Essentials of Process Control phần 1 ppt

Bailey and Oiiis: Biochcnlicul Ertgiwcrirrg l;rtrrtltrrilcrltlrlsBennett and Myers: Momentum, Heat, and Mass Transfer Brodkey and Hershey: Transport Phenomena: A Un$ed Approach Carberry:

Essentials of Process Control

McGraw-Hill Chemical Engineering Series

Editorial Advisory Board

James J Cat-berry, Pro~ssor of Cltc~tttiutl Ett,qitwcrittg, iJttil~cr.si!\~ of No,rc I~rrttw

James R Fair, Professor of Cltctttical Engitwcrittg, Univcrsi~y of l?.w.s, Austitt

Eduardo D Glandt, Prof~~ssor ~~/‘Cltcmitul Ettgittwrittg, Utriv~r.si!\~ (?f Pottt.s~~I~Vtttitr

Michael T Klein, Prof~~ssot- o~‘Chcttric~tl Ettgirtwrittg, Utti\~c~rsity of’l~c~lttwtrc

Matthew Tirrell, Profc~s.sor of Chcttticai Ettgitrwrittg, Utti\rt-.sity of‘Mitmc.sortr

Emeritus Advisory Board

Max S Peters, Retired Professor of Chemical Engineerittg, Univer.sity of Colorado

William P Schowalter, Dean, School of Engineering, University of 1Ilinoi.s

James Wei, Dean, School (?f’Engineering, Prittceton University

.

Building the Literature of a Profession

Fifteen prominent chemical engineers first met in New York more than 60 years ago

to plan a continuing literature for their rapidly growing profession From industrycame such pioneer practitioners as Leo H Baekeland, Arthur D Little, Charles L.Reese, John V N Dorr, M C Whitaker, and R S McBride From the universitiescame such eminent educatdrs as William H Walker, Alfred H White, D D Jackson,

J H James, Warren K Lewis, and Harry A Curtis H C Parmelee, then editor

of Chemical and Metallurgical Engineering, served as chairman and was joinedsubsequently by S D Kirkpatrick as consulting editor

After several meetings, this committee submitted its report to the McGraw-HillBook Company in September 1925 In the report were detailed specifications for acorrelated series of more than a dozen texts and reference books which have sincebecome the McGraw-Hill Series in Chemical Engineering and which became thecornerstone of the chemical engineering curriculum

From this beginning there has evolved a series of texts surpassing by far thescope and longevity envisioned by the founding Editorial Board The McGraw-HillSeries in Chemical Engineering stands as a unique historical record of the devel-opment of chemical engineering education and practice In the series one finds themilestones of the subject’s evolution: industrial chemistry, stoichiometry, unit oper-ations and processes, thermodynamics, kinetics, and transfer operations

Chemical engineering is a dynamic profession, and its literature continues toevolve McGraw-Hill, with its editor B J Clark and its consulting editors, remainscommitted to a publishing policy that will serve, and indeed lead, the needs of thechemical engineering profession during the years to come

.-

Bailey and Oiiis: Biochcnlicul Ertgiwcrirrg l;rtrrtltrrilcrltlrls

Bennett and Myers: Momentum, Heat, and Mass Transfer Brodkey and Hershey: Transport Phenomena: A Un$ed Approach Carberry: Chemical and Cutulytic Reaction Engineering

Constantinides: Applied Numerical Methods with Personal Computers Coughanowr: Process Systems Analysis and Control

de Nevers: Air Pollution Control Engineering

de Nevers: Fluid Mechanics for Chemical Engineers Douglas: Conceptual Design of Chemical Processes Edgar and Himmelblau: Optimization of Chemical Processes Gates, Katzer, and Schuit: Chemistry of Catalytic Processes Holland: Fundamentals of Multicomponent Distillation Katz and Lee: Natural Gas Engineering: Production and Storage King: Separation Processes

Lee: Fundamentals of Microelectronics Processing Luyben: Process Modeling, Simulation, and Control for Chemical Engineers Luyben and Luyben: Essentials of Process Control

McCabe, Smith, and Harriott: Unit Operations of Chemical Engineering Marlin: Process Control: Designing Processes and Control Systems .for Dynamic Pe$ormance

