Why horizontal tanks are nonlinear
Lay a cylinder on its side and the liquid surface becomes a chord of the circle. Near the bottom and top of the tank that chord is short — little volume per centimetre of depth — while at the middle it spans the full diameter. The exact liquid cross-section is a circular segment:
At half depth the tank is exactly half full; at a quarter depth it holds only 19.6%; at three-quarters depth, 80.4%. Any gauge, transmitter or dip stick on a horizontal tank needs this correction — it is exactly what the "strapping" or tank characterization function in a DCS does.
Worked example
Diameter 2 m, length 5 m, dip reading 0.5 m:
- r = 1 m; (r − h) = 0.5
- A = 1²·cos⁻¹(0.5) − 0.5·√(2×0.5 − 0.25) = 1.0472 − 0.4330 = 0.6142 m²
- V = 0.6142 × 5 = 3.071 m³ — 19.6% of the 15.71 m³ capacity, at 25% of the depth
Field notes
- Program the segment curve, not a straight line, into any level-to-volume conversion — most transmitters and DCS blocks have a 20+ point characterizer for exactly this.
- Dished ends add volume (typically 2–6% total depending on head type) concentrated at low and high levels. This calculator's flat-end result slightly underestimates a real dished-end vessel.
- Tank tilt matters: a horizontal tank installed with a drain slope reads differently at each end — dip at the datum point specified on the tank drawing.
Frequently asked questions
What is the formula for a partially filled horizontal cylinder?
V = L × [r²·cos⁻¹((r−h)/r) − (r−h)·√(2rh − h²)], where r is the radius, h the liquid depth and L the cylinder length. It is the circular-segment area times the length.
Is a horizontal tank half full at half depth?
Yes — exactly 50% at h = D/2, by symmetry. But everywhere else the relationship is nonlinear: at 25% of depth the tank holds only 19.6% of its volume.
Why does my level gauge disagree with the delivered volume?
On a horizontal tank, equal increments of level are NOT equal increments of volume — the middle of the tank holds far more per centimetre than the top or bottom. Any level-to-volume conversion must use the segment formula or a strapping chart.
Does this include dished ends?
No — the formula covers the cylindrical shell with flat ends. Dished (torispherical/elliptical) heads add several percent of extra volume; use vendor data when that accuracy matters.
Provided for reference and education. Verify independently before use in safety-critical or custody-transfer work. See our disclaimer.