modeling of blast furnace with layered cohesive zonemodeling of blast furnace with layered cohesive...

Modeling of Blast Furnace with Layered Cohesive Zone

X.F. DONG, A.B. YU, S.J. CHEW, and P. ZULLI

An ironmaking blast furnace (BF) is a moving bed reactor involving counter-, co-, and cross-current flows of gas, powder, liquids, and solids, coupled with heat exchange and chemicalreactions. The behavior of multiple phases directly affects the stability and productivity of thefurnace. In the present study, a mathematical model is proposed to describe the behavior of fluidflow, heat and mass transfer, as well as chemical reactions in a BF, in which gas, solid, andliquid phases affect each other through interaction forces, and their flows are competing for thespace available. Process variables that characterize the internal furnace state, such as reductiondegree, reducing gas and burden concentrations, as well as gas and condensed phase temper-atures, have been described quantitatively. In particular, different treatments of the cohesivezone (CZ), i.e., layered, isotropic, and anisotropic nonlayered, are discussed, and their influenceon simulation results is compared. The results show that predicted fluid flow and thermo-chemical phenomena within and around the CZ and in the lower part of the BF are different fordifferent treatments. The layered CZ treatment corresponds to the layered charging of burdenand naturally can predict the CZ as a gas distributor and liquid generator.

DOI: 10.1007/s11663-009-9327-y� The Minerals, Metals & Materials Society and ASM International 2009

I. INTRODUCTION

AN ironmaking blast furnace (BF) is a multiphasereactor involving counter-, co- and/or, cross-currentflows of gas, powder, liquid, and solid phases.[1] In thisprocess, iron-bearing materials and coke are charged atthe top of the furnace. Hot gas (blast) enters the furnacethrough the tuyeres in the lower part and combustscarbonaceous materials (coal, coke) to form reducinggas. As this gas ascends, it reduces and melts the iron-bearing materials to form liquid iron and slag in thecohesive zone (CZ). The liquid percolates through thecoke bed to the hearth. If pulverized coal injection orother injection technology is practiced, then unburntcoal (or other injectants) may leave the raceway regionat high injection rates through gas entrainment as adistinct powder phase.[2] These characteristics demon-strate the complexity of the BF operation and thedifficulties in understanding the physical and chemicalphenomena in a BF.

Since the 1960s, intensive research has been under-taken to characterize the internal state of a BF withdifferent techniques such as dissection studies, in situmeasurements, physical experimentation, and mathe-matical modeling. Among them, numerical modelingdemonstrates an increasing capability to providedetailed information about fluid flow, heat and mass

transfer, as well as chemical reactions throughout thefurnace. One-, two-, and three-dimensional continuum-based mathematical models have been developed insequence, as summarized in recent reviews.[3–5] Thedevelopment of numerical models originated from theunderstanding of macroscopic phenomena at the earlystage toward the simulation of critical operationalconditions and flow phenomena on a microscale. Cor-responding to this trend, the CZ has attracted muchattention in recent years. Through dissection studies, theexistence of the CZ as a layered structure where burdenmaterials undergo dramatic physiochemical change hasbeen demonstrated. Within the CZ, the iron-bearingmaterials experience softening and melting and reactwith the reducing gases to produce wustite and iron.Therefore, this zone comprises alternate layers of cokeand semifused masses of slag and iron. Through thiszone, the layered structure of solid ore and coke in theupper part of the BF gradually transits to the lower zonewhere only the solid coke remains alongside gases andliquids. The resistance to upward gas flow is great in theCZ because the ore layers are relatively impermeable,which results in most of the gas traveling through theintermediate coke layers. The shape and position of theCZ can change significantly with operational condi-tions[1,6] and, because of its restricted permeability androle as a gas and liquid distributor, largely determinesthe BF performance and operational stability. Tounderstand this zone, several numerical and physicalexperiments have been conducted to study the high-temperature properties of iron-bearing materials,[7–14]

reaction mechanisms,[15] fluid flow behavior, and tem-perature distributions[6,16–20] within the CZ. Thesestudies highlight the complex softening–melting beha-vior, which include the variation of permeability andstrong horizontal gas and liquid flows in the CZ.Moreover, efforts have been made to establish the

X. F. DONG, Research Associate, and A. B. YU, FederationFellow and Scientia Professor, are with the Lab for Simulation andModelling of Particulate Systems, School of Materials Science andEngineering, University of New South Wales, Sydney, NSW 2052,Australia, Contact e-mail: [email protected] S. J. CHEW, PrincipalResearch Engineer, and P. ZULLI, Senior Principal ResearchEngineer, are with BlueScope Steel Research, P.O. Box 202, PortKembla, NSW 2505, Australia.

Manuscript submitted February 17, 2009.Article published online January 5, 2010.

330—VOLUME 41B, APRIL 2010 METALLURGICAL AND MATERIALS TRANSACTIONS B

relationship among permeability, fluid flow, and heattransfer in this zone,[21–23] which resulted in the devel-opment of several empirical and semiempirical CZmodels. Among them, the so-called ZAP model[24]

shows the capability to predict the shape and positionof the CZ partially based on online measured data andon summarized relationships between process parame-ters and CZ characteristics.

Apart from the intensive fundamental studies dis-cussed, the prediction of the CZ in global BF modelinghas proven a long-standing challenging task. Differentnumerical treatments have been used to simulate the CZstructure in BF modeling, which generally can besummarized as nonlayered and layered CZ treatments.For a nonlayered CZ treatment, the CZ is treated as amixed region of iron-bearing materials and coke whereisotropic or anisotropic permeability distributions canbe applied in steady or unsteady simulation.[1] For alayered CZ treatment, the CZ is numerically modeled asa region comprising impermeable or low-permeabilityiron-bearing material layers and highly permeable cokelayers.[25–30] The existence of this layered structuremakes modeling more difficult. For example, to treatliquid flow in the layered CZ, the so-called force balancemodel[18,31,32] had to be modified to allow the liquid totraverse the impermeable cohesive layers. NonlayeredCZ treatments simplify the modeling and can facilitateearly application and development. However, ignoringthe existence of alternate layers of impermeable iron-bearing materials and permeable coke windows canmake it difficult for a nonlayered CZ treatment toreproduce the fluid flow, temperature, and concentra-tion fields realistically within and around the CZ.Moreover, this treatment often induces a potentiallyartificial level of permeability in the CZ to yield asolution to the gas flow field. In contrast, a layered CZtreatment explicitly considers the stratified structure ofiron-bearing materials and coke, which is consistent

with a realistic CZ structure. Unfortunately, one majorshortcoming associated with most past studies that useda layered CZ treatment is that the CZ shape andposition were specified in advance and may have beeninconsistent with other operational parameters.Although the recent MOGADOR model[24,33] has cap-tured the overall CZ position considering the layeredburden distribution in the simulation, a relatively highpermeability is maintained in the simulated fused orelayer of the CZ,[33] which indicates that the layeredstructure in the CZ is still considered implicitly.To better demonstrate the effect of impermeable fused

ore layers in the CZ on the process variables, this studyattempts to establish a BF modeling system consideringthe fluid flow, heat transfer, and chemical reactions topredict the layered CZ explicitly. The characteristics ofthe flow and thermochemical behavior in a BF will bedemonstrated through a comparison of process model-ing with different CZ treatments.

