23-1

��������� 23.1 Option (a):

• Production rate set via setpoint of wA flow controller • Level of R1 controlled by manipulating wC • Ratio of wB to wA controlled by manipulating wB • Level of R2 controlled by manipulating wE • Ratio of wD to wC controlled by adjusting wD

Options (b)-(e) are developed similarly. See table below.

Option Production

Rate Set With Control Loop #

Type of Controller

Controlled Variable

Manipulated Variable

a wA 1 Flow wA,m wA (V1) wA 2 Ratio wB,m wB (V2) wA 3 Level HR1 wC (V3) wA 4 Ratio wD,m wD (V4) wA 5 Level HR2 wE (V5)

b wB 1 Flow wB,m wB (V2) wB 2 Ratio wA,m wA (V1) wB 3 Level HR1 wC (V3) wB 4 Ratio wD,m wD (V4) wB 5 Level HR2 wE (V5) c wC 1 Flow wC,m wC (V3) wC 2 Ratio wB,m wB (V2) wC 3 Level HR1 wA (V1) wC 4 Ratio wD,m wD (V4) wC 5 Level HR2 wE (V5)

d wD 1 Flow wD,m wD (V4) wD 2 Ratio wC,m wC (V3) wD 3 Level HR1 wA (V1) wD 4 Ratio wB,m wB (V2) wD 5 Level HR2 wE (V5) e wE 1 Flow wE,m wE (V5) wE 2 Ratio wD,m wD (V4) wE 3 Level HR2 wC (V3) wE 4 Ratio wB,m wB (V2) wE 5 Level HR1 wA (V1)

• Subscript m denotes “measurement”.

Solution Manual for Process Dynamics and Control, 2nd edition,

Copyright © 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp

23-3

23.2

a) The level in the distillate (HD) will be controlled by manipulating the recycle flow rate (D), and the level in the reboiler (HB) via the bottoms flow rate (B).

Thus, HD and HB are pure integrator elements.

Closed-loop TF development assuming PI controller:

2

1( )

( ) 1

ICL

II

c p

sG s

s sK K

τ += τ + τ +

Relations:

2I

c pK K

τ = τ 2Iτ = ζτ 1pK = − for both HD and HB loops.

����� = 1 for a critically damped response Initial settings:

0.4

10c

I

K = −τ =

Final tuning: changed to proportional control only to obtain a faster response

1cK = −

b) The distillate composition (xD) will be controlled by manipulating the reflux flow rate (R), and the bottoms composition (xB) via the vapor boilup (V). Use a step response to determine an approximate first-order model (calculations are shown on last page).

0.0012

2.33 10.000372

2.08 1

D

B

x

R sx

V s

=+

−=+

23-4

Using the Direct Synthesis method:

( )1

1( ) (1 )c

c

KG s

s

G sK s

=τ +

τ= +τ τ

Choosing 1

4cτ = τ , the settings are:

xD - R loop {3333.3

2.33c

I

K =τ =

xB - V loop {10649.6

2.08c

I

K = −τ =

c) The reactor level (HR) will be controlled by manipulating the flow from the

reactor (F).

HR is a pure integrator element. Using the same relations as in part a, initial controller settings are:

0.4

10c

I

K = −τ =

After tuning:

1

5c

I

K = −τ =

23-5

Figure S23.2a. Simulink-MATLAB block diagram for case a)

2

z

1

HR

flow sumSum-z

1s

Level integrator

KRz/HR

0.3408

KR

1s

Integrator z

F0z0/HR

F0z/HR

Dz/HR

DxD/HR

5

F

4

z0

3

F0

2

xD1 D

Figure S23.2b. Simulink-MATLAB block diagram for the CSTR block

23-6

4

HB

3

xB

2

xD

1

HD

0

xi

R

V

Hs

D

y (i+1)

xD

x(i-1)

HD

Top 13-20

23.5

Hs

x(i-1)

L

F

z

V

LF

H

y (i+1)

y i

xi

FeedTray 12

x(i-1)

Hs

R

F

V

B

y i

xB

HB

Bottom 1-11

6

B

5

V

4

z

3

F

2

R

1

D

23-7

0 5 10 15 20 25 30180

200

220

240

260

280

300

HD

(lb

-mol

)

0 5 10 15 20 25 30160

180

200

220

240

260

280

HB

(lb

-mol

)time (h)

0 5 10 15 20 25 30500

520

540

560

580

600

D (

lb-m

ol/h

r)

