Variant and Invariant States for Reaction Systems S. Srinivasan, J. Billeter and D. Bonvin Laboratoire d’Automatique EPFL, Lausanne, Switzerland TFMST 2013, Lyon Laboratoire d’Automatique – EPFL Variant and Invariant States 1 / 13
Outline Concept of Variants and Invariants Concept of Vessel Extents Each extent linked to the corresponding rate process Presence of outlet(s) → vessel extents Vessel extents of reaction, mass transfer, heat transfer Applications Model reduction Static state reconstruction Incremental kinetic identification Conclusions Laboratoire d’Automatique – EPFL Variant and Invariant States 2 / 13
Homogeneous reaction systems Balance equations Homogeneous reaction system consisting of S species, R independent reactions, p inlet streams, and 1 outlet stream W in , u in Mole balances for S species n ( t ) = N T r v ( t ) + W in u in ( t ) − ω ( t ) n ( t ) , ˙ n (0) = n 0 ( S ) ( S × R ) ( R ) ( S × p ) ( p ) N n r v r v ( t ) = V ( t ) r ( t ) considered as endogenous signal ω ( t ) = u out ( t ) m ( t ) n , u out Global macroscopic view Generally valid regardless of temperature, catalyst, solvent, etc. Laboratoire d’Automatique – EPFL Variant and Invariant States 3 / 13
Reaction variants and reaction invariants in the literature 1 Linear transformation using N : � y r ( t ) � � N T + � = n ( t ) with N P = 0 R × ( S − R ) y iv ( t ) P T Reaction variants y r and reaction invariants y iv describe the reactor state: T + W in u in ( t ) − ω ( t ) y r ( t ) y r ( t ) = r v ( t ) + N ˙ y r (0) = N T + n 0 T W in u in ( t ) − ω ( t ) y iv ( t ) y iv ( t ) = P ˙ y iv (0) = P T n 0 y r are reaction and flow variants y iv are reaction invariants but flow variants y r are pure reaction variants and y iv are true invariants only for batch reactors (with u in = 0 , ω = 0) Can we compute pure reaction variants and true invariants for open reactors? 1Asbjornsen et al. (1970), Chem. Eng. Sci. , 25:1627-1639. Laboratoire d’Automatique – EPFL Variant and Invariant States 4 / 13
Vessel extents and true invariants Assumption: rank ([ N T W in n 0 ]) = R + p + 1. Linear transformation: 2 3 2 3 x r ( t ) R x in ( t ) F 6 7 6 7 x ( t ) := 5 = 5 n ( t ) = T n ( t ) 6 7 6 7 c T x ic ( t ) 4 4 x iv ( t ) Q Vessel extents of reaction x r and flow x in , discounting factor x ic , and invariants x iv : T x r ( t ) = RN ˙ r v ( t ) + RW in u in ( t ) − ω ( t ) x r ( t ) x r (0) = 0 R | {z } | {z } I R 0 ˙ T x in ( t ) = FN r v ( t ) + FW in u in ( t ) − ω ( t ) x in ( t ) x in (0) = 0 p |{z} | {z } 0 I p T N T T W in x ic ( t ) = c ˙ r v ( t ) + c u in ( t ) − ω ( t ) x ic ( t ) x ic (0) = 1 | {z } | {z } 0 0 T x iv ( t ) = QN ˙ r v ( t ) + QW in u in ( t ) − ω ( t ) x iv ( t ) x iv (0) = 0 q | {z } | {z } 0 0 q = S − R − p − 1 Laboratoire d’Automatique – EPFL Variant and Invariant States 5 / 13
Four subspaces N T W in n 0 P � − 1 � x ( t ) = T n ( t ) T = PQ n 0 c T P orthogonal to N T , W in and n 0 effet of outlet invariant subspace on IC subspace 1 q x r , i ( t ) = r v , i ( t ) − ω ( t ) x r , i ( t ) ˙ x r , i (0) = 0 N T R W in F x in , j ( t ) = u in , j ( t ) − ω ( t ) x in , j ( t ) ˙ x in , j (0) = 0 reaction inlet subspace subspace x ic ( t ) = − ω ( t ) x ic ( t ) ˙ x ic (0) = 1 p R x iv = P T n ( t ) = 0 q amount that is still in the reactor S-dimensional space, R + p + 1 variants n ( t ) = N T x r ( t ) + W in x in ( t ) + n 0 x ic ( t ) 1 Bhatt et al. (2010), I&EC Research , 49:7704-7717 Laboratoire d’Automatique – EPFL Variant and Invariant States 6 / 13
Homogeneous CSTR – Experimental data Ethanolysis reaction with seven species ( S = 7), three reactions ( R = 3), two inlets ( p = 2) and one outlet Stoichiometric matrix ( N ) and inlet-composition matrix ( W in ): h w in , A h − 1 − 1 i i T 1 1 0 0 0 0 0 0 0 0 0 N = W in = 0 − 1 − 1 1 1 0 0 0 w in , B 0 0 0 0 0 0 − 1 0 − 1 0 1 1 Measured numbers of moles w in , B , u in , B w in , A , u in , A 0.5 A 0.45 B C 0.4 D E 0.35 F n ( t ) [kmol] G 0.3 0.25 A,B,C, 0.2 D,E,F, N G 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 45 50 Time [h] n , u out Reaction extents ? Laboratoire d’Automatique – EPFL Variant and Invariant States 7 / 13
