A Dissertation Entitled Transport Modeling and CFD Simulation Of

Home , Membrane gas separation

A Dissertation

Entitled

Transport Modeling and CFD Simulation of Membrane Gas Separation Materials

and Modules

Yuecun Lou

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the

Doctor of Philosophy Degree in Engineering

______Dr. Glenn Lipscomb, Committee Chair

______Dr. Maria Coleman, Committee Member

______Dr. Dong-Shik Kim, Committee Member

______Dr. Yakov Lapitsky, Committee Member

______Dr. Arunan Nadarajah, Committee Member

______Dr. Patricia R. Komuniecki, Dean

College of Graduate Studies

The University of Toledo

December 2014

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of

Transport Modeling and CFD Simulation of Membrane Gas Separation Materials and Modules

Yuecun Lou

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering

The University of Toledo

December 2014

Gas separation using polymer membranes has become a commercially attractive area during the last fifty years. It has wide application including nitrogen production from air, carbon dioxide and water removal from natural gas, and organic vapor removal from air or nitrogen.

Theoretically there are two basic parameters to characterize the performance of the polymer membranes: permeability and selectivity. Both high permeability coefficient and selectivity are desired in order to achieve savings on both capital and operating costs.

However there is a trade-off relationship between permeability and selectivity. The ability to tune the transport properties of polymer materials through changes in primary and secondary chain architecture appears to be limited by the existence of an "upper bound". Theoretical transport models and correlations have been proposed to provide guidance on structure-property relationships and the location of the upper bound.

In the first part of this dissertation a novel model is developed to predict the gas transport properties. The non-equilibrium lattice fluid theory (NELF) predicts the existence of an upper bound for solubility selectivity. The theory provides very good a priori predictions

iii of solubility and solubility selectivity. Furthermore the temperature dependence of solubility and solubility selectivity is predicted by the analysis.

The NELF theoretical analysis also can be used to predict diffusivity and diffusivity selectivity in combination with transition state theory (TST). The diffusivity upper bound is investigated using model parameters to indicate the effect of properties of both gas pairs and polymeric materials. The temperature dependence of diffusivity and diffusivity selectivity is investigated. Finally gas permeability is successfully predicted and the existence of a permeability upper bound tested with the new model. Besides the influence of temperature on the permeability and permeability selectivity is evaluated with the model.

Membrane modules are widely used for large scale gas separation process. Spacers are a critical component of membrane modules. Spacers provide mechanical support, create a uniform flow channel, enhance mass transfer coefficient and mitigate concentration polarization. However spacers generate higher pressure drop on both sides of channel and which increases energy input and operation cost.

In the second part of this dissertation computational fluid dynamics (CFD) is used to investigate the flow within a spacer-filled channel. Three dimensional simulations were performed to visualize flow within the channel and evaluate pressure drop as a function of flow rate. The simulation method is validated by comparing results with experimental measurements for nitrogen flows. Simulations are also performed to investigate the effect of spacer geometry on velocity field and associated pressure drop. The influence of spacer geometry on the membrane module performance is examined and results show a

iv promising effect of spacer geometry on membrane separation performance and associated module pressure drop.

The effect of asymmetrical spacer design also is investigated. In contrast to a symmetrical spacer, asymmetrical spacers consist of two filaments of different diameter aligned asymmetrically to the nominal flow direction. Simulation results indicate the asymmetrical design can reduce pressure drop dramatically.

Finally the multiphysics simulation is performed to study a combination of fluid flow and mass transfer process in a triple-layer spacer configuration. Effect of the density of thinner spacer adjacent to the membrane on the module performance is studied.

Dedicated to my family

Acknowledgements

First I would like to thank my advisor Dr. Lipscomb for his consistent help and guidance on my research. He always encourages me and gives me creative suggestions when I get trapped into jungles. His kindness and attitude of doing research always influence my current study and future career.

I also want to thank my co-advisor Dr. Coleman for offering me assistance on membrane preparation and permeation test. Her profound knowledge and humor make me feel very relaxing and enjoyable of my work. I need to thank Dr. Kim, Dr. Lapitsky and Dr.

Nadarajah for being my committee members and giving me valuable suggestions. I really appreciate Membrane Technology & Research Inc. especially Pingjiao and Karl Amo for their assistance on my research. I also sincerely thank U.S Department of Energy (DOE) to fund this project.

Thanks a lot to my lab mates: Rahul, Xi, Ravi, Famila and Scott. I have a wonderful time working with you. I want to thank my senior Nima and Pei for their big help when I just started my research. All my friends in Toledo should also be acknowledged and I leave so many good memories because of you. Finally I would like to thank my wife and parents who are the most important people in my life. I would not get this accomplishment without their concern, support and love.

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Table of Contents

Abstract ...... iii

Acknowledgements ...... vii

Table of Contents ...... viii

List of Tables ...... xii

List of Figures ...... xiii

List of Abbreviations ...... xxii

List of Symbols ...... xxiii

1. Introduction ...... 1

1.1 The Solution-Diffusion Model ...... 3

1.2 Significance of Permeability Upper Bound Line ...... 4

1.3 Model Analysis of Gas Transport in Polymeric Membranes ...... 5

1.4 Development of Spacer-filled Membrane Module ...... 6

1.5 Research Objectives ...... 7

1.6 Research Significance ...... 8

2. Literature Review...... 12

2.1 Gas Transport Process in Polymeric Membrane ...... 12

2.2 Upper Bound Line for Gas Separation ...... 19

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2.3 Models and Theories of Gas Transport in Glassy Polymer Films ...... 21

2.3.1 Models of Gas Sorption in Glassy Polymer Films...... 21

2.3.2 Models of Gas Diffusion in Glassy Polymer Films ...... 23

2.4 Effect of Temperature on Gas Transport Properties ...... 25

2.5 CFD Simulation of Gas Flow in Spacer-Filled Membrane Module ...... 27

3. Model Prediction of Gas Sorption in Polymeric Membranes ...... 31

3.1 Introduction ...... 31

3.2 Solubility Selectivity Prediction using NELF Model ...... 32

3.3 Model Parameters Effect on Solubility-Solubility Selectivity ...... 36

3.4 Upper Bound Relationship ...... 43

3.5 Temperature Dependence of the Solubility Upper Bound ...... 47

3.6 Conclusions ...... 51

4. Model Prediction of Gas Diffusion in Polymeric Membranes ...... 53

4.1 Introduction ...... 53

4.2 Diffusivity and Diffusivity Selectivity Prediction using NELF-TST Model 57

4.3 Model Parameters Effect on Diffusivity-Diffusivity Selectivity ...... 61

4.4 Upper Bound Relationship ...... 65

4.5 Temperature Dependence of the Diffusivity Upper Bound ...... 67

4.6 Conclusions ...... 70

5. Model Prediction of Gas Permeation in Polymeric Membranes ...... 72

5.1 Permeability and Permeability Selectivity Prediction using NELF-TST

Model ...... 72

5.2 Model Parameters Effect on Permeability-Permeability Selectivity ...... 75

5.3 Temperature Dependence of the Permeability Upper Bound ...... 78

5.4 Conclusions ...... 82

6. CFD Simulation of Gas Flow in Spacer-Filled Membrane Module ...... 84

6.1 Introduction ...... 84

6.2 CFD Method Validation ...... 88

6.3 Effect of spacer geometries on the module performances ...... 94

6.3.1 Effect of Filament Diameter (df) ...... 96

6.3.2 Effect of Distance between Filaments (L) ...... 99

6.3.3 Effect of Spacer Orientation () ...... 102

6.3.4 Effect of Angle between Spacer Filaments () ...... 105

6.3.5 Effect of number of spacer repeat units ...... 110

6.4 Membrane Module Performance of Designed Spacers for CO2 Capture

Application ...... 114

6.5 Study of Flow Distribution and Pressure Drop in Asymmetric Spacers ..... 116

6.6 Mass Transfer Simulation in Multi-layer Spacer Configurations ...... 131

6.7 Conclusions ...... 137

7. Conclusions and Future Work ...... 141

7.1 Conclusions ...... 141

7.2 Future Work ...... 145

Reference ...... 148

List of Tables

Table 3.1: NELF model parameters for selected gases and polymers ...... 34

Table 3.2: Slopes of solubility upper bound predicted by NELF model ...... 45

Table 3.3: Predicted and experimental values [78] of mT and ΔHAB for PC and TMPC for different gas pairs...... 50

Table 4.1: Kinetic diameters of selected gas penetrants...... 56

Table 4.2: Unpenetrated density and FFV of several glassy polymers [56, 115]...... 57

Table 4.3: Slopes of diffusivity upper bound predicted by NELF-TST model...... 67

Table 4.4: Predicted and experimental values [78] of xT and for PC and TMPC for CO2/CH4...... 69

Table 5.1: Predicted and experimental values [78] of rT and for PC and TMPC for

CO2/CH4...... 81

Table 6.1: Summary of spacer configurations...... 96

Table 6.2: Separation performance and pressure drop of membrane module using Spacer

1 and Spacer 5 as the sweep spacer...... 115

Table 6.3: Dimensions of asymmetric and symmetric spacer strands. For flow attack angle of asymmetric spacer 0 indicates flow is parallel to the large strands and 45 indicates the angle between nominal flow direction and small strands...... 119

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List of Figures

Figure 1-1: Plate and frame module...... 2

Figure 1-2: Solution-diffusion model...... 4

Figure 1-3: Upper bound line of O2/N2...... 5

Figure 2-1: Gas transport through a non-porous polymeric membrane []...... 13

Figure 2-2: Correlations of permeability with gas penetrant critical volume in Polysulfone

(PSF) and Polydimethylsiloxane (PDMS) ...... 17

Figure 2-3: Polymer specific volume as a function of temperature []...... 17

Figure 2-4: NELF prediction of CO2 sorption in PC membrane...... 23

Figure 3-1: Non-equilibrium lattice fluid (NELF) theory...... 32

Figure 3-2: NELF prediction of solubility selectivity for CO2/CH4 gas pair in different polymers: SR – Silicone Rubber [], PMMA – Poly (methyl methacrylate) [], PSf –

Polysulfone [], PC – Polycarbonate [], HFPC – Hexafluoropolycarbonate [115], TMPC –

Tetramethylpolycarbonate [115], PPO – Poly (phenylene oxide) [] ...... 35

Figure 3-3: NELF prediction of solubility selectivity for CH4/N2 gas pair in different polymers. The symbols represent experimental values for the same polymers as in Figure

3-2...... 36

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[119]; open circle – SR [117], open diamond – HFPC [115], open triangle – TMPC [115], open square – PPO [121] ...... 38

* Figure 3-5: ρ dependence of CO2/CH4 solubility selectivity predicted by the NELF model for fixed p* (534 MPa) and T* (755 K). The dashed line represents the effect of varying T* from Figure 3-4. The symbols represent experimental values for the same polymers as in Figure 3-4...... 39

* Figure 3-6: p dependence of CO2/CH4 solubility selectivity predicted by the NELF model. The dashed line represents the effect of varying T* from 300 to 1800 K and the solid line represents the effect of varying ρ* from 1.205 to 2.000 g/cm3 at each of indicated p* values. The symbols represent experimental values for the same polymers as in Figure 3-4...... 41

* Figure 3-7: p dependence of CH4/ N2 solubility selectivity predicted by the NELF model.

The dashed line represents the effect of varying T* from 300 to 1800 K and the solid line represents the effect of varying ρ* from 1.205 to 2.000 g/cm3 at each of the indicated p* values. The symbols represent experimental values for the same polymers as in Figure

3-4...... 45

Figure 3-8: Temperature dependence of solubility selectivity for CO2/CH4 sorption in PC

(diamond) and TMPC (triangle). The solid line represents NELF predictions and the solid symbol the prediction at 308 K. The dashed line represents values calculated from experimental correlations and the open symbol the experimental value at 308 K [78].

Both the solid and dashed lines were calculated for a temperature range of 308 to 393 K.

Note that the experimental correlation does not pass through the data at 308 K due to

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differences between the best-fit correlation obtained from all of the temperature data and the specific data for 308 K...... 49

Figure 3-9: Temperature dependence of solubility upper bound for CO2/CH4 predicted by the NELF model. The symbols represent experimental values for the same polymers as in

Figure 3-4...... 51

Figure 4-1: “Hopping” mechanism of gas diffusion in glassy polymer film using transition state theory...... 53

Figure 4-2: NELF-TST prediction of gas diffusivity in Polycarbonate film. The symbols represent data for: solid square – CH4, solid diamond – Ar, solid triangle – O2, solid circle

– CO2; open square – N2, open diamond – H2, open triangle – He...... 58

Figure 4-3: NELF-TST prediction of gas diffusivity in PMMA. The symbols represent values for: square – CH4, diamond – N2, triangle – CO2, circle – Ar, star – O2. The empty

* data represent prediction using original p2 = 560 MPa and the solid data represent

* prediction using modified p2 = 740 MPa...... 59

Figure 4-4: NELF-TST prediction of diffusivity-selectivity for the CO2/CH4 gas pair in different polymers: PSf – Polysulfone [], PC – Polycarbonate [120], TMPC –

Tetramethylpolycarbonate [], PPO – Poly (phenylene oxide) [], HFPC –

Hexafluoropolycarbonate [125]...... 60

* Figure 4-5: T dependence of CO2/CH4 diffusivity selectivity predicted by the NELF-TST model from 377.5 K to 1132.5 K. The dashed line represents the effect of varying ρ* from

1.20 g/cm3 to 2.04 g/cm3 and the solid line represents the effect of varying p* from 267

MPa to 801 MPa at each of indicated T* values. The symbols represent experimental values of CO2/CH4 in different polymers: open square - PSf [124], open diamond - PC

[120], open triangle - TMPC [125], solid diamond - HFPC [125], solid square - PPO

[126]...... 63

Figure 4-6: Temperature dependence of diffusivity selectivity for CO2/CH4 diffusion in

PC (diamond) and TMPC (triangle). The solid line represents NELF-TST predictions and the solid symbol the prediction at 308 K. The dashed line represents values calculated from experimental correlations and the open symbol the experimental value calculated at

308 K [78]. Both the solid and dashed lines were calculated for a temperature range of

308 to 393 K...... 68

Figure 4-7: Temperature dependence of diffusivity upper bound for CO2/CH4 predicted by the NELF-TST model. The symbols represent experimental values for the same polymers as in Figure 4-5...... 70

Figure 5-1: NELF-TST prediction of gas permeability in polycarbonate film. The symbols represent data for: solid triangle – N2, solid square – CH4, solid diamond – CO2; open square – O2, open diamond – He...... 73

Figure 5-2: NELF-TST prediction of permeability-selectivity for the O2/N2 gas pair in different polymers: PSf – Polysulfone [124], PC – Polycarbonate [120], TMPC –

Tetramethylpolycarbonate [120], PPO – Poly (phenylene oxide) [], HFPC –

Hexafluoropolycarbonate [125]...... 75

* Figure 5-3: T dependence of O2/N2 permeability selectivity predicted by the NELF-TST model from 400 K to 1200 K. The dashed line represents the effect of varying ρ* from

1.20 g/cm3 to 2.04 g/cm3 and the solid line represents the effect of varying p* from 267

MPa to 801 MPa at each of indicated T* values. The symbols represent experimental

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values of O2/N2 in different polymers: open square - PSf [124], open diamond - PC [120], open triangle - TMPC [120], solid diamond - HFPC [125], solid square - PPO [127]. ... 78

Figure 5-4: Temperature dependence of permeability selectivity for O2/N2 permeation in

308 K [78]. Both the solid and dashed lines were calculated for a temperature range of

308 to 393 K...... 80

Figure 5-5: Temperature dependence of permeability upper bound for CO2/CH4 predicted by the NELF-TST model. The symbols represent experimental values of CO2/CH4 in different polymers: open square - PSf [124], open diamond - PC [120], open triangle -

TMPC [120], solid diamond - HFPC [125], solid square - PPO [127]...... 82

Figure 6-1: Top view of spacer A. The flow direction is indicated by the arrow and the repeat unit lies within the dashed box...... 85

Figure 6-2: The flow domain for the Spacer A repeat unit. The inlet and outlet planes are periodic boundaries along which the velocity field is identical. The two lateral faces are also periodic boundaries along which the velocity field is identical...... 86

Figure 6-3: Meshed solution domain for the Spacer A repeat unit...... 87

Figure 6-4: Velocity field within cross-sections normal to the flow direction for spacer A at Reynolds number ~150. Velocity increases as the color changes from blue to red (Unit: m/s)...... 89

Figure 6-5: Surface pressure field of spacer A at Reynolds number ~150. Pressure increases as the color changes from blue to red (Unit: psi)...... 90

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Figure 6-6: Comparison of the simulated and experimental f versus Re of Spacer A.

Diamond indicates the experimental data and square indicates the simulation data...... 93

Figure 6-7: Dependence of mesh elements on f versus Re of spacer A. Diamond indicates

~100,000 mesh elements, square indicates ~430,000 mesh elements and triangle indicates

~1,000,000 mesh elements...... 94

Figure 6-8: The schematic of a repeat unit of the non-woven spacer...... 95

Figure 6-9: Velocity field at the plane of intersection of spacer filaments for (a). Spacer 1

(at Re ~ 175) and (b). Spacer 2 (at Re ~ 180). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right...... 98

Figure 6-10: Calculated f versus Re for spacers simulated as varied filament diameter.

Diamond indicates data of spacer 1 and square indicates data of spacer 2...... 99

Figure 6-11: Velocity field at the plane of intersection of spacer filaments for (a). Spacer

3 (at Re ~ 285) and (b). Spacer 4 (at Re ~ 280). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right...... 101

Figure 6-12: Calculated f versus Re for spacers with filament angle 30 simulated as varied inter-filament distance. Diamond indicates the data of spacer 3 and square indicates the data of spacer 4...... 102

Figure 6-13: Velocity field at the plane of intersection of spacer filaments for: (a). Spacer

6 (at Re ~ 315) and (b). Spacer 7 (at Re ~ 310). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right...... 104

Figure 6-14: Calculated f versus Re for spacers with filament angle 90 simulated as varied flow attack angle. Diamond indicates the data of spacer 6 and square indicates the data of spacer 7...... 105

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Figure 6-15: Top view of velocity field at the plane of intersection of spacer filaments:

(a). Spacer 5 (at Re ~ 120), (b). Spacer 8 (at Re ~ 80) and (c). Spacer 9 (at Re ~ 60).

Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from left to right for Spacer 5 and from bottom to top for Spacer 8 and 9. Additional small figures are added to zoom in some region inside the channel to review the flow recirculation...... 108

Figure 6-16: Calculated f versus Re for spacers with varied spacer filament angle.

