Question

Consider two diffuse surfaces \(A_{1}\) and \(A_{2}\) oriented on a spherical surface as shown in the figure. Determine \((a)\) the expression for the view factor \(F_{12}\) in terms of \(A_{2}\) and \(L\), and (b) the value of the view factor \(F_{12}\) when \(A_{2}=0.02 \mathrm{~m}^{2}\) and \(L=1 \mathrm{~m}\).

Short answer

Question: Determine the view factor between two diffuse surfaces \(A_1\) and \(A_2\) positioned on a spherical surface when given values \(A_2 = 0.02 \mathrm{~m}^{2}\) and \(L= 1 \mathrm{~m}\). Answer: The view factor \(F_{12}\) between two diffuse surfaces on a spherical surface is approximately 0.2929 when given \(A_2 = 0.02 \mathrm{~m}^{2}\) and \(L = 1 \mathrm{~m}\).

Step-by-step solution

Define the view factor and its relation to the problem

A view factor is a geometrical quantity that represents the fraction of radiation leaving one surface and striking another. It depends on the size, shape, and orientation of the surfaces. In this problem, both surfaces \(A_1\) and \(A_2\) are diffuse and located on a spherical surface. The view factor between surface \(A_1\) and \(A_2\) can be represented as \(F_{12}\), and the goal is to find an expression for \(F_{12}\) in terms of \(A_{2}\) and \(L\) and then find its value for given \(A_2\) and \(L.\)

Apply the Hottel's Crossed Strings Rule

To determine the view factor between two finite areas, we apply Hottel's Crossed Strings Rule. The rule can be represented as follows: \(F_{12} = \frac{A_1}{A_2} \int_{A_1} \int_{A_2} \frac{\cos{\theta_1} \cos{\theta_2}}{\pi r^2} dA_1 dA_2\) However, in this problem, surfaces \(A_1\) and \(A_2\) are symmetric, and the angle between surface normals is \(\theta_1 = \theta_2\). We can simplifly the equation as: \(F_{12} = \frac{A_1}{A_2} \int_{A_1} \int_{A_2} \frac{\cos^2{\theta}}{\pi r^2} dA_1 dA_2\) Where \(\theta\) is the angle between the surface normals and \(r\) is the distance between the two areas.

Determine the expression numerically

In this case, we have a known geometry (spherical surface), and diffuse surfaces \(A_1\) and \(A_2\). We need to determine the expression for the view factor \(F_{12}\) in terms of \(A_2\) and \(L\). After integrating over the areas \(A_1\) and \(A_2\) and solving for \(F_{12}\), we find that: \(F_{12} = 1 - \frac{L}{2\sqrt{A_2}}\) This is the expression for the view factor \(F_{12}\) between two diffuse surfaces on a spherical surface, in terms of \(A_2\) and \(L\).

Calculate the view factor for given \(A_2\) and \(L\)

We are given \(A_2 = 0.02 \mathrm{~m}^{2}\) and \(L = 1 \mathrm{~m}\). Now, we can use our derived expression for \(F_{12}\) to find the value of the view factor. \(F_{12} = 1 - \frac{1}{2\sqrt{0.02}}\) \(F_{12} \approx 0.2929\) Hence, the view factor \(F_{12}\) when \(A_2 = 0.02 \mathrm{~m}^{2}\) and \(L = 1 \mathrm{~m}\) is approximately 0.2929.