A large tank, with its top open to the atmosphere, is filled with liquid to a he

Question

A large tank, with its top open to the atmosphere, is filled with liquid to a height h. Near the bottom of the tank there is a hole that allows the fluid to exit to the
atmosphere. Apply Bernoulli’s equation (neglect friction) to a fluid streamline that
extends from the surface of the liquid at the top to a point in the exit stream, just outside the vessel. Show that this leads to an exit velocity of Uefflux=sqrt of 2gh where g is the gravitational acceleration. Why can you neglect the velocity of the receding fluid at the
top surface?

Explanation / Answer

Bernoulli's equation says the following

P1 + gh1 + .5v12 = P2 + gh2 + .5v22

We are told that the Pressures at points 1 and 2 are both the atmospheric pressure. Since they are the same, they cancel. We can rewrite the equation as

gh1 + .5v12 = gh2 + .5v22

Now you can see that the density () can cancel out, so a new rewrite is

gh1 + .5v12 = gh2 + .5v22

We are told that the velocity of the fluid at the top of the tank can be neglected, we aslo know that the height at the opening is zero, thus the equation becomes

gh1 = .5v22

Solve for v2

v2 = (2gh1)

Since v2 is the exit speed and h1 = it initial height of the of the liquid in the tank, then

v = (2gh)

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