Physica C 310 Ž1998. 340–344
Stability considerations of a high-temperature superconductor
tape at different operating temperatures
Jorma Lehtonen ) , Risto Mikkonen, Jaakko Paasi
Laboratory of Electricity and Magnetism, Tampere UniÕersity of Technology, P.O. Box 692, 33101 Tampere, Finland
Abstract
Stability of a Bi-2223rAg multifilamentary composite conductor against fast transport current ramps was studied by
using a numerical model. The model was based on the two-dimensional magnetic diffusion and heat conduction equations.
Calculations were carried out both in an adiabatic mode and pool boiling modes in liquid helium, hydrogen and nitrogen.
When estimating the heat load ŽAC losses., real temperature dependent current density–electric field characteristics were
used. The results computed by the finite element method are presented and discussed with special emphasis on differences of
the stability considerations between high-temperature and low-temperature superconductors. q 1998 Elsevier Science B.V.
All rights reserved.
Keywords: Stability; Transport AC loss; Bi-2223 tape; Temperature dependence; Numerical analysis
1. Introduction
Stability considerations for low-temperature superconductors ŽLTS. are well established whereas
high-temperature superconductors ŽHTS. have a few
intrinsic features which make the analysis quite different compared to their LTS counterparts. In HTS
materials current density–electric field EŽ J .-characteristics are slanted due to thermally activated flux
creep and macroscopic inhomogeneities. Therefore
the behaviour of heat generation induced by electromagnetic phenomena is very different in HTS than in
LTS. Furthermore, higher operation and critical temperatures with increased specific heat make HTS
conductors very stable against mechanical and thermal disturbances.
In this paper the stability of a HTS composite
conductor at different operating conditions is studied
by solving the two-dimensional heat conduction
equation combined with a magnetic diffusion model
for AC loss computation. The magnetic diffusion
model is based on Maxwell’s equations and a realistic non-linear constitutive relation between the current density and the electric field w1x. The computation is carried out for a typical Bi-2223rAg multifilamentary composite conductor configuration.
2. Computational model
)
Corresponding author. Fax: q358-3-3652160; E-mail:
jlehtone@alpha.cc.tut.fi
In the geometry and field directions shown in
Fig. 1 the current flows in the xy-plane and the
magnetic field is parallel to the z-axis. Supposing
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 8 5 - 7
J. Lehtonen et al.r Physica C 310 (1998) 340–344
341
Bi-2223 material follows the power law approximation rsŽ J,T . s Ec w J ny 1rJc ŽT . n x, where n is the index number of a superconductor and Ž Jc , Ec . the
point on the material EŽ J .-characteristic defining
the critical current density. A criterion Ec s 1 mV
cmy1 is used. The temperature dependence of Jc is
assumed to be piecewise linear as shown in the inset
of Fig. 2. In order to describe the whole EŽ J .-characteristic up to the normal state with a single function a Fermi-like function is used
r s Ž J ,T . s rr Ž T . r
y exp
Fig. 1. Ža. The composite geometry and the field directions in the
2D magnetic diffusion model. Žb. The boundary conditions defining transport current and cooling.
B s m 0 H, a magnetic diffusion equation for the
conductor is
= P Ž r Ž J . = Hz . s m 0
EHz
Et
,
Ž 1.
where Hz is the component of magnetic field H in
the z-direction and r Ž J . the electrical resistivity.
Electric field and current density in each point of the
2D composite as a function of time can be computed
from the solution of Eq. Ž1. by using Ampere’s law
and a relation between E and J. The relation must
be defined separately in the normal conducting matrix metal E s rmŽT . J and in superconducting filaments E s rsŽ J,T . J, where T is the temperature,
rmŽT . the matrix resistivity and rsŽ J,T . is the nonlinear resistivity of the superconductor.
The boundary conditions for Hz are shown in
Fig. 1. At the current contacts EHzrEn s 0 is required, where EHzrEn is the normal derivative of
Hz . The transport current I Ž t . is determined by
defining that H1 s H1Ž t . z and H2 s yH1Ž t . z on
the opposite surfaces AB and CD which leads to
I Ž t . s 2 H1Ž t .. In 2D model I Ž t . is defined as the
current per unit width of the composite. For the
detailed description of the magnetic diffusion model
see Ref. w1x.
Data for the resistivity of silver is found from
Ref. w2x. In the vicinity of Jc ŽT . the resistivity of
ž
½ ž
exp
Jr Ž T . y < J <
Jr Ž T .
