#
Modeling and Optimal Operating Conditions of Hollow Fiber Membrane for CO_{2}/CH_{4} Separation

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{2}concentration from 2 to 10 mol%, the feed pressure from 2.5 to 7.5 bar, and the feed temperature from 20 to 40 °C, was investigated. Depending on the solution diffusion mechanism, coupled with the Dual sorption model, a comprehensive model was implemented to predict the CO

_{2}flux through the membrane, based on resistance in the series model. Subsequently, a 2D axisymmetric model of a multilayer HFM was proposed to simulate the axial and radial diffusion of carbon dioxide in a membrane. In the three domains of fiber, the CFD technique was used to solve the equations for the transfer of momentum and mass transfer by using the COMSOL 5.6. Modeling results were validated with 27 experiments, and there was a good agreement between the simulation results and the experimental data. The experimental results show the effect of operational factors, such as the fact that temperature was directly on both gas diffusivity and mass transfer coefficient. Meanwhile, the effect of pressure was exactly the opposite, and the concentration of CO

_{2}had almost no effect on both the diffusivity and the mass transfer coefficient. In addition, the CO

_{2}recovery changed from 9% at a pressure equal to 2.5 bar, temperature equal to 20 °C, and a concentration of CO

_{2}equal to 2 mol%, to 30.3% at a pressure equal to 7.5 bar, temperature equal to 30 °C, and concentration of CO

_{2}equal 10 mol%; these conditions are the optimal operating point. The results also manifested that the operational factors that directly affect the flux are pressure and CO

_{2}concentration, while there was no clear effect of temperature. This modeling offers valuable data about the feasibility studies and economic evaluation of a gas separation unit operation as a helpful unit in the industry.

## 1. Introduction

_{2}from natural gas; however, in large-scale applications, all the membranes depend on a high-density polymer membrane. The techniques of gas isolation in this type of membrane rely on a solution-diffusion mechanism [3,4].

- -
- The material (permeability and separation factors).
- -
- The membrane structure and thickness (permeance).
- -
- The membrane configuration (e.g., flat and hollow fiber).

#### 1.1. Material and Construction of Gas Membrane

#### 1.2. Membrane Configuration

- -
- High firmness density.
- -
- Reasonable distribution of fluid.
- -
- Good stability of mechanical, thermal, and chemical properties.
- -
- Low-pressure difference.
- -
- Low-cost fabrication.
- -
- Simplicity in maintenance and running.
- -
- The potency of membrane change.
- -
- The potential of changing the system size.
- -
- The potency of decontamination.

^{3}can accommodate an active surface area of 575 m

^{2}, while the same volume of a spiral wound design can only accommodate 30 m

^{2}[15].

## 2. Theory and Mathematical Model

_{2}captured from a gas mixture using asymmetric HFM was developed by resistance in a series approach for the three domains of the membrane. Figure 1 displays the concentration profiles of CO

_{2}on both sides of the membrane, including the effect of the three resistances on the mass transfer. Carbon dioxide transfer across the membrane is governed by Fick’s law in the gas–membrane interfaces, and the thermodynamic equilibrium is existing. In the law of Fick’s, the concentration of carbon dioxide on the surface of the membrane is related to the partial pressure of this gas and is governed by the dual-mode theory.

^{3}), the equation becomes as follows [18]:

^{2}), ${C}_{iF}$ and ${C}_{iP}$ are concentration on the feed and permeate, respectively (kmol/m

^{3}), ${D}_{Mi}$ is the diffusivity of the solute in the membrane (m

^{2}/s), ${S}_{i}$ is the solute solubility in the membrane [m

^{3}(gas)/m

^{3}(membrane)], ${l}_{M}$ is the thickness of the membrane (m), and ${k}_{iF},{k}_{iP}$ are the mass transfer coefficient on the feed and permeate, respectively (m/s).