0 5 10 15 20 25 30360

380

400

420

440

460

B (

lb-m

ol/h

r)

time (h)

Figure S23.2d. Step change in V (1600-1700) at t=5

23-8

0 10 20 300.94

0.945

0.95

0.955

xD (

mol

e fr

actio

n A

)0 10 20 30

0.01

0.012

0.014

0.016

0.018

0.02

xB (

mol

e fr

actio

n A

)time (h)

0 10 20 30180

200

220

240

260

HD

(lb

-mol

)

0 10 20 30260

280

300

320

340

HB

(lb

-mol

)

time (h)

0 10 20 301050

1100

1150

1200

1250

R (

lbm

ol/h

r)

0 10 20 301550

1600

1650

1700

1750

1800

V (

lbm

ol/h

r)

0 10 20 30500

520

540

560

D (

lb-m

ol/h

r)

0 10 20 30440

460

480

500

520

B (

lb-m

ol/h

r)

time (h)

Figure S23.2e. Step change in F (960-1060) at t=5

23-10

* Calculation of First-Order Model Parameters for xD and xB Loops xD -R:

A step change in the reflux rate (R) of +10 lbmol/hr is made and the resulting response is used to fit a first-order model:

( )10.9624 0.950

0.001210

D

KG s

sx

KR

=τ +

∆ −= = =∆

�������� ���������� ����� �������

0.632( ) (0.632)(0.012) 0.007584Dx∆ = =

( 0.957584) 12.33 10 2.33Dtime xτ = = = − =

xB -V:

Similarly, a step change in the vapor boilup (V) of +10 lbmol/hr is made:

0.00678 0.01050.00372

10Bx

KV

∆ −= = = −∆

Use 63.2% of the response to find � 0.632( ) (0.632)( 0.00372) 0.00235Bx∆ = − = −

( 0.00815) 12.08 10 2.08Btime xτ = = = − =

0 5 10 15 20 250.948

0.95

0.952

0.954

0.956

0.958

0.96

0.962

0.964

0.966

Time (hr)

x D (

mol

fra

ctio

n A

)

Figure S23.2g. Responses for step change in the reflux rate R

23-11

0 5 10 15 20 256.5

7

7.5

8

8.5

9

9.5

10

10.5

11x 10

-3

Time (hr)

x B (

mol

fra

ctio

n A

)

Figure S23.2h. Responses for step change in the vapor boilup V.

23-12

23.3 The same controller parameters are used from Exercise 23.2

Figure S23.3a. Simulink-MATLAB block diagram

23-15

23.4

a) The flow controller on F, the column feed stream, should be simulated in MATLAB as a constant flow. The controller parameters used are taken from those derived in Exercise 23.2.

0 10 20 30 40 500.948

0.95

0.952

xD (

mol

e fr

actio

n A

)

0 10 20 30 40 500.01

0.0105

0.011

xB (

mol

e fr

actio

n A

)

0 10 20 30 40 50120

140

160

180

200

HD

(lb

-mol

)

0 10 20 30 40 50260

280

300

320

340

HB

(lb

-mol

)

0 10 20 30 40 502400

2600

2800

3000

HR

(lb

-mol

)

time (h)

0 10 20 30 40 501080

1100

1120

1140

R (

lbm

ol/h

r)

0 10 20 30 40 501580

1590

1600

1610

V (

lbm

ol/h

r)

0 10 20 30 40 50440

460

480

500

520

D (

lb-m

ol/h

r)

0 10 20 30 40 50440

460

480

500

520

B (

lb-m

ol/h

r)

0 10 20 30 40 50460

480

500

520

F0

(lb-m

ol/h

r)

time (h)

Figure S23.4a. Step change in F0 (+10%) at t=5

23-16

b) Using the approximate relation (23-17), a +10% step change in F0 will result in a reactor holdup of:

0

0

0.902800

1 1 1 1( ) 0.34( )

506 960

R

R

zH

kF F

≈ = =− −

lbmol

Using the exact relation (23-6, rearranged):

0 0 (506)(0.9 0.0105)2910

(0.34)(0.455)B

R

R

F z BxH

k z

− −= = = lbmol

The value taken from the graph (2910) matches up with the expected value from the equation without the approximation.

