Page 453 - Bird R.B. Transport phenomena

§14.3 Heat Transfer Coefficients for Forced Convection in Tubes 433

§143 HEAT TRANSFER COEFFICIENTS FOR
FORCED CONVECTION IN TUBES

In the previous section we have shown that Nusselt numbers for some laminar flows can
be computed from first principles. In this section we show how dimensional analysis
leads us to a general form for the dependence of the Nusselt number on various dimen-
sionless groups, and that this form includes not only the results of the preceding section,
but turbulent flows as well. Then we present a dimensionless plot of Nusselt numbers
that was obtained by correlating experimental data.
First we extend the dimensional analysis given in §11.5 to obtain a general form for
correlations of heat transfer coefficients in forced convection. Consider the steadily driven
laminar or turbulent flow of a Newtonian fluid through a straight tube of inner radius R,
as shown in Fig. 14.3-1. The fluid enters the tube at z = 0 with velocity uniform out to very
near the wall, and with a uniform inlet temperature T } (= T M). The tube wall is insulated
except in the region 0 ^ z < L, where a uniform inner-surface temperature T o is main-
tained by heat from vapor condensing on the outer surface. For the moment, we assume
constant physical properties p, /A, k, and C p. Later we will extend the empiricism given in
§14.2 to provide a fuller allowance for the temperature dependence of these properties.
We follow the same procedure used in §6.2 for friction factors. We start by writing
the expression for the instantaneous heat flow from the tube wall into the fluid in the
system described above,

(14.3-1)
which is valid for laminar or turbulent flow (in laminar flow, Q would, of course, be in-
dependent of time). The + sign appears here because the heat is added to the system in
the negative r direction.
Equating the expressions for Q given in Eqs. 14.1-2 and 14.3-1 and solving for h u we get

RdOdz (14.3-2)
r=R
Next we introduce the dimensionless quantities f = r/D,z = z/D, and t = (T - T o)/
(T h] - T o), and multiply by D/k to get an expression for the Nusselt number NUT =

(14 3 3)
- "
Thus the (instantaneous) Nusselt number is basically a dimensionless temperature gradient
averaged over the heat transfer surface.

Nozzle

Fluid enters Fluid leaves
at uniform with bulk
temperature Tj temperature T h2

Heated section
"I" with uniform surface "2"
temperature T o
Fig. 14.3-1. Heat transfer in the entrance region of a tube.