Understanding and Predicting CO2 Properties for CCS Transport, Richard Graham, University of Nottingham. Presented at CO2 Properties and EoS for Pipeline Engineering, 11th November 2014

1. Understanding and
predicting CO2 properties
Richard Graham
Tom Demetriades, Alex Cresswell, Martin Nelson,
Richard Wilkinson and Simon Preston
School of Mathematical Sciences, University of Nottingham.

2. Overview
!
•Parametric equations of state (pressure
explicit)
•Non-parametric EoS (pressure explicit
or free energy formulation).
•Molecular simulations

3. Overview
!
•Parametric equations of state (pressure
explicit)
•Non-parametric EoS (pressure explicit
or free energy formulation).
•Molecular simulations
!
•Uncertainty quantification

4. Potential applications: Avoiding pipeline
issues
Two-phase
flow
Vapour-solid
mix
Jet
Warm air
entrainment Solar heating
Solid entrainment
Solid sublimation
Snow/ dry ice Ground heat flux
Pipe rupture

5. Coexistence - Impurities

6. Coexistence - Impurities
xG: Gas composition
vG: Gas volume
Gas
Liquid
xL: Liquid composition
vL: Liquid volume

7. Coexistence - Impurities
xG: Gas composition
vG: Gas volume
Gas
CO2+N2 data
Liquid
Gas
Molar Volume (litres/mol)
Pressure (MPa)
Liquid
Gas
Pressure (MPa)
Mole fraction of impurity
Liquid
xL: Liquid composition
vL: Liquid volume

8. Uncertainty quantification

9. Uncertainty quantification

10. Uncertainty quantification
Economic
recovery!

11. Uncertainty quantification
Huge
uncertainty!

12. Uncertainty quantification
Must account for uncertainty due to:
•Incomplete data
•Measurements errors
•Model imperfection
Huge
uncertainty!

13. A generalised equation of
R
state
Peng-Robinson
This work

14. A generalised equation of
R
state
Peng-Robinson
This work
Higher order terms
enable a longer plateau
and improved critical
volume

15. A generalised equation of
R
state
Peng-Robinson
This work
Higher order
singularity provides a
sharper ‘liquid’ region
Higher order terms
enable a longer plateau
and improved critical
volume

16. Fitting method
10
-4
Fitting criterion
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
294K

17. Fitting method
10
-4
Fitting criterion
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
294K

18. Fitting method
10
-4
Fitting criterion
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
294K

19. Fitting method
10
-4
Fitting criterion
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
294K

20. Fitting method
10
-4
Fitting criterion
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
294K
Numerically minimise the
sum of these 4 quantities
over the parameters a...g

21. MCMC: an example
Markov-Chain Monte-Carlo: an example
θ
1
θ
2
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
The search algorithm explores the fitting criterion,
spending more time in regions of good fit.

22. MCMC: an example
Markov-Chain Monte-Carlo: an example
θ
1
θ
2
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
The search algorithm explores the fitting criterion,
spending more time in regions of good fit.

23. MCMC: an example
Markov-Chain Monte-Carlo: an example
θ
1
θ
2
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
The search algorithm explores the fitting criterion,
spending more time in regions of good fit.

24. MCMC: an example
Markov-Chain Monte-Carlo: an example
θ
1
θ
2
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
The search algorithm explores the fitting criterion,
spending more time in regions of good fit.

25. MCMC: an example
Markov-Chain Monte-Carlo: an example
θ
1
θ
2
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
The search algorithm explores the fitting criterion,
spending more time in regions of good fit.

26. MCMC: an example
Markov-Chain Monte-Carlo: an example
2 4 6 8 10 12 14
18
16
14
12
10
8
6
4
θ
1
θ
2
The result is samples of the probability distribution of the
parameters

27. Predictions (pure CO2)
R

28. Predictions (pure CO2)
10
-4
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
304.3K (Tc)
294K
285K
R

29. Predictions (pure CO2)
10
-4
10
-3
Molar volume [m^3/mol]
12
10
8
6
4
2
Pressure [MPa]
304.3K (Tc)
294K
285K
-4
10
Molar volume [m^3/mol]
8
6
4
Pressure [MPa]
Coexisting liquid
Coexisting vapour
R

