometrical length to the length of the time steps in conduction heat transfer problems are too weak and cannot be used in the cases considered here).

Thermal problems when applying the cryogenic stability criterion Let us start with some estimates based on the cryogenic stability criterion p n * J g < (9/(AT) X ( T c -

Tcoolant) X

P/A

(2)

where P denotes the cooled perimeter of the conductor and A its cross-section. This equation will be solved for P/A or (if width b is given) for conductor thickness d, for a given ratio J/Jc of transport to critical current density (see below). Using these assumptions, Equation (2) describes the maximum allowed thickness of the conductor for providing a ratio of cooled perimeter (or cooled surface) to conductor cross-section (or volume) large enough to deliver the generated heat completely to the coolant. Comparison of predictions based on Equation (2) with the results obtained in the computer simulations are then made to see the limitations of the simple Equation (2). Note that the application of this equation implies a permanent thermal disturbance of power Pn* Jc2 per unit volume. When using Equation (2) with this worst case assumption, a conservative approach to stability conditions is usually produced. As will be show below, this approach can fail if the heat transfer coefficient o~(AT) is not used properly, i.e. if its temperature dependence is neglected. In Equation (2) we use the input parameters p,* = 3.57 x 10 -7 ~ m, Jc(77 K) = 10 9 A m -2 and ~(AT) = 104 W m -2 K -I. To account for a reduced heat transfer from ceramic (superconducting) surfaces to LN2, these calculations have been made with a reduced heat transfer coefficient or(AT) that was set to one-third of the corresponding values measured with copper. If we use the full o~(AT) value of copper, a considerably higher transport current density J/Jc would be expected (this demonstrates the impact of an uncertainty in or(AT) on

Cryogenics 1992 Vol 32, No 3

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Design of a high temperature superconductor: A. Abeln et al.

= (AT)

critical temperature T~ for the transport current density J: T' = Tc = Tc(J). In this picture, the 'usual' critical temperature Tc is written simply as Tc = T c ( J = 0 ) , which means that T~ should be measured with probe currents as small as possible. Accordingly, T~ is a function of J. This function is already defined by Equation

(Xmax

(1) T~(J) = T~(J = 0) - (T~(J = 0) - 77 K) × J/Jc(77 K)

(3) When inserting J and Tc(J) into Equation (2), instead of the symbols Jc and To, we have Figure 5 Schematic diagram illustrating heat transfer coefficient c~(AT) for (natural) convective and boiling heat transfer. AT denotes the temperature difference between the heater (e.g. superconductor) and cooling liquid (e.g. nitrogen)

stability analysis). In particular, the value of O/(zaT) = 104 W m -2 K -~ applied here corresponds to one-third of the maximum value O/max(AT)of the O/(AT) curve observed in the (schematic) boiling heat transfer diagram of nitrogen (Figure 5). For application of Equation (2), we have adopted the convention O/(AT)~---O/max(AT) from the literature. However, although preferentially applied in the existing work on superconductors, it should be clearly stated that this is an overestimate of real heat transfer conditions because only in a very few instances (if at all) will the difference (T~ - Tcoolan t) in Equation (2) equal the value A T = AT(O/max) where O/max is observed (AT(O/max) is = 5 K for boiling heat transfer to liquid nitrogen under atmospheric pressure9'~°). This uncertainty with respect to the use of O/(AT), apparent in numerous applications of the cryogenic stability criterion, can be made still more obvious. If we consider a particular value of transport current density J we can look for the temperature T' where this transport current density equals the critical current density (Figure 6). This temperature can be considered the

Jc(T)

~J=Jc(T') J

77K

T'-Tc(J) T c (J=O)

~T

Figure 6 Schematic diagram showing temperature dependence of critical current density Jc of a superconductor v e r s u s

temperature T (the diagram is used for interpretation of the cryogenic stability criterion; see text for details)

274 Cryogenics 1992 Vol 32, No 3

pn J 2 = p n * J c ( T ' ) 2

< O/(AT) × (Tc(J)

