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My question:

A satellite powered by a Carnot engine uses heat from a nuclear reactor at a fixed temperature T0. Heat is released into outer space via thermal radiation emitted by a set of fins at temperature T located on the outside of the satellite. The fins can be considered as ideal black bodies. Find the optimum temperature of the fins for which the Carnot engine will have the maximum power output and determine the engine efficiency in this case.

My Attempted Solution:

  • The cold reservoir is a black body.
  • It therefore radiates a power $P=\sigma T^4$
  • For the fins to remain in thermal equilibrium they must also receive a power $P=\sigma T^4$
  • So in the language of a carnot engine $Q_L=\sigma T^4$
  • From the carnot engine we also have the relation $\frac{Q_H}{Q_L}=\frac{T_0}{T}$
  • Substituting in we find the $Q_H=T_{0}\sigma T^3$
  • Then the work done is $W=Q_H-Q_L=T_{0}\sigma T^3-\sigma T^4$
  • To maximise this I differentiate with respect to $T$ and set to $0$, resulting with $3T_{0}\sigma T^2-4\sigma T^3=0$
  • Therefore we have that $T=\frac{3}{4}T_0$
  • Substituting into the efficiency formula $\eta=1-\frac{T_L}{T_H}=1-\frac{\frac{3}{4}T_0}{T_0}=1-\frac{3}{4}=25$%

My problem is that this seems too low. However, I can't spot a problem in the logic. I know this is a fairly basic treatment but I don't think that the question is asking for anything much more complicated. Any help would be much appreciated.

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closed as off-topic by user36790, ACuriousMind, user10851, CuriousOne, Kyle Kanos Mar 28 '16 at 10:21

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