Question

Consider two concentric spheres with diameters \(12 \mathrm{~cm}\) and $18 \mathrm{~cm}$ forming an enclosure. The view factor from the inner surface of the outer sphere to the inner sphere is (a) 0 (b) \(0.18\) (c) \(0.44\) (d) \(0.56\) (e) \(0.67\)

Short answer

Answer: (e) 0.67

Step-by-step solution

Identify given values

The diameters of the two concentric spheres are given as \(D_1 = 12 \mathrm{~cm}\) and \(D_2 = 18 \mathrm{~cm}\). We can find their radii by dividing the diameters by \(2\).

Calculate radii of the spheres

The radii of the inner sphere and the outer sphere can be calculated as follows: \(R_1 = \frac{D_1}{2} = 6\mathrm{~cm}\) and \(R_2 = \frac{D_2}{2} = 9\mathrm{~cm}\).

Determine the geometry factor F

The geometry factor (sometimes designated as F or G) is specific to the shape of the geometries involved. The view factor for two concentric spheres is given by the following formula: \(F = \frac{R_1}{(R_2 - R_1)}\) Substitute the values of \(R_1\) and \(R_2\) in the formula: \(F = \frac{6}{(9 - 6)} = \frac{6}{3} = 2\)

Calculate the view factor using the geometry factor

The view factor between two concentric spheres can be calculated using the following formula: \(F^{(1\to2)} = 1 - \frac{1}{1 + F} = 1 - \frac{1}{1 + 2} = 1 - \frac{1}{3}\) \(F^{(1\to2)} = \frac{2}{3} = 0.67\) Now compare the calculated view factor value to the given possible answers: (a) 0 (b) 0.18 (c) 0.44 (d) 0.56 (e) 0.67

Choose the correct answer

The correct answer for the view factor from the inner surface of the outer sphere to the inner sphere is (e) \(0.67\).