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


to 8th


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


to 8th


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


to 8th


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).


to 8th


SAFT-VR SW on adsorption

4

Adsorption of hydrogen on activated carbon (semiclassical approach)

Trejos et al., Mol. Phys., 112 , 2330-2338 (2014).


to 8th


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


to 8th


SAFT-VR

6


𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿


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SAFT-VR

7


𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂


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SAFT-VR

8


𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂 + 𝑎𝐶𝐻𝐴𝐼𝑁


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SAFT-VR

9


𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂 + 𝑎𝐶𝐻𝐴𝐼𝑁 + 𝑎𝐴𝑆𝑆𝑂𝐶


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SAFT-VR

10


𝑎 = 𝑎𝐼𝐷𝐸𝐴𝐿 + 𝑎𝑀𝑂𝑁𝑂

In this work we restrict our description to only monomeric fluids


to 8th


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


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Adsorption Model

12

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


to 8th


Adsorption Model

𝑧

So

lid

Wall

λ𝑤𝜎

13

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


to 8th


Adsorption Model

𝑧

So

lid

Wall

λ𝑤𝜎

14

ADSORBED PHASE :




To guarantee the 2D approach

(monolayer formation):

0.1305 < λ𝑤 < 0.8165

del Rio and Gil Villegas., J. Phys. Chem. , 95 , 787-792 (1991).


to 8th


Adsorption Model

𝑧

So

lid

Wall

λ𝑤𝜎

15

ADSORBED PHASE :




= 𝑎2𝐷𝐼𝐷𝐸𝐴𝐿 + 𝑎2𝐷

𝑀𝑂𝑁𝑂 + 𝑎1𝐷𝑤𝑎𝑙𝑙

𝑎𝑎𝑑𝑠 = 𝑎2𝐷𝑀𝑖𝑒 + 𝑎1𝐷

𝑤𝑎𝑙𝑙


to 8th


Adsorption Model

𝑧

So

lid

Wall

λ𝑤𝜎

16

ADSORBED PHASE :




= 𝑎2𝐷𝐼𝐷𝐸𝐴𝐿 + 𝑎2𝐷

𝑀𝑂𝑁𝑂 + 𝑎1𝐷𝑤𝑎𝑙𝑙

𝑎𝑎𝑑𝑠 = 𝑎2𝐷𝑀𝑖𝑒 + 𝑎1𝐷

𝑤𝑎𝑙𝑙

2D-SAFT-VR-Mie EOS

(This work)


to 8th


2D SAFT-VR-Mie EOS

17

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).


to 8th


2D SAFT-VR-Mie EOS

18



𝑎𝐻𝐷 =9𝛾

8 1 − 𝛾−7

8ln 1 − 𝛾

𝑑𝐻𝐷(𝑇) = 1 − 𝑒−𝛽𝑢𝑀𝑖𝑒(𝑟)

𝜎

0

𝑑𝑟


Henderson, Mol. Phys., 34, 301-315 (1977).


to 8th


2D SAFT-VR-Mie EOS

19



𝑎1𝑀𝑖𝑒−2𝐷 = 𝜋𝜌2𝐷 𝑢𝑀𝑖𝑒 𝑟 𝑔𝐻𝐷 𝑟 𝑟

∞

𝜎

𝑑𝑟



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2D SAFT-VR-Mie EOS

20




∞

𝜎

𝑑𝑟


= 𝑎1𝑀𝑖𝑒−2𝐷 𝛾; 𝜎, 𝜖, λ𝑟 , λ𝑎


to 8th


2D SAFT-VR-Mie EOS

21




∞

𝜎

𝑑𝑟


= 𝑎1𝑀𝑖𝑒−2𝐷 𝛾; 𝜎, 𝜖, λ𝑟 , λ𝑎

Validation by comparison with MC simulation

results (Suaste)


to 8th


2D SAFT-VR-Mie EOS

22

𝑇∗ = 𝑘𝑇 𝜖 = 1 𝑇∗ = 𝑘𝑇 𝜖 = 2

The theory accurately reproduces the MC

simulation results


to 8th


Modelling Adsorption

23

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


to 8th


Adsorption of a LJ (12-6) fluid

24

𝑇∗ = 2


to 8th


Adsorption of CO2 on dry coal

25

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


to 8th



26

(**) 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).


to 8th



27

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


to 8th



28

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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29

Ottiger et al. , Adsorption, 14, 539-556 (2008).

𝑇 = 318𝐾


to 8th


Conclusions and Outlook

30

• 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


to 8th


Acknowledgements

31

• QCCSRC, Imperial College London and University of Guanajuato for

project funding

• University of Manchester for conference funding

Any questions?

Contact:

[email protected]

Thank you


to 8th


SAFT-VR Mie terms

S1


to 8th


SAFT-VR Mie terms

S2


to 8th


Adsorption model

S3


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Adsorption model

S4


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Analytical expression of 𝑎1𝑀𝑖𝑒−2𝐷

S5

Molecular Thermodynamics of Adsorption Using a 2D- SAFT-VR ... · Thermodynamics 2017 -5 th to 8 th...

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