# Simulation of Binary CO2/CH4 Mixture Breakthrough Profiles in MIL-53 (Al)

Authors: Luis Fernando Gomez; Renju Zacharia; Pierre Bénard; Richard Chahine
Journal: Journal of Nanomaterials
 (2015)

---

## Abstract

MIL-53 (Al) aluminum terephthalate, a commercial metal-organic framework, has been studied as a potential candidate for pressure swing adsorption separation of CO2/CH4 binary mixtures. Pure gas isotherms of CH4 and CO2 measured over 0–6 MPa and at room temperature are fitted with the Dubinin-Astakhov (D-A) model. The D-A model parameters are used in the Doong-Yang Multicomponent adsorption model to predict the binary mixture isotherms. A one-dimensional multicomponent adsorption breakthrough model is then used to perform a parametric study of the effect of adsorbent particle diameter, inlet pressures, feed flow rates, and feed compositions on the breakthrough performance. Commercial MIL-53 with a particle diameter of 20 μm renders high tortuous flow; therefore it is less effective for separation. More effective separation can be achieved if MIL-53 monoliths of diameters above 200 μm are used. Faster separation is possible by increasing the feed pressure or if the starting compositions are richer in CO2. More CH4 is produced per cycle at higher feed pressures, but the shortened time at higher pressures can result in the reduction of the CH4 purity.


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## Body

## 1. Introduction

Pressure swing adsorption (PSA) is a well-established gas purification process which has already been employed in multiple applications, including hydrogen separation and purification [1–3], air purification [4], raw natural gas purification, and CO2 capture [5, 6]. Due to its potential to purify CH4 from CO2/CH4 mixtures especially in small and medium industrial scales, PSA techniques are currently being extended to new areas like methane purification from biogas and landfill gas [7–10]. For zeolites [11] or activated carbon [12], which are the most commonly used adsorbent materials for PSA purification of biogas/landfill gas, the adsorbent regeneration is still difficult and energy consuming, leading to lower productivity and higher expenses [9, 13].Discovery of novel nanoporous materials like metal-organic frameworks (MOFs), zeolitic imidazolate frameworks (ZIFs), and covalent frameworks (COFs) has started a new chapter in adsorbent search for applications including gas storage, drug delivery [14], carriers for nanomaterials [15, 16], and gas separation and purification [17]. Due to their high porosity and large CO2 adsorption capacities, MOFs are specifically suitable for adsorptive separation and purification of CH4 from CO2/CH4 mixtures, such as those from biogas or natural gas sources. Different types of extended framework materials have been reported with high adsorption capacities for CH4, CO2, and H2 [18–26].Breakthrough performance of adsorbent columns is an important characteristic required to evaluate the potential of adsorbents for PSA applications. There have been a few experimental measurements of breakthrough performance of MOF adsorbents for separating CO2/CH4 mixtures [17, 19]. Heymans et al. used experiments and simulations to predict breakthrough performance of MIL-53 (Al) for acidic gas separation from CH4/CO2 mixture [27]. Even though they used both the experiments and simulations, their studies were restricted to a single gas mixture composition (50 : 50) at a single pressure of 1.06 bar and no parametric effects of process variables such as feed pressure, composition, and feed flow were considered. Investigating parametric effects of the process variables and their influence on the separation process is necessary to perform preliminary screening of novel adsorbents like MOFs. This work is a novel attempt in that direction. Furthermore, the isotherms of MIL-53 (Al) sample used in our studies do not have breathing behavior as for the MIL-53 (Al) reported in previous studies [17, 19]. Therefore, our work offers a comparison of the breakthrough characteristics of MIL-53 (Al) with different structural flexibility. In order to calculate the multicomponent adsorption isotherms in this work, we have used an analytical model, namely, Doong-Yang Model. This model has already been used previously by our group and has been shown to fit the experimental isotherms rather well. Use of analytical models, such as DYM, allows one to easily implement the model in the computational fluid dynamics calculations of the breakthrough performance.In this work, we present a systematic study of the parametric effects of an aluminum terephthalate MOF-MIL-53 (Al) particle size, feed pressure, flow rates, and composition of CO2/CH4 binary gas mixtures on the dynamic breakthrough separation process. This parametric study is performed using the computational fluid dynamics simulation platform COMSOL Multiphysics. This paper is organized as follows. First, we present the characterization of MIL-53 framework used in this work. Then we present the experimentally measured pure gas CO2 and CH4 isotherms on MIL-53 which are used to predict the binary mixture adsorption isotherms. A parametric study of the effects of adsorbent particle size, feed pressures, gas flow rates, and gas composition on the breakthrough profiles of CO2/CH4 mixtures and on the adsorbent bed temperatures is finally presented.


## 2. Experimental

Particle size distribution, pure gas isotherms, and adsorption isosteric heat of commercial MIL-53 (Al) aluminum terephthalate C8H5AlO5 were measured using standard methods. A JEOL Scanning Electron Microscope (JSM-5510) was used to measure the particle diameter and estimate the diameter distribution of the MOF particles. The pure gas adsorption isotherms of CO2 and CH4 were performed at room temperature in the range of pressures between 0 and 6 MPa using Sievert’s volumetric gas adsorption system. The BET specific surface area, pore size distribution, and other pore characteristics were measured by adsorbing N2 at 77 K in a Micromeritics ASAP 2020 analyzer. Gases used for the measurements are high purity gases (99.999%) supplied by Praxair Canada. Isosteric heat of adsorption and heat capacity of MIL-53 were measured using a coupled volumetric-calorimetric system. Powder X-ray diffractometer (Bruker D8 FOCUS, Cu Kα) was used to examine the crystalline structure of the MIL-53. The coefficients of diffusions for equimolar binary mixtures of CH4 and CO2 were measured using an isotope exchange system.


## 3. Theory

A one-dimensional multicomponent adsorption breakthrough model based on the approach proposed by Casas et al. is presented here [29]. This model accounts for the mass and heat transfer inside a nonisothermal adsorbent column filled with MIL-53, the heat transfer in the fluid and in the gas-phase, and the conductive and convective heat transfer between the column wall and the surroundings. The following restrictions are assumed in the model: ambient temperature is considered to be constant, radial gradients in the column are negligible, mass transfer coefficients and isosteric heat of adsorption and heat capacities of the solid phase and of the wall are constants, and axial conductivity on the wall of the column is assumed to be zero. The adsorptive mass transfer rate is expressed in the form of a linear driving force (LDF) model. This breakthrough model was extensively validated by different authors for the PSA applications with good results [5, 6, 29, 30].


### 3.1. Mass and Energy Balance

The total mass balance in the breakthrough column is given by(1)
ϵ
t
∂
c
∂
t
+
∂
u
c
∂
z
=
-
ρ
b
∑
i
=
1
n
∂
q
i
∂
t
.Mass balance for each species is given by(2)
ϵ
t
∂
c
i
∂
t
+
∂
u
c
i
∂
z
+
ρ
b
∂
q
i
∂
t
-
ϵ
b
∂
∂
z
D
L
c
∂
y
i
∂
z
=
0
,where c is the total concentration of the fluid phase, c
i the fluid phase concentration for each component, q
i the adsorbed phase concentration for each species, u the superficial gas velocity, ϵ
t the total porosity, ϵ
b the bed porosity, ρ
b the column bulk density, D
L the axial dispersion coefficient (for all components), t the time, z the longitudinal coordinate on the column, y
i the gas-phase mole fraction of the ith component, and n the number of components in the gas mixture.The pressure drop is calculated from Darcy’s law, where pressure gradient, velocity, and porosity are correlated as(3)
u
=
-
κ
μ
∇
p
;
∇
p
=
-
150
μ
1
-
ϵ
b
2
ϵ
b
3
d
p
2
u
.Here, κ is the permeability of the material, μ the dynamic viscosity, and d
p the particle diameter.The time-dependent variation of the absolute adsorption is described using the LDF adsorption kinetics model:(4)
∂
q
i
∂
t
=
k
i
q
i
∗
-
q
i
,where k
i is the mass transfer coefficient, q
∗ the solid phase concentration at equilibrium pressure, and q the solid phase concentration at time t. To describe the adsorption isotherms, we use the D-A isotherm model. The absolute adsorption in the D-A model is given by(5)
q
i
=
q
m
a
x
exp
⁡
-
R
T
ε
ln
⁡
P
s
p
n
.Here, q
i is the absolute adsorption of ith component of the mixture, q
m
a
x the maximum absolute adsorption corresponding to saturation pressure P
s, ε the characteristic energy of adsorption, n the measure of the pore heterogeneity of the microporous material [31–33], R the ideal gas constant, T the temperature, and p the gas pressure. The measured excess adsorptions of pure gases are converted into absolute adsorption using [34] (6)
n
a
b
s
=
n
e
x
c
1
-
ρ
g
a
s
/
ρ
s
a
t
,where n
a
b
s, n
e
x
c, ρ
g
a
s, and ρ
s
a
t are the absolute adsorption, the excess of adsorption, and the density of the gas phase and of the adsorbed phase, respectively.


### 3.2. State Equation (EOS)

In the range of temperature and pressures considered in this study, we note that the compressibility factors of CO2/CH4 gas mixtures (reported in the NIST REFPROP Standard Reference Database [35]) are between 0.9 and 1. Hence to describe the state of the gases, we use the equation of state of an ideal gas: (7)
c
i
=
p
y
i
R
T
.


