All Macroscopic Objects Emit a Continuous Spectrum of Radiation With a Peak at a Wavelength That

Thermal Radiation

The absorption of thermal radiation alters the temperature of the material of the detector, which can be manifested as a change in thermistor resistance, the development of an emf as thermocouple, or a change in the dipole moment of a ferroelectric crystal as the pyroelectric detector.

From: Principles of Measurement and Transduction of Biomedical Variables , 2015

Electromagnetic Methods

Xavier Dérobert , ... Jean Dumoulin , in Non-Destructive Testing and Evaluation of Civil Engineering Structures, 2018

3.3.4.1 Introduction

Thermal radiation is one of three mechanisms which enables bodies with varying temperatures to exchange energy. Thermal radiation is characterized by the emission of electromagnetic waves from the material (variation of its internal energy). Depending on the temperature of the material, it transmits radiation ranging from ultraviolet to far-field infrared. The entire body acts as an emission source of continuous thermal radiation, and also as a continuous receiver of radiation originating even from far-field bodies. However, thermal radiation is linked to the molecular structure of the transmitter, receiver, and the crossed medium. Surface radiation of a body is also linked to its capacity to transmit and store heat (specific temperatures). Further information on this can be found in specialized literature, such as [SIE 02].

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Thermal modeling, analysis, and design

Hengyun Zhang , ... Wensheng Zhao , in Modeling, Analysis, Design, and Tests for Electronics Packaging beyond Moore, 2020

3.1.1.4 Thermal radiation

Thermal radiation is energy emitted by matter that is at a nonzero temperature, transported in the form of electromagnetic waves (or alternatively, photons). The thermal radiation power from surface 1 to surface 2, as shown in Figure 3.1.4, is expressed by

Figure 3.1.4. Heat transfer from a surface at temperature T _s1 to a hemisphere ambient at temperature T _s2 by thermal radiation.

(3.1.5) $q=\epsilon \sigma A{F}_{12}\left(T_{s1}^4-T_{s2}^4\right)$

where T _s1 and T _s2 are the absolute temperature (K) of the surfaces 1 and 2, respectively, and σ is the Stefan–Boltzmann constant, σ  =   5.67   ×   10⁸  W   m⁻² K⁻⁴, ε is the surface emissivity of the surface 1, and F ₁₂ is the view factor between surfaces 1 and 2. A surface with ε  =   1 is called the blackbody, an ideal radiator. With ε values in the range 0–1, this property provides a measure of efficiency for a surface emitting energy relative to the blackbody.

The thermal radiation may be neglected in the presence of forced convection in electronic cooling. Nonetheless, it may be significant when the natural convection is the major heat transfer mode.

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High-performance sportswear

R.M. Rossi , in High-Performance Apparel, 2018

15.2.2.3 Optimization of radiant heat exchange

Thermal radiation is emitted by all matter with temperatures above the absolute zero and is transferred in the form of electromagnetic waves. The thermal radiation of an object is dependent on its emissivity. As clothing usually has a relatively high emissivity (0.7–0.95), it exchanges large amounts of heat through radiation. Therefore, different studies have proposed strategies to lower the emissivity, from metallized interlayers ( Sun, Fan, Wu, Wu, & Wan, 2013; Wang & Fan, 2014; Morrissey & Rossi, 2015) to the use of metallic nanowire-coated textiles (Hsu et al., 2015). Apart from metallic coatings, there exist a finishing technology called Coldblack developed by Clariant and Schoeller, which acts as shield for infrared radiation, especially for textiles in darker colors (McCann, 2013).

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Biomedical Transport Processes

Gerald E. Miller PhD , in Introduction to Biomedical Engineering (Third Edition), 2012

14.3.5 Thermal Radiation

Thermal radiation is electromagnetic radiation emitted from a material that is due to the heat of the material, the characteristics of which depend on its temperature. An example of thermal radiation is the infrared radiation emitted by a common household radiator or electric heater. A person near a raging bonfire will feel the radiated heat of the fire, even if the surrounding air is very cold. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is converted to electromagnetic radiation. Sunshine, or solar radiation, is thermal radiation from the extremely hot gasses of the sun, and this radiation heats the earth. The earth also emits thermal radiation but at a much lower intensity because it is cooler. The balance between heating by incoming solar thermal radiation and cooling by the earth's outgoing thermal radiation is the primary process that determines the earth's overall temperature. As such, radiation is the only form of heat transfer that does not require a material to transmit the heat. Radiative heat is transferred from surface to surface, with little heat absorbed between surfaces. However, the surfaces, once heated, can release the heat via conduction or convection to the surroundings.

