| 4.3
Underpressure in Ullage Space of Ballast Water Tank |
|
| Deoxygenation
is enhanced by the under-pressure, as can be seen from
the following simple argument. Let p
be pressure of water at a given depth in the absence of
underpressure. Let pu
be the absolute value of the negative pressure at the
top. Let Y be the weight
fraction of oxygen in the water without underpressure
and Yu - the
same weight fraction with underpressure. Then by Henry's
Law: |
|
|
|
From
this equation we conclude that solubility of oxygen is
reduced by underpressure. This factor becomes even more
significant as a bubble rises to the surface, and the
pressure inside decreases.
For example, if p=14.7
psi (the usual value at the surface of the tank) and the
absolute value of the underpressure is 2 psi, then the
solubility of oxygen is reduced by approximately 14%.
|
| 4.4
Carbon Dioxide Transfer |
|
| Since
we assumed that the pressure inside the bubble depends
only on the pressure of the liquid surrounding it, we
can write: |
|
|
(1) |
| By
definition we have nCO2
= xn. Differentiating this equation we obtain: |
|
|
(2) |
| However,
since the reaction of carbon dioxide with water is the
dominant cause of change in the chemical composition,
we can write: |
|
|
|
| Combining
this with the Equation (2) yields the following equation:
|
|
|
(3) |
| In
addition, we can solve for n =
xn+nN to obtain |
|
|
(4) |
| From
the Law of Mass Action kinetics we have: |
|
|
(5) |
For
the partial pressure of carbon dioxide we have, according
to Dalton's Law pCO2
= xp.
Combining the equations (1), (3), (4),
and (5) yields: |
|
|
|
| This
equation can be integrated to obtain: |
|
|
(6) |
| where |
|
|
|
| This
equation can be used to calculate the parameters of the
systems, including residence time of a bubble, required
to achieve the desired molar fraction of carbon dioxide
in the bubble. The latter quantity is related to the pH
and the concentration of carbon dioxide in the water,
as we shall see in the next subsection. |
| 4.5
Concentration of Carbon Dioxide in Water and pH Calculation |
|
| Concentration
of carbon dioxide in water can be determined as the ratio
of the number of moles transferred from the bubble to
the volume of the tank. The number of moles transferred
from each bubble can be determined from the value of x
as follows. By definition, we have: |
|
|
|
| Solving
for nCO2 we find: |
|
|
|
| which
gives the following answer for the concentration of carbon
dioxide in water: |
|
|
(7) |
| The
concentration of the hydrogen ions in the water can be
calculated from c by solving
the following equation for h:
|
|
|
(8) |
| The
pH can be then found by taking the -log h.
|
| |
| We
can also solve the Equation (8) for c
and substitute the result into the Equation (7). This
yields after some tedious, but straightforward algebra
the following relationship between the desired molar fraction
of carbon dioxide in the bubble and the desired concentration
of hydrogen ions in the water: |
|
|
(9) |
| The
equations (6) and (9) constitute a closed-form mathematical
model of carbon dioxide transfer usable for design of
the treatment system. |
