Studies of air dehydration by using hollow fiber modules

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2011 Studies of air dehydration by using hollow fiber modules Pingjiao Hao The University of Toledo Follow this and additional works at: Recommended Citation Hao, Pingjiao, "Studies of air dehydration by using hollow fiber modules" (2011). Theses and Dissertations This Dissertation is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Dissertation entitled Studies of Air Dehydration by Using Hollow Fiber Modules by Pingjiao Hao Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering 1 Dr. G. Glenn Lipscomb, Committee Chair 1 Dr. Bruce E. Poling, Committee Member Dr. Isabel C. Escobar, Committee Member Dr. Arunan Nadarajah, Committee Member Dr. Sasidhar Varanasi, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo December 2011

3 Copyright 2011, Pingjiao Hao This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Studies of Air Dehydration by Using Hollow Fiber Modules by Pingjiao Hao Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering The University of Toledo December 2011 Hollow fiber membrane module is widely used for commercial gas separation, such as production of high purity of nitrogen and enriched oxygen gas from air, carbon dioxide removal from methane, carbon dioxide sequestration from flue gas and hydrogen purification. Membrane gas separation for air dehydration (AD) differs from other commercial applications in several ways. First, the component to be removed (water) possesses a permeability that may be more than three orders of magnitude greater than the other components in the feed (oxygen and nitrogen). Second, the feed concentration is small, less than 1% in a molar basis. Third, the product concentration may be two orders of magnitude less than the feed concentration. This work seeks to study the hollow fiber gas separation modules for air dehydration. The work evaluated potential module performance and the effect of the inefficiencies such as fiber property variability and deviation from ideal counter-current contacting. This research investigated the effect of sweep uniformity on gas dehydration module performance by assuming the sweep around each fiber is Gaussian sweep distribution. In addition, this work explicitly calculated sweep distribution and presented the effect of iii

5 sweep distribution, effect of fiber packing variation along case. Compared to fiber size variation, non-uniform sweep distribution has little effect on module performance. Moreover, two fundamental issues for gas separation regarding the Hagen-Poiseuille law and boundary layer contributions on mass transfer coefficients were investigated. This works investigated the of the Hagen-Poiseuille law for pressure drop calculations in hollow fiber gas separation modules. In this work a two dimensional (2D) computational fluid dynamics (CFD) mathematical modeling was used to obtain the numerical approximations to the solutions of the conservation of mass and momentum equations in a single fiber that is assumed to be representative of all fibers in the fiber bundle. Hagenposieuile law is a good approximate solution of pressure drop for sufficiently low Reynolds number of the feed and permeation fraction for compressible flows. For gas separations, concentration polarization can be significant when the fast gas permeance is greater than 1000 GPU. For air dehydration modules, the overall mass transfer coefficient is calculated by summing the lumen-side, membrane, and shell-side mass transfer resistances. In this work, effective mass transfer coefficients in the lumen and shell are calculated by using computational fluid dynamics (CFD) method and analytical method. The solutions will be obtained for a single fiber that is assumed to be representative of all fibers in the fiber bundle. The lumen mass transfer coefficient for constant wall permeance is 20% greater than that for constant wall concentration. The shell mass transfer coefficient in this research is 2-3 orders magnitude greater than the Donahue equation and the differences may arise from the complexity geometry of the shell fibers and the flows within it. iv

6 A commercial air dehydration hollow fiber module was evaluated. The module properties were obtained from experiment. Experimental results are in good agreement in simulation results-sweep increases product flow rate significantly. v

7 To my husband Hao, my daughter and my parents, who always support me and encourage me with their endless love.

8 ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Glenn Lipscomb, for giving me this great opportunity to work with him and carry through this research. His encouragement, patience, and advice make this research and the completion of this dissertation possible. I would also like to appreciate my dissertation committee: Dr. Bruce E. Poling, Dr. Isabel C. Escobar, Dr. Arunan Nadarajah and Dr. Sasidhar Varanasi for their advice and contribution to this research. I want to specially thank Rob Dunmyer, who helped me a lot with lab instruments. I got many suggestions and learnt a lot from him. I gratefully acknowledge the Generon IGS and the University of Toledo for providing financial support to this research. I would also like to thank Santosh Sonalka, Yang Su, Xi Du, Rahul Patil, Sricharan Nanduri, Ashkan Iranshahi, and Yuechun Lou for helping me through my work. Finally, I thank all friends in the University of Toledo for giving me good memories during the years I am here. vi

9 Contents Abstract iii Acknowledgements vi Contents vii List of Tables x List of Figures xi List of Symbols xvii 1 Introduction Research Objectives Literature Review Polymeric Membrane Gas Separation Processes Hollow Fiber Membrane Modules Air Dehydration (AD) Modeling and Simulation on Multicomponent Gas Separation Applicability of the Hagen-Poiseuille Law in membrane gas separation Boundary Layer Resistance Contribution to Overall Mass Transfer Coefficient Structure of the Dissertation vii

10 2. Effect of Sweep Uniformity on Gas Dehydration Module Performance Gaussian Sweep Distribution Theory Results Explicit Calculation of Sweep Distribution Theory Effect of Sweep Distribution Effect of Fiber Packing Variation Along Case Applicability of the Hagen-Poiseuille Law for Pressure Drop Calculations in Hollow Fiber Gas Separation Modules Simulation Method Fluid Flow for Single Component Gas Separation Module Performance Conclusions Effect of Boundary Layer Resistance on Mass Transfer Coefficients Simulation Method Modeling for lumen side Analytical solution for Sherwood number for constant permeance boundary condition Modeling for lumen side and shell side Evaluation of Boundary Layer Coupling Effects Lumen side with shell vacuum Coupling of Boundary Layer in Lumen side and Shell Side Conclusions Evaluation of a Commercial Air Dehydration Hollow Fiber Membrane Module Experimental Method viii

11 5.2 Experiment Results Dehydration O 2 /N 2 Separation Pure Gas Permeation Air Dehydration Module Air Separation Module Comparison of Experiment and Theory O2/N2 Separation Air Dehydration Selectivity Membrane Area and Membrane Permeance ( aq ) Conclusions Conclusions and Recommendations for Future Work References ix

12 List of Tables 1.1 Comparison between hollow fiber and flat sheet for gas separation Comparison of membrane method with other traditional technology for gas dehydration Fiber and module properties and operating conditions Boundary layer resistance percentage water for the module calculating at the sweep fraction is 0.1 and no inner diameter variation Mass transfer coefficients in the lumen with shell vacuum Well developed Mass transfer coefficients in the shell Experiment conditions for the evaluation of dehydration module x

13 List of Figures 1-1 Conceptual gas transport model through a polymeric membrane Hollow fiber module for air separation Hollow fiber module Three flow patterns in hollow fiber module Membrane dehydration module with shell sweep introduced through external plumbing and valves Membrane dehydration module with lumen sweep introduced through external plumbing and valves Membrane dehydration module with shell sweep introduced through outer shell port Membrane dehydration module with lumen sweep provided by fibers with no selectivity at fiber end Membrane dehydration module with shell sweep provided by tubes extending through retentate header Membrane dehydration module with shell sweep provided by tube extending through retentate header capped by diffuser to provide better sweep distribution Module divided into N Stages for simulation: (a) countercurrent; (b) concurrent Crossflow module divided into N stages Counter-current module divided into N stages 26 xi

14 2-3 Algorithms used to set the dew point of the product User interface for the simulator Dry gas flow rate as a function of dew point for various sweep fractions Dry gas recovery as a function of dew point for various sweep fractions Dry gas flow rate as a function of dew point for a water-nitrogen selectivity of 100 and various sweep fractions Dry gas recovery as a function of dew point for a water-nitrogen selectivity of 100 and various sweep fractions Dry gas flow rate as a function of dew point for different ID variation Dry gas recovery as a function of dew point for different ID variation solid Dry gas flow rate as a function of dew point for different inner diameter variation with and without boundary layer resistance Dry gas recovery as a function of dew point for different inner diameter variation with and without boundary layer resistance Effect of sweep variation on dry gas flow rate as a function of dew point Effect of sweep variation on dry gas recovery as a function of dew point Effect of sweep and ID variation on dry gas flow rate as a function of dew point...48 xii

15 2-16 Effect of sweep and ID variation on dry gas recovery as a function of dew point Porous media model for hollow fiber module Boundary conditions used in explicit simulations of sweep distribution to evaluate module performance Three different sweep configurations used in explicit sweep distribution calculations to determine module performance Effect of sweep configuration on dry gas flow rate as a function of dew point Effect of sweep configuration on dry gas recovery as a function of dew point Variation in water concentration in the radial direction along the retentate outlet Effect of variable fiber packing near the module case on dry gas flow rate as a function of dew point Effect of variable fiber packing near the module case on dry gas recovery as a function of dew point Boundary conditions used to solve the conservation of mass and momentum equations in the retentate, membrane, and permeate domains D-2D Pressure drop difference for incompressible fluid with constant wall velocity..66 xiii

16 3-3 1D-2D Pressure drop difference for incompressible fluid with constant wall permeability D-2D Pressure drop difference for compressible fluid with constant wall flux D-2D Pressure drop difference for compressible fluid with constant wall permeance., Dependence of product flow rate on as product dew point for air dehydration Dependence of retentate recovery on as product dew point for air dehydration Product flow rate versus O 2 purity for O 2 /N 2 separation Retentate recovery versus O 2 purity for O 2 /N 2 separation Axial diffusion effect on performance Boundary conditions used to solve the conservation of mass and momentum equations in the retentate domain for a lumen fed module Concentration profile in the lumen of hollow fiber Three domain model for a single fiber in the hollow fiber membrane module Boundary conditions used to solve the conservation of mass and momentum equations in the retentate, membrane, and permeate domains Comparison of predicted relationship between Sh local and Gr with the results of Leveque solution 87 xiv

17 4-6 Comparison of predicted relationship between lumen Sh local and Gr with shell vacuum for constant wall permeance Comparison of the predicted relationship between shell Sh local and Gr with for constant wall permeance Schematic illustration of the experimental apparatus for evaluating air dehydration module performance Photograph of experimental apparatus Schematic of evaluation of dehydration module: air separation Photograph of experimental apparatus used to evaluate the oxygen/nitrogen separation performance of an air dehydration module and an air separation module Feed flow rate as a function of dew point for dehydration module with various sweep fractions Dry gas recovery as a function of dew point for dehydration module with various sweep fractions Product recovery as a function of oxygen concentration for different module Nitrogen permeation rate as a function of the pressure difference between the lumen and shell for the air dehydration module Oxygen permeation rate as a function of the pressure difference between the lumen and shell for the air dehydration module.101 xv

18 5-10 Nitrogen permeation rate as a function of the pressure difference between the lumen and shell for the air separation module Oxygen permeation rate as a function of the pressure difference between the lumen and shell for the air separation module Product recovery as a function of oxygen concentration for different module Recovery Difference between the simulation and experiment as a function of selectivity used in the simulation Dry gas recovery as a function of dew point when sweep fraction = Dry gas recovery as a function of dew point when sweep fraction = Flow rate difference between the simulation and experiment as a function of aq used in the simulation Feed flow rate as a function of dew point sweep fraction= Feed flow rate as a function of dew point sweep fraction= xvi

