Prelecture 31: Slide 5

We have just seen that there is a limit for the efficiency of a heat engine that is proportional to the temperature difference of the reservoirs.

For example, a heat engine operating between reservoirs of boiling water and freezing water has a maximum efficiency of 27%. We would now like to look at realistic engines and calculate their efficiencies to see how they compare to this Carnot limit.

In order to make this calculation, we will use the quasi-static approximation, that all changes are done slowly enough so that the gas is essentially at equilibrium at all times, meaning that it has a well-defined pressure, temperature, volume, and entropy at all times.

Making this quasi-static approximation allows us to represent the cycle on a PV diagram. That is, at any time during the cycle, the gas will have a well-defined pressure and volume so that the state of the gas at that time can be represented as a unique point in the Pressure-Volume plane.

Here we see the PV diagram for a particular engine, the Stirling cycle, that we will study in more detail in the next few slides.

For now, we will simply note that the work done by the engine during one cycle is captured in this diagram as the area enclosed by the curve. That is, the work done by the gas is always equal to the integral of P dV, so that we can simply add up the work done by the gas during each step. No work is done in Step 1 or Step 3 since the volume does not change. The engine does positive work in Step 2 equal to the area under the curve as the gas expands. The engine does negative work in Step 4 equal to the area under the curve as it is compressed.

We will look at this cycle in more detail on the next slide.