Environmental Engineering Reference
In-Depth Information
Figure 5.1
The paths of the gas elements in the Rayleigh problem for
n
D
1and
k
D
π
/
L
.
where
n
is an integer and the amplitude of the gas velocity
w
0
is assumed to be
small compared with the corresponding parameters of the gas at rest. In particular,
w
0
is small compared with the thermal velocity of the gas molecules.
The equations of motion for an element of the gas are
dx
/
dt
D
w
x
and
w
z
, where relations (5.11) give the components of the gas velocity.
We obtain
dx
/
dz
dz
/
dt
D
nz
/
L
). This equation describes the
path of the gas element. The solution of the equation is
sin(
kx
)sin
π
D
[
π
n
/(
kL
)] tan(
kx
)cot(
π
nz
L
D
C
,
(5.12)
where
C
is a constant determined by the initial conditions. This constant is bound-
ed by
1; its actual value depends on the initial position of the gas element.
Of special significance are the lines at which
C
1and
C
D
0. These lines are given by
Lp
1
n
Lp
2
n
z
D
x
D
,
,
(5.13)
where
p
1
and
p
2
are nonnegative integers. The straight lines determined by (5.13)
divide the gas into cells. Molecules inside such a cell can travel only within this cell
and cannot leave it. Indeed, (5.13) shows that the component of the gas velocity
directed perpendicular to the cell boundary is zero; that is, the gas cannot cross the
boundary between the cells. These cells are known as Benard cells [7-9].
Formula(5.13)showsthatinsideeachcellthegaselementstravelalongclosed
paths around the cell center, where the gas is at rest. Figure 5.1 shows the path of
the elements of gas in the Rayleigh problem for
n
D
1and
k
D
π
/
L
, corresponding
D
4
D
to the Rayleigh number Ra
779. In the Rayleigh problem, the Benard cells
are pyramids with regular polygons as bases; in a general case, these cells can have
a more complicated structure.
8
π
5.1.4
Convective Heat Transport
Convection is a more effective mechanism of heat transport than is thermal con-
duction. We can illustrate this by examining convective heat transport for the
