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Math Help - Volume of Spherical Cap Section

  1. #1
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    Red face Volume of Spherical Cap Section

    So,

    In calculating the volume of a partially-filled horizontal cylindrical tank which has ends similar to spherical caps, how do you calculate the volume of a partially-filled spherical cap (from the side of a sphere?)??

    The cylinder I have solved, and the volume of a full spherical cap has been easy to find but the partial volume of the cap ends is tough.

    This has been so hard for me to find information on, so I hope someone here has an idea!

    The dimensions of this particular problem are: Cylinder Length = 8.16m, Diameter = 3.7m, Cap "height" (although horizontal) = 0.53m. I hope this is enough detail...
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  2. #2
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    I'm not sure I understand the shape itself. Can you give a "real world" example of this shape? Picture would be really helpful too!
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  3. #3
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    For example the rounded end of a part-filled horizontal cylindrical tank, in this case.
    Attached Thumbnails Attached Thumbnails Volume of Spherical Cap Section-tank.bmp  
    Last edited by babryce; March 5th 2009 at 08:21 AM.
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  4. #4
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    Something like this... ?

    So the entire normal cylinder is filled, but the half-sphere cap is only partially filled?

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  5. #5
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    Just like that, only rotated in x or z axis by 90 degrees so that the cylinder is in a horizontal position. The end is not hemispherical, but is the cap of a sphere with approximately double the diameter.
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  6. #6
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    Quote Originally Posted by babryce View Post
    The end is not hemispherical
    Yikes... This one is ugly! Do you know the radius of that arc shape? Is it even a radius or is maybe part of an elliptical shape (even uglier)? If so I may have a way to find it, but I won't explain until I know whether it's radial.
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  7. #7
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    This page may be of help:

    * TankCalc
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  8. #8
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    I guess the end caps put together could be called a squashed sphere or a "spheroid ovoid" .. There are many english sweets this shape but I don't know the names of any american candy for comparison. Anyway, it's circular in 1 axis, ovoid in 2 axes. Split in half down the circular face and stuck on the end of a horizontal cylinder.

    That link looks like just the job, I've just got to pick it to pieces so I know that I can trust it now!
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  9. #9
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    Wow... unfortunately I really have no idea how to help you...

    Just an idea about how to think of it... think of it as two separate shapes. One is the cylinder. The other is the merged end caps (which would have a shape similar to an American football). I don't know if that makes it any easier or not. Sorry I couldn't be more help!
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