Diamond indicates the data of spacer 4 (30), Square indicates the data of spacer 5 (60),

Triangle indicates the data of spacer 7 (90), Solid circle indicates the data of spacer 8

(120) and empty circle indicates the data of spacer 9 (150). Dash lines indicate f versus

Re using creeping flow model from spacer 4 (30) (bottom dashed line) to spacer 9 (150)

(top dashed line)...... 109

Figure 6-17: Identify multiple spacer repeat units. The arrow A indicates the flow direction of spacer 4 and the B indicates the flow direction of spacer 8...... 111

Figure 6-18: Velocity field in the x-y plane along the center of the flow channel for 2×2 repeat units: (a). Spacer 4 (at Re ~ 160) and (b). Spacer 8 (at Re ~ 110). Velocity (m/s) increases as the color changes from blue to red. Flow is from lower left to upper right in

(a) and from lower right to upper left in (b). Inset figures correspond to enlargements of indicated regions...... 113

Figure 6-19: Calculated f versus Re for spacers with different repeat unit. Solid diamond indicates data of spacer 4 with 1×1 repeat unit, empty diamond indicates data of spacer 4 with 2×2 repeat unit, solid square indicates data of spacer 8 with 1×1 repeat unit, and empty square indicates data of spacer 8 with 2×2 repeat unit...... 114

Figure 6-20: Asymmetric spacer-filled flow channel used to perform the simulations. . 116

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Figure 6-21: Top view of a spacer-filled channel that utilizes an asymmetric spacer with an angle of 45 between the larger strands on the bottom and smaller strands on the top.

...... 117

Figure 6-22: Geometry of membrane module channel filled by symmetric spacer...... 118

Figure 6-23: Velocity fields in x-y planes near the central part of flow channel filled by asymmetric spacer for Re ~240 near (a) the top, (b) the bottom and (c) the central surfaces. Velocity (m/s) increases as the color changes from blue to red...... 121

Figure 6-24: Velocity fields in x-y planes near the center of the flow channel for a symmetric spacer with Re ~240. The images correspond to planes near the (a) top, (b) bottom, and (c) middle of the flow channel. Velocity (m/s) increases as the color changes from blue to red...... 123

Figure 6-25: Calculated f versus Re for asymmetric and symmetric spacers: diamond - asymmetric spacer; square - symmetric spacer...... 124

Figure 6-26: Transverse y-z plane (marked in red) examined within flow channels filled by the (a) asymmetric and (b) symmetric spacers...... 125

Figure 6-27: Velocity fields in y-z planes near the central part of flow channel filled by

(a) asymmetric and (b) symmetric spacer for Re ~240. Dashed arrows indicate transverse vortices...... 127

Figure 6-28: Points selected for examining the variation of the x-component of velocity along the height of the flow channel for the (a) asymmetric and (b) symmetric spacer. 128

Figure 6-29: Variation of the x-component of velocity with distance from the bottom of the flow channel for inlet Reynolds number of (a) Re ~50, (b) Re ~240, and (c) Re ~510.

The reddish-orange curve corresponds to the asymmetric spacer and the blue curve to the symmetric spacer...... 130

Figure 6-30: Configuration of a triple-layer spacer design...... 132

Figure 6-31: Simplified configuration of the triple-layer spacer shown in Figure 6-30.

The dashed region represents a symmetrical geometry used for simulation...... 133

Figure 6-32: A part of channel geometry used for simulation. Circles represent the thinner and denser spacer B adjacent to the membrane surface with distance L between the neighboring filaments...... 134

Figure 6-33: Parameters used for gas membrane module performance...... 136

Figure 6-34: Module performance curve at permeate side. RR indicates Resistance Ratio between spacer filled channel and empty channel...... 137

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List of Abbreviations

CCS ...... Carbon Capture and Sequestration CFD ...... Computational Fluid Dynamics

DMS ...... Dual-mode sorption

FFV ...... Fractional Free Volume

HFPC ...... Hexafluoropolycarbonate

MMMs ...... Mixed Matrix Membranes

NELF...... Non-Equilibrium Lattice Fluid

PBC ...... Periodic Boundary Condition PC ...... Polycarbonate PDMS ...... Polydimethylsiloxane PMMA ...... Poly (methyl methacrylate) PSf ...... Polysulfone PPO ...... Poly(phenylene oxide) PVC ...... Poly(vinyl chloride) PVT ...... Pressure-Volume-Temperature

RR ...... Resistance Ratio

SR ...... Silicone Rubber

TMPC ...... Tetramethylpolycarbonate TST ...... Transition State Theory

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List of Symbols

b Solubility upper bound intercept (-) bT Solubility upper bound intercept with temperature variation (-) C Gas concentration (cm3(STP)/cm3) df Diameter of spacer filament (m) dh Hydraulic diameter (m) dkinetic Kinetic diameter of gas penetrant (Ǻ) D Average effective diffusivity (cm2/sec) 2 D0 Pre-exponential constant for diffusion (cm /sec) 2 Dloc Binary mutual diffusion coefficient (cm /sec) Ed Activation energy for diffusion (cal/mol) Edef Deformation energy (J) Eint Polymer-penetrant interaction energy (J) Ep Activation energy for permeation (cal/mol) F Volume force (N/m3) f Friction factor (-) G Gibbs free energy (J) I Unit vector (-) J Steady-state gas permeation rate (cm3(STP)/cm2/sec) j Diffusive mass flux vector (kg/m2/sec) -1 k Gas swelling coefficient (Pa ) -23 kB Boltzmann constant (1.38110 J/K) 3 3 kD Henry’s law coefficient (cm (STP)/cm /atm) kjump Rate constant (1/sec) -34 h Planck constant (6.62610 Jsec) hsp Height of spacer-filled channel (m) L Distance between filaments (m) LC Channel length (m) Ljump Jump length (Ǻ) l Membrane thickness (m) m Solubility upper bound slope (-) mT Solubility upper bound slope with temperature variation (-) M Molecular mass (g/mol) N Number of filaments per inch (-) Nlattice The number of polymer molecule lattice sites (-) p Operating pressure (atm) p0 Vapor pressure (atm) p* Cohesive energy density in the close-packed state (MPa)

xxiii

p Pressure difference (Pa) P Permeability coefficient (Barrer) P0 Pre-exponential constant for permeation (Barrer) qT Permeability upper bound intercept with temperature variation (-) R Universal gas constant (8.314 J/mol/K) Re Reynolds number (-) r Number of lattice sites occupied by one molecule (-) rT Permeability upper bound slope with temperature variation (-) S Solubility coefficient (cm3(STP)/cm3/atm) 3 3 S0 Pre-exponential constant for sorption (cm (STP)/cm /atm) -1 Svsp Specific surface of spacer (m ) Hs Heat of sorption (cal/mol) T Absolute temperature (K) Tg Glassy transition temperature (K) T* Polymer mer-mer interaction energy (K) u Velocity vector (m/sec) v Inlet bulk velocity (m/sec) 3 V0 Occupied volume at 0 K per mass of polymer repeat unit (cm /g) 3 Vg Glassy specific volume (cm /g) 3 Vl Equilibrium volume of rubber region (cm /g) 3 Vsp Occupied volume of spacer filament (m ) 3 Vtot Total volume of channel (m ) 3 Vvdw Van der Waals volume (cm /g) v* Site molar volume (cm3/mol) dV Expanding volume (cm3) w Mass fraction of gas species (-) x Diffusivity upper bound slope (-) xT Diffusivity upper bound slope with temperature variation (-) y Diffusivity upper bound intercept (-) yT Diffusivity upper bound intercept with temperature variation (-)

α0 Pre-exponential factor for temperature dependence of solubility selectivity on solubility (-) αD Diffusivity selectivity (-) αP Permeability selectivity (-) αS Solubility selectivity (-) β0 Pre-exponential factor for temperature dependence of diffusivity selectivity on diffusivity (-) γ0 Pre-exponential factor for temperature dependence of permeability selectivity on permeability (-) ε Voidage (-) ε* Site molar interaction energy (J/mol) θ Angle between filaments (deg)  Viscosity (Pasec) ρ Density (g/cm3) ρ* Close-packed mass density (g/cm3)

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Φp Volume fraction of the polymer (-) φ Flow attack angle (deg) χ Flory-Huggins parameter (-) Ψ Binary interaction parameter (-)  Mass fraction of gas species used in mass transfer model (-)

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Chapter 1

1. Introduction

Gas separation processes are critical to the economic viability of industrial gas markets.

These markets include production of nitrogen, methane, hydrogen, and carbon dioxide.

Multiple technologies are available including adsorption, absorption and cryogenic distillation.

Over the past 30-40 years membrane based gas separation technology has become a novel and cost-effective option. The gases are separated by polymeric membranes due to their different sorption and diffusion coefficients in the polymeric materials. Industrial scale applications commonly require large amounts of membrane area. A membrane module is used as an efficient way to package the area in compact, economic units.

Several configurations of membrane module are used in the industrial membrane field.

One configuration is the plate and frame module. A typical plate and frame module is illustrated in Figure 1-1. Generally two factors have significant impact on the membrane module performance: mass transfer coefficient and pressure drop. Enhanced mass transport property and controlled pressure drop is desired to minimize required membrane area (capital costs) and parasitic energy losses (operating costs).

Figure 1-1: Plate and frame module.

In the first part of this work we focus on the gas transport through the polymeric membranes. A novel model has been developed to predict gas transport properties and selectivity of gas pairs. The key objective of this part is predicting the properties of gas permeability upper bound which is critical for membrane gas separation process. In the second part we turn to analyze the pressure drop within membrane module. As shown in

Figure 1-1 the spacers are a critical component of the membrane module. Spacers create uniform flow channel during module fabrication and operation. Spacers are also used to mix fluid within channels to enhance mass transfer coefficients. Unfortunately spacers increase pressure drop. Computational Fluid Dynamics (CFD) is used to simulate a spacer-filled channel in a membrane module. After the simulation method is validated by comparing with experimental results, the effects of spacer design on the pressure drop

and associated membrane module performance is investigated. More details of the work are mentioned in following subsections.

1.1 The Solution-Diffusion Model

Theoretically, two basic parameters characterize the performance of the polymer membranes: permeability and selectivity. Permeability indicates the rate at which gas sorbs and diffuses through the polymer structure. Permeability is mathematically expressed as the product of the solubility coefficient and diffusivity coefficient based on the well-known solution-diffusion model [1]. Shown in Figure 1-2, transport involves five steps: diffusion to the surface of membrane, sorption into membrane, diffusion through membrane, desorption from membrane and finally diffusion from the membrane surface. Selectivity indicates the separating capability of a specific polymer membrane for two different kinds of gas molecules and is defined as the ratio of permeability of more permeable gas to that for the less permeable gas. Both high permeability and selectivity are desired. The membrane area required to perform a separation decreases as permeability increases and therefore capital costs decrease. The energy required to perform the separation decreases as selectivity increases and therefore the operating costs decrease.

Figure 1-2: Solution-diffusion model.

1.2 Significance of Permeability Upper Bound Line

Unfortunately, there is trade-off relationship between the permeability and selectivity.

The ability to tune the transport properties of polymer materials through changes in primary and secondary chain architecture appears to be limited by the existence of an

“upper bound”. First reported in the literature by Robeson [2], changes in the chain architecture that increase permeability also tend to decrease selectivity, which is shown in

Figure 1-3. Furthermore, besides the obvious trade-off relationship between permeability and selectivity, the upper bound is moving upwards as more polymers with better properties are synthesized [3]. Generally polymeric materials are divided into two major categories: rubbery polymer and glassy polymer depending on the stiffness of the polymer structure which is usually characterized by glass transition temperature (Tg).

Glassy polymeric materials constitute the majority of the data points that form the upper- bound line for a specific gas pair. Hence we will focus mainly on gas transport properties of glassy polymer materials.

Figure 1-3: Upper bound line of O2/N2.

1.3 Model Analysis of Gas Transport in Polymeric Membranes

Since the location of the upper bound line is crucial to evaluating the potential of membrane gas separation performance, significant effort has been devoted to evaluating a large number of polymeric materials to establish structure-property relationships. A number of correlations are based on either the activation energy model [4] or free volume model [5]. Alentiev et al. tried to build up the tradeoff relations between gas molecules and glassy polymer membranes based on free volume model [6]. Correlations for both permeability and diffusivity were used to predict the permeability-permeability selectivity tradeoff relationship for different gas pairs. Another commonly used correlation was proposed by Freeman [7]. The theoretical correlation for this upper bound is well developed mathematically based on the activation energy model and is further explained by Robeson in his recent paper [8]. The theory provides very good estimates of the location of the upper bound and indicates that the slope of the permeability upper

bound line is a polymer-independent constant that is a function only of the kinetic diameters of the gas pair.

As described in this dissertation, the non-equilibrium lattice fluid theory (NELF) can be used to predict the existence of an upper bound for solubility selectivity. The theory clearly captures how solubility and solubility selectivity depend on gas and polymer material properties and provides very good a priori predictions of solubility and solubility selectivity. Furthermore, the NELF theoretical analysis can be used to predict diffusivity and diffusivity selectivity by using transition state theory (TST). A new model relative to the energy change associated with forming the diffusion transition state is developed and diffusivity is directly calculated. Finally gas permeability is successfully predicted and properties of permeability upper bound are investigated by the novel model based on the solution-diffusion model. The temperature dependence on the gas transport properties are also studied using the new model.

1.4 Development of Spacer-filled Membrane Module

Membrane modules utilizing membranes in sheet form are a configuration which offers low manufacturing costs, rapid scale-up, and facile process integration for gas separation process. Spacers play a very significant role in the design and fabrication of membrane modules. A variety of forms exist, but net-type spacers consisting two layers of cylindrical filaments are commonly used. Spacers in the permeate channel provide mechanical support to the membranes and help to create a uniform flow channel during module fabrication and use by preventing the membrane from being pressed together when a pressure difference is maintained across the membrane between the feed/retentate

and permeate channels. Spacers in the feed channel also help create uniform flow channels and enhance the mass transfer coefficient of fluid. However, spacers also increase the module pressure drop. Process economics require the pressure difference that drives transport across the membrane to be as small as possible due to flue gas compression costs. Therefore, the pressure drop that accompanies flow through a spacer- filled channel will reduce this driving force and represents an additional parasitic energy load on the plant. computational fluid dynamics (CFD) is used to investigate the flow behaviors and performance of membrane modules with different spacer configurations.

1.5 Research Objectives

In this research project the first objective is to develop a theoretical analysis of the gas solubility and solubility selectivity using the well-developed NELF model. Additionally, the relationship between gas permeability and permeability selectivity will be examined using a new model based on the NELF theory in combination with transition state theory, the NELF-TST model. The newly developed model can be used to give a priori predictions of diffusivity and diffusivity selectivity. Predictions of permeability and permeability selectivity are provided by combining the diffusivity and solubility analyses.

The combined analysis can be used to investigate the effects of temperature on the upper bound. Specific tasks are:

 NELF model analysis of the gas sorption upper bound in glassy polymers

 Temperature dependence of the gas solubility, solubility selectivity and gas sorption

upper bound

 NELF-TST model prediction of gas diffusivity and diffusivity selectivity in glassy

polymers

 NELF-TST model analysis of the gas diffusivity upper bound in glassy polymers

 Temperature dependence of the gas diffusivity, diffusivity selectivity and gas

diffusion upper bound

 NELF-TST model prediction of gas permeation and analysis of gas permeability

upper bound in glassy polymers

 Temperature dependence of the gas permeability, permeability selectivity and gas

permeability upper bound

The second research objective is to investigate gas flow within spacer filled membrane modules using computational fluid dynamics (CFD). Specific tasks are:

 CFD method validation

 Effect of spacer geometric parameters on velocity field and pressure drop of gas flow

within spacer-filled channel

 Membrane module performance of designed spacers for CO2 capture application

 Flow visualization and pressure drop calculation of designed asymmetrical spacer

using 3D CFD simulation

 Effect of spacer density on the membrane module performance for a triple layer

spacer configuration using multiphysics simulation

1.6 Research Significance

This project focuses on the models and simulations of gas separation in membrane module. Till now there are several different models and theoretical correlations available to study gas separation process in glassy polymer membranes and these will be reviewed

in later section. However, most of them are specific to either gas sorption or diffusion.

Thus the parameters of model are limited to only one step of the gas transport process.

Very few models are available that consistently combine analyses of sorption and diffusion in the permeation process.

Currently the most acceptable model for gas transport in glassy polymeric membranes is the dual-mode model [9]. The model indicates two specific transport mechanisms for the whole procedure and five parameters (three for sorption and two for diffusion) are required. The model provides a very good correlation of experimental data. However, one primary limitation for this model is the parameters of this model are obtained directly from experimental transport data. This makes the model unavailable for predictions of novel polymer membranes if gas sorption or permeation data for the materials are not available.

With the development of computational technology, molecular simulation is widely used in predictions of gas transport in polymers [10,11]. Based on the molecular dynamics, force-field principles and computational methods the molecular simulation successfully simulates gas sorption, diffusion and permeation in different kinds of polymer. However, there are two main drawbacks of this method: one is the computational resources required to simulate realistic materials over statistically significant time intervals. Usually the software takes hours even days to finish a single simulation. Another drawback is based on the limitation of CPU resources. Simulations are limited to polymers having very simple chain structure such as polyethylene and polypropylene due to high computational demand.

The primary gas transport properties are characterized by the permeability upper bound.

As mentioned previously, correlations based on either free volume theory or activation energy theory exist. However, for free volume theory several adjustable parameters must be determined. For the energy correlations proposed by Freeman several parameters of the model equations are empirically given by an adjustable, universal value. Thus the theory fails to capture fully the dependence of transport on the identity of the specific chemical species present.

The novel NELF-TST model developed here is applicable to gas diffusion in glassy polymers. Since all of the critical model parameters are taken from the NELF model a parameter-consistent model is thus developed that permits a priori analysis and prediction of gas permeation in glassy polymeric membranes. In the zero pressure limit used in most discussions of the permeability-permeability selectivity upper bound, all model parameters are determined from pure polymer and gas P-V-T data and do not require experimental gas transport data. Compared to time consuming molecular simulation, the gas transport data can be obtained in seconds. Also the new model can be applied to several different kinds of polymers including polycarbonate (PC), polysulfone (PSf), and poly(phenylene oxide) (PPO) and obtain good agreement experimental data.