Thermal radiation is conducted via electromagnetic waves. As such, this form of heat transfer is not only a function of the temperature difference between the two surfaces but also the frequency range of the emitted and received energy. As an example, sunlight is composed of the visible light spectrum as well as infrared energy and ultraviolet energy. Figure 14.47 depicts the effects of temperature and wavelength of the thermal energy on the heat transfer rate.

Figure 14.47. Peak wavelength and total radiated amount vary with temperature. Although the plots show relatively high temperatures, the same relationships hold true for any temperature down to absolute zero. Visible light is between 380 and 750   nm.

When radiant energy reaches a surface, the energy can be absorbed, transmitted (through), or reflected (or any combination). The sum of these three effects equals the total energy transmitted, and the parameters that describe these three phenomena are given by

$\alpha +\rho +\tau =1,$

where α represents spectral absorption factor, ρ represents the spectral reflection factor, and τ represents the spectral transmission factor.

The radiative heat transfer rate is given by the Stefan-Boltzmann law

${\text{Q}}_{\text{r}}=\mathrm{\sigma }{\text{A}\text{T}}^4$

where σ is the Boltzmann constant and A is the surface area of the radiating source. The temperature is in an absolute scale (°Kelvin, corresponding to °C, or °Rankin, corresponding to °F).

To predict the exact amount of radiative heat transfer between two surfaces, the preceding equation is expanded as

${\text{Q}}_{\text{r}}=\sigma {\text{F}}_1{\text{A}}_1({{\text{T}}_1}^4-{{\text{T}}_2}^4)$

where F is the facing factor that represents the amount of the emitting surface (1) facing the receiving surface (2) with the surface area A representing surface 1. Correspondingly, this equation can use F₂ and A₂ to represent the facing factor for the receiving surface toward the emitting surface with the surface area of 2. Boltzmann's constant and the temperature gradient are unchanged for either form of the equation. The facing factor can be approximated as a disk of radius R if the distance between the two surfaces is large, such as the earth to the sun. For shorter distances, the facing factor is a complex interaction between the angles of the two surfaces that face each other.

As can be seen, thermal radiation is affected by the frequency of the emitted energy. This is why sunscreen ointments have ultraviolet protection, since this type of energy can be damaging to skin. In addition, it is common to feel warmer on the sunny side of the street as opposed to the shady side, given the radiative heat transfer. Radiation can be a significant source of heat as compared to the other forms (conduction and convection) because radiation is composed primarily of sunlight and the respective heating of the earth.

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Thermal radiation heat transfer

José Meseguer , ... Angel Sanz-Andrés , in Spacecraft Thermal Control, 2012

5.1 Nature of thermal radiation

Thermal radiation is electromagnetic radiation emitted from all matter that is at a non-zero temperature in the wavelength range from 0.1  μm to 100   μm. It includes part of the ultraviolet (UV), and all of the visible and infrared (IR). It is called thermal radiation because it is caused by and affects the thermal state of matter. Figure 5.1 shows the regions of the electromagnetic spectrum with the thermal radiation range indicated on it. The spectrum of the solar irradiation can be found in Figure 2.2.

Figure 5.1. Electromagnetic spectrum classification according to radiation wavelength λ, showing the wavelength range corresponding to thermal radiation.

Key: GR, gamma rays; XR, X-rays; UV, ultraviolet; VI, visible; IR, infrared; TR, thermal radiation; MW, microwaves.

Thermal radiation does not require a material medium for its propagation. Although in the context of spacecraft thermal design the interest of radiation is mainly focused on solid surfaces, emission may also occur from liquids and gases. The mechanism of radiation emission is related to energy released as a result of oscillations or transitions of the electrons that constitute matter. These oscillations are sustained by internal energy, and therefore, the temperature of the matter.

