Modelling the bed characteristics in fluidised-beds for top-spray coating processes

Particuology 10 (2012) 649–662

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Modelling the bed characteristics in fluidised-beds for top-spray coating processes Mike Vanderroost a , Frederik Ronsse a , Koen Dewettinck b , Jan G. Pieters a,∗ a b

Department of Biosystems Engineering, Ghent University, Coupure links 653, 9000 Ghent, Belgium Laboratory of Food Technology and Engineering, Ghent University, Coupure links 653, 9000 Ghent, Belgium

a r t i c l e

i n f o

Article history: Received 4 November 2011 Received in revised form 4 January 2012 Accepted 20 February 2012 Keywords: Fluidised bed Modelling Voidage distribution Fluidised bed characteristics Multiphase flow Coating process

a b s t r a c t A particle sub-model describing the bed characteristics of a bubbling fluidised bed is presented. Atomisation air, applied at high pressures via a nozzle positioned above the bed for spray formation, is incorporated in the model since its presence has a profound influence on the bed characteristics, though the spray itself is not yet considered. A particle sub-model is developed using well-known empirical relations for particle drag force, bubble growth and velocity and particle distribution above the fluidised-bed surface. Simple but effective assumptions and abstractions were made concerning bubble distribution, particle ejection at the bed surface and the behaviour of atomisation air flow upon impacting the surface of a bubbling fluidised bed. The model was shown to be capable of predicting the fluidised bed characteristics in terms of bed heights, voidage distributions and solids volume fractions with good accuracy in less than 5 min of calculation time on a regular desktop PC. It is therefore suitable for incorporation into general process control models aimed at dynamic control for process efficiency and product quality in top-spray fluidised bed coating processes. © 2012 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

1. Introduction Fluidised beds are used amongst others in food and pharmaceutical industries for coating of particles (Teunou & Poncelet, 2002). Relatively little research has been performed in developing a quality model for a coating process, i.e. a model able to predict the quality of the product in terms of coating thickness and uniformity and able to predict the process in terms of the occurrence of unwanted side-effects, including agglomeration, attrition and spray loss. When a fluid is blown through a fixed powder bed, the bed demonstrates liquid-like properties. In the process of top spray coating, a powder is fluidised by heated air issuing from the bottom of the reactor (Depypere, Pieters, & Dewettinck, 2005; Ronsse, Pieters, & Dewettinck, 2009). A nozzle placed above the fluidised bed sprays dissolved coating material onto the particles in the form of small droplets. Depending on the type of nozzle, compressed atomisation air is used to assist the formation of droplets (Fig. 1). While travelling through the bed, the particles and the droplets collide and exchange heat and moisture with each other, with the air

∗ Corresponding author. Tel.: +32 9 264 61 88; fax: +32 9 264 62 35. E-mail address: [email protected] (J.G. Pieters).

and with the reactor wall (Ronsse, Pieters, & Dewettinck, 2008). The coated product quality and coating process efficiency in a fluidised bed are largely determined by the spray and bed characteristics and the particle motion. To describe and model the phases and phenomena occurring in a fluidised bed, computational fluid dynamics (CFD) are frequently applied. In CFD, two well known modelling approaches are used:

• The Euler–Euler (or multifluid) approach in which all phases are considered interpenetrating continua. This approach has been successfully applied for modelling the overall hydrodynamics of the multiphase flow in tapered fluidised bed coaters (Panneerselvam, Savithri, & Surender, 2009; Duangkhamchang, Ronsse, Depypere, Dewettinck, & Pieters, 2010, 2011). The main disadvantage is that simulations can last long. • The Euler–Lagrange approach in which at least one phase is considered a discrete phase interacting with other phases which are considered continua (Beetstra, van der Hoef, & Kuipers, 2007; Kafui, Thornton, & Adams, 2002). This approach, unlike an Euler–Euler model in which all the occurring phases are considered continua, offers the possibility to describe the hydro- and thermodynamics of individual droplets and particles in the fluidised bed coating process. Similar to the Euler–Euler approach, the main disadvantage is that the solution of the model requires

1674-2001/$ – see front matter © 2012 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.partic.2012.02.004

650

Ar b d f F G  G L M N P Q r Re T u w v z  ε    ϕ

M. Vanderroost et al. / Particuology 10 (2012) 649–662

Archimedes number = g (p − g )gdp3 /2 , penetration depth (cm) diameter (␮m or m) fraction force (N) gravitational constant gravitational force (N) length (cm) mass (kg) number pressure (bar) volumetric flow rate (m3 /h) radius (m) Reynolds number turbulent force in the freeboard (N) velocity (m/s) width (cm) velocity vector (m/s) height (cm) unknown parameter voidage spray angle (◦ ) kinematic viscosity (Pa s) density (kg/m3 ) sphericity

Superscripts 0.97 at a voidage of 0.97 bot bottom of the reactor particle p top top of the reactor total tot Subscripts as axial subdivision at atomisation b bubble bc bottom crater bottom of reactor bot c characteristic d drag eff effective entrance en exp experiment fb fluidised bed fluidisation flow ff g gas phase mf minimum fluidisation normalised n nozzle nozzle p particle r reactor radial subdivision rs sb static bed sv solids volume sim simulation subdivision sub tf turbulent fluidisation turb turbulent termination zone tz vol volumetric

0 ˛ ˇ 

at bed surface height alpha zone beta zone gamma zone

lots of computational power and time. Furthermore, when the number of particles or droplets is large, as is the case in fluidised beds, the computational requirements can significantly exceed those of the Euler–Euler based models.

The main advantage of both CFD modelling approaches is that both tend to white-box modelling with a high level of detail. At the other side of the spectrum of fluidised bed models, black-box models are sometimes used for process control purposes. They are pure input–output models describing a certain fluidised bed configuration. Since they do not (or very little) consider fundamental interactions and phenomena inside the reactor, they are not flexible when the bed configuration or the operational conditions are changed. In the middle of the spectrum, there is a broad zone of greybox models, characterised by simplifications and abstractions of the interactions between the phases. Grey-box models are less fundamental and less time consuming than CFD models. They are particularly suitable for process control purposes and have proven to be capable of accurately describing the characteristics of dynamic phenomena inside a fluidised bed (Larsen, Sonnergaard, Bertelsen, & Holm, 2003; Ronsse, Pieters, & Dewettinck, 2007; Turchiuli, Jimenèz, & Dumoulin, 2011; Vanderroost, Ronsse, Dewettinck, & Pieters, 2011). In the literature, little or no information can be found on models capable of predicting the coated product quality and coating process efficiency in fluidised beds. In the development of such a grey-box process control model, a spray sub-model has already been developed capable of predicting the air and droplet temperatures and the air humidity (Vanderroost et al., 2011). In this paper, a next step in the development of such a model is presented: a particle sub-model, able to predict the bed characteristics in terms of bed height and solids volume fraction in less than 5 min using a regular desktop PC. Such a model can then be further developed to be built in a process control model for product quality and efficiency of a coating process in a top spray fluidised bed.