### 3.3. Porosity

The porosities are determined using(8)
ϵ
t
=
1
-
ρ
b
ρ
s
k
,
ϵ
b
=
ϵ
t
-
ϵ
m
i
,
ϵ
m
i
=
V
a
×
ρ
b
,where ρ
b is the bulk density, ρ
s
k the skeletal density, ϵ
b the bed porosity, ϵ
m
i the microporosity, ϵ
t the total porosity, and V
a the micropore volume. The skeleton density ρ
s
k is determined using the helium expansion method in standard Sievert’s apparatus, ρ
b is the bulk density measured using ASTM standard procedure (ASTM D 2854-96), and the micropore volume V
a is obtained from the measurements of the pore size distribution with nitrogen at 77 K in an ASAP instrument.For describing the multicomponent adsorption isotherms, we use the Doong-Yang Model. The DYM is based on the pure gas isotherms D-A model parameters reported in Table3. The DYM adsorption model for a multicomponent mixture is given by(9)
V
=
V
0
exp
⁡
-
R
T
ε
ln
⁡
P
s
p
n
.For binary gas adsorption, the respective amount of each adsorbed component is given by (10)
V
1
=
V
01
-
V
2
exp
⁡
-
R
T
ε
1
ln
⁡
P
s
1
p
1
n
1
,
V
2
=
V
02
-
V
1
exp
⁡
-
R
T
ε
2
ln
⁡
P
s
2
p
2
n
2
.Equations (10) can be written as(11)
V
1
=
A
1
V
01
-
V
02
A
2
1
-
A
1
A
2
,
V
2
=
A
2
V
02
-
V
01
A
1
1
-
A
1
A
2by substituting(12)
A
1
=
exp
⁡
-
R
T
ε
1
ln
⁡
P
s
1
p
1
n
1
,
A
2
=
exp
⁡
-
R
T
ε
2
ln
⁡
P
s
2
p
2
n
2
.In (10)-(11) V
0
i is the limiting micropore volume of component i and V
i the volumetric amount of adsorbate for each component. For converting the experimental isotherms between molar and volume units, the following expressions are used:(13)
q
i
∗
=
V
a
V
,
(14a)
V
a
=
V
l
s
,
n
b
p
,
T
<
T
n
b
p
,
(14b)
V
a
=
V
c
-
V
c
-
V
l
s
,
n
b
p
T
c
-
T
T
c
-
T
n
b
p
,
T
n
b
p
≤
T
<
T
c
,
(14c)
V
a
=
V
c
T
r
0.6
,
T
>
T
c
.Further details of DYM are available in Doong and Yang, Rege et al. [4, 36], and the authors’ previous work [34].The energy balance equation for the column (fluid and the solid phase) is given by the following equation:(15)
ϵ
t
C
g
+
ρ
b
C
s
+
ρ
b
C
a
d
s
∂
T
∂
t
-
ϵ
t
∂
p
∂
t
+
u
C
g
∂
T
∂
z
-
ρ
b
∑
-
Δ
H
i
∂
q
i
∂
t
+
4
h
L
d
i
T
-
T
w
-
ϵ
b
∂
∂
z
K
L
∂
T
∂
z
=
0
,where C
g is the heat capacity of the gas, C
s the heat capacity of the solid, C
a
d
s the heat capacity of the adsorbed phase, Δ
H
i the isosteric heat of adsorption for each component, h
L the heat transfer coefficient (inside the column + wall), K
L the axial thermal conductivity in the fluid phase, T the temperature inside the column, T
w the temperature of the column’s external wall, and d
i the inner diameter of the tube. The energy balance is also defined for the heat exchange between the wall and the surroundings, where the effects of conduction between the column and the ambient are considered. This is given by(16)
∂
T
w
∂
t
=
2
π
C
w
A
w
h
L
R
i
T
-
T
w
-
H
w
d
o
2
T
w
-
T
a
m
b
+
1
C
w
∂
∂
z
K
w
∂
T
w
∂
Z
,where H
w is the heat transfer coefficient between the wall and the surroundings, C
w the heat capacity of the column wall, A
w the area of the cross section of the column, and d
o the column’s external diameter.


### 3.4. Boundary and Initial Conditions

The boundary conditions used in the model are described below.Inlet boundary conditions of the system (i.e., atz
=
0) are(17)
u
c
i
n
l
e
t
=
u
c
,
u
c
i
i
n
l
e
t
=
u
c
i
-
ε
b
D
L
c
∂
y
i
∂
z
,
u
C
g
T
i
n
l
e
t
=
u
i
n
l
e
t
c
g
,
i
n
l
e
t
T
-
ε
b
K
L
∂
T
∂
z
.Outlet boundary conditions (i.e., at z
=
L) are(18)
∂
c
i
∂
z
=
0
,
∂
T
∂
z
=
0
,
p
=
p
o
u
t
l
e
t
.Initial conditions at t
=
0 for 0
≤
z
≤
L are (19)
c
C
O
2
=
c
C
H
4
=
0
,
T
w
=
T
a
m
b
,
T
=
T
i
n
i
t
.The heat capacities of the fluid and the adsorbed phase in (15) are defined using(20)
C
g
=
∑
i
N
c
i
C
g
,
i
m
o
l
,
i
=
1
,
…
,
N
,
C
a
d
s
=
∑
i
N
q
i
C
g
,
i
m
o
l
,
i
=
1
,
…
,
N
,where the specific heat capacities i
C
g
,
i
m
o
l are calculated as an average over a range of temperatures from ambient temperature to the highest temperature reached in the adsorption column for each pressure under study. This assumption will add also more simplicity to the model, without affecting the accuracy of the results [29]. Note that the concentration and heat capacity of the fluid and of the adsorbed phase are temperature-dependent quantities.The heat transfer coefficienth
L is obtained from the Nusselt number, N
u: (21)
N
u
≡
h
L
2
R
i
K
L
=
η
1
R
e
η
2
exp
⁡
-
4.6
d
p
2
R
i
,where(22)
R
e
=
ρ
u
d
p
μ
.In (21) and (22), R
i is the internal radius of the column, K
L the axial thermal conductivity in the fluid phase, and R
e the Reynolds number. The values for η
1 and η
2 are calculated from the correlation of heat transfer coefficients for gases through packed tubes [37].The thermal conductivity is estimated using(23)
K
L
=
D
L
C
g
,where D
L is the axial dispersion coefficient which is calculated with the Edwards-Richardson correlation [38]:(24)
D
L
=
0.73
D
m
+
0.5
d
p
u
1
+
9.49
×
D
m
/
d
p
u
,where u is the velocity, D
m is the molecular diffusion coefficient calculated according to the Fuller method [39], and d
p is the particle diameter. The heat transfer coefficient between the wall and surrounding is calculated using (25)
h
w
=
a
d
o
3
m
-
1
T
m
a
x
-
T
m
i
n
1
/
4
,where the heat transfer parameters a and m are reported in the literature for free convection cases [40]. T
m
a
x is the maximum temperature during the adsorption process and T
m
i
n is assumed to be room temperature.The system of mass and energy balance partial differential equations is solved using the commercial software platform COMSOL Multiphysics using modules for heat transfer of porous media, heat transfer of fluids, transport of diluted species, and Darcy’s law. The default equations of COMSOL modules are redefined according to the aforementioned system of equations. Table1 lists the model parameters used in our study. Column properties used are typical values of stainless steel.Table 1
Model parameters.
Parameter
Value
Description
ρ
b
385 [kg/m3]
Bulk density of the adsorbent bed
R
8.314 [J/(mol K)]
Ideal gas constant
η
1
3.5
Nusselt number parameter
η
2
0.7
Nusselt number parameter
k
CO
2
0.8 [1/s]
CO2 lumped mass transfer coefficient
k
CH
4
0.5 [1/s]
CH4 lumped mass transfer coefficient
h
w
4.5 [J/(m2 s K)]
Heat transfer coefficient (lumping wall + heating)
D
m
1.12 × 10−6 [m2/s]
Molecular diffusion coefficient
d
p
5 × 10−6–500 × 10−6 [m]
Particle diameter
ε
to
0.719
Overall void fraction
C
s
845 [J/(kg K)]
Heat capacity of the solid
ε
b
0.573
Bed void fraction
Δ
HCO2
−23600 [J/mol]
Heat of adsorption of CO2
Δ
HCH4
−15800 [J/mol]
Heat of adsorption of CH4
R
i
3.5 [mm]
Inner column radius
R
o
4.5 [mm]
Outer column radius (lumped)
m
wCH4
16 [g/mol]
Molar mass of CH4
m
wCO2
44 [g/mol]
Molar mass of CO2
C
gCO2
[J/(mol K)]‡
Specific heat capacity of CO2
C
gCH4
[J/(mol K)]‡
Specific heat capacity of CH4
ρCO2
[kg/m3]‡
CO2 fluid phase density
ρCH4
[kg/m3]‡
CH4 fluid phase density
C
w
4 × 106 [J/(Km3)]
Lumped heat capacity of the wall
A
w
3.1416 × (R
o
2
-
R
i
2)
Cross section of the column wall
T
amb
294.15 [K]
Ambient temperature
T
in
294.15 [K]
Temperature at the inlet
P
in
0.2–2.5 [MPa]
Inlet fluid pressure
V
ini
0.013 [m/s]
Inlet velocity
‡Properties are determined from NIST REFPROP as functions of the pressure and temperature at the inlet.


