Quantitative Estimation of Gas Permeability of Softening ...

© 2015 ISIJ 544

ISIJ International, Vol. 55 (2015), No. 3, pp. 544–551

Quantitative Estimation of Gas Permeability of Softening Sinter Layer with Liquid Phase

Kazuhira ICHIKAWA,* Jun ISHII, Shiro WATAKABE and Michitaka SATO

Steel Research Laboratory, JFE Steel Corporation, 1 Kokan-cho, Fukuyama, Hiroshima, 721-8510 Japan.

(Received on June 11, 2014; accepted on November 25, 2014; originally published in Tetsu-to-Hagané,Vol. 100, 2014, No. 2, pp. 270–276)

Quantifying gas permeability of softening sinter layer was studied with the aim of increasing the preci-sion of gas permeability in cohesive zone.

In this study, the effect of liquid in packed bed on gas permeability was evaluated by a cold modelexperiment, which simulated sinter melting behavior by using sponge ball absorbed glycerin solution. Asa result, gas permeability of packed bed with liquid was expressed by formula (a), based on Sugiyama’sformula.

.......................................................(a)

Formula (a) was adapted to actual softening sinter experiment result, and was confirmed to be goodagreement.

KEY WORDS: blast furnace; CO2 emiss0ion; low coke rate operation; permeability; cohesive zone.

1. Introduction

The price of ironmaking coal has increased in recentyears, due to increase global demand for steel. The ironmak-ing industry emits a large amount of CO2, so reducing CO2

emissions is necessary from the viewpoint of preventingglobal warming. Therefore, developing a low coke rateoperating technology in the blast furnace is required, inorder to reduce coke consumption and CO2 emissions.

During low coke rate operation, the amount of coke in thefurnace decreases. This causes some problems, includingincreased pressure drop in the blast furnace,1) delayed reduc-tion reaction, and expansion of cohesive zone.2,3) In partic-ular, increasing pressure drop is remarkable in the cohesivezone, where the pressure drop of the ore layer increases asthe ore start softening. Thus, a quantitative understanding ofgas permeability in the cohesive zone and reducing pressuredrop are important.

Mixing of coke in the ore layer is a technique for achiev-ing improved gas permeability in the cohesive zone. Theeffect of mixed coke charging in the ore layer has been con-firmed in basic studies by softening and loading tests whichsimulating the softening behavior of ore in the blast fur-nace.3–5) As the mechanism responsible for this effect, theformation of local voids around the mixed coke3) or decreas-ing liquid slag holdup which containing FeO by improvingthe reduction reaction4) has been pointed out. However,these effects on gas permeability have not been estimated.

Ono et al. estimated the gas permeability of a cohesiveore layer by using formula (1) , (2) , assuming the permea-bility index K is a function of the ore contraction ratio whenusing the softening and loading test apparatus.6,7)

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

............................... (2)

Takatani et al. estimated gas permeability of softening orelayer by using formula (3). They considered void decreasingwith ore layer contraction.8,9)

......... (3)

Sugiyama et al. proposed formula (4), assuming that thegas paths through a cohesive ore layer take the shape of anorifice.10)

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

These formulas considered void fraction changing withcontraction of ore layer. But these were not considered theeffect of liquid holdup on gas permeability. According to theresults of X-ray observation of the softening behavior of sin-ter and pellets in softening and loading tests, liquid slagclosed the spaces between particles and this slag flood up tothe top of the packed bed, above 1 300°C.11–13) Based onthese observations, the slag melt is expected to effect on thepermeability of the softening ore layer.11) Thus, it is necessary

ΔP

L C h D St

=−( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ −( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 1

1 2

22

ε

ρ

ϕ p r

u

* Corresponding author: E-mail: [email protected]: http://dx.doi.org/10.2355/isijinternational.55.544

ΔΔP

LK G= ⋅ −1 7 1 0 3. .ρ μ

K K= 02 610 . σ

ΔPL

u

D

u

Dp p

=−( )

+−( )

1501

1 751

2

3 3

2ε

εμ ε

ερ

.

ΔPL Cm D Sr

u

p

= ⎛⎝⎜

⎞⎠⎟ −( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 1

1 2

2 2

ϕρ

ISIJ International, Vol. 55 (2015), No. 3

545 © 2015 ISIJ

to estimate the effect of liquid slag on the gas permeability,when we estimate the permeability of softening ore layer.

