process dynamics and control, ch. 24 solution manual
TRANSCRIPT
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24-1
1234567 9
24.1
a) i. First model (full compositions model):
Number of variables: N V = 22
w1 x R,A x2 B
w2 x R,B x2 D
w3 x R,C x4C
w4 x R,D x5 D w5 xT,D x6 D
w6 V T x7 D
w7 H T x8 D
w8
Number of Equations: N E = 17Eqs. 2-8, 9, 10, 12, 13, 15, 16, 18, 20(3X), 21, 22, 27, 28, 29, 31
Number of Parameters: N P = 4V R , k, ,
Degrees of freedom: N F = 22 17 = 5
Number of manipulated variables: N MV = 4
w1, w2, w6, w8
Number of disturbance variables: N DV = 1
x2 D
Number of controlled variables: N CV = 4
x4 A, w4, H T , x8 D
Solution Manual for Process Dynamics and Control, 2 nd edition,Copyright 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp
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24-4
Figure S24.1a. Simulink-MATLAB block diagram for first model
2
x3
1w3
[-0.5 -0.5 1 0]
stoich. factors
3000
rho*VR
330
k
generation &Accumulation
T2
T1
Suminput flows1
Suminput flows
Mux
3000
HR
m
Demux
1s
D
1s
C
1s
B
1s
A
-out+ in
+generated
6
w2
5
x2
4
w1
3
x1
2x8
1
w8
Figure S24.1b. Simulink-MATLAB block diagram for the reactor block
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4
x4
3
w4
2
x51
w5
Product5
Product4
Product3
Product2
Product1
Productm2
x31
w3
Figure S24.1c. Simulink-MATLAB block diagram for the flash block
2x7
1
w7
3x5
2w5
1
Purge Flow
Figure S24.1d. Simulink-MATLAB block diagram for purge block
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24-6
xTD
xTD-save
xRD
xRD-save
xRB
xRB-save
xRA
xRA-save
x4A
x4A-save
0.01
x2D-set
x2D
w8
w8-save
w6
w6-savew4
w4-save
w2
w2-save
w2-change
1100
w2
w1
w1-save
w1-change
1010
w1
HT
VT-save
890
Recycle Flow
110
Purge Flow
Equations 2-(32-38)
E.2-32-38
Figure S24.1e. Simulink-MATLAB block diagram for second model
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xTD
xTD-save
x4A
x4A-save
0.01
x2D
w8
w8-save
w6
w6-save
w4
w4-save
w2
w2-save
1100
w2
w1
w1-save
1010
w1
890
Recycle Flow
110
Purge Flow
HTD
HTD-save
HTB
HTB-save
HT
HT-save
HRD
HRD-save
HRB
HRB-save
HRA
HRA-save
Equations 2-(48-52)
E.2-48-52
Figure S24.1f. Simulink-MATLAB block diagram for third model
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0 10 20 30 401009
1009.5
1010
1010.5
1011
w 1
0 10 20 30 401100
1102
1104
1106
1108
1110
w 2
0 10 20 30 40109
109.5
110
110.5
111
w 6
0 10 20 30 40889
889.5
890
890.5
891
w 8
Time (hr)
0 10 20 30 409.85
9.9
9.95
10
10.05x 10
-3
x 4 A
Time (hr)
0 10 20 30 40500
600
700
800
H T
0 10 20 30 402000
2002
2004
2006
2008
w 4
0 10 20 30 400.092
0.094
0.096
0.098
0.1
0.102
x T D
Full modelReduced concentrationsReduced holdups
Figure S24.1h. Step change in w 2 (+10) at t=5
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0 10 20 30 401009
1009.5
1010
1010.5
1011
w 1
0 10 20 30 401099
1099.5
1100
1100.5
1101
w 2
0 10 20 30 40110
112
114
116
118
120
w 6
0 10 20 30 40889
889.5
890
890.5
891
w 8
Time (hr)
0 10 20 30 400.01
0.01
0.01
0.01
x 4 A
Time (hr)
0 10 20 30 40100
200
300
400
500
H T
0 10 20 30 402000
2000
2000
2000
2000
2000
w 4
0 10 20 30 400.1
0.1
0.1
0.1
0.1
x T D
Full modelReduced concentrationsReduced holdups
Figure S24.1i. Step change in w 6 (+10) at t=5
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0 10 20 30 401009
1009.5
1010
1010.5
1011
w 1
0 10 20 30 401099
1099.5
1100
1100.5
1101
w 2
0 10 20 30 40109
109.5
110
110.5
111
w 6
0 10 20 30 40890
892
894
896
898
900
w 8
Time (hr)
0 10 20 30 409.96
9.97
9.98
9.99
10
10.01x 10
-3
x 4 A
Time (hr)
0 10 20 30 40490
492
494
496
498
500
H T
0 10 20 30 402000
2002
2004
2006
2008
w 4
0 10 20 30 400.1
0.1
0.1
0.1001
0.1001
x T D
Full modelReduced concentrationsReduced holdups
