Similarity size distribution of agglomerates during their growth by coalescence in granulation or green pelletization

International Journal of Mineral Processing, 2 (1975) 187--203 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

SIMILARITY SIZE DISTRIBUTION OF AGGLOMERATES DURING THEIR GROWTH BY COALESCENCE IN GRANULATION OR GREEN PELLETIZATION

KALANADH V.S. SASTRY

The Hanna Mining Company, Research Laboratory, Nashwauk, Minn. (U.S.A.) (Received April 28, 1974; revision accepted January 23, 1975)

ABSTRACT

Sastry, K.V.S., 1975. Similarity size distributions of agglomerates during their growth by coalescence in granulation or green pelletization. Int. J. Miner. Process., 2: 187--203. A detailed investigation of the granulation or green pelletization of particulate materials by the mechanism of coalescence is presented. The main considerations included testing the validity of random coalescence model and proposing of a nonrandom coalescence kernel. Mathematical analysis of the coalescence process indicated that there exists a selfpreserving, pseudo time-independent agglomerate size distribution, which is also independent of the material properties and process conditions. Experimental observations on the size distribution of agglomerates produced from a wide variety of materials were examined for the validity of the proposed coalescence models. INTRODUCTION

The agglomeration of particulate materials by green pelletization or granulation is found to take place through the simultaneous occurrence of such mechanisms as nucleation, coalescence and snowballing (Sastry and Fuerstenau, 1973). In continuous processes the seed generation takes place by the nucleation of added feed into well-formed species followed by coalescence of these well-formed species to form stable seeds. These seeds further grow by snowballing with the added feed (Sastry and Fuerstenau, 1973). In contrast, coalescence' is observed to be the most important mechanism responsible for growth in a batch system (Kapur and Fuerstenau, 1964, 1969; Sastry and Fuerstenau, 1973). This is particularly true in the initial stages of ball formation and growth of all kinds of particulate materials and almost throughout the batch bailing of materials with naturally comminuted and wider size distributions. A detailed analysis of pellet growth by coalescence in a batch system is desirable since it makes it possible to establish the typical characteristics of the process. This basic understanding will be useful in the future investigations and development of mathematical models of bailing circuits. A general population balance model has been developed for the agglomeration process (Sastry and Fuerstenau, 1975). This model, modified for the ball

188

growth by the isolated mechanism of coalescence, will be used in the present investigation. Further, it will be assumed that the nucleation of the moist feed has taken place in the first few drum revolutions. Accordingly, the following analysis is applicable for the coalescence of well-formed species to which no addition of new feed is made. The main purpose of this paper is to analyze typical trends in the agglomeration process by coalescence and to discuss the nature of coalescence efficiency parameters and the resulting pellet size distributions. COALESCENCE EQUATION

For green pellet growth, in a batch system, by the isolated mechanism of coalescence, the population balance model is given by (Kapur and Fuerstenau, 1969; Sastry and Fuerstenau, 1975): bn(m,t) • bt

1 N(t) +

f j

. , , , X (m,m,t)n(m ,t)n(m--m ,t)dm'

o

1

(1) m

'

,

,

X*(m ', m - - m , t)n(m , t ) n ( m - - m , t ) d m

,

o

where n(m, t)dm is the number of agglomerates in the mass range m to (m +din), at time t and N ( t ) is the total number of agglomerating species and is defined by: oo

N(t) = f 0

n(m,t)dm

(2)

.J

Further, X*(m,m', t) denotes the binary collision--coalescence rate function (or simply the coalescence kernel) for the agglomerate pair with sizes m and m'. This coalescence rate function is expected to be dependent on the particulate material being agglomerated and on the operating conditions, besides the size of the species involved in the coalescence event. THE COLLISION--COALESCENCE RATE FUNCTION

In order to solve eq.1 for n(m, t) it is necessary to know the mathematical form of the coalescence kernel X*(m,m ', t). However, it is difficult to develop any meaningful expression for the coalescence from fundamental considerations because of the complex nature of the coalescence process during the agglomeration of particulate materials. Also, even if a mathematical form is available for X* the non-linear integro-partial differential equation given by eq.1 cannot, in general, be analytically solved. For such reasons, Kaput and Fuerstenau (1969) proposed that the coalescence efficiency is size-independent (i.e., the coalescence of species occurs at random), and investigated such a random coalescence process.

