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Thermal radiation

Thermal radiation is electromagnetic radiation emitted from the surface of an object which is due to the object's temperature. Infrared radiation from a common household radiator or electric heater is an example of thermal radiation, as is the light emitted by a glowing incandescent light bulb. Thermal radiation is generated when heat from the movement of charged particles within atoms is converted to electromagnetic radiation. The emitted wave frequency of the thermal radiation is a probability distribution depending only on temperature, and for a genuine black body is given by Planck’s law of radiation. Wien's law gives the most likely frequency of the emitted radiation, and the Stefan-Boltzmann law gives the heat intensity.

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Properties

There are three main properties that characterize thermal radiation:

Thermal radiation, even at a single temperature, occurs at a wide range of frequencies. How much of each frequency is given by Planck’s law of radiation (for idealized materials). This is shown by the curves in the diagram at the right.
The main frequency (or color) of the emitted radiation increases as the temperature increases. For example, a red hot object radiates most in the long wavelengths of the visible band, which is why it appears red. If it heats up further, the main frequency shifts to the middle of the visible band, and the spread of frequencies mentioned in the first point make it appear white. We then say the object is white hot. This is Wien's law. In the diagram the peak value for each curve moves to the left as the temperature increases.
The total amount of radiation, of all frequencies, goes up very fast as the temperature rises (it grows as T⁴, where T is the absolute temperature of the body). An object at the temperature of a kitchen oven (about twice room temperature in absolute terms - 600 K vs. 300 K) radiates 16 times as much power per unit area. An object the temperature of the filament in an incandescent bulb (roughly 3000 K, or 10 times room temperature) radiates 10,000 times as much per unit area. Mathematically, the total power radiated rises as the fourth power of the absolute temperature, the Stefan-Boltzmann law. In the plot, the area under each curve rises rapidly as the temperature increases.

Interchange of energy

Thermal radiation is an important concept in thermodynamics as it is partially responsible for heat exchange between objects, as warmer bodies radiate more heat than colder ones. (Other factors are convection and conduction.) The interplay of energy exchange is characterized by the following equation:

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

Here, $\alpha \,$ represents spectral absorption factor, $\rho \,$ spectral reflection factor and $\tau \,$ spectral transmission factor. All these elements depend also on the wavelength $\lambda\,$ . The spectral absorption factor is equal to the emissivity $\epsilon \,$ ; this relation is known as Kirchhoff's law of thermal radiation. An object is called a black body if, for all frequencies, the following formula applies:

$\alpha = \epsilon =1\,$

In a practical situation and room-temperature setting, objects lose considerable energy due to thermal radiation. However, the energy lost by emitting infrared heat is regained by absorbing the heat of surrounding objects. For example, a human being, roughly 2 square meter in area, and about 307 kelvins in temperature, continuously radiates about 1000 watts. However, if people are indoors, in a room of 296 K, they receive back about 900 watts from the wall, ceiling, and other surroundings, so the net loss is only about 100 watts. Clothes (having poorer thermal conductivity than human skin, therefore reducing the speed of heat loss from the human body to surrounding environment) reduce this loss still further.

If objects appear white (reflective in the visual spectrum), they are not necessarily equally reflective (and thus non-emissive) in the thermal infrared; e. g. most household radiators are painted white despite the fact that they have to be good thermal radiators. Acrylic and urethane based white paints have 93% blackbody radiation efficiency at room temperature (meaning the term "black body" does not always correspond to the visually perceived color of an object).

Calculation of radiative heat transfer between groups of object, including a 'cavity' or 'surroundings' requires solution of a set of simultaneous equations using the Radiosity method. In these calculations, the geometrical configuration of the problem is distilled to a set of numbers called view factors, which give the proportion of radiation leaving any given surface that hits another specific surface. These calculations are important in the fields of Solar thermal energy, Boiler and Furnace design and raytraced computer graphics.

Formula

Thermal radiation power of a black body per unit of area, unit of solid angle and unit of frequency $ν$ is given by

$u(\nu,T)=\frac{2 h\nu^3}{c^2}\cdot\frac1{e^\frac{h\nu}{k_BT}-1}$

This formula mathematically follows from calculation of spectral distribution of energy in quantized electromagnetic field which is in complete thermal equilibrium with the radiating object.

Integrating the above equation over $ν$ the power output given by the Stefan-Boltzmann law is obtained, as:

$W = \sigma \cdot A \cdot T^4$

Further, the wavelength $\lambda \,$ , for which the emission intensity is highest, is given by Wien's Law as:

$\lambda_{max} = \frac{b}{T}$

For surfaces which are not black bodies, one has to consider the (generally frequency dependent) emissivity correction factor $ε(υ)$ . This correction factor has to be multiplied with the radiation spectrum formula before integration. The resulting formula for the power output can be written in a way that contains a temperature dependent correction factor which is (somewhat confusingly) often called $ε$ as well:

$W = \epsilon(T) \cdot \sigma \cdot A \cdot T^4$

Constants

Definitions of constants used in the above equations:

$h \,$	Planck's constant	6.626 0693(11)×10^-34 J·s = 4.135 667 43(35)×10^-15 eV·s
$b \,$	Wien's displacement constant	2.897 7685(51)×10^–3 m·K
$k_B \,$	Boltzmann constant	1.380 6505(24)×10⁻²³ J·K^-1 = 8.617 343(15)×10⁻⁵ eV·K^-1
$\sigma \,$	Stefan-Boltzmann constant	5.670 400(40)×10⁻⁸ W·m^-2·K^-4
$c \,$	Speed of light	299,792,458 m·s^-1
$T \,$	Temperature	Average surface temperature on Earth = 288 K
$A \,$	Surface area	A_cuboid = 2ab + 2bc + 2ac; A_cylinder = 2π·r(h + r); A_sphere = 4π·r²