Dished Head Volume · Vessel Inventory · Dip Chart
Vessel & Tank Volume Calculator
Dished head volume and total or partially-filled vessel volume for horizontal
and vertical cylindrical tanks. Flat, hemispherical, 2:1 ellipsoidal, torispherical
(ASME F&D, 80:10, 80:6 or custom crown/knuckle) and conical heads. Head volume is
reported separately from the shell, with a downloadable level–volume dip chart.
Method
Horizontal shell partial volume uses the circular segment:
V = L·[R²·acos((R−h)/R) − (R−h)·√(2Rh−h²)].
Dished head pairs follow V = π·a·h²·(3R−h)/(3R), where a is the
head inside depth — exact for hemispherical (a = R) and 2:1 ellipsoidal
(a = R/2) heads. Torispherical head capacity is computed from crown-cap plus
knuckle-torus integration, matching the handbook value 0.0809·D³ per head.
Vertical tanks treat bottom dome, shell and top dome zones separately; conical bottoms are exact.
Why horizontal tank fill volume isn't linear
In a vertical tank, volume rises in direct proportion to level — a simple relationship.
A horizontal tank behaves differently: at low fill, the liquid surface is a narrow sliver
near the bottom of the circular cross-section, so volume grows slowly per millimetre of
level rise. Near the mid-line (50% level) the surface is widest, so volume grows fastest
per millimetre, then slows again as the tank narrows toward the top. That's why a dip
chart — a level-to-volume table — is essential for horizontal tanks and gauge glasses, and
why linear interpolation between two dipstick readings can be significantly wrong unless
you're near the mid-point.
Worked example
Take a horizontal vessel with 2000 mm inside diameter, 5000 mm tan-tan length,
and 2:1 ellipsoidal heads (head depth a = R/2 = 500 mm), filled to a level
of 1000 mm (50% of diameter). At exactly the mid-line, the shell segment reduces to
half the circular cross-section, so the shell contributes half its full volume:
0.5 · π·(1)²·5 ≈ 7.854 m³. The head pair at half-fill gives
V = π·a·h²·(3R−h)/(3R) = π·0.5·1²·(3−1)/3 ≈ 1.047 m³. Total liquid volume
≈ 8.90 m³ (8,900 litres), against a full tank capacity of about
17.28 m³ — confirming the 50% level mark lands almost exactly at 50% of volume for
a horizontal tank with symmetric heads, a useful sanity check when validating any dip chart.
Where this calculator is used
Tank volume and dip-chart calculations like this are used for inventory reconciliation on
fuel, diesel and chemical storage tanks, batch sizing in food, beverage and pharmaceutical
processing, custody-transfer gauge tables, and general capacity planning for pressure
vessels, reactors, day tanks and knock-out drums with ASME flanged & dished (F&D) or
2:1 ellipsoidal heads. It's a preliminary engineering estimate — for custody transfer,
regulatory reporting, or hazardous material inventory, always confirm against a certified
manufacturer strapping table, since manufacturing tolerances, weld buildup, and internal
obstructions (baffles, coils, agitators) can shift real volume by 1–3%.
Related tools on MultiCalci: control valve sizing (IEC 60534), pipe pressure drop,
orifice flow (ISO 5167), heat exchanger rating and more at
multicalci.com.
Frequently asked questions
How is the volume of a partially filled horizontal tank calculated?
The cylindrical shell uses the exact circular segment formula
V = L·[R²·acos((R−h)/R) − (R−h)·√(2Rh−h²)]. Dished head pairs use the
spheroidal relation V = π·a·h²·(3R−h)/(3R), exact for hemispherical and
2:1 ellipsoidal heads, with an equivalent-depth method for torispherical heads
(typical deviation under 1%).
What is the volume of a torispherical (ASME F&D) head?
For a standard ASME flanged & dished head with crown radius 1.0·D and knuckle
radius 0.06·D, the head volume is approximately 0.0809·D³ and the inside depth is
about 0.169·D. This calculator computes it from exact crown-and-knuckle geometry.
Can I generate a dip chart (level to volume table)?
Yes — the calculator produces a level–volume table across the full tank height for
the chosen geometry and lets you download it as a CSV for calibration or inventory use.
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