Chemical Engineering Science 66 (2011) 5938–5944
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Science
journal homepage: www.elsevier.com/locate/ces
Dynamics of CO2 adsorption on sodium oxide promoted
alumina in a packed-bed reactor
Mingheng Li Department of Chemical and Materials Engineering, California State Polytechnic University, Pomona, CA 91768, United States
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 29 June 2011
Received in revised form
25 July 2011
Accepted 7 August 2011
Available online 16 August 2011
CO2 adsorption in packed-bed reactors has potential applications in flue gas CO2 capture and adsorption
enhanced reaction processes. This work focuses on CO2 adsorption dynamics on sodium oxide
promoted alumina in a packed-bed reactor. A comprehensive model is developed to describe the
coupled transport phenomena and is solved using orthogonal collocation on finite elements. The model
predicted breakthrough curve matches very well with experimental data obtained from a pilot-scale
packed-bed reactor. Several dimensionless parameters are also derived to explain the shape of the
breakthrough curve.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
CO2 adsorption
Sodium oxide promoted alumina
Packed-bed
Dynamics
Breakthrough curve
Modeling
1. Introduction
Fossil fuels supply around 98% of energy over the entire world.
The use of fossil fuels is the major source of CO2 emissions.
According to International Energy Outlook 2010, the worldwide
consumption of fossil fuels is projected to increase by about 50%
by 2035 from 2007, primarily due to the economical growth of
non-OECD (Organization for Economic Co-operation and Development) countries (U.S. Energy Information Administration,
2011). In view that CO2 is a greenhouse gas that contributes to
global climate warming, more stringent government regulations
have been proposed to reduce the emission of CO2 from the use of
fossil fuels, e.g., The American Clean Energy and Security Act of
2009 (Waxman and Markey, 2009). This has significantly motivated the research and development of technologies for more
efficient use of fossil fuels (e.g., hydrogen fuel cell technology
Kolb, 2008) as well as cost-effective CO2 capture and sequestration techniques (Yang et al., 2008; Aaron and Tsouris, 2005).
Among existing CO2 capture techniques such as absorption
(Rao and Rubin, 2002), membrane separation (Zhao et al., 2008),
cryogenic fractionation (Hart and Gnanendran, 2009), and adsorption (Chue et al., 1995), the last one has advantages of low energy
requirement and low capital investment cost. Its working principle is based on preferential adsorption of CO2 on a solid adsorbent
Tel.: þ 1 909 869 3668; fax: þ 1 909 869 6920.
E-mail address: [email protected]
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2011.08.013
and subsequent desorption at a different condition. Through
cyclic operations over several packed-beds (e.g., pressure swing
adsorption or PSA), CO2 is separated from the mixture.
More recently, CO2 adsorption has found applications in adsorptive reaction processes where CO2 is a by-product. The adsorptive
reactor concept enables natural separation of CO2 from the product
mixture, and therefore, enhances yield and selectivity of the
desired product, leading to process intensification (Stankiewicz,
2003). One such example is adsorption enhanced reforming (AER),
which was originated by Sircar and coworkers (Carvill et al., 1996)
and further developed by several groups around the world to
generate fuel cell grade hydrogen (Ding and Alpay, 2000; Reijers
et al., 2006; Koumpouras et al., 2007; Stevens et al., 2007; Lindborg
and Jakobsen, 2009; Duraiswamy et al., 2010). The author’s group
and Intelligent Energy, Inc. (Long Beach, CA) have recently developed a bench-scale AER-based hydrogen generator. It incorporates
a novel pulsing feed concept that further improves hydrogen yield
and purity and allows the AER to be operated at low steam/carbon
ratios (Duraiswamy et al., 2010). The unit consists of four reaction
beds packed with ceria supported rhodium as the catalyst and
hydrotalcite as the adsorbent. The beds run alternately and continuously produces high-purity hydrogen for use in conjunction
with fuel cells. The system is operated around 500 1C, significantly
lower than the conventional steam methane reforming process
(about 850 1C). During reforming step, CO2 is adsorbed and both
the reforming and water gas shift (WGS) reactions are enhanced
(Li, 2008). As a result, H2 is produced with little CO and CO2
impurities. The CO2 adsorption also suppresses carbon formation
M. Li / Chemical Engineering Science 66 (2011) 5938–5944
according to the author’s thermodynamic analysis (Li, 2009).
