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Non-isothermal Multi-cell Model for pressureswing adsorption process

Anshu Shukla, Satyanjay Sahoo, Arun S. Moharir*

Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

a r t i c l e i n f o

Article history:

Received 2 September 2016

Received in revised form

29 November 2016

Accepted 30 November 2016

Available online 11 January 2017

Keywords:

PSA process

Multi-cell Model

Multi-component system

Non-isothermal effects

Hydrogen purification

* Corresponding author. Fax: þ91 22 2572 68E-mail address: [email protected] (A.S.

http://dx.doi.org/10.1016/j.ijhydene.2016.11.20360-3199/© 2016 Hydrogen Energy Publicati

a b s t r a c t

In the present study, a comprehensive generic model is developed for pressure swing

adsorption process. The model overcomes the inadequacy related to frozen solid concept,

isothermality assumption etc. and treats them as per physics of adsorption as well as

hydrodynamics. Results are presented to show that there is significant adsorption/

desorption during pressure changing steps (pressurization, blowdown), which in turn has

significant effect on process performance. Hydrogen pressure swing adsorption process

demands high purity and recovery, which can be achieved by complex cycle configuration.

It has been used as a case study. The model is validated using the reported simulation/

experimental data to the extent possible. The present study reports separation of Hydrogen

from five-component mixture using a six-bed process. The dynamics of the process were

studied and are presented incorporating non-isothermal effects. The model is easily

adaptable to different adsorption/desorption kinetics and thermodynamics due to its

effective decoupling of bed hydrodynamics from adsorption/desorption.

© 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Introduction

Adsorptive separation technologies have found a very wide

application base. Considering the separation objectives, wide

ranges of process variations have been established for use on

commercial scale. The various adsorption processes for gas

phase separation are Temperature Swing Adsorption (TSA),

Thermal Pressure Swing Adsorption (TPSA), and Pressure

Swing Adsorption (PSA) [1]. PSA is widely used because of its

advantages like low energy requirement and capital invest-

ment cost up to a certain scale of operation. Like all adsorptive

separations, the PSA process is also discrete-continuous in

nature and several cycles are required to attain Cyclic Steady

State (CSS) performance. Literature reports experimental and

theoretical studies for several industrially important gaseous

95.Moharir).00ons LLC. Published by Els

separations using PSA process. Hydrogen purification is

possibly themostmajor andwidely used PSA process in terms

of its scale [2]. A one dimensional model on Comsol platform

for multi-component system for Hydrogen purification is re-

ported by Xiao et al. [3]. Few researchers report modeling

approach for PSA process for high purity hydrogen for fuel cell

application [4,5]. Air drying, Air separation to get enriched

oxygen or nitrogen are other two widely used techniques [6].

Most PSA processes give one desired product richer in one of

the feed components. PSA processes where more than one

components of the feed are sought to be enriched are rare.

Cen and Yang reported bulk separation of equimolar

mixture of Methane (CH4) and Hydrogen (H2) using a 5-step

PSA process [2]. The five cyclic steps include pressurization,

adsorption, co-current depressurization, counter current

depressurization and purge. They report 90% purity for both

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Nomenclatures

AP Specific surface area of the adsorbent particle, m2/

kg

b Langmuir isotherm parameter of each

component, m3/mol

bref Langmuir isotherm parameter at a reference

temperature of each component, used in the

calculation of ‘b’ at any given temperature, m3/

mol

Cpads Heat transfer coefficient of adsorbents, J/kg K

Cpg Heat transfer coefficient of gas mixture, J/mol K

Cinv Valve coefficient of the inlet valve

Coutv Valve coefficient of the outlet valve

db Diameter of bed, m

dp Diameter of adsorbent particle, m

D Distribution coefficient

D Diffusivity of component, used in Eq. (34), m2/s

Hamb bulk Heat loss to bulk phase, J

Hamb Loss Heat loss to ambient, J

Hgen Heat generated in the adsorbent due to

adsorption/desorption, J/mol

kLDF LDF coefficient of each component, 1/s

Lfeed Height of feed in the bed, m

L Height of adsorbent layer, m

mads Mass of the adsorbent present in the bed, kg

m Number of cells in the bed

M Number of temporal nodes

n Number of components in the feed mixture

Nin Number of moles entering the bed through inlet

valve, gmol

Nout Number of moles exiting the bed through outlet

valve, gmol

N Number of moles present in each cell, gmol

P* Pressure in the bed after IBPE, Pa

Pin Pressure of the inlet tank, Pa

Pout Pressure of the outlet tank, Pa

Ptank Pressure of gas in a tank, Pa

P Pressure in a cell, Pa

q* Equilibrium concentration at the surface of the

particle, mol/m3

qs Monolayer saturation capacity of each

component, mol/m3

q Adsorbate concentration in adsorbent particle,

mol/m3

R Universal gas constant, J/mol K

tcycle Total cycle time of a PSA cycle, s

tstep Duration of a PSA step, s

Tin Temperature of the inlet stream, K

Tr Temperature of the raffinate stream, K

Tads Adsorbed phase temperature in each cell, K

Tbulk Bulk phase temperature in each cell, K

Tref Reference temperature at which ‘bref’ is given, K

u Velocity of inlet and outlet stream, m/s

Uamb Heat transfer coefficient for bed to ambient heat

transfer, J/m2 K s

Ubulk Heat transfer coefficient for adsorbed to bulk

phase, J/m2 K s

Vin Volume of the inlet stream, m3

Vout Volume of the outlet stream, m3

Vads Volume of the adsorbent in the bed, m3

Vbulk Volume of the bulk phase in the bed, m3

yr Mole fraction in raffinate stream

y Mole fraction in bulk phase

Greek symbol

g Cycle index, used in Eqs. (33a)e(33d)

DH Heat of adsorption, J/mol

DL Height of L-cell, m

Dq Change in moles in the adsorbent due to

adsorption/desorption, mol/m3

Dt Step time, s

DTads Change in adsorbent temperature, K

DTbulk Change in bulk phase temperature, K

DZ Height of Z-cell, m

ε Bed porosity

rmix Density of the fluid mixture, kg/m3

rp Density of the adsorbent particle, kg/m3

f Cross-sectional area of the bed, m2

J Surface area of the bed, m2

Subscripts and superscripts

amb Ambient

bulk Bulk phase

extract Extract stream

feed Feed stream

i Index for component

in Inlet stream

j Index for temporal position

k Index for cells in the bed

out Outlet stream

purgein Purge in stream

purgeout Purge out stream

r Raffinate stream

Abbreviations

CMS Carbon Molecular Sieve

CP Counter-current Pressurization step with a

customized stream

CSS Cyclic Steady State

DE Desorption to Extract

DR Desorption to Raffinate

FA Feed Adsorption

ID Incubation or Idle

PE Pressure Equalization

PF Pressurization with Feed

PP Provide Purge

PR Pressurization with Raffinate

RP Raffinate Purge

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 4 2 ( 2 0 1 7 ) 5 1 5 0e5 1 6 7 5151




the products. They concluded purge step to be an essential

step for achieving hydrogen purity and that pressurization

with H2 is more effective than pressurization with feed. The

purity of CH4 depends mainly on co-current depressurization.

With the increase in the commercial applications of PSA

process, it was realized that the process is not limited to

separation of binary components and can be extended to

separation of multi-component gas mixtures.

Doong and Yang reported separation of ternary mixture

(H2/CH4/CO2) over activated carbon as adsorbent [7]. They

restricted the studies to five cyclic steps. Effect of various

parameters such as pressure, temperature, size of adsorbent

particle and flow-rate were studied. They concluded that

Knudsen as well as surface diffusion governs the mass

transfer rates and reported that the simulation results are

comparable with experimental results. The surface diffusion

flux plays an important role and is able to predict correct dy-

namics in the bed.

Yang and Lee reported requirement of multi-layered

adsorbent bed PSA process for H2 recovery since single

adsorbent in the bed is unable to offer high purity for this

multi-component system [8]. The advantages include selec-

tivity for certain components of the feed to be removed and

prevention of any feed component getting adsorbed in an

adsorbent which displays very non-linear adsorption

isotherm and makes desorption of the same very difficult.

Difficulty in removing the adsorbed component during

regeneration steps can affect process performance drastically.

PSA technology for purification of Hydrogen often uses an

adsorbent bed with three different adsorbents stacked in

three layers in each adsorbent bed due to this consideration.

Selectivity, high adsorption capacity and ease of desorption

are all important aspects for commercial success of any PSA

process. Adsorbent development and process design have to

keep all these aspects in mind.

The recent research on PSA process is mostly on adsor-

bents. Literature reports use of adsorbents like polymeric

hollow fiber for CO2 capture. Use of MOF's for H2 purification is

also reported by Banu and co-workers [9]. PSA is often

modeled as an isothermal process. Heat effects in the bed due

to exothermic adsorption are generally ignored. This is a

reasonable simplification if the component being removed

adsorb sparingly or is present in the feed in smaller pro-

portions or the heat of adsorption is not very significant. The

heat effect studies are important especially for systems,

which do not satisfy these criteria. H2 purification is one such

system where temperature rise can affect the adsorption pa-

rameters significantly. Several authors have reported break-

through studies for hydrogen separation with heat effects in

the adsorbent bed [1,8,10,11].

