Lessons · Engineering · efficiency of a heat engine
The ceiling on a heat engine
No engine running between a hot source at T_h and a cold sink at T_c can beat η = 1 − T_c / T_h, with both temperatures in kelvin.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
A vendor promises a waste-heat generator at 60 % efficiency from 150 °C exhaust. Kelvin says the ceiling for any machine on those temperatures is about 30 %, so the promise is not optimism; it is impossible, and you can say so in the first meeting.
How to think about it
Convert both temperatures to kelvin. Divide cold by hot, subtract from one. That is the best any engine could do; a real one does perhaps two thirds of it. Lowering T_c helps more than raising T_h by the same amount.
Worked example
η_max = 1 − T_c / T_h, temperatures in kelvinThe Carnot limit.
Steam at 400 °C, river at 20 °C: T_h = 673.15 K, T_c = 293.15 KKelvin first.
η_max = 1 − 293.15 / 673.15 = 1 − 0.4355 = 0.5645, or 56.5 %The ceiling on those temperatures.
A real plant on those temperatures might make 38 %No real engine reaches the ceiling. The number tells you what is possible, not what you will get.
Your turn
T_h = 800 K, T_c = 300 K. Write the ceiling.
η = 1 − 300 / = 0.625
Solve one, graded on the server
The trap
Kelvin, always. 1 − 20/400 = 0.95 is a 95 % engine, which does not exist; in kelvin the same engine's ceiling is 56 %.