Consider a diffuse circular disk of diameter D and area Aj and a plane diffuse surface of area Ai ≪ Aj. The surfaces are parallel, and Ai is located at a distance L from the center of Aj. Obtain an expression for the view factor Fij.

Question

Consider a diffuse circular disk of diameter D and area [latex]A_{j}[/latex] and a plane diffuse surface of area [latex]A_{i} ≪ A_{j}[/latex]. The surfaces are parallel, and [latex]A_{i}[/latex] is located at a distance L from the center of [latex]A_{j}[/latex]. Obtain an expression for the view factor [latex]F_{ij}[/latex].

Accepted Answer

Known: Orientation of small surface relative to large circular disk.

Find: View factor of small surface with respect to disk, [latex]F_{ij}[/latex].

Schematic:

Assumptions:
1. Diffuse surfaces.
2. [latex]A_{i} ≪ A_{j}[/latex].
3. Uniform radiosity on surface [latex]A_{i}[/latex].

Analysis: The desired view factor may be obtained from Equation 13.1.

[latex]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}[/latex] (13.1)

Recognizing that [latex]θ_{i}, θ_{j}[/latex], and R are approximately independent of position on [latex]A_{i}[/latex], this expression reduces to

[latex]F_{ij} = \int_{A_{j}}\frac{\cos θ_{i} \cos θ_{j}}{\pi R^{2}}dA_{j}[/latex]

or, with [latex]θ_{i} = θ_{j} \equiv θ[/latex],

[latex]F_{ij} = \int_{A_{j}}\frac{\cos^{2} θ}{\pi R^{2}}dA_{j}[/latex]

With R² = r² + L², cos θ = (L/R), and [latex]dA_{j} = 2\pi r dr[/latex], it follows that

[latex]F_{ij} = 2L^{2}\int_{0}^{D/2}\frac{r dr}{(r^{2} + L^{2})^{2}} = \frac{D^{2}}{D^{2} + 4L^{2}}[/latex] (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, [latex]r_{i}[/latex], 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].

Question 13.1: Consider a diffuse circular disk of diameter D and area Aj a...

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