Although the PSA process is discrete-continuous in nature,

two or more beds are employed to ensure that the product is

produced continuously from one of the beds at any time.

Literature reports use of 2, 3 or even more beds, depending on

the relative ease of adsorption/desorption. This often is gov-

erned by the nature of adsorption thermodynamics. Non-

linear the adsorption isotherm is, more difficult is desorp-

tion as compared to adsorption and more is the required time

for bed regeneration after the productive adsorption step. This

in turn will entail use of more than 2 adsorbent beds, with one

undergoing adsorption and others undergoing various stages

of desorption. The sequence of the steps in the adsorbent bed

has to be synchronized for continuous production of the

desired product stream.

For PSA process, valves, which let the gas in or out of the

bed, play an important role. Although the valves are mostly of

on-off type, authors have studied and reported importance of

using flow control valves with the percentage opening of the

valves changing with time to implement optimum flow pro-

files [12]. PSA process involves several steps to achieve desired

process performance. The various steps are pressurization,

feed adsorption, co-current depressurization, counter current

depressurization, purge, pressure equalization, counter cur-

rent pressurization, and incubation/idle.

Non-isothermal Multi-cell Model is reported in the present

study for PSA process to understand the heat effects involved

during the various PSA process steps and their overall impact

on process performance. A Multi-cell Model for isothermal

PSA process has been discussed earlier [13]. The model is

extended here to incorporate heat effects of adsorption/

desorption aswell as heat loss-gain frombedwalls. Themodel

was also extended to support additional PSA steps such as

pressure equalization, provided purge (used in many H2 PSA

processes).With this, themodel could compare its predictions

with reported experimental results in the literature as well as

alternative modeling approaches. The present model is vali-

dated for isothermal and non-isothermal systems [7,12,13].

The values of Langmuir adsorption parameter (b) and mono-

layer saturation capacities (qs) are calculated from the

adsorption isotherm reported in the literature [8]. The values

of “b” and “qs” were used in Extended Langmuir (EL) Isotherm

to describe adsorption thermodynamics. Linear Driving Force

Model (LDFM) was used to describe mass transfer rate.

Multi-cell PSA model

Industrial PSA processes consist of complex cyclic steps such

as pressurization, adsorption, co-current depressurization,

countercurrent depressurization, purge, pressure equaliza-

tion, and counter current pressurization. The significance and

purpose of the steps is widely discussed in the literature

[7,12,13]. These andmore steps are supported by ourmodeling

approach. The steps are discussed here in brief.

In the pressurization step, the bed is pressurized either

with feed or with desired product component. In the adsorp-

tion step, feed enters the bed and effluent obtained is richer in

the desired component than in the feed. This is also called as

raffinate product. To recover more such raffinate product, co-

current depressurization step follows feed adsorption step. In

this step, the feed valve is closed and bed is allowed to

depressurize into the raffinate tank. At the end of this step, the

bed does not have capacity to adsorb the undesired compo-

nents from the feed andmust be regenerated. For this, the bed

is depressurized in counter current direction to recover

strongly adsorbed component as another product stream.

This stream is richer in the undesired components than the

feed stream and is called as extract stream. This step is often

called as blowdown or counter current depressurization step.

The bed is partly regenerated in this step as the undesired




component in the bulk phase of the bed and is mostly recov-

ered as the extract stream. Part of the undesired components

that were adsorbed in the adsorbent during earlier steps also

desorbs because low-pressure favors desorption. As a finish-

ing or cleaning step to further regenerate the bed, purge step is

used after the blowdown step. In this step, part of the raffinate

is allowed to flow through the bed in counter current direction

to the feed to collect more extract stream. The bed is then

pressurizedwith raffinate stream in counter current direction.

This fills the raffinate end of the bed with previously collected

raffinate and simultaneously pressurizes it. The bed is now

ready to undergo next PSA cycle starting with feed pressuri-

zation. Additional steps in more recent PSA processes have

also been reported [14]. These steps are conceived to improve

both purity and recovery. Some of these steps also reduce

energy footprint of separation by PSA. In addition to these

steps, incubation or idle step is used, where all the valves of a

bed are closed. This step is mainly provides finite time in-

tervals between the production and regeneration parts of the

bed to take into account finite time taken by any valve to close

or open. Without these short duration incubation steps, there

is a possibility of feed short-circuiting into product lines,

thereby degrading product purity. The pressure equalization

stepsmainly are energy saving steps. In pressure equalization

step, the high and low-pressure adsorbent beds are connected

through a valve. One of the possibilities can be where the

connection is topetop, which means raffinate end of the two

adsorbent beds are connected through a valve. The second

possibility can be top-bottom; meaning the raffinate end of

high-pressure adsorbent bed is connected with the feed (also

extract) end of the low-pressure adsorbent bed through a

valve. The third possibility can be the bottomebottom,

meaning feed ends of both the beds are connected through a

valve. The fourth possibility can be bottom-top where feed

end of high-pressure bed is connected with the raffinate end

of the low-pressure bed. Topetop connection during pressure

equalization step is more common than the other alterna-

tives. Apart from acting as an energy saving step, the pressure

equalization step also reduces the losses that occur during

blowdown step and consequently increases the recovery of

the desired product. Several simulation models assume that

the adsorbent in the beds involved in these steps are ‘frozen’

during the steps and that there is no mass transfer between

the adsorbed and bulk phase. This is however not true and the

beds are fully ‘active’ during these steps and significant

adsorption/desorption occurs having direct impact on product

purity. The present model, therefore, supports phenomeno-

logical capture of the events during these steps.

Another step, the provided purge step is incorporated in

some PSA processes, especially some designs of Hydrogen

PSA, to improve raffinate product (enriched hydrogen) purity.

In this step, the bedwith high pressure is connected to the bed

with lower pressure as in the pressure equalization step, but

with the extract valve of the low-pressure bed open. The

discharge from the lower pressure bed contributes to the

extract stream, or even collects as second extract streams that

depend on the requirement. This step helps in effectively

purging the bed without having to use raffinate for the pur-

pose. The step simultaneously improves raffinate purity as

well as recovery. This is one of the few parameters in a PSA

process that favorably affects purity and recovery both. Most

other measures give higher purity at the cost of recovery and

vice versa. Our model supports this step also.

For brevity, the acronyms used in the present study are PF,

FA, DE, DR, RP, PR, PE, PP, CP and ID for Pressurization with

Feed, Feed Adsorption, Desorption to Extract, Desorption to

Raffinate, Raffinate Purge, Pressurization with Raffinate,

Pressure Equalization, Provided Purge, Counter current

Pressurization and Idle (Incubation) respectively.

Description of Multi-cell Model

Rajasree et al. have discussed one of the possibilities for PSA

processmodel to overcome the disadvantages associatedwith

frozen solid concept in pressure changing steps [15]. In gen-

eral, there are two distinct phenomena happening in the bed

at any time. There is convective flow due to pressure differ-

ential between the two ends of the bed connected to tanks at

different pressures and simultaneous adsorption/desorption

between the bulk phase at any location in the bed and the

adsorbent at that location. These two phenomena occur at

quite different speeds with the convective flow happening

much faster than the adsorptionedesorption. Rajasree et al.

assumed instantaneous response to pressure differential

causing convective flow at the beginning of every step fol-

lowed by continuous adsorption/desorption over the time step

[15]. The difference between the two time scales is stretched

to the extreme by shrinking the convective flow due to pres-

sure differential happening instantaneously. The Multi-cell

Model carries this concept forward systematically as follows.

A PSA process could employ several beds, two and three

bed processes being quite common. However, each bed goes

through an identical sequence of steps as discussed above

with a phase lag with the other beds. Therefore, it suffices to

consider the steps implemented over one of the beds to get the

overall process performance at the cyclic steady state. Fig. 1

show a representative packed bed connected to various

tanks through valves. Valve V1 connects the feed tank and

adsorbent bed. Valve V2 connects the extract tank and

adsorbent bed. Similarly, Valves V3 and V4 connect the

adsorbent bedwith purge tank and raffinate tank respectively.

Valves used for bed-to-bed connections as required in pres-

sure equalization and provided purge steps are not shown

explicitly in the figure. These valves are presumed to provide

the necessary flow connectivity between the two bedswithout

causing any pressure drop due to flow across the valves.

Valves play an important role in any PSA process and the flow

exchange between the bed and a tank is governed entirely by

the pressure differential between the tank pressure and the

pressure at the bed end connected to the valve. These flow

rates are calculated using standard valve equations involving

valve coefficients.

To implement any of the PSA cycle steps discussed earlier,

except the PE and PP steps, two valves each at the bottom and

the top of the bed are sufficient. Such a simplified view of the

bed is taken while explaining the salient features of the Multi-

cell Model. Therefore, the bed in Fig. 1 has two valves (V1 and

V2) at one end connected to two different tanks (say Feed Tank

and Extract Tank) and two valves (V3 and V4) at the opposite

end connected similarly to the same or two different tanks.



