Spray Drying Modelling Based On Advanced Droplet Drying Kinetics.pdf

Chemical Engineering and Processing 49 (2010) 1205–1213

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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep

Spray drying modelling based on advanced droplet drying kinetics M. Mezhericher a,b , A. Levy a,∗ , I. Borde a a b

Pearlstone Centre for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel Department of Mechanical Engineering, Sami Shamoon College of Engineering, Bialik/Basel Sts., Beer-Sheva 84100, Israel

a r t i c l e

i n f o

Article history: Received 14 May 2010 Accepted 2 September 2010 Available online 15 September 2010 Keywords: Computational Fluid Dynamics Drying kinetics Spray drying Transport phenomena

a b s t r a c t A novel theoretical model of the steady-state spray drying process is presented. The model utilizes twophase flow Eulerian–Lagrangian approach and involves a comprehensive description of two-stage droplet drying kinetics. Such an approach enables prediction of mass, moisture content and temperature profiles within the spray droplets along with flow patterns of the continuous phase. The developed two-stage spray drying model has been incorporated in on Computational Fluid Dynamics (CFD) package FLUENTTM via user-defined functions and utilized for simulation of the drying process in a short-form pilot-scale spray dryer fitted with a pressure nozzle atomizer. The predicted drying behaviour of the dispersed phase and flow patterns of air velocity, temperature and humidity are compared with the data calculated using the built-in FLUENT drying kinetics model. The results of the numerical simulations demonstrate the considerable influence of the utilized drying kinetics model on the predicted heat and mass transfer in the drying chamber as well as the significant influence of particle-wall boundary conditions on the predicted particle trajectories and residence time. Therefore, a proper modelling of the droplet drying kinetics and realistic boundary conditions are crucial for the numerical representation of the actual spray dryer performance. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The computational modelling is nowadays a popular tool for theoretical investigation of different technological processes, one of them is spray drying. Literature survey demonstrates that during the last years advanced two- and three-dimensional spray drying models have been proposed by several academic and industrial groups [1–18]. Most of the developed models were based on Computational Fluid Dynamics, an approach which utilizes numerical methods and algorithms to solve and analyse with the help of computers complicated problems that involve fluid flows. The usage of CFD technique allowed the researchers to reveal the complexity of flow fields of drying agent and droplets/particles as well as to establish the common features of spray drying processes like three-dimensionality and unsteady character of the drying [11,12,14,17]. However, in spite of the performed extensive researches on comprehensive understanding of the main physical phenomena involved in the spray drying process and many detailed investigations devoted to the droplet drying kinetics [19], the existing spray drying models are lacking proper mathematical description of the complex internal transport phenomena within

∗ Corresponding author. Tel.: +972 8 6477092. E-mail addresses: [email protected] (M. Mezhericher), [email protected] (A. Levy), [email protected] (I. Borde). 0255-2701/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2010.09.002

the dispersed phase. This disadvantage is attributed to an extreme difficulty of incorporating the comprehensive droplet drying kinetics models in commercially available CFD packages or individually developed numerical codes. Overcoming this challenge would open a way for the much more realistic theoretical modelling of the spray drying process and would give a reasonably reliable prediction of the quality and morphology of the obtained final product. The aim of the present study was to develop a two-dimensional theoretical model of the steady-state spray drying process incorporating a comprehensive description of two-stage droplet drying kinetics. Such a model can be regarded as a first approximation to realistic theoretical modelling of the actually three-dimensional and unsteady spray drying process. 2. Problem setup A cylinder-on-cone spray dryer with co-current flow of drying air and spray of droplets is adopted in the current work (Fig. 1). To facilitate the model verification, the dimensions of the chamber are taken from the reports of Kieviet and Kerkhof [2,3]. A centrifugal pressure nozzle atomizer is located at the top of the drying chamber and the drying air enters the top of the chamber through an annulus without swirling at angle of 35◦ with respect to the vertical axis. The diameter and position of air outlet pipe are assumed to be equal to the corresponding data published by Huang et al. [5]. To reduce the

