# Improvement of Component Flux Estimating Model for Pervaporation Processes

^{*}

## Abstract

**:**

_{i}) estimation. The pervaporation model of Mizsey and Valentinyi, which is based on Rautenbach’s works, is further improved in this work and tested rigorously by statistical means. Until now, this type of exponential modelling was only used for alcohol–water mixtures, but in this work, it was extended to an ethyl acetate–water binary mixture as well. Furthermore, a flowchart of modelling is presented for the first time in the case of an exponential pervaporation model. The results of laboratory-scale experiments were used as the basis of the study and least squares approximation was used to compare them to the different model’s estimations. According to our results, Valentinyi’s model (Model I) and the alternative model (Model III) appear to be the best methods for PV modelling, and there is no significant difference between the models, mainly in organophilic cases. In the case of the permeation component, Model I, which better follows the exponential function, is recommended. It is important to emphasize that our research confirms that the exponential type model seems to be universally feasible for most organic–water binary mixtures. Another novelty of the work is that after PDMS and PVA-based membranes, the accuracy of the semiempirical model for the description of water flux on a PEBA-based membrane was also proved, in the organophilic case.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Pervaporation Modelling

_{i}is the concentration outside of the membrane.

- absorption of components in the membrane;
- selective diffusion of components through the length of the membrane;
- desorption and consequential evaporation to vapour phase on the permeate side.

^{*}is the reference temperature, in this case equal to 293 K or 20 °C.

_{0}) is infinitely big compared to the transport coefficient, correlating with the concept that this layer’s resistance is negligible. Thus, the Model I can be simplified as:

#### 2.2. Model Improvement

- mole fraction of the feed (x
_{i}_{1}) [mole/mole]; - mole fraction of the permeate (x
_{i}_{3}) [mole/mole]; - coefficients of the Wilson equation (A
_{ij}, A_{ji}) [cal/moleK]; - input temperature (T) [°C. K];
- constants of the Antoine equation for both components (A, B, C, D and E) [-];
- pressure on the permeate side (p
_{3}) [bar. kPa]; - partial fluxes of both components (J
_{i}) [kg/m^{2}h].

_{i}

_{1}and p

_{3}, represent the location in the membrane module: 1 is the feed side, 2 is the intermembrane plane and 3 is the permeate side.

_{ij}and Λ

_{ji}coefficients were obtained by the following formulas:

_{i}and V

_{j}are the molar volume of pure liquid. and it can be calculated as follows:

_{i}) and the added B parameter of the new models. The estimated function derives from the combination of the respective model and Equation (8):

## 3. Results

_{i}), and the added B parameter. Estimation of these parameters can be seen in Table 3 for each component.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A | membrane area [m^{2}] |

B | constant in Model I, II and III |

c | total molar concentration [mol/mol] |

c_{i} | concentration of component i [mol/m^{3}] |

D_{i} | diffusion coefficient [m^{2}/h] |

D_{i0} | diffusion coefficient of component i [ kmol/m^{2} h] |

$\overline{{D}_{i}}$ | transport coefficient of component i [kmol/m^{2} h] |

${\overline{D}}_{i,exp}$ | modified transport coefficient of component i in Model I, II and III [kmol/m^{2} h] |

$\overline{{D}_{i}^{\ast}}$ | relative transport coefficient of component i [kmol/m^{2} h] |

E_{i} | activation energy of component i [kJ/mol] |

f_{i0} | fugacity of pure i component [mbar, kPa] |

f_{i1} | fugacity of component i in the feed side [mbar, kPa] |

f_{i3} | fugacity of component i in the permeate side [mbar, kPa] |

J | total flux [kg/m^{2}h] |

J_{i} | partial flux [kg/m^{2}h] |

L | distance of diffusion [m] |

n_{i} | weight of component i [mol] |

p_{i0} | vapour pressure of pure i component [bar, kPa] |

p_{i1} | partial pressure of component i in the feed side [bar, kPa] |

p_{i2} | partial pressure of component i between the two layers of the membrane [bar, kPa] |

p_{i3} | partial pressure of component i in the permeate side [bar, kPa] |

p_{3} | pressure on the permeate side [bar, kPa] |

Q_{0} | permeability of the porous supporting layer of the membrane [kmol/m^{2} h bar] |

