0. IS THE STEKLY CRITERION ADEQUATE TO DESIGN CABLE IN CONDUIT CONDUCTORS ?
Jean-Luc DUCHATEAU (CEA Cadarache)
Association EURATOM-CEA, CEA/DSM/DRFC

1. The Stekly criterion in question Why Cable in conduit conductors ?
Outline
The Stekly criterion in question
Why Cable in conduit conductors ?
The analytical approach for heat deposition in a CICC
The particular case of the square shape disturbance
Application to a practical case

2. The Stekly criterion in question
TF ITER conductor prototype
manufactured by Nexans
a non copper section Anoncu, a copper section Acu and an helium section AHe. In a project like
ITER the optimum composition of the conductor components is calculated through the so-
called design criteria.
The recent review of the ITER project has led to some interrogation about the systematic use
of the Stekly criterion to calculate the copper section of the ITER PF NbTi coils.

3. The Stekly criterion in question
The Stekly criterion imposes that the copper section of the cable has to be adjusted such as <1 to be in the so-called well-cooled region with  being the Stekly parameter:
The Stekly criterion expresses that when the strands are taken at a temperature above Tc by a disturbance, the CICC can be stable and can recover by evacuating the power generated in copper as it is in communication through heat transfer with an infinite bath whose temperature is at T0 if the criterion is respected.
In practice in case of NbTi, the application of the Stekly criterion can lead to very high copper to non copper ratio increasing the price of conductor

4. The Stekly criterion in question
Critics
The helium reservoir is not infinite and the helium temperature is continuously increasing during the heat transfer
Anoncu is playing practically no role , the stability relying only on Acu
The shape of the disturbance Qp(t) is playing no role which is certainly not correct.
In practice ITER is reconsidering now the use of the Stekly criterion, the CICC of the EAST
Tokamak which produces its first plasmas in 2006 does not respect the Stekly criterion and
the conductors of the Japanese JT-60SA will neither respect the Stekly criterion.
New rules of design are however still missing to calculate the different component sections of
the CICC. In the following some indications will be given about a possible approach in this
direction.

5. Behaviour of cable in conduits in transients
Why cable in conduits ?
Cable in conduits have been invented in the framework of fusion programs to face a new
challenge for applied superconductivity :
To accept without quench fast heat deposition due to plasma initiation but especially to plasma disruption (50 ms –100 ms)
Cable in conduits bring :
- an helium reservoir
-a very high wetted perimeter of the strands thanks to the subdivision of the strands allowing access to the reservoir
- low level of losses thanks to cable subdivision

6. Behaviour of cable in conduits in transients
Basis of the CICC concept
To use helium enthalpy to stabilize superconducting strands
Material
Enthalpy for a temperature excursion of 2 K starting from 4.5 K
Copper
2700 J/m3
Nb3Sn
7400 J/m3
A 316 (steel)
40 kJ/m3
Helium (constant volume)
640 kJ/m3
Helium (local enthalpy)
1660 kJ/m3
Helium (enthalpy at constant pressure)
2270 kJ/m3

7. Behaviour of cable in conduits in transients
The analytical approach [1]
0 D approach Disturbance power density Qp(t)
Cable infinite (no role of the extremities)
CHe and Ccomp independent of temperature
[1] D. Ciazynski and B. Turck   »Stability criteria and critical energy in superconducting CICC » Cryogenics

8. Behaviour of cable in conduits in transients
The analytical approach [1]
Stability in well cooled regime (<1)
Simplified approach the following formulation is taken for f(Tcomp). analytical solution
 The cable is stable if the observed minimum cable temperature is under Tc.
f(Tcomp)=0 for Tcomp<Tc
f(Tcomp)=1 for Tcomp>Tc
Ecmax is the maximum energy stored in the helium reservoir, the energies are classically referred to the energy per cubic meter of composites.
Ecmax = (CcompAcomp + CHe Ahe)(Tcs –T0)  CHe Ahe(Tc –T0)
Ec is the critical energy of the CICC.
It is found that:
Ec/Ecmax=1-  (=1 the critical energy is 0 !)