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phases in the drying chamber both internal and external transport phenomena have to be taken into account. The theoretical modelling of such system is a difficult task until some simplifications are imposed. Fortunately, in the spray dryer the gas phase occupies substantially greater volume compared to the liquid and solid phases and thus a reasonable assumption is to consider the drying air as a continuous phase. Correspondingly, droplets and particles are regarded as a discrete phase. Using the above argumentation, the drying gas (atmospheric air) is treated by an Eulerian approach and a standard k–ε model is utilized for its turbulence description. Discrete Phase Model (DPM) [20,21], which is based on Lagrangian formulation, is applied to the spray of spherical droplets and particles to track their trajectories and other valuable parameters during the drying process. 3.1. Continuous phase For the continuous phase of drying air, conservation equations of continuity, momentum, energy, turbulent kinetic energy, dissipation rate of turbulence kinetic energy and species are derived [22]: Continuity: ∂ ∂ + (uj ) = Sc , ∂t ∂xj

(1)

Momentum: Fig. 1. Adopted geometry of spray dryer [5].

computational efforts, the drying chamber is considered as twodimensional and axisymmetric. Preheated atmospheric air at temperature of 468 K and absolute humidity of 0.009 kg H2 O/kg dry air is supplied into the drying chamber through a central round inlet of the flat horizontal ceiling. The air inlet velocity is 9.08 m s−1 , whereas its turbulence kinetic energy is equal to 0.027 m2 s−2 and turbulence energy dissipation rate is 0.37 m2 s−3 [2,3]. The spray of liquid droplets is obtained by atomizing the liquid feed in the nozzle. The spray cone angle is assumed to be 76◦ , the droplet velocities at the nozzle exit are assigned to be 59 m s−1 , and the temperature of the feed is set at 300 K. The distribution of droplet diameters in the spray is assumed to obey the Rosin–Rammler distribution function [16]. In this function, the mean droplet diameter is assumed to be 70.5 ␮m, the spread parameter is set at 2.09, and the corresponding minimum and maximum droplet diameters are taken as 10.0 ␮m and 138.0 ␮m, respectively. These data results in a spray mass flow rate equal to 0.0139 kg s−1 (50 kg h−1 ). The walls of drying chamber are assumed to be made of 2-mm stainless steel, and the coefficient of heat transfer through the walls is set to zero as though there is a perfect thermal insulation. Finally, the air pressure in the outlet pipe of the drying chamber is set to −150 Pa. In the framework of the present investigation, two cases of particle-wall interactions are examined: (a) all the particles hitting the chamber walls are considered as “escaped” from the drying chamber, (b) normal and tangential restitution coefficients of particle-wall collisions are assumed to be 0.9 and 0.5 correspondingly. 3. Modelling of the spray drying process The process of spray drying is a complex multi-phase flow phenomenon including gas phase (drying air), liquid phase (droplets) and solid phase (particles). In turn, each phase is not a pure substance but it is mixture of several components. In this way, for all the

∂ ∂ ∂p ∂ (ui ) + (uj uj ) = − + ∂t ∂xj ∂xj ∂xj + Upi Sc + Energy conservation: ∂ ∂ ∂ (h) + (uj h) = ∂t ∂xj ∂xj



Species conservation: ∂ ∂ ∂ (Yv ) + (uj Yv ) = ∂t ∂xj ∂xj



e ∂k k ∂xj

∂uj ∂ui + ∂xj ∂xj

 + gi

Fgp ,

(2)

− qr + Sh ,

(3)



e ∂v Y ∂xj

Turbulence kinetic energy: ∂ ∂ ∂ (k) + (uj k) = ∂t ∂xj ∂xj



e



e ∂h h ∂xj





 + Sc ,

(4)

 + Gk + Gb − ε,

(5)

Dissipation rate of turbulence kinetic energy: ∂ ∂ ∂ (ε) + (uj ε) = ∂t ∂xj ∂xj



e ∂ε ε ∂xj



+

ε (C1 Gk − C2 ε), k

(6)

The production of turbulence kinetic energy due to mean velocity gradients is equal to:



Gk = T

∂uj ∂ui + ∂xj ∂xi



∂ui . ∂xj

(7)

The production of turbulence kinetic energy due to buoyancy is given by: Gb = −ˇgj

T ∂T , T ∂xj

(8)

where ˇ is the coefficient of thermal expansion: ˇ=−

1 



∂ ∂T



, p

(9)

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The utilized constants are C1 = 1.44, C2 = 1.92, and the Prandtl numbers are equal to  k =  h =  Y =  T = 0.9 and  ε = 1.3. The effective viscosity, e , is defined as: e =  + T ,

(10)

where T is turbulent viscosity T = C 

k2 . ε

(11)

In the above expression C = 0.09. The relationship between air temperature, pressure and density is given by the ideal gas law: p=

 T. M

(12)

By setting the time derivative to zero, the above conservation equations are rendered to a steady-state formulation. Fig. 2. Two-stage droplet drying kinetics.

