Model-based experimental analysis for inter-polymer process

CMU: Weijie Lin, Lorenz T. Biegler,Annette Jacobson

NOVA Chemicals: Paul Arch, Hany Farag, Michel Berghmans, Jennifer Chen, Elliot Wolf

Outline

Project overviewPrimitive model constructionModel refinement and existing challengesFuture work and conclusion

Project overview

Model-based experimental analysisThe link between experiment and industry process

Kinetic modeling is a key technologyDesign, control, operation of manufacturing processBetter understand of the micro-scale phenomena in chemical systems

Experiment Industry process

4

Polystyrene (PS) + Polyethylene (PE)

ARCEL

TOUGH FLEXIBLE╳

Advanced packaging material

Interpenetrating polymer network product processed in a sequential way

Polymer A

Polymer B

Project overview

Inter-polymer process

Project overview

Model-based experimental analysis for inter-polymer process

Objective --- Model-Based experimental designChallenge --- Complex kinetic mechanism

Major questions were raised How to build an initial modelHow to solve the inverse problemHow to evaluate the modelHow to implement the experimental design

Primitive model construction(1) Reduce the complexity of the Inter-polymer systemActive unit concept

Compact kinetic model scheme (Free radical Polymerization)Initiation Propagation Termination

Primitive model construction(2) Inverse approach for parameter estimation

7

0

( , , , ) 0

( | , (0), (0),0)

0,

t

n m

dxf x y tdtdxf x ydt

x R y R

=

=

=

∈ ∈

1

1

( ( ))Tu uu j

Min Tr V M

e V e

θ−

−

=

∑ ∑

0

( , , , ) 0

( | , (0), (0),0) 0,

,

t

L U L U

dxf x y tdt

dxf x ydt

x x x y y y

=

=

=

≤ ≤ ≤ ≤

S.t.

Apparent kinetic reaction rates

•Gel effect•Glass effect•Cage effect

Arrhenius equation aERTK Ae−=

1( ),K t 2( ),K t 4( )K t3( ),K t …( )K time ( )K X

Parameter-estimation-assisted

Input OutputModel

Simulation Parameter estimation

8

2. juu jMin e∑ ∑

0

( , , , ) 0

( | , (0), (0),0) 0,

L U

L U

t

dxf x y tdt

x x xy y y

dxf x ydt =

=

≤ ≤

≤ ≤

=

S.t.

11

1 ,

( ) ( )NC

ii i q

q i i q

t t dyy t y hh dt

−−

=

−= + Ω∑

Discretize all the variable in time at Radau collocation points

11 ,

( ) (1)NC

i i qq i q

dyy z y hdz−

=

= + Ω∑

Enforce continuity at each finite element

Solved as large scale nonlinear optimization problem (IPOPT)

Simultaneous approach

var . 3,000≥ In each sub-model

Primitive model construction(2) Inverse approach for parameter estimation (Cont.)

Convergence and robustness problems of the nonlinear estimation algorithms

Compute implementation

250 300 350 400 450 500 550

Time ( min )

(0.9)

(1.802e8)(2.908e4)

(53.70%)

Cage effect

Glass effect

Gel effect

PS weight fractionFeeding rate speeds up

200 250 300 350 400 450 500 550 60010

20

30

40

50

60

70

80

90

Time (min)

Con

vers

ion

ratio

%

Experiment Simulation

200 250 300 350 400 450 500 550 6000

50

100

150

200

250

300

350

Time (min)

Experiment

Simulation

Mw

MnM

olec

ular

Wei

ght

(kg/

mol

)

Simulation resultsComparisons

9

Primitive model construction (2) Inverse approach for parameter estimation (Cont.)

(Result for Polymerization Stage)

Based on the first set of data

Primitive model construction (3) Hybrid methods for property simulation

Step 1: Separated Molecular weight distribution

Molecular weight distribution / Network fraction

Step 1

Step 2

Step 3

Crosslinked- B

Grafted- B

Homo- B

Original A Molecular weight distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7Log (Molecular Weight)

dWf/d

Log

M

0

Moles n

wfWeight fractionLChain Length

22

11

A

A A

A

A A

n Nn N N wf Nn Nn N N wf N

C nSfC n N wf

⋅ −⋅ + ⋅ ⋅ −⋅ −⋅ + ⋅ ⋅ −

= ≈+ ⋅

L

Moles N1. Total activated units.2. Soluble fraction for Chain Length

Step 2: Network fraction

Primitive model construction (3) Hybrid methods for simulation (Cont.)

1. Total Network

(A/ A-g-B) network + Crosslinked B

2. Combined Molecular weight

[ ( )]A g g BL M P EX M −⋅ + ⋅

[ ( )]A g g BW wf n L P EX M Sf−⋅ + ⋅ ⋅ ⋅ ⋅

A / B Non-Network

Combined Molecular Weight

Combined Mass

+ Homo-B

grafted unitsg

total units

nP

n−

−

=Grafting probability

Step 3: Combined distribution

2.5 3 3.5 4 4.5 5 5.5 6 6.5

0

0.1

0.2

0.3

0.4

0.5

0.6

Log (Molecular Weight)

dWf / dlogM Original A

Combined distribution

Combined molecular weight distribution Simulation (Step 3)

Primitive model construction (3) Hybrid methods for simulation (Cont.)

13(Result for Polymerization Stage)

Simulation results

ComparisonsUpdate the parameter estimation with new experimental data for a good fit

Time (min)

Wei

ght a

vera

ge m

olec

ular

wei

ght

kg/m

ol

Average molecular weight development

Poly

mer

mas

s in

the

syst

em k

g

Time (min)

ExperimentSimulation

ExperimentSimulation

Polymer weight fraction

Kt (

L/m

ol *

min

)

Gel effect

Model refinement and existing challenges Coefficients sub model

Inversed problem for kinetic coefficientsCoefficients model discrimination

Whether the parameter is identifiableDefine the evaluation criteria

Input Output

Model A

Model B

Model C

…

A lot of alternatives

“Best” Model ?

Model C

Model A

Model B

Which is best?

Model refinement and existing problems Model discrimination

Better fit Better model

Alternative modelsPolymer molecular theoryEmpirical formulaBlack-box model

Discrimination CriteriaPhysical acceptability Compatibility with data diagnostic plot Cp plot (Mallows,1964)AIC score (Akaike, 1974)The determinant of the parameter estimation equation (Bilardello, 1993)Posterior probability (Stewart, etal. 1998)

Model refinement and existing problems Model discrimination (Cont)

More things are worth to be consideredMulti-phase system

Phase behavior modelThermodynamic model for phase equilibriumMolecular simulation (Coarse-grained model)

Multi-Component Diffusion in polymeric liquidsFlux model

Macroscopic continuum thermodynamics and classical transport phenomenaMesoscopic or coarse-grained level for complex fluids dynamics

Model refinement and existing problems Incremental Model Development

Future work

Develop a systematic model search approach for the model identification

Goodness of fit test for the sub-model and structured model

Model-based experimental design

Conclusion

Kinetic model-based experimental design is one of the key technologies for chemical manufacturing process A primitive kinetic model is built for inter-polymer processModel refinement and discrimination is being developed with updating experimental dataA systematic approach is needed for developing the integrated model structure