The Second Law: Heat Engines, Refrigerators and the Carnot Limit

1Why not 100%?

In Unit 6 you saw that a steam power station turns only about a third of its fuel's energy into electricity, and sends about half of it to the cooling water. Engineers are not careless: no engine, however well built, could turn all the heat it takes in into work. A cup of hot tea cools to room temperature, but a cold cup never warms itself by taking heat from a cooler room, although that would not break the first law. Something beyond the conservation of energy decides which way heat flows and how much of it can become work. That is the second law of thermodynamics, the subject of this lesson, together with the engines, refrigerators and heat pumps it governs.

2Heat engines

A heat engine is a device that works in a cycle, taking heat $Q_{H}Q\_H$ from a hot reservoir at temperature $T_{H}T\_H$ (burning fuel, hot steam), turning part of it into work $WW$, and giving out the rest, $Q_{C}Q\_C$, to a cold reservoir at $T_{C}T\_C$ (the air, a river, cooling water), as in Figure 4.1. Over each complete cycle the working gas returns to its starting state, so $\mathrm{\Delta }U=0\backslash Delta\; U\; =\; 0$ and the first law gives

A refrigerator runs the cycle backwards: work $WW$ from an electric motor drives heat $Q_{C}Q\_C$ out of the cold inside of the fridge and delivers $Q_H=Q_C+WQ\_H\; =\; Q\_C\; +\; W$ to the warmer kitchen. Its usefulness is measured by its coefficient of performance, $\text{COP}=\frac{Q_C}{W}\backslash text\{COP\}\; =\; \backslash frac\{Q\_C\}\{W\}$. A heat pump is the same machine used to warm a building; then the useful output is $Q_{H}Q\_H$, and $\text{COP}=\frac{Q_H}{W}\backslash text\{COP\}\; =\; \backslash frac\{Q\_H\}\{W\}$.

Figure 4.1: Energy flows in a heat engine and in a refrigerator or heat pump. Arrow widths suggest the sizes of the flows.

3The Carnot engine: the best possible

In 1824 the French engineer Sadi Carnot asked what limits the efficiency of an engine. He imagined an ideal, frictionless, perfectly slow engine whose gas goes round a cycle of two isothermal and two adiabatic changes (Figure 4.2), now called the Carnot cycle. Heat enters only at $T_{H}T\_H$ and leaves only at $T_{C}T\_C$, and for this cycle $\frac{Q_C}{Q_H}=\frac{T_C}{T_H}\backslash frac\{Q\_C\}\{Q\_H\}\; =\; \backslash frac\{T\_C\}\{T\_H\}$, with temperatures in kelvin. Its efficiency is therefore

No engine working between the same two temperatures can be more efficient (if one could, it could be used to drive a Carnot engine backwards and move heat from cold to hot with no work, breaking the second law). Real engines, with friction, heat leaks and fast changes, always do worse. The lesson for engineers is clear: make $T_{H}T\_H$ as high and $T_{C}T\_C$ as low as the materials and surroundings allow. The best possible refrigerator has $\text{COP}=\frac{T_C}{T_H-T_C}\backslash text\{COP\}\; =\; \backslash frac\{T\_C\}\{T\_H\; -\; T\_C\}$.

Figure 4.2: The Carnot cycle: two isothermal and two adiabatic changes. The net work per cycle is the enclosed area.

4Petrol and diesel engines

Car, lorry and motorcycle engines are internal combustion engines: the fuel burns inside the cylinder, and most work in a four-stroke cycle.

Table 4.1: The four strokes of petrol and diesel engines (typical compression ratios).

For an idealised petrol (Otto) cycle with compression ratio $r=\frac{V_{\max }}{V_{\min }}r\; =\; \backslash frac\{V\_\{\backslash max\}\}\{V\_\{\backslash min\}\}$, the efficiency is $\eta =1-\frac{1}{r^{\gamma -1}}\backslash eta\; =\; 1\; -\; \backslash frac\{1\}\{r^\{\backslash gamma\; -\; 1\}\}$. The higher the compression ratio, the higher the efficiency. Diesel engines use much higher compression ratios, which is the main reason they are more efficient than petrol engines, although real engines of both kinds reach only roughly half of their ideal efficiencies.