MaCKiE Annual Seminar, May 3, 2006, Gent, Belgium
Model-Based ExperimentalAnalysis
A Systems Approach to MechanisticModeling of Reactive Systems
Wolfgang MarquardtLehrstuhl für Prozesstechnik
RWTH Aachen
1Model-Based Experimental Analysis
Modeling of Product and Process Systemsexperimental and
theoretical methods & tools
experiment / measurement techniques
modeling/ simulation
model identification
integrated method developmenttowards a work process of model-based experimental analysis
product & process systems
kinetic phenomena• diffusion• heat and mass transfer• interface phenomena• multiphase transport• (bio-)chemical reaction• nucleation ...
multi-phase reaction andseparation processes
fluid, structured &functional products
biological systems
2Model-Based Experimental Analysis
Integration of Modeling and Experimentation
• common approach in research and industrial practice– coupled phenomena– detailed models, numerical case studies– comparison of simulation and experimental results– evaluation of the model, but no model identification !
• suggested future approach– coordinated design of model and experiment– model refinement based on experimental evidence– accounting for inevitable measurement errors– identification of a valid (mechanistic) model
(structure & parameters) !
cf. J.V. Beck, Meas. Sci. Techn. 9 (1998)α,λ,κ,
µ,σ,D(x)
α,λ,κ,µ,σ,D(x)
+
model-based experimental analysis – MEXA:valid models at minimal effort
3Model-Based Experimental Analysis
Collaborative Research Centre 540
Model-based Experimental Analysisof Kinetic Phenomena in FluidMulti-phase Reactive Systems
12 research groups with cross-disciplinary expertise• biotechnolgy (Ansorge-Schuhmacher)• biochemical engineering (Büchs)• reaction engineering (Greiner, Leitner)• thermal separations (Pfennig) • transport phenomena (Kneer)• multiphase fluid dynamics (Modigell)• computational engineering science (Behr)• process systems engineering (Bardow, Marquardt)• numerical mathematics (Reusken)• scientific computing (Bischof, Bücker)• NMR imaging (Blümich, Stapf)• optical spectroscopy (Koß, Lucas, Poprawe)
Funded by DFG(Deutsche Forschungs-gemeinschaft) since 1999Director: W. Marquardt
4Model-Based Experimental Analysis
Research Agenda of CRC 540
development of a work processModel-based Experimental Analysis (MEXA)
for systematic mechanistic modeling by means of multi-method integrationand
its application to modeling of multi-phase fluid reaction processes
reaction and
multi-component
diffusion in liquids
transport and reaction
in single dispersed
liquid droplets
transport and
(bio-)reaction in
heterogeneous systems
transport and
reaction in
liquid falling films
5Model-Based Experimental Analysis
experimentexperiment measurementtechniques
experimental conditions
MEXA Methodology
experimentaldesign
numericalsimulation
computed statesand measurements
inputs, parameters,initial conditions
a-priori knowledge and intuition
mathematicalmodels
iterative model refinement
iterative improvement of experiment
extended understanding
formulation and solutionof inverse problems
model structure,parameters,
inputs, states,confidence regions
sensor modelcalibration selection
measurements
incremental model identification
6Model-Based Experimental Analysis
Simultaneous Model Identification
measurements x
model parameters !
! mass/heat balances
! thermodynamics
! reaction kinetics
! ...
Model y(x,!,t)
model parameters θ
measurements y
! mass/heat balances! thermodynamics
! reaction kinetics
! ...
model of
observations
y(x,θ,t) ( )!!==
"m
1j
2
jj
n
1i
i )t(y~)t,,(yw2
1min èxè
Overall process model y(x,θ,t)
is fitted to experimental data:
s.t. dynamic model & constraints
• What if we do not know any candidate model structure ?
• How to select a suitable model structure ?
• Is bias due to model structure defects or a lack of information content in data ?
• How to deal with very few or very many observations ?
• How to deal with convergence & robustness problems of estimation algorithm?
7Model-Based Experimental Analysis
Incremental Model Identification
measurements x
model parameters !
! mass/heat balances
! thermodynamics
! reaction kinetics
! ...
Model y(x,!,t)
model parameters θ
measurements y
! mass/heat balances! thermodynamics
! reaction kinetics
! ...
model of
observations
y(x,θ,t)
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
measurements y
model structure and parameters θ
Decomposethe model identification andselection problem into fully
transparent steps !
