Environmental Engineering Reference
In-Depth Information
Rayleigh problem. There will be a boundary layer of thickness
formed near the
walls, within which a transition occurs from zero fluid velocity at the wall itself
to the motion occurring in the bulk of the fluid. The thickness of the bound-
ary layer is determined by the viscosity of the gas, and the heat transport in
the boundary layer is accomplished by thermal conduction, so the heat flux can
be estimated to be
q
δ
. Applying the Navier-Stokes
equation (4.15) to the boundary region, one can estimate its thickness. This equa-
tion describes a continuous transition from the walls to the bulk of the gas flow.
We now add the second term in the Navier-Stokes equation,
m
(
w
r
D r
T
(
T
1
T
2
)/
δ
)
w
,which
cannot be ignored here, to the expression in (5.3). An order-of-magnitude com-
parison of separate terms in the
z
-component of the resulting equation yields
mw
z
/
2
. Hence, we find that the boundary layer
δ
F
(
T
1
T
2
)/
T
η
w
z
/
N
δ
thickness is
1/3
2
T
N
2
Fm
3
(
T
1
η
δ
.
(5.14)
T
2
)
In the context of the Rayleigh problem, we can compare the heat flux transport-
ed by a gas due to convection (
q
) and that transported due to thermal conduction
(
q
cond
). The thermal heat flux is
q
cond
D
(
T
1
T
2
)/
L
, and the ratio of the fluxes is
N
2
FmL
3
(
T
1
1/3
q
q
cond
L
δ
T
2
)
Gr
1/3
.
(5.15)
η
2
T
Here Gr, the Grashof number, is the following dimensionless combination of pa-
rameters:
N
2
FmL
3
(
T
1
gL
3
ν
T
2
)
D
Δ
T
T
D
Gr
.
(5.16)
η
2
T
2
A comparison of the definitions of the Rayleigh number (5.7) and the Grashof num-
ber (5.16) gives their ratio as
Ra
Gr
c
V
η
D
ν
D
D
Pr
.
m
where Pr is the Prandtl number.
The continuity equation (4.1), the equation of momentum transport (4.6), and
the equation of heat transport (4.27) are valid not only for a gas, but also for a liquid.
Therefore, the results we obtain are also applicable to liquids. However, a gas does
have some distinctive features. For example, estimates (4.40) and (4.13) show that
for a gas the ratio
) is of order unity. Furthermore, the specific heat capacity
c
V
of a single molecule is also close to 1. Hence, the Rayleigh number has the same
order of magnitude as the Grashof number for a gas. Since convection develops at
high Rayleigh numbers, we find that for convection Gr
η
/(
m
1. Therefore, according
to the ratio in (5.15), we find that heat transport via convection is considerably more
effective than heat transport in a motionless gas via thermal conduction.
Theratio(5.15)betweentheconvectiveandconductiveheatfluxeswasderived
for an external force directed perpendicular to the boundary layer. We can derive
