Process Dynamics and Control Solutions
Short Description
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Description
123456789 8
24.1
a)
i.
First model (full compositions model):
Number of variables: NV = 22 w1
xR,A
x2B
w2
xR,B
x2D
w3
xR,C
x4C
w4
xR,D
x5D
w5
xT,D
x6D
w6
VT
x7D
w7
HT
x8D
w8 Number of Equations: NE = 17 Eqs. 2-8, 9, 10, 12, 13, 15, 16, 18, 20(3X), 21, 22, 27, 28, 29, 31 Number of Parameters: NP = 4 VR, k, α, ρ Degrees of freedom: NF = 22 – 17 = 5 Number of manipulated variables: NMV = 4 w1, w2, w6, w8 Number of disturbance variables: NDV = 1 x2D Number of controlled variables: NCV = 4 x4A, w4, HT, x8D nd
Solution Manual for Process Dynamics and Control, 2 edition, Copyright © 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp
24-1
ii. Second model (simplified compositions model): Number of variables: NV = 14 w1
xR,A
w4
w2
xR,B
x4A
w6
xR,D
x8D
w8
xT,D
HT
x2D
VT
Number of Equations: NE = 9 Eq. 2-33 through Eq. 2-41 Number of Parameters: NP = 4 VR, k, α, ρ Degrees of freedom: NF = 14 – 9 = 5 Number of manipulated variables: NMV = 4 w1, w2, w6, w8 Number of disturbance variables: NDV = 1 x2D Number of controlled variables: NCV = 4 x4A, w4, HT, x8D iii.Third model (simplified holdups model): Number of variables: NV = 14 w1
HR,A
w4
w2
HR,B
x4A
w6
HR,D
x8D
w8
HT,B
HT
x2D
HT,D
24-2
Number of Equations: NE = 9 Eq. 2-48 through Eq. 2-56 Number of Parameters: NP = 3 VR, k, α Degrees of freedom: NF = 14 – 9 = 5 Number of manipulated variables: NMV = 4 w1, w2, w6, w8 Number of disturbance variables: NDV = 1 x2D Number of controlled variables: NCV = 4 x4A, w4, VT, x8D b)
Model 1: The first model is left in an intermediate form, i.e., not fully reduced, so the key equations for the units are more clearly identifiable. Also, such a model is easier to develop using traditional balance methods because not as much algebraic effort is expended in simplification. Models 2 and 3: Both of the reduced models are easier to simulate (fewer equations), yet contain all of the dynamic relations needed to simulate the plant. Model 3: The “holdups model” has the further advantage of being easier to analyze using a symbolic equation manipulator because of its more symmetric organization. Also, it requires one less parameter for its specification.
c)
Each model can be simulated using the equations given in Appendix E of the text. Models 2 and 3 are simulated using the differential equation editor (dee) in Matlab. An example can be found by typing dee at the command prompt. Step changes are made in the manipulated variables w1, w2, w6 and w8 and in disturbance variable x2D to illustrate the dynamics of the entire plant.
