Correct prediction of material trajectory is a pre-requisite for effective burden distribution in a blastfurnace. A mathematical model based on singleparticle approach has been developed to estimatethe material trajectory for Tata Steel’s “F” BlastFurnace, fitted with a compact bell-less top (CBLT).The model has been validated with actual in-furnacemeasurements. Initial results about the applicabilityof the model are promising; however, furtherrefinement is in the pipeline.

Key Word

Compact bell less top, Material trajectory, Stock line.

List of Symbols

hf free fall height from discharge hopper

α chute inclination angle

vci initial velocity at chute flow

a→ acceleration of the particle

Fg→ gravitational force

g gravitational acceleration

m mass of particle

k co-efficient of restitution

R radius of chute

η co-efficient of friction

θ angle of rise

ω→ angular velocity of the chute

r→ radius

ro distance between the central line of the furnace andpoint of detachment of the particle at the chute tip

ν→ velocity of the particle

Ffr→ frictional force

Vco final velocity of chute flow

VH horizontal velocity

Vv vertical velocity

Ftotal resultant force

Fcg centrifugal force

Fcor coriolis force

INTRODUCTION

The distribution of raw material burden at the stock levelof a blast furnace is crucial for its smooth operation. Theraw materials consist of particles that are multi-sized andhave varied characteristics (density, shape, size, surface

roughness etc). After detachment from the rotating chute,the particles impact the stock surface, roll, disintegrateand may even percolate (if small in size) through theparticle bed. Knowledge of the initial stock profile isessential for understanding the radial variation of ore/cokeratio, size segregation, gas distribution and burden descent,which in turn influences the formation and shape of thecohesive zone. A thorough understanding of thesephenomena can help the operators in running the furnacesmoothly, control product quality as well as increaseproductivity. As a precursor to the above, prediction ofparticle trajectories of coke, ore, sinter and flux and theirpoint of impact, radial and azimuthal velocities areimportant. Initial surface profile of burden at the stocklevel is a strong function of the impact point of the materialtrajectory on the stock level. Determination of materialtrajectory should therefore be the first step towardsmanipulating the burden profile at the stock level(1,2).

Over the years the blast furnace charging system hasgraduated from the two-bell system to the bell throatarmour (MTA/VTA) arrangement, and finally to the bell-less charging systems. Researchers have matched paceby developing mathematical models to predict the burdentrajectory in all these systems (big bell system(3-6) and bellless system(6-10)). Since the differences in designs causewide variations between the different charging systems,in terms of manner in which the material is discharged,a particular model that describes one system naturallylooses validity in other systems. Parallel with themathematical modelling efforts physical simulationof burden flow has been attempted by means of fullscale(2,10-15) and scaled down cold models(2,16-18), and alsomeasurements in actual furnaces(19-28).

Flow of granular solids is a complex phenomenon andconstitutive equations to describe the transport mechanismof the granular material through the charging system ofthe blast furnace are hardly available. Empirical co-relations(10) and quasi-mechanistic equations(8) based onexperimental data from scaled blast furnace models, havemet some success in describing the motion of burdenmaterials. Incidentally most of the mathematical treatmentshave assumed the motion of a single particle to representthe flow of the burden material. Since the burden in arunning blast furnace, at any point of time, consists ofan aggregate of particles having large variation in shape,size and other characteristics, deviation of the materialtrajectory from the theoretical prediction is not unexpected.Many reported mathematical models are either based onempirical co-relations obtained from the analysis ofexperimental observations in a particular set-up that isthe equations are not of general nature or lacking intechnical details such as mechanical arrangement of thecharging systems, dimensions etc. So for quantitativeanalysis of burden trajectory for different materials for‘F’ blast furnace of Tata Steel, formulation of amathematical model, which will effectively predict theburden trajectories for the entire operating range of chuteinclination angles, is imperative.

Mathematical Simulation of Material Trajectory forCompact Bell-less Top of ‘F’ Blast Furnace

Tata Search, 2004

Samik Nag, Vipul Mohan Koranne, Tathagata Bhattacharya, Uttam Singh andSomnath Basu

Tata Steel, Jamshedpur - 831 001, India

Tata Search, 2004

Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace 85

MODEL DEVELOPMENT

Material Flow Path

The movement of the material during charging from aPaul Wurth compact bell-less top charging system isshown schematically in Fig.1. Raw materials are chargedfrom the hopper placed on top, and its discharge iscontrolled by the plunger. Opening of the plunger causesthe material to fall vertically downward and land on therotating chute. The chute, which rotates about the centralaxis of the furnace, distributes the materialcircumferentially on the stock line. The material may notnecessarily travel the entire chute length; the actualdistance that the particle travels along the chute is calledeffective chute length. Fig. 2 illustrates the effect of chuteinclination angle on effective chute length.

