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ERT 210/4Process Control & Dynamics

DYNAMIC BEHAVIOR OF PROCESSES :

Dynamic Behavior of First-order and Second-order Processes

Chapter 5

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

- 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.

Chapter 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.

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.

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.

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.

- 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.

- 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,

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”.

- 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

Chapter 5

Chapter 5

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,

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.

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)

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.

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

- Standard form:

which has three model parameters:

Chapter 5

=

=

=

- Equivalent form:

=

- 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

Chapter 5 value of damping coefficient, :

Chapter 5 value of damping coefficient, :

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

Chapter 5 show in Figs. 5.8 and 5.9:

- Rise Time: show in Figs. 5.8 and 5.9: 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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