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 www.elsevier.com/locate/jfoodeng

Combined population balance and thermodynamic modelling of the batch top-spray fluidised bed coating process. Part I—Model development and validation F. Ronsse a

a,b,*

, J.G. Pieters a, K. Dewettinck

b

Department of Biosystems Engineering, Ghent University, Coupure links 653, B-9000 Gent, Belgium b Food Technology and Engineering, Ghent University, Coupure links 653, B-9000 Gent, Belgium Received 29 March 2005; accepted 26 September 2005 Available online 21 November 2005

Abstract A combined population balance and thermodynamic model was developed for the batch top-spray fluidised bed coating process. The model is based on the one-dimensional discretisation of the fluidised bed into different control volumes, in which the dynamic heat and mass balances for air, water vapour, core particles and coating material were established. The calculation method involves a Monte Carlo technique for the simulation of the particle exchange in combination with the first-order EulerÕs method for solving the heat and mass balances. The model enables the prediction of both the dynamic coating mass distribution and the one-dimensional thermodynamic behaviour of the fluidised bed during batch operation. The simulation results were validated using the results from tests on a Glatt GPCG-1 fluidised bed unit.  2005 Elsevier Ltd. All rights reserved. Keywords: Fluidisation; Mass transfer; Heat transfer; Simulation; Coating mass distribution; Monte Carlo

1. Introduction Coating operations have been widely employed in various industries, including the food industry, as an effective technique for altering the surface properties of solid particles and consequently, to tune the effect of functional constituents in food systems (Abe, Yamada, Hirosue, & Nakamura, 1998; Arshady, 1993; Dziezak, 1988; Jackson & Lee, 1991). Many techniques have been developed to date, and fluidised bed coating is among the most widespread methods. The benefits of enveloping or encapsulating solid particles include controlled release, protection against reactive environments, improving the processabil-

* Corresponding author. Address: Department of Biosystems Engineering, Ghent University, Coupure links 653, B-9000 Gent, Belgium. Tel.: +32 9 264 6200; fax: +32 9 264 6235. E-mail address: [email protected] (F. Ronsse).

0260-8774/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.09.030

ity, reducing dust formation, taste masking, etc. (Guignon, Duquenoy, & Dumoulin, 2002). In top-spray fluidised bed coating, the solid particles are fluidised in a stream of hot air. The coating, in most cases in the form of an aqueous solution, is applied onto the particles by spraying the solution on top of the fluidised bed. The supplied hot air also delivers the energy to evaporate the coating solution deposited on the surface of the suspended particles. The coating solution is supplied to the fluidised bed by means of a pneumatic nozzle, in which the use of compressed air results in very strong shear forces at the gas–liquid interface, producing droplets with a size ranging from 10 to 40 lm (Guignon et al., 2002; Lefebvre, 1988). While the fluidised bed coating technique originates from the pharmaceutical industries, the framework in which these fluid beds operate in the food industry is different. The fluidised bed units are required to handle large throughputs of food constituents of which the end product has a much smaller profit margin compared to pharmaceutical

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297

Nomenclature A Bi c c0 CD Cp d D DM f G g Gr h I J M _ M MW n N Nu P Pr Q R R0 R r rD Re S Sc Sh SSR T t v V W X Y

surface area, m2 Biot number, dimensionless [= apdp/kp] number of coating control volumes, dimensionless atomisation air mixing constant, dimensionless drag force coefficient, dimensionless specific heat at constant pressure, J kg1 K1 diameter or thickness, m diffusion or dispersion coefficient, m2 s1 spraying solution dry matter content, dimensionless frequency, Hz air mass flow, kg dry air s1 gravitational constant, = 9.81 m s2 Grasshof number, dimensionless ½¼ gbf q2f h3 jDT j=l2f  height, m number of cycles, dimensionless scaling factor, dimensionless mass, kg mass flow rate, kg s1 molecular mass, kg mol1 number of control volumes number of particles Nusselt number, dimensionless [= ad/kf] pressure, Pa Prandtl number, dimensionless [= lfCp,f/kf] heat, J thermal resistance, m2K W1 thermal resistance (external), m2 K W1 universal gas constant, = 8.314 J mol1 K1 particle transfer rate, s1 drying rate, kg water kg particle1 s1 Reynolds number, dimensionless [= dvfqf/lf] control volume Schmidt number, dimensionless [= lf/qfDv] Sherwood number, dimensionless [= a 0 d/D] sum of squared residuals temperature, K time, s linear velocity, m s1 volume, m3 particle moisture content, kg water kg particle1 absolute humidity, kg water kg dry air1 particle coating mass, kg DM kg particle1

