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Dynamic Process Models

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Dynamic Process Models

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    1. Dynamic Process Models Conversion between model types Nonlinear model linearization State-space models Transfer function models Matlab/Simulink examples

    2. Conversion Between Model Types

    3. Theoretical Process Models Conservation equations Fundamental basis for model development Mass, energy & momentum conservation Nonlinear ODEs Constitutive equations Semi-empirical relations required to complete model Reaction rates, heat transfer rates, etc. Nonlinear algebraic equations Parameter estimation Needed to determine unknown parameters Reaction rate constants, heat transfer coefficients, etc. Yields parameter values that best fit available data Focus of ChE 361

    4. Nonlinear Model Linearization Nonlinear ODE model Find steady-state point Linearize about steady-state point Yields linear ODE model in deviation form Covered in ChE 361

    5. Isothermal CSTR Example Nonlinear ODE model Find steady-state point

    6. Isothermal CSTR Example cont. Linearize about steady-state point Transfer function

    7. Exothermic CSTR Example Nonlinear ODE model

    8. CSTR Model Linearization Linearization Vector ODE representation Linearization can be performed in Simulink (covered later)

    9. State-Space Models General form x` is an n-dimensional state vector u` is an m-dimensional input vector y` is an p-dimensional output vector n is the system dimension Exothermic chemical reactor example

    10. Transfer Function Models One-dimensional model Two-dimensional model CSTR example

    11. Multi-Dimensional Model State-space model Laplace transform Transfer function model

    12. Chemical Reactor Example State-space model Compute inverse Perform matrix multiplication

    13. Realization Problem Given a transfer function model Construct an equivalent state-space model Difficulties State-space model is not unique Procedure can be rather involved Second-order example Not covered further in this course

    14. Matlab Model Conversion Example State-space model Convert to transfer function model >> sys = ss([0 1;-0.25 -0.5],[0; 2],[1 0],[]); >> g=tf(sys) Transfer function: 2 ------------------ s^2 + 0.5 s + 0.25

    15. Matlab Model Conversion Example cont. Convert to back to state-space model >> sys1=ss(g) a = x1 x2 x1 -0.5 -0.5 x2 0.5 0 b = u1 x1 2 x2 0 c = x1 x2 y1 0 2 d = u1 y1 0 Convert back to transfer function model >> g1=tf(sys1) Transfer function: 2 ------------------ s^2 + 0.5 s + 0.25 Transfer function model is unique; state-space model is not unique

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