Chapter 4 Dynamic Systems Higher Order Processes Prof
- Slides: 65
Chapter 4 Dynamic Systems: Higher Order Processes Prof. Shi-Shang Jang National Tsing-Hua University Chemical Engineering Dept. Hsin Chu, Taiwan May, 2013
4 -1 Non-interactive Systems – Thermal Tanks
4 -1 Non-interactive Systems –Thermal Tanks – Cont.
4 -1 Non-interactive Systems
4 -1 Non-interactive Systems
Numerical Example
4 -1 Non-interactive Systems An over-damped second order system has two negative real poles. Therefore, 2 s 2+2 s+1=( 1 s+1)( 2 s+1); hence such that =
4 -2 Interactive Systems – Thermal Tanks with Recycle
4 -2 Interactive Systems – Thermal Tanks with Recycle- Cont.
4 -2 Interactive Systems – Thermal Tanks with Recycle- Cont.
2 -5 Step Response of Second-Order Processes – Over-damped process m(s)=A/s =1 =3 =2 Inflection point
4 -2 Interactive Systems fi V 1 fo h 1 Crosssectional=A 1 f 1 V 2 h 2 Crosssectional=A 2 f 2
4 -2 Interactive Systems Fi(s) F 0(s) + + + Fi(s + ) F 0(s) H 2(s) H 1(s) H 2(s) -
Numerical Example
4 -1 Second Order Systems A second order system is of the following form: Another form: Kp is called process gain, is called time constant, is called damping factor. The roots of the denominator are the poles of the system.
4 -1 Second-Order Processes - Continued Definition 4 -1: A second order process is called over-damped, if >1; is called under-damped if <1; is called critical damped if =1. Property 4 -1: Consider the roots of denominator, in case of over-damped system, the poles of the system are all negative real numbers. Property 4 -2: The poles of a under-damped system are complex with negative real numbers. Property 4 -3: The pole of a critical damped system is a repeated negative real number.
State Space Approach Consider the following linear system with N differential equations K inputs and P sensors where X is termed the state vector and M is the input vector. The following observation equation is available:
State Space Approach _ Cont. Assume that it is desirable to realize the input/output transfer functions and neglecting the state variables.
State Space Approach _ Example Interacting Tanks
State Space Approach _ Example Interacting Tanks – Cont.
State Space to Transfer Function. MATLAB >> A=[-0. 0375; 0. 0375 -0. 075] A = -0. 0375 -0. 0750 >> B=[1; 0]; C=[0 1]; D=0; >> [num, den]=ss 2 tf(A, B, C, D, 1) num = 0 -0. 0000 0. 0375 den = 1. 0000 0. 1125 0. 0014 >> ss=den(3) ss = 0. 0014 >> num=num/ss num = 0 -0. 0000 26. 6667 >> den=den/ss den = 711. 1111 80. 0000 1. 0000 >> tf(num, den) Transfer function: -9. 869 e-015 s + 26. 67 ----------711. 1 s^2 + 80 s + 1
2 -5 Step Response of Second-Order Processes – Under-damped process B 2. Decay Ratio= 3. Rise time=tr= Response 1. Overshoot= C T A Rise time Settling time 4. Period of oscillation=T= 5. Frequency of oscillation (Natural Frequency)=1/T
Example: Temperature Regulated Reactor t=1000 s t=3060 s Feed flow rate 0. 4→ 0. 5 kg/s at t=0. 1. What is process gain? 2. What is transfer func? 3. What is rise time?
Textbook Reading Assignment and Homework Chapter 2, p 41 -49 Homework p 58, 2 -9, 2 -15, 2 -16 Due June 8 th
Non-isothermal CSTR
Non-isothermal CSTR- Cont.
Non-isothermal CSTR- Cont. Process Information Steady State Values
Non-isothermal CSTR- Cont. Cooling flow rate 0. 8771 0. 8 Energy Balance of the Reactor Inlet and Outlet Energy and Material fluxes of the Reactor Material Balance of the Reactor Inlet and Outlet Energy fluxes of the Jacket Energy Balance of the Jacket Rate Constant Heat Exchange between Jacket and Reactor
Transfer Functions Derived by Linearization
Linearization of the Reactor Example
Transfer Functions Derived by Linearization – Cont It can be shown as generated as above: [num, den]=ss 2 tf(A, B, C, D, 4) num = 0 1. 33226762955019 e-015 -2. 81748023 -1. 356898478768 den = 1. 3804 0. 3849816 0. 038454805392 >> ss=den(4) >> den=den/ss den = 26. 0045523519419 35. 8966840666205 10. 0112741717343 1 >> num=num/ss num = 0 3. 46450233194353 e-014 -73. 2673121415962 -35. 2855375273927
SIMULINK of Linear System - CSTR
Non-isothermal CSTR- Cont. Cooling flow rate 0. 8771 0. 8 Tank temperature Time (min)
Non-isothermal CSTR- Cont. Linearized Model (page 127)
The problem of nonlinearity
4 -3 Step Response of the High Order System X(s)=A/s Responses n=2 n=3 n=5 n=10 time
4 -3 Step Response of the High Order System. Continued Response Method of Reaction Curve: inflection point time
4 -3 Step Response of the High Order System. Continued Real Responses Approximate time
4 -3 Response of Higher-Order Systems – Cont.
4 -4 Other Types of Process Response Integrating Processes: Level Process
4 -4 Other Types of Process Response
4 -4 Other Types of Process Response
4 -4 Other Types of Process Response The most general transfer function is as the following: p 1, p 2, …, pn are called the poles of the system, z 1, z 2, …, zm are the zeros of the system, Kp is the gain. Note that n m is necessary, or the system is not physically realizable.
