Effect of Packing Nonuniformity at the Fiber Bundle–Case Interface on Performance of Hollow Fiber Membrane Gas Separation Modules

High-fidelity simulations of momentum and mass transfer within a hollow fiber gas separation membrane module are here reported. The simulations capture the potential detrimental effects of poor fiber packing at the bundle–case interface on fluid distribution and performance. Results are presented for both circular and planar fiber bundles. The length over which bundle–case gaps affects flow is determined. The length increases dramatically with increasing fiber packing fraction. As the packing fraction approaches 0.6, the impact extends over the entire bundle diameter for small modules (<1000 fibers). The results clearly demonstrate the detrimental effect of poor packing along the case and can be used to develop module manufacturing guidelines. To reduce computational costs, an equivalent planar bundle module approximation is developed. The approximate simulations agree well with results from full 3-D simulations and can reduce computational costs without sacrificing fidelity.


Introduction
Carbon capture technology is of growing interest due to greenhouse gas emissions from fossil fuel-fired power plants, cement plants, and the petrochemical industry. The growth in CO 2 emissions from the burning of fossil fuels has increased since 1960 [1]. While emissions dropped during the COVID pandemic, growth is expected to continue as the global economy reopens. Emission growth is predicted to lead to undesirable climate changes including temperature increases and extreme weather events [2]. Although a transition from fossil fuels to low-carbon energy is underway, a complete shift of the existing infrastructure to cleaner alternatives is not expected quickly. Thus, separation technologies that capture CO 2 emissions at concentrated point sources or directly from air will play a vital role in bridging the gap until the new energy infrastructure is realized.
Membrane gas separation processes are cost-effective options for gas separations due to reduced energy consumption; the reduction in energy costs is partially offset by an increase in capital costs relative to competing technologies. Membrane processes offer other advantages too, including smaller footprints and rapid response times. Membrane technology for carbon dioxide capture from concentrated carbon dioxide sources, such as power and cement plants, is being developed and tested at the pilot scale [3,4] and is a promising option for capture.
Gas separation membranes are commonly produced in a hollow fiber form. Gas separation modules are formed by creating fiber bundles, embedding bundle ends in A significant body of work demonstrates that in hollow fiber modules, fiber size variation can have a detrimental impact on performance and staging can partially mitigate the effect [6][7][8]. The literature also partially addresses fluid distribution in the lumen [9] and in the shell-side [10][11][12][13][14][15][16][17]. This work focuses primarily on shell-side flows and the impact of random fiber packing on shell-side mass transfer coefficients for liquid separations.
The effects of design features for introducing and removing fluid from the shell have also been examined [18][19][20]. The most common design uses a distribution collar around the fiber bundle, which communicates through a single port to an external supply or product line as illustrated in Figure 1. The collar may possess diverters or a series of openings to help promote uniform flow into or out of the bundle. However, theoretical and experimental work to characterize the effectiveness of these designs is limited.
Theoretical predictions indicate fluid distribution is uniform if fiber packing is uniform up to the external case [19,20]. However, experimental measurements [18] and theoretical simulations [20] suggest fluid bypassing between the periphery of the fiber bundle and the enclosing case can be detrimental. We are not aware of any work that addresses the impact of fiber packing adjacent to the case on module performance.
This work quantifies the effect of nonuniform packing at the fiber bundle-case interface on gas separation module performance. High-fidelity simulations of fluid flow and mass transfer within a hollow fiber membrane module are developed that capture the potential detrimental effects of non-ideal fluid distribution on module performance. The A significant body of work demonstrates that in hollow fiber modules, fiber size variation can have a detrimental impact on performance and staging can partially mitigate the effect [6][7][8]. The literature also partially addresses fluid distribution in the lumen [9] and in the shell-side [10][11][12][13][14][15][16][17]. This work focuses primarily on shell-side flows and the impact of random fiber packing on shell-side mass transfer coefficients for liquid separations.
The effects of design features for introducing and removing fluid from the shell have also been examined [18][19][20]. The most common design uses a distribution collar around the fiber bundle, which communicates through a single port to an external supply or product line as illustrated in Figure 1. The collar may possess diverters or a series of openings to help promote uniform flow into or out of the bundle. However, theoretical and experimental work to characterize the effectiveness of these designs is limited.
Theoretical predictions indicate fluid distribution is uniform if fiber packing is uniform up to the external case [19,20]. However, experimental measurements [18] and theoretical simulations [20] suggest fluid bypassing between the periphery of the fiber bundle and the enclosing case can be detrimental. We are not aware of any work that addresses the impact of fiber packing adjacent to the case on module performance.
This work quantifies the effect of nonuniform packing at the fiber bundle-case interface on gas separation module performance. High-fidelity simulations of fluid flow and mass transfer within a hollow fiber membrane module are developed that capture the potential detrimental effects of non-ideal fluid distribution on module performance. The results can be used to inform carbon capture module manufacturing specifications and guidelines.
Specifically, the effect of packing between the fiber bundle and case is evaluated using full three-dimensional models to simulate module gas separation performance for a shellfed module. The analysis focuses on the axial flow region between the crossflow regions near the distribution collars by assuming: (1) gas permeation in the crossflow regions is small relative to that in the axial flow region and (2) the axial flow is well developed before entering the axial flow region, as illustrated in Figure 1. Additionally, an equivalent planar bundle (EPB) method is developed based on a simplified three dimensional geometry for a planar bundle derived from the full circular bundle. The EPB is validated against full 3-D circular bundle simulations. Use of the EPB can dramatically reduce computational costs compared to full three-dimensional simulations. Module performance is evaluated by solving the governing equations for conservation of momentum and mass for specific module geometries over a range of feed flow rates. The conservation of momentum and continuity equations for steady flows are given by Equations (1) and (2), respectively [21]:

