Answer
$\dot{Q}_{\mathrm{gen,} \min }=12.3\text{ kW}$
Work Step by Step
The maximum COP that this refrigeration system can have is
$$
\mathrm{COP}_{\mathrm{R}, \max }=\left(1-\frac{T_0}{T_s}\right)\left(\frac{T_L}{T_0-T_L}\right)=\left(1-\frac{298 \mathrm{~K}}{368 \mathrm{~K}}\right)\left(\frac{275}{298-275}\right)=2.274
$$ Thus, $$
\dot{Q}_{\mathrm{gen,} \min }=\frac{\dot{Q}_L}{\mathrm{COP}_{\mathrm{R}, \max }}=\frac{28 \mathrm{~kW}}{2.274}=12.3 \mathrm{~kW}
$$
