Page 409 - Bird R.B. Transport phenomena

§12.4 Boundary Layer Theory for Nonisothermal Flow 391

EXAMPLE 12.4-2 In the preceding example we used the von Karman boundary layer integral expressions. Now
we repeat the same problem but obtain an exact solution of the boundary layer equations in
Heat Transfer in the limit that the Prandtl number is large—that is, for liquids (see §9.1). In this limit, the outer
Laminar Forced edge of the thermal boundary layer is well inside the velocity boundary layer. Therefore it can
Convection along a safely be assumed that v varies linearly with у throughout the entire thermal boundary layer.
Heated Flat Plate x
(Asymptotic Solution SOLUTION
for Large Prandtl By combining the boundary layer equations of continuity and energy (Eqs. 12.4-1 and 3) we get
Numbers) 5
ЛТ ( С У r)V \ AT rf-T
xj..\*L
v V di , / I au dx = a ±_L (12.4-18)
in which a = k/pC . The leading term of a Taylor expansion for the velocity distribution near
p
the wall is
(12.4-19)

in which the constant с = 0.4696/V2 = 0.332 can be inferred from Eq. 4.4-30.
Substitution of this velocity expression into Eq. 12.4-18 gives

(12.4-20)

This has to be solved with the boundary conditions that Г = T at у = 0, and T = T x at x = 0.
o
This equation can be solved by the method of combination of variables. The choice of the
dimensionless variables
cvj /2
IKTJ) = and г) = 2 (12.4-21,22)
Vl W /
makes it possible to rewrite Eq. 12.4-20 (see Eq. C.l-9) as

(12.4-23)

Integration of this equation with the boundary conditions that П = 0 at r\ = 0 and П —»1 as
V ~^ °° gives
exp(-rj )drj exp(-rj )drj
3
3
Jo Jo (12.4-24)
exp(-7/ : Г©
for the dimensionless temperature distribution. See §C4 for a discussion of the gamma func-
tion T(n).
For the rate of heat loss from both sides of a heated plate of width W and length L, we get

Q = 2l J q^dxdz

= 2W\ -fcV-

dx
= (2WL)(T
0 xn
x
(12.4-25)

which is the same result as that in Eq. 12.4-17 aside from a numerical constant. The quantity
within brackets equals 0.677, the asymptotic value that appears in Table 12.4-1.