The spacer is a significant part of a membrane module and past studies have focused on the relationship between space geometry and module performance, specifically pressure drop and mass transfer coefficient. Computational Fluid Dynamics (CFD) is used commonly to study the flow distribution and pressure drop within spacer-filled channels inside a membrane module. However the initial simulations are mostly two dimensional studies and results may differ from that obtained with real three-dimensional spacers.

Additionally most of the work has focused on liquid separations such as reverse osmosis and application in gas separation processes is rarely discussed. This dissertation describes the development of a three dimensional CFD simulation to analyze gas flow within a spacer filled membrane module. The CFD method is validated by comparison with experimental measurements. The influence of spacer geometric parameters on the velocity field and pressure drop along the channel is investigated. Membrane module performance for designed spacers is evaluated. The design of an asymmetrical spacer also is studied. Finally, the effect of spacer filament spacing or density on membrane module performance for a triple layer spacer configuration is analyzed using multiphysics simulation.

Chapter 2

2. Literature Review

2.1 Gas Transport Process in Polymeric Membrane

For non-porous polymeric membranes, gas transport typically is analyzed in terms of the

“solution-diffusion model”. First mentioned Graham [12] and then Mitchell [13] gas permeation through the membrane structure consists of two steps: 1) sorption and 2) diffusion. A correlation of permeation flux with pressure and thickness was proposed using Equation (2-1) [14] :

(2-1) where J is the steady-state gas permeation rate. ∆p is the pressure difference across the membrane and l is the thickness of the membrane. P, defined as the permeability coefficient, is a polymer specific material property that characterizes the gas transport properties of a particular gas species. Figure 2-1 shows the film of thickness l generally separates two regions filled with a single gas. We define the gas pressure on the upstream side of film as p2 and the pressure on the downstream side as p1, at steady state Fick’s law

[15] can be used to express the gas flux within the membrane:

(2-2)

where C is the gas concentration, x is the distance across the film, w is the mass fraction of gas in the polymer, and Dloc is the binary mutual diffusion coefficient of the gas in the polymer.

Figure 2-1: Gas transport through a non-porous polymeric membrane [16].

Integrating Equation (2-2) across the film (from x=0 to x=l) yields the following equation for the flux:

(2-3)

where C1 and C2 are the concentrations of gas at upstream and downstream of polymer film respectively which are in equilibrium with the contacting gas phases at pressures p1 and p2, respectively. Defining the average effective diffusivity D as:

(2-4)

and substituting Equation (2-4) in Equation (2-3), gives:

(2-5)

Substituting Equation (2-5) in Equation (2-1) yields the following expression for the permeability:

(2-6)

Please note that all the equations are derived for a “pure-gas”; for gas mixtures the total upstream pressure p2 and downstream pressure p1 should be replaced with the corresponding partial pressures of the specific component at both upstream and downstream sides of membrane.

Commonly, permeation measurements are made by evacuating the low pressure

(downstream) side and monitoring the pressure increase. For such measurements, the upstream pressure and concentration are much larger than the downstream values, i.e., p2>>p1 and C2>>>C1. This allows simplification of Equation (2-6) to:

(2-7)

Defining the ratio of concentration of gas dissolved in the polymer to the pressure of the gas as the equilibrium solubility coefficient:

(2-8) and substituting Equation (2-8) in Equation (2-7) gives:

(2-9)

Equation (2-9) indicates that the overall permeability coefficient P depends on a thermodynamic term S related to dissolved gas concentration in the polymer and a kinetic term D related to the rate at which the gas diffuses through the polymer film. In the SI unit system, following units are used for permeability coefficient:

Alternatively, a more widely used unit for P is:

Permeability coefficients span a wide range of values from 10-3 to 104 barrer or more. For solubility S the commonly used unit is cm3 (STP)/cm3 (Polymer)/atm and diffusivity D is cm2/s.

Experimental techniques used to determine the P, D and S are reviewed by Felder et al.

[17] and Lin et al. [18]. Generally there are three different methods: A common method is to measure the permeability in steady state and based on the permeability data the diffusivity D can be determined by the time-lag method [19]. Solubility S can be calculated as the ratio of P to D based on the solution-diffusion model. A more widely used and accurate method is independently taking measurements of P in permeation test and S in sorption test [20, 21, 22], by using this method D is evaluated by using solution- diffusion model. The last method is S and D are independently measured and then P is obtained by multiplying experimental S and D together. The last method is not used very often.

Generally there are two different kinds of polymer membrane materials: rubbery polymer membranes and glassy polymer membranes according to the glass transition temperature

(Tg). A rubbery polymer possesses an amorphous structure with its operating temperature above Tg. Models of gas transport in rubbery polymer films are straight forward. For sorption the Henry’s law [23] is used when the gas concentration is low:

(2-10)

3 3 where kD is the Henry’s law coefficient with unit of cm (STP)/cm -polymer/atm. For

rubbery polymer with low gas concentration usually diffusion coefficient is a fixed value thus the permeability can be expressed as:

(2-11)

Equation (2-11) indicates at the low concentration the permeability coefficient of rubbery polymer films is independent of feed pressure. For rubbery polymers with the high activity gas the Flory-Huggins equation provides a good estimation of gas penetrant solubility [24, 25]:

(2-12)

0 where p is the penetrant pressure, p is the vapor pressure of the penetrant, Φp is volume fraction of the polymer and χ is Flory-Huggins parameter.

A glassy polymer, instead, is an amorphous polymeric material with its operating temperature below Tg. In contrast to rubbery polymers, glassy polymers have dramatically reduced movement of polymer chains and chain segments [26]. Glassy polymer membranes commonly are used for gas separation because they are more size sieving than rubbery polymers due to their highly constrained molecular motion. Glassy polymers can show selective permeation for gas pairs that differ only slightly in size such as oxygen and nitrogen. Consequently, solubility typically controls permeation selectivity in rubber polymer films while diffusivity controls selectivity for glassy polymer films.

Figure 2-2 shows the permeability coefficients of several different kinds of gas penetrants in the rubbery polymer film polydimethylsiloxane (PDMS) and glassy polymer film polysulfone (PSF) [27]. Permeability coefficients in PSF span a much wider range than

PDMS (almost five orders of magnitude versus one) and are at least two orders of magnitude lower in value.

Figure 2-2: Correlations of permeability with gas penetrant critical volume in

Polysulfone (PSF) and Polydimethylsiloxane (PDMS).

The non-equilibrium excess free volume (“unrelaxed” free volume) in glassy polymers plays a critical role in gas transport. The characterization of this excess free volume is illustrated in Figure 2-3.

Figure 2-3: Polymer specific volume as a function of temperature [28].

In the glassy region a deviation exists between the actual glassy specific volume Vg and equilibrium volume estimated by extrapolating the rubber specific volume Vl. This excess volume is due to the "non-equilibrium" chain conformations introduced by the loss of chain mobility upon quenching into the glassy state. These conformations relax over time scales much longer than typical experimental time scales and therefore are essentially frozen in.

The non-equilibrium structure of the glassy state plays a significant role in gas transport.

Generally the fractional free volume, FFV, is used to characterize the free volume.

Defined as the fraction of the total polymer specific volume which is not occupied by polymer molecules, fractional free volume can be estimated using group contribution method as in Equation (2-13):

(2-13)

where V is specific volume of polymer which can be measured experimentally [29, 30].

V0 is the occupied volume at 0 K per mole of repeat unit of polymer and can be estimated using Equation (2-14):

(2-14) where Vvdw is van der Waals volume and can be evaluated using group-contribution method [31]. Fractional free volume is a crucial property of glassy polymer and more discussion of its effect on the gas transport properties will be provided in later chapters.

Another significant parameter characterizing gas transport is selectivity. The ideal selectivity is defined as:

(2-15)

where PA and PB are the permeability of more permeable gas A and less permeable gas B, respectively. The selectivity is defined such that it is larger than 1. Combining

Equation (2-15) and Equation (2-9) gives:

(2-16)

where and are solubility selectivity and diffusivity selectivity, respectively.

Control of both solubility selectivity and diffusivity selectivity is critical to the development of high performance gas separation materials. Equation (2-16) suggests that polymeric membrane materials may be classified based on whether the permeability selectivity is controlled primarily by the solubility or diffusivity selectivity. The magnitude of the selectivity depends sensitively on the gas pair investigated and the range can span with several orders of magnitude.

2.2 Upper Bound Line for Gas Separation

It is well-known that high permeability and high selectivity are desirable to improve the economic viability of membrane gas separation processes. Unfortunately there is a tradeoff between permeability and selectivity. This “tradeoff” behavior of the transport parameters was not well understood until the concept was formalized by Robeson. From

Figure 1-3 the upper bound can move with time as efforts to refine polymer structure have improved material properties. Interestingly, the line has moved but the slope has remained virtually unchanged. A mathematical expression for the line corresponding to the upper bound is given by Equation (2-17):

(2-17)

where λAB and βAB are model parameters. Most of the permeability data are taken from the pure-gas permeation test and processed in the range of ambient temperature. Additionally most of the data are used under low pressure range. A log-log plot is used to illustrate the correlation between permeability and selectivity and graphically determine the line that represents the upper bound. λAB is the slope of this line and is correlated with the kinetic diameters of the gas molecules:

(2-18)

where dB and dA are kinetic diameters of larger gas molecule A and smaller gas molecule

B. The kinetic diameters of several commonly used gases are summarized by Robeson et al. [8]. Equation (2-18) shows strong relationship between the size of gas penetrants and the selectivity of glassy polymers. The position of the upper bound line can be estimated theoretically from the following equation for βAB [16]:

(2-19)

where SA and SB are solubility coefficients of more permeable gas A and less permeable gases B. a and b are from activation energy correlations of gas diffusivities (taken as constants for glassy polymers, a = 0.64 and b = 11.5) and f is an adjustable parameter related to the energy required to create a diffusive gap between polymer chains (f is set to

12600 cal/mol). The main drawback for this correlation is most of the model parameters are empirically determined and lack an explicit relationship to polymer structure and material properties.

2.3 Models and Theories of Gas Transport in Glassy Polymer Films

During decades of research on theoretical analysis on gas transport in polymers, large quantities of models and correlations are available for both gas sorption and diffusion.

Generally gas solubility depends on factors such as penetrant condensability (normal boiling point Tb, critical temperature Tc, or Lennard-Jones energy parameter ε/k ) [32, 33], polymer-penetrant interaction, and polymer morphology such as crystallinity and orientation [34]. Gas diffusivity depends more on the ability of gas molecules to “jump through” the polymer structure [35]. During this process the size of gas molecules, polymer morphology and segmental dynamics are significant factors. Since glassy polymers are more commonly used due to their superior permeation properties, the following model review focuses on glassy polymer films.

2.3.1 Models of Gas Sorption in Glassy Polymer Films

For the solubility of gases in glassy polymers, the most successful and accepted model is dual-mode sorption model (DMS) [36]. Two independent sorption mechanisms exist described by Henry’s law and Langmuir's law. The total sorption is expressed as the sum of the two contributions in Equation (2-20):

(2-20)

where kD and p are Henry’s constant and penetrant pressure introduced previously and and b are the Langmuir capacity constant and affinity constant, respectively. The

Langmuir contribution arises from sorption in the excess free volume in the glassy state and is dependent on the fractional free volume present. Due to the physical significance

of the model parameters the model has been used extensively for a wide variety of gas- polymer systems [37-41].

Nonetheless, a limitation of the model is that the three parameters must be evaluated from experimental sorption results. Additionally, the data from the gas sorption test cannot directly support the existence of the two sorption mechanism. Further studies are required using more fundamental approaches [42-46]. Other investigators have proposed the use of different parameters to characterize the excess free volume (order parameters) [47-48].

However, the order parameters in most of these models still must be determined from gas sorption measurements and fail to capture the dependence of solubility on sorption conditions over broad pressure ranges.

The non-equilibrium lattice fluid theory (NELF) is an alternative model that has been used to predict gas solubility in glassy polymeric materials. The theory is based on the lattice fluid theory first developed by Sanchez and Lacombe [49-51]. The expression for the mixture Gibbs free energy is obtained by using equilibrium thermodynamics. The lattice fluid theory was modified by Doghieri and Sarti [52-54] to predict gas solubility in glassy polymeric membranes for the non-equilibrium conditions of the glassy state. The density of the polymer is taken as the order parameter to characterize departure from equilibrium; a pseudo-equilibrium state is assumed to exist for which a pseudo- equilibrium chemical potential can be calculated. Equating the chemical potential for the gas phase to that for the gas-polymer mixture phase yields an expression for gas solubility. Baschetti et al. [55] used two correlations for the unknown polymer density to calculate swelling with gas sorption. Angelis et al. [56] developed an expression for the infinite dilution gas solubility which is the basis for the work described later to predict

the existence of a solubility upper bound. The NELF model provides very good predictions of gas solubility in a variety of polymers with only one order parameter to characterize departure from equilibrium: initial polymer density. Figure 2-4 shows CO2 sorption data in the polycarbonate (PC) membrane predicted by NELF model. An excellent fit exists between predictive results (solid line) and experimental ones (empty squares) [57].

Figure 2-4: NELF prediction of CO2 sorption in PC membrane.

2.3.2 Models of Gas Diffusion in Glassy Polymer Films

For theoretical analysis of gas diffusion there are mainly three types of models available:

Free volume models [58-64] correlates polymer free volume with gas diffusive jump. In general it is very difficult in evaluating all parameters in free volume theory equations.

However the theory indicates the distribution of free volume is constrained in the glassy state and thus gas diffusion occurs by molecules hopping from one free volume site to another.

The second model is the dual mode model. In the previous section the dual mode model for gas sorption (DMS) was introduced. The dual mode model also provides the following expression for gas diffusion coefficient D in glassy polymers:

(2-21)

where K=C’Hb/kD , F=DH/DD and DH and DD are the gas diffusivities for molecules sorbed in the Langmuir and Henry’s law states, respectively. From this model, gas diffusivity increases with increasing gas concentration or pressure which can result in a significant pressure dependence of gas transport properties in glassy polymer films. As for gas sorption, dual mode model diffusion parameters must be evaluated by fitting experimental data.

Finally the third type of model is molecular model and diffusivity is calculated by using the activation energy ED and pre-exponential factor D0 in Equation (2-22) [65]:

(2-22)

Both intermolecular and intramolecular contributions to the activation energy must be evaluated. The pre-exponential term D0 may be estimated using Barrer’s theory [66]. The energy calculated from the model often compares poorly to experimental measurements.

Several modifications have been proposed [67-72] to improve predictions that have led to good agreement with experimental measurements for a number of gas-polymer systems.

Significant differences still exist for some systems and the relationships with the polymer material properties that control sorption are not clear.

Relatively recently, a model based on transition state theory [73] was proposed by Gray-

Weals et al. [74]. The model was developed based on the computer simulation predictions by Greenfield et al. [75] with the intent to avoid the large scale molecular simulations required to calculate diffusivity from atomistic models. Assuming a “gas penetrant hopping” mechanism they calculated the rate coefficient which was a function of the energy required to create an opening for a diffusive hop and the partition function ratio between the transition and reactant states. The polymer was considered as a linear elastomer and its bulk properties were directly used for calculation. The model provides satisfactory predictions of diffusion coefficients for several different gases in PVC polymer [76]. However the model is not applied to other systems and the relationship of model parameters to sorption models is not clear.

2.4 Effect of Temperature on Gas Transport Properties

Membrane gas separation applications at temperature other than ambient are of increasing interest. Hence investigations of the temperature dependence of gas transport are necessary and attractive. For thermally activated processes such as diffusion, the temperature dependence of the diffusion coefficient is described by the Arrhenius equation, Equation (2-22). Gas diffusivity increases with temperature if the polymer does not degrade or undergo other structural changes. The diffusivity selectivity typically decreases as temperature increases for glassy polymer materials [77]. For gas sorption the

temperature dependence of gas solubility in polymers is typically described by the van’t

Hoff equation [78]:

(2-23)

where S0 is a pre-exponential constant and Hs is the enthalpy of sorption which can be further written as Equation (2-24):

(2-24)

Equation (2-24) indicates the sorption process of a gas penetrant into a polymer material involves two steps: condensation of the gas penetrant to a liquid-like state and mixing of the gas with the polymer. Finally the temperature dependence of gas permeability in glassy polymer films is written as a combination of Equation (2-22) and (2-23):

(2-25)

where P0 is exponential factor for permeation and is equal to S0D0. Ep is the activation energy of permeation and is equal to the sum of the activation energy of diffusion ED and enthalpy of sorption HS. Since for small gas molecules and strong size-sieving glassy polymer films diffusivity is more temperature sensitive than solubility, gas permeability commonly increases with temperature while permeability selectivity decreases. Hence the temperature can influence the location of the permeability upper bound line. Recently

Rowe et al. [79] explored the influence of temperature on the upper bound by using theoretical correlations and compared the predictions to experimental results. In this dissertation, the influence of temperature on the permeability upper bound also will be investigated using the newly developed NELF-TST model.

2.5 CFD Simulation of Gas Flow in Spacer-Filled Membrane Module

The earliest studies of spacers were conducted by Chilton et al. [80, 81, 82] and Sieder et al. [83] in 1930s. An excellent review of the large body of work that followed on different types of spacers is provided by Schwinge et al. [84]. Both permeate and feed spacers may be used to mix fluid within the channels to eliminate concentration polarization and increase mass transfer rates. Unfortunately, spacers also increase pressure drop within permeate and feed flow channels. This increase is strongly dependent on spacer geometry.

Computational fluid dynamics (CFD) has proven to be a very useful tool for investigation of the flow distribution within spacer-filled channels and numerous studies have been reported in the literature since the late 1990s. Cao et al. [85, 86] used CFD to examine spacers with filaments attached to the channel wall and the results agree with previous experimental measurements well. A computer program was developed by Varol et al. [87] to study the flow resistance and recirculation regions in a channel with the thin fins adjacent to the top wall. Geraldes et al. [88, 89] presented a numerical model to simulate the flow in a narrow slit channel with semi-permeable membrane at the bottom side and impermeable wall at the top. Flow permeation was taken into consideration and they found good agreement between simulation and experimental results. Their later work focused on the simulation of ladder-type spacers [90, 91, 92]. The flow pattern and concentration distribution in the feed channel were investigated and numerical predictions matched experimental data well. Additionally the effect of geometry of ladder-type filaments on the flow patterns and pressure drop was studied and critical

Reynolds number causing the unsteady flow was identified. Schwinge et al. [93, 94, 95]

implemented a CFD study for flow in narrow spacer-filled channels. The hydrodynamics and enhanced mass transfer rate were reported and the unsteady flow was also evaluated as a function of Reynolds number. Ahmad and Lau et al. [96, 97] focused on the study of unsteady hydrodynamics in the spiral wound membrane channel. Effect of the different spacer filaments within a certain range of Reynolds number was investigated and the conclusion was made the cylindrical spacer was the better option for use as a feed spacer.

