Environmental Engineering Reference
In-Depth Information
The parameters in the present problem are used in the last equation. The
z
-axis is
taken to be perpendicular to the plates.
We transform the first term in the first equation in (5.3) with first-order accuracy,
and obtain
1
(
p
0
C
p
0
)
p
0
N
p
0
N
p
0
N
N
N
0
D
F
N
0
N
p
0
N
r
D
r
C
r
r
C
r
.
N
C
N
0
According to the equation of state (4.12) for a gas,
N
D
p
/
T
,wehave
N
0
D
NT
0
/
T
. Inserting this relation into the second equation in (5.3),
we can write the set of equations in the form
@
N
/
@
T
)
p
T
0
D
(
p
0
N
F
T
0
r
T
η
Δ
w
L
c
V
N
(
T
2
T
0
. (5.4)
div
w
D
0,
D
0,
w
z
D
T
1
)
Δ
N
We can reduce the set of equations (5.4) to an equation of one variable. For this goal
we first apply the div operator to the second equation in (5.3) and take into account
the first equation in (5.3). Then we have
0
p
0
N
Δ
F
@
T
D
0.
(5.5)
T
@
z
Here we assume that
T
1
T
1
. Therefore, the unperturbed gas parameters
do not vary very much within the gas volume. We can ignore their variation and
assume the unperturbed gas parameters to be spatially constant.
We take
w
z
from the third equation in (5.4) and insert it into the
z
component of
the second equation. Applying the operator
T
2
Δ
to the result, we obtain
T
0
N
@
1
F
Δ
L
c
V
N
2
(
T
2
η
p
0
2
T
0
D
z
Δ
C
T
1
)
Δ
0.
@
T
p
0
and
T
0
,weobtain[1]
Using the relation (5.5) between
Δ
L
4
z
2
T
0
,
2
Ra
@
3
T
0
D
Δ
Δ
(5.6)
@
where the dimensionless combination of parameters
T
2
)
c
V
FN
2
L
3
η
(
T
1
Ra
D
(5.7)
T
is called the Rayleigh number.
The Rayleigh number is fundamental for the problem we are considering. We
can rewrite it in a form that conveys clearly its physical meaning. Introducing the
kinetic viscosity
ν
D
η
/
D
η
/(
Nm
), where
is the gas density, the thermal
diffusivity coefficient is
D
/(
Nc
V
), and the force per unit mass is
g
D
F
/
m
,the
Rayleigh number then takes the form
T
1
T
2
gL
3
ν
Ra
D
.
(5.8)
T
