L28.The Stirling Engine This activity provides a practical application of thermodynamics and heat engines. Applying theory is sometimes more difficult than the theory itself. That's because you have to 1) see how the equipment works and 2) understand how the theory relates to the operation of the equipment. There are also inherent errors that make the results less certain than a strictly theoretical problem. So you should expect to spend more time than a typical theory assignment delving into how the device described below works. In that respect, this assignment is more like a lab than a problem assignment. That's why we've labeled it as a lab.

 Goals

1. to learn how a Stirling Engine works
2. to analyze the operation of the engine using thermodynamics principles

 Prelab

1. Do the assigned readings in Ch. 18 and complete all the associated WebAssigns.
2. Study the Introduction below.
3. Submit answers to the prelab assessment on WebAssign, L28PL.

 Introduction

Visual aids

The Stirling engine, invented by Robert Stirling in 1816, is an external combustion engine that uses air as a working medium between two pistons. Like all engines, a Stirling engine uses a temperature difference between two heat reservoirs, partially converting the heat transferred between the reservoirs into useful mechanical energy. A key component of a Stirling engine is the regenerator, a device that sits between the two pistons and allows heat to be reused (that would otherwise be wasted) by temporarily storing thermal energy during the cycle. The thermal conductivity of the regenerator is low enough that there is not appreciable heat conduction between the two sides of the regenerator.

We consider a simplified Stirling engine in which only one of the pistons--called the power piston--delivers power to the outside environment; the other piston--called the displacer piston--serves as the regenerator and is used to move air between the hot and cold sides of the engine cylinder. The two pistons are connected by an external linkage so that they oscillate up and down 90° out of phase with each other. Here are some visual aids to help in seeing the components of the engine and how they move.

Animation in cross section: This simplified cross-sectional view of a Stirling engine shows the motion of the power and displacer pistons inside the cylinder and the location of the external hot and cold reservoirs. In practice, the cold reservoir may simply be the surrounding room-temperature air, and the hot reservoir may be a cup of steaming hot water. Unlike the simplified view shown, a typical cylinder is much wider than it is high.

Animation showing the external linkage:  This view shows the top of an actual Stirling engine.

NCSSM Video (WMV or MP4):  This short clip shows the side view of the engine in action. The engine sits atop a hot cup of water. The power piston is on the right side of the apparatus. The displacer is a piece of yellow foam inside the cylinder.

Youtube video:  This video shows a Stirling engine that is of the same type as the one above. The action of the power piston is clearly visible on the side of the cylinder nearest the viewer. The action of the displacer is also easy to see. The temperature difference between the palm of the hand and the outside air is enough to make this engine run. Here's another good video.

The Ideal Stirling Cycle

Real Stirling engines come closer than any other type of engine to reaching the theoretical (Carnot) maximum efficiency. Let's look at 4 distinct parts of the heat engine's cycle and the thermodynamic processes occurring in each part.

In the idealized Stirling engine depicted in the diagrams below, the air cycles between four states. Refer to the following to help in interpreting the diagrams. Note that the external linkage that drives the pistons is not shown.

• The 4 states of the process are indicated by 1 to 4.
• The 4 processes (1 to 2, 2 to 3, etc) of the complete cycle are shown one after the other. For each process, there's a P-V diagram of the cycle and a cross-sectional representation of the engine.
• The positions of the power (displacer) piston at the beginning and end of the cycle are shown in green (yellow). The dotted green (yellow) rectangle represents the piston at the beginning of the cycle and the solid green (yellow) rectangle represents the piston at the end of the cycle. If there is no dotted green (yellow) rectangle, that means the piston moves very little during the process.
• The direction of motion of the piston during the process is indicated by a black arrow.
• The direction of heat flow into or out of the air is indicated by a red arrow.
• The temperatures of the hot and cold reservoirs are represented by T_C and T_H.

Descriptions of each process are given below the diagram.

We'll take state 1 as the starting point of the cycle. The power piston is at its lowest position, and the displacer piston is at its highest.

Process 1 → 2:  The heated gas increases in pressure and pushes the power piston upward. This process is called the power stroke of the engine. This is an approximately isothermal expansion, during which heat Q_H is absorbed by the air from the hot reservoir at temperature T_H. The displacer piston doesn't move much during this process.

Process 2 → 3:  Due to motion of the external linkage, the displacer piston now moves downward, shunting the gas to the cold end of the cylinder. During this process, air is forced through  the displacer/regenerator from the hot to the cold side, giving up an amount of heat Q_R to the regenerator. This process takes place at practically constant volume. The power piston doesn't move much during this process.

Process 3 → 4:  Due to continued motion of the outside linkage, the power piston moves down, compressing cold air while in contact with the cold reservoir and therefore causing heat Q_C to leave into the cold reservoir. This is an approximately isothermal compression during which the displacer piston doesn't move much. The work done on the gas during this compression process is less than the work the gas did during the expansion process 1 → 2, since the gas’s pressure dropped when it was cooled.

Process 4 → 1:  Finally, the external linkage causes the displacer piston to move up, forcing most of the gas adjacent to the hot reservoir again. The air is forced through the regenerator from the cold side to the hot side. To accomplish this, the regenerator supplies heat Q_R to the air, approximately the same amount of heat as was given to the regenerator in step 2 → 3. This process takes place at constant volume.

