Three-dimensional Lagrangian Tracer Modelling in Wadden Sea ...

CHAPTER 2. THEORY 15
Diffusion processes that obey this relationship are called Fickian processes,
and Eq. (2.2.7) is called Fick’s law. The diffusive flux �q is expressed in units
kg/(m2 s). In order to compute a total mass flux rate m with units kg/s the
normal component of the diffusive flux vector has to be integrated over a
surface area A
� �
m = �q · �n dA (2.2.8)
A
where �n is the unit vector normal to the surface element dA. Although Fick’s
law gives an expression for the flux of mass due to diffusion, an equation that
predicts the change in concentration of the diffusing mass over time at a point
(i.e. ∂C
∂t
) is still needed. Such an equation can be derived by determining
the change in concentration with time C of a dissolved substance in a fixed
control volume CV (i.e. ∆x, ∆y, ∆z = ∆V = const.) together with the law
of conservation of mass as depicted in Fig. 2.2.2
� � �
∂
∂t
C dV. (2.2.9)
V
It should be noted that a change in C only occurs due to mass flux through
the surface areas (i.e. no sources or sinks are present in CV ). For the
x-direction the net flux qx according to Fick’s law is
� �
∂C �
qx = qx,in − qx,out = −D � −
∂x
∂C
� �
�
� .
∂x
(2.2.10)
� x=1
Here, x = 1 and x = 2 are the locations of the left and right surface areas
of CV in x-direction, respectively. The net mass flux rate m for CV is Eq.
(2.2.8) and conservation of mass requires that
∂
∂t
� � �
V
� �
CdV +
A
� x=2
�q · �n dA = 0. (2.2.11)
By bringing the time differentiation inside the volume integral and applying
the divergence theorem of Gauss
� � �
� �
∇ · �q dV = �q · �n dA (2.2.12)
V
A
to the surface integral, Eq. (2.2.11) can be written as
� � � � �
∂C
+ ∇ · �q dV = 0.
∂t
(2.2.13)
V
Since this equation holds for an arbitrary volume CV , the integrand must
vanish thus leading to
∂C
∂t
+ ∇ · �q = 0. (2.2.14)