• computationally efficient (minutes rather than days)• numerically robust and fully transparent• a-priori knowledge can be integrated into the identification process• complex and interacting kinetic phenomena can be identified
8Model-Based Experimental Analysis
Incremental Identification – Some Examples
• differential methods in reaction kinetics, e.g. Connors (1990) → estimate reaction rate by FD, then estimate kinetic parameters
• hybrid modeling, e.g. Psichogios & Ungar (1992), Tholudor & Ramirez (1999)→ combine first-principles models with neural nets
• inverse problems in population balances, Mahoney, Doyle & Ramkrishna (2002) → calculate growth rate as model-based data, correlate with states
• derived from intuition and physical insight• often ad-hoc methods • problem-specific solutions
a generice principle ?
9Model-Based Experimental Analysis
Incremental Model Identification
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model structure and parameters !
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
[ ] )t()t()t(Fdt
)t(d)t(V rin
fccc
+!=balances
Nrf )t()t(V)t(r =stoichiometry
Incremental Model Development
( )ècr ),t(f)t( =kinetic law
)t(rf
)t(r
è
Illustratrion with a CSTR Illustratrion with reaction kinetcis idenfication in a CSTR
(Marquardt, 1995)(Marquardt, 1998)
10Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model structure and parameters !
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
[ ] )t()t()t(Fdt
)t(d)t(V rin
fccc
+!=balances
Nrf )t()t(V)t(r =stoichiometry
( )ècr ),t(f)t( =kinetic law
)t(rf
)t(r
è
WhatWhat areare thethe ingredientsingredients forfor implementationimplementation ? ?
high-resolutionhigh-resolution in-situin-situ measurementsmeasurements
inversioninversion algorithmsalgorithms forfor operatoroperator equationsequations
parameterparameter and and structurestructure identificationidentification forfor algebraicalgebraic modelsmodels
model-basedmodel-based designdesign of of
experimentsexperiments
Incremental Model Identification
11Model-Based Experimental Analysis
High Resolution Measurement Techniquesnon-invasive, in-situ measurements of field data• observation of qualitative behavior• quantitative characterisation of kinetic phenomena
interpretation of the primary measurement data,calibration, quantification of measurement errors !
particle size distribution (FBRM)Kail, Briesen, Marquardt et al., LPT
r
Lc
LC
Nk
wave number
post
ion
conencentrations on a line,Raman spectroscopy,
Koß, Lucas et al., CRC 540
velocity profiles in alevitated droplet, NMR
imaging, Blümich etal., CRC 540
12Model-Based Experimental Analysis
Replacement of measurement device or varying temperatures
Problems of Established Analysis Methods
Non-linear effects due to molecular interaction
Reactive mixtures: restricted extrapolability
Most established spectral analysis methods such as
• PCA, PLS or• classical least squares
are linear and cannot model all nonlinear effects that occur in realmixtures:
13Model-Based Experimental Analysis
Indirect Hard Modeling – General Idea
Development of a rigorous mathematical model of the spectrumConsideration of physical effects of the spectrum throughphenomenological modeling
Generation of pure component models using automatic peak fitting algorithm (Alsmeyer et al., Applied Spectroscopy, 58 (8), 2004)
Step 1: Modeling of pure component spectra (during calibration)
!=i
iiP
pureVS ),(),( "## è
Indirect Hard Modeling: a nonlinear spectral analysis approach
14Model-Based Experimental Analysis
Model of the Mixture Spectrum
= ! 1" ! 2
"+
Step 2: Mixture spectrum is modeled as:Linear combination of parameterized non-linear pure component models
All non-linear effects are modeled phenomenologically:
Baseline variations Spectral shifts Peak variations
! "+=k
kSkP
pure
kkB SBS ),,()()( ,, ##$%#$
kS,!
kP,!
B!
1,pureS 2,pureSmixS
Full spectral model:
15Model-Based Experimental Analysis
Calibration
j
ij,i
j
i
x
xK=
!
!
!="
k i,ki
ki
K
x1
1
#
#
Linear, physically motivatedcalibration model(Non-linear effects are correctedby the spectral model)
S~ spectral
model ù
measurement C B
A
x
concentration
calibrationmodel
Highly reduced amount of calibration measurementsTheory: One calibration measurementPractice: Few measurements
Good extrapolability:Calibration in concentration subspace is possible
16Model-Based Experimental Analysis
SpecTool
SpecTool:• Matlab-based GUI• facilitates automatic and
manual spectral analysis
17Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model structure and parameters !
