iron ore granulation model supposing the granulation

1. Introduction

More than 50% of iron ore resources used in Japan de-pend upon Australian ores. This trend will increase moreand more, due to the low transportation costs. As to theAustralian ores, supply of coarse Block-man ores contain-ing low phosphorous elements, which is the main ores usedin Japanese sinter plants, tends to decrease because of theshortage of the resource. Instead of the Block-man ores,relatively fine Mara-mamba ores containing crystallizedwater ranging from 4 to 6 mass% will be supplied to Japanin future by their large amounts of ore deposits.1) Since theMara-mamba ores are high iron grade as compared to otherAustralian ores, the ores can contribute to produce iron oresinter containing low silica, which is evaluated as highquality burdens for blast furnace operation with high-pul-verized coal injection.2) However, recent fundamental stud-ies reported that in addition to the fine size, granulationcharacteristics of them are inferior to other ores, due to theweak adhesion properties between the fine particles.3)

Previously, maximum productivity of sinter operation de-pended mainly on the combustion efficiency of added finecoke in raw materials. Presently, permeability in the sinterbed limits the productivity. It is derived from some ad-vanced technologies, such as the method of fine coke coat-ing on the surface of quasi-particles4,5) and introduction of

segregative charging system of fine coke in the depth direc-tion of sinter bed.6) These technologies have reduced thecoke consumption remarkably in the sinter operation. Thesetechnological trends represent that it is important in the fu-ture to improve the bed permeability in the sinter bed, inorder to achieve high productivity. To keep the permeabilityusing fine ores, it is necessary to improve granulation of thefine ores before sintering. The purpose of the present studyis to propose an advanced mathematical model for simulat-ing iron ore granulation phenomena in a disc-pelletizer ordrum mixer.

2. Previously Proposed Granulation Models

To keep high permeability of granulated material bed ina grate process, granulation of iron ore fines is quite impor-tant for succeeding sintering process. Many mathematicalmodels have been proposed for simulating the granulationphenomena, in addition to the empirical ones on the basisof granulation experiments.4)

Rumpf7) clarified firstly granulation mechanism of finepowders. He concluded that the mechanism depends oncontact points, bridges and capillary forces between thepowders. He also made clear that maximum shearing ortension forces between the powders influence the strengthof the granulated materials. Ouchiyama et al.8) proposed a

ISIJ International, Vol. 42 (2002), No. 8, pp. 834–843

© 2002 ISIJ 834

Iron Ore Granulation Model Supposing the GranulationProbability Estimated from Both Properties of the Ores and Their Size Distributions

Noboru SAKAMOTO

Materials and Processing Research Center, NKK Corporation, Kokancho, Fukuyama, Hiroshima 721-8510 Japan.

(Received on February 8, 2002; accepted in final form on May 17, 2002 )

It is important to reduce fluctuation of the ore granulation for a stable operation of iron ore sinter plant,because the fluctuation affects directly productivity and productive yield of the operation. To clarify the gran-ulation phenomena in a rotating granulator, an advanced mathematical model that is different from previous-ly proposed ones was developed considering the probability theory. Mathematical characteristics of themodel and granulation simulation by the model are summarized as follows:

(1) Since it is difficult to analyze the phenomena by the models based on the motion of quasi-particlesdue to the complicated movement of large amounts of the quasi-particles, the present model treated it as akind of probability phenomena.

(2) This model is basically composed of matrix algebras defining following two granulation parameters,1) overall granulation probability resulted from ore properties, and 2) granulation and disintegration probabili-ties at each size range of the ores.

(3) This model is useful for evaluation of the granulation phenomena, because the simulation resultsusing the appropriate granulation parameters agree with those from the granulation operation.

(4) It is important to make clear the granulation conditions for defining the granulation parameters, be-cause the parameters depend on the conditions of the granulation dynamics and the size of the granulator.

