Question

Give examples of radiation effects that affect human comfort. 13-84 A thin aluminum sheet with an emissivity of \(0.15\) on both sides is placed between two very large parallel plates, which are maintained at uniform temperatures \(T_{1}=900 \mathrm{~K}\) and \(T_{2}=650 \mathrm{~K}\) and have emissivities \(\varepsilon_{1}=0.5\) and \(\varepsilon_{2}=0.8\), respectively. Determine the net rate of radiation heat transfer between the two plates per unit surface area of the plates and compare the result with that without the shield.Give examples of radiation effects that affect human comfort.

Short answer

Answer: The aluminum shield significantly reduces the net rate of radiation heat transfer between the two parallel plates. With the shield, the heat transfer is 7,691.9 W/m², while without the shield, the heat transfer is 16,314.8 W/m².

Step-by-step solution

Determine the Spatial Resistance for Radiation

To determine the net rate of radiation heat transfer between the two plates per unit surface area, we first need to find their spatial resistance for radiation. Spatial resistance is given by: \[R_{s} = \dfrac{1}{\varepsilon} - 1\] For plate 1: \<^latexCode╗ \( R_{s1} = \dfrac{1}{\varepsilon_1} - 1 = \dfrac{1}{0.5} - 1 = 1 \) For plate 2: \( R_{s2} = \dfrac{1}{\varepsilon_2} - 1 = \dfrac{1}{0.8} - 1 = 0.25 \) For the shield: \( R_{s\text{shield}} = \dfrac{1}{0.15} - 1 = 5.67 \)

Calculate the Net Radiation Heat Transfer with the Shield

To find the net rate of radiation heat transfer between the two plates with the shield, use the formula: \[Q = \dfrac{\sigma(T_{1}^4 - T_{2}^4)}{R_{s1} + R_{s\text{shield}} + R_{s2}}\] where σ is the Stefan-Boltzmann constant, which is \(5.67 \times 10^{-8} \mathrm{W/m^2K^4}\). Plug in the given values to find net radiation heat transfer: \[Q = \dfrac{5.67 \times 10^{-8}(900^4 - 650^4)}{1 + 5.67 + 0.25} = 7691.9 \mathrm{W/m^2}\]

Calculate the Net Radiation Heat Transfer without the Shield

To find the net rate of radiation heat transfer between the two plates without the shield, use the same formula without the shield's spatial resistance term: \[Q' = \dfrac{\sigma(T_{1}^4 - T_{2}^4)}{R_{s1} + R_{s2}}\] Plug in the given values: \[Q' = \dfrac{5.67 \times 10^{-8}(900^4 - 650^4)}{1 + 0.25} = 16314.8 \mathrm{W/m^2}\]

Compare the Net Radiation Heat Transfer with and without the Shield

Now, we can compare the net rate of radiation heat transfer between the two plates per unit surface area with and without the shield: - With the shield: \(Q = 7691.9 \mathrm{W/m^2}\) - Without the shield: \(Q' = 16314.8 \mathrm{W/m^2}\) From the results, we can see that the aluminum shield significantly reduces the net rate of radiation heat transfer between the two plates.

Key concepts

Emissivity

Emissivity is a measure of an object's ability to emit thermal radiation. It is represented by the symbol \( \varepsilon \), and its value ranges between 0 and 1. An emissivity of 1.0 is considered a perfect emitter, also known as a blackbody, which means it emits the maximum amount of radiation possible at its temperature. Conversely, an emissivity closer to 0 indicates poor emission capabilities, meaning the object reflects most of the radiation.
Materials with high emissivity are good at radiating energy and losing heat, whereas low-emissivity materials are better at retaining heat. For example, in our problem, the thin aluminum sheet has an emissivity of 0.15, indicating it is not very efficient at emitting thermal radiation. This feature makes aluminum a suitable material for applications like thermal insulation, where minimizing heat transfer is desired.
Understanding emissivity is fundamental when designing systems or structures where controlling heat transfer through radiation is critical. Factors like temperature, surface texture, and wavelength of the emitted radiation can influence a material's emissivity.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermal physics used to calculate the power radiated from a black body in terms of its temperature. This law states that the power emitted per unit area by a perfect black body is proportional to the fourth power of its temperature. It is expressed by the equation:
\[ E = \sigma T^4 \]where \( E \) is the radiant energy emitted per unit area, \( T \) is the absolute temperature in Kelvin, and \( \sigma \) is the Stefan-Boltzmann constant, approximately equal to \( 5.67 \times 10^{-8} \ \text{W/m}^2\text{K}^4 \).
This principle is extensively used in applications involving heat transfer and thermal radiation. In our exercise, it helps calculate the net radiation heat transfer rate between two plates held at different temperatures, with and without an intervening aluminum sheet.
The power radiated by each surface influences how we quantify and manage heat transfer, particularly in designs that leverage radiative properties to improve energy efficiency or regulate environmental comfort.

Thermal Radiation

Thermal radiation is the emission of electromagnetic waves from all objects that possess heat. This form of radiation does not require any medium to travel, meaning it can occur in vacuum environments. Any object with a temperature above absolute zero emits thermal radiation, with key characteristics dependent on the object's temperature and emissivity.

• Higher temperatures result in more intense radiation and shorter wavelengths, shifting from infrared towards visible light.
• The nature of thermal radiation allows it to be absorbed, reflected, or transmitted by other materials.

For the scenario in our exercise, thermal radiation is the primary mode through which energy is transferred between the two plates. By placing an aluminum shield between the plates, we manipulate the path and extent of this heat transfer by radiation. The shield introduces additional spatial resistance, thus decreasing the net heat transfer rate.
Understanding thermal radiation is important for designing both comfortable living spaces and efficient heat management systems in various engineering applications. By carefully selecting materials and structures that control radiation, we can significantly influence the heat dynamics in our environment.