# Optimal Design of Experiments for Liquid–Liquid Equilibria Characterization via Semidefinite Programming

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Motivation

- (i)
- choosing the appropriate thermodynamic model according to the nature of the mixture and the region of operation;
- (ii)
- defining the methodology to estimate parameters that are able to describe well the experimental data;
- (iii)
- iterative computation of LLE and stability analysis to prevent spurious phase predictions;
- (iv)
- refinement of existing models, by planning further experiments; and
- (v)
- designing experiments prior to the laboratory experiments.

^{4}constraints [35].

#### Paper Organization and Nomenclature

## 2. Background

#### 2.1. Liquid–Liquid Equilibria: Mathematical Representation

#### 2.2. Optimal Design of Experiments

#### 2.3. Semidefinite Programming

## 3. SDP Formulations for Finding Optimal Design of Experiments

#### 3.1. FIM Construction

#### 3.2. SDP Formulations for D-, A- and E-Optimal Designs

#### 3.3. Implementation Aspects

^{−7}and 1 × 10

^{−8}, respectively.

`ADiMat`code [64].

`cvx`[65] or Picos [66], that automatically transform the semidefinite constraints in Equations (16b) and (16e) and the objective functions into a series of LMIs before passing them to SDP solvers such as

`SeDuMi`[67] or

`Mosek`[68]. This is possible when ${\Phi}_{\delta}$ is SDr, which is true for our design criteria. In our work, we solved all SDP problems using the

`cvx`environment combined with the solver

`Mosek`that uses an efficient interior point algorithm [69].

^{−5}in all problems.

## 4. Results

#### 4.1. Example 1—Ternary System of Type 1, Class I; $\mathbf{\theta}$ Including Dimensionless BIPs

#### 4.2. Example 2—Ternary System of Type 1, Class II; $\mathbf{\theta}$ Including Dimensionless BIPs

#### 4.3. Example 3—Ternary System of Type 1, Class I; $\mathbf{\theta}$ Including Dimensionless and Non-Random BIPs

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. NRTL Model

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**Figure 1.**D-optimal design of experiments for system WAT-TOL-ANL: (

**a**) candidate initial mixtures and corresponding tie-lines; and (

**b**) tie-lines corresponding to D-optimal design (symbols: o, Phase 1 composition; *, Phase 2 composition; ∇, initial mixture composition; tie lines in red).

**Figure 2.**D-optimal designs for system WAT-TCE–KET: (

**a**) candidate initial mixtures and corresponding tie-lines; and (

**b**) tie-lines corresponding to D-optimal design (symbols: o, Phase 1 composition; *, Phase 2 composition; ∇, initial mixture composition; tie lines in red).

Criterion | Optimal Design (Initial Composition) | Optimum | CPU (s) |
---|---|---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccc}0.4500& 0.5500& 0.7500& 0.8500& 0.9500\\ 0.0500& 0.0500& 0.0500& 0.0500& 0.0500\\ 0.5000& 0.4000& 0.2000& 0.1000& 0.0000\\ 0.1895& 0.1858& 0.2187& 0.2410& 0.1650\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $2.30\times {10}^{-3}$ | 32.23 |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccccc}0.4500& 0.5500& 0.7500& 0.7500& 0.8500& 0.9500\\ 0.0500& 0.0500& 0.0500& 0.1500& 0.0500& 0.0500\\ 0.5000& 0.4000& 0.2000& 0.1000& 0.1000& 0.0000\\ 0.0871& 0.1993& 0.3798& 0.1964& 0.0678& 0.0697\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $1.675\times {10}^{5}$ | 32.69 |

E-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccccc}0.4500& 0.5500& 0.7500& 0.7500& 0.8500& 0.9500\\ 0.0500& 0.0500& 0.0500& 0.1500& 0.0500& 0.0500\\ 0.5000& 0.4000& 0.2000& 0.1000& 0.1000& 0.0000\\ 0.0827& 0.2471& 0.4285& 0.1858& 0.0328& 0.0231\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $1.1\times {10}^{-5}$ | 32.06 |

