Molecular Thermodynamics of Adsorption Using a 2D- SAFT-VR ... · Thermodynamics 2017 -5 th to 8 th...
Transcript of Molecular Thermodynamics of Adsorption Using a 2D- SAFT-VR ... · Thermodynamics 2017 -5 th to 8 th...
Molecular
Thermodynamics of
Adsorption Using a 2D-
SAFT-VR-Mie Approach
Gerardo Campos, Jonatan Suaste, Andrew Haslam, George
Jackson and Alejandro Gil-Villegas
Thermodynamics 2017 -5th
to 8th
September 2017, Edinburgh, UK
Thermodynamics 2017 -5th
to 8th
September 2017, Edinburgh, UK
Outline
• Adsorption
• Statistical Associating Fluid Theory (SAFT-VR)
• Adsorption Model: The quasi-two dimensional fluid approach
• 2D SAFT-VR-Mie EOS
• Running the Machinery: Modelling of Adsorption of Carbon Dioxide
on Dry Coal
• Conclusions and Outlook
1
Thermodynamics 2017 -5th
to 8th
September 2017, Edinburgh, UK
Adsorption
2
Theoretical Modelling
𝜃 =𝐾𝑃𝑔
1 + 𝐾𝑃𝑔
Langmuir, I ., J. Am. Chem. Soc., 37 , 1139-1167 (1915).
More accurate platforms are
available, e.g.:
• Molecular Simulation at
different length-scales
• Quasi-two dimensional
approach via SAFT
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR SW on adsorption
3
Model and predicts asphaltene precipitation and adsorption
Su
rfa
ce
co
ve
rag
e
Bulk packing fraction (reduced density)
Bedford limestone at 298𝐾
Castro et al., Fluid Phase Equilibria, 286 , 113-119 (2009).
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR SW on adsorption
4
Adsorption of hydrogen on activated carbon (semiclassical approach)
Trejos et al., Mol. Phys., 112 , 2330-2338 (2014).
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
5
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂 + 𝑎𝐶𝐻𝐴𝐼𝑁 + 𝑎𝐴𝑆𝑆𝑂𝐶
Gil-Villegas et al., J. Chem. Phys., 106 , 4168 (1997).
𝑎 =𝐴
𝑁𝑘𝑇
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
6
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
7
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
8
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂 + 𝑎𝐶𝐻𝐴𝐼𝑁
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
9
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂 + 𝑎𝐶𝐻𝐴𝐼𝑁 + 𝑎𝐴𝑆𝑆𝑂𝐶
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
SAFT-VR
10
Free-energy Equation of State based on Statistical Mechanical Perturbation Theory
𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂
In this work we restrict our description to only monomeric fluids
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
Adsorption Model
𝑧
So
lid
Wall
λ𝑤𝜎
11
BULK PHASE
ADSORBED PHASE
Martinez et al., J. Chem. Phys., 126 , 074707 (2007).
The adsorption isotherms are
calculated by solving the
thermodynamic equilibrium
between the two phases
Thermodynamics 2017 -5th
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Adsorption Model
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BULK PHASE :
Described with SAFT-VR-Mie
𝑢𝑀𝑖𝑒 = 𝐶𝜖𝜎
𝑟
λ𝑟−𝜎
𝑟
λ𝑎
Lafitte et al., J. Chem. Phys., 139 , 154504 (2013).
𝑎𝑏𝑢𝑙𝑘 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂
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Adsorption Model
𝑧
So
lid
Wall
λ𝑤𝜎
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ADSORBED PHASE :
Treated as a quasi-2D system
𝑢𝑎𝑑𝑠 = 𝑢2𝐷𝑀𝑖𝑒 𝑥, 𝑦 + 𝑢1𝐷
𝑤𝑎𝑙𝑙 𝑧
𝑢2𝐷𝑀𝑖𝑒
𝑢1𝐷𝑤𝑎𝑙𝑙
𝑢1𝐷𝑤𝑎𝑙𝑙 𝑧 =
∞ −𝜖𝑤0
𝑧 ≤ 0 0 < 𝑧 ≤ λ𝑤𝜎 𝑧 > λ𝑤𝜎
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
Adsorption Model
𝑧
So
lid
Wall
λ𝑤𝜎
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ADSORBED PHASE :
Treated as a quasi-2D system
𝑢2𝐷𝑀𝑖𝑒
𝑢1𝐷𝑤𝑎𝑙𝑙
To guarantee the 2D approach
(monolayer formation):
0.1305 < λ𝑤 < 0.8165
del Rio and Gil Villegas., J. Phys. Chem. , 95 , 787-792 (1991).
