Environmental Engineering Reference
In-Depth Information
This expression is given in terms of the primary physical parameters that deter-
mine the development of convection: the relative difference of temperatures (
T
1
T
2
)/
T
1
, the specific force of the field
g
, a typical system size
L
, and also the transport
coefficients taking into account the types of interaction of gas flows in the course
of convection. Note that we use the specific heat capacity at constant volume
c
V
be-
cause of the condition that exists in our investigation. If equilibrium is maintained
instead at a fixed external pressure, it is necessary to use the specific heat capacity
at constant pressure in the above equations.
Equation (5.6) shows that the Rayleigh number determines the possibility of
the development of convection. For instance, in the Rayleigh problem the bound-
ary conditions at the plates are
T
0
D
0and
w
z
D
0. Also, the tangential forces
η
(
@
w
x
/
@
z
)and
η
(
@
w
y
/
@
z
) are zero at the plates. Differentiating the equation
div
w
D
0withrespectto
z
and using the conditions for the tangential forces, we
findthatattheplates
2
w
z
/
z
2
@
@
D
0. Hence, we have the boundary conditions
2
w
z
@
0,
@
T
0
D
0,
w
z
D
D
0.
z
2
We denote the coordinate of the lower plate by
z
D
0 and the coordinate of
the upper plate by
z
D
L
. A general solution of (5.6) with the stated boundary
conditions at
z
D
0 can be expressed as
T
0
D
C
exp[
i
(
k
x
x
C
k
y
y
)] sin
k
z
z
.
(5.9)
The boundary condition
T
0
D
0at
z
D
L
gives
k
z
L
D
π
n
,where
n
is an integer.
Inserting the solution (5.9) into (5.6), we obtain
(
k
2
L
2
2
n
2
)
3
C
π
Ra
D
,
(5.10)
k
2
L
2
where
k
2
k
y
. The solution (5.9) satisfies all boundary conditions.
Equation (5.10) shows that convection can occur for values of the Rayleig
h
num-
ber not less than Ra
min
,whereRa
min
refers to
n
k
x
C
D
/(
L
p
2). The
D
1and
k
min
D
π
numerical value of Ra
min
is [6]
4
/4
Ra
min
D
27
π
D
658 .
The magnitude of Ra
min
can vary widely depending on the geometry of the problem
and the boundary conditions, but in all cases the Rayleigh number is a measure of
the possibility of convection.
5.1.3
Convective Movement of Gases
To gain some insight into the nature of convective motion, we consider the simple
case of motion of a gas in the
xz
plane. Inserting solution (5.9) into the equation
div
w
D @
w
x
/
@
x
C @
w
z
/
@
z
D
0, the components of the gas velocity are
w
0
cos(
kx
)sin
π
nz
L
D
π
kL
w
0
sin(
kx
)cos
π
n
nz
L
w
z
D
,
w
x
,
(5.11)
