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Identification of thermal zones and population balance modelling of fluidized bed spray granulation
Powder Technology 208 (2011) 542–552
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Identification of thermal zones and population balance modelling of fluidized bed spray granulation C. Turchiuli a,b,⁎, T. Jimenèz a,c, E. Dumoulin a a b c
UMR GenIAl 1145, Agroparistech, Massy, F-91744, France Univ Paris-Sud, Orsay, F-91405, France Univ. de las Américas (UDLA-P), Puebla, M-72820, Mexico
a r t i c l e
i n f o
Available online 19 September 2010 Keywords: Agglomeration Population balance Particle processing Fluidization Modelling
a b s t r a c t Air temperature measurements in a fluidized bed of glass beads top sprayed with water showed that conditions for particles growth were fulfilled only in the cold wetting zone under the nozzle which size and shape depended on operating conditions (liquid spray rate, nozzle air pressure, air temperature, and particles load). Evolution of the particle size distribution during agglomeration was modelled using population balance and representing the fluidized bed as two perfectly mixed reactors exchanging particles with particle growth only in the one corresponding to the wetting zone. The model was applied to the agglomeration of non-soluble glass beads and soluble maltodextrin particles spraying respectively an acacia gum solution (binder) and water. Among the three adjustable parameters, identified from experimental particle size distributions evolution during glass beads agglomeration, only one describing the kinetics of the size distribution evolution depended on process variables. The model allowed satisfying simulation of the evolution of the particle size distribution during maltodextrin agglomeration. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Fluidized bed agglomeration is widely used in food and pharmaceutical industry to improve handling properties of powders (flowability, wettability, density, and reduction of dust formation) by changing the physical properties of individual particles (size, shape, density, porosity, and structure) [1]. Agglomeration is obtained by fluidizing particles with hot air, to allow their individualization and circulation, and spraying a liquid (solvent or binder solution) to wet their surface and render it sticky. If particles are soluble, the spray of pure solvent is sufficient to get sticky particles due to some local dissolution. If particles are non-soluble, a binder solution must be used. Collisions between wet particles occur due to the high mixing provided by the fluidized bed and allow the viscous layers at their surface to come into contact and to form dynamic pendular liquid bridges between them. Depending on the liquid viscosity, bridges will manage or not to dissipate the relative kinetic energy due to the collision preventing or not rebound of the
⁎ Corresponding author. UMR GenIAl 1145, Agroparistech, Massy, F-91744, France. E-mail addresses: [email protected] (C. Turchiuli), [email protected] (T. Jimenèz), [email protected] (E. Dumoulin). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.08.057
colliding particles [2]. In the first case, solvent evaporation by the hot fluidizing air will lead to a consolidated solid bridge between particles. And the repetition of the different steps of wetting, collision and drying will give progressively rise to larger and larger agglomerates (agglomeration growth). In the second case, the binder layer at the particle surface will solidify and finally coat each particle (layer growth). Anyway, if drying is insufficient, a high humidity is generated and the bed risks collapse due to wet quenching. Conversely, when drying is too intense the solvent evaporates either before wetting particles or before collision between particles without any agglomeration. In the fluid bed, particles are also subjected to rupture and/or abrasion due to collisions between them or with the equipment walls [3,4]. The growth(s) mechanism(s) involved will depend on the apparatus, process and product parameters [5] such as fluidizing air temperature and flow rate; nature, concentration and feed rate of the sprayed solution; spraying system and particle properties. A proper control of the obtained powder properties would require understanding of the mechanisms prevailing in the process. But it is impossible to expect that these mechanisms (e.g. wetting, collision, consolidation, and rupture) will occur singly or simultaneously or sequentially. Population balance equations (PBE) are convenient to describe the particle size distribution evolution during granulation process. They allow taking into account the different growth modes (layering, agglomeration) and rupture mechanisms. Momentum and
C. Turchiuli et al. / Powder Technology 208 (2011) 542–552
to the wetting of the fluidized particles by the liquid sprayed and the evaporation of the solvent. In this region, symmetrical to the nozzle axis, air temperature increases from the centre sideways. 2. The isothermal zone, near the walls and around the wetting-active zone. In this region there is equilibrium between heat and mass transfer and air temperature is homogeneous. 3. The heat transfer zone, situated right above the bottom hot air distributor plate. In this narrow area, the hot air temperature decreases drastically due to the energy absorbed by the colder particles coming from the upper zones.
Twin-fluid nozzle 6 cm Hfb= 24 cm
26 cm Cold
2 1
Hot
6 cm
3 14 cm
Hot air Fig. 1. Thermal zones in the conical fluidized bed granulator. 1. Wetting-active zone, 2. Isothermal zone, 3. Heat transfer zone. —— Thermal zones boundary. -··-·· Measurement region boundary.
heat balances are implicit. For an open well-mixed system where agglomeration, layering and rupture occur, the population balance equation can be written as Eq. (1):
1 dðV ⋅Ψðv; t ÞÞ 1 = ⋅ ½Q in ðt Þ⋅Ψin ðv; t Þ−Qout ðt Þ⋅Ψout ðv; t Þ V ⋅ ffl{zfflfflfflfflfflfflfflfflfflfflfflffl dt V |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl} Accumulation
ð1Þ
Input−Output
∂ dv Ψ ð v; t Þ − + F|fflffl{zffl −E ⋅ ffl} ∂v dt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ⇑ Layering
543
+ C −D |fflffl{zfflffl} Rapture
The size of the three zones varied with the operating parameters also affecting agglomeration efficiency, especially that of the wetting-active zone, depending on the diameter and penetration depth of the spray into the bed of particles. Particles will grow only if they circulate through this zone where conditions for agglomeration (wet sticky particles, collisions) are fulfilled [12]. The size of this zone and the rate of transfer of particles to this part of the fluidized bed will therefore determine particles growth and agglomeration efficiency. In this study, air temperature measurements throughout the fluidized bed are performed at steady state, spraying water on inert glass beads to measure the fraction of the bed volume occupied by the wetting-active zone for different process conditions (air temperature, liquid spray rate, nozzle air pressure and initial particle load). A model of the granulator is then proposed and combined with population balance equation to simulate particles agglomeration. Parameters of the model are identified using results of experiments with model glass beads (non-soluble) and maltodextrin particles (soluble) agglomerated respectively with acacia gum solution and water. 2. Materials and methods
Agglomeration
where V is the system volume, Ψ(v,t) is the number density function at time t for particles with volume v, Qin and Qout are the volume flow rates of particles at the inlet and outlet of the considered volume V, F and E are the birth and death terms due to particles agglomeration, C and D are the birth and death terms due to rupture. Due to bulk mixing of the particles, the fluidized bed is often represented as a fully mixed bed. But, when liquid is sprayed at the top of the bed, it is assumed that an “active” zone is formed, close to the surface, where both deposition of the spray on particles and solvent evaporation take place. Significant temperature gradients measured in that region of the bed confirmed the existence of this zone near the nozzle. And, air temperature distributions measured in fluidized bed equipment with different scales (laboratory, pilot) and geometries (conical, cylindrical) with top spraying of liquid (water or methanol) by means of different nozzles (single or twin fluid) led to consider three regions in the fluidized bed (Fig. 1) [6–11]: 1. The wetting-active zone, low temperature and high humidity region, near the spraying nozzle at the topmost part of the bed. It is characterized by high humidity and temperature gradients due
Experiments were performed in a pilot batch fluidized bed of conical shape UNI-GLATT (Glatt GmbH Process Technology, Germany) (Fig. 1). Liquid (20 °C) was top sprayed by means of a bi-fluid nozzle in a full cone. Model inert glass beads (d50 = 160 μm, ρ = 2490 kg m− 3, DUP, France) and soluble maltodextrin particles (d50 = 180 μm, ρ = 1424 kg m− 3, Glucidex 12, Roquette, France) were fluidized and heated with a constant air flow rate (157 m3 h− 1 for glass beads and 120 m3 h− 1 for maltodextrin particles) allowing keeping the same fluidized bed height (hfb ≈ 24–25 cm) with a constant gap (1–2 cm) between the top of the fluidized bed and the tip of the nozzle. For the measurement of air temperature profiles, at steady state, in the central zone of the fluidized bed, glass beads were top sprayed with water (20 °C, spray angle between 20 and 40° (Table 1)). Three series of six thermocouples were positioned at 0, 3 and 6 cm from the axis of the apparatus and, for each position, at 6, 10, 16, 20, 22 and 24 cm from the air distribution grid. Assuming symmetry of temperatures around the bed axis, it allowed the characterization of a cylindrical measurement region right below the nozzle representing 32% of the fluidized bed volume (Fig. 1). For agglomeration trials, the initial load of particles was first heated by the hot fluidizing air (10 min, constant temperature). The
Table 1 Spray angle, bed average temperature Tb and wetting-active zone characteristics (volume fraction of fluid bed α, height and shape) (reference: 500 g glass beads, air 70 °C, water 5.33 ml min− 1, 1 bar). Variables
Spray angle (°) Tb (°C) α (%) Height (cm) Shape
Reference
38 52 29 16 2
250
750
60
80
2
3
2.65
7.75
g
g
°C
°C
bars
bars
ml min− 1
ml min− 1
38 53 N 31 17.5 1
38 53 24 12 2
38 46 N30 18 1
38 60 22 14.5 2
27 52 18 17 2
23 52 28 18 2
34 54 14 14 2
39 50 29 17 2
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liquids sprayed (20 °C) were an acacia gum aqueous solution (IRX 61410, CNI, France) for glass beads and water for maltodextrin particles. They were sprayed with constant flow rate and spraying pressure till reaching a constant agglomerates size. Particles growth was followed along the trial taking samples (≈5 g) in the fluidized bed (8 cm above the air distribution plate and 6 cm from the container wall). Particle size distribution was obtained by manual sieving, gently shaking a series of 15 sieves (diameter 5 cm) with openings following a 21/3 progression between 100 and 2500 μm. Size distribution of the droplets in the spraying conditions studied was measured by laser diffraction (Spraytec, Malvern, France) placing the nozzle outside the chamber.
