Distributed Two-phase Reaction Systems Diogo Rodrigues, Julien - - PowerPoint PPT Presentation

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Incremental Model Identification of Distributed Two-phase Reaction Systems

Diogo Rodrigues, Julien Billeter, Dominique Bonvin

Laboratoire d’Automatique Ecole Polytechnique Fédérale de Lausanne Switzerland

ADCHEM June 8, 2015, Whistler

SLIDE 2

Outline

• Kinetic Identification Methods
• Incremental Model Identification (using PDE)

with Material Balance Equations in (z,t) domain

• Simplified Incremental Identification (using ODE)

with Material Balance Equations in τ domain

• Simulated Example
• Conclusion and Perspectives

2

SLIDE 3

Kinetic Identification Methods

From data to rates/extents

3 ① Simultaneous approach ② Incremental approach (rate-based) ③ Incremental approach (extent-based) Can one identify Distributed Reaction Systems via incremental approach?

SLIDE 4

Incremental Model Identification (PDE)

Material balance equations in (z,t) domain

• Two-phase plug-flow reactor (1 dimension, without diffusion) with

– sf species – rf reactions – m mass transfers

described by a set of sf Partial Differential Equations (PDE)

4

Accumulation Transport by advection Reaction Mass Transfer , T ,

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

) ( ( )

f f f m f m f z f f f t f f f

z t z t z t z t

v ε ε ε ε

∂ ∂ ∂ ∂

+ = ± c c N r E             φ

Structural information: Nf (rf × sf) stoichiometry, Em,f (sf × m) mass-transfer matrix (+ for L, − for G) State variables: cf (z,t) (sf × 1) concentrations, εf := εf (z,t) volumetric fraction, vf := vf (z,t) velocity of phase F Time-variant signals: rf (z,t) (rf × 1) reaction rates, φm,f (z,t) (m × 1) mass-transfer rates

, ,

( , ) ( , ( ) ) ( )

( ), ( ), F {L, G}, { , }

f f i f f n

z z t t

f l g

= =

= = c c c IC c BC

SLIDE 5

Incremental Model Identification (PDE)

Step 1: Transformation to extents (1/2)

• The effect of advection on IC and BC in absence of other dynamic effects

is computed using cibc,f (sf × 1) and the velocity profile vf (z, t)

5 , ,

( , ) ( , )

( ) ( )

f

f f f t z ibc f ibc f s

z t z t

v ε ε

∂ ∂ ∂ ∂

+ = c c

, , , ,

( , ) ( , ( ) ( ) )

( ), ( )

f ibc f i f in bc f

z z t t = =

c C c c c I BC

• Removing the effect of advection on IC and BC via δ cf := cf – cibc,f yields

Reaction Ma T ss Tran , s , fer

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

( ) ( )

f f f f f f m f z f f t f f m

z t z t z t z t

v ε δ ε ε ε δ

∂ ∂ ∂ ∂

+ = ± c r E c N        φ

( , ) ( , )

f

s f f

z t

δ δ = = c c

SLIDE 6

Incremental Model Identification (PDE)

Step 1: Transformation to extents (2/2)

• If , then transforms δ cf in

6

T ,

rank([ ])

f m f f

r m ± = + N E

1 T ,

: [ ]

f m f f f −

± = N E Τ P

• Tf splits the material balance in 3 sets of PDE:

Advection of material produced by the reactions Advection of material exchanged between phas

, , , , , ,

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

( ) ( ) ( ) ( )

f

f f f f r r f r f f r m f m f r f f t z f f f t z

z t z t z t z t z t z t

v v ε ε ε ε ε

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ = = = + x x r x x x x    

es

, , ( , ) ,

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

f f

f m f m f m f m iv f s r m

z t z t z t

ε

− +

= = = = x x x    φ

Reaction variants Mass-transfer variants Invariants

, , ,

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

r f m f iv f f f

z t z t z t z t

δ     =     c T x x x

, T , ,

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

f m r f m f f ibc f

z t z t z t z t ±

= + N x x c c E

SLIDE 7

Incremental Model Identification (PDE)

Step 2: Model identification

• Incremental identification of the reaction rates rf

7

( )