Greek symbols a convective heat transfer coefficient, W m2 K1 0 a mass transfer coefficient, m s1 b volume expansion coefficient, K1 c surface tension, N m1 d bubble voidage fraction, dimensionless Dt simulation time step, s

e U g h j k ko l l0 p q r s u w X

porosity, dimensionless heat transfer rate, J s1 fraction of heat loss originating from particles compared to total heat loss emissivity, dimensionless mixing constant heat transfer coefficient, W m1 K1 heat transfer coefficient for stagnant gas in the emulsion phase, W m1 K1 viscosity, Pa s viscosity at particle surface, Pa s constant, = 3.1415926535 mass density, kg m3 Stefan–Boltzmann constant, = 5.669 · 108 W m2 K4 equivalent gas film thickness around particle, dimensionless relative humidity, dimensionless sphericity, dimensionless output variable (sensitivity analysis)

Superscripts and subscripts a air at atomisation air aw air at the wall region b bubble bed bed c coating cond conduction conv convection d drag e external f fluidum eq equivalent exp experimental ew particle gas film at wall region in input lat latent heat of evaporation loss loss mf minimum fluidisation out output p particle packet particle emulsion packet pcd particle conduction rad radiation real referring to real (physical) system being simulated sat saturated sim simulated sol coating solution surf surface v water vapour w wall

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compounds (Teunou & Poncelet, 2002). Therefore, the proper design and application of process controllers is essential for minimising both operation costs and out-ofspecification waste products (Haley & Mulvaney, 1995). With respect to fluidised bed coating, a multitude of problems and side-effects are likely to be encountered. First, liquid bridge formation between core particles could result in agglomeration and ultimately, in wet quenching of the fluidised bed (Kage, Takahashi, Yoshida, Ogura, & Matsuno, 1998; Saleh, Cherif, & Hemati, 1999; Smith & Nienow, 1983). Second, depending on the droplet travel distance and process air evaporative capacity, premature droplet evaporation is likely to occur before droplet adhesion onto the surface of the core particles (Jones, 1985; Smith & Nienow, 1983; Dewettinck & Huyghebaert, 1999). Whether agglomeration or premature droplet occurs strongly depends on the factors such as the evaporative capacity of the fluidised bed and spraying rate of the coating solution (Dewettinck & Huyghebaert, 1998). Finally, fragmentation of friable core particles and the crumbling of the coating by attrition could give rise to reduced process yields (Guignon et al., 2002; Guignon, Regalado, Duquenoy, & Dumoulin, 2003; Liu & Litster, 1993). Other important aspects in controlling the fluidised bed coating process include controlled product temperature and controlled coating film growth. When applying fluidised bed coating to heat sensitive products, the primary aim should be to keep the product temperature below a certain threshold value in order to avoid unnecessary product degradation. Furthermore, by controlling both the product temperature and the product surface humidity, the drying rate could be kept constant which is essential in maintaining the coating film quality throughout the process (Larsen, Sonnergaard, Bertelsen, & Holm, 2003). Controlled growth of coating film and limited variance thereof could be a necessity in the preparation of coated particles with modified release properties. Narrow distributions of the coating film thickness are often required, because small deviations in the coating film thickness could alter the release properties (Abe et al., 1998; Watano et al., 1995). In order to understand the impact of the different input variables on process efficiency and to translate this knowledge into improved control strategies or new reactor designs, the use of process models proves useful (Dewettinck, De Visscher, Deroo, & Huyghebaert, 1999). In modelling the coating process in a fluidised bed, different approaches were adopted: In a first approximation, the reactor is considered a black box in which particles and process air are assumed to be perfectly mixed. Ebey (1987) and later Dewettinck et al. (1999) presented a steady-state thermodynamic model for the aqueous film coating process. A simpler, but dynamic model for process control of aqueous film coating of pharmaceutical substrates was built by Larsen et al. (2003). To model the coating thickness or—in the case of agglomeration modelling—particle size evolution, entire reactor popula-

