Powder Technology 203 (2010) 371–383

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Three-dimensional modelling of pneumatic drying process

M. Mezhericher a,b, A. Levy a,⁎, I. Borde a

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

⁎ Corresponding author.E-mail addresses: mezher@bgu.ac.il (M. Mezheriche

borde@bgu.ac.il (I. Borde).

0032-5910/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.powtec.2010.05.032

a b s t r a c t

a r t i c l e i n f o

Article history:Received 11 September 2009Received in revised form 16 May 2010Accepted 27 May 2010Available online 4 June 2010

Keywords:CFDDrying kineticsHeat and mass transferPneumatic drying

Steady-state three-dimensional calculations of heat and mass transfer in vertical pneumatic dryer wereperformed. The theoretical model of the drying process is based on two-phase Eulerian–Lagrangian approachfor gas-particles flow and incorporates advanced drying kinetics for wet particles. The model was utilized forsimulation of the drying process of wet PVC and silica particles in a large-scale vertical pneumatic dryer. Theinfluence of wall thermal boundary conditions was investigated by assuming either known value of the walltemperature or adiabatic flow in the dryer. Analyzing the predicted particle drying kinetics, an unevenproduct quality was predicted due to non-uniform drying conditions in the central and peripheral zones ofthe pneumatic dryer. Moreover, for the case of non-insulated chamber walls such quality unevenness wasestimated to be substantially greater than for the case with thermally insulated drying chamber. Theexamination of the predicted temperature profiles within the silica and PVC wet particles showed that thelatter is subjected to higher temperature gradients potentially resulting in the greater rate of thermally-degraded final product.

r), avi@bgu.ac.il (A. Levy),

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Pneumatic (flash) drying is one of the widely used technologicalprocesses in food, chemical, agricultural and pharmaceutical indus-tries. Despite its apparent simplicity, the process of pneumatic dryingis a complex multi-phase transport phenomenon involving turbu-lence flow of humid compressed gas and multicomponent wetparticles, heat and mass transfer between the wet particles anddrying gas, drying kinetics of the wet particles and thermal andmechanical stresses in the dried particulate material.

Due to growing demand for the pneumatic dryer units and toughdesign requirements regarding their efficiency and low resourceconsumption, extensive experimental and theoretical studies of thepneumatic drying process have been performed during the last years[1–10]. Two-fluid theory is a typical approach used to model thepneumatic drying. The two-fluid model is based on Eulerian–Eulerianformulation which considers both the drying gas and the wet particlesas two pseudo continuous phases occupying each point of thecomputational domain with their own volume fractions. Anotherway to model the pneumatic drying is utilizing an Eulerian–Lagrangian approach like Discrete Element Modelling (DEM) [11,12]or Discrete PhaseModel (DPM) [13–16]. The Discrete PhaseModel is akind of Discrete Particle Model [17–22] and it is suitable for systems

with large amounts of particles owing to the usage of the concept ofparcels: each parcel contains a number of identical discrete particleswith the same parameters simultaneously injected into the domain. Incontrast to the Eulerian–Eulerian formulation, the Eulerian–Lagrang-ianmodels allow particle trajectories tracking by treating the particlesas situated in discrete points of the domain whilst the drying gas isassumed to be a continuous phase.

The aim of this study was to develop a steady-state three-dimensional theoretical model of the pneumatic drying process usingDiscrete Phase Model and Computational Fluid Dynamics technique.The drying kinetics of wet particles is described with the help of anadvanced theoretical model validated for single wet particle drying[22,23]. For comprehensive literature survey on drying of droplets andwet particles the reader is referred to the paper [24].

2. Problem setup

For the purposes of the theoretical study the geometry of Baeyenset al. [1] experimental setup is adopted. Hot dry air and wet particlesare supplied to the bottom of vertical pneumatic dryer with 1.25 minternal diameter and 25 m height (see Fig. 1).

The following assumptions are used in the study:

• steady-state drying process is considered;• gas-particle flow is dilute;• mass, momentum and heat transfer between particles themselvesare negligible;

• continuous phase is an ideal mixture of vapour and dry gas;

Fig. 1. Schematic illustration of vertical pneumatic dryer [1].

372 M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

• continuous phase behaves like an ideal gas;• dryer walls are either thermally insulated or wall temperaturedistribution is known.