#### 2.1. Physical Properties of the Gas Mixture

_{i}and x

_{j}are the mole fractions of species i and j, respectively. Where ${M}_{i}$ and ${M}_{j}$ are the molecular weight of species i and j, respectively (kg/mol), and ${\mu}_{i}$ and ${\mu}_{j}$ are the viscosity of pure gases i and j.

#### 2.2. Gas Diffusivity in the Membrane Regions

^{2}s

^{−1}) and B are constants. The FFV is estimated utilizing the Bondi method [23], as follows:

#### 2.3. Mass Transfer Coefficients

## 3. CFD Simulation Model

_{2}captured from the gas mixture using asymmetric HFM is developed by deriving and solving the governing equation for the three domains of the membrane. Several flow sub-models have been developed to account for flow through different module geometries. Figure 3 shows the shell, membrane, and tube domains, as well as the flow pattern and fiber side feed.

- -
- Steady-state and isothermal conditions.
- -
- Fick’s law was used to describe the diffusion mechanism.
- -
- Ideal gas behavior.
- -
- The Newtonian-type fluid.
- -
- Neglecting the support layer (ignoring the resistance).
- -
- Two-dimensional flow patterns.
- -
- The driving force in the model is the pressure difference.
- -
- All fibers have uniform outer and inner diameters.

#### 3.1. Material Balance

#### 3.1.1. Feed Side (Tube Side)

_{1}

#### 3.1.2. Membrane Part

_{1}

_{2}

#### 3.1.3. Permeate Side

_{3}

#### 3.1.4. Feed Side (Tube Side)

#### 3.1.5. Shell Side

#### 3.2. Numerical Procedure

## 4. Experimental Work

#### 4.1. Materials and Experimental Design

#### 4.2. Lab Scale System and Gas Analyzers

## 5. Results and Discussion

#### 5.1. Effect of Pressure and Temperature on Diffusion Coefficients

_{2}in the feed at constant pressure at 5 bar and temperature at 303 K.

#### 5.2. Effect of Temperature and Pressure on Mass Transfer Coefficients

_{2}in the feed at constant pressure at 5 bar and temperature at 303 K. From this figure, the behavior of the coefficient of mass transfer is very similar to the behavior of diffusivity when changing the proportion of CO

_{2}in the feed.

#### 5.3. The Diffusion Coefficient of Gases in the Dense Membrane

^{−8}cm

^{2}/s and 3.08508 × 10

^{−9}cm

^{2}/s for each carbon dioxide and methane, respectively, calculated from Equation (8). On the other hand, the temperature change within the range used in this study on the diffusion of gases was investigated using Equation (10). Figure 13 represents the effect of temperature on the diffusivity of the gases that consist of the feed mixture.

^{−9}for CO

_{2}and 2.93641 × 10

^{−10}for CH

_{4}.

#### 5.4. Model Validation

_{2}flux were compared with experimental results. There is a fine accordance between the CFD model and experimental results, with an utmost um relative mistake of 7.73%. Table 7 shows the fluxes obtained from both the CFD model and the mathematical model based on experimental data.

#### 5.4.1. Velocity Field

_{2}gas permeation. The average velocity for any step of the Z value from the total length of the fiber can be calculated from the following formula:

^{−4}(m/s) at the closed end of the shell, while the average velocity at the exit was 0.084923 (m/s). The increase in this speed is due to the flux of carbon dioxide gas through the membrane from the feed side to the shell side during the separation process.

#### 5.4.2. The Concentration Distribution of Gas in the Membrane

_{2}concentration gradient in the tube, the membrane, and the shell of the hollow fiber membrane.

_{2}transmits to the membrane due to the concentration gradient. Considering Figure 16, at z = 0 where the gas inflows the HFM, the CO

_{2}concentration has a peak value amounting to 11.909 $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$, while on the permeate side of the membrane, the average concentration of CO

_{2}is equal to 11.723 $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$ at z = L. The mechanisms of mass transfer in the tube and the shell are diffusion and convection. Since the flow is in the z-direction, the mass is transmitted by convection. In the radial direction, diffusion performs the main role in the phenomena of mass transfer. Carbon dioxide gas flows by the diffusion mechanism, where it is absorbed on the surface of the membrane and transferred to the other side [32]. Figure 23 illustrates a 2D concentration difference with the overall flow vectors of CO

_{2}. Moreover, a 3D concentration gradient of CO

_{2}is seen in Figure 24, only for the best conception of the transfer of mass.

