Using the Bernoulli equation to derive Toricelli's equation

Posted in math on Saturday, May 31 2014

Bernoulli's equation for fluid flow along a streamline is a really useful go-to equation for explaining various fluid mechanics phenomena, and if you extend it by taking into account friction you end up with the engineering Bernoulli equation or a mechanical energy balance, which is practically the default starting point when dealing with fluid flow in piping and process.

In this example, problem 3B.14 from Transport Phenomena, I use the Bernoulli equation to derive Toricelli's equation for efflux from a tank.


First we imagine the level of the tank isn't moving over the time-scale that we are looking at, this is the pseudo-steady-state or quasi-steady-state assumption. Additionally we assume that we can neglect the velocity of the fluid at the free surface of the tank. This is usually a good approximation if the diameter of the tank is large relative to the orifice.

Then we can drop pressure since the pressure at the free surface is equal to the pressure at the orifice, atmospheric pressure. Re-arranging we get Toricelli's equation: $ v_{2} = \sqrt {2 g h} $