Math CentralQuandaries & Queries


Question from Todd:

Hello I am looking for a formula to figure out the fertilizer volume in a hopper bottom bin not only when it is full but part full as well. When you are filling it is heaped up in the middle to make a cone and when you are emptying the bin the cone is inverted so it would be nice to be able to quickly figure out the tonnes partly filled and when full.
Lets say the bin is 32 feet high from top of bin where you fill to the bottom where the product goes out and it is 16 feet in diameter. I know how to calculate the cylinder it is the cones on the top and bottom of the bin I have the main question on.


I'm going to need more information to do this. First for the bin itself. You gave us an example of a bin height and diameter but my calculations will need the cylinder height and cone height.


You say "When you are filling it is heaped up in the middle to make a cone...". I think the fertilizer cone height in the diagram below depends on the material. I have never worked with fertilizer but from some work I did years ago with sand I know that the cone height depended on the type of sand and the moisture content.

bin with fertilizer

Can you tell me the fertilizer cone height for the fertilizer? It will depend on the diameter of the bin but the slope of the side of the fertilizer cone should depend only on the material and hence if you can tell me the fertilizer cone height and diameter of the bin I should be able to estimate the fertilizer cone height for bins of different diameters.

When you want to make a calculation what measurement will you use? Will it be the fertilizer height in the diagram above? The calculations I have in mind will give the volume of fertilizer in the bin so to determine the tonnage you will need the density of the fertilizer.

I am sure there will be other questions but I think this is possible with the right information.


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