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Practical criteria for the thermal stability of a unidimensional superconductor
B. Bonzi, M. El Khomssi, H. Lanchon-Ducauquis
To cite this version:
B. Bonzi, M. El Khomssi, H. Lanchon-Ducauquis. Practical criteria for the thermal stability of a unidimensional superconductor. Journal de Physique III, EDP Sciences, 1994, 4 (4), pp.653-657.
�10.1051/jp3:1994154�. �jpa-00249133�
Classification Physic-s Absn.acts
02.90 41.90
Practical criteria for the thermal stability of a unidimensional
superconductor
B. Bonzi (*), M. El Khomssi and H. Lanchon-Ducauquis
L.E.M.T.A., 2 avenue de la fordt de Haye, B.P. 160, 54504 Vandmuvre les Nancy Cedex, France (Receii<ed J5 July J993, rei'ised J5 October J993, accepted 9 November J993)
Abstract. The criterion of « equal area » [4, 5] is known as a necessary and sufficient condition
m order that a unidimensional superconductor of infinite length, submitted to fixed current density and magnetic field, could not evolve towards a resistive state after an accidental local
overheating this criterion relates only to the normalized source term of heating (competition between the heat power generated by joule effect and the one which is absorbed by the cryogenic bath). We give here an optimal criterion, valid for any length (case of an uncooled region situated
between two well cooled ones) it allows the engineers to play upon a non dimensional grouping of parameters (characterizing the intensity of the source term), related to the conductor, in order to extend the field of its superconducting properties. We give also some intermediate complementary
criteria which are only sufficient but much
more easy to check.
1. Non dimensional mathematical model for studying the temperature field.
Let be
L, the length of a part of a unidimensional superconductor situated between two well cooled parts
Tb and T2, respectively the temperature of the cryogenic bath and that, beyond which the
source term of heating (competition between the heat power generated by Joule effect and the
one which is absorbed by the cryogenic bath) is permanently negative ;
k, the heat conductivity of the superconductor, first considered as a constant one (cf.
generalization below)
C the heat capacity of the superconductor by unit volume.
Then, if we consider the following dimensional reductions :
x T-T~
x = and 6
=
,
(1)
L T2-Tb
(*) now Fac. des Sc. et Tech. Ouagadougou, Burkina Fasso, Africa.
654 JOURNAL DE PHYSIQUE III N° 4
(respectively for the space variable X, and the temperature field n, the natural characteristic time which appears is
~~2
r~ = (2)
and, for any thermal perturbation To T~, the non dimensional heat diffusion equation can be written
To T~
~~~' ~~ ~°~~~' ~°
T~ T~ ~~~
where
L~ p~ j~
A
= (6)
k(T~ T~)
and C(6 ), characterize respectively the intensity and the normalized form of the heat power received by the superconductor (cf. Fig. I). The term
PC "
PNj~,
represents in fact the characteristic heat power by unit volume, obtained from the normal electrical resistivity p~ of the conductor and, the current density j which is carried
2. Objective and notations.
For any given curve lf (compatible with the realistic laws g(6 ), q(6 ), respectively exchange with the bath and thermal dissipation in the superconductor (Fig. la)), we want to be able to
adjust A (that is to say, the other parameters) in order that, the temperature field of the superconductor come back to the stationary state 6
= 0 (imposed a prior"I by the cryogenic bath) after any thermal perturbation 60(x).
Let introduce for that (with for illustration the Fig. lb) the practical notations
a(a, p = j~ C(s)ds (7)
(which represents the algebric area included between C(6) and the « e-axis », for 6 e la, p [) and also
a = a(0, 1), aj
=
a(0, 6j), a~ = a(6j, 1) (8)
w defined by a (0, w
= 0 when a > 0 (9)
3. Known result : the criterion of equal area.
r~ k For an infinite length (that is L »
-, for a characteristic time chosen by the engineer), a
C
heat cower terms Incomoetifion
(n°rrna'~e°) q(e)
I gie
~ ~ ~ i o
c i N
g(6) = heat power generated by Joule effccis
q(6) = heat power absorbed bythecryogenic bath a)
c(e) = gie) q(e)
a~
l0i ai
b)
Fig. I. Form of the source term of heat power.
necessary and sufficient condition so that the only stationary state be 6
= 0, is that :
a<o thatis a2<aj. ~~~~
This criterion is independent of A which is not defined for an infinite lenght.
