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Energy efficiency of gas separation by Pressure Swing adsorption McLean, Christopher Ross 1996

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Energy Efficiency of Gas Separation Pressure Swing Adsorption by Christopher Ross M cLean B. Ap. Sc., The University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1996 © Christopher Ross M cLean, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of / V | -IfhrQV C A L - ^ J O f e ) A) i A) C^- , The University of British Columbia Vancouver, Canada Date Pcrr-oSgg 2.5", I C DE-6 (2/88) 11 Abstract Pressure Swing Adsorption (PSA) is a method of separating a mixture of gases into its various components. Cyclic pressure and flow variations, in the presence of a selectively adsorbent material, are used to concentrate one species or group of species at one end of an adsorbent filled vessel, while the other species or group of species is concentrated at the other end. When PSA is used in separating gases, the necessity of gas pressurization and depressurization implies that the process can become very energy intensive. This is especially true in low capacity systems that require small compressors and/or vacuum pumps. There are many ways in which traditional PSA processes have been modified in order to reduce the amount of pressurization energy that is lost. One method is to use high pressure gas from one adsorbent bed to pressurize another adsorbent bed. This "equalization" recovers some of the energy used to initially compress the gas. However, as the gas is throttled from one bed to the other, irreversibilities are introduced into the process. In this thesis, the irreversibilities that are due to throttling are separated from those which are inherent in the PSA process and cannot be removed. The work required to produce a certain amount of gas by various simple PSA cycles is compared to the reversible work required to produce that amount of gas, based on the availability (or exergy) of the gas. The ratio of the reversible work to the actual work required for the PSA cycle is defined as the second law efficiency, and is compared for three cycles: the Four-Step cycle, the Ideal Four-Step cycle, and the Ideal Three-Step cycle. iii The irreversible expansion of gas through throttling valves is shown to account for the majority of the energy losses of the Four-Step cycle. Useful work (represented by the increase in availability of the product and exhaust) is found to be very small compared with the work required by the cycles. The true bed losses, inherent in the PSA process, are found to be similar in magnitude to the useful work, but much less than the energy lost by the throttling irreversibilities. The work required per mole of product to separate the gases decreases as the pressure ratio increases, and the second law efficiency increases with pressure ratio. For the cycle with no energy recovery, the second law efficiency varies widely with the selectivity ratio. A high selectivity ratio (implying a low separation factor) implies more work is required for the separation and the second law efficiency is lower. For the cycles with full recovery of the expansion energy, the work required and the second law efficiency are relatively independent of the selectivity ratio. The equilibrium based semi-analytical results are confirmed by the use of a numerical "Multiple-Cell" model. This model is also used to show that diffusion does not affect the second law efficiency of a cycle when energy recovery is present. iv Table Of Contents A B S T R A C T I I T A B L E O F C O N T E N T S I V L I S T O F T A B L E S V I L I S T O F F I G U R E S V I I L I S T O F S Y M B O L S X A C K N O W L E D G E M E N T X I V D E D I C A T I O N X V 1. I N T R O D U C T I O N 1 1.1 OVERVIEW 1 1.2 C O M M E R C I A L APPLICATIONS OF P S A 3 1.3 SCOPE A N D O U T L I N E OF THESIS 5 2. R E V I E W O F P R E S S U R E S W I N G A D S O R P T I O N 8 2.1 REVERSIBLE W O R K OF G A S SEPARATION 8 2.2 ADSORPTION 20 2.3 B E D D Y N A M I C S 25 2.4 PREVIOUS STUDIES OF E N E R G Y LOSS A N D P S A P E R F O R M A N C E 33 3. M O D E L I N G P S A C Y C L E E N E R G Y C O N S U M P T I O N 38 3.1 INTRODUCTION 38 3.2 FOUR-STEP C Y C L E 39 3.2.1 Introduction 39 3.2.2 Feed Step 43 3.2.3 Blowdown Step 51 3.2.4 Purge Step 55 3.2.5 Pressurization Step 64 3.2.6 Expansion of Product Gas and Net Work 65 3.2.7 Recovery of Species B 68 3.2.8 Discussion of Net Work for the Four-Step Cycle 72 3.3 T H E IDEAL FOUR-STEP C Y C L E 77 3.3.1 Introduction 77 3.3.2 Feed Step 79 3.3.3 Blowdown Step 79 3.3.4 Purge Step 80 3.3.5 Pressurization Step 82 3.3.6 Expansion of Product Gas and Net Work 83 3.3.7 Discussion of Net Work for the Ideal Four-Step Cycle 85 3.4 T H E IDEAL THREE-STEP C Y C L E 89 3.4.1 Introduction 89 3.4.2 Feed Step 91 3.4.3 Evacuation Step 91 V 3.4.4 Pressurization 93 3.4.5 Expansion of Product Gas and Discussion of Net Work 94 4. COMPARISON OF MODEL WITH PREVIOUS STUDIES 97 4.1 INTRODUCTION 97 4.2 C O M P A R I S O N WITH BANERJEE ET A L . , 1990 97 4.2.1 Results of Banerjee et al., 1990 ; 97 4.2.2 Four-Step Cycle 104 4.2.3 Ideal Four-Step Cycle 108 4.2.4 Ideal Three-Step Cycle . 112 4.3 V A C U U M C Y C L E E X A M P L E 115 4.3.1 Introduction 115 4.3.2 Vacuum Four-Step Cycle 115 4.3.3 Vacuum Ideal Four-Step Cycle 117 4.3.4 Vacuum Ideal Three-Step Cycle 119 4.4 APPLICATION TO K A Y S E R A N D K N A E B E L , 1986 120 5. MULTIPLE-CELL MODEL OF A PSA SYSTEM 127 5.1 INTRODUCTION 127 5.2 DERIVATION OF T H E M O D E L 129 5.3 M U L T I P L E - C E L L M O D E L RESULTS 138 6. CONCLUSIONS 141 7. RECOMMENDATIONS 146 BIBLIOGRAPHY 147 APPENDIX A : WORK IN DEPRESSURIZING AND PRESSURIZING ADSORBENT BEDS 149 vi List of Tables Chapter 2 Table 2.1 Equilibrium Adsorption Isotherm Slopes (ICA and kB), and Selectivity Ratio ((3), for Nitrogen and Oxygen on Zeolite 5A 37 Chapter 3 Table 3.1: Summary of Work Terms for the Four-Step Cycle 67 Table 3.2 Summary of Work Terms for the Ideal Four-Step Cycle 84 Chapter 4 Table 4.1 Zeolite 5A Adsorbent Properties 99 Table 4.2 Cycle Properties used in the Energy Comparison 100 Table 4.3 Gas Quantity and Composition 101 Table 4.4 Comparison of Separation Cases 106 Table 4.5 Experimental Parameters: Knaebel and Hil l , 1986 120 Table 4.6 Experimental Results: Knaebel and Hil l , 1986 121 Table 4.7 Ideal Four-Step Theoretical Results for Five Experimental Runs of Kayser and Knaebel, 1986 122 Chapter 5 Table 5.1 Comparison of Multiple-Cell Model (150 Cells) with the Semi-Analytical Results 139 Table 5.2 Effect of Reducing the Number of Cells in the Multiple-Cell-Model 140 List of Figures Chapter 2 Figure 2.1 (0) Mixed Reference System, and (1) Separated Gases 10 Figure 2.2 Reversible Work of Gas Separation Done by the System as a Function of yo 13 Figure 2.3 The Flows in and out of a PSA Gas Separation System 14 Figure 2.4 Reversible Work done By the System in Concentrating One Litre Of Oxygen From Air 17 Figure 2.5 Total Product Pressure vs. Purity for constant O2 Partial Pressure of 3 arm 18 Figure 2.6 Reversible Work to Concentrate and Deliver Product with Oxygen Partial Pressure of 3 atm 19 Figure 2.7 Equilibrium Adsorption Isotherms for Nitrogen and Oxygen on Zeolite 5A 20 Figure 2.8 Movement of Gas Molecules Through an Adsorbent Bed 27 Figure 2.9 Propagation of a Concentration Shock Wave.... 28 Figure 2.10 Formation of a Concentration Shock Wave 30 Figure 2.11 Formation of a Simple Wave During Purge 31 Chapter 3 Figure 3.1 Four-Step Cycle: Pressurization With Product (After Knaebel and Hil l , 1985, Figure 1) 40 Figure 3.2 Propagation of the Shock Wave: Pressurization With Product (After Knaebel and Hi l l , Fig. 2) 42 Figure 3.3 Feed Step for the Four-Step Cycle Utilizing Pressurization with Product 43 Figure 3.4 Piston, Gas, and Shock Wave Velocities During Constant Pressure Feed 44 Figure 3.5 Mole Fraction in the Bed at the (a) Beginning and (b) Ending of the Blowdown Step. 51 Figure 3.6 Bed Mole Fraction During Blowdown at (a) P = P H , (b) P = Po, and (c) P = PL 52 Figure 3.7 Formation of a Simple Wave During Purge 56 Figure 3.8 Steps for Purge Work Calculation 62 Figure 3.9 Expansion of the Product to Recover Energy and Deliver Product at Atmospheric Pressure 65 viii Figure 3.10 Recovery vs. Pressure Ratio for p = 0.1 69 Figure 3.11 Recovery vs. Pressure Ratio for p = 0.9 70 Figure 3.12 Recovery vs. Pressure Ratio for Oxygen Concentration; yo = 0.78, P = 0.582 71 Figure 3.13 Four-Step Cycle: Net Work per Mole of Product Oxygen (wi) Done by System (y0 = 0.78, p = 0.582) 72 Figure 3.14 Four-Step Cycle: Second Law Efficiency 74 Figure 3.15 Net Work per Mole of Product Oxygen for Oxygen Concentration Using the Four-Step Cycle (w )^ 75 Figure 3.16 Second Law Efficiency for Oxygen Concentration Using the Four-Step Cycle 75 Figure 3.17 Recovery for Oxygen Concentration Using the Four-Step Cycle 76 Figure 3.18 Ideal Four-Step Cycle: Flows and Energy Recovery 78 Figure 3.19 Reversible Turbine used in Blowdown 79 Figure 3.20 Reversible Expansion of Purge Gas and Reversible Expansion/ Compression of Purged Gas 80 Figure 3.21 Recovery of Work During Pressurization 82 Figure 3.22 Net Ideal Four-Step Cycle Work per Mole of Product Oxygen (WH) for Oxygen Concentration; y 0 = 0.78, P = 0.582 85 Figure 3.23 Ideal Four-Step Cycle Second Law Efficiency for Oxygen Concentration; y 0 = 0.78, p = 0.582 85 Figure 3.24 Net Ideal Four-Step Cycle Work per Mole of Product Oxygen (WH) for Oxygen Concentration 87 Figure 3.25 Ideal Four-Step Cycle Second Law Efficiency for Oxygen Concentration 87 Figure 3.26 Ideal Three-Step: Cycle Work and Molar Flows 90 Figure 3.27 Evacuation of the Adsorbent Bed 91 Figure 3.28 Expansion of Product Gas to Atmospheric Pressure 94 Figure 3.29 Net Work per Mole of Product Oxygen for the Ideal Three-Step Cycle (WB) 96 Figure 3.30 Second Law Efficiency for the Ideal Three-Step Cycle 96 Chapter 4 Figure 4.1 System used by Banerjee et al. with Adiabatic Compressor and Aftercooler 98 Figure 4.2 Isothermal System to Compare the Banerjee et al. Exergy Analysis to the Current Analysis 98 Figure 4.3 Grassman Diagram for the Four-Step Cycle (Banerjee et al., 1990) with Product a tP H 103 Figure 4.4 Four-Step Cycle: Energy and Molar Flows 104 Figure 4.5 Grassman Diagram of Four-Step Cycle with Expansion of the Product gas to P L 105 Figure 4.6 Ideal Four-Step Cycle: Energy and Molar Flows 108 Figure 4.7 Grassman Diagram for the Ideal Four-Step Cycle 109 Figure 4.8 Ideal Four-Step Cycle: Energy and Mass Flows with all Turbines on One Shaft 110 Figure 4.9 Grassman Diagram for the Ideal Four-Step Cycle with all Turbines on One Shaft I l l Figure 4.10 Ideal Three-Step Cycle: Energy and Molar Flows 112 Figure 4.11 Grassman Diagram for the Ideal Three-Step Cycle 113 Figure 4.12 Grassman Diagram for the Vacuum Four-Step Cycle 116 Figure 4.13 Grassman Diagram for the Vacuum Ideal Four-Step Cycle 117 Figure 4.14 Grassman Diagram for the Vacuum Ideal Three-Step Cycle 119 Figure 4.15 Grassman Diagram for Run 1 (Kayser and Knaebel, 1986) 123 Figure 4.16 Grassman Diagram for Run 2 (Kayser and Knaebel, 1986) 123 Figure 4.17 Grassman Diagram for Run 3 (Kayser and Knaebel, 1986) 124 Figure 4.18 Grassman Diagram for Run 4 (Kayser and Knaebel, 1986) 124 Figure 4.19 Grassman Diagram for Run 5 (Kayser and Knaebel, 1986) 125 Chapter 5 Figure 5.1 Cell ' i ' of the CSTR Model (Overall Mole Balance) 129 Figure 5.2 Flow Regimes for Cell 'i 133 Figure 5.3 Cell ' i ' of the CSTR Model (Species A Mole Balance) 134 Appendix A Figure A . l Extraction and Pressurization of dN Moles from P to Px 150 Figure A.2 Gas Velocity in an Adsorbent Bed 153 Figure A.3 Mole Fraction as a Function of Pressure: (a) Initial Condition y = yB, and (b) Initial Condition y = 0 154 List of Symbols A = total cross sectional area of the bed {m2} ruo = molar enthalpy of species A at the reference state {J/mol} rui = molar enthalpy of species A at state (1) {J/mol} kA = isotherm slope for species A {-} kB = isotherm slope for species B {-} LB = length of the bed {m} N - number of moles {mol} nA = moles of species A adsorbed per unit adsorbent volume {mol/ m3} N A = total number of moles of species A in the vessel or adsorbent bed {mol} NA,adsorbed = number of adsorbed moles of species A in the bed {mol} NA, g as = number of moles of species A in the gas phase in the bed {mol} N B = total number of moles of species B in the vessel or adsorbent bed {mol} NB,adsorbed = number of adsorbed moles of species B in the bed {mol} NB / g as = number of moles of species B in the gas phase in the bed {mol} NBD = moles that leave the bed during blowdown {mol} NBDA = moles of species A that leave the bed during blowdown {mol} NBDB = moles of species B that leave the bed during blowdown {mol} N E = moles of gas in the total exhaust or evacuation step {mol} NEA = number of moles of species A in the total exhaust or evacuation step {mol} NEB = number of moles of species B in the total exhaust or evacuation step {mol} N F = number of moles of feed gas {mol} NFA = number of moles of species A in the feed gas {mol} NFB = number of moles of species B in the feed gas {mol} Npi = number of product moles delivered during the feed step {mol} Npu = number of moles of pure light product used to purge the bed {mol} NPR = number of moles of pure light product used to pressurize the bed {mol} Np2 = final number of product moles delivered during the feed step {mol} N w = moles of gas purged from the adsorbent bed during the purge step {mol} NWA = number of moles of species A in the purged gas {mol} NWB = number of moles of species B in the purged gas {mol} P = total gas pressure {Pa} Po = atmospheric pressure = 101325 Pa PA = partial pressure of species A in the gas phase {Pa} PB = partial pressure of species B {Pa} P H = high pressure limit of the cycle {Pa} PL = lower pressure limit of the cycle {Pa} R = universal gas constant = 8.3144 J/mol K SAO = molar entropy of species A at state (0) {J/mol K} SAI = molar entropy of species A at state (1) {J/mol K} T - temperature {K} To = reference temperature {K} t = time {s} tB = time for the leading edge of the simple wave to reach the entrance of the bed during purge {s} tc = time for point C on the concentration wavefront to travel the length of the bed {s} tF = time for the shock wave to just reach the end of the bed {s} tpu = time required for the purge step {s} u = interstitial gas velocity {m/ s} ui = velocity at point 1 in the bed or at the entrance of the bed {m/s} ui(t) = velocity at the entrance of the bed during the purge step {m/s} U2 = velocity at point 2 in the bed or at the exit of the bed {m/s} uc = velocity of point C on the concentration wavefront {m/ s} us = velocity of the shock wave at constant pressure {m/s} V = volume {m3} VB = A L B = total volume of the adsorbent bed {m3} VFI = volume of the feed piston at atmospheric pressure {m3} VF2 = volume of the feed piston at the beginning of Step 2-3 of the feed step{m3} Vp2 = volume of final amount of product gas (Np2) at the high pressure {m3} Vp3 = volume of final amount of product gas (Np2) at atmospheric pressure {m3} Vpu = volume of light gas used to purge the bed at P = PL {m3} Vwi = volume of the purged gas at the low pressure {m3} Vw2 = volume of the purged gas at atmospheric pressure {m3} W B = work done by the system as the blowdown gas leaves the system {J} WBI = work done by the system as the blowdown gas leaves the system (P>Po) {J} WB2 = work done by the system as the blowdown gas leaves the system (P<Po) {J} W E = work done by the system during the evacuation step {J} WEI = work done by the system during the evacuation step when P > Po {J} WE2 = work done by the system during the evacuation step when P < Po {J} W F = work done by the system during the feed step {J} WPR = work recovered by expanding the pressurization gas through a reversible turbine during the pressurization step {J} Wpu = work recovered by expanding the purge gas through a reversible turbine during the purge step {J} W R = work done by the system in expanding the product gas from P H to Po {J} Ww = work done by the system in extracting the purged gas from the system {J} W4 = net work done by the Four-Step cycle {J} Ww = net work done by the Ideal Four-Step cycle {J} W B = net work done by the Ideal Three-Step cycle {J} Wrev = reversible work of gas separation {J} WBI = WBI/NP2 {J/mol} WB2 = WB2/NP2 {J/mol} WE = W E / N P 2 {J/mol} WEI = WEI/NP2 {J/mol} WE2 = WE2/NP2 {J/mol} WF = WF/NP2 {J/mol} WPR = W P R / N P 2 {J/ mol} wpu = Wpu/Np2 {J/mol} WR = WR/NP2 {J/mol} w w = Ww/Np2 {J/mol} w 4 = W 4 / N P 2 {J/mol} W14 = Wi4/Np2 {J/mol} W B = W D / N P 2 {J/mol} Wrev = W r e v /NP2 {J/mol} y = mole fraction of the gas mixture {-} yo = mole fraction of the feed gas {-} yi = mole fraction at point 1 in the bed or at the entrance of the bed {-} yi(t) = mole fraction at the entrance of the bed during the purge step {-} y 2 = mole fraction at point 2 in the bed or at the exit of the bed {-} xiii VA = mole fraction of species A {-} yB = mole fraction of species B {-} yB = mole fraction of the gas inside the bed at the end of the blowdown step {-} yBD = mole fraction of the blowdown gas {-} yw = mole fraction of the purged gas {-} yE = mole fraction of the exhaust {-} yp = mole fraction of the product {-} z = axial displacement in the bed {m} Greek Symbols (3 = (3A / PB = separation ratio = ratio of the fraction of component A in the gas phase in the bed to the fraction of component B in the gas phase in the bed {-} PA = fraction of species A in the bed that is in the gas phase {-} PB = fraction of species B in the bed that is in the gas phase {-} s = void fraction in the adsorbent bed {-} \|/AI = molar availability of species A at state (1) {J/ mol} I|/BI = molar availability of species B at state (1) {J/mol} i|/p = molar availability of the product per mole of desired gas {J/ mol} \(/E = molar availability of the product per mole of desired gas {J/ mol} f l = pressure ratio {-} xiv Acknowledgement I would like to thank everyone who has helped me go this far: my parents, Fred and Veronica M c Lean, who continue to give me all that they have; my dearest friend, Lynore Melville, who makes up for my imperfections; Steven Rogak, my supervisor and the one who guided my thoughts; Gary Schajer, who has never let my words go unheard; Bowie Keefer, Matt Babicki, Dave Doman, Carl Hunter, Stevo Kovacevic, Sharon MacLellan, Martin Rump, and Aleksander Sljivic, the people of Highquest, who have inspired and supported all that I have done; and all my friends, who embrace me as I am, and lift me up. Dedication For within her is a spirit inteUigent, holy, unique manifold, subtle, mobile, incisive, unsullied, lucid, invulnerable, benevolent, shrewd, irresistible, beneficent, friendly to human beings, steadfast, dependable, unperturbed, almighty, all-surveying, penetrating all inteUigent, pure and most subtle spirits. For Wisdom is quicker to move than any motion; she is so pure, she pervades and permeates all tilings. She is a breath of the power of God, pure emanation of the glory of the Almighty; so nothing impure can find its way into her. For she is a reflection of the eternal light, untarnished mirror of God's active power, and image of his goodness. Wisdom I loved and searched for from my youth; I resolved to have her as my bride, I fell in love with her beauty. She enhances her noble birth by sharing God's life, for the Master of A l l has always loved her. Indeed, she shares the secrets of God's knowledge, and she chooses what he wil l do. If in this life wealth is a desirable possession, what is more wealthy than Wisdom whose work is everywhere? Or if it be the intellect that is at work, who, more than she, designs whatever exists? Or if it be uprightness you love, why, virtues are the fruit of her labours, since it is she who teaches temperance and prudence, justice and fortitude; nothing in life is more useful for human beings. Or if you are eager for wide experience, she knows the past, she forecasts the future; she knows how to turn maxims, and solve riddles; she has foreknowledge of signs and wonders, and of the unfolding of the ages and the times. Wisdom 7:22-26, 8:2-8 This thesis is dedicated to the glory of God. 1 1. Introduction 1.1 Overview Pressure Swing Adsorption (PSA) is a method of separating a mixture of gases into its various components. Cyclic pressure and flow variations, in the presence of a selectively adsorbent material, are used to concentrate one species or group of species at one end of an adsorbent filled vessel, while the other species or group of species is concentrated at the other end. A n adsorbent is a material to which different gases are attracted with varying degrees. These materials generally take the form of very high surface area, porous pellets or beads. When a gas is brought into contact with an adsorbent, some of the molecules of gas are attracted to the surface of the adsorbent and held there by van der Waals and electrostatic forces. This is termed physical adsorption, and differs from chemical adsorption which involves electron transfer and much higher forces. Chemical adsorption may occur with a specific gas and a specific adsorbent, but is generally undesirable in PSA processes as the bonds are too strong to be easily broken. The amount of a particular gas that is physically adsorbed on an adsorbent generally increases as the partial pressure increases. The presence of other gases in a mixture also influences the amount of gas adsorbed, but for this thesis, the interspecies effects on adsorption wil l be ignored; only the effect of partial pressure wi l l be considered. The amount adsorbed is also a function of temperature, and generally decreases with increasing temperature. This effect of increased adsorption at higher partial pressure is used in PSA processes to effect the separation of the mixture. This makes PSA an inherently energy 2 intensive process, as the mixture of gases must be pressurized and depressurized. Many methods of recovering some of the energy have been developed, and gas separation by PSA is competitive with cryogenic distillation at low to moderate product flow (Ruthven, 1994, p 7). Nevertheless, due to irreversibilities, the work required for PSA gas separation is still much greater than the thermodynamic limit (reversible work) for gas separation. These irreversibilities enter the system through many mechanisms: throttling, mass transfer and other frictional losses; inadvertent remixing of separated gases; mechanical friction; and irreversibilities associated with temperature and concentration gradients. Some of these irreversibilities can be removed, but others are intrinsic and perhaps essential to the separation process. Of these losses, the losses associated with throttling and the losses intrinsic to the cycle wil l be considered in this work. To the author's knowledge, the difference between these losses has not been previously quantified. As many PSA cycles contain the steps analyzed in this thesis, the results and methods can be extended to the analysis of other cycles. 