Invariants - Chemical Reaction Systems Introduction Motivation Definitions Heterogeneous Invariant Relationships for Heterogeneous Reaction Systems Chemical Reaction Systems in Open Reactors System Description Transformation to Vessel Extents Sriniketh Srinivasan , Julien Billeter and Dominique Bonvin Linear Transformation Laboratoire d’Automatique Invariant Relationships EPFL, Lausanne, Switzerland Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 1 / 20
Outline Invariants - Chemical Introduction 1 Reaction Systems Motivation Definitions Introduction Motivation Heterogeneous Reaction Systems 2 Definitions System Description Heterogeneous Reaction Systems Transformation to Vessel Extents 3 System Description Linear Transformation Transformation to Vessel Invariant Relationships Extents Linear Transformation Application 4 Invariant Relationships Data Reconciliation Application Data Reconciliation Conclusion 5 Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 2 / 20
Motivation Invariants - Chemical Reaction Systems Consider the following homogeneous reaction system - Introduction Motivation Hydrodealkylation process Definitions Heterogeneous Reaction C 7 H 8 + H 2 → C 6 H 6 + CH 4 Systems System Description 2 C 6 H 6 → C 12 H 10 + H 2 Transformation to Vessel Extents Reaction system is operated in a batch reactor Linear Transformation Invariant Relationships Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 3 / 20
Motivation Invariants - Chemical Reaction Systems The material balance equations can be written as: Introduction Motivation n C 7 H 8 = − V r 1 ˙ n C 7 H 8 (0) = n C 7 H 8 , 0 Definitions Heterogeneous ˙ = − V r 1 + V r 2 n H 2 (0) = n H 2 , 0 Reaction n H 2 Systems System n C 6 H 6 = V r 1 − 2 V r 2 ˙ n C 6 H 6 (0) = n C 6 H 6 , 0 Description Transformation to Vessel n CH 4 ˙ = V r 1 n CH 4 (0) = n CH 4 , 0 Extents Linear Transformation n C 12 H 10 = V r 2 ˙ n C 12 H 10 (0) = n C 12 H 10 , 0 Invariant Relationships Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 4 / 20
Motivation Invariants - Chemical Reaction Systems In a batch reactor, we have: Introduction n C 7 H 8 ( t ) + ˙ ˙ n CH 4 ( t ) = − Vr 1 + Vr 1 = 0 Motivation Definitions n C 7 H 8 ( t ) + n CH 4 ( t ) remains constant: Heterogeneous Reaction Systems System n C 7 H 8 ( t ) + n CH 4 ( t ) = n C 7 H 8 , 0 + n CH 4 , 0 Description Transformation to Vessel Similarly, we can get other invariant relationships: Extents Linear Transformation 2 n H 2 ( t ) + n C 6 H 6 ( t ) − n C 7 H 8 ( t ) = 2 n H 2 , 0 + n C 6 H 6 , 0 − n C 7 H 8 , 0 Invariant Relationships n C 7 H 8 ( t ) + n C 12 H 10 ( t ) − n H 2 ( t ) = n C 7 H 8 , 0 + n C 12 H 10 , 0 − n H 2 , 0 Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 5 / 20
Motivation Invariants - Chemical Reaction Invariant relationships are straightforward for an Systems homogeneous reaction system operated in batch mode Introduction Motivation What are the invariant relationships for heterogeneous Definitions Heterogeneous reaction systems with mass transfer between phases? Reaction Systems System Description What are the invariant relationships when the reactor has Transformation inlet and outlet streams ? to Vessel Extents Linear Transformation This contribution gives a systematic procedure for deriving Invariant Relationships the invariant relationships Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 6 / 20
Definitions Invariants - Chemical Reaction Systems Reaction & Flow Variants: Variables that vary with time due to the effects of chemical reactions and physical flows Introduction Motivation Definitions Flow Variants (but Reaction Invariants): Variables that Heterogeneous Reaction vary with time due to other rate processes (mass transfer, Systems System inlets/outlet) but are independent of chemical reactions Description Transformation to Vessel Extents Reaction & Flow Invariants: Variables that do not vary Linear Transformation with time and stay constant during the course of the Invariant Relationships reaction Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 7 / 20
