Page 462 - Bird R.B. Transport phenomena

442 Chapter 14 Interphase Transport in Nonisothermal Systems

consistent with boundary layer theory 6 for creeping flow with RePr > > 1. The latter re-
striction gives Nu > > 1 corresponding to a thin thermal boundary layer relative to
D /(1 - е)ф. This asymptote represents the creeping-flow mass-transfer data for liquids 3
p
very well.
The exponent § in Eq. 14.5-3 is a high-Pr asymptote given by boundary layer theory
for steady laminar flows 6 and for steadily driven turbulent flows. 7 This dependence is
consistent with the cited data over the full range Pr > 0.6 and the corresponding range of
the dimensionless group Sc for mass transfer.

514.6 HEAT TRANSFER COEFFICIENTS FOR
FREE AND MIXED CONVECTION 1
Here we build on Example 11.4-5 to summarize the behavior of some important sys-
tems in the presence of appreciable buoyant forces, first by rephrasing the results ob-
tained there in terms of Nusselt numbers and then by extension to other situations: (1)
small buoyant forces, where the thin-boundary-layer assumption of Example 11.4-5
may not be valid; (2) very large buoyant forces, where turbulence can occur in the
boundary layer, and (3) mixed forced and free convection. We shall confine ourselves to
heat transfer between solid bodies and a large quiescent volume of surrounding fluid,
and to the constant-temperature boundary conditions of Example 11.4-5. Discussions of
other situations, including transient behavior and duct and cavity flows, are available
elsewhere. 1
In Example 11.4-5 we saw that for the free convection near a vertical flat plate, the
principal dimensionless group is GrPr, which is often called the Rayleigh number, Ra. If
we define the area mean Nusselt number as Nu, w = hH/k = ^ H/A:(T - T ), then Eq.
avg
0
x
11.4-51 may be written as
Nu m = C(GrPr) 1/4 (14.6-1)
where С was found to be a weak function of Pr. The heat transfer behavior at moderate
values of Ra = GrPr is governed, for many shapes of solids, by laminar boundary layers
of the type described in Example 11.4-5, and the results of those discussions are normally
used directly.
However, at small values of GrPr direct heat conduction to the surroundings may
invalidate the boundary layer result, and at sufficiently high values of GrPr the mecha-
nism of heat transfer shifts toward random local eruptions or plumes of fluid, producing
turbulence within the boundary layer. Then the Nusselt number becomes independent
of the system size. The case of combined forced and free convection (normally referred
to as mixed convection) is more complex: one must now consider Pr, Gr, and Re as inde-
pendent variables, and also whether the forced and free convection effects are in the
same or different directions. Only the former seems to be at all well understood. The de-
scription of the behavior is further complicated by lack of abrupt transitions between the
various flow regimes.

6
W. E. Stewart, AIChE Journal, 9, 528-535 (1963); R. Pfeffer, Ind. Eng. Chem. Fund., 3, 380-383 (1964);
J. P. Sdrensen and W. E. Stewart, Chem. Eng. Sci., 29, 833-837 (1974). See also Example 12.4-3.
W. E. Stewart, AIChE Journal, 33, 2008-2016 (1987); corrigenda 34,1030 (1988).
7
1
G. D. Raithby and K. G. T. Hollands, Chapter 4 in W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho,
eds., Handbook of Heat Transfer, 3rd edition, McGraw-Hill, New York (1998).