Question

Five identical thin aluminum sheets with emissivities of \(0.1\) on both sides are placed between two very large parallel plates, which are maintained at uniform temperatures of \(T_{1}=800 \mathrm{~K}\) and \(T_{2}=450 \mathrm{~K}\) and have emissivities of \(\varepsilon_{1}=\) \(\varepsilon_{2}=0.1\), respectively. Determine the net rate of radiation heat transfer between the two plates per unit surface area of the plates, and compare the result to that without the shield.

Short answer

Answer: To assess the effect of adding the aluminum sheets as shields, the net rate of radiation heat transfer per unit surface area with and without the shield needs to be calculated using the radiosity equation. After performing the required calculations, compare the heat transfer rates with and without the shield and discuss the improvement or degradation in heat transfer due to the presence of the aluminum sheets.

Step-by-step solution

Understand the given quantities and symbols

In this exercise, we have the following quantities given: - Emissivities of the aluminum sheets (\(\epsilon_{sh} = 0.1\)) - Emissivities of the parallel plates: \(\epsilon_{1} = \epsilon_{2} = 0.1\) - Temperatures of the parallel plates: \(T_{1}=800 \mathrm{~K}\) and \(T_{2}=450 \mathrm{~K}\). We need to calculate the net rate of radiation heat transfer per unit surface area, denoted as \(q\).

Calculate surface emissive power of the plates using the Stefan-Boltzmann Law

Stefan-Boltzmann law states that the emitted power per unit area (\(E\)) is proportional to the fourth power of the absolute temperature (\(T\)). The constant of proportionality is the Stefan-Boltzmann constant (\(\sigma = 5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\)). Calculate the emitted power of the plates: \(E_1 = \sigma T_{1}^4\) and \(E_2 = \sigma T_{2}^4\) where \(E_1\) and \(E_2\) are the emitted power of the plates 1 and 2, and \(T_{1}\) and \(T_{2}\) are the temperatures of the plates 1 and 2.

Calculate the radiosity of the parallel plates and aluminum sheets

Radiosity (denoted as \(J\)) is the total energy leaving a surface, which consists of both emitted and reflected energy. Calculate the radiosity of the parallel plates: \(J_1 = E_1 \epsilon_1 + R_1 (1-\epsilon_1)\) and \(J_2 = E_2 \epsilon_2 + R_2 (1-\epsilon_2)\) Calculate the radiosity of the aluminum sheets: \(J_{sh1} = E_{sh1} \epsilon_{sh} + R_{sh1} (1-\epsilon_{sh})\), \(J_{sh2} = E_{sh2} \epsilon_{sh} + R_{sh2} (1-\epsilon_{sh})\), ..., \(J_{sh5} = E_{sh5} \epsilon_{sh} + R_{sh5} (1-\epsilon_{sh})\) where \(J_1\) and \(J_2\) are the radiosity of the plates 1 and 2, \(E_{1}\) and \(E_{2}\) are the emitted power of the plates 1 and 2, \(\epsilon_{1}\) and \(\epsilon_{2}\) are the emissivities of the plates, and \(R_1\) and \(R_2\) are the reflected powers of the plates 1 and 2. \(J_{sh1}\), \(J_{sh2}\), ..., \(J_{sh5}\) are the radiosity of the aluminum sheets, \(E_{sh1}\), \(E_{sh2}\), ..., \(E_{sh5}\) are the emitted power of the aluminum sheets, and \(R_{sh1}\), \(R_{sh2}\), ..., \(R_{sh5}\) are the reflected powers of the aluminum sheets. Note that \(R_{sh1}=J_1\), \(R_{sh2}=J_{sh1}\), and so on, such that \(R_{sh5}=J_{sh4}\) and \(R_2=J_{sh5}\).

Determine the heat transfer rate per unit surface area by applying the radiosity equation

The radiosity equation states that the heat transfer rate per unit surface area \(q\) is the difference in radiosity between the parallel plates divided by the sum of the resistances between plates, including the aluminum sheets. Calculate \(q\) with the shield: \(q = \frac{J_1 - J_2}{1+\frac{\epsilon_{sh} -1}{\epsilon_{1}}+\frac{\epsilon_{sh} -1}{\epsilon_{2}}+4(\frac{\epsilon_{sh} -1}{\epsilon_{sh}})}\) Calculate \(q\) without the shield: \(q_{wo} = \frac{J_1 - J_2}{1+\frac{\epsilon_{1}-1}{\epsilon_{2}}}\)

Perform calculations and compare the results

Use the given temperatures, emissivities, and the Stefan-Boltzmann constant to calculate the emitted power, radiosity, and heat transfer rates with and without the shield. Then, discuss the differences between the two rates and explain if using the aluminum sheets improved heat transfer or not. Top tip: Pay attention to the units of the calculations and compare the values in the same unit system.