Though we have shown the net flow of energy and entropy, we haven't proposed a more specific mechanism for the heat engine. The most basic cycle is known as the Carnot cycle, and is simple if not completely accurate for a real engine. Still, it is beneficial to see a simplified picture to understand the basic concepts.
The Carnot cycle consists of four phases. Refer to as we trace the steps of the cycle. At point A, the gas (it needn't be a gas necessarily) is at temperature τh with entropy σL where the latter represents the lowest entropy attained by the system during the cycle and is distinct from σl. The gas is then expanded at constant temperature and the entropy is increased to σH at point B. The expansion is isothermal, that is, performed at a constant temperature.
Now, the gas is expanded further, but at constant entropy. The temperature falls to τl during this isentropic process and arrives at point C. The gas is then compressed isothermally to point D, and is compressed isentropically back to point A, thus completing one cycle.
The total work accomplished by the system can be written from our previous results as W = Δτ×σh. Looking at the figure again, we see that this is merely the area enclosed by the rectangle. This yields a nice graphical method of understanding a simple version of a heat engine.
We have stressed throughout that knowing well the energy identities makes problem solving much easier, and we have seen this in many of the problems we have tackled. It appears again here, as we discuss processes performed on a gas.
For an isothermal expansion or compression, we wish to deal with an energy where τ appears as a differential. Conventionally, the Helmholtz free energy is used. Barring any diffusive exchange, we can see that dF gives us dU - dQ, which is exactly the work done on the system.