Maths: Time required to drain a tank

Just after I finished university and started working, a fellow graduate asked me how to calculate how long it would take to drain a tank. It’s fairly standard for most projects that we work on to have some kind of design criteria for how quickly a tank needs to be able to be drained for maintenance. I put together an quick spreadsheet to do an iterative time-step based calculation, which gave a perfectly acceptable answer… but it gnawed at me that it wasn’t completely correct.

So I did some calculus and came up with the equation below (for an atmospheric tank draining from height h_tank above the drain outlet:

Tank Draining, derivation_22

For a tank to drain from height h_tank above the drain, down to a second level h_B above the drain, use the following:

Tank Draining, derivation_24

The above equations use a combined loss coefficient k, which should include entry loss, valve friction loss, and pipe/bend friction losses on the drain nozzle.

Years later, and we were discussing the same problem for another project, I pulled out the equation, which I had stored away in my digital library. Then came the tricky bit… someone asked me how I came up with the equation, and to prove that it’s right…

Years out of university, and my maths isn’t as sharp as it was when I was a fresh graduate.

Dunce.

So, while on holiday recently, I had some downtime; I put up my feet, had a cup of tea, and did some maths. ahhhhhhh.

Shut up, it was fun.

So yes, while a google search can give you the basic equation for the time required to drain a tank, here’s a reference which:

  • Shows the derivation (so you understand where it comes from and how to modify it to suit your needs),
  • Tanks into account friction losses through valves and fittings, and
  • Includes a formula that takes into account non-atmospheric pressure inside the tank.

You can check the PDF linked below if you care about draining non-atmospheric tanks.

Here’s a copy of the derivation for anyone who cares: Tank Draining, derivation

And here’s a copy of the spreadsheet that I used to check the equations against a numerical solution: tank draining verification

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