Figure S24.1j. Step change in w 8 (+10) at t=5
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0 10 20 30 401009
1009.5
1010
1010.5
1011
w 1
0 10 20 30 401099
1099.5
1100
1100.5
1101
w 2
0 10 20 30 40109
109.5
110
110.5
111
w 6
0 10 20 30 40889
889.5
890
890.5
891
w 8
Time (hr)
0 10 20 30 40
0.01
0.0105
x 4 A
Time (hr)
0 10 20 30 40500
505
510
515
520
H T
0 10 20 30 401999
1999.5
2000
2000.5
w 4
0 10 20 30 400.1
0.12
0.14
0.16
x T D
Full modelReduced concentrationsReduced holdups
Figure S24.1k. Step change in x 2D (+0.005) at t=5
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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 thetable below). The values calculated are:
Gain Matrix w1 w2 w6 w8 w4 1.93 2.26 E-2 0 6.25 E-3
x8 D 8.8 E-4 -7.62 E-4 0 5.68 E-6 x4 A 2.57 E-5 -1.14 E-5 0 -3.15 E-6 H T -0.918* 0.973* -1* -6.25 E-3** For integrating variables: Gain = the slope of the variable vs. timedivided by the magnitude of the step change.
RGA w1 w2 w6 w8 w4 0.9743 0.0135 0 0.0122
x8 D 0 0.9737 0 0.0263 x4 A 0.0257 0.0128 0 0.9615 H T 0 0 1 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 loopto slow down the fast algebraic equations, which otherwise yield largeoutput spikes. These blocks are of the form of a first-order filter.:
1( )
0.001 1 f G s
s=
+
In principle, ratio control can provide tighter control of all variables.However, it is clear from the x2 D results that it offers no advantage forthis disturbance variable. For a step change in production rate, w4, onewould anticipate a different situation because a change in manipulated
variable w1 is induced. Using ratio control, w2 does change along with w 1 to maintain a satisfactory ratio of the two feed streams. Thus, ratiocontrol does provide enhanced control for the recycle tank level, H T, andcomposition, xTD, but not for the key performance variables, w4 and x4 A.This characteristic is likely a result of the particular features of therecycle plant, namely the use of a splitter (instead of a flash unit) and thelack of holdup in that vessel.
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0 5 10 15 20 25 30 350.0099
0.01
0.0101
x 4 A
Time (hr)
0 5 10 15 20 25 30 351996
1998
2000
2002
2004
w 4
No ratio controlRatio control
0 5 10 15 20 25 30 350.09
0.1
0.11
0.12
x T D
Figure S24.3a. Step change in x 2D (+0.02) at t=5 (Corresponds to Fig.24.7)
0 5 10 15 20 25 30 350.00995
0.01
0.01005
0.0101
0.01015
x 4 A
Time (hr)
0 5 10 15 20 25 30 351950
2000
2050
2100
2150
w 4
No ratio controlRatio control
0 5 10 15 20 25 30 350.098
0.1
0.102
0.104
x T D
Figure S24.3b. Step change in w 4 (+100) at t=5 (Corresponds to Fig.24.8)
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24.4
a) A simple modification of the controller pairing is needed.The settings for the modified controller setup are:
Loop Gain Integral Time (hr) xTD-w6 -5000 1 H T - R 0.002 1w4-w1 2 1
x4 A-w8 -500000 1
(See Figure S24.4a)
b) The RGA shows that flowrate w6 will not directly affect the compositionof 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 andassist the xTD-w6 loop.
c) The manipulated variable, w6, is the rate of purge flow. Purging a streamdoes not affect the compositions of its constituent species, only the totalflowrate. Therefore, purging the stream before the recycle tank will onlyaffect the level in the tank and not its compositions. The resulting RGAyields 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 setupwith one open feedback loop defined by this exercise, however, goes outof control. The xTD-w6 loop requires the interaction of the other loops tomaintain stability. When the x4 A-w8 loop is broken, the system will nolonger remain stable.
