Scale-iip of Bioprocesses
501
11.3.6. Flow in stirred tank reactors
All physical processes depend on the flow pattern and energy dissipation in the reactor. These
quantities are described by partial non-linear differential equations obtained from conservation of
mass and momentum. The flow pattern is fully described if the velocity, which is a vector
property u = u(x,y,z), and the pressure which is a scalar property, p = p(x,y,z), is known at each
point in the reactor. The numercial treatment to determine the flow field is the subject of
computational fluid dynamics, CFD, which will not be described here (see e.g Kuipers and van
Swaaij, 1998 for an introduction). However, it is useful to understand the basis for these
calculations and their limitations. The fundamental relation between the pressure, velocity and
the material properties viscosity and density, is given by the Navier-Stokes equation. This
equation states that the time rate of change of momentum in a fluid volume is the sum of
pressure forces, viscous forces and gravity forces acting on that volume. For a one-phase
incompressible flow, i.e. when V u = 0, the equation can be compactly written using vector
notation as:
-Vp + ?7fV2ii + p;g
(11.25
where
p
is the pressure, g is the gravity vector, V is the gradient operator, and V2 is the Laplacian
operator. (For translation of these operators into the correct component equations for the
respective coordinate system see e.g. Bird et al., 2002). The left hand side of the equation, the so-
called material derivative, can be expanded into:
Du
Dt
8u
dt
+ u- Vu
(11.26)
The Navier-Stokes equation can be transformed into dimensionless form, in which the familiar
Reynolds number and
Froude number
,
Fr,
occur. The choice of characteristic length depends on
the geometry of the system studied. For a stirred tank reactor, the impeller diameter is often
chosen. The dimensionless form of the Navier-Stokes equation reads:
where
Du
+ — W
1
h-----]
Dt
Fr
u
g
^
d N 2
u
=
-—
g
= — ;
Fr =
Nds
g
g
(11.27)
(11.28)
From Eq. (11.27), the Reynolds number is seen to be a parameter in the governing flow equation,
which further justifies its appearance in simplified correlations for flow related phenomena. In
baffled reactors, the influence of the Froude number on the flow pattern is normally small. For
flow in stirred tank bioreactors, two important complications arise for obtaining a numerical