Heterogeneous Reaction System: Mass Balance Equations Invariants - Chemical Reaction Systems Consider a multiphase reaction system Introduction For phase F containing S f species, R f reactions, p m mass Motivation Definitions transfers, p f inlet streams and one outlet stream, the mole Heterogeneous balance equation can be written as: Reaction Systems System Description Mole balances for S f species Transformation to Vessel Extents u out , f ( t ) n f ( t ) = N T ˙ f V f ( t ) r f ( t )+ W m , f ζ ( t )+ W in , f u in , f ( t ) − m f ( t ) n f ( t ) , n f (0) = n f 0 Linear Transformation Invariant Relationships ( S f × 1) ( S f × R f ) ( R f × 1) ( S f × p m ) ( S f × p f ) Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 8 / 20
Decoupling to Vessel Extents Invariants - Chemical Reaction Bhatt et al. introduced a linear transformation to convert Systems the number of moles to vessel extents Introduction Recently, the transformation of Bhatt et al. has been Motivation Definitions explicited as a decoupling system inversion Heterogeneous Reaction Systems The transformation generates: System Description - an extent (variant) for each of the rate processes Transformation to Vessel - a number of invariants Extents Linear Transformation Invariant Relationships Application N. Bhatt, M. Amrhein and D. Bonvin, Incremental Identification of Reaction and Mass - Transfer Kinetics Data Reconciliation Using the Concept of Extents, Industrial & Engineering Chemistry Research, 50(23), 12960 - 12974 (2011) Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 9 / 20
Decoupling to Vessel Extents Invariants - Chemical Reaction Systems Assumption: Introduction rank([ N T W m , f W in , f n f 0 ]) = R f + p m + p f + 1 f Motivation Definitions Heterogeneous Linear transformation T f from n f ( t ) to extents: Reaction Systems System x r , f ( t ) Description Transformation x m , f ( t ) to Vessel Extents x in , f ( t ) = T f n f ( t ) Linear x ic , f ( t ) Transformation Invariant Relationships x iv , f ( t ) Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 10 / 20
Decoupling to Vessel Extents Invariants - Then the system reduces to: Chemical Reaction Systems x r , f ( t ) = r v , f ( t ) − u out , f ( t ) ˙ x r , f ( t ) x r , f (0) = 0 R f m f ( t ) Introduction x m , f ( t ) = ζ ( t ) − u out , f ( t ) ˙ x m , f ( t ) x m , f (0) = 0 p m Motivation Definitions m f ( t ) Heterogeneous x in , f ( t ) = u in , f ( t ) − u out , f ( t ) Reaction ˙ x in , f ( t ) x in , f (0) = 0 p f Systems m f ( t ) System Description x ic , f ( t ) = − u out , f ( t ) ˙ x ic , f ( t ) x ic , f (0) = 1 Transformation m f ( t ) to Vessel Extents x iv , f ( t ) = 0 q f . Linear Transformation Invariant Relationships The number of invariants is: q f = S f − R f − p m − p f − 1 Application The number of moles can be reconstructed as: Data Reconciliation Conclusion n f ( t ) = N T f x r , f ( t ) ± W m , f x m , f ( t ) + W in , f x in , f ( t ) + n f 0 x ic , f ( t ) Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 11 / 20
Invariant Relationships Invariants - Chemical Reaction Systems The transformation matrix is computed as Introduction f W m , f W in , f n f 0 P f ] − 1 T f = [ N T Motivation Definitions Heterogeneous where Reaction Systems T = 0 q f × ( R f + p m + p f +1) System P f [ N T f W m , f W in , f n f 0 ] T Description Transformation to Vessel The invariant relationships is Extents Linear Transformation P f n f ( t ) = 0 q f T Invariant Relationships Application Data Reconciliation Conclusion Laboratoire d’Automatique – EPFL Invariants - Chemical Reaction Systems 19th November, 2014 12 / 20
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