Conduction shape factors for thermal analysis of energy walls under varying boundary conditions

Embedded retaining walls in the buildings can be converted to energy walls by incorporating embedded heat exchanger pipes connected to a heat pump system, which can assist in decarbonising heating. To support the effective design of such systems, fast and reliable models are required to predict the thermal performance of energy walls. Analytical shape factors provide a convenient and computationally efficient method for estimating steady-state heat transfer rates within the wall. These shape factors are mathematical expressions that relate the temperature difference between surfaces to the resulting heat flux under steady conditions. While analytical shape factor equations have been successfully applied to other types of ground heat exchangers, their application and validations for energy walls, which have more complex thermal boundary conditions, remain unexplored. This study investigates the suitability of shape factor equations, originally developed for fuel transportation pipelines and other applications which share similar geometries, to the thermal analysis of energy walls. Given that energy walls may encounter varying thermal boundary conditions, e.g. air-exposed vs. fully embedded, both scenarios are analysed. The results present the first systematic parametric validation across realistic energy wall geometric variations (pipe spacing, diameter, wall thickness and cover depth) and thermal conductivity ratios to evaluate shape factors for practical design applications. The performance of analytical shape factors is benchmarked against results from steady-state and transient numerical models. The findings demonstrate that, with the developed methodology, existing shape factor equations can be successfully extended to energy walls and other energy geostructures exhibiting planar thermal behaviour.