numerical and experimental aspects of temporal analysis of
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Laboratory for Chemical Technology, Ghent University
http://www.lct.UGent.be
Numerical and experimental aspects of Temporal Analysis of Products: a tool
for understanding and designing heterogeneous catalysts
Guy B. Marin
1
NEXTLAB 2014 Rueil-Malmaison - 2-4 April 2014
Overview
2
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
3
Steady-state versus transient
Tamaru K (1964), Adv. Catal., 15, 65‒90
A B C D
steady-state experiment transient ex
A B
only rate-determining step manifested
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Relaxation methods in heterogeneous catalysis
Pulse or step in carrier gas flow
Pulse response – TAP No carrier gas flow
A B Stimulus Response Reactor
A A,B
A
Rob J. Berger et al., Appl. Catal. A: General, 342 (2008) 3 4
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TC
Microreactor
Catalyst
Vacuum (10-8 torr)
Temporal Analysis of Product (TAP) Pulse Response Experiment
0.0 time (s) 0.5
Exi
t flo
w (F
A)
Inert
Reactant
Product
Reactant mixture
Pulse valve
Mass spectrometer
5
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Temporal Analysis of Products (TAP)
6
Types of experiments
6
Size ~ 1013 molecules
C7H8 C7H8
CO2
t/s t/s
Knudsen diffusion regime No change in the state of the catalyst
t/s
CO2
t/s
C7H8
Size > 1014 molecules/pulse Molecular diffusion regime Deliberate change in
the state of the catalyst
Single pulse experiment
Multi pulse experiment
Alternating pulse experiment
O2
CO2 CO2
t/s
C7H8
Two pulses of different gases By varying the time
delay of the pulses, lifetime of active surface species can be determined
C7H8
t/s
∆t O2
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Temporal Analysis of Products (TAP)
7
Pump-Probe experiment
7
Alternating pulse experiment
Two pulses of different gases By varying the time delay
of the pulses, lifetime of active surface species can be determined
C7H8
t/s
∆t
O2
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
8
Temporal Resolution of TAP
Reactor bed
If reactant pulses small enough: Knudsen diffusion
A
A
B
A B
∑+−∂
∂=
∂
∂jjga
gcat
gcat kCk
xC
Dt
Cθε 2
2
Pulse response
LI LII LIII
Inert zone I Catalyst zone
Inert zone II
jj R
t=
∂
∂θ
8
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Temporal Analysis of Products (TAP) setup
9
TAP-3E
TAP-1
TAP-2
1988
1997
2014
J.T. Gleaves, et al. Studies in Surface Science and Catalysis, 38 (1988) 633-644
John T. Gleaves, et al. Appl.Cat. A: General, 160 (1997) 55-88
John T. Gleaves, et al. J. Mol. Cat. A: Chem, 315(2010) 108-134
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E
10
Diffusion pump
Main chamber
Mass spectrometer chamber Main gate
valve
Manifold, Reactor Extractor
Set of pneumatic cylinders: - Vacuum/Pressure
experiments (By slide valve)
- Heating system
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E The TAP-3E menu of experiments includes: • Multireactors system (carousel). Reactor can be easily and rapidly removed from the system without venting the vacuum chambers. Easily switching between 6 reactors (in situ) within 1 minute.
• Vacuum pulse-responses. The four pulse valves can be triggered
simultaneously or in a programmed alternating sequence
• Atmospheric pressure steady-state, step-transient, and SSITKA experiments, temperature programmed desorption (TPD), and temperature programmed reaction (TPR) experiments;
• Highly automated instrument that can be operated either locally or
remotely via the Internet (with limited intervention on site for the initial experiment configuration)
John T. Gleaves, et al., J. Mol. Catal. A: Chemical 315 (2010) 108. 11
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E
12
Reactor carousel (6 reactors)
Reactor
Reactor extractor
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Reactor carousel and reactor extractor
13
Turbo pump (Attached to Mass spectrometer chamber)
Carousel
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Manifold
14
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E The TAP-3E menu of experiments includes: • Multireactors system (carousel). Reactor can be easily and rapidly removed from the system without venting the vacuum chambers. Easily switching between 6 reactors (in situ) within 1 minute.
