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Inverted Pyramid Storage Tables

Known volume and surface area values (or alternatively the volume and depth) are enough to fully-specify the dimensions of an inverted pyramid. From these it is possible to derive a piecewise-linear storage table that approximates such a shape. This is available in the editor’s right-click menu in KalixGUI.

If hmax and Vmax are known:

Amax=hmax3Vmax

If Amax and Vmax are known:

hmax=Amax3Vmax

If Amax and hmax are known:

Vmax=3Amaxhmax

When using a table to represent an inverted-pyramid storage:

  • Linear interpolation between levels will result in small errors compared to an exact pyramid. This is because height, area, and volume are not proportional to each other for such a shape (i.e. width is proportional to height, and area goes as height^2, and volume goes as height^3).

  • Therefore we need to define a number of table rows, n, that has enough resolution to keep the interpolation error acceptably low. It turns out that if we build a table with n=5 heights h evenly spaced between min and max height, our relative error in area is not more than ~10 (except at the very bottom of the table).

  • Across-sectional=(hmaxh)2Amax

  • *Height refers to height above the min height (if it is not already = 0).