3. Theoretical model

The continuous phase (drying air) is assumed to be an ideal gasand it is treated by an Eulerian approach using the k-ε model forturbulence description. The discrete phase of spherical particles isconsidered using DPM Lagrangian formulation.

3.1. Continuous phase

For the continuous phase of drying air, three-dimensional steady-state conservation equations of continuity, momentum, energy,turbulent kinetic energy, dissipation rate of turbulence kinetic energyand species are applied [25]:

– continuity

∂ρ∂t +

∂∂xj

ðρujÞ = Sc: ð1Þ

Here themass source term Sc = −npdmp

dt , where np andmp are num-ber density and single particlemass of the discrete phase respectively.

– momentum

∂∂t ðρuiÞ +

∂∂xj

ðρujuiÞ = − ∂p∂xi

+∂∂xj

μ e∂ui

∂xj+

∂uj

∂xi

!" #+ Δρgi

+ UpiSc + ∑Fgp ð2Þ

– energy conservation

∂∂t ðρhÞ +

∂∂xj

ðρujhÞ =∂∂xj

μeσh

∂h∂xj

!−qr + Sh; ð3Þ

where the energy source term Sh=hSc.

– species conservation

∂∂t ðρYvÞ +

∂∂xj

ðρujYvÞ =∂∂xj

μe

σY

∂Yv∂xj

!+ Sc: ð4Þ

– turbulence kinetic energy

∂∂t ρkð Þ + ∂

∂xjρujk� �

=∂∂xj

μe

σk

∂k∂xj

!+ Gk + Gb−ρε ð5Þ

– dissipation rate of turbulence kinetic energy

∂∂t ρεð Þ + ∂

∂xjρujε� �

=∂∂xj

μe

σε

∂ε∂xj

!+

εk

C1Gk−C2ρε� � ð6Þ

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

Gk = μT∂ui

∂xj+

∂uj

∂xi

!∂ui

∂xj: ð7Þ

The production of turbulence kinetic energy due to buoyancy isgiven by:

Gb = −βgjμTσT

∂T∂xj

; ð8Þ

where β is the coefficient of thermal expansion:

β = −1ρ

∂ρ∂T

� �p; ð9Þ

Here the model constants are equal to: C1=1.44, C2=1.92,σk=σh=σY=σT=0.9 and σε=1.3. The effective viscosity, μe, iscalculated by:

μe = μ + μT; ð10Þ

where μ T is turbulent viscosity

μT = Cμρk2

ε: ð11Þ

In the above expression Cμ=0.09.The relationship between humid air temperature, pressure and

density is given by the ideal gas law:

p =ρM

ℜT : ð12Þ

3.2. Discrete phase

The motion equation of the discrete phase of wet particles is asfollows:

d→Up

dt=

→g +

∑→Fp

mp: ð13Þ

Fig. 2. Two-stage drying kinetics of wet particle. D0— initial particle diameter,Dp— finalparticle diameter, Di — wet core diameter.

373M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

Here ΣF→p is the sum of the forces acting on the considered wet particle

from the gas phase and from other droplets/particles and walls of thedrying chamber as well. Since we deal with steady-state pneumaticdrying of non-charged micron-sized spherical particles with inletvelocity close to terminal one, we can neglect all the forces arisingfrom field gradients in the gas phase as well as from particle rotation,acceleration and electric charging. Therefore:

∑→Fp =

→FD +

→FA +

→FB +

→FC; ð14Þ

where F→D is the drag force, F

→A is the virtual (added)mass force, F

→B is the

buoyancy force and F→C is the contact force.

The drag force is determined by [26]:

→FD =

π8ρCDj→u−

→Upj →u−

→Up

� �d2p: ð15Þ

The drag coefficient, CD, is evaluated according to empiricalcorrelations [25].

The components of added mass (virtual mass) force, required toaccelerate the gas surrounding the droplet/particle, are determined asfollows [27]:

→FA = ρ

πd3p12

D→uDt

−D→Up

Dt

0@

1A: ð16Þ

The buoyancy force is given by:

→FB = −ρ

πd3p6

→g: ð17Þ

The contact force due to collisions of a droplet/particle with thewalls of the drying chamber, F

→C, is calculated in terms of the

corresponding normal and tangential coefficients of restitution [15].

