Rightmire, R. A., Jones, A. L., in Proceedings of 21st Annual Power Sources Conference, PSC Publications Committee, Red Bank, N. J., 1967. Seiger, H. N., Charlip, S., Lyall, A. E., Shair, R. C., in Proceedings of 21st Annual Power Sources Conference, PSC Publications Committee, Red Bank, N. J., 1967. Shimotake, H., Cairns, E. J., CITCE Meeting, Detroit, Mich., September 1968. Shimotake, H., Cairns, E. J., Electrochemical Society Meeting, Dallas, May 1967, Abstr. 143; Extended Abstrs. Ind. Electrolytic Diu. 3,4 (1967a). Shimotake, H., Cairns, E. J., Intersociety Energy Conversion Engineering Conference, Miami Beach, Fla., August 1967 ; “Advances in Energy Conversion Engineering,” p. 951, American Society of Mechanical Engineers, New York, 1967b.
Shimotake, H., Rogers, G. L., Cairns, E. J., Electrochemical Society Meeting, Chicago, October 1967, Abstr. 18; Extended Abstrs. J - l Battery Diu. 12, 42 (1967). Tantram, A. D. S., Tseung, A. C. C., Harris, B. S., in “Hydrocarbon Fuel Cell Technology,” B. S. Baker, ed., Academic Press, New York, 1965. Wilcox, H. A., in Proceedings of 21st Annual Power Sources Conference, PSC Publications Committee, Red Bank, N. J., 1967. RECEIVED for review April 22, 1968 ACCEPTEDJuly 15, 1968 Division of Fuel Chemistry, 155th Meeting, ACS, San Francisco, Calif., April 1968. Work performed under the auspices of the U. S. Atomic Energy Commission.
A COALESCENCE MODEL FOR GRANULATION P. C. KAPUR AND D. W . FUERSTENAU
Department of Mineral Technology, Unioersity of California, Berkeley, Calif.
94720
In the nuclei and transition regions of granulation the pellets or granules grow in size by coalescence with one another. A random coalescence model has been proposed which characterizes the size distributions of pellets in these two regions. The interrelationships between the various parameters of the size distributions generated by the random coalescence model are in agreement with the experimental data for particulate systems of different surface areas and water contents.
far, granulation or wet pelletization kinetics and pellet Tsize . distributions have not been adequately treated. Such HUS
a description, apart from formalizing the postulated pellet growth mechanisms, is required for appropriate simulation and control models for industrial balling circuits. The first systematic investigation of the growth rates and size distributions of the pellets, made by agglomerating closely sized sands and sand-silt mixtures in a laboratory batch balling drum, was carried out by Newitt and Conway-Jones (1958). They postulated that the pellets grow by direct coalescence, but made no attempt to analyze the empirical size distributions in terms of this model. More recently, Capes and Danckwerts (1965a, b) studied the mechanism of pellet growth by a technique involving the granulation of two sands of contrasting colors. They concluded that, if sufficient water is present in the system, the small nuclei pellets grow in size by direct coalescence. This growth mode ceases when the pellets reach such a size that the torque tending to separate the “twin” formed at the instance of collision between the two pellets is too large to allow a permanent bond to be formed. Subsequent growth, according to Capes and Danckwerts, takes place by crushing of the smallest pellets in the rotating charge, the resulting fragments being distributed over the remaining pellets. Based on this preferential crushing and layering mechanism, they formulated a relationship for the size distributions of pellets, which is self-preserving in the sense that successive distributions become coincident when plotted in the specified dimensionless form. Capes and Danckwerts worked exclusively with comparatively coarse and closely sized sands and did not extend their analysis to comminuted poly-dispersed systems which are normally encountered in industrial pelletization operations. Using comminuted limestone as a model system, Kapur and Fuerstenau (1964) carried out detailed studies of pelletiza56
I&EC PROCESS D E S I G N A N D DEVELOPMENT