189

The random coalescence model

Based on the size-discretized analog of eq.1, Kaput and Fuerstenau (1969) using the generating function technique (Hermans, 1965), obtained an expression for the size distribution of agglomerates. When all the species are initially of unit size, the solution is given by:

nk

N

N )k-~

(1--No

(a)

where n k is the number of species with mass mk. This mathematical form of n k is referred to as geometric distribution. It is evident from eq.3 that the unit-size species are the most abundant at all stages of the agglomeration process, or specifically nl ) n2 ) n3 ) • • • This is hardly the situation in an actual batch balling operation. The initial as well as other smaller size nuclei are depleted very fast by coalescing with larger size species. Thus, the random coalescence assumption does n o t truly represent an actual agglomeration process and other non random-coalescence kernels need to be developed. Functional form o f a non-random coalescence kernel

Based on experimental observations and intuitive arguments, some of the requirements to be satisfied by the coalescence function can be specified. The coalescence efficiency is a complex function of the particulate material properties and of process conditions such as water content and the size of the participating species. Furthermore, it is an u n k n o w n function of time, possibly either increasing or decreasing as pelletization proceeds. For example, growth and simultaneous compaction of granules tend to make them less deformable and result in retarded growth rate. At the same time, compaction expels water onto the granule surface where moisture is likely to lead to increased efficiency of coalescence. Combination of these t w o and other factors is expected to bring a b o u t changes in the coalescence efficiency. It is also expected that a true functional form for ~* should be written in terms of geometrical properties (such as volume and surface area) rather than mass. One essential simplifying assumption is that the apparent density of the species does n o t vary with size. Let the pellets of mass m and m' have volume v and v', and surface area s and s', respectively, at time t. The following properties are expected to be required of a coalescence rate function: (a) The coalescence kernel can be represented by a multiple of t w o functions, one time-dependent and the other size-dependent: ~ * ( m , m ' , t ) = ~*o ( t ) ~ ( m , m ' )

(4a)

The time-dependent function, X* (t), is regarded to take into account the dependence of coalescing efficiency on the operating conditions and the particulate material characteristics.

190

(b) The function is symmetric:

h(m,m') = h(m',m)

(4b)

(c) The efficiency of coalescence of two equal-size species decreases with increasing size:

),(m,m) <~. (m',m') if m > m'

(4c)

(d) Whenever an agglomerate encounters two larger-size species, it tends to coalesce with the larger:

h(m,m') < (m,m") i f m < m ' < m "

(4d)

(e) The rate of growth of large species by coalescence with extremely small-size nuclei is proportional to the exposed surface area of the large species, i.e.:

X(m,m')~s(m) as m' ~ 0

(4e)

and finally, (f) these relationships are equally valid when one considers the volumes v, v' of the participating species, and they can happen only if X(v,v') has similar functional form as ~,(m, m'). This requires homogeneity of the coalescence kernel.

An empirical coalescence kernel It may be possible to find several empirical functional forms satisfying the requirements stated above. One particular function is chosen in the following so as to indicate the type of analysis that can be made. In formulating this em. pirical coalescence kernel, it is considered that the driving force or potential for coalescence is determined by the surfaces, and that resistance for further deformation is offered by the volumes of the participating species. Thus, the larger the surface area of an agglomerate, the more potential it has to grow, while at the same time the more resistance it offers. Based on these ideas, the following functional form that satisfies most of the above described criteria is chosen:

~*(m,m',t) = ~o (t)(s + s')(l +1 )

(5)

where s and v are the surface area and volume of the pellets, respectively. Given p, the agglomerate apparent density, eq.5 becomes: t h (m,m,t)=Xo(t)(36np)l/3(m2/3 +m,2/3)(~_+1 m-~l)

(6)

Comparison of eq.6 with eq.4a gives:

~(m,m')=(m 2/3 + m ' 2 / 3 ) ( 1 + 1 )

(7)

191

and:

(8)

~*0 (t) = ~0 (t) (36np) 1/3 Note that h ( m , m ' ) given by eq.7 is homogeneous and has degree (--1/3). SIMILARITY DISTRIBUTIONS