II. NUMERICAL MODELING

In this section, the governing equations used in thesimulation are described first. The relevant modelparameters then are discussed in detail, and the numer-ical treatments of the CZ, i.e., layered and nonlayered,are described. Finally, numerical techniques in regard tothe discretization approach, solution algorithm, andmultiphysics field coupling are described.

A. Governing Equations

Table I summarizes the governing equations for fluidflow, heat and mass transfer, as well as chemicalreactions used in this work. Gas was described by thewell-established volume-averaged, multiphase, Navier–Stokes equations.[2,5] Solids were assumed to be a

Table I. Governing Equations

Equations Description

Mass conservation r� eiqiuið Þ ¼ Si; where Si ¼ �P

k

bi;kR�k

Momentum conservationGas r� egqgugug

� �¼ r�sg � egrpþ qgeggþ Fs

g þ Fl;dg

sg ¼ eglg rug þ rug� �T

h i� 2

3eglg r�ug� �

I

Solid r� esqsususð Þ ¼ r�ss � esrps þ qsesg

ss ¼ esls rus þ rusð ÞTh i

� 23esls r�usð ÞI

Liquid Fgl;d þ Fs

l;d þ el;dqlg ¼ 0

Heat and species conservation r� eiqiui/i;m

� ��r� eiCir/i;m

� �¼ S/i;m

if /i;m is Hi,m, Ci ¼ kicp;i,

S/i;m¼ dihija Ti � Tj

� �þ cp;iTidi

P

k

P

l

bk;lR�k þ gi

P

k

R�k �DHkð Þ

if /i;m is xi,m, Ci ¼ qiDi, S/i;m¼P

k

ai;m;kR�k, where

/i;m ¼ xg;co;xg;co2 ;xs;Fe2O3;xs;Fe3O4

;xs;FeO;xs;flux

Phase volume fractionP

i

ei ¼ 1

State equation p ¼P

i

yiMið ÞRTg=Vg

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 41B, APRIL 2010—331

continuous phase that can be modeled based on thetypical viscous model used in multiphase flow model-ing.[34] Liquids flow as rivulets or droplets under theinfluence of gravity, gas drag, and bed resistance, whichcan be described by the so-called force balanceapproach.[18,35] General convection–diffusion equationswere applied to describe heat and mass transfer amongthe phases.

B. Momentum Transfer, Chemical Reactions,and Transport Coefficients

The following correlation:

Fsg ¼ � afqg usg

��þ bf

� �usg ½1�

is an Ergun-type equation[5] that is used to describe theinteraction force between gas and solid phases. Interac-tion forces between gas and liquid Fl;d

g as well as betweensolid and liquid Fl;d

s were described by the recentcorrelations of Chew et al.[35] These correlations aresummarized in Table II. As is shown in Table III,chemical reactions are taken into account, which includedirect and indirect reduction of iron ore by coke and

CO,[36] solution loss,[37] and melting of Fe and FeO.[38,39]

Because the verification of chemical reactions is beyondthis study, reaction rates applied in other BF modelingwere adopted. Furthermore, because the hydrogenproportion in the current simulation system was small,hydrogen reduction and gas–water reactions were notconsidered. The transport coefficients that determineheat and mass transfer within and between phases wereestimated and described in the following sections.

1. Effective Diffusion CoefficientsThe effective gas diffusion coefficient was determined

by the relationship between the Peclet Number and theReynolds number.[1] Diffusion coefficients for solid andliquid phases were ignored.

2. Effective Conductivity CoefficientsGas conductivity[1] was determined by the following

equation:

kgn ¼ cpqDegn ½2�

The effective solid conductivity coefficient was deter-mined by the effective heat conductivity caused bycontact between particles and by radiation between

Table II. Empirical Correlations for the Interaction Forces Between Phases

Phases Interaction Forces Ref.

Gas–solid, G–S Fsg ¼ �Fg

s ¼ � afqg usg

��þ bf

� �usg

[5]

Gas–liquid, G–L Fl;dg ¼ �F

gl;d ¼ �

hl;ddlþ Asl;d

6

� �150

esþhl;tdw

� �lg þ 1:75qg Ug

��

h iUg

e3gwhere [35]

dl ¼ max dl;g; dl;h� �

dl;g ¼ max �6:828signffiffiffiffiffiffiXp

p� 0:891

� � ffiffiffiffiffiffiXp

p� 0:891

� �2; 0

n oþ 0:695

h i=ffiffiffiffiffiffiffiffiffiffiffiffiqlg=r

p

dl;h ¼max 6:828signðf1Þðf1Þ2;0f gþ 0:695½ �ffiffiffiffiffiffiffiffi

qlg=rp

f1 ¼ maxln

hl;thl;to

� �;0

n o

0:513

2

4

3

5

12:642

� 0:891

Xp ¼ DpeDxqlgð Þ qlgu

2d2sre2s

0:3

1þ cos hð Þ�0:5

Liquid–solid, L–S Fsl;d ¼ 150

36 ll

A2sl;d

h3l;d

þ 1:756 ql

Asl;d

h3l;d

Ulj j� �

Ul[35]

Table III. Chemical Reactions

Reaction Formula Reaction Rate Reference

Fe2O3(s)+CO(g) fi Fe(s)+CO2(g)R�1 ¼

12noreeoreP yco�y�coð Þ

8:314Tsð Þ

d2ore=Deg;co 1�foð Þ�

13�1

� �þdore

�kl 1þð1=KlÞð Þ

��1

[36]

FeO(l)+C(s) fi Fe(l)+CO(g) R�2 ¼ k2Ac

VbaFeO

[1]

