koff - LEPA
koff - LEPA
koff - LEPA
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Annexe 2: Laminar flow. The Grätz problem<br />
We shall consider here a solution flowing parallel to two plane onto which molecules are<br />
adsorbed on a section of length l as shown in Figure 13. The problem is analogous to the current<br />
obtained in a band electrode in a flow channel.<br />
The equation of flux conservation is given by<br />
!c<br />
!t<br />
Fig. 13. Adsorption in a flow channel<br />
= " divJ = D# 2 c " v·gradc A2.1<br />
In steady state, for a 2D system where the x axis is parallel to the electrode and the y axis<br />
perpendicular to it, we have only<br />
v x<br />
!c<br />
!x + v !c<br />
y<br />
!y = D !2c !y 2 + !2c !x 2<br />
" %<br />
$ '<br />
# &<br />
( D !2c !y 2<br />
A2.2<br />
by neglecting the longitudinal diffusion. The continuity equation for an incompressible fluid is<br />
classically given by<br />
!vx !x + !vy !y<br />
= div(!v) = 0 A2.3<br />
We can solve this equation by doing a series development on y, to obtain<br />
and<br />
with<br />
vx = vx (y = 0) + y !v " x %<br />
#<br />
$<br />
!y &<br />
vy = vy(y = 0) + y !v " y %<br />
#<br />
$<br />
!y &<br />
= 1 2 y2 !<br />
"<br />
!y #<br />
$<br />
!v y<br />
!y<br />
= ( 1 2 y2 ! "<br />
!x #<br />
$<br />
%<br />
&<br />
'<br />
y=0<br />
!v x<br />
!y<br />
%<br />
&<br />
' y=0<br />
+... = y!(x) A2.4<br />
"<br />
' +<br />
y=0<br />
1 2 y2 !2vy !y 2 $<br />
#<br />
%<br />
'<br />
&<br />
+... = ( 1 2 y2 ! "<br />
!y #<br />
$<br />
y=0<br />
!v x<br />
!x<br />
+...<br />
%<br />
&<br />
' y=0<br />
' +... = (<br />
y=0<br />
1 2 y2 ! '(x) +...<br />
+...<br />
A2.5<br />
28
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