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Seda
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Homework Statement
Consider a Carnot Cycle that absorbs heat from the hot reservoir at Thw and expels to the cold reservoir at Tcw just below Th and above Tc respectively.
a.) Derive an equation that relates the four temperatures.
b.) Using the first and second law, write down an expression for power entirely in terms of the four temperatures (and the constant K) and use the results of (a) to eliminate Tcw.
c.) Show that the expression found in (b) has a maximum value at Thw=.5(Th + sqrt[ThTc]). Find a corresponding expression for Tcw.
d.) Show that the efficiency of the engine is 1-sqrt(Tc/Th)
Homework Equations
Qh/t = K(Th-Thw)
Qc/t = K(Tcw-Tc)
K is the proportionality constant are the same for both processes.
The Attempt at a Solution
a.) I solved this knowing that the cycle's change in entropy is zero using S=Q/T.
My answer for a is: Qh[(1/Thw)-(1/Th)]+Qc[(1/Tc)-(1/Tcw)]=0
b.) This is where I'm stuck. I can solve the equation above for Tcw=1/[(Qh/Qc)[(1/Thw)-(1/Th)]+(1/Tc)]
I think I can substitute this monster above into this equation for power for Tcw
P=K(Th-Thw) - K(Tcw-Tc)
to solve part (b).
However, once i start taking the derivative of this to find a max for part c, the math gets ridiculous, I believe I have made an error or can somehow simplify this before attempting part c. Any tips?