tion kinetics and concluded that the growth mechanism depends on the size of the growing pellets. Initially three-phase air-water-solid nuclei are formed which grow by random coalescence of the rotating pellets. When the nuclei are compacted to such an extent that the over-all porosity of the pellet is approximately equal to its water content, the integrity of the two-phase pellet is barely maintained by surface tension of the surrounding liquid film (Rumpf, 1962) and its pronounced dilatant behavior (Newitt and Conway-Jones, 1958). I n this transition region, the pellets are readily deformable and possess a high degree of surface plasticity. As a result, the growth mechanism retains its coalescence character. Indeed, the coalescence rate may be appreciably accelerated, particularly if the system is comprised of relatively fine particles. This may be seen in Figure 1, which shows pictures of conglomerated pellets formed at a very rapid coalescence rate in the transition region. Finally, in the two-phase water-solid ball growth region, where the comparatively large pellets are held together by negative capillary potential, the growth mechanism acquires a highly complex character based upon one or more of the following modes (Capes and Danckwerts, 1965a; Kapur and Fuerstenau, 1964; Newitt and Conway-Jones, 1958): random coalescence, preferential coalescence between balls of different sizes, surface abrasion and material transfer from one ball to another, and crushing and layering. Capes and Danckwerts found that, for 97-micron diameter sand, coalescence essentially ceases when the pellets grow to a size of the order of 6 mm. However, the size range over which the coalescence mechanism might prevail will depend, coupled with the dynamics of the balling device, on the strength, plasticity, and allied rheological properties of the pellet, which may vary with the degree of compaction of the particulate assembly as well as the thermodynamic potential (suction) of water in the pellet matrix.
ing the coordination number of a pellet surrounded by polydispersed spheresin acascading bed. Published empirical data (Capes and Danckwerts, 1965b; Kapur and Fuerstenau, 1966; Newitt and Conway-Jones. 1958) on the size distributions of pellets made fr'om widely different material systems show that most of the pellets tend to be distributed linearly with respect to curnulatit 'e size distribu~tion as a function of size. Further, the ratio 0.c *1-- I"..--". Lu smallest pellet size in the linear range is constant and does not exceed 3. Since there exists an approximate dimensional similarity between the pellet size distributions, it implies that the packing characteristics of pellets in the tumbling bed will not be appreciably altered with the agglomeration time. I n this connection, careful study of the photographs of the pellet bed through the front transparent cover of the ballins- drum shows that th,e pellets are well mixed, and as far as could be determined, t here is no segregation; irrespective of size, the pellets are coordinated by five to seven nearest neighbors in the plane of the p,icture. It is then re asonable to conclude that, in a pellet assembly of z x 2 fir.+ snnmuimslt~nn thn l.llll.l._l.., ...".-.rr ._I.....__Y.. relatively nan..PAW &TP rlicner~inn number of collisions undergone by pellets belonging to any size group, i, is given by: LLlr
..
Figure 1. Conglomerates of pellets formed b y rapid coolercence in the transition region 0.71 rq. meterlgram limestone with 50.1 volume % water. 3 X
Kapur and Fuerstenau (1966) analyzed the kinetics of nuclei growth in terms of a fixed rate random coalescence hypothesis. They derived a semiempirical relationship between a characteristic parameter of the size distribution and the agglomeration time. This model has limitations, in that it provides no information an the hizher moments of the size distribution. and moreover. because of the h e d rate constant assumption, the motle1 tends to break down f or powders of comparatively large specific surface areas. Our present work provides an alternatwe approacn, Iormuiated a,n the basis of random coalescence with a time-variable coalesl:ence rate. This model leads to the conclusion that the size d istributions of pellets are selfpreserving and unique functionit1 relationships exist between the parameters of the distributi ons irrespective of the finenf 1 - r w a t m rnntr -___ :nt, coalescence rate, and ag-.the n=-.a.~ t i r_._", glomeration time. .
.._""..._
.
?
