Fluid Mechaincs - Draining Sphere

[bige1027]

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?

 

[FredGarvin]

You're going to have to come up with a ODE describing the height in the tank as a function of time.

Try taking a look here and see the process that is followed.

http://www.krellinst.org/UCES/archive/modules/cone/cone/node1.shtml

 

[bige1027]

I was going along those same lines...thank you very much for that link. It helped me get past my sticking point.