incremental model identification of reaction systems cpac...incremental model identification of...

§Incremental Model Identification

of Reaction Systems

Julien Billeter

Ecole Polytechnique Fédérale de LausanneLaboratoire d’Automatique

Switzerland

CPAC / ATOCHEMIS Workshop 25 – 27 March 2013, Rome – Italy

Outline

o Motivation

o Concept of vessel extent

o Homogeneous reaction systems

o Incremental model identification

o Fluid-Fluid reaction systems

o Extensions to calorimetry and spectroscopy

o Perspectives: distributed reaction systems

2

Model reductionSeparate fast / slow dynamics

o Discard redundant (invariant) states• What is the minimal number of states (variants) ?• Batch reactors: S → R extents• Open reactors: S → R+p+1 vessel extents• Open G-L reactors: S → R+pm+p+1 vessel extents

o Separate fast/slow dynamics• Rates are fast/slow, not individual concentrations→ work with extents, not concentrations…

3Bhatt et al, Comp. & Chem. Eng. (2011), submittedM. Amrhein, PhD dissertation n°1861 (1998), EPFL, Switzerland

Outline

Motivation

o Concept of vessel extent






4

DefinitionsExtent vs vessel extent of reaction

o Extent of the i-th reaction :number of moles produced by the i-th reaction

5

o Vessel extent of the i-th reaction : number of moles produced by the i-th reaction still in vessel

( ) ( ) ( ),

1, , , 0 0

s ir i s v i r it n r tνξ ξ= = =

Differential extent of reaction

( ),r i tξ

( ) ( ) ( )( ) ( ) ( ), ,, , 0 0out

r i v i ru t

r im t ixx t r t xt= − =

Differential vessel extent of reaction

( ),r ix t

N. Bhatt, PhD dissertation n°5028 (2011), EPFL, Switzerland

DefinitionsExtent vs vessel extent of mass transfer

o Extent of the j-th mass transfer :mass transferred by the j-th mass transfer

6

o Vessel extent of the j-th mass transfer : mass transferred by the j-th mass transfer still in vessel

( ) ( ) ( ), , 0 0m j j m jt tξ ζ ξ= =Differential extent of mass transfer

( ),m j tξ

( ) ( ) ( )( ) ( ) ( ), ,, 0 0outu t

m j j m jjm t mx t t t xxζ −= =

Differential vessel extent of mass transfer

( ),m jx t


General concept of vessel extent

A vessel extent indicates the amount of material (number of moles, mass, volume), associated with one phenomenon, that is still in the vessel…

7

( ) ( ) ( )( ) ( ) ( ), ,, , 0 0outu t

f im tf i f i f ix tx t t xξ −= =

Differential vessel extent (i-th phenomenon f )


Outline

Motivation

Concept of vessel extent






8

Homogeneous reaction systemsMole balance equations

9

( ) ( ) ( ) ( )( ) ( ) ( )T

0, 0outiv i mn

u tn tt t t t= + − =n n nr uN W n

( ) ( ) ( ) ( )( ) ( ) ( )

( )T , , m t tS w t V tt t V t tρ= = = nm 1 M n c

(S × R) (S × p)(R × 1) (p × 1)(S × 1)

Homogeneous reaction system consisting of S species,R independent reactions, p independent inlets and one outlet

Mole balances for S species

Mass, volume and concentrations

( ), in in tW u

( ) ( ), outt u tn

N ( )v tr0n

(S × 1)

Amrhein et al, AIChE J. 56 (2010) 2873

Homogeneous reaction systemsDecomposition in variants and invariants

o Condition:

o Vessel extents (variants) and q = S – R – p –1 redundant information (invariants)