KEY WORDS: granulation; simulation model; iron ore sinter; probability theory; Markov chain.

granulation model in which fine particles coalesce mutuallyby both external and friction forces during rotating motion.Suzuki et al.9) analyzed granulation phenomena formulat-ing mass balance of iron ore fines in a drum mixer and disc-pelletizer. They concluded that size of quasi-particle is de-termined by total rotating distance in the granulators andgranulation characteristics of a disc-pelletizer are superiorto those by a drum mixer. Lister et al.10) presented a popula-tion balance equation in which quasi-particles grow by thecoating of fine ores on the surface of nuclei coarse ores.This equation made clear that it is important to control sizeof fine ores for size enlargement of the quasi-particles.Kanoh et al.11) analyzed theoretically the granulation phe-nomena during rotating motion in a drum mixer, consider-ing kinetic and colliding energy. They pointed out throughthe analysis that the granulation obtained in small-scaledtests does not agree with that in commercial plant opera-tion. With recent advance of capacity of computers, analy-ses of granulation dynamics will progress remarkably intro-ducing some advanced computational methods.11)

3. Proposal of Granulation Model Based on the Proba-bility Theory

When fine ores are granulated in a disc pelletizer or

drum mixer, it is necessary to obtain information of thegranulated quasi-particles considering both the ore proper-ties and granulation conditions in a granulation operation.Figure 1 shows a relation among raw material conditions,granulation parameters and quasi-particle conditions. Itmust be considered for the constitution of the granulationmodel that size distribution of fine ores influencing thegranulation characteristics, ore properties, binder, and gran-ulation mode. In addition to information of average size ofthe quasi-particles, size distribution of the quasi-particlesalso needs for their evaluations in the succeeding sinterplant operation, because permeability of the sinter bed de-pends mainly on them.

3.1. Growing Phenomena of Quasi-particles in a Ro-tating Granulation Process

It is difficult to clarify quantitatively the growing phe-nomena of quasi-particles during rotation in a granulatingdevice under moisturizing conditions. Figure 2 shows typi-cal granulation phenomena in a rotating drum mixer. Fineores, which partially remain unchanged during the rotation,grow through a unit granulating operation to quasi-particlesby coalescence of fine ores and coating of fine ores on thesurface of nuclei ores. In addition to growing of the quasi-particles, a portion of the quasi-particles reduces their sizes

ISIJ International, Vol. 42 (2002), No. 8

835 © 2002 ISIJ

Fig. 1. Factors on raw materials and granulation operation for constituting the granulation model.

Fig. 2. Conceptual representation of the typical granulation forms in a rotating disc-pelletizer.

because of the disintegration. Quasi-particles substantiallyincrease in their sizes by progressing the granulation times.The average size and size distribution of the quasi-particles,therefore, depend upon the contributing ratio determined bythese granulation and disintegration motions.

3.2. Formulation of Mathematical Model SimulatingGranulation Phenomena

Since complicated granulation occurs simultaneously ina granulation system, there is a limit to formulate a mathe-matical model taking account of the motion based on themutual action between particles. The author tries to developa probability model supposing granulation probabilities in adisc pelletizer or drum mixer, because it is difficult to deter-mine the size and size distribution by the previous modelsdue to the limit mentioned above. The same approach as theprobability model was applied for analyzing coal grindingmechanism by Broadbent et al.12) They clarified the mecha-nism introducing a grinding matrix and elements constitut-ing the matrix.

Figure 3 shows a concept of the present granulationmodel on the basis of the probability theory. Initial fine oresare separated to granulated and un-granulated portions aftera unit rotating operation. Size distribution of the fine ores,F(�G0) and the granulated quasi-particles, GN after N-times of the rotation are expressed using column vectors ac-cording to the size distribution as follows:

F�G0�( f1, f2, f3, · · · , fi, · · · , fn) .................(1)

GN�(g1, g2, g3, · · · , gi, · · · , gn) ....................(2)

Assuming the overall granulation probability of ore, p ,which depends on the ore properties defined to the unit ro-tating operation, Fig. 3 indicates that finest portion of theinitial ores, f1 changes to following granulated and un-gran-

ulated sections after the unit rotating operation:

p · f1 (Granulated section) ..........................(3)

(1�p) · f1 (Un-granulated section) ............(4)

Equation (3) gives the granulated section regarded as massof quasi-particles that includes various sized quasi-parti-cles. In case that the initial ores grow to the quasi-particlesby the rotating operation, the size distribution of the quasi-particles depends on the granulation probability character-ized by each size range of the ore. If the probability is de-fined as qi1(�), it means the probability elements of eachsized quasi-particles in which the finest ore portion, f1 isgranulated to the quasi-particle having size of i-th sievemesh. The granulation probabilities, qij, accordingly rangefrom 0 to 1.0(�).