_{2}was only on the first surface of the membrane and at a concentration of 8.8605 $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$. After 5% of the tube length, there was a noticeable transfer of carbon dioxide gas to the permeate side of the membrane, where the gas concentrations on the first and second surfaces of the membrane were 9.35 and 7.1083 $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$, respectively. Table 8 presents the concentrations of carbon dioxide on the sorption side and desorption side in the membrane.

#### 5.5. Analysis of CO_{2} Flux and Recovery

_{2}in the membrane at a pressure range of 2–10 bar. It is clear in Figure 26 that the flux increases with increasing pressure, due to increasing the equilibrium concentration of carbon dioxide on the surface of the membrane. The increase in the concentration difference on the two surfaces of the membrane represents an increase in the driving force in the law of flux.

_{2}concentration, while at a pressure of 2.5 bar, 293.57 K, and 2% CO

_{2}, flux and recovery would be at non-optimal conditions.

## 6. Conclusions

_{2}through the membrane. The flux was calculated, taking into consideration the effect of pressure, temperature, and concentration on the properties of the gas mixture, such as density, viscosity, diffusivity, solubility, and mass transfer coefficient. Through the results, the two most controlling factors in the mass transfer were pressure and concentration. Then, a 2D axisymmetric model of a multilayer hollow fiber composite membrane for CO

_{2}segregation was suggested. The model considers the axial and radial diffusion in the HFM. CFD mechanisms were adopted to work out the model equations including continuity and momentum equations. The CFD model predicted the two-dimensional velocity, pressure, and concentration profiles in three domains of the fiber. Modeling forecastings were supported by the experimental results, and a reasonable harmony between them was observed. The relative error between the results of the mathematical model and the CFD model in calculating the flux ranged from 0.391 to 7.73%. The results also showed the direct effect of each of the pressures and the concentration of carbon dioxide in the feed on the flux, while the feed temperature had no obvious effect. The developed model can be used to predict the performance of membranes made of different polymers as well as other operational conditions. The limitation of this upgraded model is its use of low carbon dioxide concentrations as well as pressures that do not exceed 15 bar for the feed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**A schematic diagram for the hollow fiber membrane & cross-sectional area of permeate side [32].

**Figure 7.**Influence of the gas temperature on coefficients of diffusion in feed side at P

_{f}= 5 bar, CO

_{2}= 6 mol%, and CH

_{4}= 94 mol%.

**Figure 8.**Influence of the pressure on coefficients of gases in feed side at T

_{f}= 303 K, CO

_{2}= 6 mol%, and CH

_{4}= 94 mol%.

**Figure 9.**Effect of the percentage of CO

_{2}on coefficients of diffusion of gases in feed side at P

_{f}= 5 bar and T

_{f}= 303 K.

**Figure 10.**Influence of the temperature on the coefficient of mass transfer of gases in feed side at P

_{f}= 5 bar, CO

_{2}= 6 mol%, and CH

_{4}= 94 mol%.

**Figure 11.**Influence of the pressure on coefficient of mass transfer of gases in feed side at T

_{f}= 303 K, CO

_{2}= 6 mol%, and CH

_{4}= 94 mol%.

**Figure 12.**Effect of the percentage of CO

_{2}on mass transfer coefficient of gases in fees side at P

_{f}= 5 bar and T

_{f}= 303 K.

**Figure 14.**Field of velocity in the feed-side of the HFMC. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}= 303 K, and P

_{f}= 5 bar.

**Figure 15.**Profile of velocity in the feed-side across the length of membrane; Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 16.**The contour of velocity in the feed-side of the HFM. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 17.**3D Profile of velocity in the feed-side across the membrane length. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 18.**Field of velocity in the shell side of the HFMC. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 19.**Profile of velocity in the shell side across the membrane length. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar. Each color represents a different section from length along the fiber.