A More Realistic Cycle

The engineering of a practical Stirling engine results in departures from the ideal cycle described above. See this Wikipedia article for a cycle that more nearly describes the cycle of the engines in the videos and animations that you viewed previously. The relevant sections of the article are Piston Motion Variations and Pressure vs. Volume Graph.

 Prelab Assessment

The following questions are checks on what you learned from the previous information and the associated visual aids. Submit answers on the WA assessment, L28PL.

In practice, real Stirling engines depart from the ideal process described in the section Ideal Stirling Cycle above. For example, the motions of the pistons in this slowed down version of the IWP animation differ in some ways from the description of the ideal cycle. Compare the ideal Stirling engine to that shown in the animation.  Note that in the animation the four positions of the power piston are indicated with numbers corresponding to those on the P-V diagram above. Click on Show Graph to see the position vs. time graphs of the power piston and displacer.

1. Assume that any hot air inside the engine cylinder is at the same temperature as the hot water reservoir while any cold air in the cylinder is at the same temperature as the air in the room. Also assume that when the air in the cylinder is at room temperature, the pressure inside the cylinder is atmospheric pressure. In what state (1, 2, 3 or 4) is the air inside the cylinder at room temperature and atmospheric pressure?

2. For what part of the ideal cycle (indicate with 1 → 2, 2 → 3, etc.) is the temperature increasing? Use the ideal gas law in explaining your answer.

3. Does the system gain or lose heat in part 4 → 1 of the ideal cycle? Explain using the first law of thermodynamics.

4. Describe how the motions of the power and displacer pistons shown in the animation differ from those described for the ideal Stirling cycle. Be as specific as possible. The Wikipedia article should be helpful.

5. Draw a P-V diagram of the ideal Stirling cycle. Then, on top of that graph, superimpose a graph of the Stirling cycle that corresponds to the animation. This, of course, will be partly guesswork, but you need to make it clear how the animation cycle differs from the ideal cycle.

6. How do you expect the efficiency of a practical (real) Stirling engine to compare to that of an ideal Stirling engine operating between the temperatures of the cold and hot reservoirs? Consider Carnot's theorem in your answer, whether the cycle of a real Stirling engine is reversible, and what the maximum efficiency of a Carnot engine is.

 Problems

Answer the following on WA assessment L28.

Let’s calculate the values of some of the state variables (P,V and T) of the air inside the cylinder. Assume throughout that the air is an ideal gas. Let’s also assume that any hot air inside the engine cylinder is at the same temperature as the water while any cold air in the cylinder is at the same temperature as the air in the room. The measured temperatures of the air in the room and hot water are given to be 23 °C and 77 °C, respectively. (Note that these values will be randomized by WebAssign.) Finally, let’s assume that when all of the air in the cylinder is at room temperature, the pressure inside the cylinder is atmospheric pressure.

The following values are given (some of these will be randomized by WebAssign):

D_engine = Diameter of engine cylinder = 9.15 cm
H_engine = Height of engine cylinder = 1.92 cm
D_regen = Diameter of displacer/regenerator = 8.5 cm
H_regen = Height of displacer/regenerator = 1.00 cm
D_power = Diameter of power piston = 1.90 cm
H_power = Maximum distance that power piston gets pushed up = 0.60 cm

1. For state 3, calculate the volume of air in the cylinder. Give your answer to the nearest 0.1 cm³. Assume that air does not occupy the volume inside the displacer piston. Do include the volume of the power piston displacement as shown in this graphic.

2. For state 3, calculate the number of gas molecules inside the engine cylinder.

3. Calculate the change in volume of the air when the power piston moves either up or down (processes 1 → 2 and 3 → 4).

4. Assume that the air inside the engine is an ideal, monatomic gas. Note we are also making the usual assumptions that all of these processes are reversible (no frictional energy losses) and quasistatic (system remains in equilibrium at all times). Do the following calculations:

   1. volume for each state to the nearest 0.1 cm³

   2. pressure for each state to the nearest kilopascal

   3. change in internal energy for each process in the engine cycle (1 to 2, 2 to 3, 3 to 4, 4 to 1) to the nearest 0.001 J

   4. work done by the system on the environment for each process to the nearest 0.001 J

   5. heat transferred to the system during each process to the nearest 0.001 J

5. Use the definition of efficiency for a heat engine (see p. 601) to calculate the efficiency of this engine.

   1. total work done in a complete cycle to the nearest 0.001 J

   2. efficiency to 2 significant figures

6. Calculate the maximum possible efficiency of an engine when operating between the temperatures given for this engine to 2 significant figures. That is, use the Carnot efficiency formula.

7. Your answers to items 5 and 6 should have been the same or very nearly so. Why should you expect this?

8. Is there any way that this engine could be made 100% efficient (that is, could the engine convert 100% of the heat gained from the hot water into useful mechanical work)? Explain.

9. When an engine is running on a heat source, the Carnot formula determines how much of the heat can become useful work. In this engine, as in most engines, the majority of the heat doesn’t become useful work. What happens to this heat? Identify three specific loss mechanisms.

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