A heater for boiling a saturated liquid consists of two concentric stainless steel tubes packed with dense boron nitride powder. Electrical current is passed through the inner tube, creating uniform volumetric heating q (W/m³). The exposed surface of the outer tube is in contact with the liquid and the boiling heat flux is given as

It is feared that under high-power operation the stainless steel tubes would severely oxidize if temperatures exceed T_ss,x or that the boron nitride would deteriorate if its temperature exceeds T_bn,x' Presuming that the saturation temperature of the liquid (T_sat) and the boiling surface temperature (T.) are prescribed, derive expressions for the maximum temperatures in the stainless steel (ss) tubes and in the boron nitride (bn). Express your results in terms of geometric parameters (r₁, r₂, r₃, r₄), thermal conductivities (k_ss, k_bn), and the boiling parameters (C, T_sat, T_s).

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Boron nitride powder Stainless steel tube, à Stainless steel boiling surface