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# Encyclopedia

Hot metalwork from a blacksmith. The yellow-orange glow is the visible part of the thermal radiation emitted due to the high temperature. Everything else in the picture is glowing with thermal radiation as well, but less brightly and at longer wavelengths that the human eye cannot see. A far-infrared camera will show this radiation (See Thermography).
This diagram shows how the peak wavelength and total radiated amount vary with temperature. Although this plot shows relatively high temperatures, the same relationships hold true for any temperature down to absolute zero. Visible light is between 380 to 750 nm.

If the object is a black body in thermodynamic equilibrium, the radiation is termed black-body radiation[1]. The emitted wave frequency of the black body thermal radiation is described by a probability distribution depending only on temperature, and for a genuine black body in thermodynamic equilibrium 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 radiant intensity.[2]

## Properties

There are four 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 right.
• The main frequency (or color) range of the emitted radiation includes higher and higher frequencies as the temperature increases. For example, a red hot object radiates enough in the long wavelengths (red and orange) of the visible band to see, which is why it appears red. If it heats up further, it also begins to emit discernible amounts of green and blue light, and the spread of frequencies mentioned in the first point make it appear white. We then say the object is white hot. However, even at a "white-hot" temperature of 2000 K, 99% of the energy of the radiation is still in the infrared. This is related to Wien's displacement 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 at 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. The total radiative intensity in a cavity that contains a black body in thermodynamic equilibrium 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.
• The rate of thermal radiation of a particular kind of electromagnetic wave is proportional to the amount of absorption that the same type of wave experiences. Thus, a surface that absorbs more red light thermally radiates more red light. This principle applies to all properties of the wave, including wavelength (color), direction, polarization, and even coherence, so that it is quite possible to have thermal radiation which is polarized, coherent, and directional, though polarized and coherent forms are fairly rare in nature.

These properties apply if the distances considered are much larger than the wavelengths contributing to the spectrum (most significant from 8-25 micrometres at 300 K). Indeed, thermal radiation here takes only travelling waves into account. A more sophisticated framework involving electromagnetics has to be used for lower distances (near-field thermal radiation).

°C Subjective colour [2]
480 faint red glow
580 dark red
730 bright red, slightly orange
930 bright orange
1100 pale yellowish orange
1300 yellowish white
> 1400 white (yellowish if seen from a distance through atmosphere)

## Interchange of energy

Radiant heat panel for testing precisely quantified energy exposures at National Research Council, near Ottawa, Ontario, Canada.

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, humans lose considerable energy due to thermal radiation. However, the energy lost by emitting infrared heat is partially regained by absorbing the heat of surrounding objects (the remainder resulting from generated heat through metabolism). Human skin has an emissivity of very close to 1.0 .[3] Using the formulas below then shows 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, surrounded by surfaces at 296 K, they receive back about 900 watts from the wall, ceiling, and other surroundings, so the net loss is only about 100 watts. These heat transfer estimates are highly dependent on extrinsic variables, such as wearing clothes (decreasing total thermal "circuit" conductivity, therefore reducing total output heat flux.) Only truly "grey" systems (relative equivalent emissivity/absorptivity and no directional transmissivity dependence in all control volume bodies considered) can achieve reasonable steady-state heat flux estimates through the Stefan-Boltzmann law. Encountering this "ideally calculable" situation is virtually impossible (although common engineering procedures surrender the dependency of these unknown variables and "assume" this to be the case). Optimistically, these "grey" approximations will get you close to real solutions, as most divergence from Stefan-Boltzmann solutions is very small (especially in most STP lab controlled environments).

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[4] (meaning the term "black body" does not always correspond to the visually perceived color of an object). These materials that do not follow the "black color = high emissivity/absorptivity" caveat will most likely have functional spectral emissivity/absorptivity dependence.

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 Planck's law as:

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

or

$u(\lambda,T)=\frac{\beta}{\lambda^5}\left(\frac1{e^{hc/k_BT\lambda}-1}\right)$

where β is a constant.

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:

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

where the constant of proportionality σ is the Stefan–Boltzmann constant and A is the radiating surface area.

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 factor ε(υ). This factor has to be multiplied with the radiation spectrum formula before integration. If it is taken as a constant, the resulting formula for the power output can be written in a way that contains ε as a factor:

$P = \epsilon \cdot \sigma \cdot A \cdot T^4$

This type of theoretical model, with frequency-independent emissivity lower than that of a perfect black body, is often known as a gray body. For frequency-dependent emissivity, the solution for the integrated power depends on the functional form of the dependence, though in general there is no simple expression for it. Practically speaking, if the emissivity of the body is roughly constant around the peak emission wavelength, the gray body model tends to work fairly well since the weight of the curve around the peak emission tends to dominate the integral.

### 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

### Variables

Definitions of variables, with example values:

 $T \,$ Temperature Average surface temperature on Earth = 288 K $A \,$ Surface area Acuboid = 2ab + 2bc + 2ac; Acylinder = 2π·r(h + r); Asphere = 4π·r2

## References

1. ^ K. Huang, Statistical Mechanics (2003), p278
2. ^ K. Huang, Statistical Mechanics (2003), p280
3. ^ R. Bowling Barnes (24 May 1963). "Thermography of the Human Body Infrared-radiant energy provides new concepts and instrumentation for medical diagnosis". Science 140 (3569): 870–877. doi:10.1126/science.140.3569.870.
4. ^ S. Tanemura, M. Tazawa, P. Jing, T. Miki, K. Yoshimura, K. Igarashi, M. Ohishi, K. Shimono, M. Adachi, Optical Properties and Radiative Cooling Power of White Paints,[1] ISES 1999 Solar World Congress