56
4 Dimensioning Droplet Microfluidic Networks
4.3 Automatic Dimensioning
An automatic validation of a specification clearly supports the designer. Nevertheless, it does not relieve the designer from the burden to make choices until a proper
specification has been found. Moreover, it may even be possible that, using the
existing set of modules as well as the current arrangements of channels/modules,
no proper specification is possible, i.e. independent of the choices of the designer,
one of the objectives mentioned above might always fail. For example, this can
easily happen if the flow rates/pressure gradients produced by pumps are too low so
that the timing objectives cannot be ensured.
Using the 1D analysis model as described in Sect. 3.2 also allows to aid the
designer in these issues. In fact, by leaving the channels’ resistances free, still a
result can be obtained from the equation system. As this however may again include
values violating one of the objectives from above, the equation system has to be
extended by further equations.
First, reasonable resistances are ensured for all unspecified channels c. Therefore,
equations are added enforcing a minimum R min and a maximum resistance R max ,
i.e.
R min ≤ R c ≤ R max .
(4.3)
Example 4.6 Consider again the partial specification as shown in Fig. 4.1. For
all unspecified channels, the method enforces a minimum and maximum resistance
using the inequality 0.15 ≤ R c ≤ 2.57 (given in mbar/(μl/min)).
Then, the objectives for obtaining a proper specification are added: For Objective 1, equations for all channels and modules enforcing the intended flow direction
are added, i.e.
Q c ≥ 0 and Q m ≥ 0.
(4.4)
For Objective 2, an equation enforcing that droplets will always pass a channel/module below/above a given timing threshold (namely T ) is added. To this
end, the equation for determining the time a droplet takes to pass/execute a
channel/module is employed.
For calculating the speed and the time a droplet requires to pass a channel, the
section of the channel (i.e., its width and height) is needed. This section is usually
fixed over the whole device in order to minimize the (production) complexity (see,
e.g., [13]). This allows to express the length l c of a channel using its resistance
(e.g., for a rectangular channel the resistance is given by Eq. 3.2 of Sect. 3.2). By
inserting this equation transformation into Eqs. 4.1 and 4.2, the required time can be
restricted by
R c w 2
c h 4
c
α μ Q c
≤ T or
R c w 2
c h 4
c
α μ Q c
≥ T .
(4.5)
4 Dimensioning Droplet Microfluidic Networks
4.3 Automatic Dimensioning
An automatic validation of a specification clearly supports the designer. Nevertheless, it does not relieve the designer from the burden to make choices until a proper
specification has been found. Moreover, it may even be possible that, using the
existing set of modules as well as the current arrangements of channels/modules,
no proper specification is possible, i.e. independent of the choices of the designer,
one of the objectives mentioned above might always fail. For example, this can
easily happen if the flow rates/pressure gradients produced by pumps are too low so
that the timing objectives cannot be ensured.
Using the 1D analysis model as described in Sect. 3.2 also allows to aid the
designer in these issues. In fact, by leaving the channels’ resistances free, still a
result can be obtained from the equation system. As this however may again include
values violating one of the objectives from above, the equation system has to be
extended by further equations.
First, reasonable resistances are ensured for all unspecified channels c. Therefore,
equations are added enforcing a minimum R min and a maximum resistance R max ,
i.e.
R min ≤ R c ≤ R max .
(4.3)
Example 4.6 Consider again the partial specification as shown in Fig. 4.1. For
all unspecified channels, the method enforces a minimum and maximum resistance
using the inequality 0.15 ≤ R c ≤ 2.57 (given in mbar/(μl/min)).
Then, the objectives for obtaining a proper specification are added: For Objective 1, equations for all channels and modules enforcing the intended flow direction
are added, i.e.
Q c ≥ 0 and Q m ≥ 0.
(4.4)
For Objective 2, an equation enforcing that droplets will always pass a channel/module below/above a given timing threshold (namely T ) is added. To this
end, the equation for determining the time a droplet takes to pass/execute a
channel/module is employed.
For calculating the speed and the time a droplet requires to pass a channel, the
section of the channel (i.e., its width and height) is needed. This section is usually
fixed over the whole device in order to minimize the (production) complexity (see,
e.g., [13]). This allows to express the length l c of a channel using its resistance
(e.g., for a rectangular channel the resistance is given by Eq. 3.2 of Sect. 3.2). By
inserting this equation transformation into Eqs. 4.1 and 4.2, the required time can be
restricted by
R c w 2
c h 4
c
α μ Q c
≤ T or
R c w 2
c h 4
c
α μ Q c
≥ T .
(4.5)