3.2. Discrete phase The motion of the droplets/particles is described by the Newton’s Second Law:



p Fp dU = g + . mp dt

(13)



Fp is sum of the forces exerted on given spray particle Here by the gas phase, by other particles and walls of the spray drying chamber. Neglecting for simplicity all the forces arising from field gradients in the gas phase as well as from particle rotation, acceleration and electric charging, we get:



Fp = FD + FA + FB + FC ,

(14)

where FD is the drag force, FA is the virtual (added) mass force, FB is the buoyancy force and FC is the contact force. The drag force is determined by [23]:   −U  p |(U  −U  p )d2 , FD = CD |U p 8

(15)

where the drag coefficient, CD , is calculated according to known empirical correlations [22]. The added mass (or virtual-mass) force is required to accelerate the gas film surrounding the droplet/particle and it is evaluated as follows [24]: FA = 

dp3



12

p  dU DU − Dt dt



.

(16)

The buoyancy force is given by: FB = −

dp3 6

g .

case of single droplet drying in still air. The adopted drying kinetics model of spherical droplet containing solid fraction is briefly described below. The drying process of droplet containing solids is divided in two drying stages. In the first stage of drying, an excess of moisture forms a liquid envelope around the droplet solid fraction, and unhindered drying similar to pure liquid droplet evaporation results in the shrinkage of the droplet diameter. At a certain moment, the moisture excess is completely evaporated, droplet turns into a wet particle and the second stage of a hindered drying begins. In this second drying stage, two regions of wet particle can be identified: layer of dry porous crust and internal wet core. The drying rate is controlled by the rate of moisture diffusion from the particle wet core through the crust pores towards the particle outer surface. As a result of the hindered drying, the particle wet core shrinks and the thickness of the crust region increases. The particle outer diameter is assumed to remain unchanged during the second drying stage. After the point when the particle moisture content decreases to a minimal possible value (determined either as an equilibrium moisture content or as a bounded moisture that cannot be removed by drying), the particle is treated as a dry nonevaporating solid sphere. The concept of two-stage droplet drying kinetics is illustrated by Fig. 2. To conserve the space, only basic equations of the two-stage drying kinetics model are presented in the current report. For detailed description and validation of the drying kinetics model, see Mezhericher et al. [25,26].

(17)

The contact force, FC , arises from collision interactions between the given droplet/particle with other droplets/particles and walls of the drying chamber. The droplet–droplet and particle–particle collisions are neglected in the current stage for the simplicity of numerical solution, and they will be incorporated into the model in further works taking into account our previous studies [15–17]. Even so, the force exerted by the chamber walls on the hitting particle is calculated using the Newton’s Second Law and predefined values of normal and tangential restitution coefficients [16]. 3.3. Drying kinetics model The internal transport phenomena within the spray of droplets/particles are described with the help of two-stage drying kinetics model [25], which was successfully validated by comparison with the published experimental and theoretical data in the

3.3.1. First drying stage In the first drying stage, the temperature of droplet is assumed to be uniformly distributed. The corresponding equation of energy conservation is given by: ˙ v + cp,d md hfg m

dTd = h(Tg − Td )4Rd2 . dt

(18)

Here the coefficient of heat transfer, h, is calculated as the sum of convection and radiation heat transfer coefficients: h = hc + hr . In turn, the radiation heat transfer coefficient, hr , is evaluated according to the Stefan’s law of thermal radiation based on corresponding emissivities of liquid and solid fractions. The time-change of droplet outer radius is determined by: 1 dRd ˙ v. m =− dt d,w 4(Rd )2

(19)

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The mass transfer rate from the droplet surface is calculated using the law of mass convection: ˙ v = hD (v,s − v,∞ )4(Rd )2 . m

(20)

The coefficients of convective heat and mass transfer, hc and hD , are evaluated in terms of the corresponding Nusselt and Sherwood numbers: Nud = Shd =







(1 + B)−0.7 ;

(21)

2 + 0.6Red Sc 1/3 (1 + B)−0.7 .

