Page 377 - Bird R.B. Transport phenomena

§11.5 Dimensional Analysis of the Equations of Change for Nonisothermal Systems 359

We now restate the problem in dimensionless form, using / = h. We use the dimensionless
0
quantities listed under Case В in Table 11.5-1, and we select the reference temperature T to be
l(T + T ), so that
0 A
Continuity: (V • v) = 0 (11.5-20)
(V- v)
== 0
2
2 2
GrPr (
Motion: Щ = - W + PrV v - - Hi) l(f - \) (11.5-21)
v
£
r
V
P
)(T -
Dt
DT 2 :If
V
Energy: ^ = = V T (11.5-22)
Dt
with dimensionless boundary conditions
0
B.C1: at z = 0, v V = = 0 f - 0 (11.5-23)
= 0
f
=
B.C. 2: at z = 1, v V = = 0 f = 1 1 (11.5-24)
0
f
=
B.C. 3: at r = R/h, v V = = 0 df/dr = 0 0 (11.5-25)
df/dr
0
If the above dimensionless equations could be solved along with the dimensionless boundary
conditions, we would find that the velocity and temperature profiles would depend only on
Gr, Pr, and R/h. Furthermore, the larger the ratio R/h is, the less prominent its effect will be,
and in the limit of extremely large horizontal plates, the system behavior will depend solely
on Gr and Pr.
If we consider only steady creeping flows, then the term Dv/Dt may be set equal to zero.
Then we define a new dimensionless pressure difference as Ф* = РгФ. With the left side of Eq.
11.5-21 equal to zero, we may now divide by Pr and the resulting equation contains only one
2
dimensionless group, namely the Rayleigh number 1 Ra = GrPr = p gfi(T } - T )IQC /цк,
0
P
whose value will determine the behavior of the system. This illustrates how one may reduce
the number of dimensionless groups that are needed to describe a nonisothermal flow system.
The preceding analysis suggests that there may be a critical value of the Rayleigh num-
ber, and when this critical value is exceeded, fluid motion will occur. This suggestion has
2 3
been amply confirmed experimentally ' and the critical Rayleigh number has been found to
be 1700 ± 51 for R/h»\. For Rayleigh numbers below the critical value, the fluid is station-
ary, as evidenced by the observation that the heat flux across the liquid layer is the same as
that predicted for conduction through a static fluid: c\ = k{T - TJ/h. As soon as the critical
z
Q
Rayleigh number is exceeded, however, the heat flux rises sharply, because of convective en-
ergy transport. An increase of the thermal conductivity reduces the Rayleigh number, thus
moving Ra toward its stable range.
The assumption of creeping flow is a reasonable one for this system and is asymptoti-
cally correct when Pr —> <». It is also very convenient, inasmuch as it allows analytic solutions
4
of the relevant equations of change. One such solution, which agrees well with experiment,
is sketched qualitatively in Fig. 11.5-2. This flow pattern is cellular and hexagonal, with up-
flow at the center of each hexagon and downflow at the periphery. The units of this fascinat-
5
ing pattern are called Benard cells. The analytic solution also confirms the existence of a
critical Rayleigh number. For the boundary conditions of this problem and very large R/h it
4
has been calculated to be 1708, which is in excellent agreement with the experimental result
cited above.
1
The Rayleigh number is named after Lord Rayleigh (J. W. Strutt), Phil. Mag., (6) 32, 529-546 (1916).
2
P. L. Silveston, Forsch. Ingenieur-Wesen, 24, 29-32,59-69 (1958).
3
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Instability, Oxford University Press (1961);
Т. Е. Faber, Fluid Dynamics for Physicists, Cambridge University Press (1995), §8.7.
4
A. Pellew and R. V. Southwell, Proc. Roy. Soc, A176, 312-343 (1940).
5
H. Benard, Revue generale des sciences pures et appliquees, 11,1261-1271,1309-1328 (1900); Annales de
Chimie et de Physique, 23, 62-144 (1901).