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
[ ] )t()t()t(Fdt
)t(d)t(V rin
fccc
+!=balances
Nrf )t()t(V)t(r =stoichiometry
( )ècr ),t(f)t( =kinetic law
)t(rf
)t(r
è
WhatWhat areare thethe ingredientsingredients forfor implementationimplementation ? ?
high-resolutionhigh-resolution in-situin-situ measurementsmeasurements
inversioninversion algorithmsalgorithms forfor operatoroperator equationsequations
Incremental Model Identification
18Model-Based Experimental Analysis
Inversion Problems are Ill-Posed
19Model-Based Experimental Analysis
Issues in Flux Estimation
• decide on a set of measurements for best identifiablility– at least as many measurements as unknown fluxes (Hirschorn, 1979) methods for quantification of identifiability (e.g. Asprey, 2003)
• balance information content of measurements and resolution of the fluxparameterization– choose spatial and temporal resolution of flux function adaptive discretization methods (e.g. Binder et al. 2000)
• compromise between bias and variance in estimates– balanced choice of discretization, early stopping and regularization– systematic methods for the selection of regularization operators
and multiple regularization parameters (e.g. Ascher, Haber, 2001,Engl et al., 1996, Belge et al. 2002)
20Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model structure and parameters !
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
[ ] )t()t()t(Fdt
)t(d)t(V rin
fccc
+!=balances
Nrf )t()t(V)t(r =stoichiometry
( )ècr ),t(f)t( =kinetic law
)t(rf
)t(r
è
WhatWhat areare thethe ingredientsingredients forfor implementationimplementation ? ?
inversioninversion algorithmsalgorithms forfor operatoroperator equationsequations
parameterparameter and and structurestructure identificationidentification forfor algebraicalgebraic modelsmodels
Incremental Model Identification
21Model-Based Experimental Analysis
Flux Model Selection Problem
find the most appropriate functional representation
for the correlation of fluxes and states: structure and parameters
parameter estimation and model structure selection
• error-in-variables formulation (Britt & Lücke, 1978, Boggs et al. 1992)
• inference approach, decision tree & statistical tests (Verheijen, 2003)
• Bayesian a-posteriori probability tests (Stewart et al., 1998)
• combinatorial search (McKay et al., 1997, Skrifvars et al., 1998)
generate candidate model structures
• experience-based, qualitative reasoning, kinetic power laws (Schaich et al., 2001)
• molecular scale modeling (Barrett & Prausnitz, 1975, Liu et al., 1998)
• multivariate regression & data mining (Bates & Watts, 1988, Hastie et al. 2001)
22Model-Based Experimental Analysis
Function Approximation
M
1i
d
ii }RR)y~,{(S =!"= x
( )!=
"#+$
N
1i
2
iiNVf
)f(y~)(fM
1min
NN
%x
Tikhonov regularization termenforcing smoothness of fN
!+= )(fy~ ii x
Recover unknown function f∈V from data S "as good as possible"
Restriction to finite dimen-sional subspace VN
Given: noisy data set S
Binder et al. (2000)Ascher and Haber (2001)
Desirable properties
• linear scaling with number of data points (avoid „curse of dimensionality“)
• properly exploit information content in data (avoid under-/overfitting)
23Model-Based Experimental Analysis
• hierarchical d-dimensional finite-element discretization of unknown function with d-linear hat functions, Yserentant (1992)
• significant reduction of number of parameters by successive approximation on subgrids and subsequent linear combination to a sparse grid approximation
• approximation quality close to full grid approximation
[ ]1,4f [ ]2,3f [ ]3,2f [ ]4,1f
[ ]3,1f[ ]2,2f [ ])c(
4,4
)c(
4 ff ![ ]1,3f
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[4 1]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[2 3]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[1 4]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[3 2]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[1 3]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[2 2]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[3 1]
0
0.5
1
0
0.5
1-5
0
5
10
15
x2
x1
f[4 4]
[c)
(Garcke et al., 2001; Brendel & Marquardt, 2003)
Sparse Grid Approximation
24Model-Based Experimental Analysis
Calculate cross-validation error for
initial grid.
Calculate cross-
validation error forextended grids
Generate a set ofextended grids.
Select extended
grid with lowestcross-validation
error.
no
Solution found.
yes
Further
refinement?
Incremental Grid RefinementInitial grid:
Extended grids:
-0.1 -0.05 0 0.05 0.177.5
78
78.5
79
79.5
80
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
y
x1
x2
(1) level= -1 -1, gammaopt= 0, mrmsevalopt= 78.6378
Level: [-1 -1]
-0.1 -0.05 0 0.05 0.1-1
-0.5
0
0.5
1
1.5
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
x1
x2
y
(2) level= 0 -1, gammaopt= 0, mrmsevalopt= 0.26326
Level: [0 -1]
-0.1 -0.05 0 0.05 0.1-0.5
0
0.5
1
1.5
2
gamma
Validation error
(3) level= -1 0, gammaopt= 0, mrmsevalopt= 0.53218
CV
0
0.5
1
0
0.5
1-0.5
0
0.5
1
x1
x2
yLevel: [-1 0]
25Model-Based Experimental Analysis
Calculate cross-validation error for
initial grid.
Calculate cross-
validation error forextended grids
Generate a set ofextended grids.