I
~
Random Coalescence Model
Appearance-DisappearaneeKinetics Equation. T h e pellets in a balling device, such as the balling drum, tumble in loosely packed layers with continuous intermking of the pellets between the layers. At any instant, the number of pellets coordinated around any given pellet is independent of the total number of pellets in the drum. I n other words, any increase in the mass of the granulating charge will simply lead to a proportional expansion of the pelletizing zone of the drum. in so far as the averaee collision freauency of pellets is concerned. At the instant of collision between tiNO pellets a twin is formed, and there is some probability that tllis unstable configuration will be sheared apart again by the rest of the moving mass, and as such, the surface of a pellet is n ot a perfect absorber. n-$L.- L
^^
* I . -
.-..I-..-
^C ..alle*"
.._
-
l . l l
I
k n&)
St
where n&) is the number of pellets of size V , uy vulume. AT time zero, all nuclei pellets are taken to be monodispersed with normalized size of unit volume. This of course means that a pellet of size V , contains i units of the original nuclei formed a t time zero. Under the constraints of perfect mixing and completely random collisions between the pellets, the mean number of collisions between oellets belonrinr to anv two GP groups i a
where N ( t), is the total number of pellets in the system a t time t. Any given collision has only a finite probability, ,9(t), of ending ir1 a successful coalescence. This urobabilitv, is an unknown function of the intensity of the collision, porosity of the pelleit, its size, deformability, and surface plasticity or stickiness, because these factors determine the strength of the I . >...:- II. .... :- . ._c *.~.~~~.-, ~ ~ _ bond a t tuc IICLK 111 u c ~ w i i icviiiigurauuri IULIIISU ar. me mStant of collision. I n this sense, the probability term, P ( t ) , incorporates some measure of the rheological structure of the pelleit which may change with the agglomeration time. In1
i =1
(11)
A set of simultaneous equations such as Equation IO, written in terms of all pellet sizes, is readily solved by the use of generating function, F(x,t), defined as:
Multiplying Equation 10 by x' and summing the resultant expression over all pellet sizes gives
and
Here K l ( t ) corresponds to the mean, K&) is the variance of the normalized pellet size distribution, and K&) is its third central moment. By repeating this procedure it is possible to calculate all higher moments of the distribution. Further, the solution of Equation 23 is:
Kl(t) = KI(0) e x p [ M ) I
(26)
Solving Equations 23, 24, and 25 gives the interrelationships among the first, second, and third cumulants, or Here, the term [ (F(x,t)I2arises from the convolution property of the term under the summation sign of Equation 10. Since the initial condition from Equation 11 in terms of the generating function is
F(x,O)
= x
(14)
the solution of Equation 13 is obtained by straightforward integration, or
where
Kz(t) =
[Kl(t)12
- Kl(0) K l ( t )
and
K 3 ( t ) = 2 [ K l ( t )l3 - 3K1(0)[ K d t )l2
(27)
+ [K1(0)I2K d t )
(28)
At this stage it is necessary to take cognizance of the fact that pellets in the nuclei stage are compacted continuously, and consequently the pellet size scale must be suitably modified. I n Figure 2 the change in pellet porosity is shown as the function of agglomeration time for some representative systems. Thus, a typical correction term will range from 1 at time zero to about 0.6 at the most. The theoretical normalized pellet size, Vt, may be related to a measured normalized size, Vi,,by an empirical function, P ( t ) , or
(16)
V' =
vi,P(t)
i
io P ( t )
and hence By expanding both sides of Equation 15 in polynomials of x, the desired expression for the number fraction of pellets can now be extracted by equating the coefficients of x i , or
fdt)
=
11 - e x ~ [ - + ( t ) I } ~ e- x~ p [ - d t ) I
(17)
I t can be readily checked that (Jolley, 1961)
Cfdt)= e x p [ - d t ) I
il =
58
i=l
11
- e ~ p [ - # J ( t ) l } ~ - ' =1
l&EC PROCESS D E S I G N A N D DEVELOPMENT
Here, it has been assumed that all pellets in a size distribution are compacted to the same extent. Incorporating this correction term in Equation 20 gives log
m
m
(18)
=
zio(t) = io ~
( tlog{ ) 1 - exp[-+(t)l
1
(31)
A similar correction can be introduced into Equations 26, 27, and 28.
PULV. LIMESTONE 0.96
FLOC POROSITY % v WATER 8
A 0
40.6 44.6 462
AVE. N U C L E I DIAM.
