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Chapter 13: Problem 60

Two very large parallel plates are maintained at uniform temperatures of \(T_{1}=600 \mathrm{~K}\) and \(T_{2}=400 \mathrm{~K}\) and have emissivities \(\varepsilon_{1}=0.5\) and \(\varepsilon_{2}=0.9\), respectively. Determine the net rate of radiation heat transfer between the two surfaces per unit area of the plates.

Short Answer

Expert verified
Answer: The net rate of radiation heat transfer between the two surfaces per unit area of the plates is approximately \(10026.4 \mathrm{~W/m^2}\).

Step by step solution

01

Identify the given variables

We have the following variables given: - Temperature of plate 1, \(T_{1} = 600 \mathrm{~K}\) - Temperature of plate 2, \(T_{2} = 400 \mathrm{~K}\) - Emissivity of plate 1, \(\varepsilon_{1} = 0.5\) - Emissivity of plate 2, \(\varepsilon_{2} = 0.9\)
02

Apply the formula for net radiation heat transfer between two surfaces

To find the net rate of radiation heat transfer between the two surfaces per unit area, we can use the following formula: $$Q = \sigma \frac{\varepsilon_1 \varepsilon_2}{\varepsilon_1+\varepsilon_2-\varepsilon_1 \varepsilon_2} (T_1^4 - T_2^4)$$ Where: - \(Q\) is the net rate of radiation heat transfer per unit area (\(W/m^2\)) - \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} W \cdot m^{-2} K^{-4}\)) - \(T_1\) and \(T_2\) are the temperatures of the two surfaces in Kelvin, and - \(\varepsilon_1\) and \(\varepsilon_2\) are the emissivities of the two surfaces.
03

Substitute the values and solve

Now, we can substitute the given values in the formula: $$Q = \frac{(5.67 \times 10^{-8})(0.5)(0.9)}{0.5+0.9-(0.5)(0.9)} (600^4 - 400^4)$$ After evaluating the expression, we get: $$Q \approx 10026.4 \mathrm{~W/m^2}$$
04

Final answer

The net rate of radiation heat transfer between the two surfaces per unit area of the plates is approximately \(10026.4 \mathrm{~W/m^2}\).

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Most popular questions from this chapter

A car mechanic is working in a shop whose interior space is not heated. Comfort for the mechanic is provided by two radiant heaters that radiate heat at a total rate of \(4 \mathrm{~kJ} / \mathrm{s}\). About 5 percent of this heat strikes the mechanic directly. The shop and its surfaces can be assumed to be at the ambient temperature, and the emissivity and absorptivity of the mechanic can be taken to be \(0.95\) and the surface area to be \(1.8 \mathrm{~m}^{2}\). The mechanic is generating heat at a rate of \(350 \mathrm{~W}\), half of which is latent, and is wearing medium clothing with a thermal resistance of \(0.7 \mathrm{clo}\). Determine the lowest ambient temperature in which the mechanic can work comfortably.

A spherical tank of diameter \(D=2 \mathrm{~m}\) that is filled with liquid nitrogen at \(100 \mathrm{~K}\) is kept in an evacuated cubic enclosure whose sides are \(3 \mathrm{~m}\) long. The emissivities of the spherical tank and the enclosure are \(\varepsilon_{1}=0.1\) and \(\varepsilon_{2}=0.8\), respectively. If the temperature of the cubic enclosure is measured to be \(240 \mathrm{~K}\), determine the net rate of radiation heat transfer to the liquid nitrogen. Answer: \(228 \mathrm{~W}\)

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) \(0.89\)

Two aligned parallel rectangles with dimensions \(6 \mathrm{~m} \times\) \(8 \mathrm{~m}\) are spaced apart by a distance of \(2 \mathrm{~m}\). If the two parallel rectangles are experiencing radiation heat transfer as black surfaces, determine the percentage of change in radiation heat transfer rate when the rectangles are moved \(8 \mathrm{~m}\) apart.

Consider an enclosure consisting of \(N\) diffuse and gray surfaces. The emissivity and temperature of each surface as well as the view factors between the surfaces are specified. Write a program to determine the net rate of radiation heat transfer for each surface.
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