Page 411 - Bird R.B. Transport phenomena

§12.4 Boundary Layer Theory for Nonisothermal Flow 393

grating the continuity equation with the boundary condition v = 0 at the surface to obtain v .
y y
These results are valid for Newtonian or non-Newtonian flow with temperature-independent
density and viscosity. 10
By a procedure analogous to that used in Example 12.4-2, one obtains a result similar to
that given in Eq. 12.4-24. The only difference is that 77 is defined more generally as 17 = y/8 ,
r
where 8 is the thermal boundary layer thickness given by
T
1 ( P r~ -Y
\l/3
.
.
.
-^= . 9a . Vh /3h h dx . . (12.4-29)
/ f
z
x
z
ПпВ\ J -v,(z) /
and x (z) is the upstream limit of the heat transfer region. From Eqs. 12.4-24 and 25 the local
}
surface heat flux q and the total heat flow for a heated region of the form x^(z) < x < x {z),
2
Q
z <z < z are
2
}
fc(T - T.)
q = 0 (12.4-30)
0
CHZ)
C^(
3 1 / 3 « T - T J fz 2 / f-v 2 (2) \2/3
0
Vh^h h dx) dz (12.4-31)
J x z
z, VA-,( 2 ) /
This last result shows how Q depends on the fluid properties, the velocity profiles, and the
geometry of the system. We see that Q is proportional to the temperature difference, to
2/3
к/а из = к р ду , з and to the 5-power of a mean velocity gradient over the surface.
]/3
Show how the above results can be used to obtain the heat transfer rate from a heated
sphere of radius R with a viscous fluid streaming past it in creeping flow 11 (see Example
4.2-1 and Fig. 2.6-1).
SOLUTION The boundary-layer coordinates x, y, and z may be identified here with тг - 0, r - R, and ф of
Fig. 2.6-1. Then stagnation occurs at 0 = тг, and separation occurs at 0 = 0. The scale factors
are h = R, and h = R sin 0. The interfacial velocity gradient /3 is
x z
= |ysin0 (12.4-32)
Insertion of the above into Eqs. 12.4-29 and 31 gives the following results for forced convec-
tion heat transfer from an isothermal sphere of diameter D:
1 / f° V
2
2
8 = , = -9a Vh> x sin OR sin в dS)
T
2
Vh> sin 0\ К )
x
'DCReW-^-'^f ^ (12.4-33)
sin 0
3 A:(T - — T ) C Г - f ° ,
1/3
T J
2
27T
/3
k(Tn
V
1
0
2/3
= 6 KKi j i o J - V^T^^eR 2 sin edв) dф
Q 4
(12.4-34)
;
The constant in brackets is 0.991.
The behavior predicted by Eq. 12.4-33 is sketched in Fig. 12.4-3. The boundary layer
thickness increases steadily from a small value at the stagnation point to an infinite value at
separation, where the boundary layer becomes a wake extending downstream. The analysis
here is most accurate for the forward part of the sphere, where 8 is small; fortunately, that is
T
Temperature-dependent properties have been included by Acrivos, loc. cit.
10
The solution to this problem was first obtained by V. G. Levich, loc. cit. It has been extended to
11
somewhat higher Reynolds numbers by A. Acrivos and T. D. Taylor, Phys. Fluids, 5, 387-394 (1962).