ISIJ International, Vol. 50 (2010), No. 7, pp. 1023–1031

1. Introduction

A blast furnace process is modeled as a continuously op-erating reactor, where a stream of solid particles (coke andiron ore) and a stream of blast gas interact through heattransfer, fluid flow and chemical reactions. The furnace canbe classified as a counter-current moving bed reactor wherethe bed of the solid particles of iron ore and coke movesdownward while the blast gas flows upward. At the top of ablast furnace, layers of coke and iron ore are charged alter-nately, as illustrated in Fig. 1. The iron ore arrives at the co-hesive zone after being heated and partially reduced, whilethe hot blast gas which continuously interacts with neigh-boring solids moves upward from the tuyere to the top ofthe furnace. Temperature of the cohesive zone is usuallyover 1 300 K and solid iron ore starts to melt, thereby form-ing the hot metal and slag, which drip down through acoke-packed bed.

In the lower part of a blast furnace, coke particles arepiled up in the center of the furnace while being combustedwith limited oxidizer available in the blast. Several reac-tions such as melting of ash, carbon dissolution, coke silicareduction and silicon transfer reaction occur while themolten iron flows downward. Coke silica reduction andtransfer of silicon from silica to the hot metal are importantsecondary processes occurring in the lower furnace. Silicaenters the furnace as a constituent of coke ash and ferrousgangue, and exits as either molten silica in slag or dissolved

Si in the hot metal. Silica reduction is an endothermic reac-tion, which may alter the heat transfer in the lower furnace,thus affecting the hot metal temperature. Therefore, the pre-dicted temperature in the lower part is higher than meas-ured one in an actual furnace, if the reactions by silica areignored. This affects distribution of chemical species andflow rate of molten species in the lower furnace.

Many studies for the chemical reactions and its associ-ated heat transfer have been conducted for modeling the in-

Dripping Liquid Metal Flow in the Lower Part of a Blast Furnace

Hongjong JIN,1) Sangmin CHOI,1) Jun-ichiro YAGI2) and Jinkyung CHUNG3)

1) Department of Mechanical Engineering, KAIST, Daejeon, 305-701 Korea. E-mail: smchoi@kaist.ac.kr2) Tohoku University, Sendai, Miyagi 980-8577 Japan. 3) Technical Research Laboratories, POSCO, Pohang,Gwangyang, 545-711 Korea.

(Received on October 30, 2009; accepted on February 2, 2010 )

Numerical simulation of blast furnace phenomena has significantly contributed to the better understand-ing of iron making process. Recent interest on minimizing fuel consumption and reducing environmentalproblems have also benefitted from the development of comprehensive simulation models based on physi-cal principles. One of the under-developing fields, however, is related with the internal phenomena in thelower part of the blast furnace under the cohesive zone, where the liquid phase of metal and slag flowsdownward over the bed of solid coke particles. Hot flow of sluggish liquid phase is further complicated bythe chemical reactions including the transfer of silica into the silicon in the hot metal. Silica enters the fur-nace as a constituent of coke ash and ferrous gangue, and exits as either molten silica in slag or dissolved Siin the hot metal. Silica reduction is an endothermic reaction, which would alter the heat transfer in the lowerfurnace, thus affecting the hot metal temperature.

Effective flow in the dripping zone is important for stable operation of the blast furnace with high produc-tivity of iron. Study of the liquid flow behavior and secondary reactions in a packed bed allows to investigatethe effect of various operational changes in the dripping zone. In this research, a systematic numerical ap-proach for the liquid flow is presented where the flow behavior is solved along with heat transfer associatedwith physic-chemical reactions among representative components.

KEY WORDS: blast furnace; liquid flow; cohesive zone; coke silica reduction; numerical simulation.

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Fig. 1. Schematic diagram of species in the lower part.

ternal state of a blast furnace including the lower part. Re-actions such as coke silica reduction and silicon transfer re-action are studied experimentally and numerically. Trans-formation reaction of SiO2 in coke are clarified kinetically1)

and mathematical models of silicon transfer throughgaseous SiO in the lower part were developed.2,3) In addi-tion, multi-dimensional transient mathematical model basedon multi-fluid and kinetic theories considers not only theiron ore reduction but also silica reduction and silicontransfer reaction in the lower part.4) However, many previ-ous researchers do not consider in their model for the lowerfurnace the variation of cohesive zone shape which is con-trolled by the operating conditions, even though they havebeen analyzed and many parameters have been identified.

