Environmental Engineering Reference
In-Depth Information
9.3
1D Steady State Analytical Solution
Steady-state or stationary means time-independent. In real systems steady state
is achieved when the relevant timescale for the problem is too long compared to the
internal time scale.
The transport equation for the steady state is simply obtained by omitting the
storage term on the left side, which includes the time-derivative
@c=@t . The right
hand side of the 1D transport equation has thus been set to zero:
@x D @ c
@
@x v @ c
@x þq ¼
0
(9.6)
Equation ( 9.6 ) is an ordinary differential equation, as only space derivatives in
a single direction ( x ) appear. For constant parameter values, the solutions of such
an ordinary differential equation can be written explicitly. In the first step to obtain
a solution we neglect sources and sinks, i.e. the last term in ( 9.6 ) is omitted: q ¼
0.
If there is a Dirichlet condition at one side and a Neumann condition at the other
side of a finite system with length L
0
Þ¼c in
@c=@xðLÞ¼
0
(9.7)
the solution is obviously given by the constant function c ¼ c in , because all
derivatives vanish. For constant parameters D and v and Dirichlet boundary
conditions on both sides
0
Þ¼c in
cðLÞ¼c 1
(9.8)
the solution is:
1
exp
ðvx=DÞ
cðxÞ¼c in þ c 1 c in
ð
Þ
(9.9)
ðvx=DÞ
1
exp
or in dimensionless form:
c ðxÞ c in
c 1 c in ¼
1
exp
ðPexÞ
(9.10)
1
exp
ðPeÞ
, dimensionless P ´ clet number Pe ¼ vL=D for
dimensionless concentration on the left side of ( 9.10 ). Note that we here allow the
P ´ clet number to have a sign depending on flow direction. Results for selected
values of the P ´ clet number are shown in Fig. 9.4 . In all cases a steeper gradient
can be observed near the outflow boundary; i.e. for negative Pe on the left, and
for positive Pe on the right boundary. The deviation in the slope between inflow
and outflow side increases with the absolute value of Pe . The linear profile, which
with dimensionless space variable
x
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