POWDER TECHNOLOGY Powder Technology 82 (1995) 37-49 ELSEVIER Population balance modelling of drum granulation of materials with wide size distribution A.A. Adetayo a, J.D. Litster a,*, S.E. Pratsinis b, B.J. Ennis c aDepartment of Chemical Engineering, University of Queensland, Qld. 4072, Australia b Department of Chemical Engineering, University of Cincinnati, Cincinnati, OH 4522L USA c Du Pont de Nemours & Co., Wilmington, DE 19880-0402, USA Received 1 January 1992; in revised form 14 August 1993 Abstract A population balance model is developed to describe the drum granulation of feeds with a broad size distribution (e.g. recycled fertiliser granules). Granule growth by coalescence is modelled with a sequential two-stage kernel. The first stage of granulation falls within a non-inertial regime as defined by Ennis et al. (Powder Technol., 65 (1991) 257-272), with growth occurring by random coalescence. The size distribution is observed to narrow and quickly reach an equilibrium size distribution. Further growth then occurs within a second inertial stage of granulation in which the granule size distribution broadens and requires a size-dependent kernel. This stage is much slower and granule deformation is important. Non-linear regression is used to fit the model to the experimental data of Adetayo et al. (Chem Eng. Sci., 48 (1993) 3951-3961) for granulation of ammonium sulfate, mono-ammonium phosphate and di-ammonium phosphate for a range of moisture contents, granulation times and initial size distributions. The model accurately describes the shape of the granule size distributions over the full range of data. The extent of granulation occurring within the first stage is given by kit1; the extent of growth klt~ is proportional to the fractional liquid saturation of the granule, Ssat, and increases with binder viscosity. Here, kl represents the rate constant for the first stage of growth and tl represents the time required to reach the final equilibrium size distribution for the first stage. Changes to the initial size distribution affect k~tl by changing granule porosity and, therefore, liquid saturation. A critical saturation, Sm,, is necessary for the second stage of granulation to occur, leading to further growth. For S s a t ~ S c r i t , a final equilibrium size distribution is reached before 5 min of granulation time. For S,t > S~t, granules are sufficiently deformable to continue growing for up to 25 min. S=it decreases with increasing binder viscosity. This model is suitable for use in dynamic simulation of granulation circuits where both moisture content and recycle size distribution may vary significantly with time. Keywords: Granulation; Size distribution; Population balance; Modelling 1. I n t r o d u c t i o n Granulation is a key process in the pharmaceutical, food, ore processing and fertiliser industries. Drum granulation is one of the most commonly used granulation processes for its simplicity and ease of operation. Fig. 1 shows a schematic diagram of a typical industrial drum granulation process or circuit. Recycled seed granules are fed to the granulation drum. Fresh feed (slurry, solution or melt) is sprayed onto the seed granules and the granules grow. Granules leaving the granulation drum are first dried and then screened to separate out the product size. Product size specification * Corresponding author. 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0032-5910(94)02896-V is often very strict, e.g. 90% - 3 mm + 1 mm. Oversize granules are crushed and recycled with fines. Fertiliser granulation circuits are difficult to optimise and control. Often only a small fraction of granules leaving the granulation drum is in the product range. Recycle ratios can be as high as 5 or 6 to 1 [1]. Limit cycle behaviour, surging and drifting in mass flow rates and size distributions are common, sometimes leading to catastrophic results. Dynamic simulation of the granulation circuit may be an aid to circuit optimisation and control. However, a reliable granulation model to predict the granule size distribution exiting the drum as a function of process variables is a prerequisite to such a simulation. Population balance modelling has been used extensively for modelling agglomeration in many systems: 38 A.A. Adetayo et al. / Powder Technology 82 (1995) 37-49 F~feed the coalescence rate constants and extent of granulation are discussed. 2. Theory 2.1. Mechanisms of granule formation DRIER [IDry granules -~ SCREENS ~_ Oversize Product ~ T CRUSHER Underside I ~ Recycleseedgranules Fig. 1. Schematic diagram of a typical fertiliser process [1]. aerosols [2,3], pelletisation [4-6] and crystallisation [7,8]. Pelletisation is similar to fertiliser granulation. However, there are some differences. Many fertilisers are highly soluble, so that the amount of liquid binder and its properties will vary significantly with fertiliser chemistry and process conditions. Most fundamental studies of pelletisation have used fairly narrow initial size distributions of fine powders. Pellets produced from narrowly sized feed powder are relatively easily broken or squashed to the unit particle size. An exception to this work is that of Linkson et al. [9] who investigated the dependence of growth mechanisms on feed size distribution. For batch drum granulation, narrowly sized feeds grew indefinitely by crushing and layering due to weak granule strength, whereas wide feeds grew by coalescence and obtained an equilibrium size distribution due to high granule strength. In the case of fertiliser granulation studied here, the recycle stream entering a continuous drum has a very broad size distribution of hard rigid particles, ranging from the desired product size (or even larger) to fine powder, and so the predominant size enlargement mechanism is coalescence [1]. Adetayo et al. [10,11] studied fertiliser granulation experimentally. Laboratory scale batch