Torricelli's Law

Torricelli's law is theorem about the relation between the exit velocity of a fluid from a hole in a reservoir to the height of fluid above the hole.

Learning Objective

Infer the exit velocity through examining the Bernoulli equation

Key Points

Torricelli's law applies to an inviscid, incompressible fluid ("ideal" fluid).
You can ascertain results from applying the Bernoulli equation between the top of the reservoir and the exit hole.
The relationship arises from an exchange of potential energy at the top of the reservoir to kinetic energy at the exit.
The final kinetic energy is equivalent to what a solid body would acquire when falling from height h.

Term

inviscid
A fluid with zero viscosity (internal friction). In reality viscosity is always present. However, it is often very small compared with other forces (e.g. gravity, pressure) and for common fluids (water and air) the fluid can be approximated as having zero viscosity.

Full Text

Torricelli's law is theorem in fluid dynamics about the relation between the exit velocity of a fluid from a sharp-edged hole in a reservoir to the height of the fluid above that exit hole . This relationship applies for an "ideal" fluid (inviscid and incompressible) and results from an exchange of potential energy,

Torricelli's Principle

A brief introduction to Torricelli's Principle for students studying fluids.

$mgh$, for kinetic energy,

$\frac{1}{2}\rho v^2$, at the exit.

This relationship can be derived by applying the Bernoulli equation between the top of the reservoir and the exit hole . Applying Bernoulli between the top of a reservoir and an exit hole at a height h below the top of the reservoir results in,

Exchange of Energy

Potential energy at the top of the reservoir becomes kinetic energy at the exit.

$p_t + \frac{1}{2}\rho v_t^2 + \rho gh_t = p_e + \frac{1}{2}\rho v_e^2 + \rho gh_e$

where subscript t implies evaluation at the top of the reservoir and subscript e implies evaluation at the exit. If we assume both the top of reservoir and the exit are open to the atmosphere, the zero for potential energy is at the exit hole, and the fluid velocity at the top of the reservoir is essentially zero (large reservoir, small hole), we arrive at

$\rho gh_t = \frac{1}{2}\rho v_e^2$

This can be solved for the exit velocity, resulting in,

$v_e = \sqrt{2gh_t}$

where again h_t is the height difference between the top of the reservoir and the exit hole. Due to the assumption of an ideal fluid, all forces acting on the fluid are conservative and thus there is an exchange between potential and kinetic energy. The result is that the velocity acquired by the fluid is the same that a body would acquire when simply dropped from the height h_t^.

A simple experiment to test Torricelli's law involves filling a soda bottle with water and puncturing the bottom with a small hole (about 1 cm in diameter). As the height in the reservoir decreases, the exit velocity will decrease as well. The exit velocity can be increased by capping the top of the reservoir and pressurizing it .

Toricelli's Law

The exit velocity depends on the height of the fluid above the exit hole.

Ideal Fluid

Applies to an ideal fluid (inviscid, incompressible)

[ edit ]

Prev Concept

Application of Bernoulli's Equation: Pressure and Speed

Surface Tension

Next Concept