Fig. 1 e Schematic of an adsorbent bed with ‘m’ number of cells for FA step.


We have shown only Raffinate tank also used as Purge tank in

the figure. By suitably keeping none, one or more of these

valves open, all PSA steps can be implemented (except PE and

PP steps).

Fig. 1 specifically shows the schematic of a bed during FA

step of any PSA process. The closed valves are grayed for that

purpose. With only valves connected to the Feed tank and

Raffinate tank open, FA step is implemented. The schematic

also shows the bed divided into ‘m’ slices (or cells) and the

corresponding nomenclature for various state parameters of

the cell such as pressure, temperatures in the bulk and

adsorbed phases, mole fractions of individual species in the

bulk phase and the adsorbed phase concentrations of the

species.

Each such axial cell or slice has static adsorbent in contact

with the bulk phase in that cell. For better understanding of

the model, we have chosen to call the adsorbent part of the

cell as Z-cell and the bulk phase part as L-cell. The choice of

nomenclature is simply because both Z and L are often used as

nomenclature for the bed length. The difference between the

Z-cells and L-cells is that while the former is static, the gas in

the latter can move as feed enters and/or product leaves the

bed. The gas in the L-cells can also move axially due to

adsorptionedesorption which changes the net moles of gas in

the L-cells and changes their pressure. This is what could

happen even during the incubation step when there is no

inflow of feed in or outflow of product.

The Multi-cell Model works in principle as follows. The bed

has certain known bulk and adsorbed phase concentration

and temperature profiles as well as pressure profile at any

point of time. What happens over the next incremental time

step is visualized as taking place as a series of two distinct

events. One of them is Instantaneous Bulk phase Pressure

Equalization (IBPE) that happens at the beginning of the time

step and the other is continuous adsorptionedesorption be-

tween adsorbent and the bulk phase in contact with it. The gas

moves only during the IBPE step. During this step, entire gas

inflow that would take place over the time step is deemed to

take place instantaneously. The amount of gas inflow is

decided by the valve equation of the valve at the feed end of

the bed that relates the flow rate to instantaneous difference

between the valve upstreamand downstreampressures at the

beginning of the time step. Similarly, valve equation pertain-

ing to the valve at the product end decides the amount of

product gas that would instantaneously leave the bed. Amass

balance decides the moles that remain in the bed. This gas in

the bed is assumed to undergo instantaneous pressure

equalization leaving a uniform pressure in the bed valid over

the entire time step. The gas in the L-cellsmove as isolated gas

plugs because of this inflow, outflow and pressure



Fig. 2 e Distribution matrix for ‘m’ cells in the bed.


equalization during the IBPE. These gas plugs are considered

to move as isolated plugs without intermingling. As a result of

this movement, the bulk phase in contact with any Z-cell in

the bed would be constituted by fractions of one or more gas

plugs which represented gas in the L-cells inherited at the

beginning of the step just before IBPE. Depending on the

amount of feed that has entered and the volume it has occu-

pied at the feed end of the bed after IBPE, it could also

contribute entirely or partly to the bulk phase against any Z-

cell. What constitutes bulk phase against any Z-cell is the new

L-cell and its composition and temperature govern the

adsorptionedesorption over the time step. Implementation of

IBPE and definition of new L-cell composition and tempera-

ture prior to adsorptionedesorption is an important part of

the model. Once IBPE is implemented, the adsorp-

tionedesorption in each Z-cell and the new L-cell in contact

with it takes place independently and there is no axial bulk

phase movement during this continuous part of the two-step

approximation. Adsorptionedesorption changes the compo-

sition and temperature of the adsorbed phase and bulk phase

in each Z-cell and L-cell pair. The bulk phase pressure in each

cell also changes depending on the net mass transfer between

the adsorbed and bulk phases. The bedwall against each L-cell

is assumed to be in thermal equilibrium with the L-cell. The

heat transfer between the bed wall and the constant ambient

temperature is also incorporated into the model if the bed is

not insulated. The state of the Z-cells and L-cells at the end of

all these phenomena over the time step is inherited as their

initial state for implementing the next incremental time step.

The heights of Z-cell 1, Z-cell 2 to Z-cellm are designated as

DZ1, DZ2 to DZm. Similarly, heights of L-cell 1, L-cell 2 to L-cell

m are designated as DL1, DL2 to DLm. As shown in Fig. 1, every

L-cell has its own pressure, temperature and gas composition

in terms of mole fractions and total number of moles. Simi-

larly, each Z-cell will have its own adsorbed phase concen-

tration and temperature. The adsorbent particle is considered

to have a uniform temperature in any Z-cell. Thus, for L-cell 1

and Z-cell 1, themole fraction of ith component, temperature,

pressure, total number of moles, adsorbed phase concentra-

tion of ith component and adsorbent temperature are desig-

nated as y1i;j;T

1bulk j P

1j ; N

1j ; q

1i; j;T

1ads j for jth time step. Themole

fractions and adsorbed phase concentrations are for all

components.

Reconstitution of L-cells after IBPE step

The IBPE step causes movement of feed gas into the bed,

product gas out of the bed, and resultant movement of gas in

the inherited L-cells. The bulk phase standing against each Z-

cell will thus be constituted by fractions of gas in the inherited

L-cells. The gas in the bulk phase against each Z-cell will

attain new pressure, temperature and composition calculated

as their mixed cup values after the composition of the cell in

terms of various gas plugs contributing to it are known. What

one needs to know is how the feed plug and the gas in

inherited L-cells prior to IBPE distribute themselves in the

various slices of the bed. This general distribution is quanti-

tatively captured in the distribution matrix. The feed plug and

the relocated gas plugs in inherited L-cells move in axial di-

rection without intermingling as discussed earlier. After IBPE,

what fractions of each such gas plug are against static phys-

ical demarcations of the bed into ‘m’ cells are the information

content of the distribution matrix. General structure and

nomenclature of thematrix for a general case ofm divisions is

as shown in Fig. 2.

In Fig. 2, first row of the matrix (from bottom) shows frac-

tions of feed plug and the relocated gas plugs belonging to the

inherited L-cells that would contribute to the new recon-

stituted L-cell 1 that occupies the first cell in the bed along

with Z-cell 1. Similarly, the elements in the second row show

fractions of feed plug and the relocated gas plugs belonging to

the inherited L-cells that would contribute to the new recon-

stituted L-cell 2 that occupies the second cell in the bed along

with Z-cell 2. Subsequent rows similarly capture the consti-

tution of L-cells 3 to L-cell m in terms of the feed plug and

relocated gas plugs of inherited cells. The last row of the

matrix shows the fraction of feed and the gas in the inherited

L-cells that end up in the raffinate. Being fractions of various

gas plugs, the elements of thematrix are normalized and their

values are between 0 and 1. Viewed alternately, the first col-

umn in the matrix corresponds to feed distribution in recon-

stituted L-cells standing against Z-cells and the raffinate

stream. Similarly, the second column captures the distribu-

tion of gas in the inherited L-cell 1 in the reconstituted L-cells

standing against Z-cells and the raffinate stream. The next

column shows the distribution of gas in inherited L-cell 3 in

the reconstituted L-cells standing against Z-cell and the raf-

finate stream, and so on.

It can be appreciated that the sum of all elements in every

column of the matrix is unity because it simply captures the

fractions of each distinct gas plug (feed and gas in inherited L-

cells) in the various reconstituted L-cells and the raffinate

stream. It can also be seen that the matrix size is

(m þ 1) � (m þ 1) if the bed is divided into ‘m’ cells.

With this understanding about the distribution matrix

structure and nomenclature, we can now see how the matrix

elements can be calculated after the IBPE step. Once that is

done, the composition, pressure and temperature of the

reconstituted L-cells can be easily calculated as mixed cup

properties. The adsorptionedesorption between each pair of a

Z-cell and corresponding L-cell is then executed. Execution of

the IBPE and generation of the distribution matrix are quite

complex in nature.




Distribution matrix for 2 cells in the bed

For the purpose of conceptual clarity, the structure of distri-

bution matrix is explained considering that the bed is divided

into two cells (m ¼ 2) as shown in Fig. 3 at “jth” time step. The

distributionmatrix will then be a 3� 3matrix in this case. The

composition, pressure and temperature in the Z-cells and the

corresponding L-cells which govern adsorption/desorption

can be calculated using the distributionmatrix. Fig. 4(a) shows

the distribution matrix for a bed divided into two cells at the

beginning of the jth time step as well as at the end of IBPE

during this step. After the completion of the jth time step, the

Z-cells and the L-cells have the same size. Their specifications

are inherited for the simulation of the next, i.e. (j þ 1)th time

step. The feed is yet to enter and the raffinate yet to leave. At

this stage, all the elements in the first column and the last row

of the distribution matrix are zero as shown in Fig. 4(a). The

height of each Z-cell is same as the height of the corre-

sponding L-cell and only the elements corresponding to each

combination are 1 in the inherited distribution matrix. The

IBPE step for the (j þ 1)th time step begins with this status.

