Chapter 3 : Simple Process Dynamics and Transfer Function Professor Shi-Shang Jang Department of...

Chapter 3 : Chapter 3 : Simple Process Simple Process Dynamics and Dynamics and Transfer FunctionTransfer Function

Professor Shi-Shang JangDepartment of Chemical EngineeringNational Tsing-Hua UniversityHsinchu, TaiwanMarch, 2013

Motive of Developing First Motive of Developing First Principle ModelsPrinciple Models

Improve understanding of the process

Train plant operating personnelDevelop a control strategy for a new

processOptimize process operating

conditions

3-1 Introduction3-1 Introduction

Theoretical (First Principle) models are developed using the principles of chemistry , physics, and biology.

Theoretical models offer process insight into process behavior, and they are applicable over wide ranges of conditions

They trend to be expensive and time-consuming to develop

TT23

TC23

Stack gases

Fuel

Air

Air

FY23

QY23

FT24

Set point

Example - Industrial Example - Industrial FurnaceFurnace

CV: temperature of the furnaceMV: fuel flow rate to the furnace

Figure 1-1

Temperature Profile of Temperature Profile of TT23TT23

0 5 10 15 20 25 30 35 40 45 50219

219.5

220

220.5

221

221.5

222

222.5

223

Time (min)

Plant Plant DynamicsDynamics

PlantMV CV

DV

time

tem

pe

ratu

re

0 1 2 3 4 5 6 7 8 9 10

80

80.2

80.4

80.6

80.8

81

81.2

81.4

time

tem

pe

ratu

re

0 10 20 30 40 50 60 70 80 90 100500

502

504

506

508

510

512

514

516

518

520

time

Flo

w

rate

0 10 20 30 40 50 60 70 80 90 10079.5

80

80.5

81

81.5

82

82.5

83

time

Flo

w

rate

0 1 2 3 4 5 6 7 8 9 1020

20.5

21

21.5

22

22.5

23

23.5

24

24.5

25

The Concept of Deviation The Concept of Deviation VariablesVariables

PlantMV CV

DV

time

tem

pe

ratu

re

time

tem

pe

ratu

re

time

Flo

w

rate

time

Flo

w

rate

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100-0.5

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

20

yd=y - ys

Simple SystemsSimple Systems

The Essence of Process The Essence of Process Dynamics - Dynamics - ContinuedContinuedThe feedback process control needs to

understand the relationships between CV and MV, on the other hand, feedforward process control needs to understand the relationships between DV and CV. The relationships are called process models.

For the ease of mathematical analyses, the process modeling only implements a linear model and Laplace transform instead of direct use of time domain process model. Implementation of deviation variables is needed as indicated below.

3-1 Introduction- 3-1 Introduction- ContinuedContinuedEmpirical models are obtained by fitting

experimental data.Empirical models typically do not

extrapolate well, and their range is typically small.

Empirical models are frequently used in the industrial environment since a theoretical model is basically not precisely available.

3-1 Introduction- Continued3-1 Introduction- ContinuedSemi-empirical models are a combination

of the models of theoretical and empirical models; the numerical values of the parameters in a theoretical model are calculated from experimental data.

Semi-empirical models can (i) incorporate theoretical knowledge, (ii) extrapolate wider range than empirical range, (iii) require less effort than theoretical models.

3-2 General Modeling 3-2 General Modeling Principles Principles

comsumedenergy

mass/ of ratenet

producedenergy

mass/ of ratenet

outenergy

mass/ of rate

in

mass/ of rate

y mass/energ of

onaccumulati of rate

3-2 General Modeling 3-2 General Modeling Principles:Principles:

Constitution Equations

Heat Transfer ： Reaction Rate ：Flow Rate ： Equation of State ：

Phase Equilibrium:

ThAQ

ARTE

A Cekr /0

2/1/ PCF v

nRTPV

ii Kxy

3-3 Transfer Functions - 3-3 Transfer Functions - ConventionsConventions

－ on the top of a variable= steady state of a variable, example:

Capital = deviation variable, example:

Capital with (s)= Laplace transform of a variable to the deviation variable, example:

( ) ( )X t x t x

: nominal state of ( )x x t

( ) ( )X s X tL

3-3. Transfer Function3-3. Transfer Function

( )( )

( )

Y sG s

X s

• Transfer function is a mathematical representation of the relation between the input and output of a system.

