Title: Chapter%203%20:%20%20Simple%20Process%20Dynamics%20and%20Transfer%20Function
1Chapter 3 Simple Process Dynamics and
Transfer Function
- Professor Shi-Shang Jang
- Department of Chemical Engineering
- National Tsing-Hua University
- Hsinchu, Taiwan
- March, 2013
2Motive 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
33-1 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
4Example - Industrial Furnace
CV temperature of the furnace MV fuel flow rate
to the furnace
Figure 1-1
5Temperature Profile of TT23
6Plant Dynamics
Flow rate
Flow rate
time
time
temperature
temperature
time
time
7The Concept of Deviation Variables
Flow rate
Flow rate
time
time
temperature
temperature
ydy - ys
time
time
8Simple Systems
9The 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.
103-1 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.
113-1 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.
123-2 General Modeling Principles
133-2 General Modeling Principles
- Constitution Equations
- Heat Transfer
- Reaction Rate
- Flow Rate
- Equation of State
- Phase Equilibrium
143-3 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
153-3. 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-
16More 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
173-3 Transfer Functions Example Thermal Process
Ts
Inputs f(t), Ti(t),Ts(t) Output T(t)
183-3 Transfer Functions Cont.
Let f be a constant ?V constant, CvCp
193-3 Transfer Functions Cont.
Let f be a constant ?V constant, CpiCp
?time constant
203-3 Transfer Functions Cont.
where, Gp(s) is call the transfer function of the
process, in block diagram
21Step Response of a First Order System
22Process Identification
23Example 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
24Solution
((1) ?9sec)
25Comparisons
Fit 1
Fit 2
Fit 3
263-3 Transfer Functions Cont.
- By including the effect of surrounding
temperature
273-3 Transfer Functions Cont.
28Numerical Data
29Examples (1) Thermal Process- Continued
303-3 Transfer Functions-An Example
f0
313-3 Example Non-Interactive Tanks
323-3 Example Non-Interactive Tanks Cont.
333-4 Dead Time
343-4 Dead Time Cont.
353-4 Dead Time Cont.
363-4 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
373-4 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
383-5 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
393-5 Transfer Functions and Block Diagrams Cont.
403-5 Transfer Functions and Block Diagrams Cont.
413-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.2)
423-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.3)
433-5 Transfer Functions and Block Diagrams Cont.
(Example 3-5.3)
443-6 Gas Process Example
453-6 Gas Process Example Cont.
463-6 Gas Process Example Cont.
473-6 Gas Process Example Cont.
483-7 Chemical Reactor
493-7 Chemical Reactor Cont.
503-7 Chemical Reactor Cont.
513-8 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
523-8 Effects of Process Nonlinearity Cont.
533-9 Additional Comments
- Reading assignment
- P96-98
54Homework
- 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)
55Process-Supplemental Material
Inputs f(t), Ti(t),Ts(t) Output T(t)
Ts
563-3 Transfer Functions Cont.
Gp3(s)
F(s)
57Thermal Process-Supplemental Material