a mathematical model to characterise effects of liquid hold-up on bosh silicon transport in the...

Applied Mathematical Modelling 35 (2011) 4208–4221

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Applied Mathematical Modelling

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A mathematical model to characterise effects of liquid hold-up onbosh silicon transport in the dripping zone of a blast furnace

S.K. Das ⇑, Amrita Kumari, D. Bandopadhay, S.A. Akbar, G.K. MondalNational Metallurgical Laboratory, Council of Scientific and Industrial Research, Jamshedpur 831 007, India

a r t i c l e i n f o

Article history:Received 26 March 2010Received in revised form 8 February 2011Accepted 23 February 2011Available online 15 March 2011

Keywords:Mathematical modelBlast furnaceLiquid hold-upDripping zoneBosh siliconConservation equations

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.02.045

⇑ Corresponding author. Tel.: +91 0657 2345085;E-mail address: [email protected] (S.K. Das).

a b s t r a c t

A first principle based mathematical model has been developed to characterise the effect oftotal liquid hold-up on the bosh silicon distribution behaviour in the dripping zone of ablast furnace. Two specific cases of hold-up behaviour have been investigated, namely,hold-up in the absence and in the presence of counter current gas flow conditions. Themodel exemplifies coupled phenomenon of chemical kinetics, transport processes andliquid hold-up to characterise the silicon behaviour in the dripping zone. The present mod-elling investigation shows that the bosh silicon level diminishes with the enhanced liquidhold-up in the dripping zone. Further, the influence of counter current gas flow on thehold-up is not significant. However, it has been observed that the liquid phase temperaturereduces with increased liquid hold-up in dripping zone under steady state operating con-ditions. The model predictions of bosh silicon distribution have been validated with thepublished literatures (bulk values) and found to be in good agreement.

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1. Introduction

With the demands placed on quality iron and steel production, the technology associated with the blast furnace (BF) hasacquired new dimensions. Quality steel production at low cost demands superior hot metal quality with low silicon and sul-phur. In addition, silicon content decides the charge balance in the basic oxygen furnace (BOF) during steelmaking. Unifor-mity and prior knowledge of silicon content in the hot metal provides better stability in BOF operations [1]. Hence, therehave been concerted efforts to estimate and lower the silicon content in the hot metal.

The transfer of silicon is one of the most vital reactions in the blast furnace operation. The degree of silicon transfer notonly determines the silicon content of hot metal but also affects other slag metal reactions such as the desulphurisation reac-tion and the reduction of manganese oxide [2]. The primary source of silicon in the hot metal is silica, which enters the fur-nace through the burden. Silica is reduced by carbon at the high temperatures prevailing at the bosh and hearth of the blastfurnace. The extent of reduction of silica depends on the blast pressure and temperature, slag composition, and partial pres-sure of SiO (g) and gasification rate of silica to SiO (g) in the raceway [3–7]. In principle, higher metal temperature impliespresence of higher level of silicon in the hot metal.

Reduction of silica present in the coke and injected coal ash in the raceway zone leads to generation of SiO (g), whichinvokes the gas- metal reaction in the bosh region. The SiO2 content in ores may vary considerably but those in coke andcoal ash are closely similar. The formation of SiO (g) from fuel ash and slag in the blast furnace followed by its reductionto transport silicon in the hot metal is considered as the major route for the silicon transfer in the blast furnace. Fig. 1 showsa schematic of various zones in an operating blast furnace.

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List of symbols

Ags effective interfacial area between gas and solid (m2 m�3 bed �1)Agw effective interfacial area between gas and wall (m2 m�3 bed �1)Agm effective interfacial area between gas and metal (m2 m�3 bed �1)Ams effective interfacial area between metal and solid (m2 m�3 bed �1)Cpg specific heat of gas (J kg�1 K�1)Cps specific heat of solid (J kg�1 K�1)Cpm specific heat of metal (J kg�1 K�1)Dp diameter of the coke particle (mm)Pd pressure drop in non-irrigated bed (Pa)Pw pressure drop in irrigated bed (Pa)FSiO2

constant value of SiO2 conc. in ash of coke (mol-SiO2 kg �1 coke)Gg mass velocity of gas (ms�1)g acceleration due to gravity (m s�2)hto total liquid hold-up in the absence of counter gas flow (%)hs static liquid hold-up (%)hd dynamic liquid hold-up (%)htg total liquid hold-up in the presence of counter current gas flow (%)DH1 heat of reaction for the 1st reaction (kJ mol�1)DH2 heat of reaction for the 2nd reaction (kJ mol�1)PSiO partial pressure of SiO (Pa)R universal gas constant (g mol�1 K�1)R1 rate of reaction in the dripping zones (kmol m�3 s �1) Eq. (1)R2 Rate of reaction of 2nd reaction (kmol m�3 s �1) Eq. (2)Tg temperature of gas (K)Ts temperature of solid (K)Tm temperature of metal (K)Tw temperature of the wall of the furnace (K)Vm descending velocity of metal in the melting zone (ms�1)Vl superficial liquid velocity based on empty column (ms�1)V!

