Page 429 - Bird R.B. Transport phenomena

§13.4 Temperature Distribution for Turbulent Flow in Tubes 411

where / is the Prandtl mixing length introduced in Eq. 5.4-4. Note that this expression
2
predicts that Pr (0 = 1. The Taylor vorticity transport theory gives Pr (0 = \.
{t)
EXAMPLE 13.3-1 Use the Reynolds analogy (v {t) = a ), along with Eq. 5.4-2 for the eddy viscosity, to estimate
the wall heat flux q for the turbulent flow in a tube of diameter D = 2R. Express the result in
0
An Approximate terms of the temperature-difference driving force T - T , where T is the temperature at
o
o
R
Relation for the Wall the wall (y = 0) and T is the time-smoothed temperature at the tube axis (y = R).
Heat Flux for Turbulent R
Flow in a Tube SOLUTION
v)
The time-smoothed radial heat flux in a tube is given by the sum of q) and q\! :
]
Л) AT
dr 1Г> dr
(13.3-4)
Here we have used Eq. 13.3-1 and the Reynolds analogy, and we have switched to the coordi-
nate y, which is the distance from the wall. We now use the empirical expression of Eq. 5.4-2,
which applies across the viscous sublayer next to the wall:

q =- 1+Pr| dy (13.3-5)
y
where q r = -q y has been used.
If now we approximate the heat flux q in Eq. 13.3-5 by its wall value q , then integration
0
y
from у = 0 to у = R gives
'Jo Г (13.3-6)
For very large Prandtl numbers, the upper limit R in the integral can be replaced by °°, since
the integrand is decreasing rapidly with increasing y. Then when the integration on the left
side is performed and the result is put into dimensionless form, we get

Pr i/3 = 1 (13.3-7)
3V3 ( v*\ Re
\(v z )
- f R ) 2TK14.5) ()J 17.5
in which Eq. 6.1-4a has been used to eliminate v* in favor of the friction factor.
The above development is only approximate. We have not taken into account the change
of the bulk temperature as the fluid moves axially through the tube, nor have we taken into
account the change in the heat flux throughout the tube. Furthermore, the result is restricted
to very high Pr, because of the extension of the integration to у = oo. Another derivation is
given in the next section, which is free from these assumptions. However, we will see that at
large Prandtl numbers the result in Eq. 13.4-20 simplifies to that in Eq. 13.3-7 but with a differ-
ent numerical constant.

§13.4 TEMPERATURE DISTRIBUTION FOR
TURBULENT FLOW IN TUBES
In §10.8 we showed how to get the asymptotic behavior of the temperature profiles for
large z in a fluid in laminar flow in a circular tube. We repeat that problem here, but for a
fluid in fully developed turbulent flow. The fluid enters the tube of radius R at an inlet
temperature T^ For z > 0 the fluid is heated because of a uniform radial heat flux q 0 at
the wall (see Fig. 13.4-1).

G. I. Taylor, Proc. Roy. Soc. (London), A135, 685-702 (1932); Phil Trans., A215,1-26 (1915).
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