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ISIJ International, Vol. 38 (1 998), No, 1o, pp. 1062-1068Process Simulation of CupolaN. N. VISWANATHAN.. N. SRINIVASAN1)nd A. K. LAHIR12)Formerly Department of Metallurgy, Indian Institute of Science. Nowat Division of Theoretical Metallurgy.Metallurgy, Royal Institute of Technology. Stockholm, S-1 0044, Sweden.E-mail: vichu~IJmetailurgi,kth.se1)Departmentof Mechanical Engineering, Indian Institute of Science, Bangalore, India-560012.2) Departmentof Metallurgy, Indian Institute of Science, Bangalore, India-560012.

(Received on Alpril 6. 1998. accepted in final form onMay26. 1998)

Department of

Cupola is a counter current shaft reactor for production of cast iron. Thesolid charge consisting of coke,pig iron, steel scrap and flux is fed from the top of the reactor and blast is blown radially through the tuyeres.One-dimensional (1 -D) models, wherein the governing equations are solved along the axiai direction, havebeen reported in the lite~ature. However,becauseof radial entry of blast through the tuyeres, incorporationof boundary conditions at the tuyere level poses a major difficulty in I -D models. In this paper, a pseudo2-D model for cupola has been proposed in which, the governing equations are employed in 2-D at thetuyere level and I -D in the remaining portion. The solution of 2-D equations at the tuyere level generatesthe appropriate boundary conditions for the remaining I -D portion above the tuyeres. In order to evaluatethe performance of the pseudo2-D model, a 2-D modelwasalso developed. Further, the two models havebeen validated using reported experimentai data, The study shows that the overall temperature andcomposition distributions obtained from the pseudo 2-D model are quite comparable with that obtainedfrom the 2-D model. Also, the pseudo2-D modelwasfound to be computationally faster comparedto the2-D model. Hence, for practical design and operation exercise, pseudo2-D model can be effectively used.KEYWORDS:upola; counter current; shaft; reactor; mathematical; model; simulation; pseudo2-D blast;tuyere; melting; cast iron.

1. IntroductionCupola is basically a counter current shaft type reactorfor production of cast iron. A schematic diagram of acupola is shownin Fig. 1. Thesolid charge consisting ofcoke, pig iron, steel scrap and flux is fed from the topof the reactor and blast is blown radially through the

tuyeres. The tuyeres are arranged in one or more rowsalong the periphery of the reactor at lower end of theshaft. Thevelocity of the inflowing air stream is relative-ly low ( 50ms~ l). Thecombustion coke at the vicinityof tuyeres generates heat andmelts the iron. Theheightbelow the tuyeres acts as hearth or reservoir of metal.Hot ascending gas from the reaction zone preheats thedescending charge making the reactor a thermally andchemically efficient unit.Numerousmprovements in cupola design and opera-tion such as incorporation two rows of tuyeres, hot blast,oxygen enrichment etc. have been well established andwidely applied in pract ice. However, the mechanismbehind these improvementsl) are not well understoodyet. Mathematical modelling can help not only in betterunderstanding of the phenomenaut also in developinga rational approach towards cupola design and opera-tion.Manymodels have been developed for typical shaftfurnace like blast furnace.2) Howeverfor cupola, modeldeveloped by Nyamekyet al.3) is the only one available

in the literature. Recently, the present authors have de-veloped a 3-D mathematical for cupola.4) The modelwasdeveloped by solving the 3-D heat andmassbalanceequations to predict temperature and composition pro-Spark arrester

ChargeLevel

~Charging door

Taphole

Tuyeres

Slag hole

C 1998 ISIJ 1062Fig. 1. Aschematic of cupola.

