Glossary for Radiative Heat Transfer

Absorption: The process of converting radiation intercepted by matter to internal thermal energy.
Absorptivity: Fraction of the incident radiation absorbed by matter. Modifiers: directional, hemispherical, spectral, total. The spectral directional absorptivity reads

\[\alpha_{\lambda, \theta}(\lambda, \theta, \phi) \equiv \frac{I_{\lambda, i, \mathrm{abs}}(\lambda, \theta, \phi)}{I_{\lambda, i}(\lambda, \theta, \phi)}\]

For most engineering calculation, a spectral, hemispherical absorptivity \(\alpha_{\lambda}(\lambda)\) is defined as

\[\alpha_{\lambda}(\lambda) \equiv \frac{G_{\lambda, \mathrm{abs}}(\lambda)}{G_{\lambda}(\lambda)}\]

Finally The total, hemispherical absorptivity, \(\alpha,\) represents an integrated average over both direction and wavelength. It is defined as the fraction of the total irradiation absorbed by a surface

\[\alpha \equiv \frac{G_{\mathrm{abs}}}{G} = \frac{\int_{0}^{\infty} \alpha_{\lambda}(\lambda) G_{\lambda}(\lambda) d \lambda}{\int_{0}^{\infty} G_{\lambda}(\lambda) d \lambda}\]
Blackbody: The ideal emitter and absorber. Modifier referring to ideal behavior. Denoted by the subscript \(b\)
Diffuse: Modifier referring to the directional independence of the intensity associated with emitted, reflected, or incident radiation
Directional: Modifier referring to a particular direction. Denoted by the subscript \(\theta\)
Directional distribution: Refers to variation with direction.
Emission: The process of radiation production by matter at a non-zero temperature. Modifiers: diffuse, blackbody, spectral.
Emissive power: Rate of radiant energy emitted by a surface in all directions per unit area of the surface, \(E\left(\mathrm{W} / \mathrm{m}^{2}\right)\) Modifiers: spectral, total, blackbody.
Emissivity: Ratio of the radiation emitted by a surface to the radiation emitted by a blackbody at the same temperature. Modifiers: directional, hemispherical, spectral, total.
Gray surface: A surface for which the spectral absorptivity and the emissivity are independent of wavelength over the spectral regions of surface irradiation and emission.
Hemispherical: Modifier referring to all directions in the space above a surface.
Intensity: Rate of radiant energy propagation in a particular direction, per unit area normal to the direction, per unit solid angle about the direction, \(I\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr}\right)\) Modifier: spectral.
Irradiation: Rate at which radiation is incident on a surface from all directions per unit area of the surface, \(G\left(\mathrm{W} / \mathrm{m}^{2}\right)\) Modifiers: spectral, total, diffuse.
Kirchhoff’s law: Relation between emission and absorption properties for surfaces irradiated by a blackbody at the same temperature.
Planck’s law: Spectral distribution of emission from a blackbody. The blackbody spectral intensity reads

\[I_{\lambda, b}(\lambda, T)=\frac{2 h c_{o}^{2}}{\lambda^{5}\left[\exp \left(h c_{o} / \lambda k_{B} T\right)-1\right]}\]

where \(h=6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\) and \(k_{B}=1.381 \times 10^{-23} \mathrm{J} / \mathrm{K}\) are the universal Planck and Boltzmann constants, respectively, \(c_{o}=2.998 \times 10^{8} \mathrm{m} / \mathrm{s}\) is the speed of light in vacuum, and \(T\) is the absolute temperature of the blackbody (K). since the blackbody is a diffuse emitter, it follows that its spectral emissive power is

\[E_{\lambda, b}(\lambda, T)=\pi I_{\lambda, b}(\lambda, T)=\frac{C_{1}}{\lambda^{5}\left[\exp \left(C_{2} / \lambda T\right)-1\right]}\]