Reference experimental conditions were chosen for glass beads: 500 g of initial particles, inlet air temperature 70 °C, liquid feed rate 5.33 ml min− 1 and relative air spraying pressure 1 bar. Eight other conditions were tested changing one process variable at a time: initial particle load 250 and 750 g, inlet air temperature 60 and 80 °C, liquid feed rate 2.65 and 7.55 ml min− 1 and relative air spraying pressure 2 and 3 bars. For agglomeration trials two concentrations of the binder solution were tested (20 and 30%w/ w). For maltodextrin particles agglomeration trials, the initial particle load was changed to 280 g in order to approximately get the same volume of particles in the bed. Results are averages of at least two experiments.
3. Results and discussion 3.1. Thermal zones in the fluidized bed
Height (cm)
The temperature profiles measured confirmed the existence of a low temperature region, near the spraying nozzle, partly covering the top part of the bed and extending inwards towards the air distributor plate (Fig. 2). In this region, air temperatures measured were 20 to 40 °C below the fixed inlet air temperature T, with the lower gas temperature located right below the nozzle. Temperature rose steeply away from the centre of the zone leading to narrow isotherms corresponding to high temperature gradients. This region corresponds to the wetting-active zone, where the wetting of the particles takes place and most of the water evaporates. In the outer part of this zone, air temperatures were higher with much smaller temperature gradient (temperature variation not exceeding 1 °C in the measurement region). In this isothermal zone extending to the granulator walls, the average air temperature measured can be used as
C2
C2 P= 3 bars
c
d
Height (cm)
Reference
C1 T= 60°C
Distance from the centre (cm)
C2 P= 2 bars
Distance from the centre (cm)
Fig. 2. Typical air temperature profiles measured and particle circulation patterns (C1: funnel-like (c), C2: bell-like (a, b, d)) (Reference: 500 g glass beads, air 70 °C, water 5.33 ml min− 1, 1 bar).
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the bed average temperature Tb. Tb was about 15 to 20 °C below the fixed inlet air temperature T for all the conditions studied. It mainly depended on the inlet air temperature and on the liquid feed rate and was not influenced by the initial particle load and the spraying pressure (Table 1). The average bed temperature Tb is a good indicator of the drying conditions in the fluidized bed and could be used as a control process parameter. In the lower part of the fluidized bed, above the air distributor plate, an important temperature gradient was observed (about 10 °C for 6 cm height). It corresponds to the heat transfer zone where cold particles coming down from the upper zones are heated by the entering hot air. To agglomerate, particles must be wet enough for liquid bridges to appear between them when they collide [12]. It is therefore proposed that agglomeration takes place in the wetting-active zone. And, the larger it is, the more the number of particles subjected to agglomeration is important. To estimate the volume fraction α of the fluidized bed occupied by this zone, its boundary was supposed to correspond to the isotherm with a temperature 2 °C below the measured bed average temperature Tb. From this two-dimensional boundary isotherm, axial symmetry of temperatures round the bed axis was assumed to estimate the wetting-active zone volume and deduce α. The volume fraction of the fluidized bed occupied by the wetting-active zone varied from 14% for a low spraying rate to about 30% for highest liquid feed rates (Table 1). When the initial particle load was increased (250 to 750 g), the probability for the sprayed liquid droplets to collide with a solid particle increased due to a higher solid concentration in the fluidized bed (116 instead of 38.3 g L− 1) and therefore lower bed porosity (0.95 instead of 0.98). This resulted in a decrease of α from 31 to 24%. The inlet air temperature determines the air capacity to evaporate the liquid sprayed. When it was low (60 °C), the volume of the wetting-active zone was large (N30% of the fluidized bed volume) whilst when it was high (80 °C) α decreased down to 22% due to fast liquid evaporation. The spraying air pressure influences the angle of the sprayed liquid jet and the speed and diameter of the sprayed liquid droplets. The higher the spraying air pressure, the smaller the jet angle and the liquid droplets diameter and the higher the droplets speed [13]. When it was increased from 1 to 2 bars, the volume of the wetting-active zone decreased (29 to 18% of the bed volume) mainly due to smaller liquid droplets (19 μm instead of 35 μm) drying rapidly and to a smaller jet angle (27° instead of 38°) leading to a narrower zone. When the relative spraying pressure was increased to 3 bars, a drastic change in the shape of the isotherms was observed and the wetting-active zone was divided into two regions (Fig. 2b). The first one was situated in the upper part of the bed, under the nozzle as in other experiments. The second one appeared in the bottom part of the bed, at the centre, about 10 cm above the air distributor plate. This modification in the temperature profile may be attributed to a lower probability of collision of the liquid droplets with the solid particles in the upper part of the bed due to a higher speed of the droplets when entering the particle bed. The air sprayed may also create some turbulence in the chamber modifying particles recirculations. As a result, the estimated volume of the wetting-active zone was high (28% of the bed volume). The liquid feed rate influences both the angle of the sprayed jet and the droplets size, and modifies the drying conditions within the fluidized bed. When it was increased from 2.65 to 7.75 ml min− 1, the liquid jet angle increased (33° to 40°) as well as the diameter of the liquid droplets (35 to 45 μm) that were therefore more difficult to dry. As a result, the fraction of the bed occupied by the wetting-active zone increased from 14 to 29%. From the outline of the wetting-active zone, it was possible to identify two typical shapes depending on the particles circulation in the conical fluidized bed fixed by the process conditions [14,15]. The first one called “funnel” like (C1 — Fig. 2c) was observed for the smaller particle load (250 g) and the lower inlet air temperature (60 °C). Both conditions corresponded to a high humidity in the fluidized bed (insufficient drying) and therefore a high probability of wet quenching. In this case, particles rise up at the centre. Arriving in the upper part of the bed they are wetted by the liquid droplets and then wet particles go downwards along the container walls where they can stick. These two conditions were therefore no more considered for the rest of the study. For the seven other conditions tested, the outline of the low temperature region formed a “bell” like profile (C2 — Fig. 2a, b, d) corresponding to an upward movement of the particles at the sides, near the container walls, and downwards through the centre after wetting by the liquid droplets. This second circulation pattern is more favorable to control agglomeration since only “dry” particles rise up along the container walls. 3.2. Model The study of the thermal zones in the fluidized bed imposed to consider the wetting-active zone as a distinct region, exchanging particles with the bulk of the population. The fluid bed reactor was then represented by two perfectly mixed reactors exchanging particles (Fig. 3) with reactor
Qbinder Cbinder
Reactor A Wetting-active zone A (v,t) NA (t) = α NT (t) q(t)
q(t)
Reactor B Isothermal + heat transfer zones B (v,t) NB (t) = (1-α) NT (t) Fig. 3. The model.