Simulation
where ρ is the density, → u is the velocity vector, p is the absolute pressure, → F is the volume force vector, ↔ I is the unit tensor δ ij and ↔ K is the stress tensor. The conservation of mass equation is given by Equation (3): where → J i is the relative mass flux vector of component i, and ω i is the mass fraction.

Constitutive Laws
The stress tensor in Equation (1) in the case of a fluid with internal friction only due to shear stress is given by: where µ is the dynamic viscosity.
Equations (5)-(8) are used to calculate the mass flux → J i based on the Stefan-Maxwell diffusion model: where n c is the number of components, D ik are the binary diffusivities for species i and k in the multicomponent mixture, → d k is the diffusional driving force, x k is the mole fraction, M k is the molar mass, and M n is the average molar mass. Figure 2 illustrates the domain and boundary conditions used for simulations of fiber bundles. At the inlet, the velocity and gas composition are specified as the laminar, well-developed axial velocity, u z0 (x, y), and a fixed, uniform feed mole fraction, x Fi , respectively:

Domain Setup and Boundary Conditions
where u z is the retentate axial velocity and x i is the mole fraction of component i in the retentate. No-slip, zero normal flux boundary conditions are applied along the case (boundary 1), and symmetry boundary conditions are applied to give lines of symmetry for the planar and circular bundles (boundary 2). For the planar bundle, symmetry reduces the solution domain to a single column of fibers as shown in Figure 2a. For the circular bundle, symmetry reduces the solution domain to one-eighth of the bundle as shown in Figure 2b. For planar bundles, two fiber configurations are considered: square and equilateral triangular.
where is the number of components, are the binary diffusivities for species and in the multicomponent mixture, ⃗ is the diffusional driving force, is the mole fraction, is the molar mass, and is the average molar mass. Figure 2 illustrates the domain and boundary conditions used for simulations of fiber bundles. At the inlet, the velocity and gas composition are specified as the laminar, welldeveloped axial velocity, ( , ) , and a fixed, uniform feed mole fraction, , respectively:

