19.4 Analytical Method Different analytical and semi empirical schemes have been developed to estimate the thermal conductivity of composite materials [4, 7, 8]. Among those, the Lewis-Nielson (L-N) [4], Maxwell Eucken-EMA (ME-EMA) [8], and Hamilton and Cheng-Vachon model [5] are widely used. Most of the models consider the geometry of the filler, the volume fraction of the filler, the thermal conductivity of both the matrix and the filler, and thermal resistance at the junction between the filler and the matrix (Kapitaz Resistance). In the current study the Lewis-Nielson model is used. The detail analysis of the model can be found elsewhere [4], and here it is briefly described below. 19.4.1 Lewis-Nielson (LN) Model The Lewis-Nielson (LN) model is adopted from the Halpin-Tsai (HT) mechanical model. It has a semi-empirical equation, which considers the effect of particle size, particle shape (aspect ratio), the volume fraction of the particle and the critical or maximum packing of particles. The effective thermal conductive of the composite can be shown as Eq. 19.2: Ke ¼Km 1þε η φcp 1 ϕ η φcp ! ð19:2Þ Where, ϕ = 1þ 1 φm ð Þ φm 2 ð Þ φcp and η = Kf Km Kf þKm∗ε " # and Km ¼thermal conductivity of the matrix, Kf ¼thermal conductivity of the particle and the interface layer, and is given as Kf ¼1 φp Kf φp þVl " #þ φl Km φp þφl " # " # ð19:3Þ and φp is the volume fraction of the particle and φl is the volume fraction of the interface layer, and they are related as φl ¼φp rp þt 3 rp 3 1 " #and φcp ¼φp∗δ ð19:4Þ where δ = 1þt rp = Þ 3 , rp is the radius of the particle, φcp is the combined volume fraction of the particle and the interface layer. And ɛ ¼1.5 (aspect ratio factor) and φm ¼0.637 (maximum packing) for the spherical particle. Fig. 19.2 Schematic diagram of unidirectional/linear heat transfer apparatus 19 The Effect of Particles Size on the Thermal Conductivity of Polymer Nanocomposite 153
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