9. Behaviour of cable in conduits in transients The analytical approach
Stability in poor cooled regime (<1)
In poor cooled regime (>1) there is however some stability if the disturbance is not
taking the cable above Tcs. This kind of stability is providing in practice sufficient
energy margin to face the disturbances.
In this case it is possible to solve also the equations and find a new criterion for a given Qp(t).
In particular, a simple solution can be given in the frequent case of a square shape of duration
t supposed to be far smaller that the thermal time constant of the conductor comp:
comp = CcompAcomp/(hP) (typical value at 4.5 K : 1=0.2 ms)
He = CHeAHe/(hP)

10. Behaviour of cable in conduits in transients
The particular case of the square shape disturbance
Qp(t)=Q0 for a duration t
Using Laplace transforms it is possible to find an analytical solution for a disturbance of
duration t.
He =CHeAHe/hP
Tcomp reaches the highest temperature at the end of the disturbance
Tcomp= T0+ AcompQo(He + t)/ AHeCHe
Ecmax= AHeCHe(Tcomp- T0)
Ec= AcompQot
Ec/ Ecmax= 1/(1+ He /t)
Ec/Ecmax=1- 0

11. Behaviour of cable in conduits in transients
The particular case of the square shape disturbance
A new criterion in poor cooled regime
>0 zone 1
Stability if Tcomp< Tcs
<0 zone 2
Stability if Tcomp< Tc
All critical energies are given in J/m3 of composite (as usual)

12. The smaller 0, the large the critical energy for
The important role of 0
The smaller 0, the large the critical energy for
He =CHeAHe/hP
AHe = Avoid/(1-void) P=4kA/d k coefficient of wetted perimeter
He =CHed void/(4hk(1-void))
He is only depending on d and h (for a given void)
The critical energy is only depending on d and h (for a given void)
and on (Tcs-Top)

13. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
Problem : a given section Acomp  is allocated to carry I at Bmax and Top
What is the most favourable distribution between copper and NbTi for disturbances of duration 100 ms
The problem is solved for :
I=25000 A P=1.25 m Top=4.8 K Acomp =300 mm2 AHe=161.5 mm2 (void fraction 35 %)
Ccomp=2000 J/m3K CHe= J/m3K d=0.8 mm P=1.25 m

14. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
Critical properties of NbTi according to Bottura fit:
Bc20= T Tc0= 8.7 K C0=
 = = =1.94

15. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
The problem is examined for three values of h according for a typical disturbance of
150 mJ/cm3 (of composite), in view of not exceeding a Tcs of 6 K.
Ecmax =216 mJ/ cm3 (of composite)
h (W/m2 K)
min
He (s) 0
comp (s)
100
5.3
0.43
0.80
0.0048
500
1.06
0.087
0.46
1000
0.53
0.043
0.3
As 0 < min the critical energy is driven by the formula of zone 1

16. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
Composite temperature during a square shape disturbance of duration 100 ms
150 mJ/cm3 (of composite)

17. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
Influence of copper content (at given composite section) on critical energy

18. Behaviour of cable in conduits in transients
The particular case of square shape disturbance
A practical application
Influence of Stekly parameter (at given composite section) on critical energy

19. Behaviour of cable in conduits in transients
The particular case of exponential disturbance
A practical application
Similarly the equations can be solved for an exponential disturbance.
Composite temperature during an exponential disturbance of time constant 100 ms
150 mJ/cm3 (of composite)

20. Behaviour of cable in conduits in transients
A comparison of the model with gandalf
Using gandalf with the same h , gave very similar tendencies, the length of the disturbance plays a role showing that cooling by conduction is playing a role

21. Application to ITER PF4
Problem : a given section Acomp  is allocated to carry I at Bmax and Top
What is the most favourable distribution between copper and NbTi for disturbances of duration 100 ms
The problem is solved for coil PF4 of ITER (old design):
I=45000 A P=1.65 m Top=5 K Acomp =361 mm2 AHe=161.5 mm2 (void fraction 34 %)
Ccomp=2000 J/m3K CHe= J/m3K d =0.73 mm
ITER DDD September 2004 Acu/ANbTi=6.9
ANbTi= 45.7 mm2 ACu= mm2 Total section of composites : 361 mm2

22. Application to ITER PF4 0=0.27 for h=1000 W/m2 K

23. Application to ITER PF4

24. Rational for a new more economical design
Application to ITER PF4
Rational for a new more economical design
New proposal Acu/ANbTi=1.6 Same section of NbTi
ANbTi= 45.7 mm2 ACu= 73.1 mm2 Total section of composites : mm2
A factor of 3 of gain in price !!!
Additional segregate copper can exist in the cable for hot spot associated with safety discharge.
Ec = 177 mJ/cm3 composite similar to DDD version of September 2004

25. Conclusion
No, Stekly criterion is not adequate to design conductors for fusion application.
Contrary to Stekly criterion, it has been demonstrated that, for a given composite allocation,
the stability limit in energy for a disturbance (100 ms, long length of CICC) is a
decreasing function of the copper content :
The less copper, the highest the stability limit ! 
The particular role of He and comp is highlighted thanks to a simplified approach which
demonstrates that the critical energy is essentially linked to the current sharing temperature in
poor cooled regime. The crucial role of h is linked to these two parameters.
Copper is necessary for intrinsic dynamic stability but also for short (1 ms) mechanical
disturbances applied to small length of CICC (1 cm).