-

-

Tcoolant)X P/A (4)

T~(J = 0) = 94 K and Let us assume that T~oolant= 77 K; accordingly, the maximum possible interval of AT is

AT = (T~(J) - Tcoolant)~ 94 K - 77 K = 17 K Thus the value of O/(AT) is by no means fixed to O/max(AT). Instead, a variation of J between 0 < J _< J~ (77 K) leads to a variation of about 1 _ AT < 17 K, which in turn induces a strong variation of O/(AT) ( = 2 0 0 0 - 4 0 000 W m -2 K -], the interval between natural convection and maximum boiling heat transfer with LN2). To make the consequences more obvious, we will consider two cases: 1 Let J/J~(77 K) be large. In this case, To(J) is close to 77 K (Figure 6), because J is close to J~(77 K) (remember: heat production occurs with this scenario only if J _> Jc(T)). Accordingly, A T = T c ( J ) - 77 K is small, which implies that O/(AT) is also small, e.g. the lower limit (2000 W m -2 K-~) of the O/(AT) interval given above. In other words, cooling plays a weak part in the cryogenic stability criterion if the transport current is large. 2 If J is small, T~(J) is close to 9 4 K and AT > 5 K -- ATmax- In this case, the AT values are located in the film boiling region. However, it is not at all obvious that stable film boiling really can be established, and it is not known how long it would take to establish a stable vapour film on the superconductor. In summary, application of O/(AT) taken as a constant and at AT(o/max) in Equations (2) or (4) leads to, except for certain specific transport currents J, an overestimate of cooling conditions. Consequently, J/Jc values estimated from Equation (4) (as, for example, in Reference 12) are too large. Nevertheless, we provisionally followed these conventions in order to see how they compare with computer simulations of the transient behaviour of superconductor temperatures. Still bearing in mind the cryogenic stability picture, and with the conventions explained above, we then have, from Equation (4), d < - 2 b / ( 2 - a b ) using a = pn*Jc(T')E/(o/max(AT) X (Tc(J) - 77 K)), P = 2(b +

Design of a high temperature superconductor: A. Abeln et al. Results: c o m p a r i s o n simulations

d IJm)

1

15

10

5-

0 0

6.2

O.s

o,4

J/Jc('77

Figure 7 Maximum conductor thickness d versus ratio J/Jc of transport to critical current density (at 77 K) in a high Tc thin film superconductor ( ) calculated by application of the cryogenic stability criterion, e , Results from the transient analysis

d) and A = bd. In Equation (4) it has been assumed that the superconductor is wetted by the coolant over its full perimeter P. The results from the cryogenic stability criterion have been plotted in Figure 7 (solid curve). Obviously, the required conductor thickness depends strongly on J/Jc, and vice versa.

with computer

So far we have considered the application of the simple cryogenic stability criterion. The stability criterion will now be approached using computer simulations. Figure 8 shows calculated transient temperatures T(N, r) (where r = time) for element N = K/2 (K = 100, element length 100 mm) for different cooling conditions and for a (strong) thermal disturbance of length AT = 1 ms. A constant thickness d = 0.8/zm has been assumed in the calculations. Some of the curves given in this figure (and others calculated with a different conductor thickness) show a maximum temperature Tmaxreached by element N under given operating conditions at the end of the heating interval Ar (see the solid and dashed curves indicated by J/Jc(77 K) = 0.44 and 0.45, respectively, in Figure 8). When the disturbance is switched off, this does not imply automatically that the resistance of the conductor is again zero: the temperature of the conductor may have become so high that it exceeds To. The conductor temperature will then behave according to either case (a) or (b). Case (a): TN(r)< TN(r'+Ar) for all r _ r' + Ar because the cooling power is sufficiently high to bring the temperature of the conductor down even if TN(r' + At) >- To, in the presence of the transport current. Case (b): TN(r) continues to increase because heat production by the transport current flowing through the resistive conductor element is too high. In most cases, the conductor temperature will then increase to very high values. However, it appears that stationary states could also be possible (see dashed curve in Figure 8 indicated by J/J~(77 K) = 0.5). In the following, we will consider only those curves that follow case (a). The corresponding values of Tmaxare given in Figure 9. The point of intersection of the Tmax(J/J~) curves in Figure 9 with the horizontal line T = Tc = 94 K may be interpreted as the maximum permissible ratio

100 ! T(K) /

~.