19 List of Symbols A...active membrane area D...diffusivity Dh... hydraulic diameter f...flow rate g...distribution function Gz...Graetz number J...permeation flux k...mass transfer coefficient K...porous medium hydraulic permeability L...module or stage length n...number of gas components N...number of stages Nf...number of fibers p...pressure P...permeate flow rate Q...gas permeance r... radius R... retentate flow rate Rg... ideal gas constant Re...Reynolds number, (2r)vρ/η Sc... Schmidt number, η/(ρd) Sh...Sherwood number, kl/d T... temperature u... superficial velocity xvii

20 v... bulk velocity x...retentate mole fraction y... permeate mole fraction z... coordinate along flow direction Greek η... gas viscosity φ...sweep flow rate around a fiber ρ... gas density σ... standard deviation of sweep distribution Subscripts and superscripts h...high pressure i... gas component number or fiber inner diameter j... stage number l... low pressure or lumen side m... module max... maximum value min...minimum value o...fiber outer diameter s... shell side xviii

21 Chapter 1 Introduction 1.1 Research Objectives The objectives of this work were to evaluate experimentally and theoretically the relationships between the design and performance of membrane air dehydration modules. Specific objectives include: 1. develop a simulation of counter-current module performance which includes the effects of a Gaussian distribution in fiber size and sweep gas distribution 2. develop a simulation which includes how sweep is introduced into the module and is distributed across the fiber bundle based upon a solution of the conservation of mass and momentum equations in the hollow fiber module 3. evaluate the effect of gas compressibility on the pressure drop that occurs within the fiber lumen and the applicability of the Hagen-Poiseuille law 4. evaluate the coupling that occurs between the lumen and shell side mass transfer boundary layers and associated mass transfer coefficients 5. experimentally evaluate the performance of a commercial air dehydration module The results may be used to develop improved module designs and identify sources of inefficiency in current designs. 1

22 1.2 Literature Review Membrane separation processes are used in a variety of industries from biotechnology to water treatment. These processes commonly use polymeric membranes in flat sheet, spiral wound, tubular and hollow fiber form [1] in a wide range of applications which include water filtration, dialysis, and gas separations. The use of membrane gas separation processes has grown tremendously during the past few decades. The largest commercial application is production of high purity of nitrogen and enriched oxygen gas from air [2-4], however membranes are being considered for other separations including carbon dioxide removal from methane [5-6], carbon dioxide capture from flue gas [7], and hydrogen purification [8]. Air dehydration is a small but growing membrane gas separation market. Applications requiring moisture-free air include the operation of pneumatic controls laboratory instruments such as infrared spectrophotometers and X-ray diffractometers, and spray painting. Furthermore, dry and ultrapure gases such as extremely dry air, are required in the production of vacuum tubes, condensers, semiconductors, hermetically sealed components, radar equipment and other electronic communication systems. The primary competition for dehydration applications is condensation by cooling with a chiller. Relative to condensation, membrane dehydration offers the advantages of reduced energy consumption, reduced maintenance requirements, and simpler operation [9]. This work seeks to study the hollow fiber gas separation modules used for air dehydration. The theoretical simulations reported in this dissertation will permit the evaluation of potential module performance and the effect of inefficiencies such as fiber property 2

23 variability and deviation from ideal counter-current flows. In addition, two fundamental issues regarding the Hagen-Poiseuille law and boundary layer contributions to mass transfer coefficients are investigated. Finally, a commercial dehydration module is evaluated experimentally and the results are compared to the simulation results Polymeric Membrane Gas Separation Processes Polymeric membrane gas separation processes are used commercially for a number of separations. The largest current market is nitrogen production from air. Other significant applications include air dehydration, carbon dioxide removal from tertiary oil recovery operations, hydrogen production, and upgrading natural gas. Transport through dense, nonporous membranes can be described in terms of a solutiondiffusion mechanism [1], i.e. Permeability (Q) = Solubility (S) Diffusivity (D) The solution diffusion mechanism is illustrated in Figure 1-1. The process consists of five steps in which a gas molecule (red circle): (1) diffuses to the membrane surface from the high pressure gas phase, (2) dissolves in the membrane, (3) diffuses a length l through the membrane, (4) desorbs from the membrane, and (5) diffuses away from the membrane surface into the low pressure gas phase. 3

24 z= l 5 Figure 1-1. Conceptual gas transport model through a polymeric membrane. The rate of permeation is given by J DS Q = ( xph ypl ) = p (1-1) l l Where J is the gas flux (moles/area/time), D the gas diffusion coefficient in the membrane, S the gas solubility, x the mole fraction of the gas in the high pressure phase, p h the high pressure, y the mole fraction in the low pressure phase, and p l the low pressure. The product of D and S is called the permeability (Q) while the permeability divided by the effective membrane thickness is called the permeance. The ratio of the permeabilities for two different gases is defined as the selectivity: α = (D A S A )/(D B S B ). Materials with high selectivity and high permeability are desired to efficiently separate a gas mixture. Polymeric membranes can be classified either as symmetric or asymmetric. An asymmetric structure consists of a thin (commercially ~0.1 µm) selective skin layer on a highly porous (100 to 200 µm) supportive layer [10]. The very thin layer is the actual membrane. Its separation characteristics are determined by the nature of the polymer while the mass transport rate is determined by the membrane thickness, since the mass transfer rate is inversely proportional to the thickness of the actual barrier layer. The porous layer serves only as a support for the thin and fragile skin and ideally does not affect the separation characteristics or mass transfer rate across the membrane [10]. 4

25 1.2.2 Hollow Fiber Membrane Modules To apply membranes commercially, large membrane areas are required. A number of module designs are possible and all are based on two forms of the membrane: 1) flat and 2) tubular. Plate-and-frame and spiral-wound modules use flat membranes in sheet form whereas tubular, capillary and hollow fiber modules use tubular membranes [11]. For gas dehydration applications, the hollow fiber form is preferred. Table 1.1 compares the characteristics of hollow fiber and spiral modules. In comparison to spiral wound modules, hollow fiber membrane modules contain more membrane surface area per unit volume thereby reducing the size of the module. Additionally, hollow fiber module manufacturing costs are lower and hollow fiber designs easily permit permeate sweep. Table 1.1: Comparison between hollow fiber and flat sheet for gas separation. Spiral Hollow Fiber Production Cost [12] $/m $/m 2 Surface area/volume [1] ~1000m 2 / m 3 ~5000 m 2 / m 3 Cross Flow Easy Easy Counter Flow Difficult, but possible Easy Sweep Difficult, but possible Easy Figure 1-2 illustrates the design of a typical hollow fiber membrane module for air separation. The module consists of a bundle of hollow fibers enclosed in a case. In Figure 5

26 1-2, compressed air is fed to the lumens of the fibers. As the air flows through the fibers, oxygen selectively permeates across the fiber wall and is collected from the shell, the space outside the fibers, as the permeate. The gas that remains in the fiber lumens is enriched in nitrogen and is collected as the retentate from the end of the modules opposite the feed. Figure 1-2. Hollow fiber module for air separation [13]. As illustrated in Figure 1-3, tubesheets are formed at either end of the fiber bundle. The tubesheets seal the fibers together and force a fluid in contact with the tubesheet into the fiber lumens. The outer edge of the tubesheet is sealed to the cased using an o-ring or other sealant. The sealed tubesheet allows one to control separately the flows inside and outside the fibers. Such a design is the mass transfer equivalent of a traditional shell and tube heat exchanger. A hollow fiber module commonly has 2 ports on the shell and 2 on 6

27 the lumen (tube) side. It can be either shell or lumen fed. Lumen feed is preferred when operating pressures are not too high due to better low distribution in the module. Shell access port Distribution collar for shell fluid Bolts to attach header to case Tubesheet Fiber Bundle Lumen access port Case Lumen header Figure 1-3. Hollow fiber module. The module can be operated in three different modes: co-current, cross flow or counter-current. The three flow patterns are illustrated in Figure 1-4. In co-current flow, the permeate flows in the same direction as the feed and retentate. In cross flow, the permeate flows perpendicularly to the feed and retentate while in counter-current flow the permeate flows in the opposite direction. The counter-current flow pattern gives the best performance as the driving force for transport is maximized along the module length. One can produce an arbitrarily high purity retentate product with the counter-current design but the maximum permeate purity is limited by the intrinsic separation properties of the membrane [14]. 7

28 Lumen Counter-current Co-current Cross-flow Figure 1-4. Three flow patterns in hollow fiber module Air Dehydration (AD) Compared to traditional technologies (condensation, absorption, and adsorption) for air dehydration, membranes offer the advantages of simpler operation and lower energy consumption. Table 1.2 compares membranes to other competing technologies for AD [15]. 8

29 Table 1.2: Comparison of gas dehydration technologies. Technology adsorption Condensation absorption Membrane Separation principle adsorption condensation absorption diffusion and absorption Dew point range of -30 to 50 0 to 20 0 to to 40 product/ o C Space for equipment large large Large Small Maintenance medium medium difficult Easy Production Scale medium to large small to large large small to large Principle adsorption condenser, heat absorption hollow fiber equipments tower, heat exchanger tower, pump, module, filter, exchanger, heat exchanger heat exchanger, blower vacuum pump Membrane gas separation for air dehydration differs from other commercial gas separation applications in several ways. First, the component to be removed (water) possesses a permeability that may be more than three orders of magnitude greater than the other components in the feed (oxygen and nitrogen). Second, the feed concentration is 9

30 small, less than 1% on a molar basis. Third, the product concentration may be two orders of magnitude less than the feed concentration. These conditions contrast starkly with those encountered in the more common process of nitrogen enriched air (NEA) production. The component to be removed in NEA production (oxygen) possesses a permeability that is generally less than one order of magnitude greater than the other components. The feed concentration is ten times greater than in air dehydration. Finally, the product concentration most commonly is only one order of magnitude less than the feed concentration. These process differences impact module design and process operation. The much higher membrane selectivity and lower product concentration in AD typically require the use of sweep to achieve separation and productivity economically. Additionally, the area required for the separation and the effective membrane selectivity are strongly influenced by gas phase concentration boundary layers. Boundary layers are negligible in most gas separation processes. The unique aspects of AD are documented in the literature. The Doctoral thesis of Wang [16, 17] demonstrates that for a lumen fed AD module the membrane resistance to water transports is negligible the overall mass transfer coefficient is controlled by lumen and shell side concentration boundary layers. For the experimental conditions examined, mass transfer boundary layers constitute no more than 10% of the overall mass transfer coefficient for permeances up to 50 GPU. Since oxygen and nitrogen permeances are less than this, their overall mass transfer coefficients are dominated by the membrane resistance. 10

31 Wang simulated the effect of using a portion of the retentate product as a permeate sweep on module performance. The amount of water removed increased as the fraction of the retentate used as sweep increased (sweep fraction). For low sweep fractions, the increased water removal is due to an increased water partial pressure driving force. For larger sweep fractions, water removal increases because of a reduction in the shell side mass transfer resistance. Boundary layer resistances decrease as the fluid velocity increases. For a fixed product flow rate and purity, increasing the sweep fraction led to a dramatic drop in the required membrane area and a concomitant increase in feed air lost as sweep. The lost feed air increased for sweep up to 0~0.15 but the membrane area decreased by over an order of magnitude. Consequently, a sweep fraction of was speculated to be near optimal. Higher sweep fractions increase the purity at the cost of air loss. Simulation results also indicated that feed air losses decreased monotonically with an increase in overall membrane selectivity while holding the sweep constant; the dependence is most sensitive for selectivity values less than 50. In contrast to the monotonic change in feed air loss, the required membrane area passed through a minimum at ~10 as selectivity increased. The minimum was attributed to a reduction in water driving force due to reduced air permeation. Wang also examined whether it was preferable to use sweep or reduce selectivity to improve process performance. She compared two cases: (1) a fixed, low air permeability with sweep and (2) a fixed, high air permeability without sweep. In case (1) the sweep is introduced at the retentate product end of the module and is active along the entire length 11