C(s)+CO2(g) fi 2CO(g) R�3 ¼6ncokeecokepyco2 = 8:314Tsð Þdcoke=kfþ6= qcokeEfk3ð Þ

[1]

FeO(s) fi FeO(l) R�4 ¼Ti�Tmin;sm

Tmax;sm�Tmin;sm

D E1

0

HxsmuiqieidA

MsmVolcell

[38]

Flux(s) fi slag(l)


noncontacting particles.[1] The corresponding estima-tion equation is expressed as follows:

kese ¼1� eg� �

1ksþ 1

kes

� �þ egkes

½3�

where kes ¼ 2:29� 10�7dsT3s . Effective thermal conduc-

tivities of liquid slag and hot metal were determinedaccording to the literature data.[38]

3. Heat Transfer Coefficients Between PhasesThe heat transfer coefficient between gas and

solid particles was described by the Ranz–Marshallcorrelation,

Nu ¼ 2:0þ 0:6ðPrÞ0:333ð9RegÞ0:5 ½4�

where hgs ¼ chegs; hegs ¼ Nu

kgds; and c = 0.3. As the per-

meability of the softening phase is lost within the CZ,heat transfer from the gas phase gradually changesfrom a gas-particle heating mode to a gas-slab heatingmode.[40,41] The heat transfer coefficient between gasand the interface of softening and melting phase wasassumed to be expressed as follows:

hg-slab ¼ 0:203Re0:33g Pr0:33 þ 0:220Re0:8g Pr0:4� �

kg=ds

½5�

The heat transfer coefficients between gas and liquid andbetween solid and liquid are available elsewhere.[1,34]

4. Heat Loss Through the Furnace WallFor the wall boundary condition, heat transfer could

be expressed using Newton’s Law of Cooling. Thetemperature gradient normal to the furnace wall—calculated using the wall thickness and temperatures atthe inner and outer surfaces—determined heat lossthrough the wall. Based on the assumed thickness andrefractory materials, the heat conduction coefficient inthe furnace wall was expressed as 5 W 9 m�1 9 K�1.

C. CZ Treatments

The softening and melting zone within the BF, i.e., theso-called CZ, contributed significantly to the processcomplexity. This zone was of critical importance forefficient operation of the BF and because its shape andposition determined the permeability, fluid flow, gas use,thermal and chemical efficiency, and hot-metal quality inthe furnace. Within this region, iron-bearing materialswere gradually transformed from the lumpy, to thesoftening, to the half-molten states before finally meltingdown. Instead of passing through the low permeabilityportion of this region, the reducing gas can flow radiallythrough adjacent coke layers, which formed a low-resistance path between dripping and lumpy zones. Thisfinding implies a possible retardation of heating andreduction rates for iron-bearing materials in the CZ, withdirect gas contact only occurring at the iron-bearingmaterials-coke interface. Therefore, careful consider-ation must be given to the internal structure of the CZ.

Physically, the CZ may be subdivided into the follow-ing different states as shown in Figure 1(a)[42]: molten,softening, lumpy, and coke-only. According to theanalysis of Kanbara et al.,[43] in Figure 1(a), distribu-tions of metallic iron and slag were present, whereas inFigure 1(b), iron-bearing materials were about 70 pctreduced and bonded together. Their bonding was madepossible by the metallic iron produced on the surfaces ofores or by the slag transuding from their inside; in part C,iron-bearing materials existed in a lumpy state withoutsoftening and agglomerating. With reference to the workdone by Gudenau et al.[22,23] and Bakker et al.,[44]

Fig. 1—Schematic shows the (a) internal state of the cohesivezone[42]; (b) temperature-based definition of the nonlayered cohesivezone; and (c) defined layered cohesive zone. Note that Ts refers tosolid phase temperature even for the solid which is in the meltingstate.


corresponding to the complex structure of the CZ, thepermeability variation exhibited the following charac-teristics: at the early stage of softening and melting,although the iron-bearing materials experienced a sig-nificant deformation, the macroporosity of the bedremained open and a pressure drop did not significantlyincrease; as softening and melting continued, once theFeO-containing melt was exuded from the individual bedcomponents into the macroporosity of the bed, a markedincrease occurred in the pressure drop over the bed; atthe final stage, the melt started to drip from the moltenportion so that much of the region was occupied by theliquid phase, and the gas only could flow through to thelumpy zone through the coke windows.

The numerical treatment of a CZ exerted a stronginfluence in BF modeling. Therefore, in the currentsimulation, three CZ treatments were used and com-pared with each other, i.e., isotropic and anisotropicnonlayered treatments, and a layered treatment, asshown in Figures 1(b) and (c). For the first twotreatments, which represent the traditional modelingapproach,[1] the CZ was treated as a mixed region ofiron-bearing materials and coke particles. For the thirdtreatment, the CZ was treated as a layered structure. Tomodel the structure of Figure 1(a) according to BFdissection studies, the stratified structure of coke andiron-bearing material (ore) layers was first calculatedbased on the solid flow field; then, the CZ was defined tostart and finish within the temperature range 1473–1673 K. Finally, the CZ region was divided intoalternate coke as well as softening and melting layers.In BF practice, the state of the softening and meltinglayer was dependent on the high temperature propertiesof iron-bearing materials and the specific operatingconditions. However, for simplicity, in the current studythe CZ state was based on previous fundamental studies.With reference to the work done by Gudenau et al.[22,23]

and Bakker et al.,[44] the following three states werespecified: (1) state I, 0.7<Sh�r <1.0 corresponded to theportion with molten state and liquid source in which theore layer voidage was occupied fully by the liquid phase;(2) state II, 0.5 < Sh�r £ 0.7 corresponded to the

combined portion with softening and melting of iron-bearing materials in which the pressure drop may haveincreased significantly; (3) state III, 0.0 < Sh�r £ 0.5corresponded to the softening stage in which themacropores of the bed remained open so the variationof the pressure drop was limited. Solid conduction andthe gas–solid heat transfer coefficient should be specifiedaccording to the different heat and mass transfermechanisms in each of the states. Note that propertyvariations for iron-bearing materials in the CZ corre-sponding to these states also were considered in non-layered treatments.The detailed comparisons for these three treatments

are summarized in Table IV, which shows the differ-ences in dealing with solid volume fraction, particle size,solid heat conductivity, and gas resistance in the CZ.These differences may affect gas and liquid flow mod-eling significantly, although the solid phase had tomaintain some mixed properties of iron-bearing mate-rials and coke in the CZ to guarantee the overall heatand mass conservation. These different CZ treatmentswere applied separately in the simulation with the sameboundary conditions and the same numerical techniquesdescribed in the following section.