**Figure 20.**Velocity Contour in the sell side of the HFM. Flow of gas = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 21.**3D profile of velocity in the shell side across the membrane length. Gas flow rate = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}=303 K, and P

_{f}= 5 bar.

**Figure 22.**The concentration gradient of CO

_{2}in the model domains at feed flow = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}= 30 °C, P

_{f}= 5 bar, and X

_{CO2}= 0.06.

**Figure 23.**Vectors of overall flux in the sections of the model at feed flow = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}= 30 °C, P

_{f}= 5 bar, and X

_{CO2}= 0.06.

**Figure 24.**The 3D concentration gradient of CO

_{2}in all domains at feed flow = 3.333 × 10

^{−5}(m

^{3}/s), T

_{f}= 30 °C, P

_{f}= 5 bar, and X

_{CO2}=0.06.

Gas | A (cm^{2} s^{−1}) | B |
---|---|---|

CO_{2} | $2.08\times {10}^{-5}$ | 1.09 |

CH_{4} | $5.24\times {10}^{-6}$ | 1.19 |

**Table 2.**The pre-exponential factor and apparent activation energy of gas components [28].

Gas | A (cm^{2} s^{−1}) | ${\mathit{E}}_{\mathit{d}}$ (Kcal/Mol) |
---|---|---|

CO_{2} | $0.02$ | 8.3 |

CH_{4} | $0.074$ | 10 |

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

Temperature of a gas mixture | 303 | $\mathrm{K}$ | %CO_{2} in Feed | 6 | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{\%}$ |

Pressure of a gas mixture | 5 | $\mathrm{b}\mathrm{a}\mathrm{r}$ | %CH_{4} in Feed | 94 | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{\%}$ |

Feed Flow Rate | 3.33 × 10^{−5} | ${\mathrm{m}}^{3}/\mathrm{s}$ | The density of feed gas | 3.59 | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |

Inlet Conc. of CO_{2} | 1.20 × 10^{1} | $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$ | The viscosity of the feed gas | 1.15 × 10^{−5} | $\mathrm{g}/\mathrm{c}\mathrm{m}.\mathrm{s}$ |

Inlet Conc. of CH_{4} | 1.83 × 10^{2} | $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^{3}$ | The inner radius of the fibre | 90 | $\mathsf{\mu}\mathrm{m}$ |

Inlet gas velocity in fibre | 0.345 | $\mathrm{m}/\mathrm{s}$ | The outer radius of the fibre | 150 | $\mathsf{\mu}\mathrm{m}$ |

Number of fibres | 3800 | - | The thickness of the dense layer | 20 | $\mathsf{\mu}\mathrm{m}$ |

Scale | 300 | - | Length of fibre | 28 | $\mathrm{c}\mathrm{m}$ |

Diffusion Coef. of CO_{2} in Tube Side | 3.39 × 10^{−5} | ${\mathrm{m}}^{2}/\mathrm{s}$ | Partition Factor of CO_{2} | 0.801 | - |

Diffusion Coef. of CH_{4} in Tube Side | 3.36 × 10^{−6} | ${\mathrm{m}}^{2}/\mathrm{s}$ | Partition Factor of CH_{4} | 0.441 | - |

Diffusion of CO_{2} in Membrane | 2.29 × 10^{−8} | ${\mathrm{m}}^{2}/\mathrm{s}$ | Density of permeate gas | 0.87 | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |

Diffusion of CH_{4} in Membrane | 3.1 × 10^{−9} | ${\mathrm{m}}^{2}/\mathrm{s}$ | Viscosity of the permeate gas | 1.24 × 10^{−5} | $\mathrm{g}/\mathrm{c}\mathrm{m}.\mathrm{s}$ |

Diffusion Coefficients of CO_{2} in Permeate Side | 1.72 × 10^{−5} | ${\mathrm{m}}^{2}/\mathrm{s}$ | Diffusion Coefficients of CH_{4} in Permeate Side | 1.72 × 10^{−5} | ${\mathrm{m}}^{2}/\mathrm{s}$ |