The objective of this study is to simulate dripping liquidflow through a packed bed of particles where gas flows inthe counter-current flow direction while the gas, solid andliquid phases are interacting through a set of chemical reac-tions. Selected cases cover the operating conditions of ablast furnace focusing on the lower part of a blast furnace.Computational domain covers the region below the cohe-sive zone which was separately simulated for the selectedcases. This study is an extension from the preceded studyreported by the authors which simulated the isothermalflow in the lower part.5) Investigation of non-isothermalflow provided significantly meaningful information, and es-tablished a fundamental base for the simultaneous simula-tion of flow, heat transfer and reactions.

2. Mathematical Modeling

2.1. Governing Equations

Modeling system for the lower part of a blast furnace isshown in Fig. 2. The calculation domain is limited to thelower part below the cohesive zone, which consists of twoparts: the deadman and the dripping zone. The deadmanhas a conical shape of a quasi-stationary packed bed ofcoke. The dripping zone is between the lower boundary ofthe cohesive zone and the deadman, where the gas and theliquid flow counter-currently in the slowly moving bed ofcoke particles. The raceway is assumed to be an annularring cavity, in the present axi-symmetric 2-dimensionalsimulation.

The materials in the computational domain consist ofthree phases; packed solid particles of coke, the gas phaseand the liquid phase of hot metal and slag. In the liquidphase, only the hot metal is considered for simplification.

Three phases of liquid, solid and gas are described mathe-matically with governing equations for continuum flow ofthe mass and momentum. The unsteady-state terms in thegoverning equations are neglected for analyzing the flowphenomena in the steady state operation. Two-dimensionalaxial symmetry is assumed for computational efficiency.Mathematical modeling is composed of constructing sys-tem equations in the conservation form based on the as-sumption that the gas, liquid and solid are continuum.

Momentum equations for liquid and gas flow consist ofconvection, diffusion, pressure gradient and source termsfor interactions with the other phases. The interactionsamong three phases are correlated from the experimentaland numerical analysis. Non-isothermal liquid flow is basedon the preceded study reported by the authors.5) The gov-erning equations for each phase are described below.

Mass conservation of the gas phase:

.................................(1)

Momentum conservation of the gas phase:

...(2)

The momentum equation of the gas phase consists ofconvection, diffusion, pressure gradient and two sourceterms for interactions with the solid and the liquid phase.Table 1 summarizes the correlations for the interactions.The effect of the solid phase on the gas flow is determinedby using the Ergun’s relation.6) Effect of the liquid on thegas phase is calculated from the contact area, drag coeffi-cient, the diameter of liquid droplet2) and the velocity dif-ference between the gas and the liquid.7,8)

Mass conservation of the liquid phase:

..................................(3)∇⋅r

Gl l�ψ

∇⋅ ∇⋅ ∇ ∇( ) ( )ε ρ ε μ εg g g g g g g g g gs

glr r r r r

u u u P F F� � � �

∇⋅r

Gg g�ψ

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Fig. 2. Diagram of the computational domain and the modeling concept for the lower part in a blast furnace.

Table 1. Phase and species considered in the model.

Momentum conservation of the liquid phase:

.........(4)

The equations for the liquid phase are similar to those forthe gas phase except for the gravity force in the momentumequation.

Momentum conservation of the solid phase is describedon the basis of potential flow assumption for the solid phaseflow. The gradient of stream function is proportional to themass velocity. Detailed correlations for interaction and con-tact area are described elsewhere.5)

Mass conservation of the solid phase:

.................................(5)

Momentum conservation of the solid phase is given as apotential function:

...................................(6)

The cohesive zone is determined in the computation bythe specified solid temperature region. For this purpose,solid temperature distribution is transferred from the ironore reduction model developed by Yang,9) which considersgaseous reduction of iron ores and the other chemical reac-tions in the blast furnace together with fluid flow, heat andmass transfer. It is assumed that iron ore starts to melt at theupper boundary of the cohesive zone and the melting com-pletes at the lower boundary of the cohesive zone. No solidiron ore exists below the cohesive zone. Below the cohesivezone, uniform size is assumed for the coke particles in thewhole packed bed, which is treated as a porous media. Thevoid fraction of the coke bed is estimated from the sam-pling data of coke particles in the lower part of a blast fur-nace. The shape of the deadman located in the center of thelower part is expressed by a simple quadratic polynomialequation.

