© CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

Constitutive Relations

2 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

What are constitutive relations?

Relate conserved extensive quantities to intensive variables

Help define physico-chemical quantities (e.g. enthalpies, densities, viscosities ,…)

Define transfer rates (mass, energy, …) Other relations to “constitute” the model

3 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

How do constitutive relations arise ?

qJt

Related to the terms in the conservation equations for mass, energy and momentum

• Constraints on the system (control relations)

• Convective flow terms (process streams)

• Molecular flow streams (fluxes)• Internal processes

• Defining intensive variables in terms of extensive quantities and other physico-chemical properties

4 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

Balance volumes, flows and system processes

Hot oil in

Hot oil out

EM ,1S

EM ,2S

Hot oil in

Hot oil out

EM ,1S

EM ,2S

E3S

Hot o il in

Hot o il out

Co ld feed in

Heated liquid ou t

EM ,1S

EM ,2S

Q hQ rloss

Q hloss

H ot o il in

H ot o il out

C old feed in

H eated liqu id ou t

EM ,1S

EM ,2S

E3S

Q rloss Q h

Q hloss

Q r

5 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

CSTR Example

in-flow

out-flow

f, C Ai

f, C A , C B

HfHfdtdH

rVffdt

dm

i

AAA

i

ˆˆ

AA

AiA

P

iPi

AA

ARTE

fCf

fCfTcH

TcH

VCmCekr

i

ˆ

ˆ

0A BConstant V

6 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

Classes of Relations

ConstitutiveEquations Property definitions

Balance volumerelations

Reaction rates

Equipment andcontrol constraints

Transfer relations

7 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

1. Transfer Relations

General form

Particular forms

mass transfer

heat transfer

)()(),(),( rprprprate

GGG

CCKj *

TUAqCV

8 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

CSTR Example

in-flow

out-flow

f, C Ai

f, C A , C B

HfHfdtdH

rVffdt

dm

i

AAA

i

ˆˆ

AA

AiA

P

iPi

AA

ARTE

fCf

fCfTcH

TcH

VCmCekr

i

ˆ

ˆ

0

A B

Qloss

Qloss = UA(T-Tamb)

- Qloss

9 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

2. Reaction rates

Reaction rate (batch reactor only)

General reaction expression

dtdn

Vr i

i

1

RTE

A

BAAA

ekk

CCfkr

0

,...,

10 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

CSTR Example

in-flow

out-flow

f, C Ai

f, C A , C B

HfHfdtdH

rVffdt

dm

i

AAA

i

ˆˆ

AA

AiA

P

iPi

AA

ARTE

fCf

fCfTcH

TcH

VCmCekr

i

ˆ

ˆ

0A B

11 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

3. Thermodynamic relations

Property relations (density, viscosity, …)

Equilibrium relations Raoult’s law Relative volatility, K-value Activity coefficient

),,(iL

xTPf

12 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

CSTR Example

in-flow

out-flow

f, C Ai

f, C A , C B

HfHfdtdH

rVffdt

dm

i

AAA

i

ˆˆ

AA

AiA

P

iPi

AA

ARTE

fCf

fCfTcH

TcH

VCmCekr

i

ˆ

ˆ

0A B

13 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

Thermodynamic properties

Enthalpy

linear

nonlinear

T

T pR RdTTcThTh )()()(

VAPp

p

TcThTcTh

)()(

T

T p

p

dTTcThTh

TaTaac

0)()()(

...

0

2

210

14 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

CSTR Example

in-flow

out-flow

f, C Ai

f, C A , C B

HfHfdtdH

rVffdt

dm

i

AAA

i

ˆˆ

AA

AiA

P

iPi

AA

ARTE

fCf

fCfTcH

TcH

VCmCekr

i

ˆ

ˆ

0A B

15 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

Thermodynamic properties

Equations of state

ideal gas

cubic EoS

– Soave Redlich Kwong– Peng Robinson– NRTL

nRTPV

),( TVfP

16 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

4. Balance volume relations

Relations between phases

LGVVV

VG

VL

17 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5. Control systems

F1 F2

F3

E,

1

F102V

L101

LC

LTLS

LT

LS

LO

L

LSP

XC

XS

XT

xS xTxO

xB

xBSP

CW in CW out

TSTT

TCTV101

T

TSTT

TOTSP

18 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5a. Sensors

Sensors

)(

)(~

)(~)(

TTdtdT

TTMc

AUdtdT

TTAUdt

TMcddtdU

f

f

p

f

p

TT f

Fluid

19 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5b. Transmitters Transmitters (4-20mA, 20-100kPa)

gain theisG zero theis

signalinput theis

)(

0

0min

zI

GzIOO

p

ppp

20 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5c. Controllers

Traditional (P, PI, PID)

dtdKdtKKBO

dtKKBO

KBOSKBO

DC

I

C

CC

I

C

CC

CPPCC

)(

21 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5d. Actuators

factor damping andconstant time,gainactuator

1)-(0movement stem

22

2

2

a

a

GS

IGSdtdS

dtSd

x

Pdiaphragm area, A

stem velocity, v

stem packing

plug and seat

22 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences

PROCESS MODELLING AND MODEL ANALYSIS

5e. Valves

Static valves

Control valves characteristics

PCFV

PScCFV

)(

root square )(

percentage equal )(

linear )(1

SSc

aSc

SScS