Using automated techniques to generate reduced mechanisms

Louise Whitehouse

University of LeedsDepartment of Chemistry

Introduction

• Why reduce mechanisms?• Techniques used in mechanism reduction

– Sensitivity analysis– Quasi-steady-state analysis– Species lumping

• Reduction of the Master Chemical Mechanism (MCM) v2.0

• Development and improvement of automatic reduction methods.

• Reduction of the isoprene subset of MCM.• Reduction of the isoprene subset of the

Common Reactive Intermediates (CRI) mechanism.

Reasons for mechanism reduction

• Large mechanisms such as the full MCM contain large numbers of reactions and species.

• Running scenarios is very computationally expensive – 5 day run can take several hours to simulate.

• By reducing the dimension of the system– it is possible to accurately represent species without

having to deal with a large and expensive mechanism.

– this will allow the investigation of a wider range of conditions in a shorter time period.

Automation of mechanism reduction

Reasons for automation• Large mechanisms make it unfeasible to carry

out analysis by hand.• Automation allows selection of tolerance

parameter and automatic generation of reduced mechanism

Techniques that can be used to do this:• sensitivity analysis

– removal of redundant species– removal of redundant reactions

• removal of quasi-steady-state species• species lumping

Sensitivity analysis - removal of redundant species• Important species

– the ones we are interested in e.g. HOx, NOx, NO3, PAN, O3, H2O2, RO2, HCHO, CH3CHO, MVK, MACR

• Necessary species– Must be included to produce accurate results for the important

species.– Coupled to the important species via significant reactions.

• Redundant species– Not significantly coupled to the set of important and necessary

species.– Can therefore be removed from the mechanism.

• Identification of redundant species– Effect of change in concentration of species i on the rate of

production of an N-membered group of important species.– Given by the sum of squares of normalised Jacobian elements

2

1 ln

ln

N

n i

ni c

fB (1)

Sensitivity Analysis (2) - Removal of redundant reactions

•Rate sensitivity analysis•effect of a perturbation of a rate parameter on the rate of production of a necessary or important species.

•The overall sensitivity measure Fj is given by:

)2(

2

i i

j

j

ij f

k

k

fF

•Any reaction j for which Fj < threshold can be removed from the mechanism.

•fi is the right-hand side of the ith rate equation•kj is the rate of the jth reaction

Quasi-steady-state analysis

•This assumption will cause an error Δci and this can be found for a species by calculating,

•If the QSSA error is small the species can be removed from the mechanism by applying the approximation (3).

•Sensitivity analysis tends to lead to the removal of slow time-scales.•Large numbers of fast and intermediate time-scales still remain.•Species associated with these fast time-scales can be identified using quasi-steady-state-analysis (QSSA).•The QSSA involves assuming that for a species ci,

dt

dc

Jc i

iii

1

0dt

dci (3)

(4)

Species Lumping

Formation of selected species.

Lumped Reactions

•A common approach to removing species with intermediate timescales is species lumping.

Lumped Species

Lumped Products

Products divided

can be replaced by:22

11

1

1

PR

PRk

k

22111 PPR k

lump •This leads to reduction in the dimension of the system.

•A group of species can be represented in the mechanism by a single variable.

Reduction of the Master Chemical Mechanism

• Automation of these techniques required to reduce MCM v2.

• MCM contains ~11000 reactions and 3500 species– Impossible to conduct the

analysis by hand.• 90 trajectories were

examined representing polluted UK conditions.

•Using PUMA data from Birmingham and Middlesbrough•0ppb < total NOx < 300 ppb•0.25 < VOC/NOx < 3

•The scenarios were run with emissions and initial conditions designed to simulate the required range of conditions.

Reduction of the MCM (1)

• Sensitivity and quasi-steady-state analysis stages of reduction removed many species with fast and slow time-scales.

Fast <10-4s

Slow >105s (~1 day)

Reduction of the MCM (2)

Intermediate

Total

•At stage 5 of the reductions the reduced mechanism contained 1969 species and 6168 reactions.

•These techniques leave a large block of species with intermediate time-scales which all contribute in some way to the formation routes of the important and necessary species chosen.

•The number of species with intermediate time-scales can be reduced by the application of species lumping.

Select new

species Si

Does Si have

the same

or similar lifetime to S1?

Does Si react in

the same

number of

reactions as S1

YES

Selecting species lumping groups

NONO Do the

reactions have rate

coefficients within a given

percentage of the same

reaction rate for S1?

YES

NO

In reactions with the

same rate coefficients

does Si react with the same

species as S1?