Mathematical Approach

Three distinct zones can be identified along the path ofmaterial movement.

★ Descent of material from the discharge hopper

★ Material movement along the rotating chute

★ Free fall from the chute tip

Descent of material from discharge hopper: The particlefalls from the plunger opening freely under the influenceof gravity and hits the surface of the rotating chute. Heightof free fall depends on the geometry of the charging systemand the chute inclination angle. So, the component ofinitial velocity of the particle, along the chute axis,immediately after striking the chute surface, can be givenas

vci = k 2ghf sinα � ...(1)

where k is coefficient of restitution. It has been assumedthat the collision is nonelastic i.e. at the point of contactbetween the particle and the chute the particle looses allits velocity along the direction normal to the chute surface.The free fall heights of the particle at different chuteinclination angles for charging system of ‘F’ blast furnaceis shown in Fig. 3.

Flow through the rotating chute: When a point massslides in a rotating field, it is subjected to centrifugal andcoriolis forces, in addition to the gravitational force. Theequation of motion can be expressed in the vector formas :

ma→

= Fg

→− m ω

→× ω

→× r

→− 2m ω

→× v

→− Ffr

→...(2)

Where m = mass of the particle,

a→ = acceleration of the particle,

Fg

→= gravitational force,

ω→ = angular velocity of the chute,

r→ = radius,

ν→ = velocity of the particle,

and Ff→ = frictional force.

On the basis of the coordinate system shown in the Fig.4 the governing differential equations can be written inthe following form:

d2z

dt2= gcosα +ω 2 zsinα − R cosθ cosα( )sinα

cosθdθdt

−η

dz

dtmv

N–2ωRsinα ...(3)

Fig. 1 : Material flow path in compact bell less top

Chute Inclination angle (deg.)

Fre

e fa

ll h

eig

ht

(m)

Fig. 3 : Free fall heights of particle at different inclinationangles

Fig. 2 : Effective chute length at different inclination anglesfor chute of ‘F’ blast furnace

Chute Inclination angle (deg.)

Eff

ecti

ve C

hu

te L

eng

th (

m)

Total chute length = 3.5m

3

2

1

015 30 45

Tata Search, 2004

86 Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace

Rd2θdt2 = −gsinα sinθ +ω 2 z cosα + Rsinα cosθ( )

×sinθ sinα + 2ω sinα cosθdz

dt−η

dθdtmv

N ...(4)

−ω2 cosθ sinα z cosα + Rcosθ sinα( )

cosαdθdt

2ωR+ +2ω sinα sinθdz

dt

+ Rdθdt

2

gsinα cosθ +ω2RN = m ×

...(5)

v =dz

dt

2

+ Rdθdt

2

...(6)

This set of simultaneous ordinary differential equations(3 – 6) were solved numerically using the 4th order Runge- Kutta method. It is essential to know the initial positionof the particle in the chute, its velocity at that point,coefficient of friction of the particle with the chute surfaceand effective length of the chute available to the particle,in order to solve these four equations. The initial velocityof the particle along the chute axis can be obtained fromequation (1).

Free fall from chute tip: By solving the equations (3-6),it is possible to estimate the velocity and position of theparticle at the chute tip. When the particle reaches thechute tip, S in Fig. 5, it has a velocity Vco along the slope.On leaving the chute tip, S, the particle travels to the

point P on the stock line. The location of the point P canbe calculated from the magnitude of horizontal and verticalcomponents of particle velocity, VV and VH respectively.These may be expressed as -

VH = Vco sin α ...(7)

VV = Vco cos α ...8)

The time of flight of the particle, for decent through aheight h, is given by:

t =VV

2 + 2gh – VV

g...(9)

and

Distance of point P from the furnace central line =

r = ro + VHt ...(10)

The tangential component of the velocity at the chute tipand the effect of the ascending gas have not been includedin this analysis.

PLANT TRIALS

During the filling-in of the ‘F’ blast furnace of Tata Steel,it was possible to observe the actual particle trajectory bythe use of the arrangement shown in Fig.6. Two referencebars were inserted inside the furnace. Reference bar #1was inserted from the opening of the above burden probe,and Ref. bar #2 was suspended parallel to it, with thehelp of seven chains. All these chains were 500 mm apartstarting from 200 mm away from the furnace wall. All thechains were 1m long. The lengths of the ref. bars weresuch that the free ends of the bars were just at the centreof the furnace. With this arrangement the striking positionof the trajectory can be measured at two different levels.Using two cameras the material flow was captured. Thecamera #1 was used to take pictures of the burden materialas it flowed through the chute. The camera #2 was usedto capture the striking point of the falling material on therods. Camera #2 was placed in such a way that it wasperpendicular to the plane of two Ref. bars (Fig. 6).Arrangements were made for sufficient light.

