Environmental Engineering Reference
In-Depth Information
(
3.19
). Concerning the mathematical type both differences do not change the type
of the differential equation: it remains second order in space and first order in
time, and is thus of parabolic type. There is a first time derivative on the left side of
the equation, and on the right side there are three space derivative terms: the second
order term, representing diffusion and dispersion, a first order term, representing
advection, and a source term which usually does not contain derivatives and is
thus of zero order.
3.4 Dimensionless Formulation
Without the source term one often reads of the
advection-diffusion-equation
,in
connection with the differential equations derived in the preceding sub-chapter.
Depending on how the processes are understood, one may also speak of advection-
dispersion, convection-diffusion or convection-dispersion-equation. These terms
can be found for the transport equations for mass (Eq.
3.19
) or for heat (Eq.
3.30
).
For a better illustration of the sensitivity of the solutions, it is often appropriate
to use the dimensionless formulation of these transport equations. In the sequel, we
consider the situation with constant coefficients and no sources or sinks. The
transport equation in dimensionless form is:
@o
@t
¼
@
Pe
@o
1
@x
@o
(3.31)
@x
@x
cc
0
x
vt
L
and dimensionless P´clet
7
-
with dimensionless variables
o ¼
c
in
c
0
;x ¼
L
;t ¼
vL
D
number
Pe ¼
for the mass transport equation,and with dimensionless variables
TT
0
x
L
;t ¼
ykvt
and dimensionless P´clet-number
Pe ¼
ykvL
o ¼
D
T
for the heat
transport equation. The advantage of formulation (
3.31
) is obvious. There is only
one parameter left, which is the P´clet number. The behavior of the solutions can
thus be explored by the variation of that single parameter, and can often be
visualized nicely in a single diagram (see Chap. 5.3 for an example).
T
in
T
0
;x ¼
L
3.5 Boundary and Initial Conditions
In the preceding part of the topic, the fundamental theoretical and empirical laws
are presented and it is shown how these are combined to yield differential
equations. For most differential equations several functions can be found which
fulfil the equation. The equation
@u=@s ¼uðsÞ
for example is fulfilled by the
7
Jean Claude Eug`ne P´clet (1793-1857), French physicist.