## 3.1. Mass and Energy Balance

The total mass balance in the breakthrough column is given by(1)
ϵ
t
∂
c
∂
t
+
∂
u
c
∂
z
=
-
ρ
b
∑
i
=
1
n
∂
q
i
∂
t
.Mass balance for each species is given by(2)
ϵ
t
∂
c
i
∂
t
+
∂
u
c
i
∂
z
+
ρ
b
∂
q
i
∂
t
-
ϵ
b
∂
∂
z
D
L
c
∂
y
i
∂
z
=
0
,where c is the total concentration of the fluid phase, c
i the fluid phase concentration for each component, q
i the adsorbed phase concentration for each species, u the superficial gas velocity, ϵ
t the total porosity, ϵ
b the bed porosity, ρ
b the column bulk density, D
L the axial dispersion coefficient (for all components), t the time, z the longitudinal coordinate on the column, y
i the gas-phase mole fraction of the ith component, and n the number of components in the gas mixture.The pressure drop is calculated from Darcy’s law, where pressure gradient, velocity, and porosity are correlated as(3)
u
=
-
κ
μ
∇
p
;
∇
p
=
-
150
μ
1
-
ϵ
b
2
ϵ
b
3
d
p
2
u
.Here, κ is the permeability of the material, μ the dynamic viscosity, and d
p the particle diameter.The time-dependent variation of the absolute adsorption is described using the LDF adsorption kinetics model:(4)
∂
q
i
∂
t
=
k
i
q
i
∗
-
q
i
,where k
i is the mass transfer coefficient, q
∗ the solid phase concentration at equilibrium pressure, and q the solid phase concentration at time t. To describe the adsorption isotherms, we use the D-A isotherm model. The absolute adsorption in the D-A model is given by(5)
q
i
=
q
m
a
x
exp
⁡
-
R
T
ε
ln
⁡
P
s
p
n
.Here, q
i is the absolute adsorption of ith component of the mixture, q
m
a
x the maximum absolute adsorption corresponding to saturation pressure P
s, ε the characteristic energy of adsorption, n the measure of the pore heterogeneity of the microporous material [31–33], R the ideal gas constant, T the temperature, and p the gas pressure. The measured excess adsorptions of pure gases are converted into absolute adsorption using [34] (6)
n
a
b
s
=
n
e
x
c
1
-
ρ
g
a
s
/
ρ
s
a
t
,where n
a
b
s, n
e
x
c, ρ
g
a
s, and ρ
s
a
t are the absolute adsorption, the excess of adsorption, and the density of the gas phase and of the adsorbed phase, respectively.


## 3.2. State Equation (EOS)

In the range of temperature and pressures considered in this study, we note that the compressibility factors of CO2/CH4 gas mixtures (reported in the NIST REFPROP Standard Reference Database [35]) are between 0.9 and 1. Hence to describe the state of the gases, we use the equation of state of an ideal gas: (7)
c
i
=
p
y
i
R
T
.


## 3.3. Porosity

The porosities are determined using(8)
ϵ
t
=
1
-
ρ
b
ρ
s
k
,
ϵ
b
=
ϵ
t
-
ϵ
m
i
,
ϵ
m
i
=
V
a
×
ρ
b
,where ρ
b is the bulk density, ρ
s
k the skeletal density, ϵ
b the bed porosity, ϵ
m
i the microporosity, ϵ
t the total porosity, and V
a the micropore volume. The skeleton density ρ
s
k is determined using the helium expansion method in standard Sievert’s apparatus, ρ
b is the bulk density measured using ASTM standard procedure (ASTM D 2854-96), and the micropore volume V
a is obtained from the measurements of the pore size distribution with nitrogen at 77 K in an ASAP instrument.For describing the multicomponent adsorption isotherms, we use the Doong-Yang Model. The DYM is based on the pure gas isotherms D-A model parameters reported in Table3. The DYM adsorption model for a multicomponent mixture is given by(9)
V
=
V
0
exp
⁡
-
R
T
ε
ln
⁡
P
s
p
n
.For binary gas adsorption, the respective amount of each adsorbed component is given by (10)
V
1
=
V
01
-
V
2
exp
⁡
-
R
T
ε
1
ln
⁡
P
s
1
p
1
n
1
,
V
2
=
V
02
-
V
1
exp
⁡
-
R
T
ε
2
ln
⁡
P
s
2
p
2
n
2
.Equations (10) can be written as(11)
V
1
=
A
1
V
01
-
V
02
A
2
1
-
A
1
A
2
,
V
2
=
A
2
V
02
-
V
01
A
1
1
-
A
1
A
2by substituting(12)
A
1
=
exp
⁡
-
R
T
ε
1
ln
⁡
P
s
1
p
1
n
1
,
A
2
=
exp
⁡
-
R
T
ε
2
ln
⁡
P
s
2
p
2
n
2
.In (10)-(11) V
0
i is the limiting micropore volume of component i and V
i the volumetric amount of adsorbate for each component. For converting the experimental isotherms between molar and volume units, the following expressions are used:(13)
q
i
∗
=
V
a
V
,
(14a)
V
a
=
V
l
s
,
n
b
p
,
T
<
T
n
b
p
,
(14b)
V
a
=
V
c
-
V
c
-
V
l
s
,
n
b
p
T
c
-
T
T
c
-
T
n
b
p
,
T
n
b
p
≤
T
<
T
c
,
(14c)
V
a
=
V
c
T
r
0.6
,
T
>
T
c
.Further details of DYM are available in Doong and Yang, Rege et al. [4, 36], and the authors’ previous work [34].The energy balance equation for the column (fluid and the solid phase) is given by the following equation:(15)
ϵ
t
C
g
+
ρ
b
C
s
+
ρ
b
C
a
d
s
∂
T
∂
t
-
ϵ
t
∂
p
∂
t
+
u
C
g
∂
T
∂
z
-
ρ
b
∑
-
Δ
H
i
∂
q
i
∂
t
+
4
h
L
d
i
T
-
T
w
-
ϵ
b
∂
∂
z
K
L
∂
T
∂
z
=
0
,where C
g is the heat capacity of the gas, C
s the heat capacity of the solid, C
a
d
s the heat capacity of the adsorbed phase, Δ
H
i the isosteric heat of adsorption for each component, h
L the heat transfer coefficient (inside the column + wall), K
L the axial thermal conductivity in the fluid phase, T the temperature inside the column, T
w the temperature of the column’s external wall, and d
i the inner diameter of the tube. The energy balance is also defined for the heat exchange between the wall and the surroundings, where the effects of conduction between the column and the ambient are considered. This is given by(16)
∂
T
w
∂
t
=
2
π
C
w
A
w
h
L
R
i
T
-
T
w
-
H
w
d
o
2
T
w
-
T
a
m
b
+
1
C
w
∂
∂
z
K
w
∂
T
w
∂
Z
,where H
w is the heat transfer coefficient between the wall and the surroundings, C
w the heat capacity of the column wall, A
w the area of the cross section of the column, and d
o the column’s external diameter.


