Powder Technology, 65 (1991) 251-272 257

A microlevel-based characterization of granulation phenomena

Bryan J. Ennis

E.I. du Pont de Nemours and Company, Experimental Station, E402/21 IO, Agricultural Products, Wilmington, DE 1988ObO402 (U.S.A.)

and Gabriel Tardos and Robert Pfeffer

Department qf Chemical Engineering, The City College of the City Uniaersity of New York, New York, NY 10031 (U.S.A.)

Abstract

Past granulation research has been essentially restricted to a macroscopic study of the impact of operating variables on granule morphology. While fundamental groundwork regarding agglomeration forces has been laid by pioneers such as Rumpf, little progress towards an a priori characterization of microlevel phenomena in terms of macroscopic process variables has been achieved. The present work centers on this microscale and introduces a classification of granulation mechanisms based on the collisional dissipation of relative particle kinetic energy.

The mechanism of granule coalescence is, in part, a function of a dimensionless binder Stokes’ number St,, which is a measure of the ratio of granule collisional kinetic energy to the viscous dissipation brought about by interstitial binder. The Stokes’ number provides a convenient classification of granulation regimes. For small St,, coalescence hinges on the presence and distribution of binder and is independent of particle kinetic energy and binder viscosity. In this regime, binder viscosity controls the rate of granule consolidation and ultimate granule voidage. For the case where the maximum St, is of the order of St,*, increases in binder viscosity increase coalescence rate as traditionally expected. Here St: is a critical Stokes’ number, being a known function of the volume of binder deposited on the bed. Finally, for large St,, only granule coating is possible. Fluid-bed granulation and defluidization experiments supporting this simple classification of granulation regimes is presented. Implications regarding successful granula- tion operation are drawn.

Introduction

Powder granulation, as the name implies, is a particle size enlargement process. The term typi- cally encompasses a variety of processing tech- niques to include solution, slurry, melt, and binder granulation (see, e.g., [2]). In particular, binder granulation is a process where a binder such as a sugar or polymeric solution is atomized onto a well-mixed bed of one or more powders. Alternatively, the binder is often present as one of the powders and solvent alone is atomized onto the moving bed which is contained by a vessel such as a fluid-bed, a rotating drum or pan or a high-speed mixer. As solvent evaporates from the deposited binder solution, the particles undergo various relative displacements with respect to one another of which some coalesce as they move about the bed. Resulting interparticle pendular bridges and capillary regions eventually solidify, leaving behind homogeneous granules composed

of the original powders. While the present work is largely restricted to binder granulation, the underlying principles to be outlined are certainly extendable to other granulation and size enlarge- ment processes.

Binder granulation is a heavily utilized unit operation of a host of industries [3] ranging from the granulation of detergents to food prod- ucts to agricultural chemical products such as pesticides and fertilizers. It is often used as an intermediate process by, for example, the metal- lurgical industries for the balling of iron ore prior to firing or the pharmaceutical industry to provide homogeneous granules consisting of a mixture of inert and active ingredients suitable for subsequent pelletization and coating. Addi- tionally, forms of growth which bear a funda- mental similarity to binder granulation often occur unintentionally such as the coagulation of fine powders [4] or high-temperature defluidiza- tion [5].

0032-5910/91/$3.50 #Q Elsevier Sequoia/Printed in The Netherlands

258

The engineering popularity of the granulation process as a size enlargement option has led to a thorough experimental investigation of the influ- ence of operating variables and material parame- ters on granule growth rate and morphology. At the same time, however, the work has remained somewhat disjointed, being spread throughout en- gineering literature. The influence of such vari- ables as angle or rotary speed in pan granulation has been thoroughly investigated for large pri- mary particles by the iron ore industry [6] and, much less so, for fine primary powder by the agricultural industry. The fluidized-bed granula- tion of fine powders is typical of the pharmaceuti- cal industry due to the ability to subsequently coat granules in the same apparatus. Such vari- ables as drop size; atomizer location and type; binder solution viscosity; surface tension and wettability; binder and atomization air flow rates; and excess gas velocity and temperature [7- 1 l] to name but a few have all been investigated. The application of high-speed mixer granulation is increasing in popularity due to its short operating time; however, even less is known about the influ- ence of impeller speed and the effect of other variables on this process [ 121. Despite the wealth of such granulation research, its macroscopic black-box approach has essentially led to little understanding of the underlying physics of the granulation process. The results of such work tend to be very product and apparatus sensitive and even small changes in operating conditions can lead to counterintuitive results. Granulation scale-up is an often formidable, haphazard under- taking.

On the microlevel, early efforts have identified the various agglomeration mechanisms as well as some typical governing forces [ 1, 13- 151. While fundamental in nature, such work was not linked in a strong way to the macroscopic process vari- ables (e.g., drum speed) and, hence, was of limited engineering utility for actually controlling the granulation process. Some systematic work has demonstrated the existence of distinct regimes for both drum [16] and fluid-bed granulation [ 111. The transition between regimes is some function of granule size and the intensity of bed hydrody- namics represented by, for example, excess gas velocity. Such transitions, however, have not been related in a fundamental way to any a priori granulation parameters or variables.

From the early coagulation work of Friedlander and others [ 17, 181, a separate tack of granulation research involving population balance techniques has evolved [ 19-211. Such work provides an inter- esting mathematical framework for modeling gran- ulation, particularly in regard to predictions of self-preserving growth and average growth rates

via similarity transforms. The growth laws, how- ever, depend on unknown granule coalescence and breakage functions. Such functions depend in turn on the microlevel physics of the granulation pro- cess such as the probability of particle coalescence. Since the coalescence and breakage functions are not provided by the population balance techniques alone, little progress is again made towards pre- dicting granule morphology.

Given such a complex intertwining of operat- ing and material parameters of the granulation process, the question arises as to whether or not particular regimes of granulation exist. The estab- lishement of such regimes would allow us to un- ravel the effect of the various operating and material parameters within each regime on gran- ule morphology. The present work attempts to establish such a classification of granulation from forces on the microlevel and, in particular, is restricted to the influence of dynamic pendular bridge strength on granule coalescence and con- solidation. Since the classification evolves out of an analysis of the physics of the microlevel phe- nomena, it is based on first principles and, hence, is a priori in nature. Previous granulation research has provided undue focus on predicting granule growth rate and, as a result, masked the overall physics of the process. The present approach is more powerful in that one can directly see the effect of, for example, granule kinetic energy or binder viscosity on granule morphology within each regime. Hence, the approach is of significant engineering utility, providing us with a framework to address such questions as whether one should increase or decrease fluid-bed gas velocity, pan angle, or mixer impeller speed to increase granule size and for what reasons.

