# Analysis of Influencing Factors on the Gas Separation Performance of Carbon Molecular Sieve Membrane Using Machine Learning Technique

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

#### 2.1. Data Collection

_{2}, CH

_{4}, N

_{2}, O

_{2}, and H

_{2}), a quantitative index, named characteristic distance (d), was proposed in this work. Figure 1 shows the concept of d, which was the vertical distance of a black legend away from the Robeson’s upper bound line of a specific gas pair [53,54]. When the legend was above the upper bound line, the d was a positive value; otherwise, the d was negative. The d, as a quantitative index, could measure the gas separation performance of CMS membranes, including permeability and selectivity simultaneously.

^{−10}cm

^{3}(STP)·cm·cm

^{−2}·s

^{−1}·cmHg

^{−1}) is the permeability coefficient of fast gas; α is the selectivity coefficient; and n and lgk are the slope and intercept of the upper bound line, respectively. In order to make a consistent evaluation standard to different gas pairs, the n and lgk, as shown in Tables S2 and S5 in the Robeson upper bound lines [54], were selected for all gas pairs in the sample.

#### 2.2. SVR Theorem

**x**and dependent variable y, the loss is calculated only if the difference between y and the results predicted from the independent variable f(

**x**) are greater than ε [13]. When |f(

**x**) − y| < ε, the value of |f(

**x**) − y| is treated as zero; otherwise, the value is |f(

**x**) − y| − ε. It is equivalent to building a 2ε wide belt centered on f(

**x**) and the prediction is correct when the sample falls into the belt. The f(

**x**) is called hyperplane, whose geometry is related to the dimension of

**x**. However, the hyperplane may not be constructed in practice due to the characteristics of the sample. Under such circumstances, the samples could be mapped from the original sample space to a higher dimensional feature space in order to realize the linear correlation. The γ is called breadth, which affects the range of action and thereby affects the generalization performance.

^{T}is the transpose of w), and b is the bias term. When the sample needs mapping in order to construct a hyperplane, the original independent

**x**would map into the eigenvector Φ(

**x**) and the hyperplane function becomes [13,28]:

^{2}/2 subject to y

_{i}·(w

^{T}x

_{i}+ b) ≥ 1. In practice, however, the hyperplane is difficult to appear in a linear form. There are two ways to make the sample linear regression: adding a penalty coefficient to the objective function and mapping the samples to the higher dimensional space. Then, the objective function of SVR hyperplane becomes:

_{i}and $\widehat{{\xi}_{i}}$ are called slack variables, which indicate that the relaxation degree on both sides of the space might be different.

_{i}and ${\hat{\alpha}}_{i}$ are the Lagrange multipliers. The w would, therefore, be solved and used in the comparison of each influencing factor.

**x**

_{i},

**x**

_{j}), which is a symmetric and positive definite in the sample space, is constructed to replace the inner product of the mapping eigenvectors:

**x**

_{i}), which is hidden in the kernel function, is unnecessary to calculate explicitly [28,55]. Selecting the kernel function would be helpful to correctly construct the hyperplane in feature space. In this work, the SVR models with linear function, polynomial function, radial basis function (RBF), and Sigmoid function in S.6 (Equations (S13)–(S16)) were built as the kernel functions, and the best model was chosen to analyze the influences of the multiple factors on the gas separation performance of CMS membranes.

## 3. Analyzing Process of Influencing Factors

## 4. Results and Discussion

#### 4.1. Correlation Analysis on the Independent Variables

_{i}, x

_{j}) is the covariance of the independent variables x

_{i}and x

_{j}, σ

^{2}(x

_{i}) and σ

^{2}(x

_{j}) and their variations) were calculated, and one of the variables with large correlation coefficient was removed. The value of the correlation coefficient, which assessed whether to remove the variable, was between 0.8 and 0.9 in the references [28,29,56,57,58]. In this work, the value was selected as 0.8, which is also the criterion for judging whether two variables are highly correlated [59].

#### 4.2. Model Reliability and Parameter Optimization

^{2}, Equation (9)), root mean square error (RMSE, Equation (10)), and mean absolute error (MAE, Equation (11)), were calculated in order to compare the regression effect of each method.

_{i}and f(x

_{i}) are, respectively, the actual dependent variable and the calculated result based on the independent variable of the ith sample; $\stackrel{\u2014}{y}$ is the mean value of the actual dependent variable; and m is the number of samples. The R

^{2}, RMSE, and MAE of the gas permeabilities between the experimental data and the predicted values by SVR and MLR methods were calculated, and the accuracies of these methods were determined by the calculated results.

