Page 406 - Bird R.B. Transport phenomena

388 Chapter 12 Temperature Distributions with More Than One Independent Variable

ity boundary layer, whereas for Pr < 1 the relative thicknesses are just reversed (keep in
mind that for gases Pr is about f, whereas for ordinary liquids Pr > 1 and for liquid met-
als Pr < < 1).
In §4.4 we showed that the boundary layer equation of motion could be integrated
formally from у = 0 to у = °°, if use is made of the equation of continuity. In a similar
fashion the integration of Eqs. 12.4-1 to 3 can be performed to give

Momentum pv (v - v )dy p(v - v )dy
ду ax J Q x e x ax J о e x
pg^T-TJdy (12.4-4)

Energy рСрО (Т„ - T)dy (12.4-5)
dx х
Equations 12.4-4 and 5 are the von Kdrmdn momentum and energy balances, valid for
forced-convection and free-convection systems. The no-slip condition v y = 0 at у = 0 has
been used here, as in Eq. 4.4-4; nonzero velocities at у = 0 occur in mass transfer systems
and will be considered in Chapter 20.
As mentioned in §4.4, there are two approaches for solving boundary layer prob-
lems: analytical or numerical solutions of Equations 12.4-1 to 3 are called "exact bound-
ary layer solutions," whereas solutions obtained from Eqs. 12.4-4 and 5, with reasonable
guesses for the velocity and temperature profiles, are called "approximate boundary
layer solutions." Often considerable physical insight can be obtained by the second
method, and with relatively little effort. Example 12.4-1 illustrates this method.
Extensive use has been made of the boundary layer equations to establish correla-
tions of momentum- and heat-transfer rates, as we shall see in Chapter 14. Although in
this section we do not treat free convection, in Chapter 14 many useful results are given
along with the appropriate literature citations.

EXAMPLE 12.4-1 Obtain the temperature profiles near a flat plate, along which a Newtonian fluid is flowing, as
shown in Fig. 12.4-1. The wetted surface of the plate is maintained at temperature T and the
Heat Transfer in temperature of the approaching fluid is T . o
Laminar Forced K
Convection along a SOLUTION
Heated Flat Plate
(von Kdrmdn Integral In order to use the von Karman balances we first postulate reasonable forms for the velocity
Method) and temperature profiles. The following polynomial form gives 0 at the wall and 1 at the
outer limit of the boundary layer, with a slope of zero at the outer limit:

^~ — 2. I — — Z — у < 8{x)
(12.4-6, 7)
^- = 1 у => 8(x)

T -T
o
у =£ 8 T(x)
To-T (12.4-8, 9)
T - У ^ 8 (х)
т
n
That is, we assume that the dimensionless velocity and temperature profiles have the same
form within their respective boundary layers. We further assume that the boundary layer
thicknesses 8(x) and 8 (x) have a constant ratio, so that A = 8 (x)/8(x) is independent of x.
T
T
Two possibilities have to be considered: A < 1 and A > 1. We consider here A < 1 and rele-
gate the other case to Problem 12D.8.