D. Numerical Technique

A proprietary code was developed to solve theconservation equations based on the finite volumemethod, which is also the basis of most commercialCFD codes such as Fluent,[45] CFX,[46] and Star-CD[47]

in use today. The solution domain was subdivided intocomputational cells (control volumes) using a structuredand collocated grid where all variables were defined inthe center of a cell.[48] To avoid checkerboard pressure–velocity decoupling, Rhie–Chow interpolation wasapplied,[49] which uses a momentum-based interpolationfor the cell face mass fluxes in the continuity equationthat closely imitates the staggered practice by forcingmass conservation to be expressed in terms of massfluxes across cell interfaces. Therefore, velocity–pressuredecoupling cannot occur.

Table IV. Comparison of Three CZ Treatments

Treatments

Nonlayered treatments

Layered TreatmentIsotropic Anisotropic

VariablesSolid volume fraction es ¼ noreeore þ ncokeecoke

/s ¼/ore for ore layer

/coke for coke layer

(

where/ ¼ e; d; k

Solid particle size ds ¼ noredoreþ ncoke

dcoke

� ��1

Solid heat conductivity ks ¼ norekoreþ ncoke

kcoke

� ��1

Vertical to the layerGas flow resistance in CZ af ¼ a1�es

dsaf ¼ noreaore þ ncokeacoke For ore layer af ¼ aore; bf ¼ bore

bf ¼ norebore þ ncokebcoke

Parallel to the layer

bf ¼ blg 1�esð Þ2

d2s esaf ¼ noreffiffiffiffiffiffi

aorep þ ncokeffiffiffiffiffiffiffi

acokep

� ��2For coke layer af ¼ acoke; bf ¼ bcoke

bf ¼ noreboreþ ncoke

bcoke

� ��1


Convective terms were discretized by the upwindscheme in which quantities at cell faces were determinedby assuming that the cell-center value of any fieldvariable represented a cell-average value and heldthroughout the entire cell. For the discretization ofviscous transport terms in the momentum equations, thecentral finite difference approximation with second-order spatial truncation errors was used. The Green–Gauss theorem was used to compute the gradient of thescalar at the cell center. Strong under-relaxation wasapplied in the temperature and concentration fieldcalculations. A linearization of source terms related tothe generation of phase species and reaction heat had tobe introduced for physical boundness, i.e., maxima orminima of process variables was within a reasonablerange.[50] The strongly implicit procedure[51] was used tosolve the discretized equations with the large sparselinear matrix. The deadman boundary for the solid flowfield was determined with a solid isovelocity curve.[52]

The SIMPLE algorithm[50] was used to couple velocityand pressure drop of continuous phases. The mainprocedure used to calculate liquid flow is detailedelsewhere.[53]

As shown in Figure 2, the well-established sequentialsolution procedure was employed in the current study tocalculate the coupled fluid flow, heat transfer, andchemical reactions. First, the (initial) gas, solid, andliquid flow, temperature, as well as concentration fieldswere determined under the boundary conditions at theinlets, outlets, and walls. The layered structure wasdetermined based on the burden distribution at the BFtop (e.g., the ore and coke batch weights as well as theore–coke ratio) and the timelines of solid flow—atechnique used in the previous studies of gas–solid flowin a BF.[30,52] Then, the flow, temperature, and concen-tration fields were calculated without considering thechemical reactions until an approximate convergence(i.e., high tolerance) in terms of mass and energyresiduals was obtained. Finally, thermalchemical behav-ior was taken into account, which lead to the CZdetermination, and with this information, fluid flow,heat and mass transfer, as well as chemical reactionswere recalculated. This procedure was repeated until theCZ position converged. In this study, the criterion forconvergence was set to 10 pct of the relative CZpositions in two consecutive iterations. The error wassomehow large compared with those for the convergenceof gas, liquid, and solid flows in the computation. Thiswas because the estimated CZ tended to oscillate undersome conditions. But considering the complexity of theproblem, the resulting error in BF performance predic-tion was small.

III. SIMULATION CONDITIONS

Numerical simulations were performed under condi-tions similar to a BF of ~1000 m3 inner volume (hearthdiameter of 7.2 m, height of 25 m). Assuming thesymmetrical distribution of process variables, only halfthe BF geometry was considered in the simulation andwas treated as a two-dimensional slot model for

simplicity. Generally, the starting thickness of a CZore layer was around 30 cm. To model the CZ at asufficient level of detail, the whole computationaldomain was divided into 298 9 38 nonuniform controlvolumes in the Cartesian coordinates. Each computa-tional cell was then around 8 cm 9 9 cm. The corre-sponding computational domain and an enlargement ofthe cells in lower region are shown in Figure 3. Gas wasinjected to the BF through a lateral inlet (tuyere)occupying four consecutive cells at the wall in thefurnace lower zone. Solids (ore, coke, and flux) werecharged from the top of the furnace with a uniform

Fig. 2—Solution procedure in which a rectangular-shaped box repre-sents a calculation step and a diamond-shaped box represents a deci-sion step.


downward velocity along the horizontal direction. Thebasic operating data of the modeled BF—calculatedusing a heat and mass balance—are summarized inTable V. During the simulation, the productivity in aunit volume was first calculated based on the dataprovided in Table V, then consistent gas, ore, coke, andflux rates were determined that assumed a two-dimensional profile with unit thickness. The liquids,hot metal, and slag were assumed to be generated at thelower side of the melting subregion within the CZ, withflow rates determined from the total amount of liquid,inlet area, and ore–coke ratio. For typical conditions,the gas mass flow rate was 11.7 kg 9 m�2 9 s�1 at thetuyere; solid flow rate was 1.22 kg 9 m�2 9 s�1; andliquid slag and hot metal flow rates were dependent onthe liquid inlet area calculated during the simulation.Contemporary intensive BF operation required a hightop pressure, with a constant 2 atm assumed for allsimulations. Average solid particle sizes and ore–cokeratio were as shown in Table V, with a nonuniformdistribution of these variables imposed along thehorizontal direction, as discussed later.

