Get power balance from final simulations

[MatthewGrim]

I've re-run the code, taking only results for energy, current, radius pairs where the escape ratio is above 0.95. This should mean that the results are all valid. The time step was 1e-9 * radius because simulations that I ran in #27 showed time step did not change my results. I used results with the following values:

Energies: [5.0, 10.0, 50.0, 100.0, 500.0] Radius = [0.1, 1.0, 5.0, 10.0] Currents: [1e3, 5e3, 1e4, 5e4, 1e5]

These values gave the following coefficients with the variances in the results as below:

Where parameters a, b, c, d in the first row correspond to:

t_conf = a * I b * R * c K d

These values are similar to theoretical values [3.7e-7, 0.5, 1.0, -0.75], except for a kink in the current parameter. Observing plots for the current trends, the plots for different current values are a factor of 10 from where theory puts them, as shown below:

They conflict with previous empirical studies which gave [5e-7, 0.5, 1.5, -0.75]. I do not know how these approximate values were generated, but I am thinking that they were generated in a different way to how mine were, as the paper says the data was obtained by empirically fitting N(t) rather than the mean confinement time itself. I am not sure if this is true.

Purpose of this issue

In any case, this issue aims to close the work done on the Polywell single particle motion by applying these empirically obtained results to assess power losses in Polywell devices.

[MatthewGrim]

Gummersall's original paper suggests that the power loss is given by:

with:

This gives:

I'm not sure if this is fair, because it assumes that the electron number density is not offset by the number of ions. For a particular ion density, n_i, the total electron density, n_e, will be:

n_e = n_i + n_max

Where n_max is the same as defined above. I'm going to try to explore what both these assumptions mean for polywell break-even concepts.

[MatthewGrim]

Initial results using Gummersall power loss term

In the power balance I've started here, I am using the Bosch-Hale reactivities. These are not perfect for describing the fusion power balance but are close, as the original Nevins 1990s work described. The results for the power balance are shown below, assuming the power loss as in Gummersall's paper:

The plot above is the best case within the range of simulated cusps. The break even point is at n=1e15 which is not as bad as I would have expected. I still don't think this assumption is correct though. If I increase the power loss relative to the actual number density, it will be a lot worse.

[MatthewGrim]

It's worth noting that the power loss equations I use in the script committed are volumetric values, hence the change in the power of a to a value a -3.5 rather than a -0.5

[MatthewGrim]

Results with coefficients from simulations

The results do not change significantly. The boundary is between 1e15 and 1e16 ions per metre cubed

[MatthewGrim]

Results including ion densities for Gummersall

[MatthewGrim]

Results including the ion densities are less promising. The ion density only overcomes the electron cusp losses at densities above 1e20.

[MatthewGrim]

Results of my simulations including ion densities

The change in radius power makes little to no difference.

[MatthewGrim]

So some questions:

• Are these densities the ones that the Polywell community talks about?
• How does the collitional timescales compare to the loss timescale at these densities? Will the electrons be thermalised?

[MatthewGrim]

Validation of Fusion Power

The power being generated seems astronomically high. I cannot tell if this is because I don't have a clear perspective on the real power requirements. To give some indication, Park's paper from 2015 gives an indication of the power produced by a DT mixture.

Conditions Energy: 30keV Power: 1.9GW Electron number density: 2 x 10^15 -> the difference between this and the ion density is negligible

How does my implementation compare.