G,4 • TRANSPIRATION-COOLED BOUNDARY LAYER 

 absorbed by the coolant, one gets 



k [^ = PwywCp(T„ - To) (4-7) 



Since the fluid layers adjacent to the wall are at rest, the heat flow from 

 the hot gas to the wall must be transferred by conduction through these 

 layers. This is represented by the term on the left-hand side of Eq. 4-7. 

 The term on the right-hand side of Eq. 4-7 is the heat absorbed by the 

 coolant. In the case where variation of wall temperature in the direction 

 of flow is considered, an additional term representing the heat flow in the 

 metal must be added in Eq. 4-7. Hence the thermal conductivity of the 

 metal enters into the energy balance equation. 



At the suggestion of the author, Ness [8\ made a theoretical investi- 

 gation of the temperature distribution along a semi-infinite porous flat 

 plate under the condition of uniform coolant injection. A heat-balance 

 differential equation of the second order, including a term containing the 

 physical parameters of the plate, is used in conjunction with the solution 

 of the equations of continuity, momentum, and energy. The temperature 

 distribution along the plate is obtained for the respective cases of thermal 

 conductivity not equal to, and equal to, zero. Results show that the 

 inclusion of the thermal conductivity term in the heat-balance equation 

 eliminates the infinite temperature gradient at the leading edge. 



The total heat flow to the plate can be obtained by integrating Eq. 4-7 

 over the entire plate of length I. The relation between the wall temper- 

 ature and the amount of coolant needed is then determined as follows: 



T„ - T^ 105 (vX fUl 



T^ - To 2 \U 



19 , 5 27 1 (1 + \y 



1 + X ' (1 + X)2 2 1 -{- 3X + 3X^ 



H- 35 V3 tan-i 2 Vs (x 4- ^ j 



(4-8) 



where X; can be determined from the curve X vs. | for a corresponding 

 value of ^i, i.e. {Ul/v){v^/Uy. For a predesignated wall temperature and 

 given Prandtl number and Reynolds number, the amount of coolant re- 

 quired per unit time can be determined from Eq. 4-8, provided that the 

 temperature of the hot fluid and of the coolant are known. The expression 

 in Eq. 4-8 is derived for Pr = 1 and [7] should be consulted for Pr 9^ 1. 

 In Fig. G,4b the ratio of the temperature difference, {T^ — Tw)/ 

 (Tw — To) is plotted against the coolant velocity ratio v^/U for Pr = 1. 

 The influence of the Reynolds number on the coolant discharge and the 

 wall temperature is rather appreciable. As the Reynolds number increases 

 the coolant discharge decreases for a given wall temperature. The oppo- 

 site is found to be the case for the Prandtl number. The above phenomena 

 can be explained by the fact that heat transfer from a hot gas to the wall 



<441 ) 



