Page 333 - Process Modelling and Simulation With Finite Element Methods

320 Process Modelling and Simulation with Finite Element Methods

Species transport including electrophoresis:

(9.6)

Charge balance: (9.7)

The electric field satisfying Eq 9.7 must also satisfy Gauss’ law (c.f. equation
7.1), which becomes an equation determining charge density as a function of
position in the flow. In the typical conditions of electrokinetic flow, the charge
density may be taken as negligible for purposes of both charge conservation (Eq
9.7) and the momentum balance (Eq 9.5). The electrical conductivity and zeta
potential may depend on concentration of species Y and linear relations are
assumed here: 0 = 1 + Or (1 - Y) and [ = -1 - c, (1 - Y), where subscript
‘r’ indicates the ratio of the property in the two pure solutions involved in the
flows considered.
Boundary conditions at the flow inlets are that electric potential, pressure
and species concentration must be specified, and at flow outlets electric potential
and pressure must be specified. Species concentration is not known at the
boundary and an approximation regarding species diffusion, the only term that
connects the species field within the domain to the species distribution on the
outflow boundary, is required. As usual, the species diffusion is neglected at the
outflow boundary, i.e. a Neumann boundary condition just on the diffusion part
of the flux term r is used.
The electric field is taken as quasi-steady, that is the electric field adjusts
practically instantly to changes in the velocity and concentration. The above
equations represent a generic problem providing a test of the numerical
implementation which when verified may allow computation of any particular
electrokinetic flow conditions. For the test implementation, suggested
coefficient values are 1/Re = 30, 1 / Pe = 0.03, c, = 1 (no variation in wall
zeta potential), z = 0 (no charge on species Y) and 0, = 1 (no variation in
electrical conductivity).

9.3.2 Problem set up

The basic problem one can solve is the propagation of a concentration front
along a channel. Initially, a sharp front is placed at mid channel and