Heat and Mass Transfer of Single Droplet Particle Drying

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Heat and Mass Transfer and Breakage of Particlesin Drying Processes

M. Mezhericher, A. Levy, and I. BordePearlstone Centre for Aeronautical Engineering Studies, Department of Mechanical Engineering,

Ben-Gurion University of the Negev, Beer-Sheva, Israel

An advanced theoretical model of unsteady coupled heat andmass transfer and breakage of wet particles in two-stage dryingprocesses is presented. The numerical simulations of drying of silicaslurry droplets have shown that both temperature and mechanicalstresses emerge in the wet particle during drying. It has been foundthat mechanical stresses play a substantial role at the beginning of

the second drying stage, whereas the thermal stresses are much moreconsiderable at the end of drying. In addition, tangential stresses inthe crust of wet silica particles are predominant over the radial com-ponent (approximately five times greater). Compared to the pro-posed breakage criterion, the model predicts that the total stressescan be a reason for wet particle cracking/breakage and this dependson granule diameter, drying agent temperature, and size of primaryparticles. To prevent granule breakage at given drying conditions,the slurry droplets with primary particles as small as possible arerecommended for drying.

Keywords Breakage; Drying; Heat transfer; Mass transfer;Particle

INTRODUCTION

Formation processes involving drying of dropletscontaining small solid particles are used today in manyindustries such as food and dairy manufacturers, chemicaland biochemical industries, the pharmaceutical industry,ceramics production, and others.[1] Theoretical modelingof the process is based on descriptions of simultaneous heatand mass transfer of individual droplets that turn into wetporous particles during drying.

Though many studies have been devoted to transportphenomena in droplets,[2] coupled heat and mass transferand breakage of wet particles still remain broad areas forresearch. Generally, complicated phenomena occurring

within the single droplet during drying involve diffusionof liquid through small solid particles (also called primaryparticles) toward the droplet surface and diffusion andagglomeration of primary particles leading to formation

of a porous solid crust at the droplet outer surface. Afterthe onset of crust formation (when droplet is turned intowet particle), the drying process is governed by diffusionand flow of liquid from inside the wet particle throughthe porous crust. Unfortunately, the present experimentalmethods do not allow measuring of temperature distribu-

tions and concentration and pressure profiles within thedried droplet=wet particle. However, this information isnecessary for the analysis of wet particle cracking=breakage due to internal thermal and mechanical stresses,[3,4]

prediction of thermal degradation, and structure of thefinal product. On the other hand, most of the availableexperimental data were obtained only for low or moderatetemperatures of droplet drying (up to about 200C), andthere is a lack of experimental results for higher dryingtemperatures that are typical in some industries (e.g., inceramics[1]). Therefore, the development of an advancedtheoretical model of heat and mass transfer and the study of wetparticles breakage in drying processesis an essentialtask.

OBJECTIVES OF THE PRESENT STUDY

The objective of the present work was to study heat andmass transfer and breakage of particles during the transientprocess of droplet=wet particle drying. A theoretical studywas performed using the developed drying model of singledroplet, containing small solid particles.[4] Numerical simu-lations allowed us to investigate the mechanisms of heatand mass transfer, time evolutions of vapor mass fraction,and pressure profiles within the pores of wet silica particle.Applying the theory of elasticity and solid mechanics, acriterion of particle cracking=breakage was proposed andthe studies of silica particle breakage under different drying

conditions were performed.

HEAT AND MASS TRANSFER OF WETPARTICLE DRYING

In the present study, two-stage drying in atmospheric airof droplets containing small solid particles is considered. Inthe first drying stage, the process is controlled byevaporation of pure liquid. The second drying stage beginswhen droplet moisture content reduced to the critical value

Correspondence: A. Levy, Pearlstone Centre for AeronauticalEngineering Studies, Department of Mechanical Engineering,Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva84105, Israel; E-mail: [email protected]

Drying Technology, 27: 870–877, 2009

Copyright# 2009 Taylor & Francis Group, LLC

ISSN: 0737-3937 print=1532-2300 online

DOI: 10.1080/07373930902988163

870



and a thin layer of a dry porous solid crust is formed at thedroplet surface. From this point onwards, the droplet isreferred to as a wet particle with permanently growingcrust and shrinking wet core regions. In the second stage,liquid evaporation takes place inside the wet particle atthe receding interface between the crust and wet core (seeFig. 1a). The vapor, generated over the interface, diffuses

through the crust pores toward the particle outer surfaceand forms a thin boundary layer. From the surface, vaporis taken away by the convection of drying air. The processof particle drying stops when the desired value of moisturecontent is obtained.