Domain Setup and Boundary Conditions
where is the retentate axial velocity and is the mole fraction of component in the retentate. No-slip, zero normal flux boundary conditions are applied along the case (boundary 1), and symmetry boundary conditions are applied to give lines of symmetry for the planar and circular bundles (boundary 2). For the planar bundle, symmetry reduces the solution domain to a single column of fibers as shown in Figure 2a. For the circular bundle, symmetry reduces the solution domain to one-eighth of the bundle as shown in Figure 2b. For planar bundles, two fiber configurations are considered: square and equilateral triangular. Assuming a solution-diffusion transport mechanism [22], the membrane is treated as a boundary condition (boundary 3) in the simulation at which the total gas flux across the membrane is calculated from: where is the transmembrane molar flux of component , is the permeance, is the permeate mole fraction, is the retentate (high) absolute pressure, and is the permeate (low) absolute pressure. Along the outer diameter of each fiber, the normal wall velocity = ⁄ and the normal total mass flux (convective plus diffusive) of To focus on the effects of flow distribution in the shell only and reduce computational complexity, the lumen pressure is assumed to be sufficiently low to neglect relative to in Equation (11). Assuming a solution-diffusion transport mechanism [22], the membrane is treated as a boundary condition (boundary 3) in the simulation at which the total gas flux across the membrane is calculated from: where q i is the transmembrane molar flux of component i, Q i is the permeance, y i is the permeate mole fraction, p h is the retentate (high) absolute pressure, and p l is the permeate (low) absolute pressure. Along the outer diameter of each fiber, the normal wall velocity u n = M n q T /ρ and the normal total mass flux (convective plus diffusive) of component i = (j i + ρω i u) n = M i q i . To focus on the effects of flow distribution in the shell only and reduce computational complexity, the lumen pressure is assumed to be sufficiently low to neglect y i p l relative to x i p h in Equation (11).

Numerical Methodology
COMSOL Multiphysics ® (CFD and TCS modules) was used to solve the governing continuity, conservation of momentum, and conservation of mass equations for the steady, laminar, isothermal flow of a Newtonian fluid. The length of the fiber is 15 cm and the radius of the fiber is 150 µm. Triangular and quadrilateral finite elements were generated in Membranes 2022, 12, 1139 5 of 17 our model. The local grid densities near the inlet and outlet are higher than other sections. A direct solver was used to solve those equations.

Mesh Independence Studies
COMSOL utilizes the finite element method to discretize the solution domain, using a polygonal mesh, and obtains an approximate numerical solution to the governing partial differential equations. Mesh independence studies are required to ensure that the results are not affected by the chosen spatial discretization of the computational domain. Typically, they consist of calculating quantities of interest in the problem at hand with different mesh resolutions, and identifying a resolution that is computationally feasible and bounds deviations from more refined (but computationally costly) mesh resolutions by an acceptable amount, e.g., 1-5%. Figure 3 illustrates the change in the calculated retentate recovery (R) with increasingly finer meshes and associated degrees of freedom. The results change by less than 1% when the degrees of freedom exceed 6,000,000 for each circular model or 500,000 for each planar model.

Numerical Methodology
COMSOL Multiphysics ® (CFD and TCS modules) was used to solve the governing continuity, conservation of momentum, and conservation of mass equations for the steady, laminar, isothermal flow of a Newtonian fluid. The length of the fiber is 15 cm and the radius of the fiber is 150 µm. Triangular and quadrilateral finite elements were generated in our model. The local grid densities near the inlet and outlet are higher than other sections. A direct solver was used to solve those equations.

Mesh Independence Studies
COMSOL utilizes the finite element method to discretize the solution domain, using a polygonal mesh, and obtains an approximate numerical solution to the governing partial differential equations. Mesh independence studies are required to ensure that the results are not affected by the chosen spatial discretization of the computational domain. Typically, they consist of calculating quantities of interest in the problem at hand with different mesh resolutions, and identifying a resolution that is computationally feasible and bounds deviations from more refined (but computationally costly) mesh resolutions by an acceptable amount, e.g., 1-5%. Figure 3 illustrates the change in the calculated retentate recovery (R) with increasingly finer meshes and associated degrees of freedom. The results change by less than 1% when the degrees of freedom exceed 6,000,000 for each circular model or 500,000 for each planar model.

Darcy's Permeability
To validate the calculations of fluid flow through axial fiber bundles, the axial Darcy's permeability in the absence of gas permeation is calculated from: and compared to values reported in the literature [15], where is the superficial axial fluid velocity, < > is the mean-squared radius of the fibers, and / is the pressure gradient.

Darcy's Permeability
To validate the calculations of fluid flow through axial fiber bundles, the axial Darcy's permeability in the absence of gas permeation is calculated from: and compared to values reported in the literature [15], where u z is the superficial axial fluid velocity, R f 2 is the mean-squared radius of the fibers, and dp/dz is the pressure gradient.

