Optimal size of a reactor

Reference reactor - try 1

What size should a reactor be? It would be useful to have some kind of typical reactor configuration to use as a yardstick when investigating this, something that can be scaled. Let our reference reactor have a central electrode of length `2 a` and set the maximum size of the orbit to `a`. The central electrode can have diameter `k_i a` where `k_i` is constant, and have a charge of `Q = a \lambda`. This setup preserves the voltage with respect to `a` as the reactor is scaled (see below).

It would also be useful to have a simple orbit that can be used a reference betweeen scalings. The simplest orbits are circular orbits around the equator which are achieved by balancing the tangential velocity and the radius. Setting `r = a` provides a handy reference orbit.

Electrode charge

What affect does varying the size of the reactor have on the charge of the central electrode?

The electric field of an infinite line charge is given by Guass' Law for lines:
`E = \lambda / (2 \pi r \epsilon_0)`
where `E` is the electric field, `\lambda` the charge per unit length, `r` the radius of the cylinder, and `\epsilon_0 = 8.85 \times 10^{12}` F/m (see this page on hyperphysics).

Let `V` be the voltage, then:
`V = \int E dr = \int \lambda / (2 \pi \epsilon_0 r) dr = \lambda / (2 \pi \epsilon_0) [ln |r|]_j^i`
Rearranging:
`\lambda = 2 \pi \epsilon_0 V [ln |r|]_j^i`
If `i=1` and `j=0.05` then `[ln |r|]_j^i \approx 2.996`. If we divide `i` and `j` by the same amount (as would be reasonable when scaling the reactor) then `[ln |r|]_j^i` is constant. So if `V` is constant throughout the scaling process then `Q \prop a`

Consider a reactor with a pole of length `2 a`, centred at `z=0`. For simplicity we'll look at circular orbits where `z=0`, although in practice we want all orbital paths to be unchanged, with only linear scaling of the particle positions taking taking place. Note the orbital frequency will be allowed to change during scaling.

The radial force towards the pole is:
`F_r = k/(2 a r) ( (z-a)/(sqrt(r^2 + (z-a)^2)) - (z+a)/(sqrt(r^2 + (z+a)^2)) )`
where `k = k_e q Q` and `q` is the charge on a particle. Holding the voltage constant throughout scaling is probably the easiest thing to do and `V prop lambda` so substituting `Q = a lambda` we get:
`F_r = (k_e q lambda)/(2 r) ( (z-a)/(sqrt(r^2 + (z-a)^2)) - (z+a)/(sqrt(r^2 + (z+a)^2)) )`

When `z=0`:
`F_r = -(k_e q lambda a)/(r sqrt(r^2 + a^2))`

In order to achieve a circular orbit we need `F_r = (m v^2)/r`. Equating the two equations for force gives us:
`v^2 = - (k_e q lambda a)/(m sqrt(r^2 + a^2))`

The sign here can be ignored. It can be seen there are multiple circular orbits given by different `(r,v)` pairs. Larger radii lead to slower particles, although presumably such particles would have higher potential energy.

We wish the paths followed by particles to all be scaled linearly with respect to `a`. For our circular prototype orbit let `c a = r` where `c` is a constant. This gives us:
`v^2 = (k_e q lambda)/(m sqrt(c^2 + 1))`
This give us our main result that `v` independent of `a` provided the voltage is held constant.

The constancy of the velocity is very helpful for analysing these systems. As the reactor shrinks nothing changes at the tiny scale of particle encounters - the starting velocities and ending velocities of particles involved during encounters are identical as for large reactors. Consequently the ratio of fusion events to particles lost are independent of `a`.

The frequency of orbits is inversely proportional to `a`, and thus the output of a single reactor is inversely proportional to `a`. The next section, which looks at the capacity of the reactor, confirms this finding, as do computer simulations.

The cross-section for a trip through the plasma is:
`sigma = (pi b^2 n) / V = (pi b^2 n) / (2 pi a^3 k_V) = (b^2 n) / (2 a^3 k_V)`
`b` is the furthest approach distance deemed to be a collision. `V` is the effective volume of the plasma (the main body of the plasma, excluding the parts where the plasma is very diffuse), and `k_V` is the proportion of the total reactor volume the contains the effective volume of plasma. `n` is the number of particles in the plasma. Let `n = 6.242 \times 10^{18} a \lambda k_\lambda` where `k_\lambda` is the plasma charge proportional to the central electrode charge (negated) and `\lambda` is the charge per length on the central electrode.

The collision frequency is:
`X = sigma n v` ` = (b^2 n^2 v) / (2 k_V a^3)` ` = (b^2 v (6.242 \times 10^{18} \lambda k_\lambda)^2 ) / (2 k_V a)`
Now `\lambda` is being held constant with respect to `a` and from the section above `v` (the average velocity and indeed the velocity distribution) is constant with respect to `a`. Thus:
`X \prop 1/a`
The number of collisions increases inversely with respect to `a`.

If an array of reactors is used then per volume then the output would be proportional to `1/a^4`, i.e. `X prop 1/a^4`.

Reference reactor - try 2

Well all this is very interesting, but it all rests on the assumption the voltage can be held constant when scaling, but it be?

Electrodes will breakdown in a vacuum when a certain voltage is achieved. That maximum voltage appears to be somewhere in the vicinity of 20MV/m. Note that the voltage needs to be scaled linearly with respect to the radius.

Apparently smaller electrodes are more resistant to breaking down than large electrodes as the surface area is smaller and there are therefore fewer imperfections, but the gain is fairly small. For the present purposes assume a linear relationship. If we assume `V \prop a` then `lambda = k_V a ` for some constant `k_V` and:
`X = sigma n v` ` = (b^2 v (6.242 \times 10^{18} \lambda k_\lambda)^2 ) / (2 k_V a)`
` = (b^2 v (6.242 \times 10^{18} k_V k_\lambda a)^2 ) / (2 k_V a)`
`X \prop v a`
Now `v \prop V` so `v \prop a` giving:
`X \prop a^2`

One of the great things about "try 1" was that the particles preserved their energies as the reactor scaled down. Unfortunately as we have to change the voltage now we lose this invariant. Instead we can take advantage of the fact that number of fusions in D-D climb fairly linearly up to energies of about 200kV, after which the increase drops off. The expected yield `P` equals `X` times the fusion rate per collision (disregarding escaped particles). For energies under 200kV:
`P \prop a^3` (approx)
Energies above 200kV will be less favourable as the fusion rate increases below a linear rate.

A 3-D array of small reactors would provide the same amount of energy as a single large reactor if the overall volume occupied by the reactors were held constant. From these rough calculations it appears that there is little advantage to scaling down the reactor.

Smaller reactors do have a couple of benefits: they are more resistant to breakdown, and losses due to escaping particles are less. The losses should reduce linearly with respect to the voltage and hence to `a`. The losses are not the problem here though.

Conclusion

I could have made mistakes in my maths, so don't take this stuff as gospel.

To me it appears that reducing the size of the trap may give small gains, but not enough by itself to produce net power. If very high breakdown voltages could be achieved using carbon nanotubes or something similar then perhaps something might be possible. At very small scales cyclotron radiaton is likely to cause problems.

Otherwise we need to find a way to significantly increase the density of the plasma. One option might be introduce magnets, which I am trying to avoid. But I have some other ideas to try out first.