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
Adsorption Model
𝑧
So
lid
Wall
λ𝑤𝜎
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ADSORBED PHASE :
Treated as a quasi-2D system
𝑢2𝐷𝑀𝑖𝑒
𝑢1𝐷𝑤𝑎𝑙𝑙
= 𝑎2𝐷𝐼𝐷𝐸𝐴𝐿 + 𝑎2𝐷
𝑀𝑂𝑁𝑂 + 𝑎1𝐷𝑤𝑎𝑙𝑙
𝑎𝑎𝑑𝑠 = 𝑎2𝐷𝑀𝑖𝑒 + 𝑎1𝐷
𝑤𝑎𝑙𝑙
Thermodynamics 2017 -5th
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September 2017, Edinburgh, UK
Adsorption Model
𝑧
So
lid
Wall
λ𝑤𝜎
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ADSORBED PHASE :
Treated as a quasi-2D system
𝑢2𝐷𝑀𝑖𝑒
𝑢1𝐷𝑤𝑎𝑙𝑙
= 𝑎2𝐷𝐼𝐷𝐸𝐴𝐿 + 𝑎2𝐷
𝑀𝑂𝑁𝑂 + 𝑎1𝐷𝑤𝑎𝑙𝑙
𝑎𝑎𝑑𝑠 = 𝑎2𝐷𝑀𝑖𝑒 + 𝑎1𝐷
𝑤𝑎𝑙𝑙
2D-SAFT-VR-Mie EOS
(This work)
Thermodynamics 2017 -5th
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2D SAFT-VR-Mie EOS
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BH perturbation expansion up to first order term
𝑎2𝐷𝑀𝑂𝑁𝑂 = 𝑎𝐻𝐷 + 𝛽𝑎1
𝑀𝑖𝑒−2𝐷
𝛽 = 1 𝑘𝑇
J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 (1976).
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2D SAFT-VR-Mie EOS
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𝑎2𝐷𝑀𝑂𝑁𝑂 = 𝑎𝐻𝐷 + 𝛽𝑎1
𝑀𝑖𝑒−2𝐷
𝑎𝐻𝐷 =9𝛾
8 1 − 𝛾−7
8ln 1 − 𝛾
𝑑𝐻𝐷(𝑇) = 1 − 𝑒−𝛽𝑢𝑀𝑖𝑒(𝑟)
𝜎
0
𝑑𝑟
BH perturbation expansion up to first order term
Henderson, Mol. Phys., 34, 301-315 (1977).
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2D SAFT-VR-Mie EOS
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𝑎2𝐷𝑀𝑂𝑁𝑂 = 𝑎𝐻𝐷 + 𝛽𝑎1
𝑀𝑖𝑒−2𝐷
𝑎1𝑀𝑖𝑒−2𝐷 = 𝜋𝜌2𝐷 𝑢𝑀𝑖𝑒 𝑟 𝑔𝐻𝐷 𝑟 𝑟
∞
𝜎
𝑑𝑟
BH perturbation expansion up to first order term
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2D SAFT-VR-Mie EOS
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𝑎2𝐷𝑀𝑂𝑁𝑂 = 𝑎𝐻𝐷 + 𝛽𝑎1
𝑀𝑖𝑒−2𝐷
𝑎1𝑀𝑖𝑒−2𝐷 = 𝜋𝜌2𝐷 𝑢𝑀𝑖𝑒 𝑟 𝑔𝐻𝐷 𝑟 𝑟
∞
𝜎
𝑑𝑟
BH perturbation expansion up to first order term
= 𝑎1𝑀𝑖𝑒−2𝐷 𝛾; 𝜎, 𝜖, λ𝑟 , λ𝑎
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2D SAFT-VR-Mie EOS
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𝑎2𝐷𝑀𝑂𝑁𝑂 = 𝑎𝐻𝐷 + 𝛽𝑎1
𝑀𝑖𝑒−2𝐷
𝑎1𝑀𝑖𝑒−2𝐷 = 𝜋𝜌2𝐷 𝑢𝑀𝑖𝑒 𝑟 𝑔𝐻𝐷 𝑟 𝑟
∞
𝜎
𝑑𝑟
BH perturbation expansion up to first order term
= 𝑎1𝑀𝑖𝑒−2𝐷 𝛾; 𝜎, 𝜖, λ𝑟 , λ𝑎
Validation by comparison with MC simulation
results (Suaste)
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2D SAFT-VR-Mie EOS
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𝑇∗ = 𝑘𝑇 𝜖 = 1 𝑇∗ = 𝑘𝑇 𝜖 = 2
The theory accurately reproduces the MC
simulation results
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Modelling Adsorption
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𝑧
So
lid
Wall
λ𝑤𝜎
𝜇𝑏 𝜌𝑏 = 𝜇𝑎𝑑𝑠 𝜌𝑎𝑑𝑠
Phase equilibrium criterion
𝜇𝑏 𝜌𝑏 =𝜕𝐴𝑏𝜕𝑁𝑏 𝑉,𝑇
𝜇𝑎𝑑𝑠 𝜌𝑎𝑑𝑠 =𝜕𝐴𝑎𝑑𝑠𝜕𝑁𝑎𝑑𝑠 𝑠,𝑇
𝜌𝑏 𝜌𝑎𝑑𝑠
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Adsorption of a LJ (12-6) fluid
