Smooth Water Flow From Tank

Classical Mechanics Level 3

This problem still under construction, needs to reword more precisely

A round pipe orifice $1$ cm in radius with a cross section area of $\pi$ cm $^2$ is at the side of a tank as shown.

Assume ideal non-viscous fluid and frictionless hydraulic flow, and disregard acceleration of water flow due to gravity.

What's the mininum cross section area of the water flow as it leaves the pipe orifice?

If the answer is $x\pi$ cm $^2$ , express $x$ to three decimal places

The answer is 0.5000.

1 solution

Michael Mendrin
May 21, 2018

Let $A$ be the cross section area of the orifice at the tank, $B$ be the final cross section of the water flow.

Let $v$ be the velocity of the water flow as it leaves the orifice, $p$ be the mass density of water, $h$ be the height of the water level above the orifice, and $g$ be the gravitational acceleration.

Then the force $F$ of water at the orifice is $Apgh$ , which is the same as the weight of a column of water with cross section $A$ and height $h$

The thrust $T$ necessary to compel the water flow is $Bpv^2$ , where $v=\sqrt{2gh}$ , the same if the water had fallen from a height of $h$ . This is the consequence of conservation of energy.

The force $F$ and thrust $T$ must be equal, so we have

$Apgh=Bp2gh$

Thus $B$ is exactly half of $A$ , and the answer is $0.500$

See Vena Contracta

Well shouldn't F at orifice be $Aphg$

Satyabrata Dash - 3 years ago

...right, thanks for the catch. Otherwise, dimensionally it wouldn't be force.

Michael Mendrin - 3 years ago

Smooth Water Flow From Tank

The answer is 0.5000.

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