Computational Fluid Dynamics Analysis of a Hollow Fiber Membrane Module for Binary Gas Mixture

The membrane gas separation process has gained significant attention using the computational fluid dynamics (CFD) technique. This study considered the CFD method to find gas concentration profiles in a hollow fiber membrane (HFM) module to separate the binary gas mixture. The membrane was considered with a fiber thickness where each component’s mass fluxes could be obtained based on the local partial pressures, solubility, diffusion, and the membrane’s selectivity. COMSOL Multiphysics was used to solve the numerical solution at corresponding operating conditions and results were compared to experimental data. The two different mixtures, CO2/CH4 and N2/O2, were investigated to obtain concentration gradient and mass flux profiles of CO2 and O2 species in an axial direction. This study allows assessing the feed pressure’s impact on the HFM system’s overall performance. These results demonstrate that the increment in feed pressures decreased the membrane system’s separation performance. The impact of hollow fiber length indicates that increasing the active fiber length has a higher effective mass transfer region but dilutes the permeate-side purities of O2 (46% to 28%) and CO2 (93% to 73%). The results show that increasing inlet pressure and a higher concentration gradient resulted in higher flux through the membrane.


Introduction
Membrane gas separation is a predominant unit operation compared to other gas separation techniques, such as distillation, adsorption, and absorption [1,2]. The membrane process is more feasible because of its smooth operation, low cost, reliability, small heat effects, and higher separation performance [3]. These advantages enhance its utilization in various industrial gas applications. Numerical methods can be used to simulate the membrane modules for process optimization [4]. Various gas separation applications deal with a gaseous mixture, such as methane recovery from carbon dioxide, nitrogen recovery from air, and helium recovery from methane [5]. The membrane module configuration is vital for gas separation performance [6]. Gas separation units are primarily applied in offshore systems because they take less space for installation [7]. The utilization of membrane separation processes for gaseous mixtures could be widely increased in 2030 [8,9].
Membrane gas separation can eliminate specific components because it is highly selective for one component to pass through it, while other components are retained at the feed side [1,10]. It is a traditional process because of its high membrane selectivity, low maintenance cost, and simple unit operation [11][12][13]. Due to high market demand, the membranes are used for commercial applications to separate gaseous mixtures and are less economical [14,15]. Membrane gas separation has many applications such as Gases 2023, 3 78 hydrogen recovery in refineries, sour gas treatment, hydrogen recovery from ammonia, air separation for oxygen purification, and carbon dioxide removal from natural for large scale [16]. In heavy oil and petrochemical industries, membrane systems have integrated with other industrial units, but the scenarios for different groups are limitless. The growth of membrane systems for gaseous mixture separation was estimated to increase widely in 2023 [8,17]. Therefore, this technology will reduce the overall process cost for gas separation and environmental impacts [18].
Numerical approaches can solve gas separation through membrane flow difficulties to improve the configuration [19]. In addition, numerical methods have predicted values at various mesh edges for different flow conditions and given more precise results [20]. Mathematical modeling is a time-and cost-saving approach for unnecessary experimental designs [21]. Therefore, CFD modeling is used for investigating the flow problems across the membrane surface [22]. CFD techniques are applied to understand the phenomena inside the membrane and the separation performance of different membrane modules [17]. CFD allows changing the parameters for different gases obtained from the concentration profiles [23]. The CFD tool is mostly applied to visualize the membrane module's fluid hydrodynamics and flow patterns on permeate and retentate streams [9,15]. It is critical because different hydrodynamic conditions affect the transport mechanism [24]. Membrane module design is also essential for improving the efficiency of gas permeation performance through the membrane [25]. The CFD technique has been used for flow behavior in the rectangular channel with different feed spacers [16,26]. Many researchers have studied the flow characteristics of membrane modules for gas separation processes.
The properties of membrane materials for designing industrial module and process design play an essential role in successfully implementing the gas separation system [27]. For industrial applications, the HFM module's design depends on the module's cost and efficiency. Therefore, the HFM module requires a maximum membrane surface area for separation, which is necessary to determine the concentration profiles and pressure drops along with the module length [24,28]. In recent studies, the asymmetric hollow fiber membrane configuration was used for gas separation, demonstrating higher flux [10]. The countercurrent flow pattern investigated the process effect on different design variables [29,30]. A three-dimensional membrane model was developed considering a non-isothermal model for evaluating temperature's effect on membrane permeance [31]. The CFD technique was used for a perovskite HFM to investigate the oxygen permeation through the membrane. It showed the effect of pressure on permeate streams in the range of 0.2 to 0.4 kPa, and different vacuum levels were used to obtain the oxygen concentration profile at the feed side [32].
The HFM module separates binary gas using this research work's finite element method (COMSOL Multiphysics simulation software). Three-dimensional geometry was developed for the separation of the binary gas mixture. The CFD tool was applied to study the flow profiles of gases in the HFM membrane module. Two binary mixture systems, N 2 /O 2 and CO 2 /CH 4 , were investigated. In this study, different membrane module parameters, fiber length (L), inlet pressure (pf), inlet concentration (xf), fiber bundle radius (Ro), and module outer diameter (Do), were optimized. Lastly, gas separation through membrane modules was studied with different module parametric effects.

