Question 13.1: Consider a diffuse circular disk of diameter D and area Aj a...
Consider a diffuse circular disk of diameter D and area A_{j} and a plane diffuse surface of area A_{i} ≪ A_{j}. The surfaces are parallel, and A_{i} is located at a distance L from the center of A_{j}. Obtain an expression for the view factor F_{ij}.
Known: Orientation of small surface relative to large circular disk.
Find: View factor of small surface with respect to disk, F_{ij}.
Schematic:
Assumptions:
1. Diffuse surfaces.
2. A_{i} ≪ A_{j}.
3. Uniform radiosity on surface A_{i}.
Analysis: The desired view factor may be obtained from Equation 13.1.
F_{ij} = \frac{1}{A_{i}}\int_{A_{i}}\int_{A_{j}}\frac{\cos θ_{i} \cos θ_{j}}{\pi R^{2}}dA_{i} dA_{j} (13.1)
Recognizing that θ_{i}, θ_{j}, and R are approximately independent of position on A_{i}, this expression reduces to
F_{ij} = \int_{A_{j}}\frac{\cos θ_{i} \cos θ_{j}}{\pi R^{2}}dA_{j}
or, with θ_{i} = θ_{j} \equiv θ,
F_{ij} = \int_{A_{j}}\frac{\cos^{2} θ}{\pi R^{2}}dA_{j}
With R² = r² + L², cos θ = (L/R), and dA_{j} = 2\pi r dr, it follows that
F_{ij} = 2L^{2}\int_{0}^{D/2}\frac{r dr}{(r^{2} + L^{2})^{2}} = \frac{D^{2}}{D^{2} + 4L^{2}} (13.8)
Comments:
1. Equation 13.8 may be used to quantify the asymptotic behavior of the curves in Figure 13.5 as the radius of the lower circle, r_{i}, approaches zero.
2. The preceding geometry is one of the simplest cases for which the view factor may be obtained from Equation 13.1. Geometries involving more detailed integrations are considered in the literature [1, 3].
