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I need someone to write me MATLAB Code for Pressure Swing Adsorption Unit for purifying hydrogen gas. The models for such a system can be seen in the attached paper.

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Congresso de Métodos Numéricos em Engenharia 2015
Lisboa, 29 de Junho a 2 de Julho 2015
c©APMTAC, Portugal 2015
SIMULATION OF A PRESSURE SWING ADSORPTION
SYSTEM: MODELING A MODULAR ADSORPTION UNIT
Márcio R. V. Neto1∗, Rafael V. Ferreira2 and Marcelo Cardoso2
1: Department of Chemical Engineering
School of Engineering
Federal University of Minas Gerais
Address: 6627 Antônio Carlos Avenue, Pampulha, Zip Code 31.270-901, Belo Horizonte/MG,
Brazil.
e-mail: marciorvneto@ufmg.br
2: Department of Chemical Engineering
School of Engineering
Federal University of Minas Gerais
Address: 6627 Antônio Carlos Avenue, Pampulha, Zip Code 31.270-901, Belo Horizonte/MG,
Brazil.
e-mail: rafaelvillela@yahoo.com.br
3: Department of Chemical Engineering
School of Engineering
Federal University of Minas Gerais
Address: 6627 Antônio Carlos Avenue, Pampulha, Zip Code 31.270-901, Belo Horizonte/MG,
Brazil.
e-mail: mcardoso@deq.ufmg.br
Keywords: Pressure Swing Adsorption, Dynamic Simulation, Numerical Solution
Abstract. Hydrogen High Purity Grade is an important compound in oil refineries due
to its capability of withdrawing sulfur impurities in gasoline and diesel. This valuable
feedstock is commonly produced in large scale by a Steam Methane Reforming process and
is purified in batteries of adsorption columns. This work deals with numerically solving
the model of a pressure-swing adsorption (PSA) column used for hydrogen purification.
Five different numerical techniques were employed: Finite Differences, Method of Lines
with Finite Differences, Method of Lines with Orthogonal Collocation, Fourier Transform
and Method of Characteristics. The goal of this study was to develop a quick and robust
application to simulate a single bed PSA unit and be incorporated in dynamic simulators.
This model had been previously validated by comparison with data available in literature.
The Method of Lines was considered to be the best numerical technique to solve a typical
PSA column.
1
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
1 INTRODUCTION
1.1 Pressure swing adsorption
Hydrogen gas plays a fundamental role in sulfur impurities removal processes in oil derivate
products, such as gasoline and diesel. Steam reforming of natural gas, or steam methane
reforming (SMR), is the most common method of hydrogen production on a large scale
in oil refineries [7]. In this process, a mixture of steam and natural gas [9], (or in certain
situations, nafta which substitutes methane [5]), reacts at a high temperature in the
presence of a catalyst to form a mixture of carbon dioxide and hydrogen, according to the
following equations [4]:
CH4(g) +H2O(g)
CO(g) + 3H2(g) (1)
CH4(g) + 2H2O(g)
CO2(g) + 4H2(g) (2)
CO(g) +H2O(g)
CO2(g) +H2(g) (3)
Before the hydrogen produced by this reforming process is delivered to the consumers,
it must be sent to a purification unit that removes any unconverted methane and steam
along with carbon monoxide and carbon dioxide [4], [3].
Pressure swing adsorption (PSA) is widely used for hydrogen purification. The impurities
are desorbed by charging a column containing the adsorbent with the gas mixture and
then pressurizing the column to a pressure sufficient to cause the adsorption of the gases.
Hydrogen is not adsorbed when the impurities are pulled from the gas stream. When
the column pressure is reduced to about atmospheric pressure, the column is evacuated
in a countercurrent direction to withdrawthe impurities from the column. The present
operation is particularly advantageous to achieve a very high level of purified hydrogen
[5], [13].
1.2 Operation and modelling of a PSA column
The PSA adsorption process is based on internal pressure modulation in the vessel where
greater or lesser pressure during an operation cycle determines the degree of gases re-
tained by the adsorption bed inside the column as well the level of impurities inside the
column. In general, a single PSA vessel passes through five elementary steps along an-
operation cycle: adsorption; concurrent depressurization; countercurrent depressurization
or blowdown; light-product purge and repressurization. On average, a single adsorption
operation takes 10 to 20 minutes per PSA.These steps and the streams are shown in the
Figure 1, adapted from [5].