• It is the Laplace transform of the output variable, y(t), divided by Laplace Transform of the input variable, x(t), with all initial conditions equal to zero.

• The term is often used exclusively to refer to linear, time-invariant systems (LTI), and non-linear, real-system are linearize to obtain their Transfer Function.

• So, Transfer Function G(s) for a system with input x(t) and output y(t) would be-

• As for previous equation, it could be said that if transfer function for the system and input to the system is known, we can obtain the output characteristics of the system.

• Transfer Function for the system could be easily obtained by dynamic study of the system and making balances for quantities like energy, mass etc.

• We take inverse Laplace Transform to obtain time-varying output characteristics of Y(s). In block diagram:

More over Transfer More over Transfer FunctionFunction

( ) ( ) ( )G s X s Y s

G (s)X(s) Y(s)

3-3 Transfer Functions –3-3 Transfer Functions – Example: Thermal Example: Thermal ProcessProcess

Rate of Energy Input- Rate of Energy Output=Accumulation

i i i

vi p i i p

d V u tf h t f h t

dt

d V C T tf C T t f C T t

dt

Inputs: f(t), Ti(t),Ts(t)Output: T(t)

Ts

3-3 Transfer Functions 3-3 Transfer Functions – – Cont.Cont.

Let f be a constant V= constant, Cv=Cp


(1)

0 (2)

(1)-(2)

set deviation variables ,

p i p v

p i p

p i i p v

i i i

v

p

dT tf C T t f C T t V C

dtf C T f C T

d T t Tf C T t T f C T t T V C

dtt T t T t T t T

d tV Ct

f C dt

i

i

t

d tt t

dt

Let f be a constant V= constant, Cpi=Cp

=time constant


Taking Laplace Transform

Consider the process input to be the inlet temperature, and the

output to be tank temperature

s 1

1

i

pi

s s s s

G ss s

where, Gp(s) is call the transfer function of the process, in block diagram:

Gp(s)i(s) (s)

Step Response of a First Step Response of a First Order SystemOrder System

632.0)1()(Notbly

1limlim

1lim)(lim

111

1

;s

0;Input Step

1

1

0

/

//11

eKA

KAss

KAsss

KAeKAt

eKAKAeKAs

KA

s

KA

ss

KAt

s

K

s

As

s

A

tAt

s

K

s

s

ts

t

tt

tt

i

i

i

LL

Process IdentificationProcess Identification

tt

et

et

yt

tyt

edt

td

ee

A

t

AK

eKAet

tt

tt

t

t

tt

ln

1

;at hence

:0at line slop The

1)(

632.011

lim

11

//

0/

0

1/

//

plotlogarithm The:3 Fit

Slope Intial of Method:2 Fit

63.2% of Method:1 Fit

Example: Mercury Example: Mercury thermometer thermometer

A mercury thermometer is registering a temperature of 75F. Suddenly it is placed in a 400F oil bath. The following data are obtained.

Time (sec)

0 1 2.5 5 8 10 15 30

Temp. (F)

75 107 140 205 244 282 328 385

Estimate the time constant of the temperature using (1)Initial slope method(2)63% response method(3)From a plot of log(400-T) versus time

Solution Solution

0 5 10 15 20 25 3050

100

150

200

250

300

350

400

time

tem

pera

ture

((1) =9sec)

ComparisonsComparisons

0 5 10 15 20 25 3050

100

150

200

250

300

350

400

Fit 1

Fit 2

Fit 3


By including the effect of surrounding temperature:

( )

( ) (1)

steady state

0 (2)

(1) (2)

p i p v

p i s p v

p i s p

p i i

dT tf C T t q t f C T t V C

dtdT t

f C T t UA T t T t f C T t V Cdt

f C T UA T T f C T

f C T t T

( )

( )

( )

s s p v

p i s p v

pvi s

p p p

d T t TUA T t T T t T f C T t T V C

dtd t

f C t UA t t f C t V Cdt

fCd tV C UAt t t

f C UA dt f C UA f C UA


1 2

1 21 2

( )

1 1

i s

i s p i p s

d tt K t K t

dtLaplacing

K Ks s s G s s G s s

s s

Gp1(s)

Gp2(s)