Fig. 1. Top spray coating process in a tapered fluidised bed reactor.

M. Vanderroost et al. / Particuology 10 (2012) 649–662

2. Theoretical background

And (uff – ˇumf ) is defined as the excess velocity ue , implying that a fraction of the fluidisation air, fbf , passes through the bed via the bubble phase.

2.1. Fluidisation regimes The fluidised bed can amongst others be characterised by the expansion of the bed with respect to its initial fixed state and by the particle distribution in the bed. Both characteristics are determined by the intensity of the fluidisation and the properties of the powder that is used. Geldart (1973) divided powders into four groups (A, B, C and D) based on the mean particle size, dp , and the density difference between the particles and the fluidising medium, p − g . Powders of groups A and B are mostly used in industrial applications since they possess the most suitable properties for fluidisation (Goossens, 2006). The intensity of the fluidisation is mainly determined by the fluidisation flow velocity, uff . When uff is gradually increased, starting from zero, the following regimes can occur sequentially (Kunii & Levenspiel, 1991): static, minimum fluidisation, bubbling fluidisation, slugging, turbulent fluidisation, and pneumatic transport. In top-spray fluidised beds, the bubbling fluidisation regime is preferred because of both good mixing and optimal heat and mass transfer between the gas and solid phases. These beneficial features are the result of the bubble-induced particle circulation within the bed (Stein, Ding, Seville, & Parker, 2000). Bubbling fluidisation occurs when uff is between the minimum fluidisation velocity, umf , and the turbulent fluidisation velocity, utf . The latter is the velocity at which turbulence starts to occur in the fluidised bed. It can be calculated as (Yang & Leu, 2008), utf =

0.834g Ar 0.487 , g dp

651

(1)

with g the kinematic viscosity of the gas phase, Ar the Archimedes number, g the density of the gas phase and dp the particle diameter. The minimum fluidisation velocity can be calculated by means of several empirical relations (Coltters & Rivas, 2004; Ergun, 1952) or theoretical relations (Beetstra et al., 2007; Hill, Koch, & Ladd, 2001).

fbf =

Qb =1− uff Abot

 u −0.38 ff umf

(4)

2. The three-phase model considers an additional phase, namely a cloud phase surrounding the bubbles in which gas recirculates (Collins, 1965; Davidson & Harrison, 1963; Stewart, 1968). 3. A four-phase model was developed by Rowe and Partridge (1997) who observed that particles are entrained by rising bubbles, creating turbulent wakes of particles at the bottom that move upwards with the bubbles. This turbulent wake can be regarded as a new additional phase, the so-called wake phase (Kunii & Levenspiel, 1968). The particle fraction inside this wake phase is denoted with the parameter fw (Baeyens & Geldart, 1973). The level of complexity of the bubble models listed above increases with increasing number of phases that are considered. 2.3. Voidage The voidage, ε, is a dimensionless variable that is defined as the volume fraction of gas in the bed. At each position in the bed, ε is determined by (Kunii & Levenspiel, 1991) ε = εb + (1 − εb )εem ,

(5)

where εb represents the contribution of the void bubble phase to the total voidage and εem is the voidage of the emulsion phase. The solids volume fraction fsv is given by (1 − ε) and is also dimensionless, and εb can be calculated using the following equation (Choi et al., 1998): εb =

uff − ˇumf , ub

(6)

with uff the fluidisation velocity, umf the minimum fluidisation velocity and ub the bubble velocity which, according to Davidson and Harrison (1963), can be calculated as 0.5

2.2. Bubble models

ub = (uff − umf ) + 0.711(gdb )

With regard to modelling the bubbling fluidisation regime that is frequently applied in the coating process, three bubble models are frequently encountered in the literature.

with g the gravity acceleration. The variable db is the bubble diameter which is a function of the height in the bed. It is calculated by means of an equation from the generalised bubble growth model of Choi et al. (1998) for non-tapered fluidised beds. According to this model, for cylindrical reactors, small bubbles are formed across the surface area of the air distributor (db  rr ). Consequently, these bubbles rise and grow quickly by an alternating process of coalescing and breaking up, while moving towards the centre of the bed (Wormsbecker, van Ommen, Nijenhuis, Tanfara, & Pugsley, 2009). This is depicted in Fig. 2. In the emulsion phase, εem is calculated as follows: at minimum fluidisation conditions, the pressure drop over a fluidised bed can be calculated by the following equation (Kunii & Levenspiel, 1991):

1. The two-phase model developed by Toomey and Johnstone (1952) divides the bubbling fluidised bed into two noninteracting phases: a bubble phase which consists of void bubbles and an emulsion phase which consists of the fluidised particles around the bubbles. This model constitutes the fundamentals for all other models. Choi, Son, and Kim (1998) derived an empirical relation for the volumetric flow rate (Qb ) of the gas passing through the bed as bubbles: Qb = (uff − ˇumf )Abot ,

(2)

with Abot the area of the bottom surface of the reactor, and ˇ is calculated by ˇ=

 u 0.62 ff umf

,

(3)

−P = (p − g )(1 − εem )g.

,

(7)

(8)

The relationship between the pressure drop and the average normalised drag force on a single particle, Fd , in the fluidised bed can be calculated as (Beetstra, 2005): −P = 18g uff

1 − εem Fd , εem d2 p,eff

(9)

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M. Vanderroost et al. / Particuology 10 (2012) 649–662

be regarded as a static and homogeneous system. As a consequence, it is to be expected that for bubbling fluidised beds a correction parameter will be needed in the right-hand side term of Eq. (10). 2.4. Freeboard

Fig. 2. Different phases, regions, heights and dimensions in the modelled fluidised bed without atomisation air.

The normalised force Fd can be calculated using the theoretically derived drag relation of Beetstra et al. (2007):



Fd =

10

+

1 − εem ε2em



+ ε2em (1 + 1.5

1 − εem )

3(1−εem )

1 + 10

Re

−(1/2)(10+4(1−εem ))

 ,

(10)

where Re = (g uem dp,eff )/g . Unlike the commonly used and wellknown combination of the empirical Ergun and, the Wen and Yu drag relations, the Beetstra equation has no discontinuity at εem = 0.8. Substitution of Eqs. (8) and (10) into Eq. (9) results in a complex relation of the form f (uem , εem ) = 0,

(11)

with both uem and εem unknown. The former can be derived from the volumetric flow rate of the air passing through the emulsion phase: uem =

 rr 0

Qff − Qb 2 (1 − εb )r dr

(12)

.

