Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 164

The exit boundary condition at ( x * ¼N )is

ðx * ¼N; 0 y * 1 Þ ¼

vc *
vx *

:

0 (5.4)

The wall conditions are

y * ¼ 0 ; 1 ¼

vc *
vy *

0 : (5.5)

The wall condition (5.5) is also the symmetry condition in the case of mixing with multiple
streams. Thus, this model can also be used for describing parallel lamination with more than two inlets.
Using separation of variables and the corresponding boundary conditions (5.3)-(5.5) , the dimen-
sionless concentration distribution in the mixing channel is
c * x * exp x *

p N
n¼

y * ¼ cos npy *

2 n 2 p 2 2 sin apn

n

; a þ p

Pe 2

n ¼ 1 ; 2 ; 3 :

4 n 2 p 2 Pe þ þ

1

(5.6)
The function of the concentration distribution (5.6) is described by a cosine function of the y *-axis
and an exponential function of the x *-axis. It is clear that complete mixing is determined by the Peclet
number in the exponential term. A large Peclet number requires a long mixing length along the x *-axis
to make the exponential term approach zero. Physically, a large Peclet number means that convection
is dominant over molecular diffusion and it would take longer for the solute to diffuse in the transversal
direction across the mixing channel.
The solution of concentration distribution (5.6) clearly shows that mixing can be improved by
operating the mixer at a small Peclet number. Since diffusion coefficient is a material property, a small
Peclet number can be achieved with a low flow velocity or a small width. Small width and a thin
striation thickness shorten the transversal transport of the solute. The simplest way to achieve thin
striations is to split the solvent and the solute into multiple mixing streams. As mentioned above,
solution (5.6) can also be extended to the case of multiple mixing streams. The wall boundary
condition is identical to the symmetry condition in the middle of each stream. Thus, the concentration
distribution (5.6) can be extended periodically along the transversal y- or y *-direction. Considering the
distance between two neighboring concentration extrema W min,max , the Peclet number is evaluated as
Pe UW min,max / D . Figure 5.2 shows the typical theoretical and experimental results in a parallel

lamination micromixer with two or three streams, respectively. The case of mixing with three streams
is often referred to as hydrodynamic focusing, where the width of the middle stream and consequently
the mixing time can be controlled by the sheath streams.
Formixing channelswith aspect ratio on the order of unity, thevelocity profile across the channel width
is parabolic and not uniform as assumed for the analytical model. The transport equation has the form

¼

vx ¼ D v 2 c

v 2 c
vz 2

vc

u ð y ; z Þ

vx 2 þ

(5.7)

where u ( y , x ) is the velocity profile of the channel cross-section. Because convective transport is
dominant in the flow direction x , diffusive term in x can be neglected. Eqn (5.7) is nonlinear and can
only be solved numerically [1] .