arbitrarily set to about 1 l. All our concentrations will sub- sequently be given in mol l
¹ 1
and all times in seconds. For Case 1 that we just described, the cell solutions
remain undersaturated with respect to calcite throughout. Case 2 has essentially the same history for the carbonate
species; however, as the CO
2 ¹ 3
concentration rises in which cell, the solution becomes saturated and a period of preci-
pitation of solid CaCO
3
occurs, followed by its dissolution, in each cell. Fig. 3 shows this episode of precipitation occur-
ring for 600 , t , 1700 s in cell 1 and occurring for 1200 , t , 2600 s in cell 2.
The first two cases illustrate the cells progressing through the transitions in the dominant dissolved carbonate species
with and without precipitation. The third case demonstrates changes in the saturation state in a basic solution with no
transitions in the dominant carbonate species. The concentra- tion histories for Case 3 are shown in Fig. 4. Initially, there is
no calcite in either cell. The inflow of Ca
2þ
is varied so that two episodes of precipitation and complete dissolution occur
in each cell at slightly staggered intervals. While this is per- haps not a realistic scenario, it provides a computational con-
trol case for distinguishing between the effects of transitions in dominant carbonate species and the effect of dissolution
precipitation on the methods for solving the implicit time stepping equations.
4.2 Details of the demonstration cases
Cases 1 and 2 are identical in all respects, except for the higher calcium concentrations used to trigger precipitation
in Case 2, and the simulation period is longer for Case 2. There are special computational difficulties associated with
the computational solution of DAE systems when the initial states do not satisfy the algebraic equations, i.e. the initial
simulation state is not in quasi-equilibrium. Since we are demonstrating effects associated with changes in chemical
regimes, we avoid initial transient difficulties by spreading the change between the inflow pH and the initial cell pH
over an initial period of 100 s. All concentrations are in units of mol l
¹ 1
, as indicated in Section 2. The initial states are the same in Cases 1 and 2; the following values
are common to both cells: •
The initial states: C
1 ,
k
¼ 10
¹ 12
CO
2 ¹ 3
] ÿ
, C
3 ,
k
¼ 10
¹ 3
H
þ
ÿ ,
C
4 ,
k
¼ 10
¹ 11
OH
¹
½ ÿ
ð Þ
, C
5 ,
k
¼ 1:585 3 10
¹ 5
HCO
¹ 3
ÿ ,
C
6 ,
k
¼ 3:165 3 10
¹ 2
H
2
CO
3
ÿ ,
P
k
¼ 0; i:e: no precipitate in either cell:
• Volumetric flow rates: v
¼ 0.008 l s
¹ 1
, v
1
¼ 0.0085 l s
¹ 1
. •
Constant inflow concentrations: C
1,in
¼ 10
¹ 5
, C
5,in
¼ 5 3 10
¹ 7
, C
6,in
¼ 5 3 10
¹ 9
. •
Variable inflow concentrations: C
3 ,
in
t ¼
1 ¹ t=100 10
¹ 3
þ t=
100 10
¹ 11
for t , 100 ¼
10
¹ 11
for t . 100:
Fig. 3.
Concentration histories of Ca
2þ
and CaCO
3
solid pre- cipitate: Case 2.
Fig. 4. Profiles for Case 3.
Effects of chemical reactions on iterative methods 339
Note: T
in
¼ AC
in
. •
Maximum time step parameter q
max
¼ 0.5, cell
sizes h
1
¼ 1, h
2
¼ 2 l.
The differences between Cases 1 and 2 are that the simu- lated time is longer 2000 and 2800 s, respectively, and the
initial and inflow concentrations of Ca
2þ
are C
2,k
¼ 10
¹ 5
in Case 1 and C
2,k
¼ 5 3 10
¹ 5
in Case 2. Because no precipita- tion occurs in Case 1, C
2,k
t values are constant; the history of C
2,k
t for Case 2 is shown in Fig. 3. For Case 3, the cells each experience two episodes of
precipitate forming and then dissolution, as shown in Fig. 4. •
Initial states are the same in both cells and are: C
1 ,
k
¼ 5 3 10
¹ 5
, C
2 ,
k
¼ 5 3 10
¹ 5
, C
3 ,
k
¼ 10
¹ 11
, C
4 ,
k
¼ 10
¹ 3
, C
5 ,
k
¼ 8:0 3 10
¹ 6
, C
6 ,
k
¼ 1:7 3 10
¹ 10
, P
k
¼ 0:
• Volumetric flow rates: v
¼ 0.005,
v
1
¼ 0.0052 l s
¹ 1
. •
Constant inflow concentrations: C
1 ,
in
¼ 5 3 10
¹ 5
, C
3 ,
in
¼ 10
¹ 11
, C
4 ,
in
¼ 10
¹ 3
, C
5 ,
in
¼ 8:0 3 10
¹ 6
, C
6 ,
k
¼ 1:7 3 10
¹ 10
: •
Variable inflow concentrations: C
2 ,
in
t ¼
max [
, 10
¹ 5
5 þ 15sin 2pt=1200
] :
• Final time ¼ 2400 s; maximum time step parameter
q
max
¼
1 3
; cell sizes h
1
¼ 2, h
2
¼ 0.5 l.
4.3 Observations about computations with the standard Newton method