experiments were performed using broad initial particle size distributions similar to those of the recycle stream in a granulation circuit. The effect of fertiliser chemistry, initial size distribution, binder properties and liquid content on the kinetics and extent of granulation were measured. A two-stage mechanism for the granulation mechanism was identified. This present paper presents a population balance model for fertiliser granulation by coalescence. A twostage coalescence kernel is used. Size distributions predicted by the model are compared with the experimental data of Adetayo et al. [10]. The effect of initial size distribution, fertiliser type and liquid content on Ennis et al. [12] investigated the forces involved in the collision of two spherical particles in order to establish an understanding of the fundamental mechanisms of granule formation. Both the capillary and viscous contributions were found to affect significantly the bonding mechanism of colliding particles. The viscous Stokes number, Sty, was defined as the ratio of the relative kinetic energy between colliding particles to the viscous dissipation brought about by the pendular bond, and is given by S t y _ 8pgrV 9/z (1) where V is the velocity of granule collision, pg is the granule density, f is the effective granule size and t~ is the viscosity of the binding fluid. Sty increases as granule size increases or binder viscosity decreases. A critical viscous Stokes number St* must be surpassed for rebound of colliding particles to occur, where where e is the particle coefficient of restitution, h the thickness of the binder layer and h, a measure of the granule's surface asperities. Three granulation regimes were defined in terms of the magnitude of Stv in comparison to St*: Stv << St* Sty = St* St~ >> St* non-inertial regime (all collisions successful) inertial regime (some collisions successful) coating regime (no collisions successful) For fine powders, growth typically begins within the non-inertial regime of granulation. As granule size and Stv increase during granulation, the process may move through the inertial regime and finally end in the coating regime. The exact demarcation between regimes depends on the velocity of collision, the sizes of colliding particles and the properties of the binder. In general, the collision velocities of granules or particles within a process are difficult to ascertain. In the case of drum granulation, possible estimates for V are ~o [12] or ~.Ro) [10], where R and ~o are the drum diameter and rotational speed, respectively, and ~ is a numerical constant. For this work, it is not necessary to estimate St* with great accuracy. The order of magnitude relationship between Sty and St* is of interest. Typically, St* ~0(1), implying that a ~ 0 ( 1 0 -2) for the materials A.A. Adetayo et al. / Powder Technology 82 (1995) 37-49 and granulation conditions of this work if we choose aReo as the characteristic velocity. Adetayo et al. [10] investigated experimentally the mechanisms involved in the granulation of fertilisers with broad initial size distributions. Two stages of granulation were identified. The first stage, corresponding to Ennis' non-inertial regime, was fast. Granule kinetic energy, granule size and binder viscosity were not important. During this stage, the granule size distribution narrowed. The second stage, corresponding to the inertial regime was slower and the granule size distribution broadened. Not all collisions were successful and granule deformation on collision was an important factor. Three fertilisers were used: mono-ammonium phosphate (MAP), di-ammonium phosphate (DAP) and ammonium sulfate (AS). All fertilisers followed the first stage of granulation. DAP followed the second stage of granulation for all moisture contents above 2%, MAP followed the second stage of granulation after 5% moisture content, while AS followed only the first stage of granulation at all moisture contents covered. To account for the differences in solubility between the fertilisers, the liquid phase ratio y was defined, after Sherrington [13], as the volume of liquid phase per volume of solid in the granule. The liquid phase ratio is given by the following equation: g(l +s)pf (3) Y= (1-gs)~h where g is the weight percent of water in the granule, s is the solubility of the fertiliser salt in water (g g-1 water), and Of and Pl are the densities of the fertiliser salt and solution, respectively. As discussed in greater depth later, one may define an extent of granulation which refers to the degree of growth occurring within the first stage of granulation. An increase in the extent of granulation represents an increase in average granule size and decrease in granule number. For all the fertilisers studied here, the extent of granulation increased with liquid phase ratio. However, for a given value of y, the extent of granulation increased in the order AS, MAP, DAP, showing that binder viscosity as well as fertiliser solubility were important. The extent of granulation was a complex function of initial particle size distribution, going through a maximum as the level of fines ( - 1 mm) was increased. All these effects were explained, at least qualitatively, in terms of the theory proposed by Ennis et al. [12] and the effect of process variables on the critical and resultant Stokes numbers. Note that the fractional saturation of the pores in the granule, S,a,, can be written in terms of y: Ssa, = y ( 1 - p ) (4) P where p is the volume fraction of pores in the granule. 39 2.2. The population balance The population balance for a well-mixed batch system undergoing coalescence alone is given by [14] On(v, t) at - 1 f g.