3.3. Drying kinetics model

The internal transport phenomena within the dried particles aredescribed with the help of original two-stage drying kinetics model[23], which was successfully validated by comparison with thepublished experimental and theoretical data in the case of singledroplet/wet particle drying in still air. The adopted drying kineticsmodel is briefly described below.

The drying process of wet particle containing solids is divided intwo drying stages. In the first stage of drying, an excess of moistureforms a liquid envelope around the particle solid fraction, andunhindered drying similar to pure liquid droplet evaporation resultsin the shrinkage of the outer diameter. At a certain time, the moistureexcess is completely evaporated and the second stage of a hindereddrying begins. In this second drying stage, two regions of wet particlecan be identified: layer of dry porous crust and internal wet core. Thedrying process is controlled by the rate of moisture diffusion from theparticle wet core through the crust pores towards the particle outersurface. As a result of the hindered drying, the particle wet coreshrinks and the thickness of the crust region increases. The particleouter diameter is assumed to remain unchanged during the seconddrying stage. After the point when the particle moisture contentdecreases to a minimal possible value (determined either as anequilibriummoisture content or as a boundedmoisture that cannot beremoved by convective drying), the particle is treated as a dry non-evaporating solid sphere. The concept of two-stage drying kinetics isillustrated by Fig. 2.

To conserve the space, only basic equations of the two-stagedrying kinetics model are presented in the present report. For moredetailed description and validation of the drying kinetics model, seeMezhericher et al. [22,23].

3.3.1. First drying stageIn the first drying stage, the temperature of wet particle is assumed

to be uniformly distributed. The corresponding equation of energyconservation is given by:

hfgmv + cp;dmddTddt

= h Tg−Td� �

4πR2d: ð18Þ

The time-change ofwet particle outer radius is determined by [28]:

dRd

dt= − 1

ρd;w4π Rdð Þ2 mv: ð19Þ

The mass transfer rate from the particle surface is determinedaccording to the mass convection law:

mv = hD ρv;s−ρv;∞� �

4π Rdð Þ2: ð20Þ

The coefficients of heat andmass transfer are calculated in terms ofthe corresponding Nusselt and Sherwood numbers that are given byfollowing modified Ranz–Marshall correlations:

Nud = 2 + 0:6Re1 = 2d Pr1=3

� �1 + Bð Þ−0:7; ð21Þ

Shd = 2 + 0:6Re1 = 2d Sc1=3

� �1 + Bð Þ−0:7

: ð22Þ

The particle mass is found by integration of Eq. (19):

md = md;0−43πρd;w Rd;0

� �3− Rdð Þ3�

: ð23Þ

Finally, the value of particle moisture content on dry basis, Xd, isequal to:

Xd = md;w =md;s = md 1 + Xd;0

� �=md;0−1: ð24Þ

The transition between the first and second drying stages occurs inthe moment when the particle moisture content reduces to a criticalvalue. In the present study, this critical moisture content is evaluatedusing the condition of formation of final particle without internal void[15]:

Xcr =ρwρs

ε1−ε

: ð25Þ

Fig. 3. Three-dimensional numerical grid.

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3.3.2. Second drying stageIn the second drying stage, the wet particle is considered as a

sphere with isotropic physical properties and temperature-indepen-dent crust thermal conductivity. The crust region is assumed to bepierced by a large number of identical straight cylindrical capillaries,and the wet core region is considered to be a sphere with liquid andsolid fractions. The equations of energy conservation for the wet coreand crust regions are as follows:

ρwccp;wc∂Twc

∂t =1r2

∂∂r kwcr

2 ∂Twc

∂r

� �;0≤ r≤ Ri tð Þ; ð26Þ

∂Tcr∂t =

αcr

r2∂∂r r 2

∂Tcr∂r

� �;Ri tð Þ≤ r≤ Rp: ð27Þ

Here Ri is the internal radius of the interface between the crust andwet core and Rp is outer particle radius.