3 1.2 Commercial Applications of PSA The first commercial pressure swing adsorption cycles were invented in 1957-1958 by Guerin de Montgareuil and Domine, and Skarstrom. Skarstrom's apparatus was called the heatless air dryer and used to remove water vapour from air (Skarstrom, 1959,1960). Since the invention of PSA, the process has been applied in many different industries. Most pressure swing adsorption systems are used in the purification of hydrogen made by the steam reformation of fossil fuels, such as natural gas. PSA can also be used to purify hydrogen made by partial oxidation reactors, or to extract hydrogen from a dilute gas-plant waste stream that would otherwise be burned for steam generation. Oxygen concentration can also be used for partial oxidation reactions, to supply a concentrated stream of oxidant to the burner. This reduces the amount of nitrogen that must be removed in order to purify the hydrogen that is generated. Oxygen concentration is also beneficial in other oxidation reactions where the rate is a function of the partial pressure of oxygen. This is true for blast furnaces and internal combustion engines, as well as fuel cells. Medical oxygen, including home oxygen concentrators, are used by people who require a constant or intermittent source of concentrated oxygen. Onboard Oxygen Generating Systems (OBOGs) are used in military and civilian aircraft to provide breathing oxygen, instead of carrying gas bottles. However, there are some markets that are not open to gas separation by PSA systems because PSA system's are still relatively large, and energy inefficient; the portable market requires systems that are small and very efficient. 4 Highquest Engineering Inc, a Vancouver based research and development firm, has developed many proprietary PSA cycles that attempt to eliminate irreversibilities, while increasing the speed of the cycles and decreasing the size of the systems. These cycles could potentially require much less energy and open up many non-traditional markets for gas separation. 5 1.3 Scope and Outline of Thesis In order to understand the throttling losses and bed losses in PSA cycles, three closely related cycles are broken down into steps and analyzed. The flows of gas and energy through the system are calculated, and the energy losses during the cycle are quantified. The total energy requirements of each cycle is compared to the thermodynamic limit (reversible work) for gas separation, in order to calculate the second law efficiency. The second law efficiency is a measure of how close one is to obtaining the best efficiency thermodynamically possible, and can be used to compare different cycles to each other. The cycles considered are as follows: 1. The Four-Step cycle, which is a modification of the well-known Skarstrom cycle. The product is considered to be pure light component, and pressurization of the adsorbent bed is accomplished using product, rather than feed. 2. A n idealized version of the Four-Step cycle delivering pure product, in which the mrottling losses are removed and all of the pressurization and depressurization is done reversibly. 3. An Ideal Three-Step cycle that attempts to simplify the PSA process into three essential steps. Again, the cycle is assumed to produce pure product. In Chapter 2, the background necessary to understand the separation of gases by PSA is developed. First, the reversible work of gas separation is derived. Then, the adsorption process is introduced, along with the equations used in this work to describe the amount of gas adsorbed on a particular adsorbent. The next step is an exploration of bed dynamics, the study of variations in flow and concentration in an adsorbent bed 6 due to the adsorption and desorption of gas. The conservation of mass equations are presented and the formation of concentration shock waves and simple waves, essential to the separation process, is described. In Chapter 3, the mass and energy flows for each step of the three cycles are derived, and it is shown how the work for the cycles depends on the cycle and adsorbent properties. The three cycles are analyzed using a mixture of analytical and numerical calculations. The Four-Step cycle and the Ideal Four-Step cycle are then compared to each other and to the reversible work of gas separation. The analysis of the Four-Step cycle reveals that this cycle has a particularly low second law efficiency. The analysis of the Ideal Four-Step cycle reveals that there are still irreversibilites aside from those introduced by mrottling. The Ideal Three-Step cycle is found to be the limit of the Ideal Four-Step cycle, with the pressure ratio approaches infinity. The results are presented for various values of feed mole fraction and selectivity ratio. In Chapter 4, the Four-Step cycle results of Chapter 3 are compared with the previous literature. The Ideal Four-Step cycle results are then used to extend the results of previous studies by dissecting what was previously known only as "bed losses." These losses are divided into the various different mrottling losses that occur in the cycles and the losses inherent to the separation process. The equations are also applied to a vacuum cycle and five experimental separations recorded in previous studies. In Chapter 5 the Ideal Four-Step cycle is analyzed using a computer model, which treats the adsorbent bed as a series of Constantly Stirred Tank Reactors (CSTRs). The effects of diffusion and dispersion can be modeled by varying the number of cells used in the model. It is shown that the amount of diffusion and dispersion present does 7 not significantly change the energy requirements and losses of the cycle. The model is also useful for testing other cycles for separation performance and energy efficiency. The conclusions of this work are presented in Chapter 6, and the recommendations are presented in Chapter 7. 8 2. Review of Pressure Swing Adsorption 2.1 Reversible Work of Gas Separation The reversible work of gas separation is the minimum work required to separate a mixture of gases, and is independent of process. It is calculated by using the second law of thermodynamics to relate the available energy of the gases in the unmixed state to the available energy of the gases in the mixed state. In subsequent chapters, this reversible work wil l be compared to the theoretical cycle work for various PSA cycles. In the following analysis, the reversible work of gas separation wil l be derived. In this study, gases are treated as ideal. That is, a known volume V (in m3), filled with N moles of gas at pressure P (in Pa) can be described by the ideal gas law: PV = NRT ( 2 1 ) Where: R = universal gas constant = 8.3144 J/mol K T = temperature {K} Suppose that the gas is an arbitrary mixture of two gases; N A moles of species A and N B moles of species B, where: NA+NB=N ( 2 2 ) The mole fraction of each species is the number of moles of that species divided by the total number of moles. In order to simplify the equations to follow, we define the "mole fraction" y of a gas mixture to be the mole fraction of species A in the gas phase. Thus, the mole fraction of the two gases can be described as: NA (2 3) N 1 ^ f l (2 4) The partial pressure P A of gas species A is the pressure at which the gas would be if the N A moles of species A were alone in the volume V. The partial pressure PB of gas species B is defined similarly. These definitions can be written as follows: P A = ^ - y P (2-5) P B = ^ = (i-y)P (2-6) The system can now be described by y and any two of the properties P , V, and N . The reversible work required to separate two gases is the change in availability that occurs as the gases are separated. Availability is defined as the maximum reversible work that can be done by a system relative to a reference state. If we take this reference state to be the state in which the two gases are mixed and in equilibrium at temperature To, we have the system labeled (0) in Figure 2.1. For convenience, the availability of each species at state (0) is considered to be zero. The mole fraction of the 10 gas is defined by yo, and the partial pressures of each species are defined by Equations (2.5) and (2.6). P ° N = N A + N B Po Po V = V A + V B N A N B y = yo V A V B PAO = yoPo y = l y = 0 PBO = (1 - yo)Po P A I = Po PBI = Po (0) (1) Figure 2.1 (0) Mixed Reference System, and (1) Separated Gases. In state (1) of Figure 2.1, the gases are separated and the partial pressure of each gas has been raised to the total pressure. On a molar basis, the avauability of each species at state (1) (ignoring velocity and gravitational terms) is defined as: WAi=(HA\ - h A o ) - T o { s A i -SAO) (2-7) ¥BX ={HB\ - h B o ) - T o { s m - s B o ) (2-8) Where: V|/AI = molar availability of species A at state (1) {J/mol} IIAI = molar enthalpy of species A at state (1) {J/mol} hAo = molar enthalpy of species A at the reference state {J/mol} To = reference temperature {K} 11 SAI = molar entropy of species A at state (1) 0 / mol K} SAO = molar entropy of species A at state (0) {J/mol K} The variables for species B are defined similarly. As enthalpy is only a function of temperature for ideal gases, the first term in the above equations is zero, so the above formulas reduce to: VAI = - T O ( S A I - S A O ) (2.9) WB\ ~ -^o( S Bl S B o ) (2.10) As gases are separated, they move from a disordered, or more probable state to a more ordered, or less probable state. This implies a decrease in entropy and an increase in availability (Equations (2.9) and (2.10) have positive values). As the gases in state (1) have positive availability relative to state (0), a reversible change from state (1) to (0) is capable of producing work. The opposite of this work is the reversible work of separation. The isothermal change in entropy of an ideal gas species A is: (P } = -RM ( P \ _o_ {PA J (2.11) The change in entropy of species B is defined analogously: fp \ -Rln f p \ (2.12) 12 Upon substituting in the expressions for partial pressure we find that the above equations become: *AX -SAO = - R W (2.13) S B I - S B O =-Rw f 1 N v l - J V (2.U) As the total pressure cancels, we end up with the ratio of the mole fractions. When we substitute Equations (2.13) and (2.14) into (2.9) and (2.10), we find that the availability of each species at state (1) is: ¥ AX =RT0\n f P (2.25) ¥BX = RTo l n f —1 (2.16) The availability is the maximum reversible work per mole that could be done by each species as it goes from state (1) to (0). The minimum reversible work necessary to change the gases from state (0) to (1) is just the opposite, and for the given N A moles of species A and N B moles of species B, the minimum reversible work necessary to separate the gases is: Wm = -RTn NA\n\ + NB ln| f 1 ^ (2.17) 13 The negative sign indicates that work is defined as work done "by the system." Thus a negative work implies that work is done on the system. Work is defined this way because we wil l eventually be using turbines to recover the energy lost during throttling. The energy recovered is then defined as positive, as it is done by the system. If we make the substitution N A = N B yo / (1 - yo), we can plot the reversible work done by the system in producing 1 litre of pure species B (1 litre is 0.0409 moles at standard temperature and pressure) from gas with initial mole fraction yo. The equation we are plotting in Figure 2.2 is then: mol Vm = -RT0{ 0.0409— In ' l ^ + ln 1 (2.28) Reversible Work of Gas Separation yo Figure 2.2 Reversible Work of Gas Separation Done by the System as a Function ofyo. 14 As can be seen in Figure 2.2, more work must be done on the system to produce a given volume of pure species B as the initial mole fraction of species A increases and there is less species B in the feed. Equation (2.18) represents the work required to separate a mixture of gases into pure species A and pure species B. However, a PSA separation system does not generally separate a mixed gas into pure light product and pure heavy exhaust; some of the desired product gas is lost to the exhaust, and the product is generally not composed entirely of the desired gas. In order to extend Equation (2.18) to PSA processes, we must look at the flows through a PSA system and develop the concepts of purity and recovery. Purity is defined as the fraction of the product stream that is made up the desired product gas (in this case species B). Recovery is defined as the fraction of species B in the feed gas that ends up in the product gas. The flows through a PSA system can be seen in Figure 2.3. Feed N F = N F A + N F B PSA Exhaust N E = N E A + N E B Product Np = N P A + N P B Figure 2.3 The Flows in and out of a PSA Gas Separation System. 15 In Figure 2.3 the pressure swing adsorption system is considered a "black box/' which divides the feed gas into two streams: one that is enriched in species B (the product stream) and one that is enriched in species A (the exhaust stream). The process is treated as a batch process, so we deal not with flows, but with a certain number of moles of each species. The subscripts F, P and E stand for Feed, Product and Exhaust, while the subscripts A and B refer to species A and B. From this diagram, the mole fractions of the feed, product and exhaust are defined as: Y = EM (2.29) NPA + NFB y = EM (2.20) NPA+NPB y = EM (2.22) NEA+NEB Where: yo yp y E = mole fraction of the feed {-} = mole fraction of the product {-} = mole fraction of the exhaust {-} 16 We also define the purity and recovery of species B: Purity Y PB N PB NPA + NPB = 1 - ^ (2.22) Recovery N PB (2.23) N FB From these definitions, the reversible work in Equation (2.17) is a specific case of the general situation in which N B moles of species B are produced at Purity = 1 and Recovery = 1. To extend this equation to the general case with arbitrary purity and recovery, we must calculate the availability of both species in both the product and exhaust streams relative to the feed stream (which is considered to be the reference stream). Again, this translates into a function of the number of moles and the ratio of the partial pressures. Without calculating the individual availabilities, we construct the equation for reversible work when feed, product and exhaust are at atmospheric pressure. W = -RT ''rev A i 0 NPA ln| + NPB ln| + NEA ln| + NEB ln| (2.24) This equation can be seen plotted in Figure 2.4 as a function of purity and recovery, for the specific case of oxygen concentration from air (yo = 0.78). If no separation takes place (purity = 0.22), no work is done by the system. As the purity of the product and the product recovery increase, more work is done on the system to effect the separation. 17 1 Figure 2.4 Reversible Work done By the System in Concentrating One Litre Of Oxygen From Air. We now extend the reversible work calculation to the case in which the product partial pressure of species B must be kept constant and greater than atmospheric. This is true when the product is used in a chemical reaction in which the rate is dependent on the partial pressure of the reactants. An example of this is the dissociation of O2 at the cathode of a fuel cell. Another example is the combustion of fossil fuels. Holding the partial pressure of species B constant implies that at low purity, the total product pressure is high, while at 100% purity, the total product pressure is the partial pressure of species B. A graph of total product pressure Pp vs. purity can be seen in Figure 2.5, in which the partial pressure of species B is kept at three atmospheres. 18 Figure 2.5 Total Product Pressure vs. Purity for constant O2 Partial Pressure of 3 atm. The work equation in this case becomes: wrev ~ RT0 NpAm\ ypi \ f V-yP) +NpBln V J V +NEAln +NEB\n (2. The graph of this equation can be seen in Figure 2.6 for the case of oxygen concentration (yo = 0.78). More work is now required at low purity than at high purity. This is because at low purity, although little separation work is required, a lot of nitrogen leaves the system as product at pressure Pp. This is exaggerated by the increase in total pressure at low purity necessary to maintain the product oxygen partial pressure. The minimum reversible work required per litre of O2 occurs at approximately 60 to 70% purity, and depends slightly on recovery. 19 Recovery Purity Figure 2.6 Reversible Work to Concentrate and Deliver Product with Oxygen Partial Pressure of 3 atm. This implies that if a specific gas partial pressure is required in a system, there is an optimum combination of straight gas compression and gas separation. In reality, this picture is greatly altered by losses in the separation process. 20 2.2 Adsorption The performance of real PSA systems depends on the kinetics and thermodynamics of adsorption. In this work, it is assumed that the gases are locally in equilibrium with the adsorbent, so only the thermodynamics need be considered. This is a reasonable assumption if the gas velocities are kept low. In order to describe the amount of a gas that is adsorbed to a specific adsorbent, an equilibrium isotherm is plotted. The example isotherm in Figure 2.7 plots the solid concentration of nitrogen and oxygen (the amount of gas that is adsorbed) per cubic meter of Zeolite 5A adsorbent vs. the concentration of the gas in the voids around the adsorbent (which is proportional to the partial pressure). Equilibrium Isotherms for N2 and O2 on Zeolite 5A -e— Nitrogen •a— Oxygen Gas Concentration {mol / m3} Figure 2.7 Equilibrium Adsorption Isotherms for Nitrogen and Oxygen on Zeolite 5A. In this example, more nitrogen is adsorbed than oxygen. Therefore, nitrogen is known as the more-adsorbed species, the heavy component, or the heavy fraction. Conversely, oxygen is known as the less-adsorbed species, the light component, or the light fraction. 