G,5 • TRANSPIRATION-COOLED PIPE FLOW 



Multiplying both sides of Eq. 5-31 by the appropriate factor, Eq. 5-31 

 can be written as the following Sturm-Liouville equation : 



^ [m(7,)M;.] + cjp(v)M, = (5-33) 



where 



m(7j) = 1)6 

 Viv) = mirj) 



a-v')+~{77,'-W + 2r,')^ 



36 



Hence Mi(rj, d) and Mj{r], Cj) for i 9^ j are orthogonal functions with re- 

 spect to the weight function p{r]) ; i.e. 



r p{'n)MiMjdr, = (5-34) 



The coeflScients of the series expansion, ^y's (Eq. 5-30) are determined 

 from the boundary condition (Eq. 5-6) applied to Eq. 5-30, which is 



Y A,Mj{r,, cj) = 1 (5-35) 



y=i 



Then multiplying both sides of Eq. 5-35 by p{r})Mj{ri, Cj) and integrating 

 from to 1 



Aj = ^, (5-36) 



]^ p{v)M'j(v,cj)dv 



From the differential equation (Eq. 5-33), it can be seen that 



/%(,)M,-(,,c,)d, = ^.t..(^)^_ (5-37) 



and 



Thus 



Aj = -r-^ (5-39) 



' \ dc /c=c„,=i 



The heat transfer coefl&cient for the flow in a pipe is usually calcu- 

 lated with the difference between the mean temperature of the fluid and 

 the wall temperature. The mean temperature over the cross section con- 

 sidered, weighted with respect to the axial velocity, may be defined by 



(467 > 



