Overlap volume of the beta type Stirling engine

In this page, I inform about the overlap volume of the beta type Stirling engine.
Figure 1 shows a model of the beta type Stirling engine. Now, a compression volume, V_C is considered. As the assumption, the movements of the pistons are sine waves. When the top dead center of the displacer piston is the crank angle,

(rad.), V_C is lead to Equation 1.

　　(1)

Where, V_SE: swept volume of the displacer piston, V_SC: swept volume of the power piston, : phase angle between the displacer piston and the power piston, V_DC: dead volume of the compression volume and V_D: the overlap volume.

When the minimized value of V_C becomes V_DC, the overlap volume, V_D is able to be lead. Eq. 1 changes to a next equation (part is omitted).

　　(2)

So, when , the minimized value of V_C is becomes V_DC.
Then,

　　(3)

The dead volume of the compression volume, V_DC is the minimized volume of the compression space, and a clearance volume when the displacer piston comes the nearest to the power piston.
You must take care the real engine design when the engine is used a normal crank mechanism, because above equations do not hold good when the movements of th epistons are not the sine waves.