The microlevel force

With the aim of establishing regimes of granu- lation, we begin by examining the process from the microlevel and restrict our present focus to the phenomena of granule coalescence and consolida- tion. The competing process of attrition of soli- dified granules is not considered. We assume that the binder solution is sufficiently distributed on a local level to allow the occasional formation of pendular liquid bridges between colliding par- ticles. Since this is a dynamic process by nature, we may not adopt well-known Laplace-Young theory for the strength of a static bridge [ 1, 31 and, instead, are led to consider the strength of a dynamic (moving) pendular liquid bridge. The strength of the dynamic bridge plays a crucial role in determining the rate of granule consolidation as well as the mechanism of coalescence.

259

The strength of a static liquid bridge has re- ceived considerable theoretical and experimental attention [22, 231. In the case of perfect wetting, surface tension induces an attractive capillary force between two particles due to a contact line force and a pressure deficiency brought about by the curvature of the gas-fluid interface. In a similar vein, the static or capillary strength of a granule due to interstitial fluid has been exten- siveiy studied by Rumpf [l] and others, e.g., [24]. Under industrially relevant conditions, however, the strength of a dynamic bridge may exceed that of the static by at least an order of magnitude due to the additional energy dissipation resulting from binder viscosity and the minute interparticle gap distance involved [ 251.

To obtain the strength of a dynamic pendular bridge, consider two nearly touching spherical particles approaching one another along their line of centers (taken to be the z-axis) each at a velocity U and separated by a dimensionless gap distance of E = 2h,/a where a is the sphere radius. The pendular bridge may conceptually be divided into two regions as illustrated in Fig. 1. Within the immediate vicinity of the minimum gap dis- tance (r = 0), a viscous inner lubrication region exists, since z is of the order of E 6 1. In the outer capillary region, where z is of the order of 1, the capillary contribution to bridge strength is domi- nant and viscous effects may be neglected at lead- ing order in capillary number Ca = pU/y where p is the fluid viscosity and y is the air-fluid inter- facial tension. Since it may be shown that the viscous pressure of the inner lubrication region acts over a radial distance of the order of E ‘I2 [26], one can imagine an approximate solution for the strength of a dynamic pendular bridge to be given by a superposition of the Laplace-Young capil- lary pressure acting in the outer region and the viscous lubrication pressure acting within the in- ner region. This simple superposition of capillary and viscous lubrication forces has been verified experimentally over a full range of capillary num- bers [25]. Furthermore, the solution has been rigorously justified via the method of matched asymptotic expansions [26] with the force found to be given by

F = Fcap + F”i, + O(Ca In E)

with

Fcap = sin2q(C, + 2)

as .a+0 (la)

(lb)

F. =g “IS 2E (ICI

where Fcap and Fvis are the capillary and viscous forces, respectively, and where the total force has been made dimensionless with a capillary scale of

a. Outer region with z-O(l) h. Inner region with z-O(&)

Fig. 1. Regions of expansion for the pendular bridge [26].

xya. The capillary force is composed of the Laplace-Young pressure deficiency C,, and a con- tact line force of 2sin2q with cp being the filling angle. Hence, the superposition of Laplace- Young and viscous lubrication pressures is valid in the asymptotic limit of E +O with Ca up to order one (Ca - o( 1)) and so the capillary num- ber need not be particularly small. Most impor- tantly, one may note from eqn. (lb) that the viscous contribution may actually dominate the capillary contribution, being up to order (l/s) as observed experimentally [25].

Requirements for granule coalescence

Let us now consider granules colliding with one another as they move about the granulation apparatus. Those having insufficient relative kinetic energy to overcome the retarding force or strength of the liquid bridge will coalesce. Given that we know the strength of such a bridge, we now address the question of whether or not we can characterize the granulation process in any meaningful way. We assume that the binder is well dispersed, at least at a local level or region, so as to establish a characteristic layer of binder. There are cases in which the binder droplet is sufficiently viscous or binder-particle wettabilities are such that this local level of dispersion is not achieved. Such situations are excluded from con- sideration in the present work.

Consider two individual granules of radius a within the granulation process. Suppose they are approaching one another at an initial relative velocity 224, and are each covered by a binder layer of thickness h as shown in Fig. 2. As the individual binder layers of the granules come into contact, a dynamic pendular bridge will form between the colliding granules now separated by a gap distance of 2h. For sufficiently large binder

260

Fig. 2. Schematic of two colliding granules each of which is covered by a viscous layer of thickness h.

viscosity, the bridge will dissipate the relative kinetic energy of the colliding granules preventing rebound. To determine the minimum velocity re- quired for particle rebound, consider the force balance on an individual granule, or

m $ = 7rya [

sin*cp(C + 2) + 5 $ + O(Cu In E) 1 (2)

where the right-hand side represents the dynamic pendular bridge force and where m and u are the granule mass and velocity, respectively. Equa- tion (2) does not possess an analytic solution in its present non-linear form. However, since we are largely concerned with viscous binders, we shall neglect the capillary contribution to the pendular bridge force for the present and retain only the singular, dominant viscous contribution. This is quite the reverse of the majority of previous ag- glomeration research having stemmed from the pioneering work of Rumpf [ 11, where only capil- lary forces brought about by interfacial tension are accounted for with viscous lubrication effects ignored.

Viscous coalescence With this modification, eqn. (2) becomes

du 3 ,dx 1 mx=jnpa xxx (3)

where we have retained only the leading order contribution to the viscous force and where x is half the dimensional gap distance. Equation (3) has as a solution [27]

u =u,[l --$ln(k)] (4)

where we have applied the initial condition of u = u0 at x = h and where St, is a viscous Stokes’

number given by

(5)

For rebound of the colliding granules to occur, the Stokes’ number must exceed a critical value St,*. In order to determine this value, we turn to eqn. (4). With an initial velocity of uO, let the velocity of the colliding granule upon reaching a gap distance of h, be u,, where h, represents some characteric length scale of surface asperities. The initial rebound velocity is then eu, where e is the particle coefficient of restitution with the presence of binder. Knowing the initial rebound velocity

eu, , we may solve for the critical Stokes’ number at which the particle velocity equals zero when x = h at the original outer edge of the binder layer. This yields the critical viscous Stokes’ num- ber of [27]

Si:=(l +a)ln($ (6)

Granules colliding with initial Stokes’ numbers less than St,* are considered to have coalesced. So for a granular material with a known measure of surface asperities h,, a coefficient of restitution e, and a given amount of deposited binder which establishes some average binder thickness, the Stokes’ number St, governs the coalescence pro- cess. This is intuitively clear from directly consid- ering the ratio of initial kinetic energy relative to the center of mass of the two colliding granules to the energy dissipated from the viscous force acting through a distance h, or

$z( 2u(J* 2mu,

2F,i,h = - 3rtpa2

= St, (7)

Hence, an appropriate generalization for the case of granules of unequal size and mass is