21 From the isotherm, one can determine the ability of a specific adsorbent to separate two gases. This ability can be divided into two main factors: the amount of heavy component adsorbed relative to the amount of light component adsorbed (selectivity), and the absolute amount of each component adsorbed (capacity). The larger the selectivity, the more effective the adsorbent is at separating the gases. The absolute amount of each component adsorbed affects the separation process in a different way. An adsorbent capable of adsorbing more gas wil l be more "productive"; it wil l generate more product per volume adsorbent per unit time. Many equations have been developed to fit isotherm curves. For this work we use the simplest model, linear isotherms, in which the amount of heavy component adsorbed is: = k A (2.26) A A R T Where: nA = moles of species A adsorbed per unit adsorbent volume {mol/ m3} k A = isotherm slope for species A {-} PA = partial pressure of species A in the gas phase {Pa} The isotherm for the less adsorbed species is defined similarly: B B R T Due to the difference between k A and kB, the composition of the adsorbed gas (which exists as a mixture on the adsorbent) is different from the composition of the gas 22 phase; there is a higher fraction of the heavy component in the adsorbed phase than in the gas phase. To express Equations (2.26) and (2.27) in terms of the total pressure P, we again define the mole fraction of a gas to be the mole fraction of species A in the gas phase (y = y A ) . This implies that yB = 1-y, and: B B R T To find the absolute number of adsorbed moles we multiply Equations (2.28) and (2.29) by the volume of adsorbent, which is related to the total volume of the adsorbent bed by the non-dimensional fraction porosity (denoted by e). Porosity is the fraction of the adsorbent bed that is empty space. Therefore, the fraction of the adsorbent bed filled with adsorbent is (1 - s), and the number of moles in the adsorbed phase is: N — Jr J V A,adsorbed ~ A yP(\-e)VB (2.30) RT „ = k (l-yW~e)VB (2.31) ^ B,adsorbed KB Rrp Where: N A . adsorbed NB, adsorbed = number of adsorbed moles of species A in the bed {mol} = number of adsorbed moles of species B in the bed {mol} 23 s = void fraction in the adsorbent bed {-} VB = total volume of the adsorbent bed {m3} The number of moles of each species in the gas phase in an adsorbent bed is found using the ideal gas law with the partial pressure of the gas and the void volume of the bed. N A,gas yPsVB RT (2.32) N B,gas RT (2.33) Where: NA, g as = number of moles of species A in the gas phase in the bed {mol} NB, gas = number of moles of species B in the gas phase in the bed {mol} The total number of moles of species A and species B in the adsorbent bed can be found by adding Equations (2.30) and (2.32), and (2.31) and (2.33), respectively: s + {l-s)k/ yPeVB RT £ + (l-£ )kB (2.34) (2.35) RT Where: N A = total number of moles of species A in the bed {mol} N B = total number of moles of species B in the bed {mol} 24 It is useful to invert the non-dimensional terms surrounded by square brackets and define them as: R = £ (2.36) P A s + (l-s)kA BB = - (2-37) £ + (l -s)kB The total number of moles inside an adsorbent bed can then be described by the following equations: = yPeVj_ (2.38) A PART n ={l-y)PeVB ( 2 3 9 ) BBRT These equations are similar in structure to the ideal gas law; they have a partial pressure term (yP or (1 - y)P), a volume term which represents the void volume in the bed (EVB), the universal gas constant, and the temperature. The additional p terms can be understood as the fraction of that species in the bed that is in the gas phase. For example, a low value of PA implies that most of the species A in the bed is in the adsorbed phase. A high value of PA implies that most of the species A in the bed is in the gas phase. A value of unity (the highest possible value) for PA implies that all of the species A is in the gas phase and no adsorption of species A occurs on that particular adsorbent. 25 2.3 Bed Dynamics PSA cycles generally consist of several steps, during which the gas flows and concentrations in the adsorbent bed vary. The study of these variations is called bed dynamics. When we graph the gas mole fraction in the bed vs. the axial displacement, the profile usually has the appearance of a shock wave or a simple wave. Manipulation of these "wavefronts" controls the PSA cycle. In this section, the concept of wavefronts wil l be developed in order to see how they are necessary for the separation process. Bed dynamics are governed by the species conservation equations and the boundary conditions imposed on the bed by external compressors, pistons, or valves. Equation (2.40) is the overall conservation equation for an isothermal adsorbent bed with negligible axial dispersion or diffusion and no radial dependence in velocity or composition. Equation (2.41) is the species A conservation equation (Equations 1 and 2 in Knaebel and Hi l l , 1985). These equations have been validated by the experimental work of many, including that of Kayser and Knaebel (1986). The equations can be solved by the method of characteristics to yield the profiles mentioned above. dP duP + RT(l-e) dn (2.40) £ = 0 \dt dz ) dt dPA duP. + RT(l-e) dt (2.41) £ + = 0 K dt dz ) Where: z = axial displacement in the bed {m} t = time {s} u = interstitial gas velocity {m/s} 26 A concentration shock wave forms when gas with a higher mole fraction enters a bed filled with gas having a lower mole fraction (mole fraction of a mixed gas is defined as the mole fraction of the more adsorbed species). In order to understand this we wil l use a simplified model to study the motion of gas molecules through an adsorbent bed at constant pressure. In Figure 2.8 we have a representational adsorbent bed. The adsorbent is shown as unshaded beads. The more adsorbed species is shown as large shaded particles and the less adsorbed species is shown as small shaded particles. In the first frame, the bed is filled only with adsorbent and the less adsorbed species. For this simple example, each adsorbent pellet is capable of adsorbing two molecules of the light component at the total bed pressure. The feed piston is full of the feed gas mixture, which has mole fraction yo (in this case 0.5), and the product piston is empty. In the second frame, some of the feed gas has been injected into the adsorbent bed. Each adsorbent particle is capable of adsorbing three molecules of the more adsorbed component at the partial pressure yoP.* As the heavy component in the feed encounters adsorbent with no adsorbed heavy component, some of the heavy component is adsorbed and removed from the gas phase. As the pressure remains constant, the partial pressure of the less adsorbed species at the left end of the bed is reduced. This causes some of the adsorbed light component molecules to desorb. These molecules join the light component molecules in the feed and both continue through the bed. * It is assumed that the adsorption of one component does not influence the adsorption of the other component; only the partial pressure affects the amount of adsorbed gas. 27 0 • • • • • 0 • ft a ® • © ® Feed Piston Adsorbent Bed Product Piston Q Adsorbent 0 More Adsorbed Species • Less Adsorbed Species Figure 2,8 Movement of Gas Molecules Through an Adsorbent Bed. 28 If we were to plot the mole fraction of the gas phase inside the adsorbent bed as a function of axial displacement, we would find the results plotted in Figure 2.9. Propagation of a Concentration Shock Wave 0.7+ Gas Flow 0.6+ Mole Fraction 0.5+ 0.4+ y H 0.3 + ti 0.2+ 0.1 + 0 r o 0.5 1 Non-Dimensional Axia l Displacement {-} Figure 2.9 Propagation of a Concentration Shock Wave. The rapid decline in mole fraction is known in the literature as the "concentration shock wave." To the left of the shock wave, the mole fraction is equal to that of the feed, and to the right it is equal to zero. As we visualize the feed gas entering the bed and the shock wave moving through the bed, we begin to see how separation of the gases takes place. The shock wave represents the furthest position in the bed reached by the heavy component, which is being adsorbed as it encounters new adsorbent. Some light component desorbs and joins the light component in the feed. These light component gas molecules continue to move toward the product end of the bed. Due to the adsorption of the heavy component, the velocity of the shock wave is less than the velocity of the gas to the left of the shock wave, and the light component is forced through the shock wave, from low concentration to high concentration. This 29 separates the gases and increases the availabiUty of the light species, which implies that work has been done on the system. The shock wave propagates through the adsorbent bed until it reaches the end. If feed continues to enter the bed, some molecules of the heavy gas wil l enter the product piston; this is termed "breakthrough," and results in remixing of the separated gases. The formation of a concentration shock wave is counter-intuitive to what we know of gas flow through a packed bed; usually dispersion and diffusion at the boundary between two gases tend to mix the gases, thereby reducing the sharp distinction between them. However, the conservation equations for the equilibrium theory of adsorption predict that gas with high mole fraction moves faster through the bed than gas with low mole fraction. This result of this can be seen in Figure 2.10, which shows a concentration shock wave forming from an initially dispersed concentration wavefront. The part of the wave with high mole fraction moves to the right faster than the part with low mole fraction, until the shock wave develops. At this point, all of the points on the wave move with the same velocity. In reality, dispersion and diffusion oppose the formation of a shock wave and tend to spread out the concentration front. Eventually, there is a balance and the concentration front propagates with a constant pattern. 30 Formation of a Concentration Shock Wave 0.7-0.6-Mole 0.5-Fraction 0.4-y H 0.3-0.2-o.i-0 Gas Flow ', t 2 t 3 ~i 1 1 r 0.5 Non-Dimensional Axia l Displacement {-} Figure 2.10 Formation of a Concentration Shock Wave. The examples given above are for the formation of a shock wave at constant pressure. If the pressure varies while feed gas enters the bed, the shape of the curves in Figure 2.10 wil l be slightly different. This case is not considered here, and the reader should refer to Knaebel and Flill (1985) or Ruthven (1994) for further information. The formation of a simple wave occurs when gas having lower mole fraction enters an adsorbent bed with higher mole fraction. In this case the concentration front tends to move away from being a shock wave and the plots of mole fraction vs. axial displacement tend to flatten out. This is shown in Figure 2.11, which superimposes the graph of mole fraction vs. axial displacement onto a picture of the adsorbent bed. 31 Figure 2.11 Formation of a Simple Wave During Purge. In this example, the gas enters the bed from the right and has mole fraction zero. The gas phase inside the bed initially has mole fraction yo. For this situation, equilibrium theory predicts that points with a certain mole fraction travel with a constant velocity, and this velocity increases with mole fraction. This can be seen in Figure 2.11, which follows the motion of three points on the wave front. If gas with mole fraction zero continues to enter the right end of the adsorbent bed, the heavy component wil l eventually be totally purged from the bed. . So far we have shown how the concentration shock wave effects the separation of a gas mixture, and how an adsorbent bed that has been used to separate gases can be regenerated by purging the bed with pure product. Complete PSA cycles generally 32 consist of a high pressure feed step, in which shocks may form, and various lower pressure steps that return the bed to its initial clean state. Chapter 3 describes several PSA cycles in detail. 33 2.4 Previous Studies of Energy Loss and PSA Performance There have been very few papers written about the subject of energy consumption of Pressure Swing Adsorption cycles. This may be due to the lack of understanding of the basic thermodynamics of gas separation. Another possible reason is that PSA has replaced separation processes that are even more energy intensive, so that predicting recovery and purity has been more important than predicting energy consumption. Sircar and Kratz (1988) studied the difference in energy consumption between producing medium purity oxygen (23% to 50% oxygen) directly with a PSA cycle, and producing the same purity oxygen by making very pure oxygen using PSA and mixing it with air. They found that making oxygen of medium purity directly with a PSA cycle was more energy efficient. A second source is Armond (1970), who studied some optimal configurations for air separation in industry. Armond notes that one can minimize the power needed to compress the feed or product by carefully choosing the correct pump and trying to limit pressure drops through pipes. He also notes that when the lower pressure of the cycle is a vacuum, increasing the particle size (up to a certain point) and decreasing the length to diameter ratio of the adsorbent bed decreases the power consumption of the cycle. The remaining source of literature is Banerjee et al., who have published two papers involving the exergetic analyses of equilibrium separations (Banerjee et al., 1990), and kinetic separations (Banerjee et al., 1992). In the 1990 paper, Banerjee takes the results of the binary linear isotherm (BLI) model developed by Knaebel and Hi l l in 1985, and completes an overall exergy analysis of the Four-Step cycle utilizing pressurization with product. This paper looks only at the overall steady state flows of the cycle, and 34 treats the PSA system as a "black box." The energy losses that are found are termed "bed losses." Banerjee's work wil l be discussed in Chapter 4. Although there have been few studies of PSA efficiency, the present work makes use of earlier modeling and experiments, which are reviewed below. The Skarstrom cycle was first modeled in 1972 by Shendalman and Mitchell. They used a linear equilibrium model and only modeled one adsorbable trace component in an inert (non-adsorbed) carrier. By the method of characteristics they were able to find analytical solutions of the equations describing the situation in which the trace component was completely removed from the carrier. Chan et al. (1981) extended Shendalman and Mitchell's work to include an adsorbable trace component in an adsorbable carrier. They found that recovery increased with the separation factor and the pressure ratio. Flores Fernandez and Kenney (1983) studied a cycle consisting of three steps: 1. Pressurization with feed from PL to PH. 2. Product delivery at PH, while continuing to add feed. 3. Countercurrent depressurization to ambient. Their theoretical analysis uses linear isotherms and develops the mass flow equations for an arbitrary feed with both components adsorbable, but numerically integrates some of the equations. The cycle was modeled and the accuracy was judged to be ~10% for concentrations, ~15% for feed flows, and ~12% for recovery. However, the actual test data cannot be found as all of the values have been non-dimensionalized. 35 Knaebel and Hi l l (1985) extend the work of Shendalman and Mitchell and Chan et al. to the case of two adsorbable components with an arbitrary feed composition, using analytical equations. The assumptions used in Knaebel and Hill 's derivation, as well as in that of Flores Fernandez and Kenny, 1983 (from Kayser and Knaebel, 1986) are: 1. Binary, ideal gas mixture. 2. Local equilibrium between the gas and solid phases. 3. Linear, uncoupled adsorption equilibrium isotherms. 4. Negligible axial dispersion. 5. Negligible axial pressure gradients. 6. Constant pressure during feed and purge steps. 7. Isothermal operation. 8. No radial dependence in velocity or composition. 9. Identical columns: identical lengths, cross-sectional areas and interstitial void fractions. 10. Complete purification of the light component using the least possible amount of adsorbent. Knaebel and Hi l l used the method of characteristics to solve the continuity equations for the Skarstrom cycle which utilizes "pressurization with feed," and the Four-Step cycle, which utilizes "pressurization with product." The authors found that in order for complete purification to occur, a minimum pressure ratio must be exceeded, 36 and that minimum pressure ratio increased as the mole fraction of the heavy component in the feed increased. They also found that the pressurization with product variant resulted in higher recoveries than the pressurization with feed variant, with the greatest difference occurring for small separation factors, large initial heavy mole fractions and large pressure ratios (the authors note that this concept is consistent with industrial practice, as mentioned by Wagner, 1969). The authors develop equations for the number of moles used to pressurize the bed, the number of moles of feed necessary to push the shock wave the length of the bed, the number of moles of product delivered, and the.number of moles of product used to purge the adsorbent bed. From these equations, the recovery of the light component as well as the enrichment of the light and heavy components can be found. It is in this paper that the current work finds a great deal of its inspiration, drawing on their analysis of bed dynamics in order to calculate the work required for each step of the Four-Step cycle. In this thesis, only the pressurization with product variant is discussed, as the equations relating flows to pressures are more easily defined. Kayser and Knaebel (1986) compare the analytical solution of Knaebel and Hi l l (1985) to an experimental situation that closely resembles the assumptions present in the analytical work. The adsorbent used is Zeolite 5A (Union Carbide 20 x 40 mesh). The adsorbent parameters experimentally determined by Kayser and Knaebel are used as the standard values in all of the examples in this thesis. The void fraction e of the adsorbent bed was determined using a displacement liquid and was found to be 0.478 ± 0.010. The isotherm slopes for nitrogen and oxygen on this adsorbent were measured at three temperatures and the results are listed in Table 2.1. 37 Six experimental runs were documented. Five of these experiments are analyzed in Chapter 4 to determine their energy consumption. Table 2.1 Equilibrium Adsorption Isotherm Slopes (ICA and ks), and Selectivity Ratio (fl),for Nitrogen and Oxygen on Zeolite 5A. Temperature k A (N2) k B (02) P {°Q {-} {-} {-} 30 9.94 5.40 0.582 45 8.24 4.51 0.593 60 7.55 3.723 0.548 Cell models of the type used in Chapter 5 have been developed previously (Cheng and Hi l l , 1983, and Kirkby and Kenney, 1987), but it does not appear that these were used to calculate the efficiency of the PSA cycles. 38 3. Modeling PSA Cycle Energy Consumption 3.1 Introduction A great number of PSA cycles, each with its own method of mampulating the concentration wavefront, have been developed in order to increase the effectiveness of the separation process. Although the cycles are unique, a great number of them contain the same basic elements. In this chapter we first analyze the Four-Step cycle, determining the pressures and flows during each step. These pressures and flows are necessary to calculate the energy required. The energy for all the steps is then summed to find the net energy for the cycle. The same analysis is then performed for the Ideal Four-Step cycle and the Ideal Three-Step cycle. As these cycles are derivatives of the Four-Step cycle, much of the analysis for the Four-Step cycle can be directly applied. 39 3.2 Four-Step Cycle 3.2.1 Introduction A pressure swing adsorption "cycle" consists of all the steps required to feed the mixed gas, generate the product, expel the exhaust gas, and return the system to its initial state. The number of steps in a cycle is somewhat independent of the number of beds, and Figure 3.1 shows a diagram of two beds operating on the Four-Step cycle, along with the pressure history of each bed. This version of the Four-Step cycle uses pure product to pressurize the adsorbent bed. From the pressure histories, it is apparent that the beds are operating on the same cycle, with a 180° phase shift. The steps are listed as follows: 1. The feed step occurs at a high constant pressure, and during this step separation of the gas mixture takes place. The feed is pushed into the first bed until the shock wave reaches the end of the bed. Some of the pure product from this step is delivered as product, while the rest of it is used to purge the second bed, which is at low pressure, and then repressurize the second bed from PL to PH. 2. Blowdown, during which the pressure in the bed decreases from P H to PL and gas enriched in the heavier component is exhausted from the bed. This partially regenerates the adsorbent bed and prepares it for the low pressure purge step. 40 P H P L Purge Product Pressurization Feed Blowdown Product Product Product X Product 0 Feed Blowdown Purge Product Pressurization Figure 3.1 Four-Step Cycle: Pressurization With Product (After Knaebel and Hill, 1985, Figure 1). 3. Purging the adsorbent bed is the last step in regenerating the bed. During this step, some of the product from the second bed is throttled down to low pressure and used to flush the rest of the exhaust gas from the bed. This step is generally carried out at low pressure for two reasons. The first is that in blowing down the bed, some of the more adsorbed gas is removed from the bed without using any 41 pure product. The second reason is that at low pressure, the density of the product used to purge the bed is less. Again, this implies that less pure product is needed to purge the bed. 4. The final step is pressurization of the adsorbent bed, during which the pressure in the bed is increased from PL to PH . The cycle we are studying uses pure product gas from the second bed. At the end of this step the bed has returned to its initial condition. In Figure 3.2, the heavy line represents the leading edge of the concentration wavefront, and the shaded portion represents the length of the bed containing the more adsorbed component. Initially, the bed is free of species A, and during the feed step the species A molecules progress to the end of the bed. During blowdown, only the feed end is open, so the velocity at the product end is zero; hence species A remains throughout the bed. During the purge step, the species A molecules are displaced from the bed by the species B molecules. There are no species A molecules in the bed during the pressurization step. 42 Axial Displacement Feed Blowdown Purge Time Pressurization Figure 3.2 Propagation of the Shock Wave: Pressurization With Product (After Knaebel and Hill, Fig. 2). In the next four sections, each step of the Four-Step cycle is analyzed. The basis for this analysis is the work of Knaebel and Hi l l , 1985. In section 3.2.6, the work done by the system to expand the product gas (which is initially delivered at PH) to atmospheric pressure is calculated. This is done so that the work required for gas separation is not confused with the work required for gas compression. 