Earlier, it was shown that the green pellet size distributions are of one parameter type (Sastry and Fuerstenau, 1972, 1975). It will be shown in the following that the green pellets produced by coalescence have a pseudo timeinvariant size distribution, given that the coalescence kernel is a homogeneous function of its arguments (Wang, 1966). A similarity theory proposed by Friedlander (1964) to account for similarities in the shapes of atmospheric aerosol spectra under coagulating conditions, was found to be applicable to a variety of systems (Swift and Friedlander, 1964; Hidy and Lilly, 1965). Recently, Wang made an excellent mathematical analysis of the similarity distributions in coagulating dispersephase systems (Wang, 1966; Friedlander and Wang, 1966; Wang and Friedlander, 1967). These studies indicate that the self-preserving spectrum is independent of the initial distribution (Wang, 1966), after sufficiently long times. The mathematical aspects of the above-cited work were found useful in the following analysis of pelletization by coalescence. Self-preserving similarity transformation

The p-th moment, pp, of the green pellet size distribution, which is required for further analysis is given by: n(m,t) mP - ~ dm

~p =

(9)

0 The first moment, P l, has the physical significance of the average pellet mass. Furthermore, it can be shown that N (O)/N (t) = Pl (t)/pl (0)

(10)

Slightly modified versions of the Friedlander similarity transformations are defined as follows: ,7 = m / u l (t)

(lla)

and:

¢(~)dv = n ( m , t ) d m / N ( t )

(11b)

where ,7 and ~ are the normalized, pseudo time-invariant size and density functions, respectively. If the general coalescence kernel h(m, m') is a homogeneous function of its arguments (Wang, 1966) of, say, degree 5, then:

192 ~ ( m , m ' ) = u +, ~(,7,~')

(12)

By applying transformation variables V and ¢ given by e q . l l to eq.1 and making use of eqs. 4 and 12 for the coalescence rate function, one obtains:

where: J~

=

f+ f+ X ( , , , ' ) ¢ ( ? ) ¢ ( , ' ) d , T d , '

0

0

(14a)

and: J~ (~) = 0

X(~,r~')¢(~')dr~'

(14b)

The equations for the rate of change of total number of pellets in the system and their average pellet diameter are obtained by integrating eq. 1 and making use of eqs.4, 10 and 12: ldN

1 dpl pt d t

N dt

_)t0*(t) 2 J~Pl-5

(15)

These transformations indicate that, if the coalescence kernel is a homogeneous function of its arguments, then the coalescence process has a selfpreserving type of size distribution. It is noted that such a self-preserving distribution may be only a particular solution and no p r o o f of its uniqueness is known. The p-th moments of the similarity distribution, Op, which are characteristic constants (time independent) of the self-preserving size distribution, are given by: P

oo

?P¢('7)d? = p p p l -p

Vp = °j

(16)

Eq.16 is obtained using the definition of pp given by eq.9 and then performing similarity transformations ( e q . l l ) . The r a n d o m coalescence process

The similarity solution for the random coalescence, i.e., when ~,* is a constant, can be easily obtained. It can be shown that eq.13 becomes, for constant ~ : r/-d~ ~ +

f'

O(r/')~(r/--rl')dT?' = 0

(17)

0

The solution of this equation, which can be found by Laplace-transformation technique, is: ¢07) = e -~

(18)

193 By using a moment-comparison technique, Wang (1966) showed that the solution given by eq.18 is independent of initial conditions. The geometric (eq.3) and negative exponential (eq.18) distributions are equivalent (Feller, 1966) except that the first refers to discrete-size variables and the latter to the continuous. The non-random coalescence process

In this section, the proposed empirical coalescence kernel (eq.6) will be applied to the generalized similarity equations (eqs.13 and 14). Since the coalescence kernel given by eq.6 is a homogeneous function (of degree --1/3) it is possible to perform a similarity transformation on eq.1. The size-dependent coalescence equation, with the empirical coalescence kernel, is eq.13 with: X(7,~') = (72/3 + r/'2/3 ) ( 1 + 1 ) j ~ = [7-1/3 +~2/3 v-1 +7 -1 v2/3 +v_1/3]

(19) (20a)

and: J~

= 2 [v2/3 v_ l + v_1/3 ]

(20b)

Some implications o f the empirical coalescence kernel If the proposed similarity transformations are valid, i.e., the coalescence process does have a self-preserving size distribution, then from eq.15:

dUl _ ~*o(t) - Ul 2/3 dt 2

j~.