Substituting uem in Eq. (11) automatically results in εem . In order to calculate the volumetric bubble flow rate, Qb in Eq. (2) and the voidage εb of the bubble phase in Eq. (6), the minimum fluidisation velocity umf is needed. umf can be determined by calculating εmf with the Broadhurst and Becker equation (Rhodes, 1998) and substituting it in Eq. (11):

 εmf = 0.586ϕp−0.72

2g g dp3 (p − g )g

0.029 



v¯ p,0 = 2 −

4



ub,0 ,

(14)

with ub,0 the bubble velocity at the bed surface (Eq. (7)).

0.413Re ((1/εem ) + 3εem (1 − εem ) + 8.4Re−0.343 ) 24ε2em

Measurements carried out by Depypere, Pieters, and Dewettinck (2009) revealed that in a tapered bubbling fluidised bed two regions can be distinguished: a dense particle region near the wall and a relatively dilute particle region in the centre part of the bed (Fig. 2). Their results were in agreement with the research of Toyohara and Kawamura (1991) and Schaafsma, Marx, and Hoffmann (2006). Above the fluidised bed, when the rising bubbles reach the bed surface and erupt, hereby ejecting particles from the bed, a dilute and turbulent particle zone is observed (Fig. 2), the so-called freeboard (Glicksman & Yule, 1995; Santana, Nauri, Acosta, García, & Macías-Machín, 2005; Yorquez-Ramirez & Duursma, 2001). Experiments carried out by Lewis, Gilliland, and Lange (1962) and Fournol, Bergouygnound, and Baker (1973) pointed out that the solid volume fraction in this freeboard decreases exponentially with the height above the bed surface, i.e. the place where bubbles erupt. The average initial velocity of the particles when they are ejected at the bed surface, v¯ p,0 , can be calculated by the following equation (Fung & Hamdullahpur, 1993):

g p

0.021 ,

(13)

with ϕp , the particle sphericity. Beetstra (2005) explicitly emphasised that Eq. (10) was derived only for homogeneous, static, monodisperse systems. A fluidised bed operating at minimum fluidisation conditions can be regarded to be such a system. Under these conditions, bubbles do not occur and all fluidisation air passes through the emulsion phase (fbf = 0 in Eq. (4)). However, when uff > umf , an increasing fraction, fbf , of the fluidisation air passes through the bed via bubbles. A bubbling fluidised bed can no longer

3. Model description In the development of a particle sub-model for the bed characteristics the choice was made to incorporate the four-phase model of Rowe and Partridge (1997), but for reasons of simplicity, without inclusion of the cloud phase to describe the bubbling fluidisation regime. The bubble phase was considered a continuum, meaning that bubbles were not considered individually. In this way, computational power required was kept relatively low while at the same time relatively high level of detail was introduced in the model. The modelled reactor was a tapered (conical) fluidised bed reactor with a small inclination of the reactor wall (Fig. 2). The minimum fluidisation and turbulent fluidisation velocities, umf and utf , respectively, were calculated in the same way as for non-tapered fluidised bed reactors (Section 2.1). The fluidisation flow velocity, uff , is based on the bottom cross-sectional area. The bubble growth model of Choi et al. (1998), developed for non-tapered reactors (Section 2.3), was adapted for the tapered reactors considered in this paper. For the model description, distinction was made between a fluidised bed with and without atomisation air supplied at the nozzle. 3.1. Model not including atomisation air 3.1.1. Voidage Based on the observations by Depypere et al. (2009), it was assumed that the bubble phase is mainly present in the centre of the fluidised bed and that its presence decreases gradually towards the boundary of the reactor according to a Gaussian profile: εb (r) = A(r)εb ,

(15)

with 2

A(r) = (1 − fw )e−(r/ ) .

(16)

The bubble phase voidage, εb , was calculated by means of Eq. (6). The parameters r and are the radial coordinate and the standard

M. Vanderroost et al. / Particuology 10 (2012) 649–662

deviation of the bubble voidage profile, respectively. The parameter (1 − fw ) can be regarded as a correction on the bubble phase voidage owing to the presence of the wake phase inside the bubbles. In Section 2.3, it was mentioned that small bubbles are formed across the surface area of the air distributor (db  rr ) and that these bubbles rise and grow quickly by an alternating process of coalescing and breaking up, while moving towards the centre of the bed. It was assumed that the standard deviation was of the same order of magnitude as the bubble diameter db and that in a region near the air distributor, a transient zone existed in which evolved from rr to db . An a priori sensitivity analysis pointed out that a ±5% variation of had no significant influence on the bed characteristics beneath and closely above the bed surface, i.e. the region where the actual coating process occurred. It was adopted from Stein et al. (2000) that bubbles do not exert any drag force on particles in the emulsion phase. Since a bubbling fluidised bed is not a static and homogeneous system, in Eq. (10), a correction parameter (1 − fbf ) was introduced in the right-hand side term. This parameter corrects the drag force for the amount of fluidisation air that passes through the bed via bubbles. From Eq. (4), it can be derived that the more fluidisation air is used to form bubbles, relatively the less fluidisation air will pass through the emulsion phase; hence the drag force will increase less than linearly with the fluidisation air flow rate. Finally, it is important to remark that, due to the inclination of the reactor wall and the dependency of ub on the height, εem and εb change with height. 3.1.2. Freeboard zone As discussed in Section 2.4, the solid volume fraction, fsv , decreases exponentially with the height above the bed surface, i.e. the place where bubbles erupt. In this model, the exponential decrease of fsv (z) was expressed by means of the following correlation: max )

fsv (z) = fsv,0 e−((z−z0 )(1−fw ))/(A(r)Lc

,

∀z > z0 .