(See Figure S24.4b)
e) With a set point change of 10%, the controllers must be detuned to keepvariables within operating constraints. The H T -w6 loop in the RGAstructure must be more conservative (gain reduced to -1) to keep thepurge flow, w
6, from hitting its lower constraint, zero. A 20% change
will create a problem within the system that these control structurescannot handle. The new set point for w4 does not allow a steady-statevalue of 0.01 for x4 A. This will make the x4 A-w8 control loop becomeunstable. This outcome results for production rate step changes largerthan roughly 12% (for this system).
(See Figure S24.4c)
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0 20 40 601000
1050
1100
1150
w 1
0 20 40 601050
1100
1150
1200
1250
w 2
0 20 40 600
50
100
150
w 6
0 20 40 60800
1000
1200
1400
1600
w 8
Time (hr)
0 20 40 600.01
0.0101
0.0102
0.0103
0.0104
x 4 A
Time (hr)
0 20 40 60440
460
480
500
520
540
H T
0 20 40 601900
2000
2100
2200
2300
w 4
0 20 40 600.094
0.096
0.098
0.1
0.102
0.104
x T D
Exercise 24.4RGA structure
Figure S24.4c. Step change in w 4 set point(+10%) at t=5 with H T controllerdetuned
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24.5
a) Using the same methods as described in solution 24.3, the resulting gainmatrix is:
GainMatrix w1 w2 w6 w8 w3
w4 5.762E-3 4.760E-3 0 5.831E-3 -1.285E-2 x8 D 5.554E-6 4.558E-6 0 5.445E-6 -9.130E-6 x4 A -2.905E-6 -2.398E-6 0 -2.944E-6 6.542E-6 H T -2.137 -1.927 -1 -10.12 8.829 H R 1 1 0 1 -1
All variables are integrating
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 withan added loop for reactor level using the w3 flow rate as the manipulatedvariable. Both P and PI controllers yield relatively constant reactor level.The quality variable, x4A, cannot be controlled as tightly however. Theresponses with P-only control are only slightly different as compared toPI control, which means that zero-offset control on the reactor volume isnot necessary for reliable plant operation.
Controller parameters used for variable reactor holdup simulation:
Loop Gain ( K c) 12345678 4 I )w4-w1 1 1
xTD-w2 -6300 1 x4 A-w8 -200000 1 H T -w6 -3.5 1 H R-w3 -10 1*
* For PI control
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0 5 10 15 20 25 30 350.0098
0.01
0.0102
0.0104
0.0106
0.0108
x 4 A
0 5 10 15 20 25 30 351980
2000
2020
2040
2060
2080
2100
2120
w 4
0 5 10 15 20 25 30 350.099
0.1
0.101
0.102
0.103
0.104
x T D
Time (hr)
Variable reactor level (P)Variable reactor level (PI)Constant reactor level
Figure S24.5a. Step change in w 4 set point (+100) at t=5
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24-21
0 10 20 30 40900
1000
1100
1200
w 1
0 10 20 30 401050
1100
1150
1200
w 2
0 10 20 30 400
50
100
150
w 6
0 10 20 30 40800
1000
1200
1400
1600
w 8
0 10 20 30 402800
3000
3200
3400
w 3
Time (hr)
0 10 20 300.0095
0.01
0.0105
0.011
x 4 A
0 10 20 30 40480
490
500
510
H T
0 10 20 30 401950
2000
2050
2100
2150
w 4
0 10 20 30 400.098
0.1
0.102
0.104
x T D
0 10 20 30 402990
3000
3010
3020
3030
H R
Time (hr)
Variable reactor level (P)Variable reactor level (PI)
Constant reactor level
Figure S24.5b. Step change in w 4 setpoint (+100) at t=5
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24.6
To simulate the flash/splitter with a non-negligible holdup, derive a massbalance around the unit. Assume that components A and C are well
mixed and are held up in the flash for an average H F / w4 amount of time.Also assume that the vapor components B and D are passed through thesplitter instantaneously.