(21)

where J ~ is given by eq.20b and is a constant for the self-preserving size spectrum. From eq.16, we have ttl/3 = vl/3 pl 1/3 ; or: dpl/3 1 dpl dt" =-~v113 p1-213 dt

(22)

Combining eqs.21 and 22: duu3= k~*o (t)/6 dt

(23)

where: k = vl/3 J ~

(24)

Given p(t), the apparent density of the species at time t, one obtains for the average diameter D :

194

~P 1/3 = (--6--) D

ut/3

(25)

or:

dplp = [~_~_pl/3 + 1 -- -2'3 dp dt

~ )1/3 5 D p 1 ~-~ ] (g

dt

(26)

From eqs.23 and 26, after substituting eq.8 for X*0 (t), one obtains:

dD dt

4

1 D dp 3 p dt

-

k X0 (t)

(27)

If the changes in apparent density are negligible compared to changes in average diameter (Kapur, 1967), then eq.27 can be approximated to give: dD dt

-

X0(t)

k

(28)

where k is given by eq.24 and is a characteristic constant of the chosen coalescence kernel. Accordingly the result given by eq.28 becomes true whenever the coalescence kernel is homogeneous with degree (--1/3). The result given by eq.28 has the important significance that a differential growth plot of D as a function of t represents the time-dependence of the coalescence efficiency [X0 (t)]. In Fig.1 the differential growth curves for taconite pellets are presented for three moisture contents. It is observed from this figure that the coalescence efficiency increases initially with time, reaches a maximum, and starts to fall off. The original concept of batch pellet growth

1"21 [ ilOl-

~ / /

I

I

I

\10.8 % WT. WATER \

TaCoNrrE

~0.2 0

I000

2000

3000

NUMBER OF DRUM REVOLUTIONS, t

Fig.1. Variation of green taconite pellet growth rate with time, which corresponds to the time-dependence of the coalescence efficiency, X0 (t).

195 in three regions (Kapur and Fuerstenau, 1964), known as nuclei, transition and ball growth, can thus be explained as a consequence of changes in coalescence efficiency. The maximum for the coalescence efficiency corresponds to the so-called transition region, where the rate of pellet growth is indeed highest. As explained earlier, the increase in ~0 (t) with time is due presumably to the increasing appearance of water on the granule surface which dominates the decrease in coalescence efficiency due to less deformability. After the transition region, the pellets are not compacted any further, and no additional water appears on the surface. In fact, water is withdrawn into the pellet interior. These facts cause a decrease in h0 (t) b e y o n d the transition region. The differential growth curves presented in Fig.1 were found to be typical of such other polydispersed particulate systems as pulverized limestone (Kaput and Fuerstenau, 1964) and ground magnesite (Spalden et al., 1970). This is in contrast to the pelletization of closely-sized materials, where the growth rate was found to be constant (Newitt and Conway-Jones, 1958; Capes, 1964; Noack, 1969). Moreover, during the agglomeration of closelysized sands porosity of the granules does not show changes with time (Capes, 1964). This indicates that no increased amounts of water appear on the granule surface and the coalescence efficiency X0 remains constant.

Numerical solution The nonrandom coalescence eq.13, with the kernel given by eq.19, cannot be solved analytically for the similarity-size distribution of ~(~). Even the numerical solution of eq.13 is not straightforward because of the u n k n o w n but constant normalized m o m e n t s v_~, v_ 1/3 and v2/3. Under these conditions, it was found useful to solve the nonrandom coalescence equation in the time domain using eqs.1 and 7. This procedure has an added advantage in that the validity of the self-preserving form can be tested. The numerical solutions were obtained by expressing eq. 1 in its corresponding size-discretized form, and then forming an infinite set of simultaneous non-linear first-order differential equations. In order to facilitate reasonably faster computations, it was assumed that a total of 400 equations are sufficient to represent the infinite set. In fact, later computations i n d i c a t e d even 100 equations would have been sufficient. These 400 equations were solved for several arbitrary initial conditions including that for all the species of unit size. A fourth-order Adams Moulton predictor-corrector m e t h o d was used for solving the set of differential equations. The starting procedure is based on Zonneveld's formulations (Zonneveld, 1964) and is of Runge-Kutta type, but provides an estimate of truncation error at each step. The step size of the independent variable is automatically set b y minimizing the error at each step. This procedure is referred to b y the name ZAM (Meissner, 1969) and is available as a standard library routine. The numerical solutions were obtained on a CDC 6400 computer at the University of California Computer Center, Berkeley.