+

+

i(s)

s(s) (s)Σ

Numerical DataNumerical Data

3

3

3

1

10 / min; 80 ; 60 ; 25

1 /

1000 /

1000

0

10 1000 1 80 605714 / min

60 25

1000 1000 163.6min

10 1000 1 5714

i s

p v

p i s p

p i

s

v

p

p

f m T C T C T C

C Kcal Kg C C

Kg m

V m

f C T UA T T f C T

f C T TUA Kcal C

T T

V C

f C UA

fCK

2

10 1000 10.6364

10 1000 1 5714

57140.3636

10 1000 1 5714

p

p

f C UA

UAK

f C UA

Examples (1) Thermal Process- Examples (1) Thermal Process- ContinuedContinued

Deviation Variables

1 2

i i i

s s s

i s

t T t T

t T t T

t T t T

dK K

dt

3-3 Transfer Functions-An 3-3 Transfer Functions-An ExampleExample

3

3 1/ 2

( )

( ) flow through valve, /

valve coefficient, /

pressure drop across valve

specific gravity of liquid

( ) ( ) '

f

f

f

P tf t C

G

f t m s

C m s kPa

P t

G

gh tP t gh t f t C C h t k h t

G

f0

Cross-sectional=A2

h2V2

f1

f2

h1V1

Cross-sectional=A1

3-3 Example Non-3-3 Example Non-Interactive TanksInteractive Tanks

1 1 11 1 0 1 1 0 1

2 2 22 2 1 2 2 1 2 1 2

d h h dHA A f k h F aH

dt dt

d h h dHA A f k h F bH aH bH

dt dt

01 1 11 1 1 0 1 1 1 0

2 2 21 2 2 2 1 2 2 2 1

1

1

FA dH dHH K F H s H s KF s

a dt a dtA dH dHa

H H K H H s H s KH sb dt b dt

3-3 Example Non-3-3 Example Non-Interactive Tanks Interactive Tanks – Cont.– Cont.

12

2

sK

11

1

sK

F0(s)

H1(s)

H2(s)

F0(s) H2(s) 11 21

21

ss

KK

3-4 Dead Time3-4 Dead Time

3-4 Dead Time – 3-4 Dead Time – Cont.Cont.Time delay:

0

0

0

3

2

distance

velocity /

flow rate (m /s)

cross-sectional area of pipe, m

length of pipe, m

, if 1/ 11

p

t st t

i

Lt

f A

f

Ap

L

s es s t e

s s

3-4 Dead Time – 3-4 Dead Time – Cont.Cont.

3-4 Causes of Dead 3-4 Causes of Dead Time - Time - Cont.Cont.

Transportation lag (long pipelines)Sampling downstream of the processSlow measuring device: GCLarge number of first-order time

constants in series (e.g. distillation column)

Sampling delays introduced by computer control

3-4 Effects of Dead-3-4 Effects of Dead-Time - Cont.Time - Cont.

Process with large dead time (relative to the time constant of the process) are difficult to control by pure feedback alone:

Effect of disturbances is not seen by controller for a while

Effect of control action is not seen at the output for a while. This causes controller to take additional compensation unnecessary

This results in a loop that has inherently built in limitations to control

3-5 Transfer Functions 3-5 Transfer Functions and Block Diagramsand Block Diagrams

Consider a general transfer function for an input X(s) and an output Y(s):

Note that the above case is always true , although many mathematical manipulating is needed as shown below:

011 1

11 1

1

1

t sm mm m

n nn n

K a s a s a s eY sG s n m

X s b s b s b s

3-5 Transfer Functions and 3-5 Transfer Functions and Block Diagrams Block Diagrams – Cont.– Cont.

3-5 Transfer Functions and 3-5 Transfer Functions and Block Diagrams Block Diagrams – Cont. – Cont. (Example 3-5.2)(Example 3-5.2)

3-5 Transfer Functions and 3-5 Transfer Functions and Block Diagrams Block Diagrams – Cont. – Cont. (Example 3-5.3)(Example 3-5.3)

3-6 Gas Process 3-6 Gas Process ExampleExample

1

1

( ) 0.16 ( )

( ) 0.00506 ( ) ( ) ( ) ( )

8 ; 40 ; 1 ; 50%

i i

o o

i o i o

f t m t

f t m t p t p t p t

f f scfm p psia p atm m m

3-6 Gas Process 3-6 Gas Process Example – Example – Cont.Cont.

