One should note that this test case is not chemically realistic.
Moreover, its numerical value comes only from activity correction if the
unknowns of the nonlinear system are the logarithms of the activity
components . Otherwise, if the unknowns are component concentrations ,
the problem becomes trivial and linear, and its solution is the total
concentration .
Phosphoric acid test
case
This test presents reactions between phosphoric acid and salt water. It
includes 4 components and 8 chemical species. We handle only acid-base
reactions: water dissociation and the 3 phosphoric acid reactions. A
table including the stoichiometric coefficients, equilibrium constants,
total concentrations and equilibrium solutions is given in appendix 2.
Gallic acid test case
This test case was proposed by Brassard and Bodurtha [20]. It
includes 3 components and 17 chemical species. It is a classical test
case, and many difficulties in convergence have been reported while
solving it by using Newton or Newton-like algorithms [13, 14, 21]. A
table including the stoichiometric coefficients, equilibrium constants,
total concentrations and equilibrium solutions is given in appendix 3.
Iron-chromium test case
This test case concerns the rehabilitation of chromium-contaminated
industrial soil using an iron-chromium reduction [2, 22]. Chromium
(VI), which is the most toxic and mobile form of chromium, is reduced by
iron (II) to yield chromium (III), which has a much lower solubility and
is less toxic [23]. This test is reported to be a very difficult one
[14, 21], so here we use some favorable testing conditions to
increase the convergence of the Newton algorithm. A table including the
stoichiometric coefficients, equilibrium constants, total concentrations
and equilibrium solutions is given in appendix 4.
Results
Study of the test case with only
activity correction during one
resolution.
We first present two scenarios for the test case with only
activity correction : one with a low ionic strength and one with a high
ionic strength. The objective is to determine the influence of the
activity correction on the Newton procedure depending on the algorithm
used. For the situation with a low ionic strength, this influence is
expected to be negligible, whereas we expect a greater impact in the
situation with a high ionic strength.
For the low ionic strength situation, the initial component activities
are 5.0 10-7 mol. L-1 for all
components. The ionic strength is 7.80 10-6 mol.
L-1, and we obtain the species concentrations and
activity values, which are given in Table 2 . Also in Table
2 , we show the first Newton steps proposed by the fixed-point
algorithms (inner and outer) and by the full Newton algorithm.
Table 2: Initial values for the
situation with a low ionic strength.