3 4 5
3 3 4 4
3 3 4 4
0
( )
( )
F
F FA A A
F C C C
dH w w w
dt d H x
w x w xdt
d H xw x w x
dt
= =
=
=
Since the holdup is constant, the flows out of the splitter can be modeledas:
5 3 3 3
4 3 5
( ) B Dw w x x
w w w
= +
=
Use the component balances and output flow equations to simulate theflash/splitter unit. This will add a dynamic lag to the unit which slowsdown the control loops that have the splitter in between the manipulatedvariable and the controlled variable. However, a 1000kg holdup onlycreates a residence time of 0.5 hr. Considering the time scale of the
entire plant, this is very small and confirms the assumption of modelingthe flash/splitter as having a negligible holdup.
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0 10 20 30 401006
1008
1010
1012
1014
w 1
0 10 20 30 401050
1100
1150
1200
w 2
0 10 20 30 40100
150
200
250
w 6
0 10 20 30 40750
800
850
900
w 8
0 10 20 30 400.01
0.01
0.01
0.01
0.01
x 4 A
Time (hr)
0 10 20 30 40495
500
505
510
H T
0 10 20 30 401999
1999.5
2000
2000.5
2001
w 4
0 10 20 30 400.098
0.1
0.102
0.104
0.106
0.108
x T D
Negligible holdup flash1000 kg holdup flash
Figure S24.6. Step change in x 2D (+0.005) at t=5
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24.7
The MPC controller achieves satisfactory results for step changes madewithin the plant. The production rate can easily be maintained within
desirable limits and large set-point changes (20%) do not cause abreakdown in the quality of this stream. A change in the kineticcoefficient ( k ), occurring simultaneously with a 50% disturbance changewill, however, initially draw the product quality (composition of theproduction stream) out of the required limits.
0 10 20 30 40 50-0.2
0
0.2
0.4
0.6
w 1
0 10 20 30 40 50-1
-0.5
0
0.5
w 4
0 10 20 30 40 500
2
4
6
8
w 2
0 10 20 30 40 50-0.5
0
0.5
1
1.5
x 4 A
0 10 20 30 40 50-8
-6
-4
-2
0
2
w 6
0 10 20 30 40 50-0.8
-0.6
-0.4
-0.2
0
H T
0 10 20 30 40 50-20
0
20
40
60
w 8
0 10 20 30 40 50-6
-4
-2
0
x T D
Figure S24.7a. Step change in x 2D (+50%). All variables are recorded in
percent deviation.
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0 10 20 30 40 50-5
-4
-3
-2
w 1
0 10 20 30 40 500
2
4
6
w 4
0 10 20 30 40 50-8
-6
-4
-2
0
2
w 2
0 10 20 30 40 50-0.5
0
0.5
1
1.5
x 4 A
0 10 20 30 40 50-25
-20
-15
-10
-5
w 6
0 10 20 30 40 50-0.4
-0.3
-0.2
-0.1
0
0.1
H T
0 10 20 30 40 50-5
0
5
10
15
w 8
0 10 20 30 40 50-1.5
-1
-0.5
0
0.5
x T D
Figure S24.7b. Step change in production rate w4 (+5%). All variables are
recorded in percent deviation.
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24-26
0 10 20 30 40 500
2
4
6
8
w 1
0 10 20 30 40 50-3
-2
-1
0
1
w 4
0 10 20 30 40 50-5
0
5
10
15
w 2
0 10 20 30 40 50-5
0
5
10
15
20
x 4 A
0 10 20 30 40 50-80
-60
-40
-20
0
20
w 6
0 10 20 30 40 50-1
-0.5
0
0.5
1
H T
0 10 20 30 40 50-50
0
50
100
150
w 8
0 10 20 30 40 50-15
-10
-5
0
x T D
Figure S24.7c. Simultaneous step changes in x 2D (+50%) and k(+20%). All
variables are recorded in percent deviation.
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0 10 20 30 40 50-20
-15
-10
-5
w 1
0 10 20 30 40 500
5
10
15
20
25
w 4
0 10 20 30 40 50-40
-30
-20
-10
0
10
w 2
0 10 20 30 40 50-2
0
2
4
6
x 4 A
0 10 20 30 40 50-100
-80
-60
-40
-20
w 6
0 10 20 30 40 50-1.5
-1
-0.5
0
0.5
H T
0 10 20 30 40 50-20
0
20
40
60
w 8
0 10 20 30 40 50-6
-4
-2
0
2
x T D
Figure S24.7d. Step change in production rate w 4 (+20%). All variables arerecorded in percent deviation.