Volumes of vertical cylindrical tank with flat sloped floor when partially filled

[agregorynz]

I am a winemaker and am trying to work out graduated dip charts for tanks in the winery I work for.

The part that is giving me the most trouble are working out graduated volumes for flat sloped floor cylinder tanks. Think of a wedge that has had a cookie cutter put through it then placed in the bottom of a cylinder of the same diameter as the cookie cutter. I am okay to calculate volumes for the other more straight forward parts of a wine tank (cones, cylinders, rectangles etc) its just the flat sloping floor that is puzzling me.

I would like to know formula for the volume of this section of a tank at different fill heights. I am guessing it might consist of two formulas for the upper and lower half of the widest point on the floor. I haven't done algebra/calculus/geometry in maaaaaannnny years and would appreciate a bit of help. It probably is related to circular segments and chords but I couldn't tell you more than that.

Thanks in advance for any help offered.

 

[tkhunny]

If it were me, I might be tempted to get a test barrel and create a chart. Fill the VERTICAL barrel to whetever percentage you want, then put it in the desired position and measure the dip stick. A few dozen dips and you should be good to go for nearly every conceivable position.

 

[Willie]

This feels like a calculus problem to me, and I do not currently know enough calculus to answer it. But I think that a clue to this is that any horizontal slice through the lower part of the tank will look like one of three shapes: a circle segment if below the widest part of the floor, a simicircle at the widest point, or the complement of a circle segment above the widest point. Calculating the area of any one of those slices is a pretty straight forward geometry problem that can be defined as a not-too-complicated function. But, since that area changes in a non-linear way as you move up the tank, calculating the volume is not so simple. The volume at any point could be thought of as the sum of an infinite number of volumes of infinitesimal slices, which I believe would be called an "integration" problem in calculus. I don't know how to solve it, but moving the question to the calculus section might give it a better chance of getting the attention of someone who does.

The volume up to the top edge of the floor is pretty easy though. It's 1/2 of the volume of a whole cylinder of the same height.

 

[Willie]

Still trying to figure out what d'heck that is...
how can something be flat, yet sloping...

I think he means that the shape of the floor is like a plane that intersects a cylinder at a non-right angle. The outline of the floor would be an ellipse.