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A corresponding to the wetting-active zone and reactor B to the isothermal and heat transfer zones. Particles were supposed to move randomly between the two regions at a constant rate q(t), equal to the volume per unit time of particles transferred through the boundary of the wettingactive zone (A). Assuming particles homogeneously distributed in the whole fluidized bed, and the volume of both reactors constant with time, the portion of the entire population enclosed in reactor A was equal to α and the number density function Ψ(v,t), for particles with volume v, in the whole fluidized bed was given by: A
B
Ψðv; t Þ = α⋅Ψ ðv; t Þ + ð1−αÞ⋅Ψ ðv; t Þ:
ð2Þ
The low air temperatures measured in the wetting-active zone A, were assumed to correspond to high humidity allowing agglomeration whilst in reactor B, temperature and humidity conditions only allowed mixing of the bulk particles with possible breakage due to collisions or abrasion, but no agglomeration. Applying PBE to reactors A and B led to: : 8 h i ∂h i A > qðt Þ dv Ψ A ðv; t Þ + F ðvÞ−EðvÞ > > dΨ ðv;t Þ = V ⋅ Ψ B ðv; t Þ−Ψ A ðv; t Þ − ⋅ > < dt A ∂v dt : > i > dΨ B ðv; t Þ qðt Þ h A > B > : = Ψ ðv; t Þ−Ψ ðv; t Þ dt VB ⋅
ð3Þ
where the first and second terms correspond respectively to particles exchange between reactors A and B and layering growth and F and E are the birth and death terms due to particles agglomeration: F ðvÞ =
1 v A A ∫ βðv−u; uÞ⋅Ψ ðv−u; t Þ⋅Ψ ðu; t Þdu 20 A
∞
A
EðvÞ = Ψ ðv; t Þ⋅ ∫ βðv; uÞ⋅Ψ ðu; t Þ⋅du
ð4Þ
ð5Þ
0
β(v,u) is the agglomeration function representing the probability of efficient collisions between two particles with volumes v and u. Assuming a constant layering growth rate G = dv/dt, the layering growth term became:
−
i ∂ dv ∂ h ∂Ψ A ðv; t Þ A A Ψ ð v; t Þ =− G⋅Ψ ðv; t Þ = −G⋅ : ⋅ ∂v dt ∂v ∂v
ð6Þ
The resulting mathematical model was a system of two integro-differential equations. It was reformulated into a system of 32 differential equations by discretizing PBEs in the particle volume space divided in 16 size classes with limits corresponding to apertures of the sieves used for particle size distribution determination. Each continuous equation was integrated on a size domain [vi; vi + 1] leading to the discretized population density function: vi+1
Φi ðt Þ = ∫ Ψðv; t Þ⋅dv:
ð7Þ
vi
The discrete layering growth term was therefore: vi +1
−G⋅ ∫ vi
h i ∂ΨA ðv; t Þ A A = G Ψ ðvi ; t Þ−Ψ vi + 1 ; t : ∂v
ð8Þ
Assuming a constant value Ψ′i(t) for Ψ(v,t) in the size class i and that Ψ(vi,t) is the arithmetical mean of Ψ(v,t) between size classes i − 1 and i [16], led to: " # Ψ0 i ðtÞ + Ψ0 i−1 ðtÞ 1 Φ0 i ðtÞ Φ0 i−1 ðtÞ + = Ψðvi ; tÞ = 2 2 vi + 1 −vi ðvi −vi−1 Þ
ð9Þ
and the discrete layering growth term was rewritten, for any size class i ≠ 1 ≠ n, as: " # A A h i 1 Φi−1 ðt Þ Φi + 1 ðt Þ A A Di ðt Þ = G Ψ ðvi ; t Þ−Ψ vi + 1 ; t = G⋅ − 2 ðvi −vi−1 Þ vi + 2 −vi + 1
" # A A h i 1 Φ1 Φ2 A A − D1 ðt Þ = G Ψ ðv1 ; t Þ−Ψ ðv2 ; t Þ = G⋅ 2 ðv2 −v1 Þ ðv3 −v2 Þ
ð10Þ
ð11Þ
and " # h i 1 ΦAn−1 ΦnA ðt Þ A A : Dn ðt Þ = G Ψ ðvn ; t Þ−Ψ vn + 1 ; t = G⋅ − 2 ðvn −vn−1 Þ vn + 1 −vn
ð12Þ
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F(v) and E(v) were discretized using the fixed pivot technique described by Kumar and Ramkrishna [17] ensuring conservation of particles number and mass. Particles belonging to a size class i, with volumes comprised between vi and vi + 1, are represented by a volume xi called “grid point” with vi b xi = (vi + vi + 1)/2 b vi + 1. For each new particle with volume v in the domain [xi ; xi + 1] two fractions a(v, xi) and b(v, xi + 1) of the particle are attributed respectively to the populations at xi and xi + 1 ensuring that: aðv; xi Þ + b v; xi
+ 1
ð13Þ
= 1:
The discrete Fi and Ei functions obtained were [18]: 1 A A 1− δj;k ⋅ηðvÞ⋅βj;k ⋅Φj ⋅Φk 2 j;k;xi−1 ≤v≤xi + 1 j≥k
Fi =
∑
A
n
ð14Þ
A
Ei = Φi · ∑ βi;k · Φk
ð15Þ
k=1
with
ηðvÞ =
8x i + 1 −v > > ; x ≤ v ≤ xi +1 > > xi + 1 −xi i <
ð16Þ
v−xi−1 > > ; x ≤v≤xi > > : xi −xi−1 i−1
and δj;k =
1; j = k : 0; j≠k
ð17Þ
The discretized system of differential equations finally obtained for the two reactors was: 8 A h i j≥k N > 1 A dΦi qðt Þ > B A A A A > ∑ 1− ⋅δj;k ⋅ηðvÞ⋅βj;k ⋅Φj ⋅Φk −Φi : ∑ βi;k ⋅Φk > < dt = V A ⋅ Φi −Φi + Di ðt Þ + 2 j;k;xi−1 ≤v≤xi +1 k=1 > > B h i > > : dΦi = qðt Þ⋅ Φ A −ΦB : i i VB dt
ð18Þ
Assuming that the binder is homogeneously distributed onto the NA(t) particles in reactor A at time t and that the layering growth rate G is independent of the particle size [18], G=
Q binder ⋅C binder ρbinder ⋅V A ⋅ ∑ ΦiA
:
ð19Þ
i
1200
Initial 3. QL = 7,75 ml/min
1000
Equilibrium
Constant rate
d50 (μm)
800
600
400
200
0
0
20
40
60
80
100
120
140
time (min) Fig. 4. Evolution of theoretical (lines — β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) d50 during glass beads agglomeration in reference conditions (500 g glass beads, air 70 °C, 20%w/w acacia gum solution 5.33 ml min− 1, 1 bar).
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Different expressions are proposed for the agglomeration function βi,k [19,20]. It was chosen here to use a “non random coalescence” kernel with three parameters since it gave satisfying results in the case of Carboxy Methyl Cellulose particles agglomerated in similar equipment [19]: βi;k = β0 ·
ðxi + xk Þκ : ðxi ·xk Þγ
ð20Þ
0.9 t=0 s
0.8
t = 600 s
Mass fraction (-)
0.7
t = 1200 s
0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) 0.9 t = 1800 s
0.8
t = 2400 s
0.7
Mass fraction (-)
t = 3000 s 0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) 0.9 t = 3600 s
0.8
t = 4200 s
Mass fraction (-)
0.7
t = 4800 s
0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) Fig. 5. Evolution of theoretical (lines — β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) particle size distribution during glass beads agglomeration in reference conditions (500 g glass beads, air 70 °C, 20%w/w acacia gum solution 5.33 ml min− 1, 1 bar).
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Table 2 β0 and SSD values obtained for glass beads agglomeration trials with k = 1.805 and γ = 0.83 and growth rate during the constant rate phase from Fig. 6. β0 × 10− 12
Conditions Reference: 5.33 ml min− 1, 1 bar, 70 °C, 20%w/w, 500 g glass beads 2.65 ml min− 1 7.75 ml min− 1 2 bar 3 bar 80 °C 30%w/w 750 g
SSD
r (μm min− 1)
9.71
0.246
8.2
4.01 17.41 14.11 4.71 10.91 16.71 4.31
0.343 0.270 0.631 0.625 0.380 0.413 0.425
2.3 13.9 9.6 9.5 7.8 14.7 5.9
Parameters κ and γ are related to the shape of the particle size distribution and β0, describing the kinetics of the particle size distribution evolution, depends on the operating conditions. Assuming that particles are dragged by rising air bubbles in the fluidized bed, the flow rate of particles from reactor B entering reactor A can be estimated from the particles turnover time tc. If this flow rate is equal to the particle flow rate exiting from reactor A to reactor B, the exchange flow rate q(t) between the two reactors is:
qðt Þ = Mp ðt Þ = tc ·ρp
ð21Þ
where tc can be calculated from the particles minimal fluidization rate Umf, the air rate Ua and the rising air bubbles rate Ub [21]: 1−U a −Umf −1 H′
⋅ : Ub 0; 6 U a −Umf
ð22Þ
1400
(a) QL= 7,75 ml/min (X)
1200
d50 (μm)
1000
Reference ( ) QL= 5,33 ml/min, P = 1 bar,T = 70°C, Ch= 500 g, CL= 20 %
T = 80°C ( )
800 600 Ch= 750 g ( ) 400
QL = 2,65 ml/min (
)
200 0 0
20
40
60
80
100
120
140
160
180
200
time (min) 600
(b) P = 2 bar ( ∗ )
500
400
d50 (μm)
tc =
300 P = 3 bar ( ) 200
100
0 0
10
20
30
40
50
60
70
80
90
100
time (min) Fig. 6. Theoretical (lines) and experimental (points) evolutions of d50 during glass beads agglomeration trials.
