G • COOLING BY PROTECTIVE FLUID FILMS 



given point inside the wall. This expression must be corrected to take 

 into account the fact that the wall consists only partially of solid metal. 

 If s denotes the porosity of the metal, then Eq. 3-1 can be rewritten as 

 follows : 



q=-k{l-s)A^^ (3-2) 



The rate of heat transfer from metal to fluid is proportional to the 

 area of contact and to the difference between the temperature t of the 

 metal and the temperature T of the fluid. Since both t and T vary along 

 the cylindrical bars, this heat transfer changes continuously along the 

 width of the porous metal. For an infinitesimally small length dz, the 

 following expression holds: 



dq - -hAirNdit - T)dz (3-3) 



where h is the heat transfer coefficient, N is the number of passages per 

 unit cross-sectional area, d is the diameter of the cylindrical pore, and 

 irNd is the total circumference at any cross section. 



The heat conduction from metal to fluid is used in raising the tem- 

 perature of the fluid, hence 



dq = Qc^dT (3-4) 



where Q is the mass flow of the cooling fluid through the cylindrical pores 

 and Cp is the specific heat of the cooling fluid at constant pressure. 



The solutions of the above three simultaneous differential equations 

 give the temperature of the metal and the fluid at any point within the 

 porous wall. The prescribed values of the temperature of the cooling fluid 

 before its entrance into the cylindrical bars and the temperatures of the 

 metal at both the hot and cold ends are used to determine the constants 

 of integration. The resulting expressions show that the temperatures of 

 the metal and the fluid become almost indistinguishable except within a 

 very narrow range near the cold end of the wall. The temperature dis- 

 tribution along the width of the porous wall is not linear, as in the case 

 of a solid metal, but is an exponential function. 



The indistinguishable difference in temperature between the cooling 

 fluid on its flow through the pores and the wall material can be realized 

 from the fact that the metal surface area in contact with the cooling fluid 

 is very great in the porous wall. The cooling fluid therefore leaves the 

 porous wall with the wall temperature T^ and with small velocity normal 

 to the surface. In passing away from the surface, the cooling fluid picks 

 up momentum from the gas flow until it finally reaches the outside gas 

 velocity. At the same time its temperature increases either by conduction 

 or by turbulent mixing until at some distance the gas temperature is 

 reached. A counterflow is thus created between the heat flowing from the 

 hot gas toward the wall and the cooling fluid flowing away from the wall. 



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