Ideal Counter-Current Module
The 3-D module performance simulations are compared on the performance of an ideal counter-current (ICC) module [5]. The ICC model represents the benchmark against which module performance is compared when evaluating the effect of non-idealities as it assumes ideal flow distribution in the lumen and shell and uniform membrane properties. Figure 4 illustrates a mass balance for a differential module length used to develop the ICC performance analysis. ICC performance is calculated from the following mass balance equations: d With the boundary conditions: where is the molar flow rate of component in the retentate, is the feed flow rate, is the permeate flow rate, is the active membrane area per unit length, and is the length of the module.  Table 1 and Figures 5 and 6 indicate the nomenclature used here to describe fiber packing in planar and circular fiber bundles. In this work, the effects of D/Rf and ɸ are studied. The first variable, D/Rf, is a measure of the distance between the case and the outside of the fiber bundle. The second variable, ɸ, is a measure of the distance between fiber centers, d. Equation (19) indicates the relationship between the two and reflects the fractional volume occupied by fibers in an infinite bundle:

Design Space Explored
The effects of packing are more readily evaluated for planar bundles due to a reduction in computational requirements. It is anticipated planar bundles will provide a good approximation of circular bundles for sufficiently large bundles where the effects of case curvature can be neglected.  With the boundary conditions: .
where R i is the molar flow rate of component i in the retentate, . F i is the feed flow rate, . P i is the permeate flow rate, a is the active membrane area per unit length, and L is the length of the module. Table 1 and Figures 5 and 6 indicate the nomenclature used here to describe fiber packing in planar and circular fiber bundles. In this work, the effects of D/R f and φ are studied. The first variable, D/R f , is a measure of the distance between the case and the outside of the fiber bundle. The second variable, φ, is a measure of the distance between fiber centers, d. Equation (19) indicates the relationship between the two and reflects the fractional volume occupied by fibers in an infinite bundle:

Symbol Definition
R f fiber radius R bundle fiber bundle radius d distance between fibers D distance from wall to center of fiber closest to wall φ = πR f 2 /d 2 fiber packing fraction [21] Symbol Definition fiber radius fiber bundle radius distance between fibers distance from wall to center of fiber closest to wall ɸ = / fiber packing fraction [21]   Sample two-dimensional axial velocity distributions for planar bundles with regular square and equilateral triangular configurations in the absence of permeation are shown in Figure 7. Fluid bypass at the fiber bundle-wall interface is evident from the higher velocities.

Symbol
Definition fiber radius fiber bundle radius distance between fibers distance from wall to center of fiber closest to wall ɸ = / fiber packing fraction [21]   Sample two-dimensional axial velocity distributions for planar bundles with regular square and equilateral triangular configurations in the absence of permeation are shown in Figure 7. Fluid bypass at the fiber bundle-wall interface is evident from the higher velocities.  The effects of packing are more readily evaluated for planar bundles due to a reduction in computational requirements. It is anticipated planar bundles will provide a good approximation of circular bundles for sufficiently large bundles where the effects of case curvature can be neglected.
Sample two-dimensional axial velocity distributions for planar bundles with regular square and equilateral triangular configurations in the absence of permeation are shown in Figure 7. Fluid bypass at the fiber bundle-wall interface is evident from the higher velocities.   Sample two-dimensional axial velocity distributions for planar bundles with regular square and equilateral triangular configurations in the absence of permeation are shown in Figure 7. Fluid bypass at the fiber bundle-wall interface is evident from the higher velocities.  Sample three-dimensional CO 2 mole fraction distributions for a square fiber packing in planar and circular bundles are illustrated in Figures 8 and 9, respectively. Sample three-dimensional CO2 mole fraction distributions for a square fiber packing in planar and circular bundles are illustrated in Figures 8 and 9, respectively.  The EPB approximation is illustrated in Figure 10 where the flow domain is divided into two sections: (1) center and (2) wall. The fiber closest to the wall in each horizontal layer is assigned to the wall section and all other fibers to the center. The value for , the distance from the wall to the closed fiber, in the EPB is calculated from Equation (20) and NP1, the number of fibers in the region P1, from Equation (21): where , is the total area occupied by fibers in section C2, is the total area of section C2, , is the total area occupied by fibers in section P2, is the total area of section P2, is the distance from the wall to the center of the closest fiber, and is the total numbers of fibers in section . Sample three-dimensional CO2 mole fraction distributions for a square fiber packing in planar and circular bundles are illustrated in Figures 8 and 9, respectively.  The EPB approximation is illustrated in Figure 10 where the flow domain is divided into two sections: (1) center and (2) wall. The fiber closest to the wall in each horizontal layer is assigned to the wall section and all other fibers to the center. The value for , the distance from the wall to the closed fiber, in the EPB is calculated from Equation (20) and NP1, the number of fibers in the region P1, from Equation (21): where , is the total area occupied by fibers in section C2, is the total area of section C2, , is the total area occupied by fibers in section P2, is the total area of section P2, is the distance from the wall to the center of the closest fiber, and is the total numbers of fibers in section . The EPB approximation is illustrated in Figure 10 where the flow domain is divided into two sections: (1) center and (2) wall. The fiber closest to the wall in each horizontal layer is assigned to the wall section and all other fibers to the center. The value for D EPB , the distance from the wall to the closed fiber, in the EPB is calculated from Equation (20) and N P1 , the number of fibers in the region P1, from Equation (21): where A F, C2 is the total area occupied by fibers in section C2, A C2 is the total area of section C2, A F, P2 is the total area occupied by fibers in section P2, A P2 is the total area of section P2, D EPB is the distance from the wall to the center of the closest fiber, and N i is the total numbers of fibers in section i.