2 1 1

1

, ,

, , , , , , , , , , , , , , , , , , , , , , , , ,

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

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

r f i

P H r f i p h p h f r f i r f i r r f i r f i p h f f f f t z r f i r f i r f i f r f i r f i r f i i f f i

z t z t z t z t z t z t

x x x x i r v r x x ε ε ε

= = ∂ ∂ ∂ ∂

− ∀ = + = = =



θ

θ θ θ θ c θ θ 



• Incremental identification of the mass-transfer rates φm,f

( )

2 1 1

1

, ,

, , , , , , , , , , , , , , , , , , , , , , , , , ,

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

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

m f j

m f j m f j m f j m P H m f j p h p h p h f f f f t z m f j m f m f j m f j m f j m j f j m f j l j f j f m g

z t z t z t z t z t z t z t

x x x j m v x x x ε ε φ ε

= = ∂ ∂ ∂ ∂

− ∀ = + = = =



θ

θ θ θ c θ θ θ c  



~

( ) denote measured quantities or variables computed from measured quantities ⋅

• This method requires measurements along the z direction of the reactor!

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Simplified Incremental Identification (ODE) Material balance equations in τ domain

• Simplifications

– Steady-state mass transfers, – Constant volumetric fractions, – Constant and identical velocity in the two phases, – Constant boundary conditions.

8

• To an observer sitting on a particle of velocity v, the state vector c(z,t) is viewed

as cp(τ ), with z = v τ and t =τ (time spent in the reactor up to position z), described by a set of (sl +sg) Ordinary Differential Equations (ODE)

, ,

T

( ) ( ) ( ), ( )

[ ]

l in g in

d p n d i m m p τ

τ τ τ

= + = = c

c

c c c N r E φ

, , , , T

T

+

, and

[ ] [ ] [ ]

g

m l l s r m p l g p g

p m

×

−

= = =

E N E c c

N E c

• Typically: measurements are only available at the reactor exit!

SLIDE 9

Simplified Incremental Identification (ODE) Step 1: Transformation to extents

• If , is applied to δ cp := cp – cin

9

T

rank([ ])

m

r m = + N E

1 T

: [ ]

m −

= N E Τ P

, , ,

( ) ( ) ( ) ( )

p r p m p iv p

τ τ τ τ

δ     =     T c x x x

• T splits the material balance in 3 sets of ODE:

T , ,

( ) ( ) ( )

p r p m p n m i

τ τ τ +

= + x x N c E c

Reaction variants Mass-transfer variants Invariants

, , , ( , , ) ( )

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

l g

p r r p m m m p p i r p m v s s r m τ τ

τ τ τ τ τ τ τ

+ − + ∂ ∂ ∂ ∂

= = = = = x r x x x x φ

SLIDE 10

Simplified Incremental Identification (ODE) Step 2: Model identification

• Model identification requires measurements

for various values of τk

10

( )

2 1

1

,

, , , , , , , , , , , , ,

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

min ,..., s.t.

r i

p p r i K p r i k k k d r i r i r i p r i p r i i d l r i

x x x i r x r

τ

τ τ τ τ

=

− ∀ = = =



θ

θ θ θ θ c 



• Incremental identification of the mass-transfer rates φm

~

( ) denote measured quantities or variables computed from measured quantities ⋅

( )

2 1

1

,

, , , , , , , , , , , , , , ,

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

min ,..., s.t.

m j

p m j p p l p m j g K p m j k k k d p m j d m j m m j m m j j j

x x x j m x

τ

τ τ τ τ τ

φ

=

− ∀ = = =



θ

c c θ θ θ θ  



• Incremental identification of the reaction rates r

( )

p k

τ

c

e k

k z

v

τ

=

( )

p k

τ

c K measurements can be obtained by measuring the concentrations at the reactor exit z = ze for K different values of the velocity

SLIDE 11

Simulated Example

Chlorination of butanoic acid

• System composed of sg = 2 species in phase G and sl = 5 species in phase L

11

2 2

1 2 2 1 2 5 1 2

1 2 2 2

. ( ) ( ) ( ) ( ) ,BA ,Cl ,M BA ( ) ( ) ( ) ( ) ,C l

R : BA Cl MBA HCl R : BA Cl DBA HCl

l l l l l l l l l l l l

c c c c r k r k r + → + = + → + =

• m = 2 steady-state mass transfers obeying the rates

2 2 2 2 Cl 2 2 Cl HCl HCl 2

Cl ,Cl Cl HCl ,HC C l H l HC ,Cl , l Cl HCl

( ), ( ),

m p l H p l H m

a c c c a c k c c k φ φ = − = = − =

   

