Purpose -The purpose of this paper is to present a method of solving a thermal conduction equation in three-zone axially-symmetrical systems. Design/methodology/approach -In the method developed, the field functions are determined in the analytical way by the superposition of states and separation of variables method. The coefficients of the field functions and eigenvalues of the boundary-initial problem are computed by the numerical method. The coefficients are the solution to the corresponding sets of equations. These sets are the result of scalar products of non-orthogonal functions at the respective zones of the cable. The eigenvalues are determined by an algorithm, which uses the field properties and elements of the golden cut method. Findings -The method made it possible to develop a mathematical model of the dynamics of the thermal field in a polymer DC cable. This model has good physical interpretation. The paper also presents the field distributions determined in an analytical form. Some arguments of the expressions derived are however computed numerically. The results obtained by the paper's method and by the finite elements methods were compared. The relative differences are less than 6 per cent.Research limitations/implications -The method concerns axially-symmetrical three-zone systems under nominal conditions. Practical implications -By means of the method important parameters of DC lines can be determined (e.g. spatial-temporal heat-up curves, admissible sustained currents, time constants). Originality/value -An analytical-numerical method of analysis of the thermal field in a three-zone axially-symmetrical system was developed. Its original element is the algorithm of determination of eigenvalues of the boundary-initial problem and coefficients of non-orthogonal field functions.
The simplified method of analysis of the thermal field in the futuristic model of a DC cable is presented. The thermal conductivity of a non-conducting layer is very small comparing with the same parameter of conducting regions. The above causes very fast heat propagation in the core and coating, which in the consequences are approximated by inert elements of the first order. Spatial changes in the field in insulation cannot be neglected, because of the significantly slow heat transfer. For that reason, insulation is treated as an element of distributed parameters. The boundary-initial parabolic problem of a non-conducting region is solved by means of Duhamel's theorem. The fundamental solutions of superposition integrals are determined by the separation of variables method. The maximal deviation of results obtained by the finite element method does not exceed 7.2%. The heating curves and results of verification are presented in a graphic form.
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