2 September 2008 DIANA — An object-oriented tool for nonlinear analysis of chemical processes Mykhaylo Krasnyk Max Planck Institute for Dynamics of Complex Technical Systems, PSPD group Otto-von-Guericke-University, IFAT U N I E V E K R S C I I T R Ä E U T G M A N G O D V E O B U T R T O G MAX−PLANCK−INSTITUT DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG
U N I V K E E R C S I I T R E Ä T U G M Outline N A G O D V E O B U T T R O G Introduction 1 Steady-state point analysis 2 Limit points continuation Higher co-dimension singularities Simulation models in Diana Parameter continuation Case Study I: Nonlinear analysis of MCFC 3 Periodic solutions continuation 4 Analysis of periodic solutions Recursive Projection Method Case Study II: Periodic solutions in MSMPR Crystallizer 5 Summary 6 OvGU, IFAT Outline 2/20
U N I V K E E R C S I I T R E Ä T U G M Introduction N A G O D V E O B U T T R O G Chemical Process Engineering OvGU, IFAT Introduction 3/20
U N I V K E E R C S I I T R E Ä T U G M Introduction N A G O D V E O B U T T R O G Chemical Process Engineering Dynamical Systems Analysis c in T , c ˙ q c OvGU, IFAT Introduction 3/20
U N I V K E E R C S I I T R E Ä T U G M Introduction N A G O D V E O B U T T R O G Chemical Process Engineering Dynamical Systems Analysis T [ K ] c in T , c ˙ q t [ s ] c OvGU, IFAT Introduction 3/20
U N I V K E E R C S I I T R E Ä T U G M Introduction N A G O D V E O B U T T R O G Chemical Process Engineering Dynamical Systems Analysis T [ K ] c in T , c ˙ q t [ s ] T [ K ] c q [ l / h ] ˙ OvGU, IFAT Introduction 3/20
U N I V K E E R C S I I T R E Ä T U G M Introduction N A G O D V E O B U T T R O G Chemical Process Engineering Dynamical Systems Analysis T [ K ] c in T , c ˙ q t [ s ] T [ K ] c q [ l / h ] ˙ 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low-order ODE, no modeling tool DIVA stationary and dynamic simulations of higher-order DAE for engineering processes, outdated FORTRAN77 code LOCA bifurcation analysis of large-scale CFD applications, lim- ited application problems OvGU, IFAT Introduction 3/20
U N I V K E E R C S I I T R E Ä T U G M Software tool Diana N A G O D V E O B U T T R O G 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 OvGU, IFAT Introduction 4/20
U N I V K E E R C S I I T R E Ä T U G M Software tool Diana N A G O D V E O B U T T R O G 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 problems higher-order singularities of steady-state curves efficient calculation of periodic solutions by reduction techniques analysis of test models OvGU, IFAT Introduction 4/20
U N I V K E E R C S I I T R E Ä T U Steady-state point continuation G M N A G O D V E O B U T T R O G Parameter continuation vs. dynamic simulation Condition for steady-state points of an autonomous system x ∈ R n , ν ∈ R p is: ! x = f ( x , ν ) ˙ =0 OvGU, IFAT Steady-state point analysis 5/20
U N I V K E E R C S I I T R E Ä T U Steady-state point continuation G M N A G O D V E O B U T T R O G Parameter continuation vs. dynamic simulation Condition for steady-state points of an autonomous system x ∈ R n , ν ∈ R p is: ! x = f ( x , ν ) ˙ =0 x x ( t , x 0 ) x s 0 t OvGU, IFAT Steady-state point analysis 5/20
U N I V K E E R C S I I T R E Ä T U Steady-state point continuation G M N A G O D V E O B U T T R O G Parameter continuation vs. dynamic simulation Condition for steady-state points of an autonomous system x ∈ R n , ν ∈ R p is: ! x = f ( x , ν ) ˙ =0 x x x ( t , x 0 ) x s x s x ( t , x 0 ) | t = ∞ → 0 t 0 λ s λ ∈ ν OvGU, IFAT Steady-state point analysis 5/20
