# Interdisciplinary Applied Mathematics

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Jt

dc    zFcDdiff

DdmM +

(Уф) c +

^convc

Ac

(18.11)

where Jt is the total flux, Ddff is the diffusion coefficient of the species, c is the concentration of the species, Fc is Faraday’s constant, z is the valence of the ion, R is the universal gas constant, T is the temperature, Ac is the cross-sectional area of the fluidic channel, and vconv is the convective velocity of the flow that arises due to the bulk flowrate, Q, given in equation (18.6):

^conv —    »    •

Ac

From equation (18.11), the total flux is the sum of the diffusive flux (given by the first term), the electrophoretic flux (given by the second term and it is zero for uncharged species), and the convective flux (given by the last term), which arises due to the bulk flow in the channel. Typically, the separation unit is designed in such a way that the convective flux and the electrophoretic flux (for charged species) dominate over the diffusive flux (Fletcher et al., 1999). Thus, assuming that the diffusive flux is negligible,

the expression for the total flux is given by

Jt

zFcDdift

RT

(W) c + ^convc

A

c

or

Jt    (Qph + Q)c->

where Q is the convective flowrate, which is computed using equation (18.6), and Qph is the electrophoretic flow rate, which is given by the expression (Fletcher et al., 1999)

Qrh =    А^Ф.

Thus, the constitutive equation, which relates the “through quantity” (electrophoretic flowrate) to the “across quantity” (electrical potential difference), in the case of electrophoretic flow, is given by

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