**Table 2.**Optimal experimental designs for ternary system WAT-TOL-ANL (compositions in equilibrium phases).

Criterion | Equilibrium Compositions in Separating Phases |
---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccc}0.6640& 0.7427& 0.8686& 0.9325& 0.9994\\ 0.0148& 0.0096& 0.0039& 0.0019& 0.0006\\ 0.3212& 0.2477& 0.1275& 0.0656& 0.0000\\ 0.4448& 0.3482& 0.1707& 0.0782& 0.0081\\ 0.0508& 0.0923& 0.2752& 0.5003& 0.9919\\ 0.5044& 0.5595& 0.5541& 0.4215& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccccc}0.6640& 0.7427& 0.8686& 0.9596& 0.9325& 0.9994\\ 0.0148& 0.0096& 0.0039& 0.0013& 0.0019& 0.0006\\ 0.3212& 0.2477& 0.1275& 0.0392& 0.0656& 0.0000\\ 0.4448& 0.3482& 0.1707& 0.0439& 0.0782& 0.0081\\ 0.0508& 0.0923& 0.2752& 0.6511& 0.5003& 0.9919\\ 0.5044& 0.5595& 0.5541& 0.3050& 0.4215& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

E-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccccc}0.6640& 0.7427& 0.8686& 0.9596& 0.9325& 0.9994\\ 0.0148& 0.0096& 0.0039& 0.0013& 0.0019& 0.0006\\ 0.3212& 0.2477& 0.1275& 0.0392& 0.0656& 0.0000\\ 0.4448& 0.3482& 0.1707& 0.0439& 0.0782& 0.0081\\ 0.0508& 0.0923& 0.2752& 0.6511& 0.5003& 0.9919\\ 0.5044& 0.5595& 0.5541& 0.3050& 0.4215& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

Criterion | Optimal Design (Initial Composition) | Optimum | CPU ($\mathbf{s}$) |
---|---|---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccc}0.1500& 0.4500& 0.5500& 0.7500& 0.7500& 0.8500& 0.9500\\ 0.0500& 0.4500& 0.3500& 0.0500& 0.1500& 0.0500& 0.0500\\ 0.8000& 0.1000& 0.1000& 0.2000& 0.1000& 0.1000& 0.0000\\ 0.1164& 0.0078& 0.1014& 0.2194& 0.1747& 0.2134& 0.1669\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $1.75\times {10}^{-4}$ | 15.45 |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccc}0.1500& 0.7500& 0.7500& 0.9500\\ 0.0500& 0.0500& 0.1500& 0.0500\\ 0.8000& 0.2000& 0.1000& 0.0000\\ 0.1886& 0.1834& 0.3946& 0.2333\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $1.16\times {10}^{7}$ | 17.58 |

E-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccccc}0.0500& 0.1500& 0.1500& 0.2500& 0.3500& 0.4500& 0.7500& 0.7500& 0.9500\\ 0.8500& 0.0500& 0.7500& 0.6500& 0.5500& 0.4500& 0.0500& 0.1500& 0.0500\\ 0.1000& 0.8000& 0.1000& 0.1000& 0.1000& 0.1000& 0.2000& 0.1000& 0.0000\\ 0.0136& 0.0726& 0.0301& 0.1355& 0.3421& 0.0055& 0.0109& 0.2791& 0.1105\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $9.4\times {10}^{-7}$ | 14.63 |

**Table 4.**Optimal experimental designs for ternary system WAT-TCE–KET (compositions in equilibrium phases).

Criterion | Equilibrium Compositions in Separating Phases |
---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccc}0.9935& 0.9970& 0.9967& 0.9944& 0.9960& 0.9951& 0.9999\\ 0.0000& 0.0001& 0.0001& 0.0001& 0.0001& 0.0001& 0.0001\\ 0.0065& 0.0029& 0.0032& 0.0055& 0.0039& 0.0048& 0.0000\\ 0.1348& 0.0095& 0.0119& 0.0989& 0.0277& 0.0691& 0.0025\\ 0.0509& 0.8123& 0.7715& 0.1830& 0.5901& 0.3187& 0.9975\\ 0.8143& 0.1782& 0.2166& 0.7181& 0.3822& 0.6122& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{cccc}0.9935& 0.9944& 0.9960& 0.9999\\ 0.0000& 0.0001& 0.0001& 0.0001\\ 0.0065& 0.0055& 0.0039& 0.0000\\ 0.1348& 0.0989& 0.0277& 0.0025\\ 0.0509& 0.1830& 0.5901& 0.9975\\ 0.8143& 0.7181& 0.3822& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

E-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccccc}0.9977& 0.9935& 0.9975& 0.9974& 0.9972& 0.9970& 0.9944& 0.9960& 0.9999\\ 0.0001& 0.0000& 0.0001& 0.0001& 0.0001& 0.0001& 0.0001& 0.0001& 0.0001\\ 0.0022& 0.0065& 0.0024& 0.0025& 0.0027& 0.0029& 0.0055& 0.0039& 0.0000\\ 0.0058& 0.1348& 0.0063& 0.0070& 0.0080& 0.0095& 0.0989& 0.0277& 0.0025\\ 0.8896& 0.0509& 0.8771& 0.8613& 0.8406& 0.8123& 0.1830& 0.5901& 0.9975\\ 0.1046& 0.8143& 0.1166& 0.1317& 0.1514& 0.1782& 0.7181& 0.3822& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

**Table 5.**Optimal designs for ternary system WAT-TOL-ANL and $\mathbf{\theta}$ including nine parameters (initial mixtures).