Fig. 5 displays one of the possible bed conditions at the end

of the IBPE step of the (j þ 1)th time step. The operational PSA

cycle step is the FA step as indicated in the figure duringwhich

the feed enters and the raffinate leaves from the bed. The bulk

phase composition, pressure and temperature in each

inherited L-cell as well as the solid phase composition and

temperature in each inherited Z-cell are

yki;j;T

kbulk j P

kj ; N

kj ; q

ki; j;T

kads j. The index ‘k’ depicts spatial posi-

tion in the bed that varies in this simplified case from 1 to 2.

The color codes for the gas in inherited L-cells in Fig. 5 are

given below along with the color code for the feed plug that

will enter as a part of IBPE step.

L-cell 1

L-cell 2

feed plug

Fig. 5 shows only one of the scenarios possible during IBPE

step. The feed has moved in but stays entirely within the bulk

phase against Z-cell 1, the gas in the inherited L-cell 1 has

moved up and straddles across both the Z-cells and the gas in

the inherited L-cell 2 has also moved up with a part occupying

position against Z-cell 2 and the remaining part moving out of

the bed contributing to the raffinate stream. There are other

possible scenarios depending upon the amounts of gas

entering and leaving and also the extent to which the gas in

the inherited cells expand/compress due to pressure equal-

ization. It can be shown that for the present case of the bed

being divided into 2 cells, the total possibilities are 6 in num-

ber. The six possibilities are shown qualitatively in Fig. 6. The

movement of the three gas plugs (feed plug and gas in

inherited L-cell 1 and L-cell 2) relative to the two static Z-cells

is shown. The figure is self explanatory. The first of the six

possibilities corresponds to Fig. 6. The only non-zero elements

in the distribution matrix for this case which help define the

gas quality in the reconstituted L-cells will be as shown in

Fig. 4(b). In general, the number of distinct possibilities for

positioning of gas in the inherited L-cells and the entering feed

plug in FA step are m � (m þ 1). Sahoo has discussed the

quantitative aspect of generation of the distribution matrix in

detail [13].

The calculations are valid for multi-component system.

The index ‘i’ depicts the component. The time index is ‘j’ and

it varies from 0 e M, M being the total number of temporal

divisions of any PSA step duration. After the IBPE part of the

(j þ 1)th time step is executed, L-cells will be reconstituted

which will attain the same position as the corresponding Z-

cells in the bed. Fig. 5 shows the movement of the gas plugs

that were occupying void space in each bed cell along with the

feed gas plug that has pushed itself up in the bed. The

nomenclature for various parameters is shown as valid for the

end of the time step j. At the end of the next time step (j þ 1),

similar nomenclature will be valid with j replaced by (j þ 1).

The following procedure is a step by step explanation to get

various parameter values at the end of time step (j þ 1) from

the values at the end of step j. It can be used recursively to

march in time starting with j ¼ 1 to j ¼ M.

Governing equations

Consider an adsorbent bed divided into ‘m’ cells performing

FA step as shown in Fig. 1. In FA step, the inlet valve (V1) and

outlet valve (V4) are open throughout the duration of the time

step. The following set of equations present the method to

simulate the happenings over the ‘j þ 1’ time step given the

conditions in the bed at the end of the jth time step. The

equations are applicable recursively over every time step into

which the PSA step is divided.

The velocities of the inlet/outlet streams are calculated by

using the valve equations as shown in Eqs. (1a) and (1b).

uinjþ1 ¼ Cin

v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPin�P1

j

rðevaluated at PinÞ

vuut (1a)

uoutjþ1 ¼ Cout

v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPmj �Pout

r�evaluated at Pm

j

�vuuut (1b)

The number of moles entering and exiting the adsorbent

bed is calculated using ideal gas law as given in Eqs. (2a) and

(2b).

Ninjþ1 ¼

uinjþ1f PinDt

RTin(2a)

Noutjþ1 ¼

uoutjþ1fP

mj Dt

RTmbulk;j

(2b)



Z-ce

ll 2

Z-ce

ll 1

Raffinate/Purge Tank

FeedTank

ExtractTank

2jbulk

T,2jads

T,2jN,2

jP,2jiq,2

jiy

1jbulk

T,1jads

T,jN,jP,jiq,jiy 1111

V1 V2

V4

V3

Fig. 3 e Schematic of a bed divided in 2 cells at jth time step.

Fig. 4 e (a) Initial distribution matrix at the beginning of a

typical time step (b) Distribution matrix after IBPE.


The number of moles inside the bed at the end of the

previous time step is known. Simple mole balance gives the

number of moles left in the bed after the instantaneous entry

and exit of gas. The pressure that would result in the bed due

Fig. 5 e Indicative gas plug locations after IBPE at the beginning

to instantaneous pressure equalization is calculated as in Eq.

(3). It is presumed that the moving gas plugs retain their in-

dividual temperatures.

P*jþ1 ¼

�Nin

jþ1Tin �Nout

jþ1Tmbulk;j þ

Pmk¼1 N

kj T

kbulk;j

�R

Vbulk(3)

Volume of gas that is entering the bed (Vin) is calculated

using ideal gas law as in Eq. (4a). The height of the bed up to

which the feed plug occupies the bed (Lfeed) is calculated using

flow cross sectional area and bed voidage as in Eq. (4b).

Vin ¼ ðNinjþ1RT

inÞP�jþ1

(4a)

of the ‘(jþ1)th’ time step during FA step of the PSA cycle.



Z-ce

ll 2

Z-ce

ll 1

Fig. 6 e Six possibilities for FA step in the 2 cells.


Lfeed ¼ Vin

fε(4b)

The height occupied by the gas in each inherited L-cell at

the uniform bed pressure is similarly calculated using the

pressure it inherited at the beginning of the current time step

when it occupied the same height as the corresponding Z-cell

and the newly calculated uniform bed pressure. This involves

a simple application of the ideal gas law as in Eq. (5). Once the

heights Lfeed, DL1, DL2 …. DLm are known, the distribution

matrix can be developed.

DLk ¼ DZkPkj

P*jþ1

(5)

Although the algorithm for the calculation of the distri-

bution matrix is very general and allows for the entering feed

gas plug to reach to any level in the bed including reaching

into the raffinate, it is presumed that the choice of Dt is such

that the entering moles do not fill more than the void volume

available in the entry cell (i.e. Lfeed < DZ1) and the number of

moles leaving the bed are not more than what the last cell

contained at the beginning at that time step. These conditions

are stated below explicitly in Eqs. (6a) and (6b).

Vin�Tin; P*

jþ1

�<

DZ1

LVbulk (6a)

Vout�Tm

bulk;j; Pmj

�¼ ðNout

jþ1 RTmbulk;jÞ

Pmj

<DZm

LVbulk (6b)

This precautionwas taken to decide themaximum Dt in all

simulations.

After IBPE, the L-cells are re-constituted. The gas occupying

the same bed slice as each Z-cell constitutes the new L-cell.

After generating the distribution matrix, the bulk phase

composition, temperature and pressure of the gas in each new

L-cell occupying the same bed slice as the corresponding Z-

cell are calculated using mixed-cup concept. The number of

moles present in the reconstituted L-cells at the end of IBPE

part of the (jþ1)th time step is calculated using Eq. (7). The

bulk phase composition and the temperature at (j þ 1)th time

step are calculated using Eqs. (8) and (9) respectively.

Nkjþ1 ¼ DkfN

injþ1 þ

Xmq¼1

DkqNqj q (7)

yki;jþ1 ¼

DkfNinjþ1y

fi þ

Pmq¼1 Dkq Nq

j yqi;j

Nkjþ1

Where; i ¼ 1;……n (8)

Tkbulk;jþ1 ¼

DkfNinjþ1T

in þPmq¼1 DkqN

qj T

qbulk;j

Nkjþ1

(9)

The heat loss to the ambient and the resultant temperature

of the bulk phase in each L-cell are calculated using Eqs. (10)

and (11) respectively. This change in temperature alters the

pressure in the L-cells. The updated pressure in L-cells is

calculated using Eq. (12).

Hambient; loss ¼ Uambient

�DZk

LJ

��Tk

bulk;jþ1 � Tamb�Dt (10)

Tkbulk;jþ1 ¼ Tk

bulk;jþ1 ��Hambient; loss

.�Nk

jþ1Cpg

��(11)

Pkjþ1 ¼

Nkjþ1RT

kbulk;jþ1

DZk

L Vbulk

(12)

The isotherm parameters valid for the current step are

calculated using Van't Hoff Equation as given in Eq. (13).

