D • CONDUCTION OF HEAT 



representing /i, f^, and f^iQ) and observing further from Fig. D,15a that 

 these fits appear to be quite reasonable, it follows that the method of 

 average thermal properties cannot lead to excessive errors. 



Indeed this conclusion can be verified exactly in the case of small Biot 

 numbers, for in the latter case it is easy to obtain an exact solution. In 

 Fig. D,15b a comparison is shown between the exact solution, in a range 

 ^ r ^ Tm = -f^g, and the solution based upon an average value of c 

 at T = -gTg with pc = \. The variation of conductivity does not matter 

 in this limiting case. 



As seen from Fig. D,15a, somewhat better results might be obtained 

 by the use of line B instead of line A in relation to the curve representing 

 iziOf). However, on the basis of the above results and in view of the in- 

 accuracies involved in the given data it is concluded that the method of 

 average properties is adequate. 



D,16. Surface Melting and Erosion. The phenomenon of erosion in 

 rocket nozzles, occurring usually in the immediate neighborhood of the 

 throat, is of much importance but is rather complex and does not lend 

 itself readily to analysis. There are many causes at the root of this phe- 

 nomenon but it appears that surface melting may be a major one. With 

 respect to the latter, the problem of interest is the rate of melting. Thus, 

 in the case of nozzle inserts, say of the plastic type, the latter begin to 

 disintegrate at a rather low temperature and the problem of rate of 

 destruction is of importance in design, in estimating the duration time. 



The following is an approximate expression for the rate of surface 

 melting, as given in [7,24,25] : 



Let V = rate of melting (velocity of surface recession) 



L = latent heat of fusion or, more generally, the heat energy ab- 

 sorbed during a change of state. 

 H = a constant heat input rate at the flame side wall or the sur- 

 face which just begins to melt. 



At the surface just defined, the boundary condition representing the 

 heat energy balance is 



H = —k-T-r before melting 



ot 



= —k-rr-\-pL-T: after melting 

 ot at 



where ds/dt = V(t) is the rate of melting. Then the steady state rate of 

 melting is given by 



H 



V 



p[L -t- c{T^ - To)] 

 < 284 ) 