Criterion | Optimal Design (Initial Composition) | Optimum | CPU ($\mathbf{s}$) |
---|---|---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccc}0.4500& 0.5500& 0.6500& 0.7500& 0.7500& 0.8500& 0.9500\\ 0.0500& 0.0500& 0.0500& 0.0500& 0.1500& 0.0500& 0.0500\\ 0.5000& 0.4000& 0.3000& 0.2000& 0.1000& 0.1000& 0.0000\\ 0.1734& 0.1833& 0.0068& 0.2091& 0.1014& 0.1922& 0.1338\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $1.02\times {10}^{-3}$ | 32.50 |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccccc}0.3500& 0.4500& 0.4500& 0.5500& 0.6500& 0.7500& 0.7500& 0.8500& 0.9500\\ 0.1500& 0.0500& 0.1500& 0.0500& 0.0500& 0.0500& 0.1500& 0.0500& 0.0500\\ 0.5000& 0.5000& 0.4000& 0.4000& 0.3000& 0.2000& 0.1000& 0.1000& 0.0000\\ 0.0527& 0.1684& 0.0447& 0.1575& 0.0753& 0.1816& 0.0819& 0.1320& 0.1059\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ | $5.837\times {10}^{5}$ | 33.38 |

E-optimal | $2.5\times {10}^{-4}$ | 33.59 |

**Table 6.**Optimal designs for ternary system WAT-TOL-ANL and $\mathbf{\theta}$ including nine parameters (compositions in equilibrium phases).

Criterion | Equilibrium Compositions in Separating Phases |
---|---|

D-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccc}0.6640& 0.7427& 0.8067& 0.8686& 0.9596& 0.9325& 0.9994\\ 0.0148& 0.0096& 0.0064& 0.0039& 0.0013& 0.0019& 0.0006\\ 0.3212& 0.2477& 0.1869& 0.1275& 0.0392& 0.0656& 0.0000\\ 0.4448& 0.3482& 0.2610& 0.1707& 0.0439& 0.0782& 0.0081\\ 0.0509& 0.0923& 0.1582& 0.2752& 0.6511& 0.5003& 0.9919\\ 0.5044& 0.5595& 0.5808& 0.5541& 0.3050& 0.4215& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

A-optimal | $\left(\phantom{\rule{-5.4555pt}{0ex}}\begin{array}{ccccccccc}0.8244& 0.6640& 0.8550& 0.7427& 0.8067& 0.8686& 0.9596& 0.9325& 0.9994\\ 0.0056& 0.0148& 0.0044& 0.0096& 0.0064& 0.0039& 0.0013& 0.0019& 0.0006\\ 0.1699& 0.3212& 0.1406& 0.2477& 0.1869& 0.1275& 0.0392& 0.0656& 0.0000\\ 0.2356& 0.4448& 0.1908& 0.3482& 0.2610& 0.1707& 0.0439& 0.0782& 0.0081\\ 0.1848& 0.0509& 0.2432& 0.0923& 0.1582& 0.2752& 0.6511& 0.5003& 0.9919\\ 0.5796& 0.5044& 0.5660& 0.5595& 0.5808& 0.5541& 0.3050& 0.4215& 0.0000\end{array}\phantom{\rule{-5.4555pt}{0ex}}\right)$ |

E-optimal |

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Duarte, B.P.M.; Atkinson, A.C.; Granjo, J.F.O.; Oliveira, N.M.C.
Optimal Design of Experiments for Liquid–Liquid Equilibria Characterization via Semidefinite Programming. *Processes* **2019**, *7*, 834.
https://doi.org/10.3390/pr7110834

**AMA Style**

Duarte BPM, Atkinson AC, Granjo JFO, Oliveira NMC.
Optimal Design of Experiments for Liquid–Liquid Equilibria Characterization via Semidefinite Programming. *Processes*. 2019; 7(11):834.
https://doi.org/10.3390/pr7110834

**Chicago/Turabian Style**

Duarte, Belmiro P.M., Anthony C. Atkinson, José F.O. Granjo, and Nuno M.C. Oliveira.
2019. "Optimal Design of Experiments for Liquid–Liquid Equilibria Characterization via Semidefinite Programming" *Processes* 7, no. 11: 834.
https://doi.org/10.3390/pr7110834