• Vacuum pulse-responses. The four pulse valves can be triggered
simultaneously or in a programmed alternating sequence
• Atmospheric pressure steady-state, step-transient, and SSITKA experiments, temperature programmed desorption (TPD), and temperature programmed reaction (TPR) experiments;
• Highly automated instrument that can be operated either locally or
remotely via the Internet (with limited intervention on site for the initial experiment configuration)
John T. Gleaves, et al., J. Mol. Catal. A: Chemical 315 (2010) 108. 15
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E
16
EXTREL QMS with conical aperture for increased signal-to-noise ratio
one pulse TAP-3
TAP-1 TAP-3E
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Experimental data and Knudsen model
17
Ar He
D = 1.75 x 10-3 m2/s D = 5.02 x 10-3 m2/s
Experimental () and simulated (─) responses
298 K 298 K
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Intracrystalline diffusion
18
quartz (─) H-ZSM 5 (─)
T=298K
Ar
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3: Atmospheric pressure experiments The TAP-3E menu of experiments includes: • Multireactors system (carousel). Reactor can be easily and rapidly removed from the system without venting the vacuum chambers. Easily switching between 6 reactors (in situ) within 1 minute.
• Vacuum pulse-responses. The four pulse valves can be triggered
simultaneously or in a programmed alternating sequence
• Atmospheric pressure steady-state, step-transient, and SSITKA experiments, temperature programmed desorption (TPD), and temperature programmed reaction (TPR) experiments;
• Highly automated instrument that can be operated either locally or
remotely via the Internet (with limited intervention on site for the initial experiment configuration)
John T. Gleaves, et al., J. Mol. Catal. A: Chemical 315 (2010) 108. 19
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP-3E: Atmospheric pressure experiments
20
Reactor
Slide valve (Atmospheric pressure experiments)
Reactor carousel
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Overview
21
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
22
Mars-van Krevelen (Redox) cycle
Al2O3
CuO or CeO2
O O
C3H8
CuO-CeO2/γ-Al2O3
CO2
O2
reduction re-oxidation
V. Balcaen, et al., Catalysis Today, 157 (2010) 49-54 22
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Reaction mechanism
Al2O3
CuO or CeO2
O*,s Oads
C3H8
O2
O*,b CO2 CO2
CO2
Elementary steps + Active sites
23
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Kinetic modeling
C3H8 C3H8O*,s ?
C2H4O*,sCH3
…
CO2*,s
CO2
CO*,s
? Rates of elementary steps
Optimal description of experimental data
Kinetic model
r6= k6CC3H8O*,s
r5 = k5CC3H8CO*,s
r7 = k7CC3H8C²O*,s
? ?
24
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Propane responses Calculated
623 K
673 K
723 K
773 K
823 K
873 K
Experimental
25
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
26
Toluene: Oxygen labeling experiment
Only 10% of water formed contained H2
18O Mainly lattice oxygen participates in reaction
Unmesh Menon et al., J. Catal. 283 (2011) 1
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
27
+ 18O2 ↔ 18OL + 18Os
18Os + ↔ 18OL
+ 16OLb ↔ 16OL + b
18O in CO2 more than in H2O but still mainly 16O in CO2
Toluene: Oxygen labeling experiment
Di-Oxygen is mainly required for re-oxidation of the reduced surface metal centers
Unmesh Menon et al., J. Catal. 283 (2011) 1
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Toluene: Carbon labeling experiment
28
Activation of C-C bonds
13CO2 is formed before 12CO2
12C6H5-13CH3
12CO2 13CO2
Abstraction of methyl carbon atom followed by destruction of aromatic ring
Unmesh Menon et al., J. Catal. 283 (2011) 1
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Toluene: Hydrogen labeling experiment
29
C6H5-CD3
H2O, HDO, D2O
Peaks of H2O, HDO, D2O very close to each other
Activation of C-H bonds
Simultaneous abstraction of hydrogen from methyl and aromatic ring or scrambling of hydroxyl groups on the surface
Unmesh Menon et al., J. Catal. 283 (2011) 1
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Total oxidation mechanism: summary
30
1 2
3
4
5
O2 H2O
CO2
CO2
H2O
O2
O- O2-
Cu2+
Vo
Unmesh Menon et al., J. Catal. 283 (2011) 1
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Overview
31
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
TAP, designed for simple math
32
• Knudsen diffusion: constant diffusivity
• Tiny inlet pulse: essentially linear kinetics
• Long cylinder: 1D model suffices
• Linear equations in each zone
Θ−=∂Θ∂
Θ+−∂∂
=∂∂
ΘΘΘ
Θ
KCKt
KCKxCD
tC
C
CCC2
2
ε
inert
catalyst
inert
gas
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Laplace transform
33
• Laplace transform:
• Derivatives to algebra:
• Initial conditions and
• Determine coverages from equation:
• Transformed equations in each zone:
CsKs
xFFD
xC
CsKx
CDCs
LLLL
LLL
))(( and
i.e.,)(
1
2
2
+−=∂
∂−=
∂∂
−∂
∂=
− ε
ε
0=C 0=Θ
CKKs CLL Θ−
ΘΘ+=Θ 1)(
∫+∞ −=
0)()( dttFesF stL
)0()()(' FsFssF −= LL
D. Constales et al., Chem. Eng. Sci., 56 (2001) 133.
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
3D effects justifiably neglected
34
• In 2D or 3D, additional terms in Laplacian
• Simple math, separation of variables is possible.
