Introduction

We have all seen liquid running out of containers through holes in the bottom. The physical principles involved are the same whether the vessel is a bucket with a hole in it or an industrial tank being drained during a manufacturing process. Frequently it may be advantageous to drain a container as rapidly as possible. An obvious way to do that, of course, is to make the drain as big as possible. But other constraints usually make that impractical.

So, what is it that governs the drainage rate driven by gravity? This is not a trivial question, but instead raises a potentially significant engineering issue concerning the proper shapes of industrial vessels.

Stated more formally, given a smooth, well-rounded orifice for a drain, what is it that governs the rate at which a container will drain? Perhaps even more important is what determines the total time required for a given container to drain completely. Do all containers holding the same volume and identical drains empty at the same rate? Are the rates constant? If not, what is it that affects the rate? These are not trifling questions. In the world of commerce, time is money.

In this project, we will consider three different containers, all with exactly the same total volume and with identical drain orifices. What's more each container will have the same initial level of fluid, measured from the drain orifice in the bottom. The shapes we will consider are: 1) a circular cylinder, 2) a section of a cone with the small end down, and 3) an identical section of cone with the large end down. Other shapes can also be investigated, but the principles are illustrated with these simple cases.