Environmental Engineering Reference
In-Depth Information
bed whose thickness is denoted
h
n
and which is made up of the same particles as
the cloud. The apparent density of the layer is
ρ
s
=
φ
m
ρ
p
+(1
− φ
m
)
ρ
a
, where
φ
m
denotes the maximum random volume concentration of particles.
The surface area (per unit width) exposed to
the s
urrounding fluid is denoted
S
and
can be related to
H
and
L
as follows:
S
=
k
s
√
HL
, where
k
s
is a shape factor. Here
we assume that the cloud keeps a semi-elliptic form, whose aspect ratio
k
=
H/L
remains constant during the cloud run when the slope is constant. We then obtain
4
k
2
)
/
√
k,
k
s
=E(1
−
[2.7]
where E denotes the elliptic integral function. Similarly, we can also express the
volume
V
(per unit width) as:
V
=
k
v
HL
, where
k
v
is another shape factor for a
half ellipsis. Here we have
k
v
=
π/
4
.
[2.8]
In the following, we will also need to use the volume, height, and length growth rates:
1
√
V
dV
dx
,α
h
=
dH
dx
,α
l
=
dL
α
v
=
dx
.
[2.9]
Experimentally, it is easier to measure the growth rates by deriving the quantity at
hand by the front abscissa instead of the mass center abscissa; we will refer to these
rates as:
1
√
V
d
x
f
, α
h
=
d
H
d
V
d
x
f
, α
l
=
d
L
α
v
=
d
x
f
.
[2.10]
Note that all these quantities are interrelated. For instance, using
x
=
x
f
− L/
2,we
find:
α
h
=(d
H/
d
x
)(d
x/
d
x
f
)=
α
h
(1
−
α
l
/
2). Similarly, using the definition of
k
and
k
v
, we obtain:
k
k
v
α
h
=
α
2
α
v
2
√
kk
v
.
and
α
l
=
[2.11]
The KSB model outlined here includes three equations: volume, mass and
momentum balances. The volume variations mainly result from the entrainment of the
ambient, less dense fluid. Various mixing processes are responsible for the entrainment
of the ambient fluid into the cloud. It has been shown for jets, plumes and currents that
(i) different shear instabilities can occur at the interface between dense and less dense
fluids and (ii) the rate of growth of these instabilities is controlled by a Richardson
number, defined here:
Ri
=
g
H
cos
θ
U
2
,
[2.12]
