Torricelli’s law
In 1643, Evangelista Torricelli showed that water flowing out of a hole at the bottom of a tank leaves at the speed it would reach by falling freely from the water surface:
v = √(2 × g × h)
where h is the height of water above the hole and g = 9.81 m/s². The actual flow is smaller, because the jet contracts as it leaves the hole. A discharge coefficient Cd accounts for this: about 0.61 for a sharp-edged hole, about 0.82 for a short pipe stub and up to 0.97 for a smoothly rounded nozzle.
The drain time
As the water level falls, the outflow slows down. For a vertical cylindrical tank of cross-section A emptying through an outlet of area a, integrating the flow gives:
t = (A ÷ (Cd × a)) × √(2 × h ÷ g)
Worked example
A 3 ft diameter tank holds 4 ft of water and drains through a 1-inch sharp-edged hole:
| Result | Value |
|---|---|
| Time to empty | 17 min 39 s |
| Time to drain the first half | 5 min 10 s |
| Starting flow | about 24 gal/min |
| Starting outflow speed | 16.04 ft/s |
| Volume of water | about 212 gallons |
The first half of the water leaves in only 29% of the total time; the last few inches take the longest.
What changes the drain time
- Outlet size matters most: doubling the hole’s diameter quadruples its area and cuts the time by four.
- Tank diameter: a wider tank holds more water at the same height, so it takes longer, in proportion to its area.
- Starting height: time grows with the square root of the height, so four times the depth only doubles the time, while holding four times the water.
- Outlet shape: a rounded nozzle drains about 60% faster than a sharp-edged hole of the same diameter.
Limits
The formula assumes:
- a vertical cylinder with a constant cross-section;
- an outlet at the very bottom, with no long pipe, hose or valve after it;
- water open to the air at the top, so no vacuum forms.
Horizontal and spherical tanks, long drain hoses and partially closed valves all slow the flow. For those, measure the time to drain the first part of the tank and scale from there.
Tips
- Use this for rain barrels, water heaters (with a hose, expect longer times), aquarium sumps, or physics homework.
- To speed up draining, use the largest outlet available and keep the hose short and as straight as possible.