## 3.4. Boundary and Initial Conditions

The boundary conditions used in the model are described below.Inlet boundary conditions of the system (i.e., atz
=
0) are(17)
u
c
i
n
l
e
t
=
u
c
,
u
c
i
i
n
l
e
t
=
u
c
i
-
ε
b
D
L
c
∂
y
i
∂
z
,
u
C
g
T
i
n
l
e
t
=
u
i
n
l
e
t
c
g
,
i
n
l
e
t
T
-
ε
b
K
L
∂
T
∂
z
.Outlet boundary conditions (i.e., at z
=
L) are(18)
∂
c
i
∂
z
=
0
,
∂
T
∂
z
=
0
,
p
=
p
o
u
t
l
e
t
.Initial conditions at t
=
0 for 0
≤
z
≤
L are (19)
c
C
O
2
=
c
C
H
4
=
0
,
T
w
=
T
a
m
b
,
T
=
T
i
n
i
t
.The heat capacities of the fluid and the adsorbed phase in (15) are defined using(20)
C
g
=
∑
i
N
c
i
C
g
,
i
m
o
l
,
i
=
1
,
…
,
N
,
C
a
d
s
=
∑
i
N
q
i
C
g
,
i
m
o
l
,
i
=
1
,
…
,
N
,where the specific heat capacities i
C
g
,
i
m
o
l are calculated as an average over a range of temperatures from ambient temperature to the highest temperature reached in the adsorption column for each pressure under study. This assumption will add also more simplicity to the model, without affecting the accuracy of the results [29]. Note that the concentration and heat capacity of the fluid and of the adsorbed phase are temperature-dependent quantities.The heat transfer coefficienth
L is obtained from the Nusselt number, N
u: (21)
N
u
≡
h
L
2
R
i
K
L
=
η
1
R
e
η
2
exp
⁡
-
4.6
d
p
2
R
i
,where(22)
R
e
=
ρ
u
d
p
μ
.In (21) and (22), R
i is the internal radius of the column, K
L the axial thermal conductivity in the fluid phase, and R
e the Reynolds number. The values for η
1 and η
2 are calculated from the correlation of heat transfer coefficients for gases through packed tubes [37].The thermal conductivity is estimated using(23)
K
L
=
D
L
C
g
,where D
L is the axial dispersion coefficient which is calculated with the Edwards-Richardson correlation [38]:(24)
D
L
=
0.73
D
m
+
0.5
d
p
u
1
+
9.49
×
D
m
/
d
p
u
,where u is the velocity, D
m is the molecular diffusion coefficient calculated according to the Fuller method [39], and d
p is the particle diameter. The heat transfer coefficient between the wall and surrounding is calculated using (25)
h
w
=
a
d
o
3
m
-
1
T
m
a
x
-
T
m
i
n
1
/
4
,where the heat transfer parameters a and m are reported in the literature for free convection cases [40]. T
m
a
x is the maximum temperature during the adsorption process and T
m
i
n is assumed to be room temperature.The system of mass and energy balance partial differential equations is solved using the commercial software platform COMSOL Multiphysics using modules for heat transfer of porous media, heat transfer of fluids, transport of diluted species, and Darcy’s law. The default equations of COMSOL modules are redefined according to the aforementioned system of equations. Table1 lists the model parameters used in our study. Column properties used are typical values of stainless steel.Table 1
Model parameters.
Parameter
Value
Description
ρ
b
385 [kg/m3]
Bulk density of the adsorbent bed
R
8.314 [J/(mol K)]
Ideal gas constant
η
1
3.5
Nusselt number parameter
η
2
0.7
Nusselt number parameter
k
CO
2
0.8 [1/s]
CO2 lumped mass transfer coefficient
k
CH
4
0.5 [1/s]
CH4 lumped mass transfer coefficient
h
w
4.5 [J/(m2 s K)]
Heat transfer coefficient (lumping wall + heating)
D
m
1.12 × 10−6 [m2/s]
Molecular diffusion coefficient
d
p
5 × 10−6–500 × 10−6 [m]
Particle diameter
ε
to
0.719
Overall void fraction
C
s
845 [J/(kg K)]
Heat capacity of the solid
ε
b
0.573
Bed void fraction
Δ
HCO2
−23600 [J/mol]
Heat of adsorption of CO2
Δ
HCH4
−15800 [J/mol]
Heat of adsorption of CH4
R
i
3.5 [mm]
Inner column radius
R
o
4.5 [mm]
Outer column radius (lumped)
m
wCH4
16 [g/mol]
Molar mass of CH4
m
wCO2
44 [g/mol]
Molar mass of CO2
C
gCO2
[J/(mol K)]‡
Specific heat capacity of CO2
C
gCH4
[J/(mol K)]‡
Specific heat capacity of CH4
ρCO2
[kg/m3]‡
CO2 fluid phase density
ρCH4
[kg/m3]‡
CH4 fluid phase density
C
w
4 × 106 [J/(Km3)]
Lumped heat capacity of the wall
A
w
3.1416 × (R
o
2
-
R
i
2)
Cross section of the column wall
T
amb
294.15 [K]
Ambient temperature
T
in
294.15 [K]
Temperature at the inlet
P
in
0.2–2.5 [MPa]
Inlet fluid pressure
V
ini
0.013 [m/s]
Inlet velocity
‡Properties are determined from NIST REFPROP as functions of the pressure and temperature at the inlet.


## 4. Results and Discussions

### 4.1. Material Characterization

The XRD pattern of MIL-53 shown in Figure1(a) is similar to that of MIL-53 samples reported previously [41]. Results for the particle size and particle size distribution are shown in Figures 1(b) and 1(c). The particle size distribution histogram obtained using a bin width of 1 µm shows that most particles have diameters between 17 and 25 µm with a peak distribution at ~20 µm. Pore and surface characterization, densities, and porosities of MIL-53 are given in Table 2.Table 2
Pore and surface characterization, densities, and porosities of MIL-53.
BET surface area (m2/g)
984.2
Bulk density  (g/cm3)
0.385
Skeletal density (g/mL)
1.37
Micropore volume (cm3/g)
0.3787
Total porosity
0.719
Bed porosity
0.573Table 3
D-A model parameters for the adsorption of pure methane and carbon dioxide on MIL-53 (Al).
Parameter
CH4
CO2
n
max
⁡ [mol/kg]
6.84
11.16
ε  [J/mol]
9373
8812
P
s [MPa]
32.7
6.8
n
i
2.84
1.87Figure 1
Characterization of MIL-53. (a) Powder X-ray diffraction pattern. The inset shows the structure of nonbreathable MIL-53 [28]. (b) Particle size distribution histogram. (c) Representative SEM image of the material.
(a)
(b)
(c)Since no reported diffusion coefficients of CO2 and CH4 in MIL-53 are available yet, we used those available for MOF-5. These coefficients of diffusion k
i were measured for an equimolar mixture of CO2 and CH4 on MOF-5 using the isotope exchange technique [34]. Diffusion coefficients of CO2/CH4 on different MOFs (MIL-53, MIL-101, and Cu-BTC) are found to have similar order of magnitudes, so this approximation is not expected to cause significant errors [42, 43]. The mass transfer coefficients are listed in Table 1. The isosteric heat of CO2 and CH4 adsorption on MIL-53 is measured using a coupled volumetric-calorimetric system. The absolute adsorption required for the isosteric heat is obtained using Tóth’s adsorption model fit for the measured excess adsorption isotherms [44]. The specific heat capacity of MIL-53 was measured using a SETARAM calorimeter and is given in Table 1.


### 4.2. Pure and Mixed Gas Isotherms

Pure gas adsorption isotherms of methane and carbon dioxide on MIL-53 are given as symbols in Figure2. These measurements are made at 294.15 K for a pressure range between 0 and 6 MPa using a conventional Sieverts volumetric apparatus. The detailed description of the method is available from earlier works [34, 45].Figure 2
(a) Experimental pure gas isotherms of methane and carbon dioxide on MIL-53 measured at 294.15 K. (b) Equimolar binary gas isotherm comparison, Hamon et al., and DYM predictions.
(a)
(b)Doong and Yang Multicomponent (DYM) model is an empirical multicomponent adsorption model which has shown excellent predictive properties for multicomponent mixtures of CO2, CH4, and N2 on microporous adsorbents. We have used this model in the past to predict the isotherms of binary mixtures of CH4 and CO2 on MOF: Cu-BTC. DYM model is an extension of Dubinin-Astakhov analytical model, which accurately predicts the pure gas adsorption isotherms on microporous adsorbents over wide temperature and pressure ranges [4, 36]. One of the very important factors we need to consider when using the models is the ease of applicability of the models in computational fluid dynamics simulations. The parameters from the DYM/D-A models can be directly used to express the adsorptive mass source terms in the mass balance equation (q, (1)). Additionally, they provide an analytical expression for loading dependent-adsorption isosteric heat which can be easily implemented in the energy balance equation (Δ
H, q, (15)). This is unlike certain other models, such as multipotential theory of adsorption, which requires either the parameterization of the predicted isotherms or the use of iterative techniques within the CFD models [46, 47]. Both D-A model and DYM are based on the theory of micropore volume filling which postulates that adsorption in microporous adsorbent occurs by filling of the micropore volume.The pure gas isotherms are fitted with the D-A model and are given as lines in Figure2(a). The data are compared with CO2 and CH4 pure gas isotherms on isotypic MIL-53 (Cr) reported by Hamon et al. The structures of both Al and Cr variants of MIL-53 MOFs series are built up from similar infinite chains of corner-sharing MO4(OH)2 (M = Al3+, Cr3+) octahedra interconnected by the dicarboxylate groups. This results in a similar 3D metal-organic framework containing 1D diamond shaped channels. Isotherms of CO2 and CH4 on MIL-53 (Al) compare well with those on MIL-53 (Cr). This agrees well with earlier results on isotypic MIL-53 reported by Bourrelly et al. and Alhamami et al. [13, 48].Both pure gas isotherms are fitted with the D-A model with a standard error of estimate (SEE) of 1.05 for CO2 and of 0.626 for CH4. The corresponding fit parameters are presented in Table 3. The mixed gas isotherms on MIL-53 (Al) are constructed using the Doong-Yang Multicomponent isotherm model [36] using the pure gas isotherm regressions parameters. In Figure 2(b), the predicted binary adsorption isotherms are compared with the experimental equimolar binary adsorption isotherms on MIL-53 (Cr) measured by Hamon et al. Even though the isotherms cannot be quantitatively compared, they exhibit similar behavior for CH4 and CO2. We can conclude that our predictions are in agreement with the experimental data. The DYM isotherm equations are summarized in (9) to (14a), (14b), and (14c).The efficiency of MOF MIL-53 (Al) for the separation of a binary CH4/CO2 mixture can be analyzed and compared in terms of the sorption selectivity. The selectivity of ith component in a mixture of components i and j is defined on a molar basis as S
i
,
j
=
n
s
a
t
,
i
/
y
i
/
n
s
a
t
,
j
/
y
j. Here, we compare the selectivity of our sample to selectively remove CO2 from an equimolar CO2/CH4 mixture with the selectivities of other MOFs reported in the literature. In the pressure range below 0.5 MPa, the selectivity of our sample shown in Figure 3 decreases initially rapidly with pressure of about 0.1 MPa, after which it remains almost constant. The selectivity of MIL-53 reported by Hamon et al. on the other hand shows a step-like decrease, by a factor of ~3 at 0.6 MPa, after which it shows only a slight decrease [19]. The sample used by Hamon et al. showed two characteristic adsorption steps which were attributed to the breathing phenomenon. As the CO2 pressure increases, a step is observed at around 0.6 MPa [13] leading to larger uptake. This uptake is attributed to the change of MIL-53 from “narrow pore” to “large-pore” structure. On the other hand, the sample used in our work is a commercial material that shows no breathing phenomena. No drastic change in the selectivity is observed at around 0.6 MPa. Among all MOFs compared here, Cu-BTC [18] has the best selectivity at pressures above 0.2 MPa, while MOF-5 reported by Millward and Yaghi [21] has lowest selectivity. We conclude that MIL-53 (Al) offers a good separation for all ranges of pressure of up to 10 bar.Figure 3
Selectivity of MIL-53 (Al) towards CO2 at 303 K. For comparison purposes we included experimental data of Cu-BTC, MIL-53 (Cr), and MOF-5.