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H′ is the distance between the air distribution grid and the bottom of reactor A (bottom of wetting-active zone determined experimentally from air temperature profiles) and Ub is calculated from [14,22]:
1=2 Ub = Ua −Umf + 0; 711ð g · db Þ
ð23Þ
with the air bubble diameter: h
i1 = 3
1;21 ⋅ 1 + 0; 0684⋅H lf : db = 0; 853 1 + 0; 272 U a −Umf
ð24Þ
The discretized system obtained in Eq. (18), with 32 differential equations (16 for each of the two reactors corresponding to the 16 size classes chosen) and 3 parameters β0, κ and γ linked to the agglomeration function, was solved using the ode15s (Solve Stiff differential equations) function of MATLAB 6.1. 3.3. Glass beads agglomeration During agglomeration of glass beads with acacia gum solution, the median particle diameter d50 increased with time showing three different phases (Fig. 4): – the “initial phase” with a low increase of d50 with time, corresponding essentially to the deposit of binder at the particle surface until rendering it sticky. – the “constant rate phase” where d50 increased linearly with time, and agglomeration growth occurred. – the “equilibrium phase” where no more size increase was observed, corresponding to equilibrium between growth and destruction of agglomerates. All along the process there is a competition between the different growth and rupture mechanisms, each of the three phases identified corresponding to one of them prevailing on the others. The proposed model does not include the rupture mechanism, it was therefore only 0.4 QL = 2,65ml/min
0.3
QL= 5,33 ml/min
0.2
βo = 9,71.10-12
0.15 QL= 7,75 ml/min βo = 17,41.10-12 0.1
Massfraction (-)
Mass fraction (-)
0.35
βo = 4,01.10-12
0.25
0.25 T = 70°C βo = 9,71.10-12
0.2
T = 80°C βo = 10,91.10-12
0.15 0.1
0.05 0.05 0 101
102
103
0 101
104
102
Diameter (μm)
c 0.25
CL= 20% w/w βo = 9,71.10-12
Ch= 500 g βo = 9,71.10-12
0.2
Mass fraction (-)
Mass fraction (-)
104
d Ch= 750 g βo = 4,31.10-12
0.25
0.15
0.1
0.05
0 101
103
Diameter (μm)
0.2 CL= 30% w/w
0.15
βo = 10,91.10-12
0.1
0.05
102
103
Diameter (μm)
104
0 101
102
103
104
Diameter (μm)
Fig. 7. Influence of the sprayed solution flow rate (a), the air temperature (b), the initial particle load (c) and of the acacia gum solution concentration (d) on the value of β0 and on the agglomerates particle size distribution after 50 min.
C. Turchiuli et al. / Powder Technology 208 (2011) 542–552
20
β0= 1,1879.10-12 x r (R2 = 0,9489)
18
β0 (.10-12 m2,565.s-1)
551
16
Trial 4 (P=2bars)
14 12 10 8 6
Trial 5 (P=3 bars)
4 2 0 0
5
10
15
20
r (μm.min-1) Fig. 8. Relation between agglomeration kernel β0 and growth rate r.
adapted to the modelling of the constant rate phase where rupture was assumed to be negligible. Values of parameters β0, κ and γ were first determined by minimizing the sum squared differences (SSD) between theoretical and experimental particle size distributions at different times during the constant rate phase of reference trial using lsqnonlin function of MATLAB (Levenberg–Marquardt type algorithm). With β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83, the model gave a good representation of the evolution of the particle size distribution (SSD = 0.2459) during the constant rate phase but, as expected, did not allowed representing the equilibrium phase (Figs. 4 and 5). Knowing that κ and γ mainly depend on the agglomeration process [19], values obtained (κ=1.805 and γ=0.83) were used for all the other trials and only the β0 value was determined minimizing SSD (Table 2). Good correlation was obtained between calculated and experimental particle size distributions evolution during the constant rate phase (SSDb 0.5) except for trials with higher spraying pressure (e.g. 2 and 3 bars instead of 1 bar) (Fig. 6). In this last case, particle size distributions were bimodal (weak agglomerates) and the shape of the wetting-active zone obtained from air temperature profiles was different (Fig. 2b). For a good correlation, the model should be modified choosing another β function adapted to bimodal particle size distributions and/or dividing reactor A into two reactors in agreement with the particular shape of the wetting-active zone for these conditions. β0 appeared to greatly depend on the sprayed solution flow rate with a quasi linear increase of β0 with the solution flow rate throughout the studied range (e.g. 2.65–7.75 ml min− 1). β0 also significantly increased with the sprayed solution concentration and decreased with the initial particle load thus modifying the particle size distribution of agglomerates obtained (Fig. 7). In fact, β0, fitting parameter of the model, could be correlated to the growth rate r during the constant rate phase obtained from experimental d50 = f(t) curves of Fig. 6 (Table 2). β0 was found to increase linearly with r according to (Fig. 8): β0 = 1:19:10
−12
ð25Þ
· r:
3.4. Maltodextrin particles agglomeration Soluble maltodextrin particles were agglomerated by spraying water which was assumed to be integrally removed by hot air during drying and no layering growth was expected to occur. To simulate maltodextrin particles agglomeration, the layering growth rate of the model was therefore fixed to 0 and the particle load was assumed to be constant. κ and γ were assumed not to be modified since the same equipment was used. It was shown for glass beads that β0 can be estimated from the growth rate r for given operating conditions. The median diameter d50 evolution with time during maltodextrin agglomeration with water was different from that obtained for glass beads. Firstly, no “initial phase” was observed and secondly, the growth rate values were higher (40.5 instead of 8.2 μm min− 1 for the reference condition). Specific β0 value was therefore calculated for maltodextrin particles agglomeration assuming the same relation (25) between β0 and r. Good agreement was found between calculated and experimental particle size evolution (SSD = 0.251) indicating that the relation between β0 and r can be applied whatever the particles agglomerated (soluble or non-soluble) (Fig. 9). 0.4 0.35
t=0s
t = 1200 s
t = 600 s
t = 1800 s
t = 900 s Mass fraction (-)
0.3 0.25 0.2 0.15 0.1 0.05 0 101
102
103
Diameter (μm)
104
102
103
104
Diameter (μm)
Fig. 9. Evolution of theoretical (lines — β0 = 48 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) particle size distribution during maltodextrin particles agglomeration in reference conditions (280 g maltodextrin particles, air 70 °C, water 5.33 ml min− 1, 1 bar).
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4. Conclusion Air temperature measurements within the granulator chamber allowed estimating the volume fraction α of the fluidized bed occupied by the wetting-active zone where agglomeration was assumed to take place due to high humidity. Depending on the operating conditions, α occupied between 14 and 30% of the bed. The fluid bed granulator was then modelled as a system of two coupled perfect mixers. To simulate particle size distribution evolution during agglomeration, population balance equations were written for each mixer. Parameters (β0, κ and γ) of the model were identified for agglomeration trials with glass beads sprayed with acacia gum solution. κ and γ could be kept constant (k = 1.805 and γ = 0.83) whilst β0 varied with the operating parameters. Especially, β0 increased with the sprayed solution flow rate and concentration and decreased with the particle load and showed a linear relationship with the growth rate r in the constant rate phase of agglomeration obtained from d50 = f(t) curves. Good agreement was found between theoretical and experimental evolutions of particle size distribution during this phase except for conditions leading to bimodal populations (high air spraying pressure) for which other forms of the agglomeration function β should be used and a modified model should be proposed to take into account the particular shape of the wetting-active zone in this case. Simulation of soluble maltodextrin particles agglomeration using the results for glass beads led to satisfying correlation between theoretical and experimental particle size evolutions showing that results can be applied as well for soluble and non-soluble particles. Acknowledgements The authors would like to acknowledge the financial support of CONACyT (Mexico) for the PhD scholarship provided to Teresa Jimenez and Bertrand Broyart from Agroparistech for numerical calculations. References [1] H. Schubert, Instantization of powdered food products, International Chemical Engineering 33 (1) (1993) 28–45. [2] B.J. Ennis, G. Tardos, R. Pfeffer, A microlevel-based characterization of granulation phenomena, Powder Technology 65 (1991) 257–272.
[3] M. Rhodes, Introduction to Particle Technology, John Wiley Sons Ltd, England, 1998, pp. 267–281. [4] P.G. Smith, A.W. Nienow, Particle growth mechanisms in fluidised bed granulation-I, Chemical Engineering Science 38 (8) (1983) 1223–1231. [5] M.E. Aulton, M. Banks, Fluidised bed granulation — factors influencing the quality of the product, International Journal of Pharmaceutical Technology and Products Manufacturing 2 (4) (1981) 24–29. [6] S. Heinrich, J. Blumschein, M. Henneberg, M. Ihlow, L. Mörl, Study of dynamic multi-dimensional temperature and concentration distributions in liquid sprayed fluidized beds, Chemical Engineering Science 58 (2003) 5135–5160. [7] S.J. Maronga, P. Wnukowski, Establishing temperature and humidity profiles in fluidized bed particulate coating, Powder Technology 94 (1997) 181–185. [8] K. Saleh, Contribution à l'étude de l'enrobage des poudres en lit fluidisé. Etude expérimentale et modélisation du processus, PhD Thesis, INP Toulouse, France, 1998. [9] P.G. Smith, A.W. Nienow, On atomising a liquid into a gas fluidised bed, Chemical Engineering Science 37 (6) (1982) 950–954. [10] F. Ronsse, J.G. Pieters, K. Dewettinck, Combined population balance and thermodynamic modelling of the batch top-spray fluidised bed coating process. Part I—model development and validation, Journal of Food Engineering 78 (2007) 296–307. [11] F. Ronsse, J.G. Pieters, K. Dewettinck, Combined population balance and thermodynamic modelling of the batch top-spray fluidised bed coating process. Part II—model and process analysis, Journal of Food Engineering 78 (2007) 308–322. [12] P.D. Hede, P. Bach, A.D. Jensen, Batch top-spray fluid bed coating: scale-up insight using dynamic heat- and mass-transfer modelling, Chemical Engineering Science 64 (2009) 1293–1317. [13] T. Jimenez, C. Turchiuli, E. Dumoulin, Particles agglomeration in a conical fluidized bed in relation with air temperature profiles, Chemical Engineering Science 61 (2006) 5954–5961. [14] D. Kunii, O. Levenspiel, Fluidization Engineering, 2nd ed, Butterworth-Heinemann, Boston, 1991. [15] S.J. Maronga, P. Wnukowski, The use of humidity and temperature profiles in optimizing the size of fluidized bed in a coating process, Chemical Engineering Science 37 (1998) 423–432. [16] P. Marchal, R. David, P.K. Klein, J. Villermaux, Crystallisation and precipitation engineering — I. an efficient method for solving population balance in crystallisation with agglomeration, Chemical Engineering Science 43 (1) (1988) 59–67. [17] S. Kumar, D. Ramkrishna, On the solution of population balance equation by discretization — I. a fixed pivot technique, Chemical Engineering Science 51 (8) (1996) 1311–1332. [18] T. Jimenez, Agglomération de particules par voie humide en lit fluidise. PhD thesis, ENSIA/ABIES, France, 2007. [19] M. Peglow, J. Kumar, G. Warnecke, S. Heinrich, L. Mörl, A new technique to determine rate constants for growth and agglomeration with size- and timedependent nuclei formation, Chemical Engineering Science 61 (2006) 282–292. [20] P.C. Kapur, Kinetics of granulation by non-random coalescence mechanism, Chemical Engineering Science 27 (1972) 1863–1869. [21] P.N. Rowe, Estimation of solids circulation rate in a bubbling fluidized bed, Chemical Engineering Science 28 (1973) 979–980. [22] C.Y. Yu, Y.K. Xu, X.Z. Wang, Study of fluidized-bed spray granulation, Drying Technology 17 (9) (1999) 1893–1904.
