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
4.3 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 +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.
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
X
Np 2
P = P H X
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 Vw - V w i
P = P L y = 0
P = P L
Nw
P = P L y = 0 P = P L y = 0
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 + \V
v
n{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.
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~v̂-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
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Skarstrom, C. W. U.S. Patent 3,237,377 (1966).
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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+ dP
P, dV, dN P, dV, dN
P 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 (PL] -1 ~PH In 1 H -1
V J ~PH
V J PB
(A.ll)
If Px = PH, as it does when pressurizing the bed in the Four-Step cycle, this equation
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