A. Design Example A.1 Distillation Process
Transcript of A. Design Example A.1 Distillation Process
![Page 1: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/1.jpg)
A.1 Distillation Process
A. Design Example
Reference:[SP05] S. Skogestad and I. Postlethwaite,
Multivariable Feedback Control; Analysis and Design,Second Edition, Wiley, 2005.
[SP05, Sec. 13.4]
![Page 2: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/2.jpg)
2
Models: Understand the ProcessSTEP 1. Real Physical System
STEP 2. Ideal Physical Model
STEP 3. Ideal Mathematical Model
STEP 4. Reduced Mathematical Model
Conceptual/Schematic model(図式化・概念化)
Idealization(理想化)
Linearization(線形化)
Product(本物)
Gas
Liquid
BubbleTray
Stage
![Page 3: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/3.jpg)
3
STEP 2. Ideal Physical Model
Reflux flow rate, kmol/min.Boilup from reboiler, kmol/min.
Bottom product rate, kmol/min.Distillate (top product) rate, kmol/min.
Inputs
Outputs
Number of stages: 40
![Page 4: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/4.jpg)
4
STEP 3. Ideal Mathematical Model
Tray ’s composition dynamics can be formulated as follows:
where
Liquid holdup on theoretical tray , kmol.
Liquid mole fraction of light component on stage .
Vapor mole fraction of light component on stage .
Feed
Overheadvapor Reflux
Boilup
Bottom Flow
![Page 5: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/5.jpg)
5
STEP 4. Reduced Mathematical Model
Then, the whole system can be viewed as a first-order model.
Each tray has its own physical model.
Feed
Overheadvapor Reflux
Boilup
Bottom Flow
Assumptions
• The flow dynamics are immediate.• All trays have the same dynamic responses.
![Page 6: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/6.jpg)
Distillation Process: Problem Statement
6
Real Physical System Ideal Physical Model[SP05, pp. 100, 509-514]
: top composition: bottom composition
: reflux: boilup
Controlled Variables
Manipulated Inputs: distillate
: bottom flow: overhead vapor
Assumption • The composition dynamics are usually much slower than the flow dynamics
the simplifying assumption of perfect control of hold up and instantaneous flow responses in the column
Flow RelationshipsTop/Bottom CompositionInputs Outputs
2-Input 2-Output System
![Page 7: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/7.jpg)
7
Distillation Process: Plant Model [SP05, pp. 100, 509-514]
influences
Nominal Model
MATLAB CommandN = {87.8, -86.4; 108.2 -109.6};D = [75 1];Pnom = tf(N, D);
![Page 8: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/8.jpg)
Gain Margin :Delay Margin :
Multiplicative (Output) Uncertainty
Distillation Process
Nominal Model
Uncertain Plant Model
[min]
![Page 9: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/9.jpg)
9
Rise time 30 min
Uncertainty Weight
[rad/min]
0.035Gain Crossover
Frequency
1 min
1 rad/min
Performance Weight
Distillation Process: Performance Specifications
rad/min
rad/min
Steady state error < 0.01
rad/min
Delay Margin:
1.0
Gain Margin: 20%, 2dB
![Page 10: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/10.jpg)
10
Multivariable Feedback Control
-
-
Multi-Input Multi-Output(MIMO) System
How to design multivariable feedback controllers systematically?