where the first and second radiation constants are

\[C_{1}=2 \pi h c_{o}^{2}=3.742 \times 10^{8} \mathrm{W} \cdot \mu \mathrm{m}^{4} / \mathrm{m}^{2}, \quad C_{2}=\left(h c_{o} / k_{B}\right)=1.439 \times 10^{4} \mu \mathrm{m} \cdot \mathrm{K}\]
Radiosity: Rate at which radiation leaves a surface due to emission and reflection in all directions per unit area of the surface, \(J\left(\mathrm{W} / \mathrm{m}^{2}\right)\) Modifiers spectral, total.
Reflection: The process of redirection of radiation incident on a surface. Modifiers: diffuse, specular.
Reflectivity: Fraction of the incident radiation reflected by matter. Modifiers: directional, hemispherical, spectral, total. The spectral, directional reflectivity, \(\rho_{\lambda, \theta}(\lambda, \theta, \phi),\) of a surface is defined as the fraction of the spectral intensity incident in the direction of \(\theta\) and \(\phi\) which is reflected by the surface. We have

\[\rho_{\lambda, \theta}(\lambda, \theta, \phi) \equiv \frac{I_{\lambda, i, \mathrm{ref}}(\lambda, \theta, \phi)}{I_{\lambda, i}(\lambda, \theta, \phi)}\]

The spectral, hemispherical reflectivity \(\rho_{\lambda}(\lambda)\) is then defined as the fraction of the spectral irradiation that is reflected by the surface,

\[\rho_{\lambda}(\lambda) \equiv \frac{G_{\lambda, \text { ref }}(\lambda)}{G_{\lambda}(\lambda)}\]

which is equivalent to

\[\rho_{\lambda}(\lambda)=\frac{\int_{0}^{2 \pi} \int_{0}^{\pi / 2} \rho_{\lambda, \theta}(\lambda, \theta, \phi) I_{\lambda, i}(\lambda, \theta, \phi) \cos \theta \sin \theta d \theta d \phi}{\int_{0}^{2 \pi} \int_{0}^{\pi / 2} I_{\lambda, i}(\lambda, \theta, \phi) \cos \theta \sin \theta d \theta d \phi}\]

The total, hemispherical reflectivity \(\rho\) is then defined as

\[\rho \equiv \frac{G_{\mathrm{ref}}}{G}=\frac{\int_{0}^{\infty} \rho_{\lambda}(\lambda) G_{\lambda}(\lambda) d \lambda}{\int_{0}^{\infty} G_{\lambda}(\lambda) d \lambda}\]
Semitransparent: Refers to a medium in which radiation absorption is a volumetric process.
Solid angle: Region subtended by an element of area on the surface of a sphere with respect to the center of the sphere, \(\omega(\mathrm{sr})\)
Spectral: Modifier referring to a single-wavelength (monochromatic) component. Denoted by the subscript \(\lambda\)
Spectral distribution: Refers to variation with wavelength.
Specular: Refers to a surface for which the angle of reflected radiation is equal to the angle of incident radiation.
Stefan-Boltzmann law: Emissive power of a blackbody. The total emissive power of a blackbody \(E_b\) may be expressed

\[E_{b}=\sigma T^{4}\]

where the Stefan-Boltzmann constant has the numerical value

\[\sigma=5.670 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\]
Thermal radiation: Electromagnetic energy emitted by matter at a non-zero temperature and concentrated in the spectral region from approximately 0.1 to \(100 \mu \mathrm{m}\)
Total: Modifier referring to all wavelengths.
Transmission: The process of thermal radiation passing through matter.
Transmissivity: Fraction of the incident radiation transmitted by matter. Modifiers: hemispherical, spectral, total.

\[\tau_{\lambda}=\frac{G_{\lambda, \text { tr }}(\lambda)}{G_{\lambda}(\lambda)}\]

and integrated over the wavelength

\[\tau=\frac{G_{\mathrm{tr}}}{G}\]

The total transmissivity \(\tau\) is related to the spectral component \(\tau_{\lambda}\) by

\[\tau=\frac{\int_{0}^{\infty} G_{\lambda, \mathrm{tr}}(\lambda) d \lambda}{\int_{0}^{\infty} G_{\lambda}(\lambda) d \lambda}=\frac{\int_{0}^{\infty} \tau_{\lambda}(\lambda) G_{\lambda}(\lambda) d \lambda}{\int_{0}^{\infty} G_{\lambda}(\lambda) d \lambda}\]
Wien’s displacement law: Locus of the wavelength corresponding to peak emission by a blackbody.