St, = mu, 127@*

where fiand Li are the reduced granule mass and radius, 2u, is the initial relative velocity prior to granule contact at the viscous layer and p is some time-averaged binder viscosity of the granulation process. The generalization of the viscous force to the case of two particles of different size leads to the reduced radius [28]. Both (T and 6 are given

by

_ 01 aI a=-

a1 +a*

and

_ mlm2 m-

m, +m2 (10)

261

Capillary coalescence We can estimate the error in neglecting the

capillary contribution to dynamic bridge strength in the following manner. Neglecting the viscous contribution to bridge strength, the force balance (eqn. (2)) on the colliding granule now becomes

m 2 = 7cya [sin’cp(C(s) + 2)] (11)

Being the right-hand side of eqn. (1 l), the capil- lary force F, is a function of the granule gap distance, since both the capillary pressure C and cp vary with E. However, since F, is only a weak function of E in the sense that it goes to a finite value and does not diverge as E -0, we can take F, as a constant or F,(E = 0) = FcO, which is typi- cally its maximum value. Hence, eqn. (11) be- comes a linear equation which yields a capillary Stokes’ number of

&=&$ CAP

muo2 =---

n?aF,, h (12)

and a critical capillary Stokes’ number St: of

(13)

Now the ratio of critical Stokes’ numbers is given by

(14)

and, therefore, St,*/,%,* + 0 as both e + 1 and h/h, becomes large, indicating that the viscous Stokes’ number criterion is more restrictive in that more initial kinetic energy is required for re- bound. In fact, the capillary criterion would indi- cate that perfectly elastic particles cannot coalesce for any initial kinetic energy. This result should be expected in that energy dissipated by the capillary bridge during granule separation is exactly equal to the energy added by the attractive capillary force during granule approach. We note, however, that in a physically realistic collision, the bridge interface often has a non-equilibrium profile [25] leading to a difference in the energies of approach and separation which could, in fact, result in capillary coalescence. As a first approximation, however, the neglection of the capillary contribu- tion to bridge strength in so far as determining the coaescence process is warranted.

Bed hydrodynamics

The ability of two granules to coalesce within the granulation process has now been defined in terms of a viscous binder Stokes’ number St,. Of course, the calculation of St, presumes a knowl- edge of the interparticle collisional velocity u,,, leaving us at somewhat of an impasse. Such a collisional velocity is dictated by the hydrodynam- ics of the bed within the granulation apparatus. For example, in a fluidized bed granulator, a number of plausible alternatives for u,, present themselves. In cases where the gas distributor is of perforated plate design, as granules are entrained within the jet region, they reach some characteris- tic velocity Uj,t before colliding with the dense phase. During bubble coalescence, granules in the wake of a trailing bubble are thrown against the roof of the leading bubble at a velocity U, of order (U,, + UBq) where IJg is the isolated lead- ing bubble velocity [29]. While both ujet and U, are large, being of the order of 75 cm/s, these velocities are limited to localized regions, implying an uncontrolled rate of coalescence in the remain- der of the bed leading to defluidization. Hence, we will restrict our attention in this paper to veloc- ities which are distributed throughout the dense phase and which occur with sufficient frequency to control granule coalescence. Two plausible choices of the collisional velocity u,, to be consid- ered are the relative granule velocities arising from (i) the local mean velocity gradient, or 2a]du/dy], and (ii) the r.m.s. average of the fluctuations about this mean, or (u*)‘/*.

Mean collisional velocities, a ldu/dy 1 A reasonable choice of a characteristic mean

shear velocity for drum granulation would be the drum rotational speed o. In the case of a fluidized bed, we resort to two-phase inviscid flow theory [29] to estimate the order of the velocity gradient ]du/dy 1. With respect to a moving frame of spher- ical co-ordinates with a velocity of U, and an origin coincident with the center of the gas bubble of radius R,, the velocity potential @ of the inviscid dense phase is

(15)

This leads to the components of granule velocity of

43’ u, = u, - ( > r-3

cos 8,

RB3 ue = U, - ( > 2r3 sin 9 (16)

A typical local shear gradient would be dug /rat) m 3 UB/DB with the maximum occurring

262

along the bubble surface and the gradient decaying as ( l/r4). Extensional velocity gradients are of the same order with the maximum of au, jar N 6U,/D, occurring at the bubble nose of 8 = 0. Hence, the maximum collisional velocity would be of order 12U,a/D, and would decay from its maximum at the bubble surface as ( l/r4), leading to a similar distribution in granule Stokes’ numbers and an increased probability of granule coalescence in regions of minimum bubble activity.

To estimate the influence of surrounding bub- bles on the distribution of collisional velocities and St,, consider two bubbles separated by a distance s. The modified velocity potential correct of order l/s is given by

@12 = U,*@* - &jr @I (17a)

where

and

a+~(~)cosl?~+~[(~)qcos(q7) C

and where r’ and 8’ are defined in Fig. 3 [30]. Considering the extensional velocity gradient along the bubble centers (0 = 0) as an estimate of /du/dy/, a lengthy calculation for bubbles of equal size and velocity reveals

(18)

for the maximum occurring at the bubble tip (r = a) correct to order s -’ and where 6 = s/R, is the dimensionless bubble spacing. Similarly, the average extensional gradient in particle velocity along the line of centers is found to be -

(19)

The correction to velocity gradients due to sur- rounding bubbles would appear small. To leading order, therefore, appropriate choices of the char- acteristic mean collisional velocity for a fluid- bed would be u0 = ( 12U,a/D,) as a maximum, whereas u0 = ( 12UBa/D,h2) on average.

(22) Fluctuational collisional velocities, (u’>‘/’

A first estimate of the relative importance of If we assume typical parameters of (h/h,) = 20, velocity fluctuations to the collision and coales- viscosity p = 2 P, granule density p = 1.5 g/cm3, cence process is provided by the kinetic theories of granule radius a = 100 ,um, and a collision veloc- granular flow. Such kinetic theories, however, have been restricted to simple flows and constant

ity of u,, = aldu/dy( - 12U,a/D, - 2 cm/s, we ob- tain an R = 94, indicating the dampening of

coefficients of restitution [ 3 11. The constitutive fluctuations in velocity and the dominant contri- relations of the granular phase are expanded in bution of the characteristic mean shear velocity to

powers R, where R is the ratio of the characteris- tic mean shear velocity a)du/dy 1 to the r.m.s. of the velocity fluctuations (u2)‘i2. For the simple case of Couette flow with no gradients in fluctua- tion energy and no external body force such as gravity [32], the two velocities are related by

=j-[lO(l -e)]“2 (20)

for small values of R. However, the present results of granular kinetic theory are unfortunately of limited value within the context of granulation, since the extreme dissipative nature of the colli- sions due to viscous binder damps out fluctua- tions in velocity leading, instead, to large values of R.

