Complex Process Complex Process Stability, Networks: Stability, Networks: Control, and Optimality Control, and Optimality Kendell Jillson Kendell Jillson B. Erik B. Erik Ydstie Ydstie Department of Chemical Engineering Department of Chemical Engineering Carnegie Mellon University Carnegie Mellon University Research funded by: NSF CTS- Research funded by: NSF CTS -ITR 031 2771 ITR 031 2771 AIChE Annual Meeting AIChE Annual Meeting November 8, 2004 November 8, 2004
Objective Objective To introduce a modeling technique to To introduce a modeling technique to describe process networks describe process networks To develop methods for analyzing stability To develop methods for analyzing stability and control of process networks and control of process networks To establish a variational variational principle for principle for To establish a process networks to show optimality process networks to show optimality 2 2
Why Networks? Why Networks? Network representations provide a graphical Network representations provide a graphical interpretation of a process system interpretation of a process system Topology is important to the character of the Topology is important to the character of the network network Easy to break down into smaller subsystems Easy to break down into smaller subsystems Network descriptions could represent a wide- - Network descriptions could represent a wide range of applications… … range of applications 3 3
…From process From process flowsheets flowsheets… … … Douglas (1988) 4 4
…to Supply Chains to Supply Chains… … … Ydstie, et al. (2003) 5 5
…Chemical Reaction Networks Chemical Reaction Networks… … … Fishtik, et al (2004) 6 6
…even Bio Networks even Bio Networks … Glycosis and aromatic amino acid metabolic subsystem Ma, Guo, Zhao (2000) 7 7
Process Network Process Network Define Define P � Graph, G = (P,T,F) Graph, G = (P,T,F) � Process (node) Process (node) F Terminal Terminal T Flow Flow � State of each node, State of each node, v v j � j Extensive quantities Extensive quantities e.g. e.g. � Potential of each node, Potential of each node, w w j � j Intensive quantities Intensive quantities e.g. e.g. Potential differences act as driving forces for flow Potential differences act as driving forces for flow 8 8
Conservation and Continuity Conservation and Continuity At each node: Around each loop: At each node: Around each loop: f 2 W 12 w 2 w 1 f 1 f 3 W 21 KCL KVL 9 9
Stability and Control Stability and Control Input- -Output Control Output Control Input u y + S - C Several techniques to show system can be Several techniques to show system can be stable with or without using control stable with or without using control Multiple connected stable systems not Multiple connected stable systems not necessarily stable necessarily stable 10 10
Passivity Background Passivity Background Passivity theory is used to show that a system can be Passivity theory is used to show that a system can be stabilized stabilized � Originated from electrical circuit theory Originated from electrical circuit theory � � A feedback or parallel connected system of passive subsystems A feedback or parallel connected system of passive subsystems � is also passive is also passive Passivity inequality Passivity inequality � With With x,u,y x,u,y the states, inputs, and outputs to the network the states, inputs, and outputs to the network � � V(x V(x) ) ≥ ≥ 0, x 0, x ≠ ≠ 0; V(0) = 0 0; V(0) = 0 � Problem: To find a practical storage function Problem: To find a practical storage function 11 11
Entropy Entropy Concave function, S(v S(v), which is homogeneous ), which is homogeneous Concave function, of degree one of degree one Motivated from molecular level Boltzmann Boltzmann Motivated from molecular level relation relation nd Law of Thermodynamics follows: The 2 nd Law of Thermodynamics follows: The 2 Defines potentials at each process node Defines potentials at each process node 12 12
Concavity of entropy Concavity of entropy a 1,2 a 2,1 13 13
Storage Function for Process Storage Function for Process Networks Networks Define a storage function for network comparing Define a storage function for network comparing the states of two solutions based on potentials the states of two solutions based on potentials and inventories and inventories � Using deviation variables based on these two Using deviation variables based on these two � solutions solutions � Exploit the concavity of entropy Exploit the concavity of entropy � � At each node: At each node: � � For the whole network For the whole network � 14 14
Passivity of Process Networks Passivity of Process Networks Differentiation of storage function and use of Differentiation of storage function and use of an entropy balance over the network gives an entropy balance over the network gives The passivity inequality is developed: The passivity inequality is developed: � Provided following relationships hold: Provided following relationships hold: � Monotonic flow and production rates: Monotonic flow and production rates: Discrete Poincar Discrete Poincaré é inequality inequality 15 15
Reactor Network Example Reactor Network Example Three reactor nodes, and three terminals Three reactor nodes, and three terminals � Reaction: A+B Reaction: A+B � � C with rate = k C C with rate = k C A C B A C � B � Transport governed by diffusion Transport governed by diffusion � 9 ODE ODE’ ’s s 9 27 Algebraic constitutive equations 27 Algebraic constitutive equations 16 16
Control of Example Control of Example Objective: Control flow rate of C at T 3 Objective: Control flow rate of C at T 3 Stabilized by a PI flow controller (K = 50, 1/ τ τ = 10) to a = 10) to a Stabilized by a PI flow controller (K = 50, 1/ set- -point, f point, f c (3) = 0.05 set c (3) = 0.05 � f f C C (3) (3) � � y y L 3,3 L 3,3 of of f f C � u u C � � 17 17
Optimality Optimality Using a variational variational principle, entropy production principle, entropy production Using a of a network obeying monotonic constitutive of a network obeying monotonic constitutive expressions is shown to be minimized expressions is shown to be minimized � Define the entropy production Define the entropy production � For unperturbed and perturbed network For unperturbed and perturbed network � Perturbed network still obeys conservation laws at each node Perturbed network still obeys conservation laws at each node � Entropy production never greater in unperturbed network Entropy production never greater in unperturbed network 18 18
Optimality Result Optimality Result Minimal entropy production in unperturbed Minimal entropy production in unperturbed solution, compared to a randomly perturbed solution, compared to a randomly perturbed network network 19 19
Conclusions Conclusions Process Networks modeled as a graph with Process Networks modeled as a graph with state v v and potential and potential w w at each node at each node state Storage function, G G , used to show passivity , used to show passivity Storage function, provided flow and production rates are provided flow and production rates are monotonic and positive monotonic and positive Conditions for minimum entropy production Conditions for minimum entropy production established established Simulation example with three nodes, three Simulation example with three nodes, three terminals, and three components illustrate terminals, and three components illustrate theory theory 20 20
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Topological Result Topological Result A result similar to Tellegen Tellegen’ ’s s Theorem is Theorem is A result similar to developed developed Based only on topology of the network Based only on topology of the network Instead of power balance, provides an entropy Instead of power balance, provides an entropy balance balance Useful for further developments Useful for further developments 23 23
Network Operators Network Operators Transform variables to allow further Transform variables to allow further developments developments � Operators for flows/productions must still obey Operators for flows/productions must still obey � inventory balance inventory balance � Operators for potentials must obey loop balance Operators for potentials must obey loop balance � Examples include: Examples include: � Time averaging Time averaging � � Fourier or Laplace Transforms Fourier or Laplace Transforms � � Multiplication with constant matrices or vectors Multiplication with constant matrices or vectors � 24 24
Monotonic Flow Monotonic Flow For Joule- -Thompson Flow Thompson Flow For Joule � with potentials defined as: with potentials defined as: � Using the Gibbs- -Duhem Duhem equation, this can be equation, this can be Using the Gibbs re- -written as: written as: re � Which is strictly monotonic for positive definite Which is strictly monotonic for positive definite Λ Λ � 25 25
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