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
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
Kinetic Identification Methods From data to rates/extents Can one identify Distributed Reaction Systems via incremental approach? ① Simultaneous approach ② Incremental approach (rate-based) ③ Incremental approach (extent-based) 3
Incremental Model Identification (PDE) Material balance equations in ( z , t ) domain Two-phase plug-flow reactor (1 dimension, without diffusion) with • s f species – r f reactions – m mass transfers – described by a set of s f P artial D ifferential E quations (PDE) T ( c ) ( v c ) N r E ∂ ε + ∂ ε = ε ± ε φ ( , ) z t ( , ) z t ( , ) z t ( , ) z t t f f z f f f f f f m f , f m f , ∂ ∂ Accumulation Transport by advection Reaction Mass Transfer c c ( IC ), c c ( BC ), F {L, G}, f { , } l g = = ( , ) z ( z ) ( , t ) ( ) t = = 0 0 f f , f f i , n 0 State variables: c f ( z , t ) ( s f × 1) concentrations, ε f := ε f ( z , t ) volumetric fraction, v f := v f ( z , t ) velocity of phase F Structural information: N f ( r f × s f ) stoichiometry, E m,f ( s f × m ) mass-transfer matrix ( + for L, − for G) Time-variant signals: r f ( z , t ) ( r f × 1) reaction rates, φ m,f ( z , t ) ( m × 1) mass-transfer rates 4
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 c ibc,f ( s f × 1) and the velocity profile v f ( z , t ) ( c ) ( v c ) 0 ∂ ε + ∂ ε = ( , ) z t ( , ) z t t f ibc f , z f f ibc f , s ∂ ∂ f c c ( I C ), c c ( BC ) ( , ) z ( ) z ( , t ) ( ) t = = 0 0 ibc f , f , i bc f , f in , 0 Removing the effect of advection on IC and BC via δ c f := c f – c ibc,f yields • T ( c ) ( v c ) N r E ∂ ε δ + ∂ ε δ = ε ± ε φ ( , ) z t ( , ) z t ( , ) z t ( , ) z t t f f z f f f f f f m , f f m f , ∂ ∂ Reaction Ma ss Tran s fer c c 0 δ = δ = ( , ) z ( , ) t 0 0 f f s f 5
Incremental Model Identification (PDE) Step 1: Transformation to extents (2/2) T T If , then transforms δ c f in rank([ N E ]) r m : [ ] Τ = N E P − 1 ± = + ± • f m , f f f f m f , f x ( , ) z t r f , = T x T c c N x E x c δ = + ( , ) z t ( , ) z t z t z t z t z t ( , ) ( , ) ( , ) ( , ) ± m f , f f f f r f , m m f , ibc f , x ( , ) z t iv , f T f splits the material balance in 3 sets of PDE : • ( x ) ( v x ) r x x 0 ∂ ε + ∂ ε = ε = = ( , ) z t ( , ) z t ( , ), z t ( , ) z ( , ) t Reaction 0 0 f r f , f f r f , f f r , f r f , r t z ∂ ∂ variants f Advection of material produced by the reactions ( x ) ( v x ) x x 0 ∂ ε + ∂ ε = ε φ = = ( , ) z t ( , ) z t ( , ), z t ( , ) z ( , ) t Mass-transfer 0 0 t f m f , z f f m , f f m f , m f , m f , m ∂ ∂ variants Advection of material exchanged between phas es x 0 = Invariants ( , ) z t iv f , s ( r m ) − + f f 6
Incremental Model Identification (PDE) Step 2: Model identification Incremental identification of the reaction rates r f • P H 2 ( ) min x x i ,..., r − ∀ = ( z , t ) ( z , t , θ ) 1 r f i , , p h r f i , , p h r f i , , f θ r f i , , p h = 1 = 1 s.t. ( x ) ( v x ) r ∂ ∂ ε + ε = ε ( , , z t θ ) ( , , z t θ ) ( c ( , ), z t θ ) t f r , , f i r , , f i z f f r f i , , r f i , , f f , i f r f i , , ∂ ∂ x x = = z , θ t , θ ( , ) ( , ) 0 0 0 r f i , , r f i , , r , , f i r f i , , Incremental identification of the mass-transfer rates φ m,f • P H 2 ( ) min x x j ,..., m − ∀ = ( z , t ) ( z , t , θ ) 1 m f j , , p h m f j , , p h m f j , , θ m f j , , p h = = 1 1 s.t. ( x ) ( v x ) ∂ ε + ∂ ε = ε φ ( , , z t θ ) ( , , z t θ ) ( c ( , ), z t c ( , ), z t θ ) t f m f j , , m f j , , z f f m f j , , m f j , , f m , f , j l g m , , f j ∂ ∂ x x , θ = , θ = ( , z ) ( , t ) 0 0 0 m f j , , m f j , , m f , , j m , f j , This method requires measurements along the z direction of the reactor! • ~ ( ) denote measured quantities or variables computed from measured quantities ⋅ 7