Since energy is supplied to the granular phase via bubble activity and particle drag in the fluid bed case, we may assume no gradients in the fluctuation energy as a first approximation and that the fluctuations in velocity are given by eqn. (20) in the absence of viscous binder. In general, the fluctuational velocity will dominate the char- acteristic mean shear velocity, or (u2)‘12>aldu/ dyl, as e + 1, indicating the importance of the fluctuational velocity in determining the mechan- ics of the dense phase. In the granulation process, however, e is far from one and, in fact, must approach zero at least on a local microscale to ensure coalescence. The results of the previous section may be recast in the form of a modified coefficient of restitution e to account for the addi- tional collisional dissipation of energy due to the presence of viscous layers, or

(1 +e)ln c=e-

St, (21)

indicating further that e” is function of the colli- sional velocity. Assuming e = 1, eqn. (20) for R now becomes

263

u02

(1-3 Ren

” r’

.-._I_.z’

Fig. 3. Two coalescing inviscid bubbles.

the coalescence process. As we move away from the bubble and the mean shear velocity decreases as ( l/r4), it would appear that the fluctuations in velocity become negligible for the cohesive granu- lation process.

In the case of pan, drum, or mixer granulation, eqn. (20). is inapplicable from the start due to significant gradients in granular temperature since energy is only supplied to the bed at either the impeller or wall of the granulation vessel. Never- theless, similar arguments in regard to the fluctua- tional velocity should hold indicating the characteristic shear velocity ao as the dominant collisional velocity u0 as should be intuitively ex- pected for such devices.

In summary, eqn. (5) for St, becomes

St = 8Pw ”

91* where p is the granule density and the maximum in u, is taken as either 12U,a/D, or ao, depend- ing on the granulation apparatus, whereas u. = 12UBa/D,62 on average for a fluid-bed.

A classification of coalescence phenomena

A spatial distribution of Stokes’ numbers will exist within the granulation bed as depicted in Fig. 4. Consider the origin to lie in a region of high bubble activity or impeller intensity and for St, to decay with increasing r. St, must be com- pared with St,* to establish regions of coalescence in the bed. To obtain granule growth rates, one might then apply traditional population balance techniques with physically realistic coalesence ker- nels based on the results of the previous sections. Given the current level of granular mechanics, however, such a deterministic approach must be considered an unrealistic idealization at present.

St

IL __________.st~ SC-St** (a) _ whertiia/ w

sr N &am/a tin

X

st

(c) --- lL .____.st;” zst*- 7

_ cwt&w stc G-adatia?

X

Fig. 4. Regimes of granulation.

Therefore, at this stage of the development of granulation theory, it is more appropriate& ex- amine limiting case scenarios of the ratio St,/St,* which controls the coalescence phenomenon and where St, denotes some spatial average of St,. An appreciable degree of insight is gained by such an approach as to the significance of the various operating and material parameters on the granu- lation process. While the various regimes of gran- ulation to be described apply to a wide variety of granulation processes, we will focus for the most part and draw supporting examples from fluid- bed granulation.

Regimes of granulation _Let us first examine the regime where St,/St,* + 0 as illustrated in Fig. 4(a), where the entire distribution of St, lies well below the criti- cal Stokes’ number St,*. Considering from eqn. (23) that St, - (a2/p), two immediately apparent examples of this regime are (i) the granulation of fine powders where a is of the order of 10 pm or smaller and (ii) high-temperature agglomeration where particle surface viscosities due to material softening of the order of 10” P are common. The implication of granulation within this regime on coalescence is that all collisions are successful, assuming the viscous binder layer is present. The rate of coalescence is independent on granule kinetic energy and binder viscosity and critically hinges, instead, on the presence and distribution of binder. Hence, we are led to refer to this regime as noninertial or StN granulation. Within this noninertial regime, one would expect such operat- ing parameters as spray drop size and spray distri- bution to control the granulation of fine powders and overall bed temperature distribution to con- trol high-temperature agglomeration processes.

Such conclusions would appear to be strongly supported by current granulation literature. In the

264

100

50

40 60 80 100 120 140 160

Droplet size dS, calculated p

Fig. 5. Effect of drop size on granule size [33].

fluid-bed granulation of pharmaceuticals where spray size exceeds primary particle size, a nearly one-to-one linear relationship between granule size and drop size has been observed for the initial stages of granulation, as shown in Fig. 5. Similar conclusions have been reached by Waldie [34] by a careful comparison of single granule binder content and drop size distribution. The physical conclusion would be that as a binder droplet is deposited onto the bed it immediately adsorbs surrounding particles. Since such particles have insufficient energy to rebound or break away, the drop structure is maintained, forming a granule of the order of the drop size.

Observations regarding high temperature de- fluidization further support the existence of the noninertial granulation regime. Consider first the work of Compo et al. [5, 351, where an entire fluidized bed is heated uniformly. Below some characteristic sintering temperature, the bed re- mains fluidized with little evidence of agglomera- tion. As bed temperature is increased above the sintering temperature, particle surfaces soften and become sticky, leading to sudden defluidization. While somewhat simplistic, this surface softening is generally represented by a surface viscosity which, for the case of polymeric materials such as polyethylene, is of the order of lOI* P. For l-mm particles with a density of 2.5 gm/cm’ and &ID, = 10 s-‘, this would imply St, - IO-‘*. Since the entire bed is heated, this viscous surface layer will develop throughout the bed, leading to a simultaneous coalescence of all particles and catastrophic defluidization. This phenomenon has been observed for a host of particulate materials including polymers, inorganic salts, metals and metallic oxides, fly and coal ashes, and various ores and cracking catalysts over a wide range of excess gas velocities [35]. In contrast to heating uniformly, a localized zone of the bed may be heated by the introduction of a central hot jet as in the work of Arastoopour et al. [36]. Since

surface softening is localized, controlled agglom- eration is achieved. In this respect, the process becomes similar to the granulation of fine powder where the binder layer is locally introduced. The realization of a noninertial granulation regime explains the inherent instability of high-tempera- ture agglomeration and the apparent discrepancy between the works of Compo et al. [5] and Aras- toopour et al. [36].

Returning now to the binder granulation pro- cess, as granules grow in size, so do the granule Stokes’ numbers since St, is an increasing func- tion of a. When granules have reached sufficient size such that the largest Stokes’ number equals the critical value St, - St,* in regions of high bubble activity (Fig. 4(b)), granule kinetic energy and binder viscosity will begin to play a role. In this inertial or St’ regime of granulation, increases in binder viscosity or decreases in granule kinetic energy will increase granule coalescence as tradi- tionally expected. As granules grow still further and eventually themerage Stokes’ number equals the critical value St, - St,* (Fig. 4(c)), the distri- bution of St, will be such that local regions where St, < St,* and granule coalescence is assured will be balanced by regions of disagglomeration where St, > St,, implying that on average coalescence granule growth is not achieved. Hence, a constant state of bed hydrodynamics dictated, for example, by excess gas velocity in the case of a fluid-bed implies the existence of the theoretical growth limit by coalescence leading to granule coating alone or StC granulation regime.