24-3
Figure S24.1a. Simulink-MATLAB block diagram for first model
1 w3 Sum input flows1
2 x8
1 s
1 w8
A
3
1 s
x1
Demux
4
B
Mux
1 s
w1
2 3000
C 5 -out + in +generated
6
HR
1 s
x2
w2
x3
D
Sum input flows
[-0.5 -0.5 1 0] 330
stoich. factors
k 3000 rho*VR
Demux generation & Accumulation
T1 T2
Figure S24.1b. Simulink-MATLAB block diagram for the reactor block
24-4
Product3 4 x4
3 2
Product2
Demux
1
x3
Product
w4
w3 2
Product5
x5
1 Product1
w5
Product4
Figure S24.1c. Simulink-MATLAB block diagram for the flash block
1 Purge Flow
1 2
w7
w5
3
2
x5
x7
Figure S24.1d. Simulink-MATLAB block diagram for purge block
24-5
1010 w1
w1 w1-save
xRA xRA-save
w1-change xRB
w2
xRB-save
w2-save 1100
xRD
w2
xRD-save x2D w2-change
Equations 2-(32-38)
VT-save
0.01 x2D-set
xTD
110 Purge Flow
HT
xTD-save w6 w4 w6-save w4-save
890 x4A
Recycle Flow w8 E.2-32-38
x4A-save
w8-save
Figure S24.1e. Simulink-MATLAB block diagram for second model
24-6
w2
0.01 w4 x2D w4-save x4A w6
x4A-save
w6-save xTD xTD-save
w8 w8-save
24-7
2010
1014
2008
1013
2006
Full model Reduced concentrations Reduced holdups
w4
w1
1015
1012
2004
1011
2002
1010
0
10
20
30
40
1101
2000
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0.104
1100.5 w2
0.102 xTD
1100
1099.5 1099
0.1
0
10
20
30
40
110.5
500 HT
550
w6
111
110
450
109.5 109
400
0
10
20
30
350
40
890.5
0.0101 x 4A
0.0102
w8
891
890
0.0101
889.5 889
0.01 0.01 0
10
20
30
40
Time (hr)
Time (hr)
Figure S24.1g. Step change in w1 (+5) at t=5
24-8
1010.5
2006
1010
2004
1009.5 1009
Full model Reduced concentrations Reduced holdups
w4
2008
w1
1011
2002
0
10
20
30
2000
40
0.102
1108
0.1
1106
0.098
10
20
30
40
0
10
20
30
40
0
10
20
30
40
20
30
40
w2
1110
0
0.096
x TD
1104 1102 1100
0.094
0
10
20
30
0.092
40
111
800
110.5 HT
w6
700 110
600 109.5 109
0
10
20
30
500
40
-3
890.5
10
890
9.95
889.5 889
x 10
x 4A
10.05
w8
891
9.9
0
10
20
30
9.85
40
Time (hr)
0
10
Time (hr)
Figure S24.1h. Step change in w2 (+10) at t=5
24-9
1011
2000 Full model Reduced concentrations Reduced holdups
2000
1010.5
w4
2000
w1
1010 2000
1009.5 1009
2000 0
10
20
30
40
2000
20
30
40
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0.1 0.1
w2
1100.5 1100
0.1
1099.5
0.1
1099
10
x TD
1101
0
0
10
20
30
0.1
40
120
500
118
400 HT
116 w6
300
114 200
112 110
0
10
20
30
40
891
100
0.01
890.5 x 4A
w8
0.01 890
0.01 889.5 889
0
10
20
30
40
0.01
Time (hr)
Time (hr)
Figure S24.1i. Step change in w6 (+10) at t=5
24-10
1011
2008
w4
2006
w1
1010.5 1010
2004
1009.5 1009
2002
0
10
20
30
40
2000
0.1001
1100.5
0.1001
w2
x TD
1101
1100
0
10
20
30
40
0
10
20
30
40
10
20
30
40
10
20
30
40
0.1
1099.5 1099
Full model Reduced concentrations Reduced holdups
0.1
0
10
20
30
0.1
40
111
500 498
110.5
HT
w6
496 110
494 109.5 109
492 0
10
20
30
490
40
0 -3
10.01
898
10 9.99
w8
896 894
9.98
892
9.97
890
x 10
x 4A
900
0
10
20
30
40
9.96
0
Time (hr)
Time (hr)
Figure S24.1j. Step change in w8 (+10) at t=5
24-11
1011
2000.5 Full model Reduced concentrations Reduced holdups
1010.5 2000 w4
w1
1010
1999.5 1009.5 1009
0
10
20
30
40
1999
1101
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0.16
1100.5 w2
x TD
0.14 1100
0.12 1099.5 1099
0
10
20
30
0.1
40
110.5
515 HT
520
w6
111
110
510
109.5 109
505
0
10
20
30
500
40
891 0.0105
w8
x4A
890.5 890 0.01 889.5 889