When the finest ore portion, f1 having size of less than0.125 mm is granulated, the portion changes as follows bythe every one time of rotation:

(1�p)�pq11 (Sum of weight ratio of non-granulated particles and granulated quasi-particles less than 0.125 mm in size) ............................................(5)

pq21 (Weight ratio of quasi-particles ranged from0.125–1.00 mm in size) ...........................................(6)

pqi1 (Weight ratio of quasi-particles ranged sieve mesh from (i)-th to (i�1)-th) ............................................(7)

pqn1 (Weight ratio of quasi-particles more than the maximum size) ........................................................(8)

By the same procedure as shown in Fig. 3, Fig. 4 showsthe distribution of quasi-particles after granulation of theores ranged from 0.125 to 1.00 mm in size. Weight portionof f2, located in the secondary sieve mesh (0.125–1.00 mm


© 2002 ISIJ 836

Fig. 3. Distribution of quasi-particles after the granulation of ores less than 0.125 mm in size.

in size), changes to the ratio of quasi-particles, according tothe probability elements, qi2(�) by the same mathematicalprocedure. In this case, it is necessary to take account ofdisintegration probability of quasi-particles, as shown inFig. 2.

pq12 (Weight ratio less than 0.125 mm in size by the disintegration probability) .......................................(9)

(1�p)�pq22 (Sum of weight ratio of non-contributing granulation and incomplete granulation ranged from 0.125–1.00mm in size) ..........................................(10)

pqn2 (Weight ratio of quasi-particles more than 20.65 mm in size) ..................................................(11)

Size distribution of the quasi-particles at the first unit rota-tion operation can be determined by the similar mathemati-cal procedure from j�1 to n.

A square matrix B of (n, n), composed of the probabilityelements governing granulation and disintegration is de-fined as follows:

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

As material balance before and after the granulation mustbe conserved in Eq. (12), a following equation is obtained:

qij�1.0(�) ( j�1�n) ..................(13)

where, qij defines the probability elements constituting then�n matrix B.

A portion of quasi-particles located less than a minimumsized sieve mesh is presented by the following equationsbased on Eq. (5):

g1�f1(1�p)�f1pq11 ..................................................(14)

g2�f2(1�p)�f1pq21�f2pq22�f2(1�p)�p( f1q21�f2q22)

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

gk�fk(1�p)�f1pqk1�f2pqk2� · · ·�fmpqkm�· · ·�fkpqkk

�fk(1�p)�p fm ·qkm ........................................(16)

Equation (14) means that g1 is given by summing the termsof both non-granulating portion and granulated portion lessthan 0.125 mm in size, in spite of the granulation operation.Equation (15) also means that g2 is expressed by the sum ofnon-granulating portion and granulated portion resultedfrom f1 fraction and granulated one between 0.125 to 1.00mm in size. The same mathematical procedure as Eqs. (14)and (15) can introduce to the generalized formulation. Afollowing equation can be given from Eq. (16) using a ma-trix expression.

G�(1�p)E ·F�pB ·F ....................(17)

Where, G and F mean a column vector composed of gi andfi , respectively, and E shows a n�n unit matrix.

.....(18)

Size distribution of quasi-particles, G2, obtained after thesecondary unit granulating operation represents as followsby the same procedure described in Eq. (17):

G2�{(1�p) ·E�pB} ·G1�{(1�p) ·E�pB}2·F ...(19)

The size distribution, GN, after N-times of the unit granulat-ing operation is expressed by Eq. (20):

GN�{(1�p) ·E�pB}N ·F ..................(20)

From Eqs. (12) and (18), the first term of the right side in

G F E� � �

g

g

g

1

2

1

2

1 0 0

0 1 0

0 0 1

M M

L

L

M O M

Ln n

f

f

f

, ,

m

n

�1∑

i

n

�1∑

B�

q q q

q q q

q q q

n

n

n n nn

11 12 1

21 22 2

1 2

L

L

M O M

L


837 © 2002 ISIJ

Fig. 4. Distribution of quasi-particles after granulation of ores between 0.125 and 1.00 mm in size.

Eq. (20) is also expressed by a matrix B� of (n, n), as fol-lows:

B��{(1�p) ·E�pB}N� ...(21)

Both granulation and disintegration phenomena are consid-ered in the matrix B�, due to the composition of the matrixB. Equation (20) must satisfy the law of conservation ofmass before and after the granulation.

