Why heads matter more than people expect
Pressure vessels and most process tanks are not plain cylinders. The ends are dished, because a curved head carries pressure far more efficiently than a flat plate — and that curvature adds real volume that the familiar πr²L calculation simply omits.
For a 2 m diameter, 5 m long horizontal tank the shell holds 15.71 m³. Add 2:1 elliptical heads and total capacity becomes 17.80 m³. That is 13% more than the shell figure — the difference between an inventory calculation that reconciles and one that never quite does.
Head geometry reference
Each head type has a fixed relationship to the diameter, so volume and depth both scale with D:
| Head type | Depth | Volume | Each head, D = 2 m |
|---|---|---|---|
| Flat | 0 | 0 | 0 |
| ASME F&D | 0.169 · D | 0.0810 · D³ | 0.648 m³ |
| 2:1 elliptical | 0.25 · D | 0.1309 · D³ | 1.047 m³ |
| Hemispherical | 0.50 · D | 0.2618 · D³ | 2.094 m³ |
The 2:1 elliptical figure is exactly πD³/24. The ASME flanged and dished coefficient comes from integrating the actual torispherical profile — crown radius equal to D, knuckle radius 0.06 D — and works out to 0.0810 D³, matching published tables.
Worked example
Horizontal tank, D = 2 m, shell 5 m, 2:1 elliptical heads, filled to 1 m:
- Shell (half full) = 15.708 ÷ 2 = 7.854 m³
- Both heads at half full = π × 0.5 × 1.0² × (1 − 1/3) = 1.047 m³
- Total = 8.901 m³ = 8,901 litres, exactly 50% of the 17,802 L capacity
Half depth gives half volume here only because a horizontal cylinder is symmetrical about its centreline. At any other level the relationship is strongly non-linear — which is the entire reason dip charts exist.
The partial volume mathematics
Vertical tank, bottom head: for a level y inside an elliptical or hemispherical head of depth a and tank radius R,
Horizontal tank, both heads combined: at liquid depth H, the pair of dished ends together hold
Both reduce correctly to the full head volume at maximum fill, and both have been checked against numerical integration. The shell contribution uses the standard circular segment area for horizontal tanks. The ASME F&D head, whose profile has no tidy closed form, is integrated numerically inside the calculator.
Field notes
- Tangent-to-tangent is the length that matters. Vessel drawings state shell length between tangent lines; overall length includes the heads. Using the overall figure as shell length double-counts the head volume.
- Straight flange is usually ignored in quick calculations but adds a small cylindrical section at each end — typically 25–50 mm. For custody applications it belongs in the numbers.
- Level transmitters see head, not volume. A DP transmitter measures hydrostatic pressure, so it reports level; the volume conversion is a separate characterisation step in the control system. See the DP level calculator.
- Nozzles, internals and heating coils displace liquid — a calculated capacity is a geometric upper bound, not a measured one. Physical strapping remains the reference for custody transfer.
- For plain cylinders without heads, the simpler vertical and horizontal calculators are quicker.
Frequently asked questions
How do I calculate the volume of a 2:1 elliptical head?
A 2:1 semi-elliptical head has a depth of D/4 and a volume of πD³/24, which is 0.1309 × D³. For a 2 m diameter tank each head adds 1.047 m³ — so a pair of heads contributes over 2 m³ before the shell is counted.
What is the volume of an ASME F&D head?
For the standard torispherical geometry with crown radius equal to the diameter and knuckle radius 6% of the diameter, volume is approximately 0.0810 × D³ and depth is about 0.169 × D. It holds roughly 62% of what an equivalent 2:1 elliptical head holds.
Why does my tank hold more than the cylinder calculation says?
Because dished ends add volume the plain cylinder formula ignores. On a 2 m × 5 m horizontal tank, 2:1 elliptical heads add 2.09 m³ to a 15.71 m³ shell — over 13% more capacity than the shell alone.
What is a strapping table or dip chart?
A lookup table converting measured liquid depth to volume, used because the relationship is non-linear in horizontal tanks and inside dished heads. Control systems store it as a characterisation curve, typically with 20 or more points.
Geometric calculation for reference and education; ignores wall thickness, internals and straight flange unless stated. Not a substitute for a certified strapping table in custody transfer. See our disclaimer.