(17)

The parameter fsv,0 is the solid volume fraction of the bubble zone at the bed surface (at a height z0 above the air distributor). The parameter Lcmax is the characteristic length that represents the maximum average distance over which particles are ejected in the freeboard zone. It is multiplied by A(r) to account for the Gaussian bubble voidage profile in the bed and at the bed surface. This means that, on average, particles at the bed surface and in the centre of the bed will be ejected higher compared to particles at the bed surface but away from the centre due to higher presence of erupting bubbles in the centre. Introducing Eq. (17) in the model requires the knowledge of Lcmax and z0 . To calculate Lcmax , the following Newtonian equation of motion was considered for the ejected particles: m

dvp  + F + T , =G d dt

(18)

 the gravitational force, F the drag force on a single sphere, with G d and T an unknown turbulent force caused by vortices and eddies existing in the freeboard, as observed by Yorquez-Ramirez and Duursma (2001). To simplify further calculations, the forces Fd and T were replaced by a new force Fd,turb , i.e. the turbulent drag force. This force was then expressed in terms of the known gravitational force:  Fd,turb = Fd + T = turb G,

(19)

with  turb a dimensionless parameter that is introduced to account for the variable turbulent drag force in the freeboard zone. This means that, while moving upward, the particles are assumed to be subject to the gravitational and the turbulent drag force. At a certain height (z0 + Lcmax ), the velocity of the particles will become

653

Fig. 3. Estimation of the average fraction of particles fpf residing in the freeboard region as the ratio of the area beneath the profile starting from the height where the profile reaches its maximum (filled area) to the entire area beneath the profile.

zero. From Eqs. (18) and (19) it can be derived that this height is given by Lcmax =

v¯ 2p,0 2(1 + turb )g

,

(20)

with v¯ p,0 calculated by means of Eq. (14). The bed surface height z0 was defined as the height at which a fraction (1 − fpf ) of the total amount of particles resides in the reactor. Here, fpf is the average fraction of particles residing in the freeboard region. Its value was estimated from a one-dimensional solid volume fraction profile derived from measurements carried out by Depypere et al. (2009). The variable fpf is defined as the integral of the one-dimensional solids volume fraction profile starting at the height where the profile reaches its maximum, divided by the entire area beneath the profile, as illustrated in Fig. 3. The value of fpf was estimated to be 60% for (uff − umf )/(utf − umf ) = 1.04. When (uff − umf )/(utf − umf ) = 0, fpf is equal to zero since the bed is then in minimum fluidisation regime (no bubbles, hence no freeboard). For this model, it was assumed that fpf varies linearly for 0 ≤ (uff − umf )/(utf − umf ) ≤ 1.04. In a bubbling fluidised bed, for reasons of conservation of mass, all particles that are ejected into the freeboard eventually have to return to the bed. For each height in the bed, the bubble phase voidage εb (r) reaches a maximum in the centre (r = 0) and decreases towards the boundary of the reactor. On the other hand, the emulsion phase voidage, εem , remains constant at a certain height. Hence, at the bed surface, the solids volume fraction will increase with increasing radial distance r. In Fig. 4(a), the profile of the average distance to which particles are ejected in the freeboard and the solids volume fraction at the bed surface are shown as a function of the radial distance. As can be deduced from Eq. (17), the profile of the average distance to which particles are ejected is the same as that of the bubble phase voidage. Also from Fig. 4(a), it can easily be deduced that less particles in the centre are ejected higher and more particles away from the centre are ejected less high. The ejected particles also have a radial velocity component (Santana et al., 2005). As a result, they move towards the reactor boundary when moving upwards in the freeboard. Since the ratio of the axial to the radial velocity component is higher for particles ejected near the centre, these particles will reach the reactor wall at a higher point above the bed surface. There, they fall back to the bed along the wall through a layer with variable width, the so-called wall layer (Fig. 2). Particles ejected further away from the centre will reach the reactor wall at a lower point than particles closer to the centre. In Fig. 4(b), the width of the wall layer and the solids volume fraction in the wall layer are shown as a function of the height above the bed surface. The solids volume fraction and

654

M. Vanderroost et al. / Particuology 10 (2012) 649–662

Fig. 4. (a) The height above the bed surface to which particles at the bed surface are ejected into the freeboard (—) and the solids volume fraction at the bed surface (- - -) as a function of the radial distance, (b) the width of the layer at the wall through which particles in the freeboard return back to the bed (—) and the solids volume fraction in the wall layer (- - - -) as a function of the height above the bed surface.

the width of the layer at the wall are higher near the bed surface for reasons explained above. The layer at the wall was modelled as a wedge with the base at the bed surface lying between 0.9 and 1.0 times the radius of the reactor boundary. The top of the wedge was situated at a distance Lcmax above the bed surface at the reactor boundary. This is shown in Fig. 5.

Fig. 5. Abstraction of the wall layer in the freeboard. The wall layer is defined as the triangular dotted region with base at the bed surface between 0.9 times the radius of the reactor wall and the radius reactor wall.

Fig. 6. Different zones and heights in the modelled fluidised bed with atomisation air: (a) against results of Depypere et al. (2009) and (b) as a general scheme.

3.2. Model with inclusion of atomisation air Measurements carried out by Depypere et al. (2009) pointed out that the jet caused by the atomisation air flow creates a depression in the fluidised bed’s top surface, from now on denoted as ‘crater’. Based on these results, a new zone has been defined in the particle sub-model, the so-called crater zone. In the two-dimensional voidage plots from Depypere et al. (2009), three sub-zones could be distinguished in this crater zone (Fig. 6(a)): the ˛-zone, the ˇzone and the -zone. In Fig. 6(b), a schematic is shown of all the zones defined in the model for a fluidised bed with inclusion of atomisation air. 3.2.1. The ˛-zone In this zone, it was assumed that no bubbles nor fluidisation air occur due to the high pressure of the atomisation airflow that “pushes” the bubbles and fluidisation air away from the crater. Particles were hardly observed in this zone (Depypere et al., 2009). At the bottom of the ˛-zone, the atomisation air flow velocity was assumed to be zero, meaning that at this point all the atomisation air was diverted in the radial direction. This point was situated at a certain distance b under the bed surface, the so-called penetration depth. However, when the atomisation air velocity at the bed

M. Vanderroost et al. / Particuology 10 (2012) 649–662

surface was smaller than the bubble velocity, vb,0 , b would have a negative sign. In this case, the value of b was determined as the height above the bed surface at which the atomisation air velocity was equal to vb,0 . When b had a positive value, its value was calculated by balancing the kinetic energy of the atomisation air flow at the bed surface with the potential energy of the displaced particles due to the formation of the crater: g v¯ 2at = b(p − g )gε0 , 2

(21)

where ε0 is the voidage at the bed surface and v¯ at is the average atomisation air velocity at the bed surface height between r = 0 and the crater width r˛,0 . The crater width was defined as the width at the bottom of the ˛-zone. It was determined by the atomisation jet angle  at as follows: r˛,0 = tan(at )(znozzle − z0 − b).