U N I V K E E R C S I I T R E Ä T U Steady-state point continuation G M N A G O D V E O B U T T R O G Parameter continuation vs. dynamic simulation Condition for steady-state points of an autonomous system x ∈ R n , ν ∈ R p is: ! x = f ( x , ν ) ˙ =0 x x x ( λ ) x ( t , x 0 ) x s x s → 0 t 0 λ s λ ∈ ν OvGU, IFAT Steady-state point analysis 5/20
U N I V K E E R C S I I T R E Ä T U Steady-state point continuation G M N A G O D V E O B U T T R O G Parameter continuation vs. dynamic simulation Condition for steady-state points of an autonomous system x ∈ R n , ν ∈ R p is: ! x = f ( x , ν ) ˙ =0 x x x ( λ ) x ( t , x 0 ) x s x s → 0 t 0 λ s λ ∈ ν Stability is determined by eigenvalues of a linearized system at the steady-state point x s , ν s : V Λ = ∂ f ∂ x V OvGU, IFAT Steady-state point analysis 5/20
❍ ❍ P ▲ ▲ ▲ ▲ U N I V K E E R C S I I T R E Ä T U Limit points analysis G M N A G O D V E O B U T T R O G Example: limit and hysteresis points ( α, β, λ ∈ ν ) ❙ x λ OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/20
❍ ❍ P ▲ ▲ U N I V K E E R C S I I T R E Ä T U Limit points analysis G M N A G O D V E O B U T T R O G Example: limit and hysteresis points ( α, β, λ ∈ ν ) ❙ ▲ ▲ x λ OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/20
❍ P ▲ ▲ U N I V K E E R C S I I T R E Ä T U Limit points analysis G M N A G O D V E O B U T T R O G Example: limit and hysteresis points ( α, β, λ ∈ ν ) ❍ ❙ ▲ ▲ x α λ OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/20
P U N I V K E E R C S I I T R E Ä T U Limit points analysis G M N A G O D V E O B U T T R O G Example: limit and hysteresis points ( α, β, λ ∈ ν ) ❍ ❍ ❙ ▲ projection to ▲ α - λ plane ▲ ▲ x α α λ λ OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/20
U N I V K E E R C S I I T R E Ä T U Limit points analysis G M N A G O D V E O B U T T R O G Example: limit and hysteresis points ( α, β, λ ∈ ν ) ❍ ❍ P ❙ ▲ projection to ▲ α - λ plane ▲ ▲ x α α β λ λ OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/20
U N I V K E E R C S I I T R E Ä T Lyapunov-Schmidt reduction 1 U G M N A G O D V E O B U T T R O G Reduction definition For a system ˙ x = f ( x , ν ) with f ( x s , ν s ) = 0 and L = f x ( x s , ν s ) with dim ker L = 1 analysis of a limit points curve can be performed with a scalar equation g ( z , 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 7/20
U N I V K E E R C S I I T R E Ä T Lyapunov-Schmidt reduction 1 U G M N A G O D V E O B U T T R O G Reduction definition For a system ˙ x = f ( x , ν ) with f ( x s , ν s ) = 0 and L = f x ( x s , ν s ) 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 ) 1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I , 1985. OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/20
U N I V K E E R C S I I T R E Ä T Lyapunov-Schmidt reduction 1 U G M N A G O D V E O B U T T R O G Reduction definition For a system ˙ x = f ( x , ν ) with f ( x s , ν s ) = 0 and L = f x ( x s , ν s ) 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 ) 0 ∈ range L ⊥ is defined by an adjoint system The basis v 0 ∈ ker L and v ∗ 8 f ( x , ν ) = 0 , > < f x ( x , ν ) v 0 − β v ∗ = 0 , || v 0 || 2 = 1 , 0 f T x ( x , ν ) v ∗ || v ∗ > 0 − γ v 0 = 0 , 0 || 2 = 1 . : Reduced equation in the chosen basis { v 0 , v ∗ 0 } is g ( z , λ ) = � v ∗ 0 , f ( zv 0 + W ( zv 0 , 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 7/20
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