0.5

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60 T(ms) Figure 8 Dependence on time ~"of transient temperatures T(N, r) for conductor element N (see Figures 2a and 2b) for different cooling conditions: - - , pool boiling; forced convection under supercritical conditions of nitrogen. The curves are calculated for different ratios J/Jc of transport to critical current density (at 77 K). Length of therreal disturbance Ar is 1 ms Cryogenics 1992 Vol 32, No 3

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Design of a high temperature superconductor: A. Abeln

e t al.

Tmax(K)

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110

d (IJm) 2

10 d (gm) 10

2

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Figure 9

Maximum temperatures Trna× of conductor element N ( ) calculated for different conductor thickness d, as a function of permissible ratio J/Jc of transport to critical current density (at 77 K). Cooling conditions were pool boiling with liquid nitrogen (1 atm) and length of thermal disturbance zlr is 1 ms. Heat transfer coefficients were taken as one-third of those measured in Reference 9 with copper surfaces under stationary conditions. - - - , Full value of (stationary) heat transfer coefficient, measured with copper, for conductor thickness d = 0.8 #m

o.8

Figure 10

Same calculations as in vection, using Reynolds number critical nitrogen

Figure 9 but Re = 1.2 x

with forced con105 , for super-

TIn=x(K) AT (ms) 1

100

T O•94K

/

1 /

0.1

J/Jc(77 K). This is a conservative approach because in a few cases the calculations also showed that temperatures Tmaxexceeding Tc by even 5 K could be tolerated, depending on the instantaneous heat production and cooling conditions. The dashed curve calculated for forced convection in Figure 8 (with the label J/Jc = 0.5) could serve as an upper limit for the temperature excursion over Tc since it apparently represents a stationary state. Accordingly, the calculations showed that it was also possible to remove ohmic heat for cases of small excesses of superconductor temperature over Tc and if highly efficient cooling is supplied. If allowing only Tmax< T~, on the other hand, the simulated cooling by nitrogen will always be sufficient to bring the conductor temperature back to 77 K after the disturbance is switched off and within time intervals the length of which depend on heat capacity and cooling conditions. If in a sample calculation the full value O~c, is used instead of one-third of O~Cu(see above discussion), it is seen from Figure 9 that no limitation would be imposed upon J/Jc, at least within the limits 0 < J/Jc <- 0.5 (d = 0.8/xm). This finding illustrates the consequences that may result from uncertainties in the magnitude of

o~(zXT). A similar calculation for forced convection, assuming cooling with supercritical nitrogen, is given in Figure 10. The curves are calculated for a Reynolds number of 1.2 × 105, which implies a flow velocity of the order o f l m s -;. Figures 8 - 1 0 were calculated using a constant value of z17- = 1 ms. The impact of variable length of the interval =at on the maximum conductor temperature and on the maximum J/J~ is shown in Figure 11 (constant con-

276

" Cryogenics

t992

Vol 32,

No 3

80

60

o

O=

0.4

oe

08 -J/Jc(77K)

Figure 11

Same calculations as in Figure 70 but for different lengths of thermal disturbance A~ ( ) and for a 'dead time interval' A~' of 2 ms ( - - - ) . All curves calculated for d = 0.8 #m

ductor thickness d = 0.8 #m is assumed). It is seen immediately that the permissible J/Jc ratios shift to smaller values if z17- is increased, because of the increased magnitude of the thermal disturbance. Because of this dependence, an appropriate definition of the thermal disturbance, e.g. by length At, by the amount of energy deposited, by setting the conductor temperature to a certain value T' _> Tc at r = r', by local fluctuations in Jc or its temperature dependence or by local fluctuations in heat transfer, is mandatory for stability analysis. We have also studied the effect of a possible 'dead time' At' during which o~(zlT) = 0, which may occur at the beginning of a thermal excursion of the superconductor because the onset of natural convection in the coolant or the formation of bubbles (which initiate boiling heat