Middlemann and Hochberg: Process Engineering Analysis in Semiconductor Device Fabrication

Perry and Chilton (Editors): Perry S Chemical Engineers’ Handbook Peters: Elementary Chemical Engineering

Peters and Timmerhaus: Plant Design and Economics for Chemical Engineers Reid, Prausnitz, and Poling: Properties of Gases and Liquids

Smith: Chemical Engineering Kinetics Smith and Van Ness: Introduction to Chemical Engineering Thermodynamics Treybal: Mass Transfer Operations

Valle-Reistra: Project Evaluation in the Chemical Process Industries Wentz: Hazardous Waste Management

ESSENTIALS OF PROCESS CONTROL

2 3 4 5 6 7.8 9 0 BJE PMP 9 8 7

This book was set in Times Roman by Publication Services, Inc.

The editors were B.J Clark and John M Mom’ss;

the production supervisor was Denise L Puryear.

The cover was designed by Wanda Kossak.

Library of Congress Cataloging-in-Publication Data

Luyben, Michael L., (date)

Essentials of process control / Michael L Luyben, William L.

Luyben.

p cm.

I n c l u d e s i n d e x

ISBN o-07-039 172-6 - ISBN o-07-039 1734

1 Chemical process control I Luyben, William L II Title.

TP155.75.L89 1997

When ordering this title, use ISBN o-07-114193-6

Printed in Singapore

ABOUT THE AUTHORS

William I, Luyben has devoted over 40 years to his profession as teacher, searcher, author, and practicing engineer Dr Luyben received his B.S in ChemicalEngineering from Pennsylvania State University in 1955 He then worked for Exxonfor five years at the Bayway Refinery and at the Abadan Refinery (Iran) in planttechnical service and petroleum processing design After earning his Ph.D from theUniversity of Delaware in 1963, Dr Luyben worked for the Engineering Department

re-of Du Pont in process dynamics and control re-of chemical plants

Dr Luyben has taught at Lehigh University since 1967 and has participated inthe development of several innovative undergraduate courses, from the introductorycourse in mass and energy balance through the capstone senior design course and

an interdisciplinary controls laboratory He has directed the theses of more than 40graduate-students and has authored or coauthored six textbooks and over 130 tech-nical papers Dr Luyben is an active consultant for industry in the area of processcontrol He was the recipient of the Eckman Education Award in 1975 and the In-strumentation Technology Award in 1969 from the Instrument Society of America.William L Luyben is currently a Professor of Chemical Engineering at LehighUniversity

Michael L Luyben received his B.S in Chemical Engineering (1987) and B.S.

in Chemistry (1988) from Lehigh University While a student, he worked during eral summers in industry, including two summers with Du Pont and one summer withBayer in Germany After completing his Ph.D in Chemical Engineering at Prince-ton University in 1993, working with Professor Chris Floudas, he joined the ProcessControl and Modeling Group in the Central Research and Development Department

sev-of Du Pont His work has focused on the dynamic modeling and control sev-of chemicaland polymer plants He has worked on plant improvement studies and on the design

of new facilities Luyben has authored a number of papers on plantwide control and

on the interaction of process design and process control

Michael L Luyben is currently a research Engineer with Du Pont’s Central search and Development Department

Re-To Janet Niche! Luyhen-rnotheq wife, friend, loving grandmother, avid gardenec community volunteer; softhall queen extraordinaire-for 34 years

of love, care, level-headed.financial advice, and many pieces of Grandmother

of love, care, level-headed.financial advice, and many pieces of Grandmother Lester’s apple pie.