In practice, with the demands of smooth operationand high productivity, burden distribution often wasadjusted to improve the gas flow in the furnace, which iskey to BF operations. Good hot-metal quality alsolargely was achieved through appropriate control ofburden distribution. In the current simulations, theburden distribution and particle sizes, as the inputinformation, were assumed, as shown in Figure 4. Notethat ore particle size was assumed to be constant, butcoke particle size was assumed to change with the ore–coke volume ratio. With this information, physical

properties of the mixture of ore and coke, such asparticle diameter and porosity, could be calculated. Inthis case, the ore–coke ratio was low at the center, andthe coke particle size was small near the furnace wall.This burden distribution provided a high permeability atthe furnace center because large particles induced largevoids and presented a low surface area–volume ratio topassing fluids. Initially, the particle properties imposedat the top of the furnace were extended to the lower partso that particle size and porosity distributions in theentire furnace could be obtained. During the calcula-tion, with the identification of the CZ, coke zone, anddeadman, the properties in these regions were reesti-mated to replace the initial values based on thefollowing rules: (1) in the CZ, porosity and particle sizeof iron-bearing materials were a function of normalizedshrinkage ratio, Sh�r ; (2) in the coke zone, bed perme-ability was calculated based on coke size only; and (3) inthe deadman, coke size and porosity were assumed asds = 0.02 m and es = 0.65. In practice, because of theimportance of burden distribution to furnace opera-tions, these conditions should be reset with reference toequipment setpoints and measured data from actual BFoperations or laboratory scale tests.

IV. RESULTS AND DISCUSSION

A. Effect of CZ Treatment on Gas Flow

The CZ has a significant effect on the fluid flow, asdemonstrated by previous studies.[2–5] In particular, thelayered CZ can make the gas flow significantly change

Fig. 3—(a) Compuational domain and (b) the grid arrangement in an enlarged local region.


direction and force liquid to flow horizontally betweenthe fused layers. However, the effect of different CZtreatments on the modeling of fluid flow has not beencompared critically. To exclude the influence of otherfactors, the comparison was first carried out under asimplified condition, i.e., without considering solid andliquid flow, heat and mass transfer, as well as chemicalreactions. In this case, the position and structure oflayered and nonlayered cohesive zones were assumed.Figure 5 shows the fixed CZ structure and porositydistribution for layered and nonlayered CZ treatments.Here porosity referred to the interconnected pore spacebetween particles or through a partially fused layer inthe CZ that was accessible to gases and liquids. Notethat for anisotropic and isotropic nonlayered treat-ments, the CZ structure and porosity distribution werethe same. Boundary conditions are given in Table V.The computational domain encompassed the lumpyzone CZ, dripping zone, and solid stagnant region(deadman). Except for the CZ region, computationalconditions were the same for the three CZ treatmentsconsidered. Within the CZ, the layered structure com-prised low-permeability softening and melting layers as

Table V. Operational Conditions Considered in this Work

Variables BF

Gas:Volume flux, Nm3 9 tHM�1 1511Inlet gas components, molepercentage

34.95 pct CO0.0 pct CO2

0.81 pct H2

0.0 pct H2O64.23 pct N2

Inlet gas temperature, K 2313.6Top pressure, atm 2.0Solid:Ore, t 9 tHM�1 1.64Ore components, mass fraction Fe2O3 0.656

FeO 0.157CaO 0.065MgO 0.024SiO2 0.059Al2O3 0.029MnO 0.006P2O5 0.008

Average ore particle size, m 0.03Coke, t 9 tHM�1 0.5023Coke components, mass fraction C 0.857

Ash 0.128S 0.005H 0.005N 0.005

Average coke particle size, m 0.045Flux, t 9 tHM�1 0.0264Flux components, mass fraction CaO 0.438

MgO 0.079SiO2 0.024Al2O3 0.033CO2 in CaO 0.344CO2 in MgO 0.082

Ore voidage 0.403(100dore)0.14

Coke voidage 0.153logdcoke + 0.724Average ore/(ore+coke)volume ratio

0.5923

Liquid:Hot metal rate, t 9 day�1 2034

components, mass fraction C 0.04Si 0.004Mn 0.0045P 0.0003S 0.0003Fe 0.9509

density, kg 9 m�3 6600viscosity, kg 9 m�1 9 s�1 0.005conductivity, w 9 m�1 9 K�1 28.44surface tension, N 9 m�1 1.1

Slag rate, t 9 tHM�1 0.377components, mass fraction CaO 0.324

MgO 0.120SiO2 0.324Al2O3 0.200FeO 0.016MnO 0.009S 0.007

density, kg 9 m�3 2600viscosity, kg 9 m�1 9 s�1 1.0conductivity, w 9 m�1 9 K�1 0.57surface tension, N 9 m�1 0.47

Fig. 4—Top-burden distribution: (a) volume ratio of iron ore and(b) particle size.


well as highly permeable coke layers as shown inFigures 5(a) and (b). Porosity in the softening andmelting layers was assumed to reach a minimum of20 pct of the initial ore voidage—calculated accordingto the correlation given in Table V. Correspondingly,particle size in the softening and melting layers wasassumed to reach a minimum of 40 pct of average ore-particle size. With the same ore–coke volume ratio in theCZ and the same particle size and porosity assumptionsfor iron-bearing materials and coke as those used for theearlier layered treatment, porosity in the nonlayered CZwas determined for the mixture of iron-bearing materi-als and coke as shown in Figure 5(d).

Figure 6 shows the calculated gas flow field fordifferent CZ treatments. All cases demonstrated thatthe CZ acts as a gas distributor so that the flows deviatethrough the CZ. However, these deviations were differ-ent for the three treatments. Gas flowed more horizon-tally through the CZ for layered and anisotropicnonlayered treatments than for the isotropic nonlayeredtreatment, which is consistent with other studies.[26,34]

However, in essence, the reason why gas flowed hori-zontally is different. As shown in Figure 6(d), for thelayered CZ treatment, the lower permeability of the orelayer made the gas preferentially flow through the highpermeability coke layer. In contrast, for the anisotropicCZ treatment, gas flowed horizontally through theentire CZ because of the large axial resistance in thisregion.

