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**ERT 210 Process Control & dynamics**

CHAPTER 8 Dynamic Response Characteristics of More Complicated Processes Miss Anis Atikah binti Ahmad

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**OUTLINE Chapter 6 Chapter 6 Dynamic Response Characteristics**

1. Poles and Zeros and Their Effect on Process Response 2. Process with Time Delays 3. Approximation of Higher-Order Transfer Functions 4. Interacting and Noninteracting Processes 5. Multiple-Input, Multiple-Output (MIMO) Processes Chapter 6 Chapter 6

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**1. Poles and Zeros and Their Effect on Process Response**

General Representation of standard transfer function form: - There are two equivalent representations: (6-2) Chapter 6 Chapter 6 where {zi} are the “zeros” and {pi} are the “poles”.

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**Poles and Zeros and Their Effect on Process Response**

Consider a particular transfer function; (6-1) , where 0 < 1. The roots of these factors are The values of s that are the denominator polynomial-refer as poles Chapter 6 Chapter 6

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**Poles and Zeros and Their Effect on Process Response**

Chapter 6 Figure 6.1 Poles of G(s) plotted in the complex s plane. (X denotes a pole location)

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**Chapter 6 Summary: Effects of Pole and Zero Locations Chapter 6 Poles**

Pole in “right half plane (RHP)”: results in unstable system (i.e., unstable step responses) Example (grows with time) Imaginary axis x = unstable pole x Chapter 6 Chapter 6 x Real axis x Complex pole: results in oscillatory responses (contains sine and cosine terms) Imaginary axis Example x = complex poles x Real axis x

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Zeros Note: Zeros have no effect on system stability. Zero in RHP: results in an inverse response to a step change in the input Chapter 6 Chapter 6 Imaginary axis x Real axis inverse response y t Zero in left half plane: may result in “overshoot” during a step response (see Fig. 6.3).

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**2. Process with Time Delays**

Transportation lag/ transport delay/ dead time Ѳ= Time taken to transport fluid from point 1 to point 2 Chapter 6 Mathematical description: A time delay, between an input u and an output y results in the following expression: Transfer Function Representation:

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**2. Process with Time Delays (cont’)**

Chapter 6

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**3. Approximation of Higher-Order Transfer Functions**

Two widely used approximations are: Taylor Series Expansion: Chapter 6 Chapter 6 The approximation is obtained by truncating after only a few terms. Padé Approximations: Many are available. For example, the 1/1 approximation is,

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**3. Approximation of Higher-Order Transfer Functions (cont’)**

The 2/2 Padé Approximations approximation; (6.37) Chapter 6 Chapter 6 Note: Please refer page 138 and 139 for more explanations.

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**Example 6.5 Chapter 6 Chapter 6**

The trickle-bed catalytic reactor shown in Fig. 6.8 utilizes product recycle to obtain satisfactory operating conditions for temperature and conversion. Use of a high recycle rate eliminates the need for mechanical agitation. Concentrations of the single reactant and the product are measured at a point in the recycle line where the product stream is removed. A liquid phase first-order reaction is involved. Under normal operating conditions, the following assumption may be made: The reactor operates isothermally so that the reaction rate k is constant. All flow rates and the liquid volume V are constant. No reaction occurs in the piping. The dynamics of the exit and recycle lines can be approximated as constant time delay θ1 and θ2, as indicated in the figure. Let c1 denote the reactant concentration at the measured point. Because of the high recycle flow rate, mixing in the reactor is complete. Chapter 6 Chapter 6

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**Chapter 6 Example 6.5 (cont’) Chapter 6 Question:**

Derive an expression for the transfer function Using the following information, calculate for a step change in Chapter 6 Chapter 6 Parameter Values: V = 5 m3 α = 12 q = 0.05 m3/min θ1 = 0.9 min k = 0.04 min-1 θ2 = 1.1 min Figure 6.8: Schematic diagram of a trickle-bed reactor with recycle line. (AT: analyzer transmitter; θ1: time delay associated with material flow from reactor outlet to the composition analyzer; θ2: time delay associated with material flow from transmitter to reactor inlet.