All forms of matter emit radiation as they are at a nonzero temperature. For gases and semi-transparent matter, thermal radiation is a volumetric phenomenon. This can be of interest when studying the behaviour of lenses, for instance, as part of optical devices.

Since thermal radiation is electromagnetic radiation, the properties of the propagation of electromagnetic waves can be applied. The most relevant ones are the frequency v and the wavelength λ, which are related through λ  = c/v, where c is the speed of light in the medium. In the case of propagation in a vacuum, c  = c _o  =   2.998   ×   10⁸  m/s.

The spectral nature of thermal radiation is one of two features that make its study quite complex. The second feature is related to its directionality. A surface may have certain directions with preferential emission; therefore the distribution of the emitted radiation is directional. When the radiative properties do not depend on the direction, the surface is termed diffuse.

As already said, all surfaces emit thermal radiation. This emitted radiation will strike other surfaces and will be partially reflected, partially absorbed, and partially transmitted. Figure 5.2 shows the different thermal radiation interactions on a body's surface. The symbol Φ in the figure stands for the radiant energy per unit time, measured in W in the SI system. As can be seen, the surface emits Φ _e , receives the incident radiation Φ _i , out of which Φ _a , is absorbed, Φ _r is reflected and Φ _t is transmitted.

Figure 5.2. Thermal radiation interactions on a surface.

Key: Φ _e , emitted radiation; Φ _i , incident radiation; Φ _a , absorbed radiation; Φ _r , reflected radiation; and Φ _t , transmitted radiation.

The intensity of emitted radiation, I_λ,e, is defined as the rate at which radiant energy, $\delta \dot{Q}$ , is emitted at the wavelength λ in the (θ,ϕ) direction, per unit area of the emitting surface normal to this direction, per unit solid angle dω about this direction, and per unit wavelength interval dλ about λ, as indicated in Figure 5.3. Thus, the spectral intensity is

Figure 5.3. Intensity of emitted radiation, I_λ,e , in the (θ, ϕ) direction. dA is the emitting differential area (contained in the xy plane), and dω the solid angle unit about this direction.

(5.1) $I_{\lambda ,e}\left(\lambda ,\theta ,\varphi ,T\right)=\frac{\delta \dot{Q}}{\text{d}Acos\theta \text{d}\omega \text{d}\lambda }.$

In order to obtain the thermal interactions in all directions and wavelengths the intensity of radiation is successively integrated. Thus, the spectral hemispherical emissive power E _λ, measured in W/(m²  ·   μm) in the SI, is the rate at which radiation of wavelength λ is emitted in all directions from surface per unit wavelength interval dλ about λ and per unit surface area. It has the form

(5.2) $E_{\lambda }\left(\lambda ,T\right)={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{\pi /2}{I}_{\lambda ,e}}}\left(\lambda ,\theta ,\varphi ,T\right)cos\theta sin\theta \text{d}\theta \text{d}\varphi ,$

where the solid angle dω has been written as dω  =   sinθdθφ, according to the spherical coordinates defined in Figure 5.3.

Finally, integrating equation (5.2) over all wavelengths, the total emissive power, E, measured in W/m² in the SI system, is obtained as

(5.3) $\begin{array}{ll}E\left(T\right)\hfill & ={\displaystyle {\int }_0^{\infty }{E}_{\lambda }}\left(\lambda ,T\right)\text{d}\lambda \hfill \\ \hfill & ={\displaystyle {\int }_0^{\infty }{\displaystyle {\int }_0^{2\pi }}}{\displaystyle {\int }_0^{\pi /2}{I}_{\lambda ,e}\left(\lambda ,\theta ,\varphi ,T\right)cos\theta sin\theta \text{d}\theta \text{d}\mathit{\varphi d\lambda }}.\hfill \end{array}$

The previous definitions, given by equations (5.1), (5.2) and (5.3), refer to the radiation emitted by a surface. Analogous definitions and mathematical expressions can be established for the incident radiation on a surface, called irradiation G, and for all the radiation leaving a surface (sum of the reflected radiation and the emitted radiation), called radiosity J. Both can be defined at spectral and directional levels, at spectral hemispherical level and as total magnitudes integrated over all directions and all wavelengths.

