F,20 • AERODYNAMIC HEATING OF HIGH SPEED VEHICLES 



to the surface is negligible, (2) heat conduction along the skin is negli- 

 gible, (3) no heat transfer takes place to or from other parts of the missile, 

 (4) the specific heat of the air in the boundary layer is constant, and (5) 

 radiation emissivity and absorptivity of the skin are equal. Accordingly, 

 the differential equation for skin temperature is 



"^^^ = Stc,pMT. - TO - e{aTi - G) 



CwPwS, 



dt 



(20-1) 



where, on the left-hand side of the equation, T^, Cw, Pw, and 5w are the 

 500 



400 



300 



3 

 I- 

 CQ 



C 



o 



o 



^ 200 



a 



c 



r2 100 



u 



c 



10 



20 



30 



40 



50 



Altitude, ftX 10"^ 



Fig. F,20c. Solar and nocturnal radiation to a black body as a function of altitude. 



temperature, specific heat, density, and thickness of the skin, respec- 

 tively, and t is time. On the right-hand side, the symbols T^, e, and a 

 represent the insulated-skin (recovery) temperature, skin emissivity, and 

 the Stefan-Boltzmann radiation constant (0.173 X lO"* BTU/ft^ hr 

 (°R)^), respectively. G signifies the incident radiation from solar, terres- 

 trial, and interstellar sources. The specific heat at constant pressure of 

 the air is Cp, while the density and velocity at the outer edge of the 

 boundary layer are pe and We, respectively. St is the Stanton number for 

 either laminar or turbulent flow or mixed. Fig. F,20c shows the variation 

 of incident radiation with altitude [60]. Owing to orientation of the air- 

 craft in flight, only a fraction of such radiation is received. If heat trans- 

 fer to other parts of the vehicle is considered, then the rate of that heat 

 transfer must be subtracted from the right-hand side of Eq. 20-1 ; if heat 



<407 ) 