43 3.2.2 Feed Step The work done in compressing the feed gas and mjecting it into the adsorbent bed is found by calculating the volume of feed gas necessary to advance the shock wave the length of a given adsorbent bed. This step is broken down into two sub-steps, which can be seen in Figure 3.3. Feed Piston V F = V F I P = Po y = yo V F = V F 2 P = P H y = y° Adsorbent Bed P = P H y = 0 P = P H y = 0 P = P H P = P H Product Piston V P = 0 3 V P = 0 3 V P = V P 2 P = P H y = o V P = V P 2 Np 2 r N p u , y = 0 j N p R / y = 0 3 3 V F = 0 V P - V P 2 X Np 2 3 T P = P H Figure 3.3 Feed Step for the Four-Step Cycle Utilizing Pressurization with Product. 44 The two sub-steps are: Step 1-2 Pressurizing the feed gas from Po to PH. Step 2-3 Feeding this gas into the bed while delivering product, purge gas, and pressurization gas. Here we deal only with the work done by the feed piston; the work done by the product piston in accepting the product gas and expanding it to atmospheric pressure is calculated in section 3.2.6. Before calculating the work for Step 1-2, we must first look to Step 2-3 and calculate the volume of feed at high pressure needed to push a concentration wavefront the entire length of the bed. M i y = o 1 H L B 1 Feed Piston Adsorbent Bed Product Piston Figure 3.4 Piston, Gas, and Shock Wave Velocities During Constant Pressure Feed. The bed is initially at high pressure and full of pure, less adsorbed gas. This implies that the mole fraction y is zero.* As the mole fraction of the feed is higher than * When I speak of "y," or "mole fraction," it is implied that this means the mole fraction of the heavy component in the gas phase in the adsorbent bed. Therefore, the mole fraction of gas that only contains light gas is zero. In the example of oxygen separation, nitrogen is the heavy gas and oxygen is the light gas. The mole fraction of pure nitrogen is unity (y = 1), while the mole fraction of air is 0.78 (y = 0.78), and the mole fraction of pure oxygen is zero (y = 0). 45 the mole fraction of the gas in the bed, a concentration shock wave wil l propagate through the bed (see Figure 3.4). As this is done at constant pressure, the continuity equations (Equations (2.40) and (2.41)) can be integrated analytically to give Equation (3.1) (Knaebel and Hi l l , 1985, Equation (8)).* This indicates that the mole fraction to the left of the shock wave remains constant at the feed mole fraction, and the mole fraction to the right of the shock wave remains constant at y = 0. Consequently, the velocity of the gas in the bed to the left of the shock wave remains constant and uniform. This is also true of the gas to the right of the shock. Therefore, Equation (3.1) relates the velocity at the entrance of the bed to the velocity at the exit of the bed. " i = 1 + (B-1)y2 (3.1) u2 l + (B-l)y1 Where: ui = velocity at the entrance of the bed {m/s} U2 = velocity at the exit of the bed {m/s} yi = mole fraction at the entrance of the bed {-} y2 = mole fraction at the exit of the bed {-} P = (3A / PB = ratio of the fraction of component A in the gas phase in the bed to the fraction of component B in the gas phase in the bed {-} As y i equals the mole fraction of the feed gas (yo), and y2 = 0, this formula simplifies to: u2=Ul[l + (B-l)y0] (32) Where: yo = mole fraction of the feed gas {-} * This equation actually describes the difference in velocities between any two points in an adsorbent bed when the pressure is held constant. 46 In order to simplify the analysis, the piston areas are sized such that the velocity at the entrance of the bed (ui) is the same as the velocity of the left piston, and the velocity of the gas at the exit of the bed (u2) is the same as the velocity of the right piston. This is done by equating the piston area to the open cross-sectional area of the bed. The actual cross section of the bed is A, but the void fraction of the bed is e; the product of these values gives the open cross-sectional area. This implies that any velocities inside the bed are interstitial velocities. Now we must calculate the time it takes for the shock wave to travel through the bed. As the bed is initially clean, the time is takes is just the length of the bed divided by the velocity of the shock wave. At constant pressure, the velocity of a shock wave can be found by using Equation (3.1) and the criteria that there is no accumulation in the shock wave. This results in the following equation (Equation (17) from Knaebel and Hil l , 1985). U _ PAU\ _ PAUI (3.3) 5 l + (/?- l )y 2 l + ( / ? - l ) y i Where: PA = fraction of species A in the bed that is in the gas phase {-} ui = velocity immediately on the high side of the shock wave {m/s} u 2 = velocity immediately on the low side of the shock wave {m/ s} yi = mole fraction immediately on the high side of the shock wave {-} y 2 = mole fraction immediately on the low side of the shock wave {-} 47 As before, we substitute in yi = yo and yi = 0. U S = B A U I = ? A U \ <M> 5 1 l + (B-l)y0 Now we calculate the time it takes for the shock wave to travel the length of the bed. time = distance / velocity I L (3.5) us Where: tF = time for the shock wave to travel the length of the bed {s} LB = length of the bed {m} When we substitute in Equation (3.4), we find: PAUI This is the same as Knaebel and Hill 's Equation (34). Now we can calculate the volume swept by the first piston during this step. Again, the piston travels at the same speed as the gas entering the bed because its cross-sectional area is equal to the open cross-sectional area of the bed, eA. volume = velocity * area * time VF2 — ux • s A-tF (37) 48 Where: VF2 = volume of the feed piston at the beginning of Step 2-3 {m3} e = void fraction of the bed {-} A = total cross sectional area of the bed {m2} Upon substituting in Equation (3.6), we find: V = (3.8) PA Where: VB = A L B = volume of the adsorbent bed {m3} Now we can calculate the number of moles of feed, using the ideal gas law with P = P H and V = V F 2 . N =Z3L£ZB_ (3.9) Where: N F = number of moles of feed {mol} P H = high pressure limit of the cycle {Pa} This can be divided up into moles of heavy component and moles of light component by multiplying by the feed mole fraction of each component. N FA yOPH £ V B (3.10) RT BA NFB = 49 ( l - y o ) p H svB (3.U) RT BA Where: NFA = number of moles of species A in the feed gas {mol} NFB = number of moles of species B in the feed gas {mol} We can also calculate the volume of the feed piston at point 1 in Figure 3.3, by using the ideal gas law with N = N F and P = P 0 . 1 H 1 H yP0J PA v P „ y v , (3.22) In order to calculate the number of moles of product, we use Equation (3.2) which relates U2 to ui, along with the cross-sectional area and the time length of the feed step. Then we multiply this by the molar density, P H / RT: Nn=u2-eA-tF-^- (3-13> p. 2 RT Where: Npi = number of product moles delivered during the feed step {mol} Substituting in Equations (3.2) and (3.6), we find: As can be seen in Figure 3.3, not all of these moles enter the product piston. At some point during the feed step, the product piston volume freezes and some of the 50 product is diverted into purging and repressurizing the other bed. The amount of product that is diverted to the purge and pressurization step is enough to keep the pressure constant during the feed step and to complete the purge and pressurization of the second bed. When we calculate the number of moles used for purge and pressurization we can calculate the final number of moles still available as product. This calculation is done in section 3.2.5 after the number of moles required for purge (Npu) and pressurization (NPR) are known. We can now calculate the work done by the system during the feed step: W f = 1 W 2 + 2 W 3 (3.15) 0 )dV (3.16) Where: NFRT (3.17) Upon mtegrating and substituting we find: B (3.18) yPHJ PA 51 3.2.3 Blowdown Step During the blowdown step, the first of the two regeneration steps, the pressure in the bed is reduced from P H to PL by exhausting gas from the bed. The object of this step is to partially rid the bed of some of the heavy component and ready the bed for the low pressure purge step. During the purge step the rest of the heavy gas is removed from the bed. At the begirvning of the blowdown step the pressure is P H and the mole fraction is yo, which is uniform throughout the bed. This can be seen in Figure 3.5 (a). As the pressure decreases, the mole fraction of the gas in the bed increases. This happens because a given decrease in pressure results in the desorption of more heavy component than light component. The condition of the adsorbent bed at the end of the blowdown step can be seen in Figure 3.5 (b). Figure 3.5 Mole Fraction in the Bed at the (a) Beginning and (b) Ending of the Blowdown Step. For the Four-Step cycle, work is only required for that portion of the blowdown step during which the pressure in the bed is less than atmospheric. If the pressure in the bed is greater than atmospheric, the gas is allowed to escape to the atmosphere as the pressure decreases. Figure 3.6 shows the example where P H > Po > PL. For the first step 52 from P H to Po, no work is required, but from Po to PL, work is required as a partial vacuum is created in the bed. Figure 3.6 Bed Mole Fraction During Blowdown at (a) P = PH, (b) P = Po, and (c) P = Pi. The equations for the reversible work required to depressurize an adsorbent bed while rejecting gas at an arbitrary pressure Px are developed in Appendix A. Two cases of this work wil l be presented here: P H > Po >PL and Po > P H > PL . The third possible case, in which both P H and PL are greater than Po , requires no work. For the first case, we are rejecting gas at atmospheric pressure (Px = Po) and our limits of integration are Po and PL . WB = \, -eVB rln (3.29) dP * BB[\-^{p-\)y\~\P0J As noted above, the mole fraction y in Equation (3.19) is a function of pressure, and varies according to the following equation, which is the same as Equation (12) in Knaebel and Flill (1985). I y) i-pi .y0 - h 53 (3.20) If Po is greater than both P H and PL , work is required to evacuate the bed over the entire pressure range and the limits of integration are P H and PL. WB dP (3.21) Equation (3.21) is again solved simultaneously with Equation (3.20). This must be done numerically (by Mathcad). At the end of this step, the mole fraction in the bed is found from Equation (3.20), with P = PL. This mole fraction is important for the next step, in which the bed is purged of the remaining heavy gas. yB=y{PL) (3.22) The number of moles of heavy and light gas that leave the bed can be found by mtegrating the flow out of the bed. The change in mole fraction must be taken into account, so the following equations must be solved simultaneously with Equation (3.20). N BDA - f p, y p, sVB dP (3.23) N BDB (3.24) '" R T 0B[l + (/3-l)y] dP 54 Where: NBDA = moles of species A that leave the bed during blowdown {mol} NBDB = moles of species B that leave the bed during blowdown {mol} If we know the value of VB, we know the composition of the gas inside the bed at the end of the blowdown step. As we also know the composition of the gas inside the bed before blowdown, we can do a mass balance to find the composition of the gas that has left the bed. The results of this mass balance are the following equations: (y0PH-yBPL)£VB RT BA (3.25) N BDA [(\-y*)PH-(\-yB)PL]eyB (3.26) N BDA RT B 55 3.2.4 Purge Step During the purge step, pure product is throttled to low pressure and used to purge the remaining heavy gas from the adsorbent bed. As with the blowdown step, work is only required if PL < Po; otherwise, the gas is allowed to escape to the atmosphere. During the purge step, a simple wave develops because the mole fraction of the product is less than the mole fraction of the gas in the bed (see Figure 3.7). In this section we develop a mathematical understanding of the purge process in order to calculate the number of moles of pure product necessary to purge the bed (Npu), the number of moles that are purged out of the bed (Nw), and the work required to purge an adsorbent bed. The method of characteristics is used to solve the equations relating to the simple wave. This method is more fully developed by Knaebel and Hi l l (1985) and states that at constant pressure, the composition is constant along the characteristics. This means that the axial position in the bed with mole fraction y and distance from the product end of the bed z, moves with speed (Equation (10) in Knaebel and Hi l l , 1985): * = f*u = Constant ^ dt l + (B-\)y This includes the position at the far right of the simple wave, which has mole fraction zero (point A in Figure 3.7). It is important at this point to note that the velocity of the characteristic (the position with mole fraction y = constant) is related to, but not equal to the velocity of the gas at that axial position in the bed. For this reason, Figure 3.7 makes a distinction between these two velocities; UA is the velocity of the gas through the adsorbent bed at 56 position A, while dz/ dt | A is the velocity of the position in the bed with mole fraction y A . Figure 3.7 Formation of a Simple Wave During Purge. From this, we understand the time for purge to be the time it takes for position A (the characteristic with mole fraction zero) to travel the length of the bed. We assume that the velocity of the purge gas entering the right end of the bed is u 2 , which is constant. This velocity is arbitrary, and influences the time for purge (tpu), but not the amout of purge gas required (Npu), or the work required for purge. 57 As the composition between point A and the right end of the bed is constant (y = 0), the gas at position A also has velocity ui and we can use Equation (3.27) to calculate the velocity of position A: dz dt (3.28) A The time it takes to purge the bed is then just the length of the bed, LB, divided by the velocity of position A, Equation (3.28) (Equation (34) in Knaebel and Hi l l , 1985). f - LB (3.29) PA^I Where: tpu = time required for the purge step {s} We can then calculate the volume of light purge gas at low pressure used to purge the bed. volume = velocity * area * time VPU -u2-s A-tPU £j_30j Where: Vpu = volume of light gas used to purge the bed at P = PL {m3} When we substitute in Equation (3.29), the result is: V = (3-31> PU PA 58 Now we can calculate the number of moles of purge, using the perfect gas law with P = P L and V = V P U . N = 1^^_B_ (3.32) PU RT BA Where: Npu = number of moles of pure light product used to purge the bed {mol} This is Knaebel and Hill 's Equation (35). In order to calculate the volume of the withdrawal piston (the imaginary piston that contains all of the gas that is purged from the bed) and the work to extract the purged gas, we integrate the flow out the end of the bed.* From Figure 3.7, we find that the gas leaving the left of the bed and entering the withdrawal piston has changing velocity ui(t) and mole fraction yi(t). We must develop an equation that describes the velocity ui(t) as a function of yi(t). Once we know ui(t) we can integrate the flow from t = 0 to t = tpu to find the final volume of the imaginary withdrawal piston, Vwi, that contains all of the purged gas. Once we have Vwi we can calculate the work required for the purge step. * This can also be done by mass balance, knowing the moles of pure purge Npu, the initial conditions (P = PL , y = yB), and the final conditions (P = PL , y = 0). of the bed. We perform the integration, as it reveals the purge process. 59 To simplify this we wil l look at a specific position in the bed and then generalize the equations to any position in the bed. We look then at position C in Figure 3.7, which always has mole fraction yc. The gas at position C is moving at velocity uc, while position C itself is moving at constant velocity: dz dt BAUC (3.33) i + G*-i)y c The relation between the velocities and mole fractions of any two points in an adsorbent bed is given in Equation (3.1). Therefore, uc and yc are related to U2 and y2 (which is equal to zero) by the following expression. «c = / ' \ ( 3 3 4> l + ( / ? - l ) y c Substimting Equation (3.34) into Equation (3.33) yields: dz I t _ PAu2 (3.35) c [ l + ( / ? - l ) y c ] 2 At time t = 0, position C is at the right end of the bed, as are the positions of all values of y, and when t > 0, position C travels with velocity described by Equation (3.35). We can now calculate the time it takes for position C to reach the other end of the bed. time = distance / velocity f c = - ^ [ l + ( / ? - l ) y c ] 2 0.36; PAU2 60 By this we are saying that at time tc the mole fraction at the entrance of the bed is yc. This can be generalized for all time, with yc becoming the mole fraction at the left end of the bed, yi. f = - ^ - [ i + (A- i ) y i ] 2 w P AU2 However, until the front of the simple wave reaches the end of the bed, the gas leaves with mole fraction yB. The position of the simple wave at the end of this constant mole fraction step is labeled tB in Figure 3.7, and the equation describing this time is given below. P AU2 Equation (3.37) is valid for the remaining time from t = tB to t = tpu and can be solved for yi in terms of t. Only one root of the resulting quadratic has values of yi less than 1. The solution of yi(t) over the entire range 0 < t < tpu is presented below. yi(t) = yB ;0<t<tB (3.39) BAu2 1- ^^-t V L B yx ( 0 = ^ ; t B < t < t p u (3.40) Now that we know yi as a function of time, we can calculate ui as a function of time by relating ui and yi to u 2 and y 2 according to Equation (3.1), with y 2 = 0. u1(t)= , u \ / x 61 We can now calculate the number of moles that leave the bed and enter the withdrawal piston by mtegrating the flow out of the bed. Nw = \'Bul(t)%-eAdt+\,'aui(t)-%-eAdt ^ w Jo ]K/RT ha RT Where: Nw = moles of gas purged from the adsorbent bed during the purge step {mol} This expression can be integrated by substimting in Equation (3.41), the appropriate equations for yi(t) (Equation (3.39) or (3.40)), tB (Equation (3.38)), and tpu (Equation (3.29)): N'=jfcjH(l-i>)y> + 1] <3 43> Calculating Nw by mole balance, as mentioned earlier, provides the relative number of heavy and light moles that leave the adsorbent bed: N _ >>BPL S V B (3.44) i y WA ~ RT /}, Where: NWA = number of moles of species A in the purged gas {mol} NWB = number of moles of species J3 in the purged gas {mol} 62 At low pressure, Nw fills volume: Vwl=^[(l-6)yB+l] (3.46) We now refer to Figure 3.8 to calculate the work required for the purge step. Again, work is only required for the purge step if PL < Po. The work calculation is broken down into two steps: Step 1-2 Extraction of the purged gas at constant pressure PL. Step 2-3 Compression of the purged gas to atmospheric pressure. Vw = 0 £ Vw - Vw2 £ P = Po Nw P = P L y = y B P = P L y = 0 N P U Vw - V w i P = P L y = 0 P = P L Nw Purge Gas From Product Piston Figure 3.8 Steps for Purge Work Calculation. 63 The volume at the end of Step 1-2 is calculated using the ideal gas law with P = PL and N = Nw. Mi v; (3.47) wi The work required to withdraw the gas from the bed and compress it to atmospheric pressure is then: (3.48) (3.49) Where: w (3.50) When Equation (3.50) is substituted into Equation (3.49) and the result is integrated, we find the work required to withdraw the purged gas from the bed. (3.51) 0 ' HA 64 3.2.5 Pressurization Step The final step in the Four-Step cycle is pressurization with product, during which some more of the pure product (at P = PH) is throttled back into the bed to raise the pressure from PL to PH. No work is required for pressurization, but we must keep track of the flows during this step in order to calculate the final amount of product. The number of moles used to pressurize the bed is found by subtracting the number of moles in the bed at low pressure from the number of moles in the bed at high We can now calculate the final number of moles in the product piston and the volume of the product piston required in Section 3.2.6 to calculate the work recovered by expanding the product gas to atmospheric pressure. The final number of moles in the product piston is: pressure. {PH-PL)SVB RT fiB (3.52) PR (3.53) Upon substituting we find: [P„(y0-1) + PL C M eV, B (3.54) RT PA Now we can calculate Vp2 using the ideal gas law with P = PH-+ J , 0 - I W - I ) ^ ) PA B (3.55) 65 3.2.6 Expansion of Product Gas and Net Work To compare the work required for the Four-Step cycle to the reversible work of gas separation, we expand the product gas to atmospheric pressure, recovering the expansion energy. Figure 3.9 shows the two steps involved: product delivery at constant pressure and product expansion. Vre can be calculated using the ideal gas law with N = Np2 and P = Po. ^ L (> 'o- l ) + — = f P ^ 1 H (3.56) V P = 0 Vp = V P 2 V P = V P 3 IB P = P H N = Np 2 IB P = Po N = N P 2 1 2 3 Figure 3.9 Expansion of the Product to Recover Energy and Deliver Product at Atmospheric Pressure. The work that is recovered in this step is: WR = \[n(PH - P0)dVP + \Vvn{P-P0)dVP (3.57) Where: p=NP2RT (3.58) Upon mtegrating we find: WR=[PHbQ-l)+PL]h^(fi-l) sVB PA The net work for the cycle is: w4 = wF+wB+ww + WR If PL > Po, this simplifies to: w, = wF+ WR Table 3.1 is a summary of the work equations for the Four-Step Cycle. 