^{2}, smaller RMSE, and smaller MAE than the ones of the MLR method. This indicates that the SVR method with global optimality may correlate the influencing factors and the permeability of the CMS membrane more accurately. In addition, the models with the RBF kernel and the polynomial kernel showed similar regression effects on the influencing factors and the gas permeability of the CMS membrane, but the models with the linear kernel and the sigmoid kernel could not correlate the influencing factors and the performances well. The model with the RBF kernel, which is less complex and could realize non-linear mapping, is slightly better than the one with the polynomial kernel and is more suitable for the regression in this work.

^{2}while avoiding both over-fit and under-fit, C = 5 and γ = 0.3 were finally selected as the parameters of the SVR model with the RBF kernel and the quadric polynomial kernel, respectively.

#### 4.3. Analysis of the Influencing Factors

#### 4.3.1. Factors Influencing Gas Permeability

- 1.
- Regression results

^{2}and smaller MAE in Table 4 indicated that selecting the appropriate kernel function and optimizing the suitable parameters was important to improve the regression effect. The larger RMSEs in Table 4 was probably because a small number of data points were not predicted accurately in order to ensure the overall regression effect. The R

^{2}of both models with optimized parameters were larger than 0.8, which revealed the high correlation between the results calculated by the SVR models after parameter adjustment and the experimental data. In addition, the regression effect of the SVR model with RBF kernel was better for the regression, due to the larger R

^{2}and smaller RMSE and MAE.

- 2.
- Influencing factors analysis

_{i}|/Σ|w

_{i}|, where w

_{i}is the original weight of the independent variable i). The factors with underline, which had a negative weight value, were negatively related to the permeability according to Equations (3) and (4), and were shown in absolute value in the radar map in order to realize an intuitive comparison among the weight of each independent variable. Moreover, the weights of the independent variables corresponding to the SVR models with the RBF kernel and the quartic polynomial kernel were different.

#### 4.3.2. Factors Influencing Gas Separation Performance

- 1.
- Regression results

^{2}, RMSE, and MAE calculated from the predicted results and the experimental data were 0.932, 0.260, and 0.165, respectively, which indicated the reliability of the model.

- 2.
- Influencing factors analysis

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Permselectivity values (characteristic distance d) of the CO

_{2}/CH

_{4}gas pair based on Robeson’s upper bound [54].

**Figure 6.**Comparison between predicted results and experimental data of the gas permeability by SVR method.

**Figure 7.**The normalized weights of the influencing factors on gas permeability regressed by SVR model with (

**a**) RBF kernel; (

**b**) quartic polynomial kernel (factors with underline are negative).

**Figure 9.**The normalized weight of the influencing factors on characteristic distance (factors with underline are negative).

**Table 1.**The selected factors for analyzing the structural–performance relationship of CMS membrane.

Category | Contents |
---|---|

Precursor structure | Fractional free volume (FFV); carbon residue; fraction of sp2-hybrid carbon; fraction of carbon in aromatic rings |

Carbonation condition | Pyrolysis temperature |

Carbon microcrystal structure | Average interlayer spacing; length of carbon microcrystal; thickness of carbon microcrystal |

Properties of permeated gas molecules | Mass; kinetic diameter; van der Waals potential between gas and carbon |

Kernel Function | R^{2} | RMSE | MAE |
---|---|---|---|

RBF | 0.794 | 0.281 | 0.139 |

Polynomial | 0.730 | 0.321 | 0.181 |

Linear | 0.303 | 0.516 | 0.209 |

Sigmoid | –8.562 | 1.913 | 1.375 |

Regression Method | R^{2} | RMSE | MAE |
---|---|---|---|

Linear | 0.201 | 0.553 | 0.387 |

Ringe | 0.204 | 0.552 | 0.386 |

Lasso | –0.060 | 0.637 | 0.409 |

Kernel Function | R^{2} | RMSE | MAE |
---|---|---|---|

RBF | 0.841 | 0.413 | 0.129 |

Quartic polynomial | 0.809 | 0.419 | 0.156 |

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**MDPI and ACS Style**

Pan, Y.; He, L.; Ren, Y.; Wang, W.; Wang, T.
Analysis of Influencing Factors on the Gas Separation Performance of Carbon Molecular Sieve Membrane Using Machine Learning Technique. *Membranes* **2022**, *12*, 100.
https://doi.org/10.3390/membranes12010100

**AMA Style**

Pan Y, He L, Ren Y, Wang W, Wang T.
Analysis of Influencing Factors on the Gas Separation Performance of Carbon Molecular Sieve Membrane Using Machine Learning Technique. *Membranes*. 2022; 12(1):100.
https://doi.org/10.3390/membranes12010100

**Chicago/Turabian Style**

Pan, Yanqiu, Liu He, Yisu Ren, Wei Wang, and Tonghua Wang.
2022. "Analysis of Influencing Factors on the Gas Separation Performance of Carbon Molecular Sieve Membrane Using Machine Learning Technique" *Membranes* 12, no. 1: 100.
https://doi.org/10.3390/membranes12010100