The Optimization Algorithm rFSQP with Application to Nonlinear Model Predictive Control of Grate Sintering

with Application to Nonlinear Model

Predictive Control of Grate Sintering

Thesis by

Frode Martinsen

Submitted in partial fulllmentof the requirementsfor the degreeof

Doktor Ingenir

Department of Engineering Cybernetics

Norwegian University of Science and Technology

N-7491 Trondheim, Norway

August 2001

Report2001:9-W

Preface

This thesishasbeensubmittedto the NorwegianUniversityof Scienceand

Technology (NTNU) in partialfulllment of the requirementsfor the degree

of Doktor Ingenir.

Thedoctoralprojecthasbeenaccomplishedat theDepartment of engi-

neering cybernetics, NTNU, incooperation with Elkem Sauda, Norway, and

theDepartmentofChemicalEngineeringattheCarnegieMellonUniversity.

The ElkemASAplantat SaudaisnowapartofErametNorwayAS,asub-

sidiaryoftheErametGroup. MysupervisorshavebeenprofessorBjarneA.

Fossandassociate professorTorA.Johansen. One semesterofthedoctoral

project was spent at the Department of Chemical Engineeringat Carnegie

MellonUniversityunderthesupervisionofprofessorLorenzT.Biegler. The

work has been supported by the Norwegian Research Council (NFR) grant

#119314/221.

Acknowledgments

Duringtheworkonthisthesis,manypeoplehavebeeninvolvedinguidance,

discussions,solvingpractical problems, and motivating me. I would like to

mention some ofthem here.

First,sincerethankstoprofessorBjarneA.Fossforinitiatingtheproject,

accepting me as a doctoral student, and for his optimistic attitude, inspi-

ration, and supportthroughout the work. He also established the contact

and opened up forthe cooperation with professorLorenz T. Bieglerat the

Carnegie MellonUniversity.

DuetoprofessorLorenzT. Biegler'swell-knownexperienceinoptimiza-

tion, he taught me virtually all there is to know aboutpractical optimiza-

tion. He included me in his work-group asone of his students and shared

with me from his long scientic experience. He also provided access to his

optimization codes which has been of paramount importance in my work.

His hospitalityand friendly nature have made hima highly regarded men-

tor. A particular thanksto Ph.D.student AndreasWachter from professor

Biegler's group for his hospitality duringmy stay at Carnegie Mellon Uni-

versity,andforalwaysansweringmyquestions byemailduringthelasttwo

years.

I would also like to thank associate professor Tor Arne Johansen, for

always takingan interest indiscussingsome of themathematical details.

Iamgratefulforreceivingtheopportunityofworkingonindustrialprob-

lems within theframe of a doctoral degree. I acknowledge the cooperative

spiritand helpfulness that I have met withinElkem ASA, and I especially

wish to thank Stein Wasb and Ragnar Tronstad for their valuable sup-

port and cooperation. Personnel at the sintering plant, Marianne Bran-

dett, Bente Baugst, Leif-Idar Rossemyr 1

, Sigurd Simonsen and Gunnar

Mrkesdal all provided essential assistance during the measurement cam-

paigns. Morten Raanesat SINTEFMaterialsTechnology performed theSEM

analysisdocumentedinsection 2.6.3,whileTone Anzjnpreparedthesam-

plesforthese tests.

A good working environment including colleagues, supervisors, secre-

taries and lab personnel has been an important factor for me to succeed

with this project. I also thank the members of the process control group

at NTNU, led by professor Bjarne A. Foss, for an inspiringand social envi-

ronment. In particular, Geir Stian Landsverk who has been sharing oÆce

with me duringthe last year and a-halfis duely thanked for his insightful

commentsonthesinteringprocessandfor(always)acceptingoerstowaste

time in non-academic dialogue. Geir Stian has also checked the thesis for

typograhpical errors etc. My rst oÆce-mate, my friend Vidar Srhus, is

alsothanked. Without himIwouldneverhave startedon thisproject.

My parents and family are thanked for believing in me. My thanks to

Ida,Kent and "beste" for theirpatience. My wife, AnneBerit, has always

encouragedmeand believedinme, and forher patience,Ilove and respect

her.

Thelastthreeyears ofthiswork,myson,Magnus,hasbeenmygreatest

sourceofinspiration. Thankyou.

Frode Martinsen

Trondheim,August, 2001

1

SteinWasb,RagnarTronstad,BenteBaugstandLeif-IdarRossemyrarepresently

withErametNorwayAS.

Summary

This thesis contributesto the research on optimization algorithmsfor non-

linear programming,and to theapplication ofsuch algorithmsto nonlinear

modelpredictivecontrol.

Regardingthecontributionto research onalgorithmsfornonlinearpro-

gramming,anovelalgorithmisputforwardwithacompletetheoryforglobal

and localconvergence. Thisis the maincontributionof thethesis. The al-

gorithm,namedrFSQP,isareducedHessianFeasibleS equentialQuadratic

Programmingmethod. Itremainsfeasiblewithrespecttononlinearinequal-

itiesatallSQPiterations,butnonlinearequalityconstraintsaretreatedasin

generalreducedHessianSQPmethods. TherFSQPalgorithmisimplemented

inMATLABandtestedonanumberofsmallscaleproblemswithencouraging

results. However, the algorithm is designed for large scale problems with

few degreesof freedom. Somepreliminarytesting ofthealgorithm on large

scaleproblems areinvestigated.

Thethesisalsocontributesto theunderstandingoftherelationbetween

sequential and simultaneous reduced gradient methods, and to the under-

standing of the relationbetween discretizationmethods fordynamical sys-

temsand the choice of optimizationalgorithms.

The thesis also contributes to model based control approaches of grate

sintering. Grate sintering is a complex metallurgical process, where melt-

ing of solids and fast gas dynamics give rise to sti process models, i.e.

the "time constants" of the system dier by many decades in magnitude.

Hence, application of real-time optimization methods like nonlinear model

predictive control to the grate sintering process is challenging. The thesis

gives aframework forimplementingnonlinearmodel basedcontrolof grate

sintering bygiving a control objective, a nonlinearmodel and choosing an

appropriate discretization scheme. The thesis gives a reduced order model

whichislesscomputationallydemanding. Datafromindustrialexperiments

are usedto adaptthe modeland to assess thecontrolobjective.

