6.4 Radiation

Heat transfer by radiation is a crucial mode of energy transfer, especially at high temperatures. This section explores the fundamentals of thermal radiation, including blackbody radiation, Planck's law, and emissivity. Understanding these concepts is key to grasping how energy moves without a medium.

Radiative heat transfer calculations are essential for engineering applications. We'll dive into the Stefan-Boltzmann law and how to calculate heat exchange between surfaces. These tools help engineers design everything from solar collectors to thermal insulation systems.

Thermal Radiation Fundamentals

Blackbody Radiation and Planck's Law

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• Thermal radiation is electromagnetic radiation emitted by matter due to its temperature and is a mode of heat transfer that does not require a medium (vacuum)
• Blackbody radiation is the theoretical maximum amount of thermal radiation that can be emitted by an object at a given temperature (ideal radiator)
• The spectral distribution of blackbody radiation varies with temperature according to Planck's law
• Planck's law describes the spectral emissive power of a blackbody as a function of wavelength and temperature

Emissivity and Surface Properties

• Emissivity is a surface property that represents the ratio of the actual thermal radiation emitted by a surface to the theoretical maximum (blackbody radiation) at the same temperature
• Emissivity values range from 0 to 1, with a perfect blackbody having an emissivity of 1 (ideal absorber and emitter)
• The emissivity of a surface depends on factors such as material (metal, ceramic, polymer), surface finish (polished, rough), and wavelength of the emitted radiation (visible, infrared)
• Real surfaces have emissivity values less than 1, and their emissivity can vary with temperature and wavelength
• Emissivity is an important factor in determining the effectiveness of a surface in emitting or absorbing thermal radiation

Radiative Heat Transfer Calculations

Stefan-Boltzmann Law

• The Stefan-Boltzmann law relates the total energy emitted by a blackbody to its absolute temperature: $E = \sigma T^4$, where $E$ is the total emissive power, $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m²·K⁴), and $T$ is the absolute temperature (K)
• The net radiative heat transfer rate between two surfaces can be calculated using the Stefan-Boltzmann law and the surface properties (emissivity, area, and temperature)
• The radiative heat transfer rate between two surfaces is proportional to the difference in the fourth power of their absolute temperatures ($Q \propto (T_1^4 - T_2^4)$)
• The Stefan-Boltzmann law is used to calculate the radiative heat transfer rate in applications such as solar collectors, thermal insulation, and heat exchangers

Radiative Heat Exchange Between Surfaces

• The net radiative heat transfer rate between two surfaces depends on their temperatures, emissivities, areas, and view factors
• The radiative heat exchange between two blackbody surfaces is given by: $Q = \sigma A_1 F_{12} (T_1^4 - T_2^4)$, where $A_1$ is the area of surface 1, $F_{12}$ is the view factor from surface 1 to surface 2, and $T_1$ and $T_2$ are the absolute temperatures of the surfaces
• For gray surfaces (constant emissivity over all wavelengths), the radiative heat exchange is modified by the emissivities of the surfaces: $Q = \sigma A_1 F_{12} (\varepsilon_1 T_1^4 - \varepsilon_2 T_2^4)$, where $\varepsilon_1$ and $\varepsilon_2$ are the emissivities of surfaces 1 and 2, respectively
• Radiative heat exchange calculations are essential for designing and analyzing systems involving high-temperature processes, such as furnaces, boilers, and combustion chambers

View Factors in Radiative Heat Transfer

Definition and Reciprocity Relation

• View factors (also known as shape factors or configuration factors) represent the fraction of radiation leaving one surface that directly reaches another surface
• View factors depend on the geometry and orientation of the surfaces involved in the radiative heat transfer (parallel plates, perpendicular plates, concentric cylinders)
• The reciprocity relation states that the product of the area and view factor for two surfaces is equal: $A_1 F_{12} = A_2 F_{21}$, where $A_1$ and $A_2$ are the areas of surfaces 1 and 2, and $F_{12}$ and $F_{21}$ are the view factors from surface 1 to 2 and from surface 2 to 1, respectively
• The reciprocity relation is useful for determining view factors when one of them is known or can be easily calculated

Calculating View Factors

• View factors for common geometries, such as parallel plates, perpendicular plates, and concentric cylinders, can be found in standard heat transfer references or calculated using integral expressions
• For example, the view factor between two parallel plates of equal size separated by a distance $L$ is given by: $F_{12} = \frac{1}{\pi} \left[ \sqrt{1 + \left(\frac{W}{L}\right)^2} - \frac{W}{L} \right]$, where $W$ is the width of the plates
• View factors for complex geometries can be determined using numerical methods, such as the double area integration method or the Monte Carlo method
• Accurate view factor calculations are crucial for predicting the radiative heat transfer between surfaces in various applications, such as thermal insulation, solar energy systems, and spacecraft thermal control

Surface Properties and Radiative Exchange

Kirchhoff's Law and Surface Properties

• Surface properties, such as emissivity, absorptivity, and reflectivity, significantly influence radiative heat exchange between surfaces
• Kirchhoff's law states that, for a given surface at a given temperature and wavelength, the emissivity is equal to the absorptivity: $\varepsilon = \alpha$
• Surfaces with high emissivity and absorptivity (close to 1) are good radiators and absorbers (black surfaces), while surfaces with low emissivity and absorptivity (close to 0) are poor radiators and absorbers (white or reflective surfaces)
• Reflectivity is the fraction of incident radiation that is reflected by a surface, and it is related to emissivity and absorptivity by: $\rho = 1 - \varepsilon - \alpha$
• The sum of emissivity, absorptivity, and reflectivity for a given surface is equal to 1: $\varepsilon + \alpha + \rho = 1$

Selective Surfaces and Applications

• Selective surfaces, which have high emissivity or absorptivity in specific wavelength ranges, can be used to control radiative heat exchange in applications such as solar collectors and thermal insulation
• Solar selective surfaces have high absorptivity in the visible and near-infrared wavelengths (solar spectrum) and low emissivity in the mid- and far-infrared wavelengths (thermal radiation spectrum), maximizing solar energy absorption while minimizing thermal losses
• Thermal insulation materials, such as low-emissivity coatings and reflective foils, have low emissivity in the infrared wavelengths, reducing radiative heat transfer and improving insulation effectiveness
• Selective surfaces are also used in thermophotovoltaic systems, where a high-temperature emitter with a selective emission spectrum is used to generate electricity via photovoltaic cells
• Understanding and manipulating surface properties is essential for optimizing radiative heat transfer in various engineering applications, from energy conservation to aerospace thermal management