Page 378 - Bird R.B. Transport phenomena

360 Chapter 11 The Equations of Change for Nonisothermal Systems

Similar behavior is observed for other boundary conditions. If the upper plate of Fig.
11.5-2 is replaced by a liquid-gas interface, so that the surface shear stress in the liquid is neg-
ligible, cellular convection is predicted theoretically for Rayleigh numbers above about 1101.
3
A spectacular example of this type of instability occurs in the occasional spring "turnover" of
water in northern lakes. If the lake water is cooled to near freezing during the winter, an ad-
verse density gradient will occur as the surface waters warm toward 4°C, the temperature of
maximum density for water.
In shallow liquid layers with free surfaces, instabilities can also arise from surface-ten-
sion gradients. The resulting surface stresses produce cellular convection superficially similar
to that resulting from temperature gradients, and the two effects may be easily confused. In-
deed, it appears that the steady flows first seen by Benard, and ascribed to buoyancy effects,
may actually have been produced by surface-tension gradients. 6

EXAMPLE 11.5-3 An electrical heating coil of diameter D is being designed to keep a large tank of liquid above
its freezing point. It is desired to predict the temperature that will be reached at the coil sur-
Surface Temperature of face as a function of the heating rate Q and the tank surface temperature T . This prediction is
o
an Electrical Heating to be made on the basis of experiments with a smaller, geometrically similar apparatus filled
Coil with the same liquid.
Outline a suitable experimental procedure for making the desired prediction. Tempera-
ture dependence of the physical properties, other than the density, may be neglected. The en-
tire heating coil surface may be assumed to be at a uniform temperature T v

SOLUTION This is a free-convection problem, and we use the column labeled A in Table 11.5-1 for the di-
mensionless groups. From the equations of change and the boundary conditions, we know
that the dimensionless temperature T = (T - T^)/(T^ - T ) must be a function of the dimen-
o
sionless coordinates and depend on the dimensionless groups Pr and Gr.
The total energy input rate through the coil surface is

dS (11.5-26)
дг
Here r is the coordinate measured outward from and normal to the coil surface, S is the sur-
face area of the coil, and the temperature gradient is that of the fluid immediately adjacent to
the coil surface. In dimensionless form this relation is

dS = ф(Рг, Gr) (11.5-27)
~T )D
0
3
2
2
in which ф is a function of Pr = С ц/к and Gr = p g/3(T^ - T )D /fjL . Since the large-scale and
р
0
small-scale systems are geometrically similar, the dimensionless function S describing the
surface of integration will be the same for both systems and hence does not need to be in-
cluded in the function ф. Similarly, if we write the boundary conditions for temperature, ve-
locity, and pressure at the coil and tank surfaces, we will obtain only size ratios that will be
identical in the two systems.
We now note that the desired quantity (T ] - T ) appears on both sides of Eq. 11.5-27. If
o
we multiply both sides of the equation by the Grashof number, then (T - T ) appears only
o
}
on the right side:
= Gr • ф(?г, Gr) (11.5-28)

6
C. V. Sternling and L. E. Scriven, AIChE Journal, 5, 514-523 (1959); L. E. Scriven and С V. Sternling,
/. Fluid Mech., 19, 321-340 (1964).