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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

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30

40

0

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30

40

0

10

20

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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

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30

40

0

10

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30

40

0

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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

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50

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50

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50

0

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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

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30

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50

0

10

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30

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50

0

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0

10

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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

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40

50

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50

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50

0

10

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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

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50

10

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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

View more...
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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