During regeneration step, cathode off-gas of the fuel cell or steam
is sent from the reversed direction and CO2-rich exhaust gas exits
the reactor from the other end. The AER fuel cell system, if successfully commercialized, would have a better process efficiency and a
lower CO2 emission than the combustion engine based route for
power generation.
The commonly used CO2 adsorbents include zeolite (Chue et al.,
1995), activated carbon (Chue et al., 1995), and alkali composites,
e.g., lithium zirconate (Xiong et al., 2003), calcium oxide (Li et al.,
2009), aluminum oxide (Lee et al., 2007a) and hydrotalcite (Yang and
Kim, 2006; Yong et al., 2001). It has been shown that impregnation of
potassium or sodium oxide enhances CO2 adsorption (Xiong et al.,
2003; Lee et al., 2007a,b). This work aims to investigate the dynamics
of CO2 adsorption in packed-bed reactors, which is a part of the
author’s research effort on adsorption enhanced reaction and separation processes (Duraiswamy et al., 2010; Li, 2008, 2009; Li et al., in
press). The adsorbent chosen for this study is Sud Chemie ActiSorb s
CL2 adsorbent, which contains 85–95% alumina and 5–15% sodium
oxide. As compared to hydrotalcite used in the author’s previous
experimental work (Duraiswamy et al., 2010), sodium promoted
alumina works at a relatively low temperature and has potential
applications in adsorption enhanced methanol reforming and WGS
(Li et al., in press) and in CO2 capture from flue gases. However, the
developed modeling can be extended to other adsorbents. The paper
will first present a comprehensive mathematical model to describe
various transport phenomena in the packed-bed. Different from the
constant velocity assumption often employed in literature (e.g., Choi
et al., 2003; Stevens et al., 2007), this work explicitly accounts for its
evolution with respect to time and space. Several dimensionless
parameters are also derived to clearly reveal the kinetics and
thermodynamics of the adsorption process.
2. Mathematical model
Consider the flow through a porous packed-bed with simultaneous adsorption, the mass conservation equation of species j is
described by the following equation:
@C
@q
@ðvC j Þ
@Cj
@
Dz
þ
, j ¼ 1, . . . ,s
ð1Þ
e j þ rb j ¼ @z
@z
@t
@t
@z
where Cj is the concentration of species j, t is the time, e is the bed
porosity, respectively, z is the axial location, v is the superficial
velocity (volumetric flow Q_ ¼ vA), rb is the bulk density of the
packed material (total mass of the adsorbent in the reactor over
the reactor volume), qj is the loading of species j on the adsorbent,
Dz is the axial dispersion coefficient, and s is the total number of
species.
It is assumed that the gases follow the ideal gas law, or
C¼
P
RT
ð2Þ
P
where C is the total molar concentration (C ¼ si ¼ 1 Cj ), P is the
pressure and T is the temperature. Let yj be the fraction of species
j in the gas phase, or Cj ¼ yj C, it can be derived from Eqs. (1)
and (2) that
@y
ey @P ey C @T
@qj
@yj
@C
@v
þ rb
Cyj
¼ vC
vyj
e jþ j j
@z
@z
T @t
@t
RT @t
@t
@z
!
