Answer
a) $\dot{Q}_L=4200\ Btu/min$
b) $\dot{Q}_T=4900\ Btu/min$
Work Step by Step
For the engine:
$\eta=1-\frac{T_L}{T_H}$
Given $T_L=540°R,\ T_H=2160°R$:
$\eta=0.75$
Since $\eta = \dot{W}_e/\dot{Q}_H,\ \dot{Q}_H=700\ Btu/min$:
$\dot{W}_e=525\ Btu/min$, input for the refrigerator.
$\dot{Q}_H=\dot{W}_e+\dot{Q}_L$
$\dot{Q}_L=175\ Btu/min$
For the refrigerator:
$COP_R=\dfrac{1}{1-\frac{T_L}{T_H}}$
Given $T_L=480°R,\ T_H=540°R$:
$COP_R=8.0$
$COP_R=\dot{Q}_L/\dot{W}_i$
$\dot{Q}_L=4200\ Btu/min$
$\dot{Q}_H=\dot{W}_e+\dot{Q}_L$
$\dot{Q}_H=4725\ Btu/min$
Total heat rejection:
$\dot{Q}_T=\dot{Q}_{H,E}+\dot{Q}_{L,R}$
$\dot{Q}_T=4900\ Btu/min$
