Environmental Engineering Reference
In-Depth Information
Subtracting from this the continuity equation in the form
@
(
Nw
α
)
N
@
t
C
@
mw
α
D
0,
@
x
we obtain
m
@
w
α
@
N
@
1
P
α
@
mw
@
w
α
@
t
C
x
C
x
F
α
D
0.
(4.4)
The form of this equation for the mean particle momentum depends on the rep-
resentation of the pressure tensor
P
α
, which is determined by properties of the
system. In the simplest case of an isotropic gas or liquid it can be expressed in the
form
P
α
D
p
δ
α
,
(4.5)
where
δ
α
is the Kronecker symbol and
p
is the pressure inside the gas or liq-
uid. Insertion of this expression into (4.4) transforms the latter into the following
equation, which in vector form is given by
@
w
@
(
w
r
)
w
C
r
p
m
D
t
C
0
(4.6)
and is the Euler equation in the absence of external fields (
F
D
0) [4-8]. Here
w
is the average velocity of the gas,
mN
is the mass density, and
F
is the force
acting on a single particle from external fields.
The macroscopic equations (4.1) and (4.4) relate to a one-component gas. One
can generalize these equations for multicomponent gaseous systems. The right-
hand side of the continuity equation is the variation of the number density of parti-
cles of a given species per unit volume and per unit time due to production or loss
of this type of particles. In the absence of production of particles of one species as
a result of decomposition of particles of other species, the continuity equation has
the form of (4.2) regardless of processes involving other particles.
The right-hand side of (4.4) for a multicomponent system contains the variation
per unit time of the momentum of particles of a given species as a result of colli-
sions with particles of other types. If the mean velocities of the two types of parti-
cles are different, a momentum transfer will occur between them. The momentum
transfer between two species is proportional to the difference between their mean
velocities. Therefore, one can rewrite (4.4) as
D
w
(
s
)
w
(
q
)
P
(
q
)
α
α
w
(
q
)
m
(
q
)
N
(
q
)
@
w
(
q
)
X
@
1
@
F
m
q
D
α
α
w
(
q
)
α
t
C
x
C
x
.
(4.7)
@
@
@
τ
qs
s
Superscripts
q
and
s
denote here the particle species and
qs
is the characteristic
time for the momentum transfer from species
q
to species
s
.Sincethetransfer
τ
