# Parameter Estimation Strategies in Thermodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Parameter Estimation Approaches

#### 2.1. Least Squares

#### 2.2. Reconciliation Formulation with Constraints

#### 2.3. Reconciliation Formulation without Constraints

#### 2.3.1. Computational Aspects

#### 2.3.2. Linearization Points

## 3. Comparison of Parameter Estimation Strategies

#### 3.1. Example: Vapor Pressure

#### 3.2. Example: NRTL

## 4. Multi-Criteria Parameter Estimation

#### 4.1. General Framework

#### 4.2. Example: Distillation Column

## 5. Error Estimates

#### 5.1. Confidence-Region Estimates

#### 5.2. General Form of the Fisher Information Matrix

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Patino–Leal in Least-Squares Form

## Appendix B. Reformulated Exact Version

## Appendix C. Derivation of the Fisher Matrix

## References

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**Figure 1.**Regression results compared to data for regression runs with different initial parameters.

**Figure 2.**Comparison of activity coefficients in the system methanol (1) + water (2) for the different model parameters in Table 2, based on the investigated regression strategies.

**Figure 4.**Calculated Pareto front for the two objectives $\overline{f}$ and $\overline{t}$. The objectives can be interpreted as the average deviations between simulated and measured values given in units of standard deviations.

**Figure 7.**Illustration of confidence region and confidence intervals for a 2D example. ${\lambda}_{1},\phantom{\rule{4pt}{0ex}}{\lambda}_{2}$ are the eigenvalues of the covariance matrix C. The confidence region is an ellipse, and the confidence intervals are the projections of that ellipse onto the individual coordinate axes.

**Figure 8.**Errors of the model prediction based on regression results for the vapor pressure of methanol in the ordinary least-squares approach. The shaded areas depict the 95% confidence regions for the model prediction. Note that they are scaled by a factor of 10 in different directions for visibility reasons.

Method | Start | a | b | c | Iterations | Error | Time [ms] |
---|---|---|---|---|---|---|---|

Initial point | I | 100.896 | −7210.917 | −12.44128 | - | - | - |

II | 1 | 1 | 1 | - | - | - | |

III | 50 | −5000 | 0 | - | - | - | |

${s}^{A}$ | I | 44.5796 | −5511.94 | −2.87739 | 24 | 0.23211 | 2.35 |

II | 44.581 | −5512.01 | −2.8776 | 19 | 0.23211 | 1.87 | |

III | 44.581 | −5512.01 | −2.8776 | 15 | 0.23211 | 1.49 | |

${s}^{B}$ | I | 43.7994 | −5471.03 | −2.76444 | 192 | 0.107258 | 156.01 |

II | 43.7992 | −5471.03 | −2.76441 | 179 | 0.107258 | 145.33 | |

III | 43.7991 | −5471.02 | −2.7644 | 169 | 0.107258 | 137.41 | |

${s}^{C}$ | I | 43.7851 | −5470.32 | −2.76237 | 21 | 0.107503 | 11.93 |

II | 43.7833 | −5470.23 | −2.76211 | 31 | 0.107503 | 17.37 | |

III | 34.8895 | −5000.28 | −1.47651 | 9 | 0.133713 | 5.37 | |

${s}^{C\ast}$ | I | 43.7985 | −5470.98 | −2.76431 | 29 | 0.107258 | 228.6 |

II | 43.799 | −5471.02 | −2.76438 | 31 | 0.107258 | 239.06 | |

III | 34.8928 | −5000.36 | −1.477 | 10 | 0.133517 | 913.83 |

Method | $\mathit{\alpha}$ | ${\mathit{\tau}}_{12}$ | ${\mathit{\tau}}_{21}$ | Iterations | Error | Time [ms] |
---|---|---|---|---|---|---|

Initial | - | 1 | 1 | - | - | - |

${s}^{A}$ | 0.3 * | −0.254835 | 1.10159 | 6 | 1290.34 | 174 |

${s}^{B}$ | 0.3 * | −0.232939 | 1.07197 | 8 | 1132.79 | 58,870 |

${s}^{C}$ | 0.3 * | −0.233823 | 1.07287 | 9 | 1127.48 | 820 |

${s}^{C\ast}$ | 0.3 * | −0.234391 | 1.07364 | 9 | 1127.59 | 1050 |

Parameter | Initial Value | Lower Bound | Upper Bound |
---|---|---|---|

Total feed flow rate [kg/h] | 2330 | 2300 | 2550 |

Reflux ratio [-] | 1.4 | 0.5 | 3 |

Reboiler specification [-] | 0.0046 | 0.003 | 0.006 |

Tray efficiency [-] | 0.7 | 0.5 | 0.95 |

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## Share and Cite

**MDPI and ACS Style**

Höller, J.; Bickert, P.; Schwartz, P.; von Kurnatowski, M.; Kerber, J.; Künzle, N.; Lorenz, H.-M.; Asprion, N.; Blagov, S.; Bortz, M.
Parameter Estimation Strategies in Thermodynamics. *ChemEngineering* **2019**, *3*, 56.
https://doi.org/10.3390/chemengineering3020056

**AMA Style**

Höller J, Bickert P, Schwartz P, von Kurnatowski M, Kerber J, Künzle N, Lorenz H-M, Asprion N, Blagov S, Bortz M.
Parameter Estimation Strategies in Thermodynamics. *ChemEngineering*. 2019; 3(2):56.
https://doi.org/10.3390/chemengineering3020056

**Chicago/Turabian Style**

Höller, Johannes, Patricia Bickert, Patrick Schwartz, Martin von Kurnatowski, Joachim Kerber, Niklaus Künzle, Hilke-Marie Lorenz, Norbert Asprion, Sergej Blagov, and Michael Bortz.
2019. "Parameter Estimation Strategies in Thermodynamics" *ChemEngineering* 3, no. 2: 56.
https://doi.org/10.3390/chemengineering3020056