Chapter-2: Theoretical model of chemical process

Process dynamics and Control

1. Chapter
2
Theoretical Models of Chemical
Processes

2. Chapter
2
RationaleforDynamicModels
1.Improveunderstandingoftheprocess
2.TrainPlantoperatingpersonnel
3.Developcontrolstrategyfornewplant
4.Optimizeprocessoperatingconditions

3. Chapter
2
TypeofModels
1. Theoreticalmodel
-developed using the principles of chemistry, physics,
and biology
-applicableoverwiderangesofconditions
-expensiveandtimeconsuming
-some model parameters such as reaction rate coefficients
physical properties, or heat transfer coefficients are
unknown
2. Empiricalmodels
-obtainedfittingexperimentaldata
-range of the data is typically quite small compared to
thewholerangeofprocessoperatingconditions
-donotextrapolatewell
3. Semi-empiricalmodels
-combinationof1and2
-overcome the previously mentioned limitations to many
extents
-widelyusedinindustry

4. Chapter
2
GeneralModelingPrinciples
•The model equations are at best an approximation to the real
process.
•Adage:“Allmodelsarewrong,butsomeareuseful.”
•Modeling inherently involves a compromise between model
accuracyandcomplexityononehand,andthecostandeffort
requiredtodevelop themodel,ontheother hand.
•Processmodelingisbothanartandascience.Creativityis
requiredtomakesimplifyingassumptionsthatresultinan
appropriatemodel.
•Dynamic models of chemical processes consist of ordinary
differential equations (ODE) and/or partial differential equations
(PDE),plus relatedalgebraicequations.

5. Chapter
2
Table2.1.ASystematicApproachfor
Developing DynamicModels
1.State the modeling objectives and the end use of the model.
They determine the required levels of model detail and model
accuracy.
2.Draw a schematic diagram of the process and label all process
variables.
3.Listalloftheassumptionsthatareinvolvedindevelopingthe
model.Try for parsimony; the model should be no more
complicatedthannecessarytomeetthemodelingobjectives.
4.Determinewhetherspatialvariationsofprocessvariablesare
important.Ifso,apartialdifferentialequationmodelwillbe
required.
5.Writeappropriateconservationequations(mass,component,
energy,andsoforth).

6. Chapter
2
6.Introduceequilibriumrelationsandotheralgebraic
equations(fromthermodynamics,transportphenomena,
chemicalkinetics,equipmentgeometry,etc.).
7.Performadegreesoffreedomanalysis(Section2.3)to
ensurethatthemodelequationscanbesolved.
8.Simplifythemodel.Itisoftenpossibletoarrangethe
equationssothatthedependentvariables(outputs)appear
ontheleftsideandtheindependentvariables(inputs)
appearontherightside.Thismodelformisconvenient
forcomputersimulationandsubsequentanalysis.
9.Classifyinputsasdisturbancevariablesorasmanipulated
variables.
Table2.1.(continued)

7. Chapter
2
Table2.2.DegreesofFreedomAnalysis
1.Listallquantitiesinthemodelthatareknownconstants(or
parametersthatcanbespecified)onthebasisofequipment
dimensions,knownphysicalproperties,etc.
2.Determine the number of equations N
Eand the number of
processvariables,N
V.Notethattimetisnotconsideredtobea
process variable because it is neither a process input nor a
processoutput.
3.Calculatethenumberofdegreesof freedom,N
F=N
V-N
E.
4.IdentifytheN
Eoutputvariablesthatwillbeobtainedbysolving
theprocess model.
5.IdentifytheN
Finputvariablesthatmustbespecifiedaseither
disturbance variables or manipulated variables, in order to
utilizetheN
Fdegreesof freedom.

