Assessing Effective Medium Theories for Conduction through Lamellar Grains

A numerical finite-difference model was developed for 2D transient diffusion in lamellar structures. The focus was heat transfer, but it could be applied to mass, momentum, and electron/ion transfer as well. The control volume contains two phases. The phases have different transport properties and are arranged in grains that constitute units of coherent orientation. The effect of grain size, grain boundaries, and phase contrast on apparent transport properties was evaluated by examining a progression of increasingly complex structures. The numerical model had good agreement with analytical expressions for homogeneous, parallel, and perpendicular structures. Effective medium theories (EMT) predict apparent transport properties in the limit of many, small, randomly oriented grains. The impact of grain size and contrast on EMT prediction accuracy was investigated. The numerical model quantitatively shows EMT predictions grow poorer with increasing phase contrast. Moreover, structural specifics such as grain boundary connectivity have more significant impact in large grains. Thus, larger variability from one structure to another is observed in large grains than in small grains. Specifically, standard deviation decreased by an order of magnitude on going from 2 grains to 15 grains. This numerical approach provides some insight into regimes in which EMT approximations are appropriate and can be extended to 3D (which increases connectivity of transport pathways) and other types of structures.