velocity vector (ms�1)Vg velocity of gas (ms�1)de liquid diameter (m)Xp parameter related to pressure loss (Pa m�1)XSi concentration of Si (wt.%)Z longitudinal distance along the dripping zone (m)Xk weight fraction of kth componentPW Pressure in wetted bed (N m�2)

Greek symbolse void fraction (fraction)l1 viscosity of liquid (Pa s)ql density of liquid (kg m�3)rt surface tension (N m�1)/s shape factor of particlerm surface tension of metal (N m�1)lm viscosity of metal (kg m�1 s�1)rc surface tension of carbon (N m�1)h contact angle between solid and liquid (Radian)lg surface tension of gas (N m�1)lg viscosity of gas (kg m�1 s�1)qg density of gas (kg m�3)eg voidage (fraction)rt surface tension of liquid (N m�1)

S.K. Das et al. / Applied Mathematical Modelling 35 (2011) 4208–4221 4209

1.1. Mechanism of silicon transfer

The silicon transfer from the burden to the metal takes place through two majors routes i.e. (i) direct route in which sil-icon from molten slag is transferred to molten metal (through slag-metal reaction) and (ii) indirect route where Si gets trans-ferred predominately from coke ash to gas phase as SiO, and then from gas phase into the hot metal. Many investigations

Fig. 1. Schematic of different zones in the blast furnace.

4210 S.K. Das et al. / Applied Mathematical Modelling 35 (2011) 4208–4221

[1,4,6,8–10] have been carried out to clarify the mechanism of the silicon transfer in the blast furnace. Such researches in-clude laboratory experiments, theoretical studies pertaining to thermodynamics and kinetics of silicon transport, statisticalanalysis of plant data from commercial and experimental blast furnaces. These studies indicate that the rate of silicon trans-fer through gas-metal reaction is the predominant route for silicon transfer, and which is a strong function of temperature.

1.1.1. Gas–metal reactionsThe reaction of SiO (g) with carbon saturated iron is primarily responsible for silicon transfer in the hot metal. In the race-

way zone SiO(g) is generated from the reaction of carbon in coke/coal with silica present in the ash. Subsequently, the SiO (g)reacts with carbon saturated iron droplets as it ascends through the bosh zone of the furnace. The following reactions depictthe gas-metal mode of silicon transport.

SiO2ðcokeÞ þ CðcokeÞ ¼ SiOðgÞ þ COðgÞ; ð1ÞSiOðgÞ þ CðFeÞ ¼ SiðFeÞ þ COðgÞ: ð2Þ

1.1.2. Slag–metal reactionsSilica present in the slag combines with the carbon dissolved in metal [6,11] according to the following reactions:

SiO2ðslagÞ þ CðFeÞ ¼ SiOðgÞ þ COðgÞ: ð3Þ

At the gas–metal interface, SiO reacts with the dissolved carbon and gets reduced to silicon.

SiOðgÞ þ CðFeÞ ¼ COðgÞ þ SiðFeÞ: ð4Þ

It has been established that SiO generation rate from SiO2 in slag is to the extent of less than 3% of that generated from thecoke [1].

1.2. Effect of liquid-hold-up on the silicon pick-up

Understanding of liquid flow in the dripping zone is imperative to characterise the hold-up and its effect on silicon trans-fer to the hot metal. In packed beds irrigated with liquid, the counter-current gas flow leads to complex interactions betweengas and liquid phases. As the gas flow rate increases under a given liquid flow rate, the liquid hold-up remains almost con-stant initially, and then it begins to increase substantially after the gas flow exceeds a critical value determined by hydro-dynamic conditions. Some investigations [12,13] have been reported pertaining to the effect of top gas pressure, SiOconcentration and of raceway flame temperature on the overall silicon transport behaviour. However, reported quantitativeinvestigation of the effect of dripping zone liquid hold-up on the behaviour of silicon transfer is rather scarce. As liquid metal


holdup has substantial influence on silicon transport, an analytical modelling investigation has been carried out to charac-terise the effect of dripping zone liquid hold-up on the bosh silicon transfer. The dripping zone is immediately below thecohesive and melting zone (top position) and the metal droplets move downward the raceway zone (bottom) and subse-quently to the hearth.

