Question

The uncompressed radius of the fuel pellet of Sample Problem 43.05 is $\mathbf{20}\mathbf{}\mathit{\mu }\mathit{m}$. Suppose that the compressed fuel pellet “burns” with an efficiency of 10%—that is, only 10% of the deuterons and 10% of the tritons participate in the fusion reaction of Eq. 43-15. (a) How much energy is released in each such micro explosion of a pellet? (b) To how much TNT is each such pellet equivalent? The heat of combustion of TNT is 4.6 MJ/kg . (c) If a fusion reactor is constructed on the basis of 100 micro explosions per second, what power would be generated? (Part of this power would be used to operate the lasers.)

Short answer

a) The energy released in micro-explosionis 227 J .

b) The amount of TNT required is 49.3 mg.

c) The power generated is 22.7 kW.

Step-by-step solution

Describe the expression for energy released in micro-explosion

The mass is given by,

$$\begin{gathered} \mathit{m}\mathbf{=}\mathit{\rho }\mathit{V} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}\mathbf{\rho } \end{gathered}$$

Here, p is density, and V is volume.

There are equal numbers of ${}^{\mathbf{2}}\mathbf{H}$and${}^{\mathbf{3}}\mathbf{H}$present, therefore the expression for the number of ${}^{\mathbf{2}}\mathbf{H}$and${}^{\mathbf{3}}\mathbf{H}$is given by,

${\mathbf{N}}_{\mathbf{2}\mathbf{H}}\mathbf{=}{\mathbf{N}}_{\mathbf{2}\mathbf{H}}\mathbf{=}\frac{{\mathbf{N}}_{\mathbf{A}}\mathbf{m}}{{\mathbf{M}}_{\mathbf{2}\mathbf{H}}\mathbf{+}{\mathbf{M}}_{\mathbf{3}\mathbf{H}}}$

Each fusion reaction releases Q=17.59 MeV of energy, with 10% efficiency, then the total energy released by the pellet is given by,

$$\begin{gathered} \mathit{E}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}{\mathit{N}}_{\mathbf{2}\mathbf{H}}\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\left(\frac{N_{A}m}{M_{2H}+M_{3H}}\right)\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}\left(0.10\right){\mathbf{\pi r}}^{\mathbf{3}}\mathbf{}{\mathbf{\rho N}}_{\mathbf{A}}\mathbf{Q}}{\mathbf{3}\left(M_{2H}+M_{3H}\right)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right) \end{gathered}$$

Find the energy released in micro-explosion

(a)

The molar mass $M_{2H}$of deuterium atoms is $2.0\times {10}^{-3}\mathrm{kg}/\mathrm{mol}$, and the molar mass $m_{3H}$of tritium atoms is $3.0\times {10}^{-3}\mathrm{kg}/\mathrm{mol}$. The uncompressed radius of the fuel pellet is $20\mu m$and its density is $\rho =200\mathrm{kg}/{\mathrm{m}}^3$.

Substitute all the known values in equation (1).

$$\begin{gathered} E=\frac{4\left(0.10\right)\mathrm{\pi }{\left(20\times {10}^{-6}\mathrm{m}\right)}^3\left(200\mathrm{kg}/{\mathrm{m}}^3\right)\left(6.022\times {10}^{23}{\mathrm{mol}}^{-1}\right)\left(17.59\mathrm{MeV}\right)}{3\left(2.0\times {10}^{-3}\mathrm{kg}/\mathrm{mol}+3.0\times {10}^{-3}\mathrm{kg}/\mathrm{mol}\right)} \\ =1.42\times {10}^{15}\mathrm{MeV} \\ =\left(1.421\times {10}^{15}\mathrm{Mev}\right)\left(1.602\times {10}^{13}\mathrm{J}/\mathrm{MeV}\right) \\ \approx 227\mathrm{J} \end{gathered}$$

Therefore, the energy released in micro-explosionis 227 J.

Find the amount of TNT

(b)

The expression to calculate the amount of TNT is given by,

$m=\frac{E}{E_{TNT}}......\left(2\right)$

Substitute all the known values in equation (2).

$$\begin{gathered} m=\frac{227\mathrm{J}}{4.6\times {10}^6\mathrm{J}} \\ =4.93\times {10}^{-5}\mathrm{kg} \\ =49.3\mathrm{mg} \end{gathered}$$

Therefore, the amount of TNT required is 49.3 mg.

Find the power generated

(c)

The expression to calculate power generated is given by,

$P=\left(\frac{dN}{dt}\right)E.......\left(3\right)$

Substitute all the known values in equation (3).

$$\begin{gathered} P=\left(100s^{-1}\right)\left(227J\right) \\ =22700\mathrm{W} \\ =22.7\mathrm{KW} \end{gathered}$$

Therefore, the power generated is 22.7 kW.