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.
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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.
Establish the differential equation in , if is the water level in the reservoir counted from at time .
Neglecting the velocity of the free surface of the water, the Bernoulli theorem between surface and the output gives :
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 :
Literally express and calculate the duration of emptying of this tank.
Emptying duration is :
The numerical application gives minutes and seconds.