(22)

3.2.2. The ˇ-zone Beneath the ˛-zone, a small zone of higher particle densities can be observed. It was assumed that bubbles do not enter the crater zone and are forced to move around it. This means that in the ˇ-zone only the emulsion phase occurs. This zone was observed to stretch from the bottom of the ˛-zone to a distance of approximately 0.015 m beneath this height (Fig. 6(a)). It should be remarked that it is very likely that, to a certain extent, this distance varies for different atomisation air pressures, powders and fluidisation air flow rates. Due to lack of information, however, a value of 0.015 m was used in the model for all situations. 3.2.3. The -zone Around the ˛-zone, the -zone was defined as a relatively dilute region of particles. In this zone, the radially dissipated atomisation air was assumed to deflect 180◦ due to the resistance of the bubbles in the fluidised bed and the erupting bubbles in the freeboard. A turbulent mixture of atomisation air and fluidisation air was assumed to entrain the particles from the ˇ-zone at the bottom and carry them upward. In the radial direction, the -zone stretches from r˛,0 to r r − r˛,0 = 180 drbot ,

(23)

with  180 a dimensionless parameter to be determined.

if r < r , if r ≥ r ,

A(r) = 0; A(r) = (1 − fw )e

−((r−r )/ )2

(24)

.

By assumption, the voidage in the crater zone was determined by interpolation as follows:



ε=

1−

r r

 +



r (ε + (1 − εb )εem ) r b

.

3.2.5. Atomisation air flow In a bubbling fluidised bed reactor with particles and erupting bubbles, the spatial atomisation air velocity profile was assumed to be the same as that of a free jet flow (Vanderroost et al., 2011), except in a region near the bed surface, the so-called termination region (Dealy, 1964; Yue, 1999), where the velocity decreases rapidly (Kuang, Hsu, & Qiu, 2001). The atomisation air pressure, Pat , was accounted for by means of the atomisation air velocity at the nozzle, vat (m/s), which can be determined as follows:

vat =

(25)

The above equation states that in the centre of the crater zone the voidage is equal to unity and linearly decreases to the voidage value at the boundary r of the crater zone.

Qat 2 3600 rnozzle

,

(26)

with Qat (m3 /h) the volumetric atomisation air flow rate, and rnozzle the radius of the nozzle opening. The value of Qat can be determined by either direct measurement during operation or by an a priori derived empirical relation between the atomisation air pressure and the volumetric atomisation air flow rate (Dewettinck, 1997). The initial height of the termination region, zt , was assumed to depend on the turbulent characteristics of the bed surface (fluidisation air flow rate) and the nozzle characteristics (nozzle position and atomisation air pressure): zt = z0 + onset,tz (znozzle − z0 ),

(27)

with  onset,tz , a dimensionless parameter, that is introduced to account for the variable initial height of the termination region. Due to the formation of the crater in the fluidised bed (b > 0), the atomisation air flow will penetrate into the bed. Then the flow profile is modelled as a Gaussian flow pattern with equal to half the radius of the crater and an amplitude which linearly decreases to zero at the bottom of the crater, located at a height zbc = z0 − b.

(28)

3.3. Discretisation The fluidised bed reactor was modelled as a 2D axisymmetric volume, vertically divided in cylindrical shell volumes. The shell volumes were cut by equidistant horizontal planes (Fig. 7) which resulted in the control volumes used for the calculations. The control volumes at the reactor wall are called ‘wall segments’. All the control volumes had a predefined constant height zsub and width wsub . The number of axial subdivisions, Nas , was determined by: Nas =

3.2.4. Voidage When atomisation air is supplied to the nozzle, the voidage as calculated without the presence of atomisation air needs to be recalculated from the bottom of the crater. The bubble phase voidage, εb (r), was still determined by Eq. (15) but with a different expression for A(r) – again compared to the situation without atomisation air – since it was assumed that bubbles do not occur in the crater zone. Now,

655

zr , zsub

(29)

where zsub was defined such that Nas is an integer number. wsub was defined such that at the bottom of the reactor the number of radial subdivisions, Nrs (0), was an integer. The number of radial subdivisions for each axial subdivision above the bottom of the reactor (z > 0) was then determined as: Nrs (z) =

rr (z) , wsub

(30)

where Nrs (z) was rounded to the smallest neighbouring integer. That means that the outer control volumes had a width that could be smaller than, equal to or larger than wsub . 3.4. Model implementation The model was programmed in C++ in Microsoft Visual Studio 2005. For the discretisation of the reactor, Nas and Nrs (0) were given values of 224 and 56, respectively. For each simulation, first, the voidages were calculated for each control volume. Subsequently, the total mass of particles, Mptot , was divided over the control volumes based on the voidages, starting at the bottom of the reactor.

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Fig. 7. Discretisation of the reactor into control volumes: (a) side view and (b) top view.

The height at which all the particles were divided, was defined as tot . the total fluidised bed height zfb A sensitivity analysis was performed for the model parameters  turb ,  onset,tz and  180 . Since the values of  turb determined the characteristic length Lc , as shown in Eq. (20), they were investigated in terms of Lc because the latter gave a direct measure of the extent of the freeboard.

4. Experimental validation Results of experiments carried out by Depypere (2005) and Depypere et al. (2005, 2009) were used to validate the particle sub-model. They measured the voidage (or solids volume fraction) distribution and the bed height using two different techniques: PEPT (positron emission particle tracking) and gauge static pressure measurements. With the latter technique, the height at which the pressure drop in the reactor reached a constant value was determined (Depypere et al., 2005). From Eq. (9), it is clear that the pressure drop is related to the drag force on particles in the fluidised bed. The freeboard region above the bed is a turbulent zone in which bubbles erupt and particles are subject to entrainment. Because of the latter, the pressure drop above the bed surface will be negligible and the determined height can be interpreted as a measure for the bed surface height. The experiments were carried out in a GLATT GPCG-1 (Glatt GmbH, Germany) tapered bed reactor with a porous plate distributor. The height and the diameter at the bottom and at the top of the reactor were zr = 56.0 cm, drbot = 14.0 cm and

top

dr = 30.0 cm, respectively. The nozzle was a two-fluid nozzle (970, Düsen-Schlick, Germany) with a spray angle  at ranging from 10◦ to 40◦ . 4.1. Experiments without atomisation air In a first series of experiments (Exp 1), Depypere (2005) used PEPT to study glass beads (Microbeads©, Sovitec, ϕp ∼ 0.9) of different average particle sizes (108.69; 167.63; 196.54; 338.00 and 648.20 ␮m) that were fluidised at two different flow rates Qff (55 m3 /h and 97 m3 /h). In this experiment, they determined the solids volume fraction distribution and the fluidised bed height 0.97 zfb . The latter was defined as the height at which the voidage reaches an absolute critical value of 0.97. In a second series of experiments (Exp 2), Depypere et al. (2005) determined the static bed heights and carried out gauge static pressure measurements at different heights at the reactor wall, and for three different amounts (Mptot = 1.0, 1.5 and 2.0 kg) of nonpareil beads (Penwest Pharmaceuticals, United States) with dp = 360 ␮m and p = 1459 kg/m3 , fluidised at a low (∼39 m3 /h) and a high (∼67 m3 /h) fluidisation flow rate. The sphericity of the nonpareils was estimated to be 0.70. In a third series of experiments (Exp 3), Depypere (2005) again applied the PEPT technique to determine the solids volume fraction distribution of a fluidised bed filled with 0.75 kg of nonpareils with fluidisation rates of 55 and 97 m3 /h, respectively. In order to validate the results of the experiments with the particle sub-model for conditions in which the nozzle was turned off, the input parameters of the experiments were introduced in the model

Table 1 Input parameters of the experiments.

