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LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Extent- -based Incremental Identification based Incremental Identification Extent of Reaction Kinetics from Spectroscopic Data of Reaction Kinetics from Spectroscopic Data XIII Conference on Chemometrics in Analytical Chemistry (CAC 2012) 25 – 29 June 2012, Budapest – Hungary Julien Billeter , Sriniketh Srinivasan and Dominique Bonvin Ecole Polytechnique Fédérale de Lausanne Laboratoire d’Automatique Switzerland

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Kinetic investigation From data to rate expressions ( ) ( ) number of number of = measured species computed extents 1. Computation of extents 2. Individual identification of rate expressions Estimation of rate parameters ( ) ( ) number of number of = measured species computed rates 2/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Homogeneous reaction systems Balance equations Homogeneous reaction system containing S species, R independent reactions, p inlets and 1 outlet C , q in in W , u in in Mole balance for S species ( ) u t  ( ) ( ) ( ) ( ) ( ) ( ) T out = + − = n t N V t r t W u t n t , n 0 n in in ( ) 0 m t ( ) q t ( ) ( ) ( ) ( ) T out = + − N V t r t C q t n t in in ( ) V t R ( p ( ( ) ( ) ) ( ) ) S x 1 S x R x 1 S x p x 1 n , u out ρ Mass m , density , volume V and concentrations c q out ( ) ( ) m t = n t ( ) ( ) ( ) ( ) ( ) ( ) ( ) T = ρ = φ = m t 1 M n t , t n t , M , ρ , V t , c t S w w i ( ) ( ) ρ t V t 3/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Homogeneous reaction systems 4-way decomposition into extents ( )   = T Assumption: + + rank N W n R p 1   in 0     T x S ψ r 0     =  T ⎯⎯ → n x M n    in 0 T λ q     0 Vessel extents of reaction x r and of flow ( x in and x out ) u  ( ) T T T out x = S N V r + S W u − x x 0 = 0   r 0 0 in in r r R m I 0 u R R x p  ( ) T T T out = + − = x M N V r M W u x x 0 0     in 0 0 in in in i n p m 0 I R u p  ( ) T T T λ = + − out λ λ = q N V r q W u 0 1   0 0 in in m 0 0 1 x R 1 x p ( ) = − λ = x 1 x 0 0 out out x = 0 iv S R p − − 4/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Homogeneous reaction systems 4-way decomposition into extents ( )   = T Assumption: + + rank N W n R p 1   in 0     T x S ψ r 0     =  T ⎯⎯ → n x M n    in 0 T λ q     0 Vessel extents of reaction x r and of flow ( x in and x out ) u  ( ) = − out = x V r x x 0 0 r r r R m u  ( ) = − out = x u x x 0 0 in in in in p m u  ( ) out λ = − λ λ 0 = 1 m ( ) = − λ = x 1 x 0 0 out out ( ) ( ) ( ) ( ) T Reconstruction: = + + − n t n N x t W x t n x t 0 r in in 0 out 5/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Homogeneous reaction systems Reaction Variant (RV) form ( )   < T When rank N W n R + p + 1   in 0 • Compute x in and x out using u in , u out and m u  ( ) out x = u − x x 0 = 0 in in in in p m u  ( ) ( ) out x = 1 − x x 0 = 0 out out out m • Compute n RV (RV-form of n ) ( ) ( ) ( ) ( ) ( ) RV T = − − + = n t n t n W x t n x t N x t 0 in in 0 out r • Compute x r from n RV ( ) ( ) ( ) ( ) ( ) ( ) T+ RV T+ = = − − + x t N n t N n t n W x t n x t r 0 in in 0 out 6/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Gas-liquid reaction systems Assumptions Gas outlet Gas-liquid reaction system containing p g p gas inlets g u W , u out g , inlets and 1 outlet in the gas phase, and in g , in g , Gas phase p  inlets and 1 outlet in the liquid phase. n , m g g The two phases are connected with p m ζ mass transfer rates . By convention, a Mass transfer + ζ − ζ positive sign (+) is