196

1.0

I

I

I

RANDOM----

-o-

£0.8

0

0

I-I1: h 0.6 n,. t=J m

0

#

Q

,~

~, \

--

NONRANDOM Z~ 0

:E a

0

0

ILl N ...I

2 3

A

4

0

\ \

z 0.4--

v 0

6

D 0

8

\\°o

xo

~: 0.2--

\o

0

I1: 07

0 0.05

I 0.1

I 0.2

I

I

0.5

I.O

I %~',~--1.~ 2.0

50

I0

NORMALIZED MASS, t] Fig.2. Numerical solution for the normalized size distributions under random and nonrandom coalescence conditions for various values of ® = N / N o.

At first, the random coalescence equations were solved using the ZAM technique and were found to be in extremely good agreement with the analytical solutions (eq.18). Then the nonrandom coalescence equations with the proposed empirical kernel were solved by using the same technique. The numerical solutions for the random and nonrandom coalescence kernels are presented in Fig.2. It is noted that, for nonrandom coalescence, the agglomerate size distributions approach a self-preserving form for 0 ( = N / N o ) > 4.0 and become unimodal. EXPERIMENTAL SIZE DISTRIBUTIONS OF GREEN AGGLOMERATES

For comparison of the theoretical predictions regarding the self-preserving nature of the agglomerate size distribution and the validity of the empirical coalescence kernel, size distribution data on pellets made from different particulate materials are used. The determination of the green pellet size distribution by the photographic-counting technique yields relationships between diameter and number fraction. Since it was shown that the pellet size distributions are self-preserving and of one-parameter type, use of any one of the moments is valid. Thus, as a matter of convenience, the average pellet diameter D(t), is used as a normalizing parameter. The normalized size (~) is defined as: = diD

(29)

where d is the diameter of the agglomerate. Furthermore, by the properties

197 of the cumulative number density function, the following expressions are valid:

Y(m,t) = Y(d,t) = ~07) = q~(~)

(30)

where Y(d, t) and ~ (~) are the cumulative number fraction of pellets smaller than size d at time t, and normalized size ~, respectively. In Fig.3 the taconite pellet size distributions from different batch experiments that have been reported elsewhere (Sastry and Fuerstenau, 1972) are presented in the normalized form. This figure indicates that the taconite pellets possess a remarkable self-preserving size distribution, truly independent of the conditions under which they were made. I.C

y-

bJ

z_ u. 0 . 8 z

0

--

/~"

------ ), CON~r&NT x EO. 19

o 0.6 ¢

",,n ~-~z0.4 -

(~l

TACONITE

_~7

10.8% wl:. WATER

BENTONITE

.~

->"'

,~

~WT.

,~

~,~ 0.2j J/

0

,.3

/~

0

0.5

0.9

y

,'~

1.0

8.4

~

I. 5

6.4

o

/~

_,# l

0

0.5

OIA,.,,~

o

,~./ ,,

e,D

--

AVE.

{

o~

_

e.z

I

1.0 1.5 2.0 NORMALIZED D I A M E T E R , {

I

2.5

3.0

Fig.3. Self-preserving size distribution of taconite pellets. This figure also provides a comparison between the experimental results and the random and nonrandom coalescence models. Using the batch-agglomeration size distribution data available, similar normalized size distribution plots are made for pulverized limestone (Kapur, 1967), magnesite (Spaldon et al., 1970) and cement copper (Noack, 1969), and are presented in Figs.4, 5 and 6. These figures further support the concept of self-preserving green-pellet size distributions. It is to be mentioned that cement copper is a closely-sized material. It was not possible to check if closely-sized sands also have this property as neither Conway-Jones nor Capes provided detailed size distributions in the coalescing region of pelletization. Normalized pellet size distributions for taconite concentrates, pulverized limestone, magnesite and cement copper are plotted together in Fig.7. It is seen from this figure that the pellet size distributions are independent of the material characteristics.

