Vertical Tank Volume with Dished Heads
Calculate the total and partially filled volume of a vertical vessel with a cylindrical shell and 2:1 ellipsoidal, hemispherical or flat bottom and top heads.
- Head depth a = 500 mm; one head = ⅔πR²a = 1.047 m³
- Bottom head part: πR²x²(3a − x) ÷ 3a²; shell: πR² × height in shell; top head: head volume minus empty cap
- V = 8.9012 m³
Dished Head Volume Formulas
Partly filled bottom head to height x: V = π × R² × x² × (3a − x) ÷ (3a²)
Shell: V = π × R² × h
R is the inside radius, a the head depth (D/4 for a 2:1 ellipsoidal head, D/2 for a hemisphere) and x the liquid height within the head. Torispherical (F&D) heads hold slightly less than 2:1 ellipsoidal heads of the same depth; use vendor drawings for exact figures.
Worked example
A vertical vessel, 2 m diameter with 4 m straight shell and 2:1 heads, holds 14.66 m³ when full. With the level 3 m above the bottom of the lower head, it holds 8.90 m³.
Frequently Asked Questions
What is a 2:1 ellipsoidal head?
A head whose depth is a quarter of the diameter (the ellipse axes are in a 2 to 1 ratio). It is the most common head on pressure vessels.
Where is level measured from?
Here from the bottom of the lower head. Many datasheets measure from the bottom tangent line instead, so check the reference.
How do I handle torispherical heads?
They are close to 2:1 ellipsoidal but slightly smaller in volume. For accurate figures use the vessel drawing or a strapping table.
Learn more on the blog
Basics of level measurement
A practical, field focused guide on instrumentationblog.in that explains the theory behind this calculator and how engineers apply it on real plants.
Read Full Article