Controlled Thermonuclear Fusion — Question 1

Question 1

Both ions and electrons contribute to a fusion plasma energy inventory

A uniform fully ionized D-T plasma has total ion density $n=10^{20}\,\mathrm{m^{-3}}$, volume $V=100\,\mathrm{m^3}$, and equal electron and ion temperatures with kT=10.0 keV. Charge neutrality gives $n_e=n$. Use $1\,\mathrm{keV}=1.602\times10^{-16}$ J.

1. Calculate total thermal energy for classical nonrelativistic particles.

2. If net heating above losses is 20.0 MW, estimate the time needed to add this thermal energy and state the approximation.

Question 1 - Solution

Both ions and electrons contribute to a fusion plasma energy inventory

(a) Method and calculation

$$W=\tfrac32n_iVkT+\tfrac32n_eVkT\Rightarrow\boxed{W=3nVkT}.$$$$\boxed{W=3(10^{20})(100)(10)(1.602\times10^{-16})=48.06\,\mathrm{MJ}}.$$

(b) Interpretation and further calculation

$$\dot W=P_{\mathrm{heat}}-P_{\mathrm{loss}}=20.0\,\mathrm{MW}\Rightarrow\boxed{\Delta t=48.06/20.0=2.403\,\mathrm s}.$$$$\Delta W=\int(P_{\mathrm{heat}}-P_{\mathrm{loss}})dt\quad\text{if net heating is not constant}.$$

Physical check

Counting ions alone would miss half the thermal energy. The time estimate assumes fixed volume, density and net heating, with negligible initial thermal energy.