........................(22)

From Eq. (22), a series of following equations is given:

g1�q�11· f1�q�12· f2� · · ·�q�1n · fn

g2�q�21· f1�q�22· f2� · · ·�q�2n · fn

· · · · · ·

gn�q�n1· f1�q�n2· f2� · · ·�q�nn · fn

Equation (23) can be obtained by summing above the equa-tions:

gk� q�k1· f1� q�k2· f2� · · ·� q�kn · fn

.........................................(23)

Following relation is given by Eq. (13):

gk�f1�f2� · · ·�fn� fk ..............(24)

Equation (24) indicates that total weight of fine ores isequivalent to that of granulated quasi-particles.

3.3. Simulation of the Granulation Dynamics

Figure 5 shows a procedure for the simulation using theequations described in 3.2. For conducting the simulation,granulation conditions must be determined, such as totalnumbers of rotation times, Nmax, numbers of sieve meshproviding both the size distribution of fine ores and granu-lated quasi-particles, n�1, size distribution vector of initialores, F, overall granulation probability depending on theore properties, p , and matrix elements, composed of granu-lation and disintegration probability of every ore size range,B. Matrix calculation at each granulation operation is car-ried out using Eq. (20) under the above granulation condi-tions. The matrix calculation gives information of the newgranulation matrix, B�, corresponding Eq. (21) and the sizedistribution of quasi-particles, G at each unit granulatingoperation. Figure 4 indicates when fine ores less than 1mmin size are consumed completely by the coating on the sur-face of nuclei coarse ores, the calculation procedure ends.From a viewpoint of the simulation, this computational

treatment is based on the assumption that enlargement ofquasi-particles is controlled by the shortage of fine ores.The limit of the fine ore size, accordingly, must change,corresponding to ore properties and granulation conditions.

Figure 6 shows typical simulation results assuming thegranulation parameters of p and B. Table 1 shows sievemeshes and the size distribution of ore fines. The distribu-tion is characterized as finer ores in comparison with typi-cal ones used in other sinter plants in terms of the contentsof size less than 0.125 mm. To characterize the granulationphenomena, typical two kinds of parameters, matrix B wereselected under the same parameter of p . Table 2 indicatesthe granulation parameters, p and B for the simulation.Matrix B in Case 1 is characterized by the upper triangularmatrix bordered on the diagonal matrix. This controls thedisintegration of the quasi-particles during the granulationsimulation. Matrix B also indicates that the probability ele-ments from q4j to q6j are relatively larger than other row ele-ments. This consideration increases in the yield of quasi-particles having 3.0 to 10.3 mm in size. To widen the sizedistribution of the quasi-particles during the simulation,matrix B in Case 2 is characterized by the reversed consti-tutions of the matrix as shown in Case 1. Since these para-meters are determined as assumed values to verify the va-lidity of the mathematical model, the appropriate parame-ters must be defined by granulation tests using the targetores. The average data of initial size distribution of oresthat are used at Fukuyama No. 5 Sinter Plant were adoptedfor the simulation. This data was subdivided according tothe numbers of sieve mesh, because the original data of thesize distribution at the sinter plant is divided roughly. Case1 in Fig. 6 indicates the suitable granulation condition, be-cause fine ores are consumed quickly. As a result of thecondition, size distribution of the quasi-particles graduallyconcentrates to that from 5 to 10 mm in size. Case 2 sug-gests the unsuitable condition for the granulation, becausereduction of the fine ore portion delays, due to the assumedprobability elements caused by inferior granulation charac-teristics. Kinds of iron ores as mentioned above will influ-ence the characteristics by their different physical and min-eralogical properties, even though the size distributions are

k

n

�1∑

k

n

�1∑

k

n

�1∑

k

n

�1∑

k

n

�1∑

k

n

�1∑

G B� �

g

g

g

1

2

1

2

M M

n n

f

f

f

′ ⋅

′ ′ ′′ ′ ′

′ ′ ′

q q q

q q q

q q q

n

n

n n nn

11 12 1

21 22 2

1 2

L

L

M O M

L


© 2002 ISIJ 838

Fig. 5. Calculation flow of the granulation model.

similar. Figure 7 shows the simulation result of the influence of

the granulation characteristics of ores on the size distribu-tion of quasi-particles according to the granulation times.In this simulation, B is composed of the same matrix as de-scribed in Case 1. With increase in the number of unit gran-ulating operation, the size distribution of quasi-particles fi-nally becomes similar, independent of their granulationcharacteristics. However, there is a significant difference inthe size distributions during the granulation. Although theinitial size distribution of ores is the same in all simulationcases, granulation of finer ores delays in case of ores havinglow value of p .

