1 Introduction
2 Methods
2.1 Modelling of thermodynamic functions
2.2 Neural networks for the approximation of functions and their derivatives
2.2.1 Notation
2.2.2 Representation of thermodynamic functions with neural network variables
2.2.3 Activation function
2.2.4 Training neural networks and derivatives with resilient propagation
3 Results
3.1 A neural network for the approximation of thermodynamic functions
3.2 Approximation of the thermodynamic functions of pure iron
3.2.1 BCC
\(T_{\mathrm{tr}}\) in K | Source | |
---|---|---|
\({\text {Fe}}_{\text {BCC}\rightarrow \text {FCC}}\) | 1184 |
Bendick and Pepperhoff (1982) |
1184 |
Tsuchiya et al. (1971) | |
1184 |
Anderson and Hultgren (1962) | |
1184 |
Vollmer et al. (1966) | |
1184.08 | This work | |
\({\text {Fe}}_{\text {FCC}\rightarrow \text {BCC}}\) | 1665 |
Bendick and Pepperhoff (1982) |
1665 |
Tsuchiya et al. (1971) | |
1665 |
Anderson and Hultgren (1962) | |
1665 |
Vollmer et al. (1966) | |
1665.09 | This work | |
\({\text {Fe}}_{\text {BCC}\rightarrow \text {LIQ}}\) | 1811 |
Desai (1986) |
1809 |
Anderson and Hultgren (1962) | |
1809 |
Vollmer et al. (1966) | |
1808.99 | This work | |
\({\text {Fe}}_{\text {LIQ}\rightarrow \text {GAS}}\) | 3134 |
Haynes (2015) |
3134.0 | This work |
\(\Delta H_{\text {BCC}\rightarrow \text {FCC}}\) in \(\text {J}/\text {mol}\)\(T=1183.95\text {K}\) | \(\Delta H_{\text {FCC}\rightarrow \text {BCC}}\) in \(\text {J}/\text {mol}\)\(T=1664.95\text {K}\) | \(\Delta H_{\text {BCC}\rightarrow \text {LIQ}}\) in \(\text {J}/\text {mol}\)\(T=1808.99\text {K}\) | |
---|---|---|---|
Smith (1946) | 900 | – | – |
Wallace et al. (1960) | \(911(\pm 80)\) | – | – |
Anderson and Hultgren (1962) | \(941(\pm 80)\) | \(1088(\pm 210)\) | – |
Dench and Kubaschewski (1963) | 900 | 837 | – |
Braun and Kohlhaas (1965) | \(910(\pm 20)\) | \(850(\pm 30)\) | \(14{,}400(\pm 300)\) |
Morris et al. (1966) | – | – | \(13{,}799(\pm 418)\) |
Vollmer et al. (1966) | – | – | \(14{,}400(\pm 200)\) |
Treverton and Margrave (1971) | – | – | \(13{,}836(\pm 293)\) |
Normanton et al. (2013) | \(923(\pm 10)\) | \(765(\pm 22)\) | – |
Cezairliyan and McClure (1974) | – | 890 | – |
Rogez and Le Coze (1980) | \(900(\pm 20)\) | – | – |
Schürmann and Kaiser (1981) | 954 | 770 | 14,390 |
Bendick and Pepperhoff (1982) | 820 | 850 | – |
Johansson (1937) | 1339 | 444 | – |
Darken and Smith (1951) | 900 | 690 | 15,355 |
Kaufman et al. (1963) | 879 | 937 | – |
Braun and Kohlhaas (1965) | 900 | 950 | – |
Guillermet et al. (1985) | 1013 | 826 | 13,807 |
Desai (1986) | 900 | 850 | 13,810 |
This work | 928 | 871 | 14,145 |