32 of the fiber bundle while in case (2) the sweep is introduced nearly uniformly along the length of the fiber bundle so the sweep flow rate increases from zero at the retentate product end to its maximum value at the feed end. Simulation results indicated the first case was much better for a fixed feed air loss, introducing the sweep at the retentate product end required 2-4 times less membrane area. The patent literature describes three primary methods for introducing sweep from the retentate product into the shell of a hollow fiber module. First, Skarstrom and Kertzman [9] teach the use of conduits and valves external to the module to return a portion of the retentate product to the shell as sweep, as illustrated in Figure 1-5. The sweep may be introduced either through an external port on a case enclosing the fiber bundle or through a tube that extends from outside the module through the tube sheet into the shell. Similarly, Makino and Nakagawa [18] teach the use of valves and conduits to feed the sweep to the fiber lumens in a shell-fed module, as illustrated in Figure 1-6. Friesen et al. [19,20] teach the use of the sweep stream that is mixed with the water-containing permeate at a point generally opposite the feed of the module, preferably through a port near the retentate product (i.e., the dehydrated air product) end of the module, as shown in Figure 1-7. Second Stookey [21] teaches the use of fibers that possess reduced selectivity and increased permeance near the retentate product end, as shown in Figure 1-8. If the fibers used in the module are composite membranes, one can change their transport properties simply by removing the discriminating coating in a region near the tube sheet. This allows introduction of the sweep from the fibers themselves but does not allow control of the sweep rate. 12

33 Third, Morgan [22] et al. teach the use of a plurality of passages (fibers, tubes, or other conduits) embedded in the retentate end tube sheet that allow fluid communication between the retentate header and the shell, as illustrated in Figure 1-9. The pressure difference between the header and shell drives a portion of the retentate product back into the shell. The sweep flow rate is determined by the number and size of the passages and cannot be regulated externally. This internal sweep design is used extensively. Burban et al. [23] describe a modification in which a diffuser is used to distribute the sweep more uniformly in the shell, as shown in Figure A channel or conduit extends from the retentate header through the tubesheet into the fiber bundle. The channel end in the header is left as an open orifice while the channel end in the fiber bundle is capped by a porous diffuser. The patent literature also teaches the importance of counter-current contacting [24] and recycle configurations that improve process performance [25]. Auvil et al. [25] describe two configurations in which the wet permeate is recycled to eliminate feed air losses. If the feed is at ambient pressure, the permeate is sent to the inlet of the feed compressor. If the feed is already at pressure, a recycle compressor is used to increase pressure to the feed pressure. In both cases, water is removed from the process in the chiller/condenser that follows the compressor. More recent literature addresses other issues associated with membrane dehydration. Vallieres and Favre [26] demonstrate that the use of a permeate vacuum may be preferable to use of a product sweep in some cases. However, the changes in performance are modest and may not justify the purchase and maintenance of a vacuum pump. Metz et al. [27] reports measurements of the dependence of water permeability on water 13

34 concentration. This dependence is important if boundary layer resistances to mass transfer are comparable to or smaller than the membrane resistance. Figure 1-5. Membrane dehydration module with shell sweep introduced through external plumbing and valves [9]. Figure 1-6. Membrane dehydration module with lumen sweep introduced through external plumbing and valves [18]. 14

35 Figure 1-7. Membrane dehydration module with shell sweep introduced through outer shell port [19, 20]. Figure 1-8. Membrane dehydration module with lumen sweep provided by fibers with no selectivity at fiber end[21]. Figure 1-9. Membrane dehydration module with shell sweep provided by tubes extending through retentate header [22]. 15

36 Figure Membrane dehydration module with shell sweep provided by tube extending through retentate header capped by diffuser to provide better sweep distribution [23] Modeling and Simulation on Multicomponent Gas Separation Pan [28] describes one of the first attempts to evaluate the performance of a gas separation module for the separation of multicomponent mixtures. Pan s analysis assumes the local permeate composition, that determines the partial pressure driving force for permeation, is not equal to the bulk concentration of the permeate due to the membrane resistance. Instead the composition within the membrane support is taken as the composition produced by cross-flow of the permeate through the support. Additionally, permeate pressure drop within the support is neglected. Kovvali et al. [29] propose a simplification to Pan s solution procedure by assuming a linear relationship between permeate and retentate mole fractions for sufficiently small composition ranges. Chowdhury et al. [30] obtain more accurate solutions by using a backward difference Adams-Moulton or Gear approximation for the governing differential equations. The resulting non-linear algebraic equations are solved with a modified Powell hybrid algorithm that utilizes a finite difference approximation for the Jacobian. Coker et al. [31] avoid the assumption of a local permeate composition equal to the crossflow composition by dividing a module into a series of N well mixed stages and solving 16

37 the Weller-Steiner equations [32] for each stage, as shown in the Figure If the stages are numbered from 1 to N from the feed to the retentate product end, the retentate stream exiting stage i is the feed to stage i+1. The permeate from stage i+1 is sent to stage i as a sweep for counter-current modules; for co-current modules, the permeate from stage i 1 is used. This simplification is equivalent to the use of first order finite differences to approximate the derivatives in the governing mass balances. Kaldis et al. [33] describe an alternative solution procedure in which the governing differential equations are approximated using an orthogonal collocation algorithm. Lemanski and Lipscomb [34, 35] describe solutions of the governing conservation of mass equations for flows throughout the lumen and shell regions of a module. The solution assumes the flows are equivalent to flow through an effective porous media and explicitly accounts for the influence of inlet and outlet port placement on performance. Marriott et al. [36-38] also describe solutions to the governing conservation of mass, momentum, and energy equations for the flows within the lumen and shell regions. For counter-current and co-current flows, the rigorous axis symmetric conservation equations are solved within the lumen of an individual fiber all fibers are assumed identical. The lumen equations are coupled to the shell equations assuming uniform axial plug flow in the shell; fluid distribution to and from lumen distribution manifolds is not considered. 17

38 Figure Module divided into N Stages for simulation: (a) countercurrent; (b) concurrent [31] Applicability of the Hagen-Poiseuille Law in membrane gas separation For hollow fiber membrane, there are several major module design variables: inner/ outer diameter, permeance, selectivity, pressure ratio (i.e., the ratio of shell pressure to lumen pressure), and flow pattern. In addition, there are minor design variables such as fiber variability and flow uniformity which can have a dramatic effect on module performance and to determine the magnitude of these effects pressure drops through the shell and lumen of the module must be known. The analysis of module performance typically assumes pressure drop in the fiber lumens is described by the Hagen-Poiseuille law [39]. This expression is strictly valid only for incompressible flows without wall permeation. 18

39 While a significant body of work addresses the limits of validity of the Hagen-Poiseuille Law in filtration of incompressible liquids, little is available for gas permeation. A first principles study of mass and momentum transfer to determine the applicability limits of the Hagen-Poiseuille law is reported here. In his classic paper of 1953, Berman [40] developed solutions from the Navier-Stokes equations for fluid flow in a rectangular slit bounded by two equally porous walls. Karode [41] derived an approximate analytical solution for the pressure drop in fluid flow in a rectangular slit and cylindrical tube with porous walls for the case of constant wall velocity and permeability. In the above papers, the assumptions made include: (1) steady state conditions, (2) incompressible fluids, (3) no external forces act on the fluid, and (4) laminar flow. Berman also assumes the velocity of the fluid leaving the walls of the channel is independent of the position. Subramani et al. [42] simulated pressure drop and concentration profiles in open and spacer-filled membrane channels by a finite element model and simplified analytical models. Karode s and Berman s [40,41] results were used to evaluate the accuracy of the approximate solution in Sirijarukul s [43] work who considered transport in slit channels bounded by a flat sheet membrane with controlled variation of pore density and pore size in a direction parallel to the surface. Shao et al. [44] examined the relationship between pressure drop, a dimensionless parameter characterizing the physical properties of the fibers (e.g., fiber inner diameter, fiber length, and intrinsic permeability), and operational conditions (e. g., downstream pressure) based on a simplified analytical approach. 19

40 Lim et al. [45] developed a new pressure drop equation, referred to as the improved Hagen-Poiseuille model and compared the model with the Hagen-Poiseuille model for pure gas permeation of permanent gas and organic vapors. They concluded that the Hagen-Poiseuille equation can be used to calculate the pressure profile for permanent gases while the improved model is more suitable for predicting the pressure profile for organic vapors Boundary Layer Resistance Contribution to Overall Mass Transfer Coefficient For gas separation, concentration polarization is noticeable when the fast gas permeance and the selectivity of fast gas to slow gas is higher than 1000 GPU and 100 respectively (1GPU= kmol m -2 s -1 pa -1 ) [16, 17, 48, 49, 50]. For hollow fiber membrane air dehydration, the overall mass transfer coefficient is calculated by summing the lumen-side, membrane, and shell-side mass transfer resistances. The shell and lumen side boundary resistances are calculated using the Donohue and Leveque equations, as suggested by Wang [16, 17]. The mass transfer correlations developed for lumen and shell side concentration boundary layer assume low mass transfer rates across the membrane. Hence, radial mass transfer is purely diffusive. For highly permeable membranes, radial mass transfer may involve convection in addition to diffusion. Typically, membrane transport is assumed to be either diffusive or convective [1] and appropriate permeation relationships used. For dilute solutions, when convection is dominant, the flow rate of the majority component (i.e., solvent) is calculated from the hydraulic permeability of the membrane and the applied trans-membrane pressure 20

41 difference while transport rates of the minority components (i.e., solutes) are calculated from the product of solute concentration and solvent flow rate. Intermediate cases with significant contributions from both have not been addressed in the literature. Computational fluid dynamics (CFD) modeling method has been used to model rigorously concentration polarization through the solution of the continuity, Navier- Stokes and solute continuity equations and its effect on membrane separation processes [51, 52-55,56,57,58]. Bhattacharyya [51] developed a Galerkin finite element program to compute the concentration profiles throughout a reverse osmosis membrane module and predict the performance of the module. Bao [52-55] calculated the shell-side distribution and mass transfer coefficients for axial flow through randomly packed hollow fiber bundles. Liu [56] developed a CFD approach to describe not only velocity distribution but also concentration profile in the liquid boundary layer of a slit membrane channel. The overall mass transfer coefficients from numerical study were compared to those from experiment and literature correlations. Ahmad [57] predicted the concentration polarization profile, mass transfer coefficient and wall shear stress under different types of conditions in an empty narrow membrane channel. Li [58] performed threedimensional laminar CFD simulations for the spiral wound membrane feed channel to evaluate mass transfer coefficients and power consumption for commercial net spacers. 1.3 Structure of the Dissertation Chapter 1 (this chapter) provides an introduction to polymeric membrane gas separation processes, hollow fiber membrane modules, air dehydration, and the modeling of multicomponent gas separation. The applicability of the Hagen-Poiseuille law and 21