The boundary conditions for the above set of equations are:

∂Twc

∂r = 0; r = 0;

Twc = Tcr; r = Ri tð Þ;

kcr∂Tcr∂r = kwc

∂Twc

∂r + hfgmv

Ai; r = Ri tð Þ;

h Tg−Tcr� �

= kcr∂Tcr∂r ; r = Rp;

8>>>>>>>>>><>>>>>>>>>>:

ð28Þ

The crust–wet core interface receding rate is tracked by thefollowing relationship:

d Rið Þdt

= − 1ερwc;w4πR

2i

mv: ð29Þ

The total mass transfer rate through the crust pores is the sum ofcorresponding diffusion and forced mass flow rates:

mv = mv;diff + mv;flow = hD ρv;s−ρv;∞� �

Ap: ð30Þ

The mass flow rate of vapour diffusion is defined by:

mv;diff =8πεβDv;crMwRpRi

ℜ Tcr;s + Twc;s

� �pm−pvð Þ

pv∂pm∂r −pm

∂pv∂r

� �ð31Þ

The mass flow rate of forced vapour flow is determined as follows:

mv;flow = − Bk

μm

8πRpRiMmpm

ℜ Tcr;s + Twc;s

� � ∂pm∂r : ð32Þ

The permeability Bk is calculated according to well-knownCarman–Kozeny equation:

Bk = d2pε3= 180 1−εð Þ2h i

: ð33Þ

Similarly to the first drying stage, the coefficients of heat and masstransfer are calculated in terms of the corresponding Nusselt andSherwood numbers that are given by the modified Ranz–Marshallcorrelations (see Eqs. (21) and (22)).

The particle mass is tracked using the following expression:

mp = md;0 = 1 + Xd;0

� �1−ρwc;w = ρwc;s

� �+ 4= 3πρwc;w εR3

i + 1−εð ÞR3p

h i:

ð34Þ

Finally, the particle moisture content is given by:

Xp = mw =ms = mp 1 + Xd;0

� �=md;0−1: ð35Þ

3.3.3. Final sensible heatingWhen the particle moisture content decreases to a minimal value

attainable under the given drying conditions, the wet particle turnsinto a non-evaporating dry particle. This non-evaporating particle andsurrounding flow of the drying air continue the interaction by heattransfer:

∂Tp∂t =

αp

r2∂∂r r2

∂Tp∂r

!;0≤ r≤ Rp: ð36Þ

4. Numerical simulations

The developed theoretical model was numerically solved with thehelp of Finite Volume Method and 3D simulations of pneumaticdrying were performed using the CFD package FLUENT 12.0.7. Forthese purposes, the 3D numerical grid with 9078 distributedhexahedral/wedge cell volumes was generated in GAMBIT 2.2.30using the Cooper scheme (see Fig. 3).

The set of differential equations for drying air (continuous phase)was numerically solved utilizing the 3D pressure-based steady-statesolver of FLUENT package. In particular, the pressure–velocitycoupling was realized by means of the SIMPLE algorithm [29].Furthermore, the pressure equation was spatially discretized by thePRESTO! scheme [29]. For all transport equations, a second-orderupwind spatial discretization method was applied and the turbulenteffects were taken into consideration by the standard k-ε turbulencemodel.

The flow of wet particles in the pneumatic dryer was simulatedthrough 89 injections of spherical particles. Each injection began onthe bottom of the dryer at the centroid of one of the 89 bottom planemesh elements. The particle injections were normal to the dryerbottom plane and parallel to each other.

The numerical simulations were performed in the following way.First, the flow of drying air was simulated without the discrete phaseuntil converged solution. At the next step, wet particles were injectedinto the domain and two-way coupled simulations were performeduntil convergence.

The calculation of the discrete phase drying kinetics wasaccomplished using the concept of user defined functions (UDF)

Table 1Properties of PVC and silica particles.