### 4.3. Validation of Breakthrough Curve Model

We start by validating the breakthrough model by applying it to simulate the breakthrough of a CO2/H2 mixture in an activated carbon column and comparing the model results with those reported by Casas et al. The model parameters and boundary and initial conditions required for validating the model are also obtained from Casas et al. [6, 29]. Figure 4 shows very good agreement between our validation results and those reported by Casas et al. The model has also been extensively validated experimentally and numerically by Casas et al., for different breakthrough curves cases of CO2/CH4 gas mixtures flowing through beds of activated carbon [29] and a hybrid MOF UiO-67/MCM-41 [6].Figure 4
Validation of breakthrough model by simulating the adsorption and separation of equimolar mixture of CH4 and CO2 in an activated carbon column. Symbols represent our results while the lines represent the experimental results reported by Casas et al. [29].


### 4.4. Parametric Study of Adsorbent Particle Size, Inlet Pressure, Gas Flow Rate, and Feed Composition on the Breakthrough

We used the validated model to study the effects of particle size, inlet pressure, feed composition, and gas flow rate on the breakthrough of CO2/CH4 gas mixtures through the MIL-53 adsorbent column. The inlet and wall temperatures are set to 294.15 K. A 25 cm column length is considered for all simulations. For monitoring the evolution of temperature in the bed, four axial positions at 5, 10, 15, and 20 cm from the inlet of the column are chosen.


#### 4.4.1. Effect of Particle Diameter

In order to study the effect of MIL-53 (Al) particle size on breakthrough performance, we considered particle diameters 20, 200, 300, 500, and 1000µm. Inlet pressure is fixed at 0.2 MPa and an equimolar CO2/CH4 mixture is fed at a rate of 30 mL/min. In general, for the simulations with particle diameters lower than 20 µm, we found that the numerical model presents some limitations. An examination of the mass balance shows that the numerical results start to deviate from the mass predicted by the local pressure. This perhaps arises due to the large pressure drop caused by smaller particles, which is consistent with the general recommendation to use particle sizes of the order of 1 mm to avoid large pressure drops in gas-phase separations [6, 7, 17]. Therefore, we present the results only for 200, 500, and 1000 μm. Based on the literature, we set the particle size to 500 μm to investigate the effects of inlet pressure, flow rates, and feed concentration in further sections.In left panels of Figure5, breakthrough times for Y
i
/
Y
0
=
10
%, where Y
i is the molar fraction of the component and Y
0 is the feed concentration, are found to be around 7.9 minutes when particle sizes are 200 and 500 μm, while they are 7.7 minutes when the particle size is 1000 μm. In the right panels of Figure 5, the evolution of temperature at the positions 5, 10, 15, and 20 cm from the inlet of the column is shown. As adsorption is an exothermic process, the resulting adsorption heat is released into the bed. This increases the column temperature as the gas fronts move from the inlet to the outlet. We simulate the temperature evolution at four axial positions in the column. The temperature rises to around 333 K when 200 and 500 μm particles are used, while for the 1000 μm particles the temperature rises up to 336 K. At each position two different temperature peaks are observed; the low temperature peak corresponds to the CH4 front and the higher temperature peak to the CO2 front. From the simulation results, we find that the CH4 front moves faster than CO2 front. The peaks shape is influenced by the mass and heat transfer parameters: the initial fast abrupt front indicates fast mass transfer, whereas the shape of the tail is controlled mainly by the heat transfer from the column to the environment. The latter one is responsible for the time required to reach the feed composition at the outlet of the column, once the CO2 breakthrough is noticed. Since the temperature of the column continues to decrease until it reaches the initial temperature, more CO2 is adsorbed which finally results in a CO2 flat front.Figure 5
Breakthrough curves for an equimolar CO2/CH4 mixture, column length 25 cm, inlet pressure 0.2 MPa, for 3 different particle diameters. For simulation parameters refer to Table 1.Larger particles can be prepared either by mechanically compacting pristine MOFs to monoliths or by applying a binder, such as polyvinyl alcohol (PVA) or expanded natural graphite (ENG). Depending on the activation temperature, the preparation of monoliths by the addition of binder will cause partial pore blocking. The blocked pores reduce the adsorption capacity by as much as 19% of pristine powder MOF material. But this has minimum impact on the overall pore size distribution [17]. Binderless mechanical compaction of MOFs on the other hand causes partial collapse of frameworks, which reduces the sorption capacity by ~15% [49]. Our results are in agreement with Grande who recommends using the pellets instead of powder materials for efficient PSA separation [50].


#### 4.4.2. Effect of the Inlet Pressure

In order to study the effect of the inlet pressure on the breakthrough curves, we set the inlet pressure to 0.5, 1, and 2.5 MPa. Figure6 displays the breakthrough and temperature profiles for different inlet pressures. The particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 6, the breakthrough time decreases with increasing feed pressure. The breakthrough times of 5, 3.3, and 1.9 minutes are obtained with the feed pressures 0.5, 1, and 2.5 MPa, respectively. Furthermore, the higher the feed pressure, the higher the temperature along the column, ~358, 388, and 444 K, respectively, for 0.5, 1, and 2.5 MPa. As gas with higher inlet pressure flows through the bed, larger amounts of gases are adsorbed, leading to higher amounts of adsorption heat released into the bed.Figure 6
Effect of feed pressure on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


#### 4.4.3. Effect of the Mass Flow Rate

In order to study the impact of the mass flow rates on the breakthrough curves, we set the mass flow rate to 10, 25, and 50 mL/min. Figure7 displays the breakthrough and the temperature profiles for different mass flow rates. As in the case of previous simulations, the particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 7, the breakthrough time decreases with increasing of the mass flow rate. For mass flow rates of 10, 25, and 50 mL/min, the breakthrough times are 10.65, 4, and 1.9 minutes. Also, the higher the mass flow rate, the higher the temperature along the column, ~375, 385, and 392 K for 10, 25, and 50 mL/min. Gas with the higher mass flow rate leads to larger amount of adsorption. This leads to higher amounts of adsorption heat released into the bed.Figure 7
Effect of the mass flow rate on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


#### 4.4.4. Effect of the Feed Concentration

The influence of the CO2 concentration on the MIL-53 adsorption kinetics is studied for three CO2 concentrations: 25%, 50%, and 75% in the CO2/CH4 mixture. For each composition, we also considered three feed pressures: 0.2, 0.5, and 2.5 MPa. Figure 8 displays the breakthrough and temperature profiles corresponding to each composition and pressure. The breakthrough point for 25% CO2 composition at a 0.2 MPa inlet pressure appears at 9.8 minutes, while shorter breakthrough times of 7.8 and 6.7 minutes are observed when the CO2 feed composition is increased to 50 and 75%. In other words, we see that the larger CO2 concentration in the feed gas mixture accelerates the breakthrough time [29, 51]. Similar behavior is also observed for the 0.5 and 2.5 MPa feed pressures. Larger concentration difference between the compositions results in faster saturation of the adsorbent with one component which eventually leads to shorter breakthrough times. The lowest breakthrough time of as short as 1.55 minutes is observed for the highest feed pressure (2.5 MPa) and highest CO2 molar concentrations (75%). The behavior of the temperature evolution on the other hand shows an increase at increasing the feed concentration and the feed pressure which is attributed to the larger amount of gases adsorbed.Figure 8
Breakthrough and temperature profiles when feed pressures and CO2/CH4 mixture compositions are varied. The range of feeding pressures is between 0.2 and 2.5 MPa and a range of concentration is between 25 and 75% mole fraction of CO2.The different breakthrough times for CH4 and CO2 obtained with different feed pressures and molar compositions directly affect the amount of pure CH4 produced in each PSA cycle. The amount of pure CH4 produced in a cycle can be calculated from the outlet flow rate and time between the onset of the flow and breakthrough. Note that the reduction of adsorption capacity due to pelletization should also be accounted for while calculating the amount of pure CH4 produced in each cycle. Based on the reported adsorption capacities of monoliths, we used 15% reduction factor to calculate the amount of pure CH4 produced. As seen in Figure 9, at higher feed pressure, more CH4 is produced per cycle, even though each cycle lasts less than that for lower feed pressures (from left to right of Figure 8). On the other hand, the amount of CH4 produced at the specified purity with respect to the feed composition decreases with the increasing pressure. This means that more CH4 remains in the column by the time CO2 breaks through, as it is confirmed in a similar case for a CO2/H2 gas mixture [6, 29]. In a continuous process this is circumvented by adjusting the cycle time in such a way that the CH4 loss is minimized [29].Figure 9
Amount of pure CH4 obtained in a breakthrough prediction for a feed mixture containing equimolar CO2/CH4 mixture.The separation capacity of MIL-53 (Al) for CO2 and CH4 is between 7.8 at 0.1 MPa and 7.1 at 1.5 MPa at 294.15 K based on the selectivity correlation S
i
,
j
=
n
s
a
t
,
i
/
y
i
/
n
s
a
t
,
j
/
y
j [34]. These values are consistent with the values reported by Finsy et al. on the separation of an equimolar CH4/CO2 mixture at 303 K in a packed column with MIL-53 (Al, PVA) pellets containing 13 wt% PVA binder [17]. Selectivities calculated from pure component isotherms on the 13X zeolite [11] and the activated carbon material Norit R1 Extra [12] are 2 and 2.7 at 1 MPa compared with 5.5 for MIL-53 (Al).