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Identification of thermal zones and population balance modelling of fluidized bed spray granulation C. Turchiuli a,b,⁎, T. Jimenèz a,c, E. Dumoulin a a b c
UMR GenIAl 1145, Agroparistech, Massy, F-91744, France Univ Paris-Sud, Orsay, F-91405, France Univ. de las Américas (UDLA-P), Puebla, M-72820, Mexico
a r t i c l e
i n f o
Available online 19 September 2010 Keywords: Agglomeration Population balance Particle processing Fluidization Modelling
a b s t r a c t Air temperature measurements in a fluidized bed of glass beads top sprayed with water showed that conditions for particles growth were fulfilled only in the cold wetting zone under the nozzle which size and shape depended on operating conditions (liquid spray rate, nozzle air pressure, air temperature, and particles load). Evolution of the particle size distribution during agglomeration was modelled using population balance and representing the fluidized bed as two perfectly mixed reactors exchanging particles with particle growth only in the one corresponding to the wetting zone. The model was applied to the agglomeration of non-soluble glass beads and soluble maltodextrin particles spraying respectively an acacia gum solution (binder) and water. Among the three adjustable parameters, identified from experimental particle size distributions evolution during glass beads agglomeration, only one describing the kinetics of the size distribution evolution depended on process variables. The model allowed satisfying simulation of the evolution of the particle size distribution during maltodextrin agglomeration. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Fluidized bed agglomeration is widely used in food and pharmaceutical industry to improve handling properties of powders (flowability, wettability, density, and reduction of dust formation) by changing the physical properties of individual particles (size, shape, density, porosity, and structure) [1]. Agglomeration is obtained by fluidizing particles with hot air, to allow their individualization and circulation, and spraying a liquid (solvent or binder solution) to wet their surface and render it sticky. If particles are soluble, the spray of pure solvent is sufficient to get sticky particles due to some local dissolution. If particles are non-soluble, a binder solution must be used. Collisions between wet particles occur due to the high mixing provided by the fluidized bed and allow the viscous layers at their surface to come into contact and to form dynamic pendular liquid bridges between them. Depending on the liquid viscosity, bridges will manage or not to dissipate the relative kinetic energy due to the collision preventing or not rebound of the
⁎ Corresponding author. UMR GenIAl 1145, Agroparistech, Massy, F-91744, France. E-mail addresses: [email protected] (C. Turchiuli), [email protected] (T. Jimenèz), [email protected] (E. Dumoulin). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.08.057
colliding particles [2]. In the first case, solvent evaporation by the hot fluidizing air will lead to a consolidated solid bridge between particles. And the repetition of the different steps of wetting, collision and drying will give progressively rise to larger and larger agglomerates (agglomeration growth). In the second case, the binder layer at the particle surface will solidify and finally coat each particle (layer growth). Anyway, if drying is insufficient, a high humidity is generated and the bed risks collapse due to wet quenching. Conversely, when drying is too intense the solvent evaporates either before wetting particles or before collision between particles without any agglomeration. In the fluid bed, particles are also subjected to rupture and/or abrasion due to collisions between them or with the equipment walls [3,4]. The growth(s) mechanism(s) involved will depend on the apparatus, process and product parameters [5] such as fluidizing air temperature and flow rate; nature, concentration and feed rate of the sprayed solution; spraying system and particle properties. A proper control of the obtained powder properties would require understanding of the mechanisms prevailing in the process. But it is impossible to expect that these mechanisms (e.g. wetting, collision, consolidation, and rupture) will occur singly or simultaneously or sequentially. Population balance equations (PBE) are convenient to describe the particle size distribution evolution during granulation process. They allow taking into account the different growth modes (layering, agglomeration) and rupture mechanisms. Momentum and
C. Turchiuli et al. / Powder Technology 208 (2011) 542–552
to the wetting of the fluidized particles by the liquid sprayed and the evaporation of the solvent. In this region, symmetrical to the nozzle axis, air temperature increases from the centre sideways. 2. The isothermal zone, near the walls and around the wetting-active zone. In this region there is equilibrium between heat and mass transfer and air temperature is homogeneous. 3. The heat transfer zone, situated right above the bottom hot air distributor plate. In this narrow area, the hot air temperature decreases drastically due to the energy absorbed by the colder particles coming from the upper zones.
Twin-fluid nozzle 6 cm Hfb= 24 cm
26 cm Cold
2 1
Hot
6 cm
3 14 cm
Hot air Fig. 1. Thermal zones in the conical fluidized bed granulator. 1. Wetting-active zone, 2. Isothermal zone, 3. Heat transfer zone. —— Thermal zones boundary. -··-·· Measurement region boundary.
heat balances are implicit. For an open well-mixed system where agglomeration, layering and rupture occur, the population balance equation can be written as Eq. (1):
1 dðV ⋅Ψðv; t ÞÞ 1 = ⋅ ½Q in ðt Þ⋅Ψin ðv; t Þ−Qout ðt Þ⋅Ψout ðv; t Þ V ⋅ ffl{zfflfflfflfflfflfflfflfflfflfflfflffl dt V |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl} Accumulation
ð1Þ
Input−Output
∂ dv Ψ ð v; t Þ − + F|fflffl{zffl −E ⋅ ffl} ∂v dt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ⇑ Layering
543
+ C −D |fflffl{zfflffl} Rapture
The size of the three zones varied with the operating parameters also affecting agglomeration efficiency, especially that of the wetting-active zone, depending on the diameter and penetration depth of the spray into the bed of particles. Particles will grow only if they circulate through this zone where conditions for agglomeration (wet sticky particles, collisions) are fulfilled [12]. The size of this zone and the rate of transfer of particles to this part of the fluidized bed will therefore determine particles growth and agglomeration efficiency. In this study, air temperature measurements throughout the fluidized bed are performed at steady state, spraying water on inert glass beads to measure the fraction of the bed volume occupied by the wetting-active zone for different process conditions (air temperature, liquid spray rate, nozzle air pressure and initial particle load). A model of the granulator is then proposed and combined with population balance equation to simulate particles agglomeration. Parameters of the model are identified using results of experiments with model glass beads (non-soluble) and maltodextrin particles (soluble) agglomerated respectively with acacia gum solution and water. 2. Materials and methods
Agglomeration
where V is the system volume, Ψ(v,t) is the number density function at time t for particles with volume v, Qin and Qout are the volume flow rates of particles at the inlet and outlet of the considered volume V, F and E are the birth and death terms due to particles agglomeration, C and D are the birth and death terms due to rupture. Due to bulk mixing of the particles, the fluidized bed is often represented as a fully mixed bed. But, when liquid is sprayed at the top of the bed, it is assumed that an “active” zone is formed, close to the surface, where both deposition of the spray on particles and solvent evaporation take place. Significant temperature gradients measured in that region of the bed confirmed the existence of this zone near the nozzle. And, air temperature distributions measured in fluidized bed equipment with different scales (laboratory, pilot) and geometries (conical, cylindrical) with top spraying of liquid (water or methanol) by means of different nozzles (single or twin fluid) led to consider three regions in the fluidized bed (Fig. 1) [6–11]: 1. The wetting-active zone, low temperature and high humidity region, near the spraying nozzle at the topmost part of the bed. It is characterized by high humidity and temperature gradients due
Experiments were performed in a pilot batch fluidized bed of conical shape UNI-GLATT (Glatt GmbH Process Technology, Germany) (Fig. 1). Liquid (20 °C) was top sprayed by means of a bi-fluid nozzle in a full cone. Model inert glass beads (d50 = 160 μm, ρ = 2490 kg m− 3, DUP, France) and soluble maltodextrin particles (d50 = 180 μm, ρ = 1424 kg m− 3, Glucidex 12, Roquette, France) were fluidized and heated with a constant air flow rate (157 m3 h− 1 for glass beads and 120 m3 h− 1 for maltodextrin particles) allowing keeping the same fluidized bed height (hfb ≈ 24–25 cm) with a constant gap (1–2 cm) between the top of the fluidized bed and the tip of the nozzle. For the measurement of air temperature profiles, at steady state, in the central zone of the fluidized bed, glass beads were top sprayed with water (20 °C, spray angle between 20 and 40° (Table 1)). Three series of six thermocouples were positioned at 0, 3 and 6 cm from the axis of the apparatus and, for each position, at 6, 10, 16, 20, 22 and 24 cm from the air distribution grid. Assuming symmetry of temperatures around the bed axis, it allowed the characterization of a cylindrical measurement region right below the nozzle representing 32% of the fluidized bed volume (Fig. 1). For agglomeration trials, the initial load of particles was first heated by the hot fluidizing air (10 min, constant temperature). The
Table 1 Spray angle, bed average temperature Tb and wetting-active zone characteristics (volume fraction of fluid bed α, height and shape) (reference: 500 g glass beads, air 70 °C, water 5.33 ml min− 1, 1 bar). Variables
Spray angle (°) Tb (°C) α (%) Height (cm) Shape
Reference
38 52 29 16 2
250
750
60
80
2
3
2.65
7.75
g
g
°C
°C
bars
bars
ml min− 1
ml min− 1
38 53 N 31 17.5 1
38 53 24 12 2
38 46 N30 18 1
38 60 22 14.5 2
27 52 18 17 2
23 52 28 18 2
34 54 14 14 2
39 50 29 17 2
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liquids sprayed (20 °C) were an acacia gum aqueous solution (IRX 61410, CNI, France) for glass beads and water for maltodextrin particles. They were sprayed with constant flow rate and spraying pressure till reaching a constant agglomerates size. Particles growth was followed along the trial taking samples (≈5 g) in the fluidized bed (8 cm above the air distribution plate and 6 cm from the container wall). Particle size distribution was obtained by manual sieving, gently shaking a series of 15 sieves (diameter 5 cm) with openings following a 21/3 progression between 100 and 2500 μm. Size distribution of the droplets in the spraying conditions studied was measured by laser diffraction (Spraytec, Malvern, France) placing the nozzle outside the chamber.
