H
2
CO
3
*
(
K
a
H
2
CO
3*
) are used. The principle of the calcula-
tions can be exemplified by calculating [HCO
3
] from
TIC:
HCO
3
TIC
1
H
K
a
H
2
CO
3
*
K
a
HCO
3
H
(5)
pH Algorithm
The individual concentrations of HCO
3
(eq 5) and other
ions are used in the charge balance equation for calculat-
ing pH in the aqueous phase. The charge balance equa-
tion (eq 6) consists of all charged ions accounted for in
the model as well as the parameter [Z ], which is the total
concentration of ions such as calcium (Ca
2
), potassium
(K ), sodium (Na ), chloride (Cl ) etc., adjusted so that
the charge balance is zero at the initial condition before
the simulations begin:
H
OH
HCO
3
2
CO
3
2
NH
4
NO
2
Z
HS
2
SO
4
2
CH
3
S
CH
3
NH
3
0
(6)
The concentration of ions and compounds in equi-
libria are used to model oxidation kinetics in the BF, mass
transfer, and pH.
When changes in the concentration of compounds
occur as a consequence of mass transfer, water move-
ment, air movement, and oxidation processes, the charge
balance still needs to be zero. This is used to find pH.
Equation 6 is solved to find a [H ] that ensures the charge
balance is always zero.
This is done iteratively using the Newton–Raphson
procedure to find approximations of
x
n
1
in:
x
n
1
x
n
f x
n
f x
n
(7)
In eq 7,
f
(
x
n
) is the derivate of
f
(
x
n
) with respect to
x
n
or
[H ]. Equation 7 (shortened version), as implemented in
the model of the BF, is:
x
n
1
x
n
NH
4
NH
4
TIC/ 1
x
n
K
a
H
2
CO
3
*
K
a
HCO
3
x
n
TIC
K
a
H
2
CO
3
*
x
n
2
K
a
H
2
CO
3
*
K
a
HCO
3
x
n
2
K
a
H
2
CO
3
*
x
n
K
a
H
2
CO
3
*
K
a
HCO
3
2
· · ·)
(8)
The inaccuracy in the calculation of [H ] only using
the Newton–Raphson procedure twice is less than 10
17
,
when the first
x
n
is the [H ] from the previous iteration.
When the Newton–Raphson procedure has been run
through twice, in the second round using
x
1
to replace
x
n
or
x
0
in eq 8, [H ] is assigned the value of
x
2
and the
procedure stops.
The
K
L
a
and the Two-Film Theory
It is assumed that the mass-transfer rate or flux (
F
) be-
tween air and water can be described from the gradient
between the concentration of the compound in the air
C
i,(g)
, calculated from Henry’s law constant of the gas in
question (
H
i
, mol
atm
1
), the partial pressure of the gas
(
p
i
, atm), and the concentration in the aqueous phase
multiplied by a
K
L
a
(the specific surface area, m
2
m
3
, not
included):
F
K
L
(
H
i
p
i
C
i,aq
).