Controlled Thermonuclear Fusion — Question 10

Question 10

Helium ash dilutes fuel even while charge neutrality is maintained

Hold electron density $n_e$ and temperature fixed in a plasma containing D, T and fully ionized helium ash. Let $x=n_\alpha/n_e=.0500$ and $n_D=n_T$. Charge neutrality requires $n_D+n_T+2n_\alpha=n_e$. Reactivity is unchanged.

1. Find D and T densities relative to $n_e$ and fusion-rate reduction relative to an ash-free plasma.

2. Find total particle pressure relative to the ash-free pressure, including electrons and all ions.

Question 10 - Solution

Helium ash dilutes fuel even while charge neutrality is maintained

(a) Method and calculation

$$2n_D=n_e-2n_\alpha\Rightarrow\boxed{n_D=n_T=\tfrac12n_e(1-2x)=.450n_e}.$$$$\boxed{R/R_0=(1-2x)^2=(.900)^2=.810}.$$

(b) Interpretation and further calculation

$$n_{\mathrm{ions}}=n_D+n_T+n_\alpha=n_e(1-x),\quad p=n_e(2-x)kT.$$$$\boxed{p/p_0=(2-x)/2=.975}\quad\text{while fusion rate falls by 19 percent}.$$

Physical check

A small pressure change can conceal a much larger fuel-dilution penalty. Fixed electron density is the constraint; fixed ion density would give a different comparison.