PIC Part-A Student Version

8/3/2019 PIC Part-A Student Version

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2010PICLecture2010 Prof.T.K.Ghoshal &Prof.Smita Sadhu

1

PROCESS INSTRUMENTATIONAND CONTROL

PreparedandCompiledby

Prof.T.K.Ghoshal &Prof.Smita Sadhu

Taughtby

Prof.Smita Sadhu (SS)Deptt.ofElectricalEngg.JadavpurUniversity


2

SyllabusPartI ConceptsofModulatingandSequentialControl.StructureofModulatingControlloops.ProcessControlterminology.ProcessInstrumentationdiagrams.

ControllerImplementation:Electronicanalog,Digital,Pneumatic Controllers.Self-tuningandMultifunctionControllers,ControlValves.ProcessActuators:Electrical,Pneumatic,Hydraulic,Valvepositioners.IndustrialInstrumentationSystems:Components,structure,specification.SelftuningandAdaptivecontrollers.

Supervisorycontrol:ObjectivesandImplementation.

PartII

ConceptofProcessesandUnits:Processstatics,massandenthalpybalance. Modellingofprocessdynamics,ModellingofChemicalprocesses. Singleloopcontrolofstandardfirstorderprocessplants. P-I-Dcontrol,Controllertuning,Ziegler-Nichlol's method,Frequencydomaindesign. Feed-forwardcontrol,Multi-loopandCascadecontrol,Interactionanddecoupling. Non-lineareffectsinplantsandcontrollers. Simulationofprocesscontrolsystems.* BoilerDrumLevelControl.* DiscreteControllers:Selectionofsamplingintervals,stability analysis.*

Books: 1.PrinciplesandPracticeofAutomaticProcessControl- SmithandCorripio 2.PrinciplesofProcessControl- Patranabis 3.AutomaticProcessControl- Eckmann 4.ProcessControlSystems- Shinskey 5.ProcessSystemsAnalysisandControl- Coughanowr &Koppel 6.ChemicalProcessControl- Stephanopoulos 7.Processcontrol- Pollard


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3

PROCESS in Process Control

DomainofInterest Chemical,thermalandbiotechnical

processes

FormalDefinition: Aphysico-chemicalprocess involves

physicalorchemicalchangeofmatter,orconversionofenergy

Importance: Abovetypesofprocessesproduce

productslikefuels,plastics,cement,metals,adhesives,yarnsforcloth,

foods,beverages,medicines,

Allareimportantinourdailylife.

AChemicalPlant


4

CONTROL

in Process Control Theregulation,manipulation(or modulation)of

variablesinfluencingaprocessinsuchawayastoobtainthedesiredqualityandquantityofproductinasafeandefficientmanner.

Hencemodulatingcontrol

Manualmodulationand

controlAutomatic

control

Controllermodulatessteamflowthrough

theactuator

Operatorreadsthe

temperature


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5

About the Subject

Caveat!! Nosinglebookmaybegoodenoughforthiscourse

Solution:

RefreshyourknowledgeinControlSystem

Specially,timesresponse,BodePlot,Nyquist plot,stability

margins

Consultthereferencebooks

Solvetheexercisesintheproblemsheet

Readlecturenotesofcurrentandpreviousyears

Applyyourmindandcommonsense


6

About the Subject II Historyofintroducingthissubject

Introducedinthelateseventies

Digital(electronic)controlwasthenreplacingtraditionalpneumaticanalogcontrol

Expansionofpetrochemicalindustries expandedjobmarket.

ShrinkageintraditionalEEjobmarket.

Consultingengineeringcompaniesrequired

engineerswiththeseskillsets.

Someoffactorshavechangedinthepastdecades,butthesubjectremainsexciting.


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7

About the Subject III

Natureofthesubject Concernscontrolofchemicaland bio-chemical

processes Startedbychemicalengineers

Theirownvocabulary.

Obsessionwithfluids&valves.

FormalizedlaterbyElectricalEngineers&Shinskey Intensiveuseofelementarycontrolengg.principles

Requiresdeepappreciationoffreq.domainmethods.

Mixtureoftheoryandpractice


8

Nature of chemical processes Flowofliquidandslurries Chemicalandbiochemicalreaction Transferofmassandenthalpy MultiInput- multiOutput. Mathematicalmodelmaybeunavailable,unreliable,

complex,non-linear,etc. Typifiedbyunitoperations(connectedthroughanetwork

ofpipesandvalves). Filtration

Distillation(phasechange)

Fermentation Heatexchangers Boilers

Chemicalreactors

AheatExchanger


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9

Nature of chemical processes-II

ADistillationColumn AFractionating(Distillation)

ColumnforPetroleum


10

Nature of chemical processes-III

(a)

(b)

Acontinuousstirredtankreactor(CSTR)

(a)Schematicwithtemperaturecontrolloop

(b)Photograph

StirrerMotor

Cooling

Coils


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11

Objectives of Process Control(and justification of investment)

Ensureimprovedsafetyoftheplant bykeepingprocessvariableswithinsafe

boundaries.

Reducecapitalcost bykeepingprocessvariableswithinboundaries,

requiringlesssafetymargins.

Ensurequalityandconsistencyoftheproduct bykeepingprocessvariableswithinclosetolerance

Ensureproductivity&economyoperation. Maintainingprocessvariablesatappropriatevalues

forallthecomponentsoftheplant.