Because of space limitations, the mathematical descrip-tion of the first drying stage is omitted and the reader isreferred to our previously published papers.[3,4] Althougha more detailed explanation on the current modelling of heat and mass transfer in the second drying stage is givenin our recent work,[4] the key equations are presented herefor the reader’s convenience.

After the onset of crust formation, the wet particle in thesecond drying stage is treated as a two-region sphere withisotropic physical properties. The particle outer diameterremains unchangeable as the drying proceeds, whereasthe particle wet core shrinks due to evaporation from its

surface (called crust–wet core interface or, simply, theinterface in the present article), and the thickness of crustregion increases. The region of porous crust is assumedto be pierced by identical straight cylindrical capillariesof small diameter (see Fig. 1).

The equations of energy conservation in spherical coor-dinates for the crust and wet core regions of wet particle

are the following (the coordinate origin coincides withthe particle center):

Crust region:

@ Tcr

@ t ¼

acr

r2

@

@ r r2 @ Tcr

@ r

; RiðtÞ r Rp; ð1Þ

kcr@ Tcr

@ r ¼ kwc@ Twc

@ r þ hfg_mmv

Ai; r ¼ RiðtÞ;

Twc ¼ Tcr; r ¼ RiðtÞ;

hðTg TcrÞ ¼ kcr@ Tcr

@ r ; r ¼ Rp:

8<: ð2Þ

Wet core region:

qwccp;wc@ Twc

@ t ¼

1

r2

@

@ r kwcr2 @ Twc

@ r

; 0 r Ri tð Þ:

ð3Þ

@ Twc

@ r ¼ 0; r ¼ 0;

kcr@ Tcr

@ r ¼ kwc@ Twc

@ r þ hfg_mmv

Ai; r ¼ RiðtÞ;

Tcr ¼ Twc; r ¼ RiðtÞ:

8<: ð4Þ

The initial condition for both Eqs. (1) and (4) is the droplettemperature at the end of the first drying stage.

Generally, heat transfer from the drying air to the

droplet=wet particle occurs due to simultaneous convectionand radiation. Recently, Mezhericher et al.[5] showed thatfor small initial droplet diameters up to 0.25 mm and ambi-ent temperatures <750C as well as for initial droplet dia-meters <2.5 mm and ambient temperatures up to 200C,the effect of thermal radiation can be neglected. At thesame time, for droplets with initial diameter greater than1 mm and temperatures of the drying agent higher than400C, thermal radiation plays a substantial part in thedrying process.[5] In the present study, the drying processof 2 mm silica slurry droplet in air temperatures of 150–750C is considered, and therefore the influence of thermal radiation is taken into account. Consequently,

the coefficient of heat transfer is given by:

h ¼ hc þ hr: ð5Þ

The coefficient of convection heat transfer is determinedby means of the Nusselt number using the modifiedRanz-Marshall correlation:[6]

Nud ¼ ddhc=kd ¼ ð2 þ 0:6Re1=2d Pr1=3Þð1 þ BÞ0:7 ð6Þ

FIG. 1. Scheme of a wet particle drying in the second drying stage (a)and (b) capillary pore within the particle crust.