tion balance models were developed (Heinrich, Peglow, Ihlow, Henneberg, & Mo¨rl, 2002; Heinrich, Peglow, & Mo¨rl, 2002; Heinrich et al., 2003; Saleh, Steinmetz, & Hemati, 2003). Another way to deal with the modelling of the fluidised bed is to consider the bed divided into different zones or compartments. Previous research suggests the existence of distinguishable thermal zones in a fluidised bed coater: Smith and Nienow (1982) used X-rays to obtain temperature profiles around a spraying nozzle, submerged in a fluidised bed. Later, Maronga and Wnukowski (1997a, 1998) obtained both temperature and humidity profiles using a single probe, which scanned one half of a diametrical plane inside the reactor during steady coating regime. Their studies showed the existence of four different zones in the fluidised bed, delineated parallel to the air distributor plate according to their coating function: spraying zone, drying zone, heat transfer zone and non-active zone. A coating model was built by attributing to each zone a particle population balance describing how the number of particles in a given coating weight interval could be related to the rate at which particles enter and leave that weight interval due to the different phenomena occurring, such as deposition of coating on the particles and the migration of particles to adjacent zones (Maronga & Wnukowski, 1997b). The objective of this study was the development of a dynamic heat and mass transfer model for the description of temperature, humidity and coating mass concentration fields along the z-axis of fluidised beds for coating operations. This model will serve, in future studies, as a basis for process control and reactor design in fluidised bed coaters to further increase process efficiency and reduce out-ofspecification waste product generation. 2. Model 2.1. General To develop the model, the fluidised bed was horizontally divided into n control volumes, each having a volume Vbed/n. Fig. 1 shows a schematic representation of the model and the discretisation of the bed into different control volumes. In modelling the fluidised bed coating process, the following assumptions were made: • Particles and air in each control volume are perfectly mixed. • All particles have the same diameter dp. • Both the size of each control volume and the number of particles contained in each control volume are constant. • The rate at which particles are transferred from control volume Si towards its neighbouring control volume Si+1 equals the particle transfer rate from Si+1 towards Si. The rate at which particles are exchanged is expressed by ri, as the fraction of the particle population exchanged between control volumes Si and Si+1, per time unit.

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299

Air outlet Gout, Tout, ϕout

Sn rn-1

Coating solution Msol , Tsol , DMsol

Sn-1 Coating solution + Atomisation air

rn-2

Atomisation air Gat , Tat , ϕ at

rn-c

Sn-c rn-c-1

Fluidised bed height

r3

S3 r2

S2

Air inlet

r1

S1 Air flow

Air inlet Gin, Tin, ϕ in

Particle exchange

Fig. 1. Schematic representation of the overall model.

• Particles are mechanically inert; there is neither attrition nor agglomeration. • Distinction is made between coating and non-coating control volumes; each coating control volume receives _ sol =c coating liquid. Furthermore, it was assumed that M all spraying liquid is collected on the particles without premature droplet evaporation. The spraying liquid is uniformly deposited on all particles in each coating control volume. • The weight of the coating mass added to the particle is small compared to the weight of the particle itself. Consequently, the weight of each individual particle was assumed constant throughout the process. Similarly, the thickness of the deposited coating film is small compared to the particle diameter, therefore the particle diameter was assumed to be constant. • The mass flow of dry air is constant and is the same for all control volumes. • The air exhaust is at atmospheric pressure. The pressure drop across the fluidised bed is small compared to the overall atmospheric pressure. Consequently, the drying process was assumed to take place at constant atmospheric pressure.

2.2. Heat and mass balances In each control volume the dynamic heat and mass balances for air, particles and coating material (Fig. 2) were established and the following equations were obtained: Particles population balance in a single control volume. The population balance for the core particles in a control

Fig. 2. Detail of a control volume Si in the model.

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volume accounts for the change in particle population in Si due to outbound particle transfer from Si towards Si1 and Si+1 and inbound particle transfer from Si1 and Si+1 towards Si. Consequently, the particle population balance could be written as dN i ¼ ðri1 N bed þ ri N bed Þ  ðri N bed þ ri1 N bed Þ ¼ 0 dt

ð1Þ

dW i ¼ N bed ri1 M p W i1 þ N bed ri M p W iþ1 dt  N bed W i M p ðri1 þ ri Þ  rD;i N i M p

ð2Þ

In a coating control volume, the introduction of coating solution needs to be taken into account, resulting in: N iM p

dW i ¼ N bed ri1 M p W i1 þ N tot ri M p W iþ1 dt  N bed W i M p ðri1 þ ri Þ  rD;i N i M p 1 _ sol þ ð1  DM sol ÞM c

In the model, it was assumed that the spraying solution is uniformly divided over all coating control volumes, hence the term 1c in Eq. (3). Gas phase (air) moisture balance. The change in air moisture in a non-coating control volume Si is determined by the incoming moisture in the process air from Si1, the moisture in the process air leaving towards Si+1 and the amount of water evaporated on the particle surface. The amount of introduced atomisation air was assumed to be homogeneously divided over the upper c 0 control volumes and therefore, the air moisture balance could be written as dX i ¼ Ga;i1 X i1  Ga;i X i þ rD;i N i M p dt 1 6 i 6 n  c0 dX i 1 ¼ Ga;i1 X i1  Ga;i X i þ 0 Gat X at þ rD;i N i M p M a;i c dt n  c0 < i 6 n

ap;i d p  0:1 kp

M a;i

ð4Þ

ð5Þ

Because the volume fraction of the bed in which the atomisation air is homogeneously mixed was unknown, it was further assumed that c  c 0 , implying that both the spraying liquid and the atomisation air are divided over the same fraction of the fluidised bed. Coating mass of the particles. In the model where the particles have the same diameter, all particles located inside a single coating control volume receive an equal amount of coating mass. Hence, the equation for the coating mass balance is

