Calculating Volume of Ellipsoidal Propane Tanks

[GroundSpider]

I have a project I am working on that requires me to calculate the overall volume of propane tanks and the volume at a set distance from the bottom of the tank. The tanks come in two major configurations, ellipsoid ends with elliptical tank or hemispherical ends with cylindrical tank. I think the formula for the ellipsoid would work for the hemispherical tanks if we just set the dimensions to make a circle instead of an ellipse. I have attached an image to help clear this up.

W = Width of the ellipse
D = Depth of the ellipse
H = Overall height of tank
P = Height of propane in tank (This is used to calculate overall fill % of the tank)
E = Height of elliptical end

Essentially I need help determining the volume of a section of an ellipsoid end.

View attachment 3315

 

[MarkFL]

In order to compute these volumes, you will need to know the area of an ellipse, so that you can then use this as part of a volume element.

I would begin with the general equation of an ellipse centered at the origin:

\(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

Knowing the ellipse has symmetry across both coordinate axes, we need only find the area in the first quadrant, and then quadruple this result.

So, we need to solve the equation for $y$, and take the positive root...what do we get?

 

[Opalg]

If you are just interested in quoting the answer, there are easy formulas for the area of an ellipse and the volume of an ellipsoid. The area of an ellipse with semi-axes $a$ and $b$ is $\pi ab$. The volume of an ellipsoid with semi-axes $a$, $b$ and $c$ is $\frac43\!\pi abc$. Notice that when the semi-axes are all equal, these reduce to the familiar formulas for the area of a circle and the volume of a sphere.

Those formulas should enable you to find the volume of propane provided that the depth of propane is sufficient that it completely fills the hemi-ellipsoidal base of the tank (that is, if $P > E$ in your diagrams). If $P<E$ then the calculation will be harder. The formula given here for the volume of an ellipsoidal cap, where the semi-axes are $a$, $b$ (the horizontal axes) and $c$ (the vertical axis), filled to a depth $h$, is \(\displaystyle \frac{\pi ab}{3c^2}h^2(3c-h).\)

 

[GroundSpider]

Thank you so much Opalg! That is exactly what I needed. I am writing a specialized calculator to facilitate manufacturing of a float gauge that determines the tank's fill level.