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files within the cupola and the modelwasvalidated usingexperimental data for a 24" cupola with four tuyeresin a single row. The slmulation results from the modelshowedthat the combustion zonesemanating from eachtuyere overlaps and forms an overall combustion zoneof cylindrical shape. Further, the temperature andcomposition gradient in the circumferential direction ismuchsmaller comparedto that in the radial directionsuggesting that a 2-D model is adequate for the simula-tion of cupola.Although rlgorous, computer intensive 3-D or 2-Dmodels gives superior insight into the reactor phenom-ena, the information which can be drawn from suchsimulations maynot find effective application in manypractical design and operatlon exercises. This is especial-ly true since fiuctuations in spout temperature, exit gastemperature and composltion are commonn real sys-tems. Besides, 3-D and 2-D models call for a completeknowledge of voidage and particle size distribution inthe bed, which is well near impossible to obtain in anoperating cupola. Hence,asimple and faster modelwhichcan simulate an average or overall performance of thecupola could be pretty adequate to enable effectiveprocess andequipmentdesign in an engineering exercise.In this respect, the I-Dmodeldeveloped byNyamekyeet al.3) is the only simple model available in literature.In this model, since the primary direction of gas flow inthe reactor is along the axial direction, the governingbalance equations were solved along that direction.However, because of the radial entry of blast throughthe tuyeres, at the tuyere level the gas fiow in the radialdirection is significant. This poses difficulty in incorpo-rating appropriate boundary conditions at the tuyerelevel in a l-D model.In the 3-D model,4) it is noted that the radial gas flowat the vicinity of tuyeres results in non-uniform com-position and temperature distribution across the crosssection of the cupola. However, this non-uniformity isconfined to andprominent only in the vicinity of tuyeres.Within a short distance above the tuyeres, the tempera-ture and composition profiles becomesnearly uniformacross the cross section. This suggests that a modelemploying balance equations in 2-D in the vicinity oftuyeres and l-D above the tuyere region should sufficefor the simulation of cupola.In this paper, a 2-D model and a pseudo2-D modelhave been developed. The pseudo 2-D model employsthe governing equations in 2-D at the tuyere level andin 1-D for the remaining portion. Asmentioned earlier,the main drawback in a l-D model lies in the incor-poration of boundaryconditions at the tuyere level. Thisis eliminated in the pseudo2-D model, wherein the 2-Dcomputation domain at the tuyere level generates theboundaryconditions for the I-D domainabovethe tuyerelevel. The suitability of the pseudo 2-D model has beeninvestigated by comparing the computational perfor-manceand agreementwith the 2-D model. Thepseudo2-Dmodeldeveloped can be extended to simulate newercounter current reactors like MIDREXCOREXtc. toaid in design and operation.

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38 (1 998), No. IO2. TheModelCupolacan be viewed as a heterogeneousmoving-bedcounter current reactor in which heat transfer, masstransfer and chemical reaction occur simultaneously.Themodel is based on the following assumptions:(1) Inertial terms in the momentumquation are not

significant. Vectorial form of Ergun's equation5,6) is ad-equate to represent the gas fiow.(2) Averagevoidage and particle size are known.(3) Piston flow is assumedfor the charge descend.As the coke descends through the combustion zone, itgets depleted and this depletion of coke is not uniformacross the cross section. This can give rise to completedepletion of the coke before it reaches the bottom of thecupola at the places of high combustion. It is assumedthat whenthe coke gets consumedat a particular loca-tion, the coke neighbourhood fiows to that location oras the coke descends it is distributed uniformly acrossthe cross section of the cupola at each instant.(4) The temperature of charge is a function of po-sition alone and is independent of the type of materiali.e., whether it is coke, metallic charge or liquid iron.(From here onwards this tempeature will be referred astemperature of charge.)

(5) Heat transfer by conduction and radiation arenot significant.2.1. Governing EquationsVGg=Sgcontinuity equation for gas) ................ (1)VG~=S~ (continuity equation for solid) .............. (2)- VP=(l +f2 1GgDGg(Ergun's pressure drop equation) **.....*.....,..*.(3)V G(*) = S(i)(Massbalance equation for gas or; charge species)

.(4)

V (GgCgTg)=h(Ts~ Tg)+Hg- H~(Heat balance equations) ......... ........,.(5)V(G~C~Ts)=h(Tg-Ts)+H~-H* ..... ..........(6)

Here fl and f2 are the resistance terms in the Ergun'spressure drop equation given by

fl = 150(1 -8)2kt ..........(7)83Pg(dp)2, I.75(1 - e) .......... (8)2= 83Pgdp

The Equations (1)-(6) were solved using followingboundary conditions(1) Normal componentof gas flow rate at the sidewalls and the bottom is zero except at the tuyere regionwhere it is known.(2) Thecharge fiow rate at the top of the furnace isknown.(3) Pressure at top of the reactor is equal to theatmospheric pressure.(4) Thecharge at the top is at room temperature.