Materials and Methods
CFD deals with partial differential equations based on mass, momentum, and heat transport. The CFD technique is used for simulating different integrated parts of industrial HFM units. Therefore, the HFM membrane module is used to understand gas separation phenomena. Table 1 shows the parameters used for simulations. A numerical simulation was carried out using the COMSOL Multiphysics ® package. The 3-Dimensional geometries of the HFM module were developed in axisymmetric model. The transport of diluted species with Fick's Law model was considered for transport through the membrane. The following assumptions was required for this model.

Assumptions
The following assumptions were used for the modeling of the HFM membrane module: [38,39] a.
Steady-state condition is applied. b.
Laminar flow is considered with constant physical properties. c.
Ideal gas law is used. d.
Without axial mixing of gaseous molecules. e.
Fiber diameter in the module is uniform. f.
Fiber deformation with high inlet pressure is neglected. g.
Solution-diffusion mechanism is used for membrane gas separation.

Mass Transport Mathematical Modeling
In the mass transport phenomena, chemical species are transferred through the membrane from the shell to the permeate. The mass transport of components has occurred due to the concentration difference from the feed side to permeate side through the membrane using diffusion and the molecular sieving mechanism shown in the Figure 1. In this model, the convective mass transport occurs with a different concentration gradient from the feed side. The mass transport phenomena occur with convection with a velocity of average molecules, and diffusion occurs due to concentration differences in a fluid.

Assumptions
The following assumptions were used for the modeling of the HFM membra ule: [38,39] a. Steady-state condition is applied. b. Laminar flow is considered with constant physical properties. c. Ideal gas law is used. d. Without axial mixing of gaseous molecules. e. Fiber diameter in the module is uniform. f.
Fiber deformation with high inlet pressure is neglected. g. Solution-diffusion mechanism is used for membrane gas separation.

Mass Transport Mathematical Modeling
In the mass transport phenomena, chemical species are transferred through th brane from the shell to the permeate. The mass transport of components has occur to the concentration difference from the feed side to permeate side through the me using diffusion and the molecular sieving mechanism shown in the Figure 1 model, the convective mass transport occurs with a different concentration gradie the feed side. The mass transport phenomena occur with convection with a ve average molecules, and diffusion occurs due to concentration differences in a flui

Convection and Diffusion Models
The mass transport equation contains both the convection and diffusion models. The mass transport equation is as follows: The bulk velocity v (x, y, and z) is used for convective transport, and velocity is measured by combining two equations, mass balance and momentum equation. The mass balance equation represents a diffusion mechanism that deals with the diluted and solvent species on the feed side of the membrane. Operators show three directions of space variables in the x-, y-, and z-direction. Finally, the source term is on the right side of the equation, showing a chemical reaction that occurred in a process. The term "a" is defined as the concentration of gas on the feed side.

Diffusion
The transport of diluted species provides an environment for modeling mass transport in a chemical species. Here, the diluted species interface model is applied, showing all species in a diluted form. When a dilution model is applied, it shows that the properties of the solute are the same as a present solvent. Fick's Law also defines solution diffusion in a mixture. The transport of dilute species provides both convection and diffusion models for simulation in all dimensions.
where diffusion is represented as D AB .