Pressure drop is a primary indicator that representedthe permeability and gas resistance in the furnace.Figure 7 shows the corresponding pressure distributionsfor the three CZ treatments (NB. Pressure drop shownin the figure was relative to the top pressure). Largepressure gradients existed in the CZ region, which

indicate the sudden change of bed permeability for allthe treatments. Among the three treatments, thepressure drop for the anisotropic CZ was the highest,although little difference was observed in pressure dropbetween the layered and isotropic nonlayered treat-ments, as demonstrated by the pressure distributions atdifferent heights in Figure 8.Although, in this section, heat transfer and chemical

reactions have not been considered, the comparisonsdiscussed show that an anisotropic CZ treatment isinclined to predict high gas flow resistance because thelow permeability of iron-bearing materials dominatedthe calculation of resistance throughout the CZ in theaxial direction and strongly influenced resistance in theradial direction. In contrast, the pressure drop calcu-lated for the isotropic CZ treatment was less because theaveraged properties of the layers were less stronglyinfluenced by the low permeability ore. Although theanisotropic treatment seemed conceptually more realis-tic than the isotropic treatment, the significant increasein pressure drop through the CZ in the former caserelative to the layered treatment suggested that theformulation of gas flow resistance in the radial directionmay need to be reconsidered in the future.Nonetheless, it is clear that different CZ treatments

will result in different flow behavior and pressuredistribution, which in turn, will result in a differentinfluence on the thermal and chemical behavior includ-ing the CZ position and the other process performanceas discussed in the following.

B. Effect of CZ Treatment on Process Performance

In this section, the three CZ treatments were appliedto BF modeling with heat transfer and chemical

Fig. 5—Computational domains and porosity distributions for different CZ treatments; CZ position for (a) layered and (c) nonlayered; porositydistribution for (b) layered and (d) nonlayered.


Fig. 6—Gas flow fields for (a) layered, (b) anisotropic nonlayered, (c) isotropic nonlayered treatments, and (d) enlarged CZ, which are listed inparts (i) layered, (ii) anisotropic nonlayered, and (iii) isotropic nonlayered. Note that the reference vectors are given on the top of each subfigure.

Fig. 7—Pressure distributions for (a) layered, (b) anisotropic nonlayered, and (c) isotropic nonlayered treatments.


reactions. Instead of a fixed position, the CZ waspredicted for the three different treatments based on thesame boundary and operational conditions. In thesimulations, the CZ region was defined to be withinthe temperature range of 1473–1673 K, as described inthe numerical modeling section. The shrinkage ratio Shr,which is the most commonly used parameter to repre-sent the softening and melting status of iron-bearingmaterials, was applied to determine the particle sizeand porosity of iron-bearing materials in the CZ. Alinear relationship between the shrinkage ratio and thetemperature was used, as shown in Figure 9(a). Corre-sponding relationships between normalized porosity–particle size and shrinkage ratio were assumed, as shownin Figures 9(b) and (c). The relationships encompassedthe three states of iron-bearing materials discussed.

Fig. 8—Pressure distribution for different CZ treatments (a) at aheight of 6.5 m and (b) at tuyere level.

Fig. 9—Relationship between (a) shrinkage ratio and temperature,between (b) normalized shrinkage ratio and particle size for iron-bearing materials in the CZ, and between (c) normalized shrinkageratio and porosity for iron-bearing materials in the CZ.


Fig. 10—CZ shapes (a)–(c), porosity distributions (d)–(f), as well as pressure drop distribution and gas flow stream line (g)–(i) for the followingCZ treatments: left column, layered; middle column, anisotropic nonlayered; and right column, isotropic nonlayered. Note that the minimumporosity can reach 0.094 in the layered cohesive zone.


Fig. 11—Gas (a)–(c), solid (d)–(f), and liquid slag (g)–(i) flow field for the following three CZ treatments: left column, layered; middle column,anisotropic nonlayered; and right column, isotropic nonlayered.


Figure 10(a) shows that an inverse-V shaped CZstructure was obtained using the layered CZ treatment.In the layered CZ, iron-bearing materials and cokewere distributed alternately so that permeable cokewindows could form. Softening- and melting-ore layerscomprised the following three regions corresponding tothe defined states: (1) Region I where blockage of thebed porosity was nearly complete by the melting ofsolids; (2) Region II where a combination of softeningand melting of iron-bearing materials were present; and(3) Region III where iron-bearing materials onlyunderwent softening, which occupied the greatestportion of the CZ. Although the CZ was modeled asa mixed region of iron-bearing materials and coke forboth cases, for the nonlayered CZ treatments, thedifferent resistances along the vertical direction led todifferent CZ shapes, i.e., a horizontal CZ for theanisotropic nonlayered treatment and an inverse-V CZfor the isotropic nonlayered treatment, as shown inFigures 10(b) and (c).

Corresponding to these modeled CZs, porosity dis-tributions were as shown in Figures 10(d)–(f). In thelumpy and dripping zones, a high permeability wasobserved at the furnace center and a low permeabilityregion was observed near the wall. Within the CZ, highand low permeability regions were stratified for thelayered treatment, whereas the permeability graduallyincreased from the lower to upper part of the CZ for theother two treatments. It must be noted that a complexinteraction occurred between the gas flow and theporosity variation caused by the softening and meltingof the burden because it is the gas flowing through theburden that eventually causes iron-bearing materials tosoften and become impermeable to gas flow. Todemonstrate this relationship, the corresponding gasflow and pressure drop distributions are illustrated inFigures 10(g)–(i), where the variation of gas flow direc-tion corresponded to the porosity variation. Amongthese treatments, the variation of gas flow direction inthe CZ for the anisotropic treatment was the mostintensive and a marked increase in pressure dropoccurred in the CZ, which reflected the highest predictedresistance. However, the results show that different CZtreatments only affect the pressure drop distribution inthe lower part of the BF, and their influence becomesinsignificant in the lumpy zone.

Fluid flow fields for the three CZ treatments arecompared in Figure 11. The results show that the gasvelocity increases while gas passes through the cokewindow for the layered CZ treatment, and an intensivehorizontal gas flow occurred through the identified CZfor the anisotropic nonlayered treatment. For all treat-ments, the solid flow direction evidently changedthrough the CZ because coke particles had to replacethe space occupied by consumed ferrous materials, i.e.,lumpy or agglomerated iron ores. Solid downwardvelocity in the lumpy zone was larger than that in thedripping zone because solid flow in the dripping zonewas only driven by the coke consumption in the lowerpart of the BF. In addition, with the downward solidmovement and coke consumption in the raceway, thedeadman, i.e., the solid stagnant zone, clearly was

formed, in which permeability may deteriorate andsmall particles may accumulate.As shown in Figure 11(g), from the molten region

within the layered CZ, generated liquids flowed almostvertically in the lower part of the furnace. However,complex localized liquid flows occurred in the CZ wheresome liquid droplets or rivulets were likely to flow alongthe layers. Similar phenomena were observed by Chewet al.[17] in a cold physical model describing liquid flowin which part of the dripping liquid preferentiallyaccumulated in the lower layers of the CZ. This directlyresulted in nonuniform liquid flow in the lower part ofthe furnace. Additionally, the internal structure of thelower part of the furnace made the nonuniform distri-bution of liquid flow more evident. In contrast, theliquid patterns generated by using nonlayered treat-ments showed relatively uniform liquid (volume frac-tion) distributions proportional to the ore–coke ratio inthe CZ.For the layered treatment, the nonuniform liquid flow