## 4.1. Material Characterization

The XRD pattern of MIL-53 shown in Figure1(a) is similar to that of MIL-53 samples reported previously [41]. Results for the particle size and particle size distribution are shown in Figures 1(b) and 1(c). The particle size distribution histogram obtained using a bin width of 1 µm shows that most particles have diameters between 17 and 25 µm with a peak distribution at ~20 µm. Pore and surface characterization, densities, and porosities of MIL-53 are given in Table 2.Table 2
Pore and surface characterization, densities, and porosities of MIL-53.
BET surface area (m2/g)
984.2
Bulk density  (g/cm3)
0.385
Skeletal density (g/mL)
1.37
Micropore volume (cm3/g)
0.3787
Total porosity
0.719
Bed porosity
0.573Table 3
D-A model parameters for the adsorption of pure methane and carbon dioxide on MIL-53 (Al).
Parameter
CH4
CO2
n
max
⁡ [mol/kg]
6.84
11.16
ε  [J/mol]
9373
8812
P
s [MPa]
32.7
6.8
n
i
2.84
1.87Figure 1
Characterization of MIL-53. (a) Powder X-ray diffraction pattern. The inset shows the structure of nonbreathable MIL-53 [28]. (b) Particle size distribution histogram. (c) Representative SEM image of the material.
(a)
(b)
(c)Since no reported diffusion coefficients of CO2 and CH4 in MIL-53 are available yet, we used those available for MOF-5. These coefficients of diffusion k
i were measured for an equimolar mixture of CO2 and CH4 on MOF-5 using the isotope exchange technique [34]. Diffusion coefficients of CO2/CH4 on different MOFs (MIL-53, MIL-101, and Cu-BTC) are found to have similar order of magnitudes, so this approximation is not expected to cause significant errors [42, 43]. The mass transfer coefficients are listed in Table 1. The isosteric heat of CO2 and CH4 adsorption on MIL-53 is measured using a coupled volumetric-calorimetric system. The absolute adsorption required for the isosteric heat is obtained using Tóth’s adsorption model fit for the measured excess adsorption isotherms [44]. The specific heat capacity of MIL-53 was measured using a SETARAM calorimeter and is given in Table 1.


## 4.2. Pure and Mixed Gas Isotherms

Pure gas adsorption isotherms of methane and carbon dioxide on MIL-53 are given as symbols in Figure2. These measurements are made at 294.15 K for a pressure range between 0 and 6 MPa using a conventional Sieverts volumetric apparatus. The detailed description of the method is available from earlier works [34, 45].Figure 2
(a) Experimental pure gas isotherms of methane and carbon dioxide on MIL-53 measured at 294.15 K. (b) Equimolar binary gas isotherm comparison, Hamon et al., and DYM predictions.
(a)
(b)Doong and Yang Multicomponent (DYM) model is an empirical multicomponent adsorption model which has shown excellent predictive properties for multicomponent mixtures of CO2, CH4, and N2 on microporous adsorbents. We have used this model in the past to predict the isotherms of binary mixtures of CH4 and CO2 on MOF: Cu-BTC. DYM model is an extension of Dubinin-Astakhov analytical model, which accurately predicts the pure gas adsorption isotherms on microporous adsorbents over wide temperature and pressure ranges [4, 36]. One of the very important factors we need to consider when using the models is the ease of applicability of the models in computational fluid dynamics simulations. The parameters from the DYM/D-A models can be directly used to express the adsorptive mass source terms in the mass balance equation (q, (1)). Additionally, they provide an analytical expression for loading dependent-adsorption isosteric heat which can be easily implemented in the energy balance equation (Δ
H, q, (15)). This is unlike certain other models, such as multipotential theory of adsorption, which requires either the parameterization of the predicted isotherms or the use of iterative techniques within the CFD models [46, 47]. Both D-A model and DYM are based on the theory of micropore volume filling which postulates that adsorption in microporous adsorbent occurs by filling of the micropore volume.The pure gas isotherms are fitted with the D-A model and are given as lines in Figure2(a). The data are compared with CO2 and CH4 pure gas isotherms on isotypic MIL-53 (Cr) reported by Hamon et al. The structures of both Al and Cr variants of MIL-53 MOFs series are built up from similar infinite chains of corner-sharing MO4(OH)2 (M = Al3+, Cr3+) octahedra interconnected by the dicarboxylate groups. This results in a similar 3D metal-organic framework containing 1D diamond shaped channels. Isotherms of CO2 and CH4 on MIL-53 (Al) compare well with those on MIL-53 (Cr). This agrees well with earlier results on isotypic MIL-53 reported by Bourrelly et al. and Alhamami et al. [13, 48].Both pure gas isotherms are fitted with the D-A model with a standard error of estimate (SEE) of 1.05 for CO2 and of 0.626 for CH4. The corresponding fit parameters are presented in Table 3. The mixed gas isotherms on MIL-53 (Al) are constructed using the Doong-Yang Multicomponent isotherm model [36] using the pure gas isotherm regressions parameters. In Figure 2(b), the predicted binary adsorption isotherms are compared with the experimental equimolar binary adsorption isotherms on MIL-53 (Cr) measured by Hamon et al. Even though the isotherms cannot be quantitatively compared, they exhibit similar behavior for CH4 and CO2. We can conclude that our predictions are in agreement with the experimental data. The DYM isotherm equations are summarized in (9) to (14a), (14b), and (14c).The efficiency of MOF MIL-53 (Al) for the separation of a binary CH4/CO2 mixture can be analyzed and compared in terms of the sorption selectivity. The selectivity of ith component in a mixture of components i and j is defined on a molar basis as S
i
,
j
=
n
s
a
t
,
i
/
y
i
/
n
s
a
t
,
j
/
y
j. Here, we compare the selectivity of our sample to selectively remove CO2 from an equimolar CO2/CH4 mixture with the selectivities of other MOFs reported in the literature. In the pressure range below 0.5 MPa, the selectivity of our sample shown in Figure 3 decreases initially rapidly with pressure of about 0.1 MPa, after which it remains almost constant. The selectivity of MIL-53 reported by Hamon et al. on the other hand shows a step-like decrease, by a factor of ~3 at 0.6 MPa, after which it shows only a slight decrease [19]. The sample used by Hamon et al. showed two characteristic adsorption steps which were attributed to the breathing phenomenon. As the CO2 pressure increases, a step is observed at around 0.6 MPa [13] leading to larger uptake. This uptake is attributed to the change of MIL-53 from “narrow pore” to “large-pore” structure. On the other hand, the sample used in our work is a commercial material that shows no breathing phenomena. No drastic change in the selectivity is observed at around 0.6 MPa. Among all MOFs compared here, Cu-BTC [18] has the best selectivity at pressures above 0.2 MPa, while MOF-5 reported by Millward and Yaghi [21] has lowest selectivity. We conclude that MIL-53 (Al) offers a good separation for all ranges of pressure of up to 10 bar.Figure 3
Selectivity of MIL-53 (Al) towards CO2 at 303 K. For comparison purposes we included experimental data of Cu-BTC, MIL-53 (Cr), and MOF-5.


## 4.3. Validation of Breakthrough Curve Model

We start by validating the breakthrough model by applying it to simulate the breakthrough of a CO2/H2 mixture in an activated carbon column and comparing the model results with those reported by Casas et al. The model parameters and boundary and initial conditions required for validating the model are also obtained from Casas et al. [6, 29]. Figure 4 shows very good agreement between our validation results and those reported by Casas et al. The model has also been extensively validated experimentally and numerically by Casas et al., for different breakthrough curves cases of CO2/CH4 gas mixtures flowing through beds of activated carbon [29] and a hybrid MOF UiO-67/MCM-41 [6].Figure 4
Validation of breakthrough model by simulating the adsorption and separation of equimolar mixture of CH4 and CO2 in an activated carbon column. Symbols represent our results while the lines represent the experimental results reported by Casas et al. [29].


