Question

Say we have a standard thermal machine which operates in cycles. It absorbs energy in form of heat from a hot source, does some work and then transmits the remaining energy to the cold sink also in form of heat. The efficiency of this machine is given by:

η=1−|Q₂|/|Q₁|

Therefore, an ideal efficiency of η⟶1 could in theory be achieved if |Q₂|⟶0 (I know this is deemed impossible according to the Kelvin-Planck statement, but this impossibility is precisely what I don't understand).

Since the machine only interacts with the source and the sink through the exchange of heat, and there can only be a flow of heat ocurring naturally between two bodies if they have different temperatures (as far as I'm concerend, at least), then in order to achieve Q₂=0
we would simply need a sink whose temperature matched that of the thermal machine. This is especially simple if the cycle the machine is operating on uses isothermal processes, since the temperature of the system would be constant (although different for each process).

I guess where I'm wrong is on the fact that some extra energy would be required for the cold sink to change its temperature over time?

Answer 1

A thermal machine can never achieve an ideal efficiency of 1 because heat engines cannot have perfect conversion of heat into work due to the second law of thermodynamics. There will always be heat transfer to the cold sink, even if it has the same temperature as the thermal machine.

Step-by-step explanation:

The efficiency of a thermal machine operating in cycles is given by the equation η = 1 - |Q₂|/|Q₁|. An ideal efficiency of η→1 could be achieved if |Q₂|→0, meaning that the heat transfer to the cold sink approaches zero. However, achieving this is impossible due to the Kelvin-Planck statement of the second law of thermodynamics.

The second law states that heat engines cannot have perfect conversion of heat into work. Heat engines are designed to convert heat into work, and they require a temperature difference between the hot source and the cold sink. Therefore, even if the cold sink has the same temperature as the thermal machine, there will still be heat transfer to the sink, and the ideal efficiency of 1 cannot be achieved.

This means that there is a minimum amount of heat that cannot be used for work, and any heat that is transferred to the cold sink is considered wasted.