ð7Þ

ð8Þ

Consequently, the internal particle heat transfer resistance to conduction is negligible compared to the heat transfer resistance to convection at the particle surface, so each particle is isothermal and is represented as having a single temperature (Collier, Hayhurst, Richardson, & Scott, 2004; Janna, 2000). The enthalpy balance of the particles within any control volume Si depends on the enthalpy of the particles entering and leaving Si, the convective heat transfer between the air and the particles, the latent heat of evaporation of water on the particle surface and the heat losses through the shell of the reactor: N i C p;p M p

ð3Þ

ð6Þ

Particle heat balance. Considering the spherical particles, it holds that the Biot number (Kreith, 1998): Bii ¼

Moisture balance of the particles. In each non-coating control volume the balance of moisture on the surface of the core particles is governed by the amount of water introduced by the particles entering Si from Si1 and Si+1, the amount of water removed by the particles leaving Si to Si1 and Si+1 and the water evaporated from or condensed on the particle surface: N iM p

dY i 1 _ sol n  c < i 6 n ¼ DM sol M c dt dY i ¼0 16i6nc N iM p dt

N iM p

dT p;i ¼ N tot ri1 C p;p M p T p;i1 dt þ N tot ri C p;p M p T p;iþ1  N tot C p;p M p T p;i ðri1 þ ri Þ þ N i ap;i Ap ðT a;i  T p;i Þ  rD;i N i M p Qlat;i  gp;i Uloss;i

ð9Þ

where Qlat,i is the latent heat of vaporisation of pure water and is given by the following equation (Iguaz, Esnoz, Martinez, Lopez, & Virseda, 2003): Qlat;i ¼ 103 ð2500:6  2:364356ðT p;i  273:15ÞÞ

ð10Þ

In a fluidised bed, besides convective heat transfer between the fluidising medium (air) and the inner reactor wall, the particles also transfer heat towards the inner reactor wall through particle–wall collisions (Kunii & Levenspiel, 1991). Consequently, the total heat loss through the wall of the control volume can be split into two parts: particles-to-environment heat losses and fluidising air-toenvironment heat losses, hence the term gp,iUloss,i. Gas phase (air) heat balance. Since the diameter of the sprayed droplets is generally between 10 and 40 lm (Lefebvre, 1988), it was assumed that the droplets, travelling from the nozzle towards the particle surface reached a state of thermal equilibrium with the air in the control volume before colliding upon a particle surface. Taking this assumption into account, the equation for the enthalpy balance in the gas phase (air) within every coating control volume is given by the enthalpy of the air entering Si from Si1, the enthalpy of the air leaving form Si to Si+1, the enthalpy of the supplied atomisation air, the heat transfer between air and droplets, the heat transfer between air and particles, the heat required to heat vapour, originating from evaporated solvent (water) on the particles, to air

F. Ronsse et al. / Journal of Food Engineering 78 (2007) 296–307

temperature and the heat losses from the fluidising medium towards the environment. M a;i C p;a;i

dT a;i ¼ Ga;i1 C p;a;i1 T a;i1  Ga;i C p;a;i T a;i dt 1 1 _ þ 0 Gat C p;at T at  M sol C p;sol ðT a;i  T sol Þ c c  ap;i N i Ap ðT a;i  T p;i Þ þ rD;i N i M p C p;v;i ðT a;i  T p;i Þ  ð1  gp;i ÞUloss;i ð11Þ

In a non-coating control volume, the droplet-related terms are zero and consequently, the enthalpy balance equation is M a;i C p;a;i

dT a;i ¼ Ga;i1 C p;a;i1 T a;i1  Ga;i C p;a;i T a;i dt þ rD;i N i M p C p;v;i ðT a;i  T p;i Þ  ap;i N i Ap ðT a;i  T p;i Þ  ð1  gp;i ÞUloss;i when 1 6 i 6 n  c

ð12Þ

2.3. Wall element heat balance and heat losses To quantify the overall heat losses in each control volume Si, the reactor shell was modelled into different elements or control volumes (Fig. 3). The number of wall elements equalled the number of fluidised bed control volumes. In modelling the wall elementÕs temperature, the following heat in- and output terms were considered. Bed to inner wall heat transfer. For a more extensive overview of the different heat transfer mechanisms in bubbling fluidised beds, the reader is referred to Kunii and Levenspiel (1991). Only the basic equations applying to the study at hand are briefly discussed. In a bubbling fluidised bed, some of the rising bubbles sweep past the inner reactor wall, thereby washing away the particles located there and bringing fresh particles into direct contact with the surface. These groups of particles which are continuously being swept away along the inner

External air

Wall Rcond,i

Rconv,i

Ta,i

Tw,i

Te

R ’rad,i

1 1 1 ¼ þ Ri dw Rconv;i ð1  dw ÞRp;i

In Eq. (14) the radiative heat transfer is neglected because of the relatively low particle temperatures (Tp,i < 400 K) and low temperature differences between the inner wall and particle bed. Thus the thermal resistance from the particles to the inner wall could be simplified to Rp;i ¼ Rpcd;i þ Rpacket;i

Tp,i

Rrad,i≈∞

dw

Fig. 3. Overview of the heat losses in a control volume.