Specifications of Membrane Module | |||
---|---|---|---|

Product | Model | Material | |

Hollow fibre | MCB-1512A | Polysulfone | |

Dimensions and Weight | |||

Length | Dimension | Weight | |

360 mm | $55\mathrm{m}\mathrm{m}$ | $0.9\mathrm{k}\mathrm{g}$ | |

Fibre Specifications | |||

No. | length | OD | ID |

3800 | $280\mathrm{m}\mathrm{m}$ | $300\mathsf{\mu}\mathrm{m}$ | $160\text{\u2013}180\mathsf{\mu}\mathrm{m}$ |

Operating Conditions | |||

Pressure | Temp. (Min/Max) | Relative Humidity | Residual oil |

Max 10 bar | 5/50 $\mathrm{\mathbb{C}}$ | Less than 60% | ≤0.01 $\mathrm{m}\mathrm{g}/{\mathrm{m}}^{3}$ |

Step No. | Feed Pressure (Bar) | Feed Temp. (°C) | CO_{2} in Feed Mol% |
---|---|---|---|

1 | 2.5 | 20 | 2 |

2 | 5 | 30 | 6 |

3 | 7.5 | 40 | 10 |

Run | Feed Pressure (Bar) | Feed Temp. (°C) | CO_{2} Mol%Feed | CH_{4} Mol%Feed | Permeate Flow Rate (L/Min) | CO_{2} Mol%Permeate | CH_{4} Mol%Permeate |
---|---|---|---|---|---|---|---|