~ dfl(u'v't)n(u't)n(v~) du 0 v + ~T fl(u,v-u,t)n(u,t)n(v-u,t) du (5) 0 where n(v, t) is the number density function, fl(u, v, t) is the coalescence rate kernel, NT is the total number of particles at time t, a equals zero for free-in-space systems such as aerosols and unity for restricted-inspace systems such as granulation processes. The solution to this integro-differential equation is not a trivial matter. Known analytical solutions are only available for special forms of the coalescence kernel with an assumed initial number density distribution [15]. Numerical solutions to this equation have been obtained by various methods: moment [14], discrete [16], sectional [17] and sectional-midpoint [7] methods. Hounslow et al. [7], using volume as the particle size coordinate, divided the particle size spectrum into geometric sections (vi = 2v/_ 1). Assuming that the number density distribution in each section is constant, they proposed a sectional population balance model. The change in the granule size distribution is given by [7] dN/ 1 ( i-2 1 -~ =N--~T N'-~j'-~-~2j-/+l/3'-~dNj+ "J3/-~'/-~N2-' i--1 -N, oo 2 fl/dNj-N, 1 \ ,.j (6) ] where Ni is the number of particles in the ith interval, /3/,~ is the collision rate function (coalescence kernel) between particles in the ith and jth section. 2.3. The coalescence kernel The coalescence kernel/3i.j is an important parameter in population balance modelling. Much research has focused on determining the appropriate form of the coalescence kernel. Ouchiyama and Tanaka [18-20] attempted a derivation of the kernel by carrying out a force balance on the colliding particles. Due to the complexity and lack of adequate knowledge of the forces involved in the granulation process, they could only propose a form of coalescence kernel with semiempirical adjustable parameters. The values of these parameters depend, in part, on the degree of plasticity of the granule and they determine the order and form of the kernel. Thus, unlike aerosol systems, the form of the coalescence kernel for granulation systems is 40 A.A. Adetayo et al. / Powder Technology 82 (1995) 37--49 not completely established. The available kernels in the literature are either purely empirical or semiempirical [8,21]. It is commonly assumed [4] that the granulation kernel can be divided into two parts: 13,.j =/3o/3(v,, v,) (7) The coalescence rate constant /30 determines the rate of granulation and is a function of the granulator operating conditions, such as moisture contents, binder viscosity and drum speed. In other words, it controls the rate of change of mean of the granule size distribution. The dependence of the granulation process on the particle size is described by /3(vi, vj) which determines the shape of the granule size distribution. As two stages of granulation have been identified by Adetayo et al. [10], it is expected that a two-stage granulation kernel will be necessary to model adequately the granule size distributions over a wide range of conditions. In the first stage or non-inertial regime of granulation, the probability of successful coalescence following a collision is independent of particle size and collision velocity and, instead, depends only on binder distribution. The probability of coalescence equals the probability of encountering binder during a collision, with those collisions involving binder being successful. In addition, we assume the rate of collisions is independent of particle size; the first-stage mechanism becomes a random process [22]. This is a reasonable first approximation for a restricted-space concentrated system such as drum granulation, as opposed to freein-space dilute coalescence as in the case of aerosols. We therefore define the first-stage kernel to be a constant: /31,~}=kl (8) Growth with a size-independent kernel has previously been studied [22]. Both the total granule number and mean granule size were shown to vary as N ~o -- exp( - k,t/2) r -- = exp(k,t/6) ro (9) (10) where No is the initial particle number, and r0 and r are the initial and current mean granule size, respectively. During the second stage or inertial regime of granulation, the granule size distribution widens. Particle deformation is important and, therefore, collisions involving large granules are favoured due to their increased inertia upon impact. A size-dependent kernel is necessary to treat this stage of granulation. Empirical and semi-empirical kernels of various orders in volume have been proposed in the literature [23-28]. Though a number of first-order kernels were evaluated, the dig ferences among the most common kernels by Golovin [24]: /3!5] - = k2(v, + vj) (11) and Thompson [25]: /3}ff= k2(v,- vj)2/(v, + vj) (12) were not significant for the broad initial size distribution employed here [23]. As a result the simplest one, Eq. (11), is selected in this study. The first stage of granulation (non-inertial regime) is fast, relative to the duration of the experiments [10]. For experiments where the second stage of granulation does not occur, an equilibrium size distribution is quickly reached. For the fertilisers studied here, the second stage of granulation (inertial regime), which is slow, only occurs after coalescence in the non-inertial regime is complete. Due to the differing time scales in growth mechanisms, we propose a sequential kernel for both stages of granulation: [tiP] /3.= i /3SJ t <~tl (13) t > tl where /3P.