Comprehensive Theory of Heat Transfer in Heterogeneous Materials

For over forty years, researchers have attempted to refine the Fourier heat equation to model heat transfer in engineering materials. The equation cannot accurately predict temperatures in some applications, such as during transients in microscale (< 10^-12 s) situations. However, even in situations where the time duration is relatively large, the Fourier heat equation might fail to predict observed non-Fourier behavior. Therefore, non-Fourier models must be created for certain engineering applications, in which accurate temperature modeling is necessary for design purposes.

In this thesis, we use the Fourier heat equation to create a general non-Fourier, but diffusive, equation that governs the matrix temperature in a composite material. The composite is composed of a matrix with embedded particles. We let the composite materials be governed by Fourier's law and let the heat transfer between the matrix and particles be governed by contact conductance. After we make certain assumptions, we derive a general integro-differential equation governing the matrix temperature. We then non-dimensionalize the general equation and show that our model reduces to that used by other researchers under a special limit of a non-dimensional parameter.

We formulate an initial-boundary-value problem in order to study the behavior of the general matrix temperature equation. We show that the thermalization time governs the transition of the general equation from its small-time limit to its large-time limit, which are both Fourier heat equations. We also conclude that our general model cannot accurately describe temperature changes in an experimental sand composite.