I know entropy rules force my statement to be FALSE. But anyway I haven't been able to find the problem in the statement.
- We have a perfectly isolated cold reservoir at $T_c = 100 K$
- We have a room temperature of $T_r = 300 K$
- The optimal carnot heat engine efficiency will be: $\eta = 1- T_c/T_r = 0.66 $
Suppose we manage to achieve such an engine, with some losses, so only 60% efficiency. That means, each $100 W$ of heat we put into from room temperature. We achieve 60 W of useful energy, and 40 W of heat that go to the cold reservoir (therefore heating it up).
However, if we manage to have a very efficient heat pump (70%), with 60 W we will manage to pump out ($40 W - (0.7 * 60 W) = -2 W$). Therefore making the cold reservoir colder for free. Or generating 1.3W of energy out of room temperature.
Where is the fault in my logic?