• Known structural information

1 2 1 1 1 1 2 1

T

,

[ ]

l

− − − −

= N

1 1

,

,

[ ]

m l =

E

2 2 T

,

g ×

= 0 N

2 , m g =

E I

, ,

[ ]

m l m g

m

+ −

=

E E

E

2 2

T

T

,

[ ]

l

×

=

N

N (z,t): τ :

• r = 2 reactions in the phase L

SLIDE 12

Simulated Example

Chlorination of butanoic acid

• System composed of sg = 2 species in phase G and sl = 5 species in phase L

12

2 2

1 2 2 1 2 5 1 2

1 2 2 2

. ( ) ( ) ( ) ( ) ,BA ,Cl ,M BA ( ) ( ) ( ) ( ) ,C l

R : BA Cl MBA HCl R : BA Cl DBA HCl

l l l l l l l l l l l l

c c c c r k r k r + → + = + → + =

• m = 2 steady-state mass transfers obeying the rates

2 2 2 2 Cl 2 2 Cl HCl HCl 2

Cl ,Cl Cl HCl ,HC C l H l HC ,Cl , l Cl HCl

( ), ( ),

m p l H p l H m

a c c c a c k c c k φ φ = − = = − =

   

• Known structural information

1 2 1 1 1 1 2 1

T

,

[ ]

l

− − − −

= N

1 1

,

,

[ ]

m l =

E

2 2 T

,

g ×

= 0 N

2 , m g =

E I

, ,

[ ]

m l m g

m

+ −

=

E E

E

2 2

T

T

,

[ ]

l

×

=

N

N (z,t): τ :

• r = 2 reactions in the phase L

1

( , )

f h

z t

c

3

( , )

f h

z t

c

5

( , )

f h

z t

c

7

( , )

f h

z t

c

9

( , )

f h

z t

c

2

( , )

f h

z t

c

4

( , )

f h

z t

c

6

( , )

f h

z t

c

8

( , )

f h

z t

c

, (

)

f in h

t

c

10

( , )

h f z

t

c

1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10 k =

Stacked reactor (10 plates)

z = 1 m

e

z z = =

gives access to measurements along the z direction

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Simulated Example

Dynamic experiments in (z,t) domain (PDE)

Simulated (noise-free, –––) and experimental (2% noise) concentrations at t = 5 s

13

Modeled (with identified models, –––) and experimental (2% noise) extents at z = 0.5 m (above) and z = ze = 1 m (below)

0.2 0.4 0.6 0.8 1 0.5 1.0 1.5 2.0 z (m) cl (mol L−1) 1.0 2.0 3.0 4.0 cg (mol L−1)

Cl2(g) HCl(g) Cl2(l) BA(l) MBA(l) HCl(l) DBA(l)

1 2 3 4 5 1 1 2 t (s) xr, xm,l, xm,g (mol L−1) 1 2 3 4 5 1 1 2 t (s) xr, xm,l, xm,g (mol L−1)

• xr,1 (R1), xr,2 (R2), Cl2: –– xm,g,1, --- xm,l,1

HCl: –– xm,g,2, --- xm,l,2 xr,1 (R1), xr,2 (R2), Cl2: –– xm,g,1, --- xm,l,1 HCl: –– xm,g,2, --- xm,l,2

SLIDE 14

Simulated Example

Steady-state experiments in τ domain (ODE)

Simulated (noise-free, –––) and experimental (2% noise) concentrations

14

Modeled (with identified models, –––) and experimental (2% noise) extents

0 1 10 10 0 1

m s

[ . , ] s [ , . ] v τ ∈ ⇔ ∈

2 4 6 8 10 0.5 1.0 1.5 2.0 τ (s) cp,l (mol L−1) 1.0 2.0 3.0 4.0 cp,g (mol L−1)

Cl2(g) HCl(g) Cl2(l) BA(l) MBA(l) HCl(l) DBA(l)

2 4 6 8 10 1.5 0.5 0.5 1.5 2.5 τ (s) xp,r, xp,m (mol L−1)

xp,r,1 (R1) xp,m,2 (HCl) xp,m,1 (Cl2) xp,r,2 (R2)

SLIDE 15

Simulated Example

Incremental identification of reaction rates

15

Incremental model identification of reactions R1 and R2 based on their respective extents of reaction, with corresponding sum of squared errors (SSE)

