1. Dynamic Behavior Chapter 5
In analyzing process dynamic and process control systems, it is important to know how the process responds to changes in the process inputs.
A number of standard types of input changes are widely used for two reasons:
Chapter 5
They are representative of the types of changes that occur in plants.
They are easy to analyze mathematically.

2. Chapter 5

3. Step Input
A sudden change in a process variable can be approximated by a step change of magnitude, M:
Chapter 5
The step change occurs at an arbitrary time denoted as t = 0.
Special Case: If M = 1, we have a “unit step change”. We give it the symbol, S(t).
Example of a step change: A reactor feedstock is suddenly switched from one supply to another, causing sudden changes in feed concentration, flow, etc.

4. Example:
The heat input to the stirred-tank heating system in Chapter 2 is suddenly changed from 8000 to 10,000 kcal/hr by changing the electrical signal to the heater. Thus,
Chapter 5

5. Development of Transfer Functions
Chapter 4
Figure 2.3 Stirred-tank heating process with constant holdup, V.

6. Rearrange (5) to solve for
where
Chapter 4

7. Step Response of 1st-Order System

8. Ramp Input
Industrial processes often experience “drifting disturbances”, that is, relatively slow changes up or down for some period of time.
The rate of change is approximately constant.

9. Chapter 5 We can approximate a drifting disturbance by a ramp input:
Examples of ramp changes:
Ramp a setpoint to a new value. (Why not make a step change?)
Feed composition, heat exchanger fouling, catalyst activity, ambient temperature.

10. Ramp Response of 1st-Order System

11. Rectangular Pulse
It represents a brief, sudden change in a process variable: h
Examples:
Reactor feed is shut off for one hour.
The fuel gas supply to a furnace is briefly interrupted.

12. Response of 1st-Order System to Rectangular Pulse

13. Impulse Input It represents a short, transient disturbance.
It is the limit of a rectangular pulse for tw→0 and h = 1/tw
Examples:
Electrical noise spike in a thermo-couple reading.
Injection of a tracer dye.

14. Impulse Response of 1st-Order System

15. Sinusoidal Input
Chapter 5

16. Processes are also subject to periodic, or cyclic, disturbances
Processes are also subject to periodic, or cyclic, disturbances. They can be approximated by a sinusoidal disturbance:
Chapter 5
where: A = amplitude, ω = angular frequency
Examples:
24 hour variations in cooling water temperature.
60-Hz electrical noise (in USA!)

17. Chapter 5 For a sine input to the 1st order process: output is...
By partial fraction decomposition,

18. Inverting,
this term dies out for large t
Chapter 5
Note that the amplitude ration and phase angle is not a function of t but of t and w. For large t, y(t) is also sinusoidal, output sine is attenuated by…

19. Example of 2nd Order Process
Non-Interacting Storage System

20. Chapter 4

21. Nonlinear Example if q0 is manipulated by a flow control valve,
nonlinear element

24. Chapter 4

25. Example of 2nd Order Process
Interacting Storage System

27. Interacting Electrically Heated Stirred Tank System
Example 4.2
Interacting Electrically Heated Stirred Tank System

29. Dynamical Model and Transfer Functions

30. Chapter 5 Note that, when Ce  0, we obtain 1st order equation
(simpler model)

31. Standard Form of 2nd Order Transfer Function

32. Characteristic Equation

33. Solutions of 2nd Order Linear ODE

34. Step Response of 2nd-Order Overdamped System

35. Chapter 5

36. Step Response of 2nd-Order Underdamped System

37. Chapter 5

38. 1
Chapter 5

39. Rise Time

40. Peak Time

41. Maximum Overshoot Ratio

42. Decay Ratio

43. Second Order Step Change
Overshoot
time of first maximum
c. decay ratio (successive maxima – not min.)
d. period of oscillation
Chapter 5

45. Chapter 5