| 4. ANALYSIS
AND DESIGN EQUATIONS |
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| 4.1
Assumptions |
In this section,
we present mathematical descriptions of the deoxygenation
process and of the transfer of carbon dioxide into the
ballast water, which, in turn, leads to lowering of the
pH to the levels lethal to most ANS. We obtain closed-form
mathematical models, usable in design of a shipboard system
from any set of given specifications. The list of symbols
used in the equations is given at the end of the paper.
The system being analyzed places a
mixture of nitrogen and carbon dioxide with a relatively
small fraction of oxygen in contact with ballast water.
The oxygen level in the ballast water is assumed to have
reached equilibrium with air as a result of prolonged
contact, and therefore would contain a concentration of
oxygen sufficient to support a wide spectrum of life forms.
The objective is to reduce the oxygen content to a low
level by interchange with the gas mixture. The gas is
bubbled through the ballast water, which assures uniform
distribution of dissolved gas throughout the ballast tank.
Thus, diffusion within the tank can be neglected. Bubbles
are assumed to be small and variation of hydrostatic pressure
over the vertical dimension of a bubble is neglected.
We do not discuss here the size of
bubbles and the frequency of their generation. These two
issues are addressed in existing reference literature
(see, for example, Perry et al. 1984).
We assume that deoxygenation process
follows Henry's Law with equilibrium achieved within the
residence time of each bubble. The composition of the
mixture in the bubble changes primarily due to transfer
of carbon dioxide, a dynamic chemical process assumed
to obey the mass action kinetics. |
| 4.2 Deoxygenation
Process |
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As
trimix gas is flushed through the system, the total weight
of oxygen in the ballast water will be reduced. For the
purpose of analyzing the deoxygenation process we neglect
the presence of carbon dioxide in the trimix.
When a small quantity of gas, dQ,
is admitted, it contains an oxygen molar fraction y0.
By the time this quantity of gas leaves the system it
contains, according to Henry's Law, the molar fraction
Y/kH.
Therefore, we obtain the following differential equation:
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| Integration
of this equation yields: |
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| From
this equation it follows that pumping 5,200 m3
of gas into a 32,200 m3 tank reduces oxygen
concentration to 0.83 ppm. This level of hypoxia is lethal
to many ANS. With the flow rate of 38.2 m3/min
this can be achieved in 135 min. The relationship between
the size of the tank and the time required to deoxygenate
it is linear. Therefore, these results can be scaled to
any tank size. |
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