1 Introduction 1

1.1 Motivation . . . 1

1.2 Contributions . . . 3

1.3 Outline . . . 3

2 Sintering 5 2.1 Notation forchapter 2 . . . 5

2.2 Background . . . 8

2.2.1 Process goals . . . 9

2.2.2 Controlobjectives . . . 14

2.3 Modeling . . . 16

2.3.1 GlobalPDEmodel . . . 17

2.3.2 Modellingassumptions. . . 27

2.3.3 Comparisonwithexisting models . . . 28

2.3.4 Computationofthe controlinputv . . . 28

2.4 Experimentsand measurementcampaigns . . . 32

2.4.1 Plant and experiment description . . . 33

2.4.2 Measurementsat plant: Campaign2 . . . 34

2.4.3 Controlaction experiments: Campaign 1 . . . 37

2.5 Model adaptionand validation . . . 41

2.5.1 FullPDEmodel . . . 42

2.5.2 Reducedmodel . . . 43

2.5.3 Introductorycontrolaction simulations . . . 45

2.5.4 Model validation . . . 46

2.6 Data analysis . . . 47

2.6.1 Productivity . . . 49

2.6.2 Qualitytests . . . 50

2.6.3 Assessment oftheobjective function . . . 54

2.7 Conclusions . . . 58

3 rFSQP - a feasible SQP method 65

3.1 Background . . . 65

3.1.1 Background on optimization. . . 66

3.2 Introduction. . . 74

3.3 TherFSQP algorithm . . . 78

3.3.1 The feasibilitymechanism . . . 78

3.3.2 The feasiblereducedHessian method (rFSQP) . . . . 80

3.3.3 The BFGS updatescheme . . . 85

3.3.4 The rFSQPalgorithm . . . 87

3.4 Globalandlocalconvergence . . . 89

3.4.1 KKT conditions . . . 91

3.4.2 Globalconvergence . . . 93

3.4.3 Localconvergence . . . 99

3.5 Implementationand results . . . 107

3.5.1 Implementationdetails. . . 108

3.5.2 Results . . . 111

3.6 Conclusions . . . 117

4 NMPC 119 4.1 Introduction. . . 120

4.2 Optimizationmethods . . . 123

4.2.1 Reduced gradient methods . . . 125

4.3 Simulations . . . 129

4.3.1 Implementationissues . . . 130

4.3.2 Case1: CSTR . . . 131

4.3.3 Large scaleapplicationsof rFSQP . . . 135

4.3.4 Case2: Gratesintering . . . 138

4.4 Discussion . . . 140

4.5 Conclusion . . . 143

5 Conclusion 145 5.1 Conclusionsto thethesis. . . 145

A Appendices to chapter 2 159 A.1 Sinter objective . . . 159

A.2 Pitotmeasurements . . . 160

B Appendix to chapter 3 163 B.1 Proof oflemma 3.3 . . . 163

C Appendices to chapter 4 167

C.1 ExplicitEuler . . . 167

C.2 Runga-Kutta 4 . . . 169

C.3 Lobatto IIIC . . . 172

C.4 sSQP. . . 175

D Reprint of the CCA-paper 179

2.1 Zones 1-5inthesinteringbed . . . 10

2.2 Simpliedsinteringplant . . . 11

2.3 Sigmoidfunction . . . 14

2.4 Specic heatcapacities of Mn-oxides . . . 21

2.5 Equilibrium gasratio log 10 (p CO2 =p CO ) for reduction of Mn- oxides . . . 22

2.6 Ergun's relation. . . 31

2.7 Detailed sinteringplant . . . 34

2.8 Instrumentationofsinteringpan . . . 35

2.9 Batchwise experiments . . . 36

2.10 Bootstrapestimates . . . 38

2.11 Experiment1: Measurements . . . 39

2.12 Controlactionexperiments . . . 40

2.13 Simulation - fullmodel. . . 42

2.14 Gas concentrations- fullmodel . . . 44

2.15 Simulation - constant v . . . 45

2.16 Simulation - proledv . . . 46

2.17 Simulation - T s -proles. . . 47

2.18 Validation ofmodel. . . 48

2.19 Set-pointsforexperiments . . . 49

2.20 Retort usedforreductionexperiment . . . 53

2.21 Qualitymodel. . . 55

2.22 Standarddeviation ofqualitymodel . . . 56

2.23 In-and output data . . . 58

2.24 Assessment of theobjective function . . . 59

2.25 SEM exp.1 . . . 61

2.26 SEM exp.7 . . . 62

2.27 SEM exp.10 . . . 63

3.1 Conceptualcomparison betweenan infeasible SQPand a fea-

sibleGRGmethod . . . 69

3.2 IterationsforrSQPandrFSQPonhs12fromtheHock-Schittkowski test set. . . 114

3.3 IterationsforrSQPand rFSQPon hi3from Himmelblau. . . . 115

4.1 NMPCof CSTR . . . 132

4.2 Contoursof NMPCof CSTR . . . 133

4.3 Task managercrop image . . . 137

4.4 rFSQPon largescaleCSTR . . . 139

4.5 NMPCofreducedsintermodelwithsSQP, MAXIT=5 . . . 141

4.6 NMPCofreducedsintermodelwithsSQP, MAXIT=1 . . . 142

A.1 Experiment 1: Pitotmeasurements . . . 161

2.1 Measured freshfeedfrom thefeedbins . . . 37

2.2 Bootstrapestimates . . . 37

2.3 Productionrates . . . 51

2.4 Mechanical strength ofsinter . . . 52

2.5 Reducibilityofproduced sinter . . . 54

2.6 Assessment of theobjective function . . . 57

3.1 rFSQP on theHock-Schittkowski test set . . . 112

3.2 rFSQP on hs12. . . 116

3.3 rFSQP on hi3 . . . 117

4.1 Nonlinear MPCon aCSTR:Basic SQP . . . 134

4.2 Nonlinear MPCon aCSTR:sSQP . . . 135

4.3 Nonlinear MPCon aCSTR:rFSQP. . . 136

Introduction

1.1 Motivation

Optimality is a natural phenomenon which has engaged scientists in vari-

ous guises for centuries. The following quote illustrates its generality and

importance:

"Sincethefabricoftheuniverseismostperfect,andisthework

of a most wise Creator, nothing whatsoever takes place in the

universe in which some form of maximum and minimum does

notappear."

LeonhardEuler,1744 1

Acreekalwaystakingthesteepestpathdownhillisanexampleofmini-

mization ofenergy. Consider anotherexample; imaginea blindperson,call

him Mr. Iterate, on top of the mountain Besseggen 2

and ask him to nd

his way down on his own. By careful steps he explores thedownhill path,

going one step at a time, adaptinghis step lengths to theterrain and pos-

siblyusinghismemoryto correct zig-zagging. Eventually,he arrivesat the

saddlepoint betweenthetwolakes,and after testingforfurtherdescentdi-

rections,declaresthatheisnowatthelowestpoint. Ofcourse,ifhestrayed,

say 5 meters, towards anyof the two lakeshe would have senseda descent

direction and continued his path. However, from his point of view there

1

RestatedfromTroutman(1996),p.339.

2

AfamousmountainhikeinNorwaycrossesthisridgewhichhasasaddlepointbetween

agreyandagreencoloredlakeoneachsideofthetrekbetweentwopeaks.Itiscommonly

believedthatIbsen'scharacterPeerGyntjumpedothisridgeonhisbuck.

is nothing to gain by moving away from his present position. The terrain

seemsto beat, i.e. he hasstopped at alocal solution.

Althoughthereexistsaconsiderablebodyofoptimizationexamplesthat

can be handled by pen-and-paper calculations, computerized solutions of

optimalityproblemshaveemergedduringthelastfewdecadesasapowerful

tool forsolvinglargerand harderoptimizationproblems.

A computer program forsolving optimizationproblems commonlyiter-

atestheproblem,andtriestoimproveonthepresentsolution. Iftheproblem

is nonlinear but smooth and analytic (exists and is dierentiable), a com-

monapproach is to linearize and solve simpler subproblems. The solutions

tothese subproblemsbecome searchdirections,and itis thencustomaryto

moderate the step lengths to compensate for theerror inthe linearization.

Linearizationonlyprovideslocalinformationandextrapolatinginformation

too farcan be hazardous.

Returning to Mr. Iterate, this could be the way he chooses where to

place his next step; he perturbs ("linearizes") the terrain in front to nd

thesteepest descent and moderates his step length ifthe terrain is rugged

orvery steep. Note that Mr. Iterate does not have a look-ahead property,

buthe hasa memory. Usinghismemory hecan speedup hisdescent ifthe

steepest descent path tend to zig-zag. I.e. if he experiences zig-zagging he

canbendhisstepdirectionstowardswhatseemstobethehistoricalaverage

directiontowardstheminimum. Needlesstosay,hewouldnotbepleasedif

theterrainsuddenlyrevealed a discontinuousverticalwall.

Iftherearefencesintheterrainanditisrequiredthattheoptimalpoint

shouldbe within thefences, Mr.Iterate may considersearching withinthe

fences, orto cross them and search for the lowest topologicalpoint on the

outside,butkeepinginmindthatheshouldreturntotheinsideforthenal

point. Perhapsitisreasonablethathewouldmakeonlyconservative strolls

outsidethefences, andhesitate strayingtoo faraway from them?

Ifthere arestrongerrequirements, e.g.that theoptimalpoint shouldlie

on a fence or a trek, Mr. Iterate could be forced to follow the trek or to

stray from it. If there are many treks it could then be time consuming to

trace themall.

Summarizing the various ways and reasonings of Mr. Iterate in an al-

gorithm and implementing this in a computer program is precisely what

is undertaken in this thesis. The algorithm shall be a generic and eec-

tiveset of rulesthat appliesto all problemsfalling withina speciedsetof

assumptions. The proposed algorithm, termed rFSQP,is designed to solve

nonlinearoptimizationproblemswiththousandsofvariablessubjecttonon-

linear constraints. Specically,rFSQP is designed to always remain within

(e.g.feasible)allin-/outside fences(inequalities),butmaystrayfrom"on"-

fences (equalities) away from thenal point, i.e.it is an inequalityfeasible

algorithm fornonlinearoptimizationproblems 3

.