Select extended
grid with lowestcross-validation
error.
no
Solution found.
yes
Further
refinement?
Incremental Grid RefinementInitial grid:
Extended grids:
-0.1 -0.05 0 0.05 0.177.5
78
78.5
79
79.5
80
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
y
x1
x2
(1) level= -1 -1, gammaopt= 0, mrmsevalopt= 78.6378
Level: [-1 -1]
-0.1 -0.05 0 0.05 0.1-1
-0.5
0
0.5
1
1.5
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
x1
x2
y
(2) level= 0 -1, gammaopt= 0, mrmsevalopt= 0.26326
Level: [0 -1]
-0.1 -0.05 0 0.05 0.1-0.5
0
0.5
1
1.5
2
gamma
Validation error
(3) level= -1 0, gammaopt= 0, mrmsevalopt= 0.53218
CV
0
0.5
1
0
0.5
1-0.5
0
0.5
1
x1
x2
yLevel: [-1 0]
26Model-Based Experimental Analysis
Calculate cross-validation error for
initial grid.
Calculate cross-
validation error forextended grids
Generate a set ofextended grids.
Select extended
grid with lowestcross-validation
error.
no
Solution found.
yes
Further
refinement?
Incremental Grid RefinementCurrent grid:
Extended grids:
Level: [0 -1]
Level: [1 -1]
Level: [0 0]
-0.1 -0.05 0 0.05 0.1-1
-0.5
0
0.5
1
1.5
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
x1
x2
y
(2) level= 0 -1, gammaopt= 0, mrmsevalopt= 0.26326
100
0.255
0.26
0.265
0.27
gamma
Validation error
CV
0
0.5
1
0
0.5
1-1
0
1
2
x1
x2
y
(4) level= 1 -1, gammaopt= 1.1548e-005, mrmsevalopt= 0.25344
-0.1 -0.05 0 0.05 0.1-1
-0.5
0
0.5
1
1.5
gamma
Validation error
CV
0
0.5
1
0
0.5
1-2
-1
0
1
2
x1
x2
y
(5) level= 0 0, gammaopt= 0, mrmsevalopt= 0.18116
27Model-Based Experimental Analysis
00.5
1
0
0.51-2
0
2
y
(1)
x1
x2 0
0.51
00.51-2
0
2
x1
(2)
x2
y
00.5
1
00.5
1-1
0
1
x1
(3)
x2
y
00.5
1
00.51-2
0
2
x1
(4)
x2
y
00.5
1
00.51-2
0
2
x1
(5)
x2
y
00.5
1
00.51-2
0
2
x1
(6)
x2
y
00.5
1
00.51-2
0
2
x1
(7)
x2
y
00.5
1
00.51-5
0
5
x1
(8)
x2
y
00.5
1
00.51-2
0
2
x1
(9)
x2
y
00.5
1
00.51-2
0
2
x1
(10)
x2
y
00.5
1
00.51-5
0
5
x1
(11)
x2
y
00.5
1
00.51-2
0
2
x1
(12)
x2
y
00.5
1
00.51-5
0
5
x1
(13)
x2
y
00.5
1
00.51-2
0
2
x1
(14)
x2
y
00.5
1
00.51-5
0
5
x1
(15)
x2
y
00.5
1
00.51-5
0
5
x1
(16)
x2
y
00.5
1
00.51-5
0
5
x1
(17)
x2
y
00.5
1
00.51-5
0
5
x1
(18)
x2
y
00.5
1
00.51-5
0
5
x1
(19)
x2
y
00.5
1
00.51-2
0
2
x1
(20)
x2
y
Incremental Grid RefinementInitial grid
Final grid?
? ? ?
? ? ? ?
? ? ? ?
? ?
?• number of tested grids scales linearly with number of
dimensions instead of exponentially• curse of dimensionality in selecting the discretization is avoided• no guarantee to find optimal discretization
28Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model structure and parameters !
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
[ ] )t()t()t(Fdt
)t(d)t(V rin
fccc
+!=balances
Nrf )t()t(V)t(r =stoichiometry
( )ècr ),t(f)t( =kinetic law
)t(rf
)t(r
è
WhatWhat areare thethe ingredientsingredients forfor implementationimplementation ? ?
