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 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)
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