Heat produced by Focus Fusion and cooling

[jamesr]

I thought of using Comsol, which is a general purpose physics modeling program, that can combine thermal, electrical and many other phenomena together in one simulation. However until recently they could not easily handle plasmas. They have now just released a new module http://www.comsol.com/products/plasma/ which may make it easier.

It won’t be up to the task of modeling the later stages of the pinch & plasmoid, but may be sufficient for a mock-up of the axial & first part of the radial phase. To let a full 3D simulation run any further would need the inclusion of higher order terms in the equations and serious amounts of computing power.

[benf]

That’s really impressive software! Someone ought to get their hands on a Cray (or something similar) for you to get to work with it….Thanks for the preview, Jamesr

[Henning]

jamesr wrote:

I thought that with these very large currents the majority of the current flows on the outside of the electrode rather through the conductor anyway??

Exactly – that’s the problem. The current flows through a thin surface layer & hence all the resistive heating is concentrated there. Although it is the fast rise time (ie frequency) or the current flow that causes it to flow in the skin.

Isn’t only the charge located on the surface, being static? Why do I need on the other hand big copper wires to transport current over land lines? If that would be the case, a thin pipe of copper could replace solid copper lines, thus decreasing costs. Okay, they’re aluminium anyway…

Therefore I think the 20nm skin wouldn’t suffice. Correct me if I’m wrong. I’d love to be wrong here!

Okay, James quotes a different analysis here:

jamesr wrote:
I’m not sure I can go as far as percentages but here are a few figures from analysis done by Doug Olsen in 2003 (not verified):

For a copper anode, if the current is treated as a 1/4 sinusoidal rise to 600kA in 2us the skin depth is ~0.18mm. Integrating the current as a function of radius gives a maximum temperature rise of ~26C at the surface – for one shot. This diffuses into the bulk of the anode over time so after 40us or so the surface has dropped to 15C above its pre-shot temperature.

[Allan Brewer]

I’ve been doing some rough calculations based on what has been said so far:

Using Ohm’s law (apologies if I am being naive) for 1MA through a 20nm skin of the copper electrode does indeed give an answer, in the expected range, of around 2MW of heat from the anode and about the same from the cathodes. Beryllium has twice the resistivity so would double the heat “problem” (and increase the power required to initiate each shot?).

jamesr – I was thinking more that if the anode had a 10 to 20nm thick skin of something like carbon in some highly conductive configuration (eg graphene) then this would mitigate the surface vaporization and lessen the joule heating due to the large current flowing in the skin compared to beryllium.

The resistivity of graphene is four orders of magnitude less than copper so would theoretically reduce the electrode heat to 200Watts – easily manageable, and presumably drastically reducing the power required to initiate each shot.

Whilst looking at this I had the further wacky idea that if the electrodes were coated with a “high-Tc” superconductor and cooled with liquid nitrogen then we could eliminate the resistive heat generated completely – after all we would have more than saved enough power to drive the necessary refrigeration. (But even though superconduction is used to drive electromagnets I understand there is some problem when magnetic fields are changing , so the plasma/plasmoid presence might preclude superconduction?)

Finally just to note that thermal radiation from the plasma & plasmoid cannot be calculated using the Stefan-Boltzmann law (gives a silly answer) – I leave the physics of the emmissivity of high temperature plasma to experts.

[vansig]

Allan Brewer wrote:
Using Ohm’s law (apologies if I am being naive) for 1MA through a 20nm skin of the copper electrode does indeed give an answer, in the expected range, of around 2MW of heat from the anode and about the same from the cathodes. Beryllium has twice the resistivity so would double the heat “problem” (and increase the power required to initiate each shot?).

can you give the area and the length also, for us to check?

Allan Brewer wrote:
Finally just to note that thermal radiation from the plasma & plasmoid cannot be calculated using the Stefan-Boltzmann law (gives a silly answer) – I leave the physics of the emissivity of high temperature plasma to experts.

yes, ion temperature of 6.5×10^9 kelvin would seem to yield about 10^32 watts/m²; but, we should note that excitation isn’t exactly uniform in all directions (longitudinal >> transverse vibration, so maybe take the cube root? or the square root?), and quantization effects due to the strong magnetic field could also play into this.

[jamesr]

For the plasmoid at high temperature the dominant radiative cooling mechanism is via bremsstrahlung. The formula for which is approximated as:

P(br) = 1.7E-38*Z^2 *n_e * n_i *sqrt(Te) in W/m^3

where n_i, n_e are the ion & electron number densities in per m^3, and Te is the electron temperature in eV.

Taking the effective Z as 2 (combination of hydrogen & boron but with higher concentration of hydrogen) and ne =Z*n_i = 1e26/m^3 and Te= 1e9K = 80keV

Then I make P(br) ~ 4e17 W/m^3 for the plasmoid.

Multiplying this by the volume of the plasmoid (~10um diameter) an saying it lasts for 100ns gives only 4e-5J which seems way too small???