} and ~.,.j/~!2!are given by Eq. (8) and Eq. (11) or (12), respectively. Here tl represents the time required to reach the final equilibrium size distribution of the first non-inertial stage of granulation. Experimental data are available for granulation times from 5 to 25 min. Previous work [10], however, indicates the first noninertial stage of granulation is complete within 5 rain (i.e., t 1< 5 min), at which point the first size distributions are measured. With the present data, therefore, it is not possible to distinguish between differences in the rate of granulation and the first-stage rate constant kl. One may note from Eqs. (9) and (10), however, that the group k l t is clearly a measure of the extent of granulation and, in particular, kit1 defines the final extent of granulation occurring within the first stage of non-inertial granulation. While it is not possible to determine kl directly, it is possible to determine values of the extent of granulation, or klta. For the present work, this is achieved by arbitrarily choosing t~ = 2 min as the time for completing the first stage of granulation and for switching the form of the growth kernel. By minimising the error between experimental granule size distributions and numerical solutions to the population balance equation, as discussed later, one determines values of k, as well as klt,. Given the arbitrary choice of t,, the values of kl reported here are actually a measure of the extent of granulation at which the first stage of granulation is complete, and differences in k~ do not directly imply differences in granulation rate. Where the second stage of granulation does not occur, i.e. t~!2J= 0, the population balance solved for t > t~ gives the equilibrium granule distribution for coalescence in the non-inertial regime only. A,4. Adetayo et al. / Powder Technology 82 (1995) 37-49 2.4. Solution of the population balance and estimation of the coalescence rate constants After a systematic comparison of the various numerical solutions to known analytical solutions of the general pQpulation balance equation, Hounslow's sectional model solution was found to be adequate for modelling the fertiliser granulation process [23]. For given values of kl and k2, Eq. (6) is solved with the coalescence kernel given by Eq. (13). Twenty-two size intervals were used with the first top size being 0.25 mm. Thus, a total size range of 0.25 to 32 mm is covered ensuring there is always at least one empty size interval at the top of the size range so as to avoid finite domain error [17]. Eq. (6) is a series of 22 ODEs, which are solved using the Fehlberg fourth-fifth order Runge-Kutta method [29] for residence times up to 25 min. The predicted size distributions at 5, 15 and 25 min are compared to measured ones and the best values of kl and k2 for a given set of data are estimated by non-linear regression. The Marquardt Compromise method [30] is used. This routine combines the steepest descent and the linearisation methods, and has the advantage of fast convergence as well as being relatively robust. The 'best' parameters are estimated by minimising the sum-of-squares error between the simulated and experimental cumulative size distributions. The objective function, J, is defined by J= • ~lc--~j(t~) -cumj(ti)l 2 j (14) i where cumj(ti) is the experimental cumulative mass fraction at time ti, cumj(ti) is the simulated cumulative mass fraction, j is the number of size intervals and i is the number of time intervals. 41 This technique is used to estimate the coalescence rate parameters, kl and k2, for the experimental granulation data of Adetayo et al. [10]. 3. Results and discussion Model simulations were compared with the experimental granulation data of Adetayo et al. [10]. Data were available for three fertilisers: DAP (2, 4, 5, 6% moisture), MAP (3, 4, 5, 6% moisture) and AS (4, 6, 8% moisture). Data were collected at three granulation times: 5, 15 and 25 min. All fertilisers were granulated from the same standard initial size distribution (Type I in Table 1). This size distribution was typical of the recycle size distribution for DAP in one operating granulation plant. DAP was also granulated from two other initial size distributions containing progressively finer particles (Table 1). 3.1. Characteristics of constant and first-order kernels Before comparing the model with experimental data, it is useful to look at the characteristics of the two kernels presented above. Fig. 2 shows the effect of the choice of kernel on the shape of the predicted granule size distribution. The graph shows two size distributions, each with the same median particle size (4 mm), produced by solving the population balance with each of the two kernels. All simulations started with the same initial granule size distribution shown in Fig. 2. The constant kernel produces a much narrower size distribution than the first-order kernel. With the constant kernel, fine particles from the initial size distribution rapidly disappear with little change to the coarse end of the size distribution. In contrast, the first-order kernel broadens the initial size distribution and fails Table 1 Different initial particle size distributions Initial size distribution I II III Type standard (20% < 1 ram) fines (30% < 1 m m ) more fines (50% < 1 nun) Mass m e d i a n diameter, Dso (mm) 2.0-1-0.1 1.8+0.1 1.0+0.05 Mass m e a n diameter, D43 ( m m ) 2.53-1-0.1 1.97+0.1 2.01+0,1 Sauter m e a n diameter, D32 (ram) 1.24+0.05 0.89+0.05 0.97±0.05 Bulk porosity, pa 0.38:1:0.02 0.32±0.01 0.34-1-0.01 a M e a s u r e d for dry-packed bed of particles with this size distribution. Bulk porosity= 1 - t ~ / p s ~ P . 42 A.A. Adetayo et al. I Powder Technology 82 (1995) 37~I9 1.0 1.0 g O 03 ¢0 0.8 . "' " o., ' '""/ ' f ... / / , 0.4 /A" 0.2 ~ll~ ~'¢/ ,~ el E .~ e /f/ / ...' ' (a) 0.8 ~ 0.6 0.4 0.2 o.o w 1.o "' / ~ , ~ ....~//~,,' 0.s = L) 0.0 i 2 I 4 i 6 8 10 Diamet.er (ram) Fig. 2. Comparison of cumulative granule size distributions for three different granulation kernels: - - , fll,y=k~; - - - , 13i.y=k2(u + v ) ; - -, fl~,y=k~ and k2(u+v) for type-I initial size distribution ( . . - ) of D A P with 4% moisture, and after 25 min (&). E .9 1.0 0.6 0.4 = 0.2 o g 0.0 ~ 0.8 ~ E ._~ 0.6 ~ 0.2 0.4 ~ 0.2 /';° 0.0 1.0 (c) g o.a ~ m E .~ o.6 ~ 0.2 0,4 .. i IS? A / ' " :.." /1(," 0 2 8 Diameter ~ ..... 10 (ram) Fig. 4. Predicted and experimental data for MAP with standard initial distribution ( . - - ) at (a) 4%, (b) 5% and (c) 6% moisture contents, and after 5 min ( - - , model; ©, exp.), 15 min (. . . . , model; I-1, exp.) and 25 min ( - - - , model; A, exp.). 