486
Chapter II
is not a desirable situation, since the dispersion of gas drastically decreases at that point giving
poor gas-liquid mass transfer. At moderate aeration rates, below flooding, aeration appears to
only slightly improve mixing in the liquid phase (Nienow, 1997; Vrabel et al 2000).
Note 11.3. Micromixing and macromixing.
The turbulent mixing contributes to mixing down to the Kolmogorov size of the eddies. As discussed in
Note 10.1 this size is given by
l
m in
M l
Pt
f
V
(
1
)
Molecular diffusion is the mechanism for mixing below this length scale. The diffusion time is highly
dependent on the length over which diffusion occurs. The time required for diffusion,
t
D, into a sphere of
radius
l
can be obtained from solving Fick’s second law, which gives;
t
D
l2ß
A
(
2
)
The value of p depends on the stated requirements for completion of mixing. The requirement that all
molecules within the sphere should have been exchanged with the molecules in the surrounding, gives (3 =
0.5 (Bourne, 1997). For a power input of 1 W L'1, a viscosity of 1 mPa s (typical of water), and a density
of 1000 kg m
'3
we get
lmin
= 32 pm. To get an estimate of the time for the final diffusive micromixing of
e.g. glucose, we can assume that this occurs in a sphere with the radius equal to half
lmi„.
The time for
micromixing of glucose (D = 0,7 10
~9
m
2
s
' 1
in water at 25 °C) is thus obtained from Eq. (2) to
tD
= 0.18 s.
This time is normally negligible compared to
tm,
which shows that the final molecular diffusion is a rapid
process compared to the mixing processes by convective transport and turbulent mixing in most stirred
tank reactors.
The measurement of mixing times by tracer experiments is useful. It is, however, important to
keep in mind the following inherent limitations in these experiments:
1. Mixing is a local property, which depends on the chosen location for measuring the tracer
concentration. A more strict definition should therefore be based on the averaging over many
sensors in the reactor (Lunden et al, 1995). For practical reasons, however, this is seldom done.
2. The mixing time is dependent on the point of addition of tracer. A badly chosen point of
injection may give a significantly higher value of
tm
than a well-chosen injection point.
3. Mixing is a function on the scale of magnification, and the size of the sensor will decide what
level of mixing is studied (cf. Note 11.3).
11.3.2. Power consumption
The mixing process requires energy. The power input,
P
(W), to a stirred tank bioreactor consists
of two parts; the power for stirring and the compression power for aeration. The first term is
completely dominating for stirred tank reactors, whereas the compression power is dominant in
bubble columns and airlift reactors. Additional energy for liquid circulation may also be supplied
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