42 boundary layer resistance contributions to the overall mass transfer coefficient are reviewed as well. Chapter 2 investigates the effect of sweep uniformity on gas dehydration module performance. The first part of Chapter 2 assumes the sweep around each fiber is distributed in a Gaussian manner. After summarizing the assumptions, conservation equations, boundary equations and calculation method, predictions of module performance are presented. The effects of a Gaussian distribution in fiber size and concentration polarization also are reported. In the second part of Chapter 2, explicit calculations of sweep distribution and associated module performance are reported. Additionally, the effect of poor fiber packing adjacent to the case is examined. Chapter 3 investigates the applicability of the Hagen-Poiseuille law for pressure drop calculations in hollow fiber gas separation modules. The first part of Chapter 3 describes the Simulation Method. The second part reports results for the flow and permeation of a single component. The third part discusses gas separation module performance with more than one component. The last part is the conclusions. Chapter 4 evaluates the effect of boundary layer resistances on the overall mass transfer coefficient. The first part of this chapter describes the simulation method for coupling lumen and shell side mass transfer coefficients in hollow fiber membrane modules. The second part describes analytical and CFD simulation results for permeation from the fiber lumen to a shell held under an absolute vacuum. The last part reports CFD results for coupling lumen and shell side mass transfer coefficients in hollow fiber modules without a shell vacuum. 22

43 Chapter 5 reports the performance evaluation of a commercial AD membrane module. The first part of the chapter describes the experimental method. The second part reports experimental measurements. The last part compares experiment to theory. Chapter 6 is the conclusions and recommendations for future work. 23

44 Chapter 2 Effect of Sweep Uniformity on Gas Dehydration Module Performance 2.1 Gaussian Sweep Distribution Theory A counter-current, lumen-fed hollow fiber membrane air dehydration module is simulated. The retentate flows in the lumen and the permeate in the shell Model Assumptions This section describes the model used to simulate hollow fiber membrane air dehydration modules. The simulation is based upon mass balances for each fiber in the module as described previously [16, 17 & 47]. Key assumptions in the theoretical analysis are: 1. Gas mixtures behave ideally. 2. The shell pressure is constant. 3. The lumen pressure drop is described by the Hagen-Poiseuille equation for compressible flows. 4. Axial diffusion is negligible. 5. The process is isothermal. 24

45 6. The properties of a given fiber do not vary along its length. However, properties may vary from fiber to fiber. These assumptions permit simulation of multi-component mixtures (water, oxygen, and nitrogen) and explicitly account for changes in permeation driving force due to lumen pressure drop. In previous simulations [16, 17] the lumen pressure drop was assumed to be much smaller than the feed pressure and only binary feeds were considered Model Equations In this work, the lumen and shell mass balances for individual fibers are solved by dividing each fiber into N Weller-Steiner Case I stages along its length as illustrated in Figure 2-1. The number of stages N is varied until the results obtained by increasing N change by less than 1%. For stage j and component i, mass balances are imposed and the permeation rate is calculated from the general permeation expression. To solve the mass balance equations, the direct substitution method is used to calculate retentate (R j ) and permeate (P j ) flows, the mole fraction of each component in retentate (x j ) and permeate (y j ) and the lumen-side pressure in each of the N stages. The cross-flow results are calculated at first and used as the initial guess for the counter-current flow simulation. P, 1 yi,1 P j, y i, j P y N, i, N Feed 2, i,2 1 j R x R j x i, j 1,, 1 R j x i, j R x N, i, N N R x N, i,, N Figure 2-1. Crossflow module divided into N stages. 25

46 For unit j in crossflow (Figure 2-1), the mass balance equations and permeation relationships are: R = P + R j 1 j j (2-1) y Q p x θ /( Q + Q p + Q p (1 θ ) / ) (2-2) x i, j = i h, j i, j 1 / i l i h, j θ i (, j i, j 1 i, j = x (1 θ ) y /θ (2-3) Where θ is defined as n R j, and Q as Qi ( ph, jxi p, j l yi, j ) j 1 1 R p l and p h are the pressure of the shell and tube side respectively, the subscript i denotes the ith component in the mixture, and Q i is the permeance of the ith component. For unit j in counter-current flow (Figure 2-2), mass balance equation and permeation equality equation are as follows: P, 1 yi,1 Feed 1 P, 2 yi,2 R, 2 xi,2 P, j y, i, j + 1, j + 1 R j x i, 1, j 1 j P j y i R j, x i, j P y N, i, N R x N, i, N N Sweep R x N, i,, N Figure 2-2. Counter-current module divided into N stages. R j 1 xi, j 1 + Pj + 1yi, j + 1 = Pj yi, j + R jxi, j (2-4) P y j i, j j 1 i, j+ 1 i, j i( h j i, j l i, j P + y = AJ = AQ p x p y ) (2-5) where A is the membrane area and J the permeation flux. Summing the mass balances for each of the n components, Equation (2-4), gives the following expressions for the change in retentate and permeate flow rate. 26

47 27 = + = n i j i l j i j h i j j j y p x p A Q P P 1,,, 1 ) ( (2-6) = = n i j i l j i j h i j j j y p x p A Q R R 1,,, 1 ) ( (2-7) Pressure drops in the lumen are calculated using the Hagen-Poiseuille equation [39] modified by substituting the product of molar flow rate and molar density for the volumetric flow rate and using the ideal gas law to calculate molar density: j f i g j h R N r T R dz dp 4 2, 16 π η = (2-8) where z is distance along the module, the gas viscosity, R g the ideal gas constant, T temperature, r i the fiber inner radius, and N f the number of fibers. The integral of this equation for stage j is given by: L R R N r T R p p j j i g hj hj + = π η (2-9) where L is the length of the stage (i.e., (active fiber length)/n). Equation (2-8) is used to evaluate the pressure in each stage given the feed gas pressure to stage 1. Equations (2-4)-(2-8) are solved with an iterative solution algorithm. Using the crossflow solution as an initial guess, improved estimates for x i,j and y i,j are obtained from Equations (2-10) and (2-11), respectively, which follow from Equations (2-4) and (2-5). j h i j j j i l i j j i j j i A Q p R y A Q p x R x,, 1, 1, + + = (2-10)

48 y P y + A Q p x j + 1 i, j + 1 j i h, j i, j i, j = (2-11) Pj + AjQi pl Improved estimates for P j, R j, and p h,j are obtained using Equations (2-5), (2-6), and (2-8), respectively. The iterative procedure is stopped when the variables change by less than 1% from one iteration to the next. This direct substitution iterative algorithm proved to be robust and converged quickly for all cases considered Boundary Layer Equations The overall mass transfer coefficient for species i is calculated by summing the lumenside, membrane, and shell-side mass transfer resistances as given by Equation (2-12): 1 = r o k i kl, iri Qi ks, i (2-12) where r o is the fiber outer radius and Q the membrane permeance based on the fiber outer radius. As defined in Equation (2-12), the overall mass transfer coefficient is based on the fiber outer radius. The shell and lumen side boundary resistances are calculated using the Donohue and Leveque equations, Equations (2-12) and (2-13) respectively, as suggested by Wang [16, 17]. Sh = kl L D r i ri v = Re Sc = L DL 0.33 (2-13) Sh = k L D s = Dh Re Sc (2-14) 28

49 where k l is the lumen-side mass transfer coefficient, k s the shell-side mass transfer coefficient, v the bulk velocity, L the fiber length, D the diffusion coefficient, Re the Reynolds number, Sc the Schmidt number, and D h the hydraulic diameter calculated using equation (2-15). D h (2r ) 2 2 m f i = (2-15) 2r m N + 2N (2r ) f r i where r m is the module radius. Note the Reynolds number is calculated based on the fiber inner radius for Equation (2-13) and the fiber outer radius for Equation (2-14) Calculation Method for Gaussian Distribution of Sweep The effect of variability in fiber inner diameter and sweep flow rate is included in the theoretical analysis by writing mass balances for each fiber in the module. The literature [17, 47] has documented the strong dependence of module performance on fiber variability. Variation in inner diameter, permeance, or selectivity can decrease product recovery dramatically for a fixed product purity and increase the required membrane area. For a given degree of variation, inner diameter has the greatest effect [47]. Therefore, only variability in fiber inner diameter was included in the simulation along with variability in sweep distribution. Additionally, shell-side mixing of permeate was neglected. Shell mixing improves performance but predictions in the absence of mixing (permeate from a given fiber does not mix with permeate from other fibers) represent the worst case scenario. The inner diameter was assigned based on a Gaussian distribution with a specified average and standard deviation. To simulate the effect of deviations from counter-current contacting due to poor sweep distribution, different sweep flow rates was assigned to each fiber in the bundle. The 29

50 literature demonstrates that local flows through randomly packed bundles can vary dramatically and affect both liquid [53] and gas [59] separations. However, this work was limited to cases in which lumen and shell flow changes are small, i.e., a dilute solute is transferred across the membrane. For air dehydration, changes in flow rates can be significant so this work is not applicable. To simulate variation in local flow rate, the sweep flow rate for each fiber will be assigned based on a Gaussian distribution with a specified average and standard deviation. A Gaussian distribution of a fiber property (fiber inner diameter or sweep flow rate) is calculated using [47], 2 1 ( φ φ) g ( φ) = exp (2-16) 2 σ 2π 2σ where φ is the value of the fiber property (inner diameter or sweep flow rate), φ the mean value, and σ the standard deviation. The fraction of the fibers for which the property value falls in the interval ( φ, φ + dφ) is equal to g ( φ) dφ. For a bundle of fibers with a single variable material property, the average flow per fiber is given by φ max f = f ( φ) g( φ) dφ φ min (2-17) where f may be the retentate or permeate flow. Gauss-Hermite quadrature [60] is used to evaluate Equation (2-17) and determine the overall performance of the fiber bundle. The number of quadrature points is determined by increasing the number until the results change by less than the desired accuracy of the solution. The simulations assuming a 30

51 Gaussian sweep distribution were performed using three quadrature points based on previous work demonstrating that use of five quadrature points gave equivalent results to within less than 1% [47, 61]. Simulations with varying number of stages indicated that the results did not change by more than 5% if 200 or more stages were used. Thus, in this work all simulations were performed with 200 stages. The converged solution for each fiber or sweep distribution is used to evaluate overall module performance by using Equation (2-17) to determine average flow rates. Results obtained for a Gaussian sweep distribution are compared to results obtained for a Gaussian fiber inner radius distribution. Previous work [47, 61] has shown that variation in fiber inner radius has the largest impact on module performance when the variability in fiber properties is included in module performance simulations Specification of Product Dew Point In this work, Goff-Gratch water dew point correlations as given by Equations are used to convert dew points to water vapor concentrations [62, 63]. p v_water is the water vapor pressure, p total the pressure of the gas mixture, Dp the dewpoint, and x water the molar fraction of water vapor in the gas mixture. In the Goff-Gratch equations, the unit of pressure is millibar and dew point is K. log 10 p v _ water = ( a 1) log ( ( a 1) a = /( Dp) 1) + log ( a) ( (1 1/ a) 1) (2-18) (2-19) p = v _ water xwater ptotal (2-20) 31

