Extents for Process Monitoring Motivation Problem Statement System Description On the Use of Extents Material Balance Equations Energy Balance for Process Monitoring and Fault Diagnosis Equations Transformation to Vessel Extents Sriniketh Srinivasan , Julien Billeter and Dominique Bonvin Fault Detection Laboratoire d’Automatique Conclusion EPFL, Lausanne, Switzerland Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 1 / 19
Outline Extents for Process Monitoring Motivation 1 Problem Statement Motivation Problem Statement System Description 2 System Description Material Balance Equations Material Balance Equations Energy Balance Equations Energy Balance Equations Transformation Transformation to Vessel Extents 3 to Vessel Extents Fault Fault Detection 4 Detection Conclusion Conclusion 5 Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 2 / 19
Problem Statement Extents for Process Measurements of numbers of moles n ( t ), mass m ( t ) and Monitoring reactor temperature T ( t ) are available Motivation 30 95 0.8 Problem 0.6 90 0.4 28 Statement 85 0.2 n ( t ) (kmol) 0 T ( t ) (C) 26 m(t) ( kg ) 80 System 1.0 0.8 24 75 Description 0.6 70 0.4 22 0.2 Material Balance 65 0 20 Equations 0 20 40 60 80 100 0 20 40 60 80 100 60 Time (min) Time (min) 0 20 40 60 80 100 Time (min) Energy Balance Equations Assumption: Stoichiometry, inlet composition and initial Transformation to Vessel conditions are known but no information is available on Extents the reaction kinetics Fault Detection Can we detect faults using only data from the current Conclusion batch? The answer is Yes, using the extent-based approach... Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 3 / 19
Material Balance Equations Extents for Process Monitoring For a reaction system with S species, R reactions, p inlets and one outlet, Motivation Problem Mole balances for S species Statement System n ( t ) = N T V ( t ) r ( t ) + W in u in ( t ) − u out ( t ) Description ˙ m ( t ) n ( t ) , n (0) = n 0 Material Balance Equations Energy Balance ( S ) ( S × R ) ( R ) ( S × p ) ( p ) Equations Transformation to Vessel where, Extents Fault m ( t ) = 1 T ˙ p u in ( t ) − u out ( t ) , m (0) = m 0 , Detection Conclusion ω ( t ) = − u out ( t ) m ( t ) Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 4 / 19
Energy Balance Equations Extents for Process Monitoring The energy balance equation can be written as: Motivation Problem Statement System Heat balance Description Material Balance Equations Q ( t ) = ( − ∆ H ) T r v ( t ) + q ex ( t ) + ˇ ˙ T T in u in ( t ) − ω ( t ) Q ( t ) Q (0) = Q 0 Energy Balance Equations Transformation to Vessel where Q ( t ) = m ( t ) c p T ( t ) is the heat power Extents Fault ˇ T T in contains the specific heats of the inlet streams Detection Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 5 / 19
Balance Equations Extents for Process Monitoring Combining both equations Motivation Problem Combined material and energy balance Statement System z ( t ) = A r v ( t ) + b q ex ( t ) + C u in ( t ) − ω ( t ) z ( t ) ˙ Description Material Balance Equations Energy Balance Equations � � � � n n 0 Transformation z = and z 0 = . to Vessel Q Q 0 Extents Fault Detection � � � � � � N T 0 S W in A = , b = , C = Conclusion ˇ ( − ∆ H ) T 1 T T in Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 6 / 19
Linear Transformation Extents for Process Monitoring � − 1 gives, � The linear transformation T = A b C z 0 P Motivation Problem Statement x r ( t ) System x ex ( t ) Description Material Balance x in ( t ) = T z ( t ) Equations Energy Balance Equations x ic ( t ) Transformation x iv ( t ) to Vessel Extents Fault The matrix P describes the q -dimensional null space of Detection � � the matrix , with q = S − R − p − 1 . A b C z 0 Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 7 / 19
Linear Transformation Extents for Process Monitoring The transformed system reads x r ( t ) = r v ( t ) − ω ( t ) x r ( t ) ˙ x r (0) = 0 R Motivation Problem Statement x ex ( t ) = q ex ( t ) − ω ( t ) x ex ( t ) ˙ x ex (0) = 0 System ˙ Description x in ( t ) = u in ( t ) − ω ( t ) x in ( t ) x in (0) = 0 p Material Balance Equations x ic ( t ) = − ω ( t ) x ic ( t ) ˙ x ic (0) = 1 Energy Balance Equations x iv ( t ) = 0 q , Transformation to Vessel Extents The numbers of moles n ( t ) and the heat Q ( t ) can be Fault Detection reconstructed from the transformed variables: Conclusion � � � � � � � W in � � � n ( t ) N T 0 S n 0 = x r ( t ) + x ex ( t ) + x in ( t ) + x ic ( t ) . ˇ Q ( t ) ( − ∆ H ) T 1 T T Q 0 in Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 8 / 19