3.4. Application of the Model to the Analysis of Gran-ulation Experiment

In order to evaluate validity of the mathematical model

for simulating the granulating phenomena, the model wasverified by comparing the granulation test data.13) In the testdata, numbers of sieve mesh are divided into following sixparts; –0.125, 0.125–0.50, 0.50–1.0, 1.0–2.0, 2.0–5.0, 5.0–10.0 mm, respectively. Figure 6 indicated that the elements,qij influence the final size distributions. The distributionsdepend on the qij, located in the upper (I) or lower area (II)bordered on the diagonal matrix, qkk. Equation (17) charac-terizes that the elements in the area (I) contribute the granu-lation and those in the area (II) deteriorate it, due to the dis-integration probabilities. The data13) characterizes that thereduction of fine portion less than 0.25 mm in size and in-crease of the portion between 2.0 to 5.0 mm in size are re-markable. Since the granulation test does not give the infor-mation of the overall granulation probability, p , it is appar-ently determined as 0.4(�), shown in Eq. (25). This meansthat contribution ratio of the granulation is defined as 40%


839 © 2002 ISIJ

Fig. 6. Simulation results showing transition of the granulationphenomena based on the different granulation parame-ters.

Table 1. Size distributionof fine ores andsieve mesh for thegranulation simu-lation.

Fig. 7. Influence of granulation characteristics of ore on thegranulation transition.

Table 2. Matrix elements for thesimulation indicatinggranulation and disin-tegration probability.

by one unit operation. The underlined probability elementsin Eq. (26) show the diagonal matrix bordering on the gran-ulation and disintegration area. Thus granulation parame-ters used in the present model are determined consideringcharacteristics of the experimental test data13) as follows:

p�0.4(�) ..............................(25)

B� .....(26)

Figure 8 shows the comparison between the experimen-tal granulation result and the simulation one. Althoughnumbers of the unit granulation operation of the experimentare not clarified in the study,13) the experimental resultagrees well with the simulation one assuming the granula-tion operation of 4 times. Even if the numbers13) of the rota-tion are different from the assumption, the size distributionof the quasi-particles will attain to that in Fig. 8 after longtime granulation, due to the composition of the matrix B inEq. (26).

4. Discussion

4.1. Mathematical Evaluation of the Granulation Pa-rameters

Granulation matrix, B, composed of the granulation anddisintegration probabilities at the each size range of oresand overall ore granulation properties, p are the main para-meters in the present mathematical model. Theoretical

meaning of these parameters and the influence of the para-meters on the granulation phenomena are discussed as fol-lows:

4.1.1. Granulation Matrix (B)Following extreme cases can assume in the granulation

matrix, B, described in Eq. (12).

..................(27)

B1 shows a triangular matrix where probability elements lo-cated upward on the diagonal matrix, qkk are zero. Thismeans that granulated quasi-particles never disintegrate bythe granulation operations. B2 shows a matrix, which iscomposed of the same valued elements under the belowcondition:

r�1.0/n ................................(28)

This means granulation and disintegration of quasi-parti-cles occur equally because of the same probability elementsduring the granulation operation. B3 means a triangular ma-trix in which the elements are arranged inversely as com-pared to those of B1 matrix. This matrix means that thequasi-particles continue to be disintegrated in spite of thegranulating operation.

The parameter, p is regarded as a constant probabilityduring the granulation operation in the mathematicalmodel. This is based on an assumption that the parameter isindependent of the operation conditions, such as size distri-butions of quasi-particles after N-times of granulation orrotation times of the granulation systems. However, withprogress in the granulation, there is a possibility that pgradually decreases by changing the ratio of fines to quasi-particles. An appropriateness of the assumption must be ul-timately evaluated by experimental procedures, in additionto a clarification of the physical meanings.