Name Diameter,μm

Primaryparticlesdiameter, μm

Density,kg/m3

Specificheat,J/(kg K)

Thermalconductivity,W/(m K)

Thermaldiffusivity,10−7 m2/s

Porosity Initial moisturecontent, %(dry basis)

Critical moisturecontent (dry basis,Eq. 25)

Final moisturecontent, %(assumed)

PVC 140 [1] 0.272 1195 [1] 980 [1] 0.14 [1] 1.195 0.3 36 [1] 0.357 5Silica 140 0.272 [31] 1950 [31] 750 [31] 1.40 [32] 9.573 0.4 [31] 36 0.341 5

375M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

which were attached to FLUENT. In this way, the first drying stage andthe final heating of non-evaporating dry particles (Eqs. (18)-(25))were simulated using the FLUENT built-in UDFs responsible forevaporation and sensible heating of droplets/particles. For the seconddrying stage, Eqs. (26)–(35) were solved numerically using anoriginal numerical algorithm developed by Mezhericher [15,22,23],which is based on implicit finite differences method. The particle

Fig. 4. 3D air flow fields for the case of silica drying (adiabatic flow). (a) — velocity magnitu

temperature profile, diameter, mass, moisture content, positions ofinter-particle crust–wet core interface and material characteristicswere stored in the computer memory for each particle at everycalculation time-step. This numerical procedure was implementedand linked to FLUENT package via a set of original UDFs. Thetransitions between the various drying stages for each particle weremade automatically, based on the local values of the particle moisture

de, m/s (b) — temperature, K, (c) — vapour mass fraction, kg/kg, (d) — density, kg/m3.

376 M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

content. It is worth noting that for each drying stage, the thermo-physical properties of wet particles were tracked as dependent onboth particle temperature and moisture content.

The convergence of the numerical simulations was monitored bymeans of residuals of the transport equations. Particularly, theconverged values of the scaled residuals were ensured to be lowerthan 10−6 for the energy equation and 10−3 for the rest of equations.The convergence was also verified by negligibly small values of theglobal mass and energy imbalances.

Finally, the received numerical solution was tested for grid-independency by reducing the size of original meshes. For two gridsconsisting of 9078 (original) and 42,330 (reduced) cell volumes, themaximum difference in the calculated flow fields of air velocity,pressure, density, temperature and vapour mass fraction wasobserved to be smaller than 5%.

Fig. 5. 3D air flow fields for the case of PVC drying (adiabatic flow). (a) — velocity magnitu

5. Results and discussion

5.1. Three-dimensional drying simulations

The processes of pneumatic drying of PVC and silica particleswere simulated using the developed model. The particles of 140 μmin diameter were introduced from the bottom of the dryer via 89injections with zero initial velocity. The injections were placednormally to the bottom plane at the centroids of its grid cells. Themass flow rates of the injections were assumed to be identical andthe overall flow rate of the discrete phase was set to 1.583 kg/s.Particles hitting the chamber walls were reflected withcorresponding normal and tangential restitution coefficients set to0.9. The applied properties of PVC and silica particles are consoli-dated in Table 1.

de, m/s (b) — temperature, K, (c) — vapour mass fraction, kg/kg, (d) — density, kg/m3.

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Dry atmospheric air at 400 K was supplied to the dryer bottomwith mass flux of 10.44 kg/(m2 s). The air outlet pressure wasassumed to be a subatmospheric and equal to −25 Pa. Furthermore,the effect of air turbulence was considered by assuming 5% inletturbulence intensity and 1.25 m hydraulic diameter. Chamber wallswere assumed to be made of 2 mm steel. Finally, the ambienttemperature was set to 300 K.

The influence of wall thermal boundary conditions was investi-gated by assuming either adiabatic flow in the dryer (insulatedchamber walls) or linear drop of the wall temperature from 325 at theinlet to 320 K at the outlet (non-insulated chamber with known walltemperature) [4,6].

The predicted steady-state 3D flow fields of velocity, temperature,vapour mass fraction and density of the drying air are illustrated byFigs. 4–7 (all results are in SI units). The results demonstrate three-dimensionality and complexity of the flow in the pneumatic dryer.

Fig. 6. 3D air flow fields for the case of silica drying (known wall temperature). (a)— velocitykg/m3.

In bothdryingcases thedeveloping core of the air velocity is observed(Figs. 4a, 5a, 6a and 7a). The air temperature gradually decreases in thecore (Figs. 4b, 5b, 6b and7b)while thevapour fraction increases (Figs. 4c,5c, 6c and 7c) as a result of heat and mass transfer from the discretephase of particles. An interesting behaviour is demonstrated by the airdensity (Figs. 4d, 5d, 6d and 7d). Initially, the air density decreases due tofast increase of vapour fraction in the region of prevalent heat and masstransfer from thewet particles. Then,whenmost of themoisture contentofwet particleswasevaporated, the rate ofmass transfer reduces and thevalue of vapour fraction in the air stabilizes. However, the airtemperature decrease continues due to particles heating up toequilibrium. Such air temperature drop and constant vapour fractionvalue result in the observed rise of the air density.

