Quantities hemispherical spectral

Hemispherical spectral quantities average the radiation into all directions of the hemisphere over a surface element and so are only dependent on the wavelength. 

We will now investigate how the emitted radiation d is distributed over the spectrum of wavelengths and the directions in the hemisphere. This requires the introduction of a special distribution function, the spectral intensity Lx. It is a directional spectral quantity, with which the wavelength and directiondistribution of the radiant energy is described in detail. 

The spectral intensity Lx(X,j3, ip,T) describes the distribution of the emitted radiation flow over the wavelength spectrum and the solid angles of the hemisphere (directional spectral quantity). 

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

It belongs to the hemispherical spectral quantities. Integration of (5.26) over all wavelengths leads to... 

This relation is known as Kirchhoff s law. Equation 7.27 may be substituted into the various relationships for the integrated emissivity or absorptivity. However, it does not follow that such quantities as directional total, hemispherical-spectral, or hemispherical total emissivity and absorptivity are necessarily equal. In fact, the integrated properties are only equal if certain restrictions are met. These are given in Table 7.1. 

Hemispherical total quantities combine the radiation over all wavelengths and from all directions. They do not provide information on the spectral distribution and the directional dependence of the radiation but are frequently sufficient to provide the solution to radiative heat transfer problems. 

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 relationships between the four quantities are schematically represented and illustrated in Fig. 5.7. The spectral intensity Lx(X,f3,tp,T) contains all the information for the determination of the other three radiation quantities. Each arrow in Fig. 5.7 corresponds to an integration on the left first over the solid angles in the hemisphere and then over the wavelengths, on the right first over the wavelengths and then over the solid angles. The result of the two successive integrations each time is the emissive power M (T). 

To get an idea about the spectral and directional complexity of the rigorous modeling of radiant heat transfer the variables that must be specified for the radiative properties are introduced. A functional notation is used to give explicitly the variables upon which a quantity depends. The most fundamental variables includes dependencies on wavelength, direction, and surface temperature. A total quantity does not have a spectral dependency. A hemispherical spectral variable does not have a directional dependency. A hemispherical total quantity has only a temperature dependency. 

---