@2 yj
@2 C
@C @yj
þ Dz C 2 þyj 2 þ 2
@z @z
@z
@z
ð3Þ
The overall mass balance is then described by
s
X
j¼1
yj ¼ 1
ð4Þ
5939
It is also assumed that the adsorption follows the linear
driving force (LDF) model (Sircar and Hufton, 2000). Based on
this model, the changing rate of loading of species j on the
adsorbent is described by:
@qj
¼ kads,j ðqnj qj Þ
@t
ð5Þ
where kads,j is the effective LDF mass transfer coefficient (Yang,
1987). qnj is the loading of species j on the adsorbent at a partial
pressure of Pj when equilibrium is reached. According to the
P
Langmuir isotherm model, qnj ¼ mj bj Pj =ð1 þ sj ¼ 1 bj Pj Þ, where mj
and bj are the saturated capacity and the adsorption parameter,
respectively.
The momentum balance for flow over a pack-bed is described
using the Ergun equation (Bird et al., 1960):
0¼
@P 150mð1eÞ2
1:75ð1eÞ
v
rg v2
@z
dp e3
d2p e3
ð6Þ
where P is the pressure, dp is the equivalent diameter of the
P
packed material, and rg is the gas density. rg ¼ sj ¼ 1 Cj Mj ¼
Ps
Ps
C j ¼ 1 yj Mj ¼ ðP=RTÞ j ¼ 1 yj Mj , where Mj is the molecular
weight of species j and T is the temperature.
The energy balance equation is described as follows:
s X
@q
@T
@P
þ rb
ðerg cvg þ rb cpb Þ
DHads,j j e
@t
@t
@t
j¼1
@ðvrg cpg TÞ
@
@T
4U
kz
þ
ðTw TÞ
ð7Þ
þ
¼
@z
@z
Ds
@z
where cvg and cpg are the gas heat capacities under constant
volume and constant pressure, respectively, and cpb is the heat
capacity of the packed-bed material, DHads,j is the enthalpy
change of adsorption of species j, Ds is the diameter of the reactor
and U is the overall heat transfer coefficient between the wall and
the packed material.
Danckwerts (1953) boundary conditions are applied to the
equations describing concentration and temperature. For example, the boundary conditions to Eq. (1) or (3) are
@Cj
v0
¼ ðCj,0 Cj Þ,
@z
Dz
@Cj
¼ 0,
@z
x¼0
x¼L
ð8Þ
where v0 and Cj,0 are the superficial velocity and concentration of
species j at the inlet of the packed-bed, respectively.
The dynamic model composed by Eqs. (3)–(7) is a partial
differential-algebraic equation (PDAE) which can be solved
numerically. The independent variables will be yj (j ¼ 1, . . . ,s), P,
T and u. Note that C is a dependent variable of P and T based on
Eq. (2). Moreover, the first- and second-order spatial derivatives
of C can be explicitly described as functions of P and T using the
equations below:
@C
1 @P C @T
¼
@z
RT @z T @z
@2 C
1
¼
RT
@z2
@2 P
@C @T
@2 T
CR 2
2R
2
@z @z
@z
@z
ð9Þ
The numerical method is based orthogonal collocation on
finite elements, which has been used to solve diffusion-convection-reaction processes (Li and Christofides, 2008). The central
idea of the orthogonal collocation method is to discretize the
variables in the spatial domain based on the zeros of some
orthogonal polynomials and to transfer the PDAE to a set of DAEs.
For example, a variable xðz,tÞ (which may be yj, P or T in this work)
can be written as a sum of N finite elements within the spatial
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M. Li / Chemical Engineering Science 66 (2011) 5938–5944
P
domain, or xðz,tÞ ¼ N
i ¼ 1 li ðzÞxðzi ,tÞ at time t, where li(z) is the
Lagrange interpolation polynomial of ðN1Þ th order:
li ðzÞ ¼
N
Y
zzj
z zj
j ¼ 1,j a i i
which satisfies
(
0, i a j
li ðzj Þ ¼
1, i ¼ j
Table 1
Parameters used in the simulation.
