unsw ceic3000 w3 process modeling and analysis

unsw ceic3000 w3 process modeling and analysis - CEIC3000...

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Unformatted text preview: CEIC3000 Process Modelling and Analysis Week 3 Linear System Analysis Session 1, 2012 Assoc. Prof. Jie Bao Contents 1 Normalization 1 2 Linearization 2 2.1 Single state and single input systems . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Interpretation of Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Generalization to multivariable processes . . . . . . . . . . . . . . . . . . . . . . . 5 3 Feature Dynamics of Processes 7 3.1 Zero-Input Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Effect of Initial Condition Direction . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Stability Analysis for Linear Systems 11 1 Normalization Models typically contain a large number of parameters and variables that may differ in value by several orders of magnitude. It is often desirable, at least for analysis purposes, to develop models composed of dimensionless parameters and variables. Method 1. Find minimal and maximal possible value for x x [ x min ,x max ] (1) 2. Compute x = x x min x max x min (2) Normalized variable x 1. Normalized range: x [0 , 1] 2. Unit-less (dimensionless) 1 Unit-less/Dimensionless Model Equations: all variables and parameters are normalized the variations and values are comparable (both in engineering and computational sense) To illustrate the approach, consider the surge tank example: A dh ( t ) dt = k h ( t ) + F i ( t ) (3) It seems natural to work with a scaled liquid level. Define: h ( t ) = h ( t ) h min h max h min , F i ( t ) = F i ( t ) F i, min F i, max F i, min (4) where h min ( t ) and h max ( t ) are minimum and maximum liquid levels. We may use the tank height as h max ( t ) and h min = 0 , F i, min = 0 . Therefore, h ( t ) = h ( t ) h max , F i ( t ) = F i ( t ) F i, max (5) dh ( t ) dt = h max dh ( t ) dt . (6) A dh ( t ) dt = Ah max dh ( t ) dt = k h ( t ) h max + F i ( t ) F i, max (7) dh ( t ) dt = k h max Ah max h ( t ) + F i, max Ah max F i ( t ) (8) Sometimes, it is also natural to choose a scaled time t = t/ , where is a scaling parameter to be determined. We can use the relationship dt = dt to write: dh ( t ) dt = k h max Ah max h ( t ) + F i, max Ah max F i ( t ) (9) If h ( t ) and F i ( t ) are not explicit functions of t , then dh dt = k h max Ah max h + F i, max Ah max F i (10) A natural choice for appears to be Ah max F i, max (known as the residence time at the maximum inlet owrate and highest liquid level), so dh dt = k h max F i, max h + F i (11) 2 Linearization Most chemical processes are nonlinear: (e.g. Exponential dependence of reaction rate on tem- perature). However, only linear systems have been thoroughly studied and well understood....
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This note was uploaded on 03/26/2012 for the course CHEM ENG CEIC at University of New South Wales.

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unsw ceic3000 w3 process modeling and analysis - CEIC3000...

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