For the bulk plasma at a few thousand degrees the radiation comes from recombination and line radiation which is much harder to work out… I’ll try and find an estimate.

You also obviously have this hot, few thousand degree plasma in direct contact with the electrode surfaces – this I would expect to dominate everything.

[Allan Brewer]

vansig wrote:

Using Ohm’s law (apologies if I am being naive) for 1MA through a 20nm skin of the copper electrode does indeed give an answer, in the expected range, of around 2MW of heat from the anode and about the same from the cathodes. Beryllium has twice the resistivity so would double the heat “problem” (and increase the power required to initiate each shot?).

can you give the area and the length also, for us to check?

I used 20nm skin x circumference of 5cm diameter anode to give area, and 15cm of length. In practice the length varies during the pinch cycle, and I guess the length of feed cable from the capacitor bank will also heat, but approximate calculation only.

[Allan Brewer]

jamesr wrote: For the plasmoid at high temperature the dominant radiative cooling mechanism is via bremsstrahlung. The formula for which is approximated as:

P(br) = 1.7E-38*Z^2 *n_e * n_i *sqrt(Te) in W/m^3

where n_i, n_e are the ion & electron number densities in per m^3, and Te is the electron temperature in eV.

Taking the effective Z as 2 (combination of hydrogen & boron but with higher concentration of hydrogen) and ne =Z*n_i = 1e26/m^3 and Te= 1e9K = 80keV

Then I make P(br) ~ 4e17 W/m^3 for the plasmoid.

Multiplying this by the volume of the plasmoid (~10um diameter) an saying it lasts for 100ns gives only 4e-5J which seems way too small???

For the bulk plasma at a few thousand degrees the radiation comes from recombination and line radiation which is much harder to work out… I’ll try and find an estimate.

You also obviously have this hot, few thousand degree plasma in direct contact with the electrode surfaces – this I would expect to dominate everything.

I think for decaborane plasmoid n_e and n_i should be ~e29, derived as follows:
“…because of the very high density in the plasmoid, which is close to solid density…” (Eric L)
1cc water weighs 1g so 1m^3 water weighs 10^6 g
Relative density of decaborane is 0.9
so 1m^3 plasma weighs 0.9*10^6 g
Decaborane weighs 122g per mole
therefore 1m^3 contains 0.9*10^6 / 122 moles
each mole contains 6e23 molecules
each molecule gives 24 ions
so 1m^3 contains 6e23*24* 0.9*10^6 / 122 ions (= n_i = 1e29)
each molecule gives 64 electrons
so 1m^3 contains 6e23*64* 0.9*10^6 / 122 electrons (= n_e = 2.8e29)

This raises your P(br) to 2.24e24 W/m^3
10um cube gives 2.24e9W
100ns life gives 2.24e2W
notional 300 shots/s gives 70KWatts – how does that sound to you?

[vansig]

Allan Brewer wrote:

Using Ohm’s law (apologies if I am being naive) for 1MA through a 20nm skin of the copper electrode does indeed give an answer, in the expected range, of around 2MW of heat from the anode and about the same from the cathodes. Beryllium has twice the resistivity so would double the heat “problem” (and increase the power required to initiate each shot?).

can you give the area and the length also, for us to check?

I used 20nm skin x circumference of 5cm diameter anode to give area, and 15cm of length. In practice the length varies during the pinch cycle, and I guess the length of feed cable from the capacitor bank will also heat, but approximate calculation only.

Feed cables wont be too significant a problem because capacitors are mounted around the outside edge of a flat sheet.

okay, 5 cm circumference gives 1e-9 m² as the area. resistivity of copper ~ 16.78 nΩ·m, yielding 2.5Ω for 15cm length; beryllium ~ 36 nΩ·m, giving 5.4Ω for a 15 cm length; Yes, now i see it. but it’s important to note that skin depth changes through the pulse.

Graphene electrical resistivity may be less than 5e-4 nΩ·m; and its in-plane thermal conductivity, around 5e3 W/(m·K), is greater than diamond. I am starting to like this a lot.

Shall we put a graphene coating on everything, then? Now all we have to do is figure out how.

http://pubs.acs.org/doi/abs/10.1021/la101698j
http://nanopatentsandinnovations.blogspot.com/2010/01/fabrication-process-for-large-area.html

[Allan Brewer]

vansig wrote: [Graphene electrical resistivity may be less than 5e-4 nΩ·m; and its in-plane thermal conductivity, around 5e3 W/(m·K), is greater than diamond. I am starting to like this a lot.

Shall we put a graphene coating on everything, then? Now all we have to do is figure out how.

http://pubs.acs.org/doi/abs/10.1021/la101698j
http://nanopatentsandinnovations.blogspot.com/2010/01/fabrication-process-for-large-area.html

http://www.cheaptubes.com/carbon-nanotubes-prices.htm
These people sell single and multi-layer flakes and nanoplatelets….Since we probably only need a layer of say 20nm, equivalent to 2.4 mg for the anode if the layer is even, cost is not great, but one of us needs to start a converation with some of these experts to establish the best approach to coating, and to ask about stability at high temperatures.