ð15Þ

In Eq. (15), Rpcd,i is the heat transfer resistance through the particle–wall contact point and the surrounding thin gas layer. Rpcd,i was calculated according to Yagi and Kunii (1960): 1 Rpcd;i

¼

2kaw;i þ jw ðC p;a;i qa;i va;i Þ dp

ð16Þ

In Eq. (16) jw is the wall mixing constant. jw is generally assumed to be 0.05 (Kunii & Levenspiel, 1991). The thermal conductivity for stagnant gas in the wall region kaw;i , was calculated according to Kunii and Smith (1960): kaw;i ¼ ew ka;i þ

Rpacket;i R packet,i

ð13Þ

Particles to inner wall heat transfer. The heat transfer from the particles to the inner reactor wall is composed of two serially connected heat transfers: heat transfer by the emulsion packet and heat transfer through the combination of wall–particle contact point and the thin gas layer surrounding the wall–particle contact point. The latter is composed of conductive and radiative heat transfer, as shown in the next equation  1 1 1 Rp;i ¼ þ þ Rpacket;i ð14Þ Rpcd;i Rrad;i

1

Rpcd,i

hS,i Rcond,i-1

reactor wall are denoted as emulsion packets. To take into account the coexistence of bubbles and particle emulsion packets, the thermal resistance from bed to inner wall is expressed as

ð1  ew Þkp   k sw ka;ip þ 13

ð17Þ

where sw represents the equivalent thickness—compared to the particle diameter—of the gas film around the wall–particle contact point. The second component in Eq. (15), the thermal resistance due to the presence of emulsion packets, was calculated according to Kunii and Levenspiel (1991):

Fluidised bed

R ’conv,i

301

¼ 1:13

  0:5 ka;i qp ð1  emf ÞC p;p fb;w 1  dw

ð18Þ

where fb,w is the bubble frequency at the inner wall and was assumed to equal to the bubble frequency in the bed, fb. The bubble frequency was calculated using the surplus gas velocity, va  vmf and the gas bubble diameter which was estimated with the correlations proposed by Mori and Wen (1975). In Eq. (18), ka;i is the thermal conductivity for stagnant gas in the bed and is calculated as

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ka;i ¼ emf ka;i þ

ð1  emf Þkp   k sbed ka;ip þ 23

ð19Þ

Bubbles to inner wall heat transfer. In Eq. (13), the term dwRconv,i describes the heat transfer resistance between the gas bubbles and the inner reactor wall. It was calculated through the Nusselt number for forced convection in a vertical tube, using the Dittus–Boelter equation (Janna, 2000): 0:4 Nuw;i ¼ 0:023Re0:8 w;i Pr i

ð20Þ

Heat transfer through the reactor wall. Because of the relatively small heat transfer resistance of the reactor wall (stainless steel) compared to the heat transfer resistances from the bed towards the wall and from the wall towards the environment, the wall element was considered to have a single temperature Tw,i. Due to the geometric nature of the wall element (hS,i  dw), vertical heat conduction to or from adjacent wall elements was also taken into account and consequently, Rcond;i ¼

hS;i þ hS;iþ1 2kw

ð21Þ

Heat transfer from the wall towards the environment. This heat transfer towards the environment is composed of convective and radiative heat transfer. The convective heat transfer was approximated by calculating the Nusselt number of a vertical cylinder with height hbed (Janna, 2000): Nue ¼ 0:59ðGre Pre Þ

0:25

when 105 < Gre Pre < 109

ð22Þ

The radiative heat loss is calculated as Qrad;i ¼ rhw Aw;i ðT 4w;i  T 4e Þ

ð23Þ

Total heat transfer towards the environment. By combining all described heat transfers, the total heat balance for each wall element can be written as C p;w qw Aw;i d w

dT w;i Aw;i ðT a;i  T w;i Þ Aw;i ðT p;i  T w;i Þ ¼ þ dw Rconv;i ð1  dw ÞRp;i dt  rhw Aw;i ðT 4w;i  T 4e Þ  þ

