Page 120 - Introduction to Computational Fluid Dynamics

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4.10 APPLICATIONS
The predicted wall mass fractions for CO, O 2 , and CO 2 are also plotted in 11:7 99
Figure 4.7. The wall mass fraction ω O 2 ,w , starting from 0.232 at 800 K, decreases
rapidly to zero at T w ∼ 1,300 K. Note that ω O 2 ,w decreases more rapidly when
Reaction 1 is included, as expected. The wall mass fraction ω CO 2 ,w gradually in-
creases with temperature, peaks at T w = 1,300 K, and then rapidly falls to zero.
In this range where ω CO 2 ,w is significant, the BRR indicates a mild plateau after
an initial rapid rise with temperature. The ω CO,w , however, increases with wall
temperature. At T w > 1,300, CO evolution becomes significant, indicating domi-
nance of Reaction 3. At very high temperatures, this reaction becomes the most
dominant and combustion is now diffusion controlled with ω O 2 ,w = ω CO 2 ,w = 0
and ω CO,w → 0.406. Overall, Reaction 1 is important at low temperatures and
Reaction 3 is important at high temperatures. It must be noted that although the
tendencies predicted here are similar to the similarity solution for BRR obtained
by Kuo [38], the quality of predictions in combustion calculations greatly depends
on the accuracy of the assumed reaction-rate laws.

Entrance Region of a Pipe
We consider simultaneous development of velocity and temperature profiles in the
entrance region of a pipe of radius R. The flow is laminar (Re = 500) and the
fluid Prandtl number Pr = 0.7. An axially constant wall temperature boundary
condition is assumed. In this axisymmetric flow, the I boundary coincides with the
pipe axis and the E boundary with the pipe wall. Computations are performed with
a JN = 25 nonuniform grid with closer spacings near the wall. The axial locations
1.5
are determined from x = L (I − 1/ IM AX − 1) , where L = 0.2 × R × Re and
I is the axial step number. Figure 4.8 shows the computed variations of f × Re,
+
Nu x , and velocity u at the pipe axis with x = (x/R)/ Re / Pr. Also plotted in the
figure are previous numerical solutions for Nu x reported in [33]. It is seen that the
present solutions match perfectly with the previous solutions. The f × Re product
also varies as expected with asymptotic approach to 16.0. Similarly, the velocity
u/u at the pipe axis also reaches 2.0 at large x .
+
Similar computations are now carried out at higher Reynolds numbers
(1,000 < Re < 10,000) including the transition range. For this purpose, the LRE
model is used and computations are performed with a JN = 47 nonuniform grid.
Here, IMAX = 1,000 and L = 100 × D. Figure 4.9 shows variation of f, Nu
(Pr = 0.7 and 5.0), and u axis /u with Reynolds number in the fully developed state
(X/D = 100). It is seen that for Re < 1,600, the characteristics correspond to those
of a laminar flow (u axis /u = 2.0). Accoring to the model, transition occurs abruptly
and appears to extend up to Re ∼ 2,500, as evident from the Nu predictions. The
u axis /u ratio now drops suddenly from its laminar value of 2.0. At Re = 10,000,
u axis /u = 1.246. For Nu, the expected trend is again observed. In the laminar range,

Nu approaches the analytically derivable fully developed value of 3.667 for T w =
constant boundary condition for both Prandtl numbers. The thermal development