Answer
a) $\dot{Q}_H=61.1\ kJ/min$
b) $\dot{Q}_T=311\ kJ/min$
Work Step by Step
For the refrigerator:
$COP_R=\dfrac{1}{1-\frac{T_L}{T_H}}$
$T_H=300\ K,\ T_L=258\ K$
$COP_R=6.143$
$COP_R=\dot{Q}_L/\dot{W}_i$
$\dot{Q}_L=250\ kJ/min$
$\dot{W}_i=40.7\ kJ/min$: output of the engine
$\dot{Q}_H=\dot{W}_i+\dot{Q}_L$
$\dot{Q}_H=290.7\ kJ/min$
For the engine:
$\eta=1-\frac{T_L}{T_H}$
$T_L=300\ K,\ T_H=900\ K$
$\eta=0.667$
$\eta = \dot{W}_e/\dot{Q}_H$
$\dot{Q}_H=61.1\ kJ/min$
$\dot{Q}_H=\dot{W}_e+\dot{Q}_L$
$\dot{Q}_L=20.4\ kJ/min$
Total heat rejection
$\dot{Q}_T=\dot{Q}_{L,E}+\dot{Q}_{H,R}$
$\dot{Q}_T=311\ kJ/min$