1.2.1 Proportional and Proportional-Integral Level Control / 1.2.2 ,Temperature Control of a Three-Tank Process

. 1.4.1 Process Control Laws / 1.4.2 Languages qf Process

Control / 1.4.3 Levels of Process Control

P A R T I Time Domain Dynamics and Control

2.2.1 Linearization / 2.2.2 Perturbation Variables

2.3.1 First-Order Linear Ordinary Differential Equation / 2.3.2 Second-Order Linear ODES with Constant CoefJicients / 2.3.3 Nth-Order Linear ODES with Constant Coefficients

3.2.1 SpeciJications for Closedloop Response / 3.2.2 Load Pcrjormance

xii (‘ON’I‘IiN I‘S

3.3 Contrciler Tuning

3.3 I Rules of Thumb / 3.3.2 On-Line Trial and Error / 3.3.3 Ziegler-Nichols Method / 3.3.4 Tyreus- Luybcrr Method

3.4 Conclusions Problems

5 Interaction between Steady-State Design

and Dynamic Controllability

5.2 I Liquid Holdups / 5.2.2 Gravity-Flow Condenser

Simple Quantitative Example

5.3.1 Steady-State Design / 5.3.2 Dynamic Controllability / 5.3.3 Maximum Heat Removal Rate Criterion

Impact of Controllability on Capital Investment and Yield

5.4.1 Single-Reaction Case / 5.4.2 Consecutive Reactions Case

General Trade-off between Controllability and Thermodynamic Reversibility

Quantitative Economic Assessment of Steady-State Design and Dynamic Controllability

5.61 Alternative Approaches / 5.62 Basic Concepts of the Capacity-Based Method / 5.63 Reactor-Column-Recycle Example

Conclusion

6 Plantwide Control

6.1 Series Cascades of Units 6.2 Effect of Recycle on Time Constants 6.3 Snowball Effects in Recycle Systems

02

99

99

1.51151

152 153

165

6.4 llsc of Steady-State Sensitivity Analysis to Screen

6.4 I Control StrircYurt~s Screened

6.5 I Complete One-Pass Conversion / 6.5.2 Incomplete Conversion Case / 6.5.3 Interaction between Design and Control / 6.5.4 Stability Analysis

7 Laplace-Domain Dynamics

7.1 Laplace Transformation Fundamentals

7.1 I Dejnition / 7.1.2 Linearity Property

7.2 Laplace Transformation of Important Functions

7.2.1 Step / 7.2.2 Ramp / 7.2.3 Sine / 7.2.4 Exponential / 7.2.5 Exponential Multiplied by Time / 7.2.6 Impulse (Dirac Delta Function 6~~))

7.3 Inversion of Laplace Transforms7.4 Transfer Functions

7.4.1 Multiplication by a Constant / 7.4.2 Diflerentiation with Respect to Time / 7.4.3 Integration /

7.4.4 Deadtime

7.5 Examples7.6 Properties of Transfer Functions

7.61 Physical Realizability / 7.6.2 Poles and Zeros / 7.6.3 Steady-State Gains

7.7 Transfer Functions for Feedback Controllers7.8 Conclusion

Problems

8 Laplace-Domain Analysis of Conventional Feedback Control Systems

8.1 Openloop and Closedloop Systems

8.1.1 Openloop Characteristic Equation /

8 I 2 Closedloop Characteristic Equation and Closedloop Transfer Functions

8.2 Stability8.3 Performance Specifications

8.3 I Steady-State Per$ormance / 8.3.2 Dynamic Specifications

229 229 230

234 237

241 249

254 255 255

265 265

271 273

x i v (‘ON’I‘I1N’I‘S

8.4 Root Locus Analysis

8.4 I DeJirtition / 8.4.2 Construction of Root Locus Curves

8.5 ConclusionProblems

9 Laplace-Domain Analysis of Advanced

276

287 288

301 301 308

316

323 326

331 331

10 Frequency-Domain Dynamics

10.1 Definition10.2 Basic Theorem10.3 Representation

10.3.1 Nyquist Plots / 10.3.2 Bode Plots / 10.3.3 Nichols Plots

10.4 Computer Plotting

10.4.1 FORTRAN Programs for Plotting Frequency Response / 10.4.2 MATLAB Program for Plotting Frequency Response

10.5 Conclusion

Problems

11 Frequency-Domain Analysis

of Closedloop Systems

I 1.1 Nyquist Stability Criterion

11.1 I Proof / 11 I.? tk~mples / Il 1.3 Representation

339 339 341 344

360

369 370

372 372

I I .2 Closedloop Spccilications in the Frequency Domain

11.2 I I’Iimt~ Mtrrgirr / 11.2.2 Goin Margin /

! 1.2.-j Mtrximrtm Closedloop Log Modulus (,:I”” )