D JŽT .
D JŽT .
y1
/
q1
y1
/
q1
5
,
Ž 2.
where rr ŽT . is the normal state resistivity, Jr ŽT . the
current density corresponding to the resistivity
1r2 rr ŽT . and D J ŽT . characterizes the width of
transition w3x. For Bi-2223 r r ŽT . is 10 mV m w4x.
D J ŽT . and Jr ŽT . are found by fitting Eq. Ž2. to the
power law with n 10 in the neighbourhood of Jc ŽT .,
e.g., 0.7Jc ŽT . F J F 1.2 Jc ŽT ..
Fig. 2. The increase of average temperature as a function of the
normalized time tr ts in an adiabatic composite tape at the initial
temperature Top s 4.2 K, ts s60 s Žx., Top s 20.4 K, ts s 300 s
Žo. and Top s 77.3 K, ts s1500 s ŽI.. The piecewise linear
temperature dependence of the critical current density is shown in
the inset. The points which define the linearization are marked
with =.
J. Lehtonen et al.r Physica C 310 (1998) 340–344
342
Instantaneous average power loss in superconducting filaments Qsc Ž t . and in matrix metal Qm Ž t .
are now obtained in units of W my3 by
Qi Ž t . s
1
Ai
H
Ai
E Ž t . P J Ž t . d a, i s sc,m,
Ž 3.
where A sc and A m are the areas of the cross-sections
of superconducting filaments and matrix metal in the
xy-plane, respectively.
The thermodynamics of a HTS composite conductor can be described by a time-dependent quasi-linear heat conduction equation
ET
= P Ž lŽ T . = T . q Q Ž T . s C p Ž T .
Et
,
Ž 4.
where lŽT . is the thermal conductivity, C ŽT . the
volumetric specific heat at constant pressure and
QŽT . the volumetric heat generation. The material
properties are included in the numerical model as
piecewise linear functions of temperature. These
functions have been fitted to the measured data
according to Refs. w5–10x.
The cooling is included in the model as boundary
conditions of type ylŽT .ETrEn s q ŽT . on boundaries AB and CD, Fig. 1, where ETrEn is the normal
derivative of temperature and q ŽT . represents the
cooling power per unit cooled surface area. In an
adiabatic case q ŽT . s 0. Convective heat transfer
from the conductor surface to the cryogenic fluid is
determined from empirical pool boiling curves. Transient effects in pool boiling are neglected and steady
state data is adopted from Refs. w11–13x. The four
parts of the boiling curve—natural convection, nucleate boiling, transition boiling and film boiling—
are each fitted to the function q ŽT . s aT b , where a
and b are constants. Because the studied conductor
is long ETrEn is set to zero on boundaries AD and
BC.
In the present problem QŽT . s Qsc ŽT,t . in the
superconducting filaments and QŽT . s Qm ŽT,t . in
the matrix metal. A constant temperature distribution
T s Tk in the whole composite cross-section is assumed and Qsc and Qm are computed at l times t j ,
j s 1,2 . . . l, 0 F t j F t r and t j F t jq1 , where t r is the
duration of the current cycle. The procedure is repeated for m temperature distributions T s Tk , k s
1,2 . . . m, Tk F Tkq1. Now Q i ŽT,t . is known exactly
at the nodes ŽTk , t j .. In the interval t j F t F t jq1 ,
Tk F T F Tkq1 a polynomial approximation Q i ŽT,t .
s a1 tT q a 2 T q a3 t q a 4 is used where coefficients
a1 , a2 , a 3 and a4 can be uniquely determined from
the nodal values of Q i . In an adiabatic case the
increase of temperature during one current cycle is
negligible and therefore the time average of heat
generation is used, i.e.
Q iave Ž T . s
1
tr
H
tr
Q i Ž t ,T . d t.
Ž 5.
The results presented in this paper are computed
with a software OPERA-2d by Vector Fields w13x for
electromagnetic field evaluation and with a software
PDE2D by Sewell w14x for solving the temperature
distribution.
3. Results and discussion
The conductor current was continuously ramped
from zero to a peak value Ip , in 0.01 s, and back to
zero, in 0.01 s. In the following Ip is given with
respect to Ic which equals to the critical current of
superconducting filaments per unit width of the composite at the initial temperature. A real HTS tape
conductor with total filament number of 85 was
modeled by six filaments in the x-direction in Fig. 1.
