Carnot's Theorem & Irreversible Heat Engines

[nikolafmf]

(Part of the) Carnot's theorem:

"All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs."

is proved with irreversible heat engine driving Carnot refrigerator. In the proof (for example M. Zemansky, R. Dittman, Heat and Thermodynamics, 1997, page 175) it is used for efficiency of the refrigerator: η=W/Q_H, W is work done to refrigerator, Q_H is heat given to hotter reservoir. But this equation is true for the heat engine, where W would be work done by engine and Q_H heat taken from the hotter engine. For refrigerator, efficiency, as far as I know, is η=Q_L/W, where W is work done to refrigerator, Q_L is heat taken from colder reservoir. Where is my mistake, if any?

 

[anigeo]

a refrigerator is actually a carnot engine working in the opp. direction.hence the expression for efficiency of a refrigerator(as a heat engine not a cooling machine)is same as that of a regular carnot heat engine.

 

[nikolafmf]

And why efficiency of Carnot engine working in the opposite direction is the same as efficiency of Carnot engine working in the "normal" direction?