2. Mathematical model

A mathematical model has been proposed to predict the effect of liquid hold-up in the dripping zone on the bosh silicontransport process. The theoretical formulation of silicon transfer incorporates effects of hold-up on the silicon transfer phe-nomenon using first principle based conservation approach through a multi-physics coupling of the heat transfer, liquidflow, species transport and chemical kinetics processes.

2.1. Liquid hold up analysis

Liquid hold-up plays a key role in the hydrodynamics, transport processes and in the chemical reactions in packed bedreactors. It is an upper bound measure of the passive liquid volume fraction in the packed bed reactors. It is typically char-acterised in terms of both static and dynamic components. Static liquid hold-up is an outcome of the balance between grav-ity force, which tends to chase the liquid out of the porous medium, and capillary force, which on the contrary opposes to it.Dynamic holdup represents the fraction of flowing liquid that is suspended in the gas stream in the packed beds. Static hold-up is typically more significant than dynamic hold-up, thus, contributing more strongly to the change in bed voidage andpermeability to gas flow. The liquid contributing to hold-up remains stagnant and covers a fraction of the packed bed, thusimpending transport of the fluid, reactant and species during their movement. The key differences between the liquid flowconditions in the typical packed bed chemical reactor and the blast furnace are as follows:

� Slag/coke and metal/coke systems in the blast furnace are non-wetting while wetting conditions exist in the chemicalprocesses.� The superficial velocities of molten slag and metal in the blast furnace are very low (�0.06 to 0.08 mm/s) compared to

chemical reactors.� The packing material in the dripping zone of the blast furnaces is crushed coke, while artificial packing with much higher

porosity is commonly used in chemical systems.� The liquids are more than two times heavier as the packing (coke) in the furnace, while the packing are usually heavier

than the liquids in the conventional chemical reactor.

In view of the above factors, liquid flow and hold-up behaviour in the melting and the dripping zones of the blast furnaceare far more complex than conventional chemical reactors. Therefore, estimation of hold-up in blast furnace is usuallyaccomplished through semi-empirical approach [3,9,14].

2.1.1. Hold-up in the absence of gas flowTotal liquid hold-up in the dripping zone of a blast furnace may be expressed as the sum of static and dynamic liquid

hold-up in the absence of gas flow [13–16]. The effect of liquid properties like surface tension, viscosity, density and theshape of the packing materials influence the liquid hold up. In the cohesive and dripping zones, liquid phases (liquid ironand slag) are generated and which leads to the complex multiphase flow in the lower part of the furnace. It is extremelydifficult, if not impossible, to quantify such flow behaviour because of complex interactions of solid, liquid, gas and the com-plicated packing structure.

A comprehensive flow mapping to characterise liquid flow in the blast furnace [17] necessitates multi-phase computa-tional fluid dynamics (CFD) solution of the following complex momentum conservation equation:

r � ðqlel~ul~ulÞ ¼ �el � rlPl þ elllr~u2l þ FS

l þ elql~g: ð5Þ

The interaction term between solid and liquid is given by the modified Darcy equation as follows:

Fsl ¼ �180

e2s ll

ð/dpÞ2et

~ul: ð6Þ

CFD analysis of the complex multi-phase liquid flow phenomenon in the dripping zone of the furnace is a formidable taskbecause of several interaction parameters for which accurate data is not always available. Moreover, the liquid flow inthe lower part of the furnace, particularly within the cohesive and dripping zones, is not continuous and exhibits discreetflow characteristics [18–20].

In general, liquid hold-up is a phase property, dependent on the local physical and chemical conditions of the liquid andother phases, particularly gas and packed solids. As such, liquid hold-up usually estimated from the correlation derived fromexperiments under simulated conditions. A more practical approach to estimate hold-up is to utilise appropriate semi-empirical correlations. Some typical hold-up correlations have been reported, which may not be directly applicable to a blast


furnace and other pyro-metallurgical processes [21]. In this context, correlation reported by Fukutake and Rajakumar [11] toquantify the hold up in the dripping zone of the blast furnace has been utilised incorporating following assumptions:

(i) The cross sectional mean liquid velocity is very small.(ii) No coke is wetted by the liquid.