Being a batch process, the regeneration step of the adsorption bed after its saturation
is always essential.In order to ensure that the processes of hydrogen purification occurs
2
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Figure 1: Elementary steps of a PSA cycle operation.
continuously, a set of at least four columns are needed.The vessels in a PSA unit operate
simultaneously under different pressures and other conditions depending on which step
each column is at the moment [5].
The dynamic mathematical models of a single adsorbent bed are required to simulate an
H2-PSA unit. The simulation may be used to train operators by embedding it in Oper-
ator Training Simulator (OTS) systems, as well as to optimize the hydrogen purification
process. Different models for a single adsorbent bed have been studied and proposed
in literature. This paper deals with numerical issues involved in the Partial Differen-
tial Equation (PDE) solution of a single bed model of a H2-PSA unit [15], [12]. For
these simulations, five different numerical techniques were employed: Finite Differences
(FD), Method of Lines with Finite Differences (MOL-FD), Fast Fourier Transform (FFT),
Method of Lines with Orthogonal Collocation (MOL-OC), and Method of Characteristics
(MOC). This study aims at developing a dynamical model for a PSA column that can
be embedded into existing simulators. It is not required that the model be extremely
precise, as it is intended to assist in operators training. It is, however, necessary that its
running time be low, in order to prevent communication delays between the simulator and
the model. Too high a running time would compromise the concept of real-time dynamic
simulation which is central to any operators training software.
1.3 Mathematical modelling
The dynamic behavior of PSA columns results from the interaction of fluid dynamics,
adsorption equilibrium and mass transfer. For that reason, the mathematical modeling
of such systems requires that appropriate models for each of these components be selected.
3
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
The mass transfer model chosen was the linear driving-force model, also known as LDF.
It assumes that the rate of mass transfer is directly proportional to the mass transfer
driving-force, namely the difference between the actual concentration of adsorbed gas
in the solid phase and the theoretical concentration that would exist under equilibrium
conditions [16], [2] [3], [6], [15].
∂qi
∂t
= ki (q
∗
i − qi) (4)
When in contact with a gas mixture for enough time, an adsorbent material will adsorb a
certain amount of each component of the mixture and reach equilibrium [8]. The relation
between the amount adsorbed of a given compound and the total pressure at a fixed
temperature is called an adsorption isotherm. There are many isotherm models, one of
the most common being the Langmuir single-site model [15], which is the model chosen
for this paper:
q∗i =
qsati biPi
1 +
∑
j bjPj
(5)
qsati = a1,i +
a2,i
T
(6)
bi = b0,i exp
(
b1,i
T
)
(7)
The mass balance for the one-dimensional flow of PSA systems can be written as follows
[1]:
�
∂Ci
∂t
+ (1− �) ρs
∂qi
∂t
+
∂vC
∂x
= DL
∂2Ci
∂t2
(8)
If we were to omit the axial dispersion term (which is a reasonable assumption that greatly
simplifies the model), the resulting mass balance would be reduced to [1]:
�
∂Ci
∂t
+ (1− �) ρs
∂qi
∂t
+
∂vC
∂x
= 0 (9)
We will assume throughout this paper that the PSA column operates under isothermal
conditions. For that reason, no explicit energy balance equations are required. This
assumption was made due to the fact that by dropping PDEs, the model becomes less
4
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
complex an, therefore, gains speed. As in any viscous flow though porous media, there
occurs pressure drops due to the viscous energy losses and to the drop of kinetic energy.
A fairly common equation used to model these effects is the Ergun equation [1]:
−dP
dz
=
150µ (1− �2)
d2p�
3
v +
1.75
dp
1− �
�3
(∑
i
Mw,iCi
)
v|v| (10)
1.4 Numerical methods
Five techniques were employed for solving the model. A brief description of each of
them follows.
1. Finite Differences (FD) - from explicit algebraic approximations of the derivatives
in a rectangular grid, a linear system is obtained, from which the PDEs are solved.
2. Method of Lines with Finite Differences (MOL-FD) - the domain is spatially dis-
cretized and the PDEs are thus transformed into a system of ODEs. The so obtained
system is solved by any time-stepping scheme such as the explicit Euler’s method,
as was the case with our system.
3. Fast Fourier Transform (FFT) - upon taking the Fourier transform of the original
set of PDEs, a new set of differential equations is obtained. The dependent vari-
ables of these new equations are the Fourier transforms of the original variables.