0 5 10 15 20 25 30 35 40 45 502400

2500

2600

2700

2800

2900

3000

HR

(lb

-mol

)

time (h)

Figure S23.4b. Step change in F0 (+10%) at t=5

23-17

23.5 a),b) Feedforward control is implemented using the HR-setpoint equation:

0

0

( )( )

1 1( )

( )

R

R

z tH t

kF t F

≈−

Empirical adjustment of the feedforward equation is required because it is

not exact:

0

0

( )( ) 70

1 1( )

( )

R

R

z tH t

kF t F

≈ +−

This adjustment matches the initial values of HR (i.e., with and without

feedforward control).

Parts a and b are represented graphically.

23-18

0 10 20 30

0.95

xD (

mol

e fr

actio

n A

)

no controlFF control

0 10 20 300.0095

0.01

0.0105

0.011

xB (

mol

e fr

actio

n A

)

0 10 20 30160

170

180

190H

D (

lb-m

ol)

0 10 20 30270

280

290

300

HB

(lb

-mol

)

0 10 20 302100

2200

2300

2400

2500

HR

(lb

-mol

)

time (h)

0 10 20 301090

1100

1110

1120R

(lb

mol

/hr)

0 10 20 301585

1590

1595

1600

1605

V (

lbm

ol/h

r)

0 10 20 30470

480

490

500

510

D (

lb-m

ol/h

r)

0 10 20 30450

460

470

480

490

B (

lb-m

ol/h

r)

0 10 20 30200

300

400

500

F0 (

lb-m

ol/h

r)

time (h)

Figure S23.5a. Step change in z0 (-10%) at t=5

23-19

c) The controlled plant response is much faster with the feedforward controller (~10 hours settling time versus ~20 hours without it).

d) Advantage: Faster response.

Disadvantage: Have to measure or estimate two flow rates and one concentration, therefore significantly more expensive.

23.6

a) Use a flow controller to keep F constant (make F a constant in the simulation).

b) Use ratio control to set F. The ratio should be based on the initial steady state values (960/460). Therefore, as F0 changes, F will be controlled to the corresponding value set by the ratio.

Parts a) and b) show very different results for the two alternatives. With alternative # 3, feedforward control is necessary to keep the level in the distillate receiver from integrating. However, in alternative # 4, the control structure without the feedforward loop is superior to that with feedforward control.

Responses are displayed with controlled variables adjacent to their corresponding manipulated variable.

23-20

0 20 40 600.48

0.5

0.52

z (m

ole

frac

tion

A)

0 20 40 600.01

0.012

0.014

0.016

x B (

mol

e fr

actio

n A

)

0 20 40 600

500

1000

1500

HD (

lb-m

ol)

0 20 40 60250

300

350

HB (

lb-m

ol)

0 20 40 602200

2400

2600

2800

HR (

lb-m

ol)

0 20 40 60400

500

600

D (

lbm

ol/h

r)

0 20 40 601500

2000

2500

3000

V (

lbm

ol/h

r)

0 20 40 601000

1500

2000

2500

R (

lb-m

ol/h

r)

0 20 40 60450

500

550

B (

lb-m

ol/h

r)

0 20 40 60950

1000

1050

1100

F (

lbm

ol/h

r)

0 20 40 60460

480

500

520

F0 (

lbm

ol/h

r)

time (h)

0 20 40 600.9

0.95

1

x D (

lbm

ol/h

r)

time (h)

No control (a)FF control (b)

Figure S23.6a. Alternative #3 (with and without FF controller). Step change in F0

(+10%) at t=10

23-21

0 20 40 600.94

0.96

0.98

x D (

mol

e fr

actio

n A

)

0 20 40 600.005

0.01

0.015

0.02

x B (

mol

e fr

actio

n A

)

0 20 40 60100

200

300

400

HD (

lb-m

ol)

0 20 40 60250

300

350

HB (

lb-m

ol)

0 20 40 602000

2500

3000

HR (

lb-m

ol)

0 20 40 60450

500

550

600

D (

lbm

ol/h

r)

0 20 40 601400

1600

1800

2000

V (

lbm

ol/h

r)

0 20 40 601000

1100

1200

1300

R (

lb-m

ol/h

r)

0 20 40 60450

500

550

B (

lb-m

ol/h

r)

0 20 40 60950

1000

1050

1100

F (

lb-m

ol/h

r)

0 20 40 60460

480

500

520

F0 (

lb-m

ol/h

r)

time (h)

0 20 40 60

0.5

0.6

z (m

ol f

ract

ion

A)

time (h)

No control (a)FF control (b)

Figure S23.6b. Alternative #4 (with and without FF controller). Step change in F0

(+10%) at t=10

23-22

23.7

Parts a) and b) can be satisfied by combining two or more of the previous simulations into one to compare the results together. To compare how the alternatives match up, in terms of the snowball effect, a set of arrays has been constructed.

