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The figure below shows a beta engine configuration.

Figure 1. Click on picture for larger view

The figure below shows a gamma engine configuration. It is basically the same as a beta engine except that the compression space is “shifted” down, so that the power piston is not lined up with the displacer.

Figure 2. Click on picture for larger view

In a beta and gamma engine, the volume variation in the expansion and compression space can be calculated as follows.

In the expansion space, the volume variation can be approximated as sinusoidal, and expressed in general form as (click on formula for larger view):

In the compression space, the volume variation is more subtle and can also be approximated as sinusoidal. The general form is (click on formula for larger view):

Where:

V_e = volume in the expansion space

V_c = volume in the compression space

V_e^unswept = the unswept/dead volume in the expansion space

V_c^unswept = the unswept/dead volume in the compression space. Note that this is the volume in the compression space that is “not touched” by the power piston or displacer. This volume is not the same as the minimum volume in the compression space.

A_de = the area of the displacer on the expansion space side

A_dc = the area of the displacer on the compression space side. Note that this area is A_de minus the area of the displacer rod

A_p = the area of the power piston

X_d = the stroke length of the displacer. Note that X_d/2 is the amplitude of the displacer motion

X_p = the stroke length of the power piston. Note that X_p/2 is the amplitude of the power piston motion

θ = the crank angle (see Figure 3)

α = the phase difference between the displacer and the power piston (see Figure 4 and 5)

For beta and gamma engines the optimal phase difference α between the displacer and power piston for maximum power, is such that the phase difference between V_e and V_c is roughly 90 degrees. Figure 3 shows an example of this variation graphically as a function of crank angle θ. In this case the units used are inches.

Figure 3. Click on picture for larger view

The easiest way to determine the combination of the above engine dimensions such that the volumetric phase difference is 90 degrees, is to graph the equations for V_e and V_c in a spreadsheet, and adjust the dimensions until you get a 90 degree phase difference as shown in Figure 3. Note that the amplitude of V_e and V_c do not have to be equal for maximum power. It is only necessary that V_e and V_c have roughly a 90 degree phase difference apart.

A mechanical drive commonly used for a beta engine (as shown in Figure 1) is a rhombic drive. This is explained in Zig Herzog’s website http://mac6.ma.psu.edu/stirling/drives/index.html.

A mechanical drive commonly used for a gamma engine (as shown in Figure 2) is a crank drive, as it’s easy to construct. Figure 4 and 5 shows a crank drive for a gamma engine.

Figure 4. Click on picture for larger view

Figure 5. Click on picture for larger view

From Figure 5, R_d = X_d/2, and R_p = X_p/2.

This entry was posted on Wednesday, November 5th, 2008 at 11:54 pm and is filed under Stirling Engine. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.