Chapter 1, Problem 1.38P

An internally reversible refrigerator has a modified coefficient of performance accounting for realistic heat transfer processes of

${\text{COP}}_m=\frac{q_{in}}{\stackrel{.}{W}}=\frac{q_{in}}{q_{out}-q_{in}}=\frac{T_{e,i}}{T_{h,i}-T_{c,i}}$ where $q_{in}$ is the refrigerator cooling rate, $q_{out}$ is the heat rejection rate, $\stackrel{.}{W}$ and is the power input. Show that ${\text{COP}}_m$ can be expressed in terms of the reservoir temperatures $T_c$ and $T_h,$ the cold and hot thermal resistances $R_{t,c}$ and $R_{t,h},$ and $q_{in},$ as ${\text{COP}}_m=\frac{T_c-q_{in}{R}_{\text{tot}}}{T_h-T_c+q_{in}{R}_{\text{tot}}}$ where $R_{\text{tot}}=R_{t,c}+R_{i,h}.$ Also, show that the power input may be expressed as $\stackrel{.}{W}=q_{in}\frac{T_h-T_c+q_{in}R\text{tot}}{T_c-q_{\text{in}}{R}_{\text{tot}}}$