Total heat flux convergence in the calculation of 2d and 3d heat losses through building elements

TL;DR

This paper investigates the convergence issues of total heat flux calculations in 2D and 3D heat loss assessments through building elements, highlighting problems with benchmark standards and proposing a solution to improve reliability.

Contribution

It identifies the inadequacy of current benchmark cases for total heat flux convergence and suggests neglecting certain singular points to enhance calculation accuracy.

Findings

Total heat flux convergence is affected by singular points near boundary conditions.

Refinement around singular points does not guarantee convergence of total heat flux.

Neglecting some singular points improves compliance with standard tolerances.

Abstract

Heat losses through the building envelope is one of the key factors in the calculation of the building energy balance. If steady-state heat conduction is observed, which is commonly used to assess the heat losses in building, there is an analytical solution for one-dimensional problem. For two and three-dimensional problems, especially for the complex geometry cases, one must use numerical methods to solve the heat conduction equation. To standardise two and three-dimensional calculation of heat losses through building elements, ISO 10211 standard can be used. The standard has four benchmark examples with criteria that must be satisfied to declare a method as a high-precision calculation method. A problem occurs for Case 1 of benchmark test because the analysed problem has a singular point due to discretely assigned Dirichlet boundary conditions. The reliability of the results around the singular point could be improved by the refinement of the mesh in the area around the singular point, but as a point of interest is the total heat flux that is entering the building element, and it must converge between subdivisions, this method is not good since the reliable result cannot be reached. The problem for the convergence is in the marginal node because the temperature gradient in it increases as the temperature difference remains constant and the distance between the corresponding nodes decreases. For that reason, Case 1 from the benchmark is inadequate because even if there is a discontinuity in temperature field on the boundary, there is an interval in which this change is to happen, and the heat flux has a theoretical limit which is not infinity. From the results of this research, it is shown that one should neglect a certain number of singular points in order to achieve the tolerance given by the standard since the temperature further from the marginal node is stable for any subdivision.