0.4 0.0 1.0 ,P_ 0.8 and longer granulation times for DAP and MAP. Th6 first-order kernel (Eq. (11)) fails to predict the total 0.6 E 0.4 E .~- 0.0 1.o (b) ._~ d 0.6 (a) 0.8 E .>_e =o E 0.2 0.0 2 4 6 Diameter (ram) 8 10 Fig. 3. Predicted and experimental data for AS with standard initial distribution ( - - . ) at (a) 4%, (b) 6% and (c) 8% moisture contents, and after 5 min ( - - , model; O, exp.), 15 min ( - - - , model; 17, exp.) and 25 min ( - - - , model; /% exp.). to remove completely the fine particles even after a significant extent of granulation. 3.2. Comparison o f model simulations with experimental data None of the single kernels shown in Fig. 2 could, on its own, correctly predict the shape of the granule size distribution for the full range of data. The constant kernel (Eq. (8)) does not predict the broadening of the granule size distribution at higher moisture contents removal of the fine particles from the initial size distribution. The experimental data do, however, support the proposed two-stage mechanism: fast, random coalescence in the non-inertial regime, followed by slower preferential coalescence in the inertial regime. For example, in Fig. 2, the experimental granule size distribution for DAP at 4% moisture content after 25 min granulation time is compared to simulations with single kernels and the two-stage kernel of Eq. (13). Only the two-stage kernel can predict the correct shape of the granule size distribution. This kernel is used for all subsequent modelling. The evolution of the measured cumulative granule size distributions (symbols) along with the model predictions (lines) are shown in Figs. 3 to 7. For all moisture contents of AS (Fig. 3), 3 to 5% moisture for MAP (Fig. 4(a), (b)) and 2% moisture for DAP (Fig. 5(a)), an equilibrium size distribution is reached within 5 min residence time. The final granule size distribution is narrower than the initial size distribution with the almost total removal of the finest particles in the distribution. For MAP with 6% moisture (Fig. 4(c)) and all other DAP experiments (Figs. 5(b), (c), 6(a)-(c), 7(a)-(c)), the size distribution is narrowed after 5 min A.A. Adetayo et aL / Powder Technology 82 (1995) 37-49 c o +~ 43 c 1.0 ~-'°~ 0.8 la) 1.0 0.8 .'" ~ '...... ..... i........ 0.6 E •-~ 0.4 0.4 0.2 ": n'" = L) ~J O.O "~*~ 0.81.0 Ib) 0.0 1.0 (b) o.8 L- ~-~f~" " :........ ~ 0.6 o., t . . . . ~, .." i' 0.2 1.o (c) ~ . . . . . . , A .s" f A ..,"" A ..."" •: 0.0 , 1.0 | (el 0.8 o ..'"" 0.4 "-~ , o ........... ............... 0.8 0.8 0.6 : 0.4 o .:' 0.2 0.0 /" o A / /.... ." .0- ~: ~ ~..1~'""" 4- Diameter 6 (mm) 8 10 Fig. 5. Predicted and experimental data for D A P with standard initial distribution ( - - - ) at (a) 2%, (b) 4% and (c).6% moisture contents, and after 5 min ( - - , model; C), exp.), 15 min ( . . . . . , model; I-7, exp.) and 25 min ( - - - , model; A, exp.). 0 . .." o :/ 0.2 "~ 2 ' 0.4 o ..... ~ .......*.......... 0.0 0 2 4 6 Diameter (mm) 8 10 Fig. 6. Predicted and experimental data for D A P type-II initial distribution ( - - - ) at (a) 4%, (b) 5% and (c) 6% moisture contents, and after 5 min ( - - , model; 0 , exp.) and 25 min ( - - - , model; A, exp.). 3.3. Effect of liquid content and fertiliser type on kl and k2 granulation time, but further granule growth occurs for at least 15 min. The second stage of granulation broadens the granule size distribution. The model gives a reasonably good fit to the full granule size distributions for the complete range of data. The estimated parameters, k~ and k2, for each fertiliser are given in Tables 2 and 3. For experiments where an equilibrium size distribution is quickly reached (all AS and DAP with 2 to 3% moisture, MAP with 3 to 5% moisture) the best estimate of k2 is not statistically significantly different f r o m zero, i.e. the second stage of granulation does not occur. These simulations clearly match the narrowing of the size distribution on granulation extremely well. For other experiments, MAP with 6% moisture and DAP with more than 3% moisture, the model nicely follows the increase in mean granule size and spread of the size distribution with time. The model predicts that the rate of change of granule size will be slowed after 15 min which is in agreement with the DAP experimental results (Fig. 5(c)). This figure and Figs. 6(b), (c) and 7(c) also show the ability of the model to match the shape of the granule size distribution at a high extent of granulation. Fig. 8(a) shows the first-stage extent of granulation, kit1, for three fertiliser types as a function of liquid content, expressed as the fractional saturation of the granule, Ssa,. Shown for comparison are values for the first-stage extent kltl of limestone granule growth taken from the drum granulation work of Kapur [21]. A porosity of 0.48 was assumed for the limestone data. The extent k~tl increases linearly with fractional saturation for all fertilisers. An increase in granule saturation should increase both the extent and rate of granulation. In terms of Ennis' theory, an increase in the granule saturation will reduce the coefficient of restitution and will increase the effective liquid layer thickness around colliding particles, thus increasin~ t~::z critical Stokes number (St*) that must be exceeu,.,~ for colliding particles to break apart. This increases the size of particles that will make the viscous Stokes number (Sty) less than St* and will therefore successfully coalesce on collision, thereby increasing the extent of granulation. Increasing the fractional saturation of the granule will also increase the rate of granule encounters involving binder, thereby increasing the rate of granulation. Rate constants for the first stage of random growth have A.A. Adetayo et al. / Powder Technology 82 (1995) 37-49 44 ~ 1.0 ,,~ 0.8 25 20 .... ~a E g -.~ 15 0.6 "2 0.4 10 0.2 5 m 0.0 1.0 I 0.0 Ca) I t 0.2 0.4 0.6 Fractional saturation S,t [-] ~ I 0.8 1.0 0.8 0.6 ~ 0.4 ~ 0.2 10 15 d ~ ZX 0.0 1.0, (c) • .... 