52 The dew point of the product is specified and the feed rate of the module is unknown in this work. In the absence of sweep and fiber size variations, the algorithm in Figure 2-3 is used to calculate the feed rate for a specified dew point: 1) Estimate the minimum feed flow rate by calculating the total permeation rate for the feed composition in crossflow and assuming this rate does not change along the fiber length. 2) Set the module feed rate, Fguess, to this estimate and solve the mass balance equations 3) Calculate the dew point (Dp) of the product 4) If Dp > Dptarget then go to 5, else set Fhi = 1.2 Fguess, Dplo= Dp and Fguess = Fhi and go to Step 2. 5) Set Flo =Fguess/1.2, Dphi= Dp and Fguess = Flo. 6) Set F F hi lo F guess = ( Dptarget Dplo) + Dphi Dplo F lo 7) Calculate Dp for Fguess 8) if Dp< Dptarget Flo = Fguess, Dplo = Dp, else Fhi = Fguess, Dphi = Dp 9) Go to 6 until Dp - Dptarget <0.5 32

53 Figure 2-3. Algorithm used to set the dew point of the product. 33

54 When inner diameter (ID) variation or sweep distribution is considered in the module, the feed flow rate will vary from fiber to fiber but the pressure drop for each fiber is the same. To determine the lumen pressure drop and associated individual fiber flow rates to produce a desired dew point, the following procedure is used: 1) calculate pressure drop ( p lo ) value for the smallest ID (or sweep) fiber to produce the desired dew point product (D p =D ptarget ) 2) for a pressure drop of p lo calculate the mixing cup average dew point of the product produced by the entire module (D plo ) 3) calculate pressure drop ( p hi ) value for the largest ID (or sweep) fiber to produce the desired dew point product (D p =D ptarget ) 4) for a pressure drop of p hi calculate the mixing cup average dew point of the product produced by the entire module (D phi ) 5) estimate the pressure drop required to obtain a mixing cup average dew point of D p from p p p D D hi lo = ( pt arg et plo) Dphi Dplo + p lo 6) for a pressure drop of p calculate the mixing cup average dew point of the product produced by the entire module (D p ) 7) if D p < D ptarget, set p lo= p and D plo =D p else p hi= p and D phi =D p 8) if D p D ptarget <0.5 stop else goto Results Ideal Module Performance 34

55 In this work, air is considered to have three components: water, nitrogen and oxygen. The desired product is the dehydrated air with a specified dew point. The model equations are solved using a Visual Basic macro within a Microsoft Excel spreadsheet. Figure 2-4. is the user interface for the simulation. Users can input operating conditions and manipulate the output results in the Excel workbook containing the simulation macro. The results presented here assume the selectivity for water relative to nitrogen is 1000 and that for oxygen to nitrogen is 8 which are representative of current commercial air dehydration modules. Other fiber properties and operating conditions are tabulated in Table 2.1 and are representative of commercial hollow fiber dehydration modules. Figure 2-4. User interface for the simulator. 35

56 Table 2.1: Fiber and module properties and operating conditions. Literature correlations are used to convert dew points to water vapor concentrations [62, 63]. Fiber number 1000 Fiber length(m) 0.1 Fiber inner diameter(micron) 200 Fiber outer diameter(micron) 150 Fiber packing fraction 0.5 Selectivity of water/n Selectivity of O 2 /N 2 8 N 2 permeance(gpu) 1 Feed pressure (psi) 147 Shell side pressure(psi) 14.7 Feed dewpoint@14.7( F) 45 Operation Temperature( F) 115 Module performance is characterized by two factors: 1) the product gas flow rate and 2) the fraction of the feed gas recovered as retentate product, the retentate recovery. The required membrane area decreases as product flow rate per unit area increases. Product and energy (required to compress the feed) losses decrease as the fraction of the feed gas recovered as product increases. Therefore, high product flow rate and recovery are desirable. Figures 2-5 and 2-6 illustrate the effect of sweep fraction (i.e., fraction of the dry product returned as sweep to the shell) on module performance assuming uniform, ideal sweep distribution. For all sweep fractions, the product gas flow rate and recovery decrease as 36

57 the dew point decreases since increased water removal is accompanied by increased loss of oxygen and nitrogen. Increasing the sweep fraction dramatically increases the product gas flow rate. For example, Figure 2-5 indicates the dry gas flow rate increases by a factor of nearly 20 at a dew point of 0 F if the sweep fraction is increased to 0.2. This increase in flow rate comes at the cost of reduced recovery. For high selectivities, recovery is reduced by an amount approximately equal to the sweep fraction. Figure 2-6 indicates recovery decreases from 0.88 to 0.79 at a dew point of 0 F as the sweep fraction is increased to Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure 2-5. Dry gas flow rate as a function of dew point for various sweep fractions: solid 0; short dash 0.1; and long dash 0.2 sweep fractions. 37

58 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure 2-6. Dry gas recovery as a function of dew point for various sweep fractions: sweep fractions: solid 0; short dash 0.1; and long dash 0.2 sweep fractions. For the higher dew points, flow rates are high and significant pressure drops can occur. For the results in Figures 2-5 and 2-6, the highest pressure drops were ~8 kpa. Large pressure drops reduce the partial pressure driving force for water transport and consequently are detrimental to recovery. The changes in performance for a lower waternitrogen selectivity of 100 are illustrated in Figures 2-7 and 2-8. Relative to a selectivity of 1000, dry gas recovery decreases slightly but dry gas flow rates are significantly lower. A decrease in recovery is expected with a decrease in selectivity since more dry air will permeate along with the water. For a dew point of 0 F and sweep fraction of 0.2, the recovery decreases from 0.79 to The large drop in dry gas flow rate is due to the change in water permeance with selectivity. For the calculations, the nitrogen permeance 38

59 was held constant so the decrease in selectivity led to an equal decrease in water permeance Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure 2-7. Dry gas flow rate as a function of dew point for a water-nitrogen selectivity of 100 and various sweep fractions: solid 0; short dash 0.1; and long dash 0.2 sweep fractions. However, the flow rate change is less than the permeance change; one expects the changes to be comparable if pressure drops in the fiber lumen are small and the permeate concentration does not change. For a dew point of 0 F and sweep fraction of 0.2, the dry gas flow decreases by a factor of ~6 (from 2 to 0.3 SCFM) as the selectivity and water permeance decrease by a factor of 10. The decrease in flow rate is less than the decrease in permeance because more dry air permeates with the water. This reduces the water concentration in the permeate which in turn increases the water partial pressure difference that drives transport across the membrane. Consequently, the rate of water removal and 39

60 the amount of humid gas that can be dried increase. Similar changes with selectivity have been reported previously [16, 17]. 0.9 Dry Gas Recovery Figure 2-8. Dry gas recovery as a function of dew point for a water-nitrogen selectivity of 100 and various sweep fractions: solid 0; short dash 0.1; and long dash 0.2 sweep fractions Effect of Fiber Property Variation Figure 2-9 and figure 2-10 illustrate the effect of the inner diameter variation on the module performance. Inner diameter variability is detrimental to performance dry gas recovery and flow rate decrease as the variability increases. The dry gas flow rate decreases by a factor of two with 20% inner diameter variation for the lower product dew points. Changes in product gas recovery are smaller and do not exceed 5% at the lowest dew points considered. Dew Point ( o F, 1 atm) 40

61 Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure 2-9. Dry gas flow rate as a function of dew point for different ID variation solid no variation, short dash inner diameter variation =0.1, long dash inner diameter variation=

62 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Dry gas recovery as a function of dew point for different ID variation solid no variation, short dash inner diameter variation =0.1, long dash inner diameter variation= Effect of Boundary Layer Resistance Figures 2-11 and 2-12 illustrate the effect of lumen and shell mass transfer boundary layer resistances on the module performance. These figures compare performance with and without the inclusion of gas phase resistances. 42

63 10 Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure Dry gas flow rate as a function of dew point for different inner diameter variation with and without boundary layer resistance: solid no variation and with boundary layer resistance, short dash inner diameter variation =0.1 and with boundary layer resistance, dash dot no variation and without boundary layer resistance, long dash inner diameter variation =0.1 and without boundary layer resistance. 43

64 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Dry gas recovery as a function of dew point for different inner diameter variation with and without boundary layer resistance: solid no variation and with boundary layer resistance, short dash inner diameter variation =0.1 and with boundary layer resistance, dash dot no variation and without boundary layer resistance, long dash inner diameter variation =0.1 and without boundary layer resistance. Figure 2-11 indicates the dry gas flow rate increases by ~70% at a dew point of 0 F in the absence of gas phase mass transfer resistances for a sweep fraction of 0.1 and no inner diameter variation. Figure 2-12 indicates the dry gas recovery increases 1% for the same conditions. When inner diameter variation is considered, similar results are obtained. The dry gas flow rate increases by ~90% at a dew point of -30 F in the absence of gas phase mass transfer resistances and gas recovery increases ~3%. Gas phase resistances reduce product flows by reducing the overall mass transfer coefficient and associated gas permeation rate. Additionally, gas phase resistances reduce the effective selectivity of the permeation process as diffusion through concentration 44

65 boundary layers is significantly less selective than permeation through the membrane. Table 2.2 indicates the individual contributions to the overall mass transfer resistance for a sweep fraction of 0.1 and no property variation. Table 2.2: Boundary layer resistance percentage. Dew point ( F) Membrane Lumen side Shell-side Total Boundary Resistance % Resistance % Resistance % Resistance % % 3.9% 53% 57% % 3.8% 50% 54% % 3.7% 46% 50% 0 56% 3.6% 41% 44% 10 63% 3.4% 34% 37% 20 71% 3.1% % 30 80% 2.6% 18% 20% Effect of Sweep Variation Figures 2-13 and 2-14 illustrate the effect of sweep variation on module performance. An average sweep fraction of 0.1 was used for the calculations. Variability in the sweep flow around individual fibers has little effect on module performance. Only slight decreases in dry gas recovery and flow rate occur as the variability in sweep flow increases. The dry gas flow rate drops by less than 10% over the 45

66 range of dew points considered as the variability in sweep flow rate increases to 20%. The dry gas recovery changes by less than1%. Figures 2-15 and 2-16 compare the effects of inner diameter and sweep variation on module performance. The effect of inner diameter variability is significantly larger. The effect of sweep variability in air dehydration is comparable to that observed previously for fiber variability in nitrogen production from air [47, 61]. Significant changes in module performance are found only for inner diameter variability Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure Effect of sweep variation on dry gas flow rate as a function of dew point: solid variation=0, short dash variation=0.1, long dash variation=

67 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Effect of sweep variation on dry gas recovery as a function of dew point: solid variation=0, short dash variation=0.1, long dash variation=

68 Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure Effect of sweep and ID variation on dry gas flow rate as a function of dew point: solid sweep variation=0, short dash sweep variation=0.2, long dash ID variation=

69 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Effect of sweep and ID variation on dry gas recovery as a function of dew point: solid sweep variation=0, short dash sweep variation=0.2, long dash ID variation= Explicit Calculation of Sweep Distribution Theory To explicitly calculate the shell and lumen flow distributions, the shell-side and lumenside spaces are treated as bi-continuous, anisotropic porous media as show in figure Volume averaging the conservation equations for the lumen and shell spaces yields conservation equations in terms of volumetric average values of the field variables (velocity, pressure, and concentration) where the averaging volume is small compared to the macroscopic dimensions of the module but larger than fiber dimensions. 49

70 Figure Porous media model for hollow fiber module. For low Reynolds number flows, volume average of the conservation of mass equations yields Darcy s law as the relationship between superficial velocity and pressure: k u = p η (2-21) Where k denotes the hydraulic permeability of the porous medium and u is the volume average superficial velocity. Note that for anisotropic porous media k is a tensor. However, previous work [34] shows that for sufficiently high module aspect ratios (i.e., 50

