Controlled Thermonuclear Fusion — Question 3

Question 3

Plasma fusion gain fixes the share of heating supplied by alpha particles

Define plasma gain $Q=P_f/P_h$, where $P_h$ is external power deposited in a D-T plasma. Let alpha heating be $P_\alpha=.200P_f$ and assume all alpha energy is deposited.

1. Derive the fraction of total plasma heating supplied by alphas in terms of Q.

2. Evaluate the fraction at Q=1 and Q=10 and state the limiting value as Q becomes very large.

Question 3 - Solution

Plasma fusion gain fixes the share of heating supplied by alpha particles

(a) Method and calculation

$$f_\alpha=\frac{P_\alpha}{P_\alpha+P_h}=\frac{.200P_f}{.200P_f+P_f/Q}.$$$$\boxed{f_\alpha=\frac{Q}{Q+5}}.$$

(b) Interpretation and further calculation

$$\boxed{f_\alpha(1)=1/6=.166667,\quad f_\alpha(10)=2/3=.666667}.$$$$\boxed{\lim_{Q\to\infty}f_\alpha=1}.$$

Physical check

Q compares fusion power with deposited external heating, not electric power drawn from the grid. Dominant self-heating and net electrical output are different milestones.