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Chapter 4 Modelling and Analysis for Process Control

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1 Chapter 4 Modelling and Analysis for Process Control
Laplace Transform Definition

2 Input signals

3

4 (c) A unit impulse function (Dirac delta function)

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11 * Properties of the Laplace transform
Linearity Differentiation theorem

12 Zero initial values Proof:

13 Integration theorem

14 Translation theorem Proof:

15

16

17 Final value theorem Initial value theorem

18 Complex translation theorem
Complex differentiation theorem

19 Example 4.1 Solution:

20

21 Example 4.2 (S1)

22 (S2)

23 * Laplace transform procedure for differential equations
Steps:

24 Exercises: a second-order differential equation
(1) Laplace transform

25 Algebraic rearrangement
Zero initials (2) Transfer function

26 (3) Laplace Inversion Where

27 Inversion method: Partial fractions expansion (pp.931)
(i) Fraction of denominator and

28 (ii) Partial fractions
where

29 * Repeated roots (iii) Inversion
If r1=r2, the expansion is carried out as

30 where Inversion

31 * Repeated roots for m times
If the expansion is carried out as

32 and

33

34

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36 and A3=2 as (a) case.

37 The step response: Example 4.3

38 (S1)

39 (S3) Find coefficients s=0 Inversion

40 Example 4.4 (S1) Laplace transformation

41 (S2) Find coefficients s=0 s=1-j s=-1+j

42 (S3) Inversion and using the identity

43 Time delays: Consider Y(s)=Y1(s)e-st0 and

44 Example:

45 Input function f(t)

46

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49 * Input-Output model and Transfer Function
Ex.4.5 Adiabatic thermal process example

50 S1. Energy balance

51 S2. Under steady-state initial conditions
and define deviation variable

52 S3. Standard form where

53 S4. Transfer function (Laplace form)
@ Step change ( )

54

55 * Non-adiabatic thermal process example
S1. model S2. Under deviation variables, the standard form

56 where

57 S3. Laplace form @ Transfer functions

58 Ex. 4.6 Thermal process with transportation delay

59 @ Dead time

60 @ Transfer functions

61 ※ Transfer function (G(s))
Note: The transfer function defines the steady-state and dynamic characteristic, or total response, of a system described by a linear differential equation.

62 *Important properties of G(s)
Physical systems, Transforms of the derivation of input and output variables Steady state responses

63 * Steady-state gain ( ) Ex. Consider two isothermal CSTRs in series

64 Ans.: Steady-state gain: Final value of the reactant concentration in the second reactor:

65 ※ Block diagrams

66 @ Block diagram for

67 Example 4.7 Block diagram for

68 * Rules for block diagram

69

70

71 Example 4.8 Determine the transfer functions

72

73 Solution:

74 Example 3-4.3 Determine the transfer functions
=?

75 @ Reduced block

76

77 Example 4.9 =?

78 ◎ Answer

79 ◎ Design steps for transfer function

80 @ Review of complex number
c=a+ib

81 Polar notations

82 ※ Frequency response

83 ◎ Experimental determination of frequency response
S1. Process (valve, model, sensor/transmitter)

84 S2. Input signal S3. Output response where

85 P1. Amplitude of output signal
P2. Output signal ‘lags’ the input signal by θ. P3. Amplitude ratio (AR): AR=Y0/X0 P4. Magnitude ratio (MR): MR=AR/K P5. Phase angle (θ): if θ is negative, it is a lag angle.

86 Ex.4.7 A first-order transfer function G(s)=K/(τs+1)
* Consider a form of If the input is set as Then the output

87 *Through inverse Laplace transformation, the output response is reduced as
P2. (p.69)

88 Ex.4.8 Consider a first-order system

89 S2. Amplitude ratio and phase angle
Ex.4.9 Consider a second-order system

90 S1. s=iω to decide amplitude ratio

91 G(s)=K(1+τs) S2. Phase angle #
Ex.4.10 Consider a first-order lead transfer function G(s)=K(1+τs)

92 Ex.4 Consider a pure dead time transfer function
G(s) =e-t0s

93 Ex.5 Consider an integrator
G(s)=1/s G(i)=-(1/ )i

94 * Expression of AR and θ for general OLTF

95 ※ Bode plot A common graphical representation of AR (MR) and θ functions. Bode plot consists: (1) log AR or (log MR) vs. log ω (2) θ vs. log ω * (3) 20 log AR (db) vs. log ω

96 Ex. 5 Consider a first-order lag by Ex. 1
To show Bode plot. S1. MR1 as ω 0 S2. As ω 

97 * Types of Bode plots Gain element First-order lag Dead time Second-order lag First-order lead Integrator

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104 * Process control for a chemical reactor

105

106 Homework 2# Q4.6 Q4.10 Q4.16 Q4.18 (※Difficulty)


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