parameterparameter and and structurestructure identificationidentification forfor algebraicalgebraic modelsmodels
model-basedmodel-based designdesign of of
experimentsexperiments
Incremental Model Identification
29Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
measurements x
model structure and parameters θ
Modeling of Reaction Kinetics
• no. of reactions, stoichiometry and kinetics unknown
• isothermal semi-batch CSTR experiments
• concentration measurements (ex-situ, e.g. GC;in-situ, e.g. Raman/IR spectroscopy)
• a number of simulated semi-batch reactorexperiments, 60 min (cases: noise, sampling …)
Acetoacetylation ofPyrrole with Diketene
(Brendel, Bonvin, Marquardt, 2006)
1: P + D PAA2: D + D DHA3: D OL4: PAA + D F
30Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
candidates for
stoichiometry
candidates for
reaction kinetics
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
Incremental Model Identification
31Model-Based Experimental Analysis
inverse problem:
[ ])()()()( tccFdt
tdctVtf i
in
iir
i !!=
mass balances (semi-batch)
Concentration Measurements → Reaction Fluxesco
ncen
tratio
n [m
ol/l]
time [min]
measured concentrationtrue concentration
concentration measurements
time [min]
Kon
zent
ratio
n[m
ol/l]
estimated conc.true concentration
Zeit [min]Fl
uss
[mm
ol/m
in]
estimated fluxtrue flux
estimated reaction flux
estimated concentrationregularisation
32Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
candidates for
stoichiometry
candidates for
reaction kinetics
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
Incremental Model Identification
target factor analysis
Bonvin & Rippin, 1990
33Model-Based Experimental Analysis
Target Factor AnalysisPropose an R x S dimensional stoichiometric matrix Np with
R = number of reactions
S = number of involved species
Set up a B x S dimensional data matrix D
B = Number of observations
D = can be expressed as X*N, where X is a matrix indicating the extents of the Rreactions and N is the correct stoichiometric matrix
Discard those stoichiometries of Np that cannot be approximately described as linearcombination of the rows of Na
Singular Value Decomposition D = U*S * VT
= Xa * Na,
where the rows of Na represent a basis for the observed stoichiometric space
34Model-Based Experimental Analysis
• f r: reaction fluxes
• r: reaction rates
• N: stoichiometric matrix
• V: volume
Determination of reaction rates
Fluxes, Stoichiometry → Reaction Rates - Theory
( )
( )
( )
( )2
r
DHA
41
r
PAA
1
r
P
4321
r
D
rVf
rrVf
rVf
rrr2rVf
=
!=
!=
!!!!=
rNf Vr=
ulrrr
fNrr
!!
"=
.t.s
V
1minarg r
• BLS problem
Stoichiometry
35Model-Based Experimental Analysis
estimated reaction fluxes
Reaction Fluxes → Stoichiometry, Reaction Rates
stoichiometryTFA
lin.
combination
estimated reaction rates
estimated ratetrue rate
36Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
candidates for
stoichiometry
candidates for
reaction kinetics
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
Incremental Model Identification
37Model-Based Experimental Analysis
Structure Identification of Reaction Rate Function
)t(c
)t(r)c(r
correlation with data-driven methods
• NN with Bayesian regularization
• band of noisy data sets
• physical insight
correlation of concentrations and rates
38Model-Based Experimental Analysis
Estimated Reaction Rate Functions
Mean: 2.7%
Max: 6%
Mean: 4.2%
Max: 25%
Mean: 2.9%
Max: 5%
Mean: 167%
Max: 2958%
39Model-Based Experimental Analysis
Identification of Reaction Kinetic Laws
regression of estimated ratesand concentrationens for eachindividual reaction
)(tr
)(tc),( ckr
P + D PAA
candidate kinetic laws
?
P2D
(10)1
(10)1
2PD
(9)1
(9)1
KPD(8)1
(8)1
KD(7)1
(7)1
KP(6)1
(6)1
PD(5)1
(5)1
K(4)1
(4)1
P(3)1
(3)1
D(2)1
(2)1
(1)1
(1)1
cckrcckr
ccckrcckrcckrcckr
ckrckrckr
kr
=
=
=
=
=
=
=
=
=
=
best model structure withstatistically optimal parameters
simultaneous identification forpromising model candidates
estimated ratekinetic las 4kinetid law 5kinetic law 8
40Model-Based Experimental Analysis
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
measurements
model structure and parameters θ
Iterative Model lmprovement
Reaktionsparameter
experimentalrun
experimentdesign
experimentaldegrees of freedom
parameter-correction
optimal parametersconfidences
candidates forreaction kinetics
candidates forstoichiometry
41Model-Based Experimental Analysis
Iterative Improvement - Example
stoichiometry kinetic law
1: P + D PAA2: D + D DHA3: D OL 33
D22
KPD11
kr
ckr
ccckr
=
=
=
validation parameters
N/A-1
requested accuracy achieved
scenario: 3600 data points in each experiment (Ts=1 s), 5% noise
1: P + D PAA2: D + D DHA3: D OL
D33
K
2
D22
KPD11
ckr
cckr
ccckr
=
=
=
experimental design for model discrimination
2Parametersatz
k1, k2, k3
1: P + D PAA2: D + D DHA3: D OL D33
K
2
D22
KPD11
ckr
cckr
ccckr
=
=
=
experimental design for best model parameters
3Verbesserter
Parametersatzk1, k2, k3
42Model-Based Experimental Analysis
Incremental vs. Simultaneous Model Identification
# ModelsCPUTime
Ident. Corr.
