Scale-up of Bioprocesses
483
mixing, liquid-liquid mixing, gas-liquid mixing, solids-liquid mixing and three-phase mixing
(see Nienow et ah, 1997). A bioreactor normally contains three phases, and therefore true
uniformity cannot be achieved at the microscale. However, at very high stirring rates, the
concentration within the liquid and gas phases will be approximately constant throughout the
reactor volume. In the ideal tank reactor discussed in Chapter 9 complete mixing is assumed to
take place instantaneously, i.e. any medium component added to the reactor is assumed to be
homogeneously distributed in the system immediately. This assumption works reasonably well
for small-scale (1-2 L) intensly stirred bioreactors, with mixing times in the order of I s, since
most biological processes are relatively slow processes at a temperature of 30 - 40 °C. However,
in large-scale systems, the time for achieving homogeneity in the reactor can be in the order o f
minutes and may no longer be neglected. Mixing depends on the scale of homogeneity
considered.
Macromixing
refers to mixing on a conveniently observable scale, whereas
micromixing
refers to mixing on the molecular scale. On a macro-scale, mixing is achieved by
bulk flow convection, i.e. the distribution caused by the main flow pattern in the reactor.
Furthermore, in a turbulent flow field which is predominantly the case for stirred bioreactors, the
mixing caused by turbulent eddies is highly important for mixing down to the Kolmogorov size
(see Note 10.1.). The final micromixing below this size is achieved by molecular diffusion.
However, since the length-scale over which this final mixing occurs depends on the turbulent
eddy size, macromixing indirectly affects also micromixing (see Note 11.3).
One way of quantifying mixing is to make a pulse addition of a tracer, e.g. a dye, into the reactor,
and then monitor the gradual return of homogeneity. The degree of mixing (or degree of
homogeneity),
m,
is used to quantitatively describe such experiments. The degree of mixing is
defined by
s(t) -
(
11
.
1
)
where
s(t)
is the concentration of the tracer (at a measuring point) at time
t, so
is the initial
concentration and s«, is the concentration for
t—> ao
where
m
will approach 1. The
mixing time
,
t„,
is defined as the time needed to obtain a value of
m
larger than a specific threshold value.
This value is in principle arbitrary. Commonly used values are e.g. 0.95 or 0.99. Here we will
define
tm
as the time needed to achieve a value of m equal to 0.632 (= 1-e'1). Several
investigations have shown that mixing can be approximately regarded as a first order process.
With the definition used here,
tm
will therefore be the inverse of a first order rate constant for the
mixing process, making it convenient to compare the rate of mixing with the rate of mass
transfer, or the rate of reaction. The conversion of one definition to another is shown in Note
11.1.
For a one-phase stirred tank system with baffles, the mixing time has been found to be
approximately inversely proportional to the stirrer rate,
N
(unit s"1)1, i.e.
tm
°c—
(
1 1
-
2
)
N
The stirrer rate is often given in the unit rpm = rotations per minute
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