Tank Draining Time Calculator (Torricelli)
How long a vertical cylindrical tank takes to empty through a hole in its bottom, the level at any moment and the flow at the start, with the tank drawn to scale.
Results
The dashed line marks the value you entered.
What this tool does
Water leaves through a hole in the bottom at Torricelli's speed √(2gh), so as the level falls the flow slows down with it. The result surprises most people: the first half of the water leaves in only 29 % of the total time, and the last few centimetres take the longest — the dashed line shows how much faster it would go if the flow stayed at its starting rate. The discharge coefficient corrects for the jet narrowing as it leaves: about 0.6 for a sharp-edged hole in a thin wall, close to 1 for a smooth rounded nozzle. Friction in a long outlet pipe is not included.
Formula
t = (A ÷ (C_d a)) × √(2h₀ ÷ g) · h(t) = (√h₀ − (C_d a ÷ A) √(g ÷ 2) t)²
Variables
| Symbol | Meaning | Unit |
|---|---|---|
dt | Tank diameter | m |
h0 | Starting water level | m |
dh | Orifice diameter | mm |
cd | Discharge coefficient | — |
t | Time elapsed | min |
TM | Time to empty | min |
TH | Time to drain half | min |
QI | Flow at the start | L/s |
VO | Tank volume | L |
HT | Water level at that time | m |
VD | Volume drained by then | L |
Worked example
- Tank diameter1 m
- Starting water level1.5 m
- Orifice diameter25 mm
- Discharge coefficient0.61
- Time elapsed10 min
- Time to empty24.2 min
- Time to drain half7.1 min
- Flow at the start1.624 L/s
- Tank volume1178 L
- Water level at that time0.516 m
- Volume drained by then773 L
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.