Table 1 lists the design and operating details of the chuteused in ‘F’ blast furnace of Tata Steel. The chute can be

Fig. 4 : Coordinate system for chute and different forcesacting on the point mass Directions of components:—dzdt (View A) ≡ longitudinal component,

N (View B) ≡ normal component,

R dθdt

(View B) ≡ tangential component

Fig. 5 : Free fall from chute tip

α

View C

View A

View C

View B

View B

View A

M

O

Fg

Hch

N

Fce

M

O

R

z

Z

ω

α

R dθdt

dzdt

Tata Search, 2004

Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace 87

RESULTS AND DISCUSSIONS

The mathematical model can predict the influence ofchute inclination angle on the radial landing position(s)of coke and metallics on a given stock level. Fig. 7 comparesthe experimentally observed landing positions of coke, asa function of chute inclination angle, with the theoreticalpredictions. Fig. 8 presents the corresponding results forsinter. The two sets of data in either figure represent thetwo different stock levels (5.35m and 4.35m below thehinge point of chute). The burden stream in Tata Steel’sF blast furnace strikes the stock level with lateral spreadof 0.45-0.6m. The distance of the mid position of thestream trajectory from the centre line is considered asthe stream distance. From the figures it is evident thatthe predicted results show good agreement with the actual

observations, except for inclination angles of 17° and 23°in the case of coke. For these lower angles the free fallheight of the material was large (Fig. 3) and effective chutelength was small (Fig. 2). Therefore the burden streamunderwent a large scatter after striking the chute surface.This caused problem in proper measurement of streamdistance during in-furnace trails.

Incidentally, the numerical values of the coefficients offriction and restitution were not known a priori. Themathematical model assumed hypothetical values in therange of 0.1 - 0.9 to simulate the burden trajectory. Inthe actual furnace, chute inclination angles in the rangeof 40.8 - 42.8 were operationally critical since the burdentended to strike the wall at higher angles. In view of theoperational importance, burden trajectories at 40.8° and42.8° were measured with maximum care during in-furnace trials. Thereafter, it was observed that a value of0.5 for coefficients of both friction and restitution enabledbest match between the theoretically predicted coketrajectory and the experimental results, for the above-

kept at any inclination angle, between 2° –52°. But fordistribution of burden inside the furnace, certaininclination angles were selected. The angles correspondto certain virtual rings as shown in Table 2. As per thepractice followed by the operating personnel, coke wascharged at all the angular positions while ring numbers5-9 only were used for charging sinter.

Fig. 6 : Experimental setup inside ‘F’ blast furnace

Table 2 : Inclination angle for chute used in F blastfurnace for different ring number

Ring number 3 4 5 6 7 8 9

Angle of inclination (°) 17 23 29.8 34 38.8 40.8 42.8

Dis

tan

ce f

rom

f/c

cen

tre

line

(m)

Fig. 7 : Effect of chute inclination angle on radial cokestriking distance

Dis

tan

ce f

rom

fu

rnac

e ce

ntr

e lin

e (m

)

Fig. 8 : Effect of chute inclination angle on radial sinterstriking distance

Light

Table: 1 Design and operating data of rotating chuteof ‘F’ blast furnace

Chute Length 3.5 m

Design Inner diameter 0.382 m

information Hch (ref. to Fig. 2) 0.687 m

Chute type Fish plate type

Operating Rotating speed 8 rpm

parameters Inclination angle range 2°-52°

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88 Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace

mentioned chute angles. Unfortunately, the correspondingexperimental results for sinter were not as reliable as thatfor coke due to significantly higher dust generation duringcharging of sinter. Fig. 9 illustrates the situation; thereference bar is hardly visible during sinter charging.Therefore, the value of 0.5 has been used throughout themathematical model.

The present work ignores the effect of ascending gaseson burden trajectory. Since the experimental data wereobtained during the filling-in stage, prior to firing, ascendinggas flow was hardly significant. Reliability of predictionunder conditions of significant gas velocity, as prevalentin a running blast furnace, needs to be established.Further, the entire model is based on a single-particleapproach, even though multiple discharge points havebeen assumed in an attempt to roughly approximate theaggregate nature of the burden stream. While thissimplification leaves room for error, it has been observedthat actual stream width deviates by only a small extentfrom the theoretical approximation.

As discussed earlier, the motion of a single particle in thechute is governed by the effect of gravitational, centrifugal,coriolis and frictional forces. Each of the forces has beenresolved into longitudinal, normal and tangentialcomponents – directions explained in Fig. 4. Figs. 10 - 13show the typical variation in the components of velocitiesand accelerations of a single particle, moving along thechute surface, corresponding to chute inclination angleof 42.8°.

