D ■ CONDUCTION OF HEAT 



where the amphtudes [9] determined from the initial condition 6(03, 0) = 

 — 1 are 



r J2 



ji^ ^ — 11 = (7_9) 



1^" Rl{fXnOi)c^dc^ {Hn^yRlifJ-n^) - (nl + Nl) RIM 



Analogous expressions for temperature transients due to Newtonian 

 heat input at the outer radius of the shell, r = h, are given in [10]. All of 

 the above results can be readily deduced from the somewhat more general 

 result, i.e. the heat transfer through both boundaries, obtained in [6, 

 p. 278]. 



It is seen that the formal expressions for the hollow cylinder are con- 

 siderably more cumbersome than those for the slab. It is therefore of 

 practical importance to ascertain the range of wall thickness to radius 

 ratio, so that the problem can still be treated in a good approximation 

 by expressions for an equivalent slab. This point will be discussed in more 

 detail in the next article. 



D,8. Applications. Thermal Stresses. For a hollow cylinder of 

 rather large thickness number 12 = 2 and a slab of thickness d = h — a = 

 (U — \)a = a, the flame side temperature transients are shown in Fig. 

 D,8 for hd/k = h(b — a)/k = 1. Initially, the temperatures rise at the 

 same rate in both structures; subsequently, as the larger heat capacity of 

 the cylinder is utilized by the heat flow, the temperature falls below that 

 in the slab. Thus, in relatively thick (refractory) cylinders, use of the slab 

 formula may lead to considerable error in the calculation of temperatures. 

 However, for relatively thin cylinders (metallic walls), the heat capacities 

 of the cylinder and slab of same thickness are approximately equal, in 

 the ratio 



pc^^^^^^^:pc(6 - a) = ^ (12 + 1) (8-1) 



It is to be noted further that the effect of the larger heat capacity on the 

 temperature is to some extent counteracted by the Newtonian type of 

 heat input, which is greater at lower wall temperature. By detailed studies 

 of temperature distributions based on Eq. 7-8, it is found [9,11] that the 

 equivalent slab solution is applicable within the approximation with 

 which h and the material data can be specified in practice, provided the 

 thickness number 12 does not exceed 1.2. This result justifies the use of 

 slab formulas in application to thin-walled cylindrical structures, such as 

 rocket chambers, metallic nozzle walls, etc. 



Larger thickness numbers 12 occur for refractory shells used as inserts 

 in rocket chambers and nozzles. With such refractory liners the problem 

 of thermal stresses becomes important in design considerations. In the 

 approximation that the material behaves elastically in the temperature 



< 268 > 