Reduceabuseofenvironment. Bycontrollingandneutralizingpollutingeffluents


12

Means of attaining Process control objectivesEnsuringpredictableplantperformance

byfeedback

Ensuringplantcontrollability Choosingfinalcontrolelements(valvesand

actuators)

Structureofcontrolloop.(feedback,feedforward,cascade)

Goodpairingofinput-outputs.

Ensuringappropriateoperatingconditions choiceofsetpoints.

EnsuringCorrectsequenceofoperation SequentialControl(PLC)


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13

Components of A PROCESS CONTROL LOOP

Controller

Transducer/

Transmitter

Process

FeedforwardElement

FCE/ValveActuator

PVSPCO

Load Disturbance

Feedforward

Feedbacksignal

FCE:FinalControlElementCO:ControllerOutputPV:ProcessVariableSP:Setpoint

A physical quantity orproperty that can bemeasured and controlled

An elementwhich receives

signals in one form andconverts it to anotherform


14

Process Variable and SET POINTMostfeedbackcontrolloopsattempttomaintain

aprocessvariable(e.g.,temperature,pressure,pH,etc.)atadesiredvalue.

AninputvariablethatsetsthedesiredvalueofcontrolledvariableiscalledtheSetPoint

AControlloopisdesignedsuchthattheprocessvariable isnearlyequaltothesetpoint.

Thesetpointmaybeassigned manuallythroughlocaloperation

Manuallybutfromadistance(remotecontrol) suppliedautomaticallybyanothercontroller(cascade

control).


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15

Actuator & FCE

ProcessvariablesinChemicalprocessesaremostoftenmanipulatedandcontrolledbymodulatingsomeflow

Theflowismodulatedbycontrolvalves

ValvesarethereforecalledtheFinalControlElement(FCE) ApneumaticallyactuatedValve


16

Valves

Manually

operated

Butterfly

Valve

AplugorGlobe

Valve

Diaphragmvalve


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17

Actuator & FCE-II

Inorderthatvalvesmaybemanipulatedbyacontroller,valveactuatorsorvalvemotorsarerequired.

Valvemotorscanbe pneumaticor Electric

Valvepositionersarespecialactuatingsystemswithfeedbackofvalveposition


18

Loads and DisturbancesAprocessoraplantisdesignedtosupplyor

sustainload Ageneratorshould supplyadequatecurrent Aboilershould supplyadequateflowofsteam (loadispartofduty,notdisturbance)

ADisturbanceisexternalinfluencebutnot aload,thatsomehowinfluencethevariabletobecontrolled Oftenthedisturbanceschangewithtime.

Speedchangeintheprimemoverisadisturbancetoadcgenerator

Pressurefluctuationofinletsteamisadisturbancetoaturbine


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19

Loads and disturbances in simple processes

Process Schematicdiagram PV Load Disturbance1. Flow

throughpipe

Flowrate

Downstreampressure

Upstreampressure/fouling

2. Heatexchanger

Processfluido/ptemp

Processfluidflow

Steamtemp/steamflow

3. Levelcontrol

Level Outletflowrate

Inletflowpressure


20

Transmitter AControlLooprequiresthat

ProcessVariableandsomeloads,bemeasuredandthevaluestransmittedtothecontrollerforappropriateaction.

ATransmitter Measuresthevariablewitha

sensor, Convertsthesensorsignaltoa

standardlevelwhichmaybeused(remotely)byanindicatinginstrument,acontrollerorarecorder.

Thesensingelement,mayormaynotbephysicallyinsidethetransmitter.

Atemperaturetransmitterwithoutthe

temperaturesensor


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21

Transmitter-II

StandardoutputsignalofTransmitters DCCurrentLoop(most

common) 4 20mA 0 20mA

DCVoltage 0 10V 0 5V 15V

Pneumatic(nearlyobsolete)

0.21.0Atmosphericpressure

TransmittersaregenerallyspecifiedwithFullScale(range)

Transmitterreadingisusuallyexpressedaspercentageorperunitoffullscale.

Example: Witharangeof1250C

and40%reading, thecurrentoutputfora4-

20mAtransmitterwouldbe4+0.4*16=10.4ma,

correspondingto1250*0.4=500C


22

Transmitter-IIIPreferredStandardtransmitteroutput:

4-20mA(currentloop,suppressedzero)

Advantagesofcurrentloopsignals

Nodangerfromshortcircuit

Novoltagedropconcerns,lowcablecost

Advantageofsuppressedzero

Detectionofcircuitbreak(openloop) Possibilityofsupplyingcurrenttothe

transmitter(twowiretransmitter)


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23

Transmitter-IV

Transmitter

Transducer

DCPowerSource

CurrentLoopSignal

ToController

AFourWireTransmitter


24

Transmitter

TransducerDCPowerSource

Transmitter-V

ATwoWireTransmitter

CurrentLoopSignalToControllerTransmitter

Transducer

Transmitterdrawsa

constant4macurrentforown

powering

Transmitter-V


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25

Transmitter-VI

ATwoWireTransmitterwithSensorandController


26

Transmitter-VIIARosemount

FlowTransmitterconnectedtoatubesection.

Note:

(i)Local Indicator

(ii)pressuretappings

(iii)connectingflanges

FlowPipe

Connecting

Flanges

Pressure

Taps

Pressure

TapsPower

In

Signalout

LocalIndicator


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27

Process Instrumentation Diagram

FT

FC

PV

SP

CO

Flow

Actuator

Valve

FT:FlowTransmitter

FC:FlowController

AlsocalledPipingandInstrumentationDiagram


28

Example (PI Diagram) Identifytransmitters,controllersandFCEinthediagrambelow

Whatisthis,

BTW?

