Incremental Identification of Reaction Systems Minimal Number of Measurements J. Billeter, S. Srinivasan and D. Bonvin Laboratoire d’Automatique EPFL, Lausanne, Switzerland AIChE Annual Meeting 2012, Pittsburgh, PA Incremental identification 1 / 20
Outline Identification of reaction systems from measured data Simultaneous or incremental approach? Number of measurements for incremental identification? Minimal state representation Homogeneous w/o outlet (batch, semi-batch) → extents of reaction Homogeneous with outlet → vessel extents of reaction Gas-liquid with outlet → vessel extents of reaction and mass transfer Number of measurements for full state reconstruction Gas-liquid reaction system with outlet Conclusions Incremental identification 2 / 20
Context – Kinetic investigation Iterative procedure Incremental identification 3 / 20
Context – Kinetic investigation Iterative procedure Issues: Simultaneous or incremental approach? How many measurements for incremental approach? Incremental identification 4 / 20
Homogeneous reaction systems Balance equations Homogeneous reaction system consisting of S species, R independent reactions, p inlet streams, and 1 outlet stream Mole balances for S species W in , u in n ( t ) = N T V ( t ) r ( t ) + W in u in ( t ) − u out ( t ) ˙ m ( t ) n ( t ) , n (0) = n 0 ( S ) ( S × R ) ( R ) ( S × p ) ( p ) N n r ( t ) Mass m , volume V and molar concentrations c S M w n ( t ) , V ( t ) = m ( t ) ρ ( t ) , c ( t ) = n ( t ) m ( t ) = 1 T V ( t ) n , u out Global macroscopic view Generally valid regardless of temperature, catalyst, solvent, etc. Incremental identification 5 / 20
Gas-liquid reaction systems Balance equations Assumptions the gas and liquid phases are homogeneous the reactions take place in the liquid bulk only no accumulation in the boundary layer Liquid phase n l ( t ) = N T V l ( t ) r ( t ) + W m , l ζ ( t ) + W in , l u in , l ( t ) − u out , l ( t ) ˙ m l ( t ) n l ( t ) , n l (0) = n l 0 ( S l ) ( S l × R l ) ( R l ) ( S l × p l ) ( p l ) ( S l × p m ) ( p m ) Gas phase n g ( t ) = − W m , g ζ ( t ) + W in , g u in , g ( t ) − u out , g ( t ) ˙ m g ( t ) n g ( t ) , n g (0) = n g 0 ( S g ) ( S g × p g ) ( p g ) ( S g × p m ) ( p m ) Incremental identification 6 / 20
From measured data to rate expressions Simultaneous approach N W m,l W in,l u in,l ( t ) n l ( t ) u out,l ( t ) m l ( t ) V l ( t ) n l ( t ) numbers of moles LS problem Number of measurements n l ˆ ( t ) as required for identifiability 1 ( . ) dt Rate law Fat: data regarding the global reaction system candidates Simultaneous approach 1 Lean: data specific to a single reaction, mass transfer Experimental data flow Simulated data flow Information flow Library of Identified rate laws rate laws Incremental identification 7 / 20
From measured data to rate expressions Incremental rate-based approach N W m,l W in,l uout , l ( t ) n RMV u in,l ( t ) ˙ ( t ) = ˙ n l ( t ) − W in , l u in , l ( t ) + n l ( t ) n l ( t ) u out,l ( t ) l ml ( t ) m l ( t ) V l ( t ) u in,l ( t ) u out,l ( t ) W in,l m l ( t ) n l ( t ) . RMV . n ( t ) l n in dn l (t) numbers RMV form dt of moles LS problem at least R + p m measurements ˆ n l ( t ) S ≥ R + p m N T W m,l [ ] + V l ( t ) 1 r i ( t ) ( t ) ζ j ( . ) dt Rate LS problem ˆ r i ( t ) ˆ ζ j ( t ) 2 Rate law Fat: data regarding the global reaction system candidates Simultaneous approach 1 Lean: data specific to a single reaction, mass transfer Experimental data flow Incremental rate-based approach 2 Simulated data flow Information flow Library of Identified rate laws rate laws Incremental identification 8 / 20
From measured data to rate expressions Incremental extent-based approach R t n RMV ( t ) = n l ( t ) − n l 0 − W in , l 0 u in , l ( τ ) d τ l R t uout , l ( τ ) + n l ( τ ) d τ 0 N W m,l W in,l ml ( τ ) uout , l ( t ) n RMV u in,l ( t ) ˙ ( t ) = ˙ n l ( t ) − W in , l u in , l ( t ) + n l ( t ) n l ( t ) u out,l ( t ) l ml ( t ) m l ( t ) V l ( t ) u in,l ( t ) u out,l ( t ) W in,l m l ( t ) n l ( t ) . RMV . RMV n ( t ) n ( t ) l l n in dn l (t) n in numbers RMV form RMV form dt of moles LS problem at least R + p m measurements ˆ n l ( t ) S ≥ R + p m S ≥ R + p m N T W m,l [ ] + N T W m,l [ ] + V l ( t ) 1 ξ r,i ( t ) r i ( t ) ( t ) ξ m,j ( t ) ζ j Rate LS problem ( . ) dt Extent ˆ ξ r,i ( t ) r i ˆ ( t ) ˆ ξ m,j ( t ) ˆ ζ j ( t ) ( . ) dt 2 3 Rate law Fat: data regarding the global reaction system candidates Simultaneous approach 1 Lean: data specific to a single reaction, mass transfer Experimental data flow Incremental rate-based approach 2 Simulated data flow Information flow Incremental identification 9 / 20 3 Incremental extent-based approach Library of Identified rate laws