A valuable one-dimensional representation of actual three-dimensional air flow fields can be obtained by mass-weightedaveraging. Using this technique, the computational domain was cut

magnitude, m/s (b)— temperature, K, (c)— vapour mass fraction, kg/kg, (d)— density,

Fig. 7. 3D air flow fields for the case of PVC drying (known wall temperature). (a)— velocity magnitude, m/s (b)— temperature, K, (c)— vapour mass fraction, kg/kg, (d)— density,kg/m3.

378 M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

along z axis in XY plane by 100 slices. For each slice, mass-weightedaverages of the flow parameters were calculated. Combining thecorresponding values for all slices, 1D representations of the flow fieldswere obtained. The simulatedmass-weighted averages of 3Dflow fieldsof velocity, temperature, vapour mass fraction, density and pressure ofthe drying air are shown in Fig. 8. For the case of insulated chamber, thewall temperaturewas in the range of 376–394 K,whereas for the case ofnon-insulated chamber the wall temperature decreased linearly from325 to 320 K over the dryer length. It can be observed that these 1Dflowfields qualitatively illustrate the 3D behaviour of flow parameters alongz axis and this behaviour is in accordance with the conclusions drawnabove. At the same time, the usage of the 1D representations for exactquantitative comparison with the published predictions of 1D pneu-matic drying models [1,3,4] is inappropriate since such models do nottake into account numerous 3D space effects like turbulence and

transversal components of velocity. Moreover, the 1D averaged repre-sentations of the 3D flow fields cannot be adequately validated by thepublishedexperimental dataobtainedusing1Dor2Dsampling technique[1].Due to these reasons, thedevelopmentof proper validationprocedurefor the present 3D model is a prospect work. Nevertheless, a qualitativevalidation of the presented averaged flow fields (Fig. 8) demonstratessimilarity between the predicted trends of air velocity, temperature,vapour mass fraction, density and pressure with their behaviourcalculated by Baeyens et al. [1], Levy and Borde [3,4], Skuratovsky andLevy [5] and Skuratovsky et al. [6]. Further comparisonwith experimentalmeauserments of Baeyens et al. [1] is given in the next section.

Fig. 9 illustrates the predicted drying kinetics of two PVC particlesinjected at different locations. The first was injected close to the dryercenterline at the distance of 0.077 m from it and the second wasintroduced at the dryer peripheral zone of 0.585 m form the

Fig. 8. Mass-weighted averages of actual 3D air flow fields.

379M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

Fig. 9. Drying kinetics of PVC particles in the central and peripheral injections. (a) —

adiabatic flow, (b) — known wall temperature.

Fig. 10. Averaged drying kinetics of wet particles in pneumatic dryer. (a) — adiabaticflow, (b) — known wall temperature.

380 M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

centerline. For both drying cases, a considerable difference betweenthe central and peripheral particles temperatures is observed, reach-ing up to 10 K for the case of adiabatic flow and up to 20 K for the casewith known wall temperature. Furthermore, in the periphery regionthe particle transformation from a wet particle into a dry particleoccurs with relative displacement equal about 1 m for the case ofadiabatic flow and 2 m for the case with known wall temperature.These results are consequences of different drying conditions in thecentral and peripheral zones of the pneumatic dryer which eventuallylead to an uneven quality of the obtained product. Also not presentedhere, the same trends were observed for silica particles drying as well.

The 1D mass-weighted averages of particle drying kineticspredicted for the silica and PVC particles are shown in Fig. 10. It canbe found that the drying process of silica particles goes slower thanPVC particles. This phenomenon can be explained by greater heatcapacity of the silica solids (see Table 1).

Analyzing Figs. 9 and 10, it can be found that the average particlemoisture content remains constant and equal to the assumed finalvalue (see Table 1) in about 80% of the dryer length. Therefore theseresults indicate a possibility of increasing the initial product moisturecontent or shortening the length of the adopted drying chamber for allthe studied cases.

