Time to gravity-drain a vertical cylindrical tank through a bottom orifice or nozzle (Torricelli).
t = 2 A_T / (Cd A_o √(2g)) · (√h₁ − √h₂)
How it works
Torricelli's law says the outflow velocity is √(2gh), so the drain rate falls as the level drops. Integrating the level from h₁ to h₂ gives a closed-form drain time for a constant-cross-section tank — no simulation needed.
Worked example
A 2 m diameter tank, 3 m of water, draining through a 50 mm sharp-edged nozzle. ≈ 34 minutes to empty — and half the volume drains in the first ~10 minutes, because the head is highest at the start.
Inputs
Tank diameter (m)
Orifice diameter (mm)
Initial liquid level (m)
Final liquid level (m) — 0 = drain completely (to the orifice).
The driving force is the liquid head. As the level falls, √(2gh) falls with it, so the last metre of liquid takes much longer to drain than the first.
What discharge coefficient should I use?
≈ 0.62 for a sharp-edged orifice, ≈ 0.80 for a short pipe stub, and ≈ 0.98 for a smooth rounded nozzle. If the drain line is long, its friction dominates and this simple model underestimates the time.
Does this handle horizontal or cone-bottom tanks?
No — the closed form assumes constant cross-sectional area with height. Horizontal cylinders and cone bottoms need the area-versus-height function integrated numerically.
Assumptions
Vertical cylinder with constant cross-section; level measured above the orifice centreline.
Tank freely vented (no vacuum pulled); no inflow during draining.
Orifice small relative to the tank, so the quasi-steady Torricelli balance holds.
References
Torricelli's law / tank-draining integral — standard unit-operations derivation from Bernoulli's equation.
Related Fluid Mechanics tools
Pipe Pressure Drop — Frictional ΔP and head loss for single-phase flow in a circular pipe (Darcy–Weisbach, Swamee–Jain friction factor).
Fittings Pressure Drop (K-factors) — Minor losses through elbows, tees, and valves by the ΣK excess-head method — the companion to straight-pipe friction.
Pipe Velocity & Reynolds — Line velocity, Reynolds number, and flow regime from flow rate and pipe inside diameter.
Friction Factor (Moody) — Darcy (and Fanning) friction factor from Reynolds number and relative roughness — Colebrook–White solved exactly, with the Swamee–Jain explicit fit for comparison.