198 I.C n," tel Z " 0.8-Z 0 I.(.D

0

O.E--

,P

uJ In

PULV.

~_ 0 . 4 - -

WATER % WT.

,,P

AVE. DIAM. mm

15.0 15.0 16.5 16.5 17.3 17 3

0

~ 0.2

0

0

LIMESTONE

3.8 7.0 1.0 5.4 3.1 9.3

l I I 1.0 1.5 2.0 NORMALIZED D I A M E T E R , I[

I 05

I 2.5

3.0

I 2.5

3.0

Fig.4. Self-preserving size distributions of limestone pellets. 1.0

l

l

l

~ l ~

1

W Z

0.8

l-it" 016 b. nr MAGNESITE

tn :E :) 0.4 Z IAJ

/

~ 0.2

0

0.5

WATER %WT. • • 0 • rl 0 A

14.0 14.0 14.0 13.0 13.0 120 12.0

AVE. ~ A M . , mm

7.1 I1.1 12.5 5.4 9,0 2.2 8.7

1 I 1.5 2.0 1.0 NORMALIZED D I A M E T E R , ~¢

Fig. 5. Self-preserving size distribution of magnesite pellets.

A comparison between the experimental and theoretical size distributions is provided in Fig.3, where both random and nonrandom coalescence size spectra are included. The agreement between the experimental and nonrandom coalescence model size distributions is fair up to 0.8 cumulative fraction. In contrast the random coalescence model does n o t represent the lower end of the experimental size spectrum adequately.

199

, oOO o~

1.0

~" 0.8 Z 0 I-0

O r-I

o

%

-- O.E tL n,* t~ O

3E

8 C E M E N T COPPER

~ O.4 W >

WATER %wT.

F-

~

4

6:b ° o A o o

Ir W Z

o

02

0 I"1

&

~D O

0

0.5

AVE.

OIAM., mm

7.8 7.8 7.5 7. 5

8.5

12.0 4.5 6.0

I I I 1.0 1.5 2.0 NORMALIZED DIAMETER,

I 25

3.0

Fig.6. Self-preserving size distribution of cement copper pellets.

'

1.0

I

I'1

I

'

a,LIJ Z

"

0.8

o O°

Z 0 0

~ 0.6 U.

01•

MATERIAL

~Z 0.4 W >

iX

~ 0.2

r'l O

i=t 0

0.5

AVE. DIAM., mm

TACONITE

6. I

LIMESTONE

6.6

MAGNESITE

6.5

C E M E N T COPPER

I I I 1.0 1.5 2.0 NORMALIZ ED D I A M E T E R ,

--

_

6.0

I 2.5

3D

Fig.7. N o r m a l i z e d size d i s t r i b u t i o n f o r different particulate materials showing the c h a r a c t e r i s t i c size s p e c t r a o f green pelletization.

DISCUSSION

It has been shown that the green pellet size spectra possess a self-preserving form and are independent of the material being agglomerated. Since these observations and the overall pellet size distributions are in agreement with