The mesh length ratio described as the ratio of the distance between two neighboring spacer filaments to the channel height also was evaluated and optimized based on the influence of unsteady hydrodynamics on the development of the concentration polarization. The unsteady flow in the narrow spacer-filled channel also was studied numerically by Weihs et al. [98, 99]. The range of Reynolds number was determined for unsteady flow and enhanced mass transfer was caused by increased wall shear and inflow of lower concentration fluid. The spacer performance was also evaluated based on permeate processing cost analysis for spacer designs with multiple filament layers and different conclusions were drawn with Reynolds number above or below 200. Critical

Reynolds numbers were calculated by Alexiadis et al. [100] for the transition of laminar flow with varied equations. The different geometries were included and the good mixing conditions with no extra pressure drop were determined by critical Reynolds numbers.

All the CFD analysis mentioned above were implemented by two dimensional simulations and such simplification sometimes could not reflect the real cases of flow moving across the spacer-filled channel.

The three dimensional simulations were required and reported in the literature. Karode and Kumar [101] executed a 3D simulation to investigate the pressure loss and shear

stress by visualizing the flow distribution within spacer-filled channel. The pressure loss was due to the direction change of flow when it went through the plane of intersection of spacer filaments. Flow structure and dynamics were studied by Ranade et al. [102, 103] for spacer filled flat and curved channels. They developed an approach called “unit cell” and such method was used to evaluate the performance of different spacer shapes in two kinds of flow channels. No significant difference of flow behavior was shown between flat channel and spiral channel. Li et al. [104, 105] also evaluated a 3D unsteady flow and mass transfer properties were investigated. The non-woven spacers were simulated with different spacer geometric parameters and results were validated from experiment [106].

Saeed et al. [107] investigated the effect of feed spacer arrangement on the flow structure through a spacer filled reverse osmosis membrane. Flow visualization also was reported to illustrate the flow pattern and help guide development of novel RO membranes. The influence of spacer geometry on the flow pressure drop and mass transfer were studied by

Shakaib et al. [108, 109] by using finite volume package FLUENT. Factors such as filament spacing, filament thickness and flow attack angle were discussed and effect of

Reynolds number was included to investigate the unsteady flow region. Li et al. [110] executed a thorough study on the periodic boundary conditions (PBC) in CFD analysis and pointed out for asymmetric spacer system some PBC types reported in the literatures were improper. They also demonstrated that PBCs with certain cell numbers provided very accurate predictions of pressure drop in comparison to experimental data. The effect of geometry of spacer-filled channel on the performance of spiral wound membrane module also was investigated [111]. The biofouling and flow dynamics were systematically studied by Picioreanu and Vrouwenvelder et al. [112, 113]. The

performance of feed channel including flow pressure drop, flow distribution and liquid residence time was strongly influenced by the biomass accumulation. A comprehensive review of 3D CFD modeling works is provided by Weihs et al. [114].

Chapter 3

3. Model Prediction of Gas Sorption in Polymeric

Membranes

3.1 Introduction

The NELF theory assumes molecules of gas and polymer are arranged onto a lattice as illustrated in Figure 3-1. Vacant lattice sites (i.e. holes) also may exist. Each molecule may occupy more than one site and the portion of a molecule on a lattice is referred to as a “mer”. The NELF model parameters are: T*, the interaction energy between two mer segments of a molecule on adjacent lattice sites; p*, the molecular cohesive energy density in the close-packed state, and ρ* the molecular close-packed density. For pure components, these parameters are related to v*, the molecular closed packed volume and

ε*, the closed packed molecular energy, through Equations (3-1) and (3-2)

(3-1)

(3-2)

where r is the number of sites occupied by the molecule and M the molecular mass. The lattice fluid parameters may be determined for each component by fitting model

prediction to pressure-volume-temperature (PVT) data. A priori predictions for mixture are possible with use of an appropriate mixing rule as discussed later. Use of the NELF to predict a solubility upper bound is described in the next section. Predictions of the temperature dependence of the upper bound also are possible and compared to experimental data.

Figure 3-1: Non-equilibrium lattice fluid (NELF) theory.

3.2 Solubility Selectivity Prediction using NELF Model

The analysis here is limited to infinite dilution conditions to avoid the complexity of solubility changes arising from filling of excess free volume and swelling of the polymer matrix of higher gas concentrations. The permeability upper-bound was determined in the infinite dilution limit as well. The infinite dilution, zero pressure solubility limit is given by [56]:

(3-3)

where TSTP and pSTP are standard temperature and pressure 273.15 K and 1 atm respectively, T the operating temperature typically 308.15 K, Ψ an adjustable parameter

to account from deviations from non-ideal mixing, and the pure polymer density at the experimental temperature. The subscripts 1 and 2 denote values for the gas and polymer, respectively. The first two terms in the brackets in Equation (3-3) represent the entropic contribution to solubility and the third the energetic contribution.

The focus of this work is the large number of systems for which the ideal mixing assumption is adequate. For such systems, only pure component properties are required to predict mixture solubility. However, one could account for non-ideal systems by adjusting the binary interaction parameter based on heat of mixing data or to fit solubility data.

Values for the lattice fluid parameters have been estimated from PVT data for many pure gases and polymers. Assuming Ψ=1 (i.e. ideal mixtures) a priori predictions of gas solubility are possible if the pure polymer density is known. Values for these parameters used here are provided in Table 3.1.

Table 3.1: NELF model parameters for selected gases and polymers

* * * 3 0 T (K) p (MPa) ρ (g/cm ) ρ2 (35°C, Reference

g/cm3)

CO2 300 630 1.515 - [53]

CH4 215 250 0.500 - [53]

N2 145 160 0.943 - [53]

H2 46 37 0.078 - [53]

O2 170 280 1.29 - [56]

He 9.3 4 0.148 - [56]

Ar 190 180 1.400 - [56]

SR 560 354.6 1.200 1.100 [56]

PC 755 534 1.275 1.200 [53,115]

TMPC 761.6 446.4 1.174 1.083 [53,115]

HFPC 716.5 446 1.618 1.478 [55,115]

PSf 830 600 1.310 1.237 [56]

PPO 739 479 1.177 1.063 [56]

PMMA 695 560 1.270 1.176 [56]

For a given gas pair, solubility selectivity may be calculated from the solubilities of each individual gas. As with permeability selectivity, solubility selectivity is calculated as the ratio of the higher solubility species to the lower solubility species so it is greater or equal to unity. The literature indicates the NELF provides good predictions of gas solubility over broad pressure ranges [52-56]. Additionally the NELF provides good predictions of

the heat of sorption [116].

Comparisons of theoretical predictions of infinite dilution solubility and solubility selectivity to experimental measurements for several polymers are provided for CO2/CH4 and CH4/N2 in Figure 3-2 and Figure 3-3 respectively. The agreement between theory and experiment for the solubility of CO2 or CH4 is better than the agreement for selectivity.

Nonetheless, the a priori agreement for the broad range of polymers is good for both gas pairs.

Figure 3-2: NELF prediction of solubility selectivity for CO2/CH4 gas pair in different polymers: SR – Silicone Rubber [117], PMMA – Poly (methyl methacrylate) [118], PSf – Polysulfone [119], PC – Polycarbonate [120], HFPC –

Hexafluoropolycarbonate [115], TMPC – Tetramethylpolycarbonate [115], PPO –

Poly (phenylene oxide) [121].

Figure 3-3: NELF prediction of solubility selectivity for CH4/N2 gas pair in different polymers. The symbols represent experimental values for the same polymers as in

Figure 3-2.

3.3 Model Parameters Effect on Solubility-Solubility Selectivity

The dependence of solubility and solubility selectivity on the NELF parameters is examined by systematically varying each of the three parameters (ρ*, p* and T*) over a range of values that spans physically realistic extremes. The values for polycarbonate are taken as the base values from which the calculations are performed for the CO2/CH4 gas pair.

The range of model parameters considered extends beyond common values for polymeric materials. The range for ρ* of 1.205-2 g/cm3 is greater than that reported in the literature

for glassy polymers of 1.2-1.5 g/cm3. Similarly, the ranges for p* and T* (2500-7500 atm and 300-1800 K) are greater than those reported in the literature (3000-6000 atm and

500-850 K). These broader ranges allow assessment of how dramatic changes in polymer properties might affect solubility. Moreover, the ranges were selected such that solubility varied over comparable ranges.

Holding ρ* and p* fixed at the values for polycarbonate given in Table 3.1, the value of T* is varied from 300 to 1800 K to obtain the results presented in Figure 3-4. Values for the polymers are shown using the same symbols as in Figure 3-2. As T* is increased, the solubility of CO2 increases by more than four orders of magnitude while selectivity decreases by a factor of approximately three. The relationship is linear when plotted using logarithmic scales. The polymer mer-mer interaction apparently affects solubility to a much greater extent than solubility selectivity and its variation generates an upper bound-like curve.

* Figure 3-4: T dependence of CO2/CH4 solubility selectivity predicted by the NELF model for fixed p* (534 MPa) and * (1.275 g/cm3). The symbols represent experimental values for: filled diamond – PC [120], filled circle – PMMA [118], filled square – PSf [119]; open circle – SR [117], open diamond – HFPC [115], open triangle – TMPC [115], open square – PPO [121].

Examination of Equation (3-3) indicates that increasing the polymer T* increases the entropic contribution to solubility – the first two bracketed terms in Equation (3-3). This results from an increase in v* according to Equation (3-1) which reflects a reduction in the number of lattice sites occupied by a polymer molecule according to Equation (3-2).

The entropy of mixing of polymeric solutions is known to be smaller than that of lower molecular weight solutes due to chain connectivity so decreasing the number of effective segments increases the entropy change and solubility. Solubility selectivity decreases as the entropic contribution to solubility increases because entropic differences in solubility

are smaller than energetic differences. Thus, an upper bound like curve is generated by varying T*.

The effect of varying ρ* while holding T* and p* fixed is illustrated in Figure 3-5. Similar to the effect of T*, increasing ρ* from 1.205 to 2.000 g/cm3 leads to more than four orders of magnitude increase in CO2 solubility and a solubility selectivity decrease of less than a factor of three. The variation is nearly linear with the greatest non-linearity occurring for the highest values where both solubility selectivity and solubility decrease as ρ* increases.

The observed variation with ρ* is nearly coincident with the T* variation as the closed pack density affects solubility much more significantly than solubility selectivity.

* Figure 3-5: ρ dependence of CO2/CH4 solubility selectivity predicted by the NELF model for fixed p* (534 MPa) and T* (755 K). The dashed line represents the effect of

varying T* from Figure 3-4. The symbols represent experimental values for the same polymers as in Figure 3-4.

As for increasing T*, increasing * leads to a reduction in the number of lattice sites occupied by a polymer molecule according to Equation (3-2). Therefore, the same changes in the entropy of mixing, solubility, and solubility selectivity are expected so varying * generates an upper bound like curve similar to that generated by varying T*.

The differences are due to a reduction in the energetic contribution to solubility that accompanies increasing * especially for the highest values used.

Figure 3-6 illustrates the effect of varying p* from 2500 to 7500 atm on the relationship between solubility selectivity and solubility. For each value considered results also are shown for varying T* as the dashed line and ρ* as the solid line.

The cohesive energy density has a much greater impact on solubility selectivity. As the cohesive energy density increases by a factor of three, solubility selectivity can increase by more than an order of magnitude. The effect of varying T* and ρ* for each p* value is similar. Solubility selectivity decreases linearly with solubility as T* increases and the effect of ρ* is similar. These observations suggest that to increase solubility selectivity, changes to polymer primary and secondary structure that increase polymer cohesive

energy density are most desirable. Moreover, the maximum achievable value of cohesive energy density with polymeric materials establishes an upper bound on the solubility selectivity – solubility relationship. Movement along this upper bound is possible varying

* * T and ρ . Similar behaviors can be observed in Figure 3-7 for O2/N2 gas pair. Comparing to Figure 3-6 increase of solubility selectivity with cohesive energy density p* is smaller.

* Figure 3-7: p dependence of O2/N2 solubility selectivity predicted by the NELF model. The dashed line represents the effect of varying T* from 300 to 1800 K and the solid line represents the effect of varying ρ* from 1.205 to 2.000 g/cm3 at each of indicated p* values. The symbols represent experimental values for: filled diamond –

PC [120], filled circle – PMMA [122], filled square – PSf [123]; open diamond –

HFPC [124], open triangle – TMPC [120], open square – PPO [125].

Upward movement of the upper bound arises from an increase in the energetic

contribution to solubility. Increasing the polymer p* increases both solubility and solubility selectivity since greater differences in the energetic contribution to solubility exist between gas species. Increasing the polymer p* also leads to a reduction in the entropic contribution to solubility through the dependence of r and v* on p* given by

Equations (3-1) and (3-2). This further enhances the increase in solubility selectivity but reduces the increase in solubility that arises from enhanced energetic contributions. The overall effect is the observed upward movement of the upper bound.

3.4 Upper Bound Relationship

The linear relationship between solubility selectivity and solubility observed for T* variation and the nearly superimposed ρ* variation, for a given p*, suggest the solubility upper bound is given approximately by:

(3-4)

where is the solubility selectivity for gas pair A and B and SA is the solubility of the more soluble gas species A.

Analytical expressions for m and b are given by Equation (3-5) and (3-6), respectively.

Note that these expressions are derived by calculating from Equation (3-4) written for the two gas components. However, the resulting expression may be rearranged in

multiple ways to obtain relationship between and SA. The result given by Equations

* (3-5) and (3-6) is obtained by eliminating T2 from the expression for b. Alternatively, temperature T may be eliminated as discussed subsequently if the effect of temperature is to be eliminated from the constants m and b.

(3-5)

(3-6)

where is the ratio of pure polymer density to polymer close-packed density .

Equation (3-5) indicates m is a constant specific to a gas pair and is independent of polymer material properties. Equation (3-6) indicates b depends on gas pair material properties and the polymer and p*. The results of the previous section indicate the dependence on p* controls the location of the solubility upper bound.

The upper bound slope depends only on the close packed molar density (ρ*/M) ratio of gas pair. If this ratio is greater than one, the slope is negative as observed for the permeability selectivity dependence on permeability. However, if this ratio is less than one, the slope is positive as illustrated in Figure 3-8 for the CH4/N2 gas pair. Such a slope indicates changes in polymer architecture can increase both solubility and solubility selectivity contrary the results reported for permeability. Table 3.2 provides slopes for several common gas pairs.

Table 3.2: Slopes of solubility upper bound predicted by NELF model

Gas Pair m (NELF) - λA/B (Freeman) 1/n (Robeson)

O2/N2 -0.197 -0.107 -0.176

CO2/CH4 -0.102 -0.375 -0.379

CH4/N2 0.072 0.115 0.182

CO2/He 0.069 0.336 0.456

* Figure 3-8: p dependence of CH4/ N2 solubility selectivity predicted by the NELF model. The dashed line represents the effect of varying T* from 300 to 1800 K and the solid line represents the effect of varying ρ* from 1.205 to 2.000 g/cm3 at each of the indicated p* values. The symbols represent experimental values for the same polymers as in Figure 3-4.

Since a positive slope has not been observed for the permeability upper bound, any increase in solubility selectivity must be accompanied by a decrease in diffusivity selectivity of sufficient magnitude that the product of the solubility and diffusivity selectivities (i.e., the permeability selectivity) leads to the emergence of a permeability upper bound with a negative slope. Interestingly, polymers exist for some gas pairs which may have a permeability selectivity either greater than or less than one. For such gas pairs

(e.g., CH4/N2), if the upper bound relationship extends from permeability selectivity values greater than one to values less than one, the relationship suggests that increases in permeability can be made with an increase in permeability selectivity.

The potential for this behavior is illustrated by the upper bound relationship reported by

Robeson [3] for the N2/CH4 pair. The relationship is given by Equation (3-7)

(3-7)

and appears to extend for values of that are both greater than and less than one

since , this can be rewritten as

(3-8) which indicates that increases with for values of greater than one

(values of less than one) similar to the results presented here for solubility

selectivity. This type of behavior appears to exist only for gas pairs in which either species may be most permeable due to the relative contributions of solubility and diffusivity.

A comparison of the solubility selectivity slopes to the experimental permeability selectivity slopes [3] and the theoretical predictions of Freeman [7] based on gas

molecular diameter is provided. The slopes generally are of comparable magnitude. The differences reflect the dominant role that diffusivity selectivity often has on overall permeability.

3.5 Temperature Dependence of the Solubility Upper Bound

The ability of the NELF model to predict solubility and solubility selectivity suggests it may be used to investigate the effect of temperature on the upper bound location.

Assuming polymer density is weakly temperature dependent over the temperature range of interest, Equation (3-3) may be used to derive Equation (3-9) which gives the dependence of solubility selectivity on temperature either implicitly as a function of the temperature dependent solubility or explicitly through the difference in the heat of sorption for the two gas species.

(3-9)

The temperature independent constants mT and bT are given by Equations (3-10) and

(3-11), respectively.

(3-10)

(3-11)

Note that mT and bT are distinct from m and b given by Equation (3-5) and (3-6), respectively. As discussed previously, Equation (3-3) may be used to derive multiple

relationships between solubility selectivity and solubility. The relationship given by

Equations (3-9) – (3-11) is one in which the constants mT and bT are independent of temperature and allows calculation of temperature dependent selectivity directly from the temperature dependent solubility. Equation (3-4) – (3-6) also may be used but the temperature dependence of b must be included in the calculation. The value of mT depends solely on the properties of the gas pair. If mT is positive, temperature changes that increase solubility also increase solubility selectivity.

Expressions for the difference in heat of sorption of the gas pair, ∆HAB and α0 are given by

Equations (3-12) and (3-13), respectively.

(3-12)

(3-13)

Both constants are functions of gas and polymer properties.

The predictive capabilities of Equation (3-9) are illustrated in Figure 3-9 for CO2 and CH4 sorption in two polycarbonates for a temperature range from 308 K to 393 K [78]. Figure

3-9 indicates the NELF model captures the slope and location of the solubility selectivity curve reasonably well given the a priori nature of the prediction.