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Heat transfer in nuclear thermal hydraulics

P.L. Kirillov , H. Ninokata , in Thermal-Hydraulics of Water Cooled Nuclear Reactors, 2017

7.3.1 Basic principles

Thermal radiation is a portion of the process of energy propagation by means of electromagnetic waves. When it is in thermodynamic equilibrium with a substance, we call it an equilibrium radiation. The radiation results from changes in electronic, vibrational, and rotational states of the atoms-molecules and the emission of the radiant energy takes place as a result of irregular deceleration of charged particles (electrons, ions) in the media. The reverse process is then what occurs when radiant energy impinges on a solid surface and causes its temperature rise (absorption).

From a corpuscular point of view, the radiant energy is emitted and absorbed in discrete forms of the electromagnetic waves, which could be described as if particles with mass zero and charge zero, i.e., photons are carrying energy with the speed of light c. The photon energy is hν, where $h=6.626\times {10}^{-34}\mathrm{J}\mathrm{s}$ is the Planck constant, ν is the frequency of an electromagnetic wave. Note that the photon has momentum hν/c. The wavelength λ, characterizing an electromagnetic wave, is related to its frequency ν by the equation $c=\nu \lambda$ . The speed of light $c=c_0$ (=   299,792,458   m/s exactly) in vacuum, $c\approx c_0$ in gases, and in other media $c<c_0$ . The ratio $n=c/c_0$ is refractive index. The frequency does not change when the electromagnetic wave passes through from one medium to another, but the wavelength because the light speed changes. Distribution of radiation energy per frequency (or wavelength) is radiation spectrum.

The spectrum of equilibrium radiation is independent of the nature of the substance and determined by Planck's law of radiation. Thermal radiation field in the electromagnetic spectrum covers a range of wavelengths   ~   10^{−   7}–10^{−   4}  m (see Fig. 7.30), i.e., in the visible and infrared regions (Hackford, 1960). The visible part of the spectrum covers a wavelength of 0.40–0.76   μm. The infrared region of the spectrum consists of a near-infrared region (0.76–25   μm) and the far infrared region (25–1000   μm). A main portion of the thermal radiation energy falls on the wavelengths region of 0.76–15   μm, i.e., lies in the near infrared. Radiation in the visible region of the spectrum is significant only at very high temperatures.

Fig. 7.30. Portion of the thermal radiation in the electromagnetic spectrum.

Thermal radiation can be viewed as a surface phenomenon because thin surface layers (−   0.001 to 1   mm) on the body are involved in the radiative heat transfer.

Bodies can radiate energy of all wavelengths $\lambda =0–\infty$ (a continuous spectrum) or selectively to a specific wavelength range (selective spectrum). With a change in temperature, not only the intensity but also structure of the radiation (spectrum) varies. There are monochromatic and full (integral) radiation modes. Monochromatic radiation corresponds to that of a narrow wavelength range $\lambda \sim \left(\lambda +d\lambda \right)$ .

Integral radiation (full spectrum) is associated with the total radiant energy emitted from a body Q (W); in the entire wavelength range 0   < λ  <   ∞, the integral flux density of the (hemispherical) radiation, i.e., the radiation flux emitted from the unit area per unit time we call emittance (or emissive power) $E=dQ/dA$ (W/m²). Thus $Q={\displaystyle {\int }_{A}EdA}$ .

Emittance in an infinitesimal range of wavelengths divided by this interval is called the spectral density of radiation flux (spectral or monochromatic emittance): $J_{\lambda }=dE/d\lambda$ (W/m³).

The amount of energy, emitted per unit time in the angular direction ψ the elementary area dA, per unit solid angle ω and per unit area of the projection onto a plane perpendicular to the direction of radiation, is called the brightness (intensity) of radiation:

(7.69) $I=\frac{d^{2}Q}{d\omega d{A}_n}=\frac{d}{d\omega }\left(\frac{1}{cos\psi }\frac{dQ}{dA}\right)=\frac{d{E}_n}{d\omega },\mathrm{with}d{A}_n\text{≡}dAcos\psi \mathrm{and}{E}_n=\frac{dQ}{d{A}_n}\text{.}$

$\frac{dI}{d\lambda }=I_{\lambda }$ is called the spectral brightness.

