A Multi-Scale, Semi-Analytical Model for Transient Heat Transfer in a Nano-Composite Containing Spherical Inclusions

  • A. Dobri Nazarbayev University, 53 Kabanbay Batyr Ave, Nursultan, Kazakhstan
  • T. D. Papathanasiou Nazarbayev University, 53 Kabanbay Batyr Ave, Nursultan, Kazakhstan
Keywords: composite, heat transfer, modeling, Duhamel’s Theorem


This paper presents a semi-analytical model for transient heat conduction in a composite material reinforced with small spherical inclusions. Essential to the derivation of the model is the assumption that the size of the inclusions is much smaller than the length scale characterizing the macroscopic problem. An interfacial thermal resistance is also present between the two phases. During heating, the inclusions are treated as heat sinks within the matrix, with the coupling provided by the boundary conditions at the surface of the embedded particles. Application of Duhamel’s Theorem at the particle scale provides the local relationship between the temperature profile in a particle and the matrix that surrounds it. A simple spatial discretization at the macro-scale leads to an easily solvable system of coupled Ordinary Differential Equations for the matrix temperature, particle surface temperature and a series of ψ-terms related to the heat exchange between phases. The interfacial thermal resistance between the two phases can lead to the particle temperature lagging behind that of the surrounding matrix. The resulting transient response of the matrix temperature cannot be reproduced by a material with a single effective thermal conductivity. In the case where transient methods are used to determine effective thermal conductivity, this transient response may introduce errors into the measurement.


(1). J.C. Maxwell, Conduction through heterogeneous media, Chapter IX – A Treatise on Electricity and Magnetisma, Cambridge University Press, First published in: 1873, Print publication year: 2010, 360–373. Crossref

(2). R.H. Davis, Int. J. Thermophys. 7 (1986) 609– 620. Crossref

(3). D.P.H. Hasselman, L.F. Johnson, J. Compos. Mater. 21 (1987) 508–515. Crossref

(4). C.W. Nan, R. Birringer, D.R. Clarke, H. Gleiter, J. Appl. Phys. 81 (1997) 6692–6699. Crossref

(5). S. Lee, J. Lee, B. Ryu, S. Ryu, Sci. Rep. 8 (2018) 1–11. Crossref

(6). S. Zhai, P. Zhang, Y. Xian, J. Zeng, B. Shi, Int. J. Heat Mass Transf. 117 (2018) 358–374. Crossref

(7). M. Quintard, Heat Transf. Polym. Compos. Mater. Form. Process, Chapter 6 (2016) 175–201. Crossref

(8). M.J. Assael, K.D. Antoniadis, I. Metaxa, S.K. Mylona, Int. J. Thermophys. 36 (2015) 3083– 3105. Crossref

(9). C.P. Wong, R.S. Bollampally, J. Appl. Polym. Sci. 74 (1999) 3396–3403. Crossref

(10). C. Bin Kim, N.H. You, M. Goh, RSC Adv. 8 (2018) 9480–9486. Crossref

(11). C. Liu, J.S. Kim, Y. Kwon, J. Nanosci. Nanotechnol. 16 (2016) 1703–1707. Crossref

(12). G.C. Glatzmaier, W. Fred Ramirez, Chem. Eng. Sci. 43 (1988) 3157–3169. Crossref

(13). H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, 1959.

How to Cite
A. Dobri and T. Papathanasiou, “A Multi-Scale, Semi-Analytical Model for Transient Heat Transfer in a Nano-Composite Containing Spherical Inclusions”, Eurasian Chem. Tech. J., vol. 21, no. 2, pp. 101-105, Jun. 2019.