1 11

1 2

; 0.00263 /

(1)

0 (2)

(1) (2) yields

i o

i o

i o

o o oo o ss o o ss ss

o

i o

dnf t f t lbmole scf

dtpV nRT

V dpf t f t

RT dt

f t f t

f f ff t f m t m p t p p t p

m p p

V dPaM t C M t C

RT dt

3 1

31 211 1 1i o

P t C P t

KK KP s M s M s P s

s s s


Σ

s+1

K1

s+1

K2

s+1

K3

+-

-

Mi(s)

Mo(s)

P1(s)

P(s)


1 1

1/ 2

2 1 1

1/ 2

1/ 2

3 1 11

0.00506 0.00506 40(40 14.7) 0.1610

C 0.00506 1/ 2 2

0.00506 50 1/ 2 40(40 14.7) 2 40 14.7 0.2597

0.00506

0.00506 50 1/ 2 40(40 14.7)

oss

o

oss o

oss o

fC p p p

m

fm p p p p p

p

fC m p p p p

p

1/ 2

1 2

2 1 2

3 3 2

2

40 -0.16

0.16 / 0.6162

/ 0.6199

/ -0.6126

20= 5.2481min

10.73 (60 460) 0.00263 0.2597

K C

K C C

K C C

V

RT C

3-7 Chemical Reactor3-7 Chemical Reactor

2

0

( 1) AAi A A

A A

Ad A

dcf t c t f t c t r t V V

dt

r kc

c t c t t

3-7 Chemical Reactor – 3-7 Chemical Reactor – Cont.Cont.

2 2

Ai Ai Ai Ai Ai Ai Ai Ai

A A A A A A A A

A A A A A A A

f t c t fc f c c c f t f fc fC c F t

f t c t fc f c c c f t f fc fC c F t

r t r kc c c r kC

( 1) 2 (1)

( 1) 0 (2)

(1)-(2)

2

AAi Ai Ai A A A A A

Ai A A

AAi Ai A

dcfc fC t c F t fc fC c F t r kC V V

dt

fc fc r t V

dCV fC t c c F t f

dt

2 2 2

A

Ai AAA Ai

kV C

c cdCV fC C t F t

f kV dt f kV f kV

3-7 Chemical Reactor – 3-7 Chemical Reactor – Cont.Cont.

01 2 ;1 1

stA Ai Ad A

K KC s F s C s C s e C s

s s

3-8 Effects of Process 3-8 Effects of Process NonlinearityNonlinearity

Real processes are mostly nonlinearThe approximate linear models are only

valid in local about the nearby of the operating point

In some cases, process nonliearity may be detrimental to the control quality (e.g. high purity column)

Process nonliearity plays important role to control quality in control systems

3-8 Effects of Process 3-8 Effects of Process Nonlinearity – Nonlinearity – Cont.Cont.

3-9 Additional 3-9 Additional CommentsComments

Reading assignmentP96-98

HomeworkHomework

Page 993-1, 3-2, 3-3, 3-4, 3-9, 3-10 (April

17th), 3-13, 3-14(SIMULINK) 3-20, 3-21 (April 24th)

Process-Supplemental Process-Supplemental MaterialMaterial

Inputs: f(t), Ti(t),Ts(t)Output: T(t)

Ts

vpispp

ipi

p

pi

p

v

spipiipipv

sssiii

vssppipiiipi

vspipi

CCtUACf

UAtF

UACf

TTCt

UACf

Cft

dt

td

UACf

CV

tUAtFTCTCtCftUACfdt

tdCV

TtTtftftFTtTtTtTt

dt

tdTCVTtTUATCftfTtTUACfTCftfTtTCf

dt

tdTCVtTtTUAtTCf(t)tTCf(t)

; Assume

)(

;;;

)(

onAccumulatiOutputEnergy of Rate-InputEnergy of Rate


Gp1(s)

Gp2(s)

+

+

i(s)

s(s) (s)Σ

sFGsGssGsFs

Ks

s

Ks

s

Ks

Laplacing

tFKtKtKtdt

td

pspipsi

si

321321

321

)(111

Gp3(s)F(s)

Thermal Process-Thermal Process-Supplemental MaterialSupplemental Material

2714.1

57141100010

608011000

3636.0;6364.0

min6.63

1000;/1000

/1

50min;/10

3

21

33

3

UACf

TTCK

UACf

UAK

UACf

CfK

UACf

CV

mVmKg

CKgKcalCC

CTTmf

p

ip

pp

p

p

v

vp

i

Chapter 3 : Simple Process Dynamics and Transfer Function Professor Shi-Shang Jang Department of...

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