Looking at the equation for Carnot efficiency, I notice that as the temperature of the heat sink approaches zero, the efficiency approaches unity:
$$
\eta_{rev} = 1 – \frac{0}{T_H} = 1
$$
Seeing as the efficiency of a heat engine is the ratio between the heat it is absorbing and the work it outputs, an efficiency of 1 indicates that all heat absorbed is output as work. By first law, this implies that the engine is rejecting no heat to the low temperature sink.
This result doesn't make any sense to me. Why would a decreasing heat sink temperature result in less heat rejected?
To explain my confusion somewhat hand-wavily: if the temperature of the two reservoirs is equal, we end up with no heat transfer, and therefore $Q_L$ is zero. As we deviate from this case of reservoirs with equal temperature (which is what happens if you decrease $T_L$ while holding $T_H$ constant), why is it that we once again approach the case of $Q_L$ equals zero?
Best Answer
The answer can be found by looking at the details of how Carnot engines, with their ideal gas ($PV=NkT$) working fluid, work. Suppose we start at the end of the isothermal expansion. The next step is to adiabatically expand the piston until $T_c$ is reached. When $T_c=0$, though, that requires $V\rightarrow \infty$. We are then supposed to isothermally compress the gas until it is on the desired adiabatic curve. The ideal gas adiabats obey $PV^\gamma=\mathrm{constant}$, with $\gamma > 0$, so all of the adiabats intersect at $V=\infty$. Because of this property the isothermal compression is, effectively, cut out of the process. Doing this required that the piston have infinite volume with zero outside pressure into which it could expand, and the ability to do it in a finite amount of time without letting the internal gas reach an out of equilibrium state.
From a more realistic perspective, where $T_h \gg T_c>0$, it's just a question of the adiabatic expansion removing most of the internal energy of the working fluid, so there isn't much left to reject into the cold bath to get onto the compression adiabat.