50 -
40
I 20
I
40
I
I
I
100 200 400 1000 NUMBER OF DRUM R E V O L U T I O N S , t
2000
4( IO
Figure 2. Porosity of pulverized limestone nuclei as a function of number of drum revolutions
E W ( I
a a
-1
D I M E N S I O N L E S S P E L L E T S I Z E , Vi0
Figure 3. tions
Normalized size distributions of pellets at different drum revolu-
Plotted as per Equation 31 for 0.32 rq. meter/gram limestone with 4 9 . 7 volume
Thus, the measured normalized mean pellet size, v(t),is given by
Kl(t) = V ( t ) P ( t )
(32)
and, recalling that Kl(0) = v(0) = 1, Equation 26 becomes
P(t) = exp[+(t)I
[P(t)l-l
(33)
Again, the measured variance, u 2 ( t ) , of the normalized size distribution, from Equations 27 and 32 is ,72(t)
=
[P(t)I' - P(t)[P(t)]-l
(34)
The value of measured normalized mean pellet size, P ( t ) ,varies from unity to about lo6,and thus, except near t = 0
PO)1 2 >> P(t) and near t = 0, the porosity correction function P ( t ) + 1. Thus, Equation 34 can be approximated as
% water
Finally, the measured third central moment, p 3 ( t ) , of the normalized distribution is given by combining Equations 28 and 32, or p3(t) =
+ v(t)[P(t)]-'
2 [ P ( t ) l 3- 3 [ v ( t ) l 2[ P ( t ) ] - 1
(36)
And again approximately
The expressions for u*(t) and p3(t) as given by Equations 35 and 37, respectively, are consistent with the initial condition that both vanish at time zero. Equations 35 and 37 show that the parameters of the pellet size distributions generated by the proposed random coalescence model are uniquely interrelated, and as such, the size distributions are self-preserving. VOL. 8
NO. 1
JANUARY
1969
59
0
4 6 a IO DIMENSIONLESS P E L L E T SIZE, V i 0
'2
I2
14
Figure 4. Normalized size distributions of pellets at different drum revolutions Plotted as per Equation 31, for 0.49 sq. rneter/gram limestone with 48 volume
% water
size distribution was determined by scanning pellet photographs on a Zeiss Model TGZ3 particle size analyzer which could discriminate among successive size increments of about 10%.
I
I
0
LIME STON E 0.32 m2/gm; 4 9 . 7 % v WATEF A 0.49 m2/gm; 4 8 . 0 96 v W A T E F
Size Distribution of Pellets. I n Figure 3, size distributions of the pellets at different numbers of drum revolutions are plotted in accordance with Equation 31 of the random coalescence model. This run was made with ground limestone of 0.32 sq. meter per gram surface area and 49.7 volume % water. The pellet size scale in this figure must be multiplied by the number in the bracket for each size distribution curve in order to obtain the measured normalized pellet size, Vto. This particular system had rather high water content in relation to the fineness of limestone powder employed. Figure 4 shows size distributions of pellets with relatively low water content-Le., 48 volume 7.with 0.49 sq. meter per gram surface area limestone powder. The over-all agreement between experimental size distributions and those predicted by the random coalescence model is only fair (Figures 3 and 4). Frequently, considerable deviation occurs, especially at the extremities of the size distribution curves. This, however, is hardly surprising in view of the fact that:
Experimental Results and Discussion
The collision frequency, k , used in the development of this model was taken to be a constant for all pellet sizes. This is obviously a highly simplified assumption. Any error in measurement of the areal diameter of the pellet is magnified on taking its cube when pellet volume is computed. The coalescence rate function, X ( t ) , in the appearancedisappearance Equation 5, is highly variable, as may be judged from Figure 5 where the mean pellet size, v(t),of the normalized size distributions in Figures 3 and 4 is plotted against the agglomeration time. I n this figure the steep portion of the curves corresponds to the nuclei and transition regions, and the near-horizontal parts are for the ball region.