Reference experimental conditions were chosen for glass beads: 500 g of initial particles, inlet air temperature 70 °C, liquid feed rate 5.33 ml min− 1 and relative air spraying pressure 1 bar. Eight other conditions were tested changing one process variable at a time: initial particle load 250 and 750 g, inlet air temperature 60 and 80 °C, liquid feed rate 2.65 and 7.55 ml min− 1 and relative air spraying pressure 2 and 3 bars. For agglomeration trials two concentrations of the binder solution were tested (20 and 30%w/ w). For maltodextrin particles agglomeration trials, the initial particle load was changed to 280 g in order to approximately get the same volume of particles in the bed. Results are averages of at least two experiments.
3. Results and discussion 3.1. Thermal zones in the fluidized bed
Height (cm)
The temperature profiles measured confirmed the existence of a low temperature region, near the spraying nozzle, partly covering the top part of the bed and extending inwards towards the air distributor plate (Fig. 2). In this region, air temperatures measured were 20 to 40 °C below the fixed inlet air temperature T, with the lower gas temperature located right below the nozzle. Temperature rose steeply away from the centre of the zone leading to narrow isotherms corresponding to high temperature gradients. This region corresponds to the wetting-active zone, where the wetting of the particles takes place and most of the water evaporates. In the outer part of this zone, air temperatures were higher with much smaller temperature gradient (temperature variation not exceeding 1 °C in the measurement region). In this isothermal zone extending to the granulator walls, the average air temperature measured can be used as
C2
C2 P= 3 bars
c
d
Height (cm)
Reference
C1 T= 60°C
Distance from the centre (cm)
C2 P= 2 bars
Distance from the centre (cm)
Fig. 2. Typical air temperature profiles measured and particle circulation patterns (C1: funnel-like (c), C2: bell-like (a, b, d)) (Reference: 500 g glass beads, air 70 °C, water 5.33 ml min− 1, 1 bar).
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the bed average temperature Tb. Tb was about 15 to 20 °C below the fixed inlet air temperature T for all the conditions studied. It mainly depended on the inlet air temperature and on the liquid feed rate and was not influenced by the initial particle load and the spraying pressure (Table 1). The average bed temperature Tb is a good indicator of the drying conditions in the fluidized bed and could be used as a control process parameter. In the lower part of the fluidized bed, above the air distributor plate, an important temperature gradient was observed (about 10 °C for 6 cm height). It corresponds to the heat transfer zone where cold particles coming down from the upper zones are heated by the entering hot air. To agglomerate, particles must be wet enough for liquid bridges to appear between them when they collide [12]. It is therefore proposed that agglomeration takes place in the wetting-active zone. And, the larger it is, the more the number of particles subjected to agglomeration is important. To estimate the volume fraction α of the fluidized bed occupied by this zone, its boundary was supposed to correspond to the isotherm with a temperature 2 °C below the measured bed average temperature Tb. From this two-dimensional boundary isotherm, axial symmetry of temperatures round the bed axis was assumed to estimate the wetting-active zone volume and deduce α. The volume fraction of the fluidized bed occupied by the wetting-active zone varied from 14% for a low spraying rate to about 30% for highest liquid feed rates (Table 1). When the initial particle load was increased (250 to 750 g), the probability for the sprayed liquid droplets to collide with a solid particle increased due to a higher solid concentration in the fluidized bed (116 instead of 38.3 g L− 1) and therefore lower bed porosity (0.95 instead of 0.98). This resulted in a decrease of α from 31 to 24%. The inlet air temperature determines the air capacity to evaporate the liquid sprayed. When it was low (60 °C), the volume of the wetting-active zone was large (N30% of the fluidized bed volume) whilst when it was high (80 °C) α decreased down to 22% due to fast liquid evaporation. The spraying air pressure influences the angle of the sprayed liquid jet and the speed and diameter of the sprayed liquid droplets. The higher the spraying air pressure, the smaller the jet angle and the liquid droplets diameter and the higher the droplets speed [13]. When it was increased from 1 to 2 bars, the volume of the wetting-active zone decreased (29 to 18% of the bed volume) mainly due to smaller liquid droplets (19 μm instead of 35 μm) drying rapidly and to a smaller jet angle (27° instead of 38°) leading to a narrower zone. When the relative spraying pressure was increased to 3 bars, a drastic change in the shape of the isotherms was observed and the wetting-active zone was divided into two regions (Fig. 2b). The first one was situated in the upper part of the bed, under the nozzle as in other experiments. The second one appeared in the bottom part of the bed, at the centre, about 10 cm above the air distributor plate. This modification in the temperature profile may be attributed to a lower probability of collision of the liquid droplets with the solid particles in the upper part of the bed due to a higher speed of the droplets when entering the particle bed. The air sprayed may also create some turbulence in the chamber modifying particles recirculations. As a result, the estimated volume of the wetting-active zone was high (28% of the bed volume). The liquid feed rate influences both the angle of the sprayed jet and the droplets size, and modifies the drying conditions within the fluidized bed. When it was increased from 2.65 to 7.75 ml min− 1, the liquid jet angle increased (33° to 40°) as well as the diameter of the liquid droplets (35 to 45 μm) that were therefore more difficult to dry. As a result, the fraction of the bed occupied by the wetting-active zone increased from 14 to 29%. From the outline of the wetting-active zone, it was possible to identify two typical shapes depending on the particles circulation in the conical fluidized bed fixed by the process conditions [14,15]. The first one called “funnel” like (C1 — Fig. 2c) was observed for the smaller particle load (250 g) and the lower inlet air temperature (60 °C). Both conditions corresponded to a high humidity in the fluidized bed (insufficient drying) and therefore a high probability of wet quenching. In this case, particles rise up at the centre. Arriving in the upper part of the bed they are wetted by the liquid droplets and then wet particles go downwards along the container walls where they can stick. These two conditions were therefore no more considered for the rest of the study. For the seven other conditions tested, the outline of the low temperature region formed a “bell” like profile (C2 — Fig. 2a, b, d) corresponding to an upward movement of the particles at the sides, near the container walls, and downwards through the centre after wetting by the liquid droplets. This second circulation pattern is more favorable to control agglomeration since only “dry” particles rise up along the container walls. 3.2. Model The study of the thermal zones in the fluidized bed imposed to consider the wetting-active zone as a distinct region, exchanging particles with the bulk of the population. The fluid bed reactor was then represented by two perfectly mixed reactors exchanging particles (Fig. 3) with reactor
Qbinder Cbinder
Reactor A Wetting-active zone A (v,t) NA (t) = α NT (t) q(t)
q(t)
Reactor B Isothermal + heat transfer zones B (v,t) NB (t) = (1-α) NT (t) Fig. 3. The model.