YES

Add Si to species

lump groupYES

NO

Peracid Example

115 peracids lifetime range 6.48 ×103 –8.1 ×104s

1.08 ×103 –8.1 ×104

range of lifetimes

(s)

9.35 ×103 –1.31 ×104

1.24 ×104 –1.96 ×104

1.48 ×104 –5.29 ×104

6.48 ×103 –1.0 ×104

number of

species

59

42

3

7

4

range of reaction rate with OH (s)

4.52 ×10-11 –3.7 ×10-12

5.44 ×10-10 –3.03 ×10-11

3.00 ×10-11 –1.47 ×10-11

3.05 ×10-11 –5.35 ×10-12

6.90 ×10-11 –3.69 ×10-12

J(41)

J(24), J(41)

J(15), J(41)

J(22), J(41)

J(18), J(19), J(41)

Identical rates:

Formation of lumped equations

• The i species Rj, were originally each formed in only one reaction as follows,

productsRSS jl,j1, j with rate coefficient kj.

• If the group of species Rj satisfy the lumping criteria and all react with NO at a rate k1 then i equations,

jj PNOR can be replaced by a single equation,

iijjlump PPPNOR 11

ilump RRRRR j 21where

Relative product concentrations

iijjlump PPPNOR 11

•New lumped species Rlump will be formed through i production channels.

•The ratio between the rate at which the lumped species is formed through each channel can be used to calculate a variable coefficient for each product species in the lumped equation i.e. σj

•σj can be defined as,

]).[S]([Sk θθ

θσ jljjji

m m

jj ,,1

1

where

Un-lumped Peracid example

C6H5CO3H C6H5CO3H+ H2O

+O

2

+ OH

C6H5O2+ OH + CO2

C6H5CO3 + H2O

2-CH3C6H4CO3 2-CH3C6H4CO3H+ H2O

+O2

+ OH

2-CH3C6H4O2+OH +CO2

2-CH3C6H4CO3H + H2O

CH3CO3 + H2O CH3CO3H 2

+O2 CH3O2+ OH + CO2

CH3CO3 + H2O+ OH

6 Reactions

All have a chemical lifetime of 8.1×104s. Each species reacts with OH at a rate in the range 3.7-4.7×10-12sEach species reacts with O2 at rate represented by J(41).Each species is produced at the same rate, k.

Lumped Peracid Example• Define a lump:

LUMP1CO3H = CH3CO3H + C6H5CO3H + 2-CH3C6H4CO3HCH3CO3

C6H5CO3H

2-CH3C6H4CO3

+ H2 O

LUMP1CO3H+ H2O

+ H 2

O

+O2

+ OH

σ1 CH3O2+ σ2 C6H5O2

+ σ3 2-CH3C6H4O2OH + CO2

σ1 CH3CO3 + σ2 C6H5CO3 +

σ3 2-CH3C6H4CO3H +

H2O

2 Reactions

3

1j j

ii

θ1=k[CH3CO3]

θ2=k[C6H5CO3]

θ3=k[2-CH3C6H4CO3H]

Peracid Example continued

• 4 reactions and 2 species have been removed from the system.

• This lump was a part of a much larger lump. • Creation of the full lump leads to the removal of

– 58 species – 116 reactions.

• The reaction rate with OH varied over the whole group between 4.52×10-11 – 3.7×10-12

• These techniques can be used on a total of 68 groups.

Comparison of results with full mechanism

• 802 species lumped into 68 groups.

• 734 species removed.• 1777 reactions removed.• The final lumped mechanism

contains 4391 reactions and 1235 species

• This was shown to be able to accurately reproduce species concentrations over a wide range of conditions.

Species remaining at each level of reduction

Fast <10-4s Intermediate

Slow >105s Total

Scenario 8 – High NOx

Scenario 63 – Low NOx

Conclusions to MCM reduction

• Final reduced mechanism– Number of species reduced by 64%.– Computational time reduced by 88%.

• Errors– < 5% for many of the trajectories studied and 10% for the

majority.• Lumping strategy could be significantly improved to

increase optimality.– Current technique for identification of valid lumps still involves

manual selection and implementation.• It was decided to apply the same techniques to a subset

of the MCM v3.2 to develop them and to compare the results with mechanisms reduced by hand.

• This would also allow the development of a more automated lumping strategy.

Isoprene mechanism

• An MCM derived isoprene mechanism was constructed and run over three different scenarios.

