modeling and parameter estimation of batch solid-liquid...
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Modeling and Parameter Estimation of Batch Solid-Liquid Reactors
Yajun Wang, Lorenz T. Biegler
Carnegie Mellon University
Mukund Patel, Yisu Nie, John Wassick
The Dow Chemical Company
08/18/2016
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem StatementProblem
Statement
Problem Statement• Batch reactor
Preparation
Reaction
Solvent
Solid W
Liquid X
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Solvent and reactant materials
Reactor discharge
Vent
Agitator
Reactor jacket
Cooling water inlet
Cooling water outlet
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem StatementProblem
Statement
Solid-liquid Reactions
W(s) + X(l) Y(s/l) + Z(s)→
• Surface reaction, dissolution, diffusion -- where does the reaction occur, on the solid surface or in the liquid phase?
• Different particle shapes and sizes -- how to incorporate reaction surface?
• Product effects – where are the products, accumulating on the reaction surface or cracking off?
Solid and liquid reactants react to generate solid or liquid products
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reaction Mechanism
Reaction Mechanism 1Shrinking particle model
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Reactant reactantreactantFluid film
bic
sic
Liquid reactant diffuses onto the particle surface
Solid-liquid reaction
Reaction
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reaction Mechanism
Shrinking particle model
• Pseudo-steady state is assumed on the reaction surface.
Reaction rate Rk is a function of surface concentration of the liquid reactant
• Total surface area is related to total amount and the shape of particles.
σ is the specific surface area with the unit [m2/kg]. a is a dimensionless shape factor.
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1/ 1 1/0a a
s s sS M N Nσ −=
1(c ) 0
Kb s
dl l l lk kk
k c Rυ=
− + =∑
Reaction Mechanism 1
Fick’s Law
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reaction Mechanism
Shrinking particle model
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1/ 1 1/0 0
1 1
1/ 1 1/0 0
1 1
Solid: (c )
Liquid: (c )
aks s k
aks s k
EK Ka a ss RT
sk k sk s s s s k lk k
EK Ka a sl RT
lk k lk s s s s k lk k
dN SR M N N k edtdN SR M N N k edt
α
α
υ υ σ
υ υ σ
−−
= =
−−
= =
= =
= =
∑ ∑
∑ ∑
b ll
l
NcV
=
1(c ) 0
Kb s
dl l l lk kk
k c Rυ=
− + =∑
0
(c ) k
ak
sk k l
ERT
k k
R k
k k e
α
−
=
=
Reaction Mechanism 1
Surface reaction rate depends on surface concentration of the liquid reactant.
Surface area Reaction rate
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reaction Mechanism
Reaction Mechanism 2Dissolution model
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Reactant reactantreactant
Solid particles dissolve into solvent
Liquid – liquid reaction
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reaction Mechanism
• Assume dissolution is the rate limiting step. (Reaction is much faster than dissolution.)
• Dissolution rate constant is Arrhenius type.
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sdN kSdt
= − 1/ 1 1/0a a
s s sS M N Nσ −=
0
aERTk k e
−=
1/ 1 1/0 0
1
1/ 1 1/0 0
1
Solid:
Liquid:
as s
as s
EKa as RT
sk s s s sk
EKa al RT
lk s s s sk
dN M N N k edtdN M N N k edt
υ σ
υ σ
−−
=
−−
=
=
=
∑
∑
Reaction Mechanism 2
Surface area Dissolution rate
Dissolution rate doesn’t depend on surface concentration of the liquid reactant.
Dissolution model
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reactor Model
Batch Reactor Model Lumped parameters A, B, D, E, F and heat transfer coefficient U are to be estimated.
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Reactor Model
Batch Reactor Model
Surface concentration of liquid reactant
F=0 Dissolution modelF>0 Shrinking particle modelOne formulation for two mechanisms
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Parameter Estimation
Parameter EstimationWeighted least-square formulation (WLS)
Measured output errors
• End-point concentrations• Reactor temperatures
+ Smaller problem to solve.- Ignoring input measurement errors.- Fewer degree of freedom, less control.
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement
Parameter Estimation
Parameter Estimation
+ Considering both input and output measurement errors. + Doing parameter estimation and data reconciliation at the same time.+ More control of the problem, better data fitting. - More decision variables, larger problem to solve.
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Measured output errors Measured input errors
Errors-in-variables-measured formulation (EVM)
• End-point concentrations• Reactor temperatures• Jacket temperatures• Weights and flowrates
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results 13
Comparison of WLS & EVM
Fitting by EVM is much better than it by WLS.Accumulated squared error of EVM is smaller than 1/3 of 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
tem
pera
ture
Reactor temperature
PredictData
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 timeS
cale
d te
mpe
ratu
re
Reactor temperature
PredictData
WLS EVM
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results 14
Estimation ResultsEstimated parameter values and confidence levels by EVM
Large confidence level indicates the parameter is not estimable.
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results 15
Parameter Selection• Get rid of parameters with large confidence level D
• Simplify heat transfer coefficient
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Scaled temperature
Heat tranfer coefficient
Sca
led
U
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results
Modified estimation results
Small confidence level of all parameters.
Estimation Results
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results
Estimation Results
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Fitting of jacket temperature(measured inputs)
Fitting of reactor temperature(measured outputs)
Data fitting
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
tem
pera
ture
Reactor temperature
PredictData
0 0.2 0.4 0.6 0.8 10
0.5
1Jacket inlet temperature
PredictData
0 0.2 0.4 0.6 0.8 10
0.5
1
Scaled time
Jacket outlet temperature
Sca
led
tem
pera
ture
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Results
Estimation Results
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Scaled time
Sca
led
mol
ar a
mou
ntComponent molar amounts vs. reaction time
X (Liquid reactant)W (Solid reactant)Y (Product)
Reaction profiles W(s) + X(l) Y(s/l) + Z(s)→
ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Conclusions
Conclusion and Future work• Shrinking particle and dissolution models of solid-liquid
reactions are explored and an uniform dynamic model isimplemented for both mechanisms.
• Parameter estimation is conducted based on limitedindustrial data. EVM method leads to better data fitting ofboth jacket temperatures and reactor temperatures.
• An analysis of estimation results is presented to enhanceproblem estimability.
------------------------------------------------------------------------------------• More data is going to be collected to validate the model.• Recipe optimization and process control are going to be
implemented after model validation.
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ConclusionsResultsParameterEstimation
Reactor Model
Reaction Mechanism
Problem Statement Conclusions
References• Forryan, Claire L., et al. "Heterogeneous kinetics of the dissolution of an inorganic
salt, potassium carbonate, in an organic solvent, dimethylformamide." The Journal of Physical Chemistry B 109.16 (2005): 8263-8269.
• 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.
• Zavala, Victor M., and Lorenz T. Biegler. "Large-scale parameter estimation in low-density polyethylene tubular reactors." Industrial & engineering chemistry research 45.23 (2006): 7867-7881.
• Bard, Yonathan, and Yonathan Bard. Nonlinear parameter estimation. No. 04; QA276. 8, B3.. 1974.
• Biegler, Lorenz T. Nonlinear programming: concepts, algorithms, and applications to chemical processes. Vol. 10. SIAM, 2010.
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