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(5) The blast is fed at room temperature.(6) The heat loss through the reactor walls is cal-culated from the refractory wall thickness and conduc-tivity.2.2. Melting ModelPig iron in the cupola with its composition close to

that of iron~arbon eutectic melts at a temperature ofthe order of 1120'C. Whenhe pig iron reaches themelting point, it starts melting and forms a coating ofmolten around it. Whenhis melt coat reaches a critica]thickness it starts dripping from it. Thus the melt dropspercolating through the hot porous bedare heated furtherbefore reaching the hearth. Whereas, the temperature ofsolid pig iron remains at melting point until completemelting is over, the temperature of coke andmolten metalrises as it descends. Thusthe average temperature of solidandmolten metal is expected to increase as it descends.Tomodel the melting phenomena,t is assumedthata fraction, cc of the total heat transferred to charge isutilised for melting till the solid iron is melted completely.The fraction, o( is assumedo be O.5 times the differencebetween the normalised temperatures of the charge andthe melting point with respect to melting point.Themodel can be expressed mathematically as

AH~=0th(Tg-T~) ..... ..........(9)2.3. Chemical ReactionsThe oxidation of C to C02, carbon gasification re-action and oxidation of COo C02are the reactionsin the combustion zone:C+02~'C02 "__""""_"""_""__"""""__"'(lO)C+C02~2COCarbon Gasification Reaction)

.(1 l)C0+~02> C02 "' "_""_(12)

Oxidation of COo C02is a homogeneouseaction andproceeds at a muchfaster rate comparedto the othertwo heterogeneous reactions. 7) Hence, the concentrationof COn the presence of 02 is expected to be very small.In view of this, in the present model, it has beenassumedthat in presence of oxygen all COformed by carbongasification reaction gets oxidised to C02'The reaction rate expression used for oxidation ofcarbon (Eq. (lO)) and carbon gasification (Eq. (1 1)) ofa single particle is given by

411 dp 22 preR= xP 1 6 RgTg-+f dpkpEfPcEf' effectiveness factor is given by

Ef = 3(q) ' coth(q)) - l)/ep2q) = (dp/2.0)(pckp/D) 1/2D=6.7 x lO~ Io(T5)1'78k 2xRe 0.336Vg

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.(13)

(1)(15)(1)(1)

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38 (1998), No. 10For the oxidation of carbon (Eq. (lO))

p*.=(Po,~P'oq,~) ---""""-"---"-' .........(18)kp8)= 6.52 x 105exp(-22 OOOIT~)x (T,)o 5 ... (19)and for the carbon gasification reaction (Eq. (1 l))

p** = (Pco, ~P~qo~,) -"-"-"-"--"""""" (20)k 9) 40xl014exp(-440001T~) .........(21)have been used.The reaction rate per unit volume bed is given by~=~pXn ......... (22)wheren. is the numberof coke particles per unit volumeof bed.3. Computational SchemeThe computational domain was discretized using a

staggered grid system as suggested by Patankar.10) Aschematic of the staggered grid is shownin Fig. 2. Inthis, the scalar variables are defined at the centre andthe vector componentson the corresponding faces of thegrid. The governing equations (Eqs. (1)(6)) were dis-cretized by the finite volumescheme.For the convectiveterms in the governing equations upwind schemewasused, I o)The flow chart of the computation schemeis shownin Fig. 3. The overall computation schemeis commonfor both the 2-D and pseudo 2-D models.3.1. 2-D ModelThecomputational error wascalculated for each gridlocally as well as globally for the whole computationdomain using integral balance. The computation wasterminated when the error was within 10~5. Effect ofgrid size on the computedresults was studied to arriveat the appropriate grid structure. Thepresent computa-tions were carried out with 20 grids along the radialdirection and 100 along the axial direction. Thesolution

was insensitive to further grid refinement.3.2. Pseudo2-D Mode]In the pseudo2-D model, 2-D momentum,massandheat balance equations wereemployedat the tuyere leveland in other portions l-D massand heat balance equa-tions were used. The 2-D computation domain at thetuyere leve] has only one grid along axial direction(refer Fig. 4) and hence the namepseudo2-D. Asmen-tioned before, the 2-D domain generates the boundaryconditions for the upper l-D domain.Using the computedflow, temperature and composi-tion distribution at the 2-D domain, average values ofmass and heat flux leaving the tuyere domain werecalculated. These average fluxes serve as boundaryconditions for solving the temperature and compositionprofiles in the l-D domain.Similar to the 2-Dmodel, the local errors at both 2-Dand 1-D domainsand global errors for the whole com-putation dotnain using integral balance were computed.The computatlons were terminated whenthe error waswithin l0~5. As in the 2-D model, 20 grids along radial

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(P, T) Gr

Fig. r2. A schematic of the staggered grid.START

Input CupolaDimensions, Blast Rate, Iron to coke ratioGrid structure andapproximate values for charge rate and

temperature, gas flow andcomposition profilesFig. 4.