Figs. 11 and 12 demonstrate the capability of the utilized dryingkinetics model to predict the evolution of temperature profiles withinwet particles. In these figures the mass-weighted averages of tempera-tures of wet particle center and outer surface are illustrated in the firstand second drying stages. In the subsequent period of dry particleheating (not shown here) the temperature in the particle centerapproaches the value at the particle surface. Analyzing the calculated

data, it is concluded that the duration of the first drying stage (includinginitial heating and equilibrium evaporation periods) is negligible forboth types of the wet particles. Furthermore, it is found that themaximum temperature difference in PVC particles attains up to 12 Kwhilst in the silicaparticles this value is up to 3 K. Theseobservations areexplained by low thermal conductivity of the PVCparticles solid content(see Table 1). It is worth noting that the presence of such steeptemperature gradients in micron-sized PVC particles can lead tosignificant thermal stresses in the particles and eventually result inproduct thermal degradation, see Mezhericher et al. [23,30]. Therefore,the drying of PVC wet particles potentially results in greater rate ofthermal degradation in the final product than drying of the silica.

5.2. Comparison with experimental measurements

For the pneumatic drying of PVC particles with 140 µm diameter(PVC-140), the predicted profiles of air temperature and particleaverage moisture content can be compared to the Baeyens et al. [1]experimental data. To obtain a good agreement with the experimentaldata, Baeyens et al. [1] and Levy and Borde [3] recommend using thefollowing correlations of Nusselt and Sherwood numbers:

Nup = 0:15Re; ð37Þ

Shp = 0:15Re: ð38Þ

It isworth noting, that Baeyens et al. [1] announced 5 radial (r=0m,0.38 m, 0.48 m, 0.57 m and 0.62 m) and 8 height (z=0m, 4.5 m, 7.5 m,

Fig. 11. Averaged temperature evolution at center and surface of PVC wet particlesduring 1st and 2nd drying stages. (a) — adiabatic flow, (b) — known wall temperature.

Fig. 12. Averaged temperature evolution at center and surface of silica wet particlesduring 1st and 2nd drying stages. (a) — adiabatic flow, (b) — known wall temperature.

Fig. 13. Comparison between the predicted profiles of drying air temperature andBaeyens et al. [1] experimental points for PVC-140 (known wall temperature). Thecalculation results are given for different radial positions, solid lines — Ranz–Marshallcorrelations (Eqs. 21 and 22) and dashed lines— Baeyens et al. correlations (Eqs. 37 and38).

381M. Mezhericher et al. / Powder Technology 203 (2010) 371–383

10.5 m, 13.5 m, 16.5 m, 19.5 m and 22.5 m) sampling positions in theirexperimental setup. However, in the subsequent parts of the article [1]the measured air temperature and particle moisture content wereillustrated only for 7 experimental points located at various dryerheights (z=0m, 4.5 m, 7.5 m, 10.5 m, 13.5 m, 16.5 m and 19.5 m).Moreover, Baeyens et al. [1] did not indicate explicitly for which of thedeclared radial positions the presented experimental data wereobtained. Taking into account these uncertainties, and three-dimen-sionality of air and particle flow fields in the dryer observed in theprevious section, the predicted longitude distributions of air tempera-ture and particle moisture content at various for radial samplingpositions are compared with Baeyens et al. [1] experimental data (seeFigs. 13 and 14). It can be found that all the experimental temperaturepoints are lying within the range of the calculated upper and lowertemperature curves; however no one of these curves correspondsexactly to the measured data found in [1]. The calculated curves ofparticle moisture content are in satisfactory agreement with experi-mental points for the case when Baeyens et al. [1] correlations wereutilized (Eqs. 37 and 38) and substantially over predict the experimentswhen Ranz–Marshall Eqs. (21) and (22) were implemented. It is worthnoting that the same trends for particlemoisture contentwere reportedby Baeyens et al. [1] and Levy and Borde [3].

Summarizing Figs. 13 and 14, due to many uncertainties in theBaeyens et al. experimental data, only a satisfactory agreement betweenthe calculations and the measurements can be obtained. For bettercomparison of the model predictions with the experimental points andanswer thequestionwhich semi-empirical correlationsof heat andmass

transfer coefficients are better for 3Dpneumatic dryingmodelling,moreinformation is necessary and thus a separate validation study involvingsets of experiments on the pneumatic dryer is a prospect work.

