Estimating a Stoichiometric Solid’s Gibbs Free Energy Model by Means of a Constrained Evolutionary Strategy
Abstract
:1. Introduction
- data density,
- dispersion and abundance or lack of data sets,
- automation of the fitting procedure to avoid subjective data selection,
- correct thermodynamic model and its constraints,
- finding global minima considering the above.
2. Materials and Methods
2.1. Dealing with Different Data Densities: Weight Factors
- Experimental: Data points collected from experimental devices as they are most likely to be precise and reproducible.
- Mixed: A combination of experimental and theoretical approaches to obtain a given curve. For example, a fitted curve from a few experimental points. This is especially suitable when experimental points cannot be found but an equation from supposed experimental data is given.
- Theoretical/Table: Data obtained through simulations or theoretical formulations derived from other kind of properties. Table means that the value is from known data collection tables.
- Singular data: Singular points whose origin is unclear.
- Point: Just data points, they are often associated with experimental data and the combination of both gives the best weight possible.
- Quadratic: Quadratic (or higher order) equation or curve used to fit the data.
- Linear: the data comes in form of a linear temperature dependent fit.
- Constant: a constant value given for a range of temperatures.
- Not used: This extended flag has a weight of 0 and it is used to deactivate certain values for the fitting procedure without losing them entirely.
2.2. Evolutionary Strategy Method
- 1.
- The principle of maximum probability: The average distribution value is updated to maximize the likelihood that the previous most successful individuals are closer to the final solution. This is the selection and recombination step. The mean of the search distribution comes from the selection of the most relevant μ selected points from the sample. Writing the weight (not the same as the weight factors from the previous subpart) as ωi with the condition that:
- 2.
- Two types of temporal evolution of the average statistical distribution of the strategy are recorded. These paths contain significant information about the correlation between successive steps. In particular, the algorithm is effective if there is a large positive evolution in successive steps in the same direction. To do that, the covariance matrix, within the weighted selection mechanism is calculated with the following expression:
2.3. Definition of Constraints: Debye Model
if d cpfitted(l)/dT ≤ 0, a value pty1,l is added for 1 ≤ l ≤ 2000
if d2 cpfitted(l)/dT2 ≤ 0, a value pty2,l is added for 50 ≤ l ≤ 2000
if d2 cpfitted(l)/dT2 ≤ 0, a value pty3,l is added for 1 ≤ l ≤ 5
3. Results and Discussion
3.1. Computation of NASA9 Polynomial for Different Hydrates of Magnesium Sulfate
3.2. Validation of the Model: Heat Capacity Comparison and Vapor-Solid Equilibrium Curves
