# A Correlation of Overall Mass Transfer Coefficient of Water Transport in a Hollow-Fiber Membrane Module via an Artificial Neural Network Approach

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## Abstract

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## 1. Introduction

_{2}absorption into aqueous solutions. This model showed an outstanding performance over the predictive proposed correlations in the literature. ElShazly [16] estimated the mass transfer coefficient from the bottom of the agitated vessel and compared it with the empirical correlation expressed by Sherwood, Schmidt, and Reynolds numbers from a dimensional analysis. The author designed a network with one hidden layer and with varying neurons from 1 to 12 to optimize the structure. The ANN reached the best performance when applying three to seven neurons in the hidden layer. The model results were better than the prediction using mass transfer correlation because they showed smaller relative errors.

^{4}cases of tests that should be carried out for developing an empirical correlation. About 3 to 3.5 h were necessary to conduct a case of experiments, so the entire experiment completion would need about 2000 h. However, the deviation in the previous study [8,17] was still high in fitting the water vapor diffusion with R

^{2}from 0.85 to 0.9. Thus, the ANN model could be used to predict the mass transfer performance of the module after collecting the data from about a hundred experimental cases. This model provided a correlation based on the weights and biases of the optimized neural network with a very high coefficient of correlation (about 0.99) and low mean squared error.

## 2. Measurement and Calculation of the Overall Mass Transfer Coefficient

#### 2.1. Effectiveness Analysis for Mass Transfer Coefficient Determination

#### 2.2. Experiment Description

## 3. An ANN for Prediction of the Overall Mass Transfer Coefficient

#### 3.1. Structure of Artificial Neural Network

#### 3.2. Data Collection and Pre-Processing

#### 3.3. A Neural Network for Training Experimental Data

#### 3.4. Validation of the Trained Model with Experimental Analysis

## 4. Mass Transfer Performance Prediction of the Membrane Module

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature and Abbreviation

ANN | artificial neural network |

A | area, m^{2} |

b | bias |

C | mass transfer capacity ratio |

k | mass transfer coefficient |

ṁ | mass flow rate, kg/s |

MSE | mean squared error |

NTU | number of transfer units |

n | number of samples |

o | the output |

ō | the mean of the output |

p | pressure, kPa |

R | correlation coefficient |

T | temperature, °C |

t | the target value |

w | weight |

X | independent variable |

Y | dependent variable |

Greeks | |

p | density, kg/m^{3} |

ε | moisture exchange effectiveness |

φ | relative humidity |

ω | absolute humidity |

σ | activation function |

z | weighted sum of the input |

Subscripts | |

a | air |

d | dry |

i | inlet |

j | index, jth data point |

m | membrane |

min | minimum |

max | maximum |

o | overall |

s | saturation |

t | total |

tr | transfer |

w | wet |

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Parameters | From | To | Unit |
---|---|---|---|

Temperature | 40 | 80 | °C |

Absolute pressure | 100 | 200 | kPa |

Flow rate | 5 | 15 | slpm |

Relative humidity | 0.6 | 1 | |

Overall mass transfer coefficient | 0.00167 | 0.0122 | m/s |

Model | No. of Layers | No. of Neurons | R | MSE | Epoch |
---|---|---|---|---|---|

Network 1 * | 1 | 5 | 0.99072 | 9.58 × 10^{−4} | 10 |

Network 2 | 2 | 5 | 0.87269 | 8.32 × 10^{−3} | 2 |

Network 3 | 2 | 10 | 0.97295 | 4.19 × 10^{−3} | 7 |

Network 4 | 3 | 5 | 0.99632 | 7.42 × 10^{−4} | 20 |

Parameter | Parameters |
---|---|

Training function | trainlm |

Transferring function | tangent sigmoid function |

No. of layers | 1 |

No. of neurons | 5 |

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**MDPI and ACS Style**

Nguyen, X.L.; Trinh, N.V.; Kim, Y.; Yu, S.
A Correlation of Overall Mass Transfer Coefficient of Water Transport in a Hollow-Fiber Membrane Module via an Artificial Neural Network Approach. *Membranes* **2023**, *13*, 8.
https://doi.org/10.3390/membranes13010008

**AMA Style**

Nguyen XL, Trinh NV, Kim Y, Yu S.
A Correlation of Overall Mass Transfer Coefficient of Water Transport in a Hollow-Fiber Membrane Module via an Artificial Neural Network Approach. *Membranes*. 2023; 13(1):8.
https://doi.org/10.3390/membranes13010008

**Chicago/Turabian Style**

Nguyen, Xuan Linh, Ngoc Van Trinh, Younghyeon Kim, and Sangseok Yu.
2023. "A Correlation of Overall Mass Transfer Coefficient of Water Transport in a Hollow-Fiber Membrane Module via an Artificial Neural Network Approach" *Membranes* 13, no. 1: 8.
https://doi.org/10.3390/membranes13010008