10

( )T0rank 1in R p = + + N W n

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

1 1

T

T

T T T

T

, 0

, 0

1 , 0

0

out

out

out

R R p

p R p

R p

q p

o

R

ut

q

u tr v in in r rm t

u tin v in in in inm t

u tic v in in ic icm t

u tiv v in in ivm t

R

p

t t t t

t t t t

x t t t x t x

t t t t

×

×

× ×

× ×

= + − =

= + − =

= + − + =

= + −

I 0

0 I

0 0

0 0

x R N r R W u x x

x F N r F W u x x

q N r q W u

x Q N r Q W u

0

0

x

( ), 0 qiv =x 0

( )( )( )( )

( )( )T 0 r

in

ic

iv

tt tx tt

= −

x Rx F n nq

Qx

L


Homogeneous reaction systemsDecomposition in variants and invariants

o Condition:

o Vessel extents (variants) and q = S – R – p –1 redundant information (invariants)

o Reconstruction:

11

( )( )( )( )

( )( )T 0 r

in

ic

iv

tt tx tt

= −

x Rx F n nq

Qx

L

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )( )

( )( ) ( ) ( )

( )

, 0

,

0

0

, 0

out

ou

out

t

out

u tr rm t

u tin inm t

r

in

iu t

v R

in p

u tm t

q

ic icmc t

iv

t t t

t t t

t x t x

t

x

= − =

= − =

= − =

=

−

x x

x x

x

x u

0x

r 0

0

( )( ) ( ) ( ) ( )1

0T

0+ +inr in ict t txt−

− = x xN nn Wn L

( )T0rank 1in R p = + + N W n


Homogeneous reaction systemsOrthogonal spaces in 4-way decomposition

12

TQ Q

inW F

TN R

space of inlet extentsspace of reaction extents

space of invariants

q

p

R

T0n q

space of initial conditions

1

S-dim space of the number of moles


Homogeneous reaction systemsTransformation to RV form

o When ,

the numbers of moles are rearranged in Reaction Variant (RV) form:

o The R vessel extents of reaction are then computed as:

13

( ) ( ) ( )( )T0rank 1 and knownin oi tn uR p t u t < + + N W n u

( ) ( ) ( ) ( ) ( )( )TRV0 1i in in cr t xt t tt = = − − +N nx xn Wn


( ) ( ) ( )T RVr t t

+= nx N

Outline

Motivation


Homogeneous reaction systems





14

Kinetic investigation From measurements to rate expressions

15

① Simultaneous approach② Incremental approach (rate-based)③ Incremental approach (extent-based)

Bhatt et al, Chem. Eng. Sci. 8 (2012) 24

Incremental model identificationExtent-based method

o The kinetic problem is decomposed into sub-problems of lower complexity that are solved individually.

o The model identification proceeds in two steps:

o Transformat ion to extents (v+ iv)Computation of the contribution of each dynamic effect (reaction, inlets and outlets) as extents

o Model ident i f i ca t ion (Parameter estimation)Individual model identification of each effect from its corresponding extent with the integral methodof parameter estimation.

16Bhatt et al, Ind. & Eng. Chem. Res. 50 (2011) 12960

Extent-based model identificationModel identification and parameter estimation

A dynamic model is postulated for each extent of interestand a regression problem is solved individually using theintegral method of parameter estimation.

Example: fitting of R extents of reaction

17Bhatt et al, Ind. & Eng. Chem. Res. 50 (2011) 12960

Extent-based model identificationModel identification and parameter estimation

A dynamic model is postulated for each extent of interestand a regression problem is solved individually using theintegral method of parameter estimation.

Example: fitting of R extents of reaction

18

s.t. ( ) ( ) ( )( ) ( ), , , , ,

ˆ ˆ, , ,outu tr i r i v i r i r im tx t r t x t= −θ θ

, , ,L Ur i r i r i≤ ≤θ θ θ

( ),ˆ 0 0r ix =

1, , i R= ( ) ( ),

2

, , ,ˆmin ,r i

r i r i r ix t x t−θ

θ

Bhatt et al, Ind. & Eng. Chem. Res. 50 (2011) 12960

Homogeneous reaction systemsEthanolysis of phthalyl chloride in a CSTR

Ethanolysis of phthalyl chloride (A) comprising seven species(S = 7), three reactions (R = 3), two inlets (p = 2) and 1 outlet

