Question

The ingenious Stirling engine is a true heat engine that absorbs heat from an external source. The working substance can be air or any other gas. The engine consists of two cylinders with pistons, one in thermal contact with each reservoir (see Figure 4.7). The pistons are connected to a crankshaft in a complicated way that we'll ignore and let the engineers worry about. Between the two cylinders is a passageway where the gas flows past a regenerator: a temporary heat reservoir, typically made of wire mesh, whose temperature varies IQnl Hot reservoir T honom Cold reservoir T Regenerator Figure 4.7. A Stirling engine, shown during the power stroke when the hot piston is moving outward and the cold piston is at rest. (For simplicity, the linkages between the two pistons are not shown.) gradually from the hot side to the cold side. The heat capacity of the regenerator is very large, so its temperature is affected very little by the gas flowing past. The four steps of the engine's (idealized) cycle are as follows: i. Power stroke. While in the hot cylinder at temperature Ty, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure. ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to Te ili. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in. iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at TA (a) Draw a PV diagram for this idealized Stirling cycle. (b) Forget about the regenerator for the moment. Then, during step 2, the gas will give up heat to the cold reservoir instead of to the regenerator; during step 4, the gas will absorb heat from the hot reservoir. Calculate the efficiency of the engine in this case, assuming that the gas is ideal. Express your answer in terms of the temperature ratio T/T, and the compression ratio (the ratio of the maximum and minimum volumes). Show that the efficiency is less than that of a Carmot engine operating between the same temperatures. Work out a numerical example. (c) Now put the regenerator back. Argue that, if it works perfectly, the effi- ciency of a Stirling engine is the same as that of a Carnot engine. (d) Discuss, in some detail, the various advantages and disadvantages of a Stirling engine, compared to other engines.

Answer 1

Stirling engines have unique advantages in terms of efficiency and low emissions, but they also have limitations, such as lower power density and complexity. Their suitability depends on the specific application and requirements.

(a) To draw a PV diagram for the idealized Stirling cycle, follow these steps:

1. Start with the initial state of the gas in the hot cylinder (Step i). This corresponds to an isothermal expansion at temperature Th while the volume increases.

2. Then, represent the transfer to the cold cylinder (Step ii). During this step, the volume remains constant, but the pressure decreases as the gas cools.

3. Next, show the compression stroke in the cold cylinder (Step iii). This corresponds to an isothermal compression at temperature Tc while the volume decreases.

4. Finally, depict the transfer back to the hot cylinder (Step iv). During this step, the volume remains constant, but the pressure increases as the gas heats up.

The resulting PV diagram should resemble a figure-eight shape, as the gas undergoes these four steps in a cyclic manner.

(b) Without the regenerator, during step 2 and step 4, the gas exchanges heat directly with the cold and hot reservoirs, respectively. Calculate the efficiency of the engine using the formula for the efficiency of a heat engine:

Efficiency = 1 - (Tc / Th)

Here, Tc is the temperature of the cold reservoir (step 2), and Th is the temperature of the hot reservoir (step 4).

Now, express this efficiency in terms of the temperature ratio (Tc / Th) and the compression ratio (Vmax / Vmin), where Vmax is the maximum volume (during step iii), and Vmin is the minimum volume (during step i):

Efficiency = 1 - (Tc / Th)

Efficiency = 1 - (1 / Tc) / (1 / Th)

Efficiency = 1 - (1 / (Tc / Th))

Efficiency = 1 - (1 / (T / T))

Efficiency = 1 - (1 / (1 / compression ratio))

Efficiency = 1 - (compression ratio)

Now, compare this to the efficiency of a Carnot engine operating between the same temperatures (Tc and Th):

Efficiency_Carnot = 1 - (Tc / Th)

The efficiency of the Stirling engine without the regenerator is less than that of a Carnot engine operating between the same temperatures because it lacks the ideal regenerative heat exchange.

Let's work out a numerical example:

Suppose Tc = 300 K and Th = 600 K, and the compression ratio (Vmax / Vmin) is 5.

Efficiency = 1 - (300 / 600) = 0.5 (50%)

Efficiency_Carnot = 1 - (300 / 600) = 0.5 (50%)

In this case, the efficiency of the Stirling engine without the regenerator is 50%, which is the same as that of a Carnot engine operating between the same temperatures.

(c) When the regenerator is added and works perfectly, it allows the Stirling engine to exchange heat between the gas and the regenerator during steps 2 and 4. This means that the engine operates closer to the Carnot efficiency because it reduces heat loss to the surroundings. Therefore, the efficiency of a Stirling engine with a perfect regenerator is the same as that of a Carnot engine operating between the same temperatures.

(d) Advantages and disadvantages of a Stirling engine compared to other engines:

Advantages:

1. High Efficiency: Stirling engines can achieve high thermal efficiency, especially when equipped with a well-functioning regenerator.

2. Quiet Operation: Stirling engines operate quietly, making them suitable for applications where noise is a concern.

3. Low Emissions: They produce low emissions, making them environmentally friendly.

4. Flexibility: Stirling engines can run on various heat sources, including solar energy, biomass, and waste heat.

5. Longevity: Stirling engines have a long lifespan and low maintenance requirements.

Disadvantages:

1. Low Power-to-Weight Ratio: Stirling engines tend to have a lower power-to-weight ratio compared to internal combustion engines.

2. Slow Response Time: Stirling engines have a slower response time, making them less suitable for applications requiring rapid changes in power output.

3. Complex Design: The linkage mechanism and regenerator can be complex, which can increase manufacturing and maintenance costs.

4. Limited Market Penetration: Stirling engines have not seen widespread adoption in most automotive applications, limiting their availability.

5. Lower Power Density: Stirling engines are less suitable for high-power applications compared to some other engine types.