H,4 • BASIC LAWS FOR DISTRIBUTED RADIATORS 



depth dX is then 



dR^ = {PJX)Rl - iP^dX)R^ (4-2) 



where the first term on the right-hand side of Eq. 4-2 represents the 

 emitted spectral radiancy in dX, and the second term measures the 

 attenuation by the absorbers of radiation in dX. Since i?„ = for X = 

 it follows from Eq. 4-2 that 



R, = R^^(l - e-Po,^) (4-3) 



Eq. 4-3 is the basic phenomenological law for the emission of radiation 

 from distributed sources. It is apparent that if an external light source is 

 used such that R^ = Rl for X = 0, then Eq. 4-3 should be replaced by 

 the expression 



R^ = Rl{l - e-P.^) + Rle-^o.^ (4-4) 



In those cases where the first term in Eq. 4-4 is negligibly small (i.e. 

 negligible emission of radiation from the region under study), Eq. 4-4 re- 

 duces to the Bouguer-Lambert law of absorption 



R^ = Rle-P<o^ (4-5) 



In absorption studies it is customary to choose l/27rco as the unit of length 

 and to introduce an extinction coefficient k through the relation 



K = ^ (4-6) 



For absorbing liquids (and sometimes also for gases), a specific absorption 

 coefficient ^ may be introduced through the relation 



/3 = ^ (4-7) 



where c' is the concentration of the absorber. If c' is expressed in mole 

 cm~* and P^p in cm"^, then (3 has the dimensions mole~^ cm^ and the 

 absorption law may be referred to as Beer's law of absorption. For ideal 

 gases, sets of units involving mass absorption coefficients k^ (in cm^-g^^) 

 are sometimes used; in this case k^ = P^R'T where R' is the gas const 

 per gram. The quantity Po,X is now replaced by k^pl with p representing 

 the gas density. 



Reference to Eq. 4-3 shows that the spectral emissivity e„ of the uni- 

 formly distributed radiators is given by the relation 



6, = 1 - e-P.^ (4-8) 



Similarly, the total emissivity e used in engineering calculations on radiant 

 heat transfer is 



^ = ^ j^ Ria - e-^.^)do: (4-9) 



(493 ) 



