02_PreviewReview_for_CHE361W12 - CHE 361: PREVIEW MODELING...

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Unformatted text preview: CHE 361: PREVIEW MODELING EQUATIONS - Ordinary Differential Equation Models from Mass and Energy Balances A. Develop equations: rate of accumulation and other rates B. Operating point: steady-state and deviation variables C. Linearization D. Laplace transforms E. Concept of a transfer function F. Solution of ODE models using numerical methods: Euler, Runge-Kutta and integration packages - linear vs. non-linear models. The "state-space" formulation for dynamics: state derivatives x = f ( x , u ) nonlinear ≈ A x ˙ outputs y = g ( x ) nonlinear ≈ C x B u linear ODEs D d linear combination PROCESS CHARACTERISTICS - TRANSFER FUNCTION A. Classical first-order systems {Kp, τp, θ} and time behavior B. Classical second-order systems {Kp, τ, ζ, θ} underdamped, critically damped and overdamped and time behavior C. Block diagram relationships: Overall transfer function, interacting systems D. Concepts of poles and zeros in the complex plane, RHP vs LHP, Im and Re axes. slow vs. fast poles, "effective" second or first order systems E. Time delay and the Pade’ approximation, lead-lag elements PHYSICAL DEVICES AND DATA ACQUISITION A. Standard instrumentation signal levels B Transmitters: linear vs. nonlinear steady-state calibration and gain C. Final control elements: control valve design and operations D. Sensor dynamics: e.g. Gm(s) and use of Km in Fig 11.7 versus 11.8 PROCESS IDENTIFICATION / BEHAVIOR = Models Obtained from Experimental Data A. Fit step response data: first-order plus time-delay 1. 63.2 % ∆y’ method 2. initial nonzero slope = KM / τ 3. fraction incomplete: ln [ (y(∞) - yi) / y(∞) ] vs. t plot with slope of -1/τ 4. inflection point or Sundaresan and Krishnaswamy 2pt fit (35.3 and 85.3 % choices) B. Second-order step response: after effect of time delay is removed 1. Harriott’s method - overdamped only 2. Smith’s method C. First over second-order response: nonlinear least-squares fit with step2g.m MATLAB file D. Pulse testing to obtain Bode plot and G(s) from Bode plot (FREQ. RESPONSE below) E. ODE(s) from poles + zeros + gain or Bode plot (FREQ. RESPONSE below) FREQUENCY RESPONSE A. G(s) to G(jω) short-cut method for Re + (Im) j to calculate amplitude ratio (AR) and phase angle (φ) B. AR and φ equations for common G(s) systems C. Asymptotic behavior of AR, φ at low and high frequencies D. Physical significance of frequency response INTRODUCTION TO FEEDBACK PROCESS CONTROL CHE 361: REVIEW of Laplace Transforms http://oregonstate.edu/dept/math/CalculusQuestStudyGuides/ode/laplace/laplacemap.html MTH 256 Review: The Laplace Transform Method Mathematical Notation used in these pages & Introduction The Laplace Transform of a Function The Inverse Laplace Transform The Method of Partial Fractions The Laplace Transform Method for Solving Linear Differential Equations Use of Laplace Transforms to solve simple linear ODEs-IVP: Given one or more ODEs + initial conditions, Laplace transform ODE(s) and incorporate initial conditions. Use algebraic rearrangement to get each required Yi(s) as a function of s only. Then inverse Laplace transform to get the required functions of time (t). ∞ [ f (t) ] = ⌠ f(t) e ⌡ Definition of Laplace transform of f ( t ) : st dt = F(s) 0 1 Inverse Laplace transform of F ( s ) : dy = s Y(s) dt t ⌠ y (t ) dt ⌡ 0 2 dy 2 dt bt = e = 1 s = ⌠ F(s) e ⌡ st ds = f(t) a i∞ 1 s Y(s) s yt dy d t t=0 0 , for the unit step function 1 s a i∞ y t=0 = s 2 Y(s) S(t) [ F(s) ] = 1 2πi b C1 y1 ( t ) + C2 y2 ( t ) , exponential decay : see Table 3.1, pg 42 43 SEMD3 for others = C1 Y1 ( s ) C2 Y2 ( s ) , linearity of Laplace transform The partial fraction expansion for n pieces, where n = the number of roots of D ( s ) = 0 1 N(s) D(s) n = 1 i=1 P i(s) = n i=1 The Final Value Theorem = FVT , when a final value EXISTS !! lim y ( t ) = lim t →∞ s Y(s) lim s Y(s) s →0 The Initial Value Theorem = IVT lim y ( t ) = t →0 s →∞ p i(t) IVP-TF Handout CHE 361 Comparisons of two approaches to studying linear dynamics: IVP versus TF Chapter 3 : Linear Initial Value Problem (IVP = math nomenclature) 1.Determine inputs/outputs (states) and define the nominal steady state. 2.Write balance equation(s) : mass and energy. 3.Simplify: y, dy/dt, d 2y/dt 2, etc. and known functions of t. 4.Laplace transform ODE(s) and incorporate initial conditions. 5.Algebraic rearrangement to get Yi (s). 6.Partial fraction expansion of Yi (s) = N(s)/D(s) where N and D are polynomials in s of degree ≥ 0. Each root of D(s) = 0 yields one "piece" of the partial fraction expansion. 7.Inverse Laplace transform each "piece" and add them all together to get yi (t). For complex conjugate roots or repeated roots of D(s)=0, the Table of Laplace transforms or formulas may be used to find the combined result of two or more "pieces". Chapter 4 : Transfer Function (TF) Models = process dynamics nomenclature 1.Determine input (manipulated variable)/output (measured state) and define the nominal steady state. 2.Write balance equation(s). 3.Eliminate "intermediate" variable(s) and linearize, if necessary. 4.Subtract steady-state form of equation(s). 5.Substitute to obtain only deviation variables - deviations from nominal steady-state values, y ′ (t) = y (t) - y = the output deviation variable, for example 6.Laplace transform ODE(s). 7.Algebraic rearrangement to y′ (s) = G (s) u′ (s), where G (s) is the transfer function between the transformed deviation variables y′ (s) as output and u′ (s) as input. An alternative to steps 3-5 is the "all at once" linearization of each ODE (one per output) using partial derivatives with respect to each input and each output, when there are more than one. ∂ RHS1 at ss. u1 ' + ∂ u1 ( RHS1 at ss. = 0 ) + ∂ RHS1 at ss. y1 ' + ... ∂ y1 d y1 = RHS1(u1 , y1 ,...) ≈ dt ...
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This note was uploaded on 03/01/2012 for the course CHE 361 taught by Professor Staff during the Winter '08 term at Oregon State.

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