Figure 3-9: Temperature dependence of solubility selectivity for CO2/CH4 sorption in PC (diamond) and TMPC (triangle). The solid line represents NELF predictions and the solid symbol the prediction at 308 K. The dashed line represents values calculated from experimental correlations and the open symbol the experimental value at 308 K [78]. Both the solid and dashed lines were calculated for a temperature range of 308 to 393 K. Note that the experimental correlation does not pass through the data at 308 K due to differences between the best-fit correlation obtained from all of the temperature data and the specific data for 308 K.

Values for mT and ∆HAB are tabulated in Table 3.3 for several gas pairs. The agreement between the NELF model and experiment generally is good and consistent with the results presented in Figure 3-9.

Table 3.3: Predicted and experimental values [78] of mT and ΔHAB for PC and TMPC for different gas pairs.

Gas Pairs mT ΔHAB (cal/mol) Polymer materials NELF Experimental NELF Experimental

PC CO2/CH4 0.31 0.28 -2300 -1500

CH4/N2 0.26 0.03 -1400 -100

TMPC CO2/CH4 0.31 0.08 -2100 -400

CH4/N2 0.26 0.28 -1200 -1300

Given the observation in Figure 3-6 that the location of the solubility selectivity upper bound is determined by the maximum value of the cohesive energy density achievable with polymer materials, Equations (3-4) – (3-6) may be used to examine the effect of the temperature on the location of the upper bound.

Equation (3-5) indicates the slope of the upper bound is independent of temperature. The upper bound moves only due to the dependence of b on the temperature given by

* Equation (3-6). Figure 3-10 illustrates how the curve for p2 = 7500 atm in Figure 3-6 moves as the temperature is increased or decreased from 308 K for the CO2/CH4 gas pair.

Figure 3-10: Temperature dependence of solubility upper bound for CO2/CH4 predicted by the NELF model. The symbols represent experimental values for the same polymers as in Figure 3-4.

For a given CO2 solubility, the solubility selectivity increases by a factor of approximately 2 for a 35 K decrease in temperature. Equation (3-6) indicates the

* * solubility upper bound will move upward if pA > pB which is true if the more soluble species also possesses the higher cohesive energy density, i.e. more condensable.

3.6 Conclusions

The NELF model is used to predict solubility and solubility selectivity. A priori predictions for several gas pairs and polymers are in good agreement with experimental measurements.

Systematic variation of polymer material properties indicates an upper bound exists on solubility selectivity based on the maximum cohesive energy density (p*) achievable in polymeric materials. The upper bound moves upward due to an increase in the energetic contribution to gas solubility with increase p*. The slope and location of the upper bound are given approximately by considering the variation of solubility and solubility selectivity with polymer mer-mer interaction energy (T*).

The slope of the upper bound may be negative or positive depending on the molar density ratio of the gas pair. Such an observation contrasts with the observation of a negative slope for the dependence of the permeability selectivity upper bound on permeability for most system. This reflects the influence of diffusivity selectivity on permeability selectivity. However, for systems in which the permeability selectivity may be both greater than and less than one, solubility selectivity may become dominant for sufficiently high permeabilities. Experimental data for such systems is limited but consistent with a positive slope for the permeability upper bound as observed for the solubility upper bound.

The NELF model also provides reasonable a priori predictions of the temperature dependence of solubility selectivity on solubility. Thus the NELF model may be used to predict the movement of the solubility upper bound with temperature.

These results may be extended to multiple component (≥3) mixtures to examine competitive sorption effects. Additionally the results may be combined with a diffusion model to provide a priori predictions of the diffusivity and permeability upper bound discussed in the next chapters.

Chapter 4

4. Model Prediction of Gas Diffusion in Polymeric

Membranes

4.1 Introduction

A new model based on NELF and transition state theory (TST) is developed to predict the diffusivity of a gas penetrant in glassy polymers. From the transition state theory, initially the gas penetrant molecule resides in the reactant state as shown in Figure 4-1:

Figure 4-1: “Hopping” mechanism of gas diffusion in glassy polymer film using transition state theory.

A gas molecule tries to diffuse through the polymer structure by jumping from one cavity, the reactant state, to another, the product state. This requires creation of a free volume region of sufficient size for the gas molecule to move through by expansion of the surrounding polymer molecules. The energy required for this process to occur and the resulting rate constant is calculated as follows based on transition state theory:

(4-1)

where ∆G is the change of Gibbs free energy between reactant state and transition state.

In the current model, the change of Gibbs free energy is taken as the energy required to form the transition neck:

(4-2)

Equation (4-2) includes two contributions to the energy change. One is the change of energy required to deform the polymer chain to create enough space for the gas molecule to move. This energy change is defined as ∆Edef and can be expressed by Equation (4-3) :

(4-3)

where is interaction energy between polymer molecule sites, Nlattice is the number of

polymer molecule lattice sites deformed by gas penetrant, is the cohesive energy density and dV represents the expanding volume of transition state region when gas molecule goes through. Furthermore, FFV is the fractional free volume which was introduced in literature review section. The FFV can be expressed from NELF model:

(4-4)

where is the density of polymer at the experimental conditions and is the density of

polymer in the close-packed state. As proposed in the original transition state theory, the transition state neck is taken as a cylinder of volume:

(4-5)

where dkinetic is kinetic diameter of gas penetrant and Ljump is jump length. Another change of energy comes from the change of polymer-penetrant interaction energy, , which is calculated by polymer-penetrant mer-mer interaction energy using NELF theory. In our model we assume there are at least two sites of polymer-penetrant mer-mer interaction are lost from reactant state to transition state, thus the expression is shown as:

(4-6)

Till now based on Equation (4-3) and (4-6) the change of Gibbs free energy is obtained and rate constant is determined from Equation (4-1). Finally the diffusion coefficient of gas penetrant in glassy polymer films D is calculated using:

(4-7) where Ljump is the jump length of gas penetrant hopping from one reactant cavity to another product cavity. From transition state theory the jump length is defined as the sum of diameter of reactant cavity and Lennard-Jones diameter of polymer side group (the center-to-center between two cavities). Here we set the radius of reactant cavity as 2.5Ǻ suggested as the original theory. The main side group of all the glassy polymers used in this project except polysulfone is methyl group with Lennard-Jones diameter 3.56Ǻ. For polysulfone the main side group used is double oxygen group with Lennard-Jones diameter 2.18Ǻ.

The NELF model parameters of gas penetrants and glassy polymers are listed in Table

3.1. From Equation (4-5) the kinetic diameter of gas penetrant is also required. As mentioned previously numerous values are available for molecular diameters. Values proposed for kinetic diameter have been tabulated by Robeson [8] and are reproduced in

Table 4.1. The modified kinetic diameters proposed by Robeson are used here because they have been obtained from past comparisons of permeability upper bound predictions to experimental data. Additionally since Argon is not included in the list, the previously reported value of 3.4Å is used.

Table 4.1: Kinetic diameters of selected gas penetrants.

Gas Breck diameter L-J collision Dal-Cin Robeson’s (Å) diameter (Å) diameter (Å) analysis (Å) He 2.6 2.556 2.555 2.644 H2 2.89 2.928 2.854 2.875 CO2 3.3 4.07 3.427 3.325 O2 3.46 3.46 3.374 3.347 N2 3.64 3.71 3.588 3.568 CH4 3.8 3.817 3.882 3.817

The fractional free volume (FFV) plays a key role in gas transport properties in glassy polymer films. For gas sorption FFV can strongly influence gas solubility, and for gas diffusion FFV can reduce the energy required to deform the polymer chain structure to enable a diffusive hop. NELF theory indicates the polymer density is taken as an order parameter, which means is not only dependent on the operating temperature and pressure but also on the history of the glassy polymer sample. At low to moderate

pressures, the density of pure and unpenetrated polymer is used. The density and FFV of the glassy polymers studied here are provided in Table 4.2.

Table 4.2: Unpenetrated density and FFV of several glassy polymers [56, 115].

3 3 Polymer (g/cm ) (g/cm ) FFV PC 1.275 1.200 0.059 TMPC 1.174 1.083 0.077 HFPC 1.618 1.478 0.087 PSf 1.310 1.237 0.056 PPO 1.177 1.063 0.097 PMMA 1.270 1.188 0.065

4.2 Diffusivity and Diffusivity Selectivity Prediction using NELF-TST Model

Figure 4-2 illustrates the predicted gas diffusion coefficient for seven commonly used gas penetrants in polycarbonate film. Experimental data for Argon and hydrogen were obtained at 25 °C [126] while data for all other gases were obtained at 35 °C [120]. A log- log plot is used to present the predicted and experimental diffusivity coefficients. From the plot the new model shows very good agreement between predicted and experimental values with an average logarithmic error of only 2.1%. Generally the smaller size of gas penetrant the higher diffusivity coefficient it possesses since a lower energy barrier exists to pass through the transition state. This is true for most of the penetrants in Figure 4-2.

Figure 4-2: NELF-TST prediction of gas diffusivity in Polycarbonate film. The symbols represent data for: solid square – CH4, solid diamond – Ar, solid triangle –

O2, solid circle – CO2; open square – N2, open diamond – H2, open triangle – He.

The predictions of gas diffusivity coefficients for poly (methyl methacrylate) (PMMA) are shown in Figure 4-3. Experimental data are obtained by Min et al. [122] at 35ºC.

Significant deviations between experiment and theory exists and the average logarithmic

* error is nearly 15%. However, adjusting the cohesive energy density p2 in a least squares fit of predicted to experimental values yields a value of 740 MPa and reduces the average logarithmic error to almost 2.5% as illustrated in Figure 4-3. This suggests the evaluation of NELF model parameters from PVT data for PMMA may be in error. Nonetheless, the

NELF-TST model can provide good estimates of gas diffusivity in glassy polymer films.

Figure 4-3: NELF-TST prediction of gas diffusivity in PMMA. The symbols represent values for: square – CH4, diamond – N2, triangle – CO2, circle – Ar, star –

* O2. The empty data represent prediction using original p2 = 560 MPa and the solid

* data represent prediction using modified p2 = 740 MPa.

Predictions of diffusivity and diffusivity selectivity for several polymers are shown for

CO2/CH4 and O2/N2 in Figure 4-4 and Figure 4-5, respectively. Experimental data are in fair agreement with theoretical predictions. The average logarithmic errors are within

10%. The primary main reason for the deviations in the predictions especially for TMPC and HFPC is the lack of reliable experimental or simulated results for jump length of the gas penetrant as well as PVT data for accurate determination of NELF parameters. In summary the error of prediction of gas diffusivity-selectivity is more sensitive than that

for gas solubility-selectivity because of the much wider range of gas diffusivity values in strongly size-sieving glassy polymer films of interest.

Figure 4-4: NELF-TST prediction of diffusivity-selectivity for the CO2/CH4 gas pair in different polymers: PSf – Polysulfone [123], PC – Polycarbonate [120], TMPC –

Tetramethylpolycarbonate [124], PPO – Poly (phenylene oxide) [127], HFPC –

Hexafluoropolycarbonate [124].

Figure 4-5: NELF-TST prediction of diffusivity-selectivity for the O2/N2 gas pair in different polymers: PSf – Polysulfone [123], PC – Polycarbonate [120], TMPC –

Tetramethylpolycarbonate [124], PPO – Poly (phenylene oxide) [127], HFPC –

Hexafluoropolycarbonate [124].

4.3 Model Parameters Effect on Diffusivity-Diffusivity Selectivity

As done for solubility in the previous chapter, the NELF model parameters, ρ*, p* and T*, are varied systematically over a range of values to determine their effect on diffusivity and diffusivity selectivity. The values for polycarbonate are used as the base values and the calculations are performed for the CO2/CH4 and O2/N2 gas pair. A broad range of model parameters are investigated to evaluate the effect of polymer material properties on the diffusivity and diffusivity selectivity. Figure 4-6 and Figure 4-7 clearly show how the diffusivity upper bound depends on gas and polymer material properties through NELF

model parameters in a manner similar to that observed for the solubility upper bound.

However, in contrast to the solubility upper bound illustrated in Figure 3-6 and Figure

3-7, the polymer mer interaction energy T* controls the location of the diffusivity upper bound instead of p*. The solid line illustrates the dependence of diffusivity and diffusivity selectivity on the cohesive energy p*. As p* is increased from 267 MPa to 801 MPa, the diffusivity of CO2 in Figure 4-6 and O2 in Figure 4-7 decreases by almost four orders of magnitude but diffusivity selectivity increases only by about one order of magnitude for

* CO2 and a factor of 3 for O2 . Thus the cohesive energy density p impacts the diffusivity much greater than diffusivity selectivity and the variation creates an upper bound curve indicating a linear logarithmic relationship between diffusivity and diffusivity selectivity.

Increasing cohesive energy density p* increases the energy barrier for gas molecules to jump from one cavity to another based on transition state theory and this leads to a decrease in gas diffusivity. The reduction is greater for larger gas species so the diffusivity selectivity increases as the polymer possesses greater size-sieving capability.

* Figure 4-6: T dependence of CO2/CH4 diffusivity selectivity predicted by the NELF-

TST model from 377.5 K to 1132.5 K. The dashed line represents the effect of varying ρ* from 1.20 g/cm3 to 2.04 g/cm3 and the solid line represents the effect of varying p* from 267 MPa to 801 MPa at each of indicated T* values. The symbols represent experimental values of CO2/CH4 in different polymers: open square - PSf

[123], open diamond - PC [120], open triangle - TMPC [124], solid diamond - HFPC

[124], solid square - PPO [127].

* Figure 4-7: T dependence of O2/N2 diffusivity selectivity predicted by the NELF-

[123], open diamond - PC [120], open triangle - TMPC [124], solid diamond - HFPC

[124], solid square - PPO [125].

Figure 4-6 and Figure 4-7 also indicates the effect of varying ρ* as illustrated by the

* 3 3 dashed line. As ρ increases from 1.20 g/cm to 2.04 g/cm both CO2 and O2 diffusivity increase more than one order of magnitude and the diffusivity selectivity decreases as a factor of 3 for CO2/CH4 and 2 for O2/N2 gas pair. The variation is similar to the effect of p* and both of the model parameters impact the gas diffusivity much more than the

diffusivity selectivity. From Equation (4-4) increasing the density of polymer in the close-packed state increases the fractional free volume (FFV) of the polymer; this reduces the polymer deformation required for a gas molecule to execute a diffusive hop which increases diffusivity. Meanwhile the polymer structure becomes less size-sieving with increased fractional free volume and this leads to a decrease of gas diffusivity selectivity.

The position of the gas diffusivity upper bound is controlled by the polymer-mer interaction energy T*. From Figure 4-6 and Figure 4-7 the diffusivity upper bound moves upward as T* decreases from 1132.5 K to 377.5 K. Decreasing the polymer-mer interaction energy decreases the energy barrier to creation of an opening in the polymer structure that permits a diffusive hop from one free volume site to another and thereby increases gas diffusivity. The gas diffusivity selectivity also slightly increases with decreasing T*. Interestingly, the effect of cohesive energy density p* on the solubility selectivity shown in Figure 3-6 is much greater than the influence of T* on diffusivity selectivity for CO2/CH4 gas pair. This illustrates the challenge of modifying glassy polymer materials to achieve better separation performance.

4.4 Upper Bound Relationship

The linear logarithmic relationship between diffusivity and diffusivity selectivity established by varying p* for a specific T* (and followed closely by the results for ρ* variation) suggests an upper bound exists of the form:

(4-8)

where is diffusivity selectivity for gas pair A and B and DA is the diffusivity of the faster diffusing species A. The location of the upper bound is dictated by the highest value of T* that can be achieved through modification of polymer structure.

From Equation (4-1) and (4-2) there are two contributions to the energy change: the deformation energy and interaction energy expressed as Equation (4-3) and Equation

(4-6), respectively. In contrast Equation (3-3) used to evaluate gas solubility in which the two terms are comparable (i.e., the entropic contribution was comparable to the energetic contribution), the deformation energy contribution to gas diffusivity is much greater than the interaction energy contribution. Thus, the interaction energy term in Equation (4-2) can be neglected and the slope x of the gas diffusivity upper bound is given approximately by:

(4-9)

Table 4.3 provides the slopes calculated for the gas pairs considered in this work. Table

4.3 also contains values estimated from the graphs of diffusivity selectivity versus diffusivity. The two values are very close confirming the simplification that leads to

Equation (4-9) and provides an analytic expression for the slope of the upper bound.

Generally the molecular size of gas species B is larger than that of gas species A, thus the slope is always a negative value and this generates the diffusivity upper bound. The relationship between the diffusivity upper bound slope and the properties of gas penetrants is also consistent with the conclusions made by previous work [7]. The position of the diffusivity upper bound y depends on several parameters for both the gas penetrants and polymer material. A simple analytic expression for y does not exist.

Table 4.3: Slopes of diffusivity upper bound predicted by NELF-TST model.

2 Gas Pair x 1- (dB/dA)

CO2/CH4 -0.281 -0.318

N2/CH4 -0.159 -0.144

O2/N2 -0.125 -0.136

4.5 Temperature Dependence of the Diffusivity Upper Bound

The new NELF-TST model can be used to investigate the effect of temperature on the relationship between diffusivity selectivity and diffusivity as well as the location of the diffusivity upper bound. Over the range of temperatures considered here the polymer density is weakly dependent on temperature and its variation is neglected.

The relationship between temperature and gas diffusivity is expressed by the Arrhenius

Equation in (2-22) and the effect of temperature on gas diffusivity selectivity is given by:

(4-10)

Figure 4-8 illustrates the variation of CO2 and CH4 diffusivity in two polycarbonates materials for a temperature range from 308 K to 393 K [78]. The predictions using the

NELF-TST model are in fair agreement with experimental data.

Values for xT and are summarized in Table 4.4. Generally the results are consistent

with the ones shown in Figure 4-8. Deviation exists for evaluation in TMPC which may be due in part to values for NELF model parameters and the constant jump length specified for a diffusive hop.

Figure 4-8: Temperature dependence of diffusivity selectivity for CO2/CH4 diffusion in PC (diamond) and TMPC (triangle). The solid line represents NELF-TST predictions and the solid symbol the prediction at 308 K. The dashed line represents values calculated from experimental correlations and the open symbol the experimental value calculated at 308 K [78]. Both the solid and dashed lines were calculated for a temperature range of 308 to 393 K.

