Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The law states that the speed of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height , i.e. , where is the acceleration due to gravity. This expression comes from equating the kinetic energy gained, , with the potential energy lost, , and solving for . 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 the hydraulic energy is constant
at any two points in the flowing liquid. Here is fluid speed, is the acceleration due to gravity, is the height above some reference point, is the pressure, and is the density.
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. Since the liquid is assumed to be incompressible, is equal to and ; both can be represented by one symbol . The pressure and are typically both atmospheric pressure, so . Furthermore is equal to the height of the liquid's surface over the opening:
The velocity of the surface can by related to the outflow velocity by the continuity equation , where is the orifice's cross section and is the (cylindrical) vessel's cross section. Renaming to (A like Aperture) gives:
Torricelli's law is obtained as a special case when the opening is very small relative to the horizontal cross-section of the container :
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.
Experimental verification: Spouting can experiment
Every physical theory must be verified by experiments. 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. The lower a jet is on the tube, the more powerful it is. The fluid exit velocity is greater further down the tube.
The outflowing jet forms a downward parabola where every parabola reaches farther out the larger the distance between the orifice and the surface is. The shape of the parabola is only dependent on the outflow velocity and can be determined from the fact that every molecule of the liquid forms a ballistic trajectory (see projectile motion) where the initial velocity is the outflow velocity :
The results confirm the correctness of Torricelli's law very well.
Discharge and time to empty a cylindrical vessel
Assuming that a vessel is cylindrical with fixed cross-sectional area , with orifice of area at the bottom, then rate of change of water level height is not constant. The water volume in the vessel is changing due to the discharge out of the vessel:
Integrating both sides and re-arranging, we obtain
where is the initial height of the water level and is the total time taken to drain all the water and hence empty the vessel.
This formula has several implications. If a tank with volume with cross section and height , so that , is fully filled, then the time to drain all the water is
This implies that high tanks with same filling volume drains faster than wider ones.
Lastly, we can re-arrange the above equation to determine the height of the water level as a function of time as
where is the height of the container while is the discharge time as given above.
Discharge experiment, coefficient of discharge
The dicharge theory can be tested by measuring the emptying time or time series of the water level within the cylindrical vessel. In a lot of cases such experiments do not confirm the presented dicharge theory: When comparing the theoretical predictions of the discharge process with measurements, very large differences can be found in such cases. In reality, the tank usually drains much more slowly. Looking at the discharge formula
two quantities could be responsible for this discrepancy: the outflow velocity or the effective outflow cross section.