HEAT AND MASS TRANSFER AND PARTICLE BREAKAGE 871



The coefficient of radiation heat transfer is calculated bythe expression:[7]

hr ¼ rBerðT4g T4

dÞ=ðTg TdÞ: ð7Þ

The crust–wet core interface is shrinking with the rate: [6]

d ðRiÞd t ¼ 1eqwc;w4pR2

i

_mmv: ð8Þ

Here the rate of evaporation, _mmv is determined by massconvection from the droplet surface to the ambient. [4]

The processes of liquid evaporation from the crust–wetcore interface and subsequent vapor motion through thecrust pores toward the particle outer surface are describedby the laws of mass, momentum, species, and energy con-servation applied to the capillary pores volume. For indivi-dual straight capillary pore of the particle crust (Fig. 1b),the origin of a cylindrical coordinate system is placed atthe crust–wet core interface and the z axis coincides with

the line of pore symmetry. Air–vapor mixture is assumedto flow along the z axis only. Hence, the mass conservationlaw is given by:

@ q

@ t þ

@

@ zðqtzÞ ¼ 0; ð9Þ

where z ¼ (r Ri)=(Rp Ri) and 0 z 1.The momentum conservation equation for air–vapor

mixture in the pore reduces to well-known Darcy’s law:[8]

tz ¼ Bk

l

Atotal

Apores

@ p

@ z ¼

Bk

leb@ p

@ z : ð10Þ

By determining the vapor mass fraction of air–vapormixture as

xv ¼ qv=q; ð11Þ

and assuming that it changes along z-axis only, the diffu-sion process within the crust pore is described by equation:

q @ xv

@ t þ tz

@ xv

@ z

¼ @

@ z qDv

@ xv

@ z

: ð12Þ

Here the coefficient of vapor diffusion, Dv, is evaluated

according to:[9]

Dv ¼ 2:302 105 p0=pðT=T0Þ1:81; ð13Þ

where p0¼ 0.98 105Pa and T0¼ 256K.Because the length of the capillary pore is very short and

the diffusion movement of gas molecules is quick, the spa-tial change of the temperature within the pore is neglected.As a result, the temperature of air–vapor mixture in the

pore is assumed to be close to the temperature of the crust–wet core interface. Considering the air–vapormixture as an ideal gas, the following equation of energyconservation is derived:

qcp

d T

d t ¼

@ p

@ t þ tz

@ p

@ z : ð14Þ

The set of Eqs. (9), (10), (12), and (14) is rearranged to give:

1

cp

d p

d t ¼

Ma Mv

MaMvM

d xv

d t þ

Bk

leb@ 2p

@ z2 ð15Þ

d xv

d t ¼ Dv

@ 2xv

@ z2

Ma Mv

MaMvM

@ xv

@ z

2" #

ð16Þ

The following boundary conditions are utilized:

at z ¼ 0:

qDv

@ xv

@ z ¼ ð1 xvÞe1bqwc;w

Ri

Rp

d ðRiÞ

d t ð17Þ

xv ¼ qv;satðTwc;sÞ=q ð18Þ

p ¼ pv;satðTwc;sÞMv=ðxvMÞ ð19Þ

at z ¼ Lp:

Dv@ xv

@ z þ xvðqtzÞ

<Twc;s

pgM ¼ hDðxv xv;1Þ

Rp

Rieb

ð20Þ

pjz¼Lp¼ pg ð21Þ

MODELING OF WET PARTICLE CRACKING/BREAKAGE

Thermal Stresses

In our previous studies[3,4] it was shown that elevateddrying air temperatures resulted in steep temperature gra-dients in the particle crust. Because the crust thickness ison the order of micrometers at the beginning of the seconddrying stage and up to an order of millimeters at the end of drying, considerable thermal stresses can emerge. The

radial and tangential thermal stresses in porous crust arecalculated by assuming that the inner and outer surfacesof the particle crust have free radial strains:[10]

rT;rðr; tÞ ¼ 2aT;crEcr

ð1 n crÞr3

r3 R3i

ðR3p R3

i Þ

Z Rp

Ri

s2Tcrðs; tÞd s

"

Z r

Ri

s2Tcr s; tð Þd s

; ð22Þ

872 MEZHERICHER ET AL.



rT;hðr; tÞ ¼ aT;crEcr

ð1 n crÞr3

2r3 þ R3i

R3p R3

i

Z Rp

Ri

s2Tcrðs; tÞd s

"

þ

Z r

Ri

s2Tcrðs; tÞd s r3Tcrðr; tÞ

ð23Þ

The Young’s modulus for porous crust structure can be

determined as follows:[11]

