Prelecture 31: Slide 7

In order to determine the efficiency of the Stirling engine, we need to calculate the work done by the gas for a known heat input from the hot reservoir. To make this calculation, we will assume the working substance is an ideal gas.

In step 1, heat flows from the hot reservoir into the gas. This heat is just equal to the product of the heat capacity at constant volume and the temperature difference of the reservoirs.

In step 2, the volume expands and the gas does work. We can calculate this work since we know how the pressure changes with volume during this transition. Integrating over the change in volume gives us a result that is proportional to the product of the temperature and the log of the ratio of the volumes. Since the temperature does not change, the internal energy of the gas does not change, and the first law tells us that the heat that flows into the gas is the same as the work done by the gas.

In step 3, heat flows from the gas into the cold reservoir. Since the temperature change is just the opposite of step 1, the heat flow is also just the opposite.

In step 4, the volume is compressed and work is done on the gas. We can calculate this work since we know how the pressure changes with volume during this transition. Once again, integrating over the volume gives us the result that the work done is proportional to the product of the temperature and the log of the ratio of the volumes. As in step 2, the heat flow has the same magnitude as the work.

Putting this all together, the total work done by the gas is proportional to the product of the temperature difference and the logarithm of the ratio of the volumes. The total heat extracted from the hot reservoir is equal to the sum of the heat from steps 1 and 2.

We now substitute our expressions for the work done by the gas and the heat added to the gas into our efficiency relation. We can simplify this expression by factoring out the Carnot efficiency. The highlighted fraction, even though it looks complicated, is less than one since the denominator has to be bigger than the numerator. Therefore, the efficiency of the Stirling engine is indeed less than the Carnot efficiency.

For example, if we assume a doubling of volume for a monatomic ideal gas between reservoirs at the temperatures of boiling and freezing water, we obtain an efficiency for the Stirling engine of 17% while the Carnot efficiency for these temperatures is 27%.