Page 172 - Process Modelling and Simulation With Finite Element Methods

Extended Multiphysics 159
4.13 for both velocity vectors and concentration is indicative of the long-lived
nature of the transient intermediate approach to uniform mixing. It makes a
mockery of “steady-state’’ analysis, since it is not clear that steady state is ever
achieved in finite time nor is it clear that the uniformly mixed state will result
at all.
Fick’s law, which models the non-equilibrium transport of species, would
have us believe that the equilibrium endgame has concentration uniformly
diffused everywhere from a steady source. In fact, there are two greater
complications that preclude this. The first is that it is not concentration that is
diffusing at all, but rather chemical potential, and in an external gravitational
field. At equilibrium, these two potentials must be balanced. So a permanent
concentration gradient is maintained against a gravitational field. This fact is
responsible for the difference in composition between air at sea level and at Mile
High Stadium. In a buffer tank, it is probably meaningless, as the gradient in
concentration is minute. The second complication that is probably more
important in most chemical plants is that few solutions are exactly ideal, and
many show significant volume change on mixing. Zimmerman [7] has shown
that non-ideal solutions can have the structure of their stratification selected on
chemical equilibrium grounds, and that only ideal solutions can ever be expected
to form uniform mixtures at equilibrium.
The second observation is of the form of the velocity profile established -
recirculation layer over a current. This is exactly the form postulated by
Zimmerman [4] for which the lumped parameter model of imperfect mixing in
the buffer tank was derived, equations (4.6). Figure 4.14 shows the idealized
flow configuration for a denser current driving an upper recirculating layer. The
lumped parameter model presumes that the recirculation is strong enough
that the upper layer becomes well mixed, according to a theory of Batchelor [S],
and thus a single Concentration characterizes it. In fact, it seems that the
upper recirculation is weak, yet the concentration gradients are small in the
upper layer,

a
w oil-in-fresh r ! ~u

Fi Ci
FC
-0 0
outlet
Lx

Figure 4.14 Plug flow across the tank bottom driving an upper recirculation layer.