8. Chapter
2
ConservationLaws
Theoretical models of chemical processes are based on
conservationlaws.
ConservationofMass
ConservationofComponenti

9. Chapter
2
ConservationofEnergy
The general law of energy conservation is also called the First
Lawof Thermodynamics.It canbe expressedas:
Thetotalenergyofathermodynamicsystem,U
tot,isthesumofits
internalenergy, kineticenergy, andpotential energy:
U
tot U
int U
KE U
PE
(2-9)

10. Chapter
2
Fortheprocessesandexamplesconsideredinthisbook,it
isappropriatetomaketwoassumptions:
1.Changesinpotentialenergyandkineticenergycanbe
neglectedbecausetheyaresmallincomparisonwithchanges
ininternalenergy.
2.Thenetrateofworkcanbeneglectedbecauseitissmall
comparedtotheratesofheattransferandconvection.
Forthesereasonableassumptions,theenergybalancein
Eq.2-8canbewrittenas
U
inttheinternalenergyof
thesystem
Henthalpyperunitmass
wmassflowrate
Qrateofheat transfertothesystem
streams;therefore
(2-10)
denotes the difference
between outletand inlet
conditionsoftheflowing
-ΔwH) = rateofenthalpyoftheinlet
stream(s)-theenthalpy ofthe
outlet stream(s)

11. Chapter
2
Theanalogousequationformolarquantitiesis,
Where ෩??????istheenthalpy per mole and෥�isthemolarflowrate.
In order to derive dynamic models of processes from the general
energy balances in Eqs. 2-10 and 2-11, expressions for U
intand መ??????
or෩??????are required,whichcan bederivedfromthermodynamics.

12. Chapter
2
DevelopmentofDynamicModels
Illustrative Example:ABlendingProcess
Anunsteady-statemassbalancefortheblendingsystem:

13. Chapter
2
where w
1,w
2,andwaremassflowrates.
Theunsteady-statecomponentbalanceis:
(2-3)
For constant density, the corresponding model can be summarized
as:
or
(2-2)

dV
w
1w
2w (2-12)
dt
(2-13)
??????(????????????)
????????????
=�
1+�
2−�
??????(????????????�)
????????????
=�
1�
1+�
2�
2−��
??????
??????(??????�)
????????????
=�
1�
1+�
2�
2−��

14. Chapter
2
Equation 2-13 can be simplified by expanding the accumulation
termusing the “chainrule”for differentiationof aproduct:

dVx)
V
dx
x
dV
dt dt dt
Substitutionof(2-14)into(2-13)gives:
(2-14)
1 1 22
(2-15)
dt dt
V
dx
x
dV
wxwxwx
Substitutionof themass balance in(2-12)fordV/dtin(2-15)
gives:
Vxw
1w
2w)w
1x
1w
2x
2wx (2-16)
dx
dt
After canceling common terms and rearranging (2-12) and (2-16),
amore convenient model formis obtained:
1 2
dx

w
1
1 2
(2-17)
(2-18)
dt
dV

1
www)
dtV V
xx)
w
2
xx)

15. Chapter
2
Stirred-TankHeatingProcess
Figure2.3Stirred-tankheatingprocesswithconstantholdup,V.

16. Chapter
2
Stirred-TankHeatingProcess(cont’d.)
Assumptions:
1.Perfect mixing; thus, the exit temperature T is also the
temperatureof the tankcontents.
2.The liquid holdup V is constant because the inlet and outlet
flowrates areequal,w
i= w.
3.Thedensityand heat capacity C of the liquid are assumed to
beconstant.Thus, theirtemperature dependenceis neglected.
4.Heatlossesarenegligible.