71 module length to diameter ratios) the effect of anisotropy on module performance is negligible so one may assume the porous media are isotropic. The steady-state volume average conversation of mass equation for component i is given by Equation (2-22) (ρ u ) = ± i J i (2-22) Where ρ is the fluid density (kmol/m 3 ) and J the permeation flux defined in Equation (5-5). For a lumen fed module, the positive sign is used for the shell flow and the negative for the lumen flow. Note that mass transfer due to molecular diffusion and Taylor dispersion is neglected relative to convection as suggested in the literature [35]. Summing Equation (2-23) over all of the components yields the continuity equation for the porous media. (ρ u) = ± n J i i= 1 (2-23) The shell and lumen velocity fields are obtained by substituting Darcy s law, Equation (2-21) for the velocity and applying proper boundary conditions. An appropriate equation of state is required to calculate density from pressure. The ideal gas law is used for the relatively low pressure dehydration process considered here. 51

72 Figure Boundary conditions used in explicit simulations of sweep distribution to evaluate module performance: a) lumen side boundary conditions and b) shell side boundary conditions. Note that the boundary conditions for the shell correspond to one of the configurations used to simulate shell flows. Similar boundary conditions apply for the others. The shell extension allow establishment of uniform velocity and concentration fields as described previously [34]. Shell and lumen boundary conditions for external sweep ports are illustrated in Figure 2-18 for an axisymmetric module cross-section. Symmetry is applied along the module 52

73 centerline while the radial velocity and radial component mass flux are set to zero along the external case. The pressures along the inlet and outlet are specified for the lumen and shell regions. The lumen and shell flow rates are controlled by the magnitude of the pressure drop between inlet and outlet. Equations (2-21)-(2-23) are solved using COMSOL Multiphysics [64]. This simulation environment solves the governing conservation equations using the finite element method [65]. It also accommodates introduction of the appropriate form for the permeation flux and its dependence on gas partial pressure. The model for this simulation is a 2D model which has 2 domains: lumen and shell. In Comsol Multiphysics, the shell and lumen domains are coupled by adding a pseudoreaction term for transport as shown in Equation (2-24). R i = J i (2-24) Effect of Sweep Distribution Explicit sweep distribution calculations were performed for three different configurations as illustrated in Figure The sweep is introduced either 1) on the periphery of the fiber bundle adjacent to the tube sheet, 2) internally from within fiber bundle, or 3) on the periphery offset partially from the tube sheet. Figures 2-20 and 2-21 illustrate the performance predictions for the different configurations. The locations of the inlet and outlet regions for the sweep have little effect on performance. Dry gas flow rate and recovery decrease by less than 5% over the dew point range considered. The change in performance increases as dew point increases. Such results are consistent with the results obtained assuming a Gaussian distribution of the sweep around each fiber variations in sweep flow rate do not significantly affect module performance. 53

74 Lumen Shell Shell Internal Offset Figure Three different sweep configurations used in explicit sweep distribution calculations to determine module performance. The arrows indicate the macroscopic flow direction. The thick solid lines indicate the location of the inlet and outlet. The simulation results provide detailed concentration profiles within the module. Figure 2-22 illustrates a typical radial concentration profile for a module cross-section along the tube sheet at the retentate outlet (i.e., the dry gas product end). Clearly radial variations in concentration exist. The shell concentration increases by 600% from the periphery of the fiber bundle to the centerline while the lumen concentration increases by 60%. The difference in lumen and shell concentrations decreases by 400%. Surprisingly, the large variations in concentration that exist do not impact overall module performance. The mixing cup average concentrations calculated from the concentration and velocity fields are nearly identical to the concentrations calculated assuming uniform sweep distribution and no radial variation in the concentration or velocity fields. This 54

75 fortuitous result implies the inlet and outlet locations for the sweep are not a critical design variable Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure Effect of sweep configuration on dry gas flow rate as a function of dew point: diamond internal, circle shell, triangle offset. The solid line corresponds to uniform sweep distribution. Note the diamond and circle symbols overlap. 55

76 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Effect of sweep configuration on dry gas recovery as a function of dew point: diamond internal, circle shell, triangle offset. The solid line corresponds to uniform sweep distribution. Note the diamond and circle symbols overlap. 56

77 Molar density (mol/m 3 ) Radial distance (cm) Figure Variation in water concentration in the radial direction along the retentate outlet (i.e., the dry gas product end): solid lumen and short dash shell. The results are for operating conditions that give a dry gas dew point of -30 F Effect of Fiber Packing Variation Along Case The literature indicates that large variations in packing of spheres [66] or fabrics [67] can occur at the boundary between the packing and a solid case enclosing the packing. At the packing-case interface, the packing fraction can decrease (i.e., the void volume increase) relative to the bulk packing. Increased void volume will lead to significant increases in hydraulic permeability and preferential flow at the packing-case interface. Such preferential flow is referred to as race tracking since fluid flows are much higher than the superficial flow within the packing and can impact heat and mass transfer [68, 69]. The observation of large shell concentration gradients at the periphery of the fiber bundle suggests race tracking phenomena may have a significant impact on module performance. 57

78 To test this hypothesis, the hydraulic permeability for the shell side flow was assumed to have the following dependence on the radial coordinate: k = k 0 exp( αr) (2-25) where k 0 is the hydraulic permeability of the fiber bundle and α characterizes the extent and magnitude of the variation as α increases the extent and magnitude increase. Equation (2-25) introduces an exponentially increasing hydraulic permeability with r. The maximum value occurs at the bundle-case boundary which will lead to race tracking for sufficiently large. Figures 2-23 and 2-24 illustrate the effect of the variable hydraulic permeability on module performance. For a fixed dew point, dry gas flow rate and recovery decrease as α increases. The flow rate decreases by up to 50% for the largest value of α but recovery decreases by less than 10%. 58

79 Dry Gas Flow (SCFM) Dew Point ( o F, 1 atm) Figure Effect of variable fiber packing near the module case on dry gas flow rate as a function of dew point. The symbols correspond to different values of α: diamond 0, square 200, triangle 400, circle 600. The upper solid line corresponds to uniform sweep distribution. The lower solid line is provided as a guide for the reader. Reduced fiber packing at the bundle-case boundary is detrimental to performance. However, characterization of fiber packing in actual modules is required to determine the appropriate value for α and the impact of race tracking on performance. 59

80 Dry Gas Recovery Dew Point ( o F, 1 atm) Figure Effect of variable fiber packing near the module case on dry gas recovery as a function of dew point. The symbols correspond to different values of α: diamond 0, square 200, triangle 400, circle 600. The upper solid line corresponds to uniform sweep distribution. The lower solid lines are provided as guides for the reader. 2.3 Conclusions An analysis of sweep uniformity on the performance of hollow fiber gas dehydration modules is presented. The analysis is based on the governing conservation of mass and momentum equations and appropriate expressions for the overall mass transfer coefficient. In all cases, for a given product dew point, the dry gas flow rate increases as the fraction of the product used as permeate sweep increases. However, the dry gas recovery (the fraction of the wet feed recovered as product) simultaneously decreases. The optimal sweep fraction will depend on the trade-off between increased productivity and fractional loss of the dry, high pressure product. 60

81 The effect of non-uniform sweep distribution is examined by: 1) assuming a Gaussian variation in the sweep flow around each fiber and 2) explicitly calculating the sweep distribution within the bundle for specified sweep inlet and outlet locations. In both cases, non-uniform sweep flows has little effect on module performance. The explicit sweep distribution calculations indicate large radial concentration gradients are present in the module. Surprisingly, these concentration gradients are not detrimental to performance. If fiber packing is allowed to decrease at the interface between the fiber bundle and the enclosing case, preferential flow will occur along the interface relative to the center of the bundle. Such flow is detrimental to performance for sufficiently large variation in hydraulic permeability. To determine the magnitude of this effect in commercial modules, fiber packing variations need to be characterized. The analysis presented here will be useful in evaluating alternative methods for introducing sweep in membrane modules for both gas and liquid separations. Although uniform sweep distribution appears not to be critical for air dehydration, other applications may be more sensitive. 61

82 Chapter 3 Applicability of the Hagen-Poiseuille Law for Pressure Drop Calculations in Hollow Fiber Gas Separation Modules 3.1 Simulation Method The Hagen-Poiseuille Law, Equation 3-1, is the analytical solution to the Navier Stokes equations used to calculate the pressure drop of incompressible Newtonian fluids through straight cylindrical tubes without permeation across the tube wall. The objective of this chapter is to evaluate the applicability of the Hagen-Poiseuille Law for the compressible flows through cylindrical tubes with porous walls that are encountered in hollow fiber gas separation modules. dp 8µ Q = 4 dz πr (3-1) Most one-dimensional models of membrane module performance assume the Hagen- Poiseuille equation may be used for incompressible flows with wall permeation by allowing the flow rate Q to vary with axial distance along the module. Predictions of module performance are obtained by integrating the governing transport and pressure drop equations [40-45, 70, 71]. In some limiting cases, analytical solutions can be obtained. 62

83 For membrane gas or vapor separations, the Hagen-Poiseuille equation typically is modified to account for fluid compressibility by using the ideal gas law to relate volumetric flow rate to molar flow rate. However, any other equation of state may be used. The molar flow rate is allowed to vary within the module to account for permeation. dp 8µ Q 8µ nrgt = = (3-2) 4 4 dz πr πr P In this work two dimensional (2D) computational fluid dynamics (CFD) is used to obtain numerical approximations to the solutions of the conservation of mass and momentum equations in a single fiber that is assumed to be representative of all fibers in the fiber bundle. The result is obtained by rigorously solving the conservation of momentum and mass equations using COMSOL Multiphysics. This computational transport package uses the finite element method to transform the governing differential equations to an approximate set of non-linear algebraic equations. In this paper, lumen-fed module with shell side vacuum is investigated, as illustrated by Figure 3-1. Equation (3-3) describes mass transfer in the lumen side. In this equation c i denotes the concentration of the ith component(mol/m 3 ), D the gas phase diffusion coefficient (m 2 /s) and u the velocity (m/s) in the gas phase. ( D c + c u) = 0 i i (3-3) For momentum transport, the compressible Navier-Stokes equations and the continuity equation with constant viscosity is used. The density is calculated from the ideal gas law. ρ + ( ρu) = 0 t (3-4) 63

84 u ρ t + ρu u = p + (η( u + ( u) T ) F (3-5) Here η denotes the dynamic viscosity (kg/(m s)), ρ the fluid density (kg/m 3 ), and p the pressure (Pa). The stress tensor used in the above equation describes a Newtonian fluid. The boundary conditions for CFD simulation are as bellows. At the inlet of lumen side, the flow is assumed to be pressurized at a given pressure p 0. z = 0, p = p 0 (3-6) At the outlet of lumen side, the flow is assumed to be a zero gauge pressure. z = L, p = 0 (3-7) The wall between lumen and membrane is considered as a moving wall and the velocity is a function of the permeation rate, as shown in Equation (3-8). r = r, v N M / ρ = 1 w i i (3-8) where N i is molar flux of each component i at the wall, M i the molecular weight of the component i. At the center of lumen side, a symmetrical condition is used. r = 0, n u = 0 (3-9) At the inlet of lumen side, the concentration of each component i is specified at c j0. z = = 0, c i ci 0 (3-10) 64