FlowIndicatingController


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29

Recap-Control Loop

Wehavestudiedessentialaspectsofacontrolloop

LoopStructure

Setpoint,ProcessVariable

Valve,Actuator,FinalControlElementandValvepositioners

Transmitter

ProcessInstrumentationdiagram

Comingup:

Controller,ControlLaw,FeedbackandFeedforward


30

Controller Error=SP-PV

Transmitteroutputshouldmatchthecontrollerinputsphysically Type(saycurrentloop)

Range

Controlleroutputandactuatorinputalsoshouldmatchphysically

Signalisonesided,i.e.noreversalofsignals Biasisoftenusedtogenerate

someCOwithnoerror

MaximumCOvoltagecapabilitylimitstheelectricalload

ControlLaw Thedefiningfunctionthat

generatesCOfromallinputs

Controller

Bias FF

SP

PV

CO


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31

Industrial Controllers


32

Control Laws: Feedback & feed forwardControllawdefineshowthecontroller

output(CO)isgenerated.

Infeedback,CO=f1 (PV,SP)

Infeed-forward,CO=f2 (LOAD)

Incombined,CO=f3(PV,SP,LOAD)

ControllerPlant&Actuator

PVSP

CO

Feedback

FeedForward Load


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33

Control Laws:-II

Feedbackapproach: AdjustCOsuchthatPV SPnotwithstanding

loadanddisturbance slowbutgoodadjustmentcapability

Cantolerateinaccurateplantdescription(robustness)

Feedforwardapproach: AdjustCOtonullifytheeffectofloadonPV

fastbut

notaccuratewhenplantdescriptionisapproximate Usedasanauxiliarytofeedback


34

Controllability(in the sense of process control)

Aprocesstobecontrolledmustbecontrollable

Thetermcontrollability hasaspecialmeaninginprocesscontrolcontext.

Thisissometimescalledinput-outputcontrollability

InthesimplestSISOcasethereisonlyoneprocessvariableofinterest.

Forprocesscontrollabilityinthesimplecase: Theremustexistaninputvariable(usuallyaflow),whichmay

bemanipulatedindependently.

Theprocessvariableshouldchangeinresponsetochangeintheinputvariable.

Forastepchangeininput,theoutputshouldeithersettleataconstantvalueorkeeponincreasing(typeonehighertype)

Donotconfuse

withstatecontrollability


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35

Fly-ball governor- The First Feedback Controller

JamesWattsDrawing

Principle

Amoremodernversion(1930)


36

Controllability-II(in the sense of process control)

Formulti-outputsystems,therewouldbemorethanone(saymnumberof)processvariablesofinterest.

Forinput-outputcontrollabilityofmulti-outputcase Theremustbeatleastmnumberofinputvariableswhichcan

beindependentlymanipulated.

Itwouldbepossibletoformmnumberofinput outputpairs,

eachwithanuniqueinput(thatisinputvariablescannotberepeated)and

eachofwhichiscontrollable.

Notethat

Theremaybemorethanoneschemeofpairing Somepairingschemecanbebetterthantheother.

Responseofaprocessvariabletoitsinputpairshouldbemoresignificantthantheresponsetounpairedinput.

Thisiscalleddiagonaldominance.


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37

Modelling ElementaryProcesses


38

Modelling of elementary processFlowthroughOrifice,Venturi andPipe

FlowthroughControlValve

Dyeinjection

Heattransferinanoven

LiquidlevelSystem

Linearization

LessonslearntfromModelling


20/622


39

Flow Through Orifice

q=cd (/4)D22 [2(p1 - p2)/(1- d

4)]1/2

Whereq=volumetricflow,cd isaconstant

D2

=orifice,venturi ornozzleinsidediameter

D1=upstreamanddownstreampipediameter

d=D2 /D1 diameterratio,=densityp1 ,p2=upstreamanddownstreampressures

q=cf(P)

P=p1-p2p1 p2

D1

D2

Usedforsmalltomediumflow

measurement


40

Venturi Tube

Usedformeasurementofmedium(100litres/minute)tolargeflow


21/622


41

Common Flow equation

Thiscommonflowequationmaybeusedfororifice,venturi andlongpipesections

Theflowrateq isvolumetric

Flowisassumedtobeturbulent

Theflowcoefficientcf isstronglydependentondensityandgeometry

andmildlyonReynoldsnumber(1%-2%)

Theflowcoefficientcf isstronglydependentontheunitschosenfor

flowandpressure

Whenpressureisexpressedashead,densityeffectgetscancelled.

Formassflowrate,q istobemultipliedbydensity

q=cf(P)


42

Common Flow equation-II

Theflowequationisstronglynonlinear Forlinear,smallsignalapproximation,localslopemay

beused.

(2%changeinpressuredrop=>1%changeinflow)

qP

Pq

PPPc

PcPPcq

PPPqqq

Pcq

f

ff

f

~~

2

1

~)1

~/1(

~~

~;~

+=

+=

+=+=

=


22/622


43

Control Valve Characteristics

Flowthroughvalvealsoobeysthegeneralflowequation

P indicatespressuredropacrossthevalveTheflowcoefficientcf() becomesafunctionof

valveopening Letthevalveopeningbeexpressedbya

variable ,where =1indicatesfullopening =0indicatesthatthevalveisfullyclosed Theflowequationisvalidwithintheinterval

0.05


23/622


45

Control Valve Characteristics-III

Smallsignallinearequationscanbederivedasabove

Theoperatingpointshouldbeknown

PPcP

Pcq

PcPPcqPcq

PPPqqq

Pcq

ff

fff

f

+

=

++==

+=+=+==

})(){(})(){(

~)(

~)~(;

~)~(~

~;~

;~)(

)~

,

~

,~( Pq


46

Control Valve Characteristics-IVAcontrolvalveisusedinapipingcircuitwith

differenttypesofload

Theflowisdeterminedbythepumpcharacteristics,valveopeningandflowcharacteristicsofthepipingandtheload.