Ecr ¼ C1Es;0ð1 amTcr=Tm;sÞð1 eÞn: ð24Þ

In this study, C1¼ 1, n ¼ 2, and am¼ 0.35. The value of silica melting point is taken as Tm,s¼ 1600C and theYoung’s modulus is set to Es,0¼ 73 GPa; the crust thermalexpansion coefficient and the Poisson’s ratio are assumedto be equal to aT,cr¼ 5 107K1 and n ¼ 0.17.[12]

Mechanical Stresses

One of the features of the current drying model is its

ability to predict the pressure gradient inside the crustcapillary pores. From the pressure equilibrium require-ments it can be deduced that the pressure established atthe crust–wet core interface inside the crust pores is equalto the pressure acting on the region of the porous crustfrom the side of wet core. Hence, the region of the particlecrust can be treated as pseudo-solid spherical containerunder internal pressure. The corresponding mechanicalstresses in radial and tangential directions are estimatedaccording to:[10]

rrðr; tÞ ¼ DpðtÞjz¼0

R3i ðR3

p r3Þ

r3ðR3

p R3

i Þ; ð25Þ

rhðrÞ ¼ DpðtÞjz¼0

R3i ð2r3 þ R3

pÞ

2r3ðR3p R3

i Þ; ð26Þ

where

DpðtÞjz¼0 ¼ pðtÞjz¼0 pg: ð27Þ

Total Stresses

The total radial and tangential stresses in the crust emer-ging during wet particle drying can be found by summing

the corresponding components of thermal and mechanicalstresses:

Total radial stress:

rr;totðr; tÞ ¼ rT;rðr; tÞ þ rrðr; tÞ: ð28Þ

Total tangential stress:

rh;totðr; tÞ ¼ rT;hðr; tÞ þ rhðr; tÞ: ð29Þ

Particle Strength and Breakage Criterion

In the current work the dried slurry consists of waterand insoluble primary particles, which are on the order of micrometers or even smaller. In the second drying stage thetemperature of the crust of silica wet particle can potenti-ally exceed the value of water boiling point under atmos-pheric pressure. In spite of this, the primary particles in thecrust remain connected to each other. Therefore, in the abs-ence of special binder in slurry, it can be suggested that vander Waals forces (and not capillary liquid bridging) are thedominating binding mechanism between the primary parti-cles in the silica crust.[13] Such an agglomeration mechan-ism is typical, for example, in ceramics processing.[14] Usingthe above assumption, the tensile strength of the particlecrust can be considered to be equal to the upper bound of agglomerate strength due to van der Waals forces and thisstrength strongly depends on the size of primary particles.[13]

For brittle materials like ceramics, one of the well-known failure criteria can be utilized: either maximum nor-

mal stress criterion or Mohr’s theory.

[15]

In the present caseof wet particle drying, the largest total radial and tangentialstresses in the isotropic spherical crust coincide with theprincipal directions and, based on our previous experi-ence,[3] are expected to be tensile. Consequently, both fail-ure criteria result in the following wet particle breakagecriterion:

fmaxðrr;totÞ or maxðrh;totÞg rt; ð30Þ

where rt is particle crust tensile strength.

NUMERICAL SOLUTION AND VALIDATION

OF THE MODELA computer program was developed to solve themathematical model numerically. For both drying stages,the procedure of numerical solution was based on the algo-rithm proposed by Moyano and Scarpettini.[16] The detailson numerical method applied in the first drying stage can befound in our previous reports, see Mezhericher et al. [4,17]

To solve the set of nonlinear partial differential equationsof the second drying stage (1), (3), (15), and (16), apredictor-corrector method[18] was applied and the derivedalgebraic equations were coupled using the compatibleboundary conditions at the crust–wet core interface.

The validation of the present model was performed by

comparing the calculated evolution of the particle tempera-ture and mass with the theoretical data predicted using ourpreviously developed and validated model.[3] The results of validation showed a good agreement and consistency; seeMezhericher et al.[4] for additional details.