85 p = 0 convective flux N = QR T v i w = i i i g N M c i / ρ axial symmetry p = p 0 c i = c i 0 Figure 3-1. Boundary conditions used to solve the conservation of mass and momentum equations in the retentate domain for a lumen fed module. N i is flux of species i and M i the molecular weight of the species i. At the outlet of lumen side, the flow is assumed to be convective flux. z l, n ( D c ) = 0 (3-11) = i At the wall between lumen and membrane, the flux is calculated by Equation (3-12). r = r ) 1, Ni = n ( D ci + ciu = Qi RgTci (3-12) 65

86 The pressure drop results from the Hagen-Poiseuille law and CFD method are compared in the following sections. 3.2 Fluid Flow for Single Component Simulations were performed initially for the flow of a single component, incompressible fluid in a hollow fiber membrane with porous walls. Figures 3-2 and Figure3-3 illustrate the percentage discrepancy between the pressure drop calculated from the Hagen- Poiseuille law (modified to allow the flow rate to change with permeation) and the full 2D calculations for a constant wall velocity and constant permeability boundary condition, respectively. The Hagen-Poiseuille results were obtained by numerical integration of Equation (3-1). The difference between the two increases as the fraction of Figure D-2D Pressure drop difference for incompressible fluid with constant wall velocity. Figure D-2D Pressure drop difference for incompressible fluid with constant wall permeability. 66

87 the feed that permeates (f) increases. The difference also increases as the Reynolds number (Re) number, based on in the feed bulk velocity, increases. However, the difference is less than 10% for f < 0.8 and Re < For compressible flow in a hollow fiber membrane with constant wall flux, an analytical solution for the pressure drop can be obtained using the Hagen-Poiseuille equation modified as described previously to allow the molar flow rate to change with permeation. p = pi p 2 i 16µ LQ i 4 πr 1 f (3-13) In the equation (3-13), p i is the inlet pressure, L the length of the fiber, Q i the inlet volume flow rate, µ the viscosity, r the radius of the fiber, f the permeation fraction defined by equation (3-14). N the wall flux, and n i the molar feed rate. f = 2πrLN n i (3-14) The percentage difference between the pressure drops calculated using the modified Hagen-Poiseuille law and the full 2D calculations for the flow of compressible, single component fluid are illustrated in Figures 3-4 and 3-5 for a constant wall flux and constant permeance boundary conditions, respectively. The Hagen-Poiseuille results were obtained by numerical integration of Equation (3-2) for the constant permeance boundary condition. As for the single component, incompressible fluid results, the error increases with increasing f or Re and is less than 10% for f < 0.8 and Re <

88 Figure D-2D Pressure drop difference for compressible fluid with constant wall flux Figure D-2D Pressure drop difference for compressible fluid with constant wall permeance 3.3 Gas Separation Module Performance The air dehydration and air separation are simulated by using 1D and 2D method in this work. In the simulation, the shell side is assumed to held under an absolute vacuum and the pressure is set to zero. The product flow rate and product recovery are used to evaluate the performance of the module. Figure 3-6, Figure 3-7 illustrate module performance predictions for air dehydration obtained using the modified Hagen- Poiseuille law to calculate lumen pressure drop and the full 2D calculations. The two results are in good agreement. The differences in predicted product flow rates are less than 1% for product dew points ranging from -40 to 30 F [62,63]. For this range of operating conditions, pressure drops are less than 10% of the feed pressure. 68

89 7.00E E-07 Flow Rate (m 3 (STP)/s/fiber) 5.00E E E E E E Dew Point (1 atm, 0 F) Figure 3-6. Dependence of product flow rate on as product dew point for air dehydration: dash- 1D result; solid-2d result. 69

90 1.00 Retentate Recovery Dew Point (1 atm, 0 F) Figure 3-7. Dependence of retentate recovery on as product dew point for air dehydration: dash- 1D result; solid-2d result. Figures 3-8, 3-9 illustrate performance predictions for the N2 /O2 separation. The 1D and 2D results are in good agreement with each other. Predicted retentate recoveries differ by less than 1% over the retentate product purity range considered. 70

91 5.00E E-10 Flow rate (m 3 (STP)/s/fiber) 4.00E E E E E E E E E x(o 2 ) Figure 3-8. Product flow rate versus O 2 purity for O 2 /N 2 separation: dash-1d result; solid-2d result 71

92 Retentate Recovery x(o 2 ) Figure 3-9. Retentate recovery versus O 2 purity for O 2 /N 2 separation:dash-1d result; solid-2d result. 72

93 Retentate Recovery x(o 2 ) Figure Axial diffusion effect on performance: short dash-1d result; solid-2d result; long dash: 2D isotropic diffusion. However, while performing the simulations, an interesting observation was made. The length of the fiber used in the simulations was reduced in some cases to allow greater mesh refinement and check the effect of refinement on the solution. When the length was reduced by a factor of 10, performance predictions were much poorer as shown in Figure As the recovery decreases, the oxygen concentration approaches a limiting value greater than zero for the shorter fiber while the concentration approaches zero for the longer fiber. Additionally, the recovery is much lower for the shorter fiber than the longer fiber over the product concentration range achievable with the shorter fiber. 73

94 Upon examining the results, axial diffusion was hypothesized to be responsible for the drop in performance. To verify this, an anisotropic diffusion was used in the full 2D calculations. The diffusion coefficient in the axial direction was set to zero while the diffusion coefficient in the radial direction was not changed. The results obtained using this anisotropic diffusion coefficient are in excellent agreement with the 1D results. Therefore axial diffusion appears to be responsible for the drop in performance. Axial diffusion mixes the gas in the axial direction and makes the module perform like a single stage. This observation and explanation would not be possible without the full 2D simulation. 3.4 Conclusions The Hagen-Poiseuille law is a good approximate solution of pressure drop for sufficiently low Reynolds number of the feed and permeation fraction for compressible flows. Pressure drops are small for many gas separations so pressure drop calculation does not affect performance predictions in the absence of fiber variability. CFD 2D simulations can reveal performance changes due to non-idealities such as axial diffusion. 74

95 Chapter 4 Effect of Boundary Layer Resistance on Mass Transfer Coefficients 4.1 Simulation Method For gas separations, concentration polarization can be significant when the fast gas permeance is greater than 1000 GPU [16, 17, 48-50]. This is the case for many membrane dehydration separations such as compressed air drying. For air dehydration modules, the overall mass transfer coefficient is calculated by summing the lumen-side, membrane, and shell-side mass transfer resistances. The shell and lumen side boundary resistances commonly are calculated using theoretical and empirical correlations such as the Leveque and Donohue equations [16, 17], respectively. Virtually all correlations used for mass transfer coefficients are developed by assuming a constant wall concentration or flux (i.e., permeation rate). This is rarely true in real separation processes. However, the differences in mass transfer coefficients for the two cases can be significant [52-54]. To determine the effect of non-uniform wall concentration and flux on concentration boundary layers, numerical approximations to the solutions of the conservation of mass and momentum equations are obtained using a two dimensional (2D) computational fluid dynamics (CFD). The solutions will be obtained for a single fiber that is assumed to be 75

96 representative of all fibers in the fiber bundle. COMSOL Multiphysics is used to implement the CFD simulation. This computational transport package uses the finite element method to transform the governing differential equations to an approximate set of non-linear algebraic equations. To validate the numerical approximations, the solutions are compared to analytical solutions to the conservation of mass equation obtained for a constant wall concentration boundary condition. A new solution for constant wall permeance is derived for comparison as well Modeling for lumen side To focus on the lumen-side gas phase mass transfer resistance, simulations were performed assuming the permeate is held at zero pressure. For the lumen feed case, Figure 4-1 illustrates the model and the boundary conditions. Equation (4-1) describes mass transfer in the domain. In this equation c i denotes the concentration of the contaminant (mol/m 3 ), D the gas phase diffusion coefficient (m 2 /s) and u the velocity (m/s) in the gas phase. ( D c + c u) = 0 i i (4-1) For momentum transport, the compressible Navier-Stokes equations and the continuity equation with constant viscosity is used. The density is calculated from the ideal gas law. ρ + ( ρu) = 0 t (4-2) u ρ t + ρu u = p + (η( u + ( u) T ) F (4-3) 76

97 Here η denotes the dynamic viscosity (kg/(m s)), ρ the fluid density (kg/m 3 ), and p the pressure (Pa). The stress tensor used in the above equation describes a Newtonian fluid. p = 0 convective flux N = QR T v i w = i i i g N M c i / ρ axial symmetry p = p 0 c i = c i 0 Figure 4-1. Boundary conditions used to solve the conservation of mass and momentum equations in the retentate domain for a lumen fed module. N i is flux of species i and M i the molecular weight of the species i. The local mass transfer coefficient k is calculated from Equations (4-4) and (4-5) where N i is the normal total flux at the wall, c ib the bulk concentration of speciesi, c iw the wall concentration of species i. 77

98 k ilocal = c ib Ni c iw (4-4) c ib = rf 0 rf 2πc v rdr 0 i z z 2πv rdr (4-5) The local Sherwood number is calculated from Equation (4-6) and the average Sherwood number is calculated from Equation (4-7), where L is the length of hollow fiber. Sh Sh 2rK = D Ni c i = * local cib iw i av L = 0 Sh ilocal L dl 2r D (4-6) (4-7) Analytical solution for Sherwood number for constant permeance boundary condition Figure 4-2. Concentrationn profile in the lumen of hollow fiber. An analytical solution is available in the literature for the entry lumen-side mass transfer coefficient for a constant wall concentration boundary condition if mass transfer rates are 78

99 sufficiently low, the flow in the fiber lumen is laminar, and the velocity profile is fully developed at the start of the mass transfer region [72]. A new analytical solution is derived here for a constant wall permeance boundary condition. The qualitative behavior of the concentration distribution along the fiber is shown schematically in Fig When the feed enters the channel, the faster diffusing gas species (water) is reduced in a thin layer near the fiber wall due to permeation across the membrane. The concentration polarization boundary layer grows rapidly in the immediate neighborhood of the inlet. Beyond this entry mass transfer region, the concentration boundary layer reaches the fiber centerline and fills the channel. Additionally, the concentration decreases in a linear fashion with distance in the flow direction. For a fully developed velocity profile the mathematical problem reduces to the solution of the steady form of the convective diffusion equation for the solute concentration. vz 2 c 1 c c c = D( ( r ) + ) 2 z r r r z (4-8) The velocity profile in a circular tube is given by Equation (4-9)[39]. v z = v z,max 2 r ( 1 ) 2 R (4-9) Equation (4-8) simplifies to Equation (4-10) if axial diffusion may be neglected near the inlet as expected in the entry mass transfer region based on dimensional analysis. vz c z = D c c ( r ) r r r (4-10) The boundary conditions for a constant wall permeance are 79

100 z = 0 c = c 0 (4-11) c r = R QRgTcw = D (4-12) r c r = 0 = 0 r (4-13) Setting y = r R and c c c c * 0 =, in the developing region (y << R) the velocity profile 0 and the conservation of mass equation further simplify to Equations (4-15) and (4-16), respectively, where vis the average velocity which is defined by Equation (4-14). 1 v = ρ i c i ρ y v z = 4 R i v (4-14) (4-15) vz * c z 2 c* = D 2 y (4-16) The mass transfer boundary conditions become: z = 0 c * = 0 (4-17) * * c y = 0 QRgTc w = D (4-18) y The two-dimensional conservation of mass equation, Equation (4-16), can be transformed into a one-dimensional equation by introducing a dimensionless axial coordinate, ξ, and dimensionless radial coordinate,η, as defined in Equations (4-19) and (4-20). 3 QR g TR D z ξ = D (4-19) 4Rv R 80