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𝑇∗ = 2
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Adsorption of CO2 on dry coal
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Single-site CG model
of bulk CO2
Avendano et al. , J. Phys. Chem., 115, 11154-11169 (2011).
𝝐𝒌 𝑲 𝝈 𝒏𝒎 λ𝒓 λ𝒂
353.55 0.3741 23.0 6.66
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Adsorption of CO2 on dry coal
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(**) Sinanoğlu and Pitzer, J. Chem. Phys.,32, 1279-1288 (1960).
The parameters for the pair-potential describing the adsorbed phase
can be selected following two distinct routes:
Route a: Different parameters respect to those of the bulk phase
Classical approach followed in previous SAFT-VR SW models of
adsorption
λ𝑟𝑎𝑑𝑠 = λ𝑟
𝑏𝑢𝑙𝑘
λ𝑎𝑎𝑑𝑠
from 𝑅𝑐 =𝑇𝑐𝑎𝑑𝑠
𝑇𝑐𝑏𝑢𝑙𝑘 = 0.4 (*)
𝜖𝑎𝑑𝑠 = 0.8𝜖𝑏𝑢𝑙𝑘 (**)
(*) Machin and Ross , Proc. R. Soc. London, Ser., 265, (1962).
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Adsorption of CO2 on dry coal
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The parameters for the pair-potential describing the adsorbed phase
can be selected following two distinct routes:
Route b: Same parameters of the bulk phase
λ𝑟𝑎𝑑𝑠 = λ𝑟
𝑏𝑢𝑙𝑘
λ𝑎𝑎𝑑𝑠 = λ𝑎
𝑏𝑢𝑙𝑘
𝜖𝑎𝑑𝑠 from 𝑅𝑐
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Adsorption of CO2 on dry coal
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SOLID-FLUID INTERACTIONS:
0.1305 < λ𝑤 < 0.8165
𝜖𝑤 must be fitted to experimental data
The specific surface of the adsorbent is also required
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Adsorption of CO2 on dry coal
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Ottiger et al. , Adsorption, 14, 539-556 (2008).
𝑇 = 318𝐾
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Conclusions and Outlook
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• It has been demonstrated that changing the pair potential of the
adsorbed phase is not a requisite as it has been argued in previous
SAFT-VR (SW) approaches
• The theoretical framework presented here is suitable for a wide range
of applications, however, the particle-wall interactions have to be
fitted to experimental data
• The 2D-SAFT-VR-Mie EOS can still be enhanced by incorporating
higher-order terms in the thermal expansion
• The theory can be extended to model mixtures, reactive systems and
phase transitions of confined fluids
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Acknowledgements
31
• QCCSRC, Imperial College London and University of Guanajuato for
project funding
• University of Manchester for conference funding