Thermal radiation energy impinging on the body can be absorbed, reflected, or can pass through the body:

(7.70) $Q_A+Q_R+Q_D=Q,\mathrm{thus}\frac{Q_A}{Q}+\frac{Q_R}{Q}+\frac{Q_D}{Q}=1\mathrm{or}A+R+D=1\text{,}$

where A, R, and D correspond to the absorbance, reflectance, and transmittance (transparency) of the material body. Here note that A is a fraction of absorption and should be distinguished from the area A in m². Particular cases of this equation leads us to concepts of ideal bodies, i.e.,

A  =   1; R   =   D  =   0—absolutely blackbody;

D  =   1; A   =   R  =   0—completely transparent;

R  =   1; A   =   D  =   0—absolutely mirror.

Dry air, mono- and di-atomic gases at temperatures below 3000   K can be regarded as transparent (diathermic).

Most of the solids are opaque, so that D   =  0 and A   +   R  =   1. In this case, the reflectivity and absorptivity of the body are interconnected. A blackbody does not exist, and usually A   <  1 (gray body). In general, the absorbance depends on the wavelength. Such substances show selective absorption (see Fig. 7.31). Classification of different types of radiation is shown in Fig. 7.32. There, E is the own radiation; E_inc is the incident of radiation on the body; E_abs  =   AE_inc is the absorption of radiation; E_ref  =  (1   − A)E_inc is the reflected radiation. A sum of the own and reflected radiations is called the effective radiation E_eff  =   E   +   E_ref  =   E   +(1   − A)E_inc . It is noted that E_abs , E_ref , and E_eff are linear functions of E_inc . Properties (spectral content) of their own and the reflected radiations may be different.

Fig. 7.31. Absorption of different material bodies: — ideal blackbody; - - - gray body; -·-·-· selective absorption body.

Fig. 7.32. Classification of different types of radiation; I and II are the layers infinitesimally close to the surface.

Thermal radiation flux can be given from the heat balance. For the plane I-I (Fig. 7.32), we have:

(7.71) $q_{res}=E-E_{\mathit{abs}}=E-A\cdot E_{inc}\text{,}$

For the plane II-II in Fig. 7.32:

(7.72) $q_{res}=E_{\mathit{eff}}-E_{inc}\text{,}$

By eliminating E_inc from these equations yields:

$E_{\mathit{eff}}=q_{res}\left(1-\frac{1}{A}\right)+\frac{E}{A}$

or

(7.73) $q_{res}=\frac{E}{1-A}-\frac{A}{1-A}{E}_{\mathit{eff}}\text{.}$

More discussions on radiation heat transfer are provided in many monographs and text books, e.g., by Adrianov (1972), Blokh (1962), Cess (1964), Klimenko and Zorin (2001), Spalding and Taborek (1983), and Sparrow and Cess (1978).

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Heat and mass transport processes in building materials

M.R. Hall , D. Allinson , in Materials for Energy Efficiency and Thermal Comfort in Buildings, 2010

1.2.5 Thermal radiation

Thermal radiation is electromagnetic radiation that is emitted by a body as a result of its temperature. All objects with a temperature above absolute zero emit thermal radiation in a spectrum of wavelengths. The amount of radiation emitted by a black body at any one wavelength is described by the spectral black body emissive power distribution or Planck's Law, which may be written as:

1.28 ${\text{E}}_{\text{b}\lambda }=\frac{2\pi {\text{hc}}^2}{{10}^6{\lambda }^5\left(exp\left(\text{hc}/\lambda \text{K}T\right)-1\right)}$

Plotting the spectral emissive power for a black body against wavelength for a number of temperatures produces a series of curves, known as Planck's curves, as shown in Fig. 1.5.

