Interdisciplinary Applied Mathematics

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= ki(mA)(mB) — k2(mc),


where Qi is the flowrate of the *th species, which is computed from the fluidic transport model (or known from the design specifications), Ci is the concentration of the *th species, mi is the number of moles of the *th species present in the reaction chamber, ki is the forward reaction rate, and k2 is the backward reaction rate. A trapezoidal scheme is used to discretize the ODEs given above. The discretized equations are given by


тв) + ylmc * + mc),


(W^Wl) = QaCa ~ у (mnA+1 + m){ml+l + mnB) + Ц{т1+1 + mnc), <УП1В At П1В) = Qb°B ~ A»(m3.+ 1 + тА)(тв+1 + тв) + 7y(mS+1 +


k2


FIGURE 18.5. Chemical species A and B are transported to the reaction chamber, where they undergo a second-order reversible reaction process.


At = у (™A+1 + ™1K+1+mnB)~ у(m^+1 + ml).


Qph (-ve ions)-*


(Anode)+



Q



-(Cathode)


Qph (+ve ions)-►


FIGURE 18.6. A basic separation unit, which can separate species that are oppositely charged, have different valences or different electrophoretic mobilities.


The nonlinear equations given above are solved by employing a Newton-Raphson scheme to compute тгф+1, mB+1 and mgS1 at the (n + 1)th time step given m|, mB, and mg at the nth time step. These equations constitute the device model for the reaction module.

18.1.4 Separation: Device Model


Figure 18.6 shows a simple separation mechanism, which is repeated as the basic unit in the circular separation device reported in (Kutter, 2000). The separation unit can separate species that are either oppositely charged or have different valences or different electrophoretic mobilities. The total flux of a given species through a channel is given by the following expression:

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