## Stefan–Boltzmann Law

**Radiation heat transfer** rate, q [W/m^{2}], from a body (e.g. a black body) to its surroundings is proportional to the **fourth power** of the absolute temperature and can be expressed by the following equation:

*q = εσT*^{4}

where **σ** is a fundamental physical constant called the **Stefan–Boltzmann constant**, which is equal to** 5.6697×10**^{-8}** W/m**^{2}**K**** ^{4}**. The

**Stefan–Boltzmann constant is named**after Josef Stefan (who discovered the Stefa-Boltzman law experimentally in 1879) and Ludwig Boltzmann (who derived it theoretically soon after). As can be seen, radiation heat transfer is important

**at very high temperatures**and

**in a vacuum**.

**Stefan–Boltzmann law gives the radiant intensity of a single object**. But using the

**Stefan–Boltzmann law**, we can also determine the radiation heat transfer between two objects. Two bodies that radiate toward each other have a net heat flux between them. The net flow rate of heat between them is given by:

*Q = εσA*_{1-2}*(T*^{4}_{1}* −T*^{4}_{2}*) [J/s]*

*q = εσ(T*^{4}_{1}* −T*^{4}_{2}*) [J/m*^{2}*s]*

The area factor A_{1-2}, is the area viewed by body 2 of body 1, and can become fairly difficult to calculate.

## Blackbody Radiation

It is known that the amount of radiation energy emitted from a surface at a given wavelength depends on the **material** of the body and the condition of its **surface** as well as the surface **temperature**. Therefore, various materials emit different amounts of radiant energy even whhen they are at the same temperature. A **body** that emits the **maximum amount** of heat for its absolute temperature is called a **blackbody**.

A **blackbody** is an idealized physical body, that has specific properties. By definition, a black body in thermal equilibrium has an **emissivity** of ** ε = 1.0**. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m^{2}). **Emissivity** plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Since the **absorptivity** and the **emissivity** are interconnected by the **Kirchhoff’s Law of thermal radiation**, a **blackbody** is also a perfect absorber of electromagnetic radiation.

**Kirchhoff’s Law of thermal radiation**:

*For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.*

**emissivity ε = absorptivity α**

A **blackbody** absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Its **absorptivity** is therefore equal to unity, which is also the highest possible value. That is, a **blackbody** is a **perfect absorber **(and a **perfect emitter**).

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (**absorptivity α = 0.94**) since they strongly absorb long-wavelength radiation.

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