Multi-scale modeling with generalized dynamic discrepancy

David S. Mebane,*,§ K. Sham Bhat¶ and Curtis B. Storlie¶

§National Energy Technology Laboratory

*Department of Mechanical and Aerospace Engineering,

West Virginia University

¶Statistical Sciences Group, Los Alamos National Laboratory

SIAM Annual Meeting, July 2013

multi-scale modeling

• Multi-scale modeling is an inherently statistical problem.

– What information will be propagated to the next scale?

– What is the role of experimental data?

– How do we account for uncertainty in parameters and models?

– How do we handicap (or prevent) extrapolation?

2

TM

equilibrium problems

3

TM

0 1 2 3 4 5 6

x 104

0

0.02

0.04

0.06

0.08

0.1

time (s)

weig

ht fr

actio

n

0 1 2 3 4 5 6

x 104

300

320

340

360

380

400

tem

pe

ratu

re (

K)

weight fraction

temperature

Fig. 2 TGA output for a sample exposed to 18.5% CO2, balance N2

Like many data sets, this one has some shortcomings that

should beacknowledged, but should not otherwisediscourage

the effort to deriveuseful information from the data. The gap

in the data between 65 and 92◦C arises from a desire to focus

on temperatures likely to be encountered in the adsorber and

regenerator of an actual capturesystem. It isclear fromseveral

studies of PEI-impregnated sorbents of this type11,13,14,18,42

that the diffusion of CO2 through the polymer bulk can be

a limiting step in the reaction, especially for highly loaded

sorbents at dry conditions and low temperatures. Although

thereissomeevidenceto suggest that theremay bealimitation

of this type in the sorbent of focus for thisstudy (see theslow

uptakeof CO2 at thefirst desorption step in Fig. 2), since this

sorbent has a relatively low loading of active material (22.5

wt.-% is slightly less than half of the theoretical maximum),

it is assumed that these effects are not too severe. It is also

clear that there is a significant – though not overwhelming –

observation error present in the data.

5.2 Quantum chemistry

5.2.1 Formation of carbamic acid in alkanol- and

ethyleneamines. Most theoretical studies of the interac-

tion between amines and CO2 have focused on amino al-

cohols in aqueous solution, particularly monoethanolamine

(MEA).43–51 The alkylammonium carbamate ion pair formed

through the adsorption reaction is often assumed to associate

throughahydrogenbond between alkylammoniumand theni-

trogen site of thecarbamateanion.44,46 This isconsistent with

the often-cited termolecular mechanism for the adsorption of

CO2 in thesesystems,52,53 wherein theencounter complex be-

tween gaseous CO2 and the amine is immediately (without

310 320 330 340 350 360 370 380 3902

3

4

5

6

7

8

9

10

temperature (K)

wt.-%

CO

2

4% CO2

7.5%

10%

18.5%

100%

Fig. 3 Equilibrium TGA data for the sorbent exposed to CO2 at

various concentrations, balance N2

barrier) deprotonated by a second amine. However, another

possibility is the formation of carbamic acid, by a transfer

of a proton from the nitrogen atom of the encounter com-

plex to one of the oxygens of the CO2 molecule. This can be

accomplished either by a bending of the encounter complex

or through a catalytic proton exchange with water or another

amine.43,54,55 Another, related possibility is the formation of

an alkylammonium carbamateassociatewith an H-bonded re-

lationship at theoxygen of thecarbamategroup.

Therelevant aminechemistry for thisstudy isnot amino al-

cohols, but ethyleneamines. It proved useful nonetheless to

perform some calculations on MEA in aqueous solution in

order to compare with the results of previous studies. Us-

ing the B3LYP DFT exchange-correlation functional and a

polarizable continuum model (PCM) to approximate the ef-

fects of solvation,56 the reaction energy for the formation of

the alkylammonium carbamate associate was calculated to be

-32.6 kJ/mol, which compares well with the result of -33.4

kJ/mol arrived at by Shim, et al.46 Xie, et al.,44 another group

to report the preferential formation of alkylammonium carba-

mate in MEA aqueous solution, reported only Gibbs free en-

ergies of reaction at 298K, in which estimates of configura-

tional entropies are likely to figure prominently. Their esti-

mate for alkylammonium carbamate depended on the atomic

radii used: UAHF radii yielded -20.5 kJ/mol while Pauling

radii yielded -15.5 kJ/mol.

