Chapter 7

# Emptying of a spherical tank

Take 20 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

A spherical tank, of radius , is initially half filled with water density .

Atmospheric pressure prevails above the free surface of the water through an opening at the top of the tank.

One opens, at , a circular hole of a small cross section at the tank bottom.

Emptying of a spherical tank

## Question

Establish the differential equation in , if is the water level in the reservoir counted from at time .

### Solution

Neglecting the velocity of the free surface of the water, the Bernoulli theorem between surface and the output gives :

Where :

Torricelli's formula is found.

Since water is incompressible, the volume flow rate is conserved :

Or (see figure) :

Either after the separation of the variables :

Emptying of a spherical tank

## Question

Literally express and calculate the duration of emptying of this tank.

### Solution

Emptying duration is :

Either :

The numerical application gives minutes and seconds.

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Bernoulli's equation
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Navier-Stokes equation