D,15 • VARIABLE THERMAL PROPERTl 



In the former case, however, the effect on temperature ti 

 considerable. Accordingly we shall discuss only the form 

 To a first approximation, c and k may be considered 

 with the temperature, 



T T 



k = ko — ki-pjT pc = Co -i- ciy^ {0 < ki < ko), (0| 



The differential equation and initial and boundary conditions for a homo- 

 geneous slab (unidimensional) are 



dx 



, dT\ dT 



dx 

 dT 



h 



ex=k^^-^^^ -' ^ = 



(15-2) 







dT 



dx 



T = 



at x = d 

 at t = 



Let 



^ = 



d' 



t J _ Co + Cl .2 r, _ T 



(15-3) 



Substituting Eq. 15-1 and 15-3 into Eq. 15-2 we obtain after some alge- 

 braic manipulation 



/3(e) at ^ = 



at ^ = 1 

 e = -1 at T = (-1 ^ e ^ 0) 



(15-4) 



where 



e 



/i(e) = e(i - p,e), Me) = e(i + p.e), /3(e) = ^ _ 



^ p. = o 



/Cl 



2 /Co 



k,' 



^ Pc = o 



Ci 



2 Ci + cc 



< 



(15-5) 



In the purely mathematical range, — <^ ^ 6 ^ «= ^ the functions /i, /2, 

 and /g are, of course, nonlinear, so that in principle the differential equa- 

 tion and boundary conditions are nonlinear. However, the actual entire 

 physical range of 9 is — 1 ^0^0, and in this range it develops that in 

 the practical ranges of pk and pc the three functions are rather close to 

 straight Unes. 



< 281 ) 