H

ZlTuyere o

Table

,+ii

ilIl

r

ComputeReaction Rates

ompute(~as Flow Pattern

omputeGasComposition

O RcThe computation domain for the pseudo2-D model.1. Cupola dimensions and operating parameters.

UpdateChar

ComputeOasandSolid TemperatureandSolid Flow Pattern

No IsConverged

YesIs

CarbonBalanceSatisfied

No

eRate

RESULTS

Fig.

Diameter of cupola (m)Numberof tuyeresDiameter of tuyere (m)Blast rate (m3s~ 1) (STP)lron to coke ratioTotal height of packed bed (m)

0.6iO4(90' apart)0.0760.4717lO: 13.8

STOP3. Flow chart ofthe computation scheme.

direction and 100 along axial direction was found to beadequate for the pseudo 2-D model.4. Results and DiscussionsThe 2-D and the pseudo 2-D models were validatedusing experimental results reported by Nyarnekyeet al.3)Theymeasuredthe composition of 02, C02andCOandtemperature of gas in a 610nm(24") cupola, at the midway between two neighbouring tuyeres. They inserted

probes radially and carried out measurementsat 102and204mmrom the centre of the cupola at different levelalong the axial direction. Thedimensions and operationparameters of the cupola are given in Table I and thesamehas beenused for the computation.

4.1. 2-DModelAt the tuyere level, Nyamekyet al., I l) reported oxygenconcentration of 18 o/o. However, the initial computed

results from a 3-D model developed by the presentauthors4) showeda substantially lower oxygen concen-tration. This led to conclusion that the reaction rate isiow in the zone near the tuyeres. The presence of thislow reactive zonecan be attributed to higher bedvoidagearound the tuyeres and the reduced surface area availablefor chemical reaction due to the covering of molten ironor slag around the coke particles. Further, computationscarried out using the aforementioned 3-D model withthe surface area of coke available for reaction at thetuyere level as 5o/o of that calculated from the bedstructure, showeda fair agreementwith the experimentaldata.4) The samewas found to be true for the present2-D mode] and hence, all further computations arecarried out with the low reactive zone at the tuyere level.Iso-C02 and iso-CO contours inside the cupola com-puted using the 2-D model are shownin Fig. 5. As thegas ascends from the tuyere level, the C02concentrationincreases till the oxygen gets consumed.With furtherascend, C02concentration starts decreasing because ofcarbon gasification (Eq. (1 l)). At the tuyere level, COconcentration is insignificant in the presence of oxygenand starts increasing after the oxygen get consumed.Along the radial direction, the C02 Ievel, unlike CO,increases from the centre to the wall. It can be notedthat the high concentration gradients along radial direc-tion are confined to the tuyere region and above thetuyere region, the gradients in concentrations are low.Iso-temperature contours of gas and charge inside thecupola are shownin Fig. 6. The combustion is almost

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2.5

2~J::eo 1.5~:{

1o.5

oFig. 5.

3

o

C02, Vol Frac. CO,Vol. Frac

.1~) ,~

~) ,~~~ -~~ TuyereO. I 0.2 O O, lRadial distance, m

Iso-CO and iso-C02 contourscomputedusing the 2-D model

0.2

inside

Tuyere0.3

a 24" cupola

~;:eJO~:!

l

0.8

0.6

0.4

0,2

Tuyere O

~~~~ \ ~\

A2-DPseudo2-DExperimentalEl) r=102mmExperimental [~, r=204mm

Fig. 7.GasTemperature, K-120 ChargeTemperature, K- 800

'\ .~ o

2.5

2~~ 15~i

10.5

o

1600

-200

- 960-1120~128

~144-1600

o TuyereO, I O.2 O O. I 0,2 O.3Radial distance, mIso-temperature contours of charge and gas inside a24" cupola computedusing the 2-D model.

2

1.5

~~~ 1P~~

o.5

Fig. 6. Tuyere O

O 0.05 O. I O. 15 0.2OxygenConcentration, Vol. Fraction

Oxygenprofiles obtained from 2-D and pseudo 2-Dmodels along with the reported experimental data.1 1)