Forefficientflowcontrolinapipingcircuit acontrolvalveisgenerallyconnectedinseries when

aconstantpressure pumpisused

acontrolvalveisgenerallyconnectedinparallelorshunt whenaconstantflow pumpisused

Exercise:Drawdiagramsandwithanalogyfromelectriccircuits,explaintheabove


24/622


47

Control Valve Characteristics-V

a=quickopeningb=linear

c=squareroot

d,e =equalpercentage

f=hyperbolic

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

X/Xmax

F/Fmax

a b

c d

e f


48

Dye injection

v: Averagevelocityoffluid

L:Lengthofpipesection

Delay=L/v

q1Colourless

Fluidflowrate

vq1+q2

L

Dyeq2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

Input

IdealresponseActualresponse

T=L/v

(Delay) Percentagecomposition

=(q2/(q1+q2))*100 (q2/q1)*100=y (say)

q2

y


25/622


49

Dye injection-IIStep Response

q2 Blue:input

Red:idealresponseGreen:actualresponse

Percentagecomposition

=(q2/(q1+q2))*100

(q2/q1)*100=y(say)

y

T=L/v

(Delay)

time

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1


50

Dye injection-IIIDyeinjectionisusedroutinelyforcolouring

subsidizedkerosenetobesoldinFairPriceShops.

Processessimilartodyeinjectionarefairlycommon

Thedelayinresponseforthedyeinjectionandsimilarsystemsiscalledtransportdelay.

Transportdelayisacommonphenomenoninchemicalprocesseswhichinvolvepipelinesand

fluidtransportation.


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51

Heat transfer in an oven

H

watts

Insulation

Charge

m:heatedmass

c:specificheatA:Externalsurfacearea

hAdissipatedHeat

ambienaboveeTemperatur:

Insteadystate,Heatdissipated=Heatinput


52

Heat transfer in an oven- Equations:

Considerheatbalanceequations

mc

H

mc

hA

dt

dor

differenceetemperaturinchangehAdtdcmHor

gssurroundinand

ovenbetweendifferenceetemperaturthAcmtH

fisitactuallythough

etemperaturtoalproportionbetoassumedistransferheatsimplicityFor

CmwatttcoefficientransferHeathcmabsorptionHeat

timetJthAPO

JtHPI

n

=+

=+=

=+=

==

==

=

,

..,

....

)],([

,

/..

)(./

)(./

2 o


27/622


53

Heat transfer in an oven-Transfer function

t

H Hss

t

ss

( )( )

sHsH

sor

hAHconsttimetheishA

mcwhere

hAHmc

H

mc

hA

dt

d

ss

ss

ssss

+=

==

=+=+

1

1

/;

/&


54

Heat transfer in an oven-

non-dominant poles Ifonecarriesoutanexperimentonarealoven,the

responsecouldbeabitdifferent.

Thisisbecausesomeeffectshavebeenneglectedinthesimplemodel.

Forexample: Theovenitselfwouldhaveitsowntimeconstant,whichcanbe

muchmorethanthetimeconstantofthecharge.

Thetemperaturemeasuringdevicemayhaveitsowntimeconstant

Whentheaboveeffectsaremodelled properly,onewouldgetahigherordertransferfunction.

Sometimesonetimeconstantmaybesignificantlylargerthattheother.Thistimeconstantiscalleddominantandtheothersnon-dominant.


28/622


55

Heat transfer in an oven-non-dominant poles-II

( )( ) )1)(1)(1(

1321

sssHsH

sss

ss

+++=

t

ss

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Effectofnon-dominantpoles

appearsasadelay

1st orderresponse

3rdorderresponse

A3rd ordermodelwithtwonon-

dominanttimeconstants


56

Liquid Level Process-AA:Areaofcross-sectionofcontainer

Q1:Inletflowrate

Q2:Outletflowrate

H:Heightoffluid

CylindricalTank,LinearOutputFlowEquation

Q2

A

Q1

H

Q2 dependsonthe

pressureatthebottom-

whichisproportionaltotheheightoftheliquid

column

Inthisspecialcase,outletflow

IsassumedtobelinearwithH.(say); 22 HCQHQ f=


29/622


57

Liquid Level Process-A-II

Insteadystate

AQAHCdt

dH

QHCdt

dHAQQ

dt

dHA

HAtQtQ

f

f

//)( 1

112

21

=+

=+=+

+=

)/~(~~~~

~;

~~

121

122

ff CQHHCQQ

HHQQQ

===

===

DifferentialEquation


58

Liquid Level Process-A-IIITransferFunction

( )( )

sAs

QH

sC

Q

sQ

sH

QHACAwhere

C

QH

dt

dh

C

QH

dt

dhCA

f

f

f

f

f

+=

+=

+=

==

=+

=+

1

1.

1

)~/~(

1

1

]~

/~

/[

)/(

11

1

1

1

1

Timeconstant=timerequiredto

fillthetanktothesteadystate

height!!