17. Chapter
2
For a pure liquid at low or moderate pressures, the internal energy
isapproximatelyequaltotheenthalpy,U
intH,and H depends
only on temperature. Consequently, in the subsequent
development,weassumethatU
int= H andUˆ
intHˆwherethe
caret (^) means per unit mass. As shown in Appendix B, a
differential change in temperature, dT, produces a corresponding
changeintheinternalenergyperunitmass,dUˆ
int,
dUˆ
intdHˆCdT (2-29)
where C is the constant pressure heat capacity (assumed to be
constant).The total internal energy of the liquidin the tank is:
U
intVUˆ
int
(2-30)
Model Development -I

18. Chapter
2
An expression for the rate of internal energy accumulation can be
derivedfrom Eqs. (2-29) and (2-30):
dU
int
VC
dT
dt dt
(2-31)
Note that this term appears in the general energy balance of Eq. 2-
10.
Suppose that the liquid in the tank is at a temperature T and has an
enthalpy, Hˆ.IntegratingEq.2-29fromareferencetemperature
T
refto Tgives,
CTT
ref )HˆHˆ
ref (2-32)
whereHˆ
refisthevalueofHˆ
atT
ref.Withoutlossofgenerality,we
assumethatHˆ
ref0(seeAppendixB).Thus,(2-32)canbe
writtenas:
HˆCTT
ref ) (2-33)
Model Development -II

19. Chapter
2
(2-34)Hˆ
i CT
i T
ref )
Substituting (2-33) and (2-34) into the convection term of (2-10)
gives:
)))
ˆ
(2-35)
iref ref
wH   wCT T wCTT
  
Finally,substitutionof(2-31)and(2-35)into (2-10)
(2-36)
i
dt
VC
dT
wCT T)Q
Model Development -III
Fortheinletstream

20. Chapter
2
ThusthedegreesoffreedomareN
F=4–1=3.Theprocess
variablesareclassifiedas:
1outputvariable:T
3inputvariables:Ti,w,Q
For temperature control purposes, it is reasonable to classify the
threeinputsas:
2disturbancevariables:
1manipulatedvariable:
Ti,w
Q
Degrees of Freedom Analysis for the Stirred-Tank
Model:
3parameters:
4variables:
1equation:
V,,C
T,T
i,w,Q
Eq.2-36

21. Chapter
2
ContinuousStirredTankReactor(CSTR)
Fig.2.6.SchematicdiagramofaCSTR.

22. Chapter
2
CSTR:ModelDevelopment
Assumptions:
1.Single,irreversiblereaction,A→B.
2.Perfectmixing.
3.Theliquidvolume Viskeptconstantbyanoverflowline.
4.Themassdensitiesofthefeedandproductstreamsareequal
andconstant.They aredenoted by.
5.Heatlossesare negligible.
6.Thereactionrate forthedisappearance ofA,r,isgivenby,
r=kc
A
(2-62)
wherer=molesofAreactedperunit time, perunitvolume,c
A
istheconcentration ofA(molesperunitvolume),andkistherate
constant (unitsofreciprocaltime).
7.Therate constanthasanArrheniustemperaturedependence:
k=k
0
exp(-E/RT) (2-63)
where k
0
is the frequency factor, E is the activation energy,
andRis thethegas constant.

23. Chapter
2
•Unsteady-statecomponentbalance
.
•Unsteady-statemass balance
Becauseand Vareconstant,
themassbalanceisnotrequired.
.Thus,
CSTR:ModelDevelopment(continued)

24. AssumptionsfortheUnsteady-stateEnergyBalance:
8.
9.
10.
11.
12.

25. Chapter
2
CSTRModel:SomeExtensions
•Howwouldthedynamicmodelchangefor:
1.Multiplereactions(e.g.,A →B→ C)?
2.Differentkinetics,e.g.,2
nd
orderreaction?
3.Significantthermalcapacityofthecoolantliquid?
4.LiquidvolumeVisnotconstant(e.g.,nooverflowline)?
5.Heatlossesarenotnegligible?
6.Perfect mixingcannotbeassumed(e.g.,foravery
viscous liquid)?

26. Chapter
2
SimulationSoftwares
Matlab
Mathematica
HYSYS
Modelica
gPROMS
www.mathworks.com
www.wolfram.com
www.aspentech.com
www.modelica.org
www.psenterprise.com
Aspen Custom Modeller www.aspentech.com
Global CapeOpen www.global-cape-open.org