Biomedical Engineering Reference
In-Depth Information
Table 3.4 Tabulation of the Number of Required Pneumatic Lines and Allowed
Combinations for a Given Number of Inlets (C Denotes the Mathematical Symbol for
Combinations:
a
C
b
= a!/[(a - b)!b!])
No. of
Required
Pneumatic
Lines
(=2
X
)
Total
Number
of
Possible
Pairs
Number
of
Allowed
Octuplets
No. of
Allowed
Pairs
Number of
Allowed
Quadruplets
Total No. of
Possible
Quadruplets
Total No. of
Possible
Octuplets
No. of
Inlets
2
X
=
2 log
2
N
X
C
X
−3
×
2
X
−3
N
X
× 2
X
−1
N
C
2
X
C
X
−2
× 2
X
−2
N
C
4
N
C
8
2
2
1
1
N/A
N/A
N/A
N/A
4
4
4
6
1
1
N/A.
N/A
8
6
12
28
6
70
1
1
16
(
Figure
3.56
)
8
32
120
24
1820
8
12,780
32
10
80
496
80
35,960
40
1.05.×.10
7
64
12
192
2016
240
635,376
160
4.43.×.10
9
128
14
448
8128
672
1.07.×.10
7
560
1.43.×.10
12
channel 3 by the next one (valves 1, 2, and 5), channel 4 by valves 1, 3, and 4, and so on. One can
see that applying pressure to any three of the six air (yellow) channels will block 15 of the 16 luidic
channels and leave only one channel open; conversely, if the six air channels are initially pres-
surized, removing pressure from any three of them will open a certain luidic channel. Releasing
pressure from additional air channels will open additional luidic channels, which will give rise
to mixtures. Using six pneumatic lines to select 16 inlets, it is possible to achieve 37 combinations
(16 singles, 8 triplets, 6 quadruplets, 2 sextuplets, 4 nonuplets, and 1 with all 16 inlets on) compared
with the 81 combinations of the multiplexer shown in the previous section (
Figure 3.56
). However,
the Gulari design is very powerful in the number of single-inlet states that it can generate.
To visualize the number of combinations, it suices to see the number of ways
P
in which
M
valves can be placed in
N
channels:
P
=
N
!
/
[
M
!(
N
−
M
)!]
1
2
3
4
5
6
7
8
9 10 11 12 12 14 15 16
6
5
4
3
2
1
FIGURE 3.57
Combinatorial. multiplexer.. (From. Zhishan. Hua,. Yongmei. Xia,. Onnop. Srivannavit,.
Jean-Marie.Rouillard,.Xiaochuan.Zhou,.Xiaolian.Gao,.and.Erdogan.Gulari,.“A.versatile.microreactor.
platform.featuring.a.chemical-resistant.microvalve.array.for.addressable.multiplex.syntheses.and.
assays,”.
J. Micromech. Microeng.
.16,.1433-1443,.2006..Figure.contributed.by.Erdogan.Gulari.)
