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(OP)
I need the basic formulas for describing the shape of the various dished heads for 150 psi pressure vessels.  The purpose is to accurately calculate the volume contained in these heads.

The volume of a standard ASME head with dish radius equal to 100% of vessel diameter and knuckle radius equal to 6% of vessel diameter is V = 0.0809 * D^3.

For a 2:1 elliptical head, the volume is V = Pi / 24 * D^3.

For a hemispherical head, the volume is V = Pi / 12 * D^3.

In all of these formulas, use consistent units.  In other words, if diameter is in inches, then the volume is in inches cubed.

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