Heat Engines

A heat engine is any device that converts heat into continuous, usable mechanical work by carrying a working substance (steam, or a fuel-air mixture, for example) through a repeating cyclic process -- a sequence of steps that returns the working substance to exactly the state it started in, so that the whole sequence can then repeat itself, cycle after cycle, producing a continuous, steady output of work rather than just a single one-off burst.

In every complete cycle, a heat engine draws in a quantity of heat $Q_1$ from a hot reservoir (called the source, at temperature $T_1$), and, after doing what useful work it can with that energy, rejects a smaller quantity of heat $Q_2$ to a cold reservoir (called the sink, at a lower temperature $T_2$). Because the working substance returns to its exact original state at the end of every cycle, its internal energy is exactly the same at the end of the cycle as it was at the beginning, so $\Delta U = 0$ for the cycle taken as a whole (even though $U$ certainly changes at intermediate stages within the cycle). Applying the first law over one complete cycle, $Q_{\text{net}} = \Delta U + W$, with $\Delta U = 0$ and $Q_{\text{net}} = Q_1 - Q_2$, gives the net work done by the engine per cycle as

Efficiency. The efficiency of a heat engine is defined as the fraction of the heat drawn in from the source that actually ends up as useful work, rather than being rejected, unused, as waste heat to the sink:

Since $Q_2$ can never physically be reduced all the way to zero (the Kelvin-Planck statement of the second law forbids exactly this, Section 11.9), the efficiency of every real heat engine is always strictly less than $1$ (that is, less than 100%) -- some heat is unavoidably wasted, rejected to the sink, in every single cycle of every heat engine that has ever been built or ever could be built. …