New analytical solution for heat transfer in insulated wires

In this paper we investigate the heat transfer from pin-fins of infinite length made of a high thermal conductivity core with a low thermal conductivity coating. Considering a two-dimensional thermal field in the coating and a one-dimensional field in the core, an exact, analytical solution is presented for this geometry, previously not available in the literature. This solution is obtained in the form of an infinite series, but its first term can serve as an excellent approximation of practical problems under a wide range of conditions. In particular, the new model is appropriate to assist design engineers in calculating heat transfer losses from insulated wires, which can be conceived as infinitely long composite fins. The first term approximation is at once different and improves upon the traditional one-dimensional formula. Both approximations are compared vs. the exact series solution in order to establish their limits of validity. We have computed heat transfer losses for different combinations of core/coating pairs and typical geometric ratios pertaining to commercial electric wires. The figures show that the new solution produces better results than the traditional approach, especially for Bi < 1 × 10^?2 with relative errors below 1.35%. Thus, the new expedient proposed in this paper brings along a remarkable improvement, gaining accuracy and at the same time retaining a suitable simplicity.