Optimum insulation thickness of a circular duct subjected to external radiative heat transfer is studied for a given amount of insulation material. A circular duct through which fluid is transported from one end to the other is considered. The thickness of the insulation is assumed to be linearly varying since this is easy to implement in practice. Thus the slope of the insulation thickness is considered as the parameter for optimum insulation. Variations of the bulk temperature of the fluid as well as the temperature of the outer surface of the insulation are evaluated. It is shown that linearly decreasing insulation thickness with minimum slope provides the best insulation under the radiation heat transfer condition. Copyright (C) 2004 John Wiley Sons, Ltd.