Improved HTS models

[jonmaddock]

In GitLab by @mkovari on Dec 3, 2019, 15:17

REBCO superconductors have rather gradual superconducting transitions - in other words, the value of n, the power law exponent in the IV curve of the superconducting transition has a rather small value - of the order of 25, rather than 45 for LTS. (A low value of the exponent n corresponds to a shallow and broad transition, while a high value of the exponent n gives a sudden transition.) This leads to a non-zero electric field near the transition temperature, due to flux creep, giving non-zero joule heating in the superconductor.

$(V/V_0)=(I/I_c)^n$,
where V is the electric field (loosely the voltage), $V_0$ is the electric field criterion (usually taken as 100 µV/m), and $I_c$ is the critical current - defined as the current at which the electric field criterion is reached.

For example, in ITER, with LTS coils, the maximum allowable electrical field at the operating temperature is taken as 2 µV/m (to limit cryogenic heat loads).

To keep the electric field to 2 µV/m when $n=25$ requires the current to be 85.5% of $I_c$ or less, even before any other factors or margins are taken into account:

$$0.855^{25}=\frac{2 \mu V/m}{100 \mu V/m}$$

In any sensible operating conditions this joule heating will be much less than the nuclear heating in a reactor. However, to be on the safe side, it would be worth including it in the calculation of the power deposited in the cold mass, and especially in any calculation of the cooling circuits of the superconducting coils.

Any volunteers, @schislet @jlion ?

Checklist

After implementing issue do the following

• Run test suite
• Does it change the output? If so, give justifications.
• Does it change the baseline 2019 run? (you may need to run on Freia)
• Merge request created

[mkovari]

Reply from Stuart Wimbush:

I think it makes sense in broad terms. I could provide a line by line criticism if needed (for example if you want to publish) but the overall thrust agrees with my thinking.

I would add that HTS coils (rather than wires) almost always exhibit lower n-values, down as far as 5, but preferably 10-15.

I think this heating is always negligible (in every context, not just fusion). This is because no-one would design a magnet to operate at 100% (or even 85%) of Ic. A 50% margin would be typical. Even in cases where I've designed magnets to operate "on the edge", due to the material anisotropy, it is only a tiny fraction of the coil that is at 100% of Ic, so correspondingly only a small region that is generating any heat.

In practice, a coil Ic would never be measured with (or operated to) a 1 uV/cm electric field criterion because that would be infeasibly large along the full length of the wire. So this limitation reduces the heating again.

[mkovari]

I don't really agree with Stuart's comment that this heating is always negligible. The electric field criterion is not used to design the magnet, but just to define the critical current. For example, noting Stuart's comment that the n value for the coil as a whole is in the range 5-15 (and using n=10), and using a fusion-relevant current of 100,000 A:

Exponent			Electric field criterion	Electric field	Current	Power per unit length	Length	Power
n	I/Ic	V/V0	V0 (µV/m)	V (µV/m)	I (A)	P=VI (W/m)	L (m)	(W)
10	0.5	9.77E-04	100	0.10	1.0E+05	9.8E-03	1.0E+04	98

The 98 W is probably negligible in the fusion context.

Stuart replied:

For 100 kA, you would need a termination resistance of less than 5 nOhm (two terminations) to produce less than 100 W at the terminations. So I would say it will be comparable to the heat generated at the terminations, but with the advantage of being distributed around the whole coil rather than localised at the ends.

It's interesting to compare the situation if you did decide to operate at 90% of Ic, say. Then the 100 W becomes 35 kW.

In real life there will be joints in the coil as well as at the terminations.

So:

We do need to stop people putting in HTS coils at unrealistically low current margins. We can do this in two ways:

1. Putting in a prominent warning if the user doesn't include the current margin constraint with a maximum current ratio of no more than about 50%, or

2. Including the resistive heating as explained above.

@j-a-foster