The Carnot cycle and heat engines

A heat engine takes in heat at a high temperature, converts some of it into work, and dumps the rest at a low temperature. The obvious engineering question is how small that dumped fraction can be made. Carnot's answer, reached before anyone had written down the second law, is that it has a hard floor set by nothing but the two temperatures: efficiency at best equals 1 − T_cold/T_hot, with the temperatures in kelvin.

The crucial point is what kind of limit this is. It is not a statement about friction, turbulence or imperfect insulation, all of which are real and all of which push a practical engine further below the ceiling. Even a perfectly frictionless, perfectly insulated engine cannot exceed it. The reason is entropy: taking heat Q from the hot reservoir lowers its entropy by Q/T_hot, and the only way to balance the books is to deposit entropy somewhere cooler. The exhaust heat is not waste in the sense of a design flaw — it is the payment the second law requires.

Two consequences follow. Efficiency depends on the ratio of the temperatures, so lowering the cold side helps exactly as much as raising the hot side; power stations care about their cooling water for this reason. And a real engine matching the Carnot limit would have to run infinitely slowly, through a sequence of reversible steps, producing no power at all. Practical design is therefore always a trade between efficiency and output, never a hunt for the missing hundred per cent.