Carnot Engine - Carnot Refrigerator

[shingetsunohimitsu]

This is just a common problem solving question, which I cannot make, I suspect because I'm bad at maths. Anyway, I want to see it done, hopefully to grasp it. I am doing physics, so, it's an exercise from one of my books.

18-11. Show that the efficiency e of a Carnot engine and the perfomance coefficient K of a Carnot refrigerator are erlated by K=(1-e)/e. The engine and the refrigerator operate between the hot and cold reservoirs.

e= Th-Tc/Th
K= Tc/Th-Tc

[wisky40]

o.k. I'm going to give it a try.
e=(Th-Tc)/Th => e= (Th/Th)-(Tc/Th)=1-(Tc/Th) => Tc/Th=1-e
k=Tc/(Th-Tc) then, dividing the numerator and also the denominator by Th.
Tc/Th Tc/Th 1-e 1-e 1-e
k=------------- = ---------- = --------- = ------- = -------
(Th-Tc)/Th 1-(Tc/Th) 1-(1-e) 1-1+e e

[wisky40]

well, I was typing in a different way, but it didn't work because my Latex is not good

k=(Tc/Th)/(1-(Tc/Th)) = (1-e)/(1-1+e) = (1-e)/e