30/623


59

Liquid Level Process-B

A:Areaofcross-sectionofcontainer

Q1:Inletflowrate

Q2:Outletflowrate

H:Heightoffluid

CylindricalTank,non- LinearOutputFlowEquation

ThistimeoutletflowrateIsassumed

tobeproportionaltosquarerootofHClosertoempiricalresult.

Pressure squareofflowrate

(say); 22 HCQHQ f=Q2

A

Q1

H


60

Liquid Level Process-B-II

Insteadystate

1

12

21

QHCdt

dHA

QQdt

dHA

HAtQtQ

f =+

=+

+=

HQC

CQHHCQQ

HHQQQ

f

ff

~/

~

)/~

(~~~~

~;

~~

1

2

121

122

=

===

===


31/623


61

Liquid Level Process-B-III

Forsmallchangesaboutsteadystate

dt

dh

dt

dH

hHH

qQQ

qQQ

=

+=

+=

+=

~

~

~

222

111


62

Liquid Level Process-B-IV Hencetheprocessequationsare:

1

1

1

11

11

11

11

~2

~

,

~2

,

~~

2

~,

~~

21

~,

~~1

~,

~)

~(,

qhH

Q

dt

dhAor

qH

ghC

dt

dhAor

qQH

ghCgHC

dt

dhAor

qQH

hgHC

dt

dhAor

qQgH

hHC

dt

dhAor

qQghHCdt

dHAor

f

ff

f

f

f

=+

=+

+=++

+=

++

+=

++

+=++


32/623


63

Liquid Level Process-B-V

( )

( )

sQ

H

sq

shor

QHAwhere

Q

qHh

dt

dhor

Q

qHh

dt

dhQHA

A

qh

HA

Q

dt

dh

+

=

=

=+

=+=+

1

1.~

~2

,

]~

/~

2,[

~~

2,

~~

2)~

/~

2(~2

~

11

1

1

1

1

11

11


64

Liquid Level Process-C

Q1A:Areaoffreeliquidsurface

Q1:Inletflowrate

Q2:Outletflowrate

H:Heightoffluid

R:RadiusCylinder

HorizontalCylindricalTank,non- LinearOutputFlowEquation

(say); 22 HCQHQ f=

Q2

L

H

R

LHRHLHRRbLA222

22)(22 ===


33/623


65

Liquid Level Process-C-II

dt

dh

dt

dH

hHH

qQQ

qQQ

=

+=

+=

+=

~

~

~ changesmallgConsiderin

222

111

2

1

2

1

21

122

~~22

~

~/

~

)/~

(~

~~~

;~

;~~

statesteadyIn

HHRLAA

HQC

CQH

HCQQ

HHQQQ

f

f

f

==

=

=

==

===


66

Liquid Level Process-C-III

DeriveTransferfunctionfromabove

1

11

11

111

11

112

21

~

~~2where;~

~2

~2

~~~~

/1(~~

~/1(

~~~~

~~

~

QAHq

QHh

dtdh

qhH

Q

dt

dhAqQHhQ

dt

dhA

QHhHCdt

dhAQhHC

dt

dhA

QHCdt

dhAQQ

dt

dHA

HAtQtQ

ff

f

==+

=++=++

=++=++

=+=+

+=


34/623


67

Liquid Level Process-D

Q1:Inletflowrate

Q2:Outletflowrate

H:Heightoffluid A:Areaoffreeliquid

surface=R2;R:Radius

(D/B)=(2R/H)

ConicalTank,non- LinearOutputFlow

(say); 22 HCQHQ f=

(say)

})/)(4/{()2/(

2

2222

H

HBDBDHRA

=

===

Q1

D

H

A

B


68

Liquid Level Process-D-II

dt

dh

dt

dH

hHH

qQQ

qQQ

=

+=

+=

+=

~

~

~changesmallgConsiderin

222

111

2

1

2

1

21

122

~~

~/

~

)/~

(~

~~~

;~

;~~

statesteadyIn

HAA

HQC

CQH

HCQQ

HHQQQ

f

f

f

==

=

=

==

===

2

1

2

1

21

122

~~

~/

~

)/~

(~

~~~

;~

;~~

statesteadyIn

HAA

HQC

CQH

HCQQ

HHQQQ

f

f

f

==

=

=

==

===

2

1

2

1

21

122

~~

~/

~

)/~

(~

~~~

;~

;~~

statesteadyIn

HAA

HQC

CQH

HCQQ

HHQQQ

f

f

f

==

=

=

==

===


35/623


69

Liquid Level Process-D-III

DeriveTransferfunctionfromabove

1

3

1

11

11

111

11

112

21

~)

~(2

~

~~2

where;~

~2

~2

~~~~

/1(~~

~/1(

~~~~

~~

~

Q

H

Q

AHq

Q

Hh

dt

dh

qhH

Q

dt

dhAqQHhQ

dt

dhA

QHhHCdt

dhAQhHC

dt

dhA

QHCdt

dhAQQ

dt

dHA

HAtQtQ

ff

f

===+

=++=++

=++=++

=+=+

+=


70

Linearization

Nonlinearfirstorderdifferentialequationscanbelinearizedatequilibriumpoints(steadystatevalues)

vV

fy

Y

fVYf

dt

dy

dt

dYvVVyYY

VYf

VVYYdt

dYVYf

dt

dY

VVYYVVYY)|()|()

~,

~(

~;

~point,nominalabouttheonperturbatismallagConsiderin

0)~,

~(

~;

