The nominally one dimensional conduction of heat through a slab becomes two dimensional when one of the surfaces is rough or when the boundary conditions are spatially non-uniform. This paper develops the stochastic equations for a slab whose surface roughness or convective boundary condition is spatially correlated with correlation lengths ranging from 0 (white noise) to a length long in comparison to the slab thickness. The effect is described in terms of the standard deviation and the resulting spatial correlation of the heat flux as a function of depth into the slab. In contrast to the expectation that the effect is monotonic with respect to the correlation length, it is shown that the effect is maximized at an intermediate correlation length. It is also shown that roughness or a random convective heat transfer coefficient have essentially the same effects on the conducted heat, but that the combination results in a much deeper penetration than does each effect individually. In contrast to the usual methods of solving stochastic problems, both the case of a rough edge and a smooth edge with stochastic convective heat transfer coefficients can only be treated using direct Monte Carlo simulations.

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