Exp 1 Exp 2 Exp 3 Exp 4 Exp 5

Mptot (kg)

ϕp

p (kg/m3 )

dp (␮m)

Qff (m3 /h)

Pat (bar)

znozzle (cm)

0.75 1.0; 1.5; 2.0 0.75 0.75 0.75

0.9 0.70 0.70 0.70 0.9

2467 1459 1459 1459 2467

108.69; 167.63; 196.54; 338.00 360.00 360.00 360.00 196.54

55; 97 39; 67 55; 97 55 97

– – – 1.5; 2.0; 3.0; 4.0 1.5; 3.0

– – – 12.1; 16.8 12.1

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Fig. 8. Experimental versus model-predicted bed heights (static bed height, bed surface height and bed height where voidage = 0.97) for all experiments where no atomisation air was applied.

as boundary conditions, as given in Table 1. Subsequently, for each series of experiments,  turb was estimated to fit the simulated data to the experimental results. 4.2. Experiments with atomisation air In a fourth series of experiments (Exp 4), Depypere et al. (2005) studied the influence of the applied atomisation pressure, Pat , at the nozzle in the fluidised bed via the solids volume fraction distributions obtained from PEPT-measurements. For these experiments, 0.75 kg of nonpareils were fluidised at a fluidisation flow rate of 55 m3 /h. The nozzle was positioned at 12.1 cm above the air distributor. Four different atomisation pressures were applied: 1.5 bar (maintenance pressure), 2, 3, and 4 bar. Also, it was investigated how the nozzle position znozzle affected the crater. The nozzle was positioned at two different heights: 12.1 cm (low position) and 16.8 cm (high position) above the air distributor. For each height, the maintenance pressure of 1.5 bar and a pressure of 3 bar were applied to the nozzle. Finally, in a fifth series of experiments (Exp 5), 0.75 kg of glass beads was used instead of nonpareils. They were fluidised at an air flow rate of 97 m3 /h. The nozzle was positioned 12.1 cm above the air distributor while maintenance pressure and a pressure of 3 bar were applied at the nozzle. For all these experiments, Depypere et al. (2009) determined the height of the bottom of the crater zbc (=z0 − b) and the spray angle  at . The latter was used as an input parameter in the model. The parameters  180 and  onset,tz were estimated to fit the simulated results to the experimental results and a sensitivity analysis of the values was carried out. The input parameters for these experiments are also given in Table 1. 5. Results and discussion

Fig. 9. Two-dimensional voidage distributions for 0.75 kg glass beads (a) with particle diameter dp = 108.69 ␮m and a fluidisation air flow rate of Qff = 97 m3 /h; (b) with particle diameter dp = 338.00 ␮m and a fluidisation air flow rate of Qff = 55 m3 /h, both with experimental plots at right side, model-predicted plots at left side.

calculate the bed characteristics is two to three orders of magnitude smaller as compared to some other modelling methods (e.g. Duangkhamchang et al., 2010). For process control purposes, this can be regarded as an advantage. 5.2. Fluidisation without atomisation air 5.2.1. Bed heights In Fig. 8, experimental bed heights (static bed heights, bed surface heights and bed heights where the voidage reached a value of 0.97) are plotted against model predicted bed heights, and regression analysis was carried out. Good correlation was achieved for the three different bed heights with R2 values varying between 0.9908 and 0.9998, the slopes varying between 0.8686 and 1.0115 and the intercepts varying between −0.1223 and 2.3958. The largest deviation between model and experiment was found for the bed surface heights. This can to a certain extent be ascribed to fluctuations inherent to the gauge static pressure measurements (Depypere et al., 2005). In general, the model could predict the different bed heights with good accuracy.

5.1. Simulation time Each simulation lasted 5 min or less. In this relatively short time, the bed heights were calculated and a two-dimensional voidage plot was created. With this model, the time needed to

5.2.2. Voidage distribution For the first and the third experiments two-dimensional voidage distributions were simulated and compared with experiments (see Fig. 9), showing that the model could well predict the voidage

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M. Vanderroost et al. / Particuology 10 (2012) 649–662 Table 2 Particle ejection coefficients Cpe for glass beads. mp (10−7 kg) dp (␮m) Cpe (m) k

0.17 108.69 8.56 1.14

0.61 167.63 7.17 1.14

0.98 196.54 7.22 1.14

4.99 338.00 6.15 0.88

distribution. Deviations between model and experiment can be ascribed to the fact that the model is an abstraction and simplification of a real fluidised bed. In particular, in a region close to the air distributor at the wall of the reactor, a consistent discrepancy was noticed for all experiments. In the experiments of Depypere et al. (2009), this specific region was denoted as “dead zone”. The model presented in this study predicted voidage values of 0.5–0.7. The discrepancy in this region between model and experiment was also found and discussed by Duangkhamchang et al. (2010) who used an Eulerian computational fluid dynamics (CFD) model to simulate a gas–solid fluidised bed in a tapered reactor. As an additional and more quantitative validation, onedimensional solids volume fraction profiles were derived from the experimental voidage plots and compared with the model. The profiles were taken at radial distances r = 0 and r = 0.5rr and at axial distances z = 0.02 m, z = 0.05 m and z = 0.08 m. The results for different experiments are shown in Fig. 10. The modelled profiles predicted the general trend of the measured ones. When considering the voidage distributions and the solids volume fraction profiles, it can be concluded that the relatively simple model presented in this study was able to predict the general trend of the voidage distribution in a fluidised bed for B-powders with different particle diameters, particle densities and particle sphericities. Also, the introduction of a correction factor on the drag relation of Beetstra et al. (2007) in Eq. (10) proved to be physically relevant, as shown in Fig. 11. Voidage plots are shown for which the correction on the drag relation was not performed and are compared to the experimentally determined voidage plots and the modelled voidage plots with correction on the drag. It is clear that the plots for which no correction was performed on the drag deviate a lot from the other ones, hence it can be concluded that the correction on the drag relation was physically relevant and compensated for the inhomogeneous nature of bubbling fluidised beds. 5.2.3. Turbulent force The values of  turb determine the characteristic length Lc as shown in Eq. (20). Therefore, it was opportune to investigate the values of  turb in terms of Lc because the latter gives a direct measure of the extent of the freeboard. When plotting Lc against the ratio (uff − umf )/(utf − umf ) for each particle mass, as shown in Fig. 12, a power relation could be fitted with a particle ejection coefficient Cpe that depends on the mass of the particle: Lc = Cpe

 u − u k ff mf utf − umf

.