Like Mr. Iteratethe algorithm willconverge to a locally optimalpoint,

and closetotheoptimalpointtheconvergencewillbesuÆcientlyfast. The

computer programis testedon anumberofsmallsample problems,and on

two largerproblems. The rst largeproblem behaves nicely,i.e. itchanges

by relativelymoderate rates inall directions. The second problemis more

challenging; it is the case of grate sintering. The dynamical model of this

problem hasstrong nonlinearities and the time constants are separated by

several decades. Bothproblemsare examplesof nonlinearmodelpredictive

control(NMPC).

1.2 Contributions

The main contribution of this thesis is the optimization algorithm rFSQP.

A completeconvergenceanalysisis given, consideringbothglobalandlocal

properties, and the algorithm is implemented and tested on a number of

problems.

The maincase is grate sintering. For this case a reduced order model

is developedfrom modelsavailableintheliterature. Industrialexperiments

includingspecialmeasurementswere conducted, and thedata wasused for

modeladaptionand assessingthe controlobjective.

Various implementation issues concerning the interplay between dis-

cretization and optimization nonlinear MPC are explored. This provides

insightintohowacontinuoustimemodelmustbediscretizedto allowopti-

mization.

Appendix D gives a generic approach to representing hyperbolic PDE's

as multi-models. Due to numerical diÆculties the approach has not been

pursuedfurther.

1.3 Outline

Thischapterhaspresentedthemotivation forthepresentwork,whatisbe-

lievedto be itsmaincontribution,andplaceditinabroadercontext. More

detailedbackground, includingreferences, aregiven inthe introductions to

chapter 2 and3.

3

Such problems belong to the class of nonlinear programming problems, which is a

subclassofmathematicalprogrammingproblems.

Chapter2 presents thegrate sinteringcase, with a nonlinearPDE model,

controlobjective, industrialexperimentsand dataanalysis.

Chapter3 presentstheoptimizationalgorithmrFSQPwithglobalandlocal

convergenceanalysis,implementationdetailsandnumericalresultson

a selectionof smallsample problems.

Chapter4 appliestheoptimizationalgorithm rFSQPto twodierent non-

linear modelpredictive control(NMPC) examples of dierentcomplex-

ity. The secondexample isthegrate sinteringexample.

Chapter5 endsthethesis and givesits conclusions.

Appendices A to C providesome additionaldetailsforthevariouschap-

ters.

Appendix D is a reprint of the paper Martinsen, Johansen, and Foss

(1999).

The notation is consistent within each chapter, but to conform to the

common notation within the sintering and optimization literature, respec-

tively,the notationis not consistent between dierent chapters. The nota-

tionforchapter 2 isgiven insection2.1.

Sintering

This chapter considers the metallurgical process of grate sintering. The

purposeis to develop and assess a dynamicmodel suitable formodelbased

control. The model is adapted to industrial data from experiments con-

ducted at the sinteringplant at Saudain south-west Norway. The process

at Saudais batch-wisesintering ofmanganese ore, butmuch of thediscus-

sion, and in particular the model, is equally relevant for travelling grate

sintering. It should be noted that manganese isin manyaspects similarto

iron which ismostfrequentlyconsideredintheliterature.

This chapter starts with some background on sintering in section 2.2,

i.e.theprocess isdescribedwithsome commentsonthesinteringplantasa

whole. Theunderlyingprinciplesgoverningthequalityandproductionrate

of sintering are reviewed. In section 2.3 the model is documentedand dis-

cussed,whiletheindustrialexperimentsand measurementsaredescribedin

section 2.4. Modeladaptionto theindustrialdataand modelvalidationare

adressed in section2.5, whiledatareconciliationand analysis is considered

in section 2.6. Some concluding remarks and a summary of the contribu-

tionsofthischapterfollowsinsection2.7. Thenotationusedinthischapter

is summarized in section 2.1. Parts of this chapter have beenpublished in

Martinsen, Johansen, andFoss (1999), which isreprintedin appendixD.

2.1 Notation for chapter 2

Arabic letters

a

i

- polynomialcoeÆcient [-]

A

b

- specicsurface area[m 2

=m 3

]

A

f

- frequency factor[m=s p

K]

c

p

- specicheat capacitiesof solid[J=kgK]and gas[J=molK]

d

p

- average particlediameter[m]

D

ON

- axial gasdispersioncoeÆcient (O

2 -N

2 ) [m

2

=s]

E - activation energy [kJ=mol]

G - massow rateof gasused inequation(2.2)[m 2

t=h]

G

s

- gas volumepersintermassusedinequation (2.2) and (2.3) [m 3

=t]

F

i

- liquidfraction

h

c

- convective heattransfer coeÆcient [J=m 2

sK]

h

m

- mass transfercoeÆcient[m=s]

H - heat ofreaction [J=mol]

k

th

- thermalconductivityof gas[J=mK]

k

r

- chemicalreaction rate [m=s]

K

r

- overall combustion rateconstant [m=s]

L - height of bed [m]

L

f

- latent heat offusion[kJ=mol]

L

v

- latent heat ofevaporation[J=kg]

m

i

- exponent [-]

M

i

- molecular weight [kg=mol]

n

C

- numberof coke particles perunit bed volume [1=m 3

]

n

i

- molar ow[mol=m 2

s]

p

i

,p - (partial) pressure,dierentialpressure[Pa]

P Voice

l ;t

- Laminar or turbulent Voice gas permeability of bed (units vary ac-

cording to exponent m

1

inequation (2.1))

r

i

- reaction rate [mol=m 3

s]

R - universalgas constant 8.314[J=molK]

~

R

R ,R

R

- reaction rate forwater [mol=m 3

s]

s

i

- stoichiometricconstant [-]

T

s;g

- temperature, solidorgas [K]

T

fu

- incipientmeltingtemperatureof solids[K]

T

w

- wet-bulbtemperature[K]

v - gasvelocity[m=s]

v

0

- gasvelocity asreferredto theemptybedusedinequation (2.2) and

(2.3) [m=h]

v

w

- heat wave velocityusedinequation (2.3) [m=h]

W - actualmoisture content (=x

H2O(l )

) [kg=m 3

]

W

cr

- critical moisturecontent [kg=m 3

]

x

i

- component concentration of solids[kg=m 3

]orgas [mol=m 3

]

Greek letters

" - void fraction[ ]

' - scaling factor fortheSherwoodrelation[-]

g

- adiabaticconstant [ ]

- viscosity[kg=ms]

- density[kg=m 3

]

! - specic humidity[-]

Dimensionlessnumbers

Re - Reynoldsnumber: R e= dpG

Sh - Sherwoodnumber: Sh= hmdp

D

ON

Sc - Schmidnumber: Sc=

D

ON

Nu - Nusseltnumber: Nu= h

c d

p

k

Pr - Prandtl number: Pr = cp

k

2.2 Background

Themainpurposeof sinteringisto convertweakly-boundedgranulesintoa

partiallyfusedporoussinter cake suitableforfeedinga furnace. Inthe fur-

nacethesinteredoreis reducedwith carbon. Sinteringisan agglomeration

process inwhich negraded materialsarepartiallyfused into larger lumps

by heating the charge through coke combustion. Fusion occurs when the

solidcharge particlesundergo re-crystallizationacrosstheoldgrainbound-

aries, and possibly by simultaneous softening and partial melting. Sinter-

ing is a complex process involving ow of gas through a packed bed, heat

andmasstransferbetweengasandsolids,heterogeneouschemicalreactions,

and meltingof solids. The heat for sintering of oxidic ores like Fe

x O

y and

Mn

x O

y

is provided byheat exchange between gasand solidsand by com-

bustionof coke. Typicallythe temperature must be raised to the range of

1000 o

C 1400 o

C. Since onlyapproximately5% (weight)of coke isneeded

to raisethetemperatureto thislevel, theprocessis generallyconsideredto

be economic in terms of energy. Water is added to micro-agglomerize ner

granulesbythecapillaryforcesofwater. Micro-agglomerationincreasesthe

gas permeability of the bed allowing a larger gasow, which in turn im-

provessinteringconditions. Thecriticalwatercontentistypicallybelow8%

(weight). Typical granule sizes of the incoming material are in the range

3mm to 6mm. A large fraction of ner grades can cause low gas perme-

abilityof thebed,whilecoarser gradescan give poormicro-agglomeration.