Inherited adsorbent temperatures in each Z-cell are used for

this purpose. These parameters are used to calculate the

adsorption equilibrium concentration at the particle surface

using Eq. (14). In case an isotherm expression other than the

extended Langmuir isotherm is to be used, the temperature

dependence of appropriate isotherm parameters and the

equilibriumadsorbedphase concentration expressions should

replace Eqs. (13) and (14). TheMulti-cell Model allows this easy

plug-in of appropriate adsorption thermodynamics without

having to make major changes in the simulation algorithm.

bki;jþ1 ¼ bi;ref exp

DHR

1

Tkads; j

� 1Tref

!!(13)

q*;ki;jþ1 ¼

qsi P

kjþ1b

ki;jþ1y

ki;jþ1

RTkads; j þ

PmI¼1 b

kI;jþ1P

kjþ1y

kI;jþ1

(14)

The equilibrium concentration calculated is used in the

LDF model to calculate the adsorption/desorption in every Z-

cell and L-cell pair over the time step to arrive at the resultant

adsorbed phase concentrations at the end of time step as

given in Eq. (15a). The change in the adsorbed phase concen-

tration of each species over the current time step is calculated

as in Eq. (15b).

qki; jþ1 ¼ q*;k

i;jþ1 ��q*;ki;jþ1 � qk

i;j

�eð�kLDF

i DtÞ (15a)

Dqi;jþ1 ¼ qki; jþ1 � qk

i; j (15b)

This change in the adsorbed phase concentration will be

reflected in the corresponding change in the bulk phase




concentration. The bulk phase composition is calculated as

per Eq. (16).

yki ;jþ1 ¼

Nkjþ1y

ki;jþ1 � Dqi;jþ1

�DZk

L Vads

�Nk

jþ1 ��DZk

L Vads

�Pni¼1 Dqi;jþ1

(16)

The number ofmoles in every L-cell after themass transfer

between the Z-cell and the corresponding L-cell which sup-

ported adsorption/desorption is calculated using Eq. (17).

Nkjþ1 ¼ Nk

jþ1 ��DZk

LVads

�Xni¼1

Dqi;jþ1 (17)

The heat effect accompanying adsorption/desorption will

change the adsorbent temperature and will also cause heat

transfer between the adsorbed and bulk phases in each Z-cell

and L-cell pair. Heat generated due to adsorption in each Z-cell

is calculated using Eq. (18).

Hgen ¼ �

DZk

LVads

�Xni¼1

Dqi;jþ1DHi

!(18)

The change in the adsorbent temperature due to the net

heat generated in a Z-cell is calculated using Eq. (19).

DTkads; jþ1 ¼

Hgen

DZk

L VadsrpCpads

(19)

The resultant temperature of adsorbent is calculated using

Eq. (20).

Tkads; jþ1 ¼ Tk

ads; j þ DTkads; jþ1 (20)

It may be noted that depending on whether a particular

component is adsorbed or desorbed during a time step, heat

will be generated or absorbed. This is taken into account by

the sign of the change in the number of moles of each species

(Eq. (15b)).

The convective heat loss to the bulk phase from the adsor-

bent phase in a Z-cell, L-cell pair is calculated using Eq. (21).

Hbulk; loss ¼ Ubulk

�Ap

� �mads

�DZk=LÞ��Tk

ads;jþ1 � Tkbulk;jþ1

�Dt (21)

The change in temperature of adsorbent in a Z-cell and

bulk phase in a L-cell due to this heat transfer are calculated

using Eqs. (22) and (23).

DTads ¼ Hbulk; loss

DZk

L VadsrpCpads

(22)

DTkbulk;jþ1 ¼

Hbulk; loss

Nkjþ1Cpg

(23)

Recovery ð%Þ ¼ 100

"PMj¼1 N

rj y

rn; j

#Production step

�"PM

j¼1 Ninj y

rn; j

#Regener"PM

j¼1 Ninj y

fn; j

#Production step

The actual temperature of adsorbent in a Z-cell and bulk

phase in the corresponding L-cell after adsorption/desorption

and heat transfer between the adsorbent phase and the bulk

phase are calculated using Eqs. (24) and (25).

Tkads; jþ1 ¼ Tk

ads; jþ1 � DTads (24)

Tkbulk;jþ1 ¼ Tk

bulk;jþ1 þ DTkbulk;jþ1 (25)

The updated pressures in the L-cells after adsorption/

desorption and the accompanying heat effects which can be

used for the next time step can be calculated using Eq. (26).

Pkjþ1 ¼

Nkjþ1RT

kbulk;jþ1

DZk

L Vbulk

(26)

The number of moles leaving the bed is calculated as per

Eq. (27). The composition and the temperature of the outlet

stream are calculated using Eqs. (28) and (29) respectively.

Nrjþ1 ¼ DrfN

injþ1 þ

Xmk¼1

DrkNkj (27)

yri jþ1 ¼

DrfNinjþ1y

fi þPm

k¼1 DrkNkj y

ki;j

Nrjþ1

(28)

Trjþ1 ¼

DrfNinjþ1T

in þPmk¼1 DrkNk

j Tkbulk;j

Nrjþ1

(29)

Similar procedure is followed recursively for all the finite

difference time steps into which the FA step of a PSA cycle is

divided. This models the bed dynamics and decides the raffi-

nate properties. Although the procedure was discussed for the

FAstep, it iseasilyadapted toall other stepsofanyPSAcycle. For

example, in steps, PF,DR,DEandPR, onlyvalve at one endof the

bed is opened. Bed dynamics during these steps is simulated by

putting the valve coefficients for the closed end valve to zero.

Similarly, for incubation step, valve coefficients of the valves at

both the ends of the bed aremade zero. In the case of PSA steps

involving two beds connected to each other such as PE and PP

steps, a combined bed with double the length and double the

number of divisions is constructed and above logic is imple-

mented. The procedure given here has general applicability.

The performance parameters (Recovery, Purity, and

Throughput) are calculated at the end of every PSA cycle ac-

cording to the expressions given in Eqs. (30)e(32). It is pre-

sumed than in a multi-component system, the component

which is desired to be enriched is listed as the last component.

For example, in Hydrogen PSA, Hydrogen will be the last

component in the component list. In Nitrogen PSA, Nitrogen

will be the last component etc. Therefore, the desired

component has component index n in the equations.

ation step(30)




Purity ð%Þ ¼ 100

"PMj¼1 N

rjy

rn; j

#Production step" # (31)

Table 1 e Adsorbenteadsorbate properties.

Feed component Case 1 Case 2 and Case 3 [7]

PMj¼1 N

rj

Production step

Throughput ¼

"PMj¼1 N

inj

#Production step

VadsrpðtcycleÞ(32)

PSA cycles are simulated by simulating sequential steps

that comprise the cycle. For the purpose of process perfor-

mance evaluation, what is important is the Cyclic Steady State

(CSS) performance. Multiple criteria have been used in the

present work to decide whether the CSS is reached or not.

These are given in Eqs (33a)e(33d). In the first condition, Eq.

(33a), we compare the time averaged raffinate tank composi-

tion of two consecutive PSA cycles. If the absolute difference is

under the stipulated tolerance, the CSS is considered to have

been reached. In the second and third conditions, Eqs. (33b)

and (33c) respectively, we compare the axial bulk phase and

adsorbed phase concentration profiles in the bed for two

consecutive cycles at the completion of one major step of the

production phase of any PSA cycle. We have chosen FA step

for this purpose in this work. In the fourth condition, Eq. (33d),

we compare the fractional recovery of two consecutive cycles.

It is the fraction of desired component present in the feed that

has reached in the raffinate stream. For example, in the case

of Nitrogen PSA which enriches Nitrogen in air, what fraction

of Nitrogen in the feed stream used over a PSA cycle has been

captured in the net raffinate withdrawal in the cycle would be

the fractional purity. Typical tolerance value used in the pre-

sent work is 0.00001.

n�yrn

�gthcycle

� �yrn

�ðg�1Þth cycle

o � e (33a)

n�ykn; M

�gthcycle

��ykn; M

�ðg�1Þthcycle

o � e for all k (33b)

11000

h�qki; M

�gthcycle

��qki; M

�ðg�1Þthcycle

i� � e for all k (33c)

1100

hRecoveryð Þgthcycle � Recoveryð Þðg�1Þthcycle

i� � e (33d)

Considering that the quantities being considered in Eqs.

(33a), (33b) and (33d) (mole fraction of raffinate, mole fraction

inside L-cells and fractional recovery respectively) are

normalized, this is a fairly stringent tolerance. To approxi-

mately get the condition on adsorbed phase concentration (Eq.

(33c)) to match with the same rigor, the adsorbed phase con-

centrations were divided by 1000 considering that their typical

maximum values are in the range of 600 (mol/m3 of adsorbent)

or so for the adsorbenteadsorbate system used in this work.
Parameters O2 N2 CH4 CO2 H2
Mole fraction of

components in

the feed, y

0.21 0.79 0.333 0.333 0.334

kLDF (1/s) 0.038 0.0058 0.142 0.142 0.727

qs (mol/m3 of particle) 2640 2640 5214 9444 1472

b (m3/mol) 0.0035 0.00337 0.00258 0.00409 0.000951

Results and discussions

The model is validated using the reported models and

experimental data.We have not done experimental validation

ourselves. The cases considered in this paper have been used

to demonstrate the versatility of our model in handling

isothermal and non-isothermal operation, multi-component

systems, multi-bed systems, and complex PSA cycles

involving a large number of distinct component steps.

The following cases have been considered:

Case 1: Binary mixture, isothermal system (comparison

with reported simulated performance)

Case 2: Ternary mixture, non-isothermal system (com-

parison with reported experimental results)

Case 3: Same as Case 2 except that comparison is with

reported simulated performance

These cases are discussed in detail in the following.