1.5. Planck's isothermal curves for spectral emissive power vs. wavelength.

The total emissive power can be found by integrating Planck's law from λ  =   0 to λ  =   ∞ (which gives us the area under the curve for a particular temperature) and is known as the Stefan–Boltzmann law:

1.29 ${\displaystyle {\int }_0^{\infty }{\text{E}}_{\mathrm{b\lambda }}}d\mathrm{\lambda }={\text{E}}_{\text{b}}=\mathrm{\sigma }{T}^4$

where σ is the Stefan–Boltzmann constant (σ   =   5.669   ×   10^{−   8}  W/m²  K⁴). The ratio of the emissive power, E, of a surface to the emissive power of a blackbody, E_b, at the same temperature is known as the emissivity, ε (i.e. ε   =   E/E_b), therefore:

1.30 $\text{E}=\mathrm{\epsilon \sigma }{T}^4$

A black body is defined as a body that absorbs all incident radiation at any given temperature and wavelength. Real surfaces, however, absorb and reflect thermal radiation and may also transmit thermal radiation, as shown in Fig. 1.6, and this behaviour can vary with temperature and wavelength.

1.6. Absorption, reflection and transmission of incident thermal radiation by a material.

Kirchoff's law tells us that the amount of radiative energy emitted by a surface must equal the amount of radiative energy absorbed by that surface. The material properties of interest are therefore the absorbtivity (α) or emissivity (ε), reflectivity (ρ) and transmissivity (τ), which describe the fractions of the incident radiation that are absorbed, reflected and transmitted such that α   +   ρ   +   τ   =   1 and ε   =   α.

The net radiant heat transfer between two surfaces is dependent on their temperatures, sizes and view factors. Radiation view factors describe the fraction of the surface area of the hemispherical view from a surface that comprises the other surface. Techniques for determining view factors include mathematical, reference tables, ray tracing and fish eye lens photography and other methods that can be applied to urban areas (Grimmond et al., 2001). An example of their use would be for the long wave radiative heat exchange between a surface and the external environment. This could be divided into that between the surface and the ground, F_gnd , the surface and background objects, F_bg , and the surface and the sky, F_sky . Assuming temperature could be assigned to each of these: T_gnd , T_bg and T_sky , according to Mcclellan and Pedersen (1997) the radiative heat exchange at the surface can be simplified to:

1.31 $\begin{array}{lll}q\hfill & =\hfill & \epsilon \sigma (F_{\mathit{gnd}}\left(T_{\mathit{gnd}}^4-T_{\mathit{surface}}^4\right)+F_{\mathit{bg}}(T_{\mathit{bg}}^4-T_{\mathit{surface}}^4)\hfill \\ \hfill & \hfill & +F_{\mathit{sky}}\left(T_{\mathit{sky}}^4-T_{\mathit{surface}}^4\right))\hfill \end{array}$

where F_gnd   + F_bg   + F_sky   =   1. The determination of the ground, background and sky temperatures must also be considered. For inside surfaces such as the walls, floor and ceiling of a room, other methods have been developed such as the one described by Liesen and Pedersen (1997).

Radiation that originates from the sun, which has a black body temperature of around 6000   K, has a much shorter wavelength than that from objects at typical terrestrial temperature and for this reason is termed short wave (SW) radiation. This is a useful distinction as a material's wavelength dependent behaviour to thermal radiation can be split into LW and SW for convenience. A commonly used example is snow, which has a very low emissivity in the short wave (highly reflective) and a very high emissivity in the long wave (highly absorbent). The gas molecules and solid particulates that make up the earth's atmosphere absorb, reflect and scatter solar radiation; the intensity at the surface is therefore dependent on the sun's relative position and the atmospheric conditions. Cloud cover is an obvious example of this dependence. The solar radiation that is incident on a surface can be divided into that direct from the sun, diffuse radiation from the sky and reflected radiation from the ground and other surfaces. When the surface is transparent, such as glass, a portion of this radiation will be transmitted into the room, a portion reflected away from the surface and a portion absorbed by the glass. Different methods for calculating these effects have been developed (see, for example, CIBSE, 2006 and McClellan and Pedersen, 1997).