Outline Identification of reaction systems from measured data Simultaneous vs. incremental approach Number of measurements for incremental identificaion Minimal state representation Homogeneous w/o outlet (batch, semi-batch) → extents of reaction Homogeneous with outlet → vessel extents of reaction Gas-liquid with outlet → vessel extents of reaction and mass transfer Number of measurements for full state reconstruction Gas-liquid reaction system with outlet Application – Kinetic Identification Simultaneous approach Incremental approaches Conclusions Incremental identification 10 / 20
Homogeneous reaction systems without outlet Orthogonal spaces in three-way decomposition � S T � N T W in � + � = QQ T M T invariant space Q orthogonal to N T and W in S − R − p N T S T ˙ ξ r , i ( t ) = V ( t ) r i ( t ) ξ r , i (0) = 0 space of reaction extents W in M T . orthogonal to ˙ inlet extents ξ in , j ( t ) = u in , j ( t ) ξ in , j (0) = 0 space of inlet extents orthogonal to reaction extents ξ iv = Q T ( n − n 0 ) = 0 S − R − p R p n ( t ) = N T ξ r ( t ) + W in ξ in ( t ) S-dimensional space, R + p variants Amrhein et al. (2010), AIChE Journal, 56(11), 2873-2886. Incremental identification 11 / 20
Homogeneous reaction systems with outlets Orthogonal spaces in four-way decomposition S T 0 � + N T W in n 0 = � M T 0 Q 0 Q T n 0 q T 0 q T 0 0 space of invariant space Q 0 orthogonal to N T , W in and n 0 discounting factor 1 S − R − p − 1 x r , i = V r i − u out N T S T ˙ x r , i (0) = 0 m x r , i 0 W in M T space of vessel 0 x in , j = u in , j − u out . ˙ x in , j (0) = 0 extents of reaction m x in , j space of vessel extents of inlet flow R ˙ λ = − u out λ (0) = 1 m λ p x iv = Q T 0 n = 0 S − R − p − 1 S-dimensional space, R + p + 1 variants n ( t ) = N T x r ( t ) + W in x in ( t ) + n 0 λ ( t ) 1 Bhatt et al. (2010), I&EC Research , 49:7704-7717 Incremental identification 12 / 20
Gas-liquid reaction systems with outlets Orthogonal spaces in five-way and four-way decomposition Q l 0 Q T n l 0 q T l 0 l 0 n g 0 q T Q g 0 Q T S l − R − p m − p l − 1 g 0 space of g 0 discounting factor space of invariant space discounting factor invariant space 1 1 W m , l M T S g − p m − p g − 1 m , l 0 W in , g M T N T S T in , g 0 l 0 space of mass-transfer extents n g 0 q T space of g 0 space of gas-inlet extents reaction extents p m W in , l M T space of in , l 0 mass-transfer extents p g space of R p m liquid-inlet extents p l S g -dimensional space p m + p g + 1 variants S l -dimensional space R + p m + p l + 1 variants Dimensionality of the dynamic model: ( R + 2 p m + p l + p g + 2) and not ( S l + S g ) Bhatt et al. (2010), I&EC Research, 49(17), 7704-7717. Incremental identification 13 / 20
From measured data to rate expressions Incremental vessel-extent-based approach R t n RMV ( t ) = n l ( t ) − n l 0 − W in , l 0 u in , l ( τ ) d τ l R t uout , l ( τ ) + n l ( τ ) d τ N W m,l W in,l 0 ml ( τ ) uout , l ( t ) n RMV u in,l ( t ) ˙ ( t ) = ˙ n l ( t ) − W in , l u in , l ( t ) + n l ( t ) n l ( t ) u out,l ( t ) l ml ( t ) m l ( t ) V l ( t ) u in,l ( t ) u out,l ( t ) W in,l m l ( t ) n l ( t ) . RMV RMV . n ( t ) n ( t ) l l n in dn l (t) n in numbers RMV form RMV form dt of moles LS problem Number of n l ˆ ( t ) S ≥ R + p m S ≥ R + p m S ≥ R + p m + p l + 1 measurements N T W m,l N T W m,l [ n l0 ] + T W m,l [ ] + [ ] + N W in,l V l ( t ) 1 ξ r,i ( t ) x r,i ( t ) r i ( t ) x m,l,j ( t ) ( t ) ( t ) ζ j ξ m,j Vessel ( . ) dt Rate LS problem Extent LS problem extent ˆ ( t ) ξ r,i r i ˆ ( t ) ˆ x r,i ( t ) ˆ ξ m,j ( t ) ˆ ˆ x m,l,j ( t ) ζ j ( t ) ( . ) dt 2 3 3 ( . ) dt Rate law Fat: data regarding the global reaction system candidates Simultaneous approach 1 Lean: data specific to a single reaction, mass transfer Experimental data flow Incremental rate-based approach 2 Simulated data flow Information flow Incremental identification 14 / 20 3 Incremental extent-based approach Library of Identified rate laws rate laws
Outline Identification of reaction systems from measured data Simultaneous vs. incremental approach Number of measurements for incremental identificaion Minimal state representation Homogeneous w/o outlet (batch, semi-batch) → extents of reaction Homogeneous with outlet → vessel extents of reaction Gas-liquid with outlet → vessel extents of reaction and mass transfer Number of measurements for full state reconstruction Gas-liquid reaction system with outlet Conclusions Incremental identification 15 / 20
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