Chapter 13, Problem 13.125P

A radiant heater, which is used for surface treatment processes, consists of a long cylindrical heating element of diameter $D_1=0.005\text{m}$ and emissivity ${\epsilon }_1=0.80$ . The heater is partially enveloped by a long, thin parabolic reflector whose inner and outer surface emissivities are ${\epsilon }_{2i}=0.10$ and ${\epsilon }_{2o}=0.80$ , respectively. Inner and oilier surface areas per unit length of the reflector are each ${A^{\prime }}_{2i}={A^{\prime }}_{2o}=0.20\text{m}$ , and the average convection coefficient for the combined inner and outer surfaces is ${\bar{h}}_{2\left(i,o\right)}=2{\text{W/m}}^2\cdot \text{K}$ . The system may be assumed to be in an infinite, quiescent medium of atmospheric air at $T_{\infty }=300\text{K}$ and to be exposed to large surroundings at $T_{\text{sur}}=300\text{K}$ .

(a) Sketch the appropriate radiation circuit, and write expressions for each of the network resistances.
(b) If, under steady-state conditions, electrical power is dissipated in the heater at ${P^{\prime }}_1=1500\text{W/m}$ and the heater surface temperature is $T_1=1200\text{K}$ , what is the net rate at which radiant energy is transferred from the heater?
(c) What is the net rate at which radiant energy is transferred from the heater to the surroundings?
(d) What is the temperature, T₂, of the reflector?