(31)

The values of Cpe are shown in Table 2 for each particle mass (calculated using the effective particle diameter). It can be seen that heavier particles had lower Cpe values. This means that lighter particles were ejected higher into the freeboard, which was what may be expected. The reason for choosing (uff − umf )/(utf − umf ) as

Fig. 10. One dimensional experimental (—) and model-predicted (- - - -) solids volume fraction profiles. Axial profile (a) r = 0 and radial profile (d) z = 0.05 m for 0.75 kg nonpareils at a fluidisation air flow rate of Qff = 97 m3 /h; axial profile (b) r = 0.5rr and radial profile (c) z = 0.02 m for 0.75 kg glass beads with particle diameter of 196.97 ␮m fluidised at a fluidisation air flow rate of Qff = 97 m3 /h; radial profile (e) z = 0.08 m for 0.75 kg glass beads with particle diameter of 108.69 ␮m at a fluidisation air flow rate of Qff = 55 m3 /h.

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Fig. 11. Two-dimensional voidage plots for 0.75 kg (a) glass beads with particle diameter dp = 196.54 ␮m and a fluidisation air flow rate of Qff = 97 m3 /h; (b) nonpareils with particle diameter dp = 360.00 ␮m and a fluidisation air flow rate of Qff = 55 m3 /h, both with experimental plots at the centre, model-predicted plots without correction at the left side and model-predicted plots with correction at the right side.

the variable is that in this way the velocities are normalised to the velocity interval in which bubbling fluidisation occurs. This facilitates comparison of the behaviour for different particle masses. The relation in Eq. (31) is physically relevant since for values (uff − umf )/(utf − umf ) → 0, Lc also approaches zero. This corresponds to the fact that at minimum fluidisation conditions the freeboard zone does not exist due to the absence of bubbles. The values of Cpe and k in Eq. (31) for nonpareils were 10.1 m and 1.34, respectively. Based on the mass of a nonpareil particle, namely 3.56 × 10−7 kg, one would expect that the Lc -curve for the nonpareils is to be situated beneath the Lc -curve corresponding to the glass beads with mass 4.99 × 10−7 kg. However, this was not the case, which could indicate that the characteristic length not only depends on the mass of the particle. 5.3. Fluidisation with atomisation air

Fig. 12. Values of the characteristic length Lc as function of (uff − umf )/(utf − umf ) (the ratio of the difference between the fluidisation velocity and the minimum fluidisation velocity to the difference between the turbulent fluidisation velocity and the minimum fluidisation velocity) for nonpareils (NP) and glass beads (GB) with different particle mass.

5.3.1. Voidage distribution For the fourth and the fifth series of experiments, twodimensional voidage plots were simulated and compared with the experimental voidage plots, as shown in Fig. 13. Just as for the case in which no atomisation air was supplied to the nozzle, one-dimensional solids volume fraction profiles were derived

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Fig. 13. Two-dimensional voidage plots for 0.75 kg nonpareils fluidised at a fluidisation air flow rate of Qff = 55 m3 /h (a) with atomisation air pressure of 1.5 bar and nozzle positioned at 16.8 cm and (b) with atomisation air pressure of 3.0 bar and nozzle positioned at 12.1 cm, both with experimental plots at the right side and model-predicted plots at the left side.

from the experimental voidage plots and compared with the model results, as shown in Fig. 14. For all simulations, the best results were achieved when the parameter  180 , expressed in multiples of the diameter of the bottom of the reactor, rrbot was given a value of 0.35. From this section, it can be concluded that the model also performed well when atomisation air was supplied at the nozzle. 5.3.2. Parameter for the initial height of the termination region To discuss  onset,tz , the parameter for the initial height of the termination region of the atomisation air flow, i.e. the values of (1 −  onset,tz ) were analysed. The parameter (1 −  onset,tz ) is the normalised distance under the nozzle at which the termination zone starts. Normalisation was carried out with respect to the distance between the nozzle and the bed surface, so the results for different conditions could be compared. In Fig. 15, the values of (1 −  onset,tz ), obtained by fitting the model-predicted results to the experimental ones, are plotted against the atomisation pressure for the different nozzle positions, powders and fluidisation flow rates. For nonpareils fluidised at a fluidisation air flow rate of Qff = 55 m3 /h, the value of (1 −  onset,tz ) did not significantly change with increasing atomisation air pressure. This means that the beginning of the termination zone was

Fig. 14. One dimensional experimental (—) and model-predicted (- - - -) solids volume fraction profiles. Axial profiles at (a) r = 0 and (b) r = 0.5rr for 0.75 kg nonpareils at a fluidisation air flow rate of Qff = 55 m3 /h with atomisation air pressure of 1.5 bar and nozzle located in the highest position; radial profiles at (c) z = 0.02 m and (e) z = 0.08 m for 0.75 kg nonpareils at a fluidisation air flow rate of Qff = 55 m3 /h with atomisation air pressure of 3 bar and nozzle located in the highest position and (d) z = 0.05 m for 0.75 kg glass beads with particle diameter of 196.97 ␮m at a fluidisation air flow rate of Qff = 97 m3 /h with atomisation air pressure of 1.5 bar and nozzle located in the lowest position.

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Fig. 15. Values of the unknown parameter for the initial height of the termination region of the atomisation air flow, (1 −  onset,tz ), versus the atomisation pressure for nonpareils (NP, Qff = 55 m3 /h, nozzle positioned respectively at 12.1 cm ( ) and 16.8 cm ( ) above the air distributor) and glass beads (GB, Qff = 97 m3 /h, nozzle )). positioned at 12.1 cm above the air distributor (

