Construction of a generalized mathematical model and fast calculations of plane-parallel rotating magnetic fields in process reactors with longitudinal currents of cylindrical inductors on a graphical calculator

The object of research is a quasi-stationary rotating magnetic field (RMF) generated by cylindrical inductors with longitudinal windings in the working space of process reactors, in particular reactors designed to work with magnetic particles (MP). The RMF theory in the working space of reactors has not yet been sufficiently developed, which hinders the widespread introduction of the considered, rather complex technologies into practice. The RMF of a specific reactor can be calculated accurately and completely using modern programs based on the finite element method, but it does not replace the general theory and theoretical analysis. In the literature, special cases of circular and elliptical plane-parallel RMF in reactors of the type under consideration have been studied, however, analytical formulas for a plane-parallel RMF for the general case of m-phase cylindrical inductors of external and internal design with symmetrical longitudinal windings are not presented.In this paper, a mathematical model is constructed and generalized analytical formulas for magnetic induction are obtained, linking the characteristics of a plane-parallel RMF in the working space of reactors at idle speed with the main parameters of external and internal cylindrical inductors with an m-phase symmetric longitudinal winding. A physical analysis is carried out and the adequacy of the model is confirmed. Using the proposed formulas and a free, easy-to-use Desmos graphical calculator, quick trial calculations and analysis of RMF in several reactors with two-pole external inductors and various windings for three phases (for 6 and 42 slots) and for six phases (12 slots) are carried out. The calculation results are consistent with experimental and literary data.New analytical formulas, as well as the demonstrated methods of quick evaluation calculations, analysis and experimental studies are recommended for practical implementation in the research, development and operation of reactors of this type. To carry out the calculations, it is enough to have a laptop or smartphone connected to the Internet, the time costs are insignificant. The results of the work will be useful to technologists, engineers and developers of both the reactors of the type under consideration and other devices with a similar purpose with an RMF.