Environmental Engineering Reference
In-Depth Information
Fig. 2.4 Illustration of a
control volume in two space
dimensions
x
and
y
Control Volume
j
y+
Δ
V
j
x-
j
x+
j
y
-
Δ
y
Δ
x
z
grid
y
x
phase has to be taken into account, too.
D
x
D
y
D
z
is the volume.
c
denotes the
concentration, measured as [mass/volume]. The change of mass per time is given
by:
c
ð
x
;
t
þ
D
t
Þ
c
ð
x
;
t
Þ
D
t
y
D
x
D
y
D
z
Fluxes in
x
-direction are given across faces of the control volume:
yj
x
ðx; tÞ
D
y
D
z
and
yj
xþ
ðx; tÞ
D
y
D
z
where
j
x
denotes mass flux per area across the left face of the volume, in negative
x
-direction. Analogously
j
xþ
denotes the mass flux in
x
-direction across the right
face, in positive
x
-direction (see Fig.
2.4
). The fluxes may change spatially and
temporally which do the brackets indicate. Both fluxes are positive, if they add mass
to the control volume, and negative otherwise. The physical unit of mass flux is
[M/(L
2
·T)]. The term
y
D
y
D
z
denotes the area, through which flow takes place.
3
The balance between both flux terms is thus given by:
y j
x
ðx; tÞj
xþ
ðx; tÞ
ð
Þ
D
y
D
z
For simplicity the fluxes across the four other faces are neglected for the
derivation at this point. One may assume here that the flux components in
y
- and
3
It is generally assumed that the volumetric share and the area share compared to the entire
volume or area respectively are both quantified by the same number, here
. This is not necessarily
true. Especially in technical systems such as filters both ratios may vary significantly. The
practioner in the field is usually happy, if there is one value at all.
y