Bothcases give poorsinteringconditions.

Sintering plantsare commonly located close to a furnace, since, due to

its low mechanical strength, sinter is ill-suited for transportation and ex-

tensive handling. The mechanical strength of the sinter can be increased

byadding more coke and therebyincreasing thesinteringtemperature. In-

creased sintering temperature allows a larger fraction of the solidsto melt

thereby giving a stronger sinter. However, this also gives a glassy surface

of the sinter, which impairs its reducibilitysince the eective surface area

is reduced. Hence, thereis a compromisebetweenmechanicalstrength and

reducibilitywhich ismainlycontrolledbythecoke weight percent.

The metallurgicalprocess of sintering prepares theore to form suitable

feed for a (blast) furnace. Granulated ore and coke are mixed, moistened

with water and micro-pelletized to form the charge. The charge is loaded

onto a grate and levelled to form a bed which is ignited by a gas-fuelled

ignitionhood. Aheatwaveand cokecombustionzonetravelsdownthrough

the bed under the inuence of a suction pressure. Hot gas from the com-

bustion zone passes through moist charge deeper in the bed where water

evaporates. Theprocess canbedividedinto vesubsequentzones;heatex-

change,fusion,combustion,dryingandovermoistcharge. Thisisillustrated

in gure 2.1. A number of operations, such as feed mixing, feed charging,

crushing of the produced sinter cake, screening and recycling of nes are

needed. A simpliedoutlineof theoverallsinteringplant isshowningure

2.2. The sinteringplant as a whole is onlyconsidered insection 2.4where

the experimentsconductedat theplant aredescribed. The dynamicmodel

in section 2.3 is concerned withthe sinteringprocess itself,i.e. the process

taking place insidethesinteringpan.

The industrialplant at Sauda produces manganese alloys, such as fer-

romanganese (FeMn) and silicomanganese (SiMn) from manganese ore

in electric furnaces. FeMn is typically used as an addition in the steel

industry to produce certain steel qualities. Such steel qualities are used

in rail-way tracks, wear-plates, etc. Because of the high reduction tem-

perature the electric furnace is competitive with the blast furnace, espe-

cially forhigh purityqualitieswhere thecarbon content of FeMn mustbe

low. FeMn-alloyswithlowandmediumcarbon(LC=MC)content arepro-

ducedinasubsequentreningprocess(MOR).Limestone,CaO,iscommonly

added in ironmaking to adjust the basicity dened by the weight-ratio of

(CaO+MgO)=(SiO2 +Al2O3). Limestone is not used at the sintering

plant inSauda,and isnotincludedin themodel. Foradiscussionofbasic-

ity in the context of manganese reduction, see Rosenqvist (1983), p. 357.

General properties of the sintering process are described in Schluter and

Bitsianes(1962).

2.2.1 Process goals

Forthesinteringprocesstheoverallgoalisto producesinterataprescribed

qualityandrateat thelowestpossiblecost. Theprocessdependentoutputs

h[m]

T[ C] ^o

1

2 3

4 5

0 100 900 1200 1300

Zone 5: Overmoist Preheating of wet charge R e-condensation of w ater vapor Zone 4: Drying and preheating Water evaporates

Zone 3: Coke combustion alone Zone 2: Fusion: Melting of solids When all coke is combusted at level 4 the temperature cools off to the freezing level 5 Zone 1: Cooling

Incoming cold air heat-exchanges with the hot sinter cake

Dry air at 1 atm and room-temperature

Offgas

Figure 2.1: Zones 1-5 in the sintering bed. The gure shows a snapshot of a

verticalsliceofthesinteringbedapproximatelymid-waythroughtheprocess. The

vezones aredescribed tothe left,with thecorrespondingtemperatureproleto

the right. As time goes by the temperature prole proceeds down through the

bed, i.e.at atime earlierthanthesnap-shotin thegurethetemperatureprole

isshifted upwards,and at later timesthe proleis shifted downwardsrelativeto

theoneshownin thegure. Theogas isledthroughpipesto acyclone, andthe

pressurebelowthepanislessthan1atm. Atthetimeinstanceshowninthegure

coolairfromthesurroundingsisbeingsuckedintozone1whereitheat-exchanges

withthehotsintercake. I.e.theairispreheated,andthesinteriscooled. Atlevel

5thetemperatureis1200 o

C,whichistheapproximatefreezingpointofthesinter.

In zone 2 the temperature is above the freezing point, and the solids are partly

melted depending on the heat available for fusion. At level 4 allthe cokein the

bed has been combusted. Hence, abovethis point there are nosources of energy

andthe available energybetween level4and 5is beingused forfusion andheat-

exchange. Atlevel3theheatfrom cokecombustionhasraisedthetemperaturein

thebed to the melting point of thecharge. From level 3to 4cokeis combusted

andfusioncommences. Atlevel2theignitiontemperatureofcokeisreached,and

theheatfrom cokecombustionquicklyraisesthetemperature. Inzone 4,hotgas

from zone 3 pre-heats the charge, and evaporates water. Water vapor is being

transporteddownthroughthebedtozone5whereitmayrecondensate. Atlevel1

thetemperatureofthechargeisattheboilingpointofwater,andabovethis level

waterevaporates.

are the quality in terms of mechanical strength and reducibility (Dawson

1993), and theproductionrate.

Figure 2.2: Simplied sintering plant. Feed, consisting of amix of various ores,

coke and usually lime stone, enters to the left. The composition of the feed is

known, but not the moisture content. The feed is mixed with water and (hot)

recycleofundersizedparticles. Theadhesivecapillaryforces ofwatergivesmicro-

agglomeration of the ner particles. Only ne particles are micro-agglomerized,

andlargefractionsofcoarserparticlesareusuallyundesirable. Thetemperatureof

therecyclecanbeabovetheboilingpointofwater,henceevaporationofwatercan

makeitdiÆculttomix in theoptimal watercontent. Themix ofore, coke,water

and recycleisthenchargedonto thesinteringbedwhichis supportedunderneath

byagrate. Asuctionpressureisestablishedandairisdrawndownthroughthebed.

The sintering process commences when ignition is applied to the top of thebed.

The cokein thetop layerof thebedis ignitedand thesinteringprocessproceeds

asdescribedin gure2.1. After completion ofthe sinteringprocess theproduced

sinter cakeis crushedinto manageablelumpsby amechanicaldevice. Thesinter

isthen screened,andthenesarerecycled,whilecoarsersinter isstoredorfed to

thefurnace.

Production rate

The productivity is quantied by the stationary Voice gas permeability

P Voice

(Voice,Brooks,and Gledhill1953)inthelaminarandturbulentow

regimes:

P Voice

l

=v

L

p

1:0

= 1

150

"

3

(1 ") 2

d 2

p

P Voice

t

=v

L

p

0:5

= q

1

1:75

"

3

1 "

0:5

d 0:5

p

(2.1)

The productivityrelationthusbecomes (Olsen1997):

G= v

G

s

= P

Voice

G

s

p

L

m1

(2.2)

wherem

1

=0:5forturbulentowandm

1

=1forlaminarow. Themassair

owGthenservesasanon-linemeasureofproductionrate. Notethatwhen

signicant melting occurs, the solidsbecomes a continuous media and the

particle size d

p

is nota meaningfulparameter. Dawson (1993) suggests to

selectd

p

astheminimummeasuredsievefractionofincomingoresinthecase

of signicant melting. The process variable inuencingon productionrate

ismainly thepermeabilityP of thebed which inturn isinuenced mainly

by water content. Water is added to micro-agglomerize ner particles by

the capillary forces of water. This increases the average particle diameter

andthevoidfractionof thebed ifthewatercontent iskeptbelowacritical

value W

cr

. A large air ow is promoted by a large permeability which in

turn forces the heat wave to travel faster through the sintering bed, thus

giving a shorter batch duration and consequently an increased production

rate.