Value
Ds
L
0.076 m
1.22 m
0.57
785 kg/m3
1000 J/kg/K
1.48 bar
225 1C
18.8 slpm
0.091 m/s
0.16
225 1C
50 W/m2/K
2.2 10 2 s 1
e
rb
cpb
P
T0
Q0
v0
yA 0
TW
U
kads
ð11Þ
Based on the orthogonal collocation scheme, the collocation
elements (zi) and the Lagrange interpolation polynomial may be
determined a priori without information from the structure of the
PDE. Therefore, the partial derivatives of xðz,tÞ with respect to the
spatial coordinate can be expressed as follows:
N
N
X
X
@xðzk ,tÞ
dl ðz Þ
¼
xðzi ,tÞ i k ¼
Ak,i xðzi ,tÞ
@z
dz
i¼1
i¼1
Parameters
ð10Þ
ð12Þ
and
N
N
X
X
@x2 ðzk ,tÞ
d2 li ðzÞ
¼
xðz
,tÞ
¼
Bk,i xðzi ,tÞ
i
2
@z2
dz
i¼1
i¼1
0.2
where A and B are both constant matrices.
For collocation points 2ðN1Þ, all the spatial derivatives in
Eqs. (3)–(7) are converted to weighted sums of variables at these
collocation points. For collocation points 1 and N, the boundary
conditions (e.g., Eq. (8)) are converted to algebraic equations.
Note that all the temporal derivatives in Eqs. (3)–(7) are on the
left hand side. Therefore, the original PDAE model is converted to
a DAE as follows:
0.15
M
dx
¼ f ðx,tÞ,
dt
yCO2
ð13Þ
0.1
0.05
0
1
300
xð0Þ ¼ x0
ð14Þ
where x ¼ ½y11 , . . . ,y1N , . . . yS1 , . . . ,ySN ,T1 , . . . TN ,u1 , . . . uN ,P1 , . . . PN T
and M is a matrix. As one can see from Eq. (14), the velocity is
fully coupled with other variables. Eq. (14) can be solved by
standard solvers (e.g., ode15s in Matlab).
Even though the focus of this work is dynamic modeling, it is
worth nothing that the presented numerical method may be used
for future model reduction and optimization of adsorption-based
processes (Agarwal et al., 2009). For example, Karhunen–Loéve
expansion can be used to derive empirical eigenfunctions, which
in turn, are used as basis functions to derive reduced-order
models to capture the dominant process dynamics. The
reduced-order system is then used for model-based control and
optimization to enhance process performance. Interested readers
may refer to literature for finite-dimensional approximation and
control of process systems described by nonlinear parabolic
partial differential (Christofides and Daoutidis, 1997; Baker and
Christofides, 2000).
3. Results and discussion
The developed mathematical model can be applied to multicomponent adsorption in general. However, for sodium oxide
promoted alumina in this work, CO2 is assumed to be the only
adsorbate. The parameters used in the simulation are shown in
Table 1, which are based on a pilot-scale experimental system at the
author’s lab, and now at Intelligent Energy, Inc. (Long Beach,
California). The temperature of the packed-bed is controlled around
225 1C using an external heater. This temperature is suitable for WGS
and steam reforming of methanol with simultaneous CO2 adsorption
(Li et al., in press). At this temperature, our Thermogravimetric
analysis (TGA) study indicates that CO2 adsorption on sodium oxide
promoted alumina approximately follows the Langmuir isotherm
z/L
0.5
200
100
0 0
t/τ
Fig. 1. Spatial-temporal profile of CO2 fraction in the gas phase.
with parameters mA ¼0.39 mol/kg and bA ¼5.2 10 4 Pa 1, where
subscript A represents CO2. The packed-bed reactor is loaded with
3.8 kg adsorbent, corresponding to a bulk density of 785 kg/m3. The
residence time calculated based on the inlet superficial velocity
(t ¼ Le=v0 ) is about 7.6 s. The axial dispersion coefficient Dz is estimated to be about 2.4 10 4 m/s2 using an empirical equation
(Edwards and Richardson, 1968). The thermodynamic and transport
properties are calculated as functions of temperature and composition using formulas and coefficients based on the NASA CEA program
(Gordon and McBride, 1996). The pressure drop along the packedbed calculated from the Ergun equation is less than 100 Pa, or the
process may be considered isobaric.