4 -4 Poles and Zeros - Example Imaginary part Left Half Plane LHP Right Half Plane RHP 0 0 Real part
4 -4 Poles and Zeros - Example Response time
4 -4 Location of the Poles and Stability in a Complex Plane Im Purdy oscillatory Exponential Decay with oscillation Exponential growth with oscillation Fast Exponential growth Exponential Decay Fast Decay Slow growth Purdy oscillatory Stable (LHP) Unstable (RHP) Re
4 -4 The Stability of the linear system Definition 4 -2: A system is called stable for the initial point if given any initial point y 0, such that ∣y 0∣≦ε, there exists a upper bound , such that: Definition 4 -3: A system is called asymptotic stable if given any initial point y 0, then
4 -4 The Stability of the linear system Definition 4 -4: A system is called input output stable if the input is bound, then the output is bounded. (Bounded Input Bounded Output, BIBO) Property 4 -4: A linear system is asymptotic stable and BIBO if and only if all its poles have negative real parts.
4 -4 Stability - Example Response G 4 G 2 G 1 time m(s)=1 G 3
Open Loop Unstable Process. Chemical Reactor (text page 139)
Open Loop Unstable Process. Chemical Reactor
Homework Text p 148 4 -4, 4 -5, 4 -7, 4 -8, 4 -10, 4 -11, 4 -12 Due April
Supplemental Material Development of Empirical Models from Process Data
S-1 Introduction An empirical model is a model that is developed from experience and their parameters are found based on experimental tests. The most frequent implemented empirical models are first order, second order and/or with time delays. The input changes is basically a step or an impulse.
S-2 First Order without Time Delay Systems Using Step Input Consider a first order system with a output signal y(t) and input signal m(t), then:
First Order with Time Delay Systems Using Step Input Consider a first order system with a output signal y(t) and input signal m(t), then:
Example: A Typical Experiment Time (second) Y(temperature, o. C, 70100 o. C) Y(temperature, m. A, 4 -20 m. A) Y (temperature, %) 0. ln(1 -Y) 0 70 4 1 71. 74 4. 928 0. 058 -0. 0598 2 76. 51 7. 472 0. 217 -0. 2446 3 80. 8 9. 76 0. 360 -0. 4463 4 84. 64 11. 808 0. 488 -0. 6694 5 88 13. 6 0. 600 -0. 9163 6 90. 76 15. 072 0. 692 -1. 1777 7 93. 16 16. 352 0. 772 -1. 4784 8 94. 99 17. 328 0. 833 -1. 7898 9 96. 64 18. 208 0. 888 -2. 1893 10 97. 75 18. 8 0. 925 -2. 5903 0
Graphical Fitting Methods Fit 1: Method of 63. 2% Response Fit 2: Method of initial slope Fit 3: Method of Log plot
Example: An Experiment Plot Fit 1 G 1(s) = 1 5. 5 s +1 Fit 2 G 2 (s ) = e - 0. 5 s 6. 8 s + 1
Method of log plot Consider a First-Order Plus Dead Time Model
Method of log plot - Continued 0 0 5 10 15 -0. 5 -1 Series 1 -1. 5 Linear(Series 0) -2 -2. 5 -3 R 2 = 0. 9861 Fit 3
Method of log plot - Continued Fit 3 real Fit 1 Fit 2
S-3 Over-damped Second Order Systems Using Step Input
Smith’s Method for Second Order Systems Step 1: Get time delay by observing the response curve. Step 2: Find time t 20 such that y/y =0. 2, find t 60 such that y/y =0. 6 Step 3: Get t 20/t 60, then and From the right figure.
Example t 60=5 t 20=1. 9 t 20/t 60=0. 38 From the figure t 60/ =2. 4 =5/2. 4 =2. 1 =1. 2 2=4. 32, 2 =5. 04
- Higher order systems
- Concurrent in os
- 1st order 2nd order 3rd order neurons
- Transferered
- Introduction of business process
- Higher order functions haskell
- Hots verbs
- Affective objectives examples
- Higher order thinking iteach flashcards
- C higher order functions
- What is higher order derivatives
- Higher order structure of proteins
- Higher order thinking skills
- Factoring higher degree polynomials
- Rumus turunan parsial
- Higher order functions haskell
- Product and quotient rules and higher order derivatives
- Behaviorism ap psych
- Higher order thinking questions for vocabulary
- Huitt metacognition
- Higher order linear differential equations
- Higher
- Higher order functions haskell
- Haskell higher order functions
- Higher order derivatives
- Neutral stimulus
- Maniac magee higher order thinking questions
- Ap psych unit 6 study guide
- Dynamic partial order reduction
- Dynamic order statistics
- Real time operating system
- What is dynamic response in control system
- Valeria ttl
- Dynamic strategic activity systems help a firm to maintain
- Dynamic of systems
- Chapter 19 normal newborn processes of adaptation
- Designing and managing service processes
- Inverted and natural sentences
- First order vs second order change
- Order takers vs order getters
- Difference between zero and first order kinetics
- Metaboloism
- Law is order and good law is good order
- Order properties
- In order issue in order completion example
- In order issue in order completion example
- Arbol in order post order
- First order cybernetics and second order cybernetics
- Similarities between job order costing and process costing
- Decision support systems and intelligent systems
- Principles of complex systems for systems engineering
- Embedded systems vs cyber physical systems
- Engineering elegant systems: theory of systems engineering
- Mass of ocean water that flows from one place to another
- Chapter 3 the dynamic earth
- Chapter 3 the dynamic earth
- Dynamic chapter
- The dynamic earth chapter 3
- Chapter 3 the dynamic earth
- Chapter 3 the dynamic earth section 1 the geosphere
- The dynamic earth chapter 3
- Proximal processes
- Examples of word formation processes
- Study of behavior and mental processes
- Reading processes
- Biological weathering