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Partially Filled Horizontal Cylindrical Tank

Also known as partially full tank · tank strapping

V=L[r2cos⁡−1 ⁣(r−hr)−(r−h)2rh−h2]V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

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A dipstick in a horizontal tank does not read linearly, and the reason is geometry: the wetted cross-section is a circular segment whose area grows fastest near the centreline and slowest at top and bottom. Half-full is the one easy case — a 1 m radius, 3 m long tank at h = r holds 3 × (π/2) ≈ 4.712 m³ — but at a quarter depth the tank holds only about 19.6% of its capacity, not 25%. Every fuel-oil, propane and chemical bulk tank on site comes with a strapping chart that is nothing more than this formula tabulated.

The formula covers the straight shell only. Real tanks have dished, elliptical or hemispherical heads that add 5–15% more volume, and a slight tilt (deliberate, for drainage) skews the reading further. Note also the arccosine: enter a depth greater than the full diameter 2r and the expression has no real value, which is the calculator's way of telling you the tank is already overflowing.

Partially Filled Horizontal Cylindrical Tank
V=L[r2cos⁡−1 ⁣(r−hr)−(r−h)2rh−h2]V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]
Where
  • VV= Liquid volume
  • LL= Tank length
  • rr= Tank radius
  • hh= Liquid depth
Missing one of these? Work it out first, then come back