Environmental Engineering Reference
In-Depth Information
form [1]
dT
dz
>
mg
c
p
.
(5.1)
In particular, for atmospheric air subjected to the action of the Earth's gravitational
field, this relation gives (
g
10
3
cm/s
2
,
c
p
D
D
7/2)
dT
dz
>
10 K/km .
(5.2)
Criterion (5.1) is a necessary condition for the development of convection, but if
it is satisfied that does not mean that convection will necessarily occur. Indeed, if
we transfer an element of gas from one point to another, it is necessary to overcome
a resistive force that is proportional to the displacement velocity. The displacement
work is proportional to the velocity of motion of the gas element. If this velocity is
low, the gas element exchanges energy with the surrounding gas in the course of
motion by means of thermal conductivity of the gas, and this exchange is greater
the more slowly the displacement proceeds. From these arguments it follows that
the gas viscosity and thermal conductivity coefficients can determine the nature of
the development of convection. We shall develop a formal criterion for this process.
5.1.2
Rayleigh Problem
If a motionless gas is unstable, small perturbations are able to cause a slow move-
ment of the gas, which corresponds to convection. Our goal is to find the threshold
for this process and to analyze its character. We begin with the simplest problem
of this type, known as the Rayleigh problem [5]. The configuration of this problem
is that a gas, located in a gap of length
L
between two infinite plates, is subjected to
an external field. The temperature of the lower plate is
T
1
, the temperature of the
upper plate is
T
2
, and the indices are assigned such that
T
1
T
2
.
We can write the gas parameters as the sum of two terms, where the first term
refers to the gas at rest and the second term corresponds to a small perturbation
due to the convective gas motion. Hence, the number density of gas molecules is
N
>
T
0
,andthegasve-
locity
w
is zero in the absence of convection. When we insert the parameters in this
form into the stationary equations for continuity (4.1), momentum transport (4.6),
and heat transport (4.27), the zeroth-order approximation yields
C
N
0
, the gas pressure is
p
0
C
p
0
, the gas temperature is
T
C
r
p
0
D
F
N
,
Δ
T
D
0,
w
D
0,
where
F
is the force acting on a single gas molecule. The first-order approximation
for these equations gives
(
p
0
C
p
0
)
η
Δ
w
N
0,
w
z
(
T
2
T
1
)
c
V
N
Δ
r
T
0
.
C
N
0
C
F
D
D
(5.3)
N
C
N
0
C
L
