1,5 • ENCLOSURE OF GRAY SOURCE-SINK SURFACES 



The reason for the restriction on the validity of Eq. 5-5 or 5-6, that 

 any no-flux surfaces, if present, must be white rather than gray, needs 

 consideration. A white refractory surface reflects all incident radiation 

 without changing its quality, i.e. without changing the fractions of it for 

 which the gas will exhibit absorptivity 1 — Tx, 1 — Ty, etc. But a gray 

 refractory surface, to the extent that it absorbs and re-emits, changes the 

 quality of the radiation. If, for example, a beam of radiation incident on 

 Sr from the gas has an emissivity of ^ in one half of the energy spectrum, 

 or a total emissivity and absorptivity of ^, the resulting radiation leaving 

 Sr would be half absorbed by the gas on next passage through it if it 

 left Sr by reflection without change in character, and only ^ absorbed 

 if it left Sr by emission as black radiation. Since the derivation of /SiJFig 

 when Sr, Ss, . - ■ are present is based on attenuation by the gas in an 

 amount independent of the history of a beam of radiation, the nongray 

 gas solution represented by Eq. 5-5 applies rigorously only to systems in 

 which any no-flux surfaces present are perfect diffuse reflectors. If allow- 

 ance must be made for the grayness of any no-flux surface, it must be 

 reclassified as a source-sink surface, say S3, of unknown temperature, the 

 value of which is obtained by introducing the condition that the sum of 

 the interchanges of S3 with the other surfaces and with the gas must be 

 zero. 



The point has been emphasized that for gray systems the two terms 

 each representing one-way flux in the expression 



5'l^2 = *SliFl20'T^ — S2'^2l<^Tl 



differ only in temperature, that the S^ product may be factored out. 

 The imposing of the nongray gas condition makes this no longer true. 

 Gas absorptivity for radiation from a source at Ti is no longer necessarily 

 equal to gas absorptivity for radiation from a source at T2, except in the 

 limit as Ti approaches T2. Rather than use sequence of subscripts to indi- 

 cate direction of the radiation, it is preferable to use an arrow and retain 

 the equality of Si^u and *S23^2i as a matter of definition; but Si^i^2 does 

 not now equal Si^i^2 except in the limit. 3^i-.2 is evaluated by use of the 

 absorptivity of gas at Tg for radiation from a black or gray source at Ti; 

 {Fi<_2 uses gas absorptivity based on emission from T2. Similarly the net 

 interchange between gas and surface *Si must now be written 



g.^1 = <T(S,^^^Jt - S,^,^,Tt) (5-7) 



with Si^i^e based on the gas emissivity and *Si3^i_g based on gas absorp- 

 tivity for black or gray radiation from Ti. Plainly, however, if Tg ^ Ti, 

 JFi^g is the term to evaluate rigorously, and it may be used with small 

 error to represent both-way radiation. If T^ and Ti are not too far apart, 

 g^ig evaluated by the use of an effective emissivity given by the bracketed 

 term in Eq. 3-5 will probably suffice for both SJi^g and S'l-^g. 



< 533 ) 



