Page 207 - Process Modelling and Simulation With Finite Element Methods

194 Process Modelling and Simulation with Finite Element Methods

The phenomenon is a recurring fundamental instability in many realms.
Enhanced oil recovery, for instance the injection of dilute detergents into oil
sands or flooding with COz gas, as well as the remediation of contaminated
acquifers are common geophysical applications. Miscible displacement and the
concomitant pesky viscous fingering instability recur as well in regeneration
processes. Of special interest to chemical engineers is the flushing of catalytic
systems with solvents or oxidants that remove the impurities fouling the catalysts
or liquid chromatography columns. It was in the context of a regeneration
process, the ‘sweetening off‘ of sugar liquors displaced by water from a charcoal
packed column, that Hill [ll] recognized and first analyzed the channelling
instability. Homsy [ 121 gives the best review of the early work in this area. The
standard venues for miscible viscous fingering are porous media, which are well
described by Darcy’s Law, which is a simpler momentum equation than the
Navier-Stokes equations, typically semi-empirically based on measures of
pressure drop and superficial velocity in porous media:

P
vp = --u (5.10)
k
FEMLAB has a Darcy’s Law application mode built into the Chemical
Engineering Module. p is the pressure; u is the velocity vector; p is the
viscosity, and k is the permeability of the medium. Along with (5.10), it imposes
the conservation of mass for an incompressible fluid as
v.u=o (5.1 1)
The mixing as depicted in Figure 5.7 is due to convection and diffusion, also a
built-in application mode in the Chemical Engineering Module, with
concentration satisfying

ac ac ac 2
-+u-+v-=DV c (5.12)
at ax ay
where D is the molecular diffusivity. Additionally, in order to couple the mixing
with the momentum transport realisitically, the fluid viscosity must depend on
the concentration. The simplist model is monotonic dependence, which is a
good model for glycerol-water, a common laboratory model system for the
blending of viscous fluids:

/l =exp(R(l-c)) (5.13)

Armed with these equations, we are now ready to simulate viscous fingering
using the built-in application modes. Launch FEMLAB and bring up the Model
Navigator. select the Multiphysics tab.