Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 30

Any inhomogeneity in q is ignored; thus, D v 2 c

vq 2 ¼

0 ; and

The solute has the same velocity as the solvent.
Next, an average specie concentration is introduced:

ZZ c

1
pr 0

x ;

c ð t Þ¼ ð r ; q ; x ; t Þ

d r d q

:

(2.47)

With the as sumption of the long-time behavior, the axial concentration gradient is independent of

vx z vx Þ r ð vc = vc = . Thus, (2.46) can be reduced to:

r vc
vr

vc
vx ¼

D 1
r

v
vr

½ u ð r Þ u : (2.48)

The above equation can be solved for c by integration with respect to r :

r
r 0

2 r

r 0

4 vc
vx

r 0 u
4 D

1
2

x ; c ¼ C ð t Þþ (2.49)

where C ( x *, t ) is the function of integration. Substituting (2.49) into (2.47) results in the r -independent
function of the integration constant:

r 0 u
12 D

vc
vx :

x ; C ð t Þ¼ c (2.50)

Now, the axial concentration (2.49) can be expressed in terms of the average concentration:

r
r 0

2 r

r 0

4 vc
vx :

r 0 u
4 D

1
3

1
2

c z c þ (2.51)

In order to introduce the dispersion coefficient or the so-called effective diffusion coefficient D *,
the conservation of species (2.46) is written in the flux form as:

vJ
vx ¼

vc
vt þ

0 (2.52)

where J * is the area-averaged axial flux, which consists of both diffusive and convective
components:

J ¼D vc
vx
|{z}
diffusive

þ J conv

|{z}
convective

: (2.53)

The convective flux J conv can be evaluated as:

ZZ
½uðrÞuc d r d q:

1
pr 0

J conv ¼ (2.54)