Question

Prove that the COP of a reversible refrigerator operating between two given temperature is the maximum.

Answer 1

Considering two thermal energy reservoirs, one at a higher temperature of$T_H$ compared to the other at $T_L$ , if we operate a reversible refrigeration cycle between the reservoirs then its coefficient of performance is given by,

$COP_{R,rev}=T_L/T_H-T_L$

This is the basically the maximum COP a refrigeration cycle can achieve operating between two thermal energy reservoirs at $T_H$ and $T_L$ .Since a Carnot refrigerator is a reversible refrigeration cycle the above expression for$\space COP_{R,rev}$ is valid for the reversed Carnot cycle.

If we consider any other reversible refrigeration cycle operating between the same two temperature limits it should have a similar COP as given by the above expression.Now considering a reversed Brayton cycle which is a reversible cycle (all processes are internally reversible) operating between two TERs at $T_H$ and $T_L$.