Utilizing the Voice permeability P Voice

to estimate production rate is

impracticalsinceitdoesnotaccount fortheamount ofrecycleintheplant.

Inthepresentwork,P Voice

isdiscardedandtheproductionrateisestimated

from the calculated recycle. The recycle stream serves as an indirect on-

linemeasureof mechanical sinterquality: Poormechanicalqualitywillgive

increasedrecyclerate,which inturnreduces theproductionrate.

Quality

The sinter quality is determined by the amplitude and shape of the heat

wave. Increased coke content increases the maximum sintering tempera-

ture, T

s;max

(Venkataramana, Gupta, Kapur, and Ramachandran 1998),

butto achieve high reducibilityT

s;max

should not be too high (Toda, Sen-

zaki, Isozaki, and Kato 1984). In addition, proper ignitionis necessary to

establishinitialconditionsforsintering(DashandRose1977). Asdiscussed

abovethere isanoptimalwatercontentyieldingthehighestbedpermeabil-

ity. Hence, there is an optimum depending both on the coke (Toda et al.

1984) and water content (Hinkley, Waters, O'Dea, and Litster 1994). Due

tothelarge(hot) recycleandtime delayspresent,theprocess isconsidered

diÆcultto control, withqualityand productionrates being hardto predict

(Cummingand Thurlby1990).

Control objective

Tosummarize,thecontrolobjectiveshouldbalanceanoptimaltemperature

prole, while considering quality, against minimizingsintering batch time.

ThesinteringbatchtimeisreducedbyanincreasedairmassowGsincethe

heat wave velocity v

w

is increased by an increasing air ow. The vertical

velocity of the travelling combustion zone is assumed to follow the linear

relationship(Olsen 1997), p.30:

v

w

= v

0

G

s

b

(2.3)

where v

0

[m=h] isthegasvelocity,

b

=(1 ")

s [t=m

3

]isthe bulkdensity

of thebedand G

s [m

3

=t]is thegasvolume persintermass.

The optimal temperature prole can be quantied by integrating the

part of thesolidtemperaturethat hasvaluesabove the fusiontemperature

T

fu

. The fusion temperature T

fu

is dened by the liquidus curves of the

ore composition. Partial meltingof the solidsoccur when the temperature

is raisedabove T

fu

. Fusionisdiscussedinmore detail insection2.3.1.

Thetemperatureproleatvariouslevelsinsidethebedshouldbeevenly

distributedduringthebatchtogiveequalsinteringconditionsinthroughout

the whole sintering bed. A measure for the temperature prole at each

spatiallevelis

z

8

<

: t

end

R

0 (T

s;z T

fu

)dt ifT

s;z T

fu

0 otherwise

(2.4)

wherethesubscriptzemphasizesthespatialdistributionofthetemperature

prole and t

end

is the batch duration time. Since the fusion temperature

T

fu

is uncertain and will vary with varying ore composition, a "smooth"

switch is suggested to approximate the switch caused by equation (2.4).

The smoothingfunctionischosenas thesigmoidfunction

(T

s )=

1

1+e k(T

s T

fu )

(2.5)

This isplotted withT

fu

=1200 o

C andk =0:1 ingure2.3

Hence,thecontrolobjectiveistomaximizeanobjectivesubjectto the

nonlinear inequality constraint imposed by quality as discussed in section

2.2.1. Thisis expressedformally as

max

Ts

(T

s )=

L

R

0 w(z)

t

end

R

0 (T

s )(T

s T

fu )dtdz

s:t:

z (T

s )=

t

s

R

t=0 (T

s )(T

s T

fu

)dtq

additionalconstraints

(2.6)

where the parameterq remainsto be selected. Observethat

z

is a vector

function since T

s

is spatially distributed. Hence, must be evaluated at

1000 0 1050 1100 1150 1200 1250 1300 1350 1400 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sigmoid fcn.

T _s [ ^o C]

σ (T s )

Figure 2.3: Sigmoid function. The gure shows the sigmoid function with

parametersT

fu

=1200 o

C and k=0:1. Thesigmoidfunction is usedasasmooth

switchinthecontrolobjective.

eachspatialpositioninthesinteringbed,seeappendixA.1fordetails. The

weighting function w(z) is introduced to give varying contributionsto the

objective at dierent spatial levels. The parameter q is not chosen as a

function (of T

s

) since the uncertainty is assumed to be captured by the

smoothing inherent in the sigmoid function. The validity of the objective

function is assessedin section 2.6.3. The parameter k in can be tuned

down to reduce thenegative contributionfollowingfrom T

s

<T

fu .

2.2.2 Control objectives

Therearefewreportedresultson controlofthesinteringprocess. Kim and

Kwon (1998) considers a linear MPC scheme designed to control the burn-

throughpoint oftravellinggrate sintering,usinganidentiedinput/output

model. To the best of the authors knowledge, thisis the onlyreference on

modelpredictive controlof thesinteringprocess reported intheliterature.

Across multiple batches

Controllingthecoke andwatercontent asdiscussedabovecanonlybedone

acrossseveralbatches,sincethecompositioninsideeachsinteringpancannot

bealteredonceithasbeencharged. Theexperimentsdocumentedinsection

2.4 make some preliminary investigations of the relationship between the

cokeand water inputs,and thequalityandproductionrateoutputs. Inside

the sintering pan, the coke and water contents have a negative bias due

to the mixing of unmeasured (hot) recycle into the measured fresh feed.

Note that the accuracy of the weights measuring the fresh feed may vary

considerably. Manipulatedinputsaretheaddedcoke,ignitionenergy,water

and air ow rate,whilethefeedis regardedasa disturbance.

During one batch

Themodelandthecontrolstrategyconsideredinthefollowingonlyconsiders

the process taking place inside the sintering pan, i.e. the coke and water

concentrationsarenotconsideredascontrolinputs. Theonlycontrolaction

thatinuencesthesinteringprocessasthebatchproceedsisthegasvelocity.

Thegasvelocityiscontrolledbyadjustingthechokevalveintheogas-pipe,

thus alteringthedierentialpressure dropacross thebed. We also assume

thatthethereissuÆcientignitioninthesensethatthedurationandquality

of ignitionallowsthesinteringprocessto startatthetopofthebed. Below

a qualitative assessment of the gas velocity as a control actionis outlined,

whilea quantitative discussionis given in section2.3.4.

The heat exchange properties of the initial raw charge is much better

than the heat exchange properties of the sintered material. I.e., the heat

transfer propertiesare alteredbymeltingsince thesurfacearea isreduced,

and the heat capacity of sintered material is altered due to the change in

chemical composition. To compensate for this, a large amount of excess

air is used in the sintering process, and it is not expected that the O

2

concentration willbe rate limiting in coke combustion. There is an upper

bound on the gas velocity, since too largegas velocitiescan cause thebed

to collapse giving poorsintering conditions. The upper limit is dependent

upon the permeability P Voice

as discussed in section 2.2.1, since a larger

permeabilityallows a largergasowwithoutcollapsing thebed.

The large dierence in heat exchange properties of charge and sinter

causes thefusionzone to widen asthe batch proceeds, giving dierent sin-

teringconditionsatthevariouslayersofthebed. Commonlythisisdiscussed

intermsof "matching" ofthecombustionheatwaveand theheatexchange

wave in the literature. Adjusting the gas velocity, based on a model and

the measuredogas quantities, then controls the width of the fusion zone

at thevariouslayers of thebed. This conjecture isinvestigatedbyballistic

simulationsinsection 2.5, and nonlinearMPCbased on thiscontrolstrategy

is implemented in chapter 4. Preliminary experiments were conducted to

investigate this conjecture at the industrial plant, see section 2.4.3. The

resultsoftheseexperimentsarenotconclusivebutthehypothesiscannotbe

rejectedon thebasisof these experiments.