(iii) The liquid phase (slag and metal) is considered to have uniform properties such as viscosity and density.

The present model uses two correlations for static and dynamic hold-up incorporating physical properties of both liquidand packed bed, which are reported [14,19] to be based on extensive experimental work in the absence of gas flow.

Equation for static hold-up:

Table 1Static h

Parti

Stati

hs ¼ 20:5þ 0:263qlgu2

s d2p

rtð1þ cos hÞð1� eÞ2

" #�1

: ð7Þ

Equation for dynamic holdup:

hd ¼ 6:05qlv ldp/s

ð1� eÞll

� ��0:648 qlgd3p/

3s

ð1� eÞ3l2l

" #�:485qlgd3

p/2s

ð1� eÞ2rt

" #�0:097

ð1þ cos hÞ0:648: ð8Þ

And the total hold-up is given by the following equation:

hto ¼ hs þ hd: ð9Þ

2.1.2. Hold up with counter current gas flowThe influence of the gas flow on the total holdup [13] is addressed with incorporation of the following modifications per-

taining to gas pressure. The following assumptions need to be invoked:

� Axial dispersion of gas and solid is neglected.� Voidage in the bed is uniform.� The volume flow rate of solid particles remains invariant.

The modified hold-up correlation in the presence of gas flow is described as follows:

htg ¼ htof1þ 0:679X2pg; ð10Þ

where Xp is given by the following expression:

Xp ¼dPw

dzqlgqlgd2

p/2s

rTð1� eÞ2

" #0:3

ð1þ cos hÞ�0:5: ð11Þ

In order to calculate Xp in the above expression, the gas pressure drop needs to be estimated from the following pressuredrop analysis. Tables 1–6 show the calculated values of static and dynamic hold-ups for both slag and metal phases as a func-tion of particle sizes and angle of contact without and with counter-current gas flow conditions.

2.1.3. Gas pressure drop analysisThe pressure drop of gas through dry packed bed as applicable to a typical blast furnace dripping zone is expressed by the

following generic vectorial form of Ergun’s equation [14].

�rPd ¼ ~Vðf1 þ f2j~Vg jÞ; ð12Þ

where

f1 ¼ 150ð1� eÞ2lg

e3g d2

p

and f 2 ¼ 1:75ð1� egÞqg

e3g dp

: ð13Þ

old-up for slag in the dripping zone at different contact angles.

cle diameter (mm) 10 15 20 25 30 35 40 45 50

c hold-up for slag (%) for different contact angles h = 0� 3.19 2.76 2.39 2.07 1.79 1.56 3.19 2.76 2.39h = 45� 4.33 3.8 3.25 2.73 2.29 1.92 1.62 1.38 1.18h = 90� 3.62 2.74 2.04 1.54 1.18 0.93 0.74 0.61 0.51h = 125� 4.41 3.93 3.42 2.92 2.48 2.11 1.8 1.54 1.33

Table 2Static hold-up for metal in the dripping zone at different contact angles.

Particle diameter (mm) 10 15 20 25 30 35 40 45 50

Static hold-up for metal (%) for different contact angles h = 0� 4.40 3.92 3.40 2.90 2.46 2.09 1.78 1.52 1.31h = 45� 4.27 3.69 3.11 2.58 2.13 1.77 1.49 1.25 1.07h = 90� 3.50 2.58 1.89 1.41 1.07 0.84 0.67 0.54 0.45h = 125� 4.35 3.83 3.28 2.77 2.33 1.96 1.65 1.41 1.21

Table 3Dynamic hold-up for slag in the dripping zone at different contact angles.


Dynamic hold-up for slag (%) for different contact angles h = 0� 0.28 0.22 0.18 0.16 0.14 0.13 0.12 0.11 0.10h = 45� 0.23 0.18 0.15 0.13 0.12 0.11 0.10 0.09 0.09h = 90� 0.12 0.09 0.08 0.07 0.065 0.06 0.05 0.045 0.04h = 125� 0.26 0.20 0.17 0.15 0.13 0.12 0.11 0.10 0.10

Table 4Dynamic hold-up for metal in the dripping zone at different contact angles.


