# A method based on Delaunay triangulation to determine the amount of liquid inside an inclined tank

### Abstract

There is a problem of determining the amount of liquid inside an inclined tank that appears as an additional measurement error provided by difference in conditions when a level sensor graduation is being obtained and when the level sensor graduation is being applied. The additional error doesn’t exist only in the case if a cross-sectional area doesn’t depend on the cross-section height and if the sensor is located exactly in the geometrical center of cross-section. In the rest cases this additional error could be eliminated if the liquid amount calculation includes the tank inclination data. But until now a method how to do this kind of calculation has not been presented in scientific and technical sources. From math point of view, it is as a calculating volume of 3D figure. That is why, various manner of calculating volume are analyzed to determine which one is more suitable for solving problem of determining the amount of liquid inside an inclined tank. Most of them require participation of specialist in geometry every time when we are receiving measurement data and having need to estimate liquid amount. There is no process automatization, thus such methods are not for practical measurement. The author made a conclusion that the most appropriate method to solve the problem is using the Delaunay 3D triangulation. It is suggested to determinate the tank shape as a polyhedron that could be an exact representation of the tank shape in case if the tank shape is also polyhedron or an approximation if the tank shape is arbitrary. Then Delaunay 3D triangulation for that polyhedron should be applied to split the polyhedron into set of tetrahedrons. The equation that describes liquid surface plane could be obtained from measurement data such as an inclination angle and a level height supported by level sensor coordinates. After that it becomes possible to find points of intersection of liquid surface plane and edges of tetrahedrons. The next step is that vertexes of a new polyhedron that represents shape of figure filled by liquid must be determined. These vertexes are intersection points and all vertexes of original tank polyhedron which are beneath the liquid surface plane. The Delaunay 3D triangulation shall be applied once more time but now it is applied to the new polyhedron. New set of tetrahedrons is formed. Volumes of each tetrahedron from the new set could be calculated by simple formula and then these volumes must be summed up. This sum is the sought amount of liquid inside an inclined tank. All mentioned above action are performed by coordinate geometry methods. A full set of necessary formulas and equations are presented in the article.

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