Phase Equilibrium and Cage Occupancy Calculations of CO2 hydrates Using an Ab Initio Intermolecular

Introduction
• Methane hydrate as a potential energy source • Hydrates can be a possible solution to the CO2 problem
• Sequestration of CO2 in sediments below the ocean floor • CO2 can be stored in linkage with methane exploitation
∆0 w H π ∆ w 2 dT , ∫ HRT(T0 0) + 0
∫
0
P
∆V RT
π
w 2
dP − ln γ
w
xw
P1 P1 =P0 Yes Print P1 , T and yi
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NO
Ab initio method
• Ab initio quantum mechanics
Portions of this technical effort was performed in support of the National Energy Technology Laboratory’s on-going research in Gas Hydrates Under the RDS contract DE-AC26-04NT41817
∆µ (T , P) = −RT∑vi ln( − ∑θij ) 1
H w i =1 j =1 2 Nc
Cage Occupancy
θ ij = C ij f j (1 +
1 C ji = kT
2π π R
∑
Nc
C
j =1
ij
f j)
Fugacity calculated from EOS
Φ (r ,θ , φ , α , β , γ ) 2 exp − r sin θ drd θ d φ d α d β d γ ∫∫∫ kT 0 0 0
•
Gas hydrate thermodynamics is important
• CO2 – H2O interaction potential is critical • Cage occupancy
– Govern storage capacity – Ability to displace CH4
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Read pure components properties and temperature T
1 C ji = kT
2π π R
Φ(r ,θ C = , Calculate , φ , α Z β , γ ) 2 exp − r sin θdrdθdφdαdβdγ kT ∫∫∫ kT 0 0 0
500 600 700
Number of basis functions
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Ab initio calculations
-1.50 165 260
100 1000
Binding Energy (Kcal/mol)
-2.00 24
6-31++G(2d,2p) cc-pVQZ 10
-2.50
cc-pVDZ
ji ji
Estimate P0 , ln P = A + B + C ln T Calculate Fugacity from EOS Calculate
∆µ
H w 0 (T , P0 ) ∆ µ w0 ( T 0 , 0 ) = ∆µ (T ,0) − w RT RT 0 0 T
H ∆µ w (T , P0 )
• The CO2 – H2O structure optimized to determine the optimum basis set • The optimized structure is T-shaped structure • r(CO) = 2.788Å and experimentally determined by Peterson and Klemperer1 as r(CO) = 2.836Å
Image: USGS
http://www.cheng.es.osaka-u.ac.jp/ohgakilab/simages/Structure-I.jpg
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Statistical thermodynamic model
van der Waals and Platteeuw 1 proposed the chemical potential difference between the hypothetical empty and fully filled hydrate lattice as
Phase Equilibrium and Cage Occupancy Calculations of CO2 hydrates Using an Ab Initio Intermolecular Potential
Srinath Velaga and Brian Anderson
National Energy Technology Laboratory and West Virginia University
y’ z
r
Space sampling of the six dimensions:
r 2.4 - 6.0Å 10 points ξ -40° - 40° 5 points ϕ -40° - 40° 5 points α 0° - 120° 3 points β 0° - 360° 4 points γ 0° - 120° 3 points
∑
j =1
− ∆ E cp , j exp kT
− ∆ E cp , j − exp kT pred
QM
2
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Many-body effects
Ab initio calculations were performed on 15 water molecules of pentagonal dodecahedron with single CO2 guest Used 6-31++G(2d,2p) basis set
assumed constant
∆µ
β − L ,α
w
(T , P )
RT
=
∆µ
0 w
(T 0 ,0 ) − RT 0
T
∫
0
∆H RT
π
w 2
dT +
∫
0
P
∆V RT
π
w 2
dp − ln γ
w
xw
Reference chemical potential difference at T0 =273.15 K and P=0 bar
• An exact value for the system energy can be obtained • Exact solution to the Schrödinger equation is not computationally practical • Approximate methods- Hatree-Fock, MP2, MP4 – a trade-off between accuracy and computational cost • Gaussian 03 was used to calculate binding energies
BSSE corrected by Full CP method
aug-cc-pVQZ
cc-pV5Z cc-pV5Z
aug-cc-pV5Z
aug-cc-pVQZ
aug-cc-pV5Z
∆EBE ,CP = EH 2O−CO2 − EH 2O ,CP − ECO2 ,CP ∆EBE = EH 2O −CO2 − EH 2O − ECO2
cc-pVTZ
Binding Energy (kcal/mol)
-2.0 6-31++G (2d,2p) cc-pVQZ -2.5 6-31++G (2d,2p) -3.0 aug-cc-pVTZ cc-pVTZ -3.5 cc-pVDZ -4.0 0 100 200 300 400 aug-cc-pVTZ cc-pVQZ
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Multi dimensional potential hyper surface
z’
Carbon on CO2
Euler angles α, β, γ are used to rotate carbon dioxide w.r.t. its internal x’ y’ z’ coordinates
The 18,000 ab initio energies were fit to site-site potentials Lennard-Jones 6-12
σ 12 σ 6 Φ LJ (r ) = 4ε − r r
ε
r − r m
6
Exponential-6
6 r exp γ 1 − Φ (r ) = 1 − 6 γ rm γ
By minimizing the Boltzmann factor-weighted objective function χ
χ=
# data points
1. J. Chem. Phys. 1984, 80, 2439.
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Ab initio calculations
-1.5