Torricelli's law (fluid dynamics)

This calculator allows you to calculate the velocity of an ideal fluid flowing from a small hole based on the height of the fluid above the hole, as well as the horizontal distance covered by the jet as a function of the height of the hole above ground level.

You can find the formulas used for the calculation below the calculator.

Torricelli's law (fluid dynamics)

Speed of efflux of a fluid, m/s

Horizontal distance traveled by the jet, meters

Digits after the decimal point: 2

Torricelli's theorem

The Torricelli formula establishes the dependence of the velocity of an ideal fluid (i.e., a fluid in which there is no viscosity and heat conduction) flowing out of a small hole (i.e., a hole whose vertical dimension does not exceed 0.1...0.2 of the height of the fluid above the hole) in an open vessel (i.e., the surface of the vessel experiences atmospheric pressure) on the height of the fluid above the hole (also called head).

The formula is derived from Bernoulli's law, which states that $\rho v^{2}/2+\rho gh+p$ maintains a constant value along the line of current (the line whose tangent direction coincides with the direction of velocity of the fluid particles). That is
$\rho v^{2}/2+\rho gh+p=const$,
where
ρ is the density of the fluid;
v is the velocity of the flow;
h - height;
p - pressure;
g - acceleration of free fall.

To obtain the Torricelli formula, the expressions at the surface of the vessel and at the orifice are equated. The height at the hole is taken as zero. The velocity of the liquid at the surface is also assumed to be zero, since the liquid level at the moment decreases very slowly compared to the velocity of the liquid flowing through the hole. As a result, we obtain
$\rho v^{2}/2+p=\rho gh+p$,
whence
$\rho v^{2}/2=\rho gh$,
and finally
$v={\sqrt {2gh}}$

Ordinary kinematics will help to find the distance of the jet. Assuming that at the exit from the hole the jet has only horizontal component of velocity, we can find the time for which the jet will touch the ground, depending on the height of the hole above the surface
$h=\frac{gt^2}{2}$,
whence
$t=\sqrt{\frac{2h}{g}}$

The horizontal path traveled is time multiplied by velocity, or
$d=vt=v\sqrt{\frac{2h}{g}}$

It is worth noting that the maximum distance of the jet is achieved if the hole is located at the height of half of the vessel and is equal in value to the value of the height of the vessel.

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PLANETCALC, Torricelli's law (fluid dynamics)