Experimental support Since St, increases with increasing granule size,

the granulation of an initially fine powder should exhibit characteristics of all three granulation regimes as time progresses. As a demonstration supporting this hypothesis, a granulation com- parison was made between the polymeric binders of sodium carboxymethyl cellulose (CMC- Na( M)), polyvinylpyrolidone (PVP), and hydro- xyethyl cellulose (Klucel). Aqueous binder solu- tion properties are summarized in Table 1 and are detailed elsewhere [37]. The strength of an indus- trial binder pendular bridge increases exponen- tially with drying time due to solvent evaporation. An average bridge viscosity may be obtained by averaging the pendular bridge strength over its drying time. Average bridge viscosities of 1.7, 0.36, and 0.30 P were estimated for the CMC-Na(M), PVP, and Klucel solutions, respectively [37]. In addition from Table 1, the binders are seen to have similar solid bridge strengths. This allows us to neglect the influence of the solid bridge on granule attrition and, hence, overall growth rates to a first approximation.

265

TABLE 1. Ma&al properties of binder solutions

Viscous strengthening Average bridge Solid bridge

time constant, 5, viscosity, p strength

(min) (P, 15 “C) (dyn)

CMC-Na(M)” 1.2 PVPb 5.1 Klucel’ 4.6

“Sodium carboxymethylcellulose-M (CMC-Na(M), 250 K MW) bPolyvinylpyrolidone (PVP, 360 K MW) ‘Hydroxyethylcellulose (Klucel, 5 250 K MW)

1.7 14500 0.36 17400 0.30 6400

TABLE 2. Granulator operating conditions

Granulation time Temperature u-u,, Binder feed Initial height Bed material

8h 15 “C 0.25 m/s 3.5 ml/min at 2 wt.% 3 in 100 pm micro-beads (80-120 pm)

Fluid-bed granulation runs with 2 wt.% binder solutions of PVP and CMC-Na were conducted at the operating conditions given in Table 2. The fluid-bed granulator utilized for the present study is shown in Fig. 6. A bed of technical grade, closely sized glass microbeads with an initial di- ameter of 100 pm was supported by a sintered glass plate and fluidized by air supplied by a blower at 75 “C. With a constant excess gas veloc- ity of 0.2 m/s, binder was atomized onto the bed for a period of 8 h. A comparison of the sampled median diameter as determined by sieve for the

Rotomtttr

..__

d50@4 0 CMC-M T PVP36OK

9cn-

300 1 3 5

t&U 7

Fig. 7. Median granule diameter as a function of operating time for the CMC-Na(M) and PVP binders.

CMC-Na and PVP binders as a function of gran- ulation time is given in Fig. 7.

The three regimes of granulation are clearly observed, supporting the proposed classification of granulation mechanisms. For the initial 5 h of the granulation with average granule diameters less than 800 pm, similar rates of growth are

1 Atomization

System -

-I urrrr~nq 1 1 , :_..:*

Porour Plott Dirtrlbutor-

Heater

Fig. 6. Low-temperature fluid-bed granulator.

266

observed. This is despite the fact that the average CMC-Na viscosity exceeded that of the PVP by a factor of 5, indicating that the initial growth fell within the StN granulation regime and was inde- pendent of binder viscosity. In the latter half of the granulation when average granule diameter has exceeded 800 ,um, PVP granule growth begins to slow in comparison with that of the CMC-Na, implying a transition to St’ granulation for the PVP binder and an increasing importance of binder viscosity. Finally, PVP granule growth be- gins to level off somewhat in excess of 900 pm in the last hour, indicating a transition to granule coating or StC granulation. The CMC-Na binder maintains a nearly constant rate of granule growth throughout the eight-hour run. This im- plies that it remains within the StN regime as expected due to the higher CMC-Na viscosity.

Transitions between granulation regimes The transitions between granulation regimes

critically depend on the hydrodynamics of the bed as previously described. While such detailed knowledge of particle collisions is at best rudi- mentary, an attempt to estimate transitions will be made here for the case of a fluid-bed. Within the StN regime, relative particle kinetic energies are insufficient to cause rebound. The initial transition to the St’ regime will occur when granules have reached a sufficient size such that the largest colli- sional velocity will cause rebound and prevent coalescence. Hence, the transition is given by

112

a N-I = (24)

where St, has been equated to St,* with u0 = 12U,a/D,. For the case of the PVP granula- tion, a value of 2a, _, = 1 470 pm is estimated as compared with the actual transition of 800 ,um. Here, we have assumed an average binder viscos- ity p = 0.36 P, granule density p = 1.5 gm/cm3, e=l, U,=50cm/s, D,=2cm, h,=OSpm for the smooth surface of the glass microbeads, and h = 10 pm as indicated from SEM photography of the solidified granules. As explained earlier, the transition to granule coating or the StC regime occurs when the relative collisional velocity aver- age over the dense phase is sufficient for particle rebound. Taking u,, = 12UBa/D,d2, the transition to coating is marked by

a IdTaN+ 6 (25)

with aN _, given by eqn. (24). Assuming a bubble spacing of 6 = 2 in addition to the above assump- tions regarding the PVP granulation, 2a, _c =

0 60 120 180 240 309 :

t [minutes]

a

Fig. 8. The effect of excess gas velocity on granule growth (glass powder, 5% carbowax) [2].

2 940 pm as compared with the somewhat greater than 900 pm indicated for the PVP granules. In practice, attrition of solidfied granules will shift the transitions given by eqns. (25) and (26).

Similar transitions between granulation regimes is clearly indicated from the results of Nienow and Smith [ 111. The effect of excess gas velocity on granule growth is illustrated in Fig. 8. These results bear a strong similarity to the effect of binder viscosity as shown previously in Fig. 7, suggesting an initial non-inertial granulation regime indepen- dent of particle kinetic energy and, therefore, excess gas velocity. For the excess gas velocity of 0.65, calculations similar to above yield the inertial transition of 2a, _, = 1 103 pm and the coating transition of 2a, _ c = 2 206 pm as compared with the estimated experimental transitions of 450 and 525 pm, respectively. Here we have assumed an average binder viscosity of 0.2 P, e = 1, h/h, = 20, and p = 1.5 gm/cm3.