Table 4.4: Predicted and experimental values [78] of xT and for PC and TMPC for

CO2/CH4.

xT (cal/mol) Gas Pairs Polymer NELF- NELF- materials TST Experimental TST Experimental

PC CO2/CH4 -0.19 -0.20 -1600 -1700

TMPC CO2/CH4 -0.17 -0.37 -1300 -2500

Similar to the effect of temperature on the solubility upper bound, temperature does not affect the slope of the diffusivity upper bound; only the intercept y will change in a logarithmic plot. Figure 4-9 shows how the curve in Figure 4-6 for T* = 377.5 K moves as the temperature is varied from 273 K to 343 K for CO2/CH4 gas pair. From Figure 4-9, diffusivity selectivity increases slightly with temperature. The effect of temperature on diffusivity selectivity is much less pronounced than for solubility selectivity as shown in

Figure 3-10. Moreover, the diffusivity upper bound moves upward as temperature increases in contrast to the downward movement that occurs in the solubility upper bound.

Figure 4-9: Temperature dependence of diffusivity upper bound for CO2/CH4 predicted by the NELF-TST model. The symbols represent experimental values for the same polymers as in Figure 4-6.

4.6 Conclusions

The new NELF-TST model is used to predict gas diffusivity. The model provides excellent predictions for several gas penetrants in polycarbonate (PC). Greater deviation exists for PMMA mainly due to the evaluation of NELF model parameters from PVT data. The model also provides good predictions of gas diffusivity selectivity in several polymers. The main reason for the deviations is the lack of reliable experimental or simulated results for jump length or the PVT data used in the model.

The variation of model parameters indicates an upper bound exists for the diffusivity selectivity related to the lowest polymer-mer interaction energy (T*) achievable in polymeric materials. Variation of the polymer cohesive energy p* forms the upper bound with a linear logarithmic relationship between diffusivity and diffusivity selectivity. The slope of the diffusivity upper bound only depends on the kinetic diameter of the gas pair.

The location of the diffusivity upper bound depends on NELF-TST material properties of both the gas penetrants and polymer material and is controlled primarily by the polymer- mer interaction energy.

The temperature dependence of diffusivity and diffusivity selectivity also is predicted by the NELF-TST model. The calculated results are in fair agreement with experimental measurements. Deviations are due primarily to poor evaluation of model parameters as previously discussed.

The movement of the diffusivity upper bound with temperature is investigated. The upper bound moves upward slightly with increasing temperature. This is opposite to the effect of temperature on the solubility upper bound.

Chapter 5

5. Model Prediction of Gas Permeation in Polymeric

Membranes

5.1 Permeability and Permeability Selectivity Prediction using NELF-TST

Model

Predictions of the NELF model for gas solubility and NELF-TST model for gas diffusivity presented in the previous two chapters are used to calculate gas permeability from the solution-diffusion model expressed by Equation (2-9). Gas permeability selectivity is calculated using Equation (2-15) where A represents the more permeable gas species and B the less permeable gas species. Figure 5-1 illustrates the predicted gas permeability for several gas penetrants in polycarbonate (PC). Experimental data for all the gas species are taken at 35 °C [120]. Experimental gas permeabilities for CO2, CH4 and N2 are evaluated in the infinite dilution limit using dual mode model parameters for sorption and diffusion. Experimental data for O2 were obtained at 1 atm and He 20 atm.

Figure 5-1 shows the NELF-TST model provides an excellent fit between predicted and experimental values. Deviations may exist due to the accumulated error from predictions in gas solubility and diffusivity coefficients.

Figure 5-1: NELF-TST prediction of gas permeability in polycarbonate film. The symbols represent data for: solid triangle – N2, solid square – CH4, solid diamond –

CO2; open square – O2, open diamond – He.

Predictions of permeability and permeability selectivity of CO2/CH4 gas pair and O2/N2 gas pair are illustrated in Figure 5-2 and Figure 5-3 for several polymers. Predictions provide a fair agreement with experimental data for CO2/CH4 gas pair. Deviations mainly come from predictions of gas diffusivity because of the broad range of diffusivity and diffusivity selectivity values for CO2/CH4 gas pair in strongly size-sieving glassy polymer materials. For O2/N2 gas pair shown in Figure 5-3 theoretical predictions for PC,

PSf and PPO are in very good agreement with experimental measurements. Deviations exist in predictions of TMPC and HFPC mainly due to the errors from gas diffusivity

prediction. This indicates reliable experimental or theoretical value for jump length may be required. Also as mentioned before prediction of gas diffusivity selectivity is more sensitive than that for gas solubility selectivity because the gas diffusivities vary over a wide range of several orders of magnitude.

Figure 5-2: NELF-TST prediction of permeability-selectivity for the CO2/CH4 gas pair in different polymers: PSf – Polysulfone [123], PC – Polycarbonate [120],

TMPC – Tetramethylpolycarbonate [120], PPO – Poly (phenylene oxide) [127],

HFPC – Hexafluoropolycarbonate [124].

Figure 5-3: NELF-TST prediction of permeability-selectivity for the O2/N2 gas pair in different polymers: PSf – Polysulfone [123], PC – Polycarbonate [120], TMPC –

Tetramethylpolycarbonate [120], PPO – Poly (phenylene oxide) [128], HFPC –

Hexafluoropolycarbonate [124].

5.2 Model Parameters Effect on Permeability-Permeability Selectivity

The effect of model parameters on permeability and permeability selectivity are investigated by varying ρ*, p* and T* over a range of values. Figure 5-4 and Figure 5-5 illustrate the effect of polymer properties on the permeability and permeability selectivity for the CO2/CH4 and O2/N2 gas pair, respectively. The values for polycarbonate are used as the basis and a broad range of model parameters are investigated. Note that the relationship between permeability and permeability selectivity does not possess a linear

logarithmic relationship as observed for solubility and diffusivity. That is due to the different impact of model parameters on gas solubility and diffusivity. Movement along the upper bound toward higher selectivities occurs as p* increases or ρ* decreases; since the slope of the line generated is different for these materials properties, the upper bound is produced by simultaneously varying the two properties in a way to maximize selectivity.

Figure 5-4 and Figure 5-5 indicates the effect of model parameters on gas permeability and permeability selectivity is more similar to the influence on gas diffusivity and diffusivity selectivity shown in previous chapter. This indicates that the permeability upper bound is primarily controlled by diffusivity not solubility. The movement of the permeability upper bound with polymer material properties is limited which highlights the challenges associated with improvement of gas separation performance through polymer modification.

* Figure 5-4: T dependence of CO2/CH4 permeability selectivity predicted by the

NELF-TST model from 400 K to 1200 K. The dashed line represents permeability selectivity decreases as ρ* increases from 1.24 g/cm3 to 1.60 g/cm3 and the solid line represents permeability selectivity increases as p* increases from 267 MPa to 640

MPa at each of indicated T* values. The symbols represent experimental values of

CO2/CH4 in different polymers: open square - PSf [123], open diamond - PC [120], open triangle - TMPC [120], solid diamond - HFPC [124], solid square - PPO [127].

* Figure 5-5: T dependence of O2/N2 permeability selectivity predicted by the NELF-

TST model from 400 K to 1200 K. The dashed line represents permeability selectivity decreases as ρ* increases from 1.20 g/cm3 to 2.04 g/cm3 and the solid line represents permeability selectivity increases as p* increases from 267 MPa to 801

MPa at each of indicated T* values. The symbols represent experimental values of

O2/N2 in different polymers: open square - PSf [123], open diamond - PC [120], open triangle - TMPC [120], solid diamond - HFPC [124], solid square - PPO [128].

5.3 Temperature Dependence of the Permeability Upper Bound

The effect of temperature on the permeability and permeability selectivity is studied using NELF-TST model. Assuming the polymer density is nearly constant over the temperature interval examined, Figure 5-6 illustrates the variation of O2 and N2

permeability in two polycarbonate materials for a temperature range from 308 K to 393 K

[78]. Generally there is a good fit between theoretical values and experimental measurements for PC in both gas permeability and permeability selectivity. Some deviation exists in the prediction of gas permeability for TMPC and this is mainly due to the accumulated errors from predictions of gas diffusivity and solubility coefficients. The relationship between gas permeability and permeability selectivity can be expressed as:

(5-1)

Table 5.1 summarizes values of rT and of O2 and N2 in PC and TMPC. Generally the NELF-TST model provides an excellent prediction and results calculated are in good agreement with experimental measurements.

Figure 5-6: Temperature dependence of permeability selectivity for O2/N2 permeation in PC (diamond) and TMPC (triangle). The solid line represents NELF-

TST predictions and the solid symbol the prediction at 308 K. The dashed line represents values calculated from experimental correlations and the open symbol the experimental value calculated at 308 K [78]. Both the solid and dashed lines were calculated for a temperature range of 308 to 393 K.

Table 5.1: Predicted and experimental values [78] of rT and for PC and TMPC for

CO2/CH4.

rT (cal/mol) Gas Pairs Polymer NELF- NELF- materials TST Experimental TST Experimental

PC O2/N2 -0.346 -0.200 -1015 -1000

TMPC O2/N2 -0.353 -0.429 -857 -1200

The movement of permeability upper bound with temperature is investigated using the

NELF-TST model. Figure 5-7 illustrates how the permeability upper bound for CO2/CH4 moves as the temperature increases from 273 K to 343 K; the curve was generated by setting T* = 755 K and varying p* to calculate selectivity for a given permeability. For the permeability upper bound the curve is not a straight line in the logarithmic scale due to the different dependencies of solubility and diffusivity on polymer properties. The permeability upper bound moves upward as the temperature decreases from 343 K to 273

K such that the selectivity increases by a factor of ~2 for a 35 K decrease in temperature at a fixed permeability.

Figure 5-7: Temperature dependence of permeability upper bound for CO2/CH4 predicted by the NELF-TST model. The symbols represent experimental values of

CO2/CH4 in different polymers: open square - PSf [123], open diamond - PC [120], open triangle - TMPC [120], solid diamond - HFPC [124], solid square - PPO [128].

5.4 Conclusions

The NELF-TST model is used to predict gas permeability. The model provides good agreement between calculated permeability values and experimental measurements for several gas penetrants in PC. The prediction also is good for gas permeability selectivity in different polymer materials. Deviations exist mainly due to accumulated errors from predictions of gas solubility and diffusivity. For glassy polymer materials prediction of gas diffusivity selectivity is more sensitive than that for solubility selectivity.

The dependence of permeability and permeability selectivity on the model parameters is studied. For permeability upper bound the relationship between permeability and permeability selectivity is not linear logarithmically mainly because different model parameters control the properties of the solubility and diffusivity upper bounds.

The permeability upper bound is controlled primarily by the diffusivity upper bound for most glassy polymeric materials. Movement of the upper bound is limited. The relationship between permeability and permeability selectivity along the upper bound is complex and does not lend itself to a simple analytical expression. However, in the limits of either solubility or diffusivity control, the slope of the upper bound is expected to be a function only of the kinetic diameter ratio of the penetrants.

The temperature dependence of permeability and permeability selectivity also is investigated using NELF-TST model. The results are in fair agreement with experimental measurements. The movement of the permeability upper bound with temperature is evaluated using the new model. The permeability upper bound moves upward with decreased temperature investigated.

Chapter 6

6. CFD Simulation of Gas Flow in Spacer-Filled

Membrane Module

6.1 Introduction

Initial work was focused on simulations of spacer A being considered as a spacer candidate in the membrane module. The spacer is illustrated in Figure 6-1. The spacer A consists of two overlapping circular filaments. The filaments possess a diameter of 10 mils and are overlapped at an angle of 60 relative to each other. Inter-filament spacing is

15 filaments per inch thus the distance between filaments can be determined. The filaments are allowed to overlap each other by 1 mil to create a finite contact area and overlap 1 mil with the upper and lower bounding surfaces to create a flow channel with a height of 17 mils. The filaments overlap ~5% in the geometry used for the simulations to mimic actual commercial spacers and facilitate creation of a mesh suitable for simulation.

o 60 Flow

y x

Figure 6-1: Top view of spacer A. The flow direction is indicated by the arrow and the repeat unit lies within the dashed box.

As illustrated in Figure 6-1, a single spacer unit cell, which is a representative of the spacer structure, was chosen to perform the CFD study. The unit cell is enlarged in

Figure 6-2. In the unit cell domain, the governing equations of steady-state fluid flow are shown below:

(6-1)

(6-2) where ρ is the density of fluid, u is the velocity vector, p is pressure, μ is the viscosity of the fluid and F is the volume force vector.

Because of periodicity of the geometry, periodic boundaries can be used in the simulation. The planes that bound the repeat cell in the flow direction (y-z plane) and normal to the flow direction (the lateral or x-z plane) are two sets of periodic boundaries along which the velocity is identical as shown in Figure 6-2. The pressure is also identical along the lateral (x-z) planes. However, the pressure along the outlet plane differs from

that along the inlet plane by a specified value which drives the flow. The flow rate is determined by the value specified for the pressure difference. The domain in Figure 6-2 was meshed to solve for the velocity and pressure fields. A typical meshing is illustrated in Figure 6-3. This mesh contains approximately one million elements. Domain discretization was controlled to ensure results varied by less than 5% due to mesh size.

Figure 6-3: Meshed solution domain for the Spacer A repeat unit.

Fluid properties of nitrogen at 35C were used in the simulations. The fluid boundary conditions specified were:

1. Periodic velocity boundary: inlet and outlet flow planes

2. Periodic velocity boundary: lateral planes

3. Pressure (1 atm) specified along outlet plane

4. Pressure (1 atm plus the specified pressure drop) specified at one node along inlet plane

5. No slip (i.e., zero velocity) along spacer filament surface

6. No slip along upper and lower bounding surfaces

Laminar flow was assumed to perform the simulations because it matches the low pressure flow in the module. The creeping flow solution was calculated first and used as

the initial guess for the laminar flow solution. Since the creeping flow problem only requires solution of a set of linear equations, it is obtained easily and provides a good initial guess for solving non-linear laminar flow equations.

The simulations were performed in COMSOL Multiphysics. COMSOL is a computational fluid dynamics package which provides numerical solution to the governing conservation of mass and momentum equations by transforming the partial differential equations into a set of non-linear algebraic equations using finite element algorithm.

6.2 CFD Method Validation

Velocity and pressure fields of the spacer A are illustrated in Figure 6-4 and Figure 6-5, respectively. Figure 6-4 indicates that the velocity magnitude is lowest (dark blue color) adjacent to the upper and lower bounding surfaces of the flow channel as well as along the spacer surface where the no-slip boundary condition is applied. As the fluid approaches the point where the two filaments cross, the fluid accelerates as it moves around the filaments and the velocity is highest (dark red color). Within each of the illustrated cross-sections, significant variations in the magnitude of the velocity exist as the velocity increases with distance from the filament, reaches a maximum and then decreases in the middle of the flow channel. This is most apparent in the cross-sections before and after the filament cross where large lower velocity regions, yellow in color, are evident.

Figure 6-4: Velocity field within cross-sections normal to the flow direction for spacer A at Reynolds number ~150. Velocity increases as the color changes from blue to red (Unit: m/s).

The surface pressure field in Figure 6-5 indicates a nearly step-wise pressure drop occurs as the fluid passes below or above a filament. Along the top surface the pressure remains relatively constant in-between filaments (yellow in color) but decreases rapidly upon passing the upstream (from red to yellow) and downstream (from yellow to blue) filaments. This indicates the form drag is more significant than viscous drag in the flow channel.

Figure 6-5: Surface pressure field of spacer A at Reynolds number ~150. Pressure increases as the color changes from blue to red (Unit: psi).

The pressure drop of spacer-filled channel at different flow rate is expressed as the dependence of friction factor on the Reynolds number. The Reynolds number is generally given by Equation (6-3):

(6-3)

where  is the density of the fluid, v is the bulk velocity in the feed channel calculated as flow rate divided by the total cross-sectional area,  is the viscosity of the fluid and dh is the hydraulic diameter. Generally for a spacer-filled channel dh is determined by Equation

(6-4):

(6-4)

hsp is spacer thickness which is the height of spacer-filled channel,  is voidage and is determined by Equation (6-5):

(6-5)

Vsp is the occupied volume of spacer filament and Vtot is the total volume of channel. Svsp is the specific surface of spacer and can be calculated by Equation (6-6):

(6-6)

where df is the diameter of spacer filament. The friction factor f is calculated from

Equation (6-7):

(6-7)

where P is pressure difference between inlet and outlet of the channel and Lc is the channel length.

The computed frication factor as a function of Reynolds number is compared to experimental measurements in Figure 6-6. The experimental data were provided by colleagues at Membrane Technology and Research, Inc. (MTR). In MTR's lab, the pressure drops across small samples (4" by 6") of a spacer were experimentally measured using a pressure drop test system. The pressure drop is converted to friction factor by normalizing with the length of the flow channel and superficial bulk velocity before plotting as a function of the Reynolds number. As shown in Figure 6-6, predictions are in

good agreement with experiment results. Theoretically and experimentally, friction factor is a logarithmic linear function of Reynolds number over the low Reynolds number range and the data are curved at higher Reynolds number. The slopes of theoretical and experimental curves differ by only ~3% and interceptions differ by ~8% at lower

Reynolds number below ~100 and magnitudes differ ~6% at higher Reynolds number

~290. Generally experimental values are higher than the predicted values and the differences are attributed tentatively to pressure drops through the entrance and exit manifolds in the experimental apparatus which were not measured. The results in Figure

6-6 suggest the simulation procedure adequately captures the physics of flow through the spacer filled channel. With this validation, the simulation was used to evaluate the pressure drop characteristics of other spacer geometries.

Figure 6-6: Comparison of the simulated and experimental f versus Re of Spacer A.

Diamond indicates the experimental data and square indicates the simulation data.

Mesh dependency of the results also was determined. Figure 6-7 illustrates the effect of increasing meshing on the calculated relationship between friction factor and Reynolds number. The greatest differences between the results are observed at the highest Reynolds numbers and the solution becomes nearly mesh independent when the number of elements is greater than ~ 400,000. This level of mesh refinement is used for all calculations.

Figure 6-7: Dependence of mesh elements on f versus Re of spacer A. Diamond indicates ~100,000 mesh elements, square indicates ~430,000 mesh elements, triangle indicates ~1,000,000 mesh elements and circle indicates ~1,800,000 mesh elements.

6.3 Effect of spacer geometries on the module performances

Parametric studies of the net-type spacer were conducted using the repeat unit of a non- woven spacer illustrated in Figure 6-8. The spacer is symmetrical and characterized by the filament diameter (df), distance between filaments (L), angle between filaments (), and angle between the flow direction and the filament (). The distance between filaments is related to the number of filaments per unit length (N) by Equation (6-8):

(6-8)

Simulations were performed for flow through a series of model spacers to evaluate pressure drop. The boundary conditions of simulation model and material properties are the same as that applied to simulation of spacer A. Table 6.1 lists the dimensions of all of the spacers investigated. The effects of spacer design variables on flow and pressure drop are discussed in subsequent sections.

