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# Hopper volume

## Hopper volume

(OP)
Does anyone know the formula for the volume of a hopper which has a round top and a rectangular outlet.  Rather like the frustrum of a cone except the bottom is rectangular.

### RE: Hopper volume

If I'm envisioning this correctly I see a conical piece fastened to a rectangular piece with the joints defined by elipses, with two (opposite) wide sides and two narrow sides.  Is this correct?

### RE: Hopper volume

the larger dimension of the rectangular outlet equal to
or more than 50% the diameter of the hopper top? Usually
for the kind of hopper you've described, it is a common
practice (for ease in fabrication) to use a frustrum of a cone and provide a transition outlet (round to rectangular),
unless your case is what was described previously described.

### RE: Hopper volume

Show us the picture then I think most of use will tell you the Volume.

### RE: Hopper volume

(OP)
Thanks ve7brz for your reply.  This is a simple shape and is a transition from 2m dia to 600mm x 2m rectangle over a height of approximately 1.7m

Regards

Brian

### RE: Hopper volume

(OP)
Thanks vtl and Estassoc for your replies.  I cannot use the frustrum and transition as the products held in the hopper would get stuck.  I need to transform from round to rectangle in one piece.  This is a simple transition shape transition from 2m dia to 600mm x 2m rectangle over a height of approximately 1.7m.  The hopper is not offset in any way. Centre of top and bottom in line!

Sorry a picture is not available and a drawing would be a bit impractical at this stage.

Regards

Brian21

### RE: Hopper volume

Based on what you described:

2m diameter tranforming to 600mm X 2m rectangle-

My Solid model shows a volume of 3.611 cubic meters.  Now, I also took the liberty of accepting the defaults that control the lofted shape so this could be off a little based on your design.

Hope it helps.

jackboot

### RE: Hopper volume

(OP)
Thanks jackboot.  This sounds about right but as i am desgning the beast, i need the formula to allow me to settle on a design.

Regards

Brian21

### RE: Hopper volume

Brian21,

and I'll send you the solution.

Try this and determine the volume of each commponent:
1. Draw a the top circle (2.00 m) and the bottom
rectangular section. For simplicity, set the 4-corner
points of the rectangle on the circumference of the
circle (rectangle length is now less than 2.00m).
2. Now looking at your drawing, there are 5-sections
(1-rectangle & 4-circular segments, 2-large/2-small).
3. Calculate the area of the rectangle, 1- large circular
segment and 1-small circular segment. Use any method
you are familiar with.
4. The volume of the hopper will be the sum of the 3
areas multiplied by the height of the hopper.

The volume calculations is based on the premise that the
volume of a sliced circular segment (slicing plane is such
that the resulting sliced cross-section is a triangle whose
2-vertices lies on the rectangular outlet and 1-vertex on
the quadrant of the circle) is half of the unsliced circular
segment.

### RE: Hopper volume

Brian21,

In my last post, the premise on the sliced volume is incorrect. You have to calculate the sliced volume (a portion of a cylinder using the formulas shown in page 64 of the Machinery Hanbook 24), add all the 4 sliced volume
and subtract the total from the volume of the cylinder.
By 3D modelling, the volume is 3.932427 cu.m. By calcs,
the volume is approx. 0.02% less.

Estassoc

### RE: Hopper volume

(OP)
Thanks Estassoc thats about what I thought it was.  I did however at one time have a proper and easy to follow formula for this but I think a colleague now has it.  I was going to put it into a spreadsheet format on our intranet.

Regards

Brian

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