The formulated mathematical model is solved using Matlab.
Default error control settings are used (the relative tolerance is
0.1% and the absolute error tolerance is 10 6). The contours of
CO2 fraction in the gas phase and CO2 loading on the adsorbent
are shown in Figs. 1 and 2, respectively. The spatial-temporal
profile of CO2 behaves similar to the step response in a tubular
reactor where dispersion and convection occur simultaneously
(Li and Christofides, 2008). However, the breakthrough time is
much longer than t. For a dispersion–convection process with no
adsorption or reaction, the concentration of CO2 at the exit of the
reactor is about 50% of the feeding condition at t ¼ t (Li and
Christofides, 2008). When adsorption is coupled with dispersion
and convection, all CO2 fed to the reactor is adsorbed near the
entrance of the packed-bed at t ¼ t, and therefore, no CO2 comes
out of the reactor. As time proceeds, the loading of CO2 on the
adsorbent increases and the wave fronts of both CA and qA move
M. Li / Chemical Engineering Science 66 (2011) 5938–5944
0.1
0.4
0.3
v (m/s)
2
qCO (mol/kg)
5941
0.2
0.09
0.08
0.1
0.07
1
0
1
300
300
0.5
z/L
200
0.5
z/L
100
0 0
t/τ
200
100
0 0
t/τ
Fig. 4. Spatial-temporal profile of superficial velocity.
Fig. 2. Spatial-temporal profile of CO2 loading on the adsorbent.
0.2
Experiment
Model
232
0.15
228
yCO2, e
T (°C)
230
226
224
1
0.1
0.05
300
200
0.5
z/L
100
0 0
t/τ
0
0
Fig. 3. Spatial-temporal profile of bed temperature.
forward along the reactor. Eventually, the concentration wave
reaches the outlet of the reactor, and the breakthrough occurs.
The temperature contour of the bed is shown in Fig. 3. Because
CO2 adsorption is exothermic, the bed temperature rises and heat
is transferred from the packed-bed to the wall. As CO2 adsorption
approaches equilibrium, the heat release rate from adsorption
drops below the heat dissipation through the wall and the
temperature begins to drop. Therefore, one can see the propagation of temperature wave along the reactor. Finally, the bed
temperature equals the wall temperature when the steady state
is reached.
The superficial velocity is shown in Fig. 4. Initially, the superficial velocity suddenly drops below v0 because of CO2 adsorption
on the adsorbent. The velocity gradually increases as the CO2
adsorption rate reduces (due to smaller and smaller driving force
for adsorption). As the loading of CO2 on the adsorbent reaches its
saturated value, the superficial velocity is slightly higher than v0
because of an elevated bed temperature. As the bed temperature
becomes the same as the wall temperature at steady state, the
superficial velocity equals v0.
The model predicted and measured breakthrough curves of
CO2 mole fraction at the exit of the reactor are shown in Fig. 5. It
is seen that a very good match between mathematical modeling
and measurement can be obtained using mA ¼0.39 mol/kg instead
of 0.38 mol/kg derived from TGA measurement. The discrepancy
between TGA and packed-bed measurements is about 3%. Moreover, the adsorption kinetic parameter kads is chosen to be a
30
60
90
120
150
t/τ
Fig. 5. Comparison between model predicted breakthrough curve and experimental measurement.
constant (2.2 10 2 s 1) in the model to best match the modeling results with the experimental data. As will be shown later, a
change in kads will affect the slope of the breakthrough curve.
