Volume of a horizontal cylinder using height of liquid

Question

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank.

We need to determine the amount of liquid in a buried tank. We do this by “sticking”, that is, we insert a dipstick through an opening over the deepest part of the buried container until the dipstick touches the bottom, then we pull the stick out and read the liquid level showing on the stick.

I need a formula for converting the height shown on the stick to the volume of liquid in the tank.

Answer 1

If the length of the tank is $L$, the radius of the tank is $R$, and the height of the liquid is $h$, then if $h < R$ the volume is $$(R^2 \cos^{-1} \frac{R-h}{R} - (R - h)\sqrt{2Rh - h^2})L$$ This follows from the formula for the area of the segment of circle bounded by a chord and an arc. See, for example, the Wikipedia entry for circular segments.

Answer 2

We integrate the equation of a circle of diameter $1$, with center at $x=1/2$, with a correction factor such that the total area equals $1$: $$\frac{4}{\pi}\int_0^h\sqrt{x-x^2}dx$$ and get $$v=\frac{\arcsin(2h-1)+2(2h-1)\sqrt{h-h^2}}\pi+\frac12.$$

This formula gives the fraction of the total volume ($v$ in $[0,1]$) as a function of the fraction of the total height ($h$ in $[0,1]$).

Answer 3

Area found by integration of circle segment.

$$ y= h/R\,;\; Volume = (Length/\pi) ( \cos^{-1}(1-y)-(1-y) \sqrt { y( 2-y)}) $$

Plotted y vs Volume.

Answer 4

I had the same problem. Calculus is nice, but there's a much simpler way.

For a given horizontal cylinder:

V = pi/4 * D^2 * L * h/D

where,

V is the volume of the cylinder

D is the diameter of the cylinder

L is the length of the cylinder

h is the measured height of the liquid in the cylinder, and 0 <= h <= D

It's a simple ratio correction factor. If h = D/2, then the correction factor is 1/2 -- if the height measured was half the diameter, then you would have half the volume of a horizontal cylinder.

Simple.