G • COOLING BY PROTECTIVE FLUID FILMS 



coefficients are calculated from the boundary conditions as described pre- 

 viously. Eq. 4-1 can then be solved upon substitution of the velocity 

 profile and the expressions for mass density and viscosity. The results 

 can be expressed as follows: 



,.^t^y(x,f:.M-) (4-10) 



where 





depending on n -^ -y/T or /x '~ T«. 



The difference between Eq. 4-5 and 4-10 is that the latter contains the 

 Mach number and the ratio of the wall temperature to the hot gas tem- 

 perature which do not appear in the former equation. This is due to the 

 fact that the variation of the mass density and viscosity as functions of 

 temperature are taken into account in the solution of the momentum 

 equation in the present case. The growth in boundary layer thickness 

 with the increase of Mach number and the ratio of hot gas temperature 

 to the wall temperature can be easily interpreted from Eq. 4-10. Since 

 the effect of compressibility is to increase the heat transfer through the 

 wall, and since the amount of heat produced in the boundary layer in- 

 creases with speed, the effects of both the increase of Mach number and 

 the ratio of hot gas temperature to the wall temperature to the boundary 

 layer thickness are the same. The results as calculated from Eq. 4-10 also 

 reveal that the temperature gradient at the wall increases as the Mach 

 number increases, and decreases as Vy,/U increases. This behavior indi- 

 cates that the heat transfer through the wall increases as the compressi- 

 bility of the flow becomes more pronounced and decreases as the injec- 

 tion of coolant increases. 



From the balance between the total heat flow to the wall from the 

 hot gas and the total heat absorbed by the coolant, one obtains 



K i^) dx = pw2^wCp(Tw - T,)l (4-11) 



\oy /v, 



The temperature gradient at the wall can be obtained from Eq. 4-9 and 

 4-10. The relation between the wall temperature and the rate of coolant 

 injection is then determined by the following expression: 



where 



?7^i = Pr-Re, (^ - l) -- I / (x„ ^3, W- ) (4-12) 



W~ p^U 



The influence of variation in the physical properties of the gas across 

 the boundary layer to the transpiration cooling can be seen in Fig. G,4b. 



( 444 > 