(22)

1/2 2 + 0.6Red Pr 1/3



1/2

 8 d,w (Rd,0 )3 − (Rd )3 . 6

(23)

Finally, the value of droplet moisture content on dry basis, Xd , is given by: Xd =

md,w md = 1 + Xd,0 − 1. md,s md,0

(24)

3.3.2. Second drying stage In the second drying stage, the wet particle is considered as a sphere with isotropic physical properties and temperatureindependent crust thermal conductivity. The crust region is assumed to be pierced by a large number of identical straight cylindrical capillaries, and the wet core region is considered to be a sphere with liquid and solid fractions. The equations of energy conservation for the wet core and crust regions are as follows: 1 ∂ ∂Twc wc cp,wc = 2 ∂t r ∂r ∂Tcr ˛cr ∂ = 2 ∂t r ∂r

 r

2 ∂Tcr



kwc r

2 ∂Twc



∂r

,

0 ≤ r ≤ Ri (t).

(25)



∂r

,

Ri (t) ≤ r ≤ Rp .

(26)

The boundary conditions for the above set of equations are:

⎧ ∂Twc ⎪ = 0, r = 0; ⎪ ⎪ ∂r ⎪ ⎪ ⎪ ⎪ ⎨ Twc = Tcr , r = Ri (t);

(27)

˙v m ∂Tcr ∂Twc ⎪ kcr , r = Ri (t); = kwc + hfg ⎪ ⎪ A ∂r ∂r i ⎪ ⎪ ⎪ ⎪ ⎩ h(T − T ) = k ∂Tcr , r = R . g

cr

cr

∂r

p

The rate of crust–wet core interface recede is tracked by the following relationship: 1 d(Ri ) ˙ v. =− m dt εwc,w 4R2

(28)

i

The total mass transfer rate through the crust pores is the sum of corresponding diffusion and forced mass flow rates: ˙ v,diff + m ˙ v,flow = hD (v,s − v,∞ )Ap . ˙ v=m m

(29)

The mass flow rate of vapour diffusion is defined by: ˙ v,diff = m

8εˇ Dv,cr Mw Rp Ri (Tcr,s + Twc,s )(pm − pv )



pv

∂pm ∂pv − pm ∂r ∂r



.

(30)

The mass flow rate of forced vapour flow is determined as follows: ˙ v,flow = − m

Bk 8Rp Ri Mm pm ∂pm . m (Tcr,s + Twc,s ) ∂r

Bk =

(31)

dp2 ε3 180(1 − ε)2

.

(32)

The particle mass is tracked using the following expression: mp =



md,0 1 + Xd,0

4 

 

1 − wc,w + ␲wc,w εRi3 + (1 − ε) Rp3 . (33) wc,s

3

Finally, the particle moisture content is given by: Xp =

The droplet mass is found by integration of Eq. (19): md = md,0 −

The permeability, Bk , is calculated according to well-known Carman–Kozeny equation:

mp mw = 1 + Xd,0 − 1. ms md,0

(34)

3.3.3. Final sensible heating When the particle moisture content decreases to a minimal value attainable under the given drying conditions, the wet particle turns into a non-evaporating dry particle. This non-evaporating particle and surrounding flow of the drying air continue interaction by heat transfer: ∂Tp ˛p ∂ = 2 ∂t r ∂r



r

2 ∂Tp

∂r



,

0 ≤ r ≤ Rp .

(35)

4. Numerical solution The numerical solution of the model equations and computational simulations have been performed utilizing the 2D axisymmetric pressure-based solver incorporated in CFD package FLUENT 12.0.16, which is based on the finite volumes technique and enables a two-way coupled Euler-DPM algorithm for treatment of the continuous and discrete phases [21]. The computational domain has been meshed using a triangular scheme with 34,852 grid cells of various sizes. The cells of small size are located in the downstream direction of the spray injection (the region of high particle concentration), where the largest gradients of momentum, heat and mass transfer are anticipated. In contrast, larger grid cells are located near the top and side walls of the chamber. All the results of numerical simulations have been ensured to change negligibly if the sizes of grid cells were decreased. The spray injection form pressure nozzle is represented by 20 droplet streams with 10.0 ␮m and 138.0 ␮m minimum and maximum droplet diameters. As it was mentioned above in Section 2, the intermediate droplet diameters are given by Rosin–Rammler distribution function. The following strategy has been utilized for numerical simulations. First, the flow fields of drying air were simulated without the discrete phase until a converged solution was obtained. At the next step, spray droplets were injected into the domain and twoway coupled calculations were performed until converged steady solution. The calculation of the discrete phase drying kinetics was accomplished using the concept of User Defined Functions (UDF) which can be attached to the FLUENT. In this way, the first drying stage and the final heating of non-evaporating dry particles were simulated using the FLUENT built-in UDFs responsible for evaporation and sensible heating of droplets. For the second drying stage, Eqs. (25)–(34) were solved using the original numerical algorithm described in details in Ref. [16]. This numerical solution was implemented and linked to the FLUENT package via a set of original UDFs. The decision about which UDF should be used for the specific spray particle on the given calculation step during program run was made automatically by the FLUENT based on the current value of the particles moisture content.

