Page 180 - Process Modelling and Simulation With Finite Element Methods

Extended Multiphysics 167

Exercise 4.4

Alter the initial condition so that U(tO)= sheetu(x)+0.02*sin(31.4159265*~).
Does this oscillation grow or decay? What effect does the buffer tank have on
the oscillation?

4.5 Bioreactor Kinetics

Recall that in 33.4, we treated heterogeneous reaction in a porous catalyst pellet,
with a variation on the treatment in the Model Library [9]. In this section, we
will try a different variation. In this section, a similar approach will be used to
model reaction of a passive scalar occurring in a single cell. The reaction
lunetics will be taken as typical of bioreactors - Langmuir-Hinshelwood:

where r is the rate of disappearance by reaction, which only occurs within the
cell. d represents the finite capacity of the cell to hold the substrate
concentration, which saturates at a value controlled by this parameter. The usual
rate controlling step, however, is the transfer of the nutrient from the medium
across the cell membrane. The overall mass transfer process is usually modelled
with a first order resistance, with the flux j given by

(4.10)

At steady state, the rate of disappearance by reaction is equal to the flux of
nutrient across the cell membrance, i.e.

(4.1 1)

Thus, the boundary condition on mass transport on the cell wall involves the
concentration ci on the boundary and the concentration within the cell itself,
which is taken to be uniform. So the extended multiphysics here is to treat c, in
an additional 0-D space with reaction occurring only there, and coupling
between the two spaces through the flux into the cell and through the boundary
condition (4.10). Equation (4.11) can be seen as modeling the cell as a
continuously stirred tank reactor (CSTR) with effective influx given by the
integral, and irreversible reaction. The boundary condition (4.10) is ubiquitous
in the chemical engineering literature, nevertheless, to the authors’ knowledge,
this is the first hgher dimensional model that incorporates it as a boundary
condition in a non-trivial way. If ci is constant, (4.10) represents a simple mass