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A corresponding to the wetting-active zone and reactor B to the isothermal and heat transfer zones. Particles were supposed to move randomly between the two regions at a constant rate q(t), equal to the volume per unit time of particles transferred through the boundary of the wettingactive zone (A). Assuming particles homogeneously distributed in the whole fluidized bed, and the volume of both reactors constant with time, the portion of the entire population enclosed in reactor A was equal to α and the number density function Ψ(v,t), for particles with volume v, in the whole fluidized bed was given by: A
B
Ψðv; t Þ = α⋅Ψ ðv; t Þ + ð1−αÞ⋅Ψ ðv; t Þ:
ð2Þ
The low air temperatures measured in the wetting-active zone A, were assumed to correspond to high humidity allowing agglomeration whilst in reactor B, temperature and humidity conditions only allowed mixing of the bulk particles with possible breakage due to collisions or abrasion, but no agglomeration. Applying PBE to reactors A and B led to: : 8 h i ∂h i A > qðt Þ dv Ψ A ðv; t Þ + F ðvÞ−EðvÞ > > dΨ ðv;t Þ = V ⋅ Ψ B ðv; t Þ−Ψ A ðv; t Þ − ⋅ > < dt A ∂v dt : > i > dΨ B ðv; t Þ qðt Þ h A > B > : = Ψ ðv; t Þ−Ψ ðv; t Þ dt VB ⋅
ð3Þ
where the first and second terms correspond respectively to particles exchange between reactors A and B and layering growth and F and E are the birth and death terms due to particles agglomeration: F ðvÞ =
1 v A A ∫ βðv−u; uÞ⋅Ψ ðv−u; t Þ⋅Ψ ðu; t Þdu 20 A
∞
A
EðvÞ = Ψ ðv; t Þ⋅ ∫ βðv; uÞ⋅Ψ ðu; t Þ⋅du
ð4Þ
ð5Þ
0
β(v,u) is the agglomeration function representing the probability of efficient collisions between two particles with volumes v and u. Assuming a constant layering growth rate G = dv/dt, the layering growth term became:
−
i ∂ dv ∂ h ∂Ψ A ðv; t Þ A A Ψ ð v; t Þ =− G⋅Ψ ðv; t Þ = −G⋅ : ⋅ ∂v dt ∂v ∂v
ð6Þ
The resulting mathematical model was a system of two integro-differential equations. It was reformulated into a system of 32 differential equations by discretizing PBEs in the particle volume space divided in 16 size classes with limits corresponding to apertures of the sieves used for particle size distribution determination. Each continuous equation was integrated on a size domain [vi; vi + 1] leading to the discretized population density function: vi+1
Φi ðt Þ = ∫ Ψðv; t Þ⋅dv:
ð7Þ
vi
The discrete layering growth term was therefore: vi +1
−G⋅ ∫ vi
h i ∂ΨA ðv; t Þ A A = G Ψ ðvi ; t Þ−Ψ vi + 1 ; t : ∂v
ð8Þ
Assuming a constant value Ψ′i(t) for Ψ(v,t) in the size class i and that Ψ(vi,t) is the arithmetical mean of Ψ(v,t) between size classes i − 1 and i [16], led to: " # Ψ0 i ðtÞ + Ψ0 i−1 ðtÞ 1 Φ0 i ðtÞ Φ0 i−1 ðtÞ + = Ψðvi ; tÞ = 2 2 vi + 1 −vi ðvi −vi−1 Þ
ð9Þ
and the discrete layering growth term was rewritten, for any size class i ≠ 1 ≠ n, as: " # A A h i 1 Φi−1 ðt Þ Φi + 1 ðt Þ A A Di ðt Þ = G Ψ ðvi ; t Þ−Ψ vi + 1 ; t = G⋅ − 2 ðvi −vi−1 Þ vi + 2 −vi + 1
" # A A h i 1 Φ1 Φ2 A A − D1 ðt Þ = G Ψ ðv1 ; t Þ−Ψ ðv2 ; t Þ = G⋅ 2 ðv2 −v1 Þ ðv3 −v2 Þ
ð10Þ
ð11Þ
and " # h i 1 ΦAn−1 ΦnA ðt Þ A A : Dn ðt Þ = G Ψ ðvn ; t Þ−Ψ vn + 1 ; t = G⋅ − 2 ðvn −vn−1 Þ vn + 1 −vn
ð12Þ
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F(v) and E(v) were discretized using the fixed pivot technique described by Kumar and Ramkrishna [17] ensuring conservation of particles number and mass. Particles belonging to a size class i, with volumes comprised between vi and vi + 1, are represented by a volume xi called “grid point” with vi b xi = (vi + vi + 1)/2 b vi + 1. For each new particle with volume v in the domain [xi ; xi + 1] two fractions a(v, xi) and b(v, xi + 1) of the particle are attributed respectively to the populations at xi and xi + 1 ensuring that: aðv; xi Þ + b v; xi
+ 1
ð13Þ
= 1:
The discrete Fi and Ei functions obtained were [18]: 1 A A 1− δj;k ⋅ηðvÞ⋅βj;k ⋅Φj ⋅Φk 2 j;k;xi−1 ≤v≤xi + 1 j≥k
Fi =
∑
A
n
ð14Þ
A
Ei = Φi · ∑ βi;k · Φk
ð15Þ
k=1
with
ηðvÞ =
8x i + 1 −v > > ; x ≤ v ≤ xi +1 > > xi + 1 −xi i <
ð16Þ
v−xi−1 > > ; x ≤v≤xi > > : xi −xi−1 i−1
and δj;k =
1; j = k : 0; j≠k
ð17Þ
The discretized system of differential equations finally obtained for the two reactors was: 8 A h i j≥k N > 1 A dΦi qðt Þ > B A A A A > ∑ 1− ⋅δj;k ⋅ηðvÞ⋅βj;k ⋅Φj ⋅Φk −Φi : ∑ βi;k ⋅Φk > < dt = V A ⋅ Φi −Φi + Di ðt Þ + 2 j;k;xi−1 ≤v≤xi +1 k=1 > > B h i > > : dΦi = qðt Þ⋅ Φ A −ΦB : i i VB dt
ð18Þ
Assuming that the binder is homogeneously distributed onto the NA(t) particles in reactor A at time t and that the layering growth rate G is independent of the particle size [18], G=
Q binder ⋅C binder ρbinder ⋅V A ⋅ ∑ ΦiA
:
ð19Þ
i
1200
Initial 3. QL = 7,75 ml/min
1000
Equilibrium
Constant rate
d50 (μm)
800
600
400
200
0
0
20
40
60
80
100
120
140
time (min) Fig. 4. Evolution of theoretical (lines — β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) d50 during glass beads agglomeration in reference conditions (500 g glass beads, air 70 °C, 20%w/w acacia gum solution 5.33 ml min− 1, 1 bar).
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Different expressions are proposed for the agglomeration function βi,k [19,20]. It was chosen here to use a “non random coalescence” kernel with three parameters since it gave satisfying results in the case of Carboxy Methyl Cellulose particles agglomerated in similar equipment [19]: βi;k = β0 ·
ðxi + xk Þκ : ðxi ·xk Þγ
ð20Þ
0.9 t=0 s
0.8
t = 600 s
Mass fraction (-)
0.7
t = 1200 s
0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) 0.9 t = 1800 s
0.8
t = 2400 s
0.7
Mass fraction (-)
t = 3000 s 0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) 0.9 t = 3600 s
0.8
t = 4200 s
Mass fraction (-)
0.7
t = 4800 s
0.6 0.5 0.4 0.3 0.2 0.1 0 101
102
103
104
Diameter (μm) Fig. 5. Evolution of theoretical (lines — β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) particle size distribution during glass beads agglomeration in reference conditions (500 g glass beads, air 70 °C, 20%w/w acacia gum solution 5.33 ml min− 1, 1 bar).
C. Turchiuli et al. / Powder Technology 208 (2011) 542–552
549
Table 2 β0 and SSD values obtained for glass beads agglomeration trials with k = 1.805 and γ = 0.83 and growth rate during the constant rate phase from Fig. 6. β0 × 10− 12
Conditions Reference: 5.33 ml min− 1, 1 bar, 70 °C, 20%w/w, 500 g glass beads 2.65 ml min− 1 7.75 ml min− 1 2 bar 3 bar 80 °C 30%w/w 750 g
SSD
r (μm min− 1)
9.71
0.246
8.2
4.01 17.41 14.11 4.71 10.91 16.71 4.31
0.343 0.270 0.631 0.625 0.380 0.413 0.425
2.3 13.9 9.6 9.5 7.8 14.7 5.9
Parameters κ and γ are related to the shape of the particle size distribution and β0, describing the kinetics of the particle size distribution evolution, depends on the operating conditions. Assuming that particles are dragged by rising air bubbles in the fluidized bed, the flow rate of particles from reactor B entering reactor A can be estimated from the particles turnover time tc. If this flow rate is equal to the particle flow rate exiting from reactor A to reactor B, the exchange flow rate q(t) between the two reactors is:
qðt Þ = Mp ðt Þ = tc ·ρp
ð21Þ
where tc can be calculated from the particles minimal fluidization rate Umf, the air rate Ua and the rising air bubbles rate Ub [21]: 1−U a −Umf −1 H′
⋅ : Ub 0; 6 U a −Umf
ð22Þ
1400
(a) QL= 7,75 ml/min (X)
1200
d50 (μm)
1000
Reference ( ) QL= 5,33 ml/min, P = 1 bar,T = 70°C, Ch= 500 g, CL= 20 %
T = 80°C ( )
800 600 Ch= 750 g ( ) 400
QL = 2,65 ml/min (
)
200 0 0
20
40
60
80
100
120
140
160
180
200
time (min) 600
(b) P = 2 bar ( ∗ )
500
400
d50 (μm)
tc =
300 P = 3 bar ( ) 200
100
0 0
10
20
30
40
50
60
70
80
90
100
time (min) Fig. 6. Theoretical (lines) and experimental (points) evolutions of d50 during glass beads agglomeration trials.