was demonstrated again by the horizontal distributionof liquid flux passing through the bottom of thesimulation domain as shown in Figure 12. This liquidoutlet represented the surface of the hearth where liquidcollects in a BF. Liquid flux varied significantly alongthe horizontal direction, whereas a smooth distributionof liquid flux was observed for nonlayered treatments.The results also exhibited a maximum liquid slag fluxlocated some distance from the tuyere for all threetreatments. This behavior is believed to result from thecombined contribution of the strong blast pushing liquidaway from the tuyere and the high liquid flux tricklingdown from the lower, high ore volume fraction part ofthe CZ.To examine the effect of different CZ treatments on

heat and mass transfer, Figures 13–15 show the calcu-lated phase temperature and CO concentration distri-butions. The results show that different temperature andconcentration distributions were observed within andaround the CZ for the different treatments. The effect ofthe different treatments on heat and mass transfergradually decreased and became negligible at the fur-nace top, which indicates that the flow readjustedquickly in the lumpy zone so that the process variableswere independent of the CZ treatments in the upper partof the furnace where the all cases had the same burden

Fig. 12—Liquid slag flux distribution entering the hearth.


distribution. For all treatments, in the lumpy zone, theassumed charging pattern induced high permeability ofthe central part to achieve a central gas flow so that highgas temperatures and corresponding high solid temper-atures were predicted near the furnace center as shown

in Figures 13–15(a) and (b). At the upper-central part ofthe furnace, the CO concentration was high because ofpreferential gas flow and low ore to coke ratio.However, the effect of these treatments on process

variable profiles in the lower part of the BF was more

Fig. 13—Computed contours of gas (a), solid (b), liquid (c) temperature, and CO concentration (d) for the layered treatment.

Fig. 14—Computed contours of gas (a), solid (b), liquid (c) temperature, and CO concentration (d) for anisotropic nonlayered treatment.


evident, especially for the liquid temperature distribu-tion, as shown in Figures 13–15(c). The difference intemperature distributions also is demonstrated inFigure 16. Liquid slag and hot metal temperatures notonly were affected by the nonuniform gas and solidtemperature distributions but also closely linked to thepacking structure of the lower part of the BF. Thisfinding was demonstrated by the liquid slag temperaturegradient, which became larger within the deadmancompared with that in the dripping zone. Similar tothe liquid outlet flux distribution, liquid temperaturefluctuated for the layered treatment, whereas the liquidtemperature maintained a smooth variation for theisotropic and anisotropic nonlayered treatments.

When liquid flowed through the region adjacent to thetuyere, the strong heat exchange between gas and liquidmade the liquid temperature abruptly increase for allthree treatments. Apart from the region adjacent to thetuyere, a high-liquid temperature region was alsoobserved in the central part of the furnace where liquiddroplets experienced a longer trickling time from the CZto the hearth because of the higher CZ position in thecenter than near the wall, and high solid temperatures inthe deadman also play a role. This nonuniform temper-ature distribution made reactions between slag, coke, andgas more complex. It also implied that liquid propertieswere strongly dependent on the CZ structure becauseliquid residence time in the coke zone and deadmandirectly was related to the shape and position of the CZ.

Different CZ treatments represented different BF oper-ations and practice. For example, a layered CZ struc-ture inevitably exists for BF operation with alternate

Fig. 15—Computed contours of gas (a), solid (b), liquid (c) temperature, and CO concentration (d) for isotropic nonlayered treatment.

Fig. 16—Temperature distributions at the liquid outlet for (a) slagand (b) hot metal.


charging of iron-bearing materials and coke. Based on theprevious results, the layered CZ treatment can predict theformation of coke windows and, therefore, reasonable CZpermeability, which largely reflects the BF situation. Incontrast, for the anisotropic CZ treatment, the calculatedaxial resistance throughout the CZ region was high, whichplayed an important role in generating horizontal gas flowin the CZ. When the batch weight of iron-bearingmaterials and coke was small, i.e., narrow layer distribu-tion in the BF, the calculated results using the anisotropictreatment could be similar to those using the layered

treatment if the discrepancy in permeability between thetwo treatments could be eliminated. When the batchweight was large, the predicted pressure drop representingthe fluid resistance in the CZ by the anisotropic CZtreatment could be higher than the real value because therole of the highly permeable coke layer was consideredinadequately in the model formulation. In this case, theanisotropic CZ treatment should be adjusted for betterpermeability prediction. In the third approach, theisotropic nonlayered CZ treatment used the averagedproperties of iron-bearing materials and coke to predict

Fig. 17—Pseudo-transient process of solid flow (Dt 9 530 s).


the resistance, which weakened the effect of the lowpermeability region in the CZ on the fluid flow. From apractical viewpoint, this treatment is only suitable formodeling BF operation with mixed charging of ore andcoke.

Finally, it should be noted that a BF is a moving bedreactor should be modeled as a transient process. Asingle-layered CZ configuration represented only asnapshot of this transient operation. However, differentlayer configurations could be used to model the burdenpositions at different times in the charging cycle asshown in Figure 17. Here, a total of eight burdendistributions were used to represent a periodic burdenflow. Corresponding to these different configurations,the layered CZ and process variable distributions couldbe calculated to produce average flow and performancevariables. Because the residence time of gas and otherphases is much shorter than that of solids in a BF for agiven solid flow pattern, the gas and liquid flows, and toa large degree, temperature and concentration fieldsreasonably could be described as a steady-state process.Therefore, the distributions of the solid patterns andcorresponding process variables represent a pseudo-transient process.

Applying a simple averaging approach to the abovepseudo-transient results in the following:

�w ¼Pn

i wiwiPni wi

;

w ¼ p; eg; es;Tg;Ts; yco;

wi ¼1

n; n ¼ 8

½6�

A selection of the averaged results are shown inFigure 18. The results show that an approximate poros-ity range from 0.23 to 0.5 could be obtained in the CZ.Based on the solid temperature, the averaged CZ is alsoshown in Figure 18(d). The CZ position predicted by theaveraged results is different from that calculated withnonlayered CZ treatments (Figures 10(b) and (c)).Compared with the calculated results of the nonlayeredCZ treatments, which may exaggerate the gas flowresistance or underestimate the flow variation in the CZ,the averaged results are based on more realistic perme-ability distributions and could better represent a steadyBF operation. The application of different layer config-urations to model a periodic burden flow provided anapproach to better compare the layered CZ with thenonlayered CZ treatments and improved the applicabil-ity of the former to general BF simulation. Therefore,although obtained under steady-state conditions, theresults predicted by the present approach were useful toprocess understanding.

