1. Chapter 3 : Simple Process Dynamics and Transfer Function
Professor Shi-Shang Jang
Department of Chemical Engineering
National Tsing-Hua University
Hsinchu, Taiwan
March, 2013

2. 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

3. 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

4. Example - Industrial FurnaceTT 23 TC
Stack gases
Fuel
Air FY QY FT 24
Set point
CV: temperature of the furnace
MV: fuel flow rate to the furnace
Figure 1-1

5. Temperature Profile of TT23
Time (min)

6. Plant Dynamics Plant DV CV MV Flow rate Flow rate time time
temperature
temperature
time
time

7. The Concept of Deviation Variables
Flow rate
Flow rate
Plant MV CV DV
time
time
temperature
temperature
yd=y - ys
time
time

8. 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.

9. 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.

10. 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.

11. 3-2 General Modeling Principles

12. 3-2 General Modeling Principles:
Constitution Equations
Heat Transfer：
Reaction Rate：
Flow Rate：
Equation of State：
Phase Equilibrium:

13. 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:

14. 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-

15. 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:
G (s)
X(s)
Y(s)

16. 3-3 Transfer Functions – Example: Thermal ProcessTs
Inputs: f(t), Ti(t),Ts(t)
Output: T(t)

17. 3-3 Transfer Functions – Cont.
Let f be a constant V= constant, Cv=Cp

18. 3-3 Transfer Functions – Cont.
Let f be a constant V= constant, Cpi=Cp
=time constant

19. 3-3 Transfer Functions – Cont.
where, Gp(s) is call the transfer function of the process, in block diagram:
Gp(s)
i(s)
(s)

20. Step Response of a First Order System

21. Process Identification

22. 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) 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

23. Solution
((1) =9sec)

24. Comparisons
Fit 1
Fit 2
Fit 3

25. 3-3 Transfer Functions – Cont.
By including the effect of surrounding temperature:

26. 3-3 Transfer Functions – Cont.
Gp1(s)
Gp2(s) +
i(s)
 s(s)
(s) Σ

27. Numerical Data

28. Examples (1) Thermal Process- Continued
Deviation Variables

29. Simple Systems

30. Simple Systems

31. Block Diagram K1
1s+1
M(s)
Y1(s)
Y(s) + K2
2s+1
D(s)
Y2(s)

32. 3-3 Transfer Functions-An Example
Cross-sectional=A2 h2 V2 f1 f2 h1 V1
Cross-sectional=A1

33. 3-3 Example Non-Interactive Tanks

34. 3-3 Example Non-Interactive Tanks – Cont.
F0(s)
H1(s)
H2(s)

35. 3-4 Transfer Functions and Block Diagrams – Cont.

36. 3-4 Transfer Functions and Block Diagrams – Cont.

37. 3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.2)

38. 3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.3)

39. 3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.3)

40. 3-5 Gas Process Example

41. 3-5 Gas Process Example – Cont.

42. 3-5 Gas Process Example – Cont.Σ
s+1 K1
 s+1 K2 K3 + -
Mi(s)
Mo(s)
P1(s)
P(s)

43. 3-5 Gas Process Example – Cont.

44. 3-6 Dead Time

45. 3-4 Dead Time – Cont.
Time delay:

46. 3-4 Dead Time – Cont.

47. 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

48. 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

49. 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:

50. 3-7 Chemical Reactor

51. 3-7 Chemical Reactor – Cont.

52. 3-7 Chemical Reactor – Cont.

53. 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

54. 3-8 Effects of Process Nonlinearity – Cont.

55. 3-9 Additional Comments
Reading assignment
P96-98

56. 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)

57. Process-Supplemental Material
Inputs: f(t), Ti(t),Ts(t)
Output: T(t) Ts

58. 3-3 Transfer Functions – Cont.
Gp1(s)
Gp2(s) +
i(s)
 s(s)
(s) Σ
Gp3(s)
F(s)

59. Thermal Process-Supplemental Material