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Fig. 3. Particle traces coloured by particle surface temperature (in K). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

The convergence of the numerical solutions have been monitored by means of residuals of the resolved governing equations and in parallel by overall imbalance between the domain incoming and outgoing mass flow and heat transfer rates. Negligibly small values of the above parameters have been ascertained for the converged solutions. The successful validation of the present model for the case of no spray from nozzle (only preheated gas is supplied to the chamber) as well as for the case of drying of pure water droplets has been reported elsewhere [15–17]. 5. Results and discussion In the present study, numerical simulations of spray drying process of silica slurry have been performed. The slurry consists of amorphous silica spherical particles with median size of 272 nm and density of 1950 kg m−3 [27]. These primary particles are dispersed in water with initial moisture content of 1.4. The emissivities of liquid and solid fractions of the slurry are considered to be fixed values equal to εr,w = 0.96 and εr,s = 0.8 correspondingly [28]. The final moisture content of dried particles is assumed to be 5%. For purpose of extended analysis of the results predicted by the current spray drying model with two-stage droplet drying kinetics,

the numerical simulations of spray drying of pure water droplets have been also performed. In these simulations, the droplets were allowed to evaporate up to 57.5% of their initial mass and after that the droplets were treated as non-evaporating spherical particles; other parameters were set similar to the silica drying case. The predicted steady-state trajectories of water droplets and silica slurry droplets coloured by different parameters (temperature, moisture content, mass and particle law index) are shown in Figs. 3–6. Case “a” represents the simulations of pure water droplets drying and cases “b” and “c” demonstrate the trajectories of silica slurry droplets. Furthermore, Figs. 3a–6a and 3b–6b demonstrate the calculations when both coefficients of normal and tangential restitution for particle-wall collisions were set to zero (so-called “escaped” wall conditions, en = et = 0). In contrast, the results presented in Figs. 3c–6c were obtained for the case when restitution wall conditions have been employed by setting en = 0.9 and et = 0.5. Analysing the predicted behaviour of the spray given in Figs. 3–5, it can be found that smaller droplets are concentrated near the chamber centreline where they are rapidly dried and their temperatures quickly rise up to equilibrium with drying agent. At the same time, because of greater inertia, larger droplets travel towards the chamber periphery preserving substantial amount of moisture content. Small dry particles are predicted to leave the chamber through

Fig. 4. Particle traces coloured by particle moisture content. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

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Fig. 5. Particle traces coloured by particle mass (in g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

Fig. 6. Particle traces coloured by particle law index. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

the air outlet pipe, dry particles of mean size are expected to follow to the product outlet orifice and lager particles are hitting the chamber conical wall. For the case with zero restitution coefficients these particles “escape” from the domain. At the same time, for the

case with non-zero restitution some of the particles hitting wall are sliding down to the product outlet and others are lifted up by ascending air streams towards the chamber top and then follow the recirculating air pattern established in the periphery of the dryer.

Fig. 7. Flow fields of air temperature (in K). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

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Fig. 8. Flow fields of vapour mass fraction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

Fig. 9. Flow fields of air velocity (in m/s). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

These recirculating large dry particles have significantly long residence times and increased temperatures compared to others. It is obvious that such factors can affect the quality of the final product and be a cause of its thermal degradation. For this reason, in the theoretical modelling a proper choice of particle-wall boundary

conditions is essential to predicted particle trajectories and particle residence time as realistic as possible. Fig. 6 demonstrates the simulated droplet trajectories coloured by particle law index. The particle law index reflects the current stage of droplet/particle drying, namely “0” corresponds to the