The conductor thickness in the x-direction was 180
mm, the filament thickness 10 mm, the distance
between filaments 10 mm and the thickness of matrix metal sheath 35 mm. A similar conductor has
been used in a 5 kJ SMES in Tampere University of
Technology w15x.
The temperature rise in an adiabatic case was
investigated at the boiling points of helium, hydrogen and nitrogen when Ip was equal to Ic . The
values of Ic are Ic Ž4.2 K. s 304 A cmy1 , Ic Ž20.4
K. s 240 A cmy1 and Ic Ž77.3 K. s 48 A cmy1 . In
Fig. 2 the increase of average temperature of the
conductor is shown as a function of normalized time
trts where ts is the time when dTrdt starts to
increase. The ts is related to the onset of a thermal
runaway. At Top s 20.4 K the increase of temperature with time is linear up to 29 K and at Top s 77.3
K up to 80 K because the increasing specific heat of
J. Lehtonen et al.r Physica C 310 (1998) 340–344
materials compensates the increase of heat generation. The compensation is possible in HTS materials
because of slanted EŽ J .-characteristics and a wide
temperature margin between Top and the critical
temperature. At Top s 4.2 K dTrdt starts to increase
when T becomes higher than 26 K. When T is less
than 12 K, dTrdt is even decreasing. Values of ts
are 60 s, 300 s and 1500 s at Top s 4.2 K, 20.4 K and
77.3 K, respectively. The increase of ts with Top is
due to the increased specific heat and lower Ic at
higher initial temperatures.
In a real power network there can exist momentary overcurrents, which have peak values several
times higher than the nominal current. In Fig. 3 we
search current limits for stable operation when the
tape is immersed in cryogenic liquid: helium Ža.,
hydrogen Žb. and nitrogen Žc.. The results shown
here are computed for a tape which is in contact with
the coolant on both surfaces and thus describes
stability limits in ideal cooling conditions. In a pool
boiling mode a thermal equilibrium is reached when
the energy dissipated in the tape equals the heat
transferred away by the coolant during each cycle.
According to Fig. 3 Žb. in liquid hydrogen the thermal equilibrium, in which the temperature returns
back to the nucleate boiling region during each
cycle, is found when Ip s 11 Ic and the operation
remains stable. When Ip s 12 Ic the dissipated energy is already higher than that the coolant can
transfer and the stability is lost. Filled symbols show
the transition from nucleate boiling to transition boiling. In liquid helium and nitrogen the stable operation can be sustained with Ip s 6 Ic and Ip s 17Ic ,
respectively. When Ip s 7Ic in liquid helium the
thermal equilibrium is reached in the film boiling
region.
In LTS conductors a small thermal or mechanical
disturbance can lead to a thermal runaway w7x. Due
to the slanted EŽ J .-char-acteristics, higher Top with
the increased specific heat and the wide temperature
margin between Top and Tc , the increase of temperature is much slower in HTS than in their LTS
counterparts. In a HTS tape, temperature can rise
several degrees before a thermal runaway which is a
great advantage compared to LTS tapes. The limit of
a stable operation is determined by the available
cooling. In a pool boiling mode a HTS tape can
maintain a stable operation during current ramps
343
Fig. 3. Temperature as a function of time at a composite tape
immersed in Ža. liquid helium: with peak currents of 6 Ic ŽI. and
7Ic Žo., Žb. liquid hydrogen: with peak currents of 11 Ic ŽI. and
12 Ic Žo. and Žc. liquid nitrogen: with peak currents of 17Ic ŽI.
and 18 Ic Žo..
with a peak value several times higher than the
critical current. At each operation temperature, when
Ip leads to an unstable situation, the maximum power
dissipated during a current cycle is greater than Pmax
which denotes the power transferred from the tape at
the peak nucleate boiling. On the other hand, the
time average of the dissipated power is less than
Pmax . Therefore the whole boiling curve must be
taken into account before reliable stability estimations can be made.
344
J. Lehtonen et al.r Physica C 310 (1998) 340–344
4. Conclusions
A numerical model for stability considerations of
HTS composite tapes which exploits the real EŽ J .characteristic of HTS material has been presented.
The model allows that the time and temperature
dependencies of the transport current losses are taken
into account. The developed model is used to analyze the stability of a HTS tape in adiabatic and pool
boiling modes at different operating temperatures.
The results show that in some cases it is possible to
maintain stable operation during overcurrents with
peak values several times higher than the critical
current and the temperature rise of several degrees
can be recovered.
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