A[]AI

I

2-DPseudo2-DExperimental 11)C02, r=102mmC02, r=204mmCO, r=102mmCO, r=204mm

IL

Co

/ll1

lil l~

lllIic02

D ~~

complete at approximately 0.3mabove the tuyere levelwherein the gas temperature is the maximum.In theradial direction, from the wall to centre, there is anincreasing gas temperature. The sharp decrease in gastemperature near the wall is because of the heat loss.In the upper stack, the charge temperature is almosthomogeneouscross the cross section of the cupola.The results from the 2-D model suggest that the highconcentration and temperature gradients in the radialdirection are confined to the tuyere region. Hence, asimpler model may be adequate for the simulationcupola, especially to assess its overall performance forvarying operating conditions. Moreover, a 2-D modelneeds knowledge of distribution of particle size andvoidage of the packed bed, which is almost impossibleto obtain from an operating cupola.4.2. Pseudo2-D ModelFigure 7showsthe oxygen profiles obtained from (1)the pseudo2-Dmodel (2) 2-Dmodel averaged over theradius and (3) reported experimentall l) data. Thepseudo2-Dmodelagrees fairly well with the 2-D as well as with

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Fig. 8.

O 0.05 O. I O. 15 0.2 O.25Composition, Vol. Fraction

Carbon dioxide and carbon monoxide profiles ob-tained from 2-D and pseudo 2-D models along withthe reported experimental data.1 1)

the reported data. The oxygen level above the tuyeredrops sharply along the height of cupola. The zoneextending from the tuyere level to a height at which theoxygen concentration becomeszero is known as thecombustion zone of cupola. Present calculations showaheight of approximately 0.3mfor the combustion zonein a 24" cupola.TheC02andCOoncentration profiles obtained fromthe pseudo 2-D model are shownalong with that fromthe 2-D model and the reported experimental data isshownFig. 8. The concentration C02increases till theoxygen is consumedand afterwards because of cokegasification, starts decreasing. TheCOconcentration iszero at the vicinity of tuyere due to the presence ofoxygen. As the oxygen gets consumedcompletely, theCOevel increases with height due to carbon gasificationreaction.Figure 9showsthe gas and charge temperatures in the

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lower portion of cupola zone along with that obtainedfrom the 2-D model and reported gas temperature. Thetemperature profiles obtained from the pseudo 2-Dmodelagree well with that obtained from the 2-Dmodel.Thereported gas temperature byNyamekyet al.i l) wasmeasuredby inserting thermocouples through the sidewall. However,becauseof radiation from the charge, thethermocouple will give an average of solid and gastemperature.Computedaverage top gas and spout temperatures aswell as composition of exit gas is listed in Table 2alongwith the reported experimental results. Reported ex-perimental data showa variation of 5o/o in the spouttemperature. This is because, it is almost impossible to

32-DPseudo2-DExperimental II)r=102mmr=204mm

38 (1 998),

2.5 ~

No. IO

2~'1 15.~P~

10.5

Tuyere O

Charge1*

Aot Gas

~~

O 500 1000 1500 2000 2500Temperature,KTemperatureprofiles of gas and charge obtained from2-D and pseudo 2-D models along with the reportedexperimental data. 11)

maintain the sameexperimental conditions over a longduration. Hence, considering the scatter in experimentaldata, the overall agreementof the computedresults withthe reported experimental data can be considered assatisfactory.Table 2showsthat the pseudo2-Dmodelagrees quiteclosely with the 2-D model. Difference in exit gas tem-peratures is because of difference in heat loss. In 2-D,heat loss was incorporated using the gas temperaturenear the wall. But in pseudo2-D heat loss wascomputedusing the gas temperature, which represents the gastemperature averaged over the radius. Heat loss com-puted for pseudo2-D wasmore than that of 2-D.The comparison of rate of convergence for 2-D andpseudo2-Dmodels is depicted in here Fig. 10. Thefigureshows that the convergence in pseudo 2-D model isalmost monotonousin contrast with that of the 2-Dmodel. Thenumberof iterations for convergencefor thepseudo2-D is almost sameas that for the 2-D model.However,for the 2-Dmodel, the numberof computationsper iteration is muchmore because of the additionalpressure drop equation and additional numberof grids.Hence, the computation time for the pseudo2-Dmodel

is nearly one order less comparedto that of 2-D model.Thus, pseudo2-Dmodelis computationally less intensiveas well as provide sufficient information to aid in design

Table 2. computed resutts atong with the reported ex-perimental data.PseudoExpenmental 2Dmodel 2_Dmodel

