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

Abstract

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.

1. 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 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 counter-current 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, N₂/O₂ and CO₂/CH₄, 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.

2. 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. Properties for HFM module.

Parameters	Symbols	Hollow Fiber Membrane Module [33,34,35,36,37]		Units
Inlet pressure	$pf$	790.8	6895	Pa
Permeate pressure	$ph$	101.3	138	Pa
Inlet gas	$C_0$	0.205 O2	0.5 CO2	Mole fraction
Permeance	$P$	30.78 × 10−10	33 × 10−10	mol/m2. s.pa
Module diameter	Do	9.5 × 10−3	20 × 10−3	m
Module length	$L$	0.25	0.1	m
Membrane thickness	$\delta$	0.08	0.0004	mm

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.

2.1. Assumptions

The following assumptions were used for the modeling of the HFM membrane module: [38,39]

• Steady-state condition is applied.
• Laminar flow is considered with constant physical properties.
• Ideal gas law is used.
• Without axial mixing of gaseous molecules.
• Fiber diameter in the module is uniform.
• Fiber deformation with high inlet pressure is neglected.
• Solution-diffusion mechanism is used for membrane gas separation.

2.2. 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.

2.2.1. Convection and Diffusion Models

The mass transport equation contains both the convection and diffusion models. The mass transport equation is as follows:

$$\frac{ \partial C_a}{\partial t}+v_x\frac{\partial C_a}{\partial x}+v_y\frac{\partial C_a}{\partial y}+v_z\frac{\partial C_a}{\partial z}=D_{AB} \left[\frac{{\partial }^2{C}_a}{\partial x}+\frac{{\partial }^2{C}_a}{\partial y}+\frac{{\partial }^2{C}_a}{\partial z}\right]+R_a$$

(1)

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.

2.2.2. 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.

$${ N}_a{=-D}_{\mathrm{AB}} \left[\frac{{\partial }^2{C}_a}{\partial x}+\frac{{\partial }^2{C}_a}{\partial y}+\frac{{\partial }^2{C}_a}{\partial z}\right]$$

(2)

where diffusion is represented as $D_{\mathrm{AB}}.$

2.2.3. 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:

$$D_{\mathrm{AB}} \left[\frac{{\partial }^2{C}_a}{\partial x}+\frac{{\partial }^2{C}_a}{\partial y}+\frac{{\partial }^2{C}_a}{\partial z}\right]=0$$

(3)

The scalar product of concentration gradient $C_a$ (mole/m³) and $D_{\mathrm{AB}}$ diffusion coefficient of the diffusing species (m²/s) shows a steady-state mass transfer equation. All boundaries considered isolating conditions.

$$D_{\mathrm{AB}} \left[\frac{\partial C_a}{\partial x}+\frac{\partial C_a}{\partial y}+\frac{\partial a}{\partial z}\right]=0$$

(4)

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

$$C_1{=C}_0; C_2{=C}_{0,1}$$

(5)

2.2.4. 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 $C_{0,1}$.

$${ N}_a=\frac{D· S}{\delta }\left(C_0-C_{0,1}\right)$$

(6)

Term $\frac{D}{\delta }$ 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.

$$C=p_a\times s_a$$

(7)

The flux through the membrane is presented in terms of permeability and partial pressure:

$${ N}_a=\frac{\frac{P}{S}}{\delta }\left(pf\times s_a-ph\times s_a\right)$$

(8)

The pressure difference between both sides is used as a gradient for inlet pressure $pf$ 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.

2.3. Flowchart of Post-Processing Scheme

The numerical simulation schemes chart is represented in Figure 2.

2.4. Geometry

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

2.5. 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 13,191,478 domain elements, 2,281,754 boundary elements, and 223,387 edge elements, and degrees of freedom were solved at 1,364,033 (plus 864,222 internal DOFs). Solution time (Study 1): 230 s. (3 min, 50 s). In addition, one fiber tube consisting of 3,813,513 domain elements, 318,624 boundary elements, and 8988 edge elements and degrees of freedom was solved for 723,639 (plus 328,506 internal DOFs). The mesh for the membrane was generated using a triangular mesh structure. The results obtained from boundary layer mesh analysis are shown in

Table 2. Boundary layer mesh analysis.

Parameters	Total Elements
Solution time (Study 1)	625 s
Number of edge elements	33,311
Number of boundary elements	779,926
Number of elements	6,376,565
Minimum element quality	7.11 × 10−5

. 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.

3. 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.

3.1. Air Separation

The HFM module simulations were performed for oxygen separation from air. The flow variations were obtained from O₂ 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, C₀, and a low inlet concentration on the shell side, C₀. Figure 5a shows that the feed gas, O₂, permeates through membrane fibers, and a maximum purity of the N₂ 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 O₂ occurred through the fiber. Figure 5b indicates that the concentration variation between two fibers is more important. The two-dimensional results verified that O₂ 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.

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 O₂ gas on the permeate side. The different mesh points in the center show the different gas concentrations of O₂ 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 O₂ from the fiber side to the shell side.

3.2. 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 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 CO₂ gas concentration variation always occurred from the feed side to permeate side.

Figure 6b shows the concentration of CO₂ 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 CO₂. 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.

4. 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 membrane because the extension in length gradually diluted the permeate stream with N₂; thus, O₂ possessed a lower quantity while the permeate possessed a higher quantity of N₂. 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 O₂ concentration in the feed while keeping other parameters constant. The feed concentration analysis shown in Figure 7b demonstrated that the concentrations of O₂ and N₂ were varied in the permeate and reject side, respectively. The results showed that the increment in the concentration of O₂ on the feed side provides a higher permeate rate in the stream. A further increment in O₂ concentration in the feed stream produced a higher concentration gradient, while the N₂ gradient was reduced. For the long fiber, lower O₂ and higher N₂ purities were obtained at all specified feed concentrations. O₂ purity in the permeate and N₂ in the retentate stream as a function of the mole fraction of O₂ 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 O₂ at the permeate stream and N₂ at the retentate stream. Figure 7c describes the effect of feed pressure on the concentration of O₂ and N₂ in the permeate and 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.

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 resulted in significant methane loss, as shown in Figure 8a. The CO₂ in the feed is a more permeable component with a more significant effect of a mole fraction of CO₂ on the permeate side. On the other hand, the negative quadratic impact of the CO₂ mole fraction significantly affects the feed side. Higher fiber length directly affects the purity of CO₂ and higher methane loss on the retentate side because of its lower concentration. The maximum purity of the CO₂ 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 CO₂ purity was obtained when the fiber length was small or the number of fibers was at the minimum.

5. 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 (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 O₂ (20% to 75%) and CO₂ (65% to 85%).