0
10
20
30
40
Time (hr) Time (hr)
Figure S24.1k. Step change in x2D (+0.005) at t=5
24-12
24.2 To obtain a steady state (SS) gain matrix through the use of simulation, step changes in the manipulated variables are made. The resulting matrix should compare closely with that found in Eq. 24.1 of the text (or the table below). The values calculated are: Gain Matrix w1 w2 w6 w8 1.93 2.26 E-2 0 6.25 E-3 w4 8.8 E-4 -7.62 E-4 0 5.68 E-6 x8D 2.57 E-5 -1.14 E-5 0 -3.15 E-6 x4A -0.918* 0.973* -1* -6.25 E-3* HT * For integrating variables: “Gain” = the slope of the variable vs. time divided by the magnitude of the step change. RGA w4 x8D x4A HT
w1 0.9743 0 0.0257 0
w2 0.0135 0.9737 0.0128 0
w6 0 0 0 1
w8 0.0122 0.0263 0.9615 0
24.3 Controller parameters are given in Tables E.2.7 and E.2.8 in Appendix E of the text. A transfer function block is placed inside each control loop to slow down the fast algebraic equations, which otherwise yield large “output spikes”. These blocks are of the form of a first-order filter.: G f ( s) =
1 0.001s + 1
In principle, ratio control can provide tighter control of all variables. However, it is clear from the x2D results that it offers no advantage for this disturbance variable. For a step change in production rate, w4, one would anticipate a different situation because a change in manipulated variable w1 is induced. Using ratio control, w2 does change along with w1 to maintain a satisfactory ratio of the two feed streams. Thus, ratio control does provide enhanced control for the recycle tank level, HT, and composition, xTD, but not for the key performance variables, w4 and x4A. This characteristic is likely a result of the particular features of the recycle plant, namely the use of a splitter (instead of a flash unit) and the lack of holdup in that vessel.
24-13
2004 No ratio control Ratio control
w4
2002 2000 1998 1996
0
5
10
15
0
5
10
15
0
5
10
15
20
25
30
35
20
25
30
35
20
25
30
35
0.0101
x 4A
0.01
0.0099
Time (hr)
0.12
x TD
0.11 0.1 0.09
Figure S24.3a. Step change in x2D (+0.02) at t=5 (Corresponds to Fig.24.7)
2150 No ratio control Ratio control
w4
2100 2050 2000 1950
0
5
10
15
0
5
10
15
0
5
10
15
20
25
30
35
20
25
30
35
20
25
30
35
0.01015
x 4A
0.0101 0.01005 0.01 0.00995
Time (hr)
0.104
x TD
0.102 0.1 0.098
Figure S24.3b. Step change in w4 (+100) at t=5 (Corresponds to Fig.24.8)
24-14
24.4 a)
A simple modification of the controller pairing is needed. The settings for the modified controller setup are: Loop xTD-w6 HT-R w4-w1 x4A-w8
Gain Integral Time (hr) -5000 1 0.002 1 2 1 -500000 1
(See Figure S24.4a) b)
The RGA shows that flowrate w6 will not directly affect the composition of D in the recycle tank, xTD, but the xTD-w6 loop will cause unwanted interaction with the other control loops. The system can be controlled, however, if the other three loops are tuned more conservatively and “assist” the xTD-w6 loop.
c)
The manipulated variable, w6, is the rate of purge flow. Purging a stream does not affect the compositions of its constituent species, only the total flowrate. Therefore, purging the stream before the recycle tank will only affect the level in the tank and not its compositions. The resulting RGA yields a zero gain between xTD and w6.
