D,12 • ''THICK'' THERMAL SHIELDS 



indeed the case for rocket nozzles. For example let the allowable temper- 

 ature be ^er = T„/T^ = 0.5 at the flame side boundary of a low carbon 

 steeP wall of thickness 6,2 = 0.25 in. = 0.0208 ft with heat transfer coef- 

 ficient at the nozzle throat A = 1500 (BTU/hr)/ft2°F. The Biot number 

 is iV = A''2 = hdi/ki = 1.56, and the duration time t^ = dlrjKi, computed 

 with two terms in the Fourier series for the simple slab equation (Eq. 4-7) 

 in this case 



0(0, Td) = 0er = 1 - 0.625e-i-""^d - 0.095e-i--«^<i = 0.5 



is id = 1.25 sec. If a zirconia layer of thickness dx = 0.02 in. = 0.0017 ft 

 shields the steel, the effective Biot number is l^ilW + (Mi/fci)] = 0.50. 

 Eq. 4-7, in this case, 



0(0, r^) ^ B^^=\ - O.SSOe-^-^^r; _ o.086e-^°-«^d = 0.5 



gives for the duration time t'^ = 7.1 sec. 



The expression for h,u in Eq. 11-4 can be deduced from steady state 

 considerations given in [6, p. 15] in a discussion of thin layers of oxide, 

 grease, or scale. The apphcation to transient states is justified in the 

 approximations of Eq. 11-1. 



D,12. "Thick" Thermal Shields. Another important limiting case 

 arises when the temperature in the shielded material has no appreciable 

 gradients either because of large conductivity in the latter, k^ » ki, or 

 because of large insulation thickness rfi » (i2. By a limiting procedure for 

 ki-^ oo (i.e. iU2-^0, hin^ iJ.in{p2C2d2/piCidi), etc.) Eq. 10-7a can be re- 

 duced to an eigenvalue equation for ^ini 



where 7 is the ratio of heat capacities 



_ p2C2d2 



^ ~ piCidi 

 The temperature distribution (Eq. 10-lOa) becomes 



f 1 + y A„(cos Mm^i + &in sin Mm^Oe""-'' - 1 ^ ^1 ^ 

 = i ^ (12-2) 



( 1 -f ^ Ane-''> =1+11 Ane-'^l'^ ^ ^2 ^ 1 



3 Material data and units are given in Table D,3a. 



< 277 ) 