Membrane Model
Diffusion is an essential mechanism occurring in mass transfer problems. It mainly focuses on the diffusion through membranes for certain chemical species. The system is considered to check the permeability of gases, where the feed side of gas in the fiber-bore side and permeate is passed through a membrane collected on the shell side. The unit of the membrane module demonstrates the membrane model.
The model equation for steady-state mass transfer is as follows: The scalar product of concentration gradient C a (mole/m 3 ) and D AB diffusion coefficient of the diffusing species (m 2 /s) shows a steady-state mass transfer equation. All boundaries considered isolating conditions.
The boundary condition is considered as the fiber-bore side applied as an inlet that shows a concentration of component C 0 and the shell side considered as the reject side C 0,1 .
Boundary conditions 1 and 2 are represented as

Flow through Membrane in HFM Module
The membrane flux is investigated through the membrane in terms of diffusion D with solubility S with inlet concentration C 0 and outlet concentration C 0,1 .
Term D δ is determined as a barrier with corresponding thickness. The concentration is defined in terms of partial pressure p a and solubility s a for component a through the membrane. The term a is defined as any gas component.
The flux through the membrane is presented in terms of permeability and partial pressure: The pressure difference between both sides is used as a gradient for inlet pressure p f and permeate pressure ph. The ratio of their differences is called solubility for the binary mixture. This membrane model was applied to HFM membrane modules to find the flow profiles and flux of gases. The symbols of the equation were mentioned in the Appendix A.

Flowchart of Post-Processing Scheme
The numerical simulation schemes chart is represented in Figure 2.

Flow through Membrane in HFM Module
The membrane flux is investigated through the membrane in terms of diffusion D with solubility S with inlet concentration C 0 and outlet concentration C 0,1 .
Term is determined as a barrier with corresponding thickness. The concentration is defined in terms of partial pressure and solubility for component a through the membrane. The term a is defined as any gas component.
The flux through the membrane is presented in terms of permeability and partial pressure: The pressure difference between both sides is used as a gradient for inlet pressure and permeate pressure ℎ. The ratio of their differences is called solubility for the binary mixture. This membrane model was applied to HFM membrane modules to find the flow profiles and flux of gases. The symbols of the equation were mentioned in the Appendix.

Flowchart of Post-Processing Scheme
The numerical simulation schemes chart is represented in Figure 2.

Geometry
The sketches of geometries for HFM modules are shown in Figure 3.

Geometry
The sketches of geometries for HFM modules are shown in Figure 3.

Meshing
The finite element method was used in the HFM module to solve the mass tra problem and meshing of the membrane module in COMSOL Multiphysics. The complete mesh consisted of 13,191,478 domain elements, 2,281,754 boundary ele and 223,387 edge elements, and degrees of freedom were solved at 1,364,033 (plus 8 internal DOFs). Solution time (Study 1): 230 s. (3 min, 50 s). In addition, one fibe consisting of 3,813,513 domain elements, 318,624 boundary elements, and 8988 ed ments and degrees of freedom was solved for 723,639 (plus 328,506 internal DOF mesh for the membrane was generated using a triangular mesh structure. The resu tained from boundary layer mesh analysis are shown in Table 2. The resulting m shown in Figure 4e. The analysis of the module mesh was observed, and the inlet shows a higher density segment. Furthermore, membrane meshing is more critical CFD simulations because the inner membrane wall formed the concentration bou layer. Therefore, membrane refinement along with membrane thickness are essent For the boundary layer in the membrane, selecting wall thickness of the fib porous surface is necessary. The representative mesh for the membrane was gen using a triangular mesh structure. The mesh analysis observes a higher density ments in the feed region and near the interface region between the fluid and the p medium. This refinement is based on the numerical study which considered inne thickness. The formation of the concentration boundary layer occurs with low thic confirming the significant concern with the mesh in this region.