Consider an alternative granulation process such as a rotating drum or pan. As in the granula- tion of limestone powder as illustrated in Fig. 9 [ 131, a transition from a random coalescent to a nonrandom crushing and layering growth mecha- nism is observed as the average granule diameter exceeds some critical value. As the name implies, crushing and layering is a mechanism by which larger, heavier granules crush surrounding smaller, weaker nuclei which are subsequently layered onto the surface of the larger granules. This growth mechanism should increase in importance as larger granules form. It does not provide an explanation, however, of the abrupt change in growth rate at the critical diameter. Rather, the sharp change in growth at the critical diameter implies an abrupt end to coalescence. This might be expected since the collisional velocity distribution is of the order of 2ao throughout the drum, implying the lack of an St’ regime and a clear demarcation between the noninertial and coating regimes. With u. = 2ao,

261

DRUM REVOLUTIONS

Fig. 9. Median granule diameter as a function of the number of drum revolutions [ 131.

the transition is marked by

(26)

For the case of the volume per cent v = 49% of Fig. 9 with water as a binder, 2a,,c = 2.8 mm as compared with the observed transition of 6 mm. Here, we have assumed the water has been evenly distributed, giving h = 60 pm, e = 1, h, = 5 pm, p = 1 CP and p = 1.5 gm/cm3.

A defluidization criterion

As a final example supporting the underlying mechanics of the Stokes’ number granulation clas- sification, consider the phenomenon of the defluidization of a fluid-bed resulting from the presence of viscous layers on the individual par- ticles. An increase in the amount of binder de- posited onto the bed which leads in turn to an increase in the average thickness of the binder layer has been experimentally observed to increase the minimum fluidization velocity [38, 391. While such results may appear intuitive, defluidization is a complex phenomenon lying at the heart of our present understanding of powder types as well as the hydrodynamic stability of fluidized beds. Given the complexity of the problem, we shall restrict our attention to the case of large particles coated with a fairly viscous layer as in the work of Gluckman et al. [38]. The observed influence of the viscous layer on minimum fluidization velocity does not necessarily extend in a simple fashion to finer powders or binders of low viscosity. Further, more general considerations of the effect of inter- particle forces on the velocity of minimum

fluidization and bubbling are beyond the scope and purpose of the present work.

Begin by considering a bed of particles to be well above its minimum fluidization velocity. A prescribed amount of binder is then added to the bed, after which the gas velocity is decreased until defluidization occurs. We wish to investigate the relationship between the change in defluidization velocity and the amount of deposited binder. At the point of defluidization, the pressure drop bal- ances the weight of the bed establishing a relation- ship between the minimum fluidization velocity U,, and the minimum voidage of the bed e,f of which the Ergun equation is a typical example. The minimum fluidization voidage smf will clearly de- pend on the nature of the interparticle force as well as the manner in which the bed is set down, allowing for possible hysteresis. If the bed is defluidized at a sufficiently slow rate, the viscous pendular bridge will impede a decrease in bed voidage i.e., it will act against ‘consolidation’ of the bed. For the case of large particles we are consid- ering, this increase in minimum fluidization voidage amf leads to an increase in U,,,, to obtain the same pressure drop equivalent to the weight of the bed. Since the attained minimum voidage e,f de- pends on the interparticle kinetic energy prior to defluidization, we should expect that the change in minimum fluidization velocity, as expressed by an appropriate Stokes’ number, depends logarithmi- cally on the thickness of the binder layer. Hence, the appropriate critical defluidization Stokes’ num- ber St E, should be of the form

St * = fW( u,f - &da D

9p

[In(h) - ln(h,)] (27)

where u is a yet unknown proportionality con- stant and where we have chosen the characteristic

2.W

0 a=.14cm sh[[-l A a=.20cm

T a=.33cm * a=.67cm

2.00 -

Fig. 10. Theoretical defluidization limit as a function of binder layer thickness (experimental data from Gluckman et al. [38] for TCC catalyst coated with Ace Plastic coating material, p = 41 P).

268

relative particle velocity of U, = a(U,, - U,,). This would indicate a linear relationship between StX/a and In(h) with a slope of (1 + l/e)/cr and h-axis intercept of (1 + l/e)ln(h,)/cc, allowing us to indirectly determine ct and the characteristic particle surface roughness h, from experimental data. This is important in that a and h, each have a physical significance and are more than simply coefficients of a regression. One would expect that c( is of the order of 1, whereas h, should agree with physical observations of typical surface as- perities of the individual particles.

The experimental defluidization data of Gluck- man et al. [38] are plotted in the above fashion as illustrated in Fig. 10, confirming that the change in U,,- U,, does indeed vary linearly with In(h). In the work of these authors, the defluidization velocities of thermal cracking catalyst pellets of diameters of 1.4 to 6.7 mm was measured as a function of binder loading, where a viscous binder, “Ace plastic coating”, of p = 41 P was utilized. Assuming for simplicity a coefficient of restitution e of 1, we obtain a value of c1 = 1.9 and a characteristic surface asperity h, = 23 ,um from the slope and intercept. Such a surface asperity agrees with physical observations of the typical large-scale pores of the catalyst, whereas c( is of order 1 as desired. Hence, these results lend fur- ther experimental support to the Stokes’ number approach towards the classification of granulation regimes, since the defluidization of a fluid-bed due to viscous particle layers can be considered one sort of idealized limit of granulation.

In closing with regard to defluidization be- haviour, some remarks pertaining to our choice of u0 are in order. Recent investigations accounting solely for hydrodynamic effects would indicate that all powders exhibit a stable expanded state (see, e.g., [40]). This implies that a powder exhibits a minimum bubbling point prior to defluidization as gas velocity is decreased. Of course, for the class B powders being considered as in the work of Gluck- man et al. [38], the difference between minimum fluidization and minimum bubbling velocities is experimentally unmeasurable in the absence of binder. Our own preliminary defluidization investi- gations tend to indicate, however, that the de- posited binder induces a small but observable difference, suggesting an alteration of powder type. Regardless of the magnitude of this difference in critical velocities, the crucial point is that a transi- tional regime may exist prior to defluidization in which the mechanics of the particulate phase is not dominated by some mean shear flow as illustrated above for the bubbling bed. This would lead us to choose the fluctuational velocity previously de- scribed as an appropriate relative collisional veloc- ity, or u0 = (U ) ’ I/‘. While the analogy is not par-

titularly strong, such velocity fluctuations have been experimentally found to be linearly related to the excess gas velocity (U,, - U,,,, for the case of liquid fluidized beds [41] and, hence, some justifica- tion for our choice of a relative collisional velocity u0 = tl( U,, - U,,) for the defluidization Stokes’ number St;S (eqn. (27)) is provided. Hence, we envision a state just prior to defluidization in which small velocity fluctuations are sufficient for individ- ual particle rebound. As the gas velocity is de- creased further, velocity fluctuations are insuffic- ient for rebound, leading to a simultaneous coales- cence of all particles throughout the bed and catastrophic defluidization. An observant reader will note the difference between defluidization phe- nomena and noninertial St N granulation explained previously. The condition required to achieve StN granulation depends on the square of the particle radius (see, e.g., eqn. (24)) since the relative collisional velocity is dominated by a mean shear flow whereas in the case of defluidization, the critical Stokes’ relation (eqn. (27)) varies linearly with particle radius since velocity fluctuations dominate. This, in fact, is observed experimentally as shown in Fig. 10, since the data from all four particle radii collapse onto the same line, where the defluidization Stokes’ number is presupposed to be linearly related to the particle radius a.