Aw;i ðT w;i  T e Þ R0w;i

pd S;i d w ðT w;i1  T w;i Þ Rcond;i1

pd S;i d w ðT w;i  T w;iþ1 Þ  Rcond;i

ð24Þ

In Eqs. (9), (11) and (12), the heat loss from the fluidised bed towards the inner reactor wall was expressed by means of the variable Uloss,i, which corresponds to the first two terms of Eq. (24). Also, the fraction of heat loss from the fluidised bed, originating from the solid phase (particles), gp,i, was derived from Eq. (24): Aw;i ðT a;i  T w;i Þ Aw;i ðT p;i  T w;i Þ þ dw Rconv;i ð1  dw ÞRp;i Aw;i ðT p;i  T w;i Þ ¼ Uloss;i ð1  dw ÞRp;i

Uloss;i ¼

ð25Þ

gp;i

ð26Þ

2.4. Heat and mass transfer rates Particle/gas heat transfer rate. To estimate the convective heat transfer coefficient between the gas phase and particles, ap,i, the Nusselt number for forced convection around a sphere was used and approximated using the Whitaker equation (Sparrow, Abraham, & Tong, 2004):  1=4 1=2 2=3 2=5 la;i Nup;i ¼ 2 þ ð0:43Rep;i þ 0:06Rep;i ÞPri l0a;i 3:5 < Rep;i < 76; 000 for 0:71 < Pri < 380 1:0 <

ðla;i =l0a;i Þ

ð27Þ

< 3:2

Because of the relatively small difference between the air temperature Ta,i and the particle surface temperature Tp,i, the term la;i =l0a;i  1. Particle/gas mass transfer rate. The drying (or condensation) rate rD,i was calculated as rD;i ¼

a0p;i Ap ðP v;p;i  P v;a;i Þ   T a;i þT p;i R M p MW 2 v

ð28Þ

In Eq. (28) the term (Ta,i + Tp,i)/2 corresponds to the average film temperature (Campbell, 1977). The mass transfer coefficient a0p;i was calculated through an approximation by means of the Sherwood number. The calculation of the Sherwood number is analogous to Eq. (27) (Kreith, 1998): 1=2

2=3

2=5

Shp;i ¼ 2 þ ð0:43Rep;i þ 0:06Rep;i ÞSci

ð29Þ

The main driving factor for drying is the vapour pressure gradient between the particle surface and the air in the control volume, (Pv,p,i  Pv,a,i). The vapour pressure at the particle surface Pv,p,i is a function of the particle surface temperature and can be approximated through the following equation, assuming that the boundary gas layer at the surface of a wetted particle is saturated (Campbell, 1977): P v;sat ðT a Þ ¼ 103 expð52:57633  6790:4985T 1 a  5:02808 lnðT a ÞÞ

ð30Þ

The vapour pressure at the particle surface also depends on the particle diameter and was calculated using the Kelvin–Laplace equation (Scherer, 1998): ! P surf 2 R T qwater v;sat cwater=air ¼ ln ð31Þ 1 ðd p =2Þ P v;sat MW water However, considering the particle diameter ranging between 50 and 1000 lm, the change of the particle surface vapour pressure attributed to the particle surface curvature was negligible.

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In order to simulate particle exchange, a number of Nsim particles were chosen to fill the control volumes Si. During each simulated time step, particle exchange was accomplished by random selection of (ri + ri1)Ni particles in each control volume Si and by copying the selected particles to vacant locations in the control volumes Si1 and Si+1. In this respect, this procedure integrates the population balance equation of each control volume and the solution obtained in this manner is directly comparable to that obtained from standard integration techniques (Lin et al., 2002). As shown in Fig. 4, each particle exchange step is followed by the solving of the heat and mass balances within each control volume. These heat and mass balances are ordinary first-order differential equations and are solved using the first-order EulerÕs method. Due to computational limitations, the number of simulated particles (Nsim) is necessarily smaller than the number of particles involved in the system being simulated (Nbed).

2.5. Simulation procedure In most cases, population balances are written as partial differential equations. However, these partial differential equations in combination with all thermodynamic equations require complex numerical solving schemes and if the model needs to be modified, it usually requires complete basic remodelling. On the other hand, discrete Monte Carlo methods are easy to implement and to modify, because discretisation problems that hinder the direct integration of the partial differential equations are not an issue (Lin, Lee, & Matsoukas, 2002). However, Monte Carlo based methods often require much longer calculation time to find a solution. In this study a simulation method was chosen, combining the numerical solving of the population balance equations using a Monte-Carlo approach with the deterministic solving of the heat and mass balances for a given discrete number of particles.

Generate simulation population Nsim

Exchange ri × Ni randomly selected particles between Si and Si+1

Repeat Isim × Δt sim times

303

Repeat n times (for each control volume)

Coating deposition for each particle in S i , n-c ≤ i ≤ n

Calculate individual particle-fluidum mass and heat transfer and particledroplet heat transfer

Repeat Nsim times (for each particle)

Calculate new particle temperature and moisture content

Multiply heat and mass transfers with N real / N sim

Calculate bed heat loss and new wall element temperature Repeat n times (for each control volume) Calculate new fluidum temperature and humidity in S i

end

Fig. 4. Simulation block diagram.