• Transversally nonuniform components exhibit
diffusional decay (pseudo-reaction )
• At current aspect ratios, justifiably negligible:
extinct in less than 1/15th of peak time
2/ rDk λ=
D. Constales et al., Chem. Eng. Sci., 56 (2001) 133 and 56 (2001) 1913
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Other boundary conditions
35
• Inlet can have premixing chamber.
• Outlet condition can be finite
• Order 1 effect similar to zone length uncertainty
• Order 2 effect similar to tiny time shift
• Justifiably neglected
in current reactor setups
outout vCF = v
D. Constales et al. Chem. Eng. Sci., 61(2006)1878
RB Zone length BG Time shift
ms
mV
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Fourier transform
36
• Fourier transform:
• System is empty for past eternity
So and hence
• Inversion, from Fourier back to time:
• Example response one-zone inert:
0<t
)()(s
outout iFF=
= ωω LF
real
imaginary
∫+∞
∞−
−= dttFeF ti )())( ωω(F
∫∫+∞+∞
∞−=
0
∫∞+
∞−= ωω
πω dfetf ti )(
21))(1-(F
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Noise
no meaningful information
frequency (Hz)
modulus (mV)
sample time (ms)
signal (mV)
Di-oxygen pulse response
37
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Thin-zone TAP-reactor: Uniformity
• For not-too-fast reactions, narrow zone (∆Lcat/L < 1/20) minimizes spatial concentration and temperature non-uniformities.
• Observed kinetics can be related to a uniform catalyst composition.
Finlet – Foutlet = Scat RTZ(Cg,Cs) Cinlet = Coutlet = CTZ
g
• A Fourier-domain algorithm (the Y-Procedure) can be used to translate exit flow rates into otherwise non-observable gas concentrations CTZ
g and rates RTZ within the uniform catalytic zone. • The Y-Procedure is kinetically “model-free”, i.e. it does not require a priori assumptions about the mechanism and kinetic model of catalytic reactions.
Shekhtman et al. Chem. Eng. Sci., 2004, 59, 5493 – 5500 Yablonsky et al. Chem. Eng. Sci., 2007, 62, 6754-6767 38
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Thin-zone TAP-reactor: Info on intermediates
Uptake Release
Information flow:
Intermediates
Redekop et al. Chem. Eng. Sci., 2011, 66, 6441–6452 39
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Coverage
Thin-zone TAP-reactor: use of Y-procedure
RC
O2,
(mm
ol/k
g cat
/s)
CCO CO*, (mol/m3)(mmol/kgcat)
ER, ER+CO ads. LH, ER+LH
time
Redekop et al. Chem. Eng. Sci., In Press
O2 + 2CO → 2 CO2
Simulation results: CO oxidation via different mechanisms
(1) O2 + 2* → 2O*
(2) CO + * ↔ CO*
(3) CO + O* → * + CO2
(4) CO* + O* → 2* + CO2
Possible elementary steps:
Possible mechanisms (combinations of steps):
Eley-Rideal (ER): steps 1 and 3
Langmuir-Hinshelwood (LH): steps 1,2, and 4
ER + LH: steps 1 - 4
ER + CO ads.: steps 1,2, and 3
Analysis of rate-concentration dependencies allows for model discrimination
40
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Overview
41
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Experimental data: time series
42
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Conditions for maximum-likelihood estimates
• The independent variables contain no experimental errors
• The model is adequate • The mean experimental error is zero • All errors are normally distributed • The errors are homoskedastic • The errors are uncorrelated
43
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
44
Heteroskedasticity of TAP noise
More noise near peak than in tail
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
45
Autocorrelation of TAP noise
F
t
ε1
ε2
ε1
ε2
R. Roelant et al. Catal. Today 2007, 121, 269
[1]
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
46
Autocorrelation of TAP-noise
t
F
ε1
ε2
ε1
ε2
46
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
47
Second-order statistical regression
average experimental time series
G(aussian) H(omoskedastic) U(ncorrelated)
model-calculated time series
NLSQ
NLSQ
transformed average expt. time
series
transformed calculated time
series
linear transform
G H U
R. Roelant, D. Constales, R. Van Keer, G.B. Marin, Chem. Eng. Sci. 2008, 63, 1850
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
48
Principal Component Analysis and rescaling
Rescaling of the principal component
axes
ε1
ε2
Projection on the Principal Component Axes
48
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Estimation of diffusion coefficient
49
Second Order Statistical Regression (SOSR): realistic confidence intervals
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Overview
50
