{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

unsw ceic3000 w3 process modeling and analysis

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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/τ , where τ is a scaling parameter to be determined. We can use the relationship dt = τdt to write: dh ( t ) τdt = k h max Ah max h ( τt ) + F i, max Ah max F i ( τt ) (9) If h ( t ) and F i ( t ) are not explicit functions of t , then dh dt = τk h max Ah max h + τF i, max Ah max F i (10) A natural choice for τ appears to be Ah max F i, max (known as the residence time at the maximum inlet ﬂowrate 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. We can approximate the dynamical behavior of nonlinear processes around certain operating point. And we call this linearization. 2
2.1 Single state and single input systems Suppose we have a single state nonlinear system as: dx dt = f ( x, u ) y = g ( x, u ) (12) where y is the output and u is the input. We can write the Taylor series expansion of f ( x, u ) about certain operating point ( x 0 , u 0 ): f ( x, u ) = f ( x 0 , u 0 ) + ∂f ∂x x 0 ,u 0 ( x x 0 ) + ∂f ∂u x 0 ,u 0 ( u u 0 ) + 1 2 2 f ∂x 2 x 0 ,u 0 ( x x 0 ) 2 + 1 2 2 f ∂x∂u x 0 ,u 0 ( x x 0 ) ( u u 0 ) + 1 2 2 f ∂u 2 x 0 ,u 0 ( u u 0 ) 2 + high order terms (13)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 13

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online