Results and Discussion
The axial Darcy permeability, as a function of packing fraction, was calculated and compared with prior work [15,23]. Simulations were performed for a single unit cell representative of an infinite fiber bundle. The results are presented in Table 2. The agreement between the axial Darcy permeabilities is excellent and helps validate the simulation approach for flow through the bundle.  Table 3 contains membrane properties and module operating conditions used in the simulations. The values for carbon dioxide and nitrogen permeance correspond to values reported in the literature for commercial carbon capture membranes, and are within the region of desired membrane properties for carbon capture [3]. Module performance is characterized by the dependence of retentate recovery R (fraction of the feed recovered as the retentate product) and F (dimensionless feed flow rate) defined by: where is the CO2 permeance, * is the total surface area and is the absolute feed pressure. R is a measure of the process energy requirements (operating costs) while F is a measure of membrane area (capital costs). The values of both are determined as a function of the CO2 mole fraction in the retentate product. The effect of the wall gap is quantified by comparing module performance to that for an ideal counter-current model corresponding to an infinite, uniformly packed fiber bundle.

Parameter
Value Feed CO2 mole fraction 0.2

Results and Discussion
The axial Darcy permeability, as a function of packing fraction, was calculated and compared with prior work [15,23]. Simulations were performed for a single unit cell representative of an infinite fiber bundle. The results are presented in Table 2. The agreement between the axial Darcy permeabilities is excellent and helps validate the simulation approach for flow through the bundle.  Table 3 contains membrane properties and module operating conditions used in the simulations. The values for carbon dioxide and nitrogen permeance correspond to values reported in the literature for commercial carbon capture membranes, and are within the region of desired membrane properties for carbon capture [3]. Module performance is characterized by the dependence of retentate recovery R (fraction of the feed recovered as the retentate product) and F (dimensionless feed flow rate) defined by: where Q CO2 is the CO 2 permeance, a * L is the total surface area and p F is the absolute feed pressure. R is a measure of the process energy requirements (operating costs) while F is a measure of membrane area (capital costs). The values of both are determined as a function of the CO 2 mole fraction in the retentate product. The effect of the wall gap is quantified by comparing module performance to that for an ideal counter-current model corresponding to an infinite, uniformly packed fiber bundle. <u>/<u> ∞ is used to evaluate the effect of poor packing near the case on flow distribution within the bundle. <u> is the superficial velocity (flow rate divided by total cross-section area) through a region extending from the fiber wall through a specified number of fibers (N) as shown in Figure 5. <u> ∞ is the superficial velocity through an infinite fiber bundle without a wall for the same pressure drop. <u>/<u> ∞ reflects the increase in flow rate that occurs due to the gap between the fiber bundle and the case. The ratio is highest near the wall where the effect of the wall gap is largest and decreases toward a value of unity as N increases. Values are calculated as a function of fiber packing fraction and D/R f for the square configuration.
The results are presented in Figures 11 and 12, along with values for the average velocity in the flow channel adjacent to the wall (the region corresponding to . Q wall in Figure 5). Figure 11 indicates that the velocity ratio for the entire flow channel approaches unity when the number of fibers included in the simulation domain increases, while the ratio in the wall flow channel does not change significantly. The fiber numbers needed for the velocity ratio to be in the range 1 ± 0.05 (i.e., with 5% of the value for an infinite bundle) varied from 10 to 50. The number of fibers increases as the packing fraction increases for a fixed D/R f or as D/R f increases for a fixed packing fraction. The velocity ratios for domains containing 10 fibers as a function of packing fraction and D/R f are illustrated in Figure 12.
The results indicate both <u> total /<u> ∞ and <u> wall /<u> ∞ increase with either increasing D/R f for fixed packing fraction or increasing packing fraction for fixed D/R f . These results show that the contribution of the bypass flow to the total flow decreases as the number of fibers increases. The magnitude of the bypass flow depends strongly on the distance between the case and the outer fibers of the bundle (D/R f ). The ratio of the bypass flow to the flow through the fiber bundle also depends on fiber packing fractionfor a given D/R f , as the fiber packing increases the resistance to flow through the interior, the bundle increases, and a greater fraction of the total flow will occur in the bypass region adjacent to the case. Figure 13 illustrates the effect of fiber number on the results over a broad range of D/R f values for a packing fraction of 0.4. As for square configuration, the fiber number required for the average velocity ratio to be in the range 1 ± 0.05 increases with D/R f . The value increases from 10 for D/R f less than 2 to over 500 for D/R f greater than 6.   Figure 13 illustrates the effect of fiber number on the results over a broad range of D/Rf values for a packing fraction of 0.4. As for square configuration, the fiber number required for the average velocity ratio to be in the range 1 ± 0.05 increases with D/Rf. The value increases from 10 for D/Rf less than 2 to over 500 for D/Rf greater than 6.   Figure 13 illustrates the effect of fiber number on the results over a broad range of D/Rf values for a packing fraction of 0.4. As for square configuration, the fiber number required for the average velocity ratio to be in the range 1 ± 0.05 increases with D/Rf. The value increases from 10 for D/Rf less than 2 to over 500 for D/Rf greater than 6.   Figure 13 illustrates the effect of fiber number on the results over a broad range of D/Rf values for a packing fraction of 0.4. As for square configuration, the fiber number required for the average velocity ratio to be in the range 1 ± 0.05 increases with D/Rf. The value increases from 10 for D/Rf less than 2 to over 500 for D/Rf greater than 6.