Neither Xie, et al. nor Shim, et al. considered a car-

bamic acid associate involving a hydrogen-bonded relation-

ship betweenOH of theacid and thenitrogenof another amine

molecule, which was found to be the most stable form of ad-

sorbed CO2 in the present study, both for alkanolamines and

6 | 1–13

equilibrium problems

4

TM

0 1 2 3 4 5 6

x 104

0

0.02

0.04

0.06

0.08

0.1

time (s)

weig

ht fr

actio

n

0 1 2 3 4 5 6

x 104

300

320

340

360

380

400

tem

pe

ratu

re (

K)

weight fraction

temperature

Fig. 2 TGA output for a sample exposed to 18.5% CO2, balance N2

Like many data sets, this one has some shortcomings that

should beacknowledged, but should not otherwisediscourage

the effort to deriveuseful information from the data. The gap

in the data between 65 and 92◦C arises from a desire to focus

on temperatures likely to be encountered in the adsorber and

regenerator of an actual capturesystem. It isclear fromseveral

studies of PEI-impregnated sorbents of this type11,13,14,18,42

that the diffusion of CO2 through the polymer bulk can be

a limiting step in the reaction, especially for highly loaded

sorbents at dry conditions and low temperatures. Although

thereissomeevidenceto suggest that theremay bealimitation

of this type in the sorbent of focus for thisstudy (see theslow

uptakeof CO2 at thefirst desorption step in Fig. 2), since this

sorbent has a relatively low loading of active material (22.5

wt.-% is slightly less than half of the theoretical maximum),

it is assumed that these effects are not too severe. It is also

clear that there is a significant – though not overwhelming –

observation error present in the data.

5.2 Quantum chemistry

5.2.1 Formation of carbamic acid in alkanol- and

ethyleneamines. Most theoretical studies of the interac-

tion between amines and CO2 have focused on amino al-

cohols in aqueous solution, particularly monoethanolamine

(MEA).43–51 The alkylammonium carbamate ion pair formed

through the adsorption reaction is often assumed to associate

throughahydrogenbond between alkylammoniumand theni-

trogen site of thecarbamateanion.44,46 This isconsistent with

the often-cited termolecular mechanism for the adsorption of

CO2 in thesesystems,52,53 wherein theencounter complex be-

tween gaseous CO2 and the amine is immediately (without

310 320 330 340 350 360 370 380 3902

3

4

5

6

7

8

9

10

temperature (K)

wt.-%

CO

2

4% CO2

7.5%

10%

18.5%

100%

Fig. 3 Equilibrium TGA data for the sorbent exposed to CO2 at

various concentrations, balance N2

barrier) deprotonated by a second amine. However, another

possibility is the formation of carbamic acid, by a transfer

of a proton from the nitrogen atom of the encounter com-

plex to one of the oxygens of the CO2 molecule. This can be

accomplished either by a bending of the encounter complex

or through a catalytic proton exchange with water or another

amine.43,54,55 Another, related possibility is the formation of

an alkylammonium carbamateassociatewith an H-bonded re-

lationship at theoxygen of thecarbamategroup.