Case 1: Air separation for N2 enrichment using Carbon

Molecular Sieve (CMS) as adsorbent was studied. The adsor-

benteadsorbate properties are given in Table 1. The model

inputs and design parameters are given in Table 2. Isothermal

operation was mimicked by considering heat of adsorption as

zero in our non-isothermal PSA model. This was done to

established parity with two reported simulation models, one

being the isothermal rigorousmodel byMhaskar et al. [12] and

another being an isothermal Multi-cell Model of Sahoo [13].

The adsorbateeadsorbent properties were taken as re-

ported by Mhaskar et al. [12]. The comparative results are

tabulated in Table 3. From the results, it can be seen that the

steady state performances predicted by the two models are

comparable. A minor discrepancy is found in performance

parameters, which can be attributed to the different time step

used in the present study (0.01 s), compared to that used in the

other two studies (0.02 s). Table 3 also shows the computa-

tional times required by the three models on identical

computing machine. The results show the advantages of

using Multi-cell Model over rigorous model in terms of time

taken for simulation. For rigorous model, it took 300 min as

compared to 5 min taken by the Multi-cell Model.

Case 2: The simulated results are validated for multi-

component, non-isothermal system by comparing it with

the experimental data presented by Doong and Yang [7].

Adsorptive separation of ternary mixture of H2/CH4/CO2 using

activated carbon as adsorbent is studied and results are pre-

sented. In the present study, the flow rates for input or output

streams are controlled by valves and do not remain constant

over PSA steps. However, Doong and Yang have reported

constant flow rates over FA and RP steps. Valve co-efficient for

relevant valves in our model were adjusted to give same in-

tegrated volumetric inflow over the corresponding steps. The



Table 2 e Model inputs for simulation of PSA process.

Parameters Case 1 Case 2 Case 3

Inner diameter of bed, db (m) 0.035 0.051 0.051

Adsorbent layer height, L (m) 0.35 0.6 0.6

Duration of PF step, tPF (s) 15 e e

Duration of FA step, tFA (s) 60 180 180

Duration of DR-1 step, tDR-1 (s) e 120 215

Duration of DR-2 step, tDR-2 (s) e 270 265

Duration of DE step, tDE (s) 15 60 120

Duration of RP step, tRP (s) 60 60 120

Duration of CP step, tPR (s) e 30 30

Feed valve coefficient, Cfeedv 0.00021 0.00002 0.00016

Raffinate valve coefficient,

Crv [FA]

0.000018 0.0000015 0.0000046


Crv [DR-1]

e 0.000008 0.0000095


Crv [DR-2]

e 0.000005 0.000008

Extract valve coefficient,

Cextractv

0.006 0.009 0.00007

Purge in valve coefficient,

Cpurgeinv

0.00011 0.000156 0.00005

Purge out valve coefficient,

Cpurgeoutv

0.00014 0.0000727 0.00008

Bed voidage, ε 0.5 0.78 0.78

Feed tank pressure, Pfeed (bar) 3.039 9 26.2

Raffinate tank pressure,

Pr (bar) [FA]

1.213 6 7


Pr (bar) [DR-1]

e 3.3 6


Pr (bar) [DR-2]

e 2.3 3.5

Extract tank pressure,

Pextract (bar)

1 1.5 1.5

Purge tank pressure,

Ppurgein (bar)

1.2 2.5 2.3


Ppurgeout (bar)

1.0 1.5 2.3

Adsorption

Temperature, T (K)

303 300 300

Total cycle time, tcycle (s) 150 720 720

Density of gas mixture at 1bar,

r (kg/m3)

1.2 0.22 0.22

Density of adsorbent particle,

rp (kg/m3)

980 850 850

Length of a time step, Dt (s) 0.01 0.01 0.01

Number of cells in the bed 40 30 30

Diameter of adsorbent particle,

dp (m)

0.0032 0.00056 0.00056

Cpads (J/kg.K) e 1050 1050

Cpg (J/mol.K) e 31.2 31.2

Table 3 e Model validation with reported performanceparameters.

Purity (%) Recovery (%) Time(min)

Rigorous model [12] 94 18.6 300

Multi-cell Model [13] 93.8 18.3 5

Multi-cell Model

(This work)

93.7 18.8 5


adsorbent properties like equilibrium constant andmonolayer

saturation capacity in the Extended Langmuir model were

also calculated using the experimental adsorption isotherm

reported by Yang and Lee who used the same CMS [16].

Adsorbenteadsorbate properties are shown in Table 1.

Performance parameters were comparedwith the reported

experimental data. Equimolar feed mixture was used in the

experiment. The operating and design parameters are given in

Table 2. The process was run for five cyclic steps for the re-

covery of Hydrogen as raffinate. The experimental results for

Run 1 were considered for comparison of the results with the

present Multi-cell Model [13]. The cyclic sequence of PSA pro-

cess is as explainedbyDoongandYang [7]. In step I, i.e. CP step,

pure H2 is used for pressurization of the bed. FA step follows

the CP step. In FA step, feedmixture is allowed to flow through

the bed. The raffinate collected in FA step is expected to have

maximum H2 purity. The DR step is divided into two parts

namely DR-1 and DR-2. It is as expected to recover additional

H2 in DR-1 and a stream rich in CH4 in DR-2 step. CO2 and CH4

are recovered in the DE and RP steps as extract stream.

The results are compared with the performances obtained

in the experiments as shown in Table 4. The simulated purity

of H2 is 99%, which compares reasonably well with the re-

ported experimental results. The mismatch in the results is

attributed to the LDF model used in our simulations to define

the adsorption kinetics. Doong and Yang had suggested that

the controlling mechanism could be simultaneous surface

and Knudsen diffusion. In addition, one of the reasons can be

that our flow rates are variable as they are governed by valve

coefficient and the pressure differential between the up-

stream and downstream locations of the valve. Doong and

Yang have reported averaged flow over the steps only [7]. In

our simulations, it is found that part of CO2 is collected in

raffinate tank with H2 and CH4 in DR-2 step, which is not the

case in the reported study. This also could be due to ourmodel

not accounting for Knudsen diffusion. Effectively, we have

ignored the contribution of Knudsen diffusion. The surface

diffusion has been accounted for by equivalent LDF coefficient

as per Glueckauf relation [17]. Collectively, simultaneous

Knudsen and surface diffusion in their work possibly offers

somewhat different selectivities for the competing compo-

nents than pure surface diffusion simulated by us. We have

used Langmuir model with parameters extracted from re-

ported isotherms. Use of Loading Ratio Correlation (LRC)

model has not shown much deviation in the results.

Case 3: The model was also validated using simulated cy-

clic steady state performance and pressure-time profiles at

the feed end of the bed. The PSA cycle, adsorbent and the feed

were the same as in Case 2. As stated earlier in Case 2, the

mass transfer model used in the reported study was simul-

taneous surface and Knudsen diffusion model, whereas the

present simulation study considers linear driving forcemodel.

The model inputs are same as for the experimental studies

except that the pressure for adsorption step is 26 bar and the

valve coefficients are accordingly adjusted to obtain inte-

grated flow rates matching with those reported for the

experimental studies. The comparison is between our CSS

results and the simulation results by Doong and Yang [7]. The

CSS was attained in 10 cycles as reported by Doong and Yang



Table 4 e Comparison of simulated and experimental performance parameters.

Purity % Recovery (%)

ReportedWork [7]

This Work ReportedWork [7]

This Work

Purity/Recovery (in terms of H2) of

raffinate stream collected in FA

and DR-1 steps in terms of H2

99 99.6 83 78

Purity/Recovery (in terms of CH4)

of raffinate stream collected in

DR-2 step

90 82.1 32 30

Purity/Recovery (in terms of CO2)

of extract stream collected in

DE and RP step

60 51.26 99 76.71


whereas in the present study CSS was attained in 28 cycles.

This could be due to very stringent CSS conditions adopted by

us. As in the earlier case, the valve coefficients for the various

valves were calculated by us using the average flow rate re-

ported by Doong and Yang for different steps. The stepwise

flow rates of various streams used by them and achieved by

our adjusted valve equations are compared in Table 5. There is

marginal difference in the flow rates as seen from the table.

The CSS performances are compared in Table 6. The slight

discrepancy in the results can be attributed to the different

mass transfer rate models used in the studies. Doong and

Yang have reported surface and Knudsen diffusion model as

accurate and concluded that results are almost comparable

with their experimental results. Simulation with diffusion

models involve solution of partial differential equations

capturing the phenomena and can consume a lot of compu-

tational time in solving the equations over each time step by

finite difference technique. It can be plugged in effortlessly in

the Multi-cell Model but at the cost of prohibitive computa-

tional efforts. We have instead used a simplified approxima-

tion of the surface diffusion model in terms of Linear Driving

Force (LDF) model and taken our LDF coefficient commensu-

rate with their surface diffusivity as per Glueckauf's approxi-

mation as follows [17]. Their being no equivalent

approximation for the Knudsen diffusion, we could not

consider the same. Our simulation results deviate from those

of Doong and Yang mainly for Methane. It could be because

the Knudsen diffusion, which we have ignored, changes

selectivity of adsorbent towards methane significantly which

our surface diffusion model approximation was not able to

capture.

kLDFi ¼ 60Di

d2p

(34)

Table 5 e Comparison of flow rates for Case 3.

