A semi-analytical solution to estimate an effective thermal conductivity of the two-phase building materials with spherical inclusions

In the present work, two dimensional (2D) and three dimensional (3D) semi-analytical models are proposed to estimate an effective thermal conductivity of two-phase building materials with spherical inclusions. The proposed semi-analytical approach is based on the formulation and the solution of a boundary value problem. Further, the solution is coupled with Maxwell’s methodology to estimate effective thermal conductivity. The 2D model is simple and limited in scope while the 3D model is widely applicable and is based on a multipole expansion method. The 3D model incorporates secondary parameters like size distribution, variation in thermal conductivity among the inclusions, particle interaction and statistical spatial distribution. For the 3D model, spherical representative unit cells (SRUC) based on the face-centred cubic arrangement and trimmed spheres at the boundary are proposed. Effective thermal conductivity estimated by the proposed models is compared and validated with experimental results from the literature. The 2D model predicts the thermal conductivity of conventional concrete with reasonable accuracy where the relative error is less than ±12%. The 3D model predicts accurate results for foam concrete with a relative error of less than ±15%. Predictions of thermal conductivity of lightweight concrete by the 3D model are also in good agreement with experimental results. Finally, the comparison of proposed models with models from literature has shown that the incorporation of particle interaction has improved the accuracy of the solution. Thus, the flexibility with SRUC and incorporation of particle interactions make this 3D model widely applicable for building materials with spherical inclusions.