Stationary temperature field in multi-layered rods with discontinuous of cross section width

Abstract

Introduction. Presents a method for modeling a two-dimensional stationary temperature field in a layered rod. The peculiarity of the structure of the rod is the presence of discontinuity of the width of the cross section in the direction of heat flow and multilayer. Identification of the temperature field in such rods is a necessary step in solving the problem of thermoelasticity. The relevance of the problem lies in the development of analytical methods for analysis layered rods of complex geometric shape with thermal effects, with acceptable computational complexity and necessary accuracy. Materials and methods. For a multilayer rod, a method for constructing an approximate solution of the Dirichlet stationary heat conduction problem with a transverse heat flow direction is considered. Within each layer, the temperature distribution function is represented as a sum of two functions. The first function, linear in the direction of the heat flow, reflects the exact solution of the problem for a rectangular layered section. The second function is the correction nonlinear function of two variables. It describes the nonlinear distortions of the temperature field due to the presence of discontinuities in the width of the cross section. The correction function, according to the Fourier method, is represented as a product of a given coordinate function and the sum of the sought amplitudes caused by the width breaks. The functions of the effect of breaking the width on temperature fields in adjacent layers are introduced. An approximate formulation of the Dirichlet problem with integral conjugation conditions on interlayer boundaries is formulated. Results. The parameters of the stationary temperature field were calculated for a seven-layer section of a T-shaped form with alternating layers of carbon and steel. Testing the results of the Ansys program showed good qualitative and quantitative correspondence of two-dimensional temperature fields. Conclusions. The obtained solution satisfactorily describes the temperature field in the cross section of a layered rod in the vicinity of its geometric features. The method is characterized by acceptable laboriousness and accuracy suitable for solving the problem of thermoelasticity of a layered rod.