Note thatbyassumingaconstant velocityof theheatwave asit passes

through the bed, simpler control schemes not depending on real-time op-

timization and a complexprocess model may be considered. This has not

beenconsideredinthepresentwork,sinceitisoutsidethescopeofthethesis

whichfocuseson optimizationand nonlinearMPC.

To summarize; thecontrolobjective is to maintainthe same heat wave

shape at all layers inthe sinteringbed. Informallythe objective then is to

balance the heat wave prole spatially for product quality, while simulta-

neously maximizing the gas velocity for production rate. These issues are

revisitedanumberoftimes throughout thischapter.

2.3 Modeling

Severalmodelsofthesinteringprocessarepresentedintheliterature(Muchi

andHiguchi1972), (DashandRose1977),(HoislbauerandJaquemar1983),

(Kasai, Yagi, and Omori 1984), (Cumming and Thurlby 1990), (Patisson,

Bellot, Ablitzer, Marliere, Dulcy, and Steiler 1991), (Nath, Da Silva, and

Chakraborti 1997), (Venkataramana, Gupta, Kapur, and Ramachandran

1998). These models are presented as nonlinear PDE's, and mainly focus

on reproducing important process quantities. Noting that nonlinear PDE

modelsarediÆculttoimplementinacontrolstrategy,weseektoexploitthe

underlyingstructureof thesintering process to develop a simpliedmodel

which later on can be utilized to develop a MPC strategy for the sintering

process.

The model in this section should be a control relevant model suitable

for MPC. Reproduction of the internal states is important since a driving

hypothesis is that synchronization of the model and process through state

estimationisbenecialinanindustrialimplementation. If thestatescannot

be measured and estimation is hard, it is likely that the MPC algorithm

performs poorly.

This section will emphasize the development of the control relevant

model,andthecontrolalgorithmitselfistobeinvestigatedinchapter4. The

globalPDEmodeldiscussedinsection2.3.1iscompiledfromcitedreferences.

Modellingassumptionsaresummarizedinsection 2.3.2.

2.3.1 Global PDE model

A global PDE model is understoodas a nonlinearPDE model describing the

whole sinteringbed asingure 2.1withoutexplicitconsideration of zones.

In this section a model valid for the sintering bed is compiled from cited

literature. Adetailedreviewof thevariousphysicalandempiricalrelations

used intheliteratureisincluded.

Thefollowing statesareincluded inthemodel

x=

T

s

;T

g

;x

C

;x

H

2 O(l )

;x

O2

;x

N2

;x

CO2

;x

H

2 O(v)

i.e. temperature of solidsand gas, coke concentration insolid,liquid water

content 1

and gas compositionincludingwater vapor. The gas velocity and

pressure drop arenot included as states inthe model, see section 2.3.4 for

a discussion. ThehyperbolicPDE'sconstitutingthemodelare, forthemass

balanceof gas(Patisson etal. 1991)

"

@x

O

2

@t +v

@x

O

2

@x

= r

O2

"

@x

CO

2

@t +v

@x

CO

2

@x

= r

O

2

"

@x

H

2 O(v)

@t +v

@x

H

2 O(v)

@x

= r

H2O

"

@

g

@t +v

@

g

@x

= M

C r

O

2 +M

H

2 O

r

H

2 O

forthemass balanceof solids(Patisson etal. 1991),

@

C

@t

= M

C r

O

2

@

H

2 O(l)

@t

= M

H

2 O

r

H

2 O

@s

@t

= M

C r

O2 M

H2O r

H2O

and fortheenergy balance(Patisson et al. 1991)

"

@Tg

@t +v

@Tg

@x

= k

1 (T

s T

g )

@T

s

@t

= k

2 (T

g T

s )+k

3 g(T

s )

k

1

= A

b h

c

g c

p;g ,k

2

= A

b h

c

s c

p;s and k

3

= 1

s c

p;s

are aggregated temperature depen-

dent parameters. The gasis assumed to beideal and we assume plug-ow.

The solidstates areassumed notto move, i.e.@T

s

=@x =0etc.

Kineticparameters areonlyconsideredforcoke combustion,fusionand

drying inthemodel. The chemical reactionsconsideredare

C+O

2

! CO

2

H

2 O

(l )

$ H

2 O

(v)

1

Liquidandsolidsare lumpedinonephase.

Thekineticrelations r

i

g(T

s

)= ( H

r )r

O

2 L

v (T

s )M

H

2 O

r

H

2 O

L

f r

f

r

O

2 (T

s )= s

O

2 R

C (T

s )

r

H2O (T

s )= s

H2O R

w (T

s )

r

f (T

s )= s

f R

f (T

s )

are discussed in the following subsections. The kinetic parameters of coke

combustionr

O2

,fusionandsolidicationofsolidsr

f

andcondensationr

H2O

arenotknown indetail. The kineticmodelofcoke combustionis discussed

in Parker and Hottel (1936) and Muchi and Higuchi (1972) assuming the

reaction C+O

2

! CO

2

. The kinetics of fusion of solids is described by

empirical schemes based on slag diagrams (Patisson et al. 1991) or linear

schemes based on process experience (Cumming and Thurlby 1990). The

kineticsof condensationof water is derived from laboratory tests(Patisson

et al. 1990) or by heuristics and experience (Dash and Rose 1977), (Zou,

Huang, Yang, and Chen 1995). The heat of coke combustion is releasedto

thesolidphase,seediscussioninCumming andThurlby(1990). Limestone

isnotutilizedintheindustrialplant,and isnotincluded inthemodel. We

continue byspecifyingthemodel parametersinthenext subsections.

Introductory relations

Some introductory relations are derived. Subscriptb refers to bulksizes, s

referstosphericalparticlewhileprefersto (non-spherical)particle. Recall-

ingthatA

s

=d 2

s andV

s

=

6 d

3

s

,resembles As

Vs

= 6

ds and

As

ms

= 6

dss

= A

s

=V

s

s ,

wherem

s

=V

s

s

. Thevoid fraction"is denedby:

1 "=

volumeof solids

volumeof bed

= m=

b

AL

=1

b

a

(2.7)

where

b

is the bulk density of the bed and

a

is the granule apparent

density (see Hinkley, Waters, and Litster (1994) for details). The volume

occupied by solid (spheres) in the total volume is V

b

= (1 ")V. This

impliesfor sphericalparticles that A

b

V

= A

b

V

b

=(1 ")

=(1 ") A

b

V

b

= 6(1 ")

d

s

. For

anon-sphericalparticleof thesame densityasasphereoccupyingthesame

volume (

s

=

p

= and V

s

= V

p

), dene the mean particle diameter as

d

p

= 6m

A

p

= 6V

s

A

p

. Then dene the form factor

f

as the ratio between the

surface area of a sphere and the surface area of a particle occupying the

same volume:

f

= A

s

A

p

= d

p A

s

6V

s

= d

p

d

s 1

i.e.A

p

=A

s

=

f

= d

2

s

f A

s

. Finally,thereaction rateperunitsurfacearea

is given byr

A

b

= dn=dt

A

b

,andperunit massand unit volumebyr

m

= 6r

A

b

d

p

and r

V

=

6(1 ")r

A

b

dp

, respectively. Typical valuesof void fraction and form

factor of the present materials (prior to sintering) are " 2 [0:4 0:6] and

f

0:75, see Rosenqvist(1983),p.143. A

b

isintheorder 2000 [m 2

=m 3

].

The harmonicmean diameter of the charge particles is calculated from

meshanalysis ofthe rawchargeas(Hinkley,Waters, andLitster 1994):

1

d

p

= f

1

d

1 +

f

2

d

2

++ f

n

d

n

(2.8)

where f

n

is the fraction of particles between two sieve sizes with a mean

diameter d

n .