Therefore, in this study, the effect of liquid in packed bedon gas permeability was evaluated by a cold model experi-ment, which simulated sinter melting behavior by usingsponge ball absorbed glycerin solution.14) And formulatedformula was adapted to actual softening sinter experimentresult, and was confirmed to be good agreement.

2. Cold Model Experiment

2.1. Experimental ConditionIn order to estimate the effect of liquid in packed bed on

gas permeability, cold model experiment was performed.This cold model experiment simulated sinter melting behav-ior by using sponge balls absorbed glycerin solution. Figure1 shows a schematic drawing of cold model experimentalapparatus.14) In this experiment, ore particles were simulatedby sponge balls (manufactured by Showa Rubber Co., Ltd.),which were injected with glycerin to simulate the slag melt.Maximum liquid volume absorbed in this ball was 0.82 mL.The apparent density of the ball was 353 kg/m3. When thesponge balls were contracted, glycerin solution started soak-ing out and this solution closed spaces between the particles.This behavior was similar with liquid slag which soakedfrom softening sinter. The packed bed was composed bysponge balls and glass balls. Glass balls was used to simu-late mixed coke in a sinter layer. A diameter of sponge ball

was 14 mm, and that of glass balls was 14.5 mm. Amountof glycerin solution absorbed sponge balls was 0, 0.15 or0.30 mL. A density of glycerin solution used in this studywas 1 250 kg/m3. Viscosity of glycerin solution used in thisstudy was 0.109 Pa·s. These balls were packed into theacrylic glass case. Size of this case was 90 mm × 90 mm ×300 mm. Figure 2 shows the conditions of the packed bed.To control liquid volume in the packed bed, experimentswere performed under two conditions, i.e., a Layer condi-tion in which the packed bed comprised 100% sponge ballsand a Mix condition in which the layer included glass balls,which do not contribute to liquid soaking-out. In (a) Layercondition, the sponge balls were packed in a 6-ball x 6-ballx 4-layer configuration. In (b) Mix condition, the same 6-ball x 6-ball x 4-layer configuration was used, but glass ballswere mixed with the sponge balls at a ratio of 1 : 2. Thispacked bed was contracted in constant speed. Air wasflowed from lower part of packed bed. Air flow rate was 220L/min. Table 1 shows the experimental condition of thiscold model. In this experiment, Fr number and Re numberwere approximated those in the blast furnace. The pressuredrop of packed bed was measured by the pressure gaugesinstalled at the top and the bottom of the packed bed. Thecontracted sponge balls were photographed, as shown inFig. 3(a), and the volume of the balls was calculated bymeasuring the dimensions a, h, r, and c in Fig. 3(a). Here,a: diameter of the contact area between adjacent particles inthe vertical direction, h: height of the contracted spongeball, r: height of the spherical zone not in contact with anyparticle, and c: length of the major axis of an oval which thecontact surface with an adjacent part in the horizontal direc-tion. The minor axis of this oval can be expressed by h-2r.

By using these values, volume of the contracted spongeball was calculated. Figure 3(b) shows the image of the con-tracted sponge ball. The contracted sponge ball was divided3 parts, the upper part, the center part and the lower part.The center part was the area where contact with othersponge balls at the side. Volume of the contracted spongeball was calculated as follow.

First, the upper and the lower part of the sponge ball wasconsidered as a spherical zone. The diameter of large baseof this spherical zone was equal to the initial diameter of thesponge ball; DP, and the diameter of small base of this spher-ical zone was equal to the value of a. Volume of these partFig. 1. Schematic drawing of cold model experimental apparatus.

Fig. 2. Experimental conditions of packed bet for cold modelexperiment.

Table 1. Experimental condition for cold model.