Liquid hold-up is defined as the volume fraction of theliquid phase which is held on to the surface of the solid.The total hold-up of the liquid phase ht consists of statichold-up e l(s) accumulating on the surface of solid particlesand the dynamic hold-up e l flowing downwards.10) Thesummation of the volume fractions of gas, solid and liquidbecomes unity as shown in Eq. (7).

..............................(7)

...............................(8)

Detailed relations for the static and dynamic liquid hold-up are described elsewhere.5)

Conduction, convection, inter-phase heat transfer and re-action heat are considered for the energy conservation.Source term of each conservation equation is closely com-bined through the interactions of heat and mass transfer be-tween phases.

Energy conservation of gas phase:

...........................................(9)

Energy conservation of liquid phase:

.........................................(10)

Energy conservation of solid phase:

.........................................(11)

Mass fraction of components of each phase is consideredfor the species conservation equation. Convection as well asgeneration/extinction of each component is reflected, whilethe effect of diffusion is neglected. Mass fraction of inertspecies in each phase is obtained by subtracting those of theother species from unity because total mass fraction isunity. Species conservation in the gas, liquid and solidphases is described respectively as Eq. (12).

.....................(12)

2.2. Reactions

Many species are involved in chemical reactions of thelower part, but only 10 species are considered in this mod-eling as seen in Table 1. Coke as packed particles flows to-wards the raceway in the dripping zone while coke packedbed is kept qausi-stationary in the deadman. There are twokinds of SiO2 participating in the reduction process of thelower part. The silica is fed from the top of the furnace inthe form of ash in coke and gangue of iron ore. Ratio of sil-ica in coke is about 4–5%. Silica in coke is studied for sim-ulating reductions in the lower part, while iron ore as wellas gangue is assumed to complete the melting process inthe cohesive zone. The gangue in iron ore is known to meltbefore 1 480 K.10) Other species in ash or gangue are as-sumed to be inert for simplification except for silica in ash.Fe exists as liquid metal, because melting of Fe is assumedto complete in the cohesive zone. SiO2 in liquid phase orig-inates in the gangue of iron ore. Si is absorbed in liquidmetal through silica transfer reaction while C is absorbed inliquid metal by carbon dissolution reaction. SiO is an inter-mediate product which is generated from coke silica reduc-tion. Mass fraction of CO and SiO increases due to the re-duction in the dripping zone.

Silicon transfer to the molten metal is one of the key phe-nomena in the lower part of blast furnace, and five reactionsare considered as listed in Table 2. Gas–metal reactions via

∇⋅ ∑( ), , ,

rG Si i j j n

n

N

iω ω�

�1

react

∇⋅ ∇

∑

( )

( ) ( ) ( )

rG C T k T

h A T T h A T T R Hn n

n

N

s ps s s s

gs gs g s ls ls l s

react

�

� � � � � ��

γ Δ1

∇⋅ ∇

∑

( )

( ) ( ) ( )

rG C T k T

h A T T h A T T R Hn n

n

N

l pl l l l

ls ls l s gl gl g l

react

�

�� � � � � ��

β Δ1

∇⋅ ∇

∑

( )

( ) ( ) ( )

rG C T k T

h A T T h A T T R Hn n

n

N

g pg g g g

gs gs g s gl gl g l

react

�

�� � � � � ��

α Δ1

h st l l� �ε ε ( )

ε εg s t� � �h 1

∇ζ �r

Gs

∇⋅r

Gs s�ψ

∇⋅ ∇( )ε ρ ε ε ρl l l l l l ls

lg

l l

r r r r ru u P F F g�� � � �

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Table 2. Reactions considered in the model.

SiO is the key path to the silicon transfer, and reaction rateequations and associated rate constants have beenproposed.11) The current simulation utilizes rate equationsand constants, and reaction rate constants as summarized inTables A and B of Appendix. SiO is mainly generated fromcoke ash in high temperature region, and contribution ofmolten slag is considered to be minor. Modern operation ofpulverized coal injection (PCI) has identified additionalpath of SiO generation from ash from the pulverizedcoal,12) which is not, however, considered in the currentsimulation and is left for future work.

3. Simulation Methods

3.1. Solver

The generalized equations for conservation of mass, en-ergy and chemical species are rearranged and listed up inTable 3. The mathematical model was then solved by usingCOMSOL Multiphysics 3.4 which can solve partial differ-ential equations. The system equations are arranged so thata general partial differential equations solver can handleeach of the equations. The computational domain was di-vided into 9 362 triangular cells. After solving the continu-ity and momentum equations of the solid phase, computa-tion was continued to solve the momentum equations forthe gas and the liquid flow and then, the computation ofconservation equations on energy and chemical specieswere simultaneously repeated until convergence. For con-vergence criteria, relative precision of 1�10�3 and damp-ing factor of 1�10�6 were used.

