Princeton%20activities

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A Stability/Bifurcation Framework For Process
Design C. Theodoropoulos1, N. Bozinis2, C.
Siettos1, C.C. Pantelides2 and I.G. Kevrekidis1
1Department of Chemical Engineering,Princeton
University, Princeton, NJ 08544 2 Centre for
Process System Engineering, Imperial College,
London, SW7 2BY, UK
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Motivation

• A large number of existing scientific,
  large-scale legacy codes
• Based on transient (timestepping) schemes.
• Enable legacy codes perform tasks such as
  bifurcation/stability analysis
• Efficiently locate multiple steady states and
  assess the stability of solution branches.
• Identify the parametric window of operating
  conditions
• for optimal performance
• Locate periodic solutions
• Autonomous, forced (PSA,RFR)
• Appropriate controller design.
• RPM method of choice to build around existing
  time-stepping codes.
• Identifies the low-dimensional unstable subspace
  of a few slow eigenvalues
• Stabilizes (and speeds-up) convergence of
  time-steppers even onto unstable steady-states.
• Efficient bifurcation analysis by computing only
  the few eigenvalues of the small subspace.
• Even when Jacobians are not explicitly available
  (!)

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Recursive Projection Method (RPM)

• Treats timstepping routine, as a black-box
• Timestepper evaluates un1 F(un)

F.P.I.

• Recursively identifies subspace of slow
  eigenmodes, P

• Substitutes pure Picard iteration with
• Newton method in P
• Picard iteration in Q I-P

• Reconstructs solution u from sum of the
  projectors P and Q onto subspace P and its
  orthogonal complement Q, respectively
• u PN(p,q) QF

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gPROMSA General Purpose Package
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Mathematical solution methods in gPROMS

• Combined symbolic, structural numerical
  techniques
• symbolic differentiation for partial derivatives
• automatic identification of problem sparsity
• structural analysis algorithms
• Advanced features
• exploitation of sparsity at all levels
• support for mixed analytical/numerical partial
  derivatives
• handling of symmetric/asymmetric discontinuities
  at all levels
• Component-based architecture for numerical
  solvers
• open interface for external solver components
• hierarchical solver architectures
• mix-and-match
• external solvers can be introduced at any level
  of the hierarchy

• well-posedness
• DAE index analysis
• consistency of DAE ICs
• automatic block triangularisation

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FitzHugh-Nagumo An PDE-based Model

• Reaction-diffusion model in one dimension
• Employed to study issues of pattern formation
• in reacting systems
• e.g. Beloushov-Zhabotinski
• reaction
• u activator, v inhibitor
• Parameters
  
• no-flux boundary conditions
• e, time-scale ratio, continuation parameter
• Variation of e produces turning points
• and Hopf bifurcations

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Bifurcation Diagrams
Around Hopf
Around Turning Point
ltugt
e
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Eigenspectrum Around Hopf
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Eigenvectors
e 0.02
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Arc-length continuation with gPROMS
continuation (I) within gPROMS
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System Jacobian
Obtain correct Jacobian of leading
eigenspectrum
Cannot get correct Jacobian from augmented
system
Jacobian of the ODE
Stability matrix
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Tubular Reactor A DAE system
Dimensionless equations
(1)
(2)
Boundary Conditions
(3)
(4)
Eqns (1)-(4) system of DAEs. Can also
substitute to obtain system of ODEs.
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Bifurcation/Stability with RPM-gPROMS

• Model solved as DAE system
• 2 algebraic equations _at_ each boundary
• 101-node FD discretization
• 2 unknowns (x1,x2) per node

Hopf pt.

• State variables
• 99 (x 2) unknowns at inner nodes
• Perform RPM-gPROMs at 99-space
• to obtain correct Jacobian

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Eigenspectrum
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Stability Analysis without the Equations
SYSTEM AROUND STEADY STATE
Leading Spectrum
y(k)
Matrix-free ARNOLDI

e q
Large-scale eigenvalue calculations (Arnoldi
using system Jacobian) R.B. Lechouq A.G.
Salinger, Int. J. Numer. Meth.(2001)
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Rapid Pressure Swing Adsorption1-Bed 2-Step
Periodic Adsorption Process
t0 to T/2 Ci(z0)PfYf/(RTf) P(z0)Pf

• Isothermal operation
• Modeling Equations (Nilchan Pantelides)

Step 2 Depressurisation
Step 1 Pressurisation
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Rapid Pressure Swing Adsorption1-Bed 2-Step
Periodic Adsorption Process
q , c (tT)

• Production of oxygen enriched air
• Zeolite 5A adsorbent (300?m)
• Bed 1m long, 5cm diameter
• Short cycle
• 1.5s pressurisation, 1.5s depressurisation
• T 3s
• Low feed pressure (Pf 3 bar)
• Periodic steady-state operation
• reached after several thousand cycles

q ,c (t0)
q , c (tT/2)
Must obtain
q , c (tT) q , c (t0)
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Typical RPSA simulation results(Nilchan and
Pantelides, Adsorption, 4, 113-147, 1998)
c1(z0.5) (mol/m3)
Time (s)
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PRM-gPROMS Spatial Profiles (tT)
z
z
z
z
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Leading Eigenvectors, l0.99484
c1
q1
q1
c1
q2
c2
c2
q2
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Conclusions

• Can construct a RPM-based computational framework
  around large-scale timestepping legacy codes to
  enable them converge to unstable steady states
  and efficiently perform bifurcation/stability
  analysis tasks.
• gPROMS was employed as a really good simulation
  tool
• communication with wrapper routines through
  F.P.I.
• Both for PDE and DAE-based systems.
• Have brought to light features of gPROMS for
  continuation around turning points and
  information on the Jacobian and/or stability
  matrix at steady states of systems.
• Employed matrix-free Arnoldi algorithms to
  perform stability analysis of steady state
  solutions without having either the Jacobian or
  even the equations!
• Used the RPM-based superstructure to speed-up
  convergence and perform stability analysis of an
  almost singular periodically-forced system
• Have enabled gPROMS to trace autonomous limit
  cycles
• Newton-Picard computational superstructure for
  autonomous limit cycles.

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gPROMS

• General purpose commercial package for modelling,
  optimization and control of process systems.
• Allows the direct mathematical description of
  distributed unit operations
• Operating procedures can be modelled
• Each comprising of a number of steps
• In sequence, in parallel, iteratively or
  conditionally.
• Complex processes combination of distributed and
  lumped unit operations
• Systems of integral, partial differential,
  ordinary differential and algebraic equations
  (IPDAEs).
• gPROMS solves using method of lines family of
  numerical methods.
• Reduces IPDAES to systems of DAEs.
• Time-stepping or pseudo-timestepping.
• Jacobians NOT explicitly available.
• Cannot perform systematic bifurcation/stability
  analysis studies.

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Tracing Limit Cycles
Tracing limit cycles

continuation (I) within gPROMS
R.P.M through FORTRAN
continuation (II) through FORTRAN

Getting system Jacobian through an FPI
tracing limit cycles within gPROMS
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Tracing Limit Cycles
Tracing limit cycles
SYSTEM
Periodic Solutions
y(tT)y(t)
Period T not known beforehand