Process Dynamics and Control, Ch. 24 Solution Manual

8/13/2019 Process Dynamics and Control, Ch. 24 Solution Manual

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24-1

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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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24-5

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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24-7

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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24-9

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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24-10

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


Figure S24.1i. Step change in w 6 (+10) at t=5


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24-11

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


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


Figure S24.1k. Step change in x 2D (+0.005) at t=5


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24-13

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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24-14

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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24-15

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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24-18

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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24-19

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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24-22

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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24-23

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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24-24

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.

Documents

Process Dynamics and Control, Ch. 24 Solution Manual