Empirical relations

Parameter uncertainties are present in the global models, since essential

parameters typicallyaredeterminedfromempiricalformulas validonlyun-

der idealized conditions. In industrial sintering processes the formation of

cracks and channels leadsto areas where air passes throughwithout inter-

acting with the mass in the sinter bed. In particular, the mass, h

m , and

heat, h

c

,transfer coeÆcientsarecalculatedfrom theNusseltand Sherwood

numbers. Empiricalrelationsfor Sh andNu are statedinequations(2.10)

and (2.9) (Wakao and Kaguei1982):

Nu= h

c d

p

k

th

= 1

"

2+1:1Pr 1=3

R e 0:6

(2.9)

Sh= h

m d

p

D

ON

= 1

"

2+1:1Sc 1=3

R e 0:6

(2.10)

validforanidealizedbedwithhomogeneouspacking. TheReynolds,Schmid

and Prandtlnumbersare givenby

Re =

g vd

p

Sc =

g D

ON

Pr = 0:7

where the value Pr = 0:7 holds for diatomic gases 2

. In an industrial bed

thegasowingthroughchannelsandlargecracksdoesnotinteractwiththe

solid,and thevaluesestimated fromtheempiricalrelationsforan idealized

bed will deviate from the actual values. Various heuristics are utilized to

overcome this in the literature, i.e. altering the constants of the empirical

relations (Dash and Rose 1977), (Hoislbauer and Jaquemar 1983), (Nath

etal. 1997),andintroducingascalingfactor (CummingandThurlby1990),

(Patisson et al. 1991). According to the discussionabove the gas fraction

passingthroughpossiblelargecracksandchannelsinthesintercakedoesnot

contributetothemasstransferandshouldnotbeincludedwhencalculating

h

m

fromSh. Thereforeafactor'isintroducedtocompensatetheSherwood

relation(Schluterand Bitsianes1962):

h

m

=' D

ON

d

p

"

2+1:1Sc 1=3

R e 0:6

(2.11)

TheNusseltrelationis scaledbythesame factor '.

The heat capacity of the solid and the void fractions will change in a

complicatedwayassinteringproceeds. Thespecicheatcapacityofsintered

materialforFe-sinteris givenby(Rose and Dash1979)

c

p;s

=753+2410 3

T

s

[J=kgK] (2.12)

As seen from gure 2.4 this linear approximation is suitable for iron ore.

Formanganeseorethesituationisdierent,andtherelationforc

p;s

usedin

themodel is the dashed curve inthe right part of gure 2.4. This relation

was obtained as a linear combination of the data for the three Mn-oxides.

Thelinearcombinationwaschosenfromplantdatawheretheorecontained

47.0% MnO

2

,25.5% Mn

2 O

3

and27.5% Mn

3 O

4 .

Thismodel ofc

p;s

doesnotinclude otherelementscontainedintheore,

and it does not reect changes caused by fusion, chemical reactions and

thermaldecompositionof theMn-oxides.

The temperature dependence ofc

p;g

ismodelledas:

c

p;g

R

=a

1 +a

2 T+a

3 T

2

+a

4 T

3

+a

5 T

4

[J=kgK] (2.13)

wherea

i

valuesarespeciedinMoranandShapiro(1993), p.680. Forideal

gases c

p;g c

v;g

= R gives a similar expression for c

v;g

. The temperature

2

Adiatomicgascontainstwoatomsinitsmolecules. O2isadiatomicgas,whileH2O

(v)

hasthreeatomsinitsmolecules. Asmallerrorisintroducedbythis,butsinceH2O

(v) is

notpresentinthecombustionzone,theerrorisassumedtobeofminorimportance.

0 500 1000 1500 0

500 1000 1500

Fe−oxides

T [ ^o C]

Fe ₃ O ₄ Fe ₂ O ₃

Fe _mix

0 500 1000 1500

0 500 1000 1500

Mn−oxides/HSC

T [ ^o C]

Mn ₃ O ₄ Mn ₂ O ₃

Mn _O2 Mn _mix

Figure 2.4: Specic heatcapacities of Mn-oxides. The gureshowsspecic heat

capacitiesofFe-oxides(left)andMn-oxides(right). ThedashedlineFe

mix tothe

left isthestraightlinegivenby thelinearapproximation(2.12), whilethedashed

line Mn

mix

is computed as a linear combination of the three Mn-oxides shown.

Observethat the compound heat capacity ofMn-oxides isnot well approximated

byastraightline. Data fortheFe-oxidesaretakenfrom PerryandGreen (1984),

p.3-131/2,whilethedatafortheMn-oxidesaretakenfromHSC(Roine1997).

dependence of the viscosity is modelled by Sutherland'sformula (White

1999), p. 771:

=

0

T

g

T

0

3=2

113+273:1

113+T

g

m

2

[kg=ms] (2.14)

where

0

=1:7210 5

[kg/ms]andm

2

0:9. Thetemperaturedependency

of theaxial gasdispersioncoeÆcientD

ON

is modelledas:

D

ON

=D 0

ON

T

g

273

m

3

[m 2

=s]

where m

3

1:5. Thethermalconductivityof gask

th

isestimated from the

Prandtl number:

k

th

= c

p;g

Pr m4

where m

4

10:7 is determinedbyadaptingthe computedk

th

to tabulated

data fromPerryand Green(1984).

Coke combustion and overall heat of reaction

Anumberofdierentreactionsoccurduringsinteringofmanganeseore. The

equilibriumdiagramingure2.5showsthepossiblereactionsatgivenoper-

atingconditions. Thesereactionscontribute to theoverall heatof reaction,

H

r

. Since onlycoke combustion isconsidered to be of majorimportance

withrespectto thedynamicproperties,thekinetics of reductionofMn-ore

isneglected.

5.5 6 6.5 7 7.5 8 8.5 9 9.5

−6

−4

−2 0 2 4 6 8 10 12

MnO+CO(g)=Mn+CO ₂ (g) Mn ₃ O ₄ + CO(g) = 3MnO + CO ₂ (g) 3Mn ₂ O ₃ + CO(g) = 2Mn ₃ O ₄ + CO ₂ (g) 2MnO ₂ + CO(g) = Mn ₂ O ₃ + CO ₂ (g)

C+CO ₂ (g)=2CO(g)

10 ⁴ /T [1/K]

log 10 (p CO 2 /p CO )

Equilibrium gas ratio for reduction of Mn−oxides

Figure 2.5: Equilibrium gas ratio log

10 ( p

CO2

=p

CO

) for reduction of Mn-oxides.

TheBoudouard-line(2CO $ C+CO

2

)crossesthe MnO Mnline at approxi-

mately 1400 o

C. The measurements conducted at the Sauda plantshowthat the

temperatureinthesinteringbedisnominallybelowthispoint. Hence,reductionto

Mnisnotlikelytooccur. Theslopesoftheotheroxide-componentsslantsupward,

andreductionin thepresenceofCO willoccurendothermic.

In a (reducing) atmosphere of CO the following endothermic reactions

areobserved ingure 2.5:

MnO

2 +

1

2 CO$

1

2 Mn

2 O

3 +

1

2 CO

2

; H

r;1

100 kJ/molC

1

2 Mn

2 O

3 +

1

6 CO$

1

3 Mn

3 O

4 +

1

6 CO

2

; H

r;2

30 kJ/molC

1

3 Mn

3 O

4 +

1

3

CO $MnO+ 1

3 CO

2

; H

r;3

22 kJ/molC

where the heat of reaction is taken as the average over the temperature

rangeofinterest. These reactionsmove to theright duringcombustionand

fusion,cf. gure 2.1. However, dueto thelargeair excess duringsintering,

onlya smallamount of CO is actuallypresent. According to Olsen (1997)

theogas ratio= CO

CO+CO

2

subsequentto combustion canbecome ashigh

as 0:25 0:30. I.e. we have log

10 (

p

CO

2

p

CO

) log

10

(3) 0:47 showing by

inspection of gure 2.5 that the three reactions above are possible. The

Boudouard-reaction:

2CO $C+CO

2

; H

o

298

= 172:5kJ/molC

whichis stronglyendothermal, preventsfurtherreductionbyCO. In addi-

tion thermaldissociation at p

O2

=1atm occurs:

500 o

C:MnO

2

$ 1

2 Mn

2 O

3 +

2

12 O

2

; H

o

298

=41:8kJ/molC

800 o

C : 1

2 Mn

2 O

3

$ 1

3 Mn

3 O

4 +

1

6 O

2

; H

o

298

=16:3kJ/molC

Expressing X

CO

=

1+

X

CO2

and assuming that coal is present as pure

carbon,combustionisexpressed as:

C+(1+)O

2

+3:76(1+)N

2

+L$(1 )CO

2

+CO+3:76(1+)N

2 +L

where excess air L consisting of 79% N

2

and 21%O

2

is utilized. Choosing

=0:25givesa heatofreactionH

r;c

= 335 kJ/molC 28 MJ/kg C.