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Type of Thermodynamic Data | Reference |
---|---|
Cp | Glasser, 2007 [6] |
Cp, H0f, S0f | DeKock, 1986 [7] |
Cp | Pabalan, 1987 [8] |
Cp | Frost, 1957 [9] |
Cp | Gmelin, 1939 [10] |
Cp | Perry, 1999 [11] |
H0f, S0f | Grevel, 2009 [12] |
H0f, S0f | Wagman, 1982 [13] |
H0f, S0f | Grevel, 2012 [14] |
H0f, S0f | Dean, 1979 [15] |
H0f, S0f | Billon, 2015 [16] |
Group | Level Description | Assigned Weight Factor |
---|---|---|
Data source | Level 0—Experimental | 20 |
Level 1—Mixed | 2−1 | |
Level 2—Table/Theoretical | 2−2 | |
Level 3—Singular data | 2−3 | |
Equation type | Level 0—Point | 20 |
Level 1—Quadratic | 2−1 | |
Level 2—Linear | 2−2 | |
Level 3—Constant | 2−3 |
Type of Thermodynamic Data | Reference |
---|---|
Cp | Glasser, 2007 [6] |
Cp, H0f, S0f | DeKock, 1986 [7] |
Cp | Pabalan, 1987 [8] |
Cp, H0f, S0f | Perry, 1999 [11] |
H0f, S0f | Grevel, 2009 [12] |
Cp, H0f, S0f | Wagman, 1982 [13] |
Cp, H0f, S0f | Dean, 1979 [15] |
H0f, S0f | Billon, 2015 [16] |
Cp, H0f, S0f | Aylward, 1975 [20] |
Cp, H0f, S0f | Robie, 1995 [21] |
Cp, H0f, S0f | CRC_Handbook, 2003 [22] |
Cp, H0f, S0f | Patnaik, 2003 [23] |
H0f, S0f | Rayner, 2010 [24] |
Type of Thermodynamic Data | Reference | Substance |
---|---|---|
Cp | Glasser, 2007 [6] | MgSO4·2H2O, MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
Cp, H0f, S0f | DeKock, 1986 [7] | MgSO4·2H2O, MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
Cp, H0f, S0f | Pabalan, 1987 [8] | MgSO4·5H2O, MgSO4·6H2O, MgSO4·7H2O |
Cp | Gmelin, 1939 [10] | MgSO4·6H2O, MgSO4·7H2O |
Cp | Perry, 1999 [11] | MgSO4·6H2O, MgSO4·7H2O |
Cp, H0f, S0f | Grevel, 2009 [12] | MgSO4·2H2O, MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
Cp, H0f, S0f | Wagman, 1982 [13] | MgSO4·2H2O, MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
H0f, S0f | Grevel, 2012 [14] | MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
H0f, S0f | Dean, 1979 [15] | MgSO4·7H2O |
H0f, S0f | Billon, 2015 [16] | MgSO4·4H2O, MgSO4·5H2O, MgSO4·6H2O, MgSO4·7H2O |
H0f, S0f | Aylward, 1975 [20] | MgSO4·7H2O |
H0f, S0f | Robie, 1995 [21] | MgSO4·7H2O |
Cp, H0f, S0f | Patnaik, 2003 [23] | MgSO4·2H2O, MgSO4·4H2O, MgSO4·6H2O, MgSO4·7H2O |
H0f, S0f | Rayner, 2010 [24] | MgSO4·7H2O |
Cp, S0f | Cox, 1955 [32] | MgSO4·6H2O |
Cp, S0f | Gurevich, 2007 [33] | MgSO4·7H2O |
Minimum Initial Residual (No pty) | Maximum Final Residual (No pty) | pty0 | pty1 | pty2 | pty3 |
---|---|---|---|---|---|
1016 | 102 | 106 | 1012 | 104 | 109 |
Substance | Iterations | Population (λ) | Initial Distribution (σ) | Residual |
---|---|---|---|---|
MgSO4 | 290 | 315 | 3 | 1.72 × 102 |
MgSO4·1H2O | 280 | 315 | 3 | 3.17 × 101 |
MgSO4·2H2O | 350 | 315 | 3 | 2.75 × 10−4 |
MgSO4·4H2O | 350 | 315 | 3 | 4.26 × 103 |
MgSO4·5H2O | 350 | 315 | 3 | 6.11 × 10−26 |
MgSO4·6H2O | 350 | 315 | 3 | 1.50 × 103 |
MgSO4·7H2O | 350 | 315 | 3 | 5.79 × 103 |
MRE | NRMSD | |||||||
---|---|---|---|---|---|---|---|---|