~;0state,steadyin;),(

~;

~~;

~====

+

+==

+=+=

=

====

0

(Taylorseriesexpansion)


36/623


71

Linearization-II

GeneralCase

HQf

f

hAHCAQH

qAQQdt

dh

AHCQAQQdt

dH

~,

~111

1

121

1

|])}/)/(({)}/([{

/)(/)(

+

=

==

HQCCQH

HCQQHHQQQ

ff

f

~/

~)/

~(

~

~~~;

~;

~~ statesteadyIn

1

2

1

21122

==

=====


72

Linearization-III

Comparewithpreviousresult

hHQ

Hq

hHHH

QHQ

H

q

hHH

CHQH

q

hHHCHQHA

q

dt

dh

HQf

HQf

3

1

2

1

2

13

12

1

~,

~2/33

12

1

~,

~22

11

~2

~

~

}~~2/3

~

~~~

2{1

~

|})(~~

2{1

~

|])}/()/(({~

1

1

=

=

+=

+=

2~~:DProcessFor HA =

1

3

~)

~(2

Q

H =


37/623


73

Lessons learnt from Modelling

Simplechemicalprocessesmaybemodelledwiththehelpofunderlyingphysics.

Manychemicalprocessesaredescribedbynonlineardifferentialequations. Thenonlinearequationscanbelinearizedabout

operatingpoints.

Manychemicalprocessesmayexhibitdelayedresponse.

Thedelayscouldbedueto

Transportoffluidoverlongpipes Contributionofsmallernon-dominanttimeconstants(moreaboutitlater)


74

Step Response of ProcessPlants


38/623


75

Step response of process plants

SelfLimiting1st order

SelfLimitinghigherorder

0 0.5 1 1 .50

0.5

1

S te p Re sp o n se

Time(sec)

Amplitude

0 0.5 1 1 .50

0.5

1

S te p Re sp o n se

Time(sec)

Amplitude

0 0.5 1 1 .50

0.5

1

S te p Re sp o n se

Time(sec)

Amplitude

0 0.5 1 1 .50

0.5

1

S te p Re sp o n se

Time(sec)

Amp

litude

AlsoknownasType0system


76

Step response of process plants-II SelfLimitingOscillatory

withovershoot

SelfLimitingOscillatorywith

overshootandUndershoot

22

2

)(2

)()(

nn

n

sssG

++=

22

2

)(2

)(

2/1

2/1)(

nn

n

ssTs

TssG

++

+

=

Non-minimum

phasezerocauses

undershoot


39/623


77

Step response of process plants-III

Non-SelfLimiting(Type-1):Ramp

Non-SelfLimiting(Type-2):

Parabola

)1()(

ss

KsG

+=

)1(

)(2

ss

KsG

+

=


78

Step response of process plants-IV

SelfLimitingwithtransportdelay

)1()(

s

esG

sT

+=


40/624


79

Response of non-oscillatory higher order processes

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.50

0.5

1

StepResponse

Time(sec)

Amplitude

1/(0.1s+1)2

1/(0.1s+1)3

1/(0.1s+1)4

1/(0.1s+1)5

1/(0.1s+1)6

1/(0.1s+1)7

1/(0.1s+1)8

1/(0.1s+1)9


80

Frequency Response

Revisit


41/624


81

Frequency Response Intro

Frequencyresponsemethodisaveryconvenienttoolforanalyzinglinearsystems,because Frequencyresponsecanbeexperimentallyobtained

CanbeeasilysketchedfromTransferFunction

StabilitymarginsandapproximatespeedofresponsecanbeeasilyobtainedfromOPENLOOPfrequencyresponse

InthiscoursewewouldfrequentlyuseBodePlotandoccasionallyNyquist plot.

Bode:Magnitude,Phaseagainstfreq Nyquist:Phasor plotincomplexplane


82

Frequency Response Intro-II InBodePlot

Frequencyaxisislogarithmic

Magnitudemaybeplotted,eitherinabsolutevalueandusingalogarithmicscale

OrinDecibelvalueusinglinearscale

Wewouldoftenusetheasymptoticplotforaqualitativeanalysis Slopesandcornerswouldbeimportant

consideration

Meaningofunitslopeindecibelscale: 20dB/decade

Meaningofunitslopeinabsolutevaluescale: Onedecadeperdecade

Iffrequencyischanged(n=)10times,themagnitudewouldchange(n=)10times

Example:unitnegativeslope,frequencyincreased7.4times,magnitudewouldbe(1/7.4times)

100

100

1

100

100

1


42/624


83

Frequency Response Intro-III

Meaningofdoublepositive slopeinabsolutevaluescale: Increasefrequencyntimes,the

magnitudewouldbecomen-squaretimes.

Meaningofdoublenegative slopeinabsolutevaluescale: Iffrequencyisincreasedbyntimes,the

magnitudewouldbecome(1/n-square)times

Example:M=100@20r/s,at30r/s,themagnitudewouldbe100X(20/30)2 =44.44

Example:M=100@20r/s,themagnitudewouldbe250at20X((100/250)=

12.65r/s

100

2030

44.44

Doublenegativeslope

12.65

250


84

Bode plot of Integrator

10-1

100

101

102

Magnitude(abs)

100

101

102

103

-91

-90

-89

Phase

(deg)

Frequency(rad/sec)

( ) deg90==

K

j

KjG

GainCrossoveratK(r/s)

Unitnegative

slope

GainatunityFreq=K

Phase

constantat- 90deg


43/624


85

Double Integrator

100

101

102

10-2

10-1

100

101

102

Magnitude(abs)

BodeDiagram

Frequency(rad/sec)

( ) deg180)( 22

==

Kj

KjGDouble

negativeslope

GainCross

overatK(r/s)

Gainatunity

Freq=K

WhatisthePhaseplot??