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Heat Transfer

Fabio Gori , in Encyclopedia of Energy, 2004

2.2 Thermal Radiation

Thermal radiation is energy transfer in the form of electromagnetic waves. The microscopic mechanism can be related to the energy transport by photons released from molecules and atoms. The physical parameters that describe thermal radiation are the photon or wave velocity, c, the wavelength, λ, and the frequency, ν. The photon energy is given by the relation

(6) $E=hv,$

where h is the Planck constant, h=6.6256×10⁻³⁴ Js. Frequency and wavelength are related by

(7) $\lambda v=c.$

The thermal radiation velocity in vacuum is equal to c ₀=2.997925×10⁸ m/s. The wave velocity c in a medium is connected to c ₀ by the relation

(8) $n_0=c_0/c,$

where n ₀ is the refraction index of the medium traveled by the electromagnetic waves. Electromagnetic waves are classified by the wavelength λ, which is usually measured on the scale of 10⁻⁶ m=1 μm. Visible light is in the range 0.4–0.7 μm. The wavelengths generated by heated bodies are in the range 0.3–10 μm and are the thermal radiations of common interest. Radiation with a wavelength larger than the visible one is called infrared, whereas that with a wavelength smaller than the visible is called ultraviolet. In all energy applications except cryogenics, the characteristic dimensions of the system are large compared to the wavelengths of the thermal radiation. In cryogenics, because of the low temperatures, the radiation wavelengths are large. In the following, it is assumed that the energy transfer by radiation occurs along straight lines, excluding scattering or refraction. The radiant energy flux per unit time, dΦ, can be evaluated for an area element dA along a direction with an angle β toward the surface normal. The radiant energy flux dΦ contained in a solid angle dω within the frequency range dν is given by

(9) $d\mathrm{\Phi }=K_v\cos \beta \mathit{dA\; d}\omega \mathit{dv}=K_{\lambda }\cos \beta \mathit{dA\; d}\omega d\lambda ,$

where K _ν and K _λ are the monochromatic intensities of the radiant flux. In a nonemitting and nonabsorbing medium the intensity K _ν is constant along a ray, whereas it varies if it emits or absorbs radiation, and the radiant energy flux increase d ²Φ_e, which is an order of magnitude smaller than dΦ, is

(10) $d^2{\mathrm{\Phi }}_{\mathrm{e}}=j_v\rho \mathit{dV\; d}\omega \mathit{dv},$

where j _ν is the energy emitted per unit time and unit mass into a unit solid angle and within a unit frequency range. The energy flux absorbed along a path length ds is

(11) $d^2{\Phi }_{\text{a}}={\chi }_v\rho dsd\Phi ,$

where χ _ν is the coefficient of absorption. The decrease in the radiant energy flux due to scattering is

(12) $d^2{\mathrm{\Phi }}_{\mathrm{s}}=-{\sigma }_{\mathrm{v}}\rho \mathit{ds\; d}\mathrm{\Phi },$

where σ _ν is the coefficient of scattering. Thermal radiation is present when there is local thermodynamic equilibrium in a medium. Application of the laws of thermodynamics to media in thermodynamic equilibrium allows the following conclusions:

The monochromatic intensity emitted, K _νe, is given by K _νe=j _ν /χ _ν .

A radiation beam, traveling toward the interface between two media, with an angle β toward the surface normal is partly reflected with the same angle, with a ratio given by the reflectance or reflectivity, ρ _ν, and it partly penetrates, with an angle β′ according to the law sin β′/sin β=c′/c.

The assumption of constant velocities, c′ and c, allows one to find the final relation: j _ν c ²/χ _ν =j′ _ν c′²/χ′ _ν =constant, or a universal function, independent of the medium.

The absorbance of a medium, α _ν , is the ratio between the radiant energy flux absorbed and that approaching the interface.

The intensity of the radiation emitted by a medium to another medium, i _νe, is found in relation to the previous parameters as i _νe=α _ν K _νe.

A medium with α _ν =1 is a blackbody, and no radiation impinging on that body is reflected or transmitted but is only absorbed. For a blackbody, the intensity i _νb becomes i _νb=K _νe.

One of the conclusions of the electromagnetic theory of Maxwell was that the radiation reflected from an interface between two media exerts a pressure on that surface. The radiation pressure, which the solar radiation exerts on the surface of Earth, is very small (4×10⁻⁶ N/m²). An important application of radiation pressure is associated with space flight, for which it has been proposed as a means of propelling space vehicles. The monochromatic intensity has been derived by Planck from the quantum theory as

(13) $i_{\nu \mathrm{b}}=2h{\nu }^3/\left(c_0^2\left(\exp \left(h\nu /\text{k}T\right)-1\right)\right),$

where k=1.38054×10⁻²³ J K, the Boltzmann constant is universal. From Eq. (7), written in vacuum, it is found that