Granule consolidation

The granulated form of a powder is often of an intermediate nature as in the case of detergents or a granulated pesticide which must be subse- quently dispersed in some fluid agent such as water. The speed and quality of the dispersion is largely affected by granule voidage. For a given measure of solid bridge strength, the voidage of the granule will also control overall granule strength and. the resulting rate of granule break- age or attrition within the granulator. Hence, in addition to granule size, granule voidage is an equally crucial structural parameter and will be determined by the rates of granule consolidation within the granulation apparatus. Given the im- portance of granule voidage, it is appropriate to examine the phenomena of consolidation within the context of the previously presented framework of granulation,

Consider a formed granule composed of sev- eral primary particles interjoined by pendular bridges. Both compacting as well as break-up forces will act upon the granule as it moves about the granulation apparatus. Such forces are a result of the various collisional velocities described pre- viously and, hence, are also related to both the rate of coalescence and consolidation. The impul-

269

sive action of these forces will in general be op- posed by the binder viscosity.

To ascertain the possible rate of consolidation of a multiparticle, multibridge granule, it is expe- dient to consider the idealized rate of consolida- tion of two particles bound by a single liquid bridge. Within the noninertial St N regime of gran- ulation, particles do not have sufficient initial kinetic energy to overcome the strength of the dynamic pendular bridge. This implies that colli- sional forces are insufficient to rupture a wet granule on average and instead contribute to granule consolidation alone. Binder viscosity will act to impede the rate of such consolidation. With a certain initial kinetic energy imparted by the collisional force, the interparticle gap x will de- crease by a distance of

Ax =h(l -epstv) (28)

as obtained from eqn. (4) with a final relative velocity u = 0 and where the thickness of the binder layer h is taken as the initial gap distance. Hence, the change in gap distance Ax tends to zero for large bridge viscosities. Considering the binders of the previous section, we obtain values of Ax/h of 0.48, 0.95 and 0.99 for the CMC- Na(M), PVP and Klucel binders, respectively. Of course, the actual rate of consolidation is expected to be much less as a result of particle-particle interactions. Here, we have chosen a large relative collisional velocity such as that experienced dur- ing a wake collision or U, = U, and the primary particle as the relevant particle size so u0 - 2U,- 100 cm/s, a = 50,um and p =2.5 g/cm3. As given earlier, the average bridge viscosities cor- rected for the bed temperature are 1.7, 0.36 and 0.30 P for the CMC-Na(M), PVP and Klucel binders, respectively. This would imply that the PVP and Klucel granules would consolidate rather more quickly than the CMC granule, which would require several more interparticle collisions to achieve the same reduction in gap distance. However, the CMC granule is expected to dry at a faster rate than the PVP and Klucel granules as indicated by its viscous strengthening time con- stant of 1.2 min as compared with 4.6 and 5.7 min for the Klucel and PVP, respectivc:iy [37]. As a result, we would expect the CMC-Na granules to have the largest final voidage, followed by the Klucel and PVP granules. An examination of SEM photography of the product granules does indeed support this conclusion. As seen from Fig. 11, the CMC granule is observed to have a more open friable structure than that of the Klucel granule. A tapped bulk density of the CMC-Na granule product of 0.72 g/cm2 as com- pared to the densities of 0.84 and 0.81 for the

Klucel and PVP granules, respectively, further suggests the higher voidage of the CMC-Na granules.

In summary, we find that although the degree of coalescence may be independent of bridge vis- cosity and granule kinetic energy within the non- inertial regime of granulation, the rate of granule consolidation is not. Low bridge viscosities and high granule kinetic energies increase the rate of consolidation, leading to lower granule voidages. The actual time scale of consolidation would de- pend on the frequency of collisional impacts in addition to the degree of gap reduction achieved with each impact. Within the inertial St, regime of granulation, a sufficient number of regions exist within the bed in which granule rebound occurs subsequent to a collision. Since this is the case, the rate of granule consolidation should be rapid as compared with the noninertial StN regime. Fur thermore, the role of the collisional forces should shift from that of consolidation to that of break-

(a) CMC-Na(M) granule

(b) Klucel granule

Fig. 11. Examples of fluid-bed granule morphology (U- U,,,r = 0.25 m/s, 75 “C, 8 h granulation time).

270

up. As the Stokes’ number St, increases still fur- ther, we enter the coating or StC regime where clearly there is no consolidation and only break- up forces exist throughout the bed.

Conclusions

Previous agglomeration research of powders stemming from the early work of Rumpf [ 1] has been built around the strength of a static pendular bridge. The strength of such a bridge is due to the capillary pressure arising from the curvature of the interface and a contact line force with both contributions being a result of the air-liquid in- terfacial tension. In a dynamic powder process such as granulation, however, the dissipation of energy at particle point contacts due to bridge viscosity makes an additional and often dominant contribution to the strength of a dynamic pendu- lar bridge. The strength of the dynamic pendular bridge can exceed its static counterpart by an order of magnitude, since the viscous contribution to bridge strength is singular in the gap distance.

From a microlevel consideration of the condi- tions required for granule rebound, a classifica- tion of granulation mechanisms has been established based on a viscous Stokes’ number St,. The Stokes’ number St, measures the level of relative kinetic energy between two colliding gran- ules in comparison to the viscous dissipation brought about by the pendular bridge. In such an analysis, neglect of the static, capillary contribu- tion to bridge strength has been justified as a first approximation. Hence, the present approach of retaining only the viscous contribution to bridge strength significantly differs in direction from the previous approaches to granulation phenomena based solely on the static, capillary bridge strength.

In the noninertial StN regime of granulation where St, -c St,* throughout the granulation ap- paratus, all granule collisions successfully lead to coalescence provided binder is present. Hence, the rate of coalescence is independent of binder vis- cosity and kinetic energy and critically hinges, instead, on the distribution of binder throughout the bed. On the other hand, the rate of granule consolidation and, hence, indirectly the rate of attrition are influenced by binder viscosity and collisional kinetic energy. Large binder viscosities or low collisional kinetic energies lead to low rates of granule consolidation and high granule voidages. So within the StN regime of granula- tion, granule coalescence is primarily controlled by such operating variables as binder feed rate and distribution and spray drop size, whereas granule consolidation is additionally influenced by

binder viscosity and variables such as fluid-bed excess gas velocity, pan angle or impeller speed. The experimental influence of drop size for fine powders as well as the inherent instability of high-temperature agglomeration processes are but some examples demonstrating the existence of the noninertial regime of granulation.