86

Magnitude plot of first order system( )

( )22

11

+=

+=

K

j

KjG

BodeDiagram

Frequency (rad/ sec)

100

101

102

100

101

Magnitude(abs)

Corner

frequency1/

K

Asymptotic

plot

ActualplotUnitnegative

slope

/1)/1( 2 KKgcf =


44/624


87

Phase lag of first order term

BodeDiagram

Frequency(rad/sec)

100

101

102

-90

-60

-30

0

Phase(deg)

( )

( )

( )

( )

1

2,/1

45,/1

,/1

tan1

2

2

2

1

2

+>>

==


45/624


89

More General TF-2

( ) )1.1(

)02.01(

ss

sKsG +

+

=

100

101

102

103

104

-60

-40

-20

0

20

40

60

Magnitude(dB)

Frequency(rad/sec)

Unitnegativeslope

1st corner

@1r/s

Doublenegative

slope

Unitnegative

slope

2nd corner@50r/s


90

Standard Process Model


46/624


91

Standard type zero process model

Usuallychemicalprocessesarenotoscillatorybynature Manyplantsexhibit1 st order-likeselfregulating

response Havetransportdelayorhigherorderfastpoleswhich

canbeapproximatedasdelay Sometextbookscallthedelayas lag!! Separatemodelrequiredfornon- selfregulatingplants

( )s

eKsG

sT

p+

=

1

1Time

Constant

DelayProcessGain


92

Step Response of Standard Type Zero Process Model

( )s

eKsG

sT

p+

=

1

1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

StepResponse

Time(sec)

Amplitude

T

K1

( ) )()1( /)(1 TtueKtyTt =


47/624


93

Fitting empirical step response in the standard process model

y

K

y

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

0.26K

0.67K

t1 t2

u

t

u

u

yK

=

Att1=T+0.3

y=K(1-e-0.3)=0.26K

Att2=T+y=K(1-e-1)=0.67K

Wemaydetermine andTfromt1 andt2

ProcessReaction

Curve


94

Step Response of Standard Type Zero Process Model

( )s

KesG

sT

p

+=

1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

StepResponse

Time(sec)

Amplitude

T

K


48/624


95

Unit step response for

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System:sys

Time(sec):0.0964

Amplitude:0.1

System:sys

Time(sec):0.511

Amplitude:0.6

StepResponse

Time(sec)

Amplitude

( )( )( )ss

sGp05.015.01

1

++=

0427.0;511.0

),2(&)1(

)2(..........916.0511.0,

6.01

)1........(105.00964.0,

1.01

511.0

0964.0

==

=

=

=

=

T

From

Tor

e

Tor

e

T

T

Asmallnondominanttime

constantcanbeapproximatedas

adelay


96

Unit step response for

StepResponse

Time(sec)

Amplitude

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System:sys

Time(sec):0.1

Amplitude:0.0512

System:sys

Time(sec):0.6

Amplitude:0.628

( )( )( )205.015.01

1

sssG

p ++

=

07.0;535.0

),2(&)1(

)2(..........988.06.0,628.01

)1(..........0525.01.0,

0512.01

6.0

1.0

==

==

=

=

T

From

Tore

Tor

e

T

T

5.0=

( )

1.0205.0

05.012

+

T

Fairlygood

approximationofparameters!

Twonon-dominantand

smalltimeconstants


49/624


97

Alternative method: Graphical method

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

T

Tangenttothe

highestslope

FinalsteadyValue

Smoothentheplotfirst


98

Normalized modelIfthesteadystateO/Pwith100%inputis

alsodefinedas100%O/P,thegainKbecomesunity.

Thenormalizedprocessmodelbecomes:

( )s

esGsT

p

+=

1


50/625


99

Frequency response of plant

( )

( ) ( )

( ) ( )

( ) ( ) =

+===

=+

===

+==

+=

j

KjGejG

j

KjGejG

s

KsGandesGLet

s

KesG

Tj

Tj

sT

sT

p

1

1

1

1

21

21

21

??

?? ??

??


100

Magnitude contribution of delay( ) TjejG =1

=1forall

frequencies

1sincos == TjT


51/625


101

Phase contribution of delay

( ) TTT == cossintan1

-T

( ) TjejG = 1


102

Bode plot for delay usinglinearscale for frequency

10-1

100

101

Magnitude(abs)

0 50 100 150 200 250 300 350 400 450-540

-450

-360

-270

-180

-90

0

Phase(deg

)

BodeDiagram

Frequency(rad/sec)

-T


52/625


103

Magnitude contribution of first order term

( )( )

22

11

+=

+=

K

j

KjG

BodeDiagram

Frequency (rad/ sec)

100

101

102

100

101

Magnitude(abs)

Cornerat

1/

K

Asymptoticplot

Actualplot


104

Phase contribution of first order term

BodeDiagram

Frequency(rad/sec)

100

101

102

-90

-60

-30

0

Phase(deg)

( )

( )

( )

( )

1

2,/1

45,/1

,/1

tan1

2

2

2

1

2

+>>

==


53/625


105

Frequency Response of Standard TypeZero Process Model

sTesG=)(1

BodeDiagram

Frequency(rad/sec)

10-1

100

101

Magnitude(abs)

100

101

102

-90

-45

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

10-1

100

101

Magnitude(abs)

101

102

103

-540

-450

-360

-270

-180

-90

0

Phase(deg)

1/s

sG+

=1

1)(2

Whyistheshapelikethis?


