A rod of diameter D = 25 mm and thermal conductivity k = 60 W/m. K protrudes normally from a furnace wall that is at Tw = 200°C and is covered by insulation of thickness L ins = 200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To must be maintained below a specified operating limit of T max = 100°C. The ambient air temperature is T∞ = 25°C, and the convection coefficient is h = 15 W/m2 ∙ K.

(a) Derive an expression for the exposed surface temperature T" as a function of the prescribed thermal and geometrical parameters. The rod has an exposed length Lo, and its tip is well insulated.

(b) Will a rod with Lv = 200 mm meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce To.

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Air T. Hot furnace wall Insulation Lo