The heat of combustion for coal is experimentally determined to approxi-

mately

^

H

r;c

35MJ=kgC (Olsen 1997), i.e. 7MJ=kg more than indi-

catedbytheenthalpyofreactionstatedabove. Thisisattributedto volatile

components present inindustrialcoal. The overallheat of reaction used in

the model is H

r

=

^

H

r;c H

r;1 H

r;2 H

r;3

270 kJ=mol C

which is close the reported values for Fe-sintering in the literature. Note

that thermal dissociation is not included since it is completed before coke

combustionproceeds.

Thereactionrate ofcoke combustionisimportantduetothelargetem-

peraturegradientspresentingratesintering. Cokecombustionisaheteroge-

nousreaction,and theoverall combustionrateiscontrolledbytwo physical

phenomena;thechemicalcombustionrateand thegastransportrateto the

individualcokeparticle. Thecombustionrateforacokeparticleisgoverned

by(Parkerand Hottel1936), (Muchi and Higuchi 1972)

k

r

=A

f e

E=RTs p

s x

O

2

(2.15)

withthefrequencyfactorA

f

=6:5310 5

m=s p

KandE =185kJ=mol.

s is

thegastemperatureatthesolidsurface. Thisrelationshipisextensivelyused

intheliterature,butitassumestherstorderreactionC+O

2

!CO

2 (=

0)andthatcombustioncommencesonthecokeparticlesurfaces(Muchiand

Higuchi 1972). Both of these assumptions are inconsistent with industrial

experience. Still, equation (2.15) is used in the present model assuming

s

= T

s

. The activation energy, however, is reduced to approximately 70

kJ=mol to t themodel to the measured data. By usingthe ideal gaslaw

(2.15) can be writtenin termsof p

O

2

insteadof x

O

2 .

Sincethetransportofreactantsandproductstoandfromthecokeparti-

clesurfacecanberatelimitingatelevatedtemperatures,theoverallreaction

rate, including both chemical and transportation phenomena, is modelled

bytherelation (Muchiand Higuchi 1972):

K

r

= k

r h

m

k

r +h

m

whereh

m

[m=s]is calculatedfrom equation(2.11). PorediusionD

pd may

be includedsimilarlyaccording to

1

K

r

= 1

k

r +

1

h

m +

1

D

pd

Thisisnotincludedinthepresentmodel. Thisgivestheoverallcombustion

rateof coke:

R

r

=4(d

p;C

=2) 2

n

C K

r x

O

2

(2.16)

where n

C [1=m

3

] is the number of coke particles per unit bed volume and

d

p;C

isthe average coke particlediameter.

Fusion

ParticlegrowthduetofusionismodelledaccordingtoDashandRose(1977)

as

d

p;f

=d

p (1+k

f L

f )

where k

f

is a constant to be determined experimentally and L

f

is the en-

thalpyoffusion(melting). Ifsignicantmeltingdoesnotoccur,thescheme

of Dawson(1993) where smallerparticles attaches to largerparticles is as-

sumedtobeprevailing. Thisistermedheterogeneoustexture,i.e.themean

particle size of sintered particles is in the region of the larger sieve sizes

of the charge. However, if signicant melting occurs the situation will be

completelyaltered, withformationof larger continuousblockswhichagain

forms a homogeneous texture. This is assumed to correspond to a texture

of very ne particles, with particle sizes in the region of the nest mesh

sizes present in the charge. The transition between these two regimes is

controlled bythe sinteringtemperature (Dawson 1993). Particle growth is

notincluded inthepresent model.

Fusionitselfiscomplexandismodelledbythefollowingequations(Patis-

son etal. 1991):

r

f

=

s dF

i

dT

s

@Ts

@t

F

1

= (1 x

h )

a

0 +a

1 (T

s T

df )+a

2 (T

s T

df )

2

+a

3 (T

s T

df )

3

F

2

= F

m T

s T

fs

Ts;m T

fs

where the functionF

1

denotes theliquid fractionduringmelting, x

h is the

hematite fraction of the ternary system FeO MnO SiO

2 , T

df

is the

incipient melting temperature, and a

i

are dependent on the basicity index

and can be found from a liquidus surface diagram, see Verein Deutscher

Eisenhuttenleute (VDEh) (1995), gure 3.286. During solidication, com-

positionisassumedto beconstantand thesimpliedrelationF

2

isadopted

where F

m

istheliquidfraction at T

s;max and T

fs

isthetemperature at the

endof solidication.

Usingtheliquidussurfacediagramisimpracticaland thepresentmodel

only considers linear schemes based on process experience (Cumming and

Thurlby1990). The present model implementsa quadratic approximation

where theparameters weretuned to t thedata.

r

f

= ( ^a

1 (T

s T

df )+a^

2 (T

s T

df )

2

)

s

[mol=s]

L

f

= 255 [kJ=mol]

(2.17)

withT

df

=1250 o

C anddierentvaluesof^a

i

formeltingandfreezingfollow-

ing thescheme of Nath, Da Silva,and Chakraborti(1997). This simplied

modelgivesakinkinthetemperatureprolewhenthemodelswitchesfrom

meltingto freezing. Since these parameterswereselected withoutreference

to liquidus diagrams, the model is conceptual, and renements should be

consideredsincethesinterqualityis governed bytheshapeofthetempera-

ture prole. Note that fusionhasa signicant inuenceon the falling edge

of thetemperatureprole.

Water

Water evaporates andcondensates accordingto

H

2 O

(l )

$H

2 O

(v)

(2.18)

Thedrying modelis governed by(Patisson,Bellot, and Ablitzer1990)

~

R

R

= R

M P(W

r )

P(W

r

)= 1 (1 W

r

)(1 1:796W

r

+1:0593W 2

r )

R

M (T

w )=

A

b h

c

M

H

2 O

H

v (T

w )

R

R (T

s )=

A

b k

th

RTg (p

v;sat (T

s ) p

H

2 O

)

p

v;sat (T

s

)= exp

25:541 5211

Ts

k

th

=

hcTg

3:155pg r

(1 0:24x

lg )

1+

x

lg

7

(1 x

lg )

(2.19)

where W

r

= W=W

cr

and the given coeÆcients of P(W

r

) determined from

laboratory tests (Patisson, Bellot, and Ablitzer 1990) are valid for a Fe-

charge. CoeÆcients for Mn must be determined experimentally 3

. The

wet-bulb temperature T

w

can be calculated by solving a nonlinear equa-

tion (Patisson et al. 1990). The present approach uses the approximation

(Roseand Dash 1979)

T

w

=293:4+324:6W 594:1W 2

+292:1W 3

which is based on tabulated data from Perry and Green (1984). x

l g is the

logarithmicmean of themolar fraction of vaporin thebulkgasand at the

saturatedsurface:

x

l g

= x

v;g x

v;s

ln x

v;g

xv;s

Since x

v;s

is unknown the present model implementsx

l g

= 0:5x

v;g . The

heattransfer coeÆcient h

c

for the water-vapor system may dier from the

overall heat transfer coeÆcient from equation (2.9). This is notconsidered

inthepresent model. If R

R

>0 and W W

cr

(fallingrate) r

H

2 O

=

~

R

R . If

R

R

>0and W <W

cr

(constant rate)r

H2O

=R

R

. The heatcapacityofthe

moistogas is modelledby(Perryand Green1984), p.12-3:

c

p

=0:24+0:45!

where! =0:622 p

p p

H

2 O

. Simpliedschemesfor R

R

are foundinNath, Da

Silva, and Chakraborti (1997) and Zou, Huang, Yang, and Chen (1995).

Thelatent heatof evaporationis approximatedby(Patisson etal. 1990)

L

v (T

w

)=3:156310 6

2396:6T

w

[J=kg]

3

ThepresentmodelisimplementedwiththeFe-coeÆcients.