BEAR $2)(30$pts$)$ Background: A consistent energy source is needed to power deep space probes and rovers on Mars

$($including the recently landed Perseverance$)$ that are too far from the sun's energy. One device is the Radioisotope

Thermoelectric Generator $($pictured below$).$ These use the radioactive decay of Plutonium $238$ to produce thermal energy that

can be converted to electricity.

A simplified model is shown below $($Disclaimer: I made this up$).$ The Plutonium is encased in an aluminum heat sink

to dissipate the energy. The dimensions and properties of the Pu and heat sink are on the diagram. The heat sink

has $10$ fins that are $70\%$ efficient. Each fin has a corrected length of $0\mathrm{.}02$ m and a width of $0\mathrm{.}005$ m $.$ Thermal

energy is uniformly generated in the Pu $238$ at a rate of ${10\mathrm{.}7}^{*}{10}^6\frac{W}{m^3}.$ Assume the reactor is operating at

steady state in the given Martian environment $(T=206K$ and an overall heat transfer coefficient of $40\frac{W}{m^2}K).$

Neglect heat transfer from the ends.

a$.)$ Determine the temperature at the surface of the fuel if all the power generated is dissipated through heat

transfer.

b$.)$ Determine the temperature at the center of the fuel if all the power generated is dissipated through heat

transfer.

c$.)$ Bonus: Using your previous results, determine the maximum theoretical power that could be converted

to electricity from a device that operates between the surface of the Plutonium and the ambient

atmosphere. $($These devices are the "thermocouples" shown in the picture $-$ hard to see$)$

Aadiolsotope Thermopiectric