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Hopper volume

brian21 (Mechanical) (OP) 
20 Nov 02 7:52 
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. 

ve7brz (Industrial) 
20 Nov 02 11:29 
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? 

Just curious about the design of your hopper. Is one of 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. 

vtl (Mechanical) 
20 Nov 02 20:51 
Show us the picture then I think most of use will tell you the Volume. 

brian21 (Mechanical) (OP) 
21 Nov 02 6:33 
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 

brian21 (Mechanical) (OP) 
21 Nov 02 6:38 
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 

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 

brian21 (Mechanical) (OP) 
21 Nov 02 9:34 
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 

Brian21,
If you can't visualize what I'm describing, post your addy 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 4corner 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 5sections (1rectangle & 4circular segments, 2large/2small). 3. Calculate the area of the rectangle, 1 large circular segment and 1small 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 crosssection is a triangle whose 2vertices lies on the rectangular outlet and 1vertex on the quadrant of the circle) is half of the unsliced circular segment. 

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 

brian21 (Mechanical) (OP) 
22 Nov 02 3:42 
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 



