Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency

Now let us consider a device that uses heat transfer to do work. As noted in the previous section, such a device is called a heat engine, and one is shown schematically in Figure 8.10.3(b). Gasoline and diesel engines, jet engines, and steam turbines are all heat engines that do work by using part of the heat transfer from some source. Heat transfer from the hot object (or hot reservoir) is denoted as $Q_{\text{h}}$, while heat transfer into the cold object (or cold reservoir) is $Q_{\text{c}}$, and the work done by the engine is $W$. The temperatures of the hot and cold reservoirs are $T_{\text{h}}$ and $T_{\text{c}}$, respectively.

Because the hot reservoir is heated externally, which is energy intensive, it is important that the work is done as efficiently as possible. In fact, we would like $W$ to equal $Q_{\text{h}}$, and for there to be no heat transfer to the environment ($Q_{\text{c}}=0$). Unfortunately, this is impossible. The second law of thermodynamics also states, with regard to using heat transfer to do work (the second expression of the second law):

A cyclical process brings a system, such as the gas in a cylinder, back to its original state at the end of every cycle. Most heat engines, such as reciprocating piston engines and rotating turbines, use cyclical processes. The second law, just stated in its second form, clearly states that such engines cannot have perfect conversion of heat transfer into work done. Before going into the underlying reasons for the limits on converting heat transfer into work, we need to explore the relationships among $W$, $Q_{\text{h}}$, and $Q_{\text{c}}$, and to define the efficiency of a cyclical heat engine. As noted, a cyclical process brings the system back to its original condition at the end of every cycle. Such a system’s internal energy $U$ is the same at the beginning and end of every cycle—that is, $\mathrm{\Delta }U=0$. The first law of thermodynamics states that

where $Q$ is the net heat transfer during the cycle ($Q=Q_{\text{h}}-Q_{\text{c}}$) and $W$ is the net work done by the system. Since $\mathrm{\Delta }U=0$ for a complete cycle, we have

so that

Thus the net work done by the system equals the net heat transfer into the system, or

just as shown schematically in Figure 8.10.3(b). The problem is that in a heat-engine cycle, there is some heat transfer $Q_{\text{c}}$ to the environment—and usually a very significant amount at that.

In a heat engine, only part of the heat input is converted to work. We define conversion efficiency $\mathrm{\text{Eff}}$ to be the ratio of useful work output to the energy input (or, in other words, the ratio of what we get to what we spend). In that spirit, we define the efficiency of a heat engine to be its net work output $W$ divided by heat transfer to the engine $Q_{\text{h}}$; that is,

Since $W=Q_{\text{h}}-Q_{\text{c}}$ in a cyclical process, we can also express this as

making it clear that an efficiency of 1, or 100%, is possible only if there is no heat transfer to the environment ($Q_{\text{c}}=0$). Here $Q_{\mathrm{h}}$ and $Q_{\mathrm{c}}$ are positive amounts of heat transfer. The direction of heat transfer is indicated by a plus or minus sign. For example, $Q_{\text{c}}$ is out of the system and so is preceded by a minus sign.

With the information given in Example 8.10.1, we can find characteristics such as the efficiency of a heat engine without any knowledge of how the heat engine operates, but looking further into the mechanism of the engine will give us greater insight. Figure 8.10.4 illustrates the operation of the common four-stroke gasoline engine. The four steps shown complete the engine’s operating cycle, replacing the exhaust gases with a fresh gasoline-air mixture.

The ideal Otto cycle shown in Figure 8.10.5 models a fixed amount of gas undergoing two adiabatic and two constant-volume processes. It approximates the operation of a four-stroke internal combustion engine, although its paths do not correspond exactly to the strokes of the engine.

The adiabatic process AB corresponds to the nearly adiabatic compression stroke of the gasoline engine. In both cases, work is done on the system (the gas mixture in the cylinder), increasing its temperature and pressure. Along path BC of the Otto cycle, heat transfer $Q_{\text{h}}$ into the gas occurs at constant volume, causing a further increase in pressure and temperature. This process corresponds to burning fuel in an internal combustion engine, and takes place so rapidly that the volume is nearly constant. Path CD in the Otto cycle is an adiabatic expansion that does work on the outside world, just as the power stroke of an internal combustion engine does in its nearly adiabatic expansion. The work done by the system along path CD is greater than the work done on the system along path AB, because the pressure is greater, and so there is a net work output. Along path DA in the Otto cycle, heat transfer $Q_{\text{c}}$ from the gas at constant volume reduces its temperature and pressure, returning it to its original state. In an internal combustion engine, this part of the model represents the exhaust of hot gases and the intake of a cooler air-gasoline mixture. The actual engine carries energy out with the exhaust; the ideal cycle represents this by heat transfer from the gas.

The net work done by a cyclical process is the area inside the closed path on a PV diagram, such as that inside path ABCDA in Figure 8.10.5. A heat-engine cycle must reject some of its heat input. In the Otto cycle, this heat transfer occurs along path DA. Simply retracing the reversible adiabatic expansion would give no net work output. Lower pressure along AB reduces the compression work, while higher pressure along CD increases the expansion work. These changes increase the net work output of the Otto cycle. The maximum possible efficiency of a heat engine operating between two heat reservoirs increases when the hot reservoir is hotter or the cold reservoir is colder. In the next section, we shall see how these reservoir temperatures set the upper limit to efficiency.