67 Table 3.1: Summary of Work Terms for the Four-Step Cycle. Feed Work: (3.18) WF = PH)n PJ P Blowdown Work: (3.19) pL -eVR In 4-+ {P-M ypoJ dp • \fpH >P0>PL (3.21) (PL ~ SVR I WB = I — f , . -.In " PB[i+{P-i)y] \PJ dP ; i f P0 >PH>PL WB=0 ; ifPH >PL>P0 The Blowdown Work equation must be solved simultaneously with the following equation. 1 B (3.20) i/o 1-/9 Purged Gas Work: (3.521 w pLH PJ eV, P ; i f P0 > PL Ww=0 ; i f PL * Po Work recovered in expanding product gas to atmospheric pressure: (3.59) wR=[pH(y0-i) + PL In f P > ZJL 68 3.2.7 Recovery of Species B As the purity of the product generated by this cycle is always 100%, it is important to know the recovery that the cycle is producing. Without knowing this, the work calculation can be taken out of context. In this section we examine the factors that affect recovery. The recovery of species B is found by dividing the number of product moles (which are pure species B) by the number of moles of species B in the feed. Recovery = ^ 0.62; Upon substituting, the result is: Recovery = P " ( 1 y o ) ? L ( l - B ) $.63) PH ( i _ y o) We now introduce the pressure ratio, which is the non-dimensional ratio of the high pressure in the cycle to the low pressure. n = ?iL (3-64) If we substitute Equation (3.64) into (3.63) we find: Recovery = 3 1 z M z l ( i _ g ) (3.65) For this equation to be greater than zero, the following inequality must be satisfied: n > 1 (3.66) i - y 0 69 Two examples of recovery vs. pressure ratio and feed mole fraction are shown in Figure 3.10 and Figure 3.11. The first is for p = 0.1 and shows the effect of varying yo, the mole fraction of the feed. The pressure ratio is shown to vary from 1 to 300, which is extremely high, in order to show that for all values of yo, recovery approaches the limit of (1-P). The second figure is for the case where p = 0.9. These figures show that low values of p imply high recovery, and high values of p imply low recovery. For this reason, low values of P are desired for gas separation, and are associated with high separation factors ( k A / k B ) . Recovery vs. Pressure Ratio for Beta = 0.1 -0—y0 = 0.1 •H-y0 = 0.5 •A—yO = 0.9 0 50 100 150 200 P H / P L {-} Figure 3.10 Recovery vs. Pressure Ratio for {1= 0.1. 70 Recovery vs. Pressure Ratio for Beta = 0.9 0 50 100 150 200 PH/PL{-} Figure 3.11 Recovery vs. Pressure Ratio for (5= 0.9. In these figures it is evident that as yo increases and there is less light gas in the feed, the minimum n necessary for positive recovery increases. Positive recovery implies that after purge and repressurization, there is still some product gas left as product. Figure 3.12 graphs recovery vs. pressure ratio for oxygen concentration (yo = 0.78 and f3 = 0.582). The minimum pressure ratio for oxygen concentration using the Four-Step cycle is 4.55, and the maximum recovery possible is 41.82% (1 - p). 71 Recovery vs. Pressure Ratio for Oxygen Concentration PH/PL{-} Figure 3.12 Recovery vs. Pressure Ratio for Oxygen Concentration; yo = 0.78, f3 = 0.582. 72 3.2.8 Discussion of Net Work for the Four-Step Cycle When we add up the work done by the system in each step, we find the net work done by the system in separating the gases. When we plot this net work (Equation (3.60)) per mole of product oxygen for various values of PL and n, we obtain results of the form given in Figure 3.13. n w Figure 3.13 Four-Step Cycle: Net Work per Mole of Product Oxygen (1U4) Done by System (yo = 0.78, 0=0.582). This shows the work required to produce 1 mole of species B as a function of PL and n for our base case of oxygen concentration (yo = 0.78 and (3 = 0.582). Positive work is defined as work done by the system, so the values in this graph are negative. The low pressure varies from 0.2 arm. to 4 arm., while the pressure ratio varies from 7 to 25. At low values of pressure ratio, the net work required for the cycle increases dramatically 73 because less product is made and recovery is lower. This tends to increase the work required per mole of species B, especially near the lower limit of IT, where work is done, but almost no product is made. For a given PL , as n increases, the net work required per liter of light gas decreases and approaches a limit. For a given 1% as PL increases, the work required per liter increases. When P H > Po, this increase in work can be attributed to the irreversibilities that occurs during throttling. It is important to note that when PL < Po, although work is required for blowdown and purge (this is not the case when PL > Po), the work required is less than when PL > Po. This seems to imply that the vacuum Four-Step cycle requires less energy to produce a given amount of pure product. In reality, this would depend on the efficiency of the vacuum pumps used for the blowdown and purge steps. When we divide the reversible work to produce one mole of pure oxygen by the net Four-Step cycle work to produce the same amount of gas at the same recovery, we find the ratio depicted in Figure 3.14. This is the second law efficiency of the cycle. For example, at PL = 1 atm. and U = 7, the second law efficiency 2.7%. The second law efficiency increases as PL decreases because less compression work is lost by mrottling. As II increases, the second law efficiency also increases because recovery increases. At the lower limit of IT, the second law efficiency is zero; again because no product is made and work is done. The second law efficiency is greater for the vacuum Four-Step cycle than for the Four-Step cycle with PL > Po. 74 n { - } Figure 3.14 Four-Step Cycle: Second Law Efficiency. The next three graphs show how the net work required for our base case of oxygen concentration depends on the pressure ratio IT and the selectivity ratio p. Figure 3.15 shows the work required to produce 1 mole of oxygen from air. In this example, PL has been set to 1 atmosphere, which is typical of PSA cycles. For yo = 0.78, the minimum value of n for recovery greater than zero is 4.55. At this pressure ratio, the second law efficiency is zero. 75 NetWork For the Four-Step Cycle (y0 = 0.78) 0 5 10 15 20 P H / P L H Figure 3.15 Net Work per Mole of Product Oxygen for Oxygen Concentration Using the Four-Step Cycle (u>i). 2nd Law Efficiency For the Four-Step Cycle (yo = 0.78) 0 5 10 15 20 PH/PL{-} Figure 3.16 Second Law Efficiency for Oxygen Concentration Using the Four-Step Cycle. As Ft increases, the work required decreases. Although there are more mrottling losses, the recovery increases with a higher pressure ratio and more useful work is done. This can be seen in Figure 3.17, which shows the recovery for the cycle with yo = 0.78 and the same values of (3 used in the work graphs. From section 3.2.7, we know that 76 low values of B imply good separation and high recovery. From Figure 3.15 we can see that they also imply that less energy is needed for separation. The second law efficiency is plotted in Figure 3.16, also as a function of n and B. For low values of B the second law efficiency is high. However, even in the best cases shown here, the second law efficiencies are much less than 100%, implying that substantial energy savings are thermodynamically possible if better cycles are used. Recovery For the Four-Step Cycle (yo = 0.78) -©—Beta = 0.1 •H—Beta = 0.3 •A— Beta = 0.5 •e—Beta = 0.7 0 5 10 15 20 P H / P L {-} Figure 3.17 Recovery for Oxygen Concentration Using the Four-Step Cycle. 77 3.3 The Ideal Four-Step Cycle 3.3.1 Introduction At many points during the Four-Step cycle, gas was irreversibly expanded and availability was lost. In the Ideal Four-Step cycle, all of the throttling irreversibilities are removed by recovering the work used to pressurize the gas. The following is a list of the four steps of the Ideal Four-Step cycle and how they compare to those of the Four-Step cycle. A sketch of the cycle and the energy recovered in each step can be seen in Figure 3.18. 1. Feed. The feed step of the Ideal Four-Step cycle is the same as the feed step of the Four-Step cycle. As the pressure is constant and the gas flows are effected by pistons, there are no irreversibilities due to throttling. The irreversibility that exists in this step is due to the shock wave, and as the shock wave is necessary for separation, this irreversibility cannot be removed. 2. Blowdown. During blowdown, the pressure in the bed is reduced from P H to PL. In the Four-Step cycle we assumed that if P H > Po > PL, no energy was used or recovered as the pressure fell from P H to Po; the gas was just allowed to escape to the atmosphere, and work was only required to evacuate the bed from Po to PL. In the Ideal Four-Step cycle, we must recover the energy from the gas leaving the system when the bed pressure is above atmospheric, and use energy when the bed pressure is below atmospheric. 3. Purge. The purge gas in the Four-Step cycle was obtained by throttling gas from the product piston at P H down to PL. This tJirottiing process represents a loss of avaUability, which the Ideal Four-Step cycle recovers by expanding the purge 78 gas in the product piston from P H to PL through a turbine and then feeding it into the bed. The work equation for withdrawing the purged gas from the bed is the same for both the Four-Step cycle and the Ideal Four-Step cycle, but the equation is applied even if PL > Po. 4. Pressurization. During the pressurization step of the Four-Step cycle, gas is throttled from high pressure into the bed until the pressure in the bed has reached PH. In the ideal cycle, this energy is recovered in the same way as during purge, except that the bed pressure is constant during the purge step, and varying during the pressurization step. Feed Blowdown Purge Press. Figure 3.18 Ideal Four-Step Cycle: Flows and Energy Recovery. 79 3.3.2 Feed Step The work required to inject the feed gas and deliver product for the Ideal Four-Step cycle (WF) is the same as that for the Four-Step cycle. The derivation is not repeated and the reader is referred to Section 3.2.2. 3.3.3 Blowdown Step During the blowdown step, energy can be recovered from any gas leaving the bed while the pressure in the bed is greater than atmospheric. Any gas that leaves the bed while the pressure in the bed is less than atmospheric requires work. The work for the entire step is found by placing a reversible turbine between the bed and the atmosphere (see Figure 3.19). Figure 3.19 Reversible Turbine used in Blowdown. The work W B is found using the same equations developed in Section 3.2.3, but the limits of integration are P H and PL , regardless of what these values are with respect to Po. The two equations which must be solved simultaneously are: P y (3.67) (3.68) yy-P y-iy-P p Vy0J {yo-V 80 3.3.4 Purge Step In order to make the purge step reversible, we must remove the irreversible throttling used in the Four-Step cycle to reduce the purge gas pressure. Now, instead of throttling the gas we place a reversible turbine between the bed that is being purged and the source of the purge gas. The process is shown in Figure 3.20. Figure 3.20 Reversible Expansion of Purge Gas and Reversible Expansion/Compression of Purged Gas From Section 3.2.4 we know the number of moles of purge gas: N PU RT PA (3.69) The reversible work is then: WPU =NPU{y/H-\i/L) (3.70) Where \\IH and are the availabilities of the high pressure and low pressure gas relative to pure product at ambient pressure. Using the results of Chapter 2, the work recovered from the purge gas is: PU PLln (3.71) 81 The work to withdraw the purged gas out of the bed is the same as that developed in Section 3.2.4 and is repeated here. For the Four-Step cycle this is only used if PL < Po; if PL ^ Po the gas is allowed to escape and Ww = 0. For the Ideal Four-Step cycle we use this regardless of the value of PL with respect to Po. Ii PL ^ Po the work given by Equation (3.72) wil l be positive, and if PL < Po the work wil l be negative. The amount of purged gas (Nw), is the same as that calculated in Section 3.2.4. (3.72) 82 3.3.5 Pressurization Step Instead of throttling the gas from P H in order to pressurize the bed, we now place a reversible turbine between the pressurization piston and the bed. The pressurization piston acts as a reservoir, supplying gas at constant pressure. This is not an assumption as the pressurization gas is taken from the product as it is being delivered at PH. Figure 3.21 shows the system being considered. P = P L -> P = P H W P R Pressurization Gas N P R P = P H y = 0 Figure 3.21 Recovery of Work During Pressurization. The work recovered during pressurization is found using the equations developed in Appendix A for pressurizing a bed filled only with species B , from PL to P H , from a reservoir containing only species B at P = PH. WP* = [ In dP (3.73) This equation can be integrated: W P R = PH+PL In -1 (3.74) 83 3.3.6 Expansion of Product Gas and Net Work Again, to compare the Ideal Four-Step cycle with the Four-Step cycle and the reversible work of separation, we expand the product gas from P H to Po in and isothermal turbine to recover work WR. (see Equation (3.59)). The net work for the Ideal Four-Step cycle is: wIA = wp + wB + wPU+ww + wPR+wR ( 3 J 5 ) A summary of these work terms can be found in Table 3.2. The recovery is again found by dividing the number of product moles by the number of moles of the light component in the feed and the result is the same as that for the Four-Step cycle. The same criteria for minimum pressure ratio exists, and if the cycle is to produce any product, the inequaUty in Equation (3.66) must be true. 84 Table 3.2 Summary of Work Terms for the Ideal Four-Step Cycle. Feed Work: (3.18) f r> \ WF = PH In yp«) pA Blowdown Work: (3.67) w - r -*V» J L dP The blowdown work equation must be solved simultaneously with the following equation. P (3.68) i-p y0 - V I-P p Purge Gas Work: (3.71) Wpu = PL In eVR {PJ PA Purged Gas Work: (3.72) r r, \ Ww = PL lnl yPj PA \i-p)yB + \ Work extracted by expanding the pressurization gas: (3.73) W = PR PH+PL f In L -1 V yPH) J Work recovered in expanding gas to atmospheric pressure: (3.59) ( P } wR=[pH(y0-i) + PLU^-(p-i) V •'0 PA 85 3.3.7 Discussion of Net Work for the Ideal Four-Step Cycle We plot the net work and second law efficiency for the base case (yo = 0.78, B = 0.582) using the Ideal Four-Step cycle in the same manner as we did for the Four-Step cycle (see Figure 3.22 and Figure 3.23). Net Work For the Ideal Four-Step Cycle (y0 = 0.78) o -5000 ~ -10000 « -15000 $ -20000 O -25000 o S -30000 -35000 -40000 -45000 -50000 8 10 12 PH/PL{-} 14 16 18 20 Figure 3.22 Net Ideal Four-Step Cycle Work per Mole of Product Oxygen (wu)for Oxygen Concentration; yo = 0.78, f) = 0.582. 40 35 30 c • I H 25 u • pH 20 W % 15 rt 10 T3 C 5 ts 0 2nd Law Efficiency for the Ideal Four-Step Cycle (yo = 0.78) 10 P H / P L H 15 20 Figure 3.23 Ideal Four-Step Cycle Second Law Efficiency for Oxygen Concentration; yo = 0.78, /? = 0.582. 86 The net work required to produce 1 mole of product gas is much less than that required for the Four-Step cycle and the work is not a function of PL. This is because all of the compression energy is recovered. The work required using the Ideal Four-Step cycle is about three to six times less than that of the Four-Step cycle, depending on the low pressure used for the Four-Step cycle. As Fl increases, the work required for the Ideal Four-Step cycle approaches a limit, as with the Four-Step cycle. The second law efficiency is plotted in Figure 3.23. This is also not a function of PL, but reflects the same trend as the Four-Step cycle in that the work ratio approaches zero as n approaches 4.55, and approaches a limit as n increases. In comparison with the Four-Step cycle, the second law efficiency has increased from 2.70% to 16.87% at n = 7, and 5.27% to 36.77% at II = 25. At very high values of n, the second law efficiency approaches 43.98%. The recovery for this cycle is the same as that of the Four-Step cycle (see Figure 3.12). The next two graphs show how the work required for our base case of oxygen separation depends on n and p. 87 Net Work For the Ideal Four-Step Cycle (y0 = 0.78) o -5000 e So -10000 5? O -15000 8^ -20000 "** -25000 -30000 —e--Beta = 0.1 - B - -Beta = 0.3 —A- -Beta = 0.5 —e--Beta = 0.7 20 Figure 3.24 Net Ideal Four-Step Cycle Work per Mole of Product Oxygen (WM) for Oxygen Concentration. 2nd Law Efficiency for the Ideal Four-Step Cycle (yo = 0.78) 40 35 >> u 30 S <U 25 u !+* 20 W ^ 15 i-l 10 T3 5 CM 0 —e--Beta = 0.1 —B--Beta = 0.3 —A- -Beta = 0.5 —e--Beta = 0.7 20 Figure 3.25 Ideal Four-Step Cycle Second Law Efficiency for Oxygen Concentration. The work curves follow the same trends as those for the Four-Step cycle; approaching iruinity at the low pressure ratio limit, and approaching a limit as n increases. However, the net work is only weakly dependent on B, and the dependence is reversed; for the Four-Step cycle, low values of B required less energy, and for the Ideal Four-Step cycle, low values of P require more energy. 88 The second law efficiency also shows little dependence on the value of p, but is much higher than that of the Four-Step cycle. 89 3.4 The Ideal Three-Step Cycle 3.4.1 Introduction We have seen in the last section that the PSA second law efficiencies are less than 100% even when throttling losses are eliminated. It appears that the cycle affects the efficiency, so an even simpler cycle is analyzed below. This ideal cycle is different from the previous two cycles in that it consists of only three steps (see Figure 3.26). It is able to complete an entire cycle in three steps by combining the blowdown and purge steps of the last two cycles into one, which is termed evacuation. After the feed step, the adsorbent bed is evacuated until the pressure is zero and all of the gas has been rejected at atmospheric pressure. This removes all of the heavy component from the bed. Then, part of the product is expanded through a reversible turbine into the bed until the pressure is again P H and the bed is at its initial condition. The analysis is somewhat simpler, with most of the equations being derived earlier in this work. In fact, this cycle is just the limiting case of the previous cycle, with the pressure ratio approaching infinity. However, as no product is used for purge and the pressurization step requires more gas (the pressure in the bed must be raised from zero to P H instead of only from PL to PH), the amount of product delivered in one cycle is different. The bed is initially at P = P H and y = 0 and the steps proceed as follows: 1. Feed. The cycle begins with the same feed step as the Four-Step cycle and the first ideal cycle. Therefore, the feed work is the same as that calculated for the previous cycles. 90 2. Evacuation. In this step, the blowdown and purge steps of the previous cycles are combine into one step, with the gas from the bed being rejected at Po by a reversible vacuum turbine. At the end of the step, the bed is completely empty of gas. 3. Pressurization. In this step, some of the product gas is expanded through a reversible turbine and used to pressurize the bed. At the end of the cycle, the bed is at its initial condition (P = P H , y = 0). N F < W F P H Po 0 Feed Np2 Product > W R N P R W P R > > W E N E , Evacuation Pressurization Figure 3.26 Ideal Three-Step: Cycle Work and Molar Flows. 91 3.4.2 Feed Step The feed step for this ideal cycle is the same as for both the Four-Step cycle and the Ideal Four-Step cycle, except that product is only diverted to pressurize the bed; no product is required to purge the bed. This does not change the work done in this step, which is derived in section 3.2.2 (Equation (3.18)). 3.4.3 Evacuation Step The evacuation step is performed by a reversible turbine. Figure 3.27 shows the system used to analyze this step. Po — A . P = P H -> P = 0 / r \ / W E Figure 3.27 Evacuation of the Adsorbent Bed. The gas extracted from the bed is rejected at atmospheric pressure, so whenever the bed pressure is greater than Po, a decrease in bed pressure produces work. If the bed pressure is less than atmospheric, a decrease in bed pressure requires work. The formulas used to calculate the evacuation work are those used in previous sections for the blowdown step, with the upper limit being replaced by zero. The formulas, which must be solved simultaneously are: 92 fo -eVR P WE = \ —p , * -.In — — dp (3.76) ( 1 A \-B p (3.77) As the work done by the system is positive when P > Po, and negative when P < Po, this work equation is broken up into two components when P H > Po: E l I, In « + (/?