A distribution of Stokes’ numbers will exist throughout the granulation apparatus. If local regions of high granule kinetic energy are such that St, exceeds the critical Stokes’ number St:, the granulation will fall within the St’ or inertial regime. Here, increases in binder viscosity or de- creases in granule kinetic energy represented by, for example, fluid-bed excess gas velocity or mixer impeller speed increase the rate of granule coales- cence as traditionally expected. The influence of binder viscosity on granule consolidation should not be as critical as within the noninertial StN regime. This is implied from the fact that granules often have sufficient energy to rebound and, hence, consolidation of a formed granule should occur with relatively few collisons. Finally, in the-se where the spatially averaged Stokes’ number St, is of the order of the critical Stokes’ number St,*, we obtain the coating or StC regime of granulation in which permanent coalescence does not occur. In such a regime, further growth can occur by an onion-skin binder layering mechanism.

The granulation of an initially fine powder can successively exhibit all three regimes of granulation as the average granule diameter increases. As supported experimentally, the initial growth within the noninertial StN regime. Furthermore, from the experimental work presented, it would appear that the inertial St’ regime of granulation occurs over only a narrow range of granule sizes. This is significant in that one traditionally envisions an inertial regime of granulation where, in fact, the transition from the noninertial to the coating regime may be quite short for some processes.

The precise transitions or borders separating the regimes of granulation depend on the various collisional velocities throughout the bed and an attempt at estimating these transitions has been presented here. This attempt is preliminary at best and further progress in this area will certainly require more experimental investigation into the hydrodynamics of the dense particulate phase than exists at present.

A number of implications regarding successful granulation operation arise from the above frame- work. Consider the fluid-bed granulation of a fine powder lying within the noninertial StN regime for example. An attempt to lower granule size by increasing excess gas velocity with the miscon- ceived intent of lowering the rate of granule coa- lescence could, in fact, increase granule size. This

271

is because the degree of consolidation would in- crease, leading to a possible decrease in the rate of granule attrition. Similarly, the pan granulation of fine powders such as a pesticide’ typically requires higher pan angles than, for example, the granula- tion of a coarse iron ore. This is presumably related to the higher degree of consolidation at- tained at the higher pan angles required for the noninertial granulation of the fine pesticide. As a final example, consider the effect of a local and momentary maldistribution of binder, possibly a result of a large spray drop, on the initial stage of a noninertial granulation. A large granule will form of the order of the drop size which will segregrate towards the distributor in the case of fluid-bed granulation, causing no immediate prob- lems for the remainder of the granulation. This is not the case in pan, drum or mixer granulation, where a run away granulation can occur. Since granule collisions are unlikely to have sufficient energy to rupture the newly formed large granule within the noninertial regime, it will survive and become consolidated as time progresses. Now, the reduced Stokes’ number describing the collision between the granule and the surrounding powder is, in fact, small and is actually approximately twice that of the fine powder alone as seen from eqn. (8). (Simply let the large granule radius tend to infinity.) As a result, the large granule will act as a sink adsorbing surrounding powder. If the initial number of such granules is sufficient, the end granulation product will be a combination of large granules and partially granulated finer powder.

The present work has treated only the processes of granule consolidation and coalescence. Compet- ing phenomena of equal merit are granule attri- tion, particle wetting and the rate of particle penetration into the spray drop to name but a few. Some progress from the microlevel has been achieved in regards to these phenomena - particu- larly for the case of granule attrition [42,43]. Nevertheless, the above examples already serve to illustrate the engineering utility of a microlevel based classification of granulation regimes. The present work provides a beginning framework from which to proceed in developing a rational theory of the granulation process.

Acknowledgement

This research was supported by a grant from the International Fine Particle Research Institute (IFPRI). Insightful conversations with Professor Roland Clift, Dr. Jonathan Seville, and Dr. Mojtabu Ghadari of the University of Surrey regarding bed hydrodynamics and defluidization are acknowledged with appreciation.

List of symbols

a particle or granule radius, pm a” reduced granule size, pm (eqn. (9)) a\du/dy ( mean granule collision velocity, cm/s d

co Ca

lduidy 1

&I

e e”

F

F =P

Fvis

h

ha

ho m cl r R

UO

u mf

u mfo

X

Z

capillary pressure of pendular bridge, dyn/cm* capillary pressure of touching spheres, dyn/cm* capillary number, Ca = p-Y/y local time-averaged granule velocity gradient, 1 /s fluidized-bed gas bubble diameter, cm particle coefficient of restitution modified sticky granule coefficient of restitution, eqn. (21) dynamic pendular force, dyn (eqn. (la)) capillary contribution to bridge force, dyn (eqn. ( lb)) viscous contribution to bridge force, dyn (eqn. (1~)) binder layer thickness covering collid- ing granules, pm characteristic length scales of surface asperities, pm particle half-gap distance, pm granule mass, g reduced granule mass, g (eqn. ( 10)) radial co-ordinate ratio of mean to fluctuational granule velocities, eqn. (20) fluidized-bed gas bubble radius, cm fluidized-bed bubble spacing, cm capillary Stokes’ number, eqn. (12) defluidization Stokes’ number viscous Stokes’ number, eqn. (5) relative granule collisional velocity, cm/s initial relative granule collisional ve- locity, cm/s relative granule rebound velocity, cm/s radial dense phase velocity, cm/s angular dense phase velocity, cm/s r.m.s. average of granule fluctuational velocities, cm/s particle velocity fluidized-bed gas bubble velocity, cm/s modified minimum fluidization veloc- ity due to viscous layers, cm/s initial minimum fluidization velocity, cm/s half-gap distance between colliding granules, pm axial co-ordinate

272

Greek symbols a proportionality constant, see eqn. (27) 5’ binder air-liquid interfacial tension,

dyn/cm 6 dimensionless bubble spacing, s/R, AX collisional change in granule gap

distance, cm E dimensionless gap distance, 2h,la E mf minimum fluidization voidage % angular position in spherical co-ordi-

nate, see Fig. 3 Ll binder viscosity, P

: bridge half-filling angle, rad inviscid velocity potential

P granule density, g/cm3 w pan, drum or impeller rotational

speed, l/s

Subscripts and superscripts (except as noted above) C refers to granule coating regime I refers to inertial granulation regime N refers to noninertial granulation regime I-C transition from inertial to coating

granulation regime N+I transition from noninertial to inertial

granulation regime N-C transition from noninertial to coating

granulation regime - spatial average * critical value of associated Stokes’

number, eqns. (6) ( 13) and (27)

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