A time-stepping scheme (Euler) is then used to calculate the time evolution of the
transformed PDEs. After each iteration, the inverse Fourier transform is applied to
the transformed values, thus recovering the meaningful values. On the next itera-
tion, Fourier transform is applied again, Euler’s method evaluates the new values
for the transformed variables and, again, the inverse transform is applied, recovering
the original variables. The process is repeated for as man times as necessary. The
main advantages of the Fourier Transform is its spectral accuracy and its relatively
low running times - O (n log n). It does, however, impose tight restrictions on the
types of functions on which it may be applied, namely, the functions must be peri-
odic. The PSA differential equations do not yield, in general, periodic results. This
problem may be circumvented, however, by using their periodic extensions.
4. Method of Lines with Orthogonal collocation (MOL-OC) - much like the MOL-FD,
the original set of PDEs is thought of as a system of ODEs. However, instead of
replacing the space derivatives for finite difference approximations, the derivatives
are calculated by fitting a Lagrange polynomial to the function at each iteration. It
is a very widespread technique for solving adsorption-related problems.
5. Method of Characteristics (MOC) - It is very popular in fluid mechanics related
fields. In order to solve the PDE, the Method of Characteristics analytically seeks
5
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
the characteristic curves of the PDE (a reparametrization which allows for direct
integration) and solve it over these characteristics.
The finite difference approximations used were of the form:
[
∂f
∂x
]
i
=
fi−1 − fi
∆x
(11)
A more comprehensive discussion on FD, MOL-FD and MOL-OC can be found in [11].
As for the MOC, a good description can be found in [14]
2 METHODOLOGY
2.1 Hardware and software
The model applied to simulate the single unit evaluated in this study was developed using
the software MATLABr version R2013a, and installed on a machine withthe following
hardware configurations:
Item Configuration
Processor Intel r CoreTM i7, 2.00 GHz
Memory (RAM) 6,00 GB
System type 64 bits
Operational system Windows 8
Table 1: System hardware and operating system characteristics.
As previously mentioned, five numerical methods for solving the set of partial differential
equations were evaluated (FD, MOL-FD, FFT, MOL-OC, and MOC).In order to compare
the performance of each one, the number of points used and the time spent to perform
the calculations and data processing were the main variables investigated. The method
that is to be used as the default in the PSA unit simulations is be one that yields the best
relationship between accuracy and running time. The dynamic model was developed to
operate with several commercial dynamic process simulators. At the moment, the model
is being embedded in an ExcelTM spreadsheet which can be used to exchenge data with
our current simulator. We restricted ourselves to only simulating the adsorption stage, as
it is the most significative step in the overall operation.
2.2 Case of study
In this workthe adsorption of hydrogen impurities through a pressure swing operation
was simulated. The adsorption column studied is equipped with three different adjacent
zeolite layers, whose heights are ∆Z1, ∆Z2 and ∆Z3 respectively. The specifications for
this column are displayed in Figure 2 and in Table 2.
6
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Symbol Meaning Value
∆Zc Total column height 4800 mm
Dc Inner diameter 2000 mm
∆P Total pressure drop 0.5 kgf/cm2
∆Z1 Height of Zeolite H-1 layer 1600 mm
∆Z2 Height of Zeolite H-1-4 layer 1600 mm
∆Z3 Height of Zeolite H-2-10 layer 1600 mm
Table 2: Specifications of the PSA column to be simulated.
Figure 2: Schematic diagram of the PSA column to be simulated.
The thermodynamic parameters used are displayed in Table 3. The composition of the
gas mixture fed to the PSA column is shown in Table 4 and it was taken from historical
data.
3 RESULTS AND DISCUSSION
The results are presented as follows: a comparison of each numerical method with all
the others with respect to its accuracy and its computational time is presented and,
afterwards, the concentration profiles of each component (CO, H2, CO2, N2 and CH4)
along the reactor length is shown as calculated through the numerical method considered
best. An error measure was defined as follows:
7
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Layer Component
ai,1 x 10
3
(mol/g)
ai,2
(K)
bi,0
(mmHg)
k
(s−1)
�
(-)
dp
(m)
H-1
H2 1.24 0.36 2.2 1
CO -0.58 0.84 2.53 0.3
N2 -0.23 1.015 6.38 0.15 0.4 0.027
CO2 2.09 0.63 0.67 0.1
CH4 -0.29 1.04 6.44 0.4
H-1-4
H2 1.23 0.357 2.19 1
CO -0.56 0.81 2.57 0.15
N2 -0.198 1.017 6.37 0.2 0.4 0.027
CO2 2.075 0.624 0.659 0.05
CH4 -0.28 1.036 6.53 0.4
H-2-10
H2 4.32 0.0 6.72 1
CO 0.92 0.52 7.86 0.147
N2 -1.75 1.95 25.9 0.19 0.4 0.027
CO2 -14.2 6.63 33.03 0.046
CH4 -1.78 1.98 26.6 0.42
Table 3: Mass transfer and porous flow data for the model [?].