4. New sufficient criteria, easy to test, and valid for any lenght [1, 3].
Each of the three following situations is sufficient for the realization of the objective.
I) The local criterion, obtained from the spectral study of the Sturrn Liouville operator ([2]
chap. 3) associated to the stationary version of the problem (3, 4)
tf(6)<~
V6 e [0, 1]. (ll)
656 JOURNAL DE PHYSIQUE III N° 4
it) The criterion of equal area (see above) but now for a finite length.
ii) The weaken critetion of area :
a(6j, w w
I
Vw e j6j, Ii. (12)
Remark I. These three criteria are not equivalent and can be easily tested by order of
increasing complexity I) it) iii).
5. Concrete illustrations of the new sufficient criteria [3].
The specific form of the normalized source term lf(6) allows to show that if A
< w~ 6j, that is
~
~~ (~~ )~Sl
(Tj T~) «
then the local criterion is satisfied.
In the same way
~ 2
~~ ~ ~
l ~ij' ~~~~'~
PN(~2~~l) LJ
)~~~2
2k T~-T~ '
then the weaken criterion of area is satisfied (Tj being here the dimensional temperature corresponding to the non-dimensional one 6j w being defined by (9)).
Remark 2. These two last physical inequalities are, each of them, sufficient conditions so that the only stationary state be 6 0.
6. Optimal criterion [ii.
When any of the preceding criteria can be checked, it is useful to know the following optimal
one.
A necessary and sufficient condition to satisfy the objective is :
. either a < 0
°~ ~ ~~ ~~~ ~~~°~~ ~~ ~~° ~ ~~°"
whre J is defined by
J(w)= o ~°~~
, a(wt, w
Remark 3. This criterion (important to optimize an installation) has the inconvenience to
require some algorithms in order to compute J(w ) from the data lf that work is realized in [3].
7. Complementary results when the optimal criterion is not realized [1, 3].
I) There exists at the most, two values ~j and ~~ of
w in the interval ]w
j, I such as
J(~ )
=
),
they are the respective maxima of the two possible stationnary solutions 6~'~ and
6~~~ (strictly ordered) others than the trivial one.
ii) A computation of the two parasite solutions 6~'~ and 6~~~ is proposed for any couple iA, lf) which does not realize the optimal criterion.
iii A study of the zones of stability for the three stationnary solutions (0, 6~' , 6~~'), when these two last one exists, has been realized.
8. Realistic case where k depends on the temperature [ii.
All the previous results related to the stationnary problem can be extended to the more realistic situation where the heat conductivity k is strictly increasing with the temperature. We introduce for that the mean heat conductivity on [0, 1] :
kj
=
k(s) ds
0
and the pseudo temperature
~ ~~~~ /j 1) ~~~~ ~~'
The new problem to study is :
kj j2~
~ = g(u)
ax
u(0)
=
u(1)
= 0
where
g= foU~~
The fact that k is strictly increasing in H, implies that g is well defined and has the same form as
f ; this new problem is then the same as the stationnary version of (3, 4).
References
II Bonzi B., Etude des dquilibres thermiques d'un supraconducteur : existence et stabilitd, Thkse de l'Institut National Polytechnique de Lorraine, Nancy, France (1990) 117 p.
[2] Churchill R. V., Brown J. W., Fourier series and boundary value problems (Mc Graw-Hill Book
Company, New York, 1978) p. 271.
[3] El Khomssi M., Etude des Equations semi-lindaires paraboliques et elliptiques g6rant l'dquilibre thermique d'un supraconducteur, Thkse de l'Institut National Polytechnique de Lorraine, Nancy, France, to be defended in (1994).
[4] Maddock B. J., James G. B., Norris W. T., Superconductive composites hear transfer and steady
state stabilisation, Ciyogenics 9 (1969) 261.
[5] Meuris C., Contribution h l'Etude de la stabilit6 des supraconducteurs, Thkse de doctorat d'dtat es Sciences Physiques, prdsentde h l'lnstitut National Polytechnique de Lorraine, Nancy, France (1982) p. 216.