(14) $i_{\nu \mathrm{b}}=2C_1/\left[{\lambda }^5\left(\exp \left(C_2/\text{k}T\right)-1\right)\right],$

where C ₁=0.59548×10⁻¹⁶ W m² and C ₂=1.43879 cm K. The maximum of i _vb occurs according to the relation

(15) ${\lambda }_{\text{max}}T=C_3,$

where C ₃=0.28978 cm K. The total heat radiated by a blackbody, into a unit solid angle, in the wavelength range from 0 to ∞ is

(16) $i_{\text{b}}={\int }_0^{\infty }{i}_{\lambda b}d\lambda =\sigma T^4/\pi .$

The emissive power e _b is found by integration over the whole hemispherical space (i.e., the solid angle ω),

(17) $e_{\mathrm{b}}=i_{\mathrm{b}}\int \cos \beta d\omega =\pi i_{\mathrm{b}}=\sigma T^4,$

where σ=5.6697×10⁻⁸ W/m² K⁴. The previous equation is a good approximation of the behavior of the radiation emitted from a blackbody into a gas.

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Target phenomena in nuclear thermal-hydraulics

N. Aksan , in Thermal-Hydraulics of Water Cooled Nuclear Reactors, 2017

6.3.12.5 Radiation HT

Thermal radiation will transfer energy both from surface to surface and from surface to the two-phase flow. Exchange of thermal radiation with the two-phase flow will mainly consist of absorption in the two-phase flow, mainly in the droplets. Due to the relatively low temperature the emission of the two-phase mixture is negligible. The radiation HT becomes important in addition to other HT modes when the structure temperature locally exceeds the saturation temperature by 200  K. These kind of high temperatures are possible only in the core with pure steam, with droplet-steam mixture or with inverted annular flow regime. Typically the net radiative heat flux streams from the highest temperature regions into colder parts, being partially absorbed in the fluid.

The radiation flux from the solid surface consists of two contributions: a radiated flux which is a function of the surface temperature to the fourth power, and a partial reflection of the incoming radiation.

Steam and liquid emit radiation in principle, but the emission is small compared to that of surfaces. Both liquid and steam absorb radiation. In practice the absorption by steam is negligible. The density of liquid water is large enough to absorb all radiation in the space between fuel rods, for instance if an inverted annular flow regime with a continuous liquid core fills the flow channel. For droplet dispersed mixture only a part of the radiation is absorbed and the rest streams to the next surface.

Before emergency cooling water enters the core during an LOCA during which the core has become uncovered, the radiation HT between structures is the most significant HT balancing the local temperature differences. The net heat flux is directed from the hottest rods to the cooler ones. In the BWR bundle the channel wall finally absorbs the heat streaming from the central parts of the fuel element. This heat flux chain between different rod and channel wall creates inside a fuel bundle a temperature profile, where the central rods are in a higher temperature than the peripheral ones. Due to its strong dependence on the fuel temperature the radiation may effectively prevent the temperature rise and prevent cladding oxidation.

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Thermal performance of transportation packages for radioactive materials

F. Wille , ... M. Feldkamp , in Safe and Secure Transport and Storage of Radioactive Materials, 2015

8.2.2 Thermal radiation

Thermal radiation is the electromagnetic radiation emitted by a body with a temperature above absolute zero. The amount of energy emitted depends on the temperature of the surface and its ability to emit energy. A surface can also absorb, transmit or reflect incident radiation where the amount of energy depends, among other factors, on the material properties of surface. The ideal radiating body is the blackbody, which is the perfect absorber of incident radiation of all wavelengths. It is also the best possible emitter of radiation at every wavelength and in every direction, with an emissivity coefficient of 1. The radiation itself is not bound to a material and can occur in transmitting gases such as oxygen, as well as in a vacuum. The energy flux emitted by a surface increases, based on the Stefan–Boltzmann law, to the fourth power of the absolute temperature ( Rohsenow et al., 1998). Other surfaces can absorb the emitted energy and emit energy themselves. Normally, the resulting radiation heat flux exchange between two or more surfaces is analysed.

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Source: https://www.sciencedirect.com/topics/engineering/thermal-radiation

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