Question

The spectral transmissivity of a glass cover used in a solar collector is given as Solar radiation is incident at a rate of \(950 \mathrm{~W} / \mathrm{m}^{2}\), and the absorber plate, which can be considered to be black, is maintained at \(340 \mathrm{~K}\) by the cooling water. Determine \((a)\) the solar flux incident on the absorber plate, \((b)\) the transmissivity of the glass cover for radiation emitted by the absorber plate, and (c) the rate of heat transfer to the cooling water if the glass cover temperature is also \(340 \mathrm{~K}\).

Short answer

Question: Calculate the following for a solar collector: (a) the solar flux incident on the absorber plate, (b) the transmissivity of the glass cover for radiation emitted by the absorber plate, and (c) the rate of heat transfer to the cooling water. Answer: (a) The solar flux incident on the absorber plate is \(950 \mathrm{~W} / \mathrm{m}^{2}\). (b) The transmissivity of the glass cover for radiation emitted by the absorber plate can be calculated using Planck's law and the given spectral transmissivity formula. (c) The rate of heat transfer to the cooling water can be calculated using the Stefan-Boltzmann law and the average transmissivity found in part (b).

Step-by-step solution

(a) Calculate the solar flux incident on the absorber plate

First, we need to determine the solar flux incident on the absorber plate. We know that the solar radiation is incident at a rate of \(950 \mathrm{~W} / \mathrm{m}^{2}\). Since there is no information given about losses, we can assume that there is no loss in the absorber plate, and thus the total incident radiation is equal to the flux on the absorber plate. So the solar flux incident on the absorber plate is \(950 \mathrm{~W} / \mathrm{m}^{2}\).

(b) Calculate the transmissivity of the glass cover for radiation emitted by the absorber plate

To find the transmissivity of the glass cover for radiation emitted by the absorber plate, we will use Planck's law. The law states that the spectral distribution of blackbody radiation is given by: \(B_{\lambda}(T) = \frac{2 \pi h c^{2}}{\lambda^{5}} \frac{1}{e^{\frac{hc}{\lambda k T}} - 1}\) where \(B_{\lambda}(T)\) is the spectral radiance, \(h\) is the Planck's constant, \(c\) is the speed of light, \(\lambda\) is the wavelength, \(k\) is the Boltzmann constant, and \(T\) is the temperature. However, before we use Planck's law, we must determine the average transmissivity \((\bar{\tau})\) of the glass cover for the wavelengths of interest. Using the given spectral transmissivity formula, we can integrate over the wavelengths of interest: \(\bar{\tau} = \frac{\int \tau_{\lambda} B_{\lambda}(T) d\lambda}{\int B_{\lambda}(T) d\lambda}\) Given that the absorber plate is black and maintained at \(340 \mathrm{~K}\), we can find the transmissivity using this equation. After calculating the average transmissivity, we can apply it to the radiative heat transfer between the absorber plate and glass cover.

(c) Calculate the rate of heat transfer to the cooling water

With the average transmissivity from the previous step, we can now calculate the rate of heat transfer to the cooling water. Since the glass cover temperature is also \(340 \mathrm{~K}\), we can say that the heat transfer is due only to the radiation. We can calculate the net radiative heat transfer rate between the black absorber plate and the glass cover as follows: \(\dot{Q} = A (\sigma T^{4} - \sigma \bar{\tau} T_{glass}^{4})\) where \(\dot{Q}\) is the net radiative heat transfer rate, \(A\) is the area of the cover, \(\sigma\) is the Stefan-Boltzmann constant, \(T\) is the temperature of the absorber plate, and \(T_{glass}\) is the temperature of the glass cover. After we find the net radiative heat transfer rate, we can assume that this heat is transferred to the cooling water, so the rate of heat transfer to the cooling water is equal to \(\dot{Q}\).

Key concepts

Spectral Transmissivity

Spectral transmissivity describes how certain materials, like the glass cover in a solar collector, allow radiation to pass through them at different wavelengths. In simple terms, it is a measure of how transparent a material is to different colors of light. When dealing with solar collectors, spectral transmissivity is crucial because it helps in understanding how much of the sunlight can actually reach the absorber plate beneath the glass.

The transmissivity depends on the wavelength of the radiation and can vary significantly. For example, a glass cover may allow most visible light to pass through easily, but might reflect or absorb other wavelengths, such as infrared. This characteristic of materials is essential for designing and evaluating the performance of solar collectors.

In practical terms, to determine how effective a glass cover is at allowing sunlight to pass through, one might need to calculate an average transmissivity for the entire spectrum of radiation that is of interest, usually using an integration method. This average value helps engineers and scientists decide if the material is suitable for a specific solar application.

Solar Radiation

Solar radiation is the energy emitted from the sun in the form of electromagnetic waves. It includes a range of wavelengths, mainly encompassing ultraviolet, visible, and infrared light. When solar radiation reaches the Earth's surface, the average intensity is about 1,360 watts per square meter, although at a specific location, like in the problem here, it might be given as 950 watts per square meter.

This incident solar radiation is vital for the operation of a solar collector, which is used to capture and convert sunlight into usable energy, such as heat. The amount of solar radiation that actually reaches the absorber plate of a solar collector depends on various factors, such as the angle of the sun, weather conditions, and of course, the spectral transmissivity of any covering material like glass.

Knowing the rate of solar radiation incident on the collector provides critical input for calculating the energy conversion and efficiency potential of solar devices. Optimizing the exposure of solar collectors to maximum radiation can significantly improve their efficiency in harvesting solar energy.

Blackbody Radiation

Blackbody radiation refers to the perfect emission of thermal radiation by an object. An ideal blackbody is a theoretical construct that absorbs all incident radiation, regardless of wavelength or angle, and radiates energy at a maximum rate for a given temperature.

In the context of the exercise, the absorber plate is considered black, meaning it can be approximated as a blackbody. This assumption helps simplify calculations related to thermal radiation, allowing us to use Planck's law to determine the spectral radiance at a given temperature. In this case, the temperature is set at 340 K.

Blackbody concepts help in understanding radiative heat transfer. Since the absorber is emitting radiation due to its temperature, the amount of energy emitted can be relevant in calculating heat exchange processes, particularly when designing systems to transfer this energy efficiently to other components, like the cooling water mentioned in the problem.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a principle in physics that describes the power radiated from a blackbody in terms of its temperature. It is vital for understanding energy transfer in thermal systems. The formula expressed in the law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of the blackbody's absolute temperature:\[ \dot{Q} = \sigma T^4 \]where \(\dot{Q}\) is the energy radiated per unit time, \(\sigma\) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} \, \mathrm{W/m^2 \, K^4}\)), and \(T\) is the absolute temperature in Kelvin.

In this exercise, the Stefan-Boltzmann Law helps calculate the radiation emitted or absorbed by the absorber plate and glass cover, both at 340 K. Understanding how much radiation is emitted aids in determining the net rate of heat transfer, which is critical for evaluating the system's energy efficiency.

This law is particularly useful in predicting the behavior of thermal systems, from small devices like the solar collector described here, to large natural bodies like stars.