Environmental Engineering Reference
In-Depth Information
2.4 Continuity Equation for Mass
For the mathematical formulation of mass conservation consider the change of
mass during the small time
D
z
,
each for one direction in the three-dimensional space. There are two ways to
calculate changes of mass. One method is to consider the mass within the control
volume at the beginning and at the end of the time period and calculate the
difference. The other method is to balance all fluxes across the boundaries of the
volume. Balancing means that fluxes into the volume have to be taken as positive,
while those leaving the volume are negative. In three-dimensional space six faces of
the control volume have to be taken into account.
A simpler set-up for the one-dimensional space is depicted in Fig.
2.3
. A box
contains a certain amount of mass at the start of the time period, and a different
amount at the end. During the time period there was influx on one face and outflux
at the other. The graphical symbols in the equation at the bottom of the figure will
be replaced by mathematical formulae in the following derivation.
The mass at the beginning and the end of the period
t
and
t
+
D
t
within a control volume with spacing
D
x
,
D
y
and
D
t
is given by:
y cðx; tÞ
D
x
D
y
D
z
and
y cðx; t þ
D
tÞ
D
x
D
y
D
z
where
y
denotes the share on the total volume. In case of a saturated porous medium
y
is the
volumetric water saturation, when the aqueous phase is concerned. In the situation
in which two fluids occupy the space (for example water and oil) the share of each
denotes porosity. In the unsaturated zone, within soil for example,
y
start
end
Principle of mass conservation
Fig. 2.3 Illustration for the
derivation of the mass
conservation equation
-
=
-
