Analytical study for numerical instability of steady-state heat conduction problems in exchanger tubes using degenerate kernels in the null-field boundary integral equation method

Abstract

This paper not only derives an analytical solution for steady -state heat conduction problems in exchanger tubes but also can predict the location of numerical instability due to a degenerate scale in the boundary element method or the boundary integral equation method. Four shapes of the exchanger tubes including concentric annulus, eccentric annulus, confocal ellipses, and elliptical tube with a confocal crack are analytically studied by using degenerate kernels of polar, bipolar and elliptical coordinates, respectively. This work extends our prior research on numerical instability and its treatment for steady -state heat conduction problems in exchanger tubes using the dual boundary element method. Analytical solutions of the temperature field and conduction shape factor can be derived. Two main analytical tools, the degenerate kernel for the closed-form fundamental solution and the generalized Fourier expansion for the boundary densities in the null-field boundary integral equation are required. The analytical derivation process can clearly examine the occurring mechanism of numerical instability due to a zero denominator. The effectiveness of regularization techniques to promote the rank-deficiency by one to a full -rank system can be analytically examined in this paper.