30. Mixture modelling
CO2+N2

31. Introduction to non-parametric
methods
[6]

32. Introduction to non-parametric
methods
•Model for pressure against volume,
as with an equation of state.
•However, no need to specify terms or
parameters
•Model ‘learns’ the P(v) functional form
from the measurements [6]

33. 1
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1
x
0
f(x)
Introduction to non-parametric
methods
•Model for pressure against volume,
as with an equation of state.
•However, no need to specify terms or
parameters
•Model ‘learns’ the P(v) functional form
from the measurements
[6]
•Basic examples include splines
and other interpolation techniques
•Modern implementations are
significantly more sophisticated

34. 1
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1
x
0
f(x)
Gaussian processes
a) Generate random functions
from a distribution that favours
smooth functions

35. 1
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1
x
0
f(x)
Gaussian processes
a) Generate random functions
from a distribution that favours
smooth functions
1
0.8
0.6
0.4
0.2
0
Data Mean
Variance
0 0.2 0.4 0.6 0.8 1
x
b) Keep only the functions that
pass through the data points
f(x)
Mean of accepted functions = Model
Variance of accepted functions = Uncertainty quantification

36. A Gaussian process for pure CO2
1 0 1 2
−Pressure/(Critical Pressure)
pressure
0.2 0.4 0.6 0.8 1.0 2 −1 0 1 volume pressure
Temperature=290K
CO2 data
Gaussian Process mean.
95% confidence interval
Individual Gaussian Processes
Molar volume/(Ideal gas volume)

37. A Gaussian process for pure CO2
1 0 1 2
−Pressure/(Critical Pressure)
pressure
0.2 0.4 0.6 0.8 1.0 2 −1 0 1 volume pressure
Temperature=290K
Gaussian Process
accurately captures
the data
CO2 data
Gaussian Process mean.
95% confidence interval
Individual Gaussian Processes
Molar volume/(Ideal gas volume)

38. A Gaussian process for pure CO2
1 0 1 2
−Pressure/(Critical Pressure)
pressure
0.2 0.4 0.6 0.8 1.0 2 −1 0 1 volume pressure
Temperature=290K
Gaussian Process
accurately captures
the data
CO2 data
Gaussian Process mean.
95% confidence interval
Individual Gaussian Processes
Uncertainty is
only significant
in the
coexistence
region
Molar volume/(Ideal gas volume)

39. A Gaussian process for pure CO2
1 0 1 2
−Pressure/(Critical Pressure)
pressure
0.2 0.4 0.6 0.8 1.0 2 −1 0 1 volume pressure
Temperature=290K
Gaussian Process
accurately captures
the data
CO2 data
Gaussian Process mean.
95% confidence interval
Individual Gaussian Processes
Uncertainty is
only significant
in the
coexistence
region
Generalisation
to mixtures is
ongoing
Molar volume/(Ideal gas volume)

40. Molecular simulation
Computer
model
of
individual
molecules
within
a
small
box
of
fluid.
Can
predict:
•Pressure-­‐volume
•Coexistence
•Effect
of
impurity
•Most
other
quanBBes
of
interest
[7]

41. Molecular simulation
Computer
model
of
individual
molecules
within
a
small
box
of
fluid.
Can
predict:
•Pressure-­‐volume
•Coexistence
•Effect
of
impurity
•Most
other
quanBBes
of
interest
Can
be
used
where
experiments
are
unavailable?
[7]

42. Molecular simulation
Computer
model
of
individual
molecules
within
a
small
box
of
fluid.
Can
predict:
•Pressure-­‐volume
•Coexistence
•Effect
of
impurity
•Most
other
quanBBes
of
interest
Can
be
used
where
experiments
are
unavailable?
[7]
Can
be
used
to
derive
an
EquaBon
of
State?

