Scale-up of Bioprocesses
503
u =
l+Jt
Jo*
(
2
)
The time-averaged Navier-Stokes equation can thus be written:
Pi
^
= -TP +
u
+
p }
g -
p
:V •
(uV
)
(3)
As seen from comparing Eq (3) with Eq. (11.25), the time-averaged equation looks very similiar to the
original equation with the exception of one additional term. This additional term, called the Reynolds
stress tensor, is due to the turbulent velocity fluctuations. A very often used model is the so-called k-e
model, in which it is assumed that the Reynolds stress tensor can be represented by a term involving a
turbulent viscosity.
(4)
The turbulent viscosity,
rjh
is assumed to be a function of the velocity fluctuations, k, (m s'1), and the
energy dissipation, e, (m2 s*3) according to the following expression
with
and
£ —
pV
(5)
(
6
)
cn is a universal constant. Note that the turbulent viscosity is not related to the normal (laminar) viscosity
of the fluid, but is entirely a function of the flow field. The presence of both gas and liquid leads to the
need of modelling both phases. This can be done by solving the flow equations for each phase separately
(e.g. Morud and Hjertager, 1996), or by considering the fluid to be pseudo-homogeneous at low aeration
rates.
Simplified flow models
Solving the fundamental flow equations for a two-phase flow requires quite substantial
computing power as well as rather elaborate modeling. An alternative is therefore to use a
simpler so-called
compartment model.
These models often give a useful qualitative description
of the main flow or mass transfer phenomena. In these models, the reactor is divided into regions
(compartments) based on the qualitatively known bulk flow properties in the reactor. The
compartments can be modeled as ideally mixed (i.e. small tanks) or fully segregated (i.e. plug
flow reactors). The area surrounding the impeller can for instance be modeled as a well-mixed
region, whereas the bottom region may be regarded as a poorly mixed region (see also example
11.5). Further refinement is possible by introducing more compartments to give a better
resemblance to known flow patterns as shown in Fig 11.11.