Controlled Thermonuclear Fusion — Question 8

Question 8

Energy confinement time and particle residence time measure different losses

A steady plasma contains W=120 MJ and N=$10^{22}$ fuel ions. Its total energy loss is 60.0 MW and fuel-ion loss is $10^{21}\,\mathrm{s^{-1}}$. Define $\tau_E=W/P_{\mathrm{loss}}$ and $\tau_p=N/\dot N_{\mathrm{loss}}$.

1. Calculate both confinement times and compare them.

2. If heating is switched off while the energy loss law remains $P_{\mathrm{loss}}=W/\tau_E$ with constant $\tau_E$, find W after 4.00 s. Explain why this does not predict the remaining particle number.

Question 8 - Solution

Energy confinement time and particle residence time measure different losses

(a) Method and calculation

$$\boxed{\tau_E=120/60.0=2.00\,\mathrm s,\quad\tau_p=10^{22}/10^{21}=10.0\,\mathrm s}.$$$$\boxed{\tau_p/\tau_E=5.00}.$$

(b) Interpretation and further calculation

$$\dot W=-W/\tau_E\Rightarrow\boxed{W(4)=120e^{-4/2}=16.2402\,\mathrm{MJ}}.$$$$\dot N=-\dot N_{\mathrm{loss}}\quad\text{requires its own evolving loss law}.$$

Physical check

Heat can escape without the same fraction of particles escaping. A measured steady residence time alone does not specify a shutdown particle-loss model.