Chapter 12, Problem 12.17P

Consider a 5-mm-square, diffuse surface $\Delta A_0$ having a total emissive power of $E_0=4000W/m^2$ . The radiation field due to emission into the hemispherical space above the surface is diffuse, thereby providing a uniformintensity $I(\theta ,\varphi )$ .Moreover, if the space is a nonparticipating medium (nonabsorbing, nonscattering, and nonemitting), the intensity is independent of radius for any $(\theta ,\varphi )$ direction. Hence intensities at any points $P_1$ and $P_2$ would be equal.

(a) What is the rate at which radiant energy is emitted by $\Delta A_0{q}_{emit}$ ?
(b) What is the intensity $I_{o,e}$ of the radiation field emit- ted from the surface $\Delta A_0$ ?
(c) Beginning with Equation 12.13 and presuming knowledge of the intensity I , obtain an expression for $q_{emit}$ .
(d) Consider the hemispherical surface located at $r=R_1=0.5m$ . Using the conservation of energy requirement, determine the rate at which radiant energy is incident on this surface due to emission from $\Delta A_0$
(e) Using Equation 12.10, determine the rate at which radiant energy leaving $\Delta A_0$ is intercepted by the small area $\Delta A_2$ located in the direction (45▯, ▯) on the hemispherical surface. What is the irradiation on $\Delta A_2$ ?
(f) Repeat part (e) for the location $I(0^{\circ },\varphi )$ Are the irradiations at the two locations equal?
(g) Using Equation 12.18, determine the irradiation $G_1$ on the hemispherical surface at $r=R_1$