3.2. Computational Domain, Inlet and Boundary Con-ditions

The boundary types and input values for the computa-tional domain of momentum balance are illustrated in Fig.3(a). The inlet conditions of the three phases were based onthe practical operation data of the blast furnace No. 1 oper-ated by POSCO in Kwangyang. The inlet velocity of thesolid particles in the cohesive zone was calculated from thehot metal production and coke rate in the operating condi-tions of the furnace. Table 4 lists the operational conditionsof the furnace. The inlet velocity of the gas phase is calcu-lated considering the circumferential area of tuyere due tothe axial symmetry condition. The lower boundary of thecomputational domain is considered to be the bottom of theblast furnace, which is assumed to be the top of the liquidpool as seen in Fig. 2.

Boundary condition of simulation for temperature distri-bution is listed in Fig. 3(b). Left side of the geometry isfixed as axial symmetry. Inlet condition of solid and liquidin the cohesive zone is set by temperature 1 623 K which isfrom the definition of the cohesive zone. Liquid phase goesout from the bottom line and the solid particles combustcompletely at the upper boundary of the raceway. Influx ofgas is from the outer boundary of the raceway by 2 300 Kwhich is calculated as adiabatic flame temperature. No heatflux is allowed through the bottom for gas and solid phase.

For chemical species, initial mass fraction is set at theboundary of the geometry as seen in Fig. 3(c). Mass frac-tion of liquid metal is 74% of liquid phase in the cohesivezone, SiO2 in gangue is 9% and other components ofgangue are 17% in the liquid phase. Carbon of coke is 88%and solid silica is 4% of the solid phase in cohesive zone.Other components of ash are assumed to be inert. The liq-uid phase silica starts from zero and increases in the drip-ping zone by melting reaction of the solid silica. Mass frac-tion of gaseous components is obtained from the completecombustion of coke. Iterative calculation of gaseous COand SiO starts to react as initial value 0.411 and zero in theraceway while hydrogen and nitrogen are assumed as inertgas. Gaseous CO increases a little by silica reaction andSiO is generated in the dripping zone as an intermediateproduct.

3.3. Physical Properties

The physical properties of air, molten iron and coke par-ticles are listed in Table 5. The properties of gas and cokeparticles were obtained from a handbook.13) The surfacetension of molten iron and contact angle between coke andmolten iron were taken from the literature.6)

Heat capacity, inter-phase enthalpy and conductivity aredependent on temperature while other properties are as-sumed to be independent of temperature distribution. Heatcapacity of a species is a function of temperature and addi-tivity rule is applied to calculate the property of a mixture.Heat capacity of species used in this study is tabulated inTable C of Appendix.

Three enthalpy transfer coefficients are tabulated inTable D of Appendix, which have been used in heat bal-ance equations as convective heat transfer coefficient be-tween gas and solid. The modified Ranz–Marshall equationis used, which depends on conductivity of gas and solidparticle diameter.12,13) Reynolds and Prandtl number influ-ence heat transfer as seen in the relation. Resistance in boththe solid and the liquid phases is considered for theliquid–solid convective heat transfer coefficient. Resistancein liquid phase is a function of particle diameter, liquid con-ductivity, Reynolds number and Prandtl number of liquid.Resistance in the solid phase depends on the solid conduc-

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Table 3. Generalized equation form used in the solutionprocess.

Table 4. Operating condition of Kwangyang No. 1 blast fur-nace.

tivity, heat capacity of the solid, density and velocity differ-ence between the liquid and the solid. Convective heattransfer coefficient between the gas and the liquid is a func-tion of Reynolds number and velocity of the liquid phase.14)

Thermal conductivity of each phase is obtained from therelations listed in Table E of Appendix. Conductivity of ofgas species is calculated using Wilke’s method15) which is apolynomial equation. The constant of each term is found inthe standard engineering handbooks.16–19) Thermal conduc-tivity of the gas phase is the summation of each conductiv-ity multiplied by corresponding mass fraction. In the caseof liquid it is a linear function of temperature while con-stant is assumed for slag. Thermal conductivity of the solidphase is calculated using the effective conductivity.20) The

effective conductivity is a function of porosity which de-pends on particle diameter and solid temperature.