RESULTS AND DISCUSSION

Numerical simulations of drying of silica slurry dropletin atmospheric air have been performed. The slurry




consists of monodispersed amorphous silica spherical par-ticles (median size of primary particles 272 nm, density1950 kg=m3), deflocculated in water with initial volumefraction of 0.10.[19] No specific binder is utilized. Theemissivities of liquid and solid fractions of the slurry aretreated as fixed values equal to er,w¼ 0.96 and er,s¼ 0.8,correspondingly.[12] The studied drying air temperatures

range is between 150 and 750C, the air velocity is set to1.4m=s, the initial droplet diameter is equal to 2 mm, andthe particle crust porosity is assumed to be 0.4.

A number of the theoretical results predicted by the cur-rent model for the studied range of temperatures have beenalready presented in our previously published report.[4]

Among these results are the simulated evolution of thetemperature and mass of 2 mm silica slurry droplet, thetime-change of the pressure and vapor mass fraction over

the crust–wet core interface in the second drying stage,and typical distributions of the pressure and vapor massfraction within the capillary pores at the end of the dryingprocess. It has been demonstrated that at air temperatures>400C the temperature differences in the crust can attainvalues above 30C and thus considerable thermal stressescan emerge. Another important outcome is the calculated

behavior and values of vapor fraction and pressure in thecrust pores. Both the vapor fraction and the pressure overthe crust–wet core interface are growing and, at the sametime, they are decreasing gradually along the pore lengthtoward the particle outer surface.

Figures 2–5 illustrate the predicted time evolution of theprofiles of pressure, vapor mass fraction, and stresses in thesecond drying stage when air temperature is set toTg¼ 600C.

FIG. 2. Predicted drying evolution of typical distributions of the pres-sure (a) and vapor fraction (b) in the pores of silica wet particle(Tg¼ 600C).

FIG. 3. Predicted drying evolution of typical distributions of the radial(a) and tangential (b) mechanical stresses in the crust of silica wet particle(Tg¼ 600C).




The results presented in Fig. 2 prove that both pressureand vapor mass fraction in the crust pores gradually growas drying proceeds.

Figure 3 demonstrates that at the crust inner surface theabsolute value of radial mechanical stress increases,whereas the corresponding value of tangential stress com-ponent decreases during drying (in this figure and onwards,

s is the dimensionless time of wet particle drying in thesecond drying stage). Such behavior can be explained bysimultaneous shrinkage of the particle wet core and thecrust thickening.

The simulated evolution of the thermal stresses is illu-strated in Fig. 4. It can be observed that both radial andtangential components of the thermal stress are growingin the drying process. Moreover, the tangential componentof thermal stress is substantially greater than radial.

Comparing the data given in Figs. 3 and 4, it can be seenthat the absolute values of the radial component of bothmechanical and thermal stresses are of the same order dur-ing the whole second stage of silica droplet drying. At thesame time, the absolute values of the tangential componentof mechanical and thermal stresses are of the same orderonly at the beginning of the process, whereas at the endof drying the tangential thermal stress significantly prevails

over the tangential mechanical stress.The predicted evolution of total tangential and radial

stresses in the crust of silica wet particle is shown inFig. 5. It is observed that in the course of the dryingprocess the total tangential stress in the crust of silica wetparticle is 5–10 times greater than the corresponding radialcomponent. The maximum stress value is tensile and it islocated at the inner surface of the particle crust. Figure 5balso demonstrates that near the crust inner surface the

FIG. 4. Predicted drying evolution of typical distributions of the radial(a) and tangential (b) thermal stresses in the crust of silica wet particle

(Tg¼ 600C).

FIG. 5. Total radial (a) and tangential (b) stresses in the crust of silica

wet particle in the second drying stage (Tg¼ 600C).




calculated total tangential stress is greater that the particlestrength. Therefore, according to the breakage criterionEq. (30), the theoretical model predicts that in the simu-lated conditions the dried wet particle will be broken orat least cracked during the drying process.

The results of calculations of maximal stresses in thecrust of silica granules (wet particles with final moisturecontent) as a function of different granule diameters areillustrated in Fig. 6 (droplet initial temperature Td,0¼ 19C,initial moisture content X0¼ 4.62, final moisture contentXf ¼ 0.05, drying air velocity ug¼ 1.40m s1). The pre-sented curves were obtained for primary particles withmedian size equal to 272 nm. For primary particles with

diameter in the range 100–900 nm the calculated maximumdiscrepancy from the presented data is smaller than 0.2%.Comparing the stress curves given in Fig. 6, it can be

found that tangential total stresses in the particle crustare substantially greater than corresponding radial com-ponents (approximately five times). On the other hand,agglomerates consisting of smaller primary particles havegreater tensile strength (see curve rt in Fig. 6). For thatreason, slurries containing primary particles as small aspossible should be utilized in order to prevent wet particlecracking or breakage due to emerging tangential stresses inthe drying processes.

