Geoscience Reference
In-Depth Information
volume is permeable, allowing for advective and/or diffusive fluxes, transport also
will contribute for the evolution of the property inside the volume.
3.3.1
R
A
ATE
OF
CCUMULATION
The rate of accumulation of a property
inside an elementary volume is the ratio
between the variation of the total amount contained in the volume and the time
interval during which accumulation happened. This concept can be translated by the
algebraic equation:
B
t
+
t
t
()
β
V
()
β
V
t
In this equation, the total amount of
B
inside an elementary volume
V
in each moment
is given by the product of the specific value
If an
infinitesimal time interval is considered, the previous equation can be written in a
differential form:
β
multiplied with the volume.
dV
dt
()
β
This equation puts into evidence the physical meaning of the time derivative, showing
that it describes the rate of accumulation.
3.3.2
L AGRANGIAN F ORM OF THE E VOLUTION E QUATION
If there is no flux across the boundary of the control volume, the rate of accumulation
accounts for the sources ( S 0 ) minus the sinks ( S i ):
dV
dt
()
β
=
VS
(
S
)
o
i
This can happen in the case of a volume limited by a solid boundary or in the case of
a volume moving at the same velocity as the flow where diffusivity can be neglected.
Faecal bacteria are traditionally assumed to be unable to grow in saline water
and have a first-order decay rate. For such a variable in a volume where advective
and diffusive fluxes are null and the sink is the mortality of bacteria, the evolution
equation would be written as
dV
dt
()
β
=−
mV
β
where m is the rate of mortality.
In a general case, the total amount inside a volume V of a property with a specific value β is the integral
of β inside the volume. In the case that β is constant, the integral is the product of β times the volume.
 
 
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