Optimum shapes of reservoirs

The aim of this paper is to find the optimal shapes of reservoirs and to minimize the whole lateral surface area for a known volume V. The minimization is studied for different geometrical shapes and it is compared to the cube case. It is proved that the lateral surface area of a hollow ellipsoid with negligible or non-negligible thickness is minimized when it is transformed to a sphere. It is then proved that the minimization of the whole lateral surface area with a constant volume results in a differential equation which cannot be resolved in the general case; but only for some particular boundary values. Later, it is proved that it is the sphere for which the greatest ratio V/Sm where Sm is the minimum lateral surface area is obtained, and finally two formulas are proved for the calculation of lateral surface area of ellipsoid and the perimeter of ellipse.