Simplified Incremental Identification (ODE) Material balance equations in τ domain Typically: measurements are only available at the reactor exit ! • Simplifications • Steady-state mass transfers, – Constant volumetric fractions, – Constant and identical velocity in the two phases, – Constant boundary conditions. – To an observer sitting on a particle of velocity v , the state vector c ( z , t ) is viewed • as c p ( τ ), with z = v τ and t = τ ( time spent in the reactor up to position z ), described by a set of ( s l + s g ) O rdinary D ifferential E quations (ODE) = c [ ] T c N r E φ c c d = + = ( ) ( ) ( ), ( ) τ τ τ l in , 0 c d p m m p i n τ g in , c E [ ] [ N T ] [ + ] T c , N and E = = = p l , m l , l c 0 E − p m p g , s r × m , g g 8
Simplified Incremental Identification (ODE) Step 1: Transformation to extents T T If , is applied to δ c p := c p – c in rank([ N E ]) r m Τ : [ = N E P ] − 1 = + • m m x ( ) τ p r , = T x T c c N x E x c = + δ ( ) ( ) ( ) ( ) ( ) τ τ τ τ + τ p p r , m p m , i n p m , p x ( ) τ p iv , T splits the material balance in 3 sets of ODE : • x r x 0 ∂ = = ( ) ( ), ( ) Reaction variants τ τ τ p r , p , r r ∂ τ x x 0 ∂ = φ = ( ) ( ), ( ) Mass-transfer variants τ τ τ p m , m p m , m ∂ τ x 0 = ( ) Invariants τ p , i v ( s s ) ( r m ) + − + l g 9
Simplified Incremental Identification (ODE) Step 2: Model identification Model identification requires measurements c for various values of τ k ( ) τ • p k K measurements can be obtained by measuring the concentrations c ( ) τ p k z at the reactor exit z = z e for K different values of the velocity v = e k τ k Incremental identification of the reaction rates r • K 2 ( ) min x x i ,..., r − ∀ = ( ) ( , θ ) 1 τ τ p r i , , k p r i , , k r i , θ r i , k = 1 s.t. x r x d = = ( , θ ) ( c ( ), θ ), ( , θ ) 0 τ τ 0 d p r i , , r i , i p , l r i , p r i , , r i , τ Incremental identification of the mass-transfer rates φ m • K 2 ( ) min x x j ,..., m − ∀ = ( ) ( , θ ) 1 τ τ p m j , , k p m j , , k m j , θ m j , k = 1 s.t. x x d = φ = ( , θ ) ( c ( ), c ( ), θ ), ( , θ ) 0 τ τ τ 0 d p , m j , m j , m , j p l , p , g m , j p m j , , m j , τ ~ ( ) denote measured quantities or variables computed from measured quantities ⋅ 10
Simulated Example Chlorination of butanoic acid System composed of s g = 2 species in phase G and s l = 5 species in phase L • r = 2 reactions in the phase L • . R : BA Cl MBA HCl r k c c c 0 5 + → + = 1 ( ) l ( ) l ( ) l ( ) l l ,BA l ,Cl l ,M BA 2 1 1 2 R : BA Cl DBA HCl r k r c + → + = 2 2 2 ( ) l ( ) l ( ) l ( ) l l ,C l 2 2 2 1 2 m = 2 steady-state mass transfers obeying the rates • p k a c ( c ), c Cl φ = − = 2 m l ,Cl C l Cl ,Cl Cl H 2 2 2 Cl 2 2 2 p k a c ( c ), c φ = − = HCl m , HCl HC l HCl l ,HC l H Cl H HCl Known structural information • [ ] [ ] − − 1 2 1 0 [ ] [ ] E N T + , − − , 0 0 , 1 1 T T T τ : ( z , t ): N E N = 0 , E I N E m l = = m l , = m g = l = 0 0 1 0 0 E l , g , m − 2 2 × 2 1 2 0 1 m g , 2 2 × 0 1 0 0 11
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