A long, cylindrical, electrical heating element of diameter D = 10 mm, thermal conductivity k = 240 W/m· K, density ρ = 2700 kg/m3, and specific heat cp = 900 J/kg · K is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of 27°C and 10 m/s, respectively.
(a) Neglecting radiation, estimate the steady-state surface temperature when, per unit length of the heater, electrical energy is being dissipated at a rate of 1000 W/m.
(b) If the heater is activated from an initial temperature of 27°C, estimate the time required for the surface temperature to come within 10°C of its steady-state value.