Active micromixers - Micromixers Fundamentals Design and Fabrication - page 278

FIGURE 7.31
Concentration field in an annular micromixer based on MHD flow (b ¼

0.5, D ¼ 0) (after [44] ).

The mixing measure at a time t is defined as:

Z RþW Z 2 p

2

½ c ð r ; q ; t Þ c ð t Þ

R W 0

m ð t Þ¼ (7.71)

Z RþW Z 2 p

2 p
0

2

½ c ð r ; q ; t Þ c ð t Þ

RW 0

where c is the average concentration value over the whole annulus channel:

Z RþW Z 2 p

1
4 pRW

c ð t Þ¼ c ð r ; q ; t Þ r d q d r (7.72)

RW 0

Mixing measure will decrease from 1 to 0 ( m ). For eval-

uation of mixing effective, a mixing time t mixing is defined when the mixing measure reaches
a prescribed value M :

/ 0 when time is going to infinity t /N

m ð t mixing Þ¼ M : (7.73)

The mixing time can be made dimensionless by normalizing by R 2 / D :

t mixing
R 2

t mixing ¼

D ;

(7.74)

=

which eliminates the dependence of mixing time on radius of curvature and diffusion coefficient.
At low Peclet numbers (Pe < 7.2), the mixing process is determined by molecular diffusion. The
mixing time depends only on the geometry of the channel and the diffusion coefficient:

1

3 b 2

1
2 ln

1

t mixing z

M 1 ð Þ : (7.75)

< <

15/ b ), the mixing process is determined by Taylor

At intermediate Peclet numbers (7.2 Pe

dispersion:

1

1576 b 2

105
4

18165

t mixing z Pe 2 ln ðM 1

Þ : (7.76)