Substance | LM | LM + Weight Factor | CMA-ES | LM | LM + Weight Factor | CMA-ES | n | Number of Sources |
MgSO4 | 3.02% | 2.98% | 3.92% | 3.66% | 4.73% | 5.11% | 694 | 10 |
MgSO4·1H2O | 2.49% | 1.94% | 2.06% | 3.55% | 4.09% | 4.34% | 438 | 7 |
MgSO4·2H2O | 0.02% | 0.02% | 0.02% | 0.02% | 0.03% | 0.03% | 13 | 2 |
MgSO4·4H2O | 70.85% | 51.21% | 278.4% | 2.15% | 2.19% | 6.63% | 134 | 4 |
MgSO4·5H2O | <0.01% | <0.01% | <0.01% | <0.01% | <0.01% | <0.01% | 201 | 1 |
MgSO4·6H2O | 1.02% | 0.87% | 2.21% | 1.06% | 1.18% | 1.39% | 421 | 10 |
MgSO4·7H2O | 246.79% | 468.28% | 24.93% | 2.88% | 3.04% | 3.09% | 650 | 9 |
Substance | Iterations | Population (λ) | Initial Distribution (σ) |
---|---|---|---|
All | 100 | 55 | 5 |
Enthalpy | Entropy | |||||
---|---|---|---|---|---|---|
Substance | MRE | NRMSD | n | MRE | NRMSD | n |
MgSO4 | 0.81% | 1.85% | 15 | 0.54% | 1.41% | 14 |
MgSO4·1H2O | 0.21% | 0.29% | 10 | 1.80% | 3.93% | 8 |
MgSO4·2H2O | 0.03% | 0.04% | 4 | 10.22% | 11.17% | 2 |
MgSO4·4H2O | 0.03% | 0.04% | 9 | 1.71% | 1.88% | 6 |
MgSO4·5H2O | 0.03% | 0.03% | 2 | 1.65% | 1.65% | 2 |
MgSO4·6H2O | 0.03% | 0.04% | 10 | 0.12% | 0.28% | 9 |
MgSO4·7H2O | 0.02% | 0.03% | 14 | 0.60% | 1.28% | 14 |
Substance | a0 | a1 | a2 | a3 | a4 | a5 | a6 | a7 | a8 |
---|---|---|---|---|---|---|---|---|---|
MgSO4 | 2.41 × 10−2 | −9.67 × 10−3 | 2.69 × 100 | 3.86 × 10−2 | −3.86 × 10−5 | 1.93 × 10−8 | −3.62 × 10−12 | −1.57 × 105 | −1.43 × 101 |
MgSO4·1H2O | 2.37 × 10−2 | −1.38 × 10−2 | 6.22 × 100 | 3.36 × 10−2 | −3.19 × 10−6 | 4.44 × 10−9 | −2.31 × 10−12 | −1.97 × 105 | −3.01 × 101 |
MgSO4·2H2O | 3.77 × 10−2 | −2.15 × 10−2 | 5.61 × 100 | 5.39 × 10−2 | −9.19 × 10−6 | 1.04 × 10−8 | −4.45 × 10−2 | −2.32 × 105 | −2.83 × 101 |
MgSO4·4H2O | −1.24 × 10−1 | 3.71 × 10−1 | −3.71 × 10−1 | 1.24 × 10−1 | −5.20 × 10−5 | 3.78 × 10−9 | 1.22 × 10−12 | −3.05 × 105 | −2.21 × 100 |
MgSO4·5H2O | 3.51 × 10−9 | −1.18 × 10−9 | 1.02 × 101 | 8.93 × 10−2 | −2.09 × 10−17 | 1.37 × 10−19 | −1.34 × 10−22 | −3.43 × 105 | −4.84 × 101 |
MgSO4·6H2O | −1.76 × 10−1 | 5.27 × 10−1 | −5.28 × 10−1 | 1.76 × 10−1 | −1.19 × 10−4 | 3.54 × 10−8 | −3.88 × 10−12 | −3.78 × 105 | −2.64 × 100 |
MgSO4·7H2O | −1.84 × 10−1 | 5.51 × 10−1 | −5.51 × 10−1 | 1.84 × 10−1 | −9.80 × 10−5 | 1.93 × 10−8 | −7.53 × 10−13 | −4.15 × 105 | −2.71 × 100 |
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Grau Turuelo, C.; Pinnau, S.; Breitkopf, C. Estimating a Stoichiometric Solid’s Gibbs Free Energy Model by Means of a Constrained Evolutionary Strategy. Materials 2021, 14, 471. https://doi.org/10.3390/ma14020471
Grau Turuelo C, Pinnau S, Breitkopf C. Estimating a Stoichiometric Solid’s Gibbs Free Energy Model by Means of a Constrained Evolutionary Strategy. Materials. 2021; 14(2):471. https://doi.org/10.3390/ma14020471
Chicago/Turabian StyleGrau Turuelo, Constantino, Sebastian Pinnau, and Cornelia Breitkopf. 2021. "Estimating a Stoichiometric Solid’s Gibbs Free Energy Model by Means of a Constrained Evolutionary Strategy" Materials 14, no. 2: 471. https://doi.org/10.3390/ma14020471