D • CONDUCTION OF HEAT 

 where An, in this Umiting case, is 



2(sin Mln 4- TMln cos jUln) 



■A-n — 



[1 + (7Mln)^]Mln + [1 — (TMln)^] Sin JLH„ COS Mln + 2yHxn COS^ jUln 



(12-3) 



Eq. 12-2 shows that the shielded material is at a uniform temperature 

 equal to the interface temperature 0(0, r) at any instant. It is to be noted 

 that the result (Eq. 12-2) represents, for — <ii ^ ^i ^ 0, the solution of 

 the simple slab problem with Newtonian heat input through one boundary 



ki^ = h(T - T^) at re = -di 

 ox 



and heat transfer through the other boundary governed by 



/ci -T- = P2C2a2 -^ Sit X = 

 dx dt 



In view of the latter boundary condition the shielded material behaves 

 essentially as a thermal capacity of magnitude p2C2d2 per unit area of 

 the slab. 



D,13. Design Criterion for Minimum Weight. The considerations 

 of the preceding article can be applied to the design of uncooled com- 

 posite rocket chamber walls in which the outer material (envelope) pro- 

 vides structural strength, while the inner material acts as a thermal shield 

 to prevent the attainment of excessive temperatures in the envelope dur- 

 ing the firing period t^, which may be of the order of a minute in duration. 

 In the approximation that the composite thickness di + c?2 is small com- 

 pared with the chamber radius, and k^di ^ kid2, the envelope temper- 

 ature T2 is given by the interface temperature in Eq. 12-2 



^ = ^^ = e(0, r) (13-1) 



Because of strength requirements, the hoop stresses in the envelope must 

 not exceed a critical stress >S2, say the yield stress of the material, which is 

 specified as a function of T2, i.e. S2 = S2{T2). This requirement is ex- 

 pressed by 



pR ^ S2iT2)d2 (13-2) 



where p is the combustion pressure in the rocket chamber and R is the 

 chamber radius. A second requirement is that the composite weight per 

 unit area of the wall, 



w = pidi + p2d2 (13-3) 



shall be a minimum. Now, S2iT2) generally diminishes as T2 increases 

 and therefore it diminishes also as t increases during the firing period. 



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