Consider the conditions of Problem 7.36, but now allow for radiation exchange between the surface of the heating element \((\varepsilon=0.8)\) and the walls of the duct, which form a large enclosure at \(30^{\circ} \mathrm{C}\).

(a) Evaluate the steady-state surface temperature.

(b) If the heater is activated from an initial temperature of \(30^{\circ} \mathrm{C}\), estimate the time required for the surface temperature to come within \(10^{\circ} \mathrm{C}\) of the steadystate value.

(c) To guard against overheating due to unanticipated excursions in the blower output, the heater controller is designed to maintain a fixed surface temperature of \(275^{\circ} \mathrm{C}\). Determine the power dissipation required to maintain this temperature for air velocities in the range \(5 \leq V \leq 10 \mathrm{~m} / \mathrm{s}\).

Data From Problem 7.36:-

A long, cylindrical, electrical heating element of diameter \(D=12 \mathrm{~mm}\), thermal conductivity \(k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), density \(ho=2700 \mathrm{~kg} / \mathrm{m}^{3}\), and specific heat \(c_{p}=900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of \(30^{\circ} \mathrm{C}\) and \(8 \mathrm{~m} / \mathrm{s}\), 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 \(1000 \mathrm{~W} / \mathrm{m}\).

(b) If the heater is activated from an initial temperature of \(30^{\circ} \mathrm{C}\), estimate the time required for the surface temperature to come within \(10^{\circ} \mathrm{C}\) of its steady-state value.