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**Chapter 6 Chapter 6 Solution**

Equation 2.66 is applicable only to an isothermal stirred-tank reactor without recycle. Hence, we make component balance around the reactor. (6.39) where the concentration of the reactant is denoted by c. Equation 6.39 is linear with constant coefficients. Subtracting the steady-state equation and substituting deviation variables yields Chapter 6 Chapter 6 (6.40) Additional relations are needed for c2'(t) and c1'(t). They can be obtained from assumption (iii), which states that the exit and recycle lines can be modeled as time delays: (6.41) (6.42)

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Equations 6-40 through 6.42 provide the process model for the isothermal reactor with recycle. Taking the Laplace transform of each equation yields (6.43) (6.44) Chapter 6 Chapter 6 (6.45) where θ3 = θ1+ θ2. Substitute (6.45) into (6.43) and solve for the output C'(s): (6.46)

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**Equation 6.46 can be rearranged to the following form:**

(6.47) where (6.48) Chapter 6 (6.49) Chapter 6 Note that, in the limit as , and (6.50) hence K and can be interpreted as the process gain and time constant, respectively, of a recycle reactor with no time delay in the recycle line, which is equivalent to a stirred isothermal reactor with no recycle.

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**The desired transfer function C1'(s)/Ci' (s) is obtained by combining Eqs. 6-47 and 6-44 to obtain**

(6.51) (b) To find c1'(t) when ci'(t) = 2000 kg/m3, we multiply (6.51) by 2000/s (6.52) Chapter 6 Chapter 6 and take the Laplace transform. From (6.52), it is clear that the numerator time delay can be inverted directly; however, there is no transform in Table 3.1 that contains a time-delay term in the denominator. To obtain an analytical solution, the denominator time delay term must be eliminated by introducing a rational approximation, for example, the 1/1 Padé approximation in (6.35). (6.35)

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**Chapter 6 Chapter 6 Substituting (6.35) and rearranging yields (6.53)**

This expression can be written in the form (6.54) Chapter 6 Chapter 6 where a = θ3/2 and 1 and 2 are obtained by factoring the expression in brackets. For αKθ3 > 0, 1 and 2 will be real and distinct. The numerical parameters in (6.53) are:

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**Chapter 6 Substituting the values in (6.53), we obtain (6.55)**

Taking inverse Laplace and introducing the delayed unit step function S(t - 0.9) gives; Chapter 6 (6.56) Figure 6.9 Recycle reactor composition measured at analyzer: (a) complete response; (b) detailed view of short-term response.

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**3. Approximation of Higher-Order Transfer Functions**

In this section, we present a general approach for approximating high-order transfer function models with lower-order models that have similar dynamic and steady-state characteristics. In Eq we showed that the transfer function for a time delay can be expressed as a Taylor series expansion. For small values of s, after the first-order term provides a suitable approximation: Chapter 6 Chapter 6

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**An alternative first-order approximation consists of the transfer function,**

Chapter 6 Chapter 6 where the time constant has a value of Equations 6-57 and 6-58 were derived to approximate time-delay terms. However, these expressions can also be used to approximate the pole or zero term on the right-hand side of the equation by the time-delay term on the left side.

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**Chapter 6 Skogestad’s “half rule” Chapter 6**

Skogestad (2002) has proposed a related approximation method for higher-order models that contain multiple time constants. He approximates the largest neglected time constant in the following manner. One half of its value is added to the existing time delay (if any) and the other half is added to the smallest retained time constant. Time constants that are smaller than the “largest neglected time constant” are approximated as time delays using (6-58). Chapter 6 Chapter 6

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**Chapter 6 Example 6.4 Chapter 6 Consider a transfer function:**

Derive an approximate first-order-plus-time-delay model, Chapter 6 Chapter 6 using two methods: The Taylor series expansions of Eqs and 6-58. Skogestad’s half rule Compare the normalized responses of G(s) and the approximate models for a unit step input.

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Solution The dominant time constant (5) is retained. Applying the approximations in (6-57) and (6-58) gives: and Chapter 6 Substitution into (6-59) gives the Taylor series approximation

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(b) To use Skogestad’s method, we note that the largest neglected time constant in (6-59) has a value of three. According to his “half rule”, half of this value is added to the next largest time constant to generate a new time constant The other half provides a new time delay of 0.5(3) = 1.5. The approximation of the RHP zero in (6-61) provides an additional time delay of 0.1. Approximating the smallest time constant of 0.5 in (6-59) by (6-58) produces an additional time delay of 0.5. Thus the total time delay in (6-60) is, Chapter 6 Chapter 6

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**Chapter 6 Chapter 6 and G(s) can be approximated as:**

The normalized step responses for G(s) and the two approximate models are shown in Fig Skogestad’s method provides better agreement with the actual response. Chapter 6 Chapter 6 Figure 6.10 Comparison of the actual and approximate models for Example 6.4.

