A circular metal pipe has a wall thickness of 10 mm and an inner diameter of 10 cm. The pipe’s outer surface is subjected to a uniform heat flux of 5 kW/m² and has a temperature of 500°C. The metal pipe has a variable thermal conductivity given as k(T) = k₀ (1 + βT), where k₀ = 7.5 W/m ∙ K, β = 0.0012 K⁻¹, and T is in K. Determine the inner surface temperature of the pipe.

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T₂ T₁ Pipe wall K(T) = kŋ(1 + BT) à