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Torricelli formula and its applications

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1. Formula - what is Torricelli's theorem
2. Time required to empty a tank / vessel by gravity flow

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1. Formula - What is Torricelli's law

Torricelli formula is applied to calculate tank orifice flowrate. The following formula is Torricelli equation :

Q=Liquid flowrate at orifice (m3/s)
C=Orifice coefficient (-)
A₁=Orifice area (m2)
h=Height of liquid from top of tank to middle of orifice (m)
g=gravity acceleration (m.s^-2)

g=9.81 m.s^-2
C=0.6 for non profiled orifice

Torricelli's law calculation example

A tank has a hole of 10 cm diameter on its side following an incident such as a pipe rupture. The height of liquid above the hole is 5 m. What is the flow of liquid out of the tank ?

STEP 1 : calculate the orifice area

The hole is here circular, thus A1 = π*0.1²/4 = 0.0079 m²

STEP 2 : calculate the liquid flowrate

The liquid height above the middle of the hole is 5+0.1/2=5.05 m.

Q = 0.6*0.0079*(2*9.81*5.05)^0.5 = 0.047 m³/s = 168 m³/h

2. Time required to empty a tank / vessel by gravity flow

The Torricelli equation can be used to calculate the time required to empty a tank by gravity. Tank emptying time formula :

t_v=Time in s to discharge a tank up to the level of the orifice (s)
H=Height of liquid at t=0 (m)

Time to empty a tank calculation example

The tank mentioned in the example in paragraph 2 is considered here. The tank is square, with dimensions of 2m*2m.

STEP 1 : calculate the area A₀

The tank is square, this A₀ = 2*2 = 4 m²

STEP 2 : calculate the time required to empty the tank

The liquid height above the middle of the hole is 5+0.1/2=5.05 m. A₁ has been calculated in the previous example.

t_v = 4/(0.6*0.0079)*(2*5.05/9.81)^0.5 = 861 s

Source

Mecanique et Rheologie des fluides en genie chimique, Noel Midoux, Lavoisier Tec et Doc, page 51