The chemical equilibrium equations utilized in reactive transport modeling are complex and nonlinear, and are typically solved using the Newton-Raphson method. Although this algorithm is known for its quadratic convergence near the solution, it is less effective far from the solution, especially for ill-conditioned problems. In such cases, the algorithm may fail to converge or require excessive iterations. To address these limitations, a projected Newton method is introduced to incorporate the concept of projection. This method constrains the Newton step by utilizing a chemically allowed interval that generates feasible descending iterations. Moreover, we utilize the positive continuous fraction method as a preconditioning technique, providing reliable initial values for solving the algorithms. The numerical results are compared with those derived using the regular Newton-Raphson method, the Newton-Raphson method based on chemically allowed interval updating rules, and the bounded variable least squares method in six different test cases. The numerical results highlight the robustness and efficacy of the proposed algorithm.
. [J]. Frontiers of Chemical Science and Engineering, 2024, 18(3): 27.
Hongbin Lu, Shaohui Tao, Xiaoyan Sun, Li Xia, Shuguang Xiang. A projected Newton algorithm based on chemically allowed interval for chemical equilibrium computations. Front. Chem. Sci. Eng., 2024, 18(3): 27.
2. Check the convergence conditions. If it is satisfied, go to step 9. Else, do n = 0, go to step 3.
3. (Eqs. (14)–(17))
4. Solve (Eq. (13)) for
5. (Eq. (12))
6. (Eq. (11))
7. n = n + 1
8. Check the convergence conditions. If it is satisfied, go to step 9. Else, go to step 3.
9.
10. End.
Fig.1
Algorithm 2: Projected Newton (PN) algorithm
Input:
Output:
1. (Eq. (11))
2. Check the convergence conditions. If it is satisfied, go to step 12. Else, do n = 0, go to step 3.
3. (Eqs. (14)–(17))
4. Solve (Eq. (13)) for
5. (Eq. (32))
6. (Eq. (33))
7. (Eq. (34))
8. (Eq. (35))
9. (Eq. (11))
10. n = n + 1
11. Check the convergence conditions. If it is satisfied, go to step 12. Else, go to step 3.
12.
13. End.
Test cases
Xj
Ci
J-sizea)
[log(Kmin), log(Kmax)]
Valocchi
5
7
5 × 5
[4, 8.602]
Phosphoric acid
4
8
4 × 4
[−21, −2]
Gallic acid
3
17
3 × 3
[−39.56, −4.15]
Calcium carbonate
3
9
3 × 3
[−14, 16.5]
MoMaS easy
5
12
5 × 5
[−12, 35]
Iron-chromium
7
39
7 × 7
[−83.17, 80.9]
Tab.1
Test cases
τ
δ
Valocchi
0.8
1E-23
Phosphoric acid
0.8
1E-23
Gallic acid
0.8
1E-23
Calcium carbonate
0.9
1E-15
MoMaS easy
0.8
1E-24
Iron-chromium
0.8
1E-24
Tab.2
Test cases
α
β
Max iterations
Valocchi
0.9
0.8
50
Phosphoric acid
0.9
0.8
50
Gallic acid
0.9
0.8
50
Calcium carbonate
0.9
0.8
50
MoMaS easy
0.8
0.7
300
Iron-chromium
0.8
0.7
300
Tab.3
Fig.2
Test cases
Preconditioning
NR/%
NR on CAI/%
BVLS + BT/%
PN/%
Valocchi
No
74.25
100.00
100.00
100.00
PCF
96.25
100.00
90.61
100.00
Phosphoric acid
No
0.35
100.00
53.45
100.00
PCF
0.73
100.00
36.68
100.00
Gallic acid
No
97.96
73.06
100.00
100.00
PCF
99.63
91.05
100.00
100.00
Calcium carbonate
No
33.98
100.00
89.24
100.00
PCF
62.71
100.00
100.00
100.00
MoMaS easy
No
36.55
98.64
42.09
98.54
PCF
15.08
81.33
19.52
81.16
Iron-chromium
No
0.00
3.35
3.03
3.63
PCF
0.13
1.63
1.06
2.18
Tab.4
Fig.3
Fig.4
Fig.5
Fig.6
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