Page 165 - Process Modelling and Simulation With Finite Element Methods

152 Process Modelling and Simulation with Finite Element Methods

Exercise 4.2

The “sluggishness of the buffer” tank model depends to a large extent on the
ratio FN in (4.6), which is an inverse time scale. In the FEMLAB model,
implicitly, FN was taken as unity. Explicitly add FoverV as a parameter, and
explore the transient response when varying FoverV.

4.3 Primacy of the Buffer Tank

In the previous section, the “main” physics were in the 1-D heterogeneous
reactor, and the buffer tank, due to being modelled by a lumped parameter, was
treatable by a 0-D capacitor model. Where lumped parameter models work, it is
always a boon, since the dimensionality of the model is smaller and the equations
generally simpler in form than the distributed system model that treats the
physics more exactly. It begs the question, however, of where do you get a
lumped parameter model from, and how do you get the lumped parameter
dependencies. Generally, the lumped parameter model comes from analysis and
simplification of a higher dimensional, distributed model. For instance, mass
transfer coefficients come from solving film theories of convection and diffusion
in a boundary layer flow. The lumped parameter, the mass transfer coefficient,
can be predicted from the shape of the particle and the strength of the laminar
flow. In turbulent flows, the functional form of the mass transfer coefficient is
found from empirical correlations. The buffer tank lumped parameter model of
$4.1 was developed for a specified industrial application for assessing
concentration fluctuations, and the lumped parameter was fitted from samples of
inlet and exit conditions.
Certainly to treat a specific industrial unit operation, semi-empiricism is a
reliable approach. In the case of the buffer tank that inspired [4], fluid density
varied significantly with solute concentration (salinity), and thus the
“capacitance” effect of the buffer tank was expected to be influenced by the rate
of forced convection (throughput) FN, viscous and mass diffusivity, and by the
strength of free convection causing stratification, characterized by Reynolds,
Prandtl and solutal Rayleigh numbers, respectively. Since buffer tank lumped
parameter model of $4.1 only includes the throughput effects explicitly, the
dependence of the lumped parameter E on Reynolds, Prandtl and solutal
Rayleigh numbers is unknown. It is just taken as a constant found from
representative conditions. Whether or not a lumped parameter model is
sufficient depends on the type and accuracy of the predictions required from the
process model.
If the buffer tank is small, or shocks in solute concentration fluctuations are
prevalent upstream, the lumped parameter model may be insufficient in
predictive powers. Greater detail in the modeling would then be warranted.