Page 376 - Bird R.B. Transport phenomena

358 Chapter 11 The Equations of Change for Nonisothermal Systems

and Re in the two systems provides a sufficient approximation to dynamic similarity. This is
the limiting case of forced convection with negligible viscous dissipation.
If, however, the temperature differences T x - T are large, free-convection effects may be
o
appreciable. Under these conditions, according to Eq. 11.5-15, temperature differences in the
model must be 64 times those in the large system to ensure similarity.
From Eq. 11.5-16 it may be seen that such a ratio of temperature differences will not per-
mit equality of the Brinkman number. For the latter a ratio of 16 would be needed. This con-
flict will not normally arise, however, as free-convection and viscous heating effects are
seldom important simultaneously. Free-convection effects arise in low-velocity systems,
whereas viscous heating occurs to a significant degree only when velocity gradients are very
large.

EXAMPLE 11.5-2 We wish to investigate the free-convection motion in the system shown in Fig. 11.5-2. It con-
sists of a thin layer of fluid between two horizontal parallel plates, the lower one at tempera-
Free Convection in a ture T , and the upper one at T with 7\ < T . In the absence of fluid motion, the conductive
o
u
o
Horizontal Fluid Layer; heat flux will be the same for all z, and a nearly uniform temperature gradient will be estab-
Formation of Benard lished at steady state. This temperature gradient will in turn cause a density gradient. If the
Cells density decreases with increasing z, the system will clearly be stable, but if it increases a po-
tentially unstable situation occurs. It appears possible in this latter case that any chance dis-
turbance may cause the more dense fluid to move downward and displace the lighter fluid
beneath it. If the temperatures of the top and bottom surfaces are maintained constant, the re-
sult may be a continuing free-convection motion. This motion will, however, be opposed by
viscous forces and may, therefore, occur only if the temperature difference tending to cause it
is greater than some critical minimum value.
Determine by means of dimensional analysis the functional dependence of this fluid mo-
tion and the conditions under which it may be expected to arise.

SOLUTION The system is described by Eqs. 11.5-1 to 3 along with the following boundary conditions:
B.C.I: atz = 0, v = 0 T = T (11.5-17)
0
B.C. 2: at z = h, v = 0 Т = Г, (11.5-18)
B.C. 3: at r = R, v = О дТ/дг = О (11.5-19)

Top view
Fig. 11.5-2. Benard cells
formed in the region between
two horizontal parallel plates,
with the bottom plate at a
higher temperature than the
upper one. If the Rayleigh
о о о ° о number exceeds a certain
о° о ° о °
Side view о о о ° л ъ. Z = h ° ° °о о critical value, the system
о о °°
о о
о о о° Z = 0 о о о о° "
О О ° ° О О ° ° becomes unstable and
ч Insulation hexagonal Benard cells
2R- are produced.