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**Chapter 6 Example 6.5 Chapter 6**

Consider the following transfer function: Use Skogestad’s method to derive two approximate models: A first-order-plus-time-delay model in the form of (6-60) A second-order-plus-time-delay model in the form: Chapter 6 Chapter 6 Compare the normalized output responses for G(s) and the approximate models to a unit step input.

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**Chapter 6 Chapter 6 Solution**

(a) For the first-order-plus-time-delay model, the dominant time constant (12) is retained. One-half of the largest neglected time constant (3) is allocated to the retained time constant and one-half to the approximate time delay. Also, the small time constants (0.2 and 0.05) and the zero (1) are added to the original time delay. Thus the model parameters in (6-60) are: Chapter 6 Chapter 6

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**(b) An analogous derivation for the second-order-plus-time-delay model gives:**

In this case, the half rule is applied to the third largest time constant (0.2). The normalized step responses of the original and approximate transfer functions are shown in Fig Chapter 6 Chapter 6 Figure 6.11 Comparison of the actual model and approximate models for Example 6.5. The actual and second-order model responses are almost indistinguishable.

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**4. Interacting and Noninteracting Processes**

Consider a process with several invariables and several output variables. The process is said to be interacting if: Each input affects more than one output. or A change in one output affects the other outputs. Otherwise, the process is called noninteracting. As an example, we will consider the two liquid-level storage systems shown in Figs. 4.3 and 6.13. In general, transfer functions for interacting processes are more complicated than those for noninteracting processes. Chapter 6 Chapter 6

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**Chapter 6 Chapter 6 Figure 4.3. A noninteracting system:**

two surge tanks in series. Chapter 6 Chapter 6 Figure Two tanks in series whose liquid levels interact.

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**Chapter 6 Chapter 6 Substituting (4-49) into (4-48) eliminates q1:**

Figure 4.3. A noninteracting system: two surge tanks in series. Chapter 6 Chapter 6 Mass Balance: Valve Relation: Substituting (4-49) into (4-48) eliminates q1:

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**Putting (4-49) and (4-50) into deviation variable form gives**

The transfer function relating to is found by transforming (4-51) and rearranging to obtain Chapter 6 Chapter 6 where and Similarly, the transfer function relating to is obtained by transforming (4-52). = =

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**The same procedure leads to the corresponding transfer functions for Tank 2,**

Chapter 6 Chapter 6 where and Note that the desired transfer function relating the outflow from Tank 2 to the inflow to Tank 1 can be derived by forming the product of (4-53) through (4-56). = =

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**Chapter 6 Chapter 6 or which can be simplified to yield**

a second-order transfer function (does unity gain make sense on physical grounds?). Figure 4.4 is a block diagram showing information flow for this system.

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**Block Diagram for Noninteracting Surge Tank System**

Figure 4.4. Input-output model for two liquid surge tanks in series.

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**Dynamic Model of An Interacting Process**

Chapter 6 Chapter 6 Figure Two tanks in series whose liquid levels interact. The transfer functions for the interacting system are:

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Chapter 6 Chapter 6 = = In Exercise 6.15, the reader can show that ζ > 1 by analyzing the denominator of (6-71); hence, the transfer function is overdamped, second order, and has a negative zero.

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**Model Comparison Noninteracting system = = Interacting system =**

General Conclusions 1. The interacting system has a slower response. (Example: consider the special case where t = t1= t2.) 2. Which two-tank system provides the best damping of inlet flow disturbances?

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**5. Multiple-Input, Multiple-Output (MIMO) Processes**

Most industrial process control applications involved a number of input (manipulated) and output (controlled) variables. These applications often are referred to as multiple-input/ multiple-output (MIMO) systems to distinguish them from the simpler single-input/single-output (SISO) systems that have been emphasized so far. Modeling MIMO processes is no different conceptually than modeling SISO processes. Chapter 6 Chapter 6

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**For example, consider the system illustrated in Fig. 6.14.**

Here the level h in the stirred tank and the temperature T are to be controlled by adjusting the flow rates of the hot and cold streams wh and wc, respectively. The temperatures of the inlet streams Th and Tc represent potential disturbance variables. Note that the outlet flow rate w is maintained constant and the liquid properties are assumed to be constant in the following derivation. Chapter 6 Chapter 6 (6-88)

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Chapter 6 Chapter 6 Figure A multi-input, multi-output thermal mixing process.

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Chapter 6 Chapter 6 Figure Block diagram of the MIMO thermal mixing system with variable level.

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THANK YOU Chapter 6 Chapter 6

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