1-8. Thermal Conductivity of a Layered Composite As shown in Figure P1-8, suppose that a composite solid consists of alternating layers of materials A and B, with layer thicknesses LA and LB, respectively. Both materials are isotropic, with thermal conductivities kA and kB. It is desired to predict the average, steady-state heat flux in a sample that contains many layers. The flux in the composite is written as Q=KG where Q and G are the average heat flux and temperature gradient, respectively, and is the effective thermal conductivity. With the coordinate axes aligned as shown, the effective conductivity will be of the form =xx000yy000zz. (a) Evaluate xx,yy, and zz in terms of kA,kB, and , where =LA/(LA+LB) is the volume fraction of material A. (Hint: The steady heat flux q through a slab of thickness L and thermal conductivity k is q=kT/L, where T is the temperature difference between the surfaces.) (b) Show that, in general, the vectors Q and G will not be parallel to one another. That is, the directions of the heat flux and temperature gradient will tend to differ. Figure P1-8. Composite solid consisting of alternating layers of materials A and B