1 | 2.5 | 20 | 2 | 98 | 0.334 | 23.013 | 76.982 |

2 | 2.5 | 20 | 2 | 98 | 0.3336 | 22.807 | 77.191 |

3 | 2.5 | 20 | 2 | 98 | 0.3329 | 22.43 | 77.568 |

4 | 2.5 | 30 | 6 | 94 | 0.2984 | 25.308 | 74.691 |

5 | 2.5 | 30 | 6 | 94 | 0.301 | 24.704 | 75.294 |

6 | 2.5 | 30 | 6 | 94 | 0.299 | 25.5304 | 74.467 |

7 | 2.5 | 40 | 10 | 90 | 0.2798 | 28.675 | 71.324 |

8 | 2.5 | 40 | 10 | 90 | 0.2761 | 27.324 | 72.673 |

9 | 2.5 | 40 | 10 | 90 | 0.2775 | 28.3201 | 71.678 |

10 | 5 | 20 | 2 | 98 | 0.612245 | 31.054 | 68.943 |

11 | 5 | 20 | 2 | 98 | 0.623 | 30.876 | 69.123 |

12 | 5 | 20 | 2 | 98 | 0.6204 | 31.342 | 68.655 |

13 | 5 | 30 | 6 | 94 | 0.5307 | 33.05 | 66.947 |

14 | 5 | 30 | 6 | 94 | 0.549 | 33.245 | 66.753 |

15 | 5 | 30 | 6 | 94 | 0.533 | 32.991 | 67.005 |

16 | 5 | 40 | 10 | 90 | 0.4558 | 35.343 | 64.654 |

17 | 5 | 40 | 10 | 90 | 0.459 | 34.673 | 65.326 |

18 | 5 | 40 | 10 | 90 | 0.4502 | 35.457 | 64.541 |

19 | 7.5 | 20 | 6 | 94 | 0.846 | 37.325 | 62.672 |

20 | 7.5 | 20 | 6 | 94 | 0.842 | 37.01 | 62.987 |

21 | 7.5 | 20 | 6 | 94 | 0.839 | 37.123 | 62.872 |

22 | 7.5 | 30 | 10 | 90 | 0.811 | 39.861 | 60.136 |

23 | 7.5 | 30 | 10 | 90 | 0.806 | 39.918 | 60.079 |

24 | 7.5 | 30 | 10 | 90 | 0.813 | 40.023 | 59.973 |

25 | 7.5 | 40 | 2 | 98 | 0.661 | 34.87 | 65.128 |

26 | 7.5 | 40 | 2 | 98 | 0.669 | 35.674 | 64.323 |

27 | 7.5 | 40 | 2 | 98 | 0.6605 | 34.9108 | 65.088 |

${\mathit{P}}_{\mathit{F}}$ Bar | ${\mathit{T}}_{\mathit{F}}$ K | ${\mathit{X}}_{\mathit{F}}$ $\mathbf{M}\mathbf{o}\mathbf{l}/\mathbf{M}\mathbf{o}\mathbf{l}$ | Flux. Exp $\mathbf{M}\mathbf{o}\mathbf{l}/{\mathbf{m}}^{2}\xb7\mathbf{S}$ | Flux. Com $\mathbf{M}\mathbf{o}\mathbf{l}/{\mathbf{m}}^{2}\xb7\mathbf{S}$ | Relative Error | Recovery %CO _{2} |
---|---|---|---|---|---|---|

2.5 | 293 | 2 | 7.31 × 10^{−9} | 6.93 × 10^{−9} | 5.26 | 9 |

2.5 | 303 | 6 | 2.07 × 10^{−8} | 2.00 × 10^{−8} | 3.31 | 18.65 |

2.5 | 313 | 10 | 1.04 × 10^{−6} | 9.56 × 10^{−7} | 7.73 | 19.1 |

5 | 293 | 2 | 1.42 × 10^{−8} | 1.38 × 10^{−8} | 2.63 | 19.2 |

5 | 303 | 6 | 2.24 × 10^{−6} | 2.01 × 10^{−6} | 5.94 | 21.4 |

5 | 313 | 10 | 8.96 × 10^{−6} | 8.57 × 10^{−6} | 4.36 | 23.4 |

7.5 | 293 | 6 | 9.49 × 10^{−6} | 9.22 × 10^{−6} | 2.83 | 28.3 |

7.5 | 303 | 10 | 2.39 × 10^{−5} | 2.31 × 10^{−5} | 3.33 | 30.3 |

7.5 | 313 | 2 | 2.18 × 10^{−8} | 2.17 × 10^{−8} | 0.391 | 20.6 |

Length of Fiber m | Conc. of CO_{2} at r_{1}$\mathbf{M}\mathbf{o}\mathbf{l}/{\mathbf{m}}^{3}$ |
Conc. of CO_{2} at r_{2}$\mathbf{M}\mathbf{o}\mathbf{l}/{\mathbf{m}}^{3}$ |
---|---|---|

0 | 8.8605 | 0 |

0.014 | 9.3512 | 7.1083 |

0.028 | 9.4065 | 8.4672 |

0.042 | 9.4901 | 9.089 |

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## Share and Cite

**MDPI and ACS Style**

Jasim, D.J.; Mohammed, T.J.; Harharah, H.N.; Harharah, R.H.; Amari, A.; Abid, M.F.
Modeling and Optimal Operating Conditions of Hollow Fiber Membrane for CO_{2}/CH_{4} Separation. *Membranes* **2023**, *13*, 557.
https://doi.org/10.3390/membranes13060557

**AMA Style**

Jasim DJ, Mohammed TJ, Harharah HN, Harharah RH, Amari A, Abid MF.
Modeling and Optimal Operating Conditions of Hollow Fiber Membrane for CO_{2}/CH_{4} Separation. *Membranes*. 2023; 13(6):557.
https://doi.org/10.3390/membranes13060557

**Chicago/Turabian Style**

Jasim, Dheyaa J., Thamer J. Mohammed, Hamed N. Harharah, Ramzi H. Harharah, Abdelfattah Amari, and Mohammed F. Abid.
2023. "Modeling and Optimal Operating Conditions of Hollow Fiber Membrane for CO_{2}/CH_{4} Separation" *Membranes* 13, no. 6: 557.
https://doi.org/10.3390/membranes13060557