I I 3 Frequency Response of Feedback Controllers

11.3 I t’roporiionc~l Cor~troller (I’) /

11.3.2 I’rop~~rtioncil-Intcsrul Controller (PI) / 11.3.3 I-‘rol’ortiorltrl-Intc~Srcrl-Deri~)ative

13.2.1 Engineering Judgment / 13.2.2 Singular Wue Decomposition

13.3 Selection of Manipulated Variables13.4 Elimination of Poor Pairings

13.5 BLT Tuning13.6 Load Rejection Performance

4 2 9 429

440

447

452 452

456 456 457

459 460 461 466

14.4.1 Definition / 14.4.2 Derivation of z Transforms

of Common Functions / 14.4.3 Effect of Deadtime / 14.4.4 z Transform Theorems / 14.4.5 Inversion

14.5 Pulse Transfer Functions14.6 Hold Devices

14.7 Openloop and Closedloop Systems14.8 Stability in the z Plane

14.9 Conclusion

Problems

15 Stability Analysis of Sampled-Data Systems

15.2 Frequency-Domain Design Techniques

15.2.1 Nyquist Stabiliry Criterion / 15.2.2 Rigorous Method / 15.2.3 Approximate Method / 15.2.4 Use

of MATLAB

15.3 Physical Realizability15.4 Minimal-Prototype Design15.5 Conclusion

Problems

477477

480483486

496498499509

5 1 1.511

513513521

528529535535

16.1 I Controol-Relevant Identification / 16.1.2 Frequency

Content of tlw Innut .Civnnl / Ifi I 1 Mml~l Order

16.4 I Autofutzing / 164.2 Approximate Transfer Functions

567567571572~

Index

The field of process control has grown rapidly since its inception in the INOs Directevidence of this growth in the body of knowledge is easily found by comparing thelengths of the textbooks written over this time period The first process control book(Cealgske, 1956) was a modest 230 pages The popular Coughanowr and Koppel(1965) text was 490 pages The senior author’s first edition ( 1973) was 560 pages.The text by Seborg et al (1989) was 710 pages The recently published text byOgunnaike and Ray (1994) runs 1250 pages!

It seems obvious to us that more material has been developed than can be taught

in a typical one-semester undergraduate course in process control Therefore, a shortand concise textbook is needed that presents only the essential aspects of processcontrol that every chemical engineering undergraduate ought to know The purpose

of this book is to fulfill this need

Our intended audience is junior and senior undergraduate chemical engineeringstudents The book is meant to provide the fundamental concepts and the practicaltools needed by all chemical engineers, regardless of the particular area they eventu-ally enter Since many advanced control topics are not included, those students whowant to specialize in control can go further by referring to more comprehensive texts,such as Ogunnaike and Ray (1994)

The mathematics of the subject are minimized, and more emphasis is placed

on examples that illustrate principles and concepts of great practical importance.Simulation programs (in FORTRAN) for a number of example processes are used togenerate dynamic results Plotting and analysis are accomplished using computer-aided software (MATLAB)

One of the unique features of this book involves our coverage of two ingly important areas in process design and process control The first is the interac-tion between steady-state design and control The second is plantwide control withparticular emphasis on the selection of control structures for an entire multi-unit pro-cess Other books have not dealt with these areas in any quantitative way Because

increas-we feel that these subjects are central to the missions of process design engineersand process control engineers, we devote two chapters to them

We have injected some~ examples and problems that illustrate the plinary nature of the control field Most control groups -in industry utilize the tal-ents of engineers from many disciplines: chemical, mechanical, and electrical Allengineering fields use the same mathematics for dynamics and control Designingcontrol systems for chemical reactors and distillation columns in chemical engineer-ing has direct parallels with designing control systems for F-16 fighters, 747 jumbojets, Ferrari sports cars, or garbage trucks We illustrate this in several places in thetext

interdisci-This book is intended to be a learning tool We try to educate our readers, notimpress them with elegant mathematics or language Therefore, we hope you findthe book readable, clear, and (most important) useful

xix

I The most important lesson to remember is that our focus as engineers must be

on the process We must understand its operation, ob-jectives, constraints, anduncertainties No amount of detailed modeling, mathematical manipulation, orsupercomputer exercise will overcome our ignorance if we ignore the true subject

of our work We need to think of Process control with a capital P and a small c.

