Question

Two infinitely long parallel plates of width \(w\) are located at \(w\) distance apart, as shown in Fig. P13-24. Using the Hottel's crossed-strings method, determine the view factor \(F_{12}\).

Short answer

Question: Determine the view factor between two infinitely long parallel plates with width w and separated by a distance w using Hottel's crossed strings method. Answer: The view factor, \(F_{12}\), between the two infinitely long parallel plates in the given scenario is 0.5 or 50%.

Step-by-step solution

Understand Hottel's crossed-strings method

Hottel's crossed-strings method is a graphical technique used to determine view factors between two surfaces. The method involves drawing a string from the corner of one surface to the adjacent corner of the other surface and calculating the length of the string and angles formed by the string at the corners. The lengths of the strings and angles will be used to determine the view factor between the surfaces.

Visualize the scenario

Visualize two infinitely long parallel plates with width \(w\) separated by a distance \(w\). Draw a string from the corner A of plate 1 to the corner B of plate 2, and another string from corner C of plate 2 to corner D of plate 1.

Determine the string lengths

In the scenario described, there are two strings: AB and CD. These strings are the same length since the plates are parallel and equidistant. Utilizing the Pythagorean theorem, the length of the strings can be determined: Length (AB) = Length (CD) = \(\sqrt{w^2 + w^2}\) = \(\sqrt{2w^2}\) = \(w\sqrt{2}\)

Determine the angles

To find the angles of both strings, denote the angles at A and D as \(\theta_1\) and \(\theta_2\) respectively. Using the tan function, these angles can be determined: Angle A: \(\theta_1 = \tan^{-1} \frac{w}{w} = \tan^{-1}(1)\) Angle D: \(\theta_2 = \tan^{-1} \frac{w}{w} = \tan^{-1}(1)\) The angles at B and C are also the same: Angle B: \(\tan^{-1} \frac{w}{w} = \tan^{-1}(1)\) Angle C: \(\tan^{-1} \frac{w}{w} = \tan^{-1}(1)\)

Determine the view factor using the cross-string method

Apply Hottel's crossed-string formula to calculate the view factor \(F_{12}\): For two parallel infinite plates of equal width, the formula becomes: \(F_{12} = \frac{(\theta_1 + \theta_2)}{\pi}\) Using the angles determined earlier: \(F_{12} = \frac{\tan^{-1}(1) + \tan^{-1}(1)}{\pi} = \frac{\pi/4 + \pi/4}{\pi} = \frac{\pi/2}{\pi} = \frac{1}{2}\) Thus, the view factor \(F_{12}\) is equal to 0.5 or 50% for this specific scenario. It implies that 50% of the radiant energy leaving plate 1 reaches plate 2.