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Regression of TAP-Data: Best Practices
51
The Ten Commandments:
I. Thou shalt verify Knudsen II. Correct thy baseline III. Waste nor scrimp collection time IV. Beware spectrally localized noise V. Be neither finicky nor coarse in sampling
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Regression of TAP-Data: Best Practices
52
VI. Count thy replicates VII. Thy pulse size will stray VIII. Thy kinetics be linear: pulse modestly IX. Rightfully regress X. Spare thy fancy weaving the network
The Ten Commandments (cont’d):
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
I. Knudsen diffusion regime
53
Reactor bed
A inert zone inert zone
catalyst zone
2
2j j
a i i
C CD k C k
t xε θ
∂ ∂= − +
∂ ∂ ∑
jj M
RTrDπτ
ε 234
=j
ref
refrefj M
MTTDD =
A
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
standard deviation (mV)
sample time (ms)
IV. Spectrally localized noise
frequency (Hz)
standard deviation (mV)
0 Hz
50 Hz
150 Hz
250 Hz
regression
54
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
V. Do not sample too much
sample time (ms)
1.0
0.8
0.6
0.4
0.2
0.0
0 2 4
Cor
rela
tion
coef
ficie
nt
sample time (ms)
1.0
0.8
0.6
0.4
0.2
0.0
22 24 26 28 30
Cor
rela
tion
coef
ficie
nt
neighboring samples
neighboring samples
)exp()(θ
ρ tt ∆−=∆ Gauss-Markov
stochastic process
55
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
VIII. Ensure state-defining conditions
Rate Coefficients that include catalyst state
Catalyst State
εin
∂Cg
∂t= Din
∂2Cg
∂x2∑+−∂
∂=
∂
∂jjga
gcat
gcat kCk
xC
Dt
Cθε 2
2
in the inert zones in the catalyst zones
(...) (...)a g j jRate k C k θ= +∑ ∑
56
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
IX. Second-Order Statistical Regression
57
noise needs to be whitened rendered homoskedastic
t (s)
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58
X. Parcimonious network selection
!
• Bode analysis helps bound network complexity
• In line with the experimental window on reality
Example:
58
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
XI: Be lazy, use TAPFIT code
59
Purpose • Solve the direct problem: calculate TAP pulse responses from a general 1D reactor model • Solve inverse problems: regress TAP experimental data and estimate various physical and kinetic parameters Features • Regresses the raw pulse responses (in volts) directly • Regression
• NLSQ (classical non-linear least square regression) • SOSR (second order statistical regression)
•Uses the Levenberg-Marquardt method for optimization • Integration can be performed in
• time domain • frequency domain (Fourier transformation, for linear kinetics)
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Overview
60
• Introduction • Case: total oxidation of volatile organic
components
• Mathematical analysis
• Statistical analysis
• Best practices
• Conclusions
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Conclusions
61
TAP offers a unique mix of opportunities:
• data acquisition near millisecond resolution
• insignificant perturbation of catalyst state
• robust math modeling
• both intuitive qualitative interpretation and advanced model-free analysis techniques
• challenges for catalytic, engineering and mathematical research
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Acknowledgments This work was supported by • The ‘Long Term Structural Methusalem Funding by the Flemish
Government’ • Advanced Grant of the European Research Council (N° 290793) • Marie Curie International Incoming Fellowship (N° 301703)
• Profs. D. Constales and G.S.Yablonsky • Dr. Vladimir Galvita • Dr. Evgeniy Redekop • Rakesh Batchu • Dr. Veerle Balcaen • Dr. Raf Roelant • Dr. Unmesh Menon
62
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
Glossary
63
For a linear, time-invariant system…..
Bode plot is a graph of the transfer function versus frequency, typically plotted with a log-frequency axis. Often it is a combination of magnitude plot 𝐻𝐻(2𝜋𝜋𝜋𝜋𝜋𝜋) and phase plot arg(𝐻𝐻(2𝜋𝜋𝜋𝜋𝜋𝜋)).
Transfer function 𝐻𝐻(2𝜋𝜋𝜋𝜋𝜋𝜋) is a mathematical representation in terms of temporal frequency 𝜋𝜋 of the relation between the input X(2𝜋𝜋𝜋𝜋𝜋𝜋) and output Y(2𝜋𝜋𝜋𝜋𝜋𝜋) with zero initial conditions and zero-point equilibrium such that Y(2𝜋𝜋𝜋𝜋𝜋𝜋)=H(2𝜋𝜋𝜋𝜋𝜋𝜋)·X(2𝜋𝜋𝜋𝜋𝜋𝜋)
Poles of the transfer function show up on Bode plots as breakpoints at which the slope decreases.
Zeros of the transfer function show up on Bode plots as breakpoints at which the slope increases.
NEXTLAB , IFPEN / Rueil-Malmaison - 2-4 April 2014
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