BF Cold model

Slag

Density kg/m3 3 200 1 222

Viscosity Pa·s 3 0.109

Surface tention N/m 0.41 0.648

Gas

Density kg/m3 0.650 1.260

Viscosity Pa·s 5.15 × 10–5 1.78 × 10–5

Velocity m/s 1.600 0.453

OreDiameter m 0.022 0.0145

Density kg/m3 1 500 –

Fr – 0.049 0.039

Re – 444.3 465.0

© 2015 ISIJ 546


was calculated by a, r and DP.Then, volume of the center part was calculated. Figure

3(c) shows the horizontal cross sectional image of the centerpart. The horizontal cross section area was considered as arectangle with rounded corner. This area was composed bythese parts as follow, 4 sectors which radius was (DP-c’)/2,4 rectangles which length of long side was c’ and that ofshort side was (DP-c’)/2, and 1 square which length of sidewas c’. The contact surface was considered as oval, so c’was calculated by the relationship between the major axis cand the minor axis h-2r. Volume of the center part was cal-culated by the integral of these cross sectional area forheight direction.

Volume of the contracted sponge balls was added up thevolume of the upper part, the center part and the lower part.

In this cold model experiment, the effect of packed bedcondition on gass permeability was measured in dry condi-tion. Then the effect of liquid holdup on gas permeabilitywas evaluated by wet conditions.

2.2. Pressure Drop in Dry ConditionIn order to evaluate the effect of void fraction on the per-

meability of packed bed, dry condition experiments wereperformed. Figure 4 shows relationship between the con-traction ratio and pressure drop in the Layer condition and

the Mix condition. The definition of sponge contractionratio is the value obtained by dividing the contraction lengthof packed bed by the initial height of sponge ball layer. InLayer condition, the initial height of sponge ball layer isequal to the initial height of packed bed. In Mix condition,the initial height of sponge ball layer is equal to 2/3 of theinitial height of packed bed, because 2/3 of particles in thepacked bed are sponge balls. Comparing with Layer condi-tion, pressure drop of Mix condition showed lower values.This result indicated that void fraction of mixed conditionwas higher than that of Layer condition. And this wascaused by mixed glass balls which worked as spacer. Figure5 shows the photo of contracted sponge balls during exper-iment. Continuous voids have formed between the particles.Since pressure drop in a packed bed is considered to becaused by the permeability resistance that occurs betweenthe particles and a gas, the pressure drop of the packed bedwas estimated by the Sugiyama’s formula,10) which isexpressed by formula (4). Sugiyama et al. defined the aper-ture ratio m as void area ratio in a cross section vertical rel-ative to a gas flow. In this study, m was calculated by imageanalysis. In each contraction ratio, cross sectional areas ofsponge balls and glass balls were measured by image anal-ysis, and then this area was integrated from lower part ofpacked bed to upper part of packed bed, as shown in formula(5). It is considered to represent the percent of the void vol-ume as part of the packed bed volume; in other words, m isequivalent to the void fraction of the packed bed.

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

Fig. 3. The photo and approximate form of contracted sponge ball.

Fig. 4. Measurement result of ΔP/L with sponge contraction ratioin dry condition.

Fig. 5. The photo of contracted sponge balls during experiment.

mL

S S

SL

h=

−∫

1 0

00

d


547 © 2015 ISIJ

Figure 6 shows measured results of m obtained by theimage analysis. m of Mix condition was higher than that ofLayer condition. This is because the non-contracted glassballs in the packed bed worked as a spacer and thus main-tained a larger void fraction in the packed bed.

Pressure drop of each condition was estimated, by usingm value obtained by image analysis. Formula (4) was usedto estimate pressure drop. The run-off factor C in formula(4) was defined as shown in formula (6),10) which is thesame formula as in Sugiyama et al.10)

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

Figure 7 shows a comparison of the experimental resultsof pressure drop and the values estimated by formula (4).The experimental results and the calculated results showgood agreement. It is considered that the pressure drop in apacked bed in the dry condition can be expressed by formula(4).

