Biomedical Engineering Reference
In-Depth Information
a parallel mixer with two substreams, the mixing time or the required channel length can be reduced by
a factor of
n
2
.
The simplest parallel lamination mixer is a straight channel with two inlets. The two inlets form
a Y-shape or a T-shape. Thus, this mixer is often called Y-mixers or T-mixers. The following analytical
model describes the concentration distribution inside the straight channel. The model is two dimen-
sional and assumes that the microchannel is flat (
Fig. 5.1
). The height
H
of a flat channel is much
smaller than its width
W
(
H
W
). This model was solved numerically in Section 3.6.3.
Earlier, Example 2.4 showed that different viscosities will cause different flow velocities on each
side of the mixing channel. The mismatch in velocity will certainly affect the convective transport in
the mixing channel. In the fo
ll
owing model, the mixing streams are assumed to have the same viscosity
and the same mean velocity
u
to keep the model simple and analytically solvable. The mixer is a long
channel with a width
W
. The inlets are defined with the inlet boundary on the left, while the outlet is
defined with the exit boundary on the right side of the model. The inlets consist of a solute stream and
a solvent stream. The solute stream has a concentration of
c
¼
c
0
and a mass flow rate of
m
1
. The
_
solvent stream has a concentration of
c
m
2
. At an infinite exit position, the
two streams are well mixed such that no concentration gradient exists (
vc
/
vx
¼
0 and a mass flow rate of
_
¼
0,
vc
/
vy
¼
0).
With the above assumptions, the transport Eqn (2.22) can be reduced to the steady-state two-
dimensional form:
D
v
2
c
v
2
c
vy
2
u
vc
vx
¼
vx
2
þ
(5.1)
where
D
is the diffusion coefficient of the solute in the solvent. Assuming that both streams have the
same viscosity and fluid density, the dimensionless interface location
r
is equal to the mass fraction
a
of the solvent in the final mixture:
a
¼
_
m
1
=ð
_
m
1
þ
_
m
2
Þð
0
a
1
Þ
. The mixing ratio of the solute
and the solvent is therefore
a
:(1-
a
).
Introducing the dimensionless spatial va
ri
ables
x
*
¼
x
/
W
,
y
*
¼
y
/
W
, the dimensionless concentra-
tion
c
*
¼
c
/
c
0
, and the Peclet number Pe
¼
uW
=
D
, the transport equation has the dimensionless form:
Pe
vc
*
v
2
c
*
vx
*
2
þ
v
2
c
*
vy
*
2
:
vx
*
¼
(5.2)
The corresponding boundary conditions for the inlets are
c
*
j
ðx
*
¼
0
;
0
y
*
<rÞ
¼
1 and
c
*
j
ðx
*
¼
0
;ry
*
1
Þ
¼
:
0
(5.3)
FIGURE 5.1
The steady-state two-dimensional model of concentration distribution in a parallel lamination with two inlet streams.
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