How Do You Model Water Draining from a Spherical Tank Over Time?

[bige1027]

Fluid Mechanics - Draining sphere - I need Help!

I just found this forum and it seems like a wealth of knowledge; wish I had found it sooner. Looking for some help and if anyone can, it will be appreciated more than you'll ever know.

Here's the problem:

A spherical tank of diameter D is filled with water. It has a small vent at the top to allow for atmospheric pressure within the tank. The water drains from a small drain hole at the bottom (dia=1 inch). The flow is quasi steady and inviscid. Find a function for the height of the water w.r.t. time, h(t), where 'h' is the height of the water measured from the bottom of the sphere.
Use the function to determine the water depth for D=1, 10, and 50 ft.

This is what I have:
It's a "free jet" probelm, so the water draining at the bottom leaves with a velocity of V=sqrt(2*g*h) - derived from the Bernoulli eq. with points at the top of bottom of the sphere.

The flowrate out is Q=AV=[(pi/4)*(1/12)^2]*[sqrt(2*g*h)]

Volume sphere = 4/3 *pi*R3

Here I've been stuck for a long time. Does anyone have any ideas where to go from here?

 

[Gokul43201]

So far, you're on track.

What you need to do next is be able to write down the volume of water in terms of the height.

Look at eqn (6) here and make the appropriate changes (if necessary).

Then you find the rate of chage of volume in terms of the rate of change of height and equate that to the outflow.

 

[jeff25111]

fluid mech-draining tank

i kind of have the same problem, only i am dealing with a tank, with a hole at the bottom and the water surface having a pressure of 1 atm. the question is, how long will it take for the water to drain? i have assumed that the velocity of water coming out of the small hole is>>> than the velocity of the water surface so that the latter is approx=0.
i know the rate of water drainage changes according to the pressure and height of the water level inside. i just don't know hot po put this concept in equation. any help would really be appreciated!

 

[katchum]

What you need is:

Accumulation = in - out + reaction

Is there accumulation? Yes.
Is there in? No.
Is there out? Yes.
Is there reaction? No.

You should get something like: dV/dt = 0 - A.v(t)
Just calculate V(h) and you know v(t) = sqrt(2gh(t))
So you have a diff equation with variable h(t), you integrate and you have h(t).

Then you fill in h = 0 and there it is!