Page 263 - Process Modelling and Simulation With Finite Element Methods

250 Process Modelling and Simulation with Finite Element Methods

applying over a thin control volume incorporating the interface between the
electrode and the bulk fluid leads to this electric flux boundary condition:

(7.2)

where E~ is dielectric constant of the solid constituent of the electrodes and .z0 is
the dielectric constant of the bulk fluid medium. The LHS represents the electric
flux out of the interface from the electrode side, the electric flux into the
electrode from the bulk fluid, the difference balanced by the accumulated charge
on the electrode at steady state. Rearranging (7.2) leads to the boundary
status as

(7.3)

where we shall term q’ as the charge on the electrode.
With these governing equations, we can define two related tomographic
mathematical problems.

The Forward Problem
If the firing electrode is held at unit voltage (see Figure 7.1) and the sensing
electrodes are held at ground (zero voltage), then the solution Qi to (7.1)
computing the total charge on the electrodes i
a@
(q’), = J-dQ (7.4)
an an

with known dielectric constants for the inclusions, is termed the forward
problem. Figure 7.1 (right) shows the solution to the forward problem that we
will shortly formulate in FEMLAB.

The Inverse Problem
Now suppose the same experiment is conducted, but that the dielectric field in
the duct is not known a priori. The charges qi’ are measured on the electrodes
and the permittivity field E in the duct consistent (since 0 is a solution of (7.1))
with the measurements through (7.4) is sought. This is termed the inverse
problem.
Modelling the Forward ECT Problem in FEMLAB