D,6 • APPLICATIONS 



the thicknesses of relatively thin cylindrical elements in which heat flow 

 is principally in the radial direction (cf. Art. 7). The design problem usu- 

 ally consists of the determination of d for given material constants and 

 heat transfer parameters h and Tg, subject to the condition that, at the 

 end of a specified duration time U, a given critical temperature T„ shall 

 be attained at a position x„. Thus T^, might be the melting temperature 

 and the position might be at the flame boundary x„ = 0. 



In thin-walled rocket chambers, T„ usually corresponds to a space 

 average temperature at which the strength of the material begins to de- 

 crease appreciably. From Eq. 6-1 and 6-2 we obtain for this case the 

 following expression for t^ in terms of U and d„ = T^/T^: 



ta = k In r^ (6-6) 



For small Biot numbers the above criterion yields from Eq. 6-4 



d = ^^^^^ (6-7) 



and for greater Biot numbers we have from Eq. 6-5 



d^n, '\ (6-8) 



-V In 



1 - ^cr 



The latter is applicable in the approximation that the temperature drop 

 across the slab at the time t^ as computed from Eq. 4-7 is small com- 

 pared with T„: 



no, Q - T{d, Q « r„ (6-9) 



Under many conditions the inequality (Eq. 6-9) cannot be well satis- 

 fied by use of the criteria, represented by Eq. 6-7 and 6-8, based on the 

 mean temperature formula (Eq. 6-1). Thus when h and T^ are sufficiently 

 large, there may be large temperature drops across the wall at the time t^ ; 

 then Tcr is usually specified at (or near) the flame side. Here either one or 

 two terms in the Fourier series of Eq. 4-7 or the use of Eq. 5-3 may 

 suffice to determine d. It may happen, however, that the desired duration 

 time ti is so large with given large heat transfer parameters {h, Tg) that 

 0cr will be exceeded regardless of the thickness d, i.e. Eq. 4-7 or Eq. 5-3 

 cannot be solved for d. This can be ascertained by noting that the flame 

 side temperature transient for the semi-infinite solid 0(0, r*) in Eq. 5-1 

 rises slower than that of any finite slab with the same h, Tg, and material 

 constants. In other words, when the following condition, 



OiO, r*) = 1 - e^d erfc V^ > d,, 

 { 265 ) 



