Blackbody Concepts
In addition to absorbing all incident radiation, a blackbody is a perfect radiating body. To describe the emitting capabilities of a surface in comparison to a blackbody, Kirchoff defined emissivity of a real surface as the ratio of the thermal radiation emitted by a surface at a given temperature to that of a blackbody at the same temperature and for the same spectral and directional conditions.
where ( is the StefanBoltzmann constant The heat transfer rate by radiation for a nonblackbody, per unit area is defined as:
where T_{s} is the surface temperature and T_{sur} is the temperature of the surroundings.
Although some surfaces come close to blackbody performance, all real objects and surfaces have emissivities less than 1. Nonblackbody objects are either graybodies, whose emissivity does vary with wavelength, or nongraybodies, whose emissivities vary with wavelength. Most organic objects are graybodies, with an emissivity between 0.90 and 0.95 (Figure 22). The blackbody concept is important because it shows that radiant power depends on temperature. When using noncontact temperature sensors to measure the energy emitted from an object, depending on the nature of the surface, the emissivity must be taken into account and corrected. For example, an object with an emissivity of 0.6 is only radiating 60% of the energy of a blackbody. If it is not corrected for, the temperature will be lower than the actual temperature. For objects with an emissivity less than 0.9, the heat transfer rate of a real surface is identified as:
The closest approximation to a blackbody is a cavity with an interior surface at a uniform temperature T_{s}, which communicates with the surroundings by a small hole having a diameter small in comparison to the dimensions of the cavity (Figure 23). Most of the radiation entering the opening is either absorbed or reflected within the cavity (to ultimately be absorbed), while negligible radiation exits the aperture. The body approximates a perfect absorber, independent of the cavity's surface properties.
The radiation trapped within the interior of the cavity is absorbed and reflected so that the radiation within the cavity is equally distributedsome radiation is absorbed and some reflected. The incident radiant energy falling per unit time on any surface per unit area within the cavity is defined as the irradiance If the total irradiation G (W/m^{2}) represents the rate at which radiation is incident per unit area from all directions and at all wavelengths, it follows that:
If another blackbody is brought into the cavity with an identical temperature as the interior walls of the cavity, the blackbody will maintain its current temperature. Therefore, the rate at which the energy absorbed by the inner surface of the cavity will equal the rate at which it is emitted. In many industrial applications, transmission of radiation, such as through a layer of water or a glass plate, must be considered. For a spectral component of the irradiation, portions may be reflected, absorbed, and transmitted. It follows that:
In many engineering applications, however, the medium is opaque to the incident radiation. Therefore, and the remaining absorption and reflection can be treated as surface phenomenon. In other words, they are controlled by processes occurring within a fraction of a micrometer from the irradiated surface. It is therefore appropriate to say that the irradiation is absorbed and reflected by the surface, with the relative magnitudes of depending on the wavelength and the nature of the surface.
or
where I_{e} is the total intensity of the emitted radiation, or the rate at which radiant energy is emitted at a specific wavelength, per unit area of the emitting surface normal to the direction, per unit solid angle about this direction, and per unit wavelength. Notice that is a flux based on the actual surface area, where is based on the projected area. In approximating a blackbody, the radiation is almost entirely absorbed by the cavity. Any radiation that exits the cavity is due to the surface temperature only.
where (b for blackbody) represents the intensity of radiation emitted by a blackbody at temperature T, and wavelength per unit wavelength interval, per unit time, per unit solid angle, per unit area. Also, h = 6.626 x 10^{24} J•s and k = 1.3807 x 10^{23} J•K^{1} are the universal Planck and Boltzman constants, respectively; c_{o} = 2.9979 x 10^{8} m/s is the speed of light in a vacuum, and T is the absolute temperature of the blackbody in Kelvins (K). Due to the fact that deviations appeared between experimental results and the equation, Planck suggested in 1900 a refinement that better fit reality:
It is from this equation that Planck postulated his quantum theory. A more convenient expression for this equation, referred to as the Planck distribution law (Figure 24), is:
where the first and second radiation constants are and Planck's distribution shows that as wavelength varies, emitted radiation varies continuously. As temperature increases, the total amount of energy emitted increases and the peak of the curve shifts to the left, or toward the shorter wavelengths. In considering the electromagnetic spectrum, it is apparent that bodies with very high temperatures emit energy in the visible spectrum as wavelength decreases. Figure 24 also shows that there is more energy difference per degree at shorter wavelengths.
From Figure 24, the blackbody spectral distribution has a maximum wavelength value, lmax, which depends on the temperature. By differentiating equation 2.12 with respect to and setting the result equal to zero:
where the third radiation constant, C3 = 2897.7 • K. This is known as Wien's displacement law. The dashed line in Figure 24 defines this equation and locates the maximum radiation values for each temperature, at a specific wavelength. Notice that maximum radiance is associated with higher temperatures and lower wavelengths.


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