Meshing
The finite element method was used in the HFM module to solve the mass transport problem and meshing of the membrane module in COMSOL Multiphysics. The HFM complete mesh consisted of 13,191 Table 2. The resulting mesh is shown in Figure 4e. The analysis of the module mesh was observed, and the inlet region shows a higher density segment. Furthermore, membrane meshing is more critical in the CFD simulations because the inner membrane wall formed the concentration boundary layer. Therefore, membrane refinement along with membrane thickness are essential. For the boundary layer in the membrane, selecting wall thickness of the fiber as a porous surface is necessary. The representative mesh for the membrane was generated using a triangular mesh structure. The mesh analysis observes a higher density of elements in the feed region and near the interface region between the fluid and the porous medium. This refinement is based on the numerical study which considered inner wall thickness. The formation of the concentration boundary layer occurs with low thickness, confirming the significant concern with the mesh in this region.

Results
This work was carried out using COMSOL Multiphysics software 5.2 version for the HFM module design using CFD. Furthermore, the effects of module length on the diffusive flux, total flux, and concentration variations were determined using CFD.

Results
This work was carried out using COMSOL Multiphysics software 5.2 version for the HFM module design using CFD. Furthermore, the effects of module length on the diffusive flux, total flux, and concentration variations were determined using CFD.

Air Separation
The HFM module simulations were performed for oxygen separation from air. The flow variations were obtained from O 2 gas on the shell side of the HFM module. The inlet gas enters the membrane module from the lumen side, and thus, a crossflow model is considered as the boundary condition. The boundary condition was applied for a high inlet concentration on the lumen side, C 0 , and a low inlet concentration on the shell side, C 0 . Figure 5a shows that the feed gas, O 2 , permeates through membrane fibers, and a maximum purity of the N 2 was obtained on the shell side. The concentration between the fibers was changed due to high flux on the inlet side and increased gas purity in the permeate stream. Based on these results, it was concluded that permeation of O 2 occurred through the fiber. Figure 5b indicates that the concentration variation between two fibers is more important. The two-dimensional results verified that O 2 gas diffused through the membrane due to the concentration gradient, and a variation in gas was obtained in the membrane module's center. In addition, the difference between the fibers was negligible due to the compactness of the fibers, but the permeate shell showed maximum gas diffusion from the high inlet side to the outlet side.

Carbon Dioxide Separation
The HFM module geometry consists of a bundle of fibers in a single shell compacted together to form a large surface area, but in this case, the single fiber was used for gas separation. Figure 6a shows that the concentration of the inlet gas CO2 changed on the feed side and permeate side. The gas entered from the lumen, and the crossflow model was considered for flow conditions. The fiber-bore side was considered as the inlet side boundary condition C0 at one end, and another was considered the outlet boundary condition with the low concentration C0,1. The thickness of the fiber shows as the membrane, which is a thin diffusion barrier. It was concluded that the contour expressing the CO2 gas The concentration gradient displayed in Figure 5c shows that the gas diffusion varied at different mesh points from the inlet to the permeate side. The red mesh points on the shell side show the maximum purity of O 2 gas on the permeate side. The different mesh points in the center show the different gas concentrations of O 2 purity. It was confirmed that the concentration gradient was present in the HFM module due to the partial pressure of gases, and gas transported through the membrane with specific permeability. Figure 5d shows the diffusive flux magnitude variations in the HFM module for binary gas separation. Contours display a high flux variation on the shell side due to gas diffused from the inlet side to the permeate side. The results showed no pressure drop on the retentate side, and fluxes remained unchanged in the membrane center due to the diffusion of O 2 from the fiber side to the shell side.

Carbon Dioxide Separation
The HFM module geometry consists of a bundle of fibers in a single shell compacted together to form a large surface area, but in this case, the single fiber was used for gas separation. Figure 6a shows that the concentration of the inlet gas CO 2 changed on the feed side and permeate side. The gas entered from the lumen, and the crossflow model was considered for flow conditions. The fiber-bore side was considered as the inlet side boundary condition C 0 at one end, and another was considered the outlet boundary condition with the low concentration C 0,1 . The thickness of the fiber shows as the membrane, which is a thin diffusion barrier. It was concluded that the contour expressing the CO 2 gas concentration variation always occurred from the feed side to permeate side.
Gases 2023, 3, FOR PEER REVIEW 10 shell side due to gas diffused from the inlet side to the permeate side. The results showed the zero pressure drop on the lumen side as the feed on the membrane, and fluxes remained unaffected.

