Spectral emissive power

If the emissive power E of a radiation source-that is the energy emitted per unit area per unit time-is expressed in terms of the radiation of a single wavelength X, then this is known as the monochromatic or spectral emissive power E, defined as that rate at which radiation of a particular wavelength X is emitted per unit surface area, per unit wavelength in all directions. For a black body at temperature T, the spectral emissive power of a wavelength X is given by Planck s Distribution Law ... 

In practice, it is usually more convenient to work with radiation properties averaged over all directions, called hemispherical properties. Noting that the integral of the rate of radiation energy emitted at a specified wavelength per unit surface area over the entire hemisphere is spectral emissive power, the spectral hemispherical emissivity can be expressed as... 

Hemispherical spectral emissive power and total intensity... 

The spectral intensity Lx(X,f3,p,T) characterises in a detailed way the dependence of the energy emitted on the wavelength and direction. An important task of both theoretical and experimental investigations is to determine this distribution function for as many materials as possible. This is a difficult task to carry out, and it is normally satisfactory to just determine the radiation quantities that either combine the emissions into all directions of the hemisphere or the radiation over all wavelengths. The quantities, the hemispherical spectral emissive power Mx and the total intensity L, characterise the distribution of the radiative flux over the wavelengths or the directions in the hemisphere. 

The hemispherical spectral emissive power MX, T) is obtained by integrating (5.4) over all the solid angles in the hemisphere. This yields... 

Fig. 5.6 Hemispherical spectral emissive power M (X,T) as a function of wavelength A at constant temperature T (schematic). The hatched area under the curve represents the emissive power M(T)...

The hemispherical spectral emissive power Mx(X,T) covers the wavelength dependency of the radiated energy in the entire hemisphere (hemispherical spectral quantity). 

The spectral emissive power agrees with the function Lx XyT) from Fig. 5.8 except for the factor 2.094 sr. 

No radiator exists that has a spectral intensity Lx independent of the wave length. However, the assumption that Lx does not depend on j3 and ip applies in many cases as a useful approximation. Bodies with spectral intensities independent of direction, Lx = Lx(X,T), are known as diffuse radiators or as bodies with diffuse radiating surfaces. According to (5.9), for their hemispherical spectral emissive power it follows that... 

The wavelength and temperature dependency given by (5.37) correspond to a relationship found by W. Wien [5.3] in 1896 to be approximately valid for the hemispherical spectral emissive power M S(X,T ) of an ideal radiator, a black body, with a temperature T. We will come back to the properties of black bodies in section 5.1.6 and more extensively in 5.2.2. In our example a spectral irradiance E M s has been assumed, so that its indirect dependence on T appears explicitly in (5.37). 

Hollow enclosure radiation and radiation of a black body (a x = 1) have identical properties. The black body radiates diffusely from (5.18) it holds for its hemispherical spectral emissive power that... 

A black body is defined as a body where all the incident radiation penetrates it and is completely absorbed within it. No radiation is reflected or allowed to pass through it. This holds for radiation of all wavelengths falling onto the body from all angles. In addition to this the black body is a diffuse radiator. Its spectral intensity LXs does not depend on direction, but is a universal function iAs(A,T) of the wavelength and the thermodynamic temperature. The hemispherical spectral emissive power MXs(X,T) is linked to Kirchhoff s function LXs(X,T) by the simple relationship... 

We refrain from deriving the equations for the spectral intensity and the hemispherical spectral emissive power of a black body, found by M. Planck [5.6], for... 

Fig. 5.23 Hemispherical spectral emissive power MXs(X,T) of a black body according to Planck s radiation law (5.50)...

Table 5.3 Hemispherical spectral emissive power of a black body divided by the fifth power of the temperature, according to (5.54) and fraction function F(0, AT) according to (5.60) as a function of the product AT...

Fig. 5.30 Hemispherical spectral emissive power M (A, T) of a real body compared to the hemispherical spectral emissive power M g(A, T) of a black body at the same temperature. The hatched area represents the emissive power M(T) of the real body 0...

It holds for the hemispherical spectral emissive power of a real body that... 

Fig. 5.31 Hemispherical spectral emissive power Mx(, T) = e(T) MXs(, T) of a grey Lambert radiator at a certain temperature...

Fig. 5.32 Approximately constant spectral emissivity for A > Ai as well as the pattern of the hemispherical spectral emissive power M and the spectral irradiance E, such that a grey radiator can be assumed...

A radiator emits its maximum hemispherical spectral emissive power at Amax = 2.07 /im. Estimate its temperature T and its emissive power M(T), under the assumption that it radiates like a black body. 

The oven wall is exposed to radiation from glowing coal the spectral irradiance E can be assumed to be proportional to the hemispherical spectral emissive power M s(Tk) of a black body at Tk = 2000 K. 

The concept of emissive power is used to quantify the amount of radiation emitted per unit surface area. The hemispherical spectral emissive power E is defined as the rate at which radiation of wavelength A is emitted in all directions from a surface per unit wavelength dX about A and per unit surface area. It is thus related to the spectral intensity of the emitted radiation by ... 

The emissive power E is the total radiant power exitent from the surface (per unit area) toward the hemisphere due to thermal emission, which may be obtained by integrating dq in Equation (7.3) over the hemisphere and over aU wavelengths. The spectral emissive power is the emissive power per unit wavelength interval about X. If the emitted intensity is the same in all directions, the surface is called a diffuse emitter, for which E = nl and Ex = nli. 

At any specific wavelength, the higher the temperature, the greater the spectral emissive power. At any temperature, the emissive power approaches zero as 0, increases with wavelength until... 

Figure 7.3a represents the Planck distribution for blackbody spectral emissive power with E-i p / as a function of XT. The band fraction of emitted energy in the region from 0 to XT is equal to the shaded area, which is expressed as and shown in Figure 7.3b. About a quarter of the emitted energy is at wavelengths shorter than nd nearly 95% of the emitted energy is distributed between and The spectral distribution of solar radiation can be...

The ratio of the spectral emissive power of an object to that of a blackbody at the same temperature defines the spectral emissivity ... 

FIGURE 7.3 Blackbody characteristics (a) Planck s law for spectral emissive power, and (b) band fraction for emission over the range from 0 to XT. 

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