Dynamic hold-up for metal (%) for different contact angles h = 0� 0.43 0.33 0.28 0.24 0.22 0.20 0.18 0.17 0.16h = 45� 0.36 0.28 0.23 0.20 0.18 0.17 0.15 0.14 0.13h = 90� 0.18 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.07h = 125� 0.40 0.31 0.26 0.23 0.20 0.18 0.17 0.16 0.15

Table 5Total hold-up without counter current gas flow in the dripping zone at different contact angles.


Total hold-up without counter current gas flow (%) for different contactangles angles

h = 0� 9.56 8.48 7.39 6.36 5.44 4.66 4.01 3.46 3.01h = 45� 9.19 7.96 6.74 5.65 4.73 4.00 3.36 2.87 2.47h = 90� 7.43 5.56 4.14 3.12 2.41 1.91 1.54 1.27 1.07h = 125� 9.41 8.27 7.12 6.07 5.15 4.37 3.73 3.21 2.78

Table 6Total hold-up with counter current gas flow in the dripping zone at different contact angles.

Particle diameter (mm 10 15 20 25 30 35 40 45 50

Total hold-up with counter current gas flow (%) for different contactangles

h = 0� 10.04 8.60 7.43 6.38 5.45 4.67 4.01 3.47 3.01h = 45� 9.76 8.08 6.78 5.67 4.74 4.01 3.36 2.87 2.47h = 90� 7.72 5.62 4.15 3.13 2.42 1.92 1.54 1.27 1.07h = 125� 9.88 8.39 7.17 6.08 5.15 4.38 3.74 3.21 2.78


The one-dimensional form of Ergun’s pressure drop equation along the longitudinal direction of the dripping zone withoutliquid hold-up is given as:

dPd

dz¼ 150

ð1� eÞ2lg

e3g d2

p

Vg þ 1:75ð1� egÞqg

e3g dp

V2g : ð14Þ

The gas pressure drop with irrigated liquid is greater than that without the liquid in the packed bed because the effectivespace for gas flow is reduced and the surface area per unit bed volume increased. Thus, one-dimensional pressure drop equa-tion along the longitudinal direction of the dripping zone with liquid hold-up has been appropriately modified for the wetbed as: [11]

dPw

dz¼ 150

ð1� eÞ2lg

ðeg � htgÞ3d2p

Vg þ 1:75ð1� egÞqg

ðeg � htgÞ3dp

V2g : ð15Þ

Eq. (15) can further be modified to incorporate the shape factor of particles


dPw

dz¼

150 1�eþhtde

� �2lgVg þ 1:75 ð1�eþhtÞ

deqgV2

g

� �ðe� htgÞ3

; ð16Þ

where de is defined by the following expression

de ¼ ð1� eþ htgÞð1� eÞdpus

þ htg

dl

�; ð17Þ

where dl is the virtual diameter of a liquid drop given by:

dl ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qlg=rt

p 6:828ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXp � 0:891

q� �2

þ 0:695�

; ð18Þ

where Xp is a function of pressure loss given by Eq. (11).

2.2. Silicon transport model

An one-dimensional model is formulated along the longitudinal direction of the furnace in the dripping zone based onconservation principles in a one-dimesional control volume. Silicon transfer represented by chemical Eqs. (1) and (2), is con-sidered in the present model. Conservation equations for mass, enthalpy and species (SiO and Si) are developed to incorpo-rate the effects of liquid hold-up in the formulation. The following assumptions are invoked for the present model:

(i) Any radial variation of the process variables is ignored.(ii) Silicon transfer is assumed to occur only through indirect route (through SiO (g)) and the source is considered to be the

available SiO2 present only in the fuel (coke ash and pulverised coal (PC).(iii) Any interfacial silicon transport between carbon saturated molten iron and slag under dynamic conditions is neglected

[18].(iv) Reverse reaction and other reactions having low contributions have also been neglected.(v) Velocity of metal in the dripping zone is assumed to be uniform and follow plug flow condition. For calculation of pres-

sure drop as a function of liquid flow velocity, only the linear velocity term in the Ergun pressure drop equation will beconsidered (Eqs. (15) and (16)).

(vi) Reaction mechanism for the silicon transport is assumed to be similar for both coke and PC.(vii) The solid phase is considered to be an uniform mixture of ore and coke with weighted average isotropic thermo-phys-

ical properties.(viii) For the liquid phase, no distinction between the slag and metal is made.

(ix) It is assumed that the SiO (g) is completely reacted during silicon transport. Consequently, initial concentration of thesilicon at the beginning of the dripping zone is considered to be zero.

