constitutive relations

© 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


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



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



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



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



Classes of Relations

ConstitutiveEquations Property definitions

Balance volumerelations

Reaction rates

Equipment andcontrol constraints

Transfer relations



1. Transfer Relations

General form

Particular forms

mass transfer

heat transfer

)()(),(),( rprprprate

GGG

CCKj *

TUAqCV



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



2. Reaction rates

Reaction rate (batch reactor only)

General reaction expression

dtdn

Vr i

i

1

RTE

A

BAAA

ekk

CCfkr

0

,...,



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



3. Thermodynamic relations

Property relations (density, viscosity, …)

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

),,(iL

xTPf



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



Thermodynamic properties

Enthalpy

linear

nonlinear

T

T pR RdTTcThTh )()()(

VAPp

p

TcThTcTh

)()(

T

T p

p

dTTcThTh

TaTaac

0)()()(

...

0

2

210



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



Thermodynamic properties

Equations of state

ideal gas

cubic EoS

– Soave Redlich Kwong– Peng Robinson– NRTL

nRTPV

),( TVfP



4. Balance volume relations

Relations between phases

LGVVV

VG

VL



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



5a. Sensors

Sensors

)(

)(~

)(~)(

TTdtdT

TTMc

AUdtdT

TTAUdt

TMcddtdU

f

f

p

f

p

TT f

Fluid



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

gain theisG zero theis

signalinput theis

)(

0

0min

zI

GzIOO

p

ppp



5c. Controllers

Traditional (P, PI, PID)

dtdKdtKKBO

dtKKBO

KBOSKBO

DC

I

C

CC

I

C

CC

CPPCC

)(



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



5e. Valves

Static valves

Control valves characteristics

PCFV

PScCFV

)(

root square )(

percentage equal )(

linear )(1

SSc

aSc

SScS

constitutive relations

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