Torricelli's law

Not to be confused with Torricelli's equation or Torricelli point.

Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an opening to the height of fluid above the opening.

Torricelli's law describes the parting speed of a jet of water, based on the distance below the surface at which the jet starts, assuming no air resistance, viscosity, or other hindrance to the fluid flow. This diagram shows several such jets, vertically aligned, leaving the reservoir horizontally. In this case, the jets have an envelope (a concept also due to Torricelli) which is a line descending at 45 degrees from the water's surface over the jets. Each jet reaches farther than any other jet at the point where it touches the envelope, which is at twice the depth of the jet's source. The depth at which two jets cross is the sum of their source depths. Every jet (even if not leaving horizontally) takes a parabolic path whose directrix is the surface of the water.

Torricelli's law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. $v={\sqrt {2gh}}$, where g is the acceleration due to gravity (9.81 N/kg near the surface of the earth). This last expression comes from equating the kinetic energy gained, ${\frac {1}{2}}mv^{2}$, with the potential energy lost, mgh , and solving for v.

The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

Derivation

Under the assumptions of an incompressible fluid with negligible viscosity, Bernoulli's principle states that:

${v^{2} \over 2}+gz+{p \over \rho }={\text{constant}}$

where v is fluid speed, g is the gravitational acceleration (9.81 m/s^2), z is the fluid's height above a reference point, p is pressure, and ρ is density. Define the opening to be at z=0. At the top of the tank, p is equal to the atmospheric pressure. v can be considered 0 because the fluid surface drops in height extremely slowly compared to the speed at which fluid exits the tank. At the opening, z=0 and p is again atmospheric pressure. Eliminating the constant and solving gives:

$gz+{p_{atm} \over \rho }={v^{2} \over 2}+{p_{atm} \over \rho }$

$\Rightarrow v^{2}=2gz\,$

$\Rightarrow v={\sqrt {2gz}}$

z is equivalent to the h in the first paragraph of this article, so:

$v={\sqrt {2gh}}$

Experimental evidence

Torricelli's law can be demonstrated in the spouting can experiment, which is designed to show that in a liquid with an open surface, pressure increases with depth. It consists of a tube with three separate holes and an open surface. The three holes are blocked, then the tube is filled with water. When it is full, the holes are unblocked. The lower a jet is on the tube, the more powerful it is. The fluid's exit velocity is greater further down the tube.

Ignoring viscosity and other losses, if the nozzles point vertically upward then each jet will reach the height of the surface of the liquid in the container.

Application for time to empty the container

Consider a container containing water to height h is being emptied through a tube freely. Let h be the height of water at any time. Let the velocity of efflux be $v={\sqrt {2gh}}\ ={dx \over dt}$

Now, $A\,dh=a\,dx$ where A and a are the cross sections of container and tube respectively, dh is the height of liquid in container corresponding to dx in the tube which decreases in same time dt.

${A\,dh \over \ a\,dt}={\sqrt {2gh}}$

$\Rightarrow {A\,dh \over \ a{\sqrt {2gh}}}=dt$

$\Rightarrow \int \ {A\,dh \over \ a{\sqrt {2gh}}}=\int \ dt$

$\Rightarrow {2A{\sqrt {h}}\ \over \ a{\sqrt {2g}}}=t$

$\Rightarrow t={2A{\sqrt {h}}\ \over \ a{\sqrt {2g}}}$

$\Rightarrow t={2A \over \ a{\sqrt {2g}}}({\sqrt {h_{1}}}-{\sqrt {h_{2}}})$ is the time required to empty the water from height h₁ to h₂ in the container.

Further reading

• T. E. Faber (1995). Fluid Dynamics for Physicists. Cambridge University Press. ISBN 0-521-42969-2.
• Stanley Middleman, An Introduction to Fluid Dynamics: Principles of Analysis and Design (John Wiley & Sons, 1997) ISBN 978-0-471-18209-2
• Dennis G. Zill, A First Course in Differential Equations (2005)

See also

• Pascal's law
• Fluid dynamics
• Darcy's law
• Dynamic pressure
• Fluid statics
• Hagen–Poiseuille equation
• Helmholtz's theorems
• Kirchhoff equations
• Knudsen equation
• Manning equation
• Mild-slope equation
• Morison equation
• Navier–Stokes equations
• Oseen flow
• Pascal's law
• Poiseuille's law
• Potential flow
• Pressure
• Static pressure
• Pressure head
• Relativistic Euler equations
• Reynolds decomposition
• Stokes flow
• Stokes stream function
• Stream function
• Streamlines, streaklines and pathlines
• Bernoulli's principle