Answer
a) $\dot{Q}_L=4982\ kJ/min$
b) $\dot{Q}_T=5782\ kJ/min$
Work Step by Step
For the engine:
$\eta=1-\frac{T_L}{T_H}$
Given $T_L=300\ K,\ T_H=1173\ K$
$\eta=0.744$
$\eta = \dot{W}_e/\dot{Q}_H$
With $\dot{Q}_H=800\ kJ/min$
$\dot{W}_e=592.5\ kJ/min\ \rightarrow$ input to the refrigerator
$\dot{Q}_H=\dot{W}_e+\dot{Q}_L$
$\dot{Q}_L=204.8\ kJ/min$
For the refrigerator:
$COP_R=\dfrac{1}{1-\frac{T_L}{T_H}}$
Given $T_H=300\ K,\ T_L=268\ K$
$COP_R=8.37$
$COP_R=\dot{Q}_L/\dot{W}_i$
$\dot{Q}_L=4982\ kJ/min$
$\dot{Q}_H=\dot{W}_e+\dot{Q}_L$
$\dot{Q}_H=5577.2\ kJ/min$
Total heat to the environment:
$\dot{Q}_T=\dot{Q}_{L,Engine}+\dot{Q}_{H,Refrigeator}$
$\dot{Q}_T=5782\ kJ/min$