-!) v] \Pj dP (3.78) WE2 = \ sVB l n | ^ fiB[l + (/3-l)y] U dp (3.79) The amount of gas that is removed from the bed is just the amount of gas that is in the bed at the end of the feed step. Below, this is broken down into heavy and light moles. N _yoPH S V B EA RT pA (3.80) N EB RT BB (3.81) 93 3.4.4 Pressurization Step As the pressurization step is from P = 0 to P = P H , the number of moles of product used is just the number of moles of gas in a bed at P = P H with y = 0. M _ P " S V B (3.82) R ~ RT Bg Therefore, the final number of moles in the product piston is: Np2 - Npl - NPR (3.83) Upon substituting, the number of moles of product left is: The gas used for pressurization is again taken from the product, expanded through a reversible turbine to recover the energy, and put into the bed until the pressure in the bed is PH. The pressurization work is calculated using the same formula as that used for pressurization of the Four-Step cycle and the Ideal Four-Step cycle, but the integration is done from P = 0 to P = PH. ™ Jo Bg dp (3-85) This can be integrated and results in the following expression for the work recovered in pressurizing the bed. W =PH^i- (3-86) 94 3.4.5 Expansion of Product Gas and Discussion of Net Work Again, the product gas is expanded in order to compare the work of separation to the other cycles and the reversible work of separation. The system is shown in Figure 3.28. V P = 0 3 Po V P = V P 2 P = P H N p 2 3 V P = V P 3 P = Po N p 2 3 Figure 3.28 Expansion of Product Gas to Atmospheric Pressure. Volume Vp2, the volume of the product piston after the feed step, is found by using Np2 from Equation (3.84) with the ideal gas law at P = PH. P2 PA OWoX1-/*) (3.87) The final volume Vp3 is calculated using Np2 and the ideal gas law with P = Po. ^ = f ^ ( i - y . ) ( i - / ' ) = v , 1 o PA P2 fP ^ 1 H v P 0 y (3.88) The work to deliver and expand the product gas is then: WR = C2(PH - P0)dVp + C\P-P0)dVp (3.89) Where: (3.90) 95 Upon substituting Equation (3.90) into (3.89) and integrating, we find the work recovered by expanding the product gas: xu D i ( P H \ S V B (, Vi a\ <3-91> v ro ' PA The net work for the cycle is: Wn Z=WF+WB+ WPR + WR (3.92) The recovery is found by dividing the number of product moles by the number of moles of the light component in the feed. Recovery = ^  ( 3 - 9 3 ) FB When these formulas are substituted in, the result is very simple. Recovery = (l-y#) (3.94) This is the limit of the Four-Step and Ideal Four Step cycle recovery when IT approaches infinity. The work is now neither a function of PL or FL The net work is shown in Figure 3.29 as a function of yo and p. There is only a weak dependence on p, but the work required increases as yo increases and there is less of the desired product in the feed gas. The second law efficiency is shown in Figure 3.30, also as a function of p and yo. Again, there is only a weak dependence on p, and the second law efficiency improves as yo decreases. For the example of oxygen concentration with yo =0.78 and p = 0.582, the second law efficiency is 43.98%. 96 2nd Law yo Figure 3.30 Second Law Efficiency for the Ideal Three-Step Cycle. 97 4. Comparison of Model with Previous Studies 4.1 Introduction In this chapter, two examples are given that illustrate the analysis derived in the earlier chapters of this work. The first example is used to compare the current work with the previous work done by Banerjee et al. (1990). This example also extends the previous work by dividing up what was previously known as "bed losses" into those losses due to throttling and those losses which are intrinsic to the separation process. The second example is of a vacuum cycle, in which the lower pressure of the cycle is below atmospheric. This example shows how the Vacuum Four-Step cycle is superior to the Four-Step cycle with a lower pressure of 1 arm, even when assuming linear isotherms. 4.2 Comparison with Banerjee et ah, 1990 4.2.1 Results of Banerjee et al., 1990 In the 1990 paper by Banerjee et al., an exergy analysis of the pressure swing adsorption air separation process is done. Exergy is defined as "the maximum useful work which can be obtained from the system by mteracting with the environment" (Banerjee at al., 1990). This is synonymous with the "availabiHty" of a system, which wil l be the term used rather than exergy. Two cycles are studied in Banerjee's paper: the Four-Step cycle with pressurization with product, and a simple cycle which includes an equalization. In both of these cycles, the feed gas is compressed adiabatically and then the heat of compression is removed by an aftercooler. The PSA process itself is considered to be isothermal. The system analyzed by Banerjee et al. can be seen in Figure 4.1. 98 W p Cooling Water Twi i—'-n i — i Feed N F moles Product Np2 moles mole fraction y p Exhaust mole fraction yo ^ J Compressor Aftercooler N E moles mole fraction yE Figure 4.1 System used by Banerjee et al. with Adiabatic Compressor and Aftercooler. As the current work is more concerned with the energy losses during the PSA process itself, we assume that the entire process is isothermal, which combines the function of the adiabatic compressor and the aftercooler into one isothermal compressor. This isothermal system is shown in Figure 4.2, and is similar to Figure 4.1 in that the availability of the gases at points 1, 3, 4 and 5 are identical. WF • • T - • + Feed — r -N F moles 1 mole fraction y 0 isothermal Compressor, T|T = 1 -*• Product Np2 moles mole fraction yp Exhaust N E moles mole fraction yE Figure 4.2 Isothermal System to Compare the Banerjee et al. Exergy Analysis to the Current Analysis. 99 In Figure 4.2, the efficiency of the feed compressor is set to unity. This is done to further clarify the losses in the system due only to the PSA process. In order to optimize the separation of oxygen from air with zeolite 5A, Banerjee et al. set the lower pressure of the cycle to 1 arm and then varied the higher pressure of the cycle until the compressor work per unit mole of product was minimized. For the Four-Step cycle with an adiabatic compressor, this occurred at 15.6 arm. This example cycle is summarized in the following tables. Table 4.1 lists the properties of the adsorbent bed. These properties are taken from Kayser and Knaebel (1986), who compared experimental results of air separation with the theoretical results developed by Knaebel and Hi l l in 1985. The results of these experiments agree well with the theory, and are presented in Section 4.4 along with an energy analysis of the cycles used. Table 4.1 Zeolite 5A Adsorbent Properties. s 0.478 H Bed voidage kA 9.94 H Slope of species A isotherm (nitrogen) kB 5.4 {-} Slope of species B isotherm (oxygen) PA 0.0844 {-} Fraction of species A in the gas phase in the bed PB 0.1450 {-} Fraction of species B in the gas phase in the bed P 0.5818 {-} Ratio of p A to p B V B 0.10 {m3} Volume of the adsorbent bed yo 0.78 {-} Mole fraction of the ambient gas (air) The volume is set to 0.10 m 3 , which does not affect the work per mole of product, but does affect the absolute values of energy and molar flows through the system. The mole fraction of the ambient gas used in the following analysis is 0.78. Banerjee et al. use the value of yo = 0.79, as there is 21% oxygen in air. However, for oxygen 100 concentration with 5A zeolite there is 78% more-adsorbed gas (nitrogen) and 22% less-adsorbed gas (21% oxygen and 1% argon). This makes little difference in the final analysis. Table 4.2 lists the pressure limits of the cycle used in the comparison to Banerjee's work. As the low pressure is 1 arm, no energy is required for the blowdown and purge, as is necessary when PL is less than atmospheric. Table 4.2 Cycle Properties used in the Energy Comparison. PL 101,325 {Pa} Low pressure of the cycle (atmospheric) P H 1,580,670 {Pa} High pressure of the cycle n 15.6 {-} Pressure ratio Table 4 .3 lists the number of moles in each step, as well as the average mole fractions of these gases. The inputs to the model ( N F , yo and yp) are calculated using the equations developed by Knaebel and Hi l l (1985). The equations for NBD , y B D , Nw, y w , N E , and y E have been derived earlier in this thesis. The availabihty of the gas leaving the PSA system depends on the outlet gas concentrations and molar quantities. The mole fraction of the gas leaving the system during the blowdown step is higher than air, mdicating that the blowdown gas been enriched in nitrogen. The mole fraction of the gas leaving the system during the purge step is actually less than that of air, mdicating that some of the gas exhausted during the purge step is actually enriched in oxygen. The values N E and ys represent the total quantity and composition of the exhaust gas, which is the combination of the blowdown and purged gas. The table also lists the number of moles of product generated and used in the purge and pressurization steps. Note that most of the product is used in the 101 purge and pressurization steps, with little of the original product left at the end of the cycle ( N P 2 = Npi - N P U - N P R ) . Table 4.3 Gas Quantity and Composition. N F 355.37 {mol} Number of moles in the feed yo 0.78 {-} Mole fraction of the feed (air) N B D 300.31 {mol} Number of moles of blowdown gas yBD 0.850 {-} Mole fraction of the blowdown gas N w 31.89 {mol} Number of moles of purged gas y w 0.683 H Mole fraction of the purged gas N E 332.20 {mol} Number of moles of exhaust gas y E 0.8344 H Mole fraction of the exhaust gas NPI 239.45 {mol} Number of moles of product delivered during feed step NPU 22.78 {mol} Number of moles of product used to purge bed N p R 193.50 {mol} Number of moles of product used to pressurize bed N P 2 23.17 {mol} Number of moles of product remaining as product yp 0 {-} Mole fraction of the product (pure oxygen) Rec. 29.64 {%} Recovery of oxygen Prod. 231.7 {mol/m3} Productivity of the adsorbent bed The recovery of oxygen is also given, and productivity is given in units of moles per cubic meter of adsorbent bed (productivity is usually given in terms of amount of product per unit amount of adsorbent per unit length of time, but as the analysis of Knaebel and H i l l (1985) is based on equilibrium theory, time is irrelevant). Banerjee et al. calculate the availability of the gases at the different positions in the system (see Figure 4.1 and Figure 4.2) and then produce a Grassman diagram, which represents the availability of the different streams with bars of proportional width. Figure 4.3 is the Grassman diagram for this separation example as given by Banerjee. It 102 shows the work input (which is equal to the availability of the feed gas at PH) divided into three streams by the PSA system. The middle bar, which represents the availability of the product stream, is taken to be the availability of the pure oxygen product at the pressure P H . This availability includes the difference in pressure between the product stream and the ambient air (PH = 15.6 arm and Po = 1 arm, respectively), and the difference in composition between the product stream and the ambient air (yp = 0 and yo = 0.78, respectively). The equation for this is Equation (4.1). N P2 N RT0 In P2 PH^-yp) (4.1) The bottom bar represents the availability of the exhaust stream, which Banerjee calculates to be 17,000 J/mol O2. The equation that I have used in the following section for the availabiHty of the exhaust stream per mole of product is Equation (4.2). Using this equation, I cannot arrive at the same value given by Banerjee; my calculation yields \|/B = 333 J/mol 0 2 . RTn ¥E = N P2 yy0) + NEB In 1 - J V (4.2) The first term in Equation (4.2) is positive and the second is negative, as the exhaust stream is enriched in species A and depleted in species B. Even if both terms are taken as positive, the value of I|/E is 3733 J/mol O2 which is still much lower than the value given by Banerjee. However, my analysis, which follows, is not dependent on Banerjee's results. 103 PSA Bed Inlet Work Stream Input WF , 112,300 J/mol O2 (100%) PSA Bed Loss 84,555 J/mol 0 2 (75.27%) \ Product A variability ) 10,745 J/mol O2 V (9.62%) Waste Stream Availability 17,000 J/mol 02(15.11%) Figure 4.3 Grassman Diagram for the Four-Step Cycle (Banerjee et al, 1990) with Product at PH-The top bar represents the "bed losses," which are made up of all the throttling losses as well as the losses that result from the gas separation process itself. These losses are given in Equation (4.3) (Banerjee et al., 1990, Equation 26). These losses account for 75.27% of the total input availabuity, and it is these losses that are dissected in this thesis. (4.3) N P2 N P2 104 4.2.2 Four-Step Cycle With the exception noted above, Banerjee's model is consistent with the model developed in Chapter 3 of this thesis. The semi-analytical Four-Step model of Chapter 3 is now used to analyze this cycle in more depth. A sketch of the gas and energy flows for this can be seen in Figure 4.4. The only difference with Figure 4.2 is the division of the product availability into pressure and separation terms by expanding the product to atmospheric pressure through a turbine. The avaUability of the product at atmospheric pressure per unit mole of product is given by Equation (4.4). N P2 ri-yS (4.4) Feed, N F WF T - J -* Product, Np2 W R Exhaust, N E Isothermal Compressor Figure 4.4 Four-Step Cycle: Energy and Molar Flows. The Grassman diagram for this is shown in Figure 4.5 and differs only sUghtly from that given by Banerjee in Figure 4.3. Again, the bed losses and tivrottling losses are grouped together. 105 Feed Work Input Exhaust Availability, \\in (3.59%) 333 J/mol 0 2 (0.31%) Figure 4.5 Grassman Diagram of Four-Step Cycle with Expansion of the Product gas to PL-If the work recovered by expanding the product, W R , is subtracted from the feed input work, the energy done on the system to produce 1 mole of oxygen is 106,207 - 6925 = 99,282 J/mol O2 with this cycle and pressure ratio. The reversible work required to do the same separation is just the sum of the product and exhaust availabiHties (3816 + 333 = 4149 J/mol O2). In this work, the second law efficiency is defined as the reversible work of separation divided by the actual work required for the separation. For this case, this is 4149/99,292 = 4.18%. These results, along with the results for the cases discussed below, can be seen in Table 4.4. 106 Table 4.4 Comparison of Separation Cases. Units Four-Step P L = 1 atm n = 15.6 Ideal Four-Step P L = 1 atm II = 15.6 Three-Step Ideal P L = 0 atm IT = oo Four-Step P L = 0.5 atm n = 15.6 Ideal Four-Step PL = 0.5 atm n = 15.6 Three-Step Ideal PL = 0 atm n = oo N F {mol} 355.371 355.371 355.371 177.685 177.685 177.685 y o {-} 0.78 0.78 0.78 0.78 0.78 0.78 NBD {mol} 300.307 300.307 - 150.153 150.153 -y B D H 0.85 0.85 - 0.85 0.85 -Nw {mol} 31.894 31.894 - 15.947 15.947 -yw H 0.683 0.683 - 0.683 0.683 -N E {mol} 332.201 332.201 322.674 166.1 166.1 161.337 y E H 0.834 0.834 0.859 0.834 0.834 0.859 N P 2 {mol} 23.17 23.17 32.697 11.585 11.585 16.348 y p {-} 0 0 0 0 0 0 wp J/mol O z -106,207 -106,207 -75,261 -79,411 -79,411 -56,272 WB2 J/mol 02 0 0 - -733 -733 -WE2 J/mol 02 - - -1737 - - -3452 w w J/mol 02 0 0 - -2405 -2405 -WB1 J/mol 02 0 62,540 - 0 40,629 -WEI J/mol 02 - - 44,318 - - 28,791 W p u J/mol 02 0 6808 - 0 6808 -WPR J/mol 02 0 17,089 15,938 0 17,089 15,938 WR J/mol 02 6925 6925 6925 5177 5177 5177 Net Work J/mol 02 -99,283 -12,845 -9818 -77,371 -12,845 -9818 V|/P J/mol 02 3816 3816 3816 3816 3816 3816 J/mol 02 333 333 502 333 333 502 l|/p+\|/E J/mol 02 4149 4149 4318 4149 4149 4318 Bed Loss J/mol 02 -95,133 -8696 -5500 -73,222 -8696 -5500 2nd Law Efficiency {%} 4.18% 32.30% 43.98% 5.36% 32.30% 43.98% Recovery {%} 29.64% 29.64% 41.82% 29.64% 29.64% 41.82% Prod. mol/m 3 231.7 231.7 327.0 115.8 115.8 163.5 107 Note that the work terms are defined as work done by the system. Therefore, the feed work is negative. The product and exhaust availabiHties are positive, as they can be seen as work done by the system. The net work (sum of all the work terms, excluding the product and exhaust availabilities) is negative. The bed losses are also shown as negative, mdicating that they are work put into the system that results in no useful output. 108 4.2.3 Ideal Four-Step Cycle The second cycle analyzed here is the Ideal Four-Step cycle, in which the throttling losses are eliminated. This allows us to calculate the actual bed losses, which are inherent to the separation process. The number of moles of gas that move through each part of the cycle, and the position of the turbines that would recover the expansion energy are shown in Figure 4.6. The results of the gas flow and work calculations can be seen in the second column of Table 4.4. N P U + N P R w p u + WPR Feed, N F WR N P I WF Isothermal Compressor N P 2 Product, N P 2 W B + w w —• Exhaust, N E Figure 4.6 Ideal Four-Step Cycle: Energy and Molar Flows. The associated Grassman diagram is shown in Figure 4.7. The work recovered during the blowdown, purge, pressurization and expansion of product steps are shown. As the low pressure is atmospheric, no energy is required to blowdown the bed or withdraw the purged gas from the bed. Over half of the work input (58.89%) is recovered during the blowdown step. This is high, because the pressure ratio is quite high. The work recovered during pressurization (16.09% of the feed work) is the second largest amount of energy recovered. This is also high because of the large pressure ratio. Expanding the purge gas (6.41%) and expanding the pressurization gas (6.52%) 109 are smaller contributors to the losses. Again, the product availability (3.59%) and the exhaust availability (0.31%) are very small compared with the availability of the feed. PSA Bed Inlet Feed Work Stream WF 106,207 J/mol 0 2 (100%) Blowdown, WB 62,500 J/mol 0 2 (58.89%) Purge, WPU 6808 J/mol 0 2 (6.41%) Pressurization, WPR 17,089 J/mol 0 2 (16.09%) Expanding Product, W R 6925 (6.52%) Bed Loss 8696 J/mol 0 2 (8.19%) Product Availability, v|/p 3816 J/mol 0 2 (3.59%) Exhaust Availability, \|/E 333 J/mol Oz (0.31%) Figure 4.7 Grassman Diagram for the Ideal Four-Step Cycle. The bottom bar in Figure 4.7 is the true bed losses. These are the bed losses that are not associated with the energy loss during throttling, or with friction of any kind. These losses can truly be called the bed losses, as they occur entirely within the bed. In this example, they account for 8.19% of the feed availability. This is indeed a small fraction of the work input to the cycle, and indicates that the major source of energy loss 110 is tiirottlirig. If these losses can be removed, the PSA process can be made much more efficient. If the compressor and all the turbines in Figure 4.6 are put onto a common shaft, the system would resemble that of Figure 4.8. In this situation, all of the recovered energy is used to power the feed compressor, and only the net work is required to operate the system. W I 4 Feed, N F Product, Np2 N p u + N P R n t-j N p i Np2 Exhaust, N E Figure 4.8 Ideal Four-Step Cycle: Energy and Mass Flows with all Turbines on One Shaft. The Grassman diagram for this is shown in Figure 4.9. The recovered energy loops back into the system and the net work required to power the system ( W B ) , is now just equal to the availability of the product and exhaust plus the bed losses. I l l 8696 J/mol C , 333 J/mol G 2 (0.31%) (8.19%) Figure 4.9 Grassman Diagram for the Ideal Four-Step Cycle with all Turbines on One Shaft. The second law efficiency of this cycle is found by dividing the reversible work of separation (\\ip + VJ/E = 4149 J/mol O2) by the work required to power the cycle (w« = 12,845 J/mol O2). The second law efficiency is 32.30%. This is a significant reduction in work required to produce oxygen product (from 99,282 J/mol 0 2), and a significant increase in the second law efficiency (from 4.18%). The results of this cycle are also listed in Table 4.4. 112 4.2.4 Ideal Three-Step Cycle The next cycle analyzed is the Ideal Three-Step cycle. In this cycle, the low-pressure is zero atm, which implies that the pressure ratio is infinite. Although the high pressure P H chosen does not affect the overall work done per mole of product oxygen, it does affect the absolute values of work done, and must be included in the calculations. The system energy and molar flows for this cycle are shown in Figure 4.10. N P R W P R Feed, N F W R N P I N p 2 Product, N P 2 W F Isothermal Compressor W E —• Exhaust, N E Figure 4.10 Ideal Three-Step Cycle: Energy and Molar Flows. The blowdown and purge work have been combined into the evacuation work. Also, product gas is only diverted for use in the pressurization step; for this reason there is more product gas left at the end of the cycle. Both the recovery and the productivity are much higher than those of the Four-Step cycle. The results of the calculations can be seen in the third column of Table 4.4, and the Grassman diagram is shown in Figure 4.11. 