Design of a high temperature superconductor: A. Abeln et al. transfer) require a measurable time step. In the case of forced convection, as studied in Figure 11, the dead time may correspond to a possible short term fluctuation of fluid flow close to the conductor surface. It is seen from this figure that only in cases of very small thermal disturbances will a dead time of the cooling function have no influence on J/Jc. Maximum allowable J/J~ values from Figure 9 (when accepting only Tmax< T~) for a given conductor thickness d have been inserted into Figure 7 (filled circles) for comparison with the predictions from cryogenic stability. If we had also accepted those Tmax values that exceeded T~ by only a little ( < 1 K in case of pool boiling) the J/Jc values would be larger by 5% at the most, which can be estimated from the slope dTmJd(J/Jc) at T = T~ of the curves in Figure 9. The comparison is made for a thermal disturbance length of 1 ms. It is immediately seen that the cryogenic stability criterion, if used with too high a heat transfer coefficient, is necessarily weaker: it permits higher values of J/J~ compared with those of the transient analysis, although application of Equations (2) or (4) implies a permanent (Az -- co) thermal disturbance. Even with this worst case assumption, the application of a(ATm,x) in the cryogenic stability criterion leads to overestimates of the permissible pn*J 2 values. A correct formulation of the heat balance thus has to take into account exactly those ~(AT) that really occur, i.e. we need the AT = T(r) - 77 K temperature history of the superconductor, which can be obtained only in a transient analysis. Since the thickness of CVD thin film superconductors applied for devices in energy technology will probably be below 2/zm, the simple cryogenic stability criterion, used in the manner it frequently is, could lead to unacceptable uncertainties in J/J~; the differences (horizontal distances) between the filled circles and the solid curve in Figure 7 increase with increasing J/Jc. If, on the other hand, a J/Jc value is prescribed, e.g. from safety considerations, application of the same criterion will lead to an overestimate in permissible film thickness.

Comparison with other transient analyses Calculations in the previous section have been made assuming isotropic transport properties (current, heat). It is to be expected that the results given in the previous figures will need to be corrected if anisotropic transport properties of the new ceramic superconductors are taken into account. While a simple criterion like the cryogenic stability model cannot account for anisotropic transport properties, these properties can be taken into account in computer simulations. A first step in this direction has been made in References 4 and 5. Assuming a thermal disturbance, these references calculate T(x, y, r) or T(x, y, z, T) in an anisotropically heat conducting superconductor. Corresponding to this temperature distribution, there is a distribution of Jc[T(x, y, r)]. Note that the ratio Jc[T(x, y, r)]/Jc(77 K) is a function of time. However, both references have applied constant (and uniform) values of offAT), which is not permissible, as has been demonstrated above.

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Figure 12

Stability function Cmax versus aspect ratio b/d of width b to thickness d of the conductor, for a thermal disturbance of length Ar = 1 ms and with temperature (i.e. time) dependent Blot numbers (see text for details). The curve is calculated using the filled circles in Figure 7

