Transport involving conducting fibers in a non-conducting matrix

Several transport models are investigated in the context of two-dimensional fiber composites. We compare the thermal and electrical transport through the fiber composites as a function of the conductivity ratio and the fiber density. Three models will be considered and compared: 1) discretized solutions, 2) equivalent resistance, and 3) effective medium approximation. In the case of electrical transport, where the conductivity of the fiber is presumably many orders of magnitude larger than the matrix, the second model provides a fast and reliable way to predict conductance of the combined system. However, if the two materials are similar in conductivity, the second model fails to accurately capture the conductivity. Thermal transport is predicted using the discretized model because the conductivity ratio is non-negligible. The third model is an analytic approximation based on Maxwell's equation and is used to predict both types of transport through a compound with inclusions of ellipsoidal geometry. The analytic model works well for lower conductivity ratios and all area densities but under-predicts conductivity for high-conductivity ratios.