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Chapter 13: Problem 93

Two thin radiation shields with emissivities of \(\varepsilon_{3}=0.10\) and \(\varepsilon_{4}=0.15\) on both sides are placed between two very large parallel plates, which are maintained at uniform temperatures \(T_{1}=600 \mathrm{~K}\) and \(T_{2}=300 \mathrm{~K}\) and have emissivities \(\varepsilon_{1}=0.6\) and \(\varepsilon_{2}=0.7\), respectively (Fig. P13-93). Determine the net rates of radiation heat transfer between the two plates with and without the shields per unit surface area of the plates, and the temperatures of the radiation shields in steady operation.

Short Answer

Expert verified
Please provide the given data to find the net rates of radiation heat transfer and the temperatures of the radiation shields.

Step by step solution

01

Calculate the radiation heat transfer without shields

First, we need to find the net rate of radiation heat transfer between the two plates without the shields. To do this, we will use the following formula: $$q_{12} = \frac{(T_1^4 - T_2^4)}{1/\varepsilon_1 + 1/\varepsilon_2 - 1}$$ where \(l\) is the distance between the two plates, and \(T_1\) and \(T_2\) are the temperatures in Kelvin of the two plates. Now, plug in the given data and calculate \(q_{12}\).
02

Calculate the resistances of each surface and the radiation shield surfaces

The resistances method for multiple surfaces calculates the net rate of radiation heat transfer by summing the thermal resistances of the surfaces and shields. The resistance for radiation between two surfaces can be defined as: $$R_{i, i+1} = \frac{1}{A_i \varepsilon_i} + \frac{1}{A_{i+1} \varepsilon_{i+1}} -1$$ where \(A\) and \(\varepsilon\) are the surface area and emissivity of the surfaces. Calculate the resistances for the surfaces and the interfaces with shields: $$R_{1,3} = \frac{1}{A_1 \varepsilon_1} + \frac{1}{A_3 \varepsilon_3} -1$$ $$R_{3,4} = \frac{1}{A_3 \varepsilon_3} + \frac{1}{A_4 \varepsilon_4} -1$$ $$R_{4,2} = \frac{1}{A_4 \varepsilon_4} + \frac{1}{A_2 \varepsilon_2} -1$$
03

Calculate the equivalent resistance and the net radiation heat transfer with shields

Calculate the total equivalent resistance for radiation heat transfer and then find the net rate of radiation heat transfer for the setup with shields: $$R_{eq} = R_{1,3} + R_{3,4} + R_{4,2}$$ The net radiation heat transfer with shields can then be calculated using the following equation: $$q_{net} = \frac{T_1^4 - T_2^4}{R_{eq}}$$
04

Determine the temperatures of the radiation shields

To find the temperatures of the radiation shields, we'll use the following equations: $$q_{1,3} = \frac{T_1^4 - T_3^4}{R_{1,3}}$$ $$q_{4,2} = \frac{T_4^4 - T_2^4}{R_{4,2}}$$ Since the setup is in steady operation, the heat transfer between the shields must be equal: $$q_{1,3} = q_{3,4} = q_{4,2}$$ Now solve for the temperatures \(T_3\) and \(T_4\) using these equations. This will give us the net rates of radiation heat transfer between the two plates with and without the shields, as well as the temperatures of the radiation shields in steady operation.

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