G,5 • TRANSPIRATION-COOLED PIPE FLOW 



of fluid injection on the velocity distribution of a fully developed turbu- 

 lent flow in a circular pipe was shown. The theoretical curves were calcu- 

 lated from Eq. 5-56. The experimental data were taken from a porous 

 stainless steel pipe with a 5-inch diameter and a length of 20 inches. The 

 range of Reynolds numbers was from 10^ to 3 X 10^ The comparison 

 between the theoretical curve and the experimental data shows close 

 agreement. 



The momentum integral equation for the turbulent flow through a 

 pipe with fluid injection at the wall is obtained by integrating Eq. 5-46 

 over the cross section from r = to r = R. This yields 



R 



Cf 



dp 



dx 



d 



[ fe) {i)^{i) 



(5-58) 



2t 



pul ~ pui uid{x/R) L' 



The integral in Eq. 5-58 can be evaluated by substituting the ex- 

 pression for the velocity profile from Eq. 5-56. After a very tedious inte- 

 gration, the resulting expression for the skin friction coefficient becomes 



Cf 





2 -M - 



d 



< d{x/R) 



32 w. 



+ 



15Z Wc K 





+ 



2.159 



\ul "^ 2 uj 



3.235 Up Vy 



Wo Uc 



(5-59) 



Temperature distribution and heat transfer. In order to achieve an 

 approximate solution for the temperature distribution in a fully devel- 

 oped turbulent flow in a pipe with coolant injection at the wall it is 

 necessary to make some simplifications of Eq. 5-49. It is assumed that 

 V = —Vy,r/R and that in the region close to the wall 



dT[ T 

 dr^^ R 



holds. The temperature distribution may be written 



According to the mixture length theory, the eddy heat transfer term 



vT = -KKR - r)' 



dT 



dr 



(5-61) 



Following the above hypothesis and assumptions Eq. 5-49 can be simpli- 

 fied, after integration, as follows: 



ipCp 



+ K\{R - r)' 



du 

 dr 



dT 

 dr 



{i)'^ = 4rlr^' (5-62) 



The condition at r = gives the zero constant of integration. 



<473 > 



