Mass Transfer
431
is the
gas holdup
in the dispersion.
The specific interfacial area is a function of the bubble size distribution in the gas liquid dispersion,
and
a
is obtained from
6e
(1
~£)d,
(10.17)
where
dmean
is the average bubble diameter, which may be calculated as the first moment of the
bubble size distribution function. A surface-averaged diameter, the so-called
mean Sauter diameter,
is, however, often used:
d
Sauter
Z
n id h
Z
n id l i
(10.18)
In a bioreactor, three main processes interact to determine the bubble size distribution (Fig. 10.3):
1.
Bubble formation,
determined by the breakup into discrete bubbles of the gas stream as it is
sparged into the liquid phase.
2.
Bubble breakup
, determined by the competition between the stabilizing effect of the surface
tension and the destabilizing effect of inertial forces.
3.
Bubble coalescence,
i.e., fusion of bubbles, determined by the properties of the gas-liquid
interface.
A bubble is formed at the orifices of the sparger when the
buoyancy force
on the bubble exceeds the
surface tension acting on the periphery of the orifice. Thus the initial bubble diameter
dbi
is
determined from a force balance:
or
= T dliS (P t~ P g)
O
lb,i
6ori0
3
g(Pi
-
Ps )_
(10.19)
(
10
.
20
)
where a is the surface tension of the liquid phase (unit: N m '1),
d0
is the orifice diameter, and g is
the acceleration of gravity. Equation (10.20) holds only up to a certain gas flow rate over which the
diameter of the bubbles starts to increase with the gas flow rate. For much higher gas flow rates,
swarms of bubbles and finally an almost continuous jet flow will be formed. For viscous media,
liquid viscosity rather than bubble surface tension provides the predominant resistance to bubble
formation. For such systems an empirical correlation for the initial bubble diameter can be used
(see Example 10.3).
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