4. Results and Discussion

Chemical species and temperature distribution of gas,liquid and solid are calculated for the reference case of theisothermal flow which was reported previously.5) Radialprofiles and mass flow rates through the deadman and thedripping zone are used for evaluating the results. Radialprofile at the height of 7.5 m is selected to see the change ofthe liquid flow. This location is chosen simply to representone cross-section of the furnace. The results can be ex-plained by comparing to the results of the isothermal flowbecause the flow field does affect the chemical reactionsand its associated heat transfer. Mass flow rate of impuritiessuch as carbon and silicon can be calculated, and the totalmass of each species at the bottom is discussed. Prelimi-nary results are calculated with the reference operating con-dition, where the proportion of coke charged in the centralcolumn is 80%. Particle diameter is 0.0477 m and 0.03 m inthe dripping zone and in the deadman, respectively. Viscos-ity is 0.339 Pa · s which is corresponding to the correlatedvalues at 1 873 K. The temperature and chemical speciesare visualized by contour diagram. Chemical species areexpressed by mass fraction in each phase.

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Fig. 3. Boundary conditions for (a) momentum, (b) energy and (c) chemical species.

Table 5. Physical properties of the three phases.

4.1. Temperature Distribution and Chemical Compo-sition

Temperature distribution of the solid phase shows similarpattern to that of the gas phase over the computational do-main under the cohesive zone as seen in Fig. 4. Tempera-ture profiles of gas, solid and liquid at the height of 7.5 mfrom the bottom are shown in Fig. 5. Temperature at thecentral axis region is higher (i.e., T increases as r/R ap-proaches 0), while gas temperature is higher than that ofsolid phase in the whole domain. Temperature of the liquidphase increases more sharply around the central axis regionthan in the outer wall region. Gas flow rate is higher in thecentral region because more coke is charged in the centralaxis region to ensure high permeability of gas.

Temperature of the solid phase increases from the cohe-sive zone and reaches its maximum around the axis in thedeadman as moving down. On the other hand, temperatureof the gas phase decreases from the upper boundary of theraceway as going up. Thermal energy of the gas phase istransferred to the solid and the liquid phases. Isothermalcontour lines of the solid, liquid and gas phases appear tofollow the shape of the cohesive zone in the dripping zone.Contour of liquid temperature in the deadman shows thatthe temperature is dependent on the liquid flow.

Mass fraction of gaseous species CO and SiO are shownin Fig. 6 using iso-concentration contour lines. CO isformed through coke silica reduction and SiO is increasedas the balance of coke silica reduction and Si forming reac-tion. However, reaction rate of the silicon forming reaction

is less than that of the coke silica reduction. Concentrationsof both CO and SiO increase in the dripping zone, due tothe coke silica reduction. There is no change of mass frac-tion in the gas phase, because coke silica reduction does notoccur in the deadman. Mass fraction of SiO is zero in theraceway, because it was assumed that no mass fraction re-mains in the combustion gas of raceway. It increases up to3% in the gas phase at around the central axis of the drip-ping zone. Mass fraction of CO was 41.1% in the racewayfrom the composition of the gas phase. The species CO in-creases also by 2% while flowing upward.

Mass fraction of the liquid phase is illustrated in Fig. 7as a contour diagram. Liquid species such as carbon, siliconand liquid silica undergo reactions of carbon dissolution,melting and silicon transfer reaction. The mass fractions ofcarbon and silicon increase while that of Fe decreases.Fraction of liquid silica is 9% at the cohesive zone and de-creases as flowing downward because all species in iron oreare assumed as liquid phase in the cohesive zone. The de-crease of liquid silica is about 0.3% in the deadman. Car-bon absorbed in metal increases up to 4% at around thecentral axis in the deadman, while silica increases to about1.5%.

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Fig. 4. Temperature distribution of solid and liquid; (a) gas, (b) solid and (c) liquid [K].

Fig. 5. Temperature profile at the height of 7.5 m from the bot-tom.

Fig. 6. Mass fraction of gas species; (a) CO and (b) SiO.

4.2. Effect of Secondary Reactions in the Lower Parton the Temperature Distribution

Coke silica reduction is an endothermic reaction whichoccurs in the lower part of a blast furnace. Temperature dis-tribution of liquid and solid phases is illustrated in Fig. 8,which considers the endothermic reactions. Temperaturedecrease of the gas and the solid phases is about 20 K afterthose reactions. The temperature change of the liquid phasedue to the endothermic reactions is small in the peripheralregion as seen in Fig. 8. The coke silica reduction occurs inthe dripping zone as seen in Fig. 6 and it occurs substan-tially in the central region. Mass fraction increase ofgaseous species is high around the central axis while carbonabsorbed in liquid metal increases near the central axis.Therefore, the temperature decrease of liquid phase ishigher in the central region than in the peripheral region.

