Torricelli's law

of efflux of a fluid through a sharp-edged hole in the wall of the tank filled to a height

above the hole is the same as the speed that a body would acquire in falling freely from a height

The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643.

Under the assumptions of an incompressible fluid with negligible viscosity, Bernoulli's principle states that the hydraulic energy is constant at any two points in the flowing liquid.

In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening.

(A like Aperture) gives: Torricelli's law is obtained as a special case when the opening

: Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels.

The spouting can experiment consists of a cylindrical vessel filled up with water and with several holes in different heights.

It is designed to show that in a liquid with an open surface, pressure increases with depth.

at the bottom, then rate of change of water level height

The water volume in the vessel is changing due to the discharge

is the total time taken to drain all the water and hence empty the vessel.

Lastly, we can re-arrange the above equation to determine the height of the water level

In many cases, such experiments do not confirm the presented discharge theory: when comparing the theoretical predictions of the discharge process with measurements, very large differences can be found in such cases.

In 1738 Daniel Bernoulli attributed the discrepancy between the theoretical and the observed outflow behavior to the formation of a vena contracta which reduces the outflow cross-section from the orifice's cross-section

For rectangular openings, the discharge coefficient can be up to 0.67, depending on the height-width ratio.

be the vertical height traveled by a particle of jet stream, we have from the laws of falling body where

, then the horizontal distance traveled by the jet particle during the time duration

we obtain and the maximum range A clepsydra is a clock that measures time by the flow of water.

It consists of a pot with a small hole at the bottom through which the water can escape.

This can be attained by letting a constant stream of water flow into the vessel, the overflow of which is allowed to escape from the top, from another hole.

Thus having a constant height, the discharging water from the bottom can be collected in another cylindrical vessel with uniform graduation to measure time.

We want to find the radius such that the water level has a constant rate of decrease, i.e.

The instantaneous rate of change in water volume is From Torricelli's law, the rate of outflow is From these two equations, Thus, the radius of the container should change in proportion to the quartic root of its height,

Likewise, if the shape of the vessel of the outflow clepsydra cannot be modified according to the above specification, then we need to use non-uniform graduation to measure time.

Evangelista Torricelli's original derivation can be found in the second book 'De motu aquarum' of his 'Opera Geometrica'.

[5] He starts a tube AB (Figure (a)) filled up with water to the level A.

Then a narrow opening is drilled at the level of B and connected to a second vertical tube BC.

Due to the hydrostatic principle of communicating vessels the water lifts up to the same filling level AC in both tubes (Figure (b)).

When performing such an experiment only the height C (instead of D in figure (c)) will be reached which contradicts the proposed theory.