• These looked at the time series of the 13 most important compounds: – HOx, NOx, NO3, PAN, O3, H2O2 and RO2 ,HCHO, CH3CHO,

MVK and MACR.• Three scenarios were chosen with different [VOC]/[NOx].• Isoprene was kept constant along the five day run

scenario CO O3 SO2 H2 H2O2 C5H8 NO NO2[VOC]/ [NOx]

H 120 50 1 500 2 1 120 30 0.01

I 120 50 1 500 2 1 3 7 0,1

L 120 50 1 500 2 0.5 0.001 0.01 83.3

•All concentrations are ppb.

Reduction of the isoprene subset

• This mechanism had previously been reduced by removing redundant reactions and species and QSSA species by hand.

• These reduction techniques were repeated using the automated routines.

• In addition the techniques used for species lumping were developed to make the identification of potential lumps simpler.

• It should then be possible to compare the size and accuracy of the mechanisms using the two strategies.

Implementation of species lumping

• Many reactions in the MCM have the same rate coefficient.

• This method of species lumping exploits this by identifying the number of reactions each species reacts in and the rate coefficient of that reaction.

• This information is stored in a matrix and then sorted to automatically generate groups of species which react at the same rate with the same species.

• Each group can then be automatically replaced in the reaction and species source files by a single lumped species.

• Unfortunately it is not yet possible to tell which species or potential lumps will cause accuracy problems.

• This is currently under investigation.

Development of lumping techniques

• Reactions with the same reactants and fractional rates are combined to form a single reaction.– 2.4×10-12× 0.8 × RO2

ISOPAO2→ISOPAO– 2.4×10-12× 0.1 × RO2 ISOPAO2→HC4ACHO– 2.4×10-12× 0.1 × RO2 ISOPAO2→ISOPAOH

become a single equation of the form,– 2.4×10-12 × RO2

ISOPAO2 → 0.8 ISOPAO + 0.1 HC4ACHO+ 0.1ISOPAOH

• This allows more reactions to be considered when automatically searching for potential lumps.

Species removed using lumping

Number of species

Example species

Reactions +

L1 7 ISOPAO2 HO2 NO NO3

L2 10 ISOPBOOH OH

L4 2 CH2OO CO

L5 6 CH3CO3 HO2 NO NO2

L7 2 MACRO2 HO2 NO

L8 2 PRONO2AO2 HO2

Scenario Number lumps

Species

involved removed

H 17 83 66

I 17 75 58

L 12 58 46

e.g. Low

Results

  

full 

 reduced mechanisms 

H I L

species 224 83 120 112

reactions 703 221 306 313

  

full 

 reduced mechanisms 

H I L

species 224 147 166 111

reactions 703 461 384 279

Automatic results after QSSA

  

full 

 reduced mechanisms 

H I L

species 224 83 67 71

reactions 703 285 218 206

Automatic results after lumping

Manual results after QSSA

Results

high intermediate low0

50

100

150

200

250

300

350

400

450 Roberto After QSSA After lumping

Nu

mb

er

of S

pe

cie

s

Scenario

Numbers of reactions remaining after reduction

high intermediate low0

20

40

60

80

100

120

140

160 Roberto after QSSA after lumping

Nu

mb

er

of S

pe

cie

s

Scenario

Numbers of species remaining after reduction

•After QSSA significantly more species and reactions have been removed by the manual method•This suggests that the automatic method needs to be adjusted to allow greater levels of reduction.•After lumping is applied

•species numbers are lower for all scenarios•reactions numbers are lower for the I and L scenarios.

Lifetime range plots

slow intermediate fast0

20

40

60

80

100

120

full lumped

Num

ber

of S

peci

es

Scenario

High

slow intermediate fast0

20

40

60

80

100 full lumped

Num

ber

of S

peci

es

Scenario

Intermediate

slow intermediate fast0

20

40

60

80

100

120

full lumping

Num

ber

of S

peci

es

Scenario

Low

After automatic lumping many slow lifetimes still remain.Again this indicates that the automatic sensitivity analysis stage could be improved.Fast time-scales are almost entirely eliminated.

Fast Slow

Time (s) <10-4 >105 (1 day)

Results - High NOx Conditions

0 1x105 2x105 3x105 4x1050

20

40

60

80

100

120

140

160

(ppb

)

Time (s)

NO2

0 1x105 2x105 3x105 4x1050

5

10

15

20

25

30

35

40

(ppb

)

Time (s)

NO

0 1x105 2x105 3x105 4x1050.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

1.4x10-4

1.6x10-4

1.8x10-4

(ppb

)

Time (s)

OH

0 1x105 2x105 3x105 4x1050

20

40

60

80

100

(ppb

)

Time (s)

O3

Results - Intermediate NOx Conditions

0 1x105 2x105 3x105 4x1050.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8(p

pb)