113 76,998 J/mol (100%) Evacuation Work Input, WE2 1737 J/mol 0 2 Evacuation, WEI 44,318 J/mol 0 2 (57.56%) Pressurization, WPR 15,938 J/mol 0 2 (20.70%) v Expanding Product, WR / 6925 J/mol 0 2 (8.99%) Product Availability, vj/p 3816 J/mol 0 2 (4.96%) Exhaust Availability, YJ/E 502 J/mol 0 2 (0.65%) Bed Loss 5500 J/mol 0 2 (7.14%) Figure 4.11 Grassman Diagram for the Ideal Three-Step Cycle. In this diagram, the evacuation work has been broken up into two parts: WEI represents the work recovered during the evacuation step as the pressure falls from P H to Po and W E 2 represents the work required during the evacuation step to remove the gas as the pressure falls from Po to zero. For this reason, W E 2 is shown as an energy input to the system. When this diagram is compared with the Grassman diagram for the Ideal Four-Step cycle (Figure 4.7), we can see that the work input required to effect the same product availabiHty is much less for the Ideal Three-Step cycle. The bed losses are also much less for the Ideal Three-Step cycle. As we know, the performance of the Four-Step 114 cycle increases with increasing pressure ratio. Therefore, the Ideal Three-Step cycle can be seen as the limiting case of the Ideal Four-Step cycle. If all of the recoverable expansion energy is fed into the input compressor, the second law efficiency is 43.98%. 115 4.3 Vacuum Cycle Example 4.3.1 Introduction This is an example of a cycle with a lower pressure that is below atmospheric. This is different from the previous example, as work is required to remove the gas from the adsorbent bed when the pressure is below Po. This work shows up on the left side of the Grassman diagrams, and is added to the work required to power the cycle. For comparison with the previous cases, the pressure ratio is again fixed at 15.6, but the lower pressure is set at 0.5 arm. 4.3.2 Vacuum Four-Step Cycle The system analyzed in this section is the same as that in Figure 4.4. However, as the lower pressure is reduced, the number of moles and energy flowing through the system are different. As both the upper and lower pressure limits are halved, and the pressure ratio remains constant, the molar values in Table 4.4 are exactly halved (this includes the productivity, which has units of mol/m 3). The mole fractions remain the same as before, along with the recoveries. The Grassman diagram is shown in Figure 4.12. In comparing this with the Four-Step cycle depicted in the Grassman diagram of Figure 4.5, we see that although there are extra components of work input, the total work input per mole is less for the vacuum Four-Step cycle (82,549 J/mol O2 vs. 106,207 J / m o l O2 in Figure 4.5). Furthermore, once the energy recovered by expanding the product is subtracted, the net work for the separation is only 77,371 J/mol 0 2 . The reversible work of separation (i|/p + V|/E = 4149 J/mol O2) remains the same, so the second law efficiency is higher for the vacuum case than for the case with PL = 1 arm (5.36% vs. 4.18%). Feed, W F 79,411 J/mol 0 2 (96.20%) Total Work Input 82,549 J/mol 0 2 (100%) Blowdown, W B 733 J / mol O2 Purge, w w (0.89%) 2405 J/mol O z (2.91%) PSA Bed Loss and Throttling Loss 73,222 J/mol O2 (88.70%) Expanding Product, W R J> 5177 J/mol 0 2 (6.27%) Product Availability, i|/p 3816 J/mol O2 Exhaust A variability, \ | /E ) 333 J/mol 0 2 (0.40%) • °' Figure 4.12 Grassman Diagram for the Vacuum Four-Step Cycle. 117 4.3.3 Vacuum Ideal Four-Step Cycle The system analyzed in this section is the same as that in Figure 4.6. Again though, energy is needed to complete the blowdown and compress the purge gas to atmospheric pressure. The Grassman diagram is shown in Figure 4.13. Feed, W F 79,411 J/mol 0 2 (96.20%) Blowdown, WB2 733 J/mol O z (0.89%) Total Work Input 82,549 | J/mol 0 2 (100%) Purge, ww 2405 J/mol 0 2 (2.91%) Blowdown, WBI 40,629 J/mol 0 2 (49.22%) Purge, wpu 6808 J/mol 0 2 (8.25%) Pressurization, WPR 17,089 J/mol 0 2 (20.70%) Expanding Product, W R 5177 (6.27%) Product Availability, i | / P 3816 J/mol 0 2 (4.62%) Exhaust Availability, \\>E Bed Loss 333 J/mol 0 2 (0.40%) 8696 J/mol 0 2 (10.53%) Figure 4.13 Grassman Diagram for the Vacuum Ideal Four-Step Cycle. In comparing this to the Grassman diagram of Figure 4.7, we see again that less total work is required per mole of product for the vacuum cycle. However, as less work is also recovered during blowdown and expansion of the product gas, the net work input for separation is the same as that for the Ideal Four-Step cycle with PL = 1 atm. If 118 all of the expansion turbines were placed on the same shaft, the Grassman diagram would look Like that of Figure 4.9, except that the bed losses would take up a larger percentage of the total work input. It is logical that the net work input is the same for the Ideal Four-Step cycle and the Vacuum Ideal Four-Step cycle (12,845 J/mol 0 2), as all the throttling energy is recovered and the pressure ratio remains the same. 119 4.3.4 Vacuum Ideal Three-Step Cycle This cycle is the same as that of Figure 4.10, and its Grassman diagram is seen in Figure 4.14. Feed, WF 56,272 J/mol 0 2 (94.22%) Total Work Input 59,724 J/mol 0 2 (100%) Evacuation Work Input, W E 2 3452 J/mol 0 2 (5.78%) Evacuation, WEI 28,791 J/mol 0 2 (48.21%) Pressurization, WPR 15,938 J/mol 0 2 (26.69%) Expanding Product, WR 5177 J/mol O2 (8.67%) Product Availability, \\ip 3816 J/mol 0 2 (6.39%) Exhaust Availability, V|/E 502 J/mol Q 2 (0.84%) Bed Loss 5500 J/mol O z (9.21%) Figure 4.14 Grassman Diagram for the Vacuum Ideal Three-Step Cycle. The Vacuum Ideal Three-Step cycle is seen to require less of a total work input than the Ideal Three-Step cycle in Figure 4.11. Again though, when the energy recovered by the turbines is subtracted from the input work, the same amount of net work is required for both (9818 J/mol 0 2). The three different vacuum cycles are summarized in Table 4.4. 120 4.4 Application to Kay ser and Knaebel, 1986 In this section, calculations are done for the Ideal Four-Step cycle, using the compositions and pressure ratios in the experimental work of Kayser and Knaebel (1986). In their experiments of oxygen concentration using zeolite 5A, the upper pressure of the cycle was maintained at approximately 3 atm, while the lower pressure was varied in order to change the pressure ratio. Six experimental runs were documented, but only the first five are analyzed here. In the first four experiments, the pressure ratio ranges from 6.48 to 23.54. In the fifth experiment, the low pressure is almost a complete vacuum (the pressure ratio is 803). No purge step was required for this run, so it closely resembles the Ideal Three-Step cycle. In order to create the low pressure of the cycle, a vacuum pump was used to evacuate the exhaust receiver and maintain it at a constant pressure. The blowdown and purge gas was then throttled into this exhaust receiver. The parameters for the experiments are listed in Table 4.5, and the results are listed in Table 4.6. Table 4.5 Experimental Parameters: Knaebel and Hill, 1986. Run T {°Q P H {arm} PL {arm} n H kA {-} kB H PA {-} PB {-} P H 1 45 3.045 0.4702 6.476 8.24 4.51 0.100 0.1688 0.593 2 45 3.092 0.4018 7.695 8.24 4.51 0.100 0.1688 0.593 3 45 3.081 0.2493 12.36 8.24 4.51 0.100 0.1688 0.593 4 45 2.989 0.1270 23.54 8.24 4.51 0.100 0.1688 0.593 5 45 3.102 0.003863 803 8.24 4.51 0.100 0.1688 0.593 121 Table 4.6 Experimental Results: Knaebel and Hill, 1986. Run Qfeed (STP) {L/min} Qexhaust (STP) {L/min} Qproduct (STP) {L/min} Qproduct O2 (STP) {L/min} Purity (0 2 + Ar) {vol. %} Rexp {-} RTH {-} 1 2.342 2.289 0.05257 0.05242 99.72 0.1017 0.1179 2 2.342 2.247 0.09495 0.09470 99.74 0.1838 0.1637 3 2.333 2.185 0.1477 0.1473 99.75 0.2870 0.2556 4 2.314 2.146 0.1680 0.1677 99.80 0.3294 0.3275 5 2.344 2.133 0.2113 0.2097 99.27 0.4067 0.4047 The results of the Ideal Four-Step calculations (from Section 3.3) can be seen in Table 4.7. It should be noted that the efficiencies pertain to the ideal cycle, in which there is no mrottling. Nevertheless, it is expected that the calculated bed losses closely match the actual bed losses because the molar flows and compositions closely match the experimental measurements. Grassman diagrams for all five runs can be seen in Figures 4.15 through 4.19. 122 Table 4.7 Ideal Four-Step Theoretical Results for Five Experimental Runs of Kayser and Knaebel, 1986. Units Run 1 Run 2 Run 3 Run 4 Run 5 N F {mol} 0.174 0.177 0.176 0.171 0.178 yo H 0.78 0.78 0.78 0.78 0.78 NBD {mol} 0.133 0.139 0.147 0.149 -y BD {-} 0.838 0.841 0.846 0.851 -Nw {mol} 0.037 0.032 0.020 0.010 -yw {-} 0.669 0.673 0.683 0.692 -N E {mol} 0.17 0.171 0.166 0.159 0.162 y E H 0.801 0.81 0.827 0.841 0.857 Np2 {mol} 0.00466 .00649 0.0100 0.0120 0.0160 yp {-} 0 0 0 0 0 WF J/mol O2 -110,241 -81,393 -52,533 -40,056 -33,411 WB2 J/mol O2 -5339 -5092 -5813 -7140 -WE2 J/mol O2 - - - - -9222 w w J/mol O2 -15,858 -11,773 -7268 -4466 -WB1 J/mol O2 38,687 28,838 18,574 13,892 -WE1 J/mol O2 - - - - 11,862 W p u J/mol O2 28,559 19,122 9497 4908 -WPR J/mol O2 32,686 25,847 19,797 17,846 17,490 WR J/mol O2 2945 2986 2977 2897 2995 Net Work WM J/mol O2 -28,560 -21,465 -14,769 -12,119 -10,286 Vj/p J/mol O2 4005 4005 4005 4005 4005 VJ/E J/mol O2 132 185 298 394 510 V|/p+Vj/E J/mol O2 4137 4190 4303 4399 4515 Bed Loss J/mol O2 -24,423 -17,275 -10,466 -7720 -5771 2nd Law Efficiency {%} 14.49% 19.52% 29.14% 36.30% 43.89% Recovery {%} 12.15% 16.68% 25.76% 32.87% 40.74% Prod. mol/m 3 14.89 20.76 31.97 39.58 50.90 123 w F = 110,241 J/mol 0 2 (83.87%) w B 2 = 5339 (4.06%) ww = 15,858 (12.07%) Total Work Input 131,438 J / mol O2I (100%) WBI = 38,687 J/mol Q 2 (29.43%) w P U = 28,559 J/mol O2 (21.73%) WPR = 32,686 J/mol O2 (24.87%) ^ w R = 2945 J/mol O2 (2.24%) i(/P = 4005 J/mol 0 2 (3.05%) i | / E = 132 J/mol O2 (0.10%) Bed Loss = 24,423 J/mol 0 2 (18.58%) Figure 4.15 Grassman Diagram for Run 1 (Kayser and Knaebel, 1986). w F = 81,393 J/mol O2 (82.84%) w B 2 = 5092 (5.18%) w w = 11,773 (11.98%) WBI = 28,838 J/mol 0 2 (29.35%) w P U = 19,122 J/mol O2 (19.46%) WPR = 25,847 J/mol 0 2 (26.31%) > w R = 2986 J/mol O2 (3.04%) i|/p = 4005 J/mol 0 2 (4.08%) i | / E = 185 J/mol O2 (0.19%) Bed Loss = 17,275 J/mol 0 2 (17.58%) Figure 4.16 Grassman Diagram for Run 2 (Kayser and Knaebel, 1986). 124 WF = 52,533 J/mol 0 2 (80.06%) WB2 = 5813 (8.86%) w w = 7268 (11.08%) WBI = 18,574 J/mol 0 2 (28.31%) WPU = 9497 J/mol 0 2 (14.47%) WPR = 19,797 J/mol 0 2 (30.17%) ]> w R = 2977 J/mol 0 2 (4.54%) v|/P = 4005 J/mol 0 2 (6.10%) VJ/E = 298 J/mol 0 2 (0.45%) Bed Loss = 10,466 J/mol 0 2 (15.95%) Figure 4.17 Grassman Diagram for Run 3 (Kayser and Knaebel, 1986). w F = 40,056 J/mol 0 2 (77.53%) w B 2 = 7140 (13.82%) ww = 4466 (8.64%) WBI = 13,892 J/mol 0 2 (26.89%) WPU = 4908 J/mol 0 2 (9.50%) WPR = 17,846 J/mol 0 2 (34.54%) ]> w R = 2897 J/mol 0 2 (5.61%) v|/P = 4005 J/mol 0 2 (7.75%) VJ/E = 394 J/mol 0 2 (0.76%) Bed Loss = 7720 J/mol 0 2 (14.94%) Figure 4.18 Grassman Diagram for Run 4 (Kayser and Knaebel, 1986). 125 w F = 33,411 J/mol 0 2 (78.37%) WE2 = 9222 . (21.63%) WEI = 11,862 J/mol O2 (27.82%) WPR = 17,490 J/mol 0 2 (41.03%) ]> w R = 2995 J/mol 0 2 (7.03%) VJ/P = 4005 J/mol O2 (9.39%) \|/B = 510 J/mol O2 (1.20%) ed Loss = 5771 J/mol O2 (13.54%) Figure 4.19 Grassman Diagram for Run 5 (Kayser and Knaebel, 1986). The pressure ratio in all of these runs was changed by varying the lower pressure of the cycle. The net work and second law efficiency of the Ideal Four-Step analysis does not depend on the lower pressure, but only the pressure ratio. Therefore, in summarizing the results of this analysis, we shall do so with respect to the pressure ratio. As the pressure ratio increases, the following conclusions can be drawn: 1. Both the recovery and productivity increase. 2. The total work input per mole of desired product decreases. 3. The net work required by the system per mole of desired product decreases. 4. The second law efficiency increases; increasing the pressure ratio from 6.48 to 23.54 increases the second law efficiency from 14.5% to 36.3%. 5. The bed losses decrease by over a factor of four between the first and last runs. 126 6. The availability of the exhaust increases, as the recovery increases and the mole fraction of the exhaust ys increases. 7. The work recovered during the purge decreases, because, although the pressure ratio is higher, the low pressure is reduced and fewer moles of purge gas (Npu) are needed to purge the bed. 8. More work is recovered during pressurization, because of the larger pressure ratio. It should be noted that the differences in net work, second law efficiency and bed loss are all present with readily attainable pressure ratios. 127 5. Multiple-Cell Model of a PSA System 5.1 Introduction In this chapter, a Multiple-Cell model using binary linear isotherms is used to visualize and confirm the results given in the previous chapter. Models such as this have been developed before, but not for the purpose of doing energy calculations. Bed dynamics may be simulated by dividing the bed into a number of "cells" of homogeneous composition. This is effectively a one-dimensional finite-difference model of the bed, but a physical, rather than a mathematical approach is used to develop the model. The derivation of the model is based on the total mole balance and species A mole balance equations for a single cell. These equations, which relate the flows in and out of a cell to the change in pressure and mole fraction, are placed into a visual basic application that keeps track of the feed, product, blowdown, purge and pressurization flows during the four steps of the Four-Step cycle. The energy required for the different steps is also calculated. A l l of the mrottling energy is recovered, so the model is most closely related to the Ideal Four-Step Cycle. A number of assumptions are made, some of which are the same as for the BLI theory used in Chapter 3: 1. Binary ideal gas mixture. 2. Local equilibrium between gas and solid phases within each cell. 3. Linear, uncoupled adsorption isotherms. 128 4. Complete mixing within each cell. 5. Negligible pressure gradients between cells. 6. Isothermal operation. The third assumption was made in order to compare the results with the work of Chapter 4. This assumption could be relaxed, as the calculations are done numerically. 129 5.2 Derivation of the Model In this model, the adsorbent bed is broken up into N c e us different small "Continuously Stirred Reactors," or CSTRs, each with a gas phase and an adsorbed phase. The number of cells used affects the similarity of the gas flow to plug flow; the more cells used, the closer the flow in the adsorbent bed resembles plug flow. In this way, diffusion can be approximated by reducing the number of cells, which tends to spread out the shock wave. In order to develop the total mole balance for one of the cells, we look to Figure 5.1. Volume = sVc {m3} • Mole Fraction = yi {-} ^ Pressure = P {Pa} • qi-i = u n s A {m 3/s} qi = U J S A {m3/s} Volume = (l-s)Vc {m3} Figure 5.1 Cell T of the CSTR Model (Overall Mole Balance). The box represents cell i , one cell (or CSTR) of the model. Entering at the left is the flow from the previous cell, q;.i, and exiting from the right is the flow to the next cell, q i . These flows are equal to the gas velocity, U i , multiplied by the open cross sectional area of the cells, eA. 130 The hatched portion of the cell represents the amount of the cell that is composed of adsorbent. This fraction of the cell is 1 - e, as s is the void fraction. This distinguishes two separate phases within the cell. The first is the gas phase; this is composed of all the gas in the void space of the cell. This gas has mole fraction yi and pressure P, and fills the volume eVc, where Vc is the volume of the cell. The second is the adsorbed phase; this is composed of all the gas adsorbed onto the adsorbent (in the adsorbed or solid phase). The amount of gas adsorbed is dependent on the pressure of the gas in the gas phase (quantified by the equilibrium isotherm), and fills the volume 1-8. There are three components to consider when completing a mole balance on cell i : the gas flow entering and leaving the cell, the gas stored in the gas phase, and the gas stored in the adsorbed phase. The number of moles accumulating in the cell due to the flow in and out is described by Equation (5.1). Where: dNm,i = moles accumulated in cell {mol} dt = incremental change in time {s} qu = flow to the left of cell i {m3/s} q, = flow to the right of cell i {m3/s} The number of moles in the gas phase is found using the ideal gas law with the void volume of the cell. 131 N „£lcP (5.2) ' _ RT Where: NG,I = number of moles in gas phase in cell {mol} V c = cell volume {m3} If we assume an incremental change in pressure dP, the change in gas phase moles wil l be: e Vc dP (c o) RT Where: dNci = incremental change in gas phase moles {mol} The total number of moles in the adsorbed phase is found by adding the two isotherm equations: _kA-y,P-(l-e)Vc ; kB-(\-yi)P-(l-e)Vc ( 5 A ) AD,i Rj, R T Where: NAD,I = moles stored in adsorbed phase {mol} The incremental change in adsorbed phase moles due to a change in mole fraction or pressure is seen in Equation (5.5). dNAD, = (AZ§TLlk* d ( y ' P ) + k ° d P - k° * M (5'5) 132 The mass balance is found by equating Equation (5.1) to Equation (5.3) plus (5.5). moles in - moles out = change in gas phase moles + change in adsorbent phase moles (5.6) RT Upon collecting terms and rearranging, we can put Equation (5.6) in the following form: dt(<lt-\ ~<lt) £ Vr £ + (1 - £ ) kE dP \\-£)kA-{\-£)kB d(ytP) (5.7) Into this equation we substitute the following non-dimensional parameters: e + (l-e)kA (5.8) £ + (\-£)kt Where: (3A = fraction of species A in the cell that is in the gas phase {-} PB = fraction of species B in the cell that is in the gas phase {-} This leaves us with the following non-dimensional equation: (5.9) dt(q,-x-(j)_ i dp ( i iy(y,P) £VC BB P \ B A BB) P (5.10) 133 The mole balance for component A is found in a similar manner, but now care must be taken as to the direction of the flows. This is explained in Figure 5.2. Cel l l Cell 2 Cel l 3 y i y 3 qo qi q2 q3 (a) Cel l l Cell 2 Cell 3 y* P-. K m y 2 J 3 qo qi q 2 (b) q 3 Cel l l Cel l 2 Cel l 3 y i K K J 3 qo qi q 2 (c) qs Cel l l Cell 2 Cel l 3 y i y 2 qo qi q 2 q 3 Figure 5.2 Flew Regimes for Cell 'i'. 134 In Figure 5.2 (a) the overall flows between the cells, and the mole fraction in the cells are noted. If we look at Cell 2, the flow of species A to the left side of the cell is y i q i , as y is the mole fraction of species A. The flow of species A to the right of Cell 2 is y 2 q 2 . If we now look at Figure 5.2 (b), the flow to the right of Cell 2 is negative, so the flow of species A is - y 3 q 2 ; instead of looking to Cell 2 for the mole fraction of species A, we look to Cell 3. A third possibility is in Figure 5.2 (c), in which both flows are negative, and the flow of species A to the left of the cell is - y 2 q i , and the flow of species A to the right of the cell is - y 3 q 2 . The fourth possibility is in Figure 5.2 (d), and has a negative flow to the left of Cell 2 and a positive flow to the right. In this case, both of the flows look to Cell 2 for the "information" regarding the amount of species A in that flow. Volume = sVc {m3} Mole Fraction = yi {-} Pressure = P {Pa} v q A , i - i = q i - i V L {m 3/s} q A , i = q i y R {m3/s} Volume = (l-s)Vc {m3} Figure 5.3 Cell 'i' of the CSTR Model (Species A Mole Balance). 