Homogeneous CSTR – Computation of extents Extents of reaction 0.35 x r , 3 0.3 x r [kmol] x r , 1 0.25 0.2 x r , 2 0.15 0.1 0.05 Numbers of moles 0 0 5 10 15 20 25 30 35 40 45 50 Time [h] 0.5 A B Extents of inlet 0.45 C 0.4 D 70 N W in n 0 E 0.35 x in , 1 60 F n [kmol] G 0.3 x in , 2 50 2 3 R 0.25 x in [kg] 40 F 0.2 6 7 30 T = 6 7 0.15 c T 20 4 5 0.1 Q 10 0.05 0 0 5 10 15 20 25 30 35 40 45 50 Time [h] 0 0 5 10 15 20 25 30 35 40 45 50 Time [h] Effect of outlet on IC 1 0.9 0.8 0.7 0.6 x ic 0.5 0.4 0.3 0.2 0.1 1 Bhatt et al. (2010), I&EC Research , 49:7704-7717 0 0 5 10 15 20 25 30 35 40 45 50 Time [h] Laboratoire d’Automatique – EPFL Variant and Invariant States 8 / 13
Extension to fluid-fluid reaction systems For one of the phases x r ( t ) = r v ( t ) − ω ( t ) x r ( t ) ˙ x r (0) = 0 R P Q n 0 c T invariant subspace effet of outlet x m ( t ) = ζ ( t ) − ω ( t ) x m ( t ) ˙ x m (0) = 0 p m on IC subspace q 1 W m M ˙ x in ( t ) = u in ( t ) − ω ( t ) x in ( t ) x in (0) = 0 p mass-transfer N T R subspace x ic ( t ) = − ω ( t ) x ic ( t ) ˙ x ic (0) = 1 p m reaction subspace W in F x iv = P T n ( t ) = 0 q R inlet subspace p n ( t ) = N T x r ( t ) + W m x m ( t ) + W in x in ( t ) + n 0 x ic ( t ) S -dimensional space R + p m + p + 1 variants 1 Bhatt et al. (2010), I&EC Research , 49:7704-7717 Laboratoire d’Automatique – EPFL Variant and Invariant States 9 / 13
Extension to reaction systems with heat balance equation � � n ( t ) x ( t ) := T dimension S + 1 m ( t ) c p T ( t ) “Decoupled” system x r ( t ) = r v ( t ) − ω ( t ) x r ( t ) ˙ x r (0) = 0 R x ex ( t ) = q ex ( t ) − ω ( t ) x ex ( t ) ˙ x ex (0) = 0 x in ( t ) = u in ( t ) − ω ( t ) x in ( t ) ˙ x in (0) = 0 p x ic ( t ) = − ω ( t ) x ic ( t ) ˙ x ic (0) = 1 x iv = 0 q Application: estimation of q ex ( t ) or identification of heat-transfer coefficients independently of any kinetic information Laboratoire d’Automatique – EPFL Variant and Invariant States 10 / 13
Model reduction Dimensionality d := R + p + 1, min( S , d ) differential equations However, transformation assumes knowledge of n 0 , i.e., S initial conditions Elimination of fast modes via singular perturbation The reactions (and not the associated numbers of moles) exhibit fast or slow dynamic behavior → transformed decoupled model is well suited for input estimation: x r , i ( t ) = r v , i ( t ) − ω ( t ) x r , i ( t ) ˙ x r , i (0) = 0 x r , i ( t ) r v , i ( t ) Dynamical system Laboratoire d’Automatique – EPFL Variant and Invariant States 11 / 13
Incremental kinetic identification via rates or extents Computation of rates and extents Rates T † � n RV � r v , i ( t ) = N i ˙ a ( t ) (at least R measured species) n RV with ˙ a ( t ) = ˙ n a ( t ) − W in , a u in ( t ) + ω ( t ) n a ( t ) → differentiation of sparse and noisy signal n a ( t ) Vessel extents T † � � x r , i ( t ) = vRV a ( t ) (at least R measured species) N i n with n vRV a ( t ) := n a ( t ) − W in , a x in ( t ) − n 0 , a x ic ( t ) x r , i ( t ) = R i n a ( t ) (at least R + p + 1 measured species) → neither integration nor differentiation of the sparse and noisy signal n a ( t ) Laboratoire d’Automatique – EPFL Variant and Invariant States 12 / 13
Conclusions Transformation of numbers of moles to “decoupled” vessel extents Transformation uses structural information N , W in , W m and knowledge of n 0 Effect of outlets is accounted for → concept of vessel extent Rates considered as time signals, e.g. r v ( t ) and not r v ( c , T ) Possible applications Homogeneous and fluid-fluid reaction systems Model reduction Static state reconstruction Incremental kinetic identification Heterogeneous catalytic reaction systems? Distributed reaction systems? Laboratoire d’Automatique – EPFL Variant and Invariant States 13 / 13
Recommend
More recommend