d)
The RGA structure handles a positive 5% step change in the production rate well, as it maintains the plant within the specified limits. The setup with one open feedback loop defined by this exercise, however, goes out of control. The xTD-w6 loop requires the interaction of the other loops to maintain stability. When the x4A-w8 loop is broken, the system will no longer remain stable. (See Figure S24.4b)
e)
With a set point change of 10%, the controllers must be detuned to keep variables within operating constraints. The HT-w6 loop in the RGA structure must be more conservative (gain reduced to -1) to keep the purge flow, w6, from hitting its lower constraint, zero. A 20% change will create a problem within the system that these control structures cannot handle. The new set point for w4 does not allow a steady-state value of 0.01 for x4A. This will make the x4A-w8 control loop become unstable. This outcome results for production rate step changes larger than roughly 12% (for this system). (See Figure S24.4c)
24-15
1060
2100
1040
2050
w4
2150
w1
1080
1020 1000
Exercise 24.4 RGA structure
2000
0
10
20
30
40
1950
1180
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
520
1160
510 HT
1140 w2
500
1120 490
1100 1080
0
10
20
30
480
40
140
0.102
x TD
120
w6
100 80
0.1
0.098
60 40
0
10
20
30
40
0.096
0.0102
1200 1100 w8
x 4A
0.0101 1000
0.01 900 800
0.0099 0
10
20
30
40
Time (hr)
Time (hr)
Figure S24.4a. Step change in w4 set point(+5%) at t=5
24-16
1100
2100
1050
2050
w4
2150
w1
1150
1000 950
2000
0
5
10
15
20
1950
600
1300
550
0
5
10
15
20
0
5
10
15
20
0
5
10
15
20
0
5
10
15
20
w2
HT
1400
1200
500
1100 1000
Exercise 24.4 RGA structure
450
0
5
10
15
400
20
160
0.115 0.11 x TD
140
0.105
w6
120
0.1 100 80
0.095 0
5
10
15
0.09
20
891
0.012
890.5 x 4A
w8
0.011 890
0.01 889.5 889
0
5
10
15
20
Time (hr)
0.009
Time (hr)
Figure S24.4b. Step change in w4 set point(+5%) at t=5 with one loop open.
24-17
1150
w1
1100
1050
1000
0
20
40
60
1250
540 520
1200 w2
500 1150 480 1100 1050
460
0
20
40
60
0
20
40
60
440
0
20
40
60
0
20
40
60
150
w6
100
50
0
0.0104
1400
0.0103
w8
x 4A
1600
1200
0.0102
1000 800
0.0101
0
20
40
60
Time (hr)
0.01
Time (hr)
24-18
24.5 a)
Gain Matrix w4 x8D x4A HT HR
Using the same methods as described in solution 24.3, the resulting gain matrix is: w1
w2
5.762E-3 4.760E-3 5.554E-6 4.558E-6 -2.905E-6 -2.398E-6 -2.137 -1.927 1 1 All variables are integrating
w6
w8
w3
0 0 0 -1 0
5.831E-3 5.445E-6 -2.944E-6 -10.12 1
-1.285E-2 -9.130E-6 6.542E-6 8.829 -1
The resulting RGA does not provide useful insight for the preferred controller pairing due to the nature of these integrating variables. b)
Results similar to those obtained in Exercise 24.3 can be obtained with an added loop for reactor level using the w3 flow rate as the manipulated variable. Both P and PI controllers yield relatively constant reactor level. The quality variable, x4A, cannot be controlled as tightly however. The responses with P-only control are only slightly different as compared to PI control, which means that zero-offset control on the reactor volume is not necessary for reliable plant operation. Controller parameters used for variable reactor holdup simulation: Loop w4-w1 xTD-w2 x4A-w8 HT-w6 HR-w3
Gain (Kc) 123456789 49 I) 1 1 -6300 1 -200000 1 -3.5 1 -10 1* * For PI control
24-19
2120 Variable reactor level (P) Variable reactor level (PI) Constant reactor level
2100 2080
w4
2060 2040 2020 2000 1980
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
20
25
30
35
0.104
0.103
x TD
0.102
0.101
0.1
0.099
Time (hr)
0.0108 0.0106
x 4A
0.0104 0.0102 0.01 0.0098
0
5
10
15
Figure S24.5a. Step change in w4 set point (+100) at t=5
24-20
24.6 To simulate the flash/splitter with a non-negligible holdup, derive a mass balance around the unit. Assume that components A and C are well mixed and are held up in the flash for an average HF/w4 amount of time. Also assume that the vapor components B and D are passed through the splitter instantaneously. dH F = w3 − w4 − w5 = 0 dt d ( H F xFA ) = w3 x3 A − w4 x4 A dt d ( H F xC ) = w3 x3C − w4 x4C dt Since the holdup is constant, the flows out of the splitter can be modeled as: w5 = w3 ( x3 B + x3 D ) w4 = w3 − w5 Use the component balances and output flow equations to simulate the flash/splitter unit. This will add a dynamic lag to the unit which slows down the control loops that have the splitter in between the manipulated variable and the controlled variable. However, a 1000kg holdup only creates a residence time of 0.5 hr. Considering the time scale of the entire plant, this is very small and confirms the assumption of modeling the flash/splitter as having a negligible holdup.