A mass balance equation may be derived to provide an
in-depth understanding of the relationship between the breakthrough curve and the adsorbent capacity. Based on the assumptions that the adsorption of carrier gas is negligible and that the
carrier mass flow rate at the exit of the reactor is the same as the
one at the feed, a mass balance may be written as follows:
Z 1
yA0
yAe ðtÞ
ðrb qnA0 þ eCA0 ÞLAc ¼
dt
ð15Þ
n_ cg,0
n_ cg,e
1yAe ðtÞ
1yA0
0
where n_ cg is the molar flow rate of the carrier gas, and Ac is the
cross-sectional area of the packed-bed. It can be readily derived
from Eq. (15) that:
Xþ1 ¼ G
ð16Þ
R1
where X ¼ rb qnA0 =eCA0 , and G ¼ ð1=tÞ 0 ½1yAe ðtÞð1yA0 Þ=yA0
ð1yAe ðtÞÞdt, which is equal to the area between 1 and the corrected
dimensionless mole fraction yAe ðtÞð1yA0 Þ=yA0 ð1yAe ðtÞÞ in Fig. 6.
A numerical integration shows that G ¼ 87:9, almost the same as
X þ 1 ¼ rb qnA0 =eCA0 þ 1 ¼ 87:8.
It is interesting to note that X is a very important parameter also
used in the author’s thermodynamic analysis of adsorption enhanced
reaction process (Li, 2008, 2009). These studies have indicated that
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M. Li / Chemical Engineering Science 66 (2011) 5938–5944
will be investigated in future work. The base case conditions in
the parametric analysis are chosen to be Pe¼200, Da ¼0.15 and
X ¼ 800. The effect of each parameter is shown in Figs. 7–9. It is
observed that Pe or Da affects the slope of the breakthrough curve.
X has the most prominent influence on the slope as well as
the taking-off location of the breakthrough curve. Based on the
assumption of a low CO2 concentration in the gas phase, G in
Eq. (16) can be simplified as:
1
0.8
0.7
0.6
0.5
G¼
0.4
1
t
Z
1
0
1
Z Z 1
yAe ðtÞð1yA0 Þ
1 1
yA ðtÞ
dt ¼
1 e
ð1C ð1,tÞ dt
dt ¼
yA0 ð1yAe ðtÞÞ
t 0
yA 0
0
ð19Þ
0.3
which is equal to the area between 1 and the breakthrough curve
in Figs. 7–9.
If the adsorption kinetics is very fast (or Da is sufficiently
large), adsorption equilibrium may be reached instantaneously.
As a result, q ¼ C , and Eq. (18) becomes:
0.2
0.1
0
0
50
100
t/τ
150
200
Fig. 6. Temporal profile of the corrected dimensionless molar fraction at the exit
of the reactor.
X uniquely determines the equilibrium composition of a reactive
system with simultaneous adsorption at a given temperature, pressure and initial composition. The larger the X, the more the enhancement in reaction. The physical meaning of X is the ratio of CO2 on the
adsorbent to the one in the gas phase at equilibrium. In this sense,
X may be considered as the dimensionless adsorbent capacity of an
adsorbent corresponding to CA0 (note that the volumetric adsorption
capacity [mol A/m3]¼ rb qnA0 ¼ XeCA0 ).
If CO2 concentration is low, it is shown that its adsorption
dynamics on a packed-bed may be well characterized using only
three dimensionless parameters including X. Note that a low CO2
concentration implies a roughly constant bed temperature. Therefore, the mathematical model is formulated as follows:
@CA ðz,tÞ
@q ðz,tÞ
@C ðz,tÞ
@2 CA ðz,tÞ
þ rb A
¼ v A
þD
@t
@t
@z
@z2
@qA ðz,tÞ
n
¼ kads ðqA qA ðz,tÞÞ
@t
@CA ðz,tÞ
s:t: vC A0 ¼ vC A ð0,tÞDz
@z z ¼ 0
@CA ðz,tÞ
¼0
@z 1 00
0
_
ð1 þ XÞC ¼ C þ
C
Pe
1 0
s:t: 1 ¼ C ð0,tÞ C ð0,tÞ
Pe
0
0 ¼ C ð1,tÞ
Pe = 100
Pe = 200
Pe = 400
0.9
0.8
0.7
0.6
0.5
0.4
0.3
e
0.2
0.1
0
0
500
ð17Þ
z¼L
1000
t/τ
1500
2000
Fig. 7. Effect of Pe on the breakthrough curve.
