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Question:

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a) What is the area of the steel plate interface (in $m^2$) with the fluid conduit if the heat enters the steel plate at a temperature (T1) of 980K? Assume rest of the conduit surface (besides the steel plate interface) is well insulated.

b) What is the Carnot efficiency of the heat engine (in %) based on the heat entering the engine at T2 if the same amount heat is input to the heat engine?

Attempt I'm a bit confused as to how to approach both (a) and (b). Since this is a heat engine, the $\Delta U = 0$ so $Q_{in} - Q_{out} = W$. So is the correct way of approaching (a) is to conclude that $Q_{in}$ at $T_2$ is $500kW$ and then utilizing the fact that $500kW = h(A)(1000K - 980K)$ to solve for $A$?

With that reasoning, the Carnot efficiency for (b) would be 20%.

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I think your thought on part a) is correct. But for part b), if we assume the plate to be of uniform temperature, then the efficiency may turn out to be 1- (300/980)= 69.4%. If we consider the finite conductivity of steel, then also I dont think the given values could be obtained.

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