D,17 • AXIAL HEAT CONDUCTION IN NOZZLE WALLS 



where T^ and To are the melting and initial temperatures respectively. 

 The above expression is approximate in several respects. Firstly, this is 

 a limiting expression corresponding to the final, or steady state, rate of 

 melting. The actual rate of melting is less, so that the above formula 

 yields conservative results. Graphs for more exact values of the melting 

 rate are given in [7]. Secondly, the result is based upon heat propagation 

 in a semi-infinite sohd. Finally, the boundary condition, up to melting, 

 is not of the convective type and corresponds approximately to the latter 

 only in the case where the melting temperature is small as compared to 

 the flame temperature. Thus the whole nature of the boundary condition 

 is such that, up to melting, the temperature gradient at the flame side 

 wall is constant. With the exception of the limiting case just referred to, 

 this type of boundary condition does not appear to correspond very well 

 to the usual boundary conditions (convection and radiation) resulting 

 from actual types of heat inputs. 



Another drawback in the use of the above expression for the melting 

 rate is the fact that data for values of L for plastic materials, such as 

 bakelites, are not readily available at present. Nevertheless the result is 

 of importance in that it yields an insight into the role of the physical 

 parameters affecting the rate of surface melting. For more detailed results 

 the reader is referred to the references cited above. 



Finally it may be added that the same type of boundary conditions 

 may be used in the study of heat conduction phenomena with general 

 changes of state, not necessarily surface melting (cf. [26]). 



D,17. Axial Heat Conduction in Nozzle Walls. Along a nozzle 

 wall, the heat transfer coefficient varies with axial position, as shown in 

 Fig. D,17a. It is known, both experimentally and on the basis of approxi- 

 mate theoretical results [2,27,28,29], that the variation in h in the neigh- 

 borhood of the throat is rather large. 



Now, to within a good approximation the problem of heat conduction 

 in the nozzle wall reduces to the problem of heat flow in a slab with a 

 variable coefficient h. The differential equation is now 



' d'-T d^T\ ^ ar 



,dx^ '^ dy^ J ~ ^'^ ~dt 

 while the boundary conditions are (see Fig. D,17b) 



k^ = hix){T - T,) y = 



= y = d 



with the initial condition 7" = at i = 0. The principal difficulty arises 

 from the boundary condition, a problem discussed at length in [30]. It 

 appears that it is difficult to obtain an explicit continuous solution in the 



< 285 > 