Fig. 11 illustrates the variation of gravitational, coriolis,centrifugal and frictional forces’ components along the

chute axis. The longitudinal component of gravitationalforce remains constant along the chute length anddominates over components of other forces. Centrifugalforce increases as the distance between the particle andaxis of rotation increases. As can be seen from Fig. 11,the relative contributions of centrifugal and coriolis forces,towards the total force on the particle, are insignificant.The highest value of the coriolis component, ever achievedalong the entire length of the chute, is less than 15% ofthe gravitational force. The corresponding value for thecentrifugal component is less than 10%. But, the frictional

Fig. 9 : Coke (A) and sinter (B) streams striking the referencebars during f/c filling-in

Vel

oci

ty (

m/s

)R

ise

ang

le (

deg

)

Fig. 10 : (A) Rise of particle along chute wall

(B) Longitudinal & tangential components of particlevelocity

— variation along chute length

Acc

eler

atio

n (

m/s

2)

Fig. 11 : Components, along chute axis, of forces on a singleparticle

Fg=gravitational force, Fcg= centrifugal force,Fcor= coriolis force,

Ffr= frictional force Ftotal = resultant force

Longitudinal component

Tangential component

Reference bar Falling sinter

Reference bar

Falling coke

Tata Search, 2004

Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace 89

force exerts significant effect and its magnitude can goalmost as much as the gravitational force. However, thelongitudinal components of gravitational force and frictionact in opposite directions and therefore the magnitude ofnet acceleration of the particle in longitudinal directionis reduced.

Fig. 12 illustrates the force components along the tangentialdirection. The centrifugal force and friction have minorcontribution to acceleration of the particle in the tangentialdirection. It can be seen from the figure that coriolis forcefavours tangential acceleration, which gravity opposes.The gravitational component in tangential direction is asine function (eq. 4) of the angle of rise of the particle. Asthe particle moves along the chute length, its angle of risealong the chute side increases progressively, and thereforethe relative dominance of gravity increases. As a result,the overall tangential acceleration of the particle changessign after travelling some distance along the chute. Thephysical significance of this would be a reduction in thetendency of the particle to rise along the side of the chute.

The component of the forces normal to the chute surfaceis illustrated in Fig. 13. The contribution of centrifugalforce along the normal is insignificant. The coriolis forcefavours the normal component and its magnitude increasesas the particle travels along the chute length. Gravity alsohas a positive effect, but with a decreasing magnitude.The gravitational component is a cosine (eq. 5) function

of the angle of rise of the particle, a function that itselfdecreases along the chute length. This explains the waningcontribution of the gravity component.

From the above discussions, it may be concluded thatcoriolis force exerts significant influence on the tangentialand normal components of the particle motion. A directeffect of coriolis force is that it helps the particle to risealong the side of the chute. Also, the normal componentpresses the particle harder against the chute surface,thus increasing the frictional force. If the effect of coriolisforce is ignored, the velocity in the longitudinal directionis over-estimated. This has been illustrated in Fig. 14,wherefrom it is evident that the error increasesprogressively with increasing chute inclination angle.

Acc

eler

atio

n (

m/s

2)

Chute length (m)

Fig. 13 : Normal components of forces on a single particle— variation along chute length : Fg =gravitational force,Fcg = centrifugal force, Fcor = coriolis force, Ffr = frictionalforce Ftotal= resultant force

CONCLUSIONS

The following conclusions can be made with regard to thescope and applicability of the mathematical model:

★ The theoretical model appears well validated againstthe experimental data available. It can be used fordetermining the trajectory of coke and metallics forany inclination angle, over the entire range of operatingstock levels. Further validation against moreexperimental data is desirable to make the modeltheoretically exhaustive.

★ The coriolis force has significant effect on the motionof the particle. It increases the frictional force and alsocauses the particle to rise along the side of the chute.

★ The single particle approach appears promising in itsability to predict the flow phenomena of granularburden materials in the blast furnace chute.

★ The reliability of prediction is somewhat reduced atsmall chute inclination angles (<23°), most probablydue to the dilation of the particle stream.

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Fig. 12 : Tangential components of accelerations — variationalong the chute length

Fg = gravitational force, Fcg = centrifugal force,Fcor = coriolis force,

Ffr = frictional force Ftotal = resultant force

Acc

eler

atio

n (

m/s

2)

Chute length (m)

Par

ticl

e ve

loci

ty a

t ch

ute

tip

(m

/s)

Fig. 14 : Effect of coriolis force on longitudinal componentof particle ejection velocity at chute tip

Chute inclination angle (deg.)

With Coriolis force

No chute rotation

Without Coriolis force

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90 Samik Nag et al. Mathematical Simulation of Material Trajectory for Compact Bell-less Top of ‘F’ Blast Furnace

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