Calculations for Beta-  and Gamma- Stirling Engines

Special thanks to Wally, who provided us with these Calculation Formulas !!!!

By choosing the 2 Variables ( Displacement Cylinder Diameter and Power Cylinder Diameter ) one can then use the following Formulas to calculate approximate ideal Values for all other Data important for the Construction of a sucessfull Gama or Beta Stirling Engine. My model, by the way is a Gamma type. In this Example I have chosen 27mm for both Variables.

Please remember that these are only approximate Values. As far as I know, there are no Formulas that will guaranty perfect Results. Currently there is a lot of Research being done to revive Stirling Engines and incorporate their Potential into environmentally friendly and renewable sources of Energy.

 
1st Variable = Diameter of Displacement Cyl. ( Cooling and Heating Unit ) = ( here our chosen 27 mm )

A = Length of Displacement Cyl. = 3 X Diameter of Displacement Cyl. = 81 mm
B = Length of Heating Zone of Displacement Cyl.=2 X Diam. of Displacement Cyl.=54 mm
C = Length of Cooling Zone of Displacement Cyl. = Diam. of Displacement Cyl. = 27 mm

D = Diameter of Displacement Piston= 0.98 X Diameter of Displacement Cyl. = 26.46 mm

E = Length of Displacement Piston = 2 X Diam. of Displacement Cyl. = 54 mm
h = Stroke of Displacement Piston = Diam. of Displacement Cyl. = 27 mm

Volume displaced = D² X Pi / 4 X h = 14839 mm³  ( here in cubic millimeters )

( 26,46 X 26,46 X 3.14 / 4 X 27 = 14839 )

2nd Variable =Diameter of Power Cyl.= ( here our chosen 27 mm )

Volume of Power Cyl. = 0.66 X Volume of Displacement Cyl. = 9883 mm³
Stroke of Power Piston ( h ) can now be calculated using the following Formula:
( h = V X 4 / D² / Pi ) where:
h = Stroke of Power Piston
V = Volume of Power Cyl.
D = Diameter of Power Cyl.
Pi = 3,14

 In this Case:
 h = 9883 mm³ X 4 / 729 mm² / 3,14 = 17,26 mm

When you use these Calculations on my Engine you get a much longer Power Piston Stroke ( approx. 45 mm ) than what I used ( 21 mm ), and with that also a considerable difference in Power Cyl. Stroke Volume.
Also the Difference between Displacement Cyl. and Dsiplacement Piston Diameters are substantial ( here in our Formula = Factor of 0.98, but in my Stirling 0.96 ). The other Criteria are ( approx. ) comparable.
 
It would be interesting to know how much more efficient in terms of Power and or Speed my Engine would have been had I used these ( guideline ) Formulas in the making of my Engine