V. CONCLUSIONS

A mathematical model was developed to describe thefluid flow, heat and mass transfer, as well as chemicalreactions in a BF. Different from the previous models,the layered CZ was considered explicitly, and a criticalcomparison of BF modeling with different CZ treat-ments, i.e., layered, isotropic nonlayered, and aniso-tropic nonlayered, was carried out. The results showedthat predicted fluid flow and thermochemical phenomena

Fig. 18—Averaged process variable distributions for (a) gas volume fraction, (b) solid temperature, (c) liquid temperature, and (d) CZ shape.


within and around the CZ as well as in the lower part ofthe BF for different treatments were different. Thisfinding implies that the CZ treatment is of paramountimportance in BF modeling.

Physically, different CZ treatments may relate todifferent burden distributions at the furnace top. Forexample, the layered CZ treatment corresponded to theBF operation with layered charging of coke and iron-bearing materials, the isotropic nonlayered treatmentcorresponded to the BF operation with the mixedcharging of coke and iron-bearing materials, and theanisotropic nonlayered CZ treatment corresponded tothe BF operation with layered charging but a smallbatch weight for coke and iron-bearing materials. Fromthis point of view, BF modeling with a layered CZtreatment facilitated the simulation of conditions thatcould not be handled explicitly by a traditional BFmodel, such as the effects of charging sequence andbatch weight. Moreover, by applying a range of layerconfigurations to achieve a pseudo-transient represen-tation of BF operation, averaged results using a layeredCZ treatment may provide a better overall picture ofsteady BF operation in terms of the permeabilityprediction when compared with the more commonnonlayered CZ treatments. In this respect, a re-evaluationof the formulation that governs radial gas flow resis-tance in the anisotropic nonlayered CZ treatment wouldseem warranted. More studies are necessary to clarifythese considerations and to extend this work to thesimultaneous consideration of BF shaft, raceway, andhearth.

ACKNOWLEDGMENTS

The authors are grateful to the AustralianResearch Council and BlueScope Steel for the finan-cial support of this work and to Professor J. Yagi atTohoku University as well as Dr. P. R. Austin andDr. D. Maldonado at BlueScope Steel Research fortheir valuable discussion.

ABBREVIATIONS

a,b Coefficients in the Ergun equation,a = 1.75, b = 150

aFeO The activity of molten wustiteAc Effective surface area of coke for reaction,

m2

Asl,d Effective contact area between solid andliquid in unit volume of bed, m2 9 m�3

cp Specific heat, J 9 kg�1 9 K�1

d Diameter of solid particle, md* Normalized particle size, d* = d/dmax

dl,g Liquid droplet diameter as affected by gasflow, m

dl,h Droplet diameter as affected by holdup, mdw Effective packing diameter, mD Diffusion coefficient, m2 9 s�1

Deg;n Effective diffusivity of component n,

n = CO, m2 9 s�1

Ef Effectiveness factors of solution lossreaction

f0 Fraction conversion of iron oreF Interaction force per unit volume,

kg 9 m�2 9 s�2

g Gravitational acceleration, m 9 s�2

h Holduphl,t Total holduphl,t0 Total holdup without gas flowhij Heat transfer coefficient between i and j

phase, W 9 m�2 9 K�1

H Enthalpy, J 9 kg�1

DH Reaction heat, J 9 mol�1

k Thermal conductivity, W 9 m�1 9 K�1

kf Gas-film mass transfer coefficient,m 9 s�1

ki Rate constant of ith chemical reactions(i = 1, 2, or 3), m 9 s�1

K1 Equilibrium constant of indirect reductionof iron ore by CO

Mi Molar mass of ith species in gas phase,kg 9 mol�1

Msm Molar mass of feo or flux in solid phase,kg 9 mol�1

Nu Nusselt number, Nu = h 9 ds 9 k�1

p Pressure, PaDpe/Dx Effective pressure gradient, Pa m�1

Pr Prandtl number, Pr = cp 9 l 9 k�1

Reg Gas Reynolds number,Reg ¼ /s � ds � qg � eg � ug � l�1

g

R Gas constant, 8.314 J 9 K�1 9 mol�1

R* Reaction rate, mol 9 m�3 9 s�1

S Source termShr Shrinkage ratio defined as the ratio of the

decreased volume, caused by softeningand melting, to the original volumeoccupied by iron-bearing materials

Sh�r Normalized shrinkage ratio,Sh�r ¼ Shr

Shr;max; Shr;max ¼ 0:7

T Temperature, Ku Interstitial velocity, m 9 s�1

Vb Bed volume, m3

Vg Gas volume, m3

Vore, Vcoke Ore, coke volume, m3

Volcell Volume of control volume, m3

Xp Dimensionless pressure dropyi Mole fraction of ith species in gas phase

GREEK

C Diffusion coefficientI Identity tensor/ General variableu Shape factora Specific surface area, m�2 9 m�3

af, bf Coefficients in Ergun Eqaore, acoke Coefficients aore = aeore/dore,

acoke = aecoke/dcokeb Mass increase coefficient of fluid phase

associated with reactions, kg 9 mol�1


bore, bcoke Coefficients, bore ¼blge

2ore

d2ore 1�eoreð Þ;

bcoke ¼blge

2coke

d2coke

1�ecokeð Þc Modification factord Distribution coefficiente Volume fractione* Normalized volume fraction, e* = e/emax

h Contact angle, degreeg Fractional acquisition of reaction heatl Viscosity, kg 9 m�1 9 s�1

q Density, kg 9 m�3

r Surface tension, N 9 m�1

s Stress tensor, Pax Mass fractionnore, ncoke Local ore, coke volume fraction

SUBSCRIPTS

e Effectiveg Gasn N direction (= x or y)i Identifier (g, s or l)i, m Mth species in i phasej Identifier (g, s or l)k Kth reactionl Liquidl,d Dynamic liquids Solidsm Feo or flux in solid phase

SUPERSCRIPTS

e Effectiveg Gasl,d Dynamic liquids SolidT Transpose

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