How do I figure out how many gallons are in a 50 gal drum at different levels?

Question

I have a 50 gal barrel laying on its side. How do I figure out how much water each inch represents. I'm sure pi r sq'd is in there somewhere. The diameter of the barrel is 24". The length is 34". I want to mark the barrel so I know how much I am using.

First answer is for the barrel standing upright. My barrel is on its side. It's not a straight division. One inch in the center equals a lot more than one inch at the top or bottom. Like this - O.

Answer 1

Hi. If you are measuring from the bottom of the barrel (it's side, previously) call this quantity, h. Let the length of the barrel be L. Let the radius of the barrel be r. The volume at a specific h is given by:

Lr^2(arcsin[(h-r)/r] + sqrt(2rh - h)(h-r)/r^2 + pi/2) angles in radians

In case this is for a practical application, e.g. it's not your homework to derive the above, I've taken the above and fit it to a polynomial for relative volume and relative height. That is, if you are at height h, then I've defined relative height as:

x = h/r (of course h and r must be of the same units)

Then the volume is (to a very good approximation)

V = 50gal*(0.0109x^4 - 0.1866x^3 + 0.4875x^2 + 0.1888x)

To make things even simpler, if you want to demark every tenth of the tank (5gallons) then the values of x (from above) are roughly

x
0 (take h=0*radius of barrel = 0, corresponds to 0 gallons)
.3 5gal (3.6")
.5 10gal (6.24")
.68 15gal (8.16")
.85 20gal (10.2")
1 25gal (h=r=12", barrel is half full)

You can then mirror these measurements to get the other half of the barrel.

Oh, and this assumes the drum is perfectly cylidrical and horizontal. In all honesty, the best way to do this is empirically, carefully measure a gallon, dump it in, mark. I say carefully, otherwise you'll be compounding error if each dump is slightly under/over a gallon.

Peace

Answer 2

50/34 = 1 16/50 = 1.32 gal/inch

Each inch along the length will represent 1.32 gallons.

Alternatively, 34/50 = .68 inch/gallon

Every gallon can be represented by .68 inch.