Abstract This study presents detailed numerical simulations predicting the effective thermal conductivity of spherical monodisperse and polydisperse coreshell particles ordered or randomly distributed in a continuous matrix. First, the effective thermal conductivity of this threecomponent composite material was found to be independent of the capsule spatial distribution and size distribution. In fact, the study established that the effective thermal conductivity depended only on the core and shell volume fractions and on the core, shell, and matrix thermal conductivities. Second, the effective medium approximation reported by Felske [1] was in very good agreement with numerical predictions for any arbitrary combination of the abovementioned parameters. These results can be used to design energy efficient composites, such as microencapsulated phase change materials in concrete and/or insulation materials for energy efficient buildings. Felske Model Felske [1] derived a model, using the selfconsistent field approximation, to predict the effective thermal conductivity of monodisperse spherical capsules randomly distributed in a continuous matrix. In the absence of contact resistance at the shellmatrix interfaces, the model can be expressed as, where k_{c}, k_{s}, and k_{m} are the thermal conductivities of the core, shell, and matrix materials, respectively. Similarly, ϕ_{c}, ϕ_{s}, and ϕ_{m} are the volume fractions of the core, shell, and matrix materials, respectively. Schematics and Assumptions The present study examined various composite representative volumes consisting of different packing arrangements of monodisperse and polydisperse spherical capsules distributed in a continuous matrix. Figure 1. Schematic and computational domain of a single unit cell consisting of capsules distributed in a continuous matrix with (a) simple, (b) bodycentered, and (c) facecentered cubic packing arrangement. Core and shell diameters and unit cell length corresponding to core and shell volume fractions ϕ_{c} and ϕ_{s} were denoted by D_{c}, D_{s}, and L, respectively.
Governing equations and boundary conditions Under the above assumptions, the local temperatures in the core, shell, and matrix denoted by T_{c}, T_{s}, and T_{m} were governed by the steadystate heat diffusion equation in each domain, given by, .
where n is the unit normal vector at any given point on the matrix/shell and shell/core interfaces, designated by subscript m/s and s/c, respectively. Results and Discussion Table 1 compares the numerically predicted effective thermal conductivity k_{eff} of ten composite microstructures to that predicted by the Felske model. Cases 1 and 2 indicate that the numerically predicted k_{eff} was the same for composites with monodisperse or polydisperse capsules for the same values of ϕ_{c}, &ϕ_{s}, k_{c}, k_{s}, and k_{m}. Table 1 also shows that k_{eff} predicted by the Felske model fell within 0.25% of numerical predictions for a wide range of constituent thermal conductivities k_{c}, k_{s}, and k_{m} and volume fractions ϕ_{c} and ϕ_{s}. Table 1. Numerical and analytical predictions of the effective thermal conductivity of composites consisting of monodisperse or polydisperse capsules randomly distributed in a continuous matrix. The average outer diameter and thickness of the shell are D_{s,avg} = 18 μm and t_{s} = 1 μm. Figure 3 shows the effective thermal conductivity k_{eff} of a composite containing monodisperse capsules as a function of (a) the core volume fraction ϕ_{c} and (b) the shell volume fraction ϕ_{s}. The desired volume fractions were imposed by either adjusting the relevant diameter (D_{c} or D_{s}) while holding the unit cell length L constant or by adjusting the unit cell length L and holding the relevant diameter constant. Figures 3a and 3b establish that k_{eff} depended only on ϕ_{c} and ϕ_{s} and not on the individual geometric parameters D_{c}, D_{s}, and L. Figure 3. Effective thermal conductivity for (a) different values of ϕ_{c} with ϕ_{s} = 0.025 and (b) different values of ϕ_{s} with Φ_{c} = 0.05. The volume fractions were varied by adjusting either the diameter or unit cell length. Here, k_{c} = 0.21 W/mK, k_{s} = 1.3 W/mK, and k_{m} = 0.4 W/mK. Predictions by the Lichtenecker, Brailsford, and Felske models are also shown. Figure 4 plots the effective thermal conductivity k_{eff} of a composite material containing monodisperse capsules as a function of the thermal conductivity of the matrix k_{m}, core k_{c}, and shell k_{s}, respectively. Figure 4 shows that the Felske model predicted the effective thermal conductivity k_{eff} of composites containing monodisperse capsules within the estimated numerical uncertainty for all volume fractions considered and for a wide range of constituent material thermal conductivities. Figure 4. Effective thermal conductivity k_{eff} of coreshell composite as a function of the thermal conductivity of the (a) continuous k_{m}, (b) core k_{c}, and (c) shell k_{s} phase obtained numerically and predicted by the Lichtenecker, Brailsford, and Felske models. The volume fractions of core and shell were ϕ_{c} = 0.2 and Φ_{s} = 0.145. Conclusion This study established that the effective thermal conductivity was independent of the capsules' spatial distribution and size distribution. The effective thermal conductivity was found to depend solely on the core and shell volume fractions and on the core, shell, and matrix thermal conductivities. The Felske model predicted the effective thermal conductivity of the composite material within numerical uncertainty for the wide range of parameters considered. This model was used to identify conditions under which the effective thermal conductivity k_{eff} of the composite materials remained smaller than that of the matrix material. This thermal design rule will be useful in developing PCMcomposite materials for energy efficient buildings. References
