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5 December 2016

Running Small-scale

Equations for plant volumes 2

The gas storage volume (V) of an underground biogas plant is a segment of a dome, or a cap cut from the top of a dome of radius (R). The horizontal radius (r) of the base of the cap is related to the height (h) of the cap by:

\[R^2 = r^2 + \left( {R - h} \right)^2\]

Which gives:

\[h = R \pm \sqrt {R^2 - r^2 }\]

Now:

\[ V = {\pi \over 3}h^2 \left( {3R - h} \right) \]

Rearranging gives:

\[ V = {\pi \over 6}h\left( {3r^2 + h^2 } \right)\]

Using the figures from the drawing, the spherical segment that forms the floor has a height of: 0.455 m, which gives a volume of: 1.21 m3.

The main tank has a height of: 1.275 m, which is the same as the radius of the floor, i.e. the top is a full hemisphere. The volume is:  4.96 m3. The total volume is: 6.16 m3, which is a bit less than the KVIC plant. However, part of that volume is used for gas storage, when the dome is full of gas.  

The internal volumes of the KVIC dum and GGC 2047 designs can be calculated using other equations.