Page 173 - Process Modelling and Simulation With Finite Element Methods

160 Process Modelling and Simulation with Finite Element Methods

As the assumptions are all qualitatively met by the finite element model, it would
seem likely that the lumped parameter model would be an acceptable
approximation in the case of purely free convection through the buffer tank.
Were the point of this chapter to verify the applicability of the lumped parameter
buffer tank model, then we could run a parametric study fitting E(Pr,Ra) for the
free convection regime. The easiest route to fit E would be to compute outlet
concentration by boundary integration over boundary 6 and fit the value of E
which best fits the predictions of (4.6) to the simulated outlet times series. Since
it is unlikely that a buffer tank would be operated under a purely free convection
regime, it would not necessarily be useful information.
A second series of profiles are shown in Figure 4.15, which differs only
from the model of Figure 4.1 1 in the boundary condition is taken as u= 1, v=O on
the inlet (boundary 1) and the Rayleigh number is five times larger (Ra=2S). It
is actually the case that the recirculation layer above is much stronger in this
model, since forced convection imparts more momentum to the upper layer than
free convection. It should not come as striking, however, that the flow
configuration for gravity driven and pressure driven flows are broadly similar.
Only at early times, while the transient flow field is still establishing, does the
forced convection flow differ from the gravity current driven flow qualitatively.
Before diffusion has had much time to act, fluid in the upper layer is just
dragged along by viscous forces, yet it is heavy enough to fall back into the
lower layer and fall out the constant pressure outlet. Once the upper layer
becomes significantly stratified, however, the fluid dragged by the current has
enough momentum to “turn the corner” and establish the upper recirculation
layer. Thereafter, the profiles look self similar for both concentration and
velocity vectors, and in qualitative agreement with the basis of the lumped
parameter model for imperfect mixing in buffer tanks. So one would expect
(4.6) to hold on average across the outlet, with E(Re,Pr,Ra) as best fit
“capacitance” constant found from time series analysis. Such analysis is beyond
the scope of this chapter, but would be fruitful for modeling systems response in
a complex flowsheet. In the next subsection, we link the 2-D model for the
buffer tank to a I-D model of the heterogeneous reactor, thereby justifying
the description of the buffer tank modeling here in a chapter on “extended
multiphysics.”

Exercise 4.3
Compute the average outlet concentration at a number of times for a pulsed inlet
concentration, i.e. U=l for tE [0,1] and U=O thereafter. Compare qualitatively
the collected data for outlet concentration to Figure 4.7. Is the behaviour closer
to perfect mixing or imperfect mixing with E=O.S?