<u>/<u>∞
Fiber Number The values for <u> wall /<u> ∞ as a function of packing fraction (from 0.3 to 0.7) and D/R f (from 1 to 8) for an equilateral triangular configuration are illustrated in Figure 14. The values depend weakly on fiber number (results not shown) and increase with either increasing D/R f or packing fraction due to an increasing relative contribution to the total flow from the wall gap region. The values for <u>wall/<u>∞ as a function of packing fraction (from 0.3 to 0.7) and D/Rf (from 1 to 8) for an equilateral triangular configuration are illustrated in Figure 14. The values depend weakly on fiber number (results not shown) and increase with either increasing D/Rf or packing fraction due to an increasing relative contribution to the total flow from the wall gap region.

Planar Square Configuration
Gas separation module performance as a function of fiber number, packing fraction, and D/Rf for regular packings was evaluated for a module with the membrane properties, operating conditions, and feed specified in Table 3. Figure 15 shows the full 3-D module performance results. For D/Rf = 8 and ɸ = 0.4, the calculated module performance is poorer than ICC performance, as the retentate recovery (R) is lower for a given retentate product composition-this implies the captured CO2 in the permeate will have a lower composition. Additionally, a reduction in recovery implies an increase in operating costs as more pressurized N2 is depressurized through permeation and lost to the permeate. This precludes partial recovery of the associated compression energy through an expander as proposed in most carbon capture processes [3].
As expected, the differences become smaller as the fiber number increases from 25 to 200. For sufficiently large bundles, the flow through the bundle will be much larger than the flow through the wall gap, and the effect of the bypass flow will not be significant. Additionally, for sufficiently large bundles, the performance will not depend strongly on the size of the gap created by removing fibers near the wall.
However, for smaller bundles, the impact of poor wall packing is greater as the bypass flow is a greater fraction of the total shell-side flow. The flow through the gap will depend on the fourth power of the equivalent hydraulic diameter of the gap, so small changes in gap size can have a large effect.
The methodology reported in this manuscript provides the quantification of the relative flow through the bundle and the wall gap as a function of fiber number, fiber packing, and gap size. This in turn is used to evaluate module performance and determine how performance is degraded relative to ICC performance for given module properties.
Changes in the dimensionless feed flow rate (F) are more significant than the changes in recovery. For a given retentate CO2 composition, F can decrease by a factor of 10 or 800 Packing fraction Figure 14. <u> wall /<u> ∞ as a function of packing fraction and D/R f for equilateral triangular configuration.