135 For this reason, Figure 5.3 defines the flow of species A into and out of the cell in terms of yi. and V R , and the accumulation of species A in the cell is: dNAJSU=dt{qi_,yL-qiyR)^ W The terms yL and yR are defined as: yL = yt-i i f qt-i > 0 (5.22; yt i f <0 yR = yt i f <it >0 (5.23) yM I F <J, < 0 The change in species A gas phase moles for a change in pressure is found by again using the ideal gas law: sVcd{yiP) (5.14) dN, AG,i RT The change in species A adsorbed phase moles is found by using the species A isotherm with the partial pressure of species A. (l-e)VckA d(ytP) (5.15) A,AD,i The mass balance is found by equating Equation (5.11) with Equation (5.14) plus (5.15). , , P s Vc d(y(P) (l-e)VckA d(ytP) (5.16) dtyqt-x yL -q,yR)jf = —^—+ 136 This equation can be rearranged to get Equation (5.17). s Vr 'e + (l-e)kA d{y,P) (5.17) Again we substitute in the non-dimensional term from Equation (5.8), and get the following non-dimensional equation: dt{q;.lyL-qiyR) 1 d{yiP) (5.18) e Vr PA P We now have the two mole balance equations necessary to define the flows of material through the system, Equations (5.10) and (5.18). We now subtract Equation (5.18) from (5.10), eliminating the d(yiP) term. dt <li-\ ~<li ±_dP_ TB P (5.19) We now solve this for q,, substituting in the Equation P = BA / PB-qi_\\ + (f3-\)yL. It eVc 1 dP dt PB P 'l + (fi-l)yR\ (5.20) If we make the further substitution, d0 = dP/P, we find the flow to the right of cell i as a function of the flow to the left of the cell and the change in non-dimensional pressure d0. q^[\^{P-l)yL\-^^rde (5.21) dt PB ) + (p-\)yR 137 In this equation, y i and VR are defined by Equations (5.12) and (5.13). The change in mole fraction of cell i , as a function of the flows in and out of the cell, and the change in pressure, is found by expanding the d(yiP) term in Equation (5.18) and solving for dyi. dyt =fiA-£r(gl-lyL-qtyR)-y,do (5-22) Equations (5.21) and (5.22) are used for a step in which the pressure is changing. If the pressure is constant, the equations simplify as follows: = [l + Qg-lK] (5.23) *' = [l + {p-l)yR] 7 r, "< I \ dy* = & 7T~v^-i yL - <it yR) £ V c (5.24) 138 5.3 Multiple-Cell Model Results The multiple cell model was programmed in visual basic. For the changing pressure steps (blowdown and pressurization), the solution algorithm was of the "shooting'' type, in which the boundary conditions at each end of the bed were known, and iterations were done until the calculated flows matched the boundary conditions. When the pressure was held constant, the outlet flow of the bed and the composition changes in the bed were calculated for a given inlet flow. In Table 5.1, the Multiple-Cell model is compared to the semi-analytical results of Chapter 4. The results are for the two cases with differing lower pressures (PL = 1 arm and PL = 0.5 arm) and the same pressure ratio (IT = 15.6). The model was run with 150 cells, and the number of feed moles was chosen to be the same as that in the semi-analytical model. The Multiple-Cell model is in close agreement with the semi-analytical results. In Table 5.2, the results of running the Multiple-Cell Model with different numbers of cells are presented. The bed losses decrease as the number of cells decreases, but the second law efficiency remains almost constant. This would appear to imply that the presence of diffusion does not affect the efficiency of the cycle. 139 Table 5.1 Comparison of Multiple-Cell Model (150 Cells) with the Semi-Analytical Results. Units Semi-Analytical P L = 1 arm n = 15.6 Multiple-Cell Model P L = 1 arm n = 15.6 Semi-Analytical PL = 0.5 arm n = 15.6 Multiple-Cell Model PL = 0.5 arm n = 15.6 N F {mol} 355.371 355.371 177.685 177.685 yo H 0.78 0.78 0.78 0.78 NBD {mol} 300.307 299.921 150.153 149.960 yBD {-} 0.85 0.85 0.85 0.85 Nw {mol} 31.894 31.664 15.947 15.832 yw H 0.683 0.665 0.683 0.665 N E {mol} 332.201 331.585 166.1 165.792 y E H 0.834 0.832 0.834 0.832 N P 2 {mol O2} 23.17 22.869 11.585 11.434 yp {-} 0 0.007 0 .007 WF {J/mol 02} -106,207 -107,695 -79,411 -80,541 WB2 {J/mol 02} 0 0 -733 -744 Ww {J/mol 02} 0 0 -2405 -2419 WBI {J/mol 02} 62,540 63,245 40,629 41,077 Wpu {J/mol O2} 6808 6913 6808 6913 W P R {J/mol O2} 17,089 17,303 17,089 17,304 W R {J/mol 02} 6925 6976 5177 5216 Net Work WI4 {J/mol 02} -12,845 -13,257 -12,845 -13,193 l|/p {J/mol 02} 3816 3711 3816 3711 M>E {J/mol O2} 333 307 333 307 V|/P + V | / E {J/mol 02} 4149 4018 4149 4018 Bed Loss {J/mol 02} -8696 -9239 -8696 -9175 2nd Law Efficiency {%} 32.30% 30.31% 32.30% 30.46% Recovery {%} 29.64% 29.25% 29.64% 29.25% 140 Table 5.2 Effect of Reducing the Number of Cells in the Multiple-Cell-Model. Ncells {cells} 150 100 75 50 N F {mol} 355.371 355.371 355.371 355.371 yo {-} 0.78 0.78 0.78 0.78 NBD {mol} 299.921 299.719 299.533 299.027 yBD H 0.85 0.850 0.849 0.846 Nw {mol} 31.664 31.437 31.266 31.038 yw {-} 0.665 .654 0.645 0.633 N E {mol} 331.585 331.156 330.799 330.065 y E H 0.832 .831 0.830 0.826 N P 2 {mol} 22.869 23.135 23.402 23.913 yp {-} 0.007 0.010 0.014 0.020 WF 0/molO 2} -107,695 -106,458 -105,240 -102,995 W B2 {J/mol 02} 0 0 0 0 W w {J/mol 02} 0 0 0 0 WBI {J/mol 02} 63,245 62,492 61,766 60,414 W p u {J/mol 02} 6913 6831 6749 6602 W P R {J/mol 02} 17,303 17,105 16,909 16,549 W R {J/mol 02} 6976 6998 7021 7068 Net Work WI4 {J/mol 02} -13,257 -13,032 -12,794 -12,362 V|/P {J/mol 02} 3711 3675 3640 3575 {J/mol 02} 307 291 276 227 VJ/P + l)/E {J/mol 02} 4018 3966 3916 3802 Bed Loss {J/mol 02} -9239 -9066 -8878 8560 2 n d Law Efficiency {%} 30.31% 30.43% 30.61% 30.75% Recovery {%} 29.25% 29.59% 29.93% 30.59% 141 6. Conclusions Previous to this work, all of the energy losses in PSA processes were grouped into what was known as PSA Bed Losses. In this work, a semi-analytical model (based on the Binary Linear Isotherm model of Knaebel and Hi l l , 1985, which has been validated by experimental results) has been developed that is capable of separating the energy losses of the Four-Step cycle into those that are the result of throttling irreversibiHties and those that are the result of reversibilities inherent to the separation process. The following are general conclusions about this semi-analytic model: 1. The irreversible expansion of gas through throttling valves accounts for the majority of the energy losses for the Four-Step cycle utilizing pressurization with product. By comparing the Ideal Four-Step cycle (in which all of the expansion energy is recovered by reversible turbines) to the Four-Step cycle, we find that in the case of oxygen concentration on Zeolite 5A (low pressure PL = 1 atm, pressure ratio n = 15.6), 87.91% of the energy input to the cycle is lost through throttling. 2. The amount of useful work done by the Four-Step cycle (represented by the increase in availabihty of the product and exhaust) is very small compared with the work input to the system. For the example mentioned above, the useful work is only 3.90% of the work input. 3. The true bed losses, which are contained within the bed and are inherent in the separation process account for 8.19% of the work input for the example mentioned above. These losses are associated with the irreversibiHties that occur 142 during the separation process, such as the movement of the concentration shock wave. 4. While keeping the low pressure, pressure ratio, feed mole fraction, and selectivity ratio constant, changing the cycle can greatly change the second law efficiency of the separation. For the example case of oxygen concentration using Zeolite 5A adsorbent (PL = 1 arm, n = 15.6, initial mole fraction yo = 0.78, and selectivity ratio P = 0.582), the second law efficiencies of the Four-Step, Ideal Four-Step, and Ideal Three-Step cycles are 4.18, 32.30 and 43.98%, respectively. 5. As the steps used in the three cycles analyzed within this thesis are typical of the steps in many PSA systems, the principles and equations developed within this model can be applied to other cycles. 6. Graphs of the net work required to separate oxygen using zeolite 5A for the Four-Step cycle and the Ideal Four-Step cycle, as a function of the lower pressure and the pressure ratio, are given in Figures 3.13 and 3.22. 7. Graphs of the second law efficiency of oxygen separation using zeolite 5A for the Four-Step cycle and the Ideal Four-Step cycle, as a function of the lower pressure and the pressure ratio, are given in Figures 3.14 and 3.23. 8. Graphs of the net work and second law efficiency of the Ideal Three-Step cycle are given in Figures 3.29 and 3.30, respectively. When the various parameters are changed the net work and the second law efficiency also change. We first examine the effect of changing the pressure ratio and the lower pressure limit of the cycle. Increasing the pressure ratio increases the recovery 143 for the Four-Step cycle and the Ideal Four-Step cycles. The pressure ratio for the Ideal Three-Step cycle is infinite, so this cycle can be seen as the limiting case of the Ideal Four-Step cycle, in which the pressure ratio approaches infinity. The conclusions with respect to changing the lower pressure and pressure ratio are: 1. Increasing the pressure ratio for the Four-Step and Ideal Four-Step cycles decreases the amount of energy required by the system per mole of product. 2. At the low pressure ratio limit for the Four Step cycle (below which no product is generated), an infinite amount of work is required per mole of product, as work is done, but no product is generated. At this pressure ratio the second law efficiency is zero, as no useful work is done. 3. For the Four-Step cycle utilizing pressurization with product, decreasing the lower pressure decreases the amount of net work necessary per mole of product. This is also true when the lower pressure is below atmospheric and work is needed to extract the blowdown gas and the purged gas from the bed. 4. When the expansion energy is recovered, as in the Ideal Four-Step cycle, the net work required and the second law efficiency are a function of the pressure ratio, but not a function of the lower pressure. 5. Energy analyses of experimental results show that the second law efficiency can vary from 14.49% to 43.89% for achievable pressure ratios. This is a wide margin and indicates that energy savings are possible, without having to go to extreme measures. 144 As the mole fraction of the more-adsorbed gas in the feed (yo) increases, there is less of the desired gas in the feed. This increases the reversible work of separation per mole of product. The following conclusion can be made regarding changing the feed mole fraction: 1. Increasing the mole fraction of the feed gas increases the work required per mole of product and decreases the second law efficiency for all of the cycles studied in the thesis. The last parameter to be varied was the selectivity ratio p. This is a ratio of the fraction of the more-adsorbed component in the bed that is in the gas phase, to the fraction of the less-adsorbed component in the bed that is in the gas phase. A low selectivity ratio implies that there is a large difference in the adsorption of the two gases, which makes the separation easier. The recovery for low values of p is higher; the limit of recovery for the cycles studied is 1 - P (as the pressure ratio increases to infinity). The conclusions with respect to varying the selectivity ratio are: 1. For the Four-Step cycle, in which the throttling losses are not recovered, the net work and the second law efficiency vary widely with selectivity ratio. High selectivity ratios require more work per mole of product and have lower second law efficiencies. 2. When the mrottling losses are recovered, the net work is only weakly dependent on the selectivity ratio and the second law efficiency is very weakly dependent on the selectivity ratio. This means that for high selectivity ratios, recovery of the energy becomes more important. 145 The Multiple-Cell model developed has confirmed the semi-analytical model results. The model was capable of representing diffusion by decreasing the number of cells used. The following conclusion can be drawn from the analysis: 1. Decreasing the number of cells did not change the second law efficiency greatly, indicating that the sharpness of the concentration wavefront during the feed step does not change the bed losses. Although the models developed have used linear isotherms, it is expected that the results wil l not vary much with the use of non-linear isotherms. Non-linear isotherms, especially those that are favorable (concave down) are expected to increase the benefits of lower pressure ratios. The Multiple-Cell model is capable of testing this theory. The models also assume isothermal conditions. As the operation of the beds and the compression and expansion becomes adiabatic (which usually occurs with larger systems) the work required per mole is expected to increase and the second law efficiency is expected to decrease. This would be consistent with the general degradation of pressure swing adsorption cycles as heat effects are introduced. However, the general trends are expected to remain the same. If the assumption of equilibrium were relaxed, it is assumed that the second law efficiency would decrease for all of the cycles studied. 146 7. Recommendations As the mrottling losses account for the majority of the energy losses in the cycles studied, methods of energy recovery must be developed. Pressure equalizations, while recovering some of the energy of compression, still create throttling irreversibilities that decrease the second law efficiency of the cycle. Research in this area continues to be done by Highquest Engineering Inc., which has developed many proprietary cycles that incorporate energy recovery techniques. The Cell-Model has already been applied to a highly efficient cycle invented by Dr. Bowie Keefer of Highquest Engineering Inc. This cycle directly recovers energy and uses it to partially power the system. In the search for more energy efficient cycles, due effort must be given to finding adsorbents that are more selective for the desired separation. As all real cycles wi l l have some energy losses, the cycle can be made more energy efficient by finding the best adsorbent. The Cell-Model should be run with non-linear isotherms to see the effect of isotherm curvature. It is expected that non-linear isotherms wil l affect the Four-Step cycle (in which there are throttling energy losses) more than they wi l l affect the Ideal Four-Step and Ideal Three-Step cycles. 147 Bibliography Armond, J. W. "The Practical Application of Pressure Swing Adsorption to Air and Gas Separation." Properties and Applications of Zeolites. Ed R. P. Townsend. Chemical Society Special Publication No. 33,1970. 92-102. Banerjee, R., K. G. Narayankhedkar, and S. P. Sukhatme. "Exergy Analysis of Pressure Swing Adsorption Processes for Air Separation." Chem. Engng. Sci. 45 (1990), 467-475. Banerjee, R., K. G. Narayankhedkar, and S. P. Sukhatme. "Exergy Analysis of Kinetic Pressure Swing Adsorption Processes: Comparison of Different Cycle Configurations." Chem. Engng. Sci. 47 (1992), 1307-1311. Chan, Y. N . I., F. B. Hi l l , and Y. W. Wong. "Equilibrium Theory of a Pressure Swing Adsorption Process." Chem. Engng. Sci. 36 (1981), 243-251. Cheng, H . C , and F. B. Hi l l . "Recovery and Purification of Light Gases by Pressure Swing Adsorption." Industrial Gas Separations. Eds T.E. Whyte Jr., C M . Yon, and E.H. Wagener. Am. Chem. Soc. Symp. Ser. 223,1983.195-211. Flores Fernandez, G., and C. N . Kenney. "Modelling of the Pressure Swing Adsorption Air Separation Process." Chem Engng. Sci. 38 (1983), 827-834. Guerin de Montgareuil, P., and D. Domine. French Patent 1,223,261 to Air Liquide. Dec. 1957. Kayser, J. C , and K. S. Knaebel. "Pressure Swing Adsorption: Experimental Study on an Equilibrium Theory." Chem. Engng. Sci. 41 (1986), 2931. Kirkby, N . F., and C. N . Kenney. "The Role of Process Steps in Pressure Swing Adsorption." Fundam. of Adsorption, New York: Engng. Foundation, 1987. 325. Knaebel, K. S. and F. B. Hi l l . "Pressure Swing Adsorption: Development of an Equilibrium Theory for Gas Separations." Chem. Engng. Sci. 40 (1985), 2351-2360. Matz, M . J. and K. S. Knaebel. "Pressure Swing Adsorption: Effects of Incomplete Purge." A.I.Ch.E. J. 34 (1988), 1486-1492. Ruthven, D. M . , S. Farooq, and K. S. Knaebel. Pressure Swing Adsorption. New York: V C H , 1994. Shendalman, L. H . and J. E. Mitchell. " A Study of Heatless Adsorption in the Model Sytem C 0 2 in He-I." Chem. Engng. Sci. 27 (1972), 1449-1458. Sircar, S. and W. C. Kratz. " A Pressure Swing Adsorption Process for Production of 23-50% Oxygen Enriched Air." Separation Sci. Technol. 23 (1988), 437-450. J 148 Skarstrom, C. W. "Use of Adsorption phenomena in automatic plant-type gas analyzers." Ann. N.Y. Acad. Sci. 72 (1959), 751-763. Skarstrom, C. W. U.S. Patent 2,944,627 "Method and Apparatus for Fractionating Gas Mixtures by Adsorption." to Esso Research and Engineering Company. (Feb. 1958). Skarstrom, C. W. U.S. Patent 3,237,377 (1966). Wagner, J. L. U.S. Patent 3,430,418 "Selective Adsorption Processes." (1969). Yang, R.T. Gas Separation by Adsorption Processes. Boston: Butterworth Publishers, 1987. 149 Appendix A Work in Depressurizing and Pressurizing Adsorbent Beds In this appendix, we develop the equations for the reversible work used to depressurize and pressurize an adsorbent bed. We restrict our analysis to the following three cases: 1. Depressurization of an adsorbent bed with uniform initial mole fraction y = y B . 2. Depressurization of an adsorbent bed with uniform initial mole fraction y = 0. 3. Pressurization of an adsorbent bed with uniform initial mole fraction y = 0, using gas with mole fraction y = 0. In the first two cases, the gas is rejected at an arbitrary pressure Px, and in the tlvird case, the gas is assumed to come from a reservoir at an arbitrary pressure Px. To develop the depressurization equations we assume that a small number of gas moles (dN) are removed from the adsorbent bed, pressurized to Px, and then exhausted. This process is shown in Figure A . l . 150 cIZ P+ dP P, dV, dN P Px, dV(P/Px), dN -» dN Figure A.l Extraction and Pressurization ofdN Moles from P to Px-The dN moles exit the bed at pressure P and fill a volume of (isothermal ideal gas): dNRT dV = —— (A.l) This can be written in terms of dN. dN = PdV RT (A.l) The incremental work required to extract dN moles from the bed is: d1W2 = PdV (A3) 151 The dN moles in the piston are then compressed from P to Px. The work required for this compression is: p rdv— dM = I ^ P'dV 1 i JdV (A.4) Where: P' = dN RT (A.5) V When Equation (A.5) is substituted into Equation (A.l) and integrated, the result is: d2W3 = dN RT\n\ (A.6) After substituting in Equation (A.2), this is written as: d2W3 = ln| PdV (A.7) The moles are then exhausted from the system, which requires the following amount of work: d,WA = -Px ( p\ dV = -PdV (A.8) The work for the entire step is: dW =d1W2+d2W3+d3Wi (A.9) 152 Upon substituting in Equations (A.3), (A.7) and (A.8), the incremental work is: rfW-ln PdV (A.W) Finally we substitute in Equation (A.l). dW = RThi dN (All) Note that dW = d 2 W 3 and that diW 2 + d 3 W 4 = 0. Now that we know the incremental work required to compress d N moles from pressure P to Px, we must calculate the change in bed pressure (dP) that results from this loss of moles. Once we know dN in terms of dP, we use Equation (A. l l ) to calculate the work, integrating between the limits of PL and PH. From Knaebel and Hill 's Equation (7) we know the velocity of the gas in an adsorbent bed when the velocity at one end is zero, and the mole fraction at axial displacement z, is y. u -z 1 dP BB[l + (B-l)ij]P dt (A.12) Therefore, the velocity of the gas at the entrance of the bed, where z = LB, is: 1 dP (A.13) 1 BB[l + (B-l)y(P)]P dt The number of moles leaving the bed is found using Knaebel and Hill 's Equation (19). dN = u, —~z£ Adt 1 RT (A.U) 153 Figure A.l Gas Velocity in an Adsorbent Bed. Substituting Equation (A.13) into Equation (A.14) yields: dN = -sVB j3B[l + (j3-l)y]RT dP (A.15) We now have a relation for dN in terms of dP that can be used with the work equation, (A. l l ) . When Equation (A.15) is substituted into Equation (A. l l ) , the result is: dW = -sVB rln dP (A.16) fiB[i+(p-i)y] \PX; If the bed is being depressurized from P H to PL, the work required is: W= J — , — , B . -.lnl f P^ dP (A.17) The mole fraction y in the above equation depends on the initial mole fraction of the gas in the bed. We now look to our two cases of depressurization, and describe how the mole fraction in the bed changes as the pressure decreases. 154 When a bed with uniform initial mole fraction y = VB is depressurized, the mole fraction in the bed increases, but remains uniform. This can be seen in Figure A.3(a). This occurs because for a given decrease in pressure, more moles of heavy component desorb than light component; this tends to increase the ratio of heavy moles to light moles in the gas phase. The mole fraction continues to increase until the pressure approaches zero and the mole fraction approaches unity. P = P H , y = yB P=PL,y>yB P ->0 , y -» l (a) P = PH,y = 0 P = PL,y = 0 P->0,y = 0 (b) Figure A.3 Mole Fraction as a Function of Pressure: (a) Initial Condition y = ys, and (b) Initial Condition y = 0. The formula that relates the mole fraction in the bed ( y ) , to the pressure (P), initial mole fraction ( y B ) , and the initial pressure (PB) is: i-p y - i (A.18) This is Equation (12) in the paper by Knaebel and Hi l l (1985). Therefore, when the initial mole fraction in the bed is greater than zero, Equations (A.17) and (A.18) must be solved simultaneously. 155 When a bed with uniform initial mole fraction y = 0 is depressurized, there are no moles of the heavy component to be desorbed, so the mole fraction remains zero. This is shown in Figure A.3(b). For this case, Equation (A.17) is simplified and has an analytical solution: W = f f In -1 -P, In L -1 V J L J (A.19) PB The third case considered is the work done by the system in pressurizing an adsorbent bed from PL to P H from a reservoir at Px. This is found exactly the same way as the work of depressurization, but with the limits of Equation (A.17) reversed. P, -eVB l n | ^ X ' dP PB[l + (p-l)y] ^Px As our reservoir has mole fraction y = 0, the above equation reduces to: f P„ - £ VR , W= \ - I n Jp, ^ PB yPx) dP (A.20) (All) This expression integrates to: w = PL f (PL] \ f 1 H \ In -1 ~PH In -1 V J V J PB (A.ll) If Px = PH, as it does when pressurizing the bed in the Four-Step cycle, this equation reduces to: PH+PL f fp } In 1 L - l V J (A.13) 

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