106

Frequency Response of Standard Type ZeroProcess Model

-20

-15

-10

-5

0

5

Magnitude(dB)

10 20 30 40 50 60 70 80 90 100

-180

-135

-90

-45

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

Frequencyscaleislinearforphaseplot

1/


54/625


107

Frequency Response of Standard Type ZeroProcess Model

-20

-15

-10

-5

0

5

Magnitude(dB)

10-1

100

101

102

-180

-135

-90

-45

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

1/


108

Frequency Response of Standard Type Zero Process Model

Comments Ordinaryfirstordersystemhas90degree

PhaseMarginandinfinitegainmargin.Suchplantsarealwaysstableinclosedloopwithconstantgain.

Standardprocessplantisnotordinarybecauseofthedelay

Delayhasgotthecapabilityofprovidinginfinitephaselag

Delayed1st orderplantsareliabletobecomeunstableinclosedloop


55/625


109

Effect of gain on magnitude plotBodeDiagram

Frequency(rad/sec)

10-1

100

101

102

103

10-1

100

101

Magnitude(abs)

K=1

K=2

K=3

gcf2

gcf3

Unitygain

1/


110

GM=1/|G(jwp

)|

PM=180deg+/G(jwg)

10-1

100

101

Magnitude(abs)

BodeDiagram

Frequency(rad/sec)

100

101

102

-180

-135

-90

-45

0

Phase(deg)

pcf=wp

gcf=wg

PhaseMargin

Gain

Margin

gcfdecreaseswithgain

pcf doesnot

depend

upongain

PMincreases

withdecrease

ingain

GMdecreases

withincreasein

gain


56/625


111

Nyquist plot

Resultantplot

sTesG

=)(1 ssG

+= 11)(2


112

Pad Approximation forDelay


57/625


113

Pad Approximation

Thedelaytermisdifficulttohandlewhilecomputingclosedloopresponse.

FrenchmathematicianHenriPad (1863-1953),formulatedthe"best"approximationofagivenfunctionbyarationalfunction ofspecifiedorder.

UsingtheabovemethodtheLaplacetransformexp(-sT)ofdelay,maybeapproximatedbytransferfunctionsofanygivenorder.

Normally,formanualcomputation1st orsecondorderapproximationsareused.

Forcomputersimulation5

th

to10th

orderapproximationsmaybeemployed.


114

Pad Approximation -II

( ) ( )

( ) ( )...

!3

2/

!2

2/2/1

...!3

2/

!2

2/2/1

32

32

2/

2/

2/2/

++++

++==

sTsTsT

sTsTsT

e

eeee

sT

sT

sTsTsT

Thesetermsarechangedslightlytoaccommodatethe

effectoftruncationofhigherorder

terms

Pade approximationmaybeappreciatedfromthis


58/625


115

First order Pad approximation

Frequencyresponse

Unitstepresponse

TimeResponseofprocessplantwithfirst

orderPad appx.fordelay

( )2/1

2/11

sT

sTesG

sT

+

=


116

Frequency response of First order Padapprox. for unit delay

BodeDiagram

Frequency(rad/sec)

10-2

10-1

100

101

102

-180

-90

0

Phase

(deg)

10-1

100

101

Magnitude(abs)

Maxphaselagis-180degree

Unitygain


59/625


117

Unit step response of 1st order Padapprox. for delay

0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1StepResponse

Time(sec)

Amplitude

100%Negativekick

CausedbyNonMinimumPhase

term


118

Time Response of standard plant with 1st order Pad

apprx. for delay

0 0.5 1 1.5 2 2.5 3-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1StepResponse

Time(sec)

Amplitude

Negative

kick reducedduetoplanttimeconstant

( )2/1

2/1

1 sT

sT

s

KsGp

+

+=

OpenLoop


60/626


119

Second order Pad approximation

Frequencyresponse

Unitstepresponse

TimeResponseofprocessplantwith

secondorderPad appx.fordelay

( )12/2/1

12/2/122

22

1

TssT

TssTesG

sT

++

+=


120

Frequency response of Second order Padapprox. for delay

10-1

100

101

Magnitude(abs)

10-1

100

101

102

-360

-270

-180

-90

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

Maxphaselag

is-360degree


61/626


121

Unit step response of 2nd order Pad approx.for delay

StepResponse

Time(sec)

Amplitude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Smallernegative

kick for2nd

order

positivekick


122

Time Response of standard plantwith 2nd order Pad appx. for delay

StepResponse

Time(sec)

Amplitude

0 1 2 3 4 5 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2 Open

Loop

Kickssubstantially

reducedduetothetimeconstant


62/62


123

Higher the order, better the approximationUnit step response

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

time

Firstorder

Secondorder

Thirdorder

Fourthorder

Puredelay

Higher the order, better the approximation

Frequency response

10-2

10-1

100

101

102

-540

-360

-180

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

10-1

100

101

102

-540

-360

-180

0

Phase(deg)

BodeDiagram

Frequency(rad/sec)

10-1

100

101

102

-540

-360

-180

0

Ph

ase(deg)

BodeDiagram

Frequency(rad/sec)

0

Firstorder

Secondorder

Thirdorder

Phasekeepsonincreasingastheorder

becomeshigher

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