19

( ), ,, in A in Aw u t

( ) ( ), outt u tn

N0n

( ), ,, in B in Bw u t

− − = − − − −

=

T,

,

1 1 1 1 0 0 0 0 1 1 1 1 0 0

0 1 0 1 0 1 1

0 0 0 0 0 00 0 0 0 0 0in A

inin B

ww

N

W

+ → ++ → ++ +

A B C DC B E DD B F G

( )v tr

Extents of reaction ?N. Bhatt, PhD dissertation n°5028 (2011), EPFL, Switzerland

Homogeneous reaction systemsEthanolysis of phthalyl chloride in a CSTR

20

T

=

RFqQ

L

0, , inN W n

Number of moles

R = 3 Extents of reaction

p = 2 Extents of inlet

1 Extents of outlet

Each extent of reaction can then be modeledindividually, that is, independently fromall the other phenomena / extents…

Outline

Motivation



Incremental model identification




21

Fluid-Fluid reaction systemsMole balance equations

22

( ) ( ) ( ) ( ) ( )( ) ( ) ( ),

, ,T

, , 0,, 0out b

bv b b in bb mu t

b b bm tb in b bt t t t t= ± + − =N Wn nWr ζ nu n(Sb × Rb) (Sb × pm)(Rb × 1) (pm × 1)(Sb × 1)

Fluid-Fluid reaction system consisting of:

o Phase L: Sℓ species with Rℓ reactions, pm mass transfers, pℓ inlets and 1 outlet

o Phase G: Sg species with Rg reactions, pm mass transfers, pg inlets and 1 outlet

Mass transfer described by various models…

Mole balances on phase B

(Sb × 1)(Sb × pb)(pb × 1)

gp inlets in G Outlet from G

out,guin, in,, g gW u

lp inlets in L

in, in,, W u

Outlet from L

out,u

Phase G

Phase L

, , g g gm Vn

, , m Vn

, g± ζ ζMass transfer

m,

m,g

WW

Rb reactions, pm mass transfers, pb inlets, 1 outlet { } { }, , ,B L b∈ ∈ G g


Fluid-Fluid reaction systemsDecomposition in variants and invariants

o Condition:

o Vessel extents (variants) and qb = Sb – Rb – pm – pb –1 invariants

( )T, 0, ,rank 1 b in b bm b bmb R pp = + + + N W nW

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ),

,

T, , ,

T, , , , , ,

, , ,

,

,

, 0

, 0

b

R R p R pb bout

m b b

m

p R p p pm b

out

b

b

b

m m

b

b

u tm

u tr b b b v b b m b

b b b v b b m b b b in b in b m b in bm

b b in b in b r b r bm

i

R

t

t

pt t t

t t t

t t

t t× ×

× ×

= ±

= ± + −

+ − =

=I 0 0

0 I 0

x R N r R W 0ζ

x M N r M W ζ M W u

R W u x

0

x

x

x

x

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )

1 1

,

1

T, , , , , ,

T T T T, , , , , , ,

, 0

1 , 00

b

p R pb p p bb m

R p pb m

out

b

ou

b

t b

b

u tn b b b v b b m b b b in in b in b in bm t

u tic b b b v b b m b b b in b in b ic b ic

p

bm t

t t t t t

x t t t t x t x× ×

× × ×

= ± + − =

= ± + − + =0 I0

0 0 0

F N r F W ζ F W u x x

q N r q W ζ q W u

0

( )( )( )( )( )

( )( )0,T

bm b

r b

b b bin

bic

biv

t

ttt

x tt

= −

x Mx R

F n nxqQx

L { },b∈ g

Fluid-Fluid reaction systemsDecomposition in variants and invariants

o Condition:

o Vessel extents (variants) and qb = Sb – Rb – pm – pb –1 invariants

( )T, 0, ,rank 1 b in b bm b bmb R pp = + + + N W nW

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )( )

