Chapter%203%20:%20%20Simple%20Process%20Dynamics%20and%20Transfer%20Function

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Chapter 3 Simple Process Dynamics and
Transfer Function

• Professor Shi-Shang Jang
• Department of Chemical Engineering
• National Tsing-Hua University
• Hsinchu, Taiwan
• March, 2013

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Motive of Developing First Principle Models

• Improve understanding of the process
• Train plant operating personnel
• Develop a control strategy for a new process
• Optimize process operating conditions

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3-1 Introduction

• Theoretical (First Principle) models are
  developed using the principles of chemistry ,
  physics, and biology.
• Theoretical models offer process insight into
  process behavior, and they are applicable over
  wide ranges of conditions
• They trend to be expensive and time-consuming to
  develop

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Example - Industrial Furnace
CV temperature of the furnace MV fuel flow rate
to the furnace
Figure 1-1
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Temperature Profile of TT23
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Plant Dynamics
Flow rate
Flow rate
time
time
temperature
temperature
time
time
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The Concept of Deviation Variables
Flow rate
Flow rate
time
time
temperature
temperature
ydy - ys
time
time
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Simple Systems
 
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The Essence of Process Dynamics - Continued

• The feedback process control needs to understand
  the relationships between CV and MV, on the other
  hand, feedforward process control needs to
  understand the relationships between DV and CV.
  The relationships are called process models.
• For the ease of mathematical analyses, the
  process modeling only implements a linear model
  and Laplace transform instead of direct use of
  time domain process model. Implementation of
  deviation variables is needed as indicated below.

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3-1 Introduction- Continued

• Empirical models are obtained by fitting
  experimental data.
• Empirical models typically do not extrapolate
  well, and their range is typically small.
• Empirical models are frequently used in the
  industrial environment since a theoretical model
  is basically not precisely available.

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3-1 Introduction- Continued

• Semi-empirical models are a combination of the
  models of theoretical and empirical models the
  numerical values of the parameters in a
  theoretical model are calculated from
  experimental data.
• Semi-empirical models can (i) incorporate
  theoretical knowledge, (ii) extrapolate wider
  range than empirical range, (iii) require less
  effort than theoretical models.

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3-2 General Modeling Principles
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3-2 General Modeling Principles

• Constitution Equations
• Heat Transfer
  
• Reaction Rate
• Flow Rate
  
• Equation of State
  
• Phase Equilibrium

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3-3 Transfer Functions - Conventions

• - on the top of a variable steady state of a
  variable, example
• Capital deviation variable, example
• Capital with (s) Laplace transform of a variable
  to the deviation variable, example

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3-3. Transfer Function

• Transfer function is a mathematical
  representation of the relation between the input
  and output of a system.
• It is the Laplace transform of the output
  variable, y(t), divided by Laplace Transform of
  the input variable, x(t), with all initial
  conditions equal to zero.
• The term is often used exclusively to refer to
  linear, time-invariant systems (LTI), and
  non-linear, real-system are linearize to obtain
  their Transfer Function.
• So, Transfer Function G(s) for a system with
  input x(t) and output y(t) would be-

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More over Transfer Function

• As for previous equation, it could be said that
  if transfer function for the system and input to
  the system is known, we can obtain the output
  characteristics of the system.
• Transfer Function for the system could be easily
  obtained by dynamic study of the system and
  making balances for quantities like energy, mass
  etc.
• We take inverse Laplace Transform to obtain
  time-varying output characteristics of Y(s). In
  block diagram

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3-3 Transfer Functions Example Thermal Process
Ts
Inputs f(t), Ti(t),Ts(t) Output T(t)
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3-3 Transfer Functions Cont.
Let f be a constant ?V constant, CvCp
 
 
 
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3-3 Transfer Functions Cont.
Let f be a constant ?V constant, CpiCp
?time constant
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3-3 Transfer Functions Cont.
where, Gp(s) is call the transfer function of the
process, in block diagram
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Step Response of a First Order System
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Process Identification
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Example Mercury thermometer
A mercury thermometer is registering a
temperature of 75?F. Suddenly it is placed in a
400?F oil bath. The following data are obtained.
Time (sec) 0 1 2.5 5 8 10 15 30
Temp. (?F) 75 107 140 205 244 282 328 385

• Estimate the time constant of the temperature
  using
• Initial slope method
• 63 response method
• From a plot of log(400-T) versus time

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Solution
((1) ?9sec)
 
 
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Comparisons
Fit 1
Fit 2
Fit 3
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3-3 Transfer Functions Cont.

• By including the effect of surrounding
  temperature

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3-3 Transfer Functions Cont.
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Numerical Data
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Examples (1) Thermal Process- Continued

• Deviation Variables

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3-3 Transfer Functions-An Example
f0
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3-3 Example Non-Interactive Tanks
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3-3 Example Non-Interactive Tanks Cont.
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3-4 Dead Time
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3-4 Dead Time Cont.

• Time delay

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3-4 Dead Time Cont.
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3-4 Causes of Dead Time - Cont.

• Transportation lag (long pipelines)
• Sampling downstream of the process
• Slow measuring device GC
• Large number of first-order time constants in
  series (e.g. distillation column)
• Sampling delays introduced by computer control

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3-4 Effects of Dead-Time - Cont.

• Process with large dead time (relative to the
  time constant of the process) are difficult to
  control by pure feedback alone
• Effect of disturbances is not seen by controller
  for a while
• Effect of control action is not seen at the
  output for a while. This causes controller to
  take additional compensation unnecessary
• This results in a loop that has inherently built
  in limitations to control

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3-5 Transfer Functions and Block Diagrams

• Consider a general transfer function for an input
  X(s) and an output Y(s)
• Note that the above case is always true ,
  although many mathematical manipulating is needed
  as shown below

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3-5 Transfer Functions and Block Diagrams Cont.
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3-5 Transfer Functions and Block Diagrams Cont.
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3-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.2)
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3-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.3)
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3-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.3)
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3-6 Gas Process Example
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3-6 Gas Process Example Cont.
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3-6 Gas Process Example Cont.
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3-6 Gas Process Example Cont.
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3-7 Chemical Reactor
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3-7 Chemical Reactor Cont.
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3-7 Chemical Reactor Cont.
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3-8 Effects of Process Nonlinearity

• Real processes are mostly nonlinear
• The approximate linear models are only valid in
  local about the nearby of the operating point
• In some cases, process nonliearity may be
  detrimental to the control quality (e.g. high
  purity column)
• Process nonliearity plays important role to
  control quality in control systems

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3-8 Effects of Process Nonlinearity Cont.
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3-9 Additional Comments

• Reading assignment
• P96-98

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Homework

• Page 99
• 3-1, 3-2, 3-3, 3-4, 3-9, 3-10 (April 17th), 3-13,
  3-14(SIMULINK) 3-20, 3-21 (April 24th)

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Process-Supplemental Material
Inputs f(t), Ti(t),Ts(t) Output T(t)
Ts
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3-3 Transfer Functions Cont.
Gp3(s)
F(s)
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Thermal Process-Supplemental Material