Incremental Method Simultaneous Method
STDV = 2%
STDV = 10%
Reso-lution # Models
Ident. Corr.
3 sec
30 sec
5 min
1
≈ 1
≈ 7
30 min
15 sec
6 sec 100 %
100 %
100 %
3 sec
30 sec
5 min
≈ 160
≈ 200
40 min
3 min
2 min 10 %
50 %
100 %≈ 100
3 sec
30 sec
5 min 100 %
100 %
100 %
3 sec
30 sec
5 min 10 %
50 %
100 %
3600
3600
3600
3600
3600
3600
1.7 d
3.8 h
1.6 d
3.9 h
1.1 h
1.7 h
CPUTime
Reso-lution
identical60 min
much faster
43Model-Based Experimental Analysis
MEXA for Investigation of Diffusive Transport
Why studying diffusion ?
• detrimental for product and process design
• very high experimental effort
• very few multi-component diffusion data available
• validity of diffusion models still a matter of debate
• a good model problem
for the development of
MEXA methodology
measurements
balancesmodel 2
model 1 balances
stoichiometry
model 3 kinetic lawstoichiometrybalances
concentrations
fluxes
model of the reaction
concentrations
concentrations
measured concentrations
fluxes
fluxes
rates
rates reaction parameters
measurements
model structure and parameters θ
Intermediate productIntermediate product
selectivity of heterogeneously catalyzed reactions(Pantelides & Urban, 2004)
44Model-Based Experimental Analysis
Incremental Model Identification
measurements y
model F(x,θ)
parameters θ
model B
model BF
model BFD
fluxmodelsbalances
model for fluxes
z
cDJVV
!
!"=
fluxmodelsbalances
model for coefficient),( !cfDV =
balancesstructure of system
z
J
t
cV
!
!"=
!
!
binary diffusion
balances diffusive flux JV(zi,tk)
coefficient DV(zi,tk)fluxmodels
diffusioncoefficient
parameterdiffusioncoefficient
model structure and parameters for diffusion
45Model-Based Experimental Analysis
x(z,t) model
t
xi(z)
A B
C
balanceequations
1D
+model
discrimination
+transport
laws
sequence of inverse problems
Ji(z,t) Dik(x, Θ)
Incremental Model Identification
model probability
constant 13,2%
linear 39,3%
quadratic 27,8%
cubic 19,7%
Modell probability
constant 13,2%
linear 39,3%
quadratic 27,8%
kubic 19,7%
robust and efficient identification of modelsexperimental results for binary and ternary diffusion
Bardow et al. (2003, 2006)
46Model-Based Experimental Analysis
Ternary Mixtures
balancesmodel B
model BF balances
2,1 ; =!
!"=
!
!i
z
J
t
cV
ii→ problem decouples→ solve binary problem twice
z
cD
z
cDJ
VVV
!
!"
!
!"= 2
12
1
111
→ coefficients not identifiable
diffusioncoefficientmodel BFD flux
modelsbalances
fluxmodels
-10.4-10.8-33.4-9.0Error [%]
D22D21D12D11Coefficient
→ error-in-variables regression
Example: simulated experiment as in Arnold & Toor (1967), noise level σ=0.01
estimation of constant diffusion coefficients from a single experimentwith good precision
47Model-Based Experimental Analysis
Optimal Experimental Design: Idea
set free variables p such that information on model parameters
is maximized
=maximize curvature of
parameter estimation objective
2
2
max!
"
#
#
p
!(p1)
!(p2)
D* D
!
!*
!(p1)
!(p2)
D* D
!
!*error level
!(p1)
!(p2)
D* D
!
!*error level
parameter estimation objective
48Model-Based Experimental Analysis
Optimal Ternary Mixture Composition
!
