G • COOLING BY PROTECTIVE FLUID FILMS 



where X = Vy,bu/v and \h = v^8h/v; 5„ and dh are thicknesses of the hydro- 

 dynamic and the thermal boundary layers, respectively. 



With the aid of Eq. 4-3 and 4-4, the analytical solutions of the momen- 

 tum and energy equations for boundary layers are obtained. The results 

 can be expressed as follows: 



^ = ^ 



31.18 + 12X-^+^n ^1+^^ 



1 -F X ' 2 1 + 3X + 3X2 



35 V2 



tan-i 2 



V3(x + |) 



(4-5) 



3 



for the momentum equation and 



^ = f(Ku, r, Pr) (4-6) 



for energy equation where ^ = {Ux/v){Vy,/Uy, f = h/8u and Pr is the 

 Prandtl number. The complete expression for Eq. 4-6 is much too com- 

 plicated to be presented here and [7] should be consulted. 



For a Prandtl number equal to unity Eq. 4-5 and 4-6 are identical. 

 It is noted that the changes of Xh for different Prandtl numbers are not 

 appreciable within the range of ^ which is of interest in the investigation 

 of transpiration cooling. 



The results calculated by Eq. 4-5 and 4-6 indicate that the relation 

 between ^ and \h is linear except in the region where ^ is less than unity. 

 In other words, a linear relationship is approached between the bound- 

 ary layer thickness (5„ or dh) and the length in the direction of flow x 

 when 8u reaches a certain value depending on the magnitude of Ww. The 

 Blasius solution reveals, for an impermeable flat plate, that the boundary 

 layer thickness is directly proportional to the square root of the length 

 in the direction of flow as well as the viscosity of the fluid. In the case of 

 flow over a flat plate with injection when 5„ reaches a certain thickness, 

 the effect of viscosity on the boundary layer becomes negligibly small, 

 and the formation of the boundary layer is mainly due to the additional 

 mass fluid injected into the main fluid. Hence the ratio of the boundary 

 layer thickness to the length in the direction of flow is linearly propor- 

 tional to the ratio of injected velocity to the main stream velocity. The 

 instability of the laminar boundary layer may be interpreted from the 

 inflection points occurring in the velocity and temperature profiles. It is 

 found that the larger the value of Vy,/U the farther the inflection points 

 move outward from the plate. A discussion on stability considerations of 

 the laminar boundary layer is given later on in this article. 



The temperature field in the laminar boundary layer described in the 

 previous paragraphs may be used to determine the amount of coolant 

 required to cool the wall to a predesignated temperature. From the bal- 

 ance between the heat flow to the wall from the hot gas and the heat 



< 440 ) 