assigned to a mass p  liquid inlets transfer from the gas to the liquid. W , u   in , in , Liquid phase Liquid outlet n  , m Assumptions:  u  out , • the gas and liquid phases are homogeneous • the reactions take place in the liquid bulk • the mass transfer is described by the two-film theory with no accumulation in the boundary layer 7/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Gas-liquid reaction systems Balance equations Mole balance in the Liquid phase ( ) u t  ( ) ( ) ( ) ( ) ( )  ( ) ( ) T o ut , = + ζ + − = n t N V t r t W t W u t n t , n 0 n         m , in , in , ( ) 0 m t  Mole balance in the Gas phase ( ) u t  ( ) ( ) ( ) ( ) ( ) out g , n t = − W ζ t + W u t − n t , n 0 = n g m g , in g , in g , ( ) g g g 0 m t g ρ  Mass , density , volumes and , and concentrations m  V  V c  g ( ) ( ) m t = n t ( ) ( ) ( )  ( )  T = = m t 1 M n t , V t , c t     ( )  ( ) S w , ρ t V t   ( ) ( ) ( ) ( ) ( ) ρ t = φ c t , M , ρ = − V t V V t      w , i g tot 8/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Gas-liquid reaction systems 5-way decomposition into extents ( )   = T + + + Assumption: rank N W W n R p p 1       m , in , 0 m     T x S  r 0     ψ T x M =    n ⎯⎯ →  m ,  m , 0  n  T  x M       in , in , 0 T λ    q    0 Vessel extents of reaction x r and of flow ( x in and x out ) u   ( ) out , x = V r − x x 0 = 0  r m r r R  u   ( ) out , = ζ − = x x x 0 0    m , m , m , p m m  u   ( ) out , = − = x u x x 0 0     in , in , in , in , p m   u   ( ) out , λ = − λ λ 0 = 1    m  ( ) = − λ = x 1 x 0 0    out , out , Reconstruction: ( ) ( ) ( ) ( ) ( ) T = + + + − n t n N x t W x t W x t n 0 x t         0 r m , m , in , in , out , 9/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Gas-liquid reaction systems Reaction & Mass-transfer Variant (RMV) form ( )   < T When rank N W W n R + p + p + 1       m , in , ,0 m • Compute and using , x x u and m  u     in , out , in , out , u   ( ) out , = − = x u x x 0 0     in , in , m in , in , p   u ( )   ( ) out , = − = x 1 x x 0 0    out , out , out , m  RMV • Compute n  (RMV-form of ) n  ( ) ( ) ( ) ( ) ( ) ( ) RMV T = − − + = + n t n t n W x t n x t N x t W x t          0 in , in , 0 out , r m , m , RMV • Compute x r and from x n   m ,  ( )  x t + +     ( ) ( ) ( ) ( ) ( ) T RMV T r = N W n t = N W n t − n − W x t + n x t       ( )          m , m , 0 in , in , 0 out , x t    m , 10/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Individual identification of reaction rates from the extents of reaction Identification of the rate expression and estimation of r i the associated kinetic parameters for each i -th reaction θ i ( ) by comparing the computed extents and the x t r i ,  ( ) simulated extents of reaction x t r i , ( ) u t  ( )    ( ) ( )  ( ) ( ) out , x = V t r θ , c t − x t x 0 = 0   ( ) r i , i i r i , r i , m t  11/23

LABORATOIRE D’AUTOMATIQUE AUTOMATIC CONTROL LABORATORY Incremental identification using spectroscopic data Homogeneous Gas-liquid reaction systems reaction systems ( ) ( ) Calibration step ϕ ϕ F = C Y , F = C , Y   prog c c prog , c , c ( ) ( ) ( ) ( ) ˆ ˆ = = Numbers of moles n t F a t n t F a t  prog v prog v   =    ( ) T x t S ˆ ( ) ( ) ( ) T n 0 ˆ r  ˆ Extents = 0 n t using x r t S n t    ( ) T  x t M       m , m , 0   ( ) x t + ˆ RV ˆ RMV   n ( ) ( ) ( ) n  T+ ˆ RV T RMV using or r ˆ = = N W n t x r t N n t     ( )   m , x t    m , Extents can subsequently be used for model identification ( ) ( ) ( ) a v t = a t V t F prog is the prognostic matrix ( S x L ) from calibration, with dimension ( L x 1)  12/23