Experimental techniques for studying pelletization kinetics on a laboratory scale have been published in detail (Kapur and Fuerstenau, 1964). Briefly, limestone powders used were prepared by grinding in a stainless steel rod mill. Another set of experiments was carried out with commercially supplied pulverized limestone with 0.18 sq. meter per gram of surface area. Agglomeration was carried out in a 12- by 12-inch batch balling drum, rotating at 35 r.p.m. Unless otherwise stated, the charge of dry limestone powder feed was held fixed at 1.75 kg. Pellet
Interrelationships between Parameters of Pellet Size Distribution. Figure 6 illustrates that the parameters of the pellet size distributions are uniquely interrelated as per Equation 35 of the random coalescence model. Here the mean and the variance of the distributions for pulverized limestone systems are plotted in accordance with Equation 35. T h e slope of 2 in Figure 6 would be in agreement with the requirement of the model. The range of water contents employed in these experiments, although seemingly narrow,
I
I
I
I
I
I
I
20 0 400 600 NUMBER OF DRUM REVOLUTIONS, t
t
Figure 5. Mean pellet size of size distributions in Figures 3 and 4 as a function of number of drum revolutions
60
I&EC PROCESS DESIGN A N D DEVELOPMENT
11,
I
I
I
I
I
I
/
I
I
I
d
2v, “i, 1
4
42.5 44.6 46.6 48.2
L I M E STON E
PULV. L I M E S T O N E X v WATER
SURFACE
x
AREA, m?gm
40.6 42.5
I
i
L
0.19 0.19 0.19 0.32 0.32 0.32 0.49 0.49 0.49
v w
;
A
I
A
h
v WATER
47.4 45.2 4 3.6 4 9.7 47.6 46.0 49.5 48.0 45.7 50.1
0.71
0
1
2
3
4
LOG
v(t)
5
6
0 0.71
7
-0
Figure 6. Relationship, as per Equation 35, between variance and mean of normalized pellet size distributions in nuclei and transition regions
SL0PE=2
7
l3
i
-N+e6 b
Y
::5
12
f
I>
-
6
4 -
0
0.19 0.19 0.19 0.32 0.32 0.32 0.49 0.49 0.49 0.71 0.71
4
5
9
a A
3 -
H
A
2 -
rn
1
2
3 LOG
8
IO
1
I
I
I
I
I
PULV L I M E S T O N E
A 4 6 . 6 % V WATER LIMESTONE 0 0.49 m2/gm,48.Oa/ov WATER
0
0.32 rnZ/Qm,49.7%VWATER
47.4 4 5.2 43.6 49.7 47.6 46.0 49.5 48.0 45.7 47.5
0 0
(t)
LIMESTONE SURFACE AREA, rn%m % v WATER
+
J
6
LOG
Pulverized and ground limestone systems
r
zr’
-z 7 1
4
Figure 8. Relationship, as per Equation 37, between third central moment and mean of normalized pellet size distributions in nuclei and transition regions
Pulverized limestone systems
I
2
47.5
6
7
8
v(t)
4
5 LOG
6
7
(t)
Figure 7. Relationship, as per Equation 35, between variance and mean of normalized pellet size distributions in nuclei and transition regions
Figure 9. Relationship, as per Equation 35, between variance and mean of the normalized pellet size distributions in the ball region
Ground limestone systems
Limestone systems
VOL. 8
NO. 1
JANUARY 1969
61
I
1
I
I
I
the variance and the mean of the pellet size distribution is preserved for granulating loads of 1240 and 2480 grams.
I
Conclusions
Random coalescence is only one of the many possible size increment modes of the particulate assemblies. Intuitively, some of the qualitative requirements for this mechanism to operate are that the pellets should be small and porous or otherwise easily deformable without fracturing, properties which are more likely to be associated with fine particulate-high water content systems. Nevertheless, experimental data seem to indicate that pellet growth in the nuclei-transition region invariably occurs by coalescence, notwithstanding the highly diverse nature of the systems studied. Recent work in this laboratory shows that size distributions of taconite-bentonite pellets also conform to the same pattern, a result which should be of some interest for devising models of the industrial circuits for iron ore concentrates balling.