( )( ) ( ) ( )

,

,

,

,

, , , ,

, , , ,

, , ,

, , ,, 0

, 0

0 0

, 0

,

out b

b

out

b

out b out b

out b

b

mb

b

b

b

u tr b v b r b r bm t

u tin b in b in b i

u tm b b

n bm tu t u t

ic b ic

m b in bm t

R

b ic bm t m t

p

p

t t t

t t t

x t x t x

t t t=

= − =

=

= −

−

=

= − − =

x ζ x

x r x x

x u x x

0

0x

0

( )( )( )( )( )

( )( )0, b

r b

in b b

m

b

bic

iv b

b

t

t tx t

t

t = −

x Mx R

x F n nqQx

L { },b∈ g

( )( ) ( ) ( ) ( ) ( )1

T,, , ,, , , 00 ,+b

m b m bb b r b in bb in ic bb bt t t t x t−

− = +±n W xx xN W nnL

( ),with biv b qt = 0x

Fluid-Fluid reaction systemsOrthogonal spaces in 5-way decomposition

25

Tb bQ Q

, bin bW F

TbbN R

space of inlet extentsspace of reaction extents

space of invariants

bq

bp

bR

T0, bbn q

space of initial conditions

1

Sb-dim space of the number of moles in phase B

, bm bW Mmp

space of mass-transfer extents


Fluid-Fluid reaction systemsTransformation to RMV form

o When

, the numbers of moles are rearranged in

Reaction Mass-transfer Variant (RMV) form:

o The Rb vessel extents of reaction and pm extents of mass transfer

are then computed as:

26

( )T, 0, ,rank 1 b in b bm b bmb R pp < + + + N W nW

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )

T T, ,

,

,, ,

,

, ,0

R

,

MV

1

r b

in b ic

m b m b

in b b

r b m b

b

bm b

b

tt

t tt

txt t

= ± = ±

= − − +

xx x

xN W N W

W nx

n

n


( )( , andin b tu

( ) ), knownout bu t

( )( ) ( )T,

,

RMV, r b

mm b b

b

tt

t+

= ±

N Wx

nx

Fluid-Fluid reaction systemsChlorination of butanoic acid in a CSTR

Chlorination of butanoic acid comprises Sℓ = 5 (BA, MBA, DBA, Cl2, HCl) and Sg = 3 (Cl2, HCl, air) species, Rℓ = 2 reactions,pℓ = 1 and pg = 1 inlets and 2 outlets

27

in,HClW ( )out , ,u tn

Phase G

Phase L

2Clζ

Cl2, HCl, Air

BA, MBA, DBACl2, HCl

HClζin ,HClu

2in,ClW2in,Clu ( )out, ,g gu tn Gas-phase

measurements

Liquid-phasemeasurements

Extents of reaction ?

+ ⎯⎯→ ++ ⎯⎯→ +

2

2

R1

2 R2

cat

cat

BA Cl MBA HCl

BA Cl DBA HCl



28

T

=

RMfqQ

L

, , 0,, , , m inN W W nNumber of moles

in liquid phase

R = 2 Extents of reaction

pm = 2 Extents of m.t.



29

o Identification of the rate expression for the main reaction R1

o Rate expression candidates

o Identified rate expression

( )

( )

( )

( )

2

2

2

2

11 ,BA ,Cl

21 ,Cl

31 ,B

1

A ,Cl ,MBA4

1 ,BA ,Cl ,

1

1

1 MBA

r c cr cr c c cr c c c

kkkk

====

( )2

41 ,BA ,Cl ,MBA1. 3543r c c c=



o Identified rate expressions

o Results of curve fitting (2% noise level)

30

( )( )

