# Modelling and Comparative Analysis of Different Methods of Liquid Membrane Separations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling of Continuous Versions of Extraction-Stripping Separation

^{3}; ${y}_{1}$ is the concentration of the passing component in the organic phase (extractant phase) in the extraction column, g/mL or g/cm

^{3}; ${x}_{2}$ is the concentration of the passing component in the aqueous stripping phase (acceptor or extract phase) in the stripping column, g/mL or g/cm

^{3}; ${y}_{2}$ is the concentration of the passing component in the organic phase in the stripping column, g/mL or g/cm

^{3}; ${a}_{1}$ and ${a}_{2}$ are the specific areas of phase contact in the extraction and stripping columns, respectively, cm

^{−1}; ${k}_{1}$ and ${k}_{2}$ are mass-transfer coefficients in the extraction and stripping columns, respectively, cm/s; ${v}_{1}$ and ${v}_{2}$ are specific flow rates of the aqueous phases in the extraction and stripping columns, respectively, cm/s; w is the specific flow rate of the organic phase in the extraction and stripping columns (in the extraction column, it is the extract phase; in the stripping column, it is the feed or the raffinate phase), cm/s; ${m}_{1}=\frac{{y}^{*}}{{x}_{1}^{*}}=constant$ and ${m}_{2}={y}^{*}/{x}_{2}^{*}=constant$ are the equilibrium distribution coefficients in the extraction and stripping columns, respectively; and z is the co-ordinate along the phase contact surface, cm ($0\le z\le L$).

**,**${x}_{2}$, and y are the concentrations of the passing component in the donor and acceptor phases and in the liquid membrane, respectively, and the symbol * stands for equilibrium conditions; and z is the co-ordinate over the phase contact surface ($0\le z\le L$). As mentioned above, the equilibrium distribution coefficients are assumed to be constant (${m}_{1}={y}^{*}/{x}_{1}^{*}$ and ${m}_{2}={y}^{*}/{x}_{2}^{*}$). The boundary conditions for Equations (1)–(17) are shown in Figure 2 and Figure 3. By solving these equations, the outlet concentration in the raffinate can be established as follows:

## 3. Modelling of Staged Versions of Extraction-Stripping Separation

_{1}/m

_{2}.

_{1}and V

_{2}are the volumes of cells; and v

_{1}, v

_{2}, and w are the volumetric flow rates.

_{1}) and stripping (v

_{2}) aqueous phases, as well as the circulating membrane phase (w), was measured, and samples of the aqueous phases were taken from each mixer-settler. The equilibrium distribution coefficients (m

_{1}= m

_{2}= 0.73) were determined in shake experiments. The results of the experiments, as well as the calculated theoretical concentrations, are given in Table 1 and Table 2.

_{1,n}) was calculated by Equations (22) and (25). The concentrations in all stripping and extraction stages were determined from material balance equations. The results in Table 1 and Table 2 demonstrate, in general, a satisfactory agreement between the prediction of mathematical modeling and the experiment, and confirm the advantages of the staged version of supported liquid membranes over the conventional extraction-stripping separation.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagrams of extraction-stripping separation processes: a—conventional conjugated extraction-stripping; b—film pertraction; c—emulsion membranes; and d—supported liquid membranes.

**Figure 4.**Comparison between efficiency of continuous schemes of extraction-stripping separation technique.

**Table 1.**Theoretical and experimental concentrations in the mixer-settlers operating in the supported liquid membrane mode (scheme (e)).

Number of Stages | x_{1} (%)Experiment | x_{1} (%)Theory | x_{2} (%)Experiment | x_{2} (%)Theory |
---|---|---|---|---|

v_{1} = 1.12 L/h, x_{1,0} = 4.8%, v_{2} = 2.36 L/h, w = 3.0 L/h | ||||

1 | 3.49 | 3.00 | 2.00 | 2.04 |

2 | 2.20 | 1.78 | 1.36 | 1.18 |

3 | 1.27 | 1.01 | 0.72 | 0.61 |

4 | 0.70 | 0.50 | 0.26 | 0.24 |

v_{1} = 1.48 L/h, x_{1,0} = 4.7%, v_{2} = 2.64 L/h, w = 6.0 L/h | ||||

1 | 2.94 | 2.96 | 2.44 | 2.37 |

2 | 1.83 | 1.79 | 1.40 | 1.40 |

3 | 0.98 | 1.00 | 0.74 | 0.74 |

4 | 0.49 | 0.47 | 0.23 | 0.30 |

v_{1} = 1.15 L/h, x_{1,0} = 4.8%, v_{2} = 2.16 L/h, w = 10.0 L/h | ||||

1 | 2.77 | 2.72 | 2.34 | 2.40 |

2 | 1.56 | 1.48 | 1.28 | 0.29 |

3 | 0.76 | 0.74 | 0.62 | 0.63 |

4 | 0.34 | 0.30 | 0.23 | 0.23 |

**Table 2.**Theoretical and experimental concentrations in the mixer-settlers operating in the conventional extraction-stripping mode (scheme (Figure 5h)).

Number of Stages | v_{1} = 1.15 L/h, x_{1,0} = 4.8%, v_{2} = 2.16 L/h, w = 10.0 L/h | |||
---|---|---|---|---|

x_{1} (%)Experiment | x_{1} (%)Theory | x_{2} (%)Experiment | x_{2} (%)Theory | |

1 | 1.84 | 1.86 | 1.83 | 1.85 |

2 | 1.43 | 1.40 | 1.81 | 1.82 |

3 | 1.37 | 1.33 | 1.71 | 1.71 |

4 | 1.36 | 1.32 | 1.40 | 1.32 |

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**MDPI and ACS Style**

Kostanyan, A.E.; Voshkin, A.A.; Belova, V.V.; Zakhodyaeva, Y.A. Modelling and Comparative Analysis of Different Methods of Liquid Membrane Separations. *Membranes* **2023**, *13*, 554.
https://doi.org/10.3390/membranes13060554

**AMA Style**

Kostanyan AE, Voshkin AA, Belova VV, Zakhodyaeva YA. Modelling and Comparative Analysis of Different Methods of Liquid Membrane Separations. *Membranes*. 2023; 13(6):554.
https://doi.org/10.3390/membranes13060554

**Chicago/Turabian Style**

Kostanyan, Artak E., Andrey A. Voshkin, Vera V. Belova, and Yulia A. Zakhodyaeva. 2023. "Modelling and Comparative Analysis of Different Methods of Liquid Membrane Separations" *Membranes* 13, no. 6: 554.
https://doi.org/10.3390/membranes13060554