To write the above PDE in a dimensionless form, the following
variables are defined: t ¼ Le=v (the characteristic time of the
reactor), Pe ¼ Lv=Dz (the Peclet number, or the rate of convection
over the one of axial dispersion), Da ¼ kads t (the Damkohler
number for adsorption, or the residence time over the time
scale for adsorption), z ¼ z=L, t ¼ t=t, C ¼ CA =CA0 , q ¼ qA =qnA0 and
X ¼ rb qnA0 =eCA0 . At a low CO2 concentration, bP A 5 1, and
qnA ¼ mbP A =ð1 þ bPA Þ mbP A . Therefore, qnA =qA0 C . As a result,
Eq. (17) can be written in a dimensionless form as follows:
1
Da = 0.075
Da = 0.15
Da = 0.3
0.9
0.8
0.7
0.6
C/C0
1 00
0
_
C þ X q_ ¼ C þ
C
Pe
q_ ¼ DaðC qÞ
1 0
s:t: 1 ¼ C ð0,tÞ C ð0,tÞ
Pe
0
0 ¼ C ð1,tÞ
ð20Þ
1
C/C0
yCO2,e (1−yCO2,0)/yCO2,0 (1−yCO2,e)
0.9
0.5
0.4
0.3
ð18Þ
0.2
It is seen from Eq. (18) that the process dynamics and the
breakthrough curve are primarily dependent on three important
dimensionless parameters: Pe, Da and X. Eq. (18) can be readily
solved using numerical methods such as finite difference and
orthogonal collocation on finite elements (Li, 2008; Li and
Christofides, 2008). Possible infinite series solution to Eq. (18)
0.1
0
0
500
1000
t/τ
1500
Fig. 8. Effect of Da on the breakthrough curve.
2000
M. Li / Chemical Engineering Science 66 (2011) 5938–5944
1
would like to thank Dr. Duraiswamy at Intelligent Energy, Inc.
(Long Beach, CA) for discussions.
Ξ = 400
Ξ = 800
Ξ = 1600
0.9
0.8
5943
References
0.7
C/C0
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
t/τ
1500
2000
Fig. 9. Effect of X on the breakthrough curve.
An approximate analytic solution of C ð1,tÞ may be derived
following an approach similar to the one in Li and Christofides
(2008). The result is as follows:
"
#
1
1t=ðX þ 1Þ
C ð1,tÞ ¼ erfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð21Þ
2
4t=ðX þ 1Þ=Pe
which implies that:
C ð1, X þ 1Þ ¼ 0:5
ð22Þ
Eq. (22) provides an efficient way to estimate X and the
capacity of the adsorbent based on the experimentally measured
breakthrough curve, provided that the adsorption is fast enough.
It is found that Eq. (22) can be used to estimate X in all cases
shown in Figs. 7–9 with a very reasonable accuracy. For the
non-dilute case presented in Fig. 6, t=t corresponding to
yAe ðtÞð1yA0 Þ=yA0 ð1yAe ðtÞÞ ¼ 0:5 is 85.9, which is also close to
the actual value of X þ 1, or 87.8.
4. Concluding remarks
A comprehensive mathematical model is formulated and
solved to describe the CO2 adsorption dynamics on sodium oxide
promoted alumina in a packed-bed reactor. The modeling result
matches the experimental data very well with mA ¼0.39 mol/kg
and kads ¼2.2 10 2 s 1 at a bed temperature around 225 1C. The
capacity data are also close to TGA derived value 0.38 mol/kg.
Several dimensionless parameters (X, Pe and Da) may be used
to explain the breakthrough curve. In particular, X, or the dimensionless adsorbent capacity, significantly affects both the takingoff location and slope of the breakthrough curve. It is explained as
the area between 1 and the corrected dimensionless mole fraction
in the breakthrough curve. The value of X may also be quickly
estimated with reasonable accuracy when the corrected dimensionless mole fraction reaches 0.5, provided that Da is large
enough. X also links this study to the author’s previous work on
thermodynamic analysis of adsorption-reaction system.
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