Thermal Efficiency

Thermal efficiency is a bookkeeping question with an uncomfortable answer: of all the heat you paid for, how much came back out as work? $\eta = W/Q_h$, a ratio of two energies, so it is dimensionless and sits between 0 and 1. The uncomfortable part is that it is never close to 1, and not because engineers have failed. A heat engine cannot convert all of $Q_h$ into work no matter how well it is built, because to run a cycle at all it has to dump some heat into a colder place. That requirement is the second law, and it is why the missing energy is not waste in the sense of carelessness — it is rent.

A 1000 MW thermal station delivering 380 MW to the grid runs at $\eta = 380/1000 = 0.38$. The other 620 MW is not lost in any physical sense; it leaves at low temperature through the condenser and has to go somewhere real — a river, a lake, or the plume off a cooling tower. That 620 MW is the reason large thermal plants are built beside water, and the reason a heat wave can force one to throttle back.

What sets the ceiling is on the neighbouring Carnot page: $\eta_{max} = 1 - T_c/T_h$, with both temperatures absolute. A steam cycle at 550 °C rejecting to a 27 °C condenser has a Carnot limit of $1 - 300/823 = 63.5\%$, so a real 38% plant is achieving about six-tenths of what thermodynamics allows. That ratio — actual over Carnot — is the honest measure of engineering quality, and it is far more flattering than the raw efficiency. It also explains why every gain in this field comes from raising $T_h$: combined-cycle plants reach 60% by putting a gas turbine at 1400 °C in front of the steam cycle and feeding the steam cycle its exhaust.

The error that matters most is comparing two efficiencies computed on different denominators. For fuel-burning equipment, $Q_h$ can be the higher heating value, which counts the energy recovered when the combustion water vapour condenses, or the lower heating value, which does not. The two differ by about 10% for natural gas. This is why condensing boilers are sometimes advertised above 100% efficiency: that figure is on LHV, and it breaks no law — it is a fraction with a deliberately small denominator. A 95% AFUE furnace and a "108%" boiler may be the same machine described twice. Always ask which heating value, especially when comparing a quoted number against one you calculated yourself.

Three more distinctions worth keeping straight. $W$ must be net work: an engine's output minus whatever the feed pumps, compressors and auxiliaries consume, or you are counting energy that never left the building. $Q_h$ is the heat that actually entered the working fluid, which for a fired system is the fuel energy less the stack loss, not the fuel energy itself. And a heat pump's coefficient of performance is not a thermal efficiency and routinely reads 3 or 4 — no contradiction, because a heat pump is not converting heat into work, it is spending work to move heat, and the heat it moves was already there. Putting a COP and an η on the same axis is comparing two different questions.