Therelevant aminechemistry for thisstudy isnot amino al-

cohols, but ethyleneamines. It proved useful nonetheless to

perform some calculations on MEA in aqueous solution in

order to compare with the results of previous studies. Us-

ing the B3LYP DFT exchange-correlation functional and a

polarizable continuum model (PCM) to approximate the ef-

fects of solvation,56 the reaction energy for the formation of

the alkylammonium carbamate associate was calculated to be

-32.6 kJ/mol, which compares well with the result of -33.4

kJ/mol arrived at by Shim, et al.46 Xie, et al.,44 another group

to report the preferential formation of alkylammonium carba-

mate in MEA aqueous solution, reported only Gibbs free en-

ergies of reaction at 298K, in which estimates of configura-

tional entropies are likely to figure prominently. Their esti-

mate for alkylammonium carbamate depended on the atomic

radii used: UAHF radii yielded -20.5 kJ/mol while Pauling

radii yielded -15.5 kJ/mol.

Neither Xie, et al. nor Shim, et al. considered a car-

bamic acid associate involving a hydrogen-bonded relation-

ship betweenOH of theacid and thenitrogenof another amine

molecule, which was found to be the most stable form of ad-

sorbed CO2 in the present study, both for alkanolamines and

6 | 1–13

5

equilibrium problems

TM

DS Mebane, KS Bhat, JD Kress, et al., Phys. Chem. Chem. Phys. 15 (2013) 4355.

6

application in amine-based CO2 sorbents

bivariate ΔH-ΔS

posterior (left) with

and (right) without

ab initio priors

TM

DS Mebane, KS Bhat, JD Kress, et al., Phys. Chem. Chem. Phys. 15 (2013) 4355.

7

equilibrium problems

(left) model +

discrepancy

predictions (right)

model-only

predictions, with

95% confidence

bounds

TM

DS Mebane, KS Bhat, JD Kress, et al., Phys. Chem. Chem. Phys. 15 (2013) 4355.

dynamic problems

8

TM

0 1 2 3 4 5 6

x 104

0

0.02

0.04

0.06

0.08

0.1

time (s)

we

ight

fra

ctio

n

0 1 2 3 4 5 6

x 104

300

320

340

360

380

400

tem

pe

ratu

re (

K)

weight fraction

temperature

Fig. 2 TGA output for a sample exposed to 18.5% CO2, balance N2

Like many data sets, this one has some shortcomings that

should beacknowledged, but should not otherwisediscourage

the effort to derive useful information from the data. The gap

in the data between 65 and 92◦C arises from a desire to focus

on temperatures likely to be encountered in the adsorber and

regenerator of an actual capturesystem. It isclear fromseveral

studies of PEI-impregnated sorbents of this type11,13,14,18,42

that the diffusion of CO2 through the polymer bulk can be

a limiting step in the reaction, especially for highly loaded

sorbents at dry conditions and low temperatures. Although

thereissomeevidenceto suggest that theremay bealimitation

of this type in thesorbent of focus for thisstudy (see the slow

uptakeof CO2 at thefirst desorption step in Fig. 2), since this

sorbent has a relatively low loading of active material (22.5

wt.-% is slightly less than half of the theoretical maximum),

it is assumed that these effects are not too severe. It is also

clear that there is a significant – though not overwhelming –

observation error present in thedata.

5.2 Quantum chemistry

5.2.1 Formation of carbamic acid in alkanol- and

ethyleneamines. Most theoretical studies of the interac-

tion between amines and CO2 have focused on amino al-

cohols in aqueous solution, particularly monoethanolamine

(MEA).43–51 The alkylammonium carbamate ion pair formed

through the adsorption reaction is often assumed to associate

through ahydrogenbond between alkylammoniumand theni-

trogen siteof thecarbamateanion.44,46 This isconsistent with

the often-cited termolecular mechanism for the adsorption of

CO2 in thesesystems,52,53 wherein theencounter complex be-

tween gaseous CO2 and the amine is immediately (without

310 320 330 340 350 360 370 380 3902

3

4

5

6

7

8

9

10

temperature (K)

wt.

