1,6 • APPLICATION OF PRINCIPLES 



the many-times-reflected radiation relatively more important. In that 

 case some other pair of Cg values may be used, such as eg and es.g, or €2.g and 

 e4.B. Seldom is it necessary to add an extra term to Eq. 5-5. 



1,6. Application of Principles. The procedures discussed above 

 permit allowance for the effects of factors often casually handled in the 

 past. The relations presented are not as easy to use as the relations of 

 convective heat transmission; but this is because the mathematics of 

 radiation in an enclosure, where every part of the system affects every 

 other part, is intrinsically more complicated than the mathematics of 

 heat transfer processes capable of expression in the form of a differential 

 equation. With a little practice in manipulation of determinants^ the 

 reader should be able to evaluate JF factors for systems of a considerable 

 degree of complexity in a reasonable time. If the higher order determi- 

 nants encountered are evaluated numerically for the specific example of 

 interest rather than algebraically to obtain results like Eq. 4-19, the time 

 required for a solution is not prohibitive. Some of the special cases en- 

 countered are used so frequently, however, that algebraic formulation of 

 their general solution is desirable. A few such cases are presented here. 



Real gas, gray sink Si, and reflecting no-flux surface Sr. The gray gas 

 solution for this case, simplified by the use of a single path length and 

 therefore a single Tg for all zone pairs, was given in Eq. 4-18 and 4-20. 

 Modification to allow for nongray gas gives 



Si^i^, = YJl X 1 (6-1) 



Sl\ei J ' €e / n . S 



1 + 



e^/x 



1 - €g/x 



with X equal to e^/(2eg — e2.g), and with eg evaluated for a path length 

 given by Table 1,3. If >SiJFi_g is wanted, agi replaces eg. 



Real gas, enclosed hy 2 gray sinks Si and S2, and no Sr. This vari- 

 ation on the previous case has interest for at least two reasons. Consider 

 a gas, a primary heat sink Si and a refractory surface, the external loss 

 from which is so large that it cannot be treated as a no-flux surface (the 

 term no-flux still refers to radiant heat transmission only), because the 

 internal gain by convection is so much less than the loss through the wall. 

 Then the refractory surface becomes a secondary heat sink and is >S2 

 rather than Sr. Or consider a gas, a heat sink Si and a no-flux surface 

 which is not justifiably classed as completely reflecting. As the discussion 



6 Evaluation of higher order determinants by mechanical computers is of course 

 feasible. The labor of evaluation with pencil and slide rule has been so greatly reduced 

 by the method of Crout [44], however, that fifth or sixth order determinants need no 

 longer be considered formidable by the engineer not equipped with the newer devices. 



< 535 > 



