Page 402 - Bird R.B. Transport phenomena

384 Chapter 12 Temperature Distributions with More Than One Independent Variable

EXAMPLE 12.2-2 Note that the sum in Eq. 12.2-11 converges rapidly for large z but slowly for small z. Develop
an expression for T{r, z) that is useful for small values.
Laminar Tube Flow
with Constant Heat SOLUTION
Flux at the Wall-
Asymptotic Solution For small z the heat addition affects only a very thin region near the wall, so that the follow-
for the Entrance Region ing three approximations lead to results that are accurate in the limit as z —» 0:
a. Curvature effects may be neglected and the problem treated as though the wall were
flat; call the distance from the wall у = R - r.
b. The fluid may be regarded as extending from the (flat) heat transfer surface (y = 0) to
У = °°-
c. The velocity profile may be regarded as linear, with a slope given by the slope of the
2
parabolic velocity profile at the wall: v (y) = v y/R, in which v 0 = (2P - ty)R /2/JLL.
o
z
0
l
This is the way the system would appear to a tiny "observer" who is located within the very
thin shell of heated fluid. To this observer, the wall would seem flat, the fluid would appear
to be of infinite extent, and the velocity profile would seem to be linear.
The energy equation then becomes, in the region just slightly beyond z = 0,
2
У дТ д Т (12.2-13)
Actually it is easier to work with the corresponding equation for the heat flux in the у direc-
tion (cjy = —к дТ/ду). This equation is obtained by dividing Eq. 12.2-13 by у and differentiat-
ing with respect to y:

V a (12.2-14)
°R dz ду \У ду
It is more convenient to work with dimensionless variables defined as
Л = az (12.2-15)

Then Eq. 12.2-14 becomes

(12.2-16)
дк
with these boundary conditions:
B.C. 1: at A = 0, (12.2-17)
B.C. 2: at 7] = 0, ф = 1 (12.2-18)
B.C. 3: aS 7 7 ^ > о (12.2-19)
This problem can be solved by the method of combination of variables (see Examples 4.1-1
and 12.1-1) by using the new independent variable x = i7/>^9A. Then Eq. 12.2-16 becomes

3
X —% + (3* - 1) ^ = 0 (12.2-20)
-
dx dx
a n
The boundary conditions are: at x = 0/ Ф = 1/ d as x ~> °°, Ф —* 1- The solution of Eq. 12.2-20
is found by first letting йф/dx = p, and getting a first-order equation for p. The equation for p
can be solved and then ф is obtained as
J
(12.2-21)