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As the particle-related transfer terms are calculated based on the number of simulated particles, these transfer terms have to be multiplied with a scaling factor J to connect them with gas phase related transfer terms. In case of a monodisperse particle population, the scaling factor equals

Fan

T

Exhaust air

Filter

J¼

N bed N sim

Atomisation air

ð32Þ

Fig. 4 shows a basic overview of the simulation procedure, which was implemented in Microsoft Visual Basic 6. The Mersenne Twister random number generator, proposed by Matsumoto and Nishimura (1998) was implemented. This random number generator has the Mersenne prime period of 219937  1 and passes several tests for randomness, including the DIEHARD testing suite, generally considered the standard for random number generator testing (Srinivasan, Mascagni, & Ceperley, 2003; Tan, 2002).

Expansion chamber

Peristaltic pump

Coating solution

F

Air distributor

heating coils Fluidisation air

T T RH F Fig. 5. Experimental set-up (Glatt GPCG-1).

3. Model validation and discussion 3.2. Calculation and procedures

3.1. Experimental validation set-up Experimental steady-state coating data, generated by Dewettinck et al. (1999) have been used to validate the model in this study. In these experiments, 0.75 kg of glass beads with a volume weighted average diameter of 365 lm (Sovitec Micropearl, B) were fluidised in a Glatt GPCG-1 fluidised bed coater with top-spray reactor insert (Glatt GmbH, D) of which the dimensions are represented in Table 1. An overview of the Glatt GPCG-1 fluidised bed unit is given in Fig. 5. For thermodynamic analysis, distilled water at ambient temperature was used as the spraying liquid. Reactor outlet air temperature was measured at the top of the reactor by means of a stainless steel sheathed T-type thermocouple (Thermocoax, F). Ambient temperature and humidity were recorded using a Testo 610 (Testo, B). The atomisation air mass flow rate Gat, was obtained through measuring the linear air velocity with a Testovent 4000 anemometer (Testo, B) while blowing compressed air through the nozzle at different pressures—between 0.5 and 3.5 bar—in a cylindrical pipe with a diameter of 0.03 m. Regression analysis resulted in the following equation (Dewettinck et al., 1999):  2 dV at 0:03 ¼p ð0:1511905P 2at þ 1:32431P at þ 0:1274167Þ 2 dt with R2 ¼ 0:9859

ð33Þ

Table 1 Glatt GPCG-1 top-spray insert dimensions and properties Height, m Top diameter, m Bottom diameter, m Wall thickness, m Wall material Nozzle tip height, m

0.56 0.30 0.14 0.002 AISI 304 0.135

Two remaining essential variables that need to be characterised in order to complete the modelling of the fluidised bed are the size of the spraying region, c/n (being the fraction of coating control volumes compared to the total number of control volumes) and the particle exchange rate, ri . Table 2 Model and operational parameters of the simulation Model parameters Control volumes Coating control volumes Number of simulated particles Relative coating volume size, % of bed volume Particle exchange rate

n c Nsim c/n r

24 3 10,008 12.5 1.397

Bed material (glass beads) Overall mass, kg Particle diameter, lm Density, kg m3 Specific heat capacity, J kg1 K1 Thermal conductivity, W m1 K1

Mbed dp qbed Cp,p kp

0.750 365 2600 837 0.8

Liquid spraying Dry matter content Solution temperature, C Atomisation air mass flow rate, kg dry air s1 Atomisation air temperature, C Atomisation relative air humidity

DM Tsol Gat Tat uat

0 20 0.00204 20 0.30

Other parameters Reactor wall thickness, m Reactor wall thermal conductivity, W m1 K1 Reactor wall emittance for far-infrared radiation Reactor wall specific heat, J kg1 K1 Reactor wall density, kg m3 Wall mixing constant Bed voidage at minimum fluidisation velocity Bubble frequency, Hz

dw kw hw Cp,w qw jw emf fb

0.002 14.6 0.16 500 8000 0.05 0.39 100

Simulation parameters Simulated time, s Simulation time step, s

tsim Dtsim

1500 0.001

F. Ronsse et al. / Journal of Food Engineering 78 (2007) 296–307

305

Table 3 Process conditions and experimental vs. model predicted values of outlet temperature

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Te (C) 20.7 21.3 19.2 21.1 20.4 21.4 19.5 21.4 20.1 21.6 20.2 20.6 21.1 21.2 22.4 20.8 20.7 20.7 19.8 20.0 18.3 19.0 18.9 18.5 19.6 20.3 20.4 21.6

ue 0.51 0.46 0.47 0.53 0.52 0.46 0.47 0.53 0.44 0.52 0.55 0.54 0.51 0.48 0.43 0.56 0.55 0.54 0.55 0.53 0.29 0.26 0.24 0.25 0.45 0.43 0.43 0.39

