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Chapter 5

ERT 210/4 Process Control & Dynamics. DYNAMIC BEHAVIOR OF PROCESSES : Dynamic Behavior of First-order and Second-order Processes. Chapter 5. COURSE OUTCOME 1 CO1) 1. Theoretical Models of Chemical Processes 2. Laplace Transform 3. Transfer Function Models

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Chapter 5

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  1. ERT 210/4Process Control & Dynamics DYNAMIC BEHAVIOR OF PROCESSES : Dynamic Behavior of First-order and Second-order Processes Chapter 5

  2. COURSE OUTCOME 1 CO1) 1. Theoretical Models of Chemical Processes 2. Laplace Transform 3. Transfer Function Models 4. Dynamic Behavior of First-order and Second-order Processes DEFINE, REPEAT, APPLY and DERIVE general process dynamic formulas for simplest transfer function: first-order processes, integrating units and second-order processes 5. Dynamic Response Characteristics of More Complicated Processes 6. Development of Empirical Models from Process Data Chapter 5

  3. 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. • Process inputs falls into two categories: • 1. Inputs that can be manipulated to control the process • 2. Inputs that are not manipulated, classified as disturbance • variables • 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.

  4. Standard Process Inputs Chapter 5

  5. 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.

  6. 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, = and Chapter 5 • Ramp Input • Industrial processes often experience “drifting disturbances”, that is, relatively slow changes up or down for some period of time with a roughly constant slope. • The rate of change is approximately constant.

  7. We can approximate a drifting disturbance by a ramp input: = • Examples of ramp changes: • Ramp a set point to a new value rather than making a step change. • Feed composition, heat exchanger fouling, catalyst activity, ambient temperature. Chapter 5 • Rectangular Pulse • It represents a brief, sudden step change in a process variable that then returns to its original value.

  8. h 0 = XRP Chapter 5 Tw Time, t • Examples: • Reactor feed is shut off for one hour. • The fuel gas supply to a furnace is briefly interrupted.

  9. Chapter 5

  10. Sinusoidal Input • Processes are also subject to periodic, or cyclic, disturbances. They can be approximated by a sinusoidal disturbance: = Chapter 5 where: A = amplitude, w = angular frequency • Examples: • 24 hour variations in cooling water temperature. • 60-Hz electrical noise arising from electrical equipment and instrumentation.

  11. Impulse Input • Here, • It represents a short, transient disturbance. • Examples: • Electrical noise spike in a thermo-couple reading. • Injection of a tracer dye. Chapter 5 • Useful for analysis since the response to an impulse input is the inverse of the TF. Thus, Here,

  12. The corresponding time domain express is: where: = Chapter 5 Suppose . Then it can be shown that: Consequently, g(t) is called the “impulse response function”.

  13. Random Inputs • Many process inputs changes with time in such a complex manner that it is not possible to describe them as deterministic functions of time. • The mathematical analysis of the process with random inputs is beyond the scope of this chapter. Chapter 5

  14. Types of Dynamic Response Chapter 5

  15. Dynamic Response Chapter 5

  16. First-Order System:(Step Response) The standard form for a first-order TF is: where: = = Chapter 5 Consider the response of this system to a step of magnitude, M: Substitute into (5-16) and rearrange,

  17. Take L-1 (cf. Table 3.1), Let steady-state value of y(t). From (5-18), = t___ 0 0 0.632 0.865 0.950 0.982 0.993 Chapter 5 Note: Large means a slow response.

  18. First-Order System:(Ramp Response) • The ramp input (Eq. 5-8): • Eq. (5-22) implies that after an initial transient period. The ramp input yields a ramp output with slope equal to Ka, but shifted in time by the process time constant . Chapter 5 (5-22)

  19. Integrating Process Not all processes have a steady-state gain. For example, an “integrating process” or “integrator” has the transfer function: Chapter 5 Consider a step change of magnitude M. Then U(s) = M/s and, L-1 Thus, y(t) is unbounded and a new steady-state value does not exist.

  20. Common Physical Example: Consider a liquid storage tank with a pump on the exit line: • Assume: • Constant cross-sectional area, A. • Mass balance: • Eq. (1) – Eq. (2), take L, assume steady state initially, • For (constant q), Chapter 5

  21. Second-Order Systems • Standard form: which has three model parameters: Chapter 5 = = = • Equivalent form: =

  22. The type of behavior that occurs depends on the numerical value of damping coefficient, : It is convenient to consider three types of behavior: Chapter 5 • Note: The characteristic polynomial is the denominator of the transfer function: • What about ? It results in an unstable system

  23. Chapter 5

  24. Chapter 5

  25. Several general remarks can be made concerning the responses show in Figs. 5.8 and 5.9: • Responses exhibiting oscillation and overshoot (y/KM > 1) are obtained only for values of less than one. • Large values of yield a sluggish (slow) response. • The fastest response without overshoot is obtained for the critically damped case Chapter 5

  26. Chapter 5

  27. Rise Time: is the time the process output takes to first reach the new steady-state value. • Time to First Peak: is the time required for the output to reach its first maximum value. • Settling Time: is defined as the time required for the process output to reach and remain inside a band whose width is equal to ±5% of the total change in y. The term 95% response time sometimes is used to refer to this case. Also, values of ±1% sometimes are used. • Overshoot: OS = a/b (% overshoot is 100a/b). • Decay Ratio: DR = c/a (where c is the height of the second peak). • Period of Oscillation: P is the time between two successive peaks or two successive valleys of the response. Chapter 5

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