Meshless boundary integral quadrature method for calculating the conduction shape factor of exchanger tubes containing slits

Abstract

In this paper, the meshless boundary integral quadrature method (MBIQM) is proposed to determine the conduction shape factor of heat exchanger tubes containing slits. The MBIQM is a meshless method of quadrature form by introducing the adaptive exact solution and Gaussian quadrature. In this way, the singular integral can be technically calculated free of the sense of Cauchy principal value in numerical implementation. When dealing with the boundary value problem containing a slit or so-called degenerate boundary, a rank-deficient influence matrix due to a degenerate boundary may occur. To overcome the rank-deficiency problem, we introduce the dual boundary integral equation (BIE) with the hypersingular BIE to obtain independent equations for collocation points on the slit. A feasible adaptive exact solution is also required for the problem with a degenerate boundary. Since the jump behavior cannot be described by the ordinary form of adaptive exact solution for the corresponding collocation point on the slit, we adopt the harmonic basis function in the elliptical coordinates to construct the new adaptive exact solution. This is the main novelty of this paper. In addition, the numerical instability due to the degenerate scale of an outer boundary is also observed. To avoid the appearance of numerical instability due to a degenerate scale, regularized techniques are employed. Accurate conduction shape factors for any size are obtained by using the proposed approach with regularized techniques.