0.8 0 to 0.0 o.6 E ~> f : o 0 .--~" .-K .-N' 0.0 2 4 6 "~0 8 Diameter (ram) Fig. 7. Predicted and experimental data for DAP type-III initial distribution ( . - - ) at (a) 4%, (b) 5% and (c) 6% moisture contents, and after 5 min ( - - , model; O, exp.) and 25 min (-- -, model; A, exp.). Table 2 Coalescence rate constants for AS and MAP Moisture content (%) t I o.o o 2 o~ o.6 o.a 0.1 0.2 0.3 Fractional saturation Ssat [-I ~.o 0.4 0.4 0.2 ~.1 E] ix [] Solution phase ratio, y Initial distrib. kl (regressed) 4 6 8 0.106 0.165 0.228 type I type I type I 2.855:0.1 3.90+0.2 4.80 + 0.2 MAP 3 4 5 6 0.068 0.090 0.115 0.135 type type type type 1.605:0.03 1.70+0.3 2.30+0.2 3.00 + 0.2 k2 (regressed) AS I I I I 0.00025:0.0006 0.0013+0.001 0.0006 + 0.0004 -0.015+0.1 -0.016+0.01 -0.006+0.002 0.012 + 0.003 Fig. 8. (a) Extent of granulation, kttl, as a function of fractional saturation and (b) extent of granulation normalised for differences in St~o as a function of fractional saturation (inset: comparison to data of Kapur [21]): O, AS; f"l, MAP; A DAP; II, limestone (Kapur [21]). MAP solutions have similar viscosities and similar rate constants. Qualitatively, the variation in extent of granulation with fertiliser type fits the theory of Ennis. The first-stage growth, as defined by Eqs. (9) and (10), is for a particular level of binding fluid. Since growth is effected byy, we can reflect this in the growth kinetics by defining kl=k*f(y) where k* is the rate of growth with all collisions occurring in the presence of binder (f(y) = 1). Replacing kl with k* in Eq. (10), the extent is given by 1 k~'tl= ln(r~) where r I is the final mean granule size of the first stage of growth. By equating Eqs. (1) and (2), the final mean granule size is given by ra = 8 ~ v S t * previously been shown to depend exponentially on liquid phase ratio, which in turn affects the fractional saturation of the granule's pore space [6]. For a given fractional saturation, the value of kit1 is the same for AS and MAP within experimental error. DAP has a significantly higher rate constant. Some properties of the three fertilisers are shown in Table 4. DAP solution has the highest viscosity and, for a given particle size, the lowest viscous Stokes number (Sty). We therefore expect DAP to show the greatest extent of growth in the non-inertial regime. AS and (15) Pg (16) Upon substitution into Eq. (15), a modified extent of granulation is given by kit1 = 6 ln(St* /Stvo)f(y) (17a) with Note that we are assuming in the present work that the volumetric mean (as used by Kapur [21]) and the harmonic mean (as used by Ennis et al. [12]) can be interchanged. For the equilibrium, narrow size distributions of the first stage of growth, this may not be a bad first approximation. 45 A.A. Adetayo et aL / P o w d e r Technology 82 (1995) 37-49 Table 3 Coalescence rate constants for D A P Initial size distribution Moisture content y S~.t kl (min -x) k2 (ram -3 min - t ) kl a (min - I ) (%) Type I 2 4 5 6 0.~5 0.~2 0.117 0.144 0.073 0.150 0.191 0.235 1.38±0.1 3.39±0.1 4.14±0.1 5.98±0.1 -0.0~±0.002 0.0043±0.001 0.~33±0.~2 0.0006±0.0004 1.2±0.1 3.3±0.1 3.9±0.1 4.5±0.1 TypelI 3 4 5 6 0.069 0.~2 0.117 0.1~ 0.113 0.202 0.255 0.312 4.1±0.3 5.2±0.2 0.0041±0.~1 0.~23±0.~I 3.0±0.1 4.1±0.1 4.6±0.1 5.8±0.1 3 4 5 6 0.069 0.092 0.117 0.1~ 0.126 0.169 0.215 0.~0 2.02±0.2 3.18 ± 0.1 4.1±0.21 5.1±0.2 0.012±0.02 0.011 ± 0.003 0.~6±0.0~ 0.006±0.002 2.1±0.3 3.4±0.1 4.2±0.1 5.2±0.1 Type III "kl values by regression with k2 fixed at zero for Ssat~<0.13 and 0.005 ( m m -3 min -1) for S~.t>0.13. Table 4 Properties of the materials Material Ix (P) p (g em -3) ~o (s -1) ro (cm) 1~Sty ln(odStv) A AS MAP DAP I D A P II D A P III Limestone [21] 0.034 0.037 0.085 0.085 0.085 0.0085 1.7 1.6 1.5 1.5 1.5 1.85 2"n'/3 2"rr/3 2~'/3 2~'/3 2~'/3 7-n-/3 0.124 0.124 0.124 0.089 0.097 0.02 0.00279 0.00323 0.00792 0.011 0.010 0.00227 - 5.8800 - 5.7349 -4.8385 -4.5070 - 4.5930 -6.0859 27.7 + 3 25.8 + 3 39.9+3 38.4+5 39.1 ± 2 21.8+2 Stvo = 8Vp~ro 9l~ (17b) One may note, in particular, that the modified extent of granulation varies with ln(tx/pg). Normalising the data of Fig. 8(a) with respect to the viscosity and granule density of DAP (see Appendix), the extents of granulation for all three fertilisers are seen to collapse onto the same line, as shown in Fig. 8(b). In addition, the limestone data of Kapur [21] are seen also to collapse onto the same common line, as illustrated in the inset of that figure. The value of k2 is zero for all fertilisers at low moisture content. A critical saturation must be reached before the second stage of granulation proceeds. The second stage of granulation relies on plastic deformation of colliding granules at the point of contact to achieve successful coalescence [10]. The role of granule plasticity on growth in drum granulators was realised as early as the work of Newitt and Conway-Jones [31]. Kristensen et al. [32] showed that moist granules showed increasing plastic behaviour as liquid saturation increased. Thus, a certain minimum moisture is required to give sufficient plastic deformation for the second stage of granulation to proceed. This is analogous to the pelletisation of closely sized powders where significant pellet growth by coalescence only occurs for a very narrow range of moisture contents (liquid saturations) [31]. The value of the critical saturation increases in the order DAP (0.13 + 0.01), MAP (0.20 + 0.02), AS (greater than 0.36). This is also the order of decreasing viscosity and increasing Sty, showing that the second stage of granulation proceeds most easily for the fertiliser solution that forms the most viscous binder. Low viscosity binders allow a greater degree of granule compaction [12] which will, in turn, impede granule deformability. For only one fertiliser, DAP, is there sufficient data to examine the effect of the fractional