Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 70

This concept is called parallel lamination where the channel length can be reduced by a factor of n 2 .
If the inlets are stretched and folded in n cycles, the mixing length is reduced to L mixing ¼ W / b n . The

base b depends on the type of mixer. In the case of sequential lamination as discussed in the next
section, the base is, for instance, b 2. The base could have a different value in the case of mixing

based on chaotic advection. The ratio between the required channel length and the channel width is:

¼

L mixer
W ¼

1
b 2 n FoPe W :

(2.201)

The above equation reveals that a very compact micromixer can be designed using sequential
lamination or chaotic advection.
In general, fast mixing can be achieved with smaller mixing path and larger interfacial area. If the
channel geometry is very small, the fluidmolecules collide most often with the channel wall and not with
other molecules. In this case, the diffusion process is called Knudsen diffusion [7] . The ratio between the
distance of molecules and the channel size is characterized by the dimensionless Knudsen number:

l
D h :

Kn ¼ (2.202)

where l is the mean free path and D h is the hydraulic diameter of the channel structure. The mean free
path for gases is given by (see Section 2.1.1 ):

k B T
p ps 2 m p l ¼ (2.203)

10 23 J/K is the Boltzmann constant, T is the absolute temperature, p is the
pressure, and s m is the molecular diameter of the diffusing species. The Knudsen number for liquid is
small, because the mean free path of liquid is on the order of a few angstroms. Thus, Knudsen diffusion
may occur only in pores with nanometer sizes. In gases, the mean free path is on the order of a hundred
nanometers to several micrometers. For example, at room condition, the mean free path of hydrogen is
0.2

where k B ¼ 1.38066

m. Knudsen diffusion may occur in microchannels with diameters on the order of a few
micrometers.
Among the above dimensionless numbers, Reynolds number Re represents the flow behavior in the
microchannel, while Peclet number (Pe) represents the ratio between advection and diffusion. Thus,
these two numbers are suitable for characterizing the operation point of a micromixer. From the
definitions (2.195) and (2.196) , the relation between Pe and Re is:

m

uL m ixing = D

uD h =v ¼

L mixing
D h

L mixing
D h

Pe
Re ¼

v
D ¼

:

Sc (2.204)

where u D , v , and Sc are the mean velocity, the diffusion coefficient, the kinematic viscosity, and the

Schmidt number (2.193) , respectively. The hydraulic diameter D h and the mixing path L mixing are
usually on the same order; therefore, we can assume L mixing / D h z

;

1. The kinematic viscosity of liquids

10 6 m 2 /s, while the diffusion coefficient ranges and the diffusion coefficient is on the order of v

¼

10 9 m 2 /s to D 10 11 m 2 /s. The Schmidt number is about 10 3 10 5 . On a Pe-Re

from D

¼ ¼ < <

Sc

diagram, the area between the two lines Pe 100,000Re represents the operation

range of micromixers. Operation points of micromixers are expected to be in this area.

z 1000Re and Pe z