Reported work [7] This work

Inlet Outlet Inlet Outlet

FA 33 17.1 32.3 16.9

DR-1 e 10.6 e 10.1

DR-2 e 3.6 e 4.14

DE, RP e, 1.9 19.6 e, 1.6 19.7

CP (H2) 14 e 16.9 e

The authors also reported the experimental pressure pro-

file for this case [7]. The pressure profiles simulated by uswere

compared with their experimental profiles. Fig. 7(a) shows the

pressure profile at CSS over the entire PSA cycle. The pressure

profile is plotted considering the pressure near the feed end of

the bed.

The model and experimental pressure profiles are in good

agreement that shows that the Multi-cell Model is able to

capture the bed hydrodynamics closely. Experimental tem-

peratureetime profiles were not reported. Our simulated

temperature profiles are as shown in Fig. 7(b).

The temperature profiles have been shown in the figure at

the bed bottom,middle, and top over the entire PSA cycle. The

durations of individual steps have been marked as distinct

color bands for a better feel. Considering that the feed tem-

perature is 289 K, the temperature rise of about 30 K due to

adsorption is significant and affects the performance signifi-

cantly. Doong andYang have also reported temperature rise of

about 40 K in their experiment. They have, however not re-

ported temperature profiles for this experimental run. Simi-

larly, during regeneration steps, significant temperature drops

are evident due to desorption. As can be observed, the bed

temperatures go below even the feed temperature to about

282 K. The overall temperature swing of the bed is thus about

37 K. Doong and Yang have not reported such temperatures

below the feed temperature in their experimental work [7].

The temperature profiles at different bed heights also show

the mass transfer zones traveling in the bed with time as

evident from the crossover of temperature profiles.

Complex PSA process

After the above exercise to validate the Multi-cell Model with

reported experimental-simulated performance data, we have

tried to use the model for a more complex PSA process with 6,

8 and 10 step PSA cycle implemented on a 3 bed PSA process.

This is discussed in the following.

Industrial PSA process designs often employ several addi-

tional steps in the PSA cycle to improve process performance.

Hydrogen PSA is one important example. To demonstrate

working of our simulation model for complex PSA processes

like this as well as to quantitatively show the effect of these

so-called minor steps on process performance, we have car-

ried out simulation of a five-component mixture for H2 puri-

fication. AdsorbenteAdsorbate properties are mentioned in



Table 6 e Performance of PSA process at steady state for Non-isothermal system.

Steps CH4 CO2 H2

Reported work [7] This work Reported work [7] This work Reported work [7] This work

FA 0.1 0.2 0 0 99.9 99.8

DR-1 3.0 10.8 0.1 0.3 96.9 88.9

DR-2 89.9 82.3 0.6 6.1 9.5 11.6

DE 37.6 37.3 54.6 53.4 7.8 9.3

RP 37.6 37.3 54.6 53.4 7.8 9.3


Table 7. The design parameters are as shown in Table 8. The

simulations were performed for various steps of PSA process

considering energy saving step like pressure equalization. The

PP step is added after DR step so that initial purging can be

done with less pure component. This helps reduce raffinate

loss and thus recovery can be enhanced without adverse

impact on purity.

Separation performances were compared for 6-step, 8-

step, and 10-step, 3-bed PSA processes for same feed

mixture, desired product, total cycle time, bed design. This

(b)

(a)

280

290

300

310

320

330

0 310 620 930

T (K

)

t (s)

L = 0.0 mL = 0.3 mL = 0.6 m

L =L =L =

===

0.0 m0.3 m0.6 m

FA DR-1 DR-2 DE RP

CP

0

5

10

15

20

25

0 310 620 930

P (b

ar)

t (s)

This WorkReported Work [7]

FA

DR-1

DR-2

DE

RP

CP

Fig. 7 e (a) Comparison of experimental and simulated

pressure profiles at CSS. (b) Steady state simulated

temperature profile against time at various bed positions.

helps in judging the relative importance of the various addi-

tional PSA steps on performance parameters, especially purity

and recovery. The 6-step PSA includes provided purge (PP)

step. The 8-step PSA includes PE as well as PP steps. In the 10-

step PSA process, we have included two short duration incu-

bation (ID) steps. Implementation of the PP step in all the three

PSA processes requires bed effluent from one bed to purge

another bed rather than using collected raffinate for the pur-

pose. For the given PSA cycle configurations, it was not

possible to use a bed in a 3-bed system to implement PP step

involving another concurrent step happening in another bed

of the same 3-bed system. A suitable bed in another identical

3-bed system is required to be used for this purpose. This

other 3-bed system follows the same PSA cycle, but with a

suitable time lag. Technically, the PSA system thus becomes a

6-bed system. This is very common in Hydrogen PSA in-

stallations. We have preferred to call it a 3-bed PSA because

the second 3-bed system essentially follows the same PSA

cycle as the first PSA system. The two 3-bed systems interact

with each other only during the PP step.

FA and DR steps in production phase collect the raffinate in

raffinate tank. PE-1 step follows the DE step. In this step, the

bed pressure is reduced and equalizes with another bed at

higher pressure operating PE-2 step. The PP step is imple-

mented on beds in one 3-bed system with a bed operating RP

step in another 3-bed PSA system. Effectively, one bed in one

system is undergoing co-current depressurization with the

effluent counter-currently purging another bed in another

system. The implementation thus called for topetop

connection of two beds in two systems.

The PE and PP steps require direct connection of two

adsorbent systems. This was achieved in simulation by con-

structing a ‘double’ bed by suitably connecting the cascade of

Z-cells in one bed to a cascade of Z-cells in another bed. This

double bed thus has double the number of Z-cells (2m). The

distribution matrix will thus be double the size and its gen-

eration is that muchmore involved. The conceptual approach

however remains the same as was discussed earlier.

Table 7 e Adsorbenteadsorbate properties for 5-components, 3-bed PSA process.

Parameter CH4 CO2 CO N2 H2

yfeed (mole fraction) 0.005 0.015 0.01 0.02 0.95

b (m3/mol) 0.0007 0.1 0.04 0.009 0.00008

kLDF (s�1) 0.001 0.00158 0.002 0.0025 0.003

qs (mol/m3) 2100 3510 3750 3920 4150

DH (J/mol) 35,119 14,350 8433 7817 246,400



Fig. 9 e Step configuration for 8-step, 3-bed PSA process.

Table 8 e Model inputs for 5-components, 3-bed PSAprocess.

Parameters 6-Step 8-Step 10-Step

Inner diameter of bed (m) 1 1 1

Adsorbent layer height (m) 4 4 4

Duration of FA step, tFA (s) 230 230 230

Duration of DR step, tDR (s) 30 30 25

Duration of ID-1 step, tID-1 (s) e e 5

Duration of PE-1 step, tPE-1 (s) e 20 20

Duration of PP-1 step, tPP-1 (s) 150 130 130

Duration of DE step, tDE (s) 120 120 120

Duration of PP-2 step, tPP-2 (s) 150 130 130

Duration of PE-2 step, tPE-2 (s) 20 20

Duration of PR step, tPR (s) 100 100 95

Duration of ID-2 step, tID-2 (s) e e 5

Feed valve coefficient ðCfeedv Þ 0.000055 0.000055 0.000055

Raffinate valve coefficient

in FA step ðCrvÞ

0.00035 0.00035 0.00035

Raffinate valve coefficient

in DR step ðCrvÞ

0.0005 0.0005 0.0005

Extract valve

coefficient ðCextractv Þ

0.0008 0.0008 0.0008

Purge in valve

coefficient ðCpurgeinv Þ

0.000001 e e

Purge out valve

coefficient ðCpurgeoutv Þ

0.000008 0.000008 0.000008

Bed voidage, ε 0.5 0.5 0.5

Feed tank pressure, Pfeed (bar) 25 25 25

Raffinate tank pressure in

FA step, Pr (bar)

24 24 24

Raffinate tank pressure in

DR step, Pr (bar)

15 15 15


Ppurgein, (bar)

5 5 5


Ppurgeout, (bar)

1.5 1.5 1.5

Extract tank pressure,

Pextract (bar)

1.5 1.5 1.5

Feed Inlet Temperature (K) 300 300 300

Total cycle time (s) 780 780 780

Density of gas mixture

at 1bar (kg/m3)

0.095 0.095 0.095

Density of adsorbent

particle (kg/m3)

850 850 850

CPg (J/mol K) 35 35 35

CPads (J/kg K) 1260 1260 1260

Ubulk (J/m2 K s) 20 20 20

Uamb (J/m2 K s) 5 5 5



The PSA cycle configuration of 6-step, 3-bed PSA process is

as shown in Fig. 8. The production phase consists of FA andDR

steps. The regeneration phase consists of PP-1, DE, PP-2, and

PR steps. The 8-step, 3-bed PSA process consists of FA and DR

steps in production phase and PE-1, PP-1, DE, PP-2, PE-2, PR

steps in regeneration phase as shown in Fig. 9. The PE-1 step

has counter-current flow causing de-pressurization in the

bed. In this step, the bed will connect to another low-pressure

bed that will get pressurized. In 10-step, 3-bed PSA process, ID

steps are additional steps considered as shown in Fig. 10. ID-1

is a part of production phase after DR step whereas ID-2 is a

part of regeneration phase after PR step. In the figure, the ID

steps are not designated to avoid cluttering. However, these

can be seen as bands between corresponding steps on either

side.

