numerical and experimental aspects of temporal analysis of

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


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


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


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


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


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


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


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


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


TAP-3E

12

Reactor carousel (6 reactors)

Reactor

Reactor extractor


Reactor carousel and reactor extractor

13

Turbo pump (Attached to Mass spectrometer chamber)

Carousel


Manifold

14


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.








TAP-3E

16

EXTREL QMS with conical aperture for increased signal-to-noise ratio

one pulse TAP-3

TAP-1 TAP-3E


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


Intracrystalline diffusion

18

quartz (─) H-ZSM 5 (─)

T=298K

Ar


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.








TAP-3E: Atmospheric pressure experiments

20

Reactor

Slide valve (Atmospheric pressure experiments)

Reactor carousel


Overview

21


components



• Best practices

• Conclusions


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


Reaction mechanism

Al2O3

CuO or CeO2

O*,s Oads

C3H8

O2

O*,b CO2 CO2

CO2

Elementary steps + Active sites

23


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


Propane responses Calculated

623 K

673 K

723 K

773 K

823 K

873 K

Experimental

25


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


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



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



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



Total oxidation mechanism: summary

30

1 2

3

4

5

O2 H2O

CO2

CO2

H2O

O2

O- O2-

Cu2+

Vo



Overview

31


components



• Best practices

• Conclusions


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


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.


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


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


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


Noise

no meaningful information

frequency (Hz)

modulus (mV)

sample time (ms)

signal (mV)

Di-oxygen pulse response

37


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


Thin-zone TAP-reactor: Info on intermediates

Uptake Release

Information flow:

Intermediates

Redekop et al. Chem. Eng. Sci., 2011, 66, 6441–6452 39


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


Overview

41


components



• Best practices

• Conclusions


Experimental data: time series

42


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


44

Heteroskedasticity of TAP noise

More noise near peak than in tail


45

Autocorrelation of TAP noise

F

t

ε1

ε2

ε1

ε2

R. Roelant et al. Catal. Today 2007, 121, 269

[1]


46

Autocorrelation of TAP-noise

t

F

ε1

ε2

ε1

ε2

46


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


48

Principal Component Analysis and rescaling

Rescaling of the principal component

axes

ε1

ε2

Projection on the Principal Component Axes

48


Estimation of diffusion coefficient

49

Second Order Statistical Regression (SOSR): realistic confidence intervals


Overview

50


components



• Best practices

• Conclusions


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


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):


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


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


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


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


IX. Second-Order Statistical Regression

57

noise needs to be whitened rendered homoskedastic

t (s)


58

X. Parcimonious network selection

!

• Bode analysis helps bound network complexity

• In line with the experimental window on reality

Example:

58


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)


Overview

60


components



• Best practices

• Conclusions


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


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


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.


numerical and experimental aspects of temporal analysis of

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