2 2 22

2

2

2

1

2

Cl

HC

1 ,BA ,Cl ,MBA

2 1 ,Cl

*Cl ,Cl Cl ,Cl

*HCl ,HCl ,HCl H ll C

s w

s w

r c c cr r c

A V M c c

A

k

k V c

kk

M c

ζζ

=== −

= −

Parameter Simulated value Estimated value 95% Confidence interval

k1 1.3577 1.3543 [1.3207 – 1.3879]

k2 0.1 0.105 [0.0884 – 0.1216]

kCl20.666·10-4 0.594·10-4 [0.514·10-4 – 0.674·10-4]

kHCl 0.845·10-4 0.813·10-4 [0.763·10-4 – 0.863·10-4]


Outline

Motivation




Fluid-Fluid reaction systems



31

Recent extensionsRank augmentation of conc. data by calorimetry



32

( ) ( )( ) ( ) ( )RV

VT

T

R T

: :a aaug aug

rr r

rQ tt

t tt

= = = − Δ

n NNH

nx x

( ) ( )( )

( )( )

( )( )

T,T T

, ,

TRMV, , , , ,,

, ,

RMV : : augrm

a m a r ra

m mma

rug Q

tttt t

tt

= = = −Δ − Δ

N W x xnx x

n NH H

( ) ( ) ( )T RVau ugg ar tt

+= nx N

( )( ) ( ) ( )RMT

,, V

,aug a

rg

mu

tt

t+

=

nxx

N

Srinivasan et al, Chem. Eng. J. 207-208 (2012) 785

Recent extensionsConstruction of calibration models in spectroscopy

33Billeter et al, Anal. Chim. Acta 767 (2013) 21

Recent extensionsPrediction of concentrations from spectral data

34Billeter et al, Anal. Chim. Acta 767 (2013) 21

Outline

Motivation




Fluid-Fluid reaction systems

Extensions to calorimetry and spectroscopy


35

PerspectivesDistributed reaction systems

36

( )*(BC, ODE) g tc

( ) (BC, ODE)tn

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )

( )

2

2T

*

0PDE: , , , , 0, (IC,0 (BC)

, (

)

BC)

d ddt dx

gt

V t

t x t x t x x

t

xt t

δ

= + ==

= n

c N r D c c cc c

c

G LFilm F(1-dim)

( ),t xc

0 δ

Bonvin et al, TFMST, Lyon, July 13–16, 2013 (to be published)

How to decouple reaction and diffusion phenomena?

x

PerspectivesDistributed reaction systems

37

( )0(BC) x t=c ( ) (BC)x L t=c

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )

2

2 0

0

TPDE: , , , , , 0, (IC)

,0 (BC), (BC)

d d dx tdt dx d

x

x L

xt x t x v t x t x

t tt c t

x x

δ

=

=

=

= − +

==

=c N r c D c c

c cc

c

0 L

Bonvin et al, TFMST, Lyon, July 13–16, 2013 (to be published)

( ),t xc

How to decouple reaction, convection and diffusion phenomena?

1-dim

x

Laboratoire d’Automatique

38

Prof. Bonvin, Prof. Longchamp, Prof. Jones, Dr. MER Karimi6 Postdocs + 20 PhD students + technical / administrative staff

Thank you for your attention

39

Model reductiono M. Amrhein, PhD dissertation n°1861 (1998), EPFL, Switzerlando Bhatt et al, Comp. & Chem. Eng. (2011), submitted

Transformation to variants/invariants (extents)o N. Bhatt, PhD dissertation n°5028 (2011), EPFL, Switzerlando Amrhein et al, AIChE J. 56 (2010) 2873o Bonvin et al, TFMST, Lyon, July 13–16, 2013 (to be published)

Incremental model identificationo Bhatt et al, Ind. & Eng. Chem. Res. 49 (2010) 7704o Bhatt et al, Ind. & Eng. Chem. Res. 50 (2011) 12960o Bhatt et al, Chem. Eng. Sci. 8 (2012) 24

Rank augmentation by calorimetric datao Srinivasan et al, Chem. Eng. J. 207-208 (2012) 785

Incremental kinetic modeling of spectroscopic datao Billeter et al, Anal. Chim. Acta 767 (2013) 21

incremental model identification of reaction systems cpac...incremental model identification of...

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