Question

The spectral emissivity function of an opaque surface at \(1000 \mathrm{~K}\) is approximated as $$ \varepsilon_{\lambda}= \begin{cases}\varepsilon_{1}=0.4, & 0 \leq \lambda<2 \mu \mathrm{m} \\ \varepsilon_{2}=0.7, & 2 \mu \mathrm{m} \leq \lambda<6 \mu \mathrm{m} \\ \varepsilon_{3}=0.3, & 6 \mu \mathrm{m} \leq \lambda<\infty\end{cases} $$ Determine the average emissivity of the surface and the rate of radiation emission from the surface, in \(\mathrm{W} / \mathrm{m}^{2}\).

Short answer

Based on the provided spectral emissivity function of an opaque surface at 1000K, we determined that the average emissivity of the surface could not be calculated using the given method due to its indefinite result. However, the rate of radiation emission from the surface was found to be \(5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2}\) using the Stefan-Boltzmann law and the given spectral emissivities.

Step-by-step solution

Define the function of spectral emissivity

The spectral emissivity function is given by: $$ \varepsilon_{\lambda}= \begin{cases}\varepsilon_{1}=0.4, & 0 \leq \lambda<2 \mu \mathrm{m} \\ \varepsilon_{2}=0.7, & 2 \mu \mathrm{m} \leq \lambda<6 \mu \mathrm{m} \\ \varepsilon_{3}=0.3, & 6 \mu \mathrm{m} \leq \lambda<\infty\end{cases} $$

Calculate the average emissivity

To find the average emissivity, we need to integrate the spectral emissivity function over the entire wavelength range and divide by the corresponding wavelength range. We can divide the integration into three intervals and add the results: $$ \bar{\varepsilon} = \frac{\int_{0}^{\infty} \varepsilon_{\lambda} d\lambda}{\int_{0}^{\infty} d\lambda} = \frac{\int_{0}^{2} 0.4 d\lambda + \int_{2}^{6} 0.7 d\lambda + \int_{6}^{\infty} 0.3 d\lambda}{\int_{0}^{\infty} d\lambda} $$ First, let's compute the spectral emissivity integrals and the total wavelength range integral separately: $$ I_1 = \int_{0}^{2} 0.4 d\lambda = 0.4 \times (2-0) = 0.8 \\ I_2 = \int_{2}^{6} 0.7 d\lambda = 0.7 \times (6-2) = 2.8 \\ I_3 = \int_{6}^{\infty} 0.3 d\lambda = 0.3 \times (\infty-6) = \infty \\ \int_{0}^{\infty} d\lambda = \infty $$ Now, let's compute the average emissivity: $$ \bar{\varepsilon} = \frac{I_1 + I_2 + I_3}{\infty} =\frac{0.8+2.8+\infty}{\infty}= \infty $$ Since the result is not finite in this case, we conclude that we can't determine the average emissivity using this method.

Calculate the rate of radiation emission

Despite the indefinite average emissivity, we can still find the rate of radiation emission using each spectral emissivity value. We use the Stefan-Boltzmann law to calculate the total radiation emission for each part of the spectral emissivity function: $$ R_{total} = \varepsilon_1 R_1 + \varepsilon_2 R_2 + \varepsilon_3 R_3 $$ Where \(R_1\), \(R_2\), and \(R_3\) are the radiation emissions for the three wavelength ranges given, and the Stefan-Boltzmann constant is \(\sigma=5.67\times10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-4}\). $$ R_1 = \sigma T^4 = 5.67\times10^{-8} (1000)^4 = 5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2} \\ R_2 = \sigma T^4 = 5.67\times10^{-8} (1000)^4 = 5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2} \\ R_3 = \sigma T^4 = 5.67\times10^{-8} (1000)^4 = 5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2} $$ Now compute the total radiation emission: $$ R_{total} = 0.4 \cdot R_1 + 0.7 \cdot R_2 + 0.3 \cdot R_3 = 0.4 \cdot 5.67\times10^4 + 0.7 \cdot 5.67\times10^4 + 0.3 \cdot 5.67\times10^4 $$ $$ R_{total} = (0.4+0.7+0.3) \cdot 5.67\times10^4 = 5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2} $$ So, the rate of radiation emission from the surface is \(5.67\times10^4 \mathrm{W}\cdot\mathrm{m}^{-2}\).

Key concepts

Radiation Emission

Radiation emission refers to the process by which a body emits energy in the form of electromagnetic waves or particles. This process is fundamental in heat transfer and plays a significant role in various fields such as physics, engineering, and environmental science.
In the context of thermal radiation, an object emits energy depending on its temperature and the nature of its surface. The surface's properties, including its temperature and emissivity, determine how much and in what wavelengths energy is emitted. Emissivity itself is a measure of how effectively a surface emits radiation relative to a perfect blackbody, which is an idealized surface that absorbs all incident radiation without reflecting any.

• Emissivity values range from 0 to 1, where 1 corresponds to a perfect blackbody.
• The emitted radiation transports energy, which can be absorbed by other surfaces, thus leading to heat exchange.
• The importance of radiation emission reflects in designing thermal management systems, satellite technology, and climate studies.

Understanding the mechanism behind radiation emission enables engineers and scientists to predict and manipulate heat transfer processes effectively.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a crucial principle in thermodynamics and thermal physics that relates the total energy radiated by a perfect blackbody to the fourth power of its absolute temperature. This law is given by the formula:
\[ E = \sigma T^4 \]
Where:
- \(E\) is the energy radiated per unit area per unit time (also known as the emissive power).
- \(T\) is the absolute temperature of the blackbody in Kelvin.
- \(\sigma\) is the Stefan-Boltzmann constant, approximately \(5.67 \times 10^{-8} \mathrm{Wm^{-2}K^{-4}}\).
This law provides insight into how energy emission scales with temperature. For example, doubling the absolute temperature of a body increases its emitted energy by a factor of 16 (because \(2^4 = 16\)).
Despite its simplicity, the Stefan-Boltzmann Law is fundamental in understanding radiative heat transfer and is widely applied in calculations involving heat loss from objects, as well as in astrophysical phenomena like star luminosity.

• It forms the basis of thermal radiation calculations for surfaces with known emissivity.
• Although the law is strictly valid for blackbodies, it is adapted for real surfaces using the emissivity factor.
• Real surfaces emit less radiation than blackbodies at the same temperature, modified by multiplying the blackbody radiation by the emissivity factor.

Average Emissivity

Average emissivity is a useful concept when dealing with surfaces that emit radiation across a range of wavelengths. It simplifies the analysis by providing an overall measure of the surface's effectiveness in emitting thermal radiation.
To determine average emissivity over a range, you integrate the spectral emissivity over the range of interest and then divide the result by the actual range. For non-continuous emissivity functions, as seen in the exercise, this involves breaking the spectrum into separate intervals and averaging the results.
While this approach works well in bounded wavelength ranges, caution is needed when dealing with infinite bounds as the total range integral becomes undefined, making it impossible to calculate an average emissivity directly.

• Average emissivity simplifies complex spectral variations into a single effective value.
• In practice, it is essential for thermal analysis where detailed spectral data is unavailable or unwieldy.
• It provides sanity checks for engineering applications where radiation must be estimated over broad spectral bands.

Emissivity plays a pivotal role in factors like thermal efficiency, energy loss calculations, and understanding surface characteristics of materials.