G • COOLING BY PROTECTIVE FLUID FILMS 



The pressure drop in the flow direction can be obtained upon the 

 substitution of Eq. 5-22 into Eq. 5-11 and 5-12, i.e. 



p(0, r) - p{x, r) _ 8 



ipu\ 



Re 



i + |x 



11 



270 



/(O) '^ ReR, 



R ^^-^^^ 



The coefficient of skin friction at the wall can also be obtained from 

 Eq. 5-24, and can be written 



Cf 



2t„ 

 pul 



Re 



^ 18 ^ 5400 



83X2 ^^ EeR 



1 4- A _ 1^' 

 "^ 12 540 



(5-27) 



In a like manner, the solution of Eq. 5-15 can be expressed for large 

 values of X (X > 1) by a power series developed near 1/X = 0. The com- 

 plete treatment of this part of the work is given in [31]. 



1.0 



0.8 



0.6 



0.4 



0.2 



0.2 



0.4 



0.6 0.8 



u/ui 



1.0 



1.2 



1.4 



Fig. G,5b. Velocity profiles vs. length in radial direction 

 for various X(i?e = lO', x/R = 10). (From [SI].) 



The velocity distributions in the main flow direction at an arbitrary 

 cross section of the pipe f or X = +1 and X = 10 are shown in Fig. G,5b. 

 It was noted that when X = 0, the profile becomes Poiseuille's paraboloid, 

 and for X > (fluid being injected through the wall) the axial velocity 

 increases and the velocity gradient at the wall increases. For X < (fluid 

 being withdrawn through the wall) both the axial velocity and the ve- 

 locity gradient at the wall decrease as compared with Poiseuille's case. 

 The above phenomenon follows the law of conservation of matter. In the 

 present case the radial velocity, which vanishes in Poiseuille's case, has a 

 finite magnitude except at the center of the pipe where it vanishes. 



< 464 ) 



