Controlled Thermonuclear Fusion — Question 2

Question 2

Alpha self-heating gives an idealized density-confinement requirement

For equal D and T densities with total ion density n, use thermal energy density $w=3nkT$, fusion rate density $R=n^2K/4$, and alpha energy $E_\alpha=3500$ keV. Define loss density as $w/\tau_E$. Let kT=10.0 keV and supplied reactivity $K=10^{-22}\,\mathrm{m^3/s}$. Assume full alpha deposition.

1. Derive the ideal Lawson self-heating condition on $n\tau_E$ without external heating.

2. Evaluate the bound and find the minimum confinement time at $n=10^{20}\,\mathrm{m^{-3}}$.

Question 2 - Solution

Alpha self-heating gives an idealized density-confinement requirement

(a) Method and calculation

$$P_\alpha/V=\frac{n^2K}{4}E_\alpha\ge\frac{3nkT}{\tau_E}.$$$$\boxed{n\tau_E\ge\frac{12kT}{KE_\alpha}}.$$

(b) Interpretation and further calculation

$$\boxed{n\tau_E\ge\frac{12(10)}{10^{-22}(3500)}=3.42857\times10^{20}\,\mathrm{m^{-3}s}}.$$$$\boxed{\tau_E\ge3.42857\,\mathrm s}\quad(n=10^{20}\,\mathrm{m^{-3}}).$$

Physical check

All losses are included in the defined energy-confinement time. This supplied fixed-temperature model omits impurities, incomplete alpha deposition and profile effects.