# How can you explicate the efficiency of a turbofan engine with Carnot's theorem?

From Carnot's theorem, we know that efficiency is

$$\eta = 1-\dfrac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ the temperature of the hot reservoir. But what do we have to consider for both?

Is $T_c$ the temperature of the air coming into the compressor from the fan or the temperature of air coming into the combustion chamber after being compressed and heated by the compressor?

Is $T_h$ the temperature of the gases in the combustion chamber (1500C°) or at the exhaust nozzle?

With that out of the way, let's look at the Carnot cycle. This cycle assumes a heat source that can provide arbitrary amounts of energy at a certain temperature $T_h$, and a heat sink that can sink arbitrary amounts of heat at a temperature $T_c$. These are the isothermal parts of the cycle. To get the working fluid to either temperature, adiabatic compression is used.
Sometimes you hear that $T_h$ is the hottest temperature in the cycle, and $T_c$ the coldest temperature. While this is often true in practice, this does not need to be the case for theoretical considerations. The only thing that matters for the Carnot efficiency is the temperature of the heat sinks/sources. You can't get more efficient than the Carnot cycle by hooking your engine up to a fridge. Indeed, the Carnot cycle is just a way to express the Second Law of Thermodynamics, which states that entropy in a closed system can never increase.