Fig. 14. Comparison between the predicted profiles of particle moisture content andBaeyens et al. [1] experimental points for PVC-140 (known wall temperature). Thecalculation results are given for periphery and center of the dryer, solid lines — Ranz–Marshall correlations (Eqs. 21 and 22) and dashed lines — Baeyens et al. correlations(Eqs. 37 and 38).

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6. Conclusions

Steady-state three-dimensional calculations of the pneumaticdrying processes in vertical dryer were performed. The presentedtheoretical model of the process is based on two-phase Eulerian–Lagrangian approach for gas-particles flow. The two-stage dryingkinetics of wet particles is described with the help of an advancedtheoretical model validated for single wet particle drying. Thedeveloped model was numerically solved using the 3D pressure-based steady-state solver of FLUENT package. The processes ofpneumatic drying of 140 μm PVC and silica particles in a large-scalevertical pneumatic dryer were numerically simulated. The influenceof wall thermal boundary conditions was investigated by assumingeither adiabatic flow in the dryer or linear drop of the walltemperature from 325 at the inlet to 320 K at the outlet. The obtained3D results demonstrated three-dimensionality and complexity of theflow in the pneumatic dryer. For qualitative illustration of the mainflow features, 1D mass-weighted averaging of 3D flow fields has beenutilized. Analyzing the predicted particle drying kinetics, a consider-able difference between the central and peripheral particles temper-ature was observed. Such a difference is a sequence of non-uniformdrying conditions in the central and peripheral zones of thepneumatic dryer and can lead to an uneven quality of the obtainedproduct. Moreover, for the case of non-insulated chamber walls thisquality unevenness is estimated to be substantially greater than forthe case with thermally insulated drying chamber. The examination ofthe predicted temperature profiles within the silica and PVC wetparticles showed that the latter are subjected to higher temperaturegradients that potentially result in greater thermal stresses and couldlead to thermal degradation in the final product.

Notation

A surface area, m2

B Spalding numberBk crust permeability, m2

CD drag coefficientcp specific heat under constant pressure, J kg−1 K−1

d diameter, mDv coefficient of vapour diffusion, m2 s−1

FA virtual mass force, N

FB buoyancy force, NFC contact force, NFD drag force, NG gravity acceleration, m s−2

h heat transfer coefficient,Wm−2 K−1; specific enthalpy, J kg−1

hD mass transfer coefficient, m s−1

hfg specific heat of evaporation, J kg−1

k thermal conductivity,Wm−1 K−1; turbulence kinetic energy,m2 s−2

m mass, kgṁv vapour mass transfer rate, kg s−1

M molecular weight, kg mol−1

np number density of discrete phaseNu Nusselt numberp pressure, PaPr Prandtl numberr radial space coordinate, mR radius, mℜ universal gas constant, J mol−1 K−1

Sc Schmidt numberSc mass source termSh Sherwood numberSh energy source termt time, sT temperature, Ku velocity of drying agent, m s−1

U→p vector of particle velocity, m s−1

u→ vector of gas velocity, m s−1

x space coordinate, mX moisture content (dry basis), kg kg−1

y space coordinate, mz space coordinate, m

Greek symbolsα thermal diffusivity, m2 s−1

β coefficient of thermal expansion, K−1; empirical coefficientε particle crust porosity; dissipation of turbulence kinetic

energy, m2 s−3

μ dynamic viscosity, kg m−1 s−1

ν kinematic viscosity, m2 s−1

ρ density, kg m−3

υ velocity, m s−1

ωv vapour mass fraction

Subscriptsa air, dry air fractionatm atmosphericcr particle crust; criticald dropletdiff diffusionf final point of drying processflow forced flowg drying agenti crust–wet core interfacem air–vapour mixturep particle, discrete phasepores crust poresr radial directions solid fraction; surfacev vapourw waterwc particle wet core0 initial point of drying process1 initial point of droplet evaporation period2 contributor∞ bulk of drying agent

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Acknowledgement

The authors appreciatively acknowledge the financial support ofthe present work from GIF — German-Israeli Foundation for ScientificResearch and Development under grant no. 952-19.10/2007.

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