Biomedical Engineering Reference
In-Depth Information
FIGURE 5.10
Transient one-dimensional model of sequential segmentation.
( c 0
0
c 0
0
t
aT
=
2
aT
=
2
<
t
T
aT
=
2
c
ð
t
;
0
Þ¼
(5.17)
T
aT
=
2
<
t
T
where c 0 , T , and a are the initial concentration of the solute, the period of the segmentation, and the
mixing ratio, respectively.
Normalizing the concentration by c 0 , the spacial variable by L , and the time by T results in the
dimensionless form of (5.16) :
v 2 c *
vx * 2
vc *
vt * ¼
vc *
vx *
1
Pe
(5.18)
where the star
denotes dimensionless variables. The Peclet number i s defined based on the char-
acteristic mixing length L and the dispersion coefficient D *asPe
)
¼ uL=D * . Because of the much
higher effective diffusion coefficient at the same flow rate and the Reynolds number, the Peclet number
of parallel lamination is about two orders of magnitude higher than the Peclet number of a sequential
segmentation. From (5.17) , the corresponding dimensionless boundary condition is
( 10 t * a= 2
0 a
c *
t *
ð
;
0
Þ¼
t *
:
(5.19)
=
2
<
1
a
=
2
t *
11
a
=
2
<
1
Solving (5.18) with (5.19) and c *(
N
)
¼
a results in the transient concentration distribution
( Fig. 5.10 ):
a þ
exp 1
2
x *
Pe
p
Pe 2
N
ð
apn
Þ
2sin
c *
ðx *
; t *
2 pt *
Þ¼<
þ
8 pn Pe i
exp
ð
pn
1
(5.20)
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