The steam generator at your local electric power plant works by boiling water to produce high-pressure steam that spins a turbine (which then spins a generator to produce electricity). That is, the steam pressure is doing work. The steam is then condensed to liquid water and pumped back to the boiler to start the process again. There are two crucial ideas here. First, the device works in a cycle, with the water returning to its initial conditions once a cycle. Second, heat is transferred to the water in the boiler, but heat is transferred out of the water in the condenser.

Car engines and steam generators are examples of what we call heat engines. A heat engine is any closed-cycle device that extracts heat $Q_{\text{H}}$ from a hot reservoir, does useful work, and exhausts heat $Q_{\text{C}}$ to a cold reservoir. A closed-cycle device is one that periodically returns to its initial conditions, repeating the same process over and over. That is, all state variables (pressure, temperature, thermal energy, entropy, and so on) return to their initial values once every cycle. Consequently, a heat engine can continue to do useful work for as long as it is attached to the reservoirs.

FIGURE 21.5 is the energy-transfer diagram of a heat engine. Unlike the forbidden “perfect engine” of Figure 21.4, a heat engine is connected to both a hot reservoir and a cold reservoir. You can think of a heat engine as “siphoning off” some of the heat that moves from the hot reservoir to the cold reservoir and transforming that heat into work—some of the heat, but not all.

FIGURE 21.5 The energy-transfer diagram of a heat engine.

Because the temperature and thermal energy of a heat engine return to their initial values at the end of each cycle, there is no net change in $E_{\text{th}}\text{:}$

Consequently, the first law of thermodynamics for a full cycle of a heat engine is $(\text{Δ}{E}_{\text{th}}){}_{\text{net}}=Q-W_{\text{s}}=0$.

Let’s define $W_{\text{out}}$ to be the useful work done by the heat engine per cycle. The first law applied to a heat engine is

This is just energy conservation. The energy-transfer diagram of Figure 21.5 is a pictorial representation of Equation 21.4.

For practical reasons, we would like an engine to do the maximum amount of work with the minimum amount of fuel. We can measure the performance of a heat engine in terms of its thermal efficiency $\eta$ (lowercase Greek eta), defined as

Using Equation 21.4 for $W_{\text{out}}$, we can also write the thermal efficiency as

FIGURE 21.6 illustrates the idea of thermal efficiency.

FIGURE 21.6 $\eta$ is the fraction of heat energy that is transformed into useful work.

A perfect heat engine would have ${\eta }_{\text{perfect}}=1$. That is, it would be 100% efficient at converting heat from the hot reservoir (the burning fuel) into work. You can see from Equation 21.6 that a perfect engine would have no exhaust $(Q_{\text{C}}=0)$ and would not need a cold reservoir. Figure 21.4 has already suggested that there are no perfect heat engines, that an engine with $\eta =1$ is impossible. A heat engine must exhaust waste heat to a cold reservoir. It is energy that was extracted from the hot reservoir but not transformed to useful work.

Practical heat engines, such as car engines and steam generators, have thermal efficiencies in the range $\eta \approx 0.1-0.5$. This is not large. Can a clever designer do better, or is this some kind of physical limitation?