9 -
8 -
+ -
7-
T @
I>
6 -
I
+1
N
A
5 -
s
PULV. LIMESTONE 46.6 Y. v WATER
LIMESTONE 0 2480 g r a m s LOAD 1240 grama LOAD
2
Ac knowledgment
The authors thank T. C. Kuykendall for his valuable assistance in the experimental part of this work. This research was sponsored by the American Iron and Steel Institute.
I t ’
1
I
1
I
2
I
3
LOG
I
4
I
5
I
6
I
7
(t)
Figure 10. Relationship, as per Equation 35, between variance and mean of the normalized pellet size distributions in nuclei and transition regions with different drum loads
was about the maximum over which satisfactory granulation could be carried out. Figure 7 shows similar correlation for pellet size distributions of ground limestone powders of different surface areas. Similarly, Figure 8, where the data for both pulverized and ground limestones are included, indicates that the third central moment of the pellet size distributions is again uniquely related to the mean pellet size in accordance with Equation 37 of the random coalescence model. I n the ball region, the random coalescence mechanism of pellet growth is not necessarily retained. This is brought out in Figure 9, where the mean and the variance of the pellet size distribution in the ball region are plotted in accordance with Equation 35. For comparatively fine ground limestone of 0.41 sq. meter per gram surface area-as well as limestone of 0.71 sq. meter per gram surface area, not shown in the figurethe random coalescence mechanism seems to be extended right into the ball region for all water contents, until finally the pellet growth ceases altogether. For relatively coarse powders, on the other hand, the pellet size spectrum becomes narrow with the balling time, followed by increased dispersion a t a later stage. Since both the fracture-layering and the random coalescence mechanisms require that the variance of the size distributions increase regularly with the mean pellet size, it is likely that the narrowing of size dispersion in the initial stages of ball region is due to preferential coalescence among the smaller pellets. I t will be seen from the development of the random coalescence model, and particularly from Equation 10, that the pellet size distributions are not influenced by any change in total mass of the tumbling charge. This is readily confirmed from Figure 10, which shows that the unique relationship between 62
l & E C PROCESS DESIGN AND DEVELOPMENT
Nomenclature = number fraction of pellets of size V f F(x, t ) = generating function = collision frequency of pellets in a tumbling charge k K l ( t ) = first cumulant of size distribution K*(t) = second cumulant of size distribution K s ( t ) = third cumulant of size distribution K(w, t ) = cumulant generating function = number of pellets belonging to size group i = total number of pellets in tumbling charge = correction factor-for pellet compaction = agglomeration time in number of balling drum revolutions = theoretical normalized volume of pellet belonging to size group i = measured normalized volume of pellet belonging to size group i = measured mean of normalized pellet size distribution = a constant in cumulant generating function = a constant in generating function = cumulative number fraction larger than pellet size Vi = cumulative number fraction larger than pellet size Vi, = probability of successful coalescence = consolidated coalescence rate function = measured third central moment of normalized size distribution = measured variance of normalized size distribution = a coalescence function defined in Equation 16
fi(t)
literature Cited
Bharucha-Reid, A. T., “Elements of the Theory of Markow Processes and Their Applications,” McGraw-Hill, New York, 1960. Capes, C. E., Danckwerts, P. V., Trans. Znst. Chem. Engrs. 43, T116 11965a). Capes,. 6, E., ’Danckwerts, P. V., Trans. Znst. Chem. Engrs. 43, T125 (196513). Jolley, L. B. W., “Summation of Series,” Dover Publications, New York, 1961. Kauur, P. C., Fuerstenau, D. W., IND. END. CHEM.PROCESS DESIGNDEVELOP. 5 , 5 (1966). Kapur, P. C., Fuerstenau, D. W., Trans. AZME 229, 348 (1964). Newitt. D. M.. Conwav-Jones, J. M., Trans. Znst. Chem. Engrs. 36, 422 (1958). Rumpf, H., “Agglomeration,” W. A . Knepper, ed., Interscience, New York, 1962. RECEIVED for review January 29, 1968 ACCEPTED May 1, 1968