Figure 9 shows the simulation results using the granula-tion matrixes described in Eq. (27) under the constant oregranulation properties, p�0.4(�). In case of using B1,granulation phenomena are influenced by initial ore sizedistribution if numbers of the granulation times are limited.The granulation gradually proceeds, with increase in num-bers of the unit granulation operation. Finally, ores are

B3

11 12 1

22 20

0 0

0 0 0

�

q q q

q q

q

n

n

nn

L L

L L

O M

M O M

L

B2 �

r r r

r r r

r r r

L L

L L

O M

M O M

L L

,

B1

21

1 2

0 0 0

0 0

0

0

�

L L

L L

M O L

M O

L L

q

q qn n

,

0 02 0 01 0 02 0 02 0 03 0 02

0 05 0 03 0 05 0 05 0 05 0 03

0 13 0 06 0 10 0 15 0 09 0 13

0 15 0 03 0 13 0 15 0 20 0 23

0 40 0 40 0 45 0 43 0 43 0 34

0 25 0 20 0 25 0 20 0 20 0 25

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .


© 2002 ISIJ 840

Fig. 8. Comparison of granulation phenomena between experi-mental and simulation results.

completely granulated due to the operations. As a result ofthe granulation simulation, size of the granulated quasi-par-ticles becomes larger more than the maximum sieve mesh,20.7 mm in size. In case of B2, all probability elements con-stituting the matrix are the same, r�0.10(�) under the con-dition described in Eq. (28). In this case, as both granula-tion and disintegration of quasi-particles occur by the si-multaneous probability during the operation, weight ratio ofeach sized quasi-particles focuses on the same value,g(i)�0.1(�). When matrix B3 that is conversely composedof B1 is used for the simulation, quasi-particles continue tobe disintegrated by progressing the operation. Size of thequasi-particles focuses on the minimum sieve meshes lessthan 0.125 mm in size. This means that quasi-particles re-duce their sizes due to the matrix B3, in spite of the granu-

lation operation. This mathematical procedure is approxi-mately the same as that of the grinding model reported byBroadbent et al.12)

The proposed model, therefore, provides size distributionof quasi-particles, G, according to introducing the granula-tion and disintegration matrix, B. There are no cases thatsize of all quasi-particles becomes larger than the maxi-mum sieve mesh, B1 or passes the minimum one, B3 in acommercial granulation plant. Actual granulation matrix Bused in the mathematical model accordingly exists betweenthe matrix B1 and B3. Broadbent et al.12) propose the empir-ical formula of the grinding matrix, B by grinding tests.Similar mathematical procedure may be introduced to thegranulation model, although the physical phenomena aredifferent. Detailed discussions are left over as a future prob-lem to be solved.

4.1.2. Granulation Property Depending on Ore Character-istics

Figure 7 clarified that the influence of overall granulationproperty of ores, p on the size distribution of quasi-parti-cles. With a decrease in the value of p , it takes relativelylong times to attain the final size distribution of quasi-parti-cles which are defined by the granulation matrix B. Thesegranulation parameters are, therefore, regarded that B de-fines the final size distribution and p determines a granula-tion rate until attaining the final distribution.

Figure 10 shows the simulation results showing influ-ence of granulation operation times on the change of matrixB� obtained from Eq. (21), using the initial granulation pa-rameters described in Case 1 of Table 2. With the increasein the numbers of operation times, the matrix showing bothgranulation and disintegration of quasi-particles tends tobecome similar. From a viewpoint of the probability theory,this tendency depends on a transition probability matrix Bunder the condition of Eq. (13). In case of the transitionprobability matrix, probability elements after the operationof (N�1) times, qij�

(N�1) is only influenced by the elementsof qij�

(N ), as described in Eq. (29) based on Eq. (20).

qij�(N�1)� qir�

(N ) ·qri�(N ) ....................(29)

r

k

�1∑


841 © 2002 ISIJ

Fig. 10. Influence of granulation operations on the change of the granulation matrix.

Fig. 9. Relation between granulation matrix and calculated sizedistribution of quasi-particles.

Equation (29) means that the matrix B� is regarded asMarkov chain,14) due to the change of the elements q�ij. Ifthe elements are controlled by Markov chain process, theelements converge on following constant ones by the proba-bility theory14):

pi�1.0(�) ...(30)

As shown in Fig. 10, when the granulation operation ex-ceeds more than 3-times (N�3), q�ik (k: constant) approach-es to a probability column vector on the basis of Eq. (29).Therefore, simulation result, such as the result in Fig. 7, bythe present probability model necessarily is focused on aconstant column vector G, according to the initial granula-tion matrix. From a viewpoint of a probability model, thepresent mathematical model leads that the simulation resultof the size distribution after a large number of granulationoperations necessarily approaches a constant value, G with-out divergence.

The simulation results made clear that there are an inti-mate correlation between the granulation parameters andgranulation characteristics. It is, accordingly, very impor-tant to establish sieving conditions of initial ores incom-pletely granulated and quasi-particles after every granula-tion operation.