[vansig]

i started dreaming up an advanced material..
perhaps: take graphene nano-ribbons, allow them to twist nicely, spin them into thread, weave this on the bias, and stretch it over the anode, like a tea cosy. then it would conduct electricity along the surface, but allow thermal penetration to the interior, and remain very strong at high temperatures.

http://www.springerlink.com/content/e092tm31km023064/

but then i realized two things:

first, graphene is basically single-layer graphite, which is stable at high temperature and used in refractory materials and high-voltage electrodes. in graphite, the layers are ~.33 nm apart, so we could use coatings of 100 layers of that, maybe, or just make the whole electrode from ordinary graphite. it’s a bit crumbly, but ought to be better in vacuum.

but, the show-stopper:
it’s going to react with the protons in the plasma, producing hydrocarbons and maybe compromising the anode integrity (not sure).

[jamesr]

I have been thinking any surface layer will need to be much thicker & more robust than a few nm skin.

If you compare with the materials & heat fluxes for tokamak divertors (the area at the bottom of the tokamak where the magnetic field sweeps any plasma that escapes the confinement down to collide with & dissapate its energy). They are either carbon fibre composite (CFC) or tungsten. We would not really want tungsten as it would absorb too much of the X-Rays, but the CFC materials they use could be a possibility, and can cope with ~20MW per m^2, and transient peaks of 500kJ/m^2 in 0.1ms or 5GW/m^2.

Even if we had a 0.1mm surface conductive/protective layer the differential thermal & pinch force stresses between it and the beryllium bulk would probably just make it all crack & fall off.

Another thought is to dope the beryllium to form an alloy with better overall electrical conductivity. After all Beryllium is pretty good anyway from most other aspects so just tweaking the properties slightly to optimize the thermal and electrical properties is probably the easiest option (although still a few years of research).

[vansig]

for a 5 MW reactor, assuming anode circumference is 5cm, that yields Area ~2e-4 m² facing the plasmoid, so we blast thru that 20 MW/m² limit by a factor of 1250.

i’m starting to think about radical designs.

if it can handle even a single transient shot, then how about triggering several anodes in a round-robin fashion? then each one gets some cool-down time. but i’m not confident that it can.

the anode is beginning to look like a big welding rod, here. maybe a design that can tolerate ablation? it gets used up in the process, just like a welding rod, or electrodes from an electric arc furnace, but we keep pushing it in from underneath.

“A typical alternating current furnace has three electrodes. Electrodes are round in section, and typically in segments with threaded couplings, so that as the electrodes wear, new segments can be added. ” — http://en.wikipedia.org/wiki/Electric_arc_furnace

graphite then becomes the most reasonable choice. though this does beg the question: carbon ions will get into the plasmoid on subsequent shots. are there any undesirable reactions we should be aware of?

[Allan Brewer]

I don’t think that graphite/carbon rod/carbon fibre is a viable electrode material because its electrical resistivity is quoted as around 5000*10^-8 ohm.metre compared with Copper 1.7*10^-8, so it would produce orders more heat than the copper electrode. (Graphene itself is a very special case with magically low resistivity.)

Referring back to the calculations on the dominant plasmoid cooling mechanism, I believe bremsstrahlung is being minimised by the design anyway, and it would be X-ray rather than thermal.

jamesr wrote: You also obviously have this hot, few thousand degree plasma in direct contact with the electrode surfaces – this I would expect to dominate everything.

I watched an arbitrary discharge today and it looks like the actual point of contact between plasma and electrode is really relatively small. To get a rough idea, if we have a 1cm^2 point of contact round the electrode (which I think would be an overestimate) and 4000 degree drop in temperature across a 3mm thick electrode we are talking about 50KW conduction – this is encouragingly very small compared to the electrode resistive heating.

That still leaves the bulk plasma thermal radiation for which James was going to try to find an estimate, but that doesn’t sound like MWatts

So its beginning to sound like the dominant source of heat would be resistive heating in the outer skin of the electrodes.

So how about a beryllium electrode to give structure without blocking X-rays, covered with a 20nm layer of graphene for electrical conductivity (to minimise heat production), topped with another very thin layer of metal to protect the graphene and conduct the last 1 or 2 nm to the plasma??

[jamesr]

Given the pulsed nature of the device, I’m thinking any stratified surface layers would be problematic unless they had very closely matched thermal expansion coefficients.

In the central well of the anode where the electron beam hits you could consider a special coating in this area (such as 20um of tungsten), but for the main shaft of the anode, and cathodes where the current is flowing then you want as low a stress due to thermal & pinch forces as possible. The overall cooling aspect I think is manageable if you pump enough helium through. It is the fatigue and cracking due to many cycles that I’m beginning to get more concerned about.

If you carefully graduated the electrical conductivity near the surface, in the opposite sense to the skin current’s normal preference so you got a more even current distribution deeper through the top 1mm of material it may help.

[Author]

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