a
2
– Henry’s constant for CO
2
degassing; X
2e
– equilibrium to atmospheric CO
2
concentration in the solution. Of course, we understand that Henry’s approximation is
good for steady gas exchange process and will give uncontrolled error at the initial
phase of saturation or degassing. Note that transitive processes of such kind were
considered in Danckwerts’ papers [3]. But the simplicity of approximation (6)
compensates for its defects.
Quasiequilibrium approximation in the model of СаСО
3
formation. Accept
that the processes in CHCS take place in quasiequilibrium way. Here we understand
that the reactions of mutual transformation of components within CHCS
7
1
6
2
X
X
X
X
+
+
;
7
3
1
X
X
X
+
;
7
1
3
X
X
X
+
(7)
take place much faster than the processes of gas exchange or crystallization
(adiabatic approximation for crystallization kinetics). Adequacy criterion for such
approximation is the agreement of simulated calculations and experimental data.
CaCO
3
formation is not available. Under these conditions from (4) we obtain
the expression for the rate of calcium ions concentration change and from (3a) – their
concentration. Take into account that at equilibrium of reactions (7) carbonates sum
can be presented by relationship,
(X7(t))
f
(t)
X
Sc(t)
2
2
=
,
+
+
=
7
X
2
2
7
X
2
1
1
7
2
γ
K
1
γ
K
1
)
(X
f
,
(8)
K
1
, K
2
– thermodynamic constants of equilibrium dissociation of carbon acid of
the first and the second degrees; γ
1
, γ
2
– activity coefficients of singly and doubly
charged ions.
Substituting (8) into (5) we obtain kinetic equation for Ca
2+
ions in
quasiequilibrium (adiabatic) approximation
)
X
(X
a
X
)
(X
f
X
X
)
(X
f
X
2e
_
2
2
7
7
2
2
2
7
2
8
+
+
=
,
(9)
f
2
’(X
7
) – derivative by X
7
from f
2
(X7).
According to equation (3a) we obtain:
)
X
,
(X
f
S
X
7
2
8
o
Ca
8
=
;
+
+
=
7
18
2
7
2
18
2
1
1
2
X
K
K
1
X
X
K
γ
K
γ
38
7
2
8
K
1
1
)
X
,
X
(
f
, (10)
7
2
1
2
1
e
1
X
γ
X
K
X
=
,
2
7
X
2
γ
γ
X
K
K
X
2
1
2
2
2
1
e
3
=
;
(11)
K
18
, k
38
– thermodynamic constants of equilibrium X
18
and X
38
dissociations, X
i
e
– equilibrium by (7) components concentrations.
Equations (9) – (11) present the laws of Ca
2+
concentration change and other
CHCS components in quasiequilibrium approximation when crystallization is not
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