Environmental Engineering Reference
In-Depth Information
Flux components in
y
- and
z
-direction can also be taken into account, based on
formulae analogous to formula for the
x
-direction. The fluxes
j
y
; j
yþ
; j
z
and
j
zþ
have to be introduced, balanced and the balances added on the right side of (
2.1
) and
(
2.2
). Taking the limits
D
y !
0 and
D
z !
0 one obtains:
y
@c
@t
¼
@
@x
yj
x
þ
@
@y
yj
y
þ
@
@z
yj
z
þ q
(2.5)
which is the generalized formulation of the mass conservation for three space
dimensions. Using the formal
-operator (speak: 'nabla'),
∇
0
@
1
A
!
in 2D,
@
@x
@
@y
@
@z
@
@x
@
@y
¼
@
@x
r¼
in 3D,
¼
in 1D
(2.6)
the equation can be written more compactly:
y
@c
@t
¼r
y
j
þ q
(2.7)
-operator the short notation of the continuity
(
2.7
) is valid in one-, two- or three-dimensional space. On the right side the
With the different forms of the
∇
-
∇
0
@
1
A
as a vector product. In the
j
x
j
y
j
z
operator is multiplied by the flux vector
y
j
¼ y
denotes the standard
vector product,
4
formulae, here and in the following, the
which in MATLAB
is applied by using the
*
multiplication and the transpose of
one column vector. Examine with the following command:
®
An advantage of the formulation (
2.7
) is that it is valid for one-, two- and three-
dimensional situations. The number of components in the flux-vector and
-
operator is equal to the number of space dimensions. In two dimensions, as
illustrated in Fig.
2.4
, the flux vector has two components. The illustration is
concerned with a fluid, for which the mass conservation principle can also be
applied, as for any other chemical species. When the fluid density is not changing,
∇
4
In three dimensions for vectors arbitrary vectors u and v:
0
@
1
A
0
@
1
A
¼ u
x
v
x
þ u
y
v
y
þ u
z
v
z
, not to be confused with the cross-product u
v;
u
x
u
y
u
z
v
x
v
y
v
z
u
v
¼
another formulation, found in the literature is:
@c
@t
¼div
j; divergence 'div' is another expression
for a vector product with the nabla-operator.