For comparison of the three different PSA cycle configu-

rations, the tank pressures, valve co-efficient of all the valves

of each bed necessary for effective implementation of all steps

are kept constant. In addition, the duration for FA step is kept

constant i.e. 230 s in all the 3 PSA cycles.

The performance parameters for the same are shown in

Table 9. The performance parameters presented in the table

are calculated using equations explained above. A low recov-

ery of 5.16% with 6-step, 3-bed PSA process can be attributed

to the high amount of gas that is discharged in extract tank

during DE and PP-2 steps. The high recovery can be obtained if

the purge-out valve co-efficient is adjusted to prevent exces-

sive release of gas in bulk phase in adsorbent bed to extract

tank. It is expected that the addition of PE step before the

blowdown step will reduce amount of blowdown as bed

pressure is moderated. This will improve recovery. The high

recovery of 73.8% for 8-step, 3-bed PSA process indicates this.

The jump in recovery is achieved with a marginal drop in

purity, which is now 98.73%. PSA processes with high-

pressure ratio (ratio of feed pressure to extract tank




0

5

10

15

20

25

0 130 260 390 520 650 780

P (b

ar)

t (s)

FA

DR

PP-1

DE

PP-2

PR

5

10

15

20

25

P (b

ar)

DR PE-1 PE-2

FA

PP-1

DE

PP-2

PR

(a)


pressure) as the one in this case get significant advantage due

to pressure equalization steps. The numbers of cycles to reach

CSS are also shown in Table 9. The CSS is achieved within 45

cycles for all simulations.

Pressure profiles for 6-step, 8-step, and 10-step, 3-bed PSA

processes over a PSA cycle are as shown in Figs. 11(a), (b) and

12(a). Comparison of pressure profiles over a PSA cycle for the

6, 8 and 10 step PSA processes clearly shows moderation of

pressure fluctuations offered by PE and PP steps which also

result in less loss of raffinate to extract during DE and RP steps.

Temperature profile over a PSA cycle at three bed positions

(bottom, middle and top) for 10-step, 3-bed PSA process are

shown in Fig. 12(b). The feed temperature was 300 K. It can be

seen from the temperature profiles that the temperatures in

the bed go through a cycle between temperatures lower than

the feed temperature attained during regeneration phase due

to desorption to temperatures higher than the feed during

production phase due to adsorption taking place. The profiles

show sharp ups and downs at changes from step to step. This

is expected because the temperatures are bulk phase tem-

peratures and change fast as the inflow and outflow over

every small time interval is considered as instantaneous in

the Multi-cell Model. The adsorbent temperature will show a

muchmoderated temperature profile although its highest and

lowest temperatures will follow similar patterns as discussed

above.

The results of non-isothermal systems for 10-step, 3-bed

PSA process were also compared for isothermal and

Table 9 e Performance parameters for multi-step 3-bedPSA process.

Purity (%) Recovery (%) Throughput(mol/kg/s)

Cycles

6-Step 99.95 5.16 0.00106 41

8-Step 98.73 73.08 0.00105 38

10-Step 98.69 73.83 0.00105 44

00 130 260 390 520 650 780

t (s)

(b)

Fig. 11 e (a) Simulated steady state pressure profile for 6-

step, 3-bed PSA process. (b) Simulated steady state

pressure profile for 8-step, 3-bed PSA process.



0

5

10

15

20

25

0 130 260 390 520 650 780

P (b

ar)

t (s)

PE-2DR PE-1

FA

PP-1DE

PP-2

PR

294

296

298

300

302

304

306

0 130 260 390 520 650 780

T (K

)

t (s)

L = 0 mL = 2 mL = 4 m

L = 0 mL = 2 mL = 4 m

FA

DR PE-1

PP-1 DE PP-2

PE-2

PR

(a)

(b)

Fig. 12 e (a) Simulated steady state pressure profile for 10-step, 3-bed PSA process. (b) Steady state simulated temperature

profile for 10-step, 3-bed PSA process.


adiabatic conditions as shown in Table 10. The objective was

to study the impact of making isothermality assumption so

commonly resorted to and justified by saying that it is a

reasonable assumption due to low heat effects associated

with physical adsorption. The results in the present case show

that the simulated performance, especially the recovery could

be significantly different depending on whether isothermal or

non-isothermal models are used. There was not much dif-

ference between non-isothermal models allowing heat

Table 10 e Simulated performance parameters forIsothermal, Non-isothermal and Adiabatic conditions for10-step, 3-bed PSA process.

Conditions Purity (%) Recovery (%) Throughput(mol/kg/s)

Cyclesto CSS

Isothermal 99.21 59.05 0.00104 22

Non-isothermal 98.69 73.83 0.00106 44

Adiabatic 98.68 74.19 0.00104 46

exchange with ambient through bed wall or considering

adiabatic operation (bed is insulated). This is understandable

because feed temperature was taken the same as the ambient

temperature in the simulation resulting in minimal heat

transfer between bed walls and ambient.

The results make a strong case for incorporating heat ef-

fects of adsorption/desorption in any PSA process modeling,

as there are significant temperature fluctuations in the bed

even over a cycle. With the Multi-cell Model, which decouples

hydrodynamics from adsorption/desorption effectively, it is

possible to consider non-isothermality without any additional

burden on computation. We observed that about 50% more

computational time was required for achieving CSS with non-

isothermal model as compared to corresponding isothermal

model. However, as seen earlier, the computational effort for

theMulti-cell Model is less by a factor of about 60 as compared

to the rigorous model. Therefore, the Multi-cell Model brings

the handling of non-isothermality within the realm of

practicality.




Conclusions

Multi-cell Model for multi-component separation, especially

H2 purification, is studied. Representative calculation steps for

one finite difference time step in one of the PSA steps over a

finite difference step along the bed are presented. The simu-

lation model is validated with reported results of isothermal

and non-isothermal experimental and simulation studies.

The comparison indicates that the model does capture es-

sentials of a complex PSA process and predicts the relevant

performance parameters such as product purity and recovery

reasonably well. The computational time was less by a factor

of 60 as compared to reported rigorous models.

A five-component system for recovery of Hydrogen using

6-step, 8-step and 10-step, 3-bed PSA processes was studied. It

was mainly to study and quantify the importance of minor

steps such as pressure equalization, provided purge and in-

cubation on process performance. The impact on purity was

observed to be marginal whereas the impact on recovery is

significant.

The temperature profile for 10-step, 3-bed PSA process for

three locations in the bed gave an understanding about the

thermal effects due to adsorption/desorption. From the re-

sults, it can be seen that even in the case of physical adsorp-

tion, temperature increase/decrease in the beds is significant

and affects recovery significantly. Incorporation of non-

isothermal effects in simulation and simulation-based

design is achievable using the Multi-cell Model as process

simulation due to its minimal computational footprint as

compared to other rigorous models.

r e f e r e n c e s

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[10] Caldwett SJ, Duri BA, Sun N, Sun CG, Snape CE, Li K, et al.Carbon di-oxide separation from nitrogen hydrogenmixtures over activated carbon beads: adsorption isothermsand breakthrough studies. Energy Fuels 2015;29:3796e807.http://dx.doi.org/10.1021/acs.energyfuels.5b00164.

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http://dx.doi.org/10.1021/i100024a047

http://dx.doi.org/10.1021/i100024a047






http://dx.doi.org/10.1016/0950-4214(90)80016-E

http://dx.doi.org/10.1016/0950-4214(90)80016-E

http://dx.doi.org/10.1002/aic.690320306

http://dx.doi.org/10.1016/S0140-6701(99)97697-X

http://dx.doi.org/10.1021/ie4011035

http://dx.doi.org/10.1021/acs.energyfuels.5b00164

http://dx.doi.org/10.1016/j.ces.2010.10.034

http://dx.doi.org/10.1016/j.ces.2010.10.034

http://dx.doi.org/10.1515/1934-2659.1574

http://refhub.elsevier.com/S0360-3199(16)33533-9/sref13



http://dx.doi.org/10.1007/s10450-013-9598-0

http://dx.doi.org/10.1007/s10450-013-9598-0

http://dx.doi.org/10.1016/s0098-1354(00)00606-2

http://dx.doi.org/10.1016/s0098-1354(00)00606-2

http://dx.doi.org/10.1021/ie960728h

http://dx.doi.org/10.1021/ie960728h

http://dx.doi.org/10.1039/TF9555101540

http://dx.doi.org/10.1039/TF9555101540



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