200

the results predicted by the coalescence models it may be concluded that various assumptions made in the mathematical model are valid. The basic assumption concerning the variable separable representation of the collisioncoalescence rate function (eq.4a) is the key to the present success, since it makes it possible to consider agglomeration by coalescence as a geometrical phenomena. It should be remembered that expressing the coalescence rate function by eq.4 and requiring its homogeneity lead to the conclusion that the agglomerate size distributions are self-preserving but the kinetics are different as reflected by h0 (t) in eq.28 (also see Fig.l). Even though the empirical coalescence kernel proposed in this paper reasonably represents the experimental pellet size spectra, it is expected to be useful in providing insight to the coalescence process in green pelletization. This is due to the mathematical complexities involved in solving the equations. However, from a viewpoint of practical application, it is suggested that the random coalescence model be used after making appropriate corrections. We are in the process of developing such a model. At this stage, it is necessary to recognize the fact that pellet growth by coalescence is a sufficient condition for observing the self-preserving size spectra and not necessarily the only one. For example, Capes and Danckwerts (1965) have shown that the breakage-and-layering mechanism causes granule size distributions to become self-preserving and they have developed a theoretical equation for such distributions. Recently, Capes (1967) has plotted the limestone pellet size distribution data (in the nuclei region where the pellet growth occurs essentially by coalescence) of Kapur and Fuerstenau (1966) and argued that it may be represented fairly closely by the theoretical expression for breakage and layering. However, Capes (1967), in his note, concluded that the overall behavior of pellet size distribution may be misleading to ascertain the growth mechanisms. In fact, according to the Capes and Danckwerts (1965) size distribution equation, the maximum number of pellets are in the region near the minimum or maximum size depending on the ratio of maximum to minimum size. But from the green pellet size distributions presented in Figs.2 through 7 (which include the data of Kapur and Fuerstenau) it is seen that the mode occurs somewhere near the median of the distribution. Thus it is noted that the self-preserving pellet size spectra shown in Figs.2 through 7 could not have resulted from breakage and layering mechanism. Based on the agreement between the theoretical and experimental size distributions (Fig.3) we come to the conclusion that the nonrandom coalescence model presented in this paper sufficiently describes the green pelletization in a batch system. In Fig.8 normalized size distributions of green pellets are plotted at various stages of a taconite pelletization experiment. It can be observed from this figure that the pellets have a self-preserving form up to 800 revolutions and for 1,600 and 3,200 revolutions the size spectrum becomes wider. Referring back to Fig.l, it can be seen that, for this experiment, beyond 1,000 revolutions the coalescence efficiency is in the declining stages, and presumably other mechanisms such as breakage and abrasion transfer come into action,

201

I.G •

t~ z

" 08 o (J h

e o

w

•~

z w

TACON ITE 10.4% WT. wATER, G5%WT. 8ENTO.

(~

I--

~ a2

~,

NO. OF

AVE.

0

REV. 50

OL4M~ n~ 0.14

o ¢

I00 400 800

0.4 1.4 4.5

3200

12.I

•

4~0

P 0

0.5

I

I

1600

7.9

I

1.0 1.5 2.0 NORMALIZED DIAMETER,

I 2.5

3.0

Fig.8. N o r m a l i z e d size d i s t r i b u t i o n o f t a c o n i t e pellets at various stages o f g r o w t h in a b a t c h pelletization experiment.

leading to wider size distributions. However, the differences in the size distributions observed in Fig.8 do n o t seem to be very important, at least from a practical viewpoint, and suggest that the self-preserving size spectra of green pellets is a characteristic of batch agglomeration. In fact, it appears very reasonable to expect a characteristic size distribution for the growing pellets as a result of their tendency to pack optimally and change their sizes accordingly. SUMMARY AND CONCLUSIONS

In this paper, we have presented a detailed analysis of agglomerate growth by the isolated mechanism of coalescence. The collision--coalescence rate function was represented as a multiple of two functions; one that is dependent on the size of the coalescing agglomerates and the other dependent on material characteristics and operating conditions prevailin~ during the bailing process. This assumption required that the green pellet size distributions will be characterized by the coalescence process alone and n o t be influenced by the material properties and operating conditions. The experimental results were found to be in agreement with this condition. Based on experimental observations and physical arguments various conditions that are to be satisfied by a collision--coalescence efficiency function were discussed and an empirical function was proposed. Under the condition that the coalescence function is homogeneous, it was shown that the green pellet size distributions possess a similarity (also known as pseudo time-invariant) distribution

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indicating that the distributions were of a single parameter type. The suitable normalizing parameter was observed to be the average pellet mass or the average pellet diameter. Experimental agglomerate size distributions from the batch bailing of taconite concentrates, limestone and magnesite powders, and cement copper were found to be normalizable and in agreement with the theoretical conclusions. Thus, it is concluded from the present analysis that there exists a unique size distribution of agglomerates in a batch pelletization system, independent of the material and processing conditions. This observation is of significant importance in developing further mathematical models of continuous agglomeration circuits. For instance, the seed size distribution can be generated when once criteria for establishing the average seed diameter is known and the additional growth of agglomerates by snowballing and changes in the overall pellet size distributions can be calculated. Work is currently under progress presently on the computer simulation of continuous balling circuits by using these concepts. ACKNOWLEDGEMENTS