saturation on k2 above the critical saturation. In this region, there is considerable scatter of the data (see Table 3). This is due, in part, to the strong correlation between the two parameters, kl and k2. During the regression process, a large change in k2 is compensated for by only a small change in the values of kl; no effect of liquid phase ratio is identified. Experimental data at a greater number of time intervals are needed to pin down this effect. A.A. Adetayo et al. / Powder Technology 82 (1995) 37-49 46 3.4. Effect of initial granule size distribution on kl and k2 alescence kernel represented as kl For DAP, granulation experiments were performed with two other initial size distributions with increasing levels of fines for moisture from 3 to 6% (see Figs. 6 and 7). At all initial size distributions, both stages of granulation were followed for moisture contents of 4 to 6%. Fig. 9 shows that the granulation extent kata goes through a maximum as the amount of fines in the initial granule size distribution increases. This corresponds to the packed bed porosity of dry particles going through a minimum (Table 1). If it is assumed that this packing density represents that of particles in a wet granule, the key effect of varying the initial size distribution for a given liquid phase ratio is to vary the degree of saturation by changing the volume of voids in the granule. Fig. 10 shows a plot of k~tl against liquid saturation, calculated from Eq. (4). A single relationship is shown for all three initial size distributions. Within experimental error, all three initial size distributions have the same critical saturation for the onset of the second stage of granulation. Above the critical saturation, there is too much scatter in ka to detect any effect of the initial size distribution. 3.5. A granulation model for all fertilisers In summary, the granulation of broad size distribution feeds is a two-stage coalescence process with the coa,- 12 ..... /,, ........... ...................... 6% -,... /x m •':" '" ,.x. -~ -D . . . . . 5% 0 10 0 i i i i 20 30 40 50 -- 60 % < 1 (ram) Fig. 9. Effect of initial size distribution off the first-stage granulation extent, k~h. • .~ 12 ,a lO g 8 [] 0 ~i,j = O; t <~ta Ssa t < Scrit k:(v~+v~); S.~, > So., t>t~ (18) In addition, the first-stage extent of granulation has been shown to be given by kit I =Ay(1 -p)/p (19) with A ~ ln(St*/St~o) (20) Here, A and Scrlt are characteristic parameters of the fertiliser system dependent on binder and granule properties (especially binder viscosity) and particle interactions. The rank of these parameters for the three fertilisers agree with that predicted by the theory of Ennis [12] and, in particular, the dependence of A on binder viscosity as illustrated by Fig. 8(b). Additional data are necessary before the individual dependences of kl and tl on operating parameters or binder and granule properties can be established. The first-stage rate constant k~ is presumably independent of binder viscosity except for possible effects due to binder wetting, whereas tl should vary as In(/,). From the work of Kapur and Fuerstenau [6], kl was shown to vary exponentially with y, whereas later work by Kapur [21] demonstrated that tl decreased with increases in y. In the present work, these combined effects result in a linear dependence of the extent of granulation klta on liquid phase ratio y. A reasonable amount of data for moisture contents above Scrlt is only available for DAP. Given the scatter of data, no effect of system parameters on the second-stage rate constant, k2, can be identified. For DAP granulation in a 0.31 m diameter drum at room temperature, the values of A, k2 and Scru can be identified. Considering the high level of interaction between the two parameters [23], controlled estimate of the parameters is justified. The average value of k2 for all data above S=it is 0.005 mm -3. The non-linear regression program was rerun to find the best values for ka for all DAP data with k2 fixed at zero for Ssat~<S , , and 0.005 mm -3 for Ss,t>S=,. These values, slightly different to the original values of kl, are given in Table 3. Linear regression of the modified ka values with fractional saturation gives a value of A = 39.0 +3 for DAP. Thus for DAP, the coalescence kernel becomes: 3 { 19.5Ssat g ~ 2 /3~.j= t~<2 min 0; Ss~t ~<0.13 0.005(v~ + vj); S~,,> 0.13 0 0.00 0.05 0.10 0.15 Fractional 0.20 saturation, 0.25 0.30 0.35 t > 2 min (21) Ssa t Fig. 10. Correlation of k~t~ with fractional saturation for all DAP data: O, SD I; rl, SD II; A SD III. The Appendix shows that the normalised extent of granulation, klta, is given by A.A. Adetayo et al. / Powder Technology 82 (1995) 37-49 6 O ~o4 "ID 3 ~2 0 0 i i i i i 1 2 3 4 5 Experimental D50 (mm) Fig. 11. Comparison of the predicted and experimental median diameter Dso for all fertilisers: (3, DAP type-l; Iq, DAP type-II; A, DAP type-III; O, AS; &, MAP. klt~ =klt~ + m ln( ~--~P_iSsat \l.t'Pg] (22) Thus, from the knowledge of the first-stage extent of granulation for DAP materials, kata, the corresponding extent of granulation for other fertilisers, kxta, of known viscosity and granule density can be determined. Fig. 11 shows a comparison of the experimental and model (Eq. (22)) predicted median granule size. Excellent agreement is achieved between the model and experimental values for all fertilisers over a wide range of moistures and initial size distributions. A model of this form is suitable for use in dynamic simulation of the granulation circuit where both moisture content and recycle size distribution may vary significantly with time. 4. Conclusions A population balance model utilising a sequential two-stage granulation kernel is developed for the drum granulation of ammonium sulfate (AS), mono-ammonium phosphate (MAP) and di-ammonium phosphate (DAP). The model matches the shape of the granule size distribution of the three