independent of the atomisation air pressure. From Fig. 15 it can also be seen that the values of (1 −  onset,tz ) decreased when the nozzle was positioned higher. This means that, for a certain atomisation air pressure, the termination zone started at a point relatively closer to the nozzle when the nozzle was positioned higher. This can be explained by the fact that the distance between the bed surface and the nozzle had been increased. For glass beads fluidised at a fluidisation air flow rate of 97 m3 /h and the nozzle in the lowest position, it was observed that (1 −  onset,tz ) increased with increasing atomisation air pressure and that the values were smaller compared to nonpareils fluidised with the nozzle in the lowest position. This can be explained by the higher turbulence at the bed surface which causes the beginning of the termination zone to move further away from the bed surface, hence closer to the nozzle. The fact that for glass beads the value of (1 −  onset,tz ) depended on the atomisation air pressure can be explained by the fact that for higher atomisation air pressures, the jet flow becomes more powerful and is able to penetrate the turbulent region at the bed surface. 5.4. Sensitivity analysis of the model parameters tot , and the height at which the The total fluidised bed height, zfb 0.97 , were considered for voidage reached the critical value of 0.97, zfb the sensitivity analysis for variations of the model parameters Lc ,  onset,tz , and  180 . The effect of the variations of these model parameters on the bed surface height, z0 , will not be discussed since z0 appeared to be insensitive for variations of the model parameters. For the model parameters Lc and  onset,tz , a ±5% variation was 0.97 imposed. In Fig. 16(a), it can be noticed that the height zfb slightly increased with increasing Lc . However, the total fluidised bed tot , clearly decreased with increasing values of L . This can height, zfb c easily be understood. An increase of Lc implies a less rapid increase 0.97 of the voidage in the freeboard (Eq. (17)). This means that zfb , the height at which the voidage will reach the critical value of 0.97, tot will decrease since relatively more particles will will increase. zfb 0.97 . From Fig. 16(b), it can be deduced that fill the region beneath zfb 0.97 tot zfb and zfb did not change significantly with a ±5% variation of  onset,tz . For  180 , an absolute variation of ±0.05 of the estimated value was considered. In Fig. 16(c), it can be seen that a decrease of  180 0.97 tot . From the above sendid not affect zfb , but slightly increased zfb sitivity analysis it can be concluded that the model is not very sensitive to small variations of the model parameters Lc ,  onset,tz ,

0.97 tot Fig. 16. Sensitivity analysis for the bed heights zfb ( ) and zfb ( ) for variations of the model parameters Lc ,  onset,tz and  180 : (a) a ±5% variation of Lc for 3 glass beads (dp = 338 ␮m and Qff = 97 m /h), (b) a ±5% variation of  onset,tz for glass beads (dp = 196 ␮m, Qff = 97 m3 /h and atomisation air pressure of 3 bar of a nozzle positioned at 12.1 cm above the air distributor), (c) an absolute ± 0.05 variation of  180 for nonpareils (Qff = 55 m3 /h and atomisation air pressure of 3 bar of a nozzle positioned at 16.8 cm above the air distributor).

and  180 . Moreover, the sensitivity of the model is limited to the top 0.97 tot . Since this and zfb region of the freeboard, characterised by zfb model is developed to describe the coating process, which mainly occurs near the bed surface height, it can be expected that a change in the dimension of the top region of the freeboard will only have a small influence on the overall process. 6. Conclusions In this paper, a particle sub-model was presented that calculates the fluidised bed characteristics in terms of bed height and particle distribution. The model was developed based on theoretical relations and empirical data from literature and complemented with assumptions and abstractions concerning the drag force in the freeboard and the behaviour of a jet flow when impacting the surface of a bubbling fluidised bed. Additional assumptions had to be made because information and data in the literature are sparse or non-existent.

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The particle sub-model presented in this paper could predict the bed characteristics with good accuracy in only a few minutes. Three model parameters  turb (Lc ),  180 and  onset,tz were introduced to model the turbulent behaviour in the freeboard and the characteristics of a jet flow impacting the surface of a bubbling fluidised bed, respectively. The values of these parameters, obtained by fitting the model to experiment, were considered consistent, relevant and could lead to physical explanation. It was clearly shown that the correction for bubble flow in bubbling fluidised beds on the drag relation of Beetstra et al. (2007) was a relevant correction resulting in satisfactory agreement between model prediction and experimentally determined voidage plots. The characteristic length over which particles were ejected into the freeboard zone increased with increasing fluidisation flow rate, as could be expected. Finally, the initial height of the termination zone was localised further away from the nozzle for higher nozzle positions. Increasing the fluidisation flow rate moved the initial location closer to the nozzle. Sensitivity analysis of the model parameters pointed out that only the dimension of the top region of the freeboard is affected by small variations of the model parameters. Since the actual coating process occurs near the bed surface, the latter is not considered a problem. The model parameters were determined for the reactor geometry considered in this study. For other geometries, the model parameters may need to be recalibrated. As shown in this study, the calibrated model parameters for a given reactor geometry, could be used to predict the bed characteristics for a broad range of particle diameters and fluidisation flow rates and for different particle sphericities, demonstrating its usefulness in process control applications. The particle sub-model presented in this study, will be further developed to describe solids motion in a fluidised bed. Subsequently, the particle sub-model and the developed spray submodel (Vanderroost et al., 2011) will be merged into one model, namely, a process control model describing coating process efficiency and product quality in a top-spray fluidised bed. Acknowledgement The authors acknowledge the financial support of the Special Research Fund (BOF) of the Ghent University. References Baeyens, J., & Geldart, D. (1973). Fluidization et ses applications. Toulouse: ENSIC. Beetstra, R., 2005. Drag force in random arrays of mono- and bidisperse spheres (Doctoral dissertation). University Twente, Twente. Beetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007). Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations. Chemical Engineering Science, 62, 246–255. Choi, J. H., Son, J. E., & Kim, S. D. (1998). Generalized model for bubble size and frequency in gas-fluidised beds. Industrial and Engineering Chemistry Research, 37, 2559–2564. Collins, R. (1965). An extension of Davidson’s theory of bubbles in fluidised beds. Chemical Engineering Science, 20, 747–755. Coltters, R., & Rivas, A. L. (2004). Minimum fluidisation velocity correlations in particulate systems. Powder Technology, 147, 34–48. Davidson, J. F., & Harrison, D. (1963). Fluidised particles. New York: Cambridge University Press. Dealy, J.M. (1964). Momentum exchange in a confined circular jet with turbulent source. Ph.D. dissertation, University of Michigan, Michigan. Depypere, F. (2005). Characterisation of fluidised bed coating and microcapsule quality: A generic approach (Doctoral dissertation). Ghent University, Gent. Depypere, F., Pieters, J. G., & Dewettinck, K. (2005). Expanded bed height determination in a tapered fluidised bed reactor. Journal of Food Engineering, 67, 353–359. Depypere, F., Pieters, J. G., & Dewettinck, K. (2009). PEPT visualisation of particle motion in a tapered fluidised bed coater. Journal of Food Engineering, 93, 324–336. Dewettinck, K. (1997). Fluidized bed coating in food technology: Process and product quality (Doctoral dissertation). Ghent University, Gent. Duangkhamchang, W., Ronsse, F., Depypere, F., Dewettinck, K., & Pieters, J. G. (2010). Comparison and evaluation of interphase momentum exchange models for

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