Page 408 - Bird R.B. Transport phenomena

390 Chapter 12 Temperature Distributions with More Than One Independent Variable

Table 12.4-1 Comparison of Boundary Layer Heat Transfer Calculations for Flow along a
Flat Plate
Value of numerical coefficient in
expression for heat transfer rate
Method in Eq. 12.4-17
Von Karman method with profiles of Eqs. 12.4-9 to 12 V148/315 = 0.685
Exact solution of Eqs. 12.4-1 to 3 by Pohlhausen 0.657 at Pr = 0.6
0.664 at Pr = 1.0
0.670 at Pr = 2.0
Curve fit of exact calculations (Pohlhausen) 0.664
Asymptotic solution of Eqs. 12.4-1 to 3 for Pr >> 1 0.677

The temperature profile is then finally given (for A < 1) by

Q
I ~l = 2[ -^ J - 2[~ ) + [1-Х (12.4-16)
\&O/ \&O/ \&O/
1 Q ~ 1 X
in which A * p ~ 1/ 3 and 8(x) = V(1260/37)(i/*/u ). The assumption of laminar flow made
r
M
5
here is valid for x < x , where x v p/ JJL is usually greater than 10 .
crit
crit x
Finally, the rate of heat loss from both sides of a heated plate of width W and length L
can be obtained from Eqs. 12.4-5,11,12,15, and 16:
rW Cl
Q = 2 \ q\ y=o dx dz
J о J о
= 2 Г Г C v (T-TJ\ dydz
Jo Jo P p x x=L
4
= 2 WpC p voc(T o - TooX^A - T!OA 3 + ^A )5 r (L)
1/3
« Vgf(2WL)(T - Tjf | jPr Re[ /2 (12.4-17)
0
in which Re L = Lv p//ji. Thus the boundary layer approach allows one to obtain the depen-
x
dence of the rate of heat loss Q on the dimensions of the plate, the flow conditions, and the
thermal properties of the fluid.
Eq. 12.4-17 is in good agreement with more detailed solutions based on Eqs. 12.4-1 to 3. The
asymptotic solution for Q at large Prandtl numbers, given in the next example, 5 has the same
form except that the numerical coefficient V148/315 = 0.685 is replaced by 0.677. The exact so-
lution for Q at finite Prandtl numbers, obtained numerically, 6 has the same form except that
the coefficient is replaced by a slowly varying function C(Pr), shown in Table 12.4-1. The
value С = 0.664 is exact at Pr = 1 and good within ±2% for Pr > 0.6.
1/3
The proportionality of Q to Pr , found here, is asymptotically correct in the limit as
Pr —> oo, not only for the flat plate but also for all geometries that permit a laminar, nonsepa-
1/3
rating boundary layer, as illustrated in the next example. Deviations from Q ~ Pr occur at
finite Prandtl numbers for flow along a flat plate and even more so for flows near other-
shaped objects and near rotating surfaces. These deviations arise from nonlinearity of the ve-
locity profiles within the thermal boundary layer. Asymptotic expansions for the Pr
dependence of Q have been presented by Merk and others. 7

5
M. J. Lighthill, Proc. Roy. Soc, A202, 359-377 (1950).
6
E. Pohlhausen, Zeits.f. angew. Math. u. Mech., 1,115-121 (1921).
7 H. J. Merk, /. Fluid Mech., 5,460-480 (1959).