Modified divergence theorem for analysis and optimization of wall reflecting cylindrical UV reactor

In this paper, the modified divergence theorem (MDT), known in earlier literature as the Gauss-Ostrogradsky theorem, was formulated and proposed as a general approach to electromagnetic (EM) radiation, especially ultraviolet (UV) radiation reactor modeling. A formulated mathematical model, based on MDT, for a multilamp UV reactor was applied to all sources in a reactor in order to obtain intensity profiles at chosen surfaces inside the reactor. Applied modification of MDT means that intensity at a real opaque or transparent surface or through a virtual surface, opened or closed, from different sides of the surface are added and not subtracted as in some other areas of physics. The derived model is applied to an example of the multiple UV sources reactor, where sources are arranged inside a cylindrical reactor at the coaxial virtual cylinder, having the radius smaller than the radius of the reactor. In this work, optimization of a reactor means maximum transfer of EM energy sources into the fluid for given fluid absorbance and fluid flow-dose product. The obtained results, for water quality known in advance, give a unique solution for an optimized model of a multilamp reactor geometry. As everyone can easily verify, MDT is a very good starting point for every reactor modeling and analysis