The Probability of Coalescence in Granulation Kinetics Norio Ouchiyama and Tatsuo Tanaka*' Nalional industrial Research institute 01 Kyushu, Tosu, Saga-ken. Japan
Lack of knowledge in estimating the probability of coalescence at present prevents further development
of the theory of the kinetics of granulation. The objective of the present paper is to theoretically formulate the probability of coalescence based on a physical model of granulation, or wet pelletization, in pan and drum type granulators.
From observations of particulate motion in operating pan and drum type granulators, two regions of motion can be distinguished-a constant angular velocity region and a tumbling down region. In these regions all the granules are subject to complex forces. However, at present there is insufficient knowledge of the forces in each region for quantitative introduction into a model of the probability of coalescence (Ouchiyama and Tanaka, 1974). It has therefore been assumed t h a t a n axial compressive force is exerted upon each pair of granules in contact within the granulator and this force causes a n increase in the surface area of contact which results in cohesion. Further, n mutually independent pairs of forces are assumed to apply perpendicular to a tangent common to the contacting granules, which tend to separate them without any change in the area. Furthermore, supposing that the former may be exerted in the constant angular velocity region and the latter in the tumbling down region, it is assumed t h a t the former is represented by a range of field forces whose distribution is independent of granule size and the latter by the forces which are proportional to the volumes of the granules in contact. The probability of coalescence can then be derived by adopting the tensile stress exerted a t the area of contact due to the moment of the separating forces as the criterion of coalescence. First, let us consider the critical bending moment which is necessary so t h a t the granules might just not be separated from each other (Figure l ) . Two idealized systems may be differentiated depending on the mechanisms of the separating forces-these are the static system and the dynamic system. In the static system the pair of granules is assumed to be pin-fixed in space a t the point 0 . The moment, M , acting between the granules is given by M = FD--D 2(Dd)'"
where 11, and I d are the moments of inertia around the centers of the granules, and Md are the masses of the granules, and k is a constant of proportionality. Replacing (pd + p/j) by q and using the initial conditions, +(o) = ~(o= ) 0, the equations can be solved with respect to p (see Appendix), Since the maximum moment determines whether or not the granules coalesce, we have, by using q m a x ,
( D d 3 I 2FdD4 + FDd4 ( 2) D + d D5+d5 T h e relationship between the moment and the tensile stress should be considered next. Since the normal stresses contribute to the moment, these can be simplified, as shown in Figure 2, into a tensile and a compressive stress which are numerically equal to each other and act uniformly upon half the area of contact. Denoting the area of contact by S and the distance between the figure centers of both half circles by 21, t h e following relationship can be obtained between the moment and the tensile stress. M = kq,,
= 2-
The dependence of the probability of coalescence on the particle size is examined in a later section. For this purpose it is necessary to evaluate the surface area of contact and the separating forces. As mentioned above, the surface of contact is brought about by a n axial compressive force, Q, having a distribution independent of granule size. S o w , considering that the surface area of contact between two spheres can be expressed as S 0: Q2I3[Dd/(D d) for elastic bodies, according to Herzian theory, and S 01 Q for plastic bodies then, the surface area of contact between the granules may, in general, be approximated as
+
Dd D + d
= Fa--d 2(Dd)'I2
(1) 2 D + d Equation 1 neglects the effect of gravitational forces and changes in the granule's configuration upon the balance of forces and this simplification will also be necessary in the case of the dynamic system. In the dynamic system the relative angular displacement between the contacting granules must be considered. Assuming. for simplicity, a proportionality between the moment and the displacement, the equations obtained for the motion around the point 0 are
2 D + d
where K , { and 9 are physical constants independent of sizes D and d . T h e separating forces, as already assumed, may be given by either one of the following proportionalities corresponding to the system. In the static system
Fa
+ +
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Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
+
D3)
(1)
F , = A(D3d3)'I2
(ID
+ L?)
(1111
F , = F , = A(L?d3)'I2
(Iv1
Fd
F , = A(d3
In the dynamic system
F, = F, = A ( d 3
To whom correspondence should be addressed at the Department of Chemical Process Engineering. Hokkaido University Sapporo Japan
(4)
Fd = A d 3 ; F , = AL? (V) where A is a constant of proportionality independent of granule size.
I t is now possible to formulate t h e probability of coalescence. Replacing t h e separating forces F , a n d F d in eq 1 or 2 by one of t h e equations given above and using eq 3 with eq 4 t h e tensile stress is given by
B
D
+
d
3r112
g(D,d)
u = K312Q3"2 (7) 2"
(5)
where B = 3 A / 4 d I 2 and the values of a a n d t h e functions g(D, d ) are given in Table I. T h e probability, P, t h a t t h e two contacting granules could withstand a pair of separating forces following a n axial compressive force can be derived from eq 5 by denoting the tensile strength by gst
Rat
0. This corresponds with the fact t h a t there is no upper limit to growth by layering or snow balling. I t is also apparent t h a t if t = 1, then u 5 1. T h a t is, there should be a characteristic limiting size for the coalescence of granules of similar size and this has been found in practice. From eq 11, t h e coalescence probability, P,, can be a p proximated by the following equation for sufficiently small d (i.e., d 4 t h e probability increases with increasing D, whereas if y < 4, the reverse is true. However, there is no experimental evidence at the present time to confirm this dependence on predicted by t h e theoretical model.
*,
Conclusion Based on theoretical analysis and adopting the tensile stress due to the bending moment as the determining factor for coalescence to occur, the probability has been formulated in a general dimensionless form and its dependence on granule size has been examined using a published empirical formula and experimental d a t a . Acknowledgment T h e authors gratefully acknowledge the assistance of Mr. Ward Selby, a post-graduate student a t the University of Melbourne, for discussions and help in preparation of t h e paper. Appendix
+
Replacing ( q d pll) by q gives the second-order ordinary differential equation.
$ where
t A2q
-
B = 0
Nomenclature k = proportionality constant. kg m K = defined by eq 4,k g l m2--v F d , Fl, = separating forces, kg I d , I,, = moments of inertia, kg m s2 M = moment, kg m M d , Ml, = masses of granule, kg s2 m - l n = number of pairs of separating forces, dimensionless Q = axial compressive force, kg $9 = surface area of contact, m2 P, = coalescence probability, dimensionless u = D / 6 , dimensionless u = d/d, dimensionless
Greek Letters = given in Table I, dimensionless = adjustable parameter, dimensionless = characteristic limiting size, m = Q.M/(Q.M- Q m ) , dimensionless {,7 = constants, see K ( = d / D , dimensionless 0 = density of granule, kg s2 m - 4 u t = tensile stress. kg/m2 gst = tensile strength, kh/m2 q = q d + q / ~dimensionless , q d , q~ = angular displacement of contacting granules. dimensionless a 7 d X
L i t e r a t u r e Cited Capes, C . E., Danckwerts, P. V . , Trans. lnst. Chem. Eng.. 43, T i 1 6 (1965). Capes, C E . . lnd. Erig. Chem.. Process Des Dev.. 6 , 390 (1967) Kapur, P. C , Chem Eng. Sci.. 27, 1863 (1972) Linkson. P. B , Glastonbury, J. R . Duffy, G. J.. T r a n s lnst. Chem. Eng.. 51, 251 (1973). Ouchiyama, N . , Tanaka, T.. lnd. Eng Chem.. Process Des. Dev.. 13, 383 (1974). Sherrington, P. J . . Can. J . Chem. Eng , 47, 308 (1969)
Receiued f o r rwieu: Octoher 9. 1971 A c c e p t e d February 17. 1975
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