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Slide 1 - Operation of a Free-Piston Stirling Convertor

Adding heat to an enclosed vessel of gas (working fluid) of mass M results in increased gas temperature, T_h, and increased gas pressure P_h. 

Removing heat results in decreased gas temperature, T_c, and decreased gas pressure, P_c.

Alternating T_h and T_c creates a pressure wave. The pressure wave moves the piston (power output).

This is not practical since the vessel cannot be heated and cooled quickly.

Slide 2 - Operation of a Free-Piston Stirling Convertor

If a displacer is added inside the vessel, it can be used to shuttle the working fluid between hot and cold spaces in the vessel. When the displacer is moved to the cold end, the working fluid is forced into the hot end, creating high pressure. Moving the displacer to the hot end forces the fluid over to the cold end of the vessel, creating lower pressure. As the fluid expands and contracts, it pushes an external piston up and down to create a power output.

Slide 3 - Operation of a Free-Piston Stirling Convertor

Three Common Configurations of Stirling Machines

Alpha Configuration

Typical for high power kinematic engines, automotive application, Rinia (4 cylinder) configuration

• High loads on kinematic linkage
• Not suitable for free-piston operation

Beta Configuration

Commonly used for free-piston & kinematic engines

• Compression & Expansion via piston
• Lightly loaded displacer drive
• Suitable configuration for coolers

Gamma Configuration

Features and applications similar to Beta configuration. Piston is offset from container.

Slide 4 - Operation of a Free-Piston Stirling Convertor

Four stage operation of the Beta Configuration - with the addition of heat exchangers Heater/Regenerator/Cooler

1. Isothermal compression
2. Constant volume displacement
3. Isothermal expansion
4. Constant volume displacement

The regenerator stores heat as the working fluid flows from the hot (expansion) space to the cool (compression) space, and  returns the heat to the fluid with the flow is reversed.

Slide 5 - Operation of a Free-Piston Stirling Power Convertor

Rhombic Drive Stirling Engine

With the appropriate areas and pressure, the displacer drive rod can become unloaded, i.e. self driving operation.

With the addition of a load, such as a linear alternator, the Stirling enine becomes a Stirling power convertor.

With the proper masses, spring rates and damping (dynamic tuning), the convertor will resonate as a Free-Piston Stirling Convertor converting heat into electric power.

Slide 6 - Dynamics of the Displacer and Power Piston

Initial conditions:

• expansion space heated
• compression space cooled
• piston & displacer at center of stroke
• working space & bounce space at equal pressure

Four cases:

These four cases show no coupling between the piston and displacer.

1. Piston is pushed in

   Working space pressure increases due to volume change pushing the piston back out

   Positive spring rate

2. Piston is pulled out

   Working space pressure decreases due to volume change pulling the piston back in

   Positive spring rate

3. Displacer is nudged in

   Working space pressure decreases due to working fluid being cooled pulling the displacer further in

   Negative spring rate

4. Displacer is nudged out

   Working space pressure increases due to working fluid being heated pushing the displacer further out

   Negative spring rate

Slide 7 - Operation of a Free-Piston Stirling Power Convertor

The unstable thermal-mechanical oscillator operates in a limit cycle. Using a relatively light displacer, here is one possible startup mode.

1. Unstable displacer moves out and raises the working space pressure
2. Piston moves out due to increased working space pressure. Work is extracted by the load (damping). Working space pressure is reduced by expansion
3. Piston moves past the point where working space pressure equals bounce space pressure.
4. Unstable displacer moves in and lowers the working space pressure.
5. Piston moves in due to decreased working space pressure. Work is extracted by the load (damping). Working space pressure is raised by volume change.
6. Piston moves past the point where working space pressure equals bounce space pressure.
7. Repeat cycle.

Slide 8 - Dynamics of a Free-Piston Stirling Convertor

Assuming linear operation, all aspects of the dynamics can be approximated with vector analysis. A phasor diagram shows an operating free-piston Stirling power convertor.

Mechanical or Gas Spring

A spring produces a force proportional to, and in the opposite direction to the displacement

Damper or Load

If the force lags the displacement, work must be added to maintain cyclic motion at frequency w.

Stirling work space

If the force leads the displacement, work must be extracted to maintain cyclic motion at frequency w.

Slide 9 - Tuning of a Free-Piston Stirling Power Convertor

Exact dynamic solution to resonance of free-piston Stirling convertors depends on:

• Non-linear gas springs
• Non-linear flow losses
• Complex interaction of loss mechanisms
• Configuration of manifolds

Convertor operates at conversion system resonant frequency

• Dominated by power piston mass
• Piston spring combination of gas spring and flexures
• Displacer mass and spring tuned to achieve proper stroke and phase angle
• Displacer spring combination of gas spring, flexures and cycle heating/cooling

These concepts are demonstrated using a phase diagram and a plot.

Slide 10 - Free-Piston Convertor Response to Change in Operating Conditions

Example 1:
Reduced heat input
Controller programmed to maintain piston stroke (voltage)

• Heater head temperature reduced in response to reduced heat input
• Reduced pressure phase angle as heating/cooling component of pressure is reduced
• Note: Pressure wave amplitude may remain relatively unchanged
• Reduced pressure phase angle leads to less power input to the power piston
• Less power into power piston leads to reduced piston stroke and output voltage
• Controller senses reduced voltage, lessens load
• Reduced load allows the power piston stroke to increase
• Controller adjusts load until power piston resumes full stroke (voltage) operation
• New steady state operating condition has resumed power piston stroke at reduced heater head temperature

Example 2:
Reduced heat input
Controller programmed to maintain heater temperature

• Heater head temperature reduced in response to reduced heat input
• Controller senses reduced heater head temperature, increases load (damping) on the power piston
• Increased load reduces the power piston stroke and output voltage
• Reduced piston stroke leads to reduced power output, reduces heat absorbed
• Reduced heat absorbed causes heater head temperature to increase
• Controller adjusts load until heater head temperature resumes set-point temperature
• New steady state operating condition has resumed heater head temperature at reduced power piston stroke

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Curator: Wayne A. Wong
  NASA Official Responsible For Content:  Richard K. Shaltens
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Last Updated: 08/01/2002