101 y η = ξ 1/3 QR T g D (4-20) and assuming the dependence of * c on ξ and η is given by: * 1/ 3 c = ξ f ( η) (4-21) From Equations (4-19) (4-21), one can demonstrate that the derivatives of respect to y and z are given by Equations (4-22) and (4-23), respectively. * c with c z df dη 3z 1/3 * 1/3 ξ η fξ = 3z (4-22) 2 c 2 y * = 2 / d f 4v ξ 1/ (4-23) 2 dη RDz Substituting Equations (4-15) and (4-22)-(4-23) into the conservation of mass equation, Equation (4-16) transforms from a partial differential equation into the following 3 ordinary differential equation, under the assumption that ξ 1 / 0. 2 d f 2 dη 2 η + 3 df ηf dη 3 = 0 (4-24) 3 The assumption that ξ 1 / 0 is appropriate in the entry mass transfer region for sufficiently small distances z or permeances Q The boundary conditions become f 0 asη df ( 0) = 1 dη (4-25) (4-26) 81

102 1/ 3 The second boundary condition also requires ξ 0. The requirement of small ξ is analogous to a slow wall reaction rate condition, the limit of which is the zero mass transfer statement c = 0. The solution for f (η ) is y 3 η 3 9 η η 9 f ( η) = 1.536( e ηe dη) 3 (4-27) η Using Equation (4-6a), the local Sherwood in the lumen side is calculated. Sh local = 1.302Gr 1/ 3 (4-28) Where Gr is Graetz number defined as: 2 2r (2r) v Gr = ReSc = L DL (4-29) Modeling for lumen side and shell side To simulate flow and mass transfer in individual fibers, the axisymmetric computational domain illustrated in Figure 4-3 is used. A fiber of specified length encloses a lumen space and is surrounded by an annular shell space. The outer shell boundary is a symmetry boundary that approximates the effect of surrounding fibers. Such an equivalent annular region has been used extensively in past analyses of heat and mass transfer and its dimensions are determined by the fiber packing fraction [73, 74]. Only countercurrent flows are considered here although the work could easily be extended to concurrent flows. Gas is transported by diffusion and convection in the two gas phases, whereas diffusion is the only transport mechanism in the membrane phase. Equations 4-30 through

103 describe mass transfer in the three domains illustrated in Figure 4-3: lumen (retentate), membrane, and shell (permeate). In these equations c j,i (mol/m 3 ) denotes the concentration of component i in the respective phase, D the gas phase diffusion coefficient (m 2 /s), D m,i the diffusion coefficient of component i in the membrane, and u the velocity (m/s) in the respective gas phase. D c1, i + c1 u) = 0 (, i ( D m c2, ) =, i i 0 (4-30) (4-31) ( D c3,, = i + c3 iu) 0 (4-32) For momentum transport in the two gas phases (retentate and permeate domains), the compressible Navier-Stokes equations and the continuity equation with constant viscosity is used. The density is calculated from the ideal gas law. ρ + ( ρu) = 0 t u ρ + ρu u = p + (η( u + ( u) t T ) F (4-33) (4-34) The boundary conditions for the calculations are illustrated in Figure

104 Model domain Ω Retentate Ω Membrane Ω Permeate Figure 4-3. Three domain model for a single fiber in the hollow fiber membrane module. 84

105 p = 0 convective flux insulation p = p 20 c i = c is 0 v J w in = v( Q, c) = J out axial symmetry v J w in = v( Q, c) = J out insulation v J w in = 0 = J out p = p 0 c i = c i 0 p = 0 convective flux insulation Figure 4-4. Boundary conditions used to solve the conservation of mass and momentum equations in the retentate, membrane, and permeate domains. 4.2 Evaluation of Boundary Layer Coupling Effects Lumen side with shell vacuum The Leveque equation, Equation (4-35), expresses the average Sherwood number dependence on the Graetz number for a constant wall concentration boundary condition. The local Sherwood number is calculated using Equation (4-36). As demonstrated 85

106 previously, the analytical solutions for average and local mass transfer coefficient for a constant wall permeance boundary condition are given by Equations (4-37) and (4-38) respectively. Table 4.1 summarizes these expressions and the available expressions for the Sherwood number in the well developed mass transfer limit, Gr>>1. Sh = 1.632Gr 0.33 (4-35) Sh l ocal d( LSh) dl = = Gr (4-36) Sh =1.95Gr 0.33 Sh local = 1.30Gr 0.33 (4-37) (4-38) Table 4.1: Mass transfer coefficients in the lumen with shell vacuum. Constant wall permenace Gr >>1 Sh local = 1.30 Gr 1/3 Constant wall concentration Sh local = 1.08 Gr 1/3 Sh av = 1.95 Gr 1/3 Sh av = 1.63 Gr 1/3 Gr << 1 Sh = 3.64 The local Sherwood number also was calculated from CFD simulations for these two cases to validate the simulation. Figure 4-5 illustrates the variation of the local Sherwood number with the Graetz number in the lumen with a shell vacuum. For Gr > 1, the Leveque solution and CFD simulations for constant wall concentration are in good agreement. For the constant wall permeance case, the local Sherwood number is 20% larger than that for the constant wall 86

107 concentration case for Gr > 1; the Sherwood number for constant wall permeance is larger than that for constant wall concentration for Gr < 1 as well but the two are comparable for Gr ~ Sh E E E E+04 Gr Figure 4-5. Comparison of predicted relationship between Sh local and Gr with the results of Leveque solution: Triangle CFD simulation for constant wall permeance; Square CFD simulation for constant wall concentration; dotted line analytical solution for constant wall permeance; solid line Leveque solution for constant wall concentration. 87

108 4.2.2 Coupling of Boundary Layer in Lumen side and Shell Side Figure 4-6 illustrates the lumen local Sherwood number dependence on Gr for constant wall permeance with and without shell vacuum. The shell vacuum may increase the driving force between the lumen and shell but has little effect on the mass transfer coefficient. 100 Sh Gr Figure 4-6 Comparison of predicted relationship between lumen Sh local and Gr with shell vacuum for constant wall permeance: dotted line CFD simulation; solid line analytical solution for shell vacuum. 88

109 Figure 4-7 illustrates the shell local Sherwood number dependence on Gr for constant wall permeance. The CFD results are nearly 2-3 orders magnitude greater than the results from the Donahue equation, but are comparable to mass transfer coefficients for constant wall concentration and constant wall flux boundary conditions in regular fiber packings in hollow fiber modules [46]. The well developed lumen Sherwood number for different wall boundary conditions are shown in the table 4.2. Table 4.2: Well developed Mass transfer coefficients in the shell. wall boundary conditions lumen Sherwood number Constant wall concentration 9.89[46] Constant wall flux 11.8[46] Constant wall permeance The large differences between the CFD results for shell mass transfer coefficients and the values obtained from the Donahue equation most likely arise from the complexity of the fiber bundle and the flows within it. This complexity arises primarily from four factors: 1) random fiber packing, 2) non-parallel fibers in the module, 3) cross-flow regions near the shell entrance and exit ports, and 4) non-uniform fluid flow from the shell distribution manifold into the fiber bundle [11]. 89

110 1.00E E+01 Sh 1.00E E E E E E E E E E+03 Gr Figure 4-7. Comparison of the predicted relationship between shell Sh local and Gr with for constant wall permeance: solid line CFD simulation; dotted line Donahue equation. 4.3 Conclusions Analytical and numerical solutions for the lumen-side mass transfer coefficient for a constant wall permeance boundary condition with shell vacuum are presented. The two results are in good agreement in the entry mass transfer region, Gr >> 1. Average mass transfer coefficients for a constant wall permeance boundary condition, Equation 4-36, are ~20% larger than the values for a constant wall concentration boundary condition, Equation In the well developed mass transfer limit, the Sherwood number for a constant wall concentration and wall permeance boundary condition are given by 3.64 and 4.3, respectively. Equation (4-39) provides a good approximation of the Sherwood number 90

111 dependence on the Graetz number for the entire range of Gr for a constant wall permeance boundary condition. 1/ /3 Sh local = (( 1.95Gr ) ) (4-39) The first term in the sum captures the dependence in the entry region while the second captures the dependence in the well-developed region. The shell mass transfer coefficient in the well-developed limit is 2-3 orders magnitude greater than the value obtained from the Donahue equation. The difference is speculated to arise from the complexity of flow into and through the fiber bundle. Additional work is required to confirm this. 91

112 Chapter 5 Evaluation of a Commercial Air Dehydration Hollow Fiber Membrane Module In this chapter, the performance of a commercial air dehydration hollow fiber membrane module is evaluated experimentally. 5.1 Experimental Method Dehydration Experiment The hollow fiber module GMD 210 used in this study was provided by Generon IGS. The experimental apparatus is illustrated in Figure 5-1. The module is lumen-fed and operated at 90 psig and permeate flows counter-currently to the retenate. The dew points of the feed, sweep, permeate and retentate are measured using a Vaisala DMT242 dew point transmitter. The flow rate of feed, sweep, permeate and retentate are measured using Top Track Model 821-1, OMEGA FSK 8129, 8130 and OMEGA FMA 766-V flow meters which were calibrated with a Precision wet test meter. The feed and the retentate pressure are measured with a standard pressure gauge. A photograph of the experimental apparatus is provided in Figure 5-2. Figure 5-1. Schematic illustration of the experimental apparatus for evaluating air dehydration module performance. 92

113 Figure 5-2. Photograph of experimental apparatus. The composition of the hollow fiber membrane, number of fibers in the module, fiber size, and permeance and selectivity were not provided by the manufacturer. However, the length of the module was estimated by direct measurement to be 0.54 m. Other experimental conditions are summarized in Table 5.1. Table 5.1: Experiment conditions for the evaluation of the air dehydration module. Feed temperature 24±2 oc Feed dew point -8±2 oc Feed pressure 90±3 psi Permeate pressure 14.7 psi Temperature 24±2 oc 93

114 5.1.2 Oxygen/Nitrogen Separation Experiment The oxygen/nitrogen selectivity of the module was determined using the experimental setup illustrated schematically in Figure 5-3. The module is lumen-fed at 90 psi and operated such that the permeate flows counter-currently to the retentate. The oxygen concentration of the feed, retentate, and permeate were measured using ESD model 600 oxygen analyzers. The flow rate of feed, sweep, and product were measured using Top Track Model 821-1, OMEGA FSK 8129, 8130 and OMEGA FMA 766-V flow meters. A photograph of the experimental apparatus is provided in Figure 5-4. A commercial air separation module (Permea inc, Model PPA-22AD) used for the production of nitrogen enriched air was evaluated as well for comparison purposes Figure 5-3 Schematic of evaluation of dehydration module: air separation. 94

115 Figure 5-4. Photograph of experimental apparatus used to evaluate the oxygen/nitrogen separation performance of an air dehydration module and an air separation module Pure Gas Permeation Experiment Pure gas permeation measurements were performed for oxygen and nitrogen. Since the membrane area is not known, only the product of the permeance and membrane area can 95