2 A steady-state analysis, although essential, is typically not sufficient to operate

a chemical process satisfactorily We must also understand something about thedynamic behavior of the individual units and the process as a whole At a mini-mum, we need to know what characteristics (deadtimes, transport rates, and ca-pacitances) govern the dynamic response of the system

3 It is always best to utilize the simplest control system that will achieve the desiredobjectives Sophistication and elegance on paper do not necessarily translate intoeffective performance in the plant Careful attention must be paid to the practicalconsequences of any proposed control strategy Our control systems must ensuresafe and stable operation, they must be robust to changes in operating conditionsand process variables, and they must work reliably

4 Finally, we must recognize that the design of a process fundamentally determineshow it will respond dynamically and how it can be controlled Considerations

of controllability need to be incorporated into the process design Sometimes thesolution to a control problem does not have anything to do with the control systembut requires some modification to the process itself

If we keep these ideas in mind, then we can apply the basic principles of processcontrol to solve engineering problems

Michael L Luyben William L Luyben

As the field of process control has matured over the last 30 years, it has become one

of the core areas in chemical engineering along with thermodynamics, heat fer, mass transfer, fluid mechanics, and reactor kinetics Any chemical engineering.graduate should have some knowledge not only of these traditional areas but also

trans-of the fundamentals trans-of process control For those trans-of us who have been part trans-of thisperiod of development, the attainment of parity with the traditional areas has beenlong overdue

The literature in process control is enormous: over a dozen textbooks and sands of papers have been published during the last three decades This body ofknowledge has become so large that it is impossible to cover it all at the undergrad-uate level Therefore, we present in this book only those topics we feel are essentialfor gaining an understanding of the basic principles of process control

thou-One of the important themes that we emphasize is the need for control engineers

to understand the process-its operation, constraints, design, and objectives Theway the plant is designed has a large impact on how it should be controlled and whatlevel of control performance can be obtained As the mechanical engineers say, youcan’t make a garbage truck drive like a Ferrari!

We present in the following section three simple examples that illustrate theimportance of dynamic response; show the structure of a single-input, single-outputconventional control system; and illustrate a typical plantwide control system.Throughout the rest of the book, many more real-life examples and problems arepresented All of these are drawn from close to 50 years of collective experience

of the authors in solving practical control problems in the chemical and petroleumindustries

EXAMPLES OF PROCESS DYNAMICS AND CONTROL

E X A M P L E 1 I Figure I 1 shows a tank into which an incompressible (constant-density)liquid is pumped at a variable rate Fo (gal/min) This inflow rate can vary with timebecause of changes in operations upstream The height of liquid in the vertical cylindricaltank is h (ft) The flow rate out of the tank is F (gal/min)

Now Fo, h, and F will all vary with time and are therefore functions of time 1.Consequently, we use the notation Fo(,), II(,), and F(,) Liquid leaves the base of the tank

via a long horizontal pipe and discharges into the top of another tank Both tanks areopen to the atmosphere

Let us look first at the steady-state conditions By “steady state” we mean the ditions when nothing is changing with time or when time has become very large Math-ematically this corresponds to having all time derivatives equal to zero or allowing time

con-to approach infinity At steady state the flow rate out of the tank must equal the flow rateinto the tank: Fo = F In this book we denote the steady-state value of a variable by anoverscore or bar

For a given F, the height of liquid in the tank at steady state h is a constant, and

a larger flow rate requires a higher liquid level The liquid height provides just enoughhydraulic pressure head at the inlet of the pipe to overcome the frictional pressure losses

of the liquid flowing down the pipe

The steady-state design of the tank involves the selection of the height and diameter

of the tank and the diameter of the exit pipe For a given pipe diameter, the tank height

must be large enough to prevent the tank from overflowing at the maximum expectedflow rate Thus, the design involves an engineering trade-off, i.e., an economic balancebetween the cost of a taller tank and the cost of a bigger-diameter pipe A larger pipediameter requires a lower liquid height, as illustrated in Fig 1.2 A conservative designengineer would probably include a 20 to 30 percent over-design factor in the tank height

to permit future capacity increases

Safety and environmental reviews would probably recommend the installation of ahigh-level alarm and/or an interlock (a device to shut off the feed if the level gets too high)

to guarantee that the tank could never overfill The tragic accidents at Three Mile land, Chernobyl, and Bhopal illustrate the need for well-designed and well-instrumentedplants