-% C

O2

4% CO2

7.5%

10%

18.5%

100%

Fig. 3 Equilibrium TGA data for the sorbent exposed to CO2 at

various concentrations, balance N2

barrier) deprotonated by a second amine. However, another

possibility is the formation of carbamic acid, by a transfer

of a proton from the nitrogen atom of the encounter com-

plex to one of the oxygens of the CO2 molecule. This can be

accomplished either by a bending of the encounter complex

or through a catalytic proton exchange with water or another

amine.43,54,55 Another, related possibility is the formation of

an alkylammonium carbamateassociatewith an H-bonded re-

lationship at theoxygen of thecarbamategroup.

Therelevant aminechemistry for thisstudy isnot amino al-

cohols, but ethyleneamines. It proved useful nonetheless to

perform some calculations on MEA in aqueous solution in

order to compare with the results of previous studies. Us-

ing the B3LYP DFT exchange-correlation functional and a

polarizable continuum model (PCM) to approximate the ef-

fects of solvation,56 the reaction energy for the formation of

the alkylammonium carbamate associate was calculated to be

-32.6 kJ/mol, which compares well with the result of -33.4

kJ/mol arrived at by Shim, et al.46 Xie, et al.,44 another group

to report the preferential formation of alkylammonium carba-

mate in MEA aqueous solution, reported only Gibbs free en-

ergies of reaction at 298K, in which estimates of configura-

tional entropies are likely to figure prominently. Their esti-

mate for alkylammonium carbamate depended on the atomic

radii used: UAHF radii yielded -20.5 kJ/mol while Pauling

radii yielded -15.5 kJ/mol.

Neither Xie, et al. nor Shim, et al. considered a car-

bamic acid associate involving a hydrogen-bonded relation-

ship between OH of theacid and thenitrogenof another amine

molecule, which was found to be the most stable form of ad-

sorbed CO2 in the present study, both for alkanolamines and

6 | 1–13

dynamic problems

9

TM

0 1 2 3 4 5 6

x 104

0

0.02

0.04

0.06

0.08

0.1

time (s)

we

ight

fra

ctio

n

0 1 2 3 4 5 6

x 104

300

320

340

360

380

400

tem

pe

ratu

re (

K)

weight fraction

temperature

Fig. 2 TGA output for a sample exposed to 18.5% CO2, balance N2

Like many data sets, this one has some shortcomings that

should beacknowledged, but should not otherwisediscourage

the effort to derive useful information from the data. The gap

in the data between 65 and 92◦C arises from a desire to focus

on temperatures likely to be encountered in the adsorber and

regenerator of an actual capturesystem. It isclear fromseveral

studies of PEI-impregnated sorbents of this type11,13,14,18,42

that the diffusion of CO2 through the polymer bulk can be

a limiting step in the reaction, especially for highly loaded

sorbents at dry conditions and low temperatures. Although

thereissomeevidenceto suggest that theremay bealimitation

of this type in thesorbent of focus for thisstudy (see the slow

uptakeof CO2 at thefirst desorption step in Fig. 2), since this

sorbent has a relatively low loading of active material (22.5

wt.-% is slightly less than half of the theoretical maximum),

it is assumed that these effects are not too severe. It is also

clear that there is a significant – though not overwhelming –

observation error present in thedata.

5.2 Quantum chemistry

5.2.1 Formation of carbamic acid in alkanol- and

ethyleneamines. Most theoretical studies of the interac-

tion between amines and CO2 have focused on amino al-

cohols in aqueous solution, particularly monoethanolamine

(MEA).43–51 The alkylammonium carbamate ion pair formed

through the adsorption reaction is often assumed to associate

through ahydrogenbond between alkylammoniumand theni-

trogen siteof thecarbamateanion.44,46 This isconsistent with

the often-cited termolecular mechanism for the adsorption of

CO2 in thesesystems,52,53 wherein theencounter complex be-

tween gaseous CO2 and the amine is immediately (without

310 320 330 340 350 360 370 380 3902

3

4

5

6

7

8

9

10

temperature (K)

wt.

