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Many processes in the chemicals industry involve the passage of gas bubbles
through liquid for the purpose of heat and/or mass transfer. Examples
include bubble columns and sparged stirred tank reactors. Typically, these
mixtures are simulated using the Eulerian multiphase model or algebraic
slip mixture model using a spherical bubble of one size (to minimize the
computational effort), despite the fact that bubbles rarely, if ever,
fit this simplistic pattern. In addition to having a range of bubble sizes
in practice, the complex behavior of gas-liquid mixtures is often compounded
by the fact that the gas bubbles tend to break apart and coalesce as they
move through the liquid. Breakup tends to occur when shear forces in the
liquid are large enough to overcome the surface tension of the bubbles.
Coalescence occurs when two or more bubbles collide and the film of liquid
between them thins and breaks. Modeling such a mixture with CFD is difficult
because many bubble sizes need to be considered, and because the bubble
size can change depending upon local conditions in the background liquid.
To account for this added complexity in gas-liquid mixtures, engineers
at Fluent have implemented two numerical models for breakup and coalescence
through user defined functions in FLUENT 5.4. The models are designed
to work in conjunction with the algebraic slip mixture model for gas-liquid
flows.
The full-field approach (Luo and Svendsen, 1996) is based on population
balance theory. Ten ranges, or bins, of bubble sizes are considered. Within
each range, the bubble diameters are described by a distribution function.
Using Boltzmann's equation, the bubbles are allowed to move among the
bins as breakup and coalescence processes occur. To do this, source terms
are constructed that follow the birth and death of bubbles in each size
group so that an overall mass balance in the gas phase is assured. In
the FLUENT implementation, ten scalar equations are required, one for
the number density distribution in each bin. Each of these has its own
set of composite source terms that account for the birth and death rates.
Turbulence is the primary mechanism for breakup; only eddies that are
smaller than the bubble diameter can act to break apart the bubble. Larger
eddies are primarily responsible for the transport of bubbles. Coalescence
depends upon turbulent fluctuations in the liquid phase that force nearby
bubbles together (Prince and Blanch, 1990). To a lesser degree, the variation
in rise velocity of differently sized bubbles can also lead to collisions,
as can macroscopic recirculation patterns that give rise to radial velocity
gradients that transport nearby bubbles at different rates.
(a)
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(b)
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(c)
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a) The ratio of k 3/2 to e, which
corresponds to the size of turbulent eddies, is shown in a typical
bubble column. Large eddies in the center of the column do little
to break up rising bubbles. At the wall, however, the eddies are
small and give rise to an increase in breakup, (i.e. generation
of smaller bubbles). As these small bubbles rise, some get caught
in recirculation patterns near the wall while others continue
to rise near the center with the larger, more buoyant bubbles.
b) The gas volume fraction (holdup) is typically large where
the bubbles are small. Here, the maximum holdup is shown, where
the small bubbles, generated by the breakup, have risen in the
column but have not yet branched into either the group that undergoes
recirculation or the group that continues to ascend.
c) Without breakup and coalescence, the bubbles are all one
size and the gas volume fraction is more uniform in the column.
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The number density approach (Millies and Mewes, 1999) is also based on
population balance theory, but it requires less computational effort than
the full-field approach. Rather than use ten distribution functions for
ten size ranges, a single distribution function of particle sizes is assumed.
An assumption is made that equilibrium exists between breakup and coalescence
over a specified range of bubble sizes. In the FLUENT implementation,
only a single scalar transport equation is required for the overall number
density distribution. The predictions of this model include the distribution
of bubble sizes based on the same mechanisms for breakup and coalescence
described above.
References:
- Luo, H. and Svendsen, H.F., AIChE Journal, Vol. 42, No. 5, pp. 1225-1233,
1996.
- Millies, M. and Mewes, D., Chemical Engineering and Processing, Vol.
38, pp. 307-319, 1999.
- Prince, M. J., and Blanch, H.W., AIChE Journal, Vol. 36, No. 10, pp.
1485-1499, 1990.
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