Modeling, estimation and control of distributed parameter systems ...
Modeling and Parameter Estimation of Batch Solid-Liquid...
Transcript of Modeling and Parameter Estimation of Batch Solid-Liquid...
Parameter estimation and model discrimination of batch solid-liquid reactors
Yajun Wang, Lorenz T. Biegler
Carnegie Mellon University
Mukund Patel, John Wassick
The Dow Chemical Company
March 2017
Enterprise-wide Optimization Meeting
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
Problem Statement & Motivation
• Batch reactor
Preparation
Solid-liquid Reaction
Solvent
Solid W
Liquid X
2
Solvent and reactant
materials
Reactor discharge
Vent
Agitator
Reactor
jacket
Cooling water
inlet
Cooling water
outlet W(s) + X(l) Y(s/l) + Z(s)
• Unknown reaction mechanism –
• Limited information –
Surface reaction, dissolution, diffusion
Different particle shapes and sizes
Product effects
Lack of concentration measurements
An over-parameterized model without enough data
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
Reaction Mechanisms
Shrinking particle model (SPM)
3
Reactant reactant Fluid
film
b
lc
s
lc
Liquid reactant diffuses onto the particle surface
Solid-liquid reaction
Solid product breaks off from reaction surface.
Dissolution model (DM)
Reactant reactant
Solid particles dissolve into solvent
Liquid–liquid reaction
Products precipitate into solid phase.
Assume dissolution is the rate limiting step.
Reaction rate of SPM depends on liquid reactant concentration while DM doesn’t.
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
Batch Reactor Model
4
• Model indicator
F=0 DM
F>0 SPM
• Parameters
1/ 1 1/
0
0
a ass s
s
aMS N N
R
Total surface area is related to total amount and the shape of particles[1].
[1] Salmi, Tapio, et al. "New modelling approach to liquid–solid reaction kinetics: From ideal particles to real particles." Chemical
Engineering Research and Design 91.10 (2013): 1876-1889.
1/ 1 1/
0 0
0
1/ 1 1/
0 0
0
(c )a
s s
a
s s
E
a a ss s RTs s s s l
s
E
a as s RTs s s
s
dN aMSR N N k e
dt R
dN aMkS N N k e
dt R
SPM
DM
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
Parameter Estimation
5
Estimability evaluation
Simultaneous estimation
Estimation quality analysis
Posterior probability share
Sensitivity coefficient matrix
QR transformation with column permutation
Rank parameters by their individual variance contribution Estimable?
YES EVM formulation > WLS formulation
• Data reconciliation
• Parameter estimation
Small Confidence
interval?
Reduce parameter set/ Simplify model
YES
NO
NO
Estimation covariance matrix
Reduced Hessian – confidence region
Model Validation
Posterior probability share
Bayes’ theorem
• Fitting
• Number of parameters
Validation by unseen batch data
Cross validation
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
Industrial Case Study
Estimability evaluation
• Parameter rankings at different points are different • All parameters are included in the estimation
Rank parameters by QRcp of scaled sensitivity matrix
6
Solid-liquid batch reactor
• Jacket temperatures • Inlet flowrates • Reactor weights
• Reactor temperatures • Concentrations
• Batch k k = 1 … NS • Measured input ukj* • Measured output ηki* • Normally distributed measurements
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
7
Measured output errors
Simultaneous estimation -- EVM
• EVM does parameter estimation and data reconciliation
simultaneously
• Better output data fitting Accumulated squared errors of EVM is
reduced by 44% compared with WLS
• Intensive computational load EVM has 15771 variables and
13830 equation constraints
Industrial Case Study
Measured input errors
• Reactor temperatures • End-point concentrations • Jacket temperatures • Weights and flowrates
0 0.2 0.4 0.6 0.8 10.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Scaled time
Sca
led
te
mp
era
ture
Reactor temperature
Predict
Data
WLS
0 0.2 0.4 0.6 0.8 10.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Scaled time
Sca
led
te
mp
era
ture
Reactor temperature
Predict
Data
EVM
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
8
Industrial Case Study Estimation quality analysis
• Set Cb,X = Cs,X
• Set F = 0 dissolution model
• Fix parameter UA
Linearly temperature-dependent U
Posterior probability share
Problem Statement & Motivation
Reactor Modeling
Significance & Potential Impact
Parameter Estimation
Industrial Case Study
9
Industrial Case Study Estimation results
Unseen data
Model Validations
1 2 3 4 5 6 7 8 90.09
0.095
0.1
0.105
0.11
0.115
0.12
Iteration
Pa
ram
ete
r A
3
Estimated value in each iteration
Average of 9 estimations
Estimated value by all data
New data 9-fold cross validation
Problem Statement & Motivation
Reactor Modeling
Conclusions & Significance
Parameter Estimation
Industrial Case Study
Conclusions & Significance
• Discussed two potential models for a solid-liquid reaction in a batch
reactor
• Built a uniform dynamic model with an indicating parameter to
discriminate two mechanisms
• Introduced a parameter estimation procedure
• Applied parameter estimation to the solid-liquid batch reactor model
• Conducted model validation to examine the model capability of
predicting system behaviors
Optimal control will be conducted based on estimated model to
reduce the batch time
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