Discussion
The HFM module performance was investigated using different parameters, such as fiber length (L), inlet pressure (pf), inlet concentration (xf), fiber bundle radius (Ro), and outer module diameter (Do). Figure 7a shows the simulation of O2/N2 separation through the membrane. The results obtained from simulations show that the increment in the length of the fiber from 5 to 25 cm reduced the purity of O2 from 15 to 10%, while the concentration of N2 in the retentate stream was enhanced by 10%. The fiber length produced a large membrane area used for separation. The higher stage cut also caused more gas to permeate through the  Figure 6b shows the concentration of CO 2 in the two-dimensional model. From Figure 6b, the inlet gas variations from the inlet channel to the permeate collector can be observed. The maximum gas concentration at the fiber surface was higher because of the higher gas concentration transported through the membrane from the retentate. The results showed that gas diffused through the membrane at a higher permeation rate.
A concentration gradient was observed in the HFM module for CO 2 . The different mesh points show that the inlet gas passes through the fiber side to the permeate side of the module, and the gradient below the fiber is higher due to the maximum gas diffusion through the membrane. The results showed that the gradient was present, and gas was diffused through the membrane. Figure 6d demonstrates the diffusive flux from the gradient applied from the lumen to the shell side. The flux was calculated with the given parameters of the membrane module. The flux magnitude variations describe binary gas separation in the HFM membrane module. Contours display a high flux variation on the shell side due to gas diffused from the inlet side to the permeate side. The results showed the zero pressure drop on the lumen side as the feed on the membrane, and fluxes remained unaffected.

Discussion
The HFM module performance was investigated using different parameters, such as fiber length (L), inlet pressure (pf), inlet concentration (xf), fiber bundle radius (Ro), and outer module diameter (Do). Figure 7a shows the simulation of O 2 /N 2 separation through the membrane. The results obtained from simulations show that the increment in the length of the fiber from 5 to 25 cm reduced the purity of O 2 from 15 to 10%, while the concentration of N 2 in the retentate stream was enhanced by 10%. The fiber length produced a large membrane area used for separation. The higher stage cut also caused more gas to permeate through the membrane because the extension in length gradually diluted the permeate stream with N 2 ; thus, O 2 possessed a lower quantity while the permeate possessed a higher quantity of N 2 . Therefore, the purity of the permeate side was decreased. The oxygen depletion from the feed side to the permeate stream was less enriched, while a higher stage cut was achieved.
The concentration effect was investigated by changing the values of O 2 concentration in the feed while keeping other parameters constant. The feed concentration analysis shown in Figure 7b demonstrated that the concentrations of O 2 and N 2 were varied in the permeate and reject side, respectively. The results showed that the increment in the concentration of O 2 on the feed side provides a higher permeate rate in the stream. A further increment in O 2 concentration in the feed stream produced a higher concentration gradient, while the N 2 gradient was reduced. For the long fiber, lower O 2 and higher N 2 purities were obtained at all specified feed concentrations. O 2 purity in the permeate and N 2 in the retentate stream as a function of the mole fraction of O 2 in the feed are presented in Figure 7b.
Furthermore, the effect of pressure on the HFM was also investigated by changing the feed pressure from 5 to 25 bar. Interestingly, the increment in feed pressure provides different behavior for O 2 at the permeate stream and N 2 at the retentate stream. Figure 7c describes the effect of feed pressure on the concentration of O 2 and N 2 in the permeate and retentate streams. It was found that the increment in feed pressure provides a higher permeate concentration of O 2 , while a further increment in feed pressure reduced the O 2 in the permeate. Increases in the feed pressure also produced an excessive mass transfer gradient for N 2 , which transported through membrane and thus reduced the concentration of O 2 in the permeate stream. Furthermore, it was also observed that the maximum purity was achieved for a long fiber length with low feed pressure, mainly because of the larger membrane surface available in permeation. retentate streams. It was found that the increment in feed pressure provides a higher permeate concentration of O2, while a further increment in feed pressure reduced the O2 in the permeate. Increases in the feed pressure also produced an excessive mass transfer gradient for N2, which transported through membrane and thus reduced the concentration of O2 in the permeate stream. Furthermore, it was also observed that the maximum purity was achieved for a long fiber length with low feed pressure, mainly because of the larger membrane surface available in permeation.  [36]. 2016, Seyed Saeid) [36]; (c) feed mole fraction of the O2/N2 in permeate and reject sides in HFM module (Reprinted with permission from Ref. [36]. 2016, Seyed Saeid) [36].
The increase in fiber length gives more losses of CH4 because the increment in fiber length provides a higher surface area of the membrane. Therefore, higher permeation  [36]. 2016, Seyed Saeid) [36]; (c) feed mole fraction of the O 2 /N 2 in permeate and reject sides in HFM module (Reprinted with permission from Ref. [36]. 2016, Seyed Saeid) [36].
The increase in fiber length gives more losses of CH 4 because the increment in fiber length provides a higher surface area of the membrane. Therefore, higher permeation resulted in significant methane loss, as shown in Figure 8a. The CO 2 in the feed is a more permeable component with a more significant effect of a mole fraction of CO 2 on the permeate side. On the other hand, the negative quadratic impact of the CO 2 mole fraction significantly affects the feed side. Higher fiber length directly affects the purity of CO 2 and higher methane loss on the retentate side because of its lower concentration. The maximum purity of the CO 2 of 80% was obtained when the fiber length was short. From Figure 8b, results show that increasing the fiber length is not always favorable in that increased length caused a less driving force for overcoming resistance. For instance, the mole fraction of the more permeable component in the permeate stream decreased from 2% to 0.6% by increasing the length of fibers from 5 to 25 cm. The same trends were obtained to increase the fiber bundle radius, which showed more methane loss on the reject side. This decrease may be explained by the fact that maximum CO 2 purity was obtained when the fiber length was small or the number of fibers was at the minimum.
8b, results show that increasing the fiber length is not always favorable in that increased length caused a less driving force for overcoming resistance. For instance, the mole fraction of the more permeable component in the permeate stream decreased from 2% to 0.6% by increasing the length of fibers from 5 to 25 cm. The same trends were obtained to increase the fiber bundle radius, which showed more methane loss on the reject side. This decrease may be explained by the fact that maximum CO2 purity was obtained when the fiber length was small or the number of fibers was at the minimum.