Planar Square Configuration
Gas separation module performance as a function of fiber number, packing fraction, and D/R f for regular packings was evaluated for a module with the membrane properties, operating conditions, and feed specified in Table 3. Figure 15 shows the full 3-D module performance results. For D/R f = 8 and φ = 0.4, the calculated module performance is poorer than ICC performance, as the retentate recovery (R) is lower for a given retentate product composition-this implies the captured CO 2 in the permeate will have a lower composition. Additionally, a reduction in recovery implies an increase in operating costs as more pressurized N 2 is depressurized through permeation and lost to the permeate. This precludes partial recovery of the associated compression energy through an expander as proposed in most carbon capture processes [3].   As expected, the differences become smaller as the fiber number increases from 25 to 200. For sufficiently large bundles, the flow through the bundle will be much larger than the flow through the wall gap, and the effect of the bypass flow will not be significant. Additionally, for sufficiently large bundles, the performance will not depend strongly on the size of the gap created by removing fibers near the wall.
However, for smaller bundles, the impact of poor wall packing is greater as the bypass flow is a greater fraction of the total shell-side flow. The flow through the gap will depend on the fourth power of the equivalent hydraulic diameter of the gap, so small changes in gap size can have a large effect.
The methodology reported in this manuscript provides the quantification of the relative flow through the bundle and the wall gap as a function of fiber number, fiber packing, and gap size. This in turn is used to evaluate module performance and determine how performance is degraded relative to ICC performance for given module properties.
Changes in the dimensionless feed flow rate (F) are more significant than the changes in recovery. For a given retentate CO 2 composition, F can decrease by a factor of 10 or more. Such a decrease will necessitate an increase in the membrane area and associated capital costs for treating a fixed feed flow.
The effect of D/R f on performance for 25 fibers and φ = 0.6 is illustrated in Figure 16. Performance decreases as D/R f increases from 1 to 8, suggesting gaps should be minimized.
more. Such a decrease will necessitate an increase in the membrane area and associated capital costs for treating a fixed feed flow.
The effect of D/Rf on performance for 25 fibers and ɸ = 0.6 is illustrated in Figure  16. Performance decreases as D/Rf increases from 1 to 8, suggesting gaps should be minimized.

Circular Square Bundle
The performance of full circular fiber bundles is evaluated as a function of fiber number, packing fraction, and D/Rf for regular packings for the conditions in Table 3. Figure 17 illustrates the full 3-D circular square configuration bundle module performance results. Results are presented as retentate recovery versus CO2 composition. The results in Figure 17 correspond to D/Rf = 3, ɸ = 0.4 or 0.6, and the number of fibers in the entire bundle is 405. For fixed D/Rf, the module performance improves as the fiber bundle packing fraction becomes smaller and approaches ideal ICC module performance. As observed for planar bundles, the impact of poor packing is greater for compact bundles (larger fiber packing fraction) because the resistance to flow between fibers increases as the inter-fiber distance decreases, while the resistance to flow in the gaps between the bundle and case does not change significantly.
The effect of D/Rf on performance is illustrated in Figure 18. The results in Figure

Circular Square Bundle
The performance of full circular fiber bundles is evaluated as a function of fiber number, packing fraction, and D/R f for regular packings for the conditions in Table 3. Figure 17 illustrates the full 3-D circular square configuration bundle module performance results. Results are presented as retentate recovery versus CO 2 composition. The results in Figure 17 correspond to D/R f = 3, φ = 0.4 or 0.6, and the number of fibers in the entire bundle is 405. For fixed D/R f , the module performance improves as the fiber bundle packing fraction becomes smaller and approaches ideal ICC module performance. As observed for planar bundles, the impact of poor packing is greater for compact bundles (larger fiber packing fraction) because the resistance to flow between fibers increases as the inter-fiber distance decreases, while the resistance to flow in the gaps between the bundle and case does not change significantly.  The effect of D/R f on performance is illustrated in Figure 18. The results in Figure 18 correspond to D/R f = 1 or 3, φ = 0.4 or 0.6, and number of fibers in the entire bundle equals 405, with the same bundle size and same packing fraction. Module performance improves as D/R f decreases and the size of the bypass region and its detrimental effect on module performance decrease.  To reduce computational costs, an equivalent planar bundle module was intro previously. Figures 19-21 show the module performance comparison between a ci bundle and an equivalent planar bundle for the conditions in Table 3. The results f circular and equivalent planar bundles agree well despite the differences in the geom The result proves that the EPB can be used to simulate performance with re computational costs and without sacrificing fidelity. To reduce computational costs, an equivalent planar bundle module was introduced previously. Figures 19-21 show the module performance comparison between a circular bundle and an equivalent planar bundle for the conditions in Table 3. The results for the circular and equivalent planar bundles agree well despite the differences in the geometry. The result proves that the EPB can be used to simulate performance with reduced computational costs and without sacrificing fidelity.