"xmax
xc
xL
xU
x1
x2
Example: acetone-benzene-methanol at xC = [0.35; 0.302; 0.348]T, (Alimadadian and Colver, 1976)
• scaled objective ζ-efficiency measures information per parameter• one Raman experiment suffices to determine ternary Fick matrix • two different experiments result in substantial improvement• experiments should be as distinct as possible (φ(2)=φ(1)+90°)
49Model-Based Experimental Analysis
Optimal Initial Conditions
mea
surin
g zo
ne
z 0 z sen
sor
xL
xU
Example: acetone-benzene-methanol at xC = [0.35; 0.302; 0.348]T, (Alimadadian and Colver, 1976)
• measurements at the wall, i.e. restricted diffusion experiments• unequal volume of both phases, almost independent of concrete mixture • short experiments are beneficial
50Model-Based Experimental Analysis
-1
-0.5
0
0.5
1
1.5
2
2.5
Dij
V [10
-9 m2/s]
D22
V
D11
V
D21
V
D12
V
Ternary Results
Diffusivities from a single Raman experiment
→ one Raman experiment gives full diffusion matrix→ currently scatter in data is still significant
-1
-0.5
0
0.5
1
1.5
2
2.5
Dij
V [10
-9 m2/s]
D22
V
D11
V
D21
V
D12
V
System: 1-Propanol - 1-Chlorobutane - n-Heptane
51Model-Based Experimental Analysis
Ternary Results
Diffusivities from two optimal Raman experiments
→ one Raman experiment gives full diffusion matrix→ good precision from 2 optimized runs→ robust & efficient measurement→ quantitative validation of design predictions
-1
-0.5
0
0.5
1
1.5
2
2.5
Dij
V [10
-9 m2/s]
D22
V
D11
V
D21
V
D12
V
-1
-0.5
0
0.5
1
1.5
2
2.5
Dij
V [10
-9 m2/s]
D22
V
D11
V
D21
V
D12
V
with reference data from Weingärtner et al. (1994)
Diffusivities from two Raman experiments
52Model-Based Experimental Analysis
Quaternary Diffusion
concentration measurements with Raman spectroscopy andmodel-based design and evaluation of diffusion experiments
(cyclohexane – toluene – dioxane – chlorobutane)
matrix of Fick‘sdiffusion coefficients
!!!
"
#
$$$
%
&
'
'
'
1 ,740 ,1110 ,296
0 ,0282 ,0660 ,019
0 ,2440 ,0861 ,708
- 9 Marquardt, LPT
com
pone
ntAco
mpo
nentB
Lösu
ng I
Lösu
ng II
thermodynamical factor – accuracy improvements – electrolytes
!"#
$%& '
sm Dij
2910
53Model-Based Experimental Analysis
Benefits of MEXA for Diffusion Problem
reduces experimental effort fewer experiments maximum precision simple design and preparation
diffusion experiments for - multi-component mixtures
- reactive systems- electrolytes
• experimentally validated forbinary, ternary, quaternaryand quinternary mixtures
• concentration dependenceof diffusion coefficients
high resolutionmeasurements
model-basedmethods
• towards reactive andelectrolytes mixtures
• further improvement of method• diffusion modeling
54Model-Based Experimental Analysis
• falling films are all around: - falling film cooler
- falling film evaporator- falling film absorber- falling film reactors
• transport phenomena are hardlyunderstood, interaction between
- fluid dynamics with free surface - heat and mass transfer - chemical reaction
• first milestone: modelling of heat transfer with effective transport coefficients
Kinetic Phenomena in Falling Film Reactors
y
x
ϕ
( )x,z,t!
Ozonolysis ofolefines in silicon oil
y
x
ϕ
!
molecular transport,3D, wavy
effective transport,
2D, flat
55Model-Based Experimental Analysis
Heat Transfer through Falling Film
),( txFTut
Tc effeff
eff
p =!!"
#$$%
&'(+
)
)*
)),...,,((),( !" txuftx effeff =
( )effeff TtxtxF !!= ),(),( "
measurement data
energy bal.model 2
model 1 energy bal.
transportlaw
model 3 constitutiveequation
transportlaw
energy bal.
temperatures
heat flux
Model structure and parameters
temperatures
temperatures
measured temperature field
heat flux
heat flux
eff. heat conduction
eff. heatconductiont
parameters
56Model-Based Experimental Analysis
Experimental Set-upeperiments:F. Al-Sibai, A. Leefken, R. Kneer, U. Renz, SFB 540
39,4
39,6°C
wall temperaturemeasurements
TW
57Model-Based Experimental Analysis
IR-camera
mT
Mathematical Problem Formulation
q
heating foillacquer
3!
f!
filmfoilq
!
2!
1!
