G,5 • TRANSPIRATION-COOLED PIPE FLOW 



The solution of Eq. 5-14 can be expressed for small values of X (X ^ 1) 

 by a power series developed near X = as follows: 



/ = /o + X/i + X2/2 + • • • + X«/. (5-16) 



and 



c = Co + Xci -f- X2c2 + • • • + X«c„ (5-17) 



where /re's and c„'s are taken to be independent of X. By substituting Eq. 

 5-16 and 5-17 into Eq. 5-14 and equating coefficients of like power of X, 

 one obtains the following set of equations: 



2/o" + /;' = Co 



zf[" + /r - /o^ + un' = ci 



zjT + /" - m[ + /o7i + /of/ = C2 



The boundary conditions to be satisfied by /„'s are 



/n(0) = m) = 0; 

 for all n 



Ml) = \ 



/n(l) =0 n ^ 1 



The second order perturbation solution of Eq. 5-14 obtained by solv- 

 ing Eq. 5-18, 5-19, and 5-20 is given as follows: 



lim V^/;'(2) = 



2^0 



(5-18) 

 (5-19) 

 (5-20) 



(5-21) 



/(.) = (, - 1 .2) + X (- ^ + |g - 1^ + IJ) 



,-2/83 19 



+ ^ (,5400^-- 



7} A- —- z^ — 2* -I — z^ — 



540 ^432 144 ^720 10800 



800 / 



(5-22) 

 (5-23) 



It is seen from the foregoing equations that the second order pertur- 

 bation solution is sufficiently accurate even for X = 1. The velocity com- 

 ponents in the axial and radial directions are obtained by substituting 

 Eq. 5-22 and 5-10 into Eq. 5-9 as follows: 



u 



1 



18 ^ 5400 



^ ReR 



1 -z-\-^(-2-\-9z-9z' + 2z') 



+ 



V 



10800 

 2X 



(166 - 7602 + 825^2 - 3OO2* + 75z^ - Qz') 



Re ^ 



z - ^z' -\- ^(-4:z + 9z' - 6z' -{- z') 



+ 



10800 



(I662 - 380^2 -1- 2752^ - 752^ + 152^ - z') 



(5-24) 



(5-25) 



(463 ) 