Circular Triangular Bundle
The performances of circular bundles with equilateral triangular configuratio compared to the square configuration for the conditions in Table 3. Figure 22 com the results for D/Rf = 3, ɸ = 0.6, and 405 fibers. The performances of bundles equilateral triangular configurations are the same as for square configurations f same conditions. This suggests the packing geometry does not affect module perform significantly for similar operating conditions.

Circular Triangular Bundle
The performances of circular bundles with equilateral triangular configuratio compared to the square configuration for the conditions in Table 3. Figure 22 com the results for D/Rf = 3, ɸ = 0.6, and 405 fibers. The performances of bundles equilateral triangular configurations are the same as for square configurations fo same conditions. This suggests the packing geometry does not affect module perform significantly for similar operating conditions.

Circular Triangular Bundle
The performances of circular bundles with equilateral triangular configuration are compared to the square configuration for the conditions in Table 3. Figure 22 compares the results for D/R f = 3, φ = 0.6, and 405 fibers. The performances of bundles with equilateral triangular configurations are the same as for square configurations for the same conditions. This suggests the packing geometry does not affect module performance significantly for similar operating conditions.

Conclusions
The performance of shell-fed hollow fiber membrane modules is simu accounting for poor fiber packing between the fiber bundle and the case. Resul presented for both planar and circular bundles with regular square and equil triangular configurations.
Simulations of flow through the bundle clearly indicate the potential for signi

Conclusions
The performance of shell-fed hollow fiber membrane modules is simulated accounting for poor fiber packing between the fiber bundle and the case. Results are presented for both planar and circular bundles with regular square and equilateral triangular configurations.
Simulations of flow through the bundle clearly indicate the potential for significant bypass between the bundle and the case. The magnitude of the bypass flow depends strongly on how close the bundle is to the case (the distance between the case and the nearest fiber) and the fiber packing fraction. Changes in flow through the bundle arising from the wall gap can extend across thousands of fibers (equivalent to a 35 cm bundle with 1000 fibers and packing fraction of 0.6). The bypass contribution increases as either the fiber packing fraction or distance between the closest fiber and the wall increase. As expected, the contribution of the bypass flow to the total flow decreases as the number of fibers increases.
For both planar and circular bundle modules, significant reductions in module performance can arise relative to an ICC module. These changes are evident in how the module recovery (retentate product flow/feed flow) and dimensionless feed flow change relative to that expected for the ICC. For a fixed retentate product composition, both performance measures are lower than for an ICC. The decrease in recovery implies higher operating costs as more N 2 is depressurized and lost to the permeate due to permeation, and the associated compression energy cannot be recovered through an expander. The decrease in dimensionless feed flow implies greater capital costs as the membrane area required to process a given feed flow is larger.
The detrimental effects increase as the packing fraction increases, D/R f increases and module size decreases, as expected due to the relative contribution of the bypass flow to the total flow. Changes in module performance for regular square and equilateral triangular configurations are comparable for the same operating conditions. An equivalent planar bundle module is also proposed for circular fiber modules. Performance predictions for the EPB are in good agreement with full 3-D circular module predictions. This suggests the EPB can be used to simulate performance with reduced computational costs without sacrificing fidelity.
This work does not address bundle defects except at the wall. One would expect some effect on performance when removing fibers from the middle of the bundle, or other non-uniformities in fiber packing, but the impact of such bundle defects is beyond the scope of the present work.
The methodology reported here can be used to evaluate the impact of poor wall packing on gas separation module performance for other gas separations. While specific results are not provided, similar trends are expected and the impact of poor packing can be quantified.