Component Molar fraction
H2 0.0400
CO 0.0243
N2 0.3000
CO2 0.1709
CH4 0.4648
Total 1.000
Table 4: PSA feed composition.
e =
tmax∫
0
L∫
0
|yMOLCO2 (z, t)− yCO2(z, t)|dzdt (12)
In this equation yMOLCO2 (z, t) denotes the CO2 concentration profile over time calculated
through the MOL-FD using a 400-point discretization over the z-axis and a 10000-point
discretization over the t-axis. This MOL-FD solution was considered to be sufficiently
close to the real solution, so that it could be used as a reference. The yCO2(z, t) term
denotes the concentration profile of CO2 over time obtained through the method being
considered. The simulation was for all methods for tmax = 55 s. Figure 3 shows a graph
comparing the accuracy (the error measure) of different methods versus the number of
8
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Method
Number of x-points yielding
log(e) = 0.724
MOL-FD 13
MOL-OC 22
FFT 64
FD 13
MOC 137
Table 5: Number of spatial discretization points for each method necessary to achieve log(e) = 0.724.
discretization points on the z-axis. It is clear that the error diminishes as the number
of points increases, as it would be expected. The curve corresponding to orthogonal col-
location stops at about 35 points due to numerical instability. It can be seen that the
most precise methods overall were MOL-FD and FD. Following these two methods were
MOL-OC (for less than 35 points) and FFT. The MOC was the least accurate method.
In order to compare the running time of each method, a benchmark error was chosen
and each algorithm was run with a number of points which would yield that same error.
The choice was carefully made in such a way that the number of points corresponding to
it was a power of 2. This is necessary for the FFT algorithm to function correctly. The
chosen value was such that log(e) = 0.724 (represented by the black horizontal line in
Figure 3). Table 5 shows the number of points that each method requires to achieve such
precision.
By running each of the methods 30 times with the number of points shown in Table 5, it
is possible to calculate the mean running time of each one of them and its corresponding
standard deviances. The values so obtained are shown both in Table 6 and in Figure 4.
By ensuring that the errors associated to each method are roughly the same, it is now
possible to make a fair comparison of their running times.
For the chosen accuracy, the fastest methods were MOL-FD and MOL-OC. Despite being
a fast algorithm, the extra work required to fit the problem to the requirements of the
FFT greatly increased its running time. FD ran slower than both MOL and OC, but
still faster than the FFT. The greatest running time is that of the MOC. Despite being
realatively simple, many points are required for the MOC to yield a satisfactory accuracy.
Figure 5 shows the concentration profiles of methane and nitrogen along the PSA bed
during the adsorption stage at three different times. The profiles were obtained through
a 22-point OC. The profiles do exhibit the expected behavior.
One important observation is that the choice of the fastest algorithm is heavily dependant
on the implementation of the methods. For instance, MATLABr solves linear systems
through a heavily optimized routine for quickly handling matrices. If another routine is
9
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Figure 3: Error versus number of spatial discretization points for each method.
Figure 4: Comparison of the running times of each method.
10
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Figure 5: Concentration profiles of methane and nitrogen along the column during the adsorption phase
at different times. Graphs obtained through a 22-point MOL-OC method.
used, the running time might change considerably.
4 CONCLUSIONS
The method considered best for the given PSA problem is the MOL-FD, since it demands
a relatively low computational time to run, is easy to implement, is accurate and handles
well varying boundary conditions. Despite the fact that the MOL-OC method required a
comparable computational time, it was also less precise. The FD method displayed the
third lowest running time. However, it also demands a considerable amount of computer
memory, which may be a problem. Even though the FFT is a very fast algorithm, the
amount of extra work required to fit the problem into its requirements made the overall
running time greatly surpass that of the algorithm itself. Moreover, the FFT is highly
sensitive to changing boundary conditions, which occur frequently in acommon PSA unit.
11
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
Method Mean running time (s) Standard deviation (s)
MOL-FD 0.28 0.03
MOL-OC 0.28 0.02
FFT 0.69 0.07
FD 0.51 0.04
MOC 1.7 0.1
Table 6: Mean simulation time per method and standard deviances.