Non-interactionSingle-Input Single-Output(SISO) System -
-
![Page 11: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/11.jpg)
11
Distillation Process: SISO Plant Model
Plant
Pole:
Zero: none
Stable System
Minimum Phase System
Re
Im
Frequency Response (Bode Plot) Step Response
0
0.1
0
Frequency [rad/min]
![Page 12: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/12.jpg)
12
Distillation Process: SISO Plant Model
Plant:Re
Im
Frequency Response (Bode Plot) Step Response
0
0.1
0
Pole:
time scaling rad/min rad/s
![Page 13: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/13.jpg)
Frequency [rad/min]
13
Auto PID tuning algorithm: pidtuneDistillation Process: SISO Controller Design
K = pidtune( Pnom, ‘pidf‘ ) ;L = series( K, P ) ;T = feedback( L, 1 ) ;figure; bode( L ) ;figure; step( T ) ;
PID:
Bode diagram
Step response
MATLAB Command
![Page 14: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/14.jpg)
14
-controller
Bode diagram Step response
Perturbed Plant ModelNominal Plant Model
(See 6th lecture)Auto tuning algorithm:
Distillation Process: SISO Plant Model
Frequency [rad/min]
![Page 15: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/15.jpg)
-controller (update)
Bode diagram Step response
(See 6th lecture)Auto tuning algorithm:
Distillation Process: SISO Plant Model
Frequency [rad/min]
![Page 16: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/16.jpg)
16
-controller
Bode diagram Step response
Perturbed Plant ModelNominal Plant Model
(Order 11 → 4)(See 6th lecture)
Auto tuning algorithm:Distillation Process: SISO Plant Model
Frequency [rad/min]
![Page 17: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/17.jpg)
-controller (update)
Bode diagram Step response
(Order 11 → 4)(See 6th lecture)
Auto tuning algorithm:Distillation Process: SISO Plant Model
Frequency [rad/min]
![Page 18: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/18.jpg)
Frequency [rad/min]Frequency [rad/min]18
Distillation Process: Evaluate SISO Controller Design*
PID
Complementary SensitivitySensitivity
0.02241 rad/min0.0227 rad/min
-controller
1.01
Mag
nitu
de [d
B]
Mag
nitu
de [d
B]
![Page 19: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/19.jpg)
19
Distillation Process: Evaluate SISO Controller Design*Complementary SensitivitySensitivity
-controller
Frequency [rad/min] Frequency [rad/min]
Mag
nitu
de [d
B]
Mag
nitu
de [d
B]
![Page 20: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/20.jpg)
20
Control of Multivariable Plants1. Diagonal Controller (decentralized control)
[SP05, pp. 91-93]
-
-
Controller
0
0.1
0
00
Nominal Plant ModelTime delay
0 1.0
NSNP
RPRS
○
×
NSNP
RPRS
○
×
×
×
○ ○
![Page 21: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/21.jpg)
21
(Input Uncertainty)
2. Two-step compensator design: dynamic decoupling
-
-
Controller
0
0.1
0
00
Inverse-based controller (decoupling control)
Nominal Plant ModelTime delay
0 1.0
NSNP
RPRS
○
×
×
○
NSNP
RPRS
○
×
×
○
![Page 22: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/22.jpg)
22
Inverse-based controller (decoupling control)2. Two-step compensator design: dynamic decoupling
(proper) (proper)
NSNP
RPRS
×
×
×
×
NSNP
RPRS
×
×
×
×
0
0.1
0
00
-
-
![Page 23: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/23.jpg)
23
-
Multivariable Feedback Control
(Input Uncertainty)
controller
NSNP
RPRS
○
○
×
○
![Page 24: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/24.jpg)
24
Control of Multivariable Plants1. Diagonal Controller (decentralized control)
[SP05, pp. 91-93]
-
-
Controller
0
0.1
0
00
Nominal Plant ModelTime delay
0 1.0
NSNP
RPRS
○
×
NSNP
RPRS
○
×
○
○
○
○
Performance Weight
![Page 25: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/25.jpg)
25
-
Multivariable Feedback Control
(Input Uncertainty)
controller
NSNP
RPRS
○
○
○
○
NSNP
RPRS
○
○
○
○
![Page 26: A. Design Example A.1 Distillation Process](https://reader031.fdocuments.in/reader031/viewer/2022012300/61e169f2767cb4181a04fd49/html5/thumbnails/26.jpg)
26
Another Example [2]
[2] R.K. Wood and M.W. Berry, “Terminal composition control of a binary distillation column,”Chemical Engineering Science, Vol. 28, No. 9, pp. 1707-1717, 1973.
Column’s diameter: 9inThe number of tray: 8The space of each tray: 12in4 bubble caps are arranged in a square patternand each size is in
(Above distillation column is interfaced with an IBM 1800)A total condenser and basket type reboiler is equipped.
Then, the above distillation process’s model was determined as followsfrom its step response.
where
Distillation of methanol