Figure 7 shows mass fractions of solid phase species Cand Si in the liquid metal. The result can be discussed withoperating conditions and its effect on the liquid metal com-position. Silicon in the liquid metal comes through reduc-tion of gaseous SiO with carbon dissolved in the liquidmetal. Therefore, silicon concentration in the liquid metaldepends on the contact time between SiO and liquid metal.Lowering the cohesive zone position would decrease thecontact time between SiO and liquid metal, and would de-crease the concentration of silicon in the liquid metal. Ad-ditionally, carbon in the liquid metal decreases due to theconsumption for reduction of SiO from the silica in cokeash.

Figure 8 shows the difference of the predicted tempera-ture, with and without reactions. This temperature differ-ence shows that the predicted temperature distribution byiron ore reduction model can be higher than the measuredvalue in the operating furnace. The iron ore reductionmodel of the author group does not consider liquid flow andreaction in the lower part of a blast furnace. The cohesivezone would also be affected by these reactions. Carbon dis-solution and silicon transfer reaction are directly relatedwith the position of the cohesive zone in the blast furnace.

5. Conclusion

Liquid metal flow in the lower part of a blast furnace issimulated by describing the dripping liquid over a packedbed of particles where gas flows in the counter-current di-rection. Axi-symmetric 2-dimensional computational do-main covers the region under the cohesive zone, which wascomputed separately and used as input condition. The simu-lations have been performed for the case of KwangyangNo. 1 blast furnace. Liquid flow is represented by the con-servation equations which accounts for the dynamic andstatic hold-up. Numerical modeling of the heat transfer andchemical reactions is incorporated into the previously re-ported iso-thermal flow simulation model.

Melting of ash, carbon dissolution, coke silica reductionand slag silica reduction are considered in the drippingzone and in the deadman. Silica enters the furnace as a con-stituent of coke ash and ferrous gangue, and exits as eithermolten silica in slag or dissolved Si in the hot metal. Flow(momentum), energy (thermal energy) and material conser-vation equations were numerically solved simultaneously.The effect of reduction reactions in the lower part on thetemperature decrease was evaluated. Additional informa-tion on the internal state of liquid, gas and solid phases inthe lower part would enhance the understanding of the op-erating conditions and its related parameters.

Acknowledgements

The authors wish to express their appreciation to POSCOand BK21 (Brain Korea 21) for the support to this research.

Nomenclature

G : Mass flow rate (kg m�2 s�1)h : Total liquid hold up, convective heat

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Fig. 7. Mass fraction of solid species; (a) C and (b) Si in liquidmetal.

Fig. 8. Temperature distribution with and without consideringreactions in the lower part; (a) solid and (b) liquid tem-perature difference.

transfer coefficient (kW m�2 K�1)P : Pressure (Pa)C : Drag coefficienta : Contact area (m2)F : Interaction parameter (N m�3)u : Velocity (m s�1)d : Diameter (m)g : Acceleration of gravity (m s�2)S : General source termRe : Reynolds numberGa : Gallilei numberCp : Capillary numberWe : Weber numberFr : Froude numberT : Temperature (K)V : Volume flow rate (m3 s�1)E : Blast energy (kg m s�1)k : Conductivity (W m�1 K�1)n : Index for reactionsR : Reaction rate (kmol m�3 s�1)DH : Heat of reaction (kJ kmol�1)

Greeke : Volume fraction of gas or solid, dynamic

liquid hold upe(s) : Static liquid hold upm : Viscosity (Pa · s)r : Density (kg m�3)q : Contact angle of liquid on the solid sur-

face (rad)y : Stream functiond : Surface tension (N m�1)f : Shape factorw : Mass fractiona : Fraction of heat absorbed by gasb : Fraction of heat absorbed by liquidg : Fraction of heat absorbed by solidz : Potential function

Subscriptst : Totalg : Gasl : Liquids : Solidp : Packed particle

Superscriptsg : Gasl : Liquids : Solid

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Appendix A

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Table B. Reaction constants.

Table A. Rate equations and constants.

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Table C. Heat capacity and heat of formation.

Table D. Inter-phase enthalpy transfer.

Table E. Thermal conductivity of each phase.