This research was carried out at the University of California as a part of the Ph.D. dissertation. The author wishes to acknowledge the financial support by the American Iron and Steel Institute. REFERENCES Capes, C.E., 1964. The Formation of Granules from Powders. Thesis, Churchill College, Univ. of Cambridge. Capes, C.E., 1967. Mechanism of pellet growth in wet pelletization. Ind. Eng. Chem. Process Design Develop., 6: 390--392. Capes, C.E. and Danckwerts, P.V., 1965. Granule formation by the agglomeration of damp powders. Part II: The distribution of granule sizes. Trans. Inst. Chem. Engr., 43: T125--T130. Feller, W., 1966. Introduction to Probability Theory, I. Wiley, New York, N.Y., 2nd ed. Friedlander, S.K., 1964. The similarity theory of the particle size distribution of the atmospheric aerosol. Aerosols, Physical Chemistry and Application. Proc. 1st Natl. Conf. on Aerosols, Liblice, October 8--13, 1962, Czech. Acad. Sci. Friedlander, S.K. and Wang, C.S., 1966. The self-preserving particle size distribution for coagulation by Brownian motion. J. Colloid Interface Sci., 22: 126--132. Hermans, J.J., 1965. Molecular weight distributions resulting from irreversible polycondensation reactions. Makromol. Chem., 87: 21--31. Hidy, G.M. and Lilly, D.K., 1965. Solutions to the equations for the kinetics of coagulation. J. Colloid Sci., 20: 867--874.. Kapur, P.C., 1967. Kinetics of Wet Pelletization. Thesis, College of Engineering, Univ. of California, Berkeley. Kapur, P.C. and Fuerstenau, D.W., 1964. Kinetics of green pelletization. Trans. AIME, 229: 348--355. Kapur, P.C. and Fuerstenau, D.W., 1966. Size distributions and kinetic relationships in the nuclei region of wet pelletization. Ind. Eng. Chem. Process Design Develop., 5: 5--10.

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Kapur, P.C. and Fuerstenau, D.W., 1969. A coalescence model for granulation. Ind. Eng. Chem. Process Design Develop., 8: 56--62. Meissner, L.P., 1969. ZAM (Zonneveld-Adams Moulton), No. D2-BKY-ZAM, Computer Center Library, Univ. of Calif., Berkeley. Newitt, D.M. and Conway-Jones, J.M., 1958. A contribution to the theory and practice of granulation. Trans. Inst. Chem. Engr., 36: 422--442. Noack, H., 1969. Aglomeracion densificacion de cementos de cobre. Thesis, Univ. Tecn. del Estado, Santiago, Chile. Sastry, K.V.S. and Fuerstenau, D.W., 1972. Ballability index to quantify agglomerate growth by green pelletization. Trans. AIME, 252: 254--258. Sastry, K.V.S. and Fuerstenau, 1973. Mechanisms of agglomerate growth in green pelletization. Powder Tech., 7 : 97--105. Sastry, K.V.S. and Fuerstenau, D.W., 1975. Population balance models in the agglomeration of particulate materials by green pelletization or granulation, in preparation. Spaldon, F., Sastry, K.V.S. and Fuerstenau, D.W., 1970. Green pelletization characteristics of magnesite powders. (unpubl. data) Swift, D.L. and Friedlander, S.K., 1964. The coagulation of hydrosols by Brownian motion and laminar shear flow. J. Colloid Sci., 19: 621--647. Wang, C.S., 1966. A Mathematical Study of the Particle Size Distribution of Coagulating Disperse Phase Systems. Thesis, Calif. Inst. of Technol., Pasadena. Wang, C.S. and Friedlander, S.K., 1967. The self-preserving particle size distribution for coagulation by Brownian motion. II. Small particle slip correction and simultaneous shear flow. J. Colloid and Interface Sci., 24: 170--179. Zonneveld, J.A., 1964. Automatic Numerical Integration (MCT-8). Mathemat. Centrum, Amsterdam.