Page 463 - Bird R.B. Transport phenomena

§14.6 Heat Transfer Coefficients for Free and Mixed Convection 443

It has been shown, however, that simple and reliable predictions of heat transfer
rates (expressed as area mean Nusselt numbers Nu ) may be obtained for this wide vari-
m
ety of flow regimes by empirical combinations of asymptotic expressions:
a. Ni4 ond , for conduction in the absence of buoyant forces or forced convection
b. Nujj , for thin laminar boundary layers, as in Example 11.4-5
m
rb
с Nu^ , for turbulent boundary layers
d. Nu^ rced / for pure forced convection
These are dealt with in the following subsections.

No Buoyant Forces
The limiting Nusselt number for vanishingly small free and forced convection is ob-
2
tained by solving the heat conduction equation (the Laplace equation, V T = 0) for con-
stant, uniform temperature over the solid surface and a different constant temperature at
infinity. The mean Nusselt number then has the general form
Nu c ond = K(shape) (14.6-2)
m
With К equal to zero for all objects with at least one infinite dimension (e.g., infinitely
long cylinders or infinitely wide plates). For finite bodies К is nonzero, and an important
case is that of the sphere for which, according to Problem 10B.1,
Nu™ nd = 2 (14.6-3)
with the characteristic length taken to be the sphere diameter. Oblate ellipsoids of revo-
lution and circular disks are discussed in Problem 14D.1.

Thin Laminar Boundary Layers

For thin laminar boundary layers, the isothermal vertical flat plate is a representative
system, conforming to Eq. 14.6-1. This equation may be generalized to
NujT = C(Pr, shape)(GrPr) 1/4 (14.6-4)

Moreover, the function of Pr and shape can be factored into the product
С = C (shape)C (Pr) (14.6-5)
1
2
with 2
0.671 (14.6-6)
l
(0.492/Pr) 9/16 |4/9
1 3
Representative values ' of Q and C are given in Tables 14.6-1 and 2, respectively. Shape
2
3 4
factors for a wide variety of other shapes are available. ' For heated horizontal flat sur-
faces facing downward and cooled horizontal flat surfaces facing upward, the following
5
correlation is recommended:
° ^ (GrPr) 1/5 (14.6-7)
(1.9/Pr) 9/10 ] 2

2
S. W. Churchill and R. Usagi, AlChE Journal, 23,1121-1128 (1972).
3
W. E. Stewart, Int. J. Heat and Mass Transfer, 14,1013-1031 (1971).
4
A. Acrivos, AIChE Journal, 6, 584-590 (1960).
T. Fujii, M. Honda, and I. Morioka, Int. J. Heat and Mass Transfer, 15, 755-767 (1972).
5