Fig. 9.Topgas temperature (K)TopgasCO'/*)Topgas C02("/.)Spout temperature (K)

63612.6ll.3l 673

84412.313.5l 720

815ll.713.8l 715

~o~,a

10-1

10-2

10-3

10-4

10-5

10-6

10-7

Fig.

sJ~A \ \ \ 1~+l olA ol eol lA i

A l~. 1

ol ol ~ol aI Io 2-D GasflowA 2-D composition2-D Temperature lpseudo2-D composition l

- - ' pseudo2-D temperature

I I l1 ol l~t 1[ Ilo ot oI o2000 3000 10000 20000Numberof Iterations

lO. Comparisonof rate ofconvergence for 2-D and pseudo2-D model.

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ISIJ International , Vol . 38 (1998), No, 10exercise.5. ConclusionsSeeking computationally less intensive model, which

can provide sufficient information for design exercise, apseudo 2-D model has been proposed in this paper.Earlier l-D models pose problems in proper incorpora-tion of boundary condition at the tuyere level, becauseof the radial entry of blast. This has been eliminated inthe pseudo 2-D model by employing the governingequations in 2-D at the tuyere level so as to generateappropriate boundaryconditions for upper l-D domain.The simulation results from the pseudo 2-D model arefound to in close correspondence with the 2-D modeldeveloped specially for the purpose. Also, the computa-tion time was also one order less for the pseudo 2-Dmodel comparedthat of 2-D model. Thus, the pseudo2-Dmodelcanbe successfully applied for practical designandoperation of cupola. Probably, the pseudo2-Dmodelcan also be extended to other counter current reactorsused in newer iron making process like MIDREX,COREXtc.NomenclatureC: Specific heat (J kg~ I K~l)D: Diffusivity (m2s~ 1)dp : Particle size (m)jl,f2 : Packedbed resistance to gas flow(ms~ 1, kg~1m2)G Massflow rate (kg m~2s~ 1)H: Height up to charge level (m)h: Volumetric heat transfer coefficient(Jm~3s~ i K~1)Ef : Effectlveness factorkp : Rate constant (s~ l)kf : Masstransfer coefficient (ms~ i)n. : Numberof coke particles per unit volumeof bed(m~ 3)P: Pressure (Pa)p*. : Driving force for reaction (Pa)Zl : Height of 2-D domainRg: Gasconstant (J mol~I K~1)

R. : Radius of cupola (m)~: Reaction rate per unit volume of bed.

(molm~3s~ l)Rp: Reaction rate of single particle (mol s~ 1)Re: Reynolds numberS: Generation per unit volume (kgm~3s~1)T: Temperature (K)V: Velocity (ms~1)o( : Parameter in melting modelAH Enthalpy/Heat (J)e: Bedvoidage,4: Viscosity ofgas (kg~1m~1s~1)p: Density (kgm~3)

SubscriptsG: Gas(i) : Solid or gas speciesm: MeltingR: RadialS Solidw: WallZ: Axial

Superscri ptEqm Equilibriuml)2)

3)4)5)6)7)

8)9)lO)11)

REFERENCESM. J. Selby: B,'. Found,yman,71 (1978), No. 1, 241.D. A. Bennett and R. Bradley: J. App!. Math. Modc'lling, 15(1991), 506.K. Nyamekyeand A. B. Draper: T,'ans. AFS, (1989), 441.N. N. Viswanathan. M. N. Srinivasan and A. K. Lahiri:I,'onmaking Steelmaking, 24 (1997), No. 6, 476.S. Ergun: Che,n. Eng Prog., 45 (1953). 89.M. Choudhary,M. Propster and J. Szekely: AIChEJ., 22 (1976),600.B. Lewis and G. von Elbe: Combustion, Flamesand Explosionof Gases, AcademicPress, Inc., NewYork, (1971).I. Muchiand J. Higuchi: T,'ans. I,'on Steel Insl. Jpn., 12 (1972), 54.J. Yagi and I. Muchi: T,'ans. I,'on Stee/ Inst. Jpn., 10 (1970), 392,S. V. Patankar: Numerical Heat Transfer and Fluid Flow,Hemisphere Publ. Cor.. McGraw-Hill Book Company,NewYork, (1980), 83,1 18.K. Nyamekye:Temperature and gas composition profiles in thecombustion zone of an operating cupola (Ph. D. thesis), U.M.1.Dissertation information service, Michigan, (1992).

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