2.2.1. Mass conservation equations2.2.1.1. Gas phase. The conservation equation in the dripping zone is:

dGg

dZ¼X

k

Xj

Xi

bk;iðjÞRi: ð19Þ

In the above equation, i represents one of the two reactions: (i) SiO gas generation, (ii) Si adsorption by the molten iron, jrefers to phases, namely, the gas and liquid, and k represents the three reacting components given in the reactions (Eqs.(1) and (2)). k,i represents mass of k component transferred from phase j by the reaction i.

The transport equation for Gaseous species is:

dXk

dZ¼

Xj

Xi

bk;iðjÞRi � Xk

Xk

Xj

Xi

bk;iðjÞRi

!,Gg : ð20Þ

2.2.1.2. Metal phase. The conservation equation for the metal phase (Si species) in the dripping zone is:

dXSi

dZ¼ bSiR2

qlhtgVm; ð21Þ

where Hm is the liquid metal hold-up in the dripping zone, the rate of reactions are given as follows:

R1 ¼ k1FSiO2

6ð1� eÞpd3

p

!; ð22Þ

R2 ¼ k2 � Agm � PSiO: ð23Þ


In the above expressions, k1 and k2 are given as:

k1 ¼ 2:0� 104 � exp�69000:0

RTS

�; ð24Þ

k2 ¼ 1:22� 102 � exp�58000:0

RTg

�; ð25Þ

PSiO, Agm, Ams, and Agw have been estimated using expressions from the published literature [19,20,22] which is not repeatedhere.

2.2.2. Thermal conservation equationsThe thermal transport equations for gas, solid and metal (liquid) phases are as follows:

2.2.2.1. Gas phase.

dTg

dz¼ ðhgsAgsðTg � TSÞ þ hwAgwðTg � TWÞ � R1DH1Þ=ðCPgGgÞ: ð26Þ

2.2.2.2. Solid phase.

dTS

dz¼ ðhgsAgsðTg � TsÞ þ hmsAmsðTm � TsÞ � R1DH1Þ=ðCPsGsÞ: ð27Þ

2.2.2.3. Metal phase.

dTm

dz¼ ð�hmsAmsðTm � TsÞ þ R1DH1Þ=ðCPmhtgVlqlÞ: ð28Þ

3. Numerical solution and code development

The initial data (initial conditions) at the start of dripping zone for integrating the equations has been taken from liter-ature and RIST diagrams which incorporates necessary blast furnace reaction kinetics [6,8,23]. The coke rate and PC rate are504 kg and 33 kg per ton of hot metal (THM). The operating data used for calculation of rate expression are as follows: (i)blast rate = 1407 Nm3/THM, (ii) blast temperature = 1273 K, (iii) humidification of blast = 2200 kg/h and (iv) oxygen enrich-ment = 2%. The initial data for the solid, liquid and gas temperatures at the top of the dripping zone are 1680 K, 1580 K, and2100 K, respectively [6,8,23]. Other pertinent input data namely particle size, average voidage in the packed bed, coke and PCcomposition have been incorporated from the published literature [19,22,23]. The dripping zone height is assumed to be2.5 m. and a typical value of Vl = 0.035 m/s has been considered [3] for numerical calculations.

The governing Eqs. (7)–(11) for the liquid hold-up have been solved using a modified Newton–Raphson iterative proce-dure. The pressure drop value is computed from Eq. (16) and incorporated in Eq. (11). The total liquid hold-up is calculated asa linear combination of static and dynamic hold-ups. The iterative process is implemented and the following convergencecriterion is used.

jXnþ1p � Xn

pj 6 10�3: ð29Þ

‘‘n’’ and ‘‘n + 1’’ represent nth and (n + 1)th iterations, respectively in the numerical scheme. The system of coupled conser-vation differential equations (19)–(21), (26), (27), (and) (28) along with associated algebraic equations have been solvedusing a 4th order Runge–Kutta in conjunction with Adams–Bashforth–Moulton predictor–corrector algorithms. The afore-said numerical procedure has been implemented in a C++ computer code for simulation runs.

