1,4 • RADIANT EXCHANGE 



factor. Plainly, it cannot depend on any system temperatures. Conse- 

 quently, if the gas and all source-sink surfaces except S\ are kept at abso- 

 lute zero, and 51-2 (which now becomes simply qi-^i) is evaluated and used 

 to determine 3^ in Eq. 4-1, that value of 3^ will be generally applicable 

 regardless of the particular combination of temperatures of the source- 

 sink surfaces. Space does not permit presentation of the detailed deri- 

 vation here [5, Chap. 4; Sit]- Briefly, there are as many unknown emissive 

 powers (due to reflection alone at all source-sink zones except one, and to 

 reflection and/or emission at the no-flux zones) as there are zones; an 

 energy balance may be written for each zone, thereby permitting a solu- 

 tion. The evaluation of fF necessitates the use of a determinant D, sym- 

 metrical about its major diagonal, and of order equal to the total number 

 of zones into which the enclosing surface has been divided. To simplify 

 the expression of D and the solution for fF, a shorthand nomenclature is 

 desirable. Let SiFirtir (which also equals SrFritri) be represented by li^ 

 (or its equivalent Rl, although the convention is adopted of mentioning 

 small numbers first, and numbers before letters); because reflectivity 

 (1 — e) appears so often, replace it (except in any final simplification) 

 by p. Then 



D = 



IR 2R SR ' • • RR — Sr RS 



IS 2S SS • • • 'RS 'SS - Ss 



(4-2) 

 With D defined, Sm'i^mn ( = Sn^nm) may be evaluated. 



Q ct: 1 ^ml^m ^n>Jn ^mn (A o„\ 



Pm Pn -L* 



Sr^^nn = - '-"-" U + "^ ^\'^ (4-3b) 



Pn \ Pn U / 



where D^„ is the cofactor of row m and column n of D, defined as (—1)'"+" 

 times the minor of D formed by crossing out the mth row and nth column. 



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