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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