F,ll • HEAT TRANSFER 



since \/t(0)/p = u^ ■\/cf/2. Because these velocity profile assumptions 

 are based on the incompressible flow experiments of Nikuradse [23], the 

 present development is essentially for incompressible flow of gases, al- 

 though, as demonstrated later, compressibility will not alter the results 

 appreciably. 



The first term of Eq. 11-5 becomes, with the aid of Eq. 11-8, 



Since 



and 





u^du^ = Pri^^u% = 25 7/ Pr\ 



Pr^ 



IJL + €,, 



M + Cm 



M + Cp 



Pf\sim Pft 



(11-10) 



(11-11) 



(11-12) 



du/dy 



it will be found that the second term of Eq. 11-5 may be written, through 

 the use of Eq. 11-8 and 11-9, as 



2 / Pr^u^du^ ^ 2 X25^ Pn In (5?^ + I 



+ 2 X 25 ^Pr, 



In (y V5) (y' 



Pn _^^yl \5 



(11-13) 



' 1 P't'Um ^ 



Now, the integral on the right-hand side of Eq. 11-13 cannot be evalu- 

 ated in closed form; however, it can be expressed accurately, according to 

 numerical calculation, by 



~3 / Pn 



In (t/V5) 



Pr\ 

 Pru, 



1 + 



y 



© 



.^Kt)^'^'- 



\Pru^ ) 



4 \Pn 



-1+6 



3 / Pn 



L4 VPn,, 



-1 -f 1 



(11-14) 



for -1 ^ PrJPn,^ - 1 ^ a,. 



Next, Eq. 11-7 is substituted into the last two terms of Eq. 11-5, and 

 the integrals are obtained by expanding the integrands by means of the 

 binomial theorem and obtaining the limit of the sum of the integrated 

 terms. The result can be represented accurately by 



Pr, 

 for Pft -^ 1, and by 



1_,|^ + ^| /|^(1_P,0 



Pr,U-< + |Jf (1-^n) 



^V|(l-Pn) 



(11-15) 



(11-16) 



< 375 ) 