Figure 6-8: The schematic of a repeat unit of the non-woven spacer.

Table 6.1: Summary of spacer configurations.

3 3 Spacer number df10 (m) L10 (m)  (deg)  (deg)

1 0.254 1.814 60 30

2 0.381 1.814 60 30

3 0.381 1.814 30 15

4 0.381 2.309 30 15

5 0.381 2.309 60 30

6 0.381 2.309 90 0

7 0.381 2.309 90 45

8 0.381 2.309 120 60

9 0.381 2.309 150 75

6.3.1 Effect of Filament Diameter (df)

Spacers 1 and 2 are used to investigate the effect of filament diameter on spacer performance. Note that increasing filament diameter necessitates an increase in the flow channel height as the flow channel height is approximately equal to twice the filament diameter. Typical simulation results for the velocity field at comparable Reynolds number range for both spacers are presented in Figure 6-9. The arrow indicates the velocity direction and the color reflects the magnitude of velocity. As shown in Figure

6-9, it is evident larger variation in the velocity exists for spacer 1 than spacer 2. Along the flow direction spacer 2 possesses better flow distribution than spacer 1 and velocity varies more uniformly when fluid moves across the spacer filaments. Also higher maximum velocity possessed by spacer 1 filled channel indicates filaments of spacer 1

have larger drag force than those of spacer 2. The flow drag was mainly due to the form drag from the spacer and kinetic loss due to directional change [129].

(a)

(b)

Figure 6-9: Velocity field at the plane of intersection of spacer filaments for (a).

Spacer 1 (at Re ~ 175) and (b). Spacer 2 (at Re ~ 180). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right.

The predicted dependence of friction factor on Reynolds number is illustrated in Figure

6-10. Spacer 2, which possesses the larger filaments, gives the lower flow resistance over the Reynolds number range simulated. The ratio of the two friction factor values increases with Reynolds number and becomes greater than 1.5 for Reynolds number in excess of ~270. This observation is expected as the pressure drop depends strongly on channel height .For an empty channel, the viscous resistance depends on the square of the channel height. The decrease in pressure drop with increasing filament diameter or

channel height is at the expense of lower membrane packing density and larger module sizes.

Figure 6-10: Calculated f versus Re for spacers simulated as varied filament diameter. Diamond indicates data of spacer 1 and square indicates data of spacer 2.

6.3.2 Effect of Distance between Filaments (L)

Spacer 3 and 4 are used to evaluate the influence of distance between filaments (or equivalently filaments per inch) on flow and pressure drop. Spacer 3 has smaller inter- filament spacing while the filament diameter and angle between filaments are identical for the two spacers. Figure 6-11 illustrates the simulated velocity field of spacer 3 and 4.

With increased distance between filaments spacer 4 provides wider openings at inlet and

outlet side respectively, as shown in Figure 6-11 (b). Additionally the open space within the channel filled with spacer 4 is larger compared to that within spacer 3. The larger open space helps to form a better flow distribution and decreases the form drag on the surface of spacer filaments when flow moves across.

(a)

100

(b)

Figure 6-11: Velocity field at the plane of intersection of spacer filaments for (a).

Spacer 3 (at Re ~ 285) and (b). Spacer 4 (at Re ~ 280). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right.

Figure 6-12 illustrates the dependence of pressure drop on fluid flow rate expressed as friction factor as a function of Reynolds number for spacers 3 and 4. For most of

Reynolds number range investigated, the resistances for the two spacers are nearly equal.

For Reynolds number between ~435 and ~500, the friction factor for spacer 4 (larger inter-filament distance) is slightly lower than that for spacer 3. A decrease in flow resistance with increasing distance between filaments is expected as less viscous and form drag due to flow around the filaments occurs. Still the influence of spacer inter-

101

filament distance on the velocity field and pressure drop is much weaker over the range considered.

6.3.3 Effect of Spacer Orientation ()

Spacers 6 and 7 are used to investigate the effect of spacer orientation by using the angle between spacer filament and flow direction (i.e., flow attack angle). Spacer 6 possesses

0 flow attack angle which corresponds to flow parallel to one pair of spacer filaments and normal to the other as shown in Figure 6-13 (a) while spacer 7 possesses 45° flow

102

attack angle as shown in Figure 6-13 (b). Rotating spacer 6 45 leads to the orientation of spacer 7.

Figure 6-13 illustrates the velocity is much more uniform with an attack angle of 0 compared to 45. The magnitude of the velocity is indicated by the length and color of the velocity vector while the direction is indicated by the orientation of the vector.

Velocity changes along the x direction before and after the transverse filaments are much smaller for an attack angle of 0 which should lead to lower form drag due to changes in velocity head.

(a)

103

(b)

Figure 6-13: Velocity field at the plane of intersection of spacer filaments for: (a).

Spacer 6 (at Re ~ 315) and (b). Spacer 7 (at Re ~ 310). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from the lower left to the upper right.

This expectation is confirmed in Figure 6-14 which illustrates the calculated friction factor as a function of Reynolds number for spacers 6 and 7. For low Reynolds number, the two spacers provide comparable pressure drop indicating viscous drag is similar.

However, for Reynolds number larger than ~140, the difference of friction factor between two spacers gradually increases and at Reynolds number ~350 the friction factor of spacer 6 is ~50% lower than that of spacer 7. Additionally for spacer 6 the friction factor is logarithmic linear with Reynolds number over the entire Re range indicating a viscous

104

laminar flow while the friction factor of spacer 7 reaches a plateau with increase of

Reynolds consistent with form drag dominated pressure drop.

Figure 6-14: Calculated f versus Re for spacers with filament angle 90 simulated as varied flow attack angle. Diamond indicates the data of spacer 6 and square indicates the data of spacer 7.

6.3.4 Effect of Angle between Spacer Filaments ()

Spacers 4, 5, 7, 8 and 9 are used to evaluate the dependence of flow and pressure drop on spacer filament angle. These five spacers possess angles that range from 30 to 150; all other parameters are identical. Typical velocity fields for spacers 8 and 9 (120 and 150, respectively) are illustrated in Figure 6-15. It clearly illustrates low velocity regions exist

105

in front of and behind the point where the two filaments touch in the center of the flow channel. The enlargements also suggest that recirculation regions are present at Reynolds number ~80 for spacer 8 and ~60 for spacer 9 in which the fluid flows back towards the inlet along a closed streamline. These recirculation regions occupy a portion of the cross- sectional area for flow and thereby force the flow through a smaller region – this increases velocities over that which would exist in the absence of recirculation. The higher velocities increase the contribution of form drag to the total flow resistance.

(a)

106

(b)

(c)

107

Figure 6-15: Top view of velocity field at the plane of intersection of spacer filaments: (a). Spacer 5 (at Re ~ 120), (b). Spacer 8 (at Re ~ 80) and (c). Spacer 9 (at

Re ~ 60). Velocity increases as the color changes from blue to red (Unit: m/s). Flow is from left to right for Spacer 5 and from bottom to top for Spacer 8 and 9.

Additional small figures are added to zoom in some region inside the channel to review the flow recirculation.

As the angle increases, the number of filaments the fluid must cross per unit length along the flow direction (x or y direction) also increases. For example, in Figure 6-15 the fluid passes two filaments for both spacers but the distance over which this occurs for the 150° spacer (spacer 9) is approximately one-half of that for the 120° spacer (spacer 8).

Therefore, the pressure drop is expected to increase with flow angle. This expectation is confirmed in Figure 6-16 which illustrates the calculated dependence of friction factor on

Reynolds number. The pressure drop increases dramatically with angle. The friction factor of the 150 spacer can be an order of magnitude larger than that of the 30 spacer.

That indicates for increased spacer filament angle the flow approaches that of flow past parallel, close packed, staggered cylinders requiring more intensive changes in flow direction and associated flow inertia. These results are consistent with the findings of Da

Costa et al. [129] in their study of spacer design for ultrafiltration. Additionally the large flow resistance possessed by spacer 8 and spacer 9 also attributes to the flow recirculation observed within the flow channel.

108

Figure 6-16: Calculated f versus Re for spacers with varied spacer filament angle.

Diamond indicates the data of spacer 4 (30), Square indicates the data of spacer 5

(60), Triangle indicates the data of spacer 7 (90), Solid circle indicates the data of spacer 8 (120) and empty circle indicates the data of spacer 9 (150). Dash lines indicate f versus Re using creeping flow model from spacer 4 (30) (bottom dashed line) to spacer 9 (150) (top dashed line).

The total flow resistance in Figure 6-16 contains contributions from both viscous and form drag. To separate the relative contributions of both, simulations were performed for low Reynolds number (i.e., Stokes or creeping) flow. In such case the inertial force is much smaller than the viscous force. Based on that Equation (6-2) can be simplified with inertia term neglected:

109

(6-9)

The calculated resistance to flow from viscous drag alone expressed as friction factor versus Reynolds number is illustrated as dashed lines in Figure 6-16. It indicates that as filament angle increases the viscous flow resistance increases. This is attributed to the increased number of filaments per unit length in the flow direction that the fluid encounters. The results from Figure 6-16 indicate that the contribution of viscous drag to the total resistance decreases with increased angle between spacer filaments. If the dashed line is extended the friction factor data points of spacer 4 and 5 are quite close to the lines indicating the viscous resistance dominates the total flow resistance. For spacer

8 and 9 the data points have larger deviation above the extending line showing a strong domination of form drag resistance.

6.3.5 Effect of number of spacer repeat units

The results presented in the previous sections were obtained for a section of the spacer corresponding to one repeat unit or unit cell. The entire spacer is generated by creating multiple copies of the unit cell and translating them along the x and y directions. The use of a single unit cell unit to represent the entire spacer requires use of periodic boundary conditions. That assumes no features of the velocity field are larger than the unit cell, e.g. a recirculation region that extends over longer distances. This greatly reduces the work required to perform the simulations.

To determine if this assumption is valid, simulations were performed with geometries that contain multiple unit cells. Figure 6-17 illustrates the portion of the spacer used to create the larger simulation geometry. The dashed region encloses four of the unit cells

110

used in all previous work. This geometry extends two unit cells in length and width (x and y directions) and is referred to as a 2x2 configuration. The geometry used previously corresponds to a 1x1 configuration.

Figure 6-17: Identify multiple spacer repeat units. The arrow A indicates the flow direction of spacer 4 and the B indicates the flow direction of spacer 8.

Results obtained for the 2x2 geometry for spacers 4 and 8 (filament angles of 30° and

120°, respectively) are illustrated in Figure 6-18. Stark differences in flow uniformity exist for these two spacers. The velocity field for the smaller filament angle spacer

(Spacer 4) is much more uniform than for the larger filament angle spacer (Spacer 8).

Large changes in velocity are evident for spacer 8 in regions near the filaments which will generate significant form drag.

The insets in Figure 6-18 (b) indicate that significant recirculation regions exist between filaments for spacer 8. These recirculation regions do not extend in length more than one unit cell and are nearly identical to those found with the 1x1 geometry. This suggests the

111

1x1 geometry adequately captures the features of the velocity field and the additional work required to perform the 2x2 simulations is not necessary.

(a)

112

(b)

Figure 6-18: Velocity field in the x-y plane along the center of the flow channel for

2×2 repeat units: (a). Spacer 4 (at Re ~ 160) and (b). Spacer 8 (at Re ~ 110). Velocity

(m/s) increases as the color changes from blue to red. Flow is from lower left to upper right in (a) and from lower right to upper left in (b). Inset figures correspond to enlargements of indicated regions.

Figure 6-19 illustrates the calculated friction factor versus Reynolds number for Spacers

4 and 8 using the 1x1 and 2x2 geometries. The calculated resistance for the 2x2 geometry is nearly identical to that for the 1x1 geometry. The 2x2 resistances are slightly higher but the difference is less than ~15% for spacer 4 and ~10% for spacer 8. These differences are comparable to the accuracy of the simulation. This provides further evidence for the validity of using a single unit cell for the simulations.

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1×1 repeat unit, and empty square indicates data of spacer 8 with 2×2 repeat unit.

6.4 Membrane Module Performance of Designed Spacers for CO2 Capture

Application

Carbon dioxide (CO2) emissions from coal-fired power plants may lead to significant global climate change if not controlled. CO2 capture and sequestration (CCS) is one of many solutions being considered to mitigate emissions.

Membrane based processes potentially are a cost-effective option for CO2 capture. The flow resistance of the spacers studied here was used to predict the pressure drop in the

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3 membrane module which was used to separate 1 m (STP)/s of gas containing 10% CO2.

Simulations were performed by the colleague at MTR. This separation is approximately the volume of flue gas produced by a 1 MWe coal-fed power plant. The amount of the membrane area to do the separation and the pressure drop were calculated. Spacer 1 and spacer 5 were used in the sweep side in two simulations while all the other parameters of the module and the operating conditions are kept at the same values. The results are listed in Table 6.2. The pressure drop is about two times lower when spacer 5 is used than spacer 1 while the separation performance is slightly better. The results indicate how spacer design can affect overall module dimensions and performance.

Table 6.2: Separation performance and pressure drop of membrane module using Spacer

1 and Spacer 5 as the sweep spacer.

Sweep side pressure Module Membrane area need (m2) drop (psi)

Module with spacer 1 in the 1856 3.71 sweep side

Module with spacer 5 in the 1750 1.31 sweep side

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6.5 Study of Flow Distribution and Pressure Drop in Asymmetric Spacers

Design of asymmetric spacers also is studied. Figure 6-20 illustrates the geometry of asymmetric spacer used. In contrast to a symmetric spacer, the asymmetric spacer consists of two groups of spacer strands: (1). large strands located at the bottom of a channel, oriented parallel to the gas flow, with a diameter of 0.017 inch and aligned in the flow direction at 7 strands per inch and (2). small strands located at the top of a channel, oriented diagonally to the gas flow, with a diameter of 0.014 inch and a pitch of 10 strands per inch.

Figure 6-20: Asymmetric spacer-filled flow channel used to perform the simulations.

Figure 6-21 illustrates a top view of a spacer filled channel that utilizes an angle of 45 between the larger and smaller strands. The feed enters from the left and flows parallel to the larger strands towards the right. The square channel is 1 inch wide, 1 inch long, and

0.03 inch high. The channel encloses 7 large spacer filaments and 13 smaller spacer filament. Such channel size is sufficiently large to eliminate the effects of the lateral solid

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boundaries on the calculated velocity field and pressure drop. The boundary conditions used to simulate the flow are summarized as:

(1). Laminar inflow with specific average velocity along inlet

(2). Laminar outflow with fixed pressure along outlet

(3). No slip along lateral surfaces (top and bottom in Figure 6-21)

(4). No slip along spacer filament surfaces

(5). No slip along upper and lower bounding surfaces

Figure 6-21: Top view of a spacer-filled channel that utilizes an asymmetric spacer with an angle of 45 between the larger strands on the bottom and smaller strands on the top.

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The performance of the asymmetric spacer is compared to that of the symmetric spacer illustrated in Figure 6-22. The channel size (width, length and height) is identical to that used for the asymmetric spacer. The detailed geometry of both spacers is summarized in

Table 6.3.

Figure 6-22: Geometry of membrane module channel filled by symmetric spacer.

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Dimensions of spacer Asymmetrical spacer Symmetrical spacer strands

Large strand (inch) 0.017 0.015 Small strand (inch) 0.013

Spacer thickness (inch) 0.03 0.03

Number of filament per Large strand – 7 inch 9 Small strand – 10

Spacer filament angle 45 90 (degree)

Flow attack angle 0/45 45 (degree)

Figure 6-23 illustrates the velocity fields within x-y planes near the top, bottom and middle of the channel for the asymmetric spacer. The results are obtained for an intermediate flow rate with Re ~240. The region shown is near the center of the flow channel where the effects of the lateral and inlet/outlet boundaries are negligible. Near the top of the flow channel, Figure 6-23 (a) indicates the flow is directed from the lower left side to the upper right side along the length of smaller filaments. In contrast, Figure

6-23 (b) indicates the flow is nearly unidirectional in the nominal flow direction, from left to right, as it moves parallel to the larger filaments near the bottom of the flow

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channel. Figure 6-23 (c) indicates the flow in the middle is similar to that near the bottom, as the larger filaments appear to exert greater control on the flow than the smaller filaments. Note that the velocity magnitude decreases as gas in the middle plane passes beneath the upper spacer filaments due to the drag exerted by these filaments.

Additionally, the highest velocities are observed in the bottom plane where the flow must accelerate as the upper filaments force gas down towards the bottom while the gas is moving from inlet to outlet.

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(a) (b)

(c)

Velocity fields within x-y planes near the top, bottom. and middle of the flow channel for the symmetric spacer are illustrated in Figure 6-24 for the same flow rate as used to obtain the results in Figure 6-23 for the asymmetric spacer. The velocity fields on the top

[Figure 6-24 (a)] and bottom [Figure 6-24 (b)] are symmetrical due to the geometry of the

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spacer. Both figures show the flow is parallel to the filament that is adjacent to the boundary: lower left to upper right in Figure 6-24 (a) and upper left to lower right in

Figure 6-24 (b). Flow near the central x-y plane, illustrated in Figure 6-24 (c), is nearly unidirectional in the nominal flow direction, left to right. Near the filaments, the velocity decreases and the direction is predominantly normal to the filaments that gas flow passes under or over; the direction is determined by the filaments that lie transverse to the filament being passed over or under because the gas flow between the transverse filaments is dominant in these regions.

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(a) (b)

(c)

Figure 6-24: Velocity fields in x-y planes near the center of the flow channel for a symmetric spacer with Re ~240. The images correspond to planes near the (a) top,

(b) bottom, and (c) middle of the flow channel. Velocity (m/s) increases as the color changes from blue to red.

Figure 6-25 illustrates the calculated friction factor as a function of Reynolds number for both asymmetric and symmetric spacers. Flow channels with the asymmetric spacer possess less flow resistance than ones filled with the symmetric spacer. The flow

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resistance for the symmetric spacer is a factor of 2 greater at the lowest flow rate (Re

~50) and 2.5 at the highest flow rate (Re ~510).

Figure 6-25: Calculated f versus Re for asymmetric and symmetric spacers: diamond - asymmetric spacer; square - symmetric spacer.

The higher resistance of the symmetric spacer arises primarily from gas recirculation and associated momentum losses. The velocity fields within the y-z planes shown in Figure

6-26 are used to illustrate the differences between the two spacers. Figure 6-27 shows the calculated velocity fields for Re ~240.