Is-Now that we have considered the traditional steady-state design aspects of this fluidflow system, we are ready to examine its dynamics What happens dynamically if wechange Fo, and how will h(,, and F(,, vary with time? Obviously, F eventually has to end

up at the new value of Fo We can easily determine from the steady-state design curve

of Fig 1.2 where h will be at the new steady state But what dynamic paths or timetrajectories will h(,, and F(,, take to get to their new steady states? Fig 1.3 shows twonncc,hlp t,-a.,,,r,~n, f-nc.,.,,\n,~,,<r /r r n I ?\ r- I- 1 1 fl t # ,

in h and F to their new steady-state values Curves 2, however, show the liquid heightrising above (“overshooting”) its final steady-state value before settling out at the newliquid level Clearly, if the peak of the overshoot in h were above the top of the tank, wewould be in trouble.

Our steady-state design calculations tell us nothing about the dynamic response ofthe system They tell us where we start and where we end but not how we get there ThisL;nr{ fif;nfnrmnt;,., :” r- ,., I-A I- 1 ’ ,- ’

T T I;,

Oil feed

t 7

Sensor

FIGURE 1.4

EXAMPLE 1.2 Consider the heat exchanger sketched in Fig 1.4 An oil stream passesthrough the tube side of a tube-in-shell heat exchanger and is heated by condensingsteam on the shell side The steam condensate leaves through a steam trap (a devicethat permits only liquid to pass through it, thus preventing “blow-through” of the steamvapor) We want to control the temperature of the oil leaving the heat exchanger To dothis, a thermocouple is inserted in a thermowell in the exit oil pipe The thermocouplewires are connected to a “temperature transmitter, ” an electronic device that converts themillivolt thermocouple output to a 4- to 20-mA “control signal.” This current signal issent to a temperature controller, an electronic, digital, or pneumatic device that comparesthe desired temperature (the “setpoint”) with the actual temperature and sends out asignal to a control valve The temperature controller opens the steam valve a little if thetemperature is too low and closes the valve a little if the temperature is too high

We consider all the components of this temperature control loop in more detail later

in this book For now we need only appreciate the fact that the automatic control of somevariable in a process requires the installation of a sensor, a transmitter, a controller, and afinal control element (usually a control valve) A major component of this book involveslearning how to decide what type of controller should be used and how it should be

“tuned,” i.e., how the adjustable tuning parameters in the controller should be set so that

EXAMPLE 1.3 Our third example illustrates a typical control scheme for a simplifiedversion of an entire chemical plant Figure 1.5 gives a sketch of the process configurationand its control system Two liquid feeds are pumped into a reactor, in which they react toform products The reaction is exothermic, and therefore heat must be removed from thereactor This is accomplished by adding cooling water to a jacket surrounding the reactor.The reactor effluent is pumped through a preheater into a distillation column that splits

it into two product streams

Traditional steady-state design procedures are used to specify the various pieces ofequipment in the plant:

Fluid mechanics: pump heads, rates, and power; piping sizes; column tray layoutand sizing; heat-exchanger tube and shell side baffling and sizing

Hear trcrmfer: reactor heat removal; preheater, reboiler, and condenser heat transferareas; temperature levels of steam and cooling water

Chenticd kinetics: reactor siLe and operating conditions (temperature, pressure,catalyst, etc.)