2.3. Pressure Drop in Wet ConditionTo evaluate the effect of liquid holdup on gas permeabil-

ity at various void fractions, the experiment was performedin the wet condition. Definition of liquid holdup in thisstudy was liquid volume ratio in the packed bed. Figures8(a) and 8(b) show measurement results of pressure dropwith sponge contraction ratio in wet condition. In the caseof liquid volume 0.15 mL, liquid started soaking at the point“▼”. In the case of liquid volume 0.30 mL, liquid startedsoaking at the point “▽”. In Layer conditions, contractionratio of soaking started was 0.65 in the case of liquid volume0.15 mL, and was 0.37 in the case of liquid volume 0.30mL. In Mix conditions, contraction ratio of soaking startwas 0.65 in the case of liquid volume 0.15 mL, and was 0.35in the case of liquid volume 0.30 mL. In each case, pressuredrop increased with increasing absorbed liquid volume, afterliquid start soaking. These pressure drop increasing werecaused by void closing by soaked liquid. Comparing Layercondition to Mix condition, pressure drop of Mix conditionshowed lower value than that of Layer condition. This resultindicated that void fraction of Mix condition was higherthan that of Layer condition. In Layer condition, liquidflooding was observed, at contraction ratio 0.76 in the caseof liquid volume 0.15 mL, and at 0.65 in the case of liquidvolume 0.30 mL. On the other hand, flooding was notobserved in Mix condition.

Because it is expected that soaked liquid closed the voidsin a packed, the voids available for gas flow was decreasedby liquid hold up. So the aperture ratio m can be expressedby the difference between the void fraction ε and liquidholdup ht, as expressed by formula (7).

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

Figure 9 shows relationship between the contraction ratioand liquid holdup ht estimated from formulae (4) and (7) by

Fig. 6. Calculation result of aperture ratio: m with sponge contrac-tion ratio.

Fig. 7. Comparison between ΔP/L(exp.) and ΔP/L(calc.) in drycondition.

C m m= − +0 597 0 011 0 432 2. . .

Fig. 8. Measurement result of ΔP/L with sponge contraction ratioin wet condition.

m ht= −ε

© 2015 ISIJ 548


using the results of image analysis of the void fraction andthe measured results of pressure drop in the Layer condition.ht started to increase at the contraction ratio of 0.65 withinjection of 0.15 mL/ball and at 0.40 with 0.30 mL/ball, andthese contraction ratios are close to the contraction ratio atwhich soaking-out started. After flooding occurred, thedecrease in liquid holdup was observed. This result indicat-ed that the amount of holdup liquid decreased as the resultof liquid flooding from the top of the packed bed.

ht was approximated by formula (8), based on the follow-ing assumptions: The changes in ht from the start of liquidsoaking until the start of flooding are a function of the con-traction ratio; there are no liquid holdup before the contrac-tion ratio Sr1, at which liquid soaking-out starts; and liquidholdup increases monotonously from the start of liquidsoaking until Sr2, at which point flooding occurs.

... (8)

By using formula (8), liquid holdup of Mix condition wasestimated. In the Mix condition, liquid holdup is less thanthat in the Layer condition, because there are glass ballswhich are not injected liquid. It is expected that the defor-mation behavior of the sponges and the amount of liquidsoaking out from the sponges are the same at the samesponge contraction ratio. Thus, the liquid holdup in the Mixcondition is expected to decrease in direct proportion to thedecrease in the liquid source, in other words, the decreasein the number of sponge balls in the bed. In the Mix condi-tion, 1/3 of the sponge balls were replaced with glass ballsin comparison with the Layer condition. Therefore, liquidholdup in the Mix condition is considered to decrease by1/3ht, and the value of liquid holdup in the Mix condition is(1 – 1/3)ht, that is, 2/3ht. Pressure drop was estimated bysubstitution into formula (9), which considers the influenceof the liquid phase, together with the void fraction obtainedby image analysis. Figure 10 shows comparison betweenΔP/L (exp.) and ΔP/L (calc.) in wet condition. Calculated

results show good agreement with experimental results. Sothis result indicated that pressure drop of packed bed withliquid could be expressed by formula (9).

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

3. Estimating Pressure Drop in Softening and LoadingTest

3.1. Experimental ConditionFormula (9) was applied to the pressure drop of actual

softening sinter obtained from softening and loading testshown in Fig. 11. The experimental apparatus comprised agas heating furnace, a holding furnace, and a reaction fur-nace. The gas used in the experiment was first preheated inthe gas heating furnace, and then, was injected into the reac-tion furnace.

A graphite crucible with an inner diameter of 50 mm anddepth of 80 mm was used. For injection of gas and drippingof the melt, there are 13 holes with diameters of 5 mm inthe bottom of the crucible. The crucible was packed with thesample material, and a maximum load of 98 kPa was appliedto the sample from the top via a grate. The exhaust gas wasmeasured by an infrared gas analyzer. Sinter with an averageparticle size of 5.5 mm was charged into the crucible. Thesinter weight was 100 g. To prevent blockage of the gas pas-sage holes, 10 g of bed coke was laid at the top and bottomof the sample layer, respectively.