58Model-Based Experimental Analysis
Optimization Algorithm (CG)
DROPS: adaptive FEM, 3D, anisotropic grids: Soemers, Groß, Reichelt, Reusken, CRC 540
temperatureupdate
direct problem
adjoint problem
sensitivityproblem
stop? end
nJ
nnJ !,"
nµ
0q
1+nq
nnn
e
npqq µ!=+1
1!+"= nnnn
pJp #
no
yes
DROPS
DROPS
DROPS
sam
e str
uctu
re a
s dire
ct pr
oblem
!
coarselevel
finelevel
0
5
10
15
20
25
Re
ch
en
ze
it
[h]
Level 3
Level 2
Level 1
nested iteration
• multi-level strategy:exploit grid hierarchy
• good initial guesses fromcoarse grid results
59Model-Based Experimental Analysis
Heat Flux Estimates from Experimental Data
measured temperature distributions estimated heat flux distribution
(Groß, Soemers, Mhamdi, Al-Sibai, Reusken, Marquardt, Renz, 2005)
60Model-Based Experimental Analysis
6.36.4
6.5
x 10-3
6.36.4
6.5
6.36.46.5
qfo
il
6.36.4
6.5
0 5 10 15 20 25 30 35 406.36.4
6.5
z
Tim
e
38.95
39
39.05
38.95
39
39.05
38.95
39
39.05
Tem
pera
ture
38.95
39
39.05
0 5 10 15 20 25 30 35 4038.95
39
39.05
z
Heat Flux Estimates from Experimental Data
61Model-Based Experimental Analysis
Towards Inverse Transport ProblemsCG method for the solution of inverse
problem embeds DROPS(Reusken u.a., SFB 540) ein.
DROPS employsadaptive multi grid methods,finite element discretization,levelset method
and facilitates the numerical simulationof multi-phase flow problems in 3D at high resolutionof the phenomena at the phase interface, efficient and error-controlledwith appropriate flexibilityfor model extensions. a droplet rising in a stagnant liquid
Pfennig, Reusken u.a., CRC 540
62Model-Based Experimental Analysis
MEXA Business Process – Evaluation
method integration has high potential !
• optimal experimental design reduces effort• structure identification leads to mechanisms
• improvement of method integration in particular for distributed problems
incremental refinement has high potential !• homogenous reactions, multi-component diffusion, diffusion & bioreaction in gels• drastic reduction of experimental and engg. effort• significantly improved transparency • further development for CFD problems
Messdaten
BilanzenModell 2
Modell 1 Bilanzen
Stöchiometrie
Modell 3 KinetikgesetzStöchiometrieBilanzen
Konzentrationen
Flüsse
Modellstruktur und Parameter !
Konzentrationen
Konzentrationen
Gemessene Konzentrationen
Flüsse
Flüsse
Raten
Raten Reaktionsparameter
Kandidaten für
Stöchiometrie
Kandidaten für
Reaktionskinetik
Versuchs-
planung
Versuchs-
durchführung
Entwurfs-
variablen
Parameter-
korrektur
opt. Parameter
Konfidenzen
Versuchs-planung
NumerischeSimulation
berechnete Zustände
und Messgrößen
Eingänge, Parameter,
Anfangsbedingungen
Vorwissen und Intuition
MathematischeModelle
Iterative Verfeinerung des Modells
Iterative Verbesserung des Experiments
Erweitertes Verständnis
Messgrößen
Formulierung und Lö-sung inverser Probleme
Modellstruktur,Parameter,
Kali- Modell-bration auswahl Konfidenz-
aussagen
Eingänge, Zustände
Versuchs-aufbau Messtechnik
Versuchs-bedingungen
our vision:
development of an integrated MEXA tool set to transfer systems methods to experimentators planning and coordination… documentation… guidance … process reengineering …
…of MEXA processes
63Model-Based Experimental Analysis
Lessons Learned
• accept interactions between kinetic phenomena in experiments,but
isolate them during identification by a suitable decomposition strategy
• high precision calibration of high-resolution measurements(PIV, LIC, LCSM, NMR imaging, Raman / IR spectroscopy etc.)often is a difficult modeling problem in itself
• statistics of measurement errors need to be included in the analysis
• flux estimation is the key to reliable identification
• tremendous improvements are possible by systematiccross-disciplinary linking of process systems and experimental skills
64Model-Based Experimental Analysis
Some Challenges for Future Work
• refinement of MEXA work process– flux estimation (estimation quality, numerical efficiency)– exploit error statistics– integrated calibration and kinetics identification– tailor optimal experimental design methods to incremental identification– assessment of identifiability– adjust model parameterization to information content of experiments
• roll-out MEXA strategy from meso- to micro- and macro-scale– hybrid modeling on macro-scale– model structure generation from molecular simulation results
• application and benchmarking of MEXA work process– complicated reaction and transport problems– population systems (crystallization, ...)– biological systems (metabolic pathways, ...)
65Model-Based Experimental Analysis
AcknowledgementFunding
Deutsche Forschungsgemeinschaft, CRC 540
Cooperating partners
F. Al-Sibai J. Koß F. Alsmeyer K. Lucas A. Bardow T. Lüttich D. Bonvin, EPFL A. Mhamdi J. Blum A. Pfennig M. Brendel U. RenzV. Göke A. ReuskenS. Groß A. Schuppert, BTSO. Kahrs M. SoemersR. Kneer S. Stapf
and others
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