Thermal constitutive matrix applied to asynchronous electrical machine using the cell method

Abstract

This work demonstrates the equivalence of two constitutive equations. One is used in Fourier’s law of the heat conduction equation, the other in electric conduction equation; both are based on the numerical Cell Method, using the Finite Formulation (FF-CM). A 3-D pure heat conduction model is proposed. The temperatures are in steady state and there are no internal heat sources. The obtained results are compared with an equivalent model developed using the Finite Elements Method (FEM). The particular case of 2-D was also studied. The errors produced are not significant at less than 0.2%. The number of nodes is the number of the unknowns and equations to resolve. There is no significant gain in precision with increasing density of the mesh.

New Thermal-Conductivity Constitutive Matrix in Fourier’s Law for Heat Transfer Using the Cell Method

“…In the present section, we obtain the analytical formulation of the matrix [M τ ] that has been developed to improve the results obtained with [M λ ]. In a steady state, and without internal heat sources, the equation of energy balance without mass transfer [10][11][12][13] is in the CM,…”

“…Monzón-Verona [12][13] analyzed the temperature distribution in a conductor disk in a transitory regime. The disk is in motion in a stationary magnetic field generated by a permanent magnet, and so the electric currents induced inside it generate heat.…”

“…Three methodologies have been formulated previously in order to obtain the constitutive thermal matrix. These methods are those by Tonti [16], Bullo [21][22], and the method proposed by Specogna [14], for problems of electrical conduction, which we have adapted to thermal conduction [10][11][12][13].…”

“…Dual barycentric cell complexes are used for both space and time domains, the latter inducing a Crank-Nicolson time integration scheme. In the present work, we propose the finite formulation (FF) [10][11][12][13][14][15], as well as the Cell Method (CM), [16,17] as an associated numerical method to analyze the numerical models proposed. In this methodology, we work with the global magnitudes associated with space-oriented elements such as volumes, surfaces, lines, and points of the discretized space, as well as to temporal elements, instead of the field magnitudes associated with independent variables with spatial and temporal coordinates [16][17][18][19][20].…”

“…González [12,13] developed a new constitutive matrix ([M λ ]) for thermal conduction in a transient thermal regime using CM. He demonstrates that this matrix is equivalent to the electrical conduction constitutive matrix in a steady state, and applies this constitutive matrix to the thermal analysis of asynchronous electric machines in a transient regime.…”