R | gas constant [kJ/kmol K] |

t | time [s, h] |

T | temperature [K, °C] |

T^{*} | reference temperature: 273 K = 20 °C |

w_{F} | feed concentration of component i [wt%] |

x_{i1} | mol fraction of component i in the feed [mol/mol] |

y_{i} | mol fraction of component i in the permeate [mol/mol] |

Greek letters: | |

β_{ij} | selectivity for component i and j |

δ | thickness of the membrane [m] |

γ_{i1} | activity coefficient of component i in the feed |

γ_{i3} | activity coefficient of component i in the permeate |

$\overline{{\gamma}_{i}}$ | average activity coefficient of component i |

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**Figure 1.**Flowchart of calculation of pervaporation modelling. (Interpretation: p_i0 means p

_{i0}, others can be interpreted the same way and avg(y_i) means $\overline{{\gamma}_{i}}$. Green parameters are the inputs of STATISTICA software. Red parameters are the parameters estimated by STATISTICA software.)

**Figure 2.**Comparison of Model I (

**- - -**), Model III (

**–––**) and experimental data (●), where the colour code means the following: blue: 50 °C, yellow: 60 °C and red: 70 °C. Objective functions (OFs) are represented in the text bubbles per model per temperature. (

**a**) MeOH-water hydrophilic (HPV) water flux; (

**b**) MeOH-water HPV MeOH flux.

Mixture | Type | Examined Temperatures [°C] | Water Content of Feed [wt%] | Membrane | Ref. |
---|---|---|---|---|---|

OPV | |||||

EtOH-water | azeotropic | 40, 50, 60, 70, 80 | 91.57–99.63 | Sulzer PERVAP 4060 | [14,15] |

iBuOH-water | azeotropic | 50, 60, 70 | 98.16–99.89 | Sulzer PERVAP 4060 | [10,16] |

EtAc-water | azeotropic | 50, 60, 70 | 98.86–99.82 | Sulzer PERVAP 4060 | [17] |

30, 40, 45, 50 | 98.93–99.80 | ZSM-5 filled PEBA | [18] | ||

HPV | |||||

MeOH-water | zeotropic | 50, 60, 70 | 1.78–3.075 | Sulzer PERVAP 1510 | [16,19] |

iBuOH-water | azeotropic | 70, 80, 90 | 4.57–36.39 | Sulzer PERVAP 1510 | [10,16] |

Component | Model I | Model II | Model III |
---|---|---|---|

OPV | |||

water | 6.0 × 10^{−4} | 0.003 | 5.7 × 10^{−4} * |

EtOH | 0.783 * | 0.800 | 0.987 |

water | 0.028 | 0.508 | 0.027 * |

iBuOH | 2.139 * | 2.142 | 2.140 |

water | 0.658 | 0.719 | 0.095 * |

EtAc | 0.084 * | 0.087 | 0.086 |

water | 1.942 | 5.327 | 1.688 * |

EtAc | n/a ^{1} | n/a ^{1} | n/a ^{1} |

HPV | |||

water | 2.385 | 6.022 | 0.274 * |

MeOH | 0.074 | 1.714 | 0.070 * |

water | 3.321 * | 6.507 | 6.493 |

iBuOH | 4.873 * | 8.077 | 4.359 ^{2} |

^{1}Source did not define enough data.

^{2}Yields unrealistic physical parameter * More accurate Model.

