G • COOLING BY PROTECTIVE FLUID FILMS 



is largely controlled by the relatively high resistance of the laminar sub- 

 layer next to the wall. 



The above concept was used by Prandtl in the investigation of turbu- 

 lent flow in a pipe. Rannie's extension of the concept [27\ to heat transfer 

 in transpiration cooling is discussed here. 



The assumptions made in this investigation are: (1) steady flow is 

 assumed and all derivatives with respect to length in the direction of flow 

 are zero, (2) the physical properties of the fluid remain constant across 

 the sublayer, (3) the gas flowing along the wall and the coolant flowing 

 through the pores are assumed homogeneous, and (4) the wall temper- 

 ature in the direction of flow is constant. 



The velocity distribution in the laminar sublayer can be determined 

 easily by integrating the Prandtl boundary layer equation with the aid 

 of a continuity equation which may be expressed in the following form: 



p-wV-w 



-^ = ""'-^ (4-28) 



"'lam ■ olam 



e ^ — 1 



where the boundary conditions at the wall (subscript w) and at the 

 boundary of the laminar sublayer and turbulent layer (subscript lam) 

 are applied. Eq. 4-28 indicates that the velocity distribution becomes linear 

 when Ww = and the wall shear decreases as the rate of injection increases. 

 In a like manner the temperature profile in the laminar sublayer can 

 be derived from the energy equation which can be written in the form 



T — T e**^— 1 



" — (4-29) 



■'■ lam -*■ w Olam 



e " — 1 



Eq. 4-29 reduces to Eq. 4-28 for Pr = 1. 



The motion of molecules in laminar flow and the motion of eddies in 

 turbulent flow by its transport of momentum are the causes of skin 

 friction; the same motions also transport heat. Therefore a relationship 

 should exist between skin friction and heat transport. Reynolds used this 

 approach to obtain the following relation between momentum transfer 

 and heat transfer across a turbulent stream: 



?lam '''lam /'A.Qfl^ 



^p(-'g ■'lam) (Wc Wlam) 



where qi^^a and Ti^m are the heat transfer and shearing stress at the bound- 

 ary between the laminar sublayer and turbulent layer and Tg and u^ are 

 the temperature and velocity at the center of the pipe. These can be 

 evaluated from Eq. 4-29 and 4-28, respectively. Upon the substitution 

 for quantities g-iam, Tum, and Tiam in Eq. 4-30, the relation between the wall 



( 454 ) 



