The problem of conduction heat transfer through an eccentric annulus with the inner surface kept at a constant temperature and the outer surface subjected to convective condition is solved by three different techniques. A perturbation analysis yields an analytical expression for temperature profile for small values of eccentricity. The two dimensional conduction problem has also been solved by a two dimensional semi-analytical technique in which the condition at the outer periphery is matched by a collocation technique. Finally, a one dimensional approximate technique namely sector method has been used to solve the same problem. The sector method does not require any numerical technique yet yields remarkable accuracy. Next, the heat flow through an eccentric insulation surrounding a circular cylinder was considered. It has been demonstrated that the sector method is effective also in determining the geometry of the critical insulation in this case over a wide range of radius ratio and Biot number. Finally, both all the three methods, i.e., the perturbation technique, the boundary collocation method and the sector method have been applied to determine the geometry of the critical as well as crossover perimeter of insulation around a circular cylinder when the insulation is provided eccentrically.

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