Determine the view factors F12 and F21 for the following geometries: 1. Sphere of diameter D inside a cubical box of length L = D. 2. One side of a diagonal partition within a long square duct. 3. End and side of a circular tube of equal length and diameter.

Question

Determine the view factors [latex]F_{12}[/latex] and [latex]F_{21}[/latex] for the following geometries:

1. Sphere of diameter D inside a cubical box of length L = D.

2. One side of a diagonal partition within a long square duct.

3. End and side of a circular tube of equal length and diameter.

Accepted Answer

Known: Surface geometries.

Find: View factors.

Assumptions: Diffuse surfaces with uniform radiosities.

Analysis: The desired view factors may be obtained from inspection, the reciprocity rule, the summation rule, and/or use of the charts.

1. Sphere within a cube:

By inspection, [latex]F_{12} = 1[/latex]

By reciprocity, [latex]F_{21} = \frac{A_{1}}{A_{2}}F_{12} = \frac{\pi D^{2}}{6L^{2}} × 1 = \frac{\pi}{6}[/latex]

2. Partition within a square duct:

From summation rule, [latex]F_{11} + F_{12} + F_{13} = 1[/latex]

where [latex]F_{11} = 0[/latex]

By symmetry, [latex]F_{12} = F_{13}[/latex]

Hence [latex]F_{12} = 0.50[/latex]

By reciprocity, [latex]F_{21} = \frac{A_{1}}{A_{2}}F_{12} = \frac{\sqrt{2}L}{L} × 0.5 = 0.71[/latex]

3. Circular tube:

From Table 13.2 or Figure 13.5, with [latex](r_{3}/L) = 0.5[/latex] and [latex](L/r_{1}) = 2, F_{13} = 0.172[/latex]

TABLE 13.2 View Factors for Three-Dimensional Geometries [4]
Geometry	Relation
Aligned Parallel Rectangles (Figure 13.4)	[latex]\overline{X} = X/L, \overline{Y} = Y/L\[/latex] [latex]F_{ij} = \frac{2}{\pi \overline{X}\overline{Y}}\left\{\ln \left[\frac{(1 + \overline{X}^{2})(1 + \overline{Y}^{2})}{1 +\overline{X}^{2}+ \overline{Y}^{2}}\right]^{1/2} + \overline{X}(1 + \overline{Y}^{2})^{1/2} \tan^{-1}\frac{\overline{X}}{(1 + \overline{Y}^{2})^{1/2}} + \overline{Y}(1 + \overline{X}^{2})^{1/2} \tan^{-1}\frac{\overline{Y}}{(1 + \overline{X}^{2})^{1/2}} - \overline{X} \tan^{-1} \overline{X} - \overline{Y} \tan^{-1} \overline{Y}\right\}[/latex]
Coaxial Parallel Disks (Figure 13.5)	[latex]R_{i} = r_{i}/L, R_{j} = r_{j}/L\ S = 1 + \frac{1 + R^{2}_{j}}{R^{2}_{i}}\ F_{ij} = \frac{1}{2} \left\{S - [S^{2} - 4(r_{j}/r_{i})^{2}]^{1/2}\right\}[/latex]
Perpendicular Rectangles with a Common Edge (Figure 13.6)	H = Z/X, W = Y/X[latex]\ F_{ij} = \frac{1}{\pi W} \left(W\tan^{-1} \frac{1}{W} + H \tan^{-1} \frac{1}{H} - (H^{2} + W^{2})^{1/2} \tan^{-1} \frac{1}{(H^{2} + W^{2})^{1/2}} + \frac{1}{4}\ln \left\{\frac{(1 + W^{2})(1 + H^{2})}{1 + W^{2} + H^{2}}\left[\frac{W^{2}(1 + W^{2} + H^{2})}{(1 + W^{2})(W^{2} + H^{2})}\right]^{W^{2}} \times \left[\frac{H^{2}(1 + H^{2} + W^{2})}{(1 + H^{2})(H^{2} + W^{2})}\right]^{H^{2}}\right\} \right)[/latex]

From summation rule, [latex]F_{11} + F_{12} + F_{13} = 1[/latex]

or, with [latex]F_{11} = 0, F_{12} = 1 - F_{13} = 0.828[/latex]

From reciprocity, [latex]F_{21} = \frac{A_{1}}{A_{2}}F_{12} = \frac{\pi D^{2}/4}{\pi DL} × 0.828 = 0.207[/latex]

Comment: The geometric surfaces may, in reality, not be characterized by uniform radiosities. The consequences of nonuniform radiosity are discussed in Example 13.3.

Question 13.2: Determine the view factors F12 and F21 for the following geo...

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