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(a) (b)

Figure 6-26: Transverse y-z plane (marked in red) examined within flow channels filled by the (a) asymmetric and (b) symmetric spacers.

Figure 6-27 (a) illustrates flow within the asymmetric spacer filled channel and indicates the flow is primarily normal to the plane (in the nominal flow direction) in the central part of the plane as the vectors appear as points in this region. The magnitude of the velocity vector is indicated by the color while the y-z components are indicated by the vectors shown in the figure. The x component is not shown but can be inferred from the total magnitude and the y-z components. The magnitude of the velocity decreases near the upper and lower bounding surfaces and a small velocity component within the plane is observed, especially in the upper part of the channel where the spacer forces flow from left to right.

Figure 6-27 (b) indicates flow within the symmetric spacer filled channel. Much larger velocities within the transverse plane are evident and represent transverse vortices or recirculation regions; the dashed lines indicate these regions. The loss in momentum due

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to the large changes in flow direction that occur within the transverse plane are responsible for the higher flow resistance of the symmetric spacer.

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(a)

(b)

Figure 6-27: Velocity fields in y-z planes near the central part of flow channel filled by (a) asymmetric and (b) symmetric spacer for Re ~240. Dashed arrows indicate transverse vortices.

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Figure 6-29 illustrates the magnitude of x-component of the velocity (the component in the nominal flow direction) along a line normal to the upper and lower boundaries at the points indicated in Figure 6-28. The selected points correspond to points in the center of the open regions within the spacers.

(a) (b)

Figure 6-28: Points selected for examining the variation of the x-component of velocity along the height of the flow channel for the (a) asymmetric and (b) symmetric spacer.

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(a)

(b)

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(c)

Figure 6-29: Variation of the x-component of velocity with distance from the bottom of the flow channel for inlet Reynolds number of (a) Re ~50, (b) Re ~240, and (c) Re

~510. The reddish-orange curve corresponds to the asymmetric spacer and the blue curve to the symmetric spacer.

The velocity profile for the symmetric spacer (blue curve) is symmetrically distributed along the channel height and passes through two maxima and one minimum. The minimum occurs at a height (the center of the flow channel) corresponding to the location where the top and bottom filaments contact each other. The gas that flows within this plane is dead-ended at the points where the filaments touch, which reduces the velocity throughout the plane. As flow rate increases, the difference between the maximum and

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minimum velocities increases. The presence of multiple maxima reflects the greater viscous and form drag of the symmetric spacer relative to the asymmetric spacer.

The velocity profile for the asymmetric spacer (red curve) is not symmetrical. The velocity passes through a single maximum, which occurs at a point in the lower half of the flow channel. The observed differences in velocity variation are consistent with the lower flow resistance of the asymmetric spacer.

6.6 Mass Transfer Simulation in Multi-layer Spacer Configurations

Mass transfer simulations are performed for channels filled with multiple layers of spacers. Such configurations have been proposed to provide greater membrane support in the presence of a trans-membrane pressure driving force but help reduce parasitic pressure drop. A concern of such a configuration is the potential for enhanced concentration polarization in the channel, especially in the spacer next to the membrane.

Mass transfer simulations were performed to determine if concentration polarization might occur.

Figure 6-30 illustrates the triple-layer spacer configuration used in the simulations. A thicker, more open spacer (spacer A) is sandwiched between identical thinner, less open spacers (spacers B). The B spacers possess a smaller inter-filament spacing to help provide mechanical support to the membrane under pressure while spacer A possesses a higher spacing to create a more open flow channel that minimizes pressure drop. Such a configuration may help reduce pressure drop but concerns exist over whether this configuration will introduce detrimental concentration polarization. The simulations

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reported here were performed to determine the effect of filament spacing in spacer B on concentration polarization and module performance.

Figure 6-30: Configuration of a triple-layer spacer design.

Two-dimensional simulations were performed initially to reduce simulation complexity and required computational resources. The simplified two-dimensional spacer configuration is illustrated in Figure 6-31. Similar configurations were used in initial studies of flow through spacer filled channels.

Relative to the thinner and denser spacer B adjacent to the membrane surface, the influence of the thicker and more open spacer A on concentration polarization is anticipated to be small. This leads to a further simplification of the spacer configuration by removing the middle layer spacer A. If concentration polarization is not observed for this configuration, it is not expected with a spacer present as the middle spacer will help force flow through the upper and lower spacers which will reduce concentration polarization. The configuration is symmetrical and only half of the geometry is needed for the simulation as a symmetry boundary condition can be applied along the top boundary of the dashed box in Figure 6-31.

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Figure 6-31: Simplified configuration of the triple-layer spacer shown in Figure

6-30. The dashed boundary encloses the symmetric domain used for simulation.

The geometry used for simulation is illustrated in Figure 6-32. The simulation is performed using COMSOL's fluid flow model coupled with the mass transfer model. The fluid flow model is expressed as the governing equations of steady-state fluid flow using

Equation (6-1) and (6-2). The mass transfer model is expressed as the mass balance equation of convection and diffusion for an individual species i at steady state:

(6-10) where  is the density of gas flow, i is the mass fraction of gas species i, u is the velocity vector and ji is the diffusive mass flux vector of gas species i.

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Figure 6-32: A part of channel geometry used for simulation. Circles represent the thinner and denser spacer B adjacent to the membrane surface with distance L between the neighboring filaments.

As shown in Figure 6-32 the density of spacer B can be adjusted by varying the inter- filament distance L. The 2D simulations can be conducted for channels 0.5 m long and

0.5 m wide; such large domains could not be used in 3D simulations due to computational resource limitations. The spacer possesses a thickness 0.02 inch with variable filament density.

The simulation addresses mass transfer in the high pressure retentate channel for a mixture of CO2/N2 with a 10% CO2 (mole fraction) feed; similar simulations could be performed for the permeate channel. The flow rate of the feed gas was varied over a broad range to evaluate the module performance. The membrane CO2 and N2 permeances were set equal to 1000 GPU and N2 25 GPU, respectively, to provide a CO2/N2 selectivity equal to 40; such values are representative of the high performance membranes being considered for carbon dioxide capture. The permeate side was specified to be a vacuum (0 atm) and the process is assumed to operate isothermally at a temperature of 35 C. The boundary conditions used for fluid flow model are summarized as:

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(1). Laminar inflow with specific average velocity along inlet

(2). Laminar outflow with fixed pressure along outlet

(3). Symmetrical boundary condition at top surface

(4). Permeation velocity (leaking wall boundary condition) at bottom surface

(5). No slip along spacer filament surfaces

The boundary conditions used for mass transfer model are summarized as:

(1). Inflow with specific concentration of gas flow along inlet

(2). Outflow with dominated convection along outlet

(3). Symmetrical boundary condition at top surface

(4). Mass flux of gas species at bottom surface

(5). No flux along spacer filament surfaces

Key parameters used to evaluate the performance of gas separation membrane module are illustrated in Figure 6-33. F, P, and R are gas molar flow rates for the feed, permeate and retentate side, respectively. z, y and x are mole fractions of the most permeable species

(CO2) for the feed, permeate and retentate side, respectively. The recovery is used as a performance figure of merit and is defined as the ratio of permeate to feed molar flow rate (P/F) when the permeate stream is the desired product. The recovery is taken as the ratio of retentate to feed molar flow rate (R/F) when the retentate stream is the desired product.

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Figure 6-33: Parameters used for gas membrane module performance.

Figure 6-34 illustrates the calculated performance curves for the CO2 permeate product.

The permeate recovery (P/F) is plotted as a function of CO2 mole fraction (yCO2) for the permeate product stream. Five cases are illustrated. The ideal case corresponds to the calculation assuming no concentration polarization exists. For the other four cases a resistance ratio (RR) is used to represent the spacer density. The resistance ratio is defined as the ratio of the flow resistance for the spacer filled channel to that for an empty channel. The flow resistance is defined as a ratio of pressure drop within the channel to the inlet flow rate. The empty channel is used to normalize the resistance so

RR=1 corresponds to absence of a spacer.

Generally for gas separation processes, concentration polarization does exist for empty channels because gas diffusivities are large relative to membrane permeances. This is confirmed in Figure 6-34 as module performance for an empty channel is virtually identical to that calculated assuming no concentration polarization. As previously mentioned the spacer density can be adjusted by the filament distance L shown in Figure

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6-32. The spacer becomes denser as L decreases and this corresponds to an increase of total flow resistance as well as RR value. Figure 6-34 shows no effect of concentration polarization on performance for resistance ratios up to ~10. This indicates that for the conditions examined the use of the spacer sandwich configuration is not expected to introduce significant concentration polarization.

Figure 6-34: Module performance curve at permeate side. RR indicates Resistance

Ratio between spacer filled channel and empty channel.

6.7 Conclusions

The velocity field, flow resistance, and pressure drop within spacer-filled channels were studied by using three dimensional computational fluid dynamics (CFD) simulations. The

137

simulation was run for spacer A to valid our CFD methods. The velocity and pressure field are visualized and pressure drop expressed as friction factor versus Reynolds number is calculated to compare with experimental data. The flow velocity is lowest adjacent to upper and lower bounding and highest when it moves across the spacer filaments. A step-wise pressure drop is observed showing dominant form drag than viscous drag. Simulation results are in good agreement with experimental measurements for spacer A.

With validation of the CFD method, a parametric study of the spacer geometry was performed. The parameters include filament diameter, distance between filaments, angle between filaments, and angle between flow direction and filament. Among the parameters, the distance between filaments has the smallest effect on spacer flow resistance within the Reynolds number range investigated. Friction factor increases dramatically with decreased filament diameter and increased angle between filaments.

Flow distribution is more uniform for larger filaments and associated channel heights.

This uniformity comes at the expense of lower membrane packing density and increased module size. Velocity changes before and after transverse filaments are smaller for a 0° flow attack angle than a 45° flow attack angle indicating lower flow resistance and pressure drop especially at higher Reynolds number.

For spacers with small filament angle very uniform flow distribution is observed. For spacers with large filament angle (120 and 150) recirculation regions exist within the channel which force fluid to flow through smaller regions with increased velocity. The higher velocities lead to an increased contribution of form drag to total flow resistance.

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Pressure drop increases with spacer filament angle over the range of Reynolds number investigated. This is attributed to an increase in the number of filaments fluid must pass per unit length which increases flow resistance. The resistance of the 150 filament angle spacer is almost an order of magnitude larger than that of the 30 angle spacer. The high resistance of spacers with large filament angle also is attributed to the presence of recirculation regions which increase total flow resistance.

The contribution of viscous drag and form drag to total flow resistance is investigated by using simulations with creeping flow model. Viscous resistance also increases with spacer filament angle. The viscous resistance has a significant contribution to the total resistance for smaller filament angle spacers especially at small Reynolds number while form drag resistance dominates for larger filament angle spacers.

Properties of flow resistance of the calculated spacers are further used by the colleague at

MTR to predict the pressure drop in the membrane module for CO2 capture application.

Membrane module with larger spacer filaments and proper angle between filaments provides better separation performance and lower pressures drops.

The effect of asymmetric spacer design also is investigated. In contrast to a symmetric spacer, asymmetric spacers consist of two filaments of different diameter aligned asymmetrically to the nominal flow direction. Simulation results indicate the asymmetric design can reduce pressure drop dramatically because of uniform flow distribution.

The multiphysics simulation is performed to study the performance of membrane module using a triple-layer spacer configuration. Effect of the density of thinner spacer adjacent to the membrane on concentration polarization is studied. Concentration polarization is

139

not observed for spacers with up to 10 times the flow resistance of an empty channel.

This indicates the use of a spacer sandwich to support the membrane and help mitigate parasitic pressure loss should not lead to significant performance loss due to concentration polarization.

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Chapter 7

7. Conclusions and Future Work

In this dissertation a new model is developed to predict gas transport properties and an upper bound on the properties for glassy polymeric materials. The gas solubility and solubility selectivity upper bound is predicted by the non-equilibrium lattice fluid

(NELF) theory. The gas diffusivity and diffusivity selectivity upper bound is predicted by using transition state theory (TST) with model parameters from NELF theory. The temperature dependence of gas transport properties is studied systematically.

Computational fluid dynamics (CFD) is used to study flow behavior in the spacer filled channels present in membrane modules. The effect of different spacer designs

(symmetric, asymmetric and triple-layer configurations) on pressure drop and module performance is investigated.

7.1 Conclusions

1. The NELF model is used to predict solubility and solubility selectivity. The

predictions for several gas pairs and polymers are in good agreement with

experimental measurements.

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2. Systematic variation of model parameters indicates an upper bound exists on

solubility selectivity based on the maximum cohesive energy density polymer

materials can achieve. The solubility and solubility selectivity data along the upper

bound can be given by the variation of polymer mer-mer interaction energy.

3. The slope of the solubility upper bound may be negative or positive depending on

the molar density ratio of the gas pair. This contrasts with the observation of a

negative slope of the permeability upper bound due to the influence of diffusivity

and diffusivity selectivity.

4. The temperature dependence of solubility selectivity on solubility is predicted using

NELF model. Predictions are reasonable and fit the experimental results well. The

NELF model can also be used to predict the movement of the solubility upper

bound with temperature.

5. A new model based on transition state theory is developed to predict gas diffusivity

coefficient using NELF model parameters. The model provides excellent

predictions for several gas penetrants in polycarbonate material. The model also

provides a good prediction of gas diffusivity selectivity in several polymer

materials.

6. The variation of the model parameters indicates a diffusivity upper bound exists and

moves upward with decreased polymer-mer interaction energy. Variation of the

polymer cohesive energy forms the upper bound with a logarithmic linear

relationship between diffusivity and diffusivity selectivity.

142

7. The slope of the diffusivity upper bound depends on the kinetic diameter of the gas

pair approximated by the NELF-TST model. The location of the diffusivity upper

bound depends on properties of gas penetrants and polymer materials.

8. The temperature dependence of diffusivity selectivity on diffusivity is evaluated

using the NELF-TST model. The calculated results fit the experimental

measurements well. The greatest deviations exist in diffusivity selectivity

prediction. The new model can also be used to predict movement of the diffusivity

upper bound with temperature. The diffusivity upper bound moves upward slightly

with increased temperature.

9. The NELF-TST model is used to predict gas permeability coefficient using

solution-diffusion model. The model provides good match between calculated

permeability and experiment measurements for several gas penetrants in

polycarbonate. Gas permeability selectivity also are predicted successfully and

deviations are mainly due to accumulated errors from predictions of gas solubility

and diffusivity.

10. Variation of model parameters is studied to investigate the relationship between

permeability and permeability selectivity. The relationship is not linear in

logarithmic scale because the different model parameters control the properties of

solubility and diffusivity upper bound respectively. Permeability upper bound is

mainly controlled by diffusivity upper bound for glassy polymer materials and the

movement of permeability upper bound is very limited.

143

11. The temperature dependence of permeability selectivity on permeability is

calculated using NELF-TST model. The calculated results fit experimental

measurements well. The movement of permeability upper bound with temperature

variation is evaluated and permeability upper bound moves upward with decreased

temperature investigated.

12. The velocity field, flow resistance, and pressure drop within spacer-filled channels

are studied by using three dimensional computational fluid dynamics (CFD)

simulations. The simulation is run for spacer A first to validate the CFD method.

The velocity and pressure field are visualized and pressure drop (expressed as

friction factor versus Reynolds number) is calculated to compare with experimental

data. The simulation results are in good agreement with experimental

measurements.

13. With the validation of CFD method, parametric study of the spacer geometry is

performed. Distance between filaments has weakest effect on the spacer flow

resistance within the Reynolds number range investigated. Friction factor increases

dramatically with decreased filament diameter and increased angle between

filaments. Spacer with 0° flow attack angle possesses lower flow resistance and

pressure drop compared to spacer with 45° flow attack angle.

14. Recirculation regions exist at specific Reynolds number for large filament angle

(120 and 150) spacers. This leads to an increase in total flow resistance. The

contribution of viscous drag and form drag to total flow resistance is investigated.

The viscous resistance has a significant contribution to the total resistance for

144

smaller filament angle spacers at low Reynolds number and form drag resistance

dominates for larger filament angle spacers.

15. Properties of flow resistance of the calculated spacers were further used to calculate

the pressure drop in the membrane module for CO2 capture application. Membrane

module with larger spacer filaments and proper angle between filaments provides

better separation performance and lower pressures drops.

16. The effect of asymmetric spacer design also is investigated. In contrast to a

symmetric spacer, asymmetric spacers consist of two filaments of different

diameter aligned asymmetrically to the nominal flow direction. Simulation results

indicate the asymmetric design can reduce pressure drop dramatically because of

uniform flow distribution.

17. The multiphysics simulation is performed to study the performance of membrane

module using a triple-layer spacer configuration. Effect of the density of thinner

spacer adjacent to the membrane on the concentration polarization is studied. No

change in module performance is observed as the spacer becomes denser. This

suggests concentration polarization should not impact module performance with use

of a three-layer spacer configuration.

7.2 Future Work

1. The new NELF-TST model is promising and can be used to predict transport

properties of gas mixtures in the glassy polymer materials. The polymer density is

taken as an order parameter and depends on the activity of more permeable gas

species. The variation can affect the transport properties of less permeable gas

145

species and real selectivity of gas mixtures can be predicted. The current model

assumes ideal mixing of gas penetrant and polymer material. Cases of non-ideal

mixing can also be studied by adjusting the interaction parameter between gas and

polymer.

2. The model also can be used to study broader class of glassy polymers especially for

ones with unusual free volume variation. Such variation is hard to account for

directly but it is averaged in NELF theory. Model parameters can be quite different

for glassy polymers even the fractional free volume they possess is nearly identical.

The model can be further developed to predict gas transport properties in polymer

composite materials such as mixed matrix membranes (MMMs). A modified

diffusion model is required to capture the transport mechanism of gas penetrants in

the structure of organic polymers with dispersed inorganic fillers. The tortuous path

of diffusion should be taken into consideration.

3. For the CFD simulation of gas flow in spacer-filled membrane module multiphysics

simulations are successfully performed to evaluate module performance. The

permeate side is currently set vacuum and future studies can be implemented in the

cases of cross and counter-current flow with sweep gas at permeate side.

4. The mass transfer simulations can be performed in three dimension based on the

methods currently used in two dimensional simulation. Results obtained from 2D

simulation are reasonable and the same models and boundary conditions can be

applied in 3D simulation to evaluate the membrane module performance.

Comparing to 2D simulation 3D simulation rigorously captures the performance of

real three-dimensional spacer geometry. Challenges may exist in the large memory

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usage of computer, much longer computational time comparing to 2D simulation and converged solutions obtained from solver.

147

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