Calculation of the effective thermal conductivity of fiber composites in non-stationary case
Natalia Rylko Akademia Pedagogiczna im. KEN, Instytut Techniki, ul. Podchorążych 2, 30-084 Kraków
Quarterly No. 4, 2005 pages 96-99
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abstract Unsteady heat conduction of the unidirectional fibers of conductivity λ 1 embedded in a host material of conductivity λ 2 is discussed when the considered composite is represented by a unit-periodicity cell. First, the local temperature field in the unit cell is modeled by the heat equation. The perfect contact between different materials is described by conjugation conditions on the boundary of the fibers. In order to determine the macroscopic conductivity λ e of the composite we perform the spatial average of the Fourier law over the unit cell. Special attention is paid to the case when the given external flux obeys the decreasing exponential law in time. Then the problem is reduced to a boundary value problem for the Helmholtz equation. The latter problem is solved for weakly inhomogeneous composites, i.e. Δ = (λ 1 – λ 2)/(λ 1 + λ 2) is sufficiently small. As a result an analytical formula for λ e is obtained. It corresponds to the classical Clausius-Mossotti approximation for the steady heat conduction. The obtained result implies that the effective conductivity of composites in unsteady case depends on time. This dependence is explicitly written in the considered case. Key words: unsteady heat conduction, fiber material, effective conductivity