Question

The variation of the spectral transmissivity of a \(0.6\)-cm-thick glass window is as given in Fig. P12-80. Determine the average transmissivity of this window for solar radiation \((T \approx 5800 \mathrm{~K})\) and radiation coming from surfaces at room temperature \((T \approx 300 \mathrm{~K})\). Also, determine the amount of solar radiation transmitted through the window for incident solar radiation of \(650 \mathrm{~W} / \mathrm{m}^{2}\).

Short answer

Answer: The steps are: 1. Calculate the average transmissivity for solar radiation, \(T_s\), by integrating the transmissivity function over the solar spectrum and dividing by the range of the spectrum. 2. Calculate the average transmissivity for room temperature radiation, \(T_r\), by integrating the transmissivity function over the room temperature spectrum and dividing by the range of the spectrum. 3. Calculate the amount of solar radiation transmitted through the window by multiplying the incident solar radiation, \(I_{incident}\), by the average transmissivity for solar radiation, \(T_s\).

Step-by-step solution

Calculate the average transmissivity for solar radiation

To determine the average transmissivity of a glass window for solar radiation, first, we need to consider the radiation emitted by the sun at approximately \(5800 \mathrm{~K}\). Due to the variation in spectral transmissivity, we need to integrate the function describing the window's transmissivity over the entire solar spectrum and divide by the total range of the spectrum. The average transmissivity for solar radiation, \(T_s\), can be represented as: \(T_s = \cfrac{\int_{\lambda_1}^{\lambda_2} T(\lambda) d\lambda}{\lambda_2 - \lambda_1}\) where \(T(\lambda)\) is the spectral transmissivity function, and \(\lambda_1\) and \(\lambda_2\) are the limits of the solar spectrum. Note: The exercise does not provide the specific function for transmissivity or the limits of the spectrum. Usually, they can be found in a textbook or given in the problem statement.

Calculate the average transmissivity for room temperature radiation

Similarly, to find the average transmissivity of the glass window for radiation coming from surfaces at room temperature, approximately \(300 \mathrm{~K}\), we will again need to integrate the transmissivity function over the relevant range of wavelengths and divide by the total range of the spectrum. The average transmissivity for room temperature radiation, \(T_r\), can be represented as: \(T_r = \cfrac{\int_{\lambda'_1}^{\lambda'_2} T(\lambda) d\lambda}{\lambda'_2 - \lambda'_1}\) where \(\lambda'_1\) and \(\lambda'_2\) are the limits of the room temperature spectrum. Note: As mentioned in Step 1, we do not have the specific function or limits provided.

Calculate the amount of solar radiation transmitted through the window

Now that we have the average transmissivity for solar radiation, \(T_s\), we can determine the amount of solar radiation transmitted through the window by multiplying the incident solar radiation, \(I_{incident}\), by the average transmissivity: \(I_{transmitted} = I_{incident} \times T_s\) Given that the incident solar radiation is \(650 \mathrm{~W} / \mathrm{m}^{2}\): \(I_{transmitted} = 650 \mathrm{~W}/\mathrm{m}^{2} \times T_s\) Note: Without the specific average transmissivities from Steps 1 and 2, we cannot provide exact numerical solutions. Make sure to use the given transmissivity function and wavelength limits for your specific problem when following these steps to calculate the average transmissivities and transmitted radiation.

Key concepts

Spectral Transmissivity

When studying the properties of materials like glass, an important concept to understand is spectral transmissivity. This term refers to the ability of a material to transmit light at different wavelengths. To put it simply, some materials are better at letting certain colors or energy levels of light pass through them than others.

Spectral transmissivity, represented by the function T(λ), is a variable depending on the wavelength λ. Thus, different wavelengths will have different transmissivities. The behavior of light when it encounters a material can be complex; various wavelengths can be absorbed, reflected, or transmitted.

To calculate the average transmissivity across a range of wavelengths, scientists perform an integration over the specific wavelengths of interest. This calculation tells how well a material transmits light overall within the specified range. In the context of solar radiation and room temperature radiation, where the wavelengths are vastly different, average spectral transmissivity helps in designing and evaluating materials for applications such as energy-efficient windows or solar panels.

Solar Radiation

Solar radiation is the stream of energy that we receive from the sun—ranging from ultraviolet, through the visible spectrum, to infrared. Understanding solar radiation is crucial because it significantly influences many aspects of our lives, from the weather we experience to the generation of solar power.

The sun is a relatively hot body, with a surface temperature of around 5800 K. This means that it emits a broad spectrum of wavelengths, predominantly in the visible part of the spectrum, which is vividly experienced as daylight. When discussing the transmissivity of materials with respect to solar radiation, we're essentially asking: How much of this sunlight can pass through a given material like glass?

For structures like buildings and vehicles, materials that transmit solar radiation effectively can help with lighting and heating. However, too much transmissivity could lead to overheating, while too little could result in inadequate natural light. Architects and engineers need to calculate the average transmissivity carefully to strike the right balance.

Radiation Heat Transfer

Radiation heat transfer is one of the three fundamental methods of heat movement, alongside conduction and convection. It refers to the transfer of energy by electromagnetic radiation, which does not require any medium to travel through, allowing heat to even move through a vacuum, such as the space between the sun and Earth.

Every object that has a temperature above absolute zero emits radiation. It's important to note that the amount and wavelength of radiation emitted from a surface increases with temperature. Objects at room temperature emit primarily infrared radiation, which we perceive as heat rather than light. The principles governing spectral transmissivity are critical for understanding radiation heat transfer as well, as materials interact with this form of energy by absorbing, reflecting, or transmitting it.

When analyzing heat transfer for materials like glass, it's important for students to consider both solar radiation and room temperature radiation because these aspects dictate how much heat enters and leaves an environment—influencing everything from energy costs to comfort levels.