Analyzing Process Dynamics: Input Changes & Responses

1. Dynamic Behavior 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. We can approximate a drifting disturbance by a ramp input: Chapter 5 • 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. h 0 Rectangular Pulse It represents a brief, sudden change in a process variable: • 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. 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. For a sine input to the 1st order process: output is... Chapter 5 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. 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