Tin (C) 50 60 70 80 90 60 70 80 70 80 80 80 80 80 80 50 50 50 50 50 80 80 80 80 50 50 50 50

_ sol (g min1) M 5.52 5.53 5.53 5.51 5.21 7.25 7.23 7.23 8.93 8.91 7.57 7.58 7.40 7.43 7.51 3.17 3.18 3.19 3.20 3.19 7.66 7.69 7.69 7.70 3.20 3.23 3.20 3.22

3

9.38 · 10 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 9.38 · 103 1.25 · 102 1.41 · 102 1.56 · 102 9.38 · 103 1.25 · 102 1.41 · 102 1.56 · 102

2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

Based on the conclusions with regards to the size of the actual spraying region in fluidised bed coating drawn by Maronga and Wnukowski (1997a), a coating volume of 12.5% was assumed, which corresponded to 3 coating control volumes out of the 24 modelled control volumes. The values for the particle exchange rates ri were based on measured axial dispersion coefficients found in literature. Both the particle exchange rate and the axial dispersion coefficient describe the axial mixing of particles in a fluidised bed. Their relation is given by Dp ¼ rh2bed

Gin (kg s1)

Pat (bar)

ð34Þ

Mostoufi and Chaouki (2001) demonstrated that the axial dispersion coefficients for sand particles with dp = 385 lm and with superficial gas velocities varying from 0.5 to 2.8 m s1, ranged between 3.3 · 103 and 5.6 · 102 m2 s1, which is close to the glass beads with dp = 365 lm and superficial gas velocities ranging from 0.65 to 1.06 m s1 used in this studyÕs experimental validation. Using Eq. (34), these axial dispersion coefficients translate into particle exchange rates, ranging between r = 0.81 s1 and 1.82 s1. The process variables, having the largest impact on the thermodynamic operation point were varied and the experimentally measured steady-state outlet air temperature was compared with the model predicted outlet air temperature. The fixed operational parameters along with the model parameters are listed in Table 2, while the varied process conditions are listed in Table 3 together with the experi-

Tout,exp (C)

Tout,sim (C)

30.4 ± 0.6 37.6 ± 0.6 45.1 ± 0.6 50.2 ± 0.6 57.9 ± 0.6 32.9 ± 0.6 39.5 ± 0.6 45.6 ± 0.6 34.2 ± 0.6 41.1 ± 0.6 45.7 ± 0.6 44.6 ± 0.6 44.7 ± 0.6 44.0 ± 0.6 43.3 ± 0.6 37.0 ± 0.6 37.2 ± 0.6 37.1 ± 0.6 36.9 ± 0.6 36.8 ± 0.6 46.0 ± 0.6 54.7 ± 0.6 57.0 ± 0.6 58.9 ± 0.6 35.9 ± 0.6 40.3 ± 0.6 41.9 ± 0.6 43.1 ± 0.6

30.72 ± 0.09 37.92 ± 0.09 44.59 ± 0.09 50.76 ± 0.09 57.90 ± 0.09 33.75 ± 0.11 40.31 ± 0.12 46.46 ± 0.11 36.26 ± 0.14 42.17 ± 0.14 42.46 ± 0.14 43.73 ± 0.14 44.22 ± 0.13 46.15 ± 0.12 46.78 ± 0.11 34.65 ± 0.06 35.27 ± 0.05 35.77 ± 0.05 36.08 ± 0.05 36.50 ± 0.05 46.45 ± 0.13 53.30 ± 0.13 55.21 ± 0.13 58.99 ± 0.13 36.63 ± 0.05 39.19 ± 0.05 40.00 ± 0.05 41.12 ± 0.05

mental versus modelled outlet air temperature during steady state coating regime. The total simulated time does not reflect the true length of the fluidised bed coating process. To save calculation time, the simulation was terminated after a thermodynamic steady-state was reached, which is approximately 1500 s.

60

Simulated outlet air temperature (˚C)

No.

55

50

45

40

35

30 30

35

40

45

50

55

60

Measured outlet air temperature (˚C) Fig. 6. Correlation between experimental and simulated outlet air temperature, using the new model (j) and the model (s) described by Dewettinck et al. (1999).

306

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Table 4 Relationship between model predicted (Tout,sim) and experimental outlet air temperature (Tout,exp) Tout,exp = aTout,sim + b Proposed model Topsim model IIa a

R2

SSR

Slope, a

95% confidence interval of slope

Intercept, b

95% confidence interval of intercept

0.9592 0.9786

63.9 119.5

0.996 0.954

[0.913; 1.079] [0.898; 1.011]

0.433 0.301

[3.145; 4.011] [2.273; 2.874]

Dewettinck et al., 1999.

3.3. Validation results

References

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