A semi-analytical solution to estimate an effective thermal conductivity of the two-phase building materials with spheri
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ORIGINAL
A semi-analytical solution to estimate an effective thermal conductivity of the two-phase building materials with spherical inclusions Prajakta Patil 1 & K. S. Reddy 1 Received: 25 October 2019 / Accepted: 6 June 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Abbreviations FCC face centred cubic ND non-dimensional diameter RUC representative unit cell SRUC spherical representative unit cell Nomenclature c Volume concentration k Thermal conductivity, W/mK kaggre Thermal conductivity of aggregate, W/mK ke Effective thermal conductivity, W/mK kest Estimated effective thermal conductivity, W/mK kest_2D Effective thermal conductivity estimated by 2D model, W/mK kest_3D Effective thermal conductivity estimated by 3D model, W/mK * K. S. Reddy [email protected] 1
Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
kexp ki kmor N nc ninc r Reff ri Rq T(r) Tdis Tfar Ti Tm U Ui Um
Experimental thermal conductivity, W/mK Thermal conductivity of inclusion, W/mK Thermal conductivity of mortar, W/mK Number of inclusions Number of complete inclusions or complete spheres Number of incomplete inclusions or spherical caps Position vector, m Radius of spherical matrix, m Radius of inclusions, m Radius
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