The Speed of Efflux
Torricelli's Law is a theorem in fluid dynamics relating the speed of fluid flowing out of an orifice to the height of fluid above that opening. Discovered by Evangelista Torricelli in 1643, the law states that the speed of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth $h$ is the exact same as the speed that a body (in this case, a drop of water) would acquire in falling freely from a height $h$.
Because it is derived directly from Bernoulli's principle, it assumes the fluid is ideal (incompressible and inviscid) and that the tank is open to the atmosphere.
Practical Fluid Flow
This principle is highly practical when dealing with draining tanks or emergency pressure leaks:
- Water Towers: If a water tower springs a leak at the very bottom, Torricelli's law calculates exactly how fast the water will jet out, determining how quickly the town loses its water supply.
- Coffee Dispensers: Have you noticed that coffee flows out of a full urn very quickly, but trickles out slowly when it's almost empty? That is Torricelli's law in action—as the height ($h$) decreases, the exit velocity decreases.
The Formula
Example Calculation
A large industrial water tank springs a leak $5 , ext{meters}$ below the current water line. What is the velocity of the water jetting out of the hole?
- Multiply $2 \cdot g \cdot h$: $2 \cdot 9.81 \cdot 5 = 98.1$.
- Take the Square Root: $\sqrt{98.1} \approx 9.9 , ext{m/s}$.
The water will shoot out of the hole at nearly $10 , ext{m/s}$ (about 22 mph) until the tank level begins to drop.