11 March 2009 ICVT, Universit¨ at Stuttgart MAX-PLANCK-INSTITUT TECHNISCHER SYSTEME MAGDEBURG DYNAMIK KOMPLEXER O O T T V O N G U E R I C K E U N IV E R S I T Ä T M A G D E B U R G Numerical nonlinear analysis in DIANA Michael Krasnyk Max Planck Institute for Dynamics of Complex Technical Systems, PSPD group Otto-von-Guericke-University, IFAT
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11 March 2009ICVT, Universitat Stuttgart
MAX−PLANCK−INSTITUT
TECHNISCHER SYSTEMEMAGDEBURG
DYNAMIK KOMPLEXER
O
O
TTV
ON
GU
ERIC
KE UNIVERSITÄT
MA
GD
EBU
RG
Numerical nonlinear analysis in DIANA
Michael Krasnyk
Max Planck Institute for Dynamics of Complex Technical Systems, PSPD groupOtto-von-Guericke-University, IFAT
O
O
TTV
ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Outline 2/24
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Introduction
Chemical Process Engineering
Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Software tool Diana
Diana — Dynamic simulation and nonlinear analysis tool
developed at MPI Magdeburg
modularization, extensibility and object-oriented architecture
equation based models
numerical solvers based on free code
enhanced scripting and visualization
Objectives of the work
generation of C++ model code for Diana
symbolic differentiation of models (Maxima package)
parameter continuation of nonlinear problemshigher-order singularities of steady-state curvesefficient calculation of periodic solutions by reduction techniques
analysis of test models
OvGU, IFAT Introduction 4/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Software tool Diana
Diana — Dynamic simulation and nonlinear analysis tool
developed at MPI Magdeburg
modularization, extensibility and object-oriented architecture
equation based models
numerical solvers based on free code
enhanced scripting and visualization
Objectives of the work
generation of C++ model code for Diana
symbolic differentiation of models (Maxima package)
parameter continuation of nonlinear problemshigher-order singularities of steady-state curvesefficient calculation of periodic solutions by reduction techniques
analysis of test models
OvGU, IFAT Introduction 4/24
O
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TTV
ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
O
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S
L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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ON
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AG
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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ON
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AG
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Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!
Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
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ON
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AG
DE
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Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
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Higher co-dimension singularities
Classification of singular points with codim g 6 3
The classification theorem guarantees existence only the following possible singu-larities of g with codim g 6 3
codim g
0
1
2
3 z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ
gzzzz = 0gλ = 0
z3 ± zλ
gzzz = 0gzλ = 0
z2 ± λ3
gzz = 0 | d3g| = 0
z3 ± λ
gzzz = 0gλ = 0
z2 ± λ2
gzz = 0 | d2g| = 0
z2 ± λ
gzz = 0 gλ = 0
equilibrium
gz = 0
type
OvGU, IFAT Steady-state point analysis/ Higher co-dimension singularities 8/24
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Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
O
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
In Diana two types of predictors are used:
tangential predictor ~y(k)
t
chord predictor ~y(k)
c
x
ν
y (k) ~y(k)
t
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
In Diana two types of predictors are used:
tangential predictor ~y(k)
t
chord predictor ~y(k)
c
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k−1)
y (k)
~y(k)
c
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k) y (k+1)
y (k+1)
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k) y (k+1)
y (k+1)
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 10/24
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 21/24
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Model: MSMPR Crystallizer
Model equations [Pathath and Kienle, 2002]
Feed
q, cin
qq , c, F
q, c, Fhp
Productclassification
qf , c, Fhf
Finesdissolver
The population balance equation is
∂F
∂t= −∂(GF )
∂L− q
V(hf (L) + hp(L))F
with the boundary condition
F (0, t) =B(c, t)
G(c, t)=
kb(c(t)− csat)b
kg (c(t)− csat)g.
The recycle ratio of the fines dissolution andthe classified product removal are
hf = R1(1− h(L− Lf )), hp = 1 + R2h(L− Lp)The mass balance of solute is
MAdc
dt= (ρ−MAc)
„q
V+
1
ε
dε
dt
«+
qMAcin
V ε− qρ
V ε
„1 + kv
Z ∞
0
(hp − 1)FL3 dL
«,
where ε is the void fraction which is given by ε = 1− kv
Z ∞
0
FL3 dL
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 22/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1 III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Summary
Simulation system Diana
Lisp-module for the modeling tool ProMoT has been developed (transformationand symbolic differentiation of ProMoT models, C++ code generator)
C++ and Python interaction with CAPE-OPEN interfaces
solver classes for linear and nonlinear problems have been implemented
Numerical nonlinear analysis
implementation of continuation methods for steady-state, limit and Hopf points
generation of adjoint systems and reduced test functions for singular points
periodic solution continuation with simple bifurcations detection
recursive projection method has been applied to speed-up computation
OvGU, IFAT Summary 24/24
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References
Golubitsky, M. and Schaeffer, D. G. (1985).Singularities and groups in bifurcation theory. Vol. I, volume 1 of Applied Math-ematical Sciences.Springer-Verlag, New York.
Mangold, M., Krasnyk, M., and Sundmacher, K. (2004).Nonlinear analysis of current instabilities in high temperature fuel cells.Chemical Engineering Science, 59(22-23):4869–4877.
Pathath, P. K. and Kienle, A. (2002).A numerical bifurcation analysis of nonlinear oscillations in crystallization pro-cesses.Chemical Engineering Science, 57(10):4391–4399.
Zeyer, K. P., Mangold, M., Obertopp, T., and Gilles, E. D. (1999).The iron(III)-catalyzed oxidation of ethanol by hydrogen peroxide: a thermoki-netic oscillator.Journal of Physical Chemistry, 103A(28):5515–5522.