Prove:β (heat pump) is always less or equal to β(Carnot HP)

[mek09e]

Hello All!

My professor in thermodynamics showed us the proof of the Carnot theory using integrals and a temp vs. entropy plot for a heat engine cycle. We haven't actually learned about entropy yet, so can someone help me understand how this translates into the coefficient of performance β for a heat pump? We were given the rule that β≤β(Carnot) for heat pumps and refrigerators, but I can't prove this is true on my own. Any explanation is appreciated :)

 

[DrClaude]

Don't you have a textbook?

Basically, you start from the definition of the coefficient of performance and use the 1st law to write it in terms of $Q_C$ and $Q_H$, the heat coming from the cold reservoir and that going to the hot reservoir, respectively. Then, you use the 2nd law to translate $Q_C$ and $Q_H$ to $T_C$ and $T_H$. This gives you the highest $\beta$ possible according to the 2nd law. Then you prove that a Carnot cycle working between $T_C$ and $T_H$ has a value of $\beta$ that is the highest possible. Therefore, $\beta \le \beta(\mathrm{Carnot})$.