Components | E_{i} [kJ/mol] | $\overline{{\mathit{D}}_{\mathit{i}}^{\ast}}[\mathbf{mol}/{\mathbf{m}}^{2}\mathbf{h}]$ | B [-] | Model |
---|---|---|---|---|

OPV | ||||

water | 31.28 | 4.94 | −0.49 | III |

EtOH | 33.09 | 77.78 | −0.04 | I |

water | 42.20 | 3.45 | −22.58 | III |

iBuOH | −18.28 | 14,879.52 | −1.83 | I |

water | 30.96 | 6.99 | −52.22 | III |

EtAc | 8.96 | 8373.44 | −4.48 | I |

water | 3.69 | 5468.59 | −0.64 | III |

EtAc | n/a ^{1} | n/a ^{1} | n/a ^{1} | n/a ^{1} |

HPV | ||||

water | 23.50 | 167.30 | −6.52 | III |

MeOH | 30.77 | 0.01 | −1.49 | III |

water | 58.25 | 0.535 | 8.12 | I |

iBuOH | 52.25 | 2.63 | −8.06 | I |

^{1}Source did not define enough data.

Mixture | Temperature [°C] | Model I | Model III | ||
---|---|---|---|---|---|

Water | Organic | Water | Organic | ||

OPV | |||||

Water-EtOH | 40 | 2.57 × 10^{−4} | n/a ^{1} | 0.290 | n/a ^{1} |

60 | 1.54 × 10^{−4} | n/a ^{1} | 0.262 | n/a ^{1} | |

80 | 1.91 × 10^{−4} | n/a ^{1} | 0.231 | n/a ^{1} | |

Water-iBuOH | 50 | 0.011 | 0.011 | 1.973 | 1.995 |

60 | 0.010 | 0.010 | 0.091 | 0.082 | |

70 | 0.006 | 0.006 | 0.075 | 0.063 | |

Water-EtAc | 50 | 0.159 | 0.007 | 0.052 | 0.056 |

60 | 0.366 | 0.077 | 0.021 | 0.022 | |

70 | 0.133 | 0.010 | 0.011 | 0.008 | |

Water-EtAc | 30 | 0.302 | 0.121 | n/a ^{1} | n/a ^{1} |

40 | 0.243 | 0.125 | n/a ^{1} | n/a ^{1} | |

45 | 0.719 | 0.634 | n/a ^{1} | n/a ^{1} | |

50 | 0.678 | 0.808 | n/a ^{1} | n/a ^{1} | |

HPV | |||||

MeOH-water | 50 | 1.003 | 0.206 | 0.032 | 0.027 |

60 | 0.720 | 0.013 | 0.017 | 0.020 | |

70 | 0.662 | 0.056 | 0.028 | 0.025 | |

iBuOH-Water | 70 | 1.147 | 2.489 | 1.937 | n/a ^{2} |

80 | 1.169 | 2.484 | 1.708 | n/a ^{2} | |

90 | 1.004 | 1.490 | 1.203 | n/a ^{2} |

^{1}Source did not define enough data.

^{2}Yields unrealistic physical parameter.

Mixture | Recommended Model for | |
---|---|---|

Aqueous Component | Organic Component | |

OPV | ||

EtOH-water | III (I) | I |

iBuOH-water | III | I (III) |

EtAc-water | III (I) | I (III) |

HPV | ||

MeOH-water | III (I) | I (III) |

iBuOH-water | I (III) | I |

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**MDPI and ACS Style**

Szilagyi, B.; Toth, A.J.
Improvement of Component Flux Estimating Model for Pervaporation Processes. *Membranes* **2020**, *10*, 418.
https://doi.org/10.3390/membranes10120418

**AMA Style**

Szilagyi B, Toth AJ.
Improvement of Component Flux Estimating Model for Pervaporation Processes. *Membranes*. 2020; 10(12):418.
https://doi.org/10.3390/membranes10120418

**Chicago/Turabian Style**

Szilagyi, Botond, and Andras Jozsef Toth.
2020. "Improvement of Component Flux Estimating Model for Pervaporation Processes" *Membranes* 10, no. 12: 418.
https://doi.org/10.3390/membranes10120418