Fault Detection Extents for Process Monitoring Objective Use extents to identify faults in: Motivation Problem Statement Outlet flowrates u out ( t ) 1 System Description Material Balance Inlet flowrates u in ( t ) 2 Equations Energy Balance Equations Heat exchange q ex ( t ) 3 Transformation to Vessel Extents Note: In order to identify faults in reactions, we need Fault Detection either historical data or a kinetic model Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 9 / 19
Fault Detection - Fault in Flowrates Extents for Process Monitoring Motivation Compute the reference mass m ref ( t ) Problem Statement System 1 T m ref ( t ) ˙ = p u in , ref ( t ) − u out , ref ( t ) m ref (0) = m ref , 0 Description Material Balance Equations Energy Balance Compare m ref ( t ) with the measured mass m ( t ) using Equations Transformation either z-test or t-test to Vessel Extents If an error is detected, fault either in u in ( t ) and/or u out ( t ) Fault Detection Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 10 / 19
Fault Detection - Fault in Flowrates Extents for Process Compute the extents by applying the linear transformation Monitoring Compute x ic , ref ( t ) Motivation Problem − u out , ref ( t ) Statement x ic , ref ( t ) = m ref ( t ) x ic , ref ( t ) System Description Material Balance Equations Compare x ic , ref ( t ) with x ic ( t ) - Error in outlet flowrate? Energy Balance Equations Compute x in , ref ( t ) Transformation to Vessel Extents u in , ref ( t ) − u out , ref ( t ) Fault x in , ref ( t ) = m ref ( t ) x in , ref ( t ) Detection Conclusion Compare x in , ref ( t ) with x in ( t ) - Error in inlet flowrates? Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 11 / 19
Fault Detection - Fault in Heat transfer Extents for Process Monitoring Motivation Problem Compute x ex , ref ( t ) Statement System Description q ex , ref ( t ) − u out , ref ( t ) Material Balance x ex , ref ( t ) = m ref ( t ) x ex , ref ( t ) Equations Energy Balance Equations Transformation Compare x ex , ref ( t ) with x ex ( t ) - Error in heat transfer? to Vessel Extents Fault Detection Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 12 / 19
Simulated Example Extents for Process Monitoring Consider the hydrodealkylation reaction system Motivation Problem Statement System C 7 H 8 + H 2 → C 6 H 6 + CH 4 Description Material Balance 2 C 6 H 6 → C 12 H 10 + H 2 Equations Energy Balance Equations Both reactions are exothermic Transformation to Vessel Extents Simplification: Hydrogen is considered as a dissolved Fault Detection species fed directly into the liquid phase Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 13 / 19
Fault Detection - Simulated Example Extents for Process Monitoring For the hydrodealkylation example, under normal operating conditions (NOC), n and T vary with time Motivation Problem The measurements are corrupted with 1% zero-mean Statement System gaussian white noise Description Material Balance Equations Energy Balance Equations 30 0.8 0.6 Transformation 28 0.4 to Vessel 0.2 Extents n ( t ) (kmol) 0 T ( t ) (C) 26 1.0 Fault 0.8 24 Detection 0.6 0.4 22 Conclusion 0.2 0 20 0 20 40 60 80 100 0 20 40 60 80 100 Time (min) Time (min) Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 14 / 19
Fault Detection - Fault in u out Extents for Process Monitoring NOC: u out ( t ) = 0 . 5 kg min − 1 AOC: u out ( t ) = 0 kg min − 1 Motivation Problem Fault introduced at time t = 30 min. Statement System Description Material Balance 0.2 Equations 0.1 90 Energy Balance Equations 0 -0.1 m ( t ) (kg) x ic ( t ) (-) Transformation 80 -0.2 to Vessel Extents -0.3 70 -0.4 Fault -0.5 Detection 60 -0.6 0 20 40 60 80 100 0 20 40 60 80 100 Time (min) Time (min) Conclusion Laboratoire d’Automatique – EPFL Extents for Process Monitoring 19th November, 2014 15 / 19
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