4.2. Concept of Application of the Model to the Con-trol of Granulation Operation

Granulation operation using a large amount of fine oresfrequently fluctuates by changing the ore pile, even underthe constant operating conditions. Sintering operation in-evitably shifts unstably by the fluctuation. It is true thatthere are many reasons for taking place it, but physical andmineralogical properties of ores seem to be one of the mainreasons. The fluctuation results from the change of mixingratio of ores having different granulation properties, even if

fluctuation of size distribution and chemical compositionsof the pile are controlled to minimize. As for the granula-tion model, the fluctuation corresponds to the change of theparameters; qij and p , used in the model. In order to reducethe fluctuation, it is necessary to introduce the operationconsidering the granulation characteristics of typical ores.From the viewpoint of simulation methods for reducing thefluctuation, the granulation parameters must be establishedto clarify quantitatively the granulation phenomena bygranulation tests. Granulation characteristics of the pile,composed of many kinds of ores, can estimate by the gran-ulation model using the mixing ratio of ores and these ob-tained parameters.

Suzuki et al. clarified experimentally the influence ofpellet feeds in the sinter mix on the size of the quasi-parti-cles by the consideration of their granulation characteris-tics.15) Figure 11 shows a concept of granulation controlmethod using the model referring to the previous study.15)

The method is based on an assumption that the granulationparameters of the pile are given from those of each mixingore and the mixing ratio. The granulation parameters, p andqij are given from following equations considering numbersof ore kinds (m) and their mixing weight ratios (w1, w2, w3,· · · , wm(�)) under the assumption of constant ore densities.

p� (wl · p l) , qij� {wl · (qij)l}

where; wl�1.0 (�) ...................(31)

Equation (31) means mathematical strictness that qij ob-tained from the sum of the product given by multiplyingscalar wi by matrix (qij)l is the same as that of Eq. (13).Mass conservation between ores and quasi-particles by thematrix calculation with Eq. (20), accordingly, is establishedbefore and after the granulation operation. Since the granu-lation characteristics of the mixed ores seem to depend onthe mixing ratio of each ore possessing the different granu-

l

m

�1∑

l

m

�1∑

l

m

�1∑

i

n

�1∑′

∞B( ) ,�

p p p

p p p

p p pn n n

1 1 1

2 2 2

L

L

M O M

L


© 2002 ISIJ 842

Fig. 11. A method of operation control of granulation plants using the granulation model.

lation parameters, the granulation parameters of main orescomposing the pile must be determined experimentally.Following experimental granulation tests will clarify the pa-rameters under the same dynamic similarity16) as that ofcommercial granulation plants: (1) Parameter, p can be de-termined by observing the change of mean size or cumula-tive distribution curve of the ore, because it is regarded asan index of the granulation rate. (2) As the quasi-particlesconverge constant size distributions after a large number ofrotation times of the mixer, parameter qij can be estimatedby the final distribution. When granulation conditions suchas addition of binder or moisture are changed, the granula-tion parameters, p l and (qij)l must be reconsidered, becauseEq. (31) can be applied under the same granulation condi-tions.

Columnar vector, G indicating size distribution of thequasi-particles at the exit of a drum mixer is estimated giv-ing F of the mixed ores and retention time which is predict-ed by charging tracers with the ores in the mixer. It is wellknown that proper addition of moisture or binder in themixed ores can improve the granulation characteristics, dueto the change of the interfacial properties of the ores. Thepresent model with modified granulation parameters, whichare obtained experimentally as mentioned above, can evalu-ate this improvement. This model, therefore, shows a possi-bility to estimate the granulation phenomena and indicatesthe direction of the suitable granulation conditions beforeusing the pile. To improve the accuracy of the model, it isnecessary in future to establish the prediction methods ofthe granulation parameters.

4.3. Problems of the Proposed Mathematical Model toBe Solved in Future

Since the proposed model deals with only ore granula-tion phenomena based on the parameters, p and qij, the ef-fects of size and operating conditions of the granulatormust be considered by another methods. If the conditions ofcommercial granulation plant operation are given, theFroude number16) is determined taking account of diameterof the granulator and rotating speed. The granulation para-meters, thus, can be estimated by experimental tests underthe same Froude number condition as that of the commer-cial operation.

