# Interdisciplinary Applied Mathematics

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effectiveness is defined as (see Section 9.4)

e

mix

1 —

4- Т1* (r- — ^PM12

N Z^i=l c‘i ci )

-L^N (r° -г-РЩ2

N 2-^i= 1 Vt ci >

where ci is the concentration at the *th node, cPM is the concentration at the *th node if the two streams (i.e., the sample and the buffer streams)

(a)    (b)

FIGURE 18.18. (a) The dependence of the flowrate on the number of split levels used. (b) The dependence of the effectiveness of the mixing (i.e., homogeneity of the mixture) on the number of split levels used.

are perfectly    mixed, and c0    is    the    concentration at    the    *th    node    if    the    two

streams do not mix at all. The analytical solution of the diffusion equation, obtained by the method of separation of variables, provides the concentration variation in the transverse direction. The mixing effectiveness, em;x, ranges from 0 to 1, with 1 indicating complete mixing and 0 indicating no mixing.

The following parameters have been used for the results shown in Figures 18.18(a-b): Аф = 100 V (the potential difference applied between the start and    the    end    of the channel,    e.g.,    in    Figure    18.17(a)    it is applied    be

tween 0 and 11); length of each level = 200 p,m; height of the initial channel = 16 ^m;    aT    =    6.2 x    10~3    C/m2;    ц = 10~3 kg/m.s;    e =    6.95    x 10~10.    The

concentration of species A at the inlet of the transport system is considered to be 0.1    mM.    The    results    in    Figures    18.18(a-b)    indicate    that    there is a

trade-off between the throughput and mixture homogeneity. However, if one uses a high value for the fluidic resistance of the initial channel, then that dominates the total fluidic resistance. As a result, the throughput is standardized and does not depend strongly on the number of splitting levels. Therefore, the device designer can control the effectiveness of the mixing process by varying the number of splitting levels.

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