D • CONDUCTION OF HEAT 



Computation of temperatures from Eq. 10-11 are generally rather 

 cumbersome. However, for two limiting cases of importance in rocket 

 engineering, i.e. thin and thick shielding layers, computations based on 

 Eq. 10-11 can be considerably simplified, as shown in the next two 

 articles. 



D,ll. The "Thin" Shield. In practice, some thermal shielding of 

 rocket walls is effected by very thin refractories or protective paints on 

 the inner boundary under conditions when, while Ni> \, the thickness 

 di is sufficiently small to give rise to the following relations in Eq. 10-7a: 



Min < M2«, Mil ^ tan /xii, jLifi « ATi = --i (11-1) 



When these relations are satisfied, Eq. 10-7a leads to an approximate 

 equation for /i2i 



X .^ hdi/ki /-, -, ON 



^" *'«' "" - 1 + M./fc. ("-2) 



Furthermore, upon substitution of these approximations into the formula 

 for A\ in Eq. 10-10, it is found that the expression 



^l(C0S M21^2 + &21 sin M21^2) 



reduces to 



4 sin M21 



2jU2i + sin 2/i2i 



cos jU2i(l - ^2) 



Hence, when the higher Fourier terms in Eq. 10-11 are negligible, the 

 temperature of the shielded material given by this equation reduces to 

 the simple slab formula (cf . Eq. 4-7 and 4-8) : 



d{^„ r2) ^ 1 - o ^X^^O cos M2l(l - ^2)6-"^^.- (11-3) 



-^^21 ~r sm Z)U2i 



where, in view of the eigenvalue equation (Eq. 11-2), the effective "Biot" 

 number is 



i.e. the shielding reduces the heat transfer coefficient to the shielded 

 material to an effective value 



K, = j^ (U-4) 



It is thus seen that the problem of the composite slab reduces, in this 

 limit, to the case of the simple slab with a reduced effective heat transfer 

 coefficient. Furthermore, Eq. 11-4 shows that even a thin thermal shield 

 can cause a large reduction in the heat input rates if h is large, which is 



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