4.2 Validating the Specification
55
• Objective 2—A droplet passes a channel too slowly/quickly: This is the case when
at least one channel/module exists in which its determined flow rate Q c /Q m
yields a droplet speed resulting in a duration t c /t m which is larger/smaller than a
time limit T .
In order to determine the durations, again, the flow rates can be used. In fact, by
dividing the flow rate by the section of the corresponding channel/module, the speed
(in m/s) of the flow can be determined, i.e. by
u c =
Q c
w c h c
or u m =
Q m
w m h m
.
(4.1)
Then, these speeds can be used to approximate 2 the duration (in s) a droplet requires
in order to pass a channel/module, i.e. by
t c =
l c
u c
or t m =
l m
u m
.
(4.2)
Example 4.4 Let’s assume that droplets should move from the heating module h 1 to
the detecting module d 1 in less than T = 500 ms (e.g., to prevent the droplet from
cooling down before it gets analyzed). As this requires the droplet to flow through
channels c 5 , c 6 , c 10 , the sum of the corresponding durations must be less than that.
Since according to Example 4.3, Q c 5 = 0.76, Q c 6 = 2.17, and Q c 10 = 3 (given in
μl/min), the respective durations are t c 5 = 39 ms, t c 6 = 14 ms, and t c 10 = 10 ms.
This sums up to 63 ms and, hence, validates that the choice by the designer yields a
specification fulfilling at least this objective.
Finally, the method allows to check whether the Reynolds number and the
Capillary number fall into the desired ranges (cf. Re ≤ 1 and Ca < 10 −2 as
described in Sect. 3.2).
Example 4.5 The maximal speed of the continuous phase is determined by dividing
the maximal flow rate (cf. 3 μl/min in Example 4.3) by the cross section of the
respective channel (50 μm · 50 μm) and is equal to 0.02 m/s. In this example, the
Reynolds number is equal to Re = 1 and the Capillary number is equal to
Ca = 4.35 · 10 −4 (the characteristic length L for a squared channel is its
width [93]), i.e. both are within the desired ranges.
Overall, the proposed method allows to formulate objectives, which are afterwards
automatically validated.
2 Note that additionally a slip factor can be considered. Furthermore, this is an approximation, since
droplets increase the resistance in channels/modules. Exact durations can afterwards be obtained
by simulating the injected droplet sequence.
55
• Objective 2—A droplet passes a channel too slowly/quickly: This is the case when
at least one channel/module exists in which its determined flow rate Q c /Q m
yields a droplet speed resulting in a duration t c /t m which is larger/smaller than a
time limit T .
In order to determine the durations, again, the flow rates can be used. In fact, by
dividing the flow rate by the section of the corresponding channel/module, the speed
(in m/s) of the flow can be determined, i.e. by
u c =
Q c
w c h c
or u m =
Q m
w m h m
.
(4.1)
Then, these speeds can be used to approximate 2 the duration (in s) a droplet requires
in order to pass a channel/module, i.e. by
t c =
l c
u c
or t m =
l m
u m
.
(4.2)
Example 4.4 Let’s assume that droplets should move from the heating module h 1 to
the detecting module d 1 in less than T = 500 ms (e.g., to prevent the droplet from
cooling down before it gets analyzed). As this requires the droplet to flow through
channels c 5 , c 6 , c 10 , the sum of the corresponding durations must be less than that.
Since according to Example 4.3, Q c 5 = 0.76, Q c 6 = 2.17, and Q c 10 = 3 (given in
μl/min), the respective durations are t c 5 = 39 ms, t c 6 = 14 ms, and t c 10 = 10 ms.
This sums up to 63 ms and, hence, validates that the choice by the designer yields a
specification fulfilling at least this objective.
Finally, the method allows to check whether the Reynolds number and the
Capillary number fall into the desired ranges (cf. Re ≤ 1 and Ca < 10 −2 as
described in Sect. 3.2).
Example 4.5 The maximal speed of the continuous phase is determined by dividing
the maximal flow rate (cf. 3 μl/min in Example 4.3) by the cross section of the
respective channel (50 μm · 50 μm) and is equal to 0.02 m/s. In this example, the
Reynolds number is equal to Re = 1 and the Capillary number is equal to
Ca = 4.35 · 10 −4 (the characteristic length L for a squared channel is its
width [93]), i.e. both are within the desired ranges.
Overall, the proposed method allows to formulate objectives, which are afterwards
automatically validated.
2 Note that additionally a slip factor can be considered. Furthermore, this is an approximation, since
droplets increase the resistance in channels/modules. Exact durations can afterwards be obtained
by simulating the injected droplet sequence.