R1 Rate expression SSE R2 Rate expression SSE xr,1(zp,th) xp,r,1(τk ) xr,2(zp,th) xp,r,2(τk ) 0.923 1.270 0.153 0.018 5.120 0.563 0.131 0.017 2.013 0.320 0.107 0.010 0.089 0.026 0.049 0.005

2

0 5 1 . ,BA ,Cl ,MBA l l l

k c c c

2

1 ,BA ,Cl ,MBA l l l

k c c c

2

1 ,BA ,Cl l l

k c c

2

1 ,Cl l

k c

2

2 2 1 ,BA ,Cl l l

k k c c

2

2 2 1 ,BA ,Cl l l

k k c c

2

2 2 1 ,BA ,Cl ,MBA l l l

k k c c c

2

2 0 5 2 1 . ,BA ,Cl ,MBA l l l

k k c c c

1 1 ( )

ˆ r

2 1 ( )

ˆ r

3 1 ( )

ˆ r

4 1 ( )

ˆ r

1 2 ( )

ˆ r

2 2 ( )

ˆ r

3 2 ( )

ˆ r

4 2 ( )

ˆ r

SLIDE 16

Simulated Example

Parameter estimation

16

Estimated rate constants for reactions R1 and R2, and mass-transfer coefficients for Cl2 and HCl, with corresponding 99% confidence intervals

0.2 0.4 0.6 0.8 1 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375

k1 = 1.3577 xr,1(zp,th) xp,r,1(τk )

Reaction R1 (r1)

SLIDE 17

Simulated Example

Parameter estimation

17

Estimated rate constants for reactions R1 and R2, and mass-transfer coefficients for Cl2 and HCl, with corresponding 99% confidence intervals

0.2 0.4 0.6 0.8 1 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375

k1 = 1.3577 xr,1(zp,th) xp,r,1(τk )

0.2 0.4 0.6 0.8 1 0.66 0.665 0.67 0.675 0.68 0.685 0.69 0.695 0.7

xr,2(zp,th) xp,r,2(τk ) k2 = 0.6788

Reaction R1 (r1) Reaction R2 (r2)

SLIDE 18

Simulated Example

Parameter estimation

18

Estimated rate constants for reactions R1 and R2, and mass-transfer coefficients for Cl2 and HCl, with corresponding 99% confidence intervals

0.2 0.4 0.6 0.8 1 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375

k1 = 1.3577 xr,1(zp,th) xp,r,1(τk )

Mass transfer Cl2 Mass transfer HCl

0.2 0.4 0.6 0.8 1 0.66 0.665 0.67 0.675 0.68 0.685 0.69 0.695 0.7

xr,2(zp,th) xp,r,2(τk ) k2 = 0.6788

2

,Cl

( )

m

φ

,HCl

( )

m

φ Reaction R1 (r1) Reaction R2 (r2)

0.2 0.4 0.6 0.8 1 0.835 0.84 0.845 0.85 0.855 0.86 0.865 xm,l,2(zp,th)

xm,g,2(zp,th) xp,m,2(τk ) = 0.8450·10-4

HCl

k

0.2 0.4 0.6 0.8 1 0.655 0.66 0.665 0.67 0.675 0.68 0.685

xm,l,1(zp,th) xm,g,1(zp,th) xp,m,1(τk ) = 0.6660·10-4

2

Cl

k

SLIDE 19

Conclusion and Perspectives

• Extend-based model identification for 2-phase distributed reaction systems

– via dynamic experiments collected at several (z,t) points → (z,t) domain → PDE – via steady-state experiments collected under various velocities → τ domain → ODE

• Extension to distributed systems with diffusion

– to identify kinetics independently from the effects of diffusion – to extract extents of diffusion

• Applications of industrial relevance

– to distributed reaction-separation systems (reactive absorption, reactive distillation) – to 3D distributed reaction systems (tubular reactors, micro-reactors)

19

SLIDE 20

Final word

20

References

• Brendel M., Bonvin D., Marquardt W., Incremental identification of kinetic models

for homogeneous reaction systems, Chem. Eng. Sci. 61 (2006) 5404

• Bhatt N., Amrhein M., Bonvin D., Incremental identification of reaction and mass-

transfer kinetics using the concept of extents, Ind. Eng. Chem. Res. 50 (2011) 12960

• Rodrigues D., Srinivasan S., Billeter J., Bonvin D., Variant and invariant states

for chemical reaction systems, Comp. Chem. Eng. 73 (2015) 23

Thank you for your attention