## 4.4. Parametric Study of Adsorbent Particle Size, Inlet Pressure, Gas Flow Rate, and Feed Composition on the Breakthrough

We used the validated model to study the effects of particle size, inlet pressure, feed composition, and gas flow rate on the breakthrough of CO2/CH4 gas mixtures through the MIL-53 adsorbent column. The inlet and wall temperatures are set to 294.15 K. A 25 cm column length is considered for all simulations. For monitoring the evolution of temperature in the bed, four axial positions at 5, 10, 15, and 20 cm from the inlet of the column are chosen.


### 4.4.1. Effect of Particle Diameter

In order to study the effect of MIL-53 (Al) particle size on breakthrough performance, we considered particle diameters 20, 200, 300, 500, and 1000µm. Inlet pressure is fixed at 0.2 MPa and an equimolar CO2/CH4 mixture is fed at a rate of 30 mL/min. In general, for the simulations with particle diameters lower than 20 µm, we found that the numerical model presents some limitations. An examination of the mass balance shows that the numerical results start to deviate from the mass predicted by the local pressure. This perhaps arises due to the large pressure drop caused by smaller particles, which is consistent with the general recommendation to use particle sizes of the order of 1 mm to avoid large pressure drops in gas-phase separations [6, 7, 17]. Therefore, we present the results only for 200, 500, and 1000 μm. Based on the literature, we set the particle size to 500 μm to investigate the effects of inlet pressure, flow rates, and feed concentration in further sections.In left panels of Figure5, breakthrough times for Y
i
/
Y
0
=
10
%, where Y
i is the molar fraction of the component and Y
0 is the feed concentration, are found to be around 7.9 minutes when particle sizes are 200 and 500 μm, while they are 7.7 minutes when the particle size is 1000 μm. In the right panels of Figure 5, the evolution of temperature at the positions 5, 10, 15, and 20 cm from the inlet of the column is shown. As adsorption is an exothermic process, the resulting adsorption heat is released into the bed. This increases the column temperature as the gas fronts move from the inlet to the outlet. We simulate the temperature evolution at four axial positions in the column. The temperature rises to around 333 K when 200 and 500 μm particles are used, while for the 1000 μm particles the temperature rises up to 336 K. At each position two different temperature peaks are observed; the low temperature peak corresponds to the CH4 front and the higher temperature peak to the CO2 front. From the simulation results, we find that the CH4 front moves faster than CO2 front. The peaks shape is influenced by the mass and heat transfer parameters: the initial fast abrupt front indicates fast mass transfer, whereas the shape of the tail is controlled mainly by the heat transfer from the column to the environment. The latter one is responsible for the time required to reach the feed composition at the outlet of the column, once the CO2 breakthrough is noticed. Since the temperature of the column continues to decrease until it reaches the initial temperature, more CO2 is adsorbed which finally results in a CO2 flat front.Figure 5
Breakthrough curves for an equimolar CO2/CH4 mixture, column length 25 cm, inlet pressure 0.2 MPa, for 3 different particle diameters. For simulation parameters refer to Table 1.Larger particles can be prepared either by mechanically compacting pristine MOFs to monoliths or by applying a binder, such as polyvinyl alcohol (PVA) or expanded natural graphite (ENG). Depending on the activation temperature, the preparation of monoliths by the addition of binder will cause partial pore blocking. The blocked pores reduce the adsorption capacity by as much as 19% of pristine powder MOF material. But this has minimum impact on the overall pore size distribution [17]. Binderless mechanical compaction of MOFs on the other hand causes partial collapse of frameworks, which reduces the sorption capacity by ~15% [49]. Our results are in agreement with Grande who recommends using the pellets instead of powder materials for efficient PSA separation [50].


### 4.4.2. Effect of the Inlet Pressure

In order to study the effect of the inlet pressure on the breakthrough curves, we set the inlet pressure to 0.5, 1, and 2.5 MPa. Figure6 displays the breakthrough and temperature profiles for different inlet pressures. The particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 6, the breakthrough time decreases with increasing feed pressure. The breakthrough times of 5, 3.3, and 1.9 minutes are obtained with the feed pressures 0.5, 1, and 2.5 MPa, respectively. Furthermore, the higher the feed pressure, the higher the temperature along the column, ~358, 388, and 444 K, respectively, for 0.5, 1, and 2.5 MPa. As gas with higher inlet pressure flows through the bed, larger amounts of gases are adsorbed, leading to higher amounts of adsorption heat released into the bed.Figure 6
Effect of feed pressure on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


### 4.4.3. Effect of the Mass Flow Rate

In order to study the impact of the mass flow rates on the breakthrough curves, we set the mass flow rate to 10, 25, and 50 mL/min. Figure7 displays the breakthrough and the temperature profiles for different mass flow rates. As in the case of previous simulations, the particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 7, the breakthrough time decreases with increasing of the mass flow rate. For mass flow rates of 10, 25, and 50 mL/min, the breakthrough times are 10.65, 4, and 1.9 minutes. Also, the higher the mass flow rate, the higher the temperature along the column, ~375, 385, and 392 K for 10, 25, and 50 mL/min. Gas with the higher mass flow rate leads to larger amount of adsorption. This leads to higher amounts of adsorption heat released into the bed.Figure 7
Effect of the mass flow rate on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


### 4.4.4. Effect of the Feed Concentration

The influence of the CO2 concentration on the MIL-53 adsorption kinetics is studied for three CO2 concentrations: 25%, 50%, and 75% in the CO2/CH4 mixture. For each composition, we also considered three feed pressures: 0.2, 0.5, and 2.5 MPa. Figure 8 displays the breakthrough and temperature profiles corresponding to each composition and pressure. The breakthrough point for 25% CO2 composition at a 0.2 MPa inlet pressure appears at 9.8 minutes, while shorter breakthrough times of 7.8 and 6.7 minutes are observed when the CO2 feed composition is increased to 50 and 75%. In other words, we see that the larger CO2 concentration in the feed gas mixture accelerates the breakthrough time [29, 51]. Similar behavior is also observed for the 0.5 and 2.5 MPa feed pressures. Larger concentration difference between the compositions results in faster saturation of the adsorbent with one component which eventually leads to shorter breakthrough times. The lowest breakthrough time of as short as 1.55 minutes is observed for the highest feed pressure (2.5 MPa) and highest CO2 molar concentrations (75%). The behavior of the temperature evolution on the other hand shows an increase at increasing the feed concentration and the feed pressure which is attributed to the larger amount of gases adsorbed.Figure 8
Breakthrough and temperature profiles when feed pressures and CO2/CH4 mixture compositions are varied. The range of feeding pressures is between 0.2 and 2.5 MPa and a range of concentration is between 25 and 75% mole fraction of CO2.The different breakthrough times for CH4 and CO2 obtained with different feed pressures and molar compositions directly affect the amount of pure CH4 produced in each PSA cycle. The amount of pure CH4 produced in a cycle can be calculated from the outlet flow rate and time between the onset of the flow and breakthrough. Note that the reduction of adsorption capacity due to pelletization should also be accounted for while calculating the amount of pure CH4 produced in each cycle. Based on the reported adsorption capacities of monoliths, we used 15% reduction factor to calculate the amount of pure CH4 produced. As seen in Figure 9, at higher feed pressure, more CH4 is produced per cycle, even though each cycle lasts less than that for lower feed pressures (from left to right of Figure 8). On the other hand, the amount of CH4 produced at the specified purity with respect to the feed composition decreases with the increasing pressure. This means that more CH4 remains in the column by the time CO2 breaks through, as it is confirmed in a similar case for a CO2/H2 gas mixture [6, 29]. In a continuous process this is circumvented by adjusting the cycle time in such a way that the CH4 loss is minimized [29].Figure 9
Amount of pure CH4 obtained in a breakthrough prediction for a feed mixture containing equimolar CO2/CH4 mixture.The separation capacity of MIL-53 (Al) for CO2 and CH4 is between 7.8 at 0.1 MPa and 7.1 at 1.5 MPa at 294.15 K based on the selectivity correlation S
i
,
j
=
n
s
a
t
,
i
/
y
i
/
n
s
a
t
,
j
/
y
j [34]. These values are consistent with the values reported by Finsy et al. on the separation of an equimolar CH4/CO2 mixture at 303 K in a packed column with MIL-53 (Al, PVA) pellets containing 13 wt% PVA binder [17]. Selectivities calculated from pure component isotherms on the 13X zeolite [11] and the activated carbon material Norit R1 Extra [12] are 2 and 2.7 at 1 MPa compared with 5.5 for MIL-53 (Al).


