Question #42557

A Carnot refrigerator extracts heat from a cold reservoir at a constant temperature $T_C$. It rejects heat to a hot environment whose temperature fluctuates harmonically as $T_H(t) = T_0 + \Delta T \cos(\omega t)$, where $T_0 > T_C$ and $\Delta T < T_0 - T_C$. The rate of heat extraction (cooling load) is actively modulated as $\dot{Q}_{in}(t) = \dot{Q}_0 [1 + \alpha \sin(\omega t + \phi)]$, where $\dot{Q}_0 > 0$ and $0 < \alpha < 1$. The cycle-averaged Coefficient of Performance ($\overline{\text{COP}}$) is defined as the ratio of the total heat extracted to the total work input over one full period $T = \frac{2\pi}{\omega}$. Determine the phase constant $\phi \in [0, 2\pi)$ that maximizes the cycle-averaged $\overline{\text{COP}}$.