Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 69

Lewis number is the ratio between heat transport and diffusive mass transport:

heat transport
diffusive mass transport ¼

a
D ¼

k
rcD :

Le ¼ (2.194)

The ratio between advective transport and momentum transport is called the Reynolds number:

advective mass transport
momentum transport ¼

ruL ch
m ¼

uL ch
v :

Re ¼ (2.195)

where u is the mean velocity in the flow direction and L ch is the characteristic length of the considered
channel. In many cases, the hydrauli c diameter of the channel is taken as the characteristic length L ch .
For typical values of L ch ¼

10 5 cm 2 /s, the typical Reynolds number is 100 m m, u ¼ 1 mm/s, and v ¼

Re 0.01. This small number means that laminar flow exists in almost all microfluidic applications.

Peclet number is the ratio between advective mass transport and diffusive mass transport:

¼

advective mass transport
diffusive mass transport ¼

uL ch
D :

Pe ¼ (2.196)

The ratio between the Peclet number and the Reynolds number is actually the ratio between
momentum transport and diffusive mass transport or the Schmidt number above. For diffusion coef-
ficients ranging from 10 5 m 2 s 1 to 10 7 cm 2 s 1 , the Peclet numbers for the typical values in the

above example are 100 10,000. This means, advective mass transport dominates over diffusive

transport in almost all microfluidic applications.
The average diffusion time t over the characteristic mixing length L mixing , also called the striation
thickness, is represented by the Fourier number [7] :

< Pe <

D t diff
L 2 mixing :

Fo ¼ (2.197)

The Fourier number is usually in the range between 0.1 and 1. For a simple T-mixer with two
streams in a microchannel of a length L mixer and a width of W , the residence time should be the same as
the average diffusion time:

t res ¼ t diff

Fo L 2 mixing
2

(2.198)

Fo W 2
D :

L mixer
U ¼

¼

Thus, the ratio between the channel length and channel width is:

L mixer
W ¼

Fo uW

D ¼

FoPe W : (2.199)

For the above typical values of Fourier number 0.1 < Fo < 1 and Peclet number based on the

channel width 10 < Pe W < 10,000, the range of this ratio is 10 < L mixer / W < 10,000. For some

applications, the required mixing channel is unacceptably long.
If the inlets are split and rejoined as n pairs of solute/solvent streams, the mixing length is reduced
to L mixing ¼ W / n . The ratio of the required channel length and channel width then becomes:

L mixer
W ¼

Fo uW

1
n 2 FoPe W : D ¼

(2.200)