43. Gibbs
ensemble
simulaBons
Gas
Liquid

44. Gibbs
ensemble
simulaBons
Two
simulaBon
boxes,
represenBng
coexisBng
phases
Gas
Liquid

45. Gibbs
ensemble
simulaBons
Two
simulaBon
boxes,
represenBng
coexisBng
phases
The
system
approaches
equilibrium
by
making
a
series
of
moves,
consistent
with
staBsBcal
mechanics
Gas
Liquid
ParBcle
displacement Volume
change
ParBcle
transfer
Once
in
equilibrium,
the
system
predicts
the
coexistence
properBes

46. M23
Molecular force-fields

47. M23
Molecular force-fields
•All
physical
proper-es
are
ulBmately
determined
by
interac-ons
between
molecules
•Force-­‐fields
that
describe
these
interacBons
are
a
key
input
to
simula-ons

48. M23
Molecular force-fields
•All
physical
proper-es
are
ulBmately
determined
by
interac-ons
between
molecules
•Force-­‐fields
that
describe
these
interacBons
are
a
key
input
to
simula-ons
•InteracBons
of
CO2
with
itself
and
with
impuri-es
must
be
specified
!

49. Semi-empirical forcefields
CO2+N2

50. Semi-empirical forcefields
CO2+N2
Simulations
using literature
force fields

51. Semi-empirical forcefields
CO2+N2
Simulations
using literature
force fields
Simulations after
optimising the
force field

52. Bubble point comparison
CO2 + 5%H2
Phase boundary
measurements
by Jie Ke, Mike
George et al

53. Simulation aids EoS development
xG: Gas composition
vG: Gas volume
Gas
Two phase region
Liquid
Gas
Molar Volume (litres/mol)
Pressure (MPa)
Liquid
Gas
Pressure (MPa)
Mole fraction of impurity
Liquid
xL: Liquid composition
vL: Liquid volume

54. Simulation aids EoS development
xG: Gas composition
vG: Gas volume
Gas
Two phase region
Liquid
Gas
Molar Volume (litres/mol)
Pressure (MPa)
Liquid
Gas
Pressure (MPa)
Mole fraction of impurity
Liquid
xL: Liquid composition
vL: Liquid volume

55. Simulation aids EoS development
xG: Gas composition
vG: Gas volume
Gas
Two phase region
Liquid
Gas
Molar Volume (litres/mol)
Pressure (MPa)
Liquid
Gas
Pressure (MPa)
Mole fraction of impurity
Liquid
xL: Liquid composition
vL: Liquid volume

56. Ab initio force fields
CO2+H2
Quantum Chemistry
calculations of CO2-
H2 interaction
Gaussian Process fit
for use in
simulations
+

57. Ab initio force fields
CO2+H2
Quantum Chemistry
calculations of CO2-
H2 interaction
Force field
computed from
first principles
Gaussian Process fit
for use in
simulations
+
Potential for accurate
predictions without
data fitting ⇒

58. Making it all work together
•Parametric equations of state
•Non-parametric EoS
•Semi-empirical molecular simulation
•Ab-initio molecular simulation

59. Making it all work together
•Parametric equations of state
•Non-parametric EoS
•Semi-empirical molecular simulation
•Ab-initio molecular simulation

60. Making it all work together
•Parametric equations of state
• Fast, flexible models for computational studies
• Fit to experiments, simulation data more advanced
EoS
•Non-parametric EoS
•Semi-empirical molecular simulation
•Ab-initio molecular simulation

61. Making it all work together
•Parametric equations of state
• Fast, flexible models for computational studies
• Fit to experiments, simulation data more advanced
EoS
•Non-parametric EoS
• Rigorous uncertainty quantification - optimise choice of
experiments
• (Somewhat) expensive but very accurate EoS
•Semi-empirical molecular simulation
•Ab-initio molecular simulation

62. Making it all work together
•Parametric equations of state
• Fast, flexible models for computational studies
• Fit to experiments, simulation data more advanced
EoS
•Non-parametric EoS
• Rigorous uncertainty quantification - optimise choice of
experiments
• (Somewhat) expensive but very accurate EoS
•Semi-empirical molecular simulation
• Accurate treatment of temperature variation
• Completes coexistence measurements to help EoS fitting
•Ab-initio molecular simulation

63. Making it all work together
•Parametric equations of state
• Fast, flexible models for computational studies
• Fit to experiments, simulation data more advanced
EoS
•Non-parametric EoS
• Rigorous uncertainty quantification - optimise choice of
experiments
• (Somewhat) expensive but very accurate EoS
•Semi-empirical molecular simulation
• Accurate treatment of temperature variation
• Completes coexistence measurements to help EoS fitting
•Ab-initio molecular simulation
• Most physically realistic but also most expensive.
• Can augment or replace experiments