Conclusions
The gas separation performance of an HFM unit was studied using the COMSOL Multiphysics Software. A CFD analysis for N2/O2 and CO2/CH4 separation was performed considering different parameters, such as fiber length (L), feed pressure (pf), feed concentration (xf), fiber bundle radius (Ro), and outer module diameter (Do). Three-dimensional geometry was developed by using COMSOL geometry, and meshing was performed using triangular mesh. It was observed that an increment in fiber length reduced the oxygen purity because of the diluted permeate stream. The increment in feed concentration resulted in increasing the purity of the permeate. The HFM unit observed that increased feed pressure results in an increased pressure gradient, higher flux, more permeability, and higher permeate purity. Similarly, an increment in feed concentration increased the permeate mole fraction, facilitating the permeate stream and producing higher purities of O2 (20% to 75%) and CO2 (65% to 85%).    [37]. 2015, Faizan Ahmed) [37]; (b) fiber length effect on methane loss in HFM module (Reprinted with permission from Ref. [37]. 2015, Faizan Ahmed) [37].

Conclusions
The gas separation performance of an HFM unit was studied using the COMSOL Multiphysics Software. A CFD analysis for N 2 /O 2 and CO 2 /CH 4 separation was performed considering different parameters, such as fiber length (L), feed pressure (p f ), feed concentration (x f ), fiber bundle radius (R o ), and outer module diameter (D o ). Three-dimensional geometry was developed by using COMSOL geometry, and meshing was performed using triangular mesh. It was observed that an increment in fiber length reduced the oxygen purity because of the diluted permeate stream. The increment in feed concentration resulted in increasing the purity of the permeate. The HFM unit observed that increased feed pressure results in an increased pressure gradient, higher flux, more permeability, and higher permeate purity. Similarly, an increment in feed concentration increased the permeate mole fraction, facilitating the permeate stream and producing higher purities of O 2 (20% to 75%) and CO 2 (65% to 85%).
Greek Symbols x x Co-ordinate y y Co-ordinate z z Co-ordinate Vx Velocity in x-direction Vy Velocity in y-direction Vz Velocity in z-direction "∇" Nabla operator Latin symbols t time (s) Ra.
Rate of reaction (mol/m 3