The problem you present is a transient one and will be dominated by conduction until the outer layer of powder begins to melt. In order to bound the problem one may make a conservative estimate of the time for the center of the pipe to reach the melt point by using the Carlsaw & Jaeger plots found in “Conduction of Heat in Solids”, Oxford University press (1959). This approach is covered in “Transport Phenomena” by Bird, Stewart & Lightfoot. Once melting has begun, the heat transfer rate to the melt will increase due to natural convection effects and the power will increase as heat of fusion is supplied.
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