For those reasons, the FFT is not recommended. The MOC displays a relatively high run-
ning time and is not as accurate as any of the other methods. Once again, it is important
to state that these results are highly dependent on how each method is implemented.
5 SYMBOLS TABLE
Symbol Meaning
b Langmuir equation parameter
C Molar concentration
dp Particle diameter
Dc Column inner diameter
ki Mass transfer coefficient of the i-th component
Pi Partial pressure of the i-th component
ρs Solid medium density
qi Adsorbed amount of the i-th component
q∗i Adsorbed amount of the i-th component at equilibrium
t Time
tmax Simulation time
T Absolute temperature
v Intersticial velocity
z Axial position in an adsorption column
y Molar fraction
∆Zc Total column height
∆Z1 Height of the H-1 Zeolite layer
∆Z2 Height of the H-1-4 Zeolite layer
∆Z3 Height of the H-2-10 Zeolite layer
µ Viscosity
Table 7: Symbols table.
12
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
REFERENCES
[1] A. Agarwal, Advanced strategies for optimal design and operation of pressure swing
adsorption processes. (2010). 216f. Thesis (Doctor of Philosophy in Chemical Engi-
neering) - Carnegie Mellon University, Pittsburgh.
[2] M. Ashkan and M. Masoud, Simulation of a Single Bed Pressure Swing Adsorp-
tion for Producing Nitrogen. (2011) In: INTERNATIONAL CONFERENCE ON
CHEMICAL, BIOLOGICAL AND ENVIRONMENT SCIENCES (ICCEBS’2011),
Bangkok.
[3] P. Biswas, S. Agarwal and S. Sinha, Modeling and Simulation for Pressure Swing
Adsorption System for Hydrogen Purification. (2010) Chemical & Biochemical Engi-
neering Quarterly, v. 24, n.4, p. 409-414.
[4] F.E. Cruz, Produção de hidrogênio em refinarias de petróleo: avaliação exergética e
custo de produção. (2010). 164p. Dissertation (Masters in Chemical Engineering) ?
Polytechnic School, São Paulo University, São Paulo.
[5] M.A. Fahim, T.A. Al-Shaaf and A. Elkilani, Fundamentals of Petroleum Refining.
(2010), 1.ed. Oxford: Elsevier B.V. 516p.
[6] Q. Huang et al., Optimization of PSA process for producing enriched hydrogen from
plasma reactor gas. (2008). Separation and Purification Technology, v.62, p.22?31.
[7] R. A. Meyers, Handbook of Petroleum Refining Processes (2003) , 3.ed. McGraw-Hill
Book Company, New York.
[8] A.P. Scheer, Desenvolvimento de um sistema para simulação e otimização do processo
de adsorção para avaliação da separação de misturas ĺıquidas. (2002), [s.n.]. Thesis
(Doctor of Philosophy in Chemical Engineering) - Faculty of Chemical Engineering,
State University of Campinas, São Paulo.
[9] J.G. Speight, Hidrogen Production. (2006) In: . The Chemistry and Technology
of Petroleum. 4.ed. Taylor & Francis Group, LLC: New York. chap. 22.
[10] M. R. Rahimpour, et al., The Enhancement of hydrogen recovery in PSA unit of
domestic petro-chemical plant. (2013) Chemical Engineering Journal, Volume 226,
15 June 2013, Pages 444-45.
[11] R. G. Rice, D. D. Do, Applied Mathematics and Modeling for Chemical Engineers.
(2012) 2.ed. New Jersey: Wiley, 400 p.
[12] D.M. Ruthven, S. Farooq, K.S. Knaebel, Pressure swing adsorption. (1994). New
York: UCH, 352p.
13
Márcio R. V. Neto, Rafael V. Ferreira and Marcelo Cardoso
[13] G. Towler, R.K. Sinnott, Chemical Engineering Design: Principles, Practice and
Economics of Plant and Process Design. (2012), 2.ed. Burlington: Butterworth-
Heinemann; Elsevier. 1320 p.
[14] E. B. Wylie, V. L. Streeter, Fluid Transients (1983). FEB Press, 384 p.
[15] J. Yang, C. H. Lee, J. W. Chang, Separation of Hydrogen Mixtures by a Two-Bed
Pressure Swing Adsorption Process Using Zeolite 5A. (1997) Sunkyong Engineering
Ind. Eng. Chem. Res., v. 36, n. 7, p. 2789-2798.
[16] Y. W. You et al., H2 PSA purifier for CO removal from hydrogen mixtures. (2012)
International Journal of Hydrogen Energy, v. 37, p. 18175-18186.
14

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