CONCLUSIONSAn advanced theoretical model of heat and mass

transfer and breakage of single wet particles during dryinghas been presented. The model enables prediction of stresses emerging in wet particle crust in the second dryingstage.

Two origins of the stresses in the wet particle have beenobserved: thermal differences and pressure differences inthe region of the particle porous crust.

In the present study it is suggested that van der Waalsforces are the dominating binding mechanism betweenthe primary particles in the crust of silica slurry dropletsduring the second drying stage. Such an assumption allowsus to propose a breakage criterion, according to which thefinal particle is predicted to be damaged if one of the totalstress components in the crust is greater than crust agglom-

erate strength determined by the van der Waals’ forcesbinding mechanism.

The numerical simulations demonstrate that mechanicalstresses play a substantial role at the beginning of thesecond drying stage, but at the end of drying the thermalstresses are much more considerable. It is also found thatthe total tangential stress in the crust of wet silica particleprevails over the radial component (approximately fivetimes greater during drying).

Comparing the stresses to the proposed breakage criter-ion of wet particles, it is concluded that the total stressescan be a reason for particle cracking=breakage and thisdepends on particle diameter, drying agent temperature,and diameter of primary particles. To prevent the granulebreakage at given drying conditions, the slurry dropletswith primary particles as small as possible are recom-mended to be dried.

NOMENCLATURE

A Surface area (m2)Apores Pores area in crust cross section (m2)Atotal Total area of crust cross section (m2)B Spalding numberBi Biot number

Bk Crust permeability (m

2

)cp Specific heat at constant pressure (Jkg1K1)

Dv Coefficient of vapor diffusion (m2s1)d Diameter (m)E Young’s modulus (Pa)Es,0 Young’s modulus of solid fraction (Pa)h Heat transfer coefficient (Wm2K1)hc Coefficient of convection heat transfer

(Wm2K1)hfg Specific heat of evaporation (Jkg1)hr Coefficient of radiation heat transfer

(Wm2K1)k Thermal conductivity (Wm1K1)

Lp Length of crust pore (m) (Rp –Ri)M Molecular weight (kg mol1)m Mass (kg)_mmv Rate of evaporation (kg s1)

Nu Nusselt numbern Empirical coefficientPr Prandtl numberp Pressure (Pa)p0 Reference pressure (Pa)

FIG. 6. Calculated maximal total stresses in the silica granules at differ-

ent temperatures of the drying air.




R Radius (m)< Universal gas constant (Jmol1K1)Re Reynolds numberr Radial coordinate (m)s Space coordinate (m)T Temperature (K)Tm Temperature of melting point (K)T0 Reference temperature (K)t Time (s)ug Velocity of drying agent (m s1)X Moisture content (dry basis) (kg kg1)z Axial coordinate (m)

Greek Letters

a Thermal diffusivity (m2s1)am Empirical coefficientaT Coefficient of thermal expansion (K1)b Empirical coefficientc Ratio of specific heatse Crust porosity

er Emissivityh Angular coordinatel Dynamic viscosity (kg m1 s1)n Poisson’s ratioq Density (kg m3)r Mechanical stress (Pa)rB Stefan-Boltzmann constant (Wm2K4)rt Particle tensile strength (Pa)rT Thermal stress (Pa)s Dimensionless time of wet particle dryingt Velocity (m s1)xv Vapor mass fraction

Subscriptsa Air, dry air fractioncr Particle crustd Dropletf Final point of drying processg Drying agenti Crust–wet core interfacep Particler Radial directions Solid fraction or surfacesat Saturatedtot Totalv Vapor, vapor fraction

w Water

wc Particle wet corez Axial direction0 Initial point of drying process1 Bulk of drying agenth Tangential direction

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