4. Result and discussion

Figs. 2 and 3 show the variation of percent of total liquid hold-up as a function of harmonic mean particle diameter fordifferent solid–liquid contact angles, namely, 0�, 45�, 90� and 125� in the absence of counter current gas flow. The liquidhold-up predictions as a function of particle size have been verified with the published literature [3,17,18] and found tobe in good agreement. In these figures, it may be noted that total liquid hold-up decreases monotonically as a function ofincreasing harmonic mean particle diameter. Further, as the solid–liquid contact angles increases the hold-up tends to de-crease up to contact angle of 90� However, the liquid hold-up again continue to increases with obtuse contact angle. Thehold-up behaviour exhibits a nearly linear behaviour at lower contact angles but becomes increasingly non-linear at highercontact angles.

Fig. 4 shows gas pressure drop characteristics in the dripping zone as a function of gas velocity for typical harmonic meanparticle diameter of 10 and 20 mm, respectively. The lower particle size in the dripping zone causes relatively greater pres-

0123456789

1011

0 5 10 15 20 25 30 35 40 45 50 55

Tota

l liq

uid

hold

-up

(%) contact angle = 0 deg.

contact angle = 45 deg.


Harmonic mean particle diametre (mm)

Fig. 2. Variation of total liquid hold-up as a function of mean particle size without counter current gas flow (contact angles = 0�, 45� and 90�).

0123456789

10

0 5 10 15 20 25 30 35 40 45 50 55Harmonic mean particle size (mm)

Tota

l liq

uid

hold

-up

(%)

Fig. 3. Variation of total liquid hold-up as a function of mean particle size without counter current gas flow (contact angle = 125�).

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1Gas velocity (m/sec)

Pres

sure

dro

p (K

Pa/m

)

Avg particle size = 10 mm

Avg particle size = 20 mm

Fig. 4. Variation of pressure drop as a function of gas velocity in the dripping zone for different particle size.


sure drop. Figs. 5 and 6 show the predicted total liquid hold-up as a function of harmonic mean particle diameter for variouscontact angles, i.e. 0�, 45�, 90� and 125� in the presence of counter current gas flow. The overall behaviour of the liquid hold-up as a function of particle size are found to be similar for both the cases (in the absence and presence of gas flow) for samecontact angles. However, the magnitude of total liquid hold-up is higher for counter current gas flow. In essence, total liquidhold-up gets enhanced by almost 5% under the influence of the gas flow. The predictions are found to be realistic and con-sistent with literature information [14,15,22].

The effect of total liquid hold-up in the dripping zone on the bosh silicon transfer behaviour has been characterised byparametric sensitivity analysis. Fig. 7 shows the variation of the gas, liquid metal and solid temperatures, as a function ofdripping zone height along the longitudinal direction with 5% liquid hold-up and the prediction are consistent with the tem-perature – silicon functional relationship as prevalent in an operating blast furnace. This means that elevated temperaturefavours more decomposition of silica and formation of SiO gas which augments silicon level in the hot metal. Fig. 8 illustrates

0123456789

1011

0 5 10 15 20 25 30 35 40 45 50 55Harmonic mean particle diametre (mm)

Tota

l liq

uid

hold

up (%

)




Fig. 5. Variation of total liquid hold-up as a function of mean particle size with counter current gas flow (contact angles = 0�, 45� and 90�).

0

2

4

6

8

10

12

0 10 20 30 40 50 60Harmonic mean particle size (mm)

Tota

l liq

uid

hold

-up

(%)

Fig. 6. Variation of total liquid hold up as a function of mean particle size with counter-current gas flow (contact angle = 125�).

15001600170018001900200021002200230024002500

0 0.5 1 1.5 2 2.5 3Dripping zone height (m)

Tem

prat

ure

(K)

Gas temperatureSolid tempeartureLiquid temperature

Fig. 7. Temperature distribution of gas, metal and solid along the dripping zone height (2.5 m) with 5% liquid hold-up.


the variation of liquid metal temperature along the dripping zone height for different liquid hold-up, namely, 5%, 7% and 9%,respectively. It is observed that there is a significant variation in the liquid metal temperature with the increasing liquidhold-up in the dripping zone. Liquid hold-up has resulted in temperature drop. This is attributed to the fact that large liquidhold-up increases the liquid residence time near the cool bosh wall, thus increasing the heat transfer and consequently metaltemperature decreases for a given steady state operating conditions.

Fig. 9 shows the variation of SiO (g) as a function of dripping zone height for total liquid hold-up of 5%. The concentrationof SiO (g) decreases in a linear fashion along the dripping zone height. This is attributed to the reaction of SiO gas with thefalling metal droplets during the upward motion of the gas. Fig. 10 shows the variation of silicon concentration along thedripping zone height with liquid hold-up 5%, 7% and 9%, respectively. The rate of silicon transfer is also a strong functionof the temperature of the metal. The silicon pickup in the liquid metal decreases with enhanced liquid hold-up with deple-

16601680170017201740

1760178018001820


Tem

pera

ture

of m

etal

(K)

5% Holdup7% Holdup9% Holdup

Fig. 8. Distribution of metal temperature along the dripping zone height for different liquid hold-up.