Figure 12 shows experimental conditions for softeningand loading test. These conditions were decided to simulateactual blast furnace conditions. Gas flow rate was 6 L/minunder an ordinary temperature and pressure.

Sinter reduction ratio FR was estimated from the outletgas composition by (10).

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

Fig. 9. Calculation result of liquid hold up: ht with sponge contrac-tion ratio in layer condition.

h Sr Sr

h Sr Sr Sr Sr Sr

t sponge

t sponge sponge

= ≤ ≤( )= − ≤ ≤(0 0

0 44

1

1 1 2. ( ) ))

Fig. 10. Comparison between ΔP/L (exp.) and ΔP/L (calc.) in wetcondition.

ΔPL C h D Sr

u

t p

=−( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ −( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 1

1 2

22

ε ϕρ

FM

v v v v tRO

CO out CO out CO in CO int

t= +( )− +( ){ }∫

1000

22 42 2

2 20. , , , , d


549 © 2015 ISIJ

3.2. ResultFigure 13 shows pressure drop, contraction ratio and

reduction ratio during softening and loading test. Pressuredrop increased over 1 150°C, and reached maximum at1 350°C. Contraction ratio started degreasing at 1 100°C.Reduction ratio increased during experiment.

3.3. Estimating Pressure Drop of Softening Sinter Con-sidering Liquid Phase

Pressure drop of softening and loading test was estimatedby using formula (9). To estimate pressure drop by usingformula (9), it is necessary to obtain the void fraction andliquid holdup. The void fraction was obtained from the vol-

ume of the packed bed and the volume of the solid phase.The estimation of liquid holdup was calculated by using thethermodynamic calculation program, FactSage®.

Packed bed volume was expressed as a function of initialpacked bed volume and contraction ratio, formula (11).

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

To obtain the volume of the solid phase in a packed bed,it is necessary to consider three factors, namely, the decreasein the porosity of the sinter due to ore contraction, the vol-umetric decrease due to progress of the reduction reaction,and the volumetric decrease of the solid phase due to for-mation of a liquid phase. For the change in sinter porosity,a uniform decrease in porosity accompanying contractionwas expected, as expressed by formula (12).

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

Density changing with proceeding reduction was calcu-lated by calculation result of reduction ratio. Solid decreas-ing with liquid generation was difference between total slagvolume and calculated liquid slag volume. Each volume wascalculated by formula (13).

........................ (13)

Liquid weight was calculated by thermo dynamic calcu-lation program called FactSage®. FeO–SiO2–CaO–Al2O3–MgO system was calculated. Weight of FeO was estimatedby reduction ratio. Table 2 shows the composition of slagused in the calculation. Weight of FeO decreased withincreasing reduction ratio. Figure 14 shows calculationresults of weight of liquid phase. Weight of liquid startincreasing over 1 250°C. Liquid volume was calculated byformula (14), by using calculated weight of liquid.

............................. (14)

Based on the results obtained here, the aperture ratio mwas estimated from the volumetric changes due to formationof the liquid phase, progress of the reduction reaction, andcontraction of the sinter. Formula (15)15) was used for thedensity of the liquid phase; this formula evaluates liquid

Fig. 11. Schematic drawing of experimental apparatus for soften-ing and loading test.

Fig. 12. Experimental conditions for softening and loading test.

Fig. 13. Experimental results of softening and loading test.

V Sr V= −( )1 0

ρ ρ εS i S i v Sr, ,= − −( ){ }0 1 1

V MS S i S i= ∑ ( ), ,ρ

V ML L L= ρ

© 2015 ISIJ 550


density on the assumption that the density of a multi-com-ponent slag system is a function of the mass fraction of ironoxide.

....................... (15)

The densities of gangue and iron were assumed to be2 600 kg/m3 and 7 600 kg/m3, respectively. Using the densi-ties obtained as described above, the volumes of each phasewere calculated, and the aperture ratio of the packed bedwas then calculated by formula (16) by dividing the resultsby the volume of the packed bed. However, at temperatureshigher than 1 300°C, in case VS + VL > V, it is assumed thatliquid flooding has occurred, and VL = 0.