24-22
1014
2001
1010
2000
w4
2000.5
w1
1012
1008 1006
Negligible holdup flash 1000 kg holdup flash
1999.5
0
10
20
30
40
1200
1999
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
0.108 0.106
1150
x TD
w2
0.104 0.102
1100
0.1 1050
0
10
20
30
40
0.098
510
200
505
w6
HT
250
150
100
500
0
10
20
30
40
495
0.01
900
0.01 w8
x 4A
850 0.01
800 0.01 750
0.01 0
10
20
30
40
Time (hr)
Figure S24.6. Step change in x2D (+0.005) at t=5
24-23
24.7 The MPC controller achieves satisfactory results for step changes made within the plant. The production rate can easily be maintained within desirable limits and large set-point changes (20%) do not cause a breakdown in the quality of this stream. A change in the kinetic coefficient (k), occurring simultaneously with a 50% disturbance change will, however, initially draw the product quality (composition of the production stream) out of the required limits.
0.6
0.5
0.4 0.2
w4
w1
0
-0.5 0
0
10
20
30
40
-1
50
8
1.5
6
1 x4A
w2
-0.2
4
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
0
10
20
30
40
50
-0.5
2
0
0
-0.2
-2 -0.4
HT
w6
10
0.5
2 0
0
-4
-0.6
-6 -8
0
10
20
30
40
50
-0.8
60
0
40 xTD
w8
-2 20
-4 0 -20
0
10
20
30
40
50
-6
Figure S24.7a. Step change in x2D (+50%). All variables are recorded in percent deviation.
24-24
6
-3
4 w4
w1
-2
-4
-5
2
0
10
20
30
40
0
50
2
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
1.5
0
1
-2 x4A
w2
0
-4
0
-6 -8
0.5
0
10
20
30
40
50
-0.5
0.1 HT
-5 -10
0
w6
-0.1 -15 -0.2 -20
0
10
20
30
40
50
-0.4
15
0.5
10
0 xTD
w8
-25
-0.3
5
-0.5
0 -5
-1
0
10
20
30
40
50
-1.5
Figure S24.7b. Step change in production rate w4 (+5%). All variables are recorded in percent deviation.
24-25
1
6
0
4
-1
w4
w1
8
2 0
-2
0
10
20
30
40
-3
50
15
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
20 15
10
x4A
w2
10 5 0 -5
0
0
10
20
30
40
-5
50
20
1
0
0.5
-20 HT
w6
5
-40
-0.5
-60 -80
0
0
10
20
30
40
-1
50
150
0
100 xTD
w8
-5 50
-10 0 -50
0
10
20
30
40
50
-15
Figure S24.7c. Simultaneous step changes in x2D (+50%) and k(+20%). All variables are recorded in percent deviation.
24-26
-5
25 20
-10 w1
w4
15 10
-15
5 -20
0
10
20
30
40
0
50
10
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
6
0 w2
4 x4A
-10 -20
0
-30
0
10
20
30
40
-2
50
-20
0.5
-40
0 HT
w6
-40
2
-60
-0.5
-80
0
10
20
30
40
50
-1.5
60
2
40
0 xTD
w8
-100
-1
20 0
-20
-2 -4
0
10
20
30
40
50
-6
Figure S24.7d. Step change in production rate w4 (+20%). All variables are recorded in percent deviation.
24-27
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