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H′ is the distance between the air distribution grid and the bottom of reactor A (bottom of wetting-active zone determined experimentally from air temperature profiles) and Ub is calculated from [14,22]:
1=2 Ub = Ua −Umf + 0; 711ð g · db Þ
ð23Þ
with the air bubble diameter: h
i1 = 3
1;21 ⋅ 1 + 0; 0684⋅H lf : db = 0; 853 1 + 0; 272 U a −Umf
ð24Þ
The discretized system obtained in Eq. (18), with 32 differential equations (16 for each of the two reactors corresponding to the 16 size classes chosen) and 3 parameters β0, κ and γ linked to the agglomeration function, was solved using the ode15s (Solve Stiff differential equations) function of MATLAB 6.1. 3.3. Glass beads agglomeration During agglomeration of glass beads with acacia gum solution, the median particle diameter d50 increased with time showing three different phases (Fig. 4): – the “initial phase” with a low increase of d50 with time, corresponding essentially to the deposit of binder at the particle surface until rendering it sticky. – the “constant rate phase” where d50 increased linearly with time, and agglomeration growth occurred. – the “equilibrium phase” where no more size increase was observed, corresponding to equilibrium between growth and destruction of agglomerates. All along the process there is a competition between the different growth and rupture mechanisms, each of the three phases identified corresponding to one of them prevailing on the others. The proposed model does not include the rupture mechanism, it was therefore only 0.4 QL = 2,65ml/min
0.3
QL= 5,33 ml/min
0.2
βo = 9,71.10-12
0.15 QL= 7,75 ml/min βo = 17,41.10-12 0.1
Massfraction (-)
Mass fraction (-)
0.35
βo = 4,01.10-12
0.25
0.25 T = 70°C βo = 9,71.10-12
0.2
T = 80°C βo = 10,91.10-12
0.15 0.1
0.05 0.05 0 101
102
103
0 101
104
102
Diameter (μm)
c 0.25
CL= 20% w/w βo = 9,71.10-12
Ch= 500 g βo = 9,71.10-12
0.2
Mass fraction (-)
Mass fraction (-)
104
d Ch= 750 g βo = 4,31.10-12
0.25
0.15
0.1
0.05
0 101
103
Diameter (μm)
0.2 CL= 30% w/w
0.15
βo = 10,91.10-12
0.1
0.05
102
103
Diameter (μm)
104
0 101
102
103
104
Diameter (μm)
Fig. 7. Influence of the sprayed solution flow rate (a), the air temperature (b), the initial particle load (c) and of the acacia gum solution concentration (d) on the value of β0 and on the agglomerates particle size distribution after 50 min.
C. Turchiuli et al. / Powder Technology 208 (2011) 542–552
20
β0= 1,1879.10-12 x r (R2 = 0,9489)
18
β0 (.10-12 m2,565.s-1)
551
16
Trial 4 (P=2bars)
14 12 10 8 6
Trial 5 (P=3 bars)
4 2 0 0
5
10
15
20
r (μm.min-1) Fig. 8. Relation between agglomeration kernel β0 and growth rate r.
adapted to the modelling of the constant rate phase where rupture was assumed to be negligible. Values of parameters β0, κ and γ were first determined by minimizing the sum squared differences (SSD) between theoretical and experimental particle size distributions at different times during the constant rate phase of reference trial using lsqnonlin function of MATLAB (Levenberg–Marquardt type algorithm). With β0 = 9.71 × 10− 12, k = 1.805 and γ = 0.83, the model gave a good representation of the evolution of the particle size distribution (SSD = 0.2459) during the constant rate phase but, as expected, did not allowed representing the equilibrium phase (Figs. 4 and 5). Knowing that κ and γ mainly depend on the agglomeration process [19], values obtained (κ=1.805 and γ=0.83) were used for all the other trials and only the β0 value was determined minimizing SSD (Table 2). Good correlation was obtained between calculated and experimental particle size distributions evolution during the constant rate phase (SSDb 0.5) except for trials with higher spraying pressure (e.g. 2 and 3 bars instead of 1 bar) (Fig. 6). In this last case, particle size distributions were bimodal (weak agglomerates) and the shape of the wetting-active zone obtained from air temperature profiles was different (Fig. 2b). For a good correlation, the model should be modified choosing another β function adapted to bimodal particle size distributions and/or dividing reactor A into two reactors in agreement with the particular shape of the wetting-active zone for these conditions. β0 appeared to greatly depend on the sprayed solution flow rate with a quasi linear increase of β0 with the solution flow rate throughout the studied range (e.g. 2.65–7.75 ml min− 1). β0 also significantly increased with the sprayed solution concentration and decreased with the initial particle load thus modifying the particle size distribution of agglomerates obtained (Fig. 7). In fact, β0, fitting parameter of the model, could be correlated to the growth rate r during the constant rate phase obtained from experimental d50 = f(t) curves of Fig. 6 (Table 2). β0 was found to increase linearly with r according to (Fig. 8): β0 = 1:19:10
−12
ð25Þ
· r:
3.4. Maltodextrin particles agglomeration Soluble maltodextrin particles were agglomerated by spraying water which was assumed to be integrally removed by hot air during drying and no layering growth was expected to occur. To simulate maltodextrin particles agglomeration, the layering growth rate of the model was therefore fixed to 0 and the particle load was assumed to be constant. κ and γ were assumed not to be modified since the same equipment was used. It was shown for glass beads that β0 can be estimated from the growth rate r for given operating conditions. The median diameter d50 evolution with time during maltodextrin agglomeration with water was different from that obtained for glass beads. Firstly, no “initial phase” was observed and secondly, the growth rate values were higher (40.5 instead of 8.2 μm min− 1 for the reference condition). Specific β0 value was therefore calculated for maltodextrin particles agglomeration assuming the same relation (25) between β0 and r. Good agreement was found between calculated and experimental particle size evolution (SSD = 0.251) indicating that the relation between β0 and r can be applied whatever the particles agglomerated (soluble or non-soluble) (Fig. 9). 0.4 0.35
t=0s
t = 1200 s
t = 600 s
t = 1800 s
t = 900 s Mass fraction (-)
0.3 0.25 0.2 0.15 0.1 0.05 0 101
102
103
Diameter (μm)
104
102
103
104
Diameter (μm)
Fig. 9. Evolution of theoretical (lines — β0 = 48 × 10− 12, k = 1.805 and γ = 0.83) and experimental (points) particle size distribution during maltodextrin particles agglomeration in reference conditions (280 g maltodextrin particles, air 70 °C, water 5.33 ml min− 1, 1 bar).
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4. Conclusion Air temperature measurements within the granulator chamber allowed estimating the volume fraction α of the fluidized bed occupied by the wetting-active zone where agglomeration was assumed to take place due to high humidity. Depending on the operating conditions, α occupied between 14 and 30% of the bed. The fluid bed granulator was then modelled as a system of two coupled perfect mixers. To simulate particle size distribution evolution during agglomeration, population balance equations were written for each mixer. Parameters (β0, κ and γ) of the model were identified for agglomeration trials with glass beads sprayed with acacia gum solution. κ and γ could be kept constant (k = 1.805 and γ = 0.83) whilst β0 varied with the operating parameters. Especially, β0 increased with the sprayed solution flow rate and concentration and decreased with the particle load and showed a linear relationship with the growth rate r in the constant rate phase of agglomeration obtained from d50 = f(t) curves. Good agreement was found between theoretical and experimental evolutions of particle size distribution during this phase except for conditions leading to bimodal populations (high air spraying pressure) for which other forms of the agglomeration function β should be used and a modified model should be proposed to take into account the particular shape of the wetting-active zone in this case. Simulation of soluble maltodextrin particles agglomeration using the results for glass beads led to satisfying correlation between theoretical and experimental particle size evolutions showing that results can be applied as well for soluble and non-soluble particles. Acknowledgements The authors would like to acknowledge the financial support of CONACyT (Mexico) for the PhD scholarship provided to Teresa Jimenez and Bertrand Broyart from Agroparistech for numerical calculations. References [1] H. Schubert, Instantization of powdered food products, International Chemical Engineering 33 (1) (1993) 28–45. [2] B.J. Ennis, G. Tardos, R. Pfeffer, A microlevel-based characterization of granulation phenomena, Powder Technology 65 (1991) 257–272.
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