-% C

O2

4% CO2

7.5%

10%

18.5%

100%

Fig. 3 Equilibrium TGA data for the sorbent exposed to CO2 at

various concentrations, balance N2

barrier) deprotonated by a second amine. However, another

possibility is the formation of carbamic acid, by a transfer

of a proton from the nitrogen atom of the encounter com-

plex to one of the oxygens of the CO2 molecule. This can be

accomplished either by a bending of the encounter complex

or through a catalytic proton exchange with water or another

amine.43,54,55 Another, related possibility is the formation of

an alkylammonium carbamateassociatewith an H-bonded re-

lationship at theoxygen of thecarbamategroup.

Therelevant aminechemistry for thisstudy isnot amino al-

cohols, but ethyleneamines. It proved useful nonetheless to

perform some calculations on MEA in aqueous solution in

order to compare with the results of previous studies. Us-

ing the B3LYP DFT exchange-correlation functional and a

polarizable continuum model (PCM) to approximate the ef-

fects of solvation,56 the reaction energy for the formation of

the alkylammonium carbamate associate was calculated to be

-32.6 kJ/mol, which compares well with the result of -33.4

kJ/mol arrived at by Shim, et al.46 Xie, et al.,44 another group

to report the preferential formation of alkylammonium carba-

mate in MEA aqueous solution, reported only Gibbs free en-

ergies of reaction at 298K, in which estimates of configura-

tional entropies are likely to figure prominently. Their esti-

mate for alkylammonium carbamate depended on the atomic

radii used: UAHF radii yielded -20.5 kJ/mol while Pauling

radii yielded -15.5 kJ/mol.

Neither Xie, et al. nor Shim, et al. considered a car-

bamic acid associate involving a hydrogen-bonded relation-

ship between OH of theacid and thenitrogenof another amine

molecule, which was found to be the most stable form of ad-

sorbed CO2 in the present study, both for alkanolamines and

6 | 1–13

dynamic problems

10

TM

dynamic problems

11

TM

dynamic problems

12

TM

dynamic discrepancy

13

TM

Bayesian smoothing spline analysis of variance (BSS-ANOVA)

BJ Reich, CB Storlie and HD Bondell, Technometrics 51 (2009) 110.

dynamic discrepancy

14

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• Input domain for the GP drastically reduced. • Computational complexity of the GP evaluation reduced. • Integration of the SDE takes place in the course of the

MCMC routine.

15

dynamic discrepancy

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Proof-of-concept with “reality” (left) creating

data for calibration of a simpler model with

dynamic discrepancy (below).

dynamic discrepancy

16

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Bivariate posteriors for two of the parameters in t. Left is a case with informative priors, and right

with uninformative priors. The blue dot is “reality.”

dynamic discrepancy

17

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Extrapolative predictions in the case of upscaling to a steady-state process model. Left is a case

with informative priors, and right with uninformative priors. The black line is “reality.”

0 50 100 150

31

0320

33

0340

35

0360

Extrapolated Temperature Profile

Time (s)

Tem

pera

ture

(K

)

0 50 100 150

31

0320

33

0340

35

0360

Extrapolated Temperature Profile

Time (s)

Tem

pera

ture

(K

)

0 50 100 150

31

03

20

33

034

03

50

36

0

Extrapolated Temperature Profile

Time (s)

Tem

pe

ratu

re (

K)

0 50 100 150

31

03

20

33

034

03

50

36

0

Extrapolated Temperature Profile

Time (s)

Tem

pe

ratu

re (

K)

dynamic discrepancy

18

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conclusions

19

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• A novel, Bayesian perspective on multi-scale modeling is introduced. • A new framework for Bayesian calibration in dynamic contexts has been

demonstrated. • The dynamic discrepancy leads to the possibility of an iterative process that

can prevent extrapolation.

thanks

20

TM

David Miller Juan Morinelly Dan Fauth Mac Gray Andrew Lee

Joel Kress

Contact: David S. Mebane david.mebane@mail.wvu.edu

Disclaimer:This presentation was prepared as an account of work sponsored by an agency of the United States Government.

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