fertilisers for a wide range of moisture contents and initial size distributions. The first stage of granulation (non-inertial regime) is by random coalescence and an equilibrium size distribution is quickly reached. The second stage (inertial regime) requires a size-dependent kernel and is much slower. Granule deformation is important in the second stage. For all fertilisers as well as limestone granulation results of Kapur [21], the first-stage granulation extent, defined as klq, is found to be proportional to the fractional liquid saturation of the granule. It increases with binder viscosity. Changes to the initial size distribution affect the extent of granulation in the firststage k~t~ by changing granule porosity and, therefore, liquid saturation. Normalising the first-stage granulation extent of the other materials by that of DAP in order to account 47 for the effect of differences in binder viscosity and particle density shows that a single correlation exists between the first-stage granulation extent kit I and the fractional saturation of the granule's pore space Ssa, for all fertilisers. A critical saturation, S=,, is necessary for the second stage of granulation to proceed. For Ssat<Scrit, an equilibrium size distribution is reached before 5 min granulation time. For Ssat> Scri, granules are sufficiently deformable to continue growing for up to 25 min. Scri, decreases with increasing binder viscosity. The proposed model sufficiently describes the granulation process and is suitable for use in the dynamic simulation of the granulation circuits where both moisture content and recycle size distribution may vary significantly with time. The model also has great potential to be extended to other granulating materials. 5. List of symbols A cumi(t,) cumAt,) 950 e g h ha k n(v, t) N, NT No P ro rl R s Ssat st, st* t ta f) ~)i V Y fitting parameter experimental cumulative mass fraction at time ti simulated cumulative mass fraction at time t, median granule diameter (mm) coefficient of restitution percentage by weight of water in granule thickness of the binder layer (/zm) measure of the granule's surface asperity (t~m) coalescence rate constants number density fraction number of particles in ith interval total number of particles at time t initial total number of particles granule porosity (cm3 cm -3) current mean granule size (/zm) initial mean granule size (~m) final mean granule size at end of first stage (gm) drum diameter (/xm) solubility of fertiliser salt in water (g g-1 water) fractional saturation (y(1-p)/p) viscous Stokes number critical viscous Stokes number granulation time (min) mechanism switching time (min) effective granule size (/zm) volume boundary of ith section particles' velocity of approach (cm s -1) solution phase ratio 48 AM..4detayo et al. / Powder Technology 82 (1995) 37-49 Greek letters parameter coalescence rate kernel collision rate function between particles in ith and jth section (coalescence kernel) coalescence rate constant binder viscosity (P) density of fertiliser salt (g cm -3) granule density (g cm -3) density of fertiliser solution (g cm -3) drum revolution (s- 1) o/ vj) fli.i ~o /x Pf Pg Pl ¢o assumed that V= acoR and, in fact, have plotted A as a function of ln(odStvo). In so doing the slope is still m; the intercept, however, m is given by m ln(St*/a). For the data of Fig. A1, we obtain a slope m=10_+3. Having determined m, it is now possible to correct the data for the extent of granulation kl tl for differences in S%. Let us take the data for DAP as a reference, with a particular value of the Stokes group (Sty ~Sty,,). Define the ratio of this group for DAP, Stv/Stvo), to that of other fertilisers, (St*/Stvo), as (St* /Stvo) K ~(Stv/Stvo) (A2) The normalised extent of granulation kit1 is then given by Acknowledgements kltl =ASsat = m In(St*/Stvo)Ssat = m ln[K(St*/St~o)]Ssat S.E.P. acknowledges an international travel grant from the US National Science Foundation, US-Australia Program, INT-9114590 and CTS-8957042. Appendix The extent of granulation kit1 has been shown to depend on both the granule saturation Ss,t and the Stokes number, which combines the effects of binder viscosity and granule inertia. In order to compare the effect of granule saturation on the extent katl for different materials, it is necessary to correct for differences in Stv. From Eqs. (19) and (20): A = m ln(St*/Stvo) (A1) where m is a constant independent of S~at and Stv, and A is the slope of the extent k l q versus fractional saturation S~,t. The initial Stokes number St~o is given by Eq. (17b). A plot of A as a function of ln(1/St~o) has a slope of m as well as an intercept of m In(St*). Fig. A1 illustrates such a plot for the fertilisers of the present work, which includes the limestone results of Kapur [21]. Estimated limestone properties, taken in part from Kapur, are summarised with fertiliser properties in Table 4. In calculating St~o for Fig. A1, we 45 40 35 A 3O 25 20 15 h I -6 -5 In (~/Stvo) Fig. A1. -4 = m ln(St*/Stvo)Ssa, +m(ln K)Ssa t =kit1 +m(In K)S~at (A3) And so, in order to compare the fertiliser and limestone data, a factor of m(ln K)S~at must be added to the unmodified extent of granulation kl ta in order to remove differences in St~o brought about by the differences in binder viscosity, granule density, drum speed and initial granule size. To normalise the data for MAP and AS, we have assumed that e and h, are the same for all fertilisers. Therefore K= /iO-g /zOg (A4) To include the data of Kapur, the differences in drum speed and initial particle size are also taken into account. The extent of granulation for the present work, normalised with respect to DAP, is given in Fig. 8(b). References [ 1] J.D. Litster and L.X. Liu, ICHEME, 5th Int. Symp. Agglomeration, Brighton, UK, 25-27 Sept., 1989, pp. 611-621. [2] J.D. Landgrebe and S.E. Pratsinis, J. 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