Time (s)

NO

0 1x105 2x105 3x105 4x1050

2

4

6

8

10

(ppb

)

Time (s)

NO2

0 1x105 2x105 3x105 4x1050.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

(ppb

)

Time (s)

OH

0 1x105 2x105 3x105 4x105

20

30

40

50

60

70

80

(ppb

)

Time (s)

O3

Results - Low NOx Conditions

0 1x105 2x105 3x105 4x1050.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

(ppb

)

Time (s)

NO

0 1x105 2x105 3x105 4x1050.000

0.005

0.010

0.015

0.020

(ppb

)

Time (s)

NO2

0 1x105 2x105 3x105 4x1050

10

20

30

40

50

(ppb

)

Time (s)

O3

0 1x105 2x105 3x105 4x1050.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

(ppb

)

Time (s)

OH

Common Reactive Intermediates Mechanism1

• The CRI mechanism is a lumped reduced mechanism.– ~570 reactions and ~250 species.

• Derived from the MCM therefore has potential for the application of species lumping.– Identification of large groups with potential for lumping should be

possible.

• The isoprene subset examined contained 160 species and 439 reactions.

• Scenarios H, I and L are again used for the reduction.• There are no emissions added.

1 Jenkin et al. Atm. Env 36 (2002) 4725-4734

Potential Lumps – CRI mechanism

•There are 6 main potential lump groups.

Number of species

Example Species

Reactions +

L1 42 NRN15O2 NO3 HO2 NO

L2 20 ETHGLY

L3 3 CARB17 OH

L4 11 C2H5NO3 OH

L5 4 RN10NO3 OH

L6 43 C2H5OOH OH

•This could potentially lead to the removal of 117 species and 297 reactions.

Lumping

• In total 128 species and 345 reactions are removed– This is due to some species gaining infinite lifetimes

when other species are lumped.– Other species have very short lifetimes and can also

be removed.• The mechanism can be reduced to

– 32 species and 94 reactions for high and intermediate NOx

– 45 species and 151 reactions for low NOx.• Lumping of CRI mechanism allows a larger

proportion of the remaining species to be lumped than in the ordinary isoprene mechanism.

Lifetime range plots

slow intermediate fast0

20

40

60

80

100

full lumped

Nu

mb

er

of S

pe

cie

s

Scenario

Intermediate

slow intermediate fast0

20

40

60

80

100

full lumped

Nu

mbe

r o

f Sp

eci

es

Scenario

Low

slow intermediate fast0

20

40

60

80

100

full lumped

Num

ber

of S

peci

es

Scenario

High Fast Slow

Time (s) <10-4 >105 (1 day)

Large number of all lifetime ranges are removed.

Crimech High NOx

1x105 2x105 3x105 4x1050

5

10

15

20O

3

(pp

b)

Time (s)1x105 2x105 3x105 4x105

0.0

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

1.2x10-5

OH

(ppb

)

Time (s)

1x105 2x105 3x105 4x10570.0

72.5

75.0

77.5

80.0

82.5

85.0

87.5

90.0NO

(pp

b)

Time (s)1x105 2x105 3x105 4x105

10

20

30

40

50

60

70

80NO

2

(pp

b)

Time (s)

Crimech Intermediate NOx

1x105 2x105 3x105 4x1050

10

20

30

40

50 O3

(ppb

)

Time (s)1x105 2x105 3x105 4x105

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4OH

(ppb

)

Time (s)

1x105 2x105 3x105 4x1050.0

0.5

1.0

1.5

2.0

2.5

3.0 NO

(ppb

)

Time (s)1x105 2x105 3x105 4x105

0

1

2

3

4

5

6

7

8NO

2

(pp

b)

Time (s)

Crimech low NOx

1x105 2x105 3x105 4x1050.000

0.002

0.004

0.006

0.008

0.010NO

2

(ppb

)

Time (s)1x105 2x105 3x105 4x105

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

NO

(ppb

)

Time (s)

1x105 2x105 3x105 4x1050.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

1.4x10-4

1.6x10-4

1.8x10-4OH

(ppb

)

Time (s)1x105 2x105 3x105 4x105

0

10

20

30

40

50O

3

(ppb

)

Time (s)

Future work

• Reduce full Crimech mechanism using species lumping.

• Improve application of automatic sensitivity analysis.

• Refine isoprene reductions to eliminate more slow species and eliminate further reactions.

• Carry out specific comparisons between reduced mechanisms produced via the two techniques.

Acknowledgements

• Thanks to Roberto and Mike Jenkin for providing me with various mechanisms.