For a rough comparison of our results and those of Reference 4 (using the curves calculated for isotropic thermal conductivity in this reference), in Figure 12 the stability function tI)ma x ~- 1 - J / J c ( 7 7 K) has been plotted versus the aspect ratio b/d using the data (filled circles) given in Figure 7 of this work. These ~max values are larger (i.e. the permissible J/Jc ratios are smaller) than those reported in Reference 4. This can be explained by the fact that Reference 4 applies a rather large (and constant) Blot number (Bi = 0.2) in the corresponding plot of ~max versus aspect ratio. In our case, we would have a range of Blot numbers of 0.001 ~< Bi <_ 0.01, using Bi = t~(AT)d/(2k), where k is the isotropic thermal conductivity (-~ 4 W m -l K -l) and 0.8 <_ d < 10/~m, if we kept or(AT) and k constant and if we used o~(A/) = C~max.In reality, both quantities are functions of temperature and thus of time, since T = T(r). Accordingly, the actual Blot numbers in our calculations are even smaller, because of the variation, with AT(z), of a(AT) ___ O(max. Since these Blot numbers indicate a considerably smaller ratio of heat transfer to the coolant and thermal conductivity than estimated in Reference 4, this explains the larger t~ma x calculated here (compare Figure 6 of Reference 4). In addition, the thermal disturbances used in our case are approximately a factor of two larger if using Ar = 1 ms, which also shifts the ~maxto larger values, i.e. J/Jc to smaller permissible values.

Conclusions Stability analysis for ceramic superconductors cannot be performed without appropriate models for temperature history and transient heat transfer (and anisotropic transport properties of heat and current). Also the source function that describes the generation of heat and the magnitude of a thermal disturbance has to be thoroughly defined. As a consequence, conductor design will not be successful if these models and the required parameter inputs are missing or incomplete. Since materials

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development has to rely on conductor design, these models are of great importance for high Tc superconductor projects. Experimental work is needed to determine the heat transfer coefficients from ceramic superconductors to liquid nitrogen. A transient analysis has been performed in this work to quantify the maximum allowable transport current in a band-like conductor, still with isotropic transport properties, and with no magnetic field included. The analysis shows that application of the simple cryogenic stability criterion is not sufficient for conductor design. A more extensive comparison between computer simulations and the other classical stability criteria, also taking into account the anisotropic transport properties of a superconductor, is in preparation. Acknowledgement H. Reiss wishes to express his gratitude to Dr M. I. Flik for friendly discussions.

References I Wilson, M.N. Superconducting Magnets, Oxford Publishers, Oxford, UK (1983)

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2 Iwasa, Y. Design and operational issues for 77 K superconducting magnets IEEE Trans Magn (1988) MAG-24 1211 - 1214 3 Ogasawara, T. Conductor design issues for oxide superconductors. Part l: criteria of magnetic stability Cryogenics (1989) 29 3 - 9 4 Flik, M.I. and Tien, C.L. Intrinsic thermal stability of anisotropic thin-film superconductors, paper presented at ASME Winter Meeting, Chicago, Illinois, USA 0988) 5 Chert, R.C. and Chu, H.S. Study on the intrinsic thermal stability of anisotropic thin film superconductors with a line heat source Cryogenics (1991) 31 749-755 6 Schmaderer, F., Huber, R., Oetzmann, H. and Wahl, G. High T¢ YBa2Cu307-~ prepared by chemical vapour deposition Appl Surface Sci (1990) 46 5 3 - 6 0 7 Gmelin, E. Thermal properties of high temperature superconductors, in: High Temperature Superconductors (Ed Narlikar, A.V.) Nova Science Publishers, New York, USA (1989) 8 Swartz, E.T. and Pobl, R.O. Thermal resistance at interfaces Appl Phys Lett (1987) 51 2200-2202 9 Bier, K. and Lambert, M. Comparison of pool boiling heat transfer coefficients for different pure liquids, paper presented at Advances in Pool Boiling Heat Transfer: Eurotherm 8, Paderborn, Germany, May (1989) l0 Riithlein, H. Autbau und Erprobung einer Apparatur zur Messung des W/irmeiibergangs von einem horizontalen Rohr an tiefsiedende Flfissigkeiten PhD Thesis University of Karlsruhe, Germany (1984) 11 Fastowski, W.G., Petrowski, J.W. and Rowinski, A.E. Kryotechnik Akademie Verlag, Berlin, Germany (1970) 12 Wipf, S. Stability of possible high T¢ oxide superconductors as current carriers Proc ICEC 12 Butterworths, Guildford, UK 0988) 931 - 935