00.020.040.060.080.1

0.120.140.160.180.2


SiO

con

cent

ratio

n(m

ol/m

3 be

d.s)

Fig. 9. Distribution of SiO concentration along the dripping zone height for different liquid hold-up.

00.10.20.30.40.50.60.70.80.9

1


Bos

h si

licon

(%)

5% Holdup7% Holdup9% Holdup

Fig. 10. Si distribution along the dripping zone (for total liquid hold up 5%, 7% and 9%).


tion in the temperature in the dripping zone. Thus lower solid–liquid contact angle may favour lesser silicon pickup for theblast furnace operations. The predictions are found to be in good agreement with the published data (bulk values) [3,6,8].However, no detailed published information is available on the longitudinal spatial distribution of bosh silicon in the drip-ping zone of a blast furnace. Further, higher dripping zone height is expected to favour bosh silicon pick-up in the hot metal.

Fig. 11 shows the 3-D visualisation of variation of the SiO gas concentration with respect to gas temperature and drippingzone height (m). It may be noted that for a variation of about 200 �C in the gas temperature during its rise upward, concen-tration of SiO drops down from 0.2 (mol/mol bed-s) to almost 0. Fig. 12 shows the 3-D map of variation of silicon content inthe hot metal with the ascending SiO gas concentration along the dripping zone height. The initial concentration of the SiOgas at the top of dripping zone is negligibly small (�0 mol/m3-s). It may be observed that for depletion of SiO from 0.2 to 0(mol/m3 bed-s), has been manifested with a bosh silicon pickup in the hot metal from 0% to 0.86%. In addition, the variation

Fig. 11. 3D visualisation of SiO (mol/m3 bed s) distribution as a function of dripping zone height and gas temperature.

Fig. 12. 3D visualisation of silicon distribution as a function of dripping zone height and SiO concentration.


Fig. 13. 3D visualisation of silicon distribution as a function of dripping zone height and metal temperature.


of SiO and Si depicted in this figure reflects realistically the situation that depletion of concentration of SiO gas leads to theenhanced concentration of silicon in the metal along the dripping zone height. This is attributed to the depletion of SiO as aconsequence of reaction given by Eq. (2). Fig. 13 depicts a 3-D plot of simultaneous variation of temperature of hot metal andsilicon along the dripping zone height. The initial value of hot metal temperature at the start of dripping zone height is con-sidered to be approximately 1700 �C. It may be observed that elevated metal temperature from 1683 �C to 1790 �C, hasshown a consistent trend of monotonic rise of silicon level in the hot metal from 0% to 0.86%. In principle, higher metal tem-perature favours greater transfer of silicon to the metal.

5. Conclusion

A mathematical model has been developed to investigate the influence of liquid hold-up on the bosh silicon transferbehaviour in the dripping zone of blast furnace. The hold-up analysis comprises of quantification of total hold-up withand without counter current gas flow using appropriate functional correlations including both static and dynamic compo-nents. In order to investigate the influence of counter current gas flow on the liquid hold-up, Ergun’s pressure drop equationin an liquid irrigated packed bed has been employed in conjunction with the hold-up correlation. The concluding remarks areas follows:

� Counter current gas flow does not have any dominant influence on the enhancement of liquid hold-up under a stable blastfurnace operating conditions.� Silicon transfer to the hot metal gets depleted with higher total liquid hold-up in the dripping zone. However, reported

information in the open literature is very scanty.� The silicon pickup in the bosh zone and its axial distribution as a function of dripping zone height decreases as the liquid

hold-up gets enhanced in the same zone.� The liquid metal temperature in the bosh depletes with enhanced hold-up which may be attributed to circumferential

heat transfer. The bosh silicon pick-up is found to be consistent with the thermal state of the liquid metal under the influ-ence of hold-up in the dripping zone.


Acknowledgements

The authors express their gratitude and further acknowledge the financial support provided by the Steel DevelopmentFund (SDF), Ministry of Steel, and Government of India for undertaking this investigation.

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a mathematical model to characterise effects of liquid hold-up on bosh silicon transport in the...

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