Keywords: mаthematical mоdel

Mathematical modeling of heat transfer through building envelopes with cracks

https://doi.org/10.58224/2618-7183-2026-9-4-4
Abstract
Cracks formed in building walls are frequently restored by injecting specially formulated repair grouts into the damaged areas. This study examines the effect of such repaired cracks on the thermal performance of building envelopes. Heat transfer within cracked wall sections was modeled by solving the two-dimensional steady-state heat conduction equation. Convective boundary conditions of the third kind were specified on the exterior surfaces to account for heat exchange between the wall and the surrounding environment. The truncated edges of the computational domain were assigned insulated (second-kind) boundary conditions, corresponding to zero normal heat flux. At the interfaces between adjacent materials, continuity of both temperature and heat flux was enforced through fourth-kind boundary conditions. The numerical analysis was carried out using the finite element method implemented in ANSYS Workbench 2024. Based on the obtained numerical results, a procedure was developed for constructing nomograms that characterize the specific heat losses associated with wall cracks repaired by injection grouting. The temperature fields of aerated concrete, clay brick, and reinforced concrete wall assemblies were investigated. The results indicate that injecting a repair grout into a crack increases the heat flux through the examined section in brick and aerated concrete walls, whereas the opposite effect is observed for reinforced concrete walls, where the heat flux decreases after crack injection. The simulations demonstrated that replacing the air-filled crack with injection grout increases the heat flux across the repaired region of the wall. Graphical nomograms were therefore established to estimate the heat flow through repaired cracks directly from two governing parameters – the crack depth and the crack opening width – without the need for a complete numerical solution of the temperature field. In addition, the paper presents a practical algorithm describing the application of these nomograms in engineering calculations. The developed graphical tools can be employed to evaluate the overall thermal resistance of building envelopes during reconstruction, rehabilitation, and major renovation of existing buildings.
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