All arrays are of the form:

0 0

0 0

R

R

HD

F F

HD

z z

where the response of D or HR is analyzed as a result of a step change in F0 or z0.

In the notation below:

S represents the occurrence of the snowball effect (>20% change in steady-state output for a 10% change in input). A represents an acceptable response (~10% change in steady-state output). B represents the best possible response (no change in steady-state output).

Alternative #1 Alternative #2

S B

S B

B S

B A

Alternative #3 Alternative #4

A A

B A

A A

B A

These results indicate that Alternative #2 still exhibits a snowballing characteristic, but in HR instead of D. Alternatives #3 and #4, on the other hand, eliminate the effect altogether.

23-23

0 20 40 600.005

0.01

0.015

0.02

xB (

mol

e fr

actio

n A

)

0 20 40 600.92

0.94

0.96

xD (

mol

e fr

actio

n A

)

0 20 40 60100

200

300

400

HD

(lb

-mol

)

0 20 40 60250

300

350

HB

(lb

-mol

)

0 20 40 602000

2500

3000

HR

(lb

-mol

)

time (h)

0 20 40 60400

500

600

700

D (

lbm

ol/h

r)

0 20 40 601400

1600

1800

2000

V (

lbm

ol/h

r)

0 20 40 601000

1200

1400

R (

lb-m

ol/h

r)

0 20 40 60450

500

550

B (

lb-m

ol/h

r)

0 20 40 600.4

0.5

0.6

z (m

ole

frac

tion

A)

0 20 40 60460

480

500

520

F0

(lb-m

ol/h

r)

0 20 40 60900

1000

1100

1200

F (

lb-m

ol/h

r)

Alternative #1Alternative #2Alternative #3Alternative #4

Figure S23.7a. Step change in F0 (+10%) at t=10

23-25

23.8

Begin with a dynamic energy balance on the reactor:

0 0

( ( ))( ) ( ) ( )

( )

R R RefP P Ref P D Ref P R Ref R

R C

d H T TC C F T T C D T T C F T T H kz Q

dt

Q UA T T

−= − + − − − − −

= −

�

�

λ

This model can be simplified using the mass balance:

0RdH

F D Fdt

= + −

And, rearranging to get an equation for modeling the reactor temperature:

0 0

1[ ( ) ( ) ( ) ]R

P R P D R R C RP R

dTF C T T DC T T UA T T H kz

dt C Hλ= − + − − − −

It is clear from the following figures that the temperature loop is much faster than the interconnected level-flow loops. This characteristic allows the reaction rate multiplier to settle before it can affect the other variables.

23-26

30 40 50 60616.4

616.41

616.42

616.43

616.44

616.45T R

(R

)

30 40 50 60590

591

592

593

594

595

596

597

T C (

R)

30 40 50 600.339

0.3395

0.34

0.3405

0.341

KR (

lb-m

ol/h

r)

30 40 50 600.4

0.45

0.5

0.55

z (m

ol f

ract

ion

A)

30 40 50 60950

1000

1050

1100

1150

1200

1250

1300

F (

lb-m

ol/h

r)

30 40 50 60450

500

550

600

650

700

D (

lb-m

ol/h

r)

Time (hr)

Thermal modelConstant temperature

Figure S23.8a. Step change in F0 (+10%) at t=30 (Constant temperature simulation does not include a thermal model)

23-27

30 40 50 60616.4

616.41

616.42

616.43

616.44

616.45T R

(R

)

30 40 50 60590

592

594

596

598

600

T C (

R)

30 40 50 600.339

0.3395

0.34

0.3405

0.341

KR (

lb-m

ol/h

r)

30 40 50 600.4

0.45

0.5

0.55

z (m

ol f

ract

ion

A)

30 40 50 60850

900

950

1000

F (

lb-m

ol/h

r)

30 40 50 60400

450

500

550

D (

lb-m

ol/h

r)

Time (hr)

Thermal modelConstant temperature

Figure S23.8b. Step change in z0 (-10%) at t=30 (Constant temperature simulation

does not include a thermal model)