Although this model seems to be effective for the evalua-tion of the granulation phenomena, there are still possibili-ties that the model limits its application by the followingreasons: (1) Possibilities of changing the granulation para-meters according to the type of granulators and their di-mensions, in spite of arranging the dimensionless number.(2) Difficulties of introducing Eq. (31) by ore characteris-tics. (3) Dependence of the granulation parameters on thegranulation time or ore characteristics, in spite of the inde-pendence, as described in Sec. 4.1.1. Both experimentaland operational granulation trials are needed for evaluatingthese possibilities.

5. Conclusion

Supposing import of large amounts of fine ores contain-ing crystallized water, ranging from 4 to 6%, occurred fromAustralia in near future, an advanced granulation model

was proposed to use these ores effectively in the sinteringprocess. The granulation model is based on the probabilitytheory assuming the granulation can be regarded as a kindof probability phenomena. Simulation results using appro-priate granulation parameters basically agree with thosefrom the granulation operation. Further prediction methodof the granulation parameters used in the model is quite im-portant for improving the precision of the granulation simu-lation.

Nomenclature

B : Square matrix of (n, n) composed of elements; qij

B� : Square matrix of (n, n) (�{(1�p) ·E�p ·B}N)E : n�n unit matrixF : Column vector showing initial size distribution of

oresG : Column vector showing size distribution of

quasi-particlesfi, gi : Elements of weight ratio at the sieve mesh, i-th,

constituting F and G (�)qij, q�ij : Granulation and disintegration probability

elements constituting matrix B and B� (�)pi : Transition probability based on Markov chain

theory (�)r : Granulation and disintegration probability

elements constituting matrix B2 (�1.0/n(�))w : Weight ratio of mixing ore in piled ores (�)p : Overall granulation probability of ores (�)

Subscriptsi, j : Sieve mesh counting from minimum size as

described in Figs. 3 and 4N : Rotation times of drum mixer or disc-pelletizer

(�)n : Division numbers of ore/quasi-particle sizes (�)

(n�1) : Numbers of sieve mesh (�)

REFERENCES

1) S. Kamikawa and K. Nagano: CAMP-ISIJ, 11 (1998), 86.2) K. Yamaguchi: Bull. Iron Steel Inst. Jpn., 4 (1999), 666.3) J. Okazaki, M. Nakano and Y. Hosotani: CAMP-ISIJ, 14 (2001), 187.4) T. Furui, M. Kawazu, K. Sugawara, T. Fujiwara, M. Kagawa, J.

Sawamura and N. Uno: Seitetsu Kenkyu, 288 (1976), 9.5) Y. Niwa, N. Sakamoto, O. Komatsu, H. Noda and A. Kumasaka:

Tetsu-to-Hagané, 78 (1992), 1029.6) T. Inazumi, M. Fujimoto, S. Kasama and K. Sato: Tetsu-to-Hagané,

77 (1991), 63.7) H. Rumpf: Agglomeration, AIME Inter-science Publishers, John

Willey & Sons, New York, (1962), 379.8) N. Ouchiyama and T. Tanaka: 5th Int. Symp. on Agglomeration, The

Inst. of Chem. Eng., RUGBY, (1989), 577.9) S. Suzuki, M. Fujimoto and K. Sato: Tetsu-to-Hagané, 73 (1987),

1932.10) J. D. Lister, A. G. Water and S. K. Nicol: 5th Int. Symp. on

Agglomeration, The Inst. of Chem. Eng., RUGBY, (1989), 33.11) J. Kano and E. Kasai: Examination Committee on Agglomeration of

Iron Ores Unsuitable for Sintering, ISIJ, Tokyo, (2001), 23.12) S. R. Broadbent and T. G. Callcott: J. Inst. Fuel, 29 (1956), 524.13) M. Fujimoto, K. Sugawara, T. Furui and R. Shimizu: Tetsu-to-

Hagané, 61 (1975), 48.14) S. Lipschutz: Probability, McGRAW-HILL-KOGAKUSHA, Tokyo,

(1981), 166.15) M. Suzuki, H. Sato, A. Sakai, H. Noda, T. Takai and S. Kishimoto:

CAMP-ISIJ, 7 (1994), 995.16) C. O. Bennett and J. E. Myers: Momentum, Heat, and Mass Transfer,

McGRAW-HILL BOOK COMPANY, New York, (1974), 176.


843 © 2002 ISIJ

iron ore granulation model supposing the granulation

Documents