Fig. 10. Flow patterns of air velocity vectors (in m s−1 ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)

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droplet initial heating, “1” means the evaporation period, “2” indicates the second drying stage and “3” denotes the final sensible heating of the particle. At the considered conditions, the utilized colour indication of the first and second drying stages demonstrates that the spray of droplets is dried in the “bird wing”-shaped area spread towards the conical part of the dryer. Moreover, the particles hitting the chamber walls are completely dry (except the largest silica particles with 138 ␮m initial diameter, see Fig. 6b) and thus they are able to rebound back into the chamber, just as it has been presumed in the case “c”. The steady-state air flow patterns predicted for the particles “escape” and “restitution” conditions are given in Figs. 7–10. These results show that in both cases of silica slurry drying the corresponding flow fields of air temperature, vapour mass fraction and velocity are similar (cases “b” and “c”). At the same time, the flow fields of pure water drying and silica slurry drying (cases “a” and “b”,“c”) are considerably different. Therefore, the utilized drying kinetics model plays important role in the prediction of heat and mass transfer and flow regimes in the drying chamber. 6. Conclusions In the present study a theoretical model of the steady-state spray drying process has been developed. This two-dimensional axisymmetric model utilizes two-phase flow Eulerian–Lagrangian approach and involves a comprehensive description of the twostage droplet drying kinetics. Three different cases of spray drying process were numerically investigated utilizing the CFD package FLUENT. In the first case, drying of water droplets was simulated using the built-in FLUENT drying kinetics model, and in the second and third cases drying of silica slurries was modelled with the help of original two-stage drying kinetics implemented in FLUENT via the set of user-defined functions. In the first and second cases particles hitting chamber walls were considered to “escape” from the domain whereas in the third case the particle rebound was assumed. The results of the numerical simulations demonstrated a considerable influence of the utilized drying kinetics model on the predicted heat and mass transfer in the drying chamber. Furthermore, a significant impact of particle-wall boundary conditions on the predicted particle trajectories and particle residence time has been observed. It is concluded that a proper modelling of the droplet drying kinetics as well as a realistic setting of boundary conditions is crucial for numerical representation of the actual spray dryer performance. Acknowledgement The authors appreciatively acknowledge the financial support of the present work from GIF: German-Israeli Foundation for Scientific Research and Development under Grant No. 952-19.10/2007. Appendix A. Nomenclature A B Bk CD cp d Dv en et FA FB FC

surface area, m2 Spalding number crust permeability, m2 drag coefficient specific heat under constant pressure, J kg−1 K−1 diameter, m coefficient of vapour diffusion, m2 s−1 normal coefficient of restitution tangential coefficient of restitution virtual mass force, N buoyancy force, N contact force, N

FD g h hD hfg k m ˙v m M Nu p Pr r R  Re Sc Sh t T u p U  u x X y Y z

drag force, N gravity acceleration, m s−2 heat transfer coefficient, W m−2 K−1 ; specific enthalpy, J kg−1 mass transfer coefficient, m s−1 specific heat of evaporation, J kg−1 thermal conductivity, W m−1 K−1 ; turbulence kinetic energy, m2 s−2 mass, kg vapour mass transfer rate, kg s−1 molecular weight, kg mol−1 Nusselt number pressure, Pa Prandtl number radial space coordinate, m radius, m universal gas constant, J mol−1 K−1 Reynolds number Schmidt number Sherwood number time, s temperature, K velocity of drying agent, m s−1 vector of particle velocity, m s−1 vector of gas velocity, m s−1 space coordinate, m moisture content (dry basis), kg kg−1 space coordinate, m vapour mass fraction, kg kg−1 space coordinate, m

Greek symbols ˛ thermal diffusivity, m2 s−1 ˇ coefficient of thermal expansion, K−1 ; empirical coefficient ε particle crust porosity or emissivity; dissipation of turbulence kinetic energy, m2 s−3  dynamic viscosity, kg m−1 s−1 kinematic viscosity, m2 s−1  density, kg m−3

Subscripts a air, dry air fraction atm atmospheric cr particle crust; critical d droplet diff diffusion f final point of drying process flow forced flow g drying agent i crust–wet core interface m air–vapor mixture p particle, primary particle r radial direction s solid fraction; surface v vapour w water wc particle wet core 0 initial point of drying process 1 initial point of droplet evaporation period 2 contributor ∞ bulk of drying agent

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