Tllerrno~i~tl~lmi~s czrlcl NULV fr~rns/tit.: operating pressure, number of plates and

3 dolple

; the

al isares

ut a

’ the

ater3me

n d aIves

j bethat

n

lfiedtion

ct to

I thector.plits:s ofyoutlsfersure,

d lilib-

re-But what procedure do we use to decide how to control this planC’l We spend most ol‘oul time in this book exploring this important design and operating problem Our studies of‘ process control are aimed a~ undcrs~anding the dynamics of processes and control sys- tems so that we can develop and design plants that operate more efficiently and safely, produce higher-quality products, arc more easily controlled, and are more cnvironmen- tally friendly.

For now let us merely say that the control system shown in Fig I .S is a typical

conventional system It is about the minimum rhat would be needed to run this plant automatically without constant operator attention Even in this simple plant with a min- imum of instrumentation, IO control loops are required We will find that most chemical engineering processes are multivariable The key to any successful control system is

1.2

SOME IMPORTANT SIMULATION RESULTS

In the preceding section we discussed qualitatively some concepts of dynamics andcontrol Now we want to be more quantitative and look at two numerical examples

of dynamic systems: The first involves level control in a series of tanks The secondinvolves temperature control in a three-tank process These processes are simple,but their dynamic response is rich enough that we can observe some very importantbehavior

1.2.1 Proportional and Proportional-Integral Level Control

The process sketched in Fig 1.6 consists of two vertical cylindrical tanks with a levelcontroller on each tank The feed stream to the first tank comes from an upstreamunit The liquid level in each of the tanks is controlled by manipulating the flowrate of liquid pumped from the corresponding tank The level signal from the leveltransmitter on each tank is sent to a level controller The output signal from eachcontroller goes to a control valve that sets the outflow rate

FO

I

FIGURE 1.6

Level control.

A clynnlnic model of this process contains two ordinary differential equations,

which arise from the lotal mass balance on each of the tanks We assume constantdensity

(1.1)

(1.2)

where A,, = cross-sectional area of izth tank

h,, = liquid height in nth tankF,, = volumetric flow rate of liquid from nth tank

Fo = volumetric flow rate of feed to processThe flow rates F1 and F2 are set by the controller output signals CO, and CO2 fromthe two level controllers The process variable signals PV, and PV2 from the twolevel transmitters depend on the two liquid levels 111 and 152 In this example weexpress these PV and CO signals as fractions of the full-scale range of the signals.The signals from transmitters and controllers are voltage, current, or pressure signals,which vary over standard ranges (0 to 10 V, 4 to 20 mA, or 3 to 15 psig)

max

where F,‘Fax = flow rate when the control valve is wide open

where h,,,span = the “span” of the level transmitter, i.e., the difference between themaximum and minimum liquid levels measured in the tank Numerical values of allparameters and the values of the variables at the initial steady-state conditions aregiven in Table 1.1 The FORTRAN program used to simulate the dynamics of theprocess is given in Table 1.2 For more background on dynamic modeling and sim-ulation methods, refer to W L Luyben, Process Modeling, Simulation and Control

TABLE 1.1

Values of parameters and steady-state variables

Diameter of tank = 10 ft Cross-sectional area of tank = 78.54 ft*

Span of level transmitters = 20 ft Maximum flow rate through control valves = 200 ft3/min Steady-state flow rates = 100 ft3/min

Steady-state levels = 10 ft Bias value of level controllers = Bias = 0.5 fraction of full scale Setpoint signals of controller = SP = 0.5 fraction of full scale Steady-state value of controller outputs = CO = 0.5 fraction of full scale Steady-state value of level transmitter outputs = PV = 0.5 fraction of full scale

data delta, mop/ 1,200./

c controller calculations to get flow- rates

c All control signals (pv, sp, and co) are in fractions of full scale

whr

Not

is tl

CIWTEK I: Introduction 9

TARt,E:1.2 ( C O N T I N U E D )

FORTRAN simulation program for PI level control

c print and store far plotting

if( time Lt tprin t)go to 30

where Bias, = a constant (the value of CO when PV is equal to SP)

,

I& = controller gain

SP, = setpoint of the controller, i.e., the desired value of PVNote that if the liquid level goes up, PV goes up, CO goes up, and F increases This

is the correct response of the level controller to an increase in level

Figure 1.7 shows the dynamic responses of the two liquid levels and the twooutflow rates when a 10 percent increase in the feed flow rate to the process occurs