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Chapter 13: Problem 147

Consider a 15-cm-diameter sphere placed within a cubical enclosure with a side length of \(15 \mathrm{~cm}\). The view factor from any of the square-cube surfaces to the sphere is (a) \(0.09\) (b) \(0.26\) (c) \(0.52\) (d) \(0.78\) (e) 1

Short Answer

Expert verified
Answer: The view factor between any of the cube's surfaces and the sphere in this case is approximately 0.26.

Step by step solution

01

Calculate critical radius of the sphere

We'll first need to find the critical radius \(r_c\) of the sphere. The critical radius is half of the diameter, and since the diameter is 15cm, the critical radius is: \(r_c = \frac{15}{2} = 7.5\mathrm{~cm}\).
02

Calculate the string lengths

Now, we will find the length of the strings connecting the cube's surface corners to the sphere's diametrically opposite points. We can call these strings A and B. Since the cube has a side length of 15cm and considering the sphere is entirely inside the cube touching all sides, the lengths of strings A and B will be equal to the cube's side length and the sphere's diameter: \(A = B = 15\mathrm{~cm}\).
03

Apply Hottel's cross string method formula

Hottel's formula for calculating view factor between a sphere and a planar surface (i.e., the cube surface) is given as: \(F_{sphere\rightarrow surface} = \frac{1}{2}\left(1 - \frac{AB}{(AB + r_c^2)^{\frac{1}{2}}}\right)\) Substitute the values of A, B, and \(r_c\) into the formula: \(F_{sphere\rightarrow surface} = \frac{1}{2}\left(1 - \frac{15\times15}{(15\times15 + 7.5^2)^{\frac{1}{2}}}\right)\)
04

Calculate the view factor

Now, we can find the numerical value of the view factor: \(F_{sphere\rightarrow surface} = \frac{1}{2}\left(1 - \frac{15\times15}{(15\times15 + 7.5^2)^{\frac{1}{2}}}\right)\approx 0.26\) Comparing the result with the given options, 0.26 is the correct answer, therefore the view factor from any of the square-cube surfaces to the sphere is (b) \(0.26\).

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Most popular questions from this chapter

Consider an enclosure consisting of 12 surfaces. How many view factors does this geometry involve? How many of these view factors can be determined by the application of the reciprocity and the summation rules?

Consider a cylindrical enclosure with \(A_{1}, A_{2}\), and \(A_{3}\) representing the internal base, top, and side surfaces, respectively. Using the length to diameter ratio, \(K=L / D\), determine (a) the expression for the view factor from the side surface to itself \(F_{33}\) in terms of \(K\) and \((b)\) the value of the view factor \(F_{33}\) for \(L=D\).

Consider a \(3-\mathrm{m} \times 3-\mathrm{m} \times 3-\mathrm{m}\) cubical furnace. The base surface is black and has a temperature of \(400 \mathrm{~K}\). The radiosities for the top and side surfaces are calculated to be \(7500 \mathrm{~W} / \mathrm{m}^{2}\) and \(3200 \mathrm{~W} / \mathrm{m}^{2}\), respectively. If the temperature of the side surfaces is \(485 \mathrm{~K}\), the emissivity of the side surfaces is (a) \(0.37\) (b) \(0.55\) (c) \(0.63\) (d) \(0.80\) (e) \(0.89\)

This experiment is conducted to determine the emissivity of a certain material. A long cylindrical rod of diameter \(D_{1}=0.01 \mathrm{~m}\) is coated with this new material and is placed in an evacuated long cylindrical enclosure of diameter \(D_{2}=0.1 \mathrm{~m}\) and emissivity \(\varepsilon_{2}=0.95\), which is cooled externally and maintained at a temperature of \(200 \mathrm{~K}\) at all times. The rod is heated by passing electric current through it. When steady operating conditions are reached, it is observed that the rod is dissipating electric power at a rate of \(8 \mathrm{~W}\) per unit of its length and its surface temperature is \(500 \mathrm{~K}\). Based on these measurements, determine the emissivity of the coating on the rod.

Consider a large classroom with 90 students on a hot summer day. All the lights with \(2.0 \mathrm{~kW}\) of rated power are kept on. The room has no external walls, and thus heat gain through the walls and the roof is negligible. Chilled air is available at \(15^{\circ} \mathrm{C}\), and the temperature of the return air is not to exceed \(25^{\circ} \mathrm{C}\). The average rate of metabolic heat generation by a person sitting or doing light work is \(115 \mathrm{~W}\) ( \(70 \mathrm{~W}\) sensible and \(45 \mathrm{~W}\) latent). Determine the required flow rate of air that needs to be supplied to the room.
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