................. (16)

Figure 15 shows comparison between aperture ratioobtained by formula (9) and aperture ratio obtained by for-mula (16). Each values show good agreement. Figure 16shows comparison between experiment result of ΔP and cal-culation result of ΔP in softening and loading test. Calculat-ed result shows good agreement with experimental result.By these results, it is considered possible to estimate the gaspermeability of softening ore layers by the estimation meth-od proposed here.

4. Conclusion

The effect of the liquid phase on pressure drop in the blastfurnace was quantified using a cold model which simulatedthe ore softening and contraction behavior of the cohesivezone considering the formation of a liquid phase. A perme-ability formula was also proposed based on the results of aload softening test with actual sinter, and the constituent ele-ments of the packed bed when coke is mixed with sinterwere discussed based on that formula. As a result, the fol-lowing conclusions were obtained.

(1) From the cold model experiment, it was confirmedthat pressure drop in a packed bed containing a liquid phasecan be expressed by the following formula:

Table 2. Calculating condition used in calculation.

Temperature[°C]

Composition [mass%]

FeO Al2O3 MgO SiO2 CaO

1 000 78.2 2.3 1.3 5.8 12.4

1 050 77.1 2.4 1.4 6.1 13.1

1 100 76.7 2.4 1.4 6.2 13.3

1 150 74.5 2.6 1.5 6.8 14.5

1 200 70.2 3.1 1.8 8.0 17.0

1 250 65.3 3.6 2.1 9.3 19.8

1 300 61.9 3.9 2.3 10.2 21.7

1 350 56.2 4.5 2.6 11.7 25.0

Fig. 14. Calculation result of weight of liquid phase.

ρL FeOX= +2490 12

m h V V Vt S L= − = − +( )ε 1

Fig. 15. Comparison between aperture ratio (calculated by Eq. (9))and aperture ratio (calculated by Eq. (16)).

Fig. 16. Comparison between experiment result of ΔP and calcula-tion result of ΔP in softening and loading test.

ΔPL C h D Sr

u

t p

=−( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ −( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 1

1 2

22

ε ϕρ


551 © 2015 ISIJ

(2) Proposed formula was applied to the results of theload softening test, and it was found that the gas permeabil-ity of an actual softening sinter layer can be estimated byusing this formula.

NomenclatureC [–]: Run off factorDp [m]: Particle diameterFR [–]: Reduction degreeG [g/cm2s]: Mass velocity of gash [m]: Height of contracted sponge ballht [–]: Liquid holdupK [–]: Air flow resistance coefficientK0 [–]: Air flow resistance coefficient before stat-

ing contractionL [m]: Height of packed bedMO [mol]: Total molar of reduced oxygenMS,i [kg]: Mass of solid (i = slag, Fe)ML [kg]: Mass of liquid (i = slag, Fe)m [–]: Aperture ratioP [Pa]: PressureS [m2]: Void areaS0 [m2]: Cross sectional areaSr [–]: Contraction rateSr1 [–]: Contraction rate liquid start soakingSr2 [–]: Contraction rate liquid start floodingSrsponge [–]: Sponge contraction ratet [s]: Reduction timet0 [s]: Time start reductionu [m/s]: Gas velocity in void towerV [m3]: Volume of packed bedV0 [m3]: Initial volume of packed bedVS [m3]: Solid volumeVL [m3]: Liquid volumevCO.in [m3/s (s.t.p.)]: Volume flow rate of CO in inlet gasvCO.out [m3/s (s.t.p.)]: Volume flow rate of CO in outlet gas

vCO2,in [m3/s (s.t.p.)]: Volume flow rate of CO2 in inlet gasvCO2.out [m3/s (s.t.p.)]: Volume flow rate of CO2 in outlet gasXFeO [–]: Mass ratio of FeO in slagγ [–]: Particle diameter ratioε [–]: Void fractionε0 [–]: Initial void fractionεv [–]: Pore fractionρ [kg/m3]: DensityρS,i [kg/m3]: Apparent density of solid (i = slag, Fe)ρ0S,i [kg/m3]: True density of solid (i = slag, Fe)ρL [kg/m3]: Density of liquidσ [–]: Correction coefficientφ [–]: Shape coefficientμ [Pa·s]: Viscosity

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