## 4.4.1. Effect of Particle Diameter

In order to study the effect of MIL-53 (Al) particle size on breakthrough performance, we considered particle diameters 20, 200, 300, 500, and 1000µm. Inlet pressure is fixed at 0.2 MPa and an equimolar CO2/CH4 mixture is fed at a rate of 30 mL/min. In general, for the simulations with particle diameters lower than 20 µm, we found that the numerical model presents some limitations. An examination of the mass balance shows that the numerical results start to deviate from the mass predicted by the local pressure. This perhaps arises due to the large pressure drop caused by smaller particles, which is consistent with the general recommendation to use particle sizes of the order of 1 mm to avoid large pressure drops in gas-phase separations [6, 7, 17]. Therefore, we present the results only for 200, 500, and 1000 μm. Based on the literature, we set the particle size to 500 μm to investigate the effects of inlet pressure, flow rates, and feed concentration in further sections.In left panels of Figure5, breakthrough times for Y
i
/
Y
0
=
10
%, where Y
i is the molar fraction of the component and Y
0 is the feed concentration, are found to be around 7.9 minutes when particle sizes are 200 and 500 μm, while they are 7.7 minutes when the particle size is 1000 μm. In the right panels of Figure 5, the evolution of temperature at the positions 5, 10, 15, and 20 cm from the inlet of the column is shown. As adsorption is an exothermic process, the resulting adsorption heat is released into the bed. This increases the column temperature as the gas fronts move from the inlet to the outlet. We simulate the temperature evolution at four axial positions in the column. The temperature rises to around 333 K when 200 and 500 μm particles are used, while for the 1000 μm particles the temperature rises up to 336 K. At each position two different temperature peaks are observed; the low temperature peak corresponds to the CH4 front and the higher temperature peak to the CO2 front. From the simulation results, we find that the CH4 front moves faster than CO2 front. The peaks shape is influenced by the mass and heat transfer parameters: the initial fast abrupt front indicates fast mass transfer, whereas the shape of the tail is controlled mainly by the heat transfer from the column to the environment. The latter one is responsible for the time required to reach the feed composition at the outlet of the column, once the CO2 breakthrough is noticed. Since the temperature of the column continues to decrease until it reaches the initial temperature, more CO2 is adsorbed which finally results in a CO2 flat front.Figure 5
Breakthrough curves for an equimolar CO2/CH4 mixture, column length 25 cm, inlet pressure 0.2 MPa, for 3 different particle diameters. For simulation parameters refer to Table 1.Larger particles can be prepared either by mechanically compacting pristine MOFs to monoliths or by applying a binder, such as polyvinyl alcohol (PVA) or expanded natural graphite (ENG). Depending on the activation temperature, the preparation of monoliths by the addition of binder will cause partial pore blocking. The blocked pores reduce the adsorption capacity by as much as 19% of pristine powder MOF material. But this has minimum impact on the overall pore size distribution [17]. Binderless mechanical compaction of MOFs on the other hand causes partial collapse of frameworks, which reduces the sorption capacity by ~15% [49]. Our results are in agreement with Grande who recommends using the pellets instead of powder materials for efficient PSA separation [50].


## 4.4.2. Effect of the Inlet Pressure

In order to study the effect of the inlet pressure on the breakthrough curves, we set the inlet pressure to 0.5, 1, and 2.5 MPa. Figure6 displays the breakthrough and temperature profiles for different inlet pressures. The particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 6, the breakthrough time decreases with increasing feed pressure. The breakthrough times of 5, 3.3, and 1.9 minutes are obtained with the feed pressures 0.5, 1, and 2.5 MPa, respectively. Furthermore, the higher the feed pressure, the higher the temperature along the column, ~358, 388, and 444 K, respectively, for 0.5, 1, and 2.5 MPa. As gas with higher inlet pressure flows through the bed, larger amounts of gases are adsorbed, leading to higher amounts of adsorption heat released into the bed.Figure 6
Effect of feed pressure on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


## 4.4.3. Effect of the Mass Flow Rate

In order to study the impact of the mass flow rates on the breakthrough curves, we set the mass flow rate to 10, 25, and 50 mL/min. Figure7 displays the breakthrough and the temperature profiles for different mass flow rates. As in the case of previous simulations, the particle size is fixed at 500 μm for all simulations. As seen in the left panels of Figure 7, the breakthrough time decreases with increasing of the mass flow rate. For mass flow rates of 10, 25, and 50 mL/min, the breakthrough times are 10.65, 4, and 1.9 minutes. Also, the higher the mass flow rate, the higher the temperature along the column, ~375, 385, and 392 K for 10, 25, and 50 mL/min. Gas with the higher mass flow rate leads to larger amount of adsorption. This leads to higher amounts of adsorption heat released into the bed.Figure 7
Effect of the mass flow rate on the breakthrough profile and temperature on MIL-53 for an equimolar CO2/CH4 mixture.


## 4.4.4. Effect of the Feed Concentration

The influence of the CO2 concentration on the MIL-53 adsorption kinetics is studied for three CO2 concentrations: 25%, 50%, and 75% in the CO2/CH4 mixture. For each composition, we also considered three feed pressures: 0.2, 0.5, and 2.5 MPa. Figure 8 displays the breakthrough and temperature profiles corresponding to each composition and pressure. The breakthrough point for 25% CO2 composition at a 0.2 MPa inlet pressure appears at 9.8 minutes, while shorter breakthrough times of 7.8 and 6.7 minutes are observed when the CO2 feed composition is increased to 50 and 75%. In other words, we see that the larger CO2 concentration in the feed gas mixture accelerates the breakthrough time [29, 51]. Similar behavior is also observed for the 0.5 and 2.5 MPa feed pressures. Larger concentration difference between the compositions results in faster saturation of the adsorbent with one component which eventually leads to shorter breakthrough times. The lowest breakthrough time of as short as 1.55 minutes is observed for the highest feed pressure (2.5 MPa) and highest CO2 molar concentrations (75%). The behavior of the temperature evolution on the other hand shows an increase at increasing the feed concentration and the feed pressure which is attributed to the larger amount of gases adsorbed.Figure 8
Breakthrough and temperature profiles when feed pressures and CO2/CH4 mixture compositions are varied. The range of feeding pressures is between 0.2 and 2.5 MPa and a range of concentration is between 25 and 75% mole fraction of CO2.The different breakthrough times for CH4 and CO2 obtained with different feed pressures and molar compositions directly affect the amount of pure CH4 produced in each PSA cycle. The amount of pure CH4 produced in a cycle can be calculated from the outlet flow rate and time between the onset of the flow and breakthrough. Note that the reduction of adsorption capacity due to pelletization should also be accounted for while calculating the amount of pure CH4 produced in each cycle. Based on the reported adsorption capacities of monoliths, we used 15% reduction factor to calculate the amount of pure CH4 produced. As seen in Figure 9, at higher feed pressure, more CH4 is produced per cycle, even though each cycle lasts less than that for lower feed pressures (from left to right of Figure 8). On the other hand, the amount of CH4 produced at the specified purity with respect to the feed composition decreases with the increasing pressure. This means that more CH4 remains in the column by the time CO2 breaks through, as it is confirmed in a similar case for a CO2/H2 gas mixture [6, 29]. In a continuous process this is circumvented by adjusting the cycle time in such a way that the CH4 loss is minimized [29].Figure 9
Amount of pure CH4 obtained in a breakthrough prediction for a feed mixture containing equimolar CO2/CH4 mixture.The separation capacity of MIL-53 (Al) for CO2 and CH4 is between 7.8 at 0.1 MPa and 7.1 at 1.5 MPa at 294.15 K based on the selectivity correlation S
i
,
j
=
n
s
a
t
,
i
/
y
i
/
n
s
a
t
,
j
/
y
j [34]. These values are consistent with the values reported by Finsy et al. on the separation of an equimolar CH4/CO2 mixture at 303 K in a packed column with MIL-53 (Al, PVA) pellets containing 13 wt% PVA binder [17]. Selectivities calculated from pure component isotherms on the 13X zeolite [11] and the activated carbon material Norit R1 Extra [12] are 2 and 2.7 at 1 MPa compared with 5.5 for MIL-53 (Al).


## 5. Conclusions

To conclude, we presented a parametric study of MIL-53 aluminum terephthalate particle size, inlet pressure, mass flow rate, and feed composition on the breakthrough of CO2/CH4 binary gas mixtures. Pure gas CO2 and CH4 adsorption isotherms on commercial MIL-53 were measured using Sieverts method and were fitted with the D-A analytical model. Using the D-A model fit parameters, binary adsorption isotherms were predicted. These isotherms agree well with the reported experimental binary isotherms measured on isotypic MIL-53 chromium terephthalate. A one-dimensional multicomponent adsorption model was used to simulate the breakthrough behavior of CO2/CH4 mixtures in a column packed with MIL-53 (Al). The model was initially validated by applying it to simulate the breakthrough of H2/CO2 mixtures reported in the literature. Experimentally measured particle size, porosity, kinetic diffusion parameters, isosteric heat, and specific heat were used in the model to increase the reliability of its predictions. In the parametric study, we considered the effect of adsorbent particle diameters (5, 20, 200, 300, 500, and 1000 µm), feed pressures (0.2, 1, and 2.5 MPa), feed flow rates (10, 25, and 50 mL/min), and inlet compositions (25%, 50%, and 75% CO2) on the breakthrough performance. As-purchased MIL-53, with a peak particle diameter of 20 µm, was found to be less effective for separation because of the higher pressure drops. Effective separation within two minutes of the onset of flow was achieved for MIL-53 monoliths of diameters above 200 µm. We found that faster separation can be made possible by increasing the feed pressure from 0.2 MPa to 2.5 MPa and also if the starting compositions are rich in CO2. As higher pressure CO2 richer stream passed through the column, more heat was generated in the column when compared with the low-feed pressure CH4 rich stream. More CH4 was produced per cycle at higher feed pressures, even though each cycle lasted less than that for lower feed pressures. On the other hand, increasing pressure decreases the CH4 recovery.



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*Source: 439382-2015-12-09.xml*