Environmental Engineering Reference
In-Depth Information
z
-direction are zero. As stated above, both formulations measure the change of mass
and thus need to be equal:
c
ð
x
;
t
þ
D
t
Þ
c
ð
x
;
t
Þ
D
t
y
D
x
D
y
D
z ¼ y j
x
ðx; tÞj
xþ
ðx; tÞ
ð
Þ
D
y
D
z
(2.1)
Division through the volume
D
x
D
y
D
z
and porosity
y
yields:
c
ð
x
;
t
þ
D
t
Þ
c
ð
x
;
t
Þ
D
t
j
x
þ
ð
x
;
t
Þ
j
x
ð
x
;
t
Þ
D
x
¼
(2.2)
From this equation a differential equation can be derived by the transition of the
finite grid spacing
D
x
and time step
D
t
to infinitesimal expressions, e.g. by the limits
D
x !
0 and
D
t !
0. It follows:
@c
@t
¼
@
j
x
(2.3)
@x
which is a differential formulation for the principle of mass conservation. The
presumption for the differentiation procedure is that the functions
c
and
j
x
, are
sufficiently smooth, mathematically speaking differentiable, which is usually
taken for granted. Equation
2.3
is valid for one-dimensional transport and is the
basis for the mathematical analysis of transport processes. The unit of the equation is
[M/(L
3
·T)].
Formulation (
2.3
) is valid if there are no internal sources or sinks for the
concerned biogeochemical species. Sources and sinks are understood here in the
most general sense: each process, which creates or destroys some species mass, can
contribute to such a source or sink. In the remainder of this volume we will see
examples, where chemical reactions and inter-phase exchange of species can be
included in that way.
Easily the given mathematical formulation can be extended to consider sources
and sinks additionally. If these are described by a source- or sinkrate
q
(
x,t
)
[M/(L
3
·T)], which may vary spatially and temporally, one simply has to add a
corresponding integral term
ð
ð
qðx; tÞdtdx
D
x
D
t
on the right side of (
2.1
) and (
2.2
). The term is positive, if mass is added (source)
and negative, if mass is removed (sink). In the derivation of (
2.3
) the integral term
had to be differentiated, which leads to the general transport equation in one space
dimension:
y
@
c
@t
¼
@
@x
yj
x
þ q
(2.4)