International Journal of Heat and Mass Transfer 59 (2013) 66–74

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Convective drying of particulate solids – Packed vs. fluid bed operation

Milan Stakic ⇑, Predrag Stefanovic, Dejan Cvetinovic, Predrag ŠkobaljLaboratory for Thermal Engineering and Energy, University of Belgrade – Institute of Nuclear Sciences ‘‘Vinca’’, Belgrade, Republic of Serbia

a r t i c l e i n f o

Article history:Received 25 September 2012Accepted 27 November 2012Available online 2 January 2013

Keywords:Fine-grained hygroscopic materialsHeat and mass transferDrying kineticsDrying coefficientModeling

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11

⇑ Corresponding author. Address: Institute of NBox 522, 11001 Belgrade, Republic of Serbia. Tel.: +38

E-mail addresses: mstakic@vinca.rs (M. Stakic), pstdeki@vinca.rs (D. Cvetinovic), p.skobalj@vinca.rs (P. Š

a b s t r a c t

The paper addresses results for the case of convective drying of particulate solids in a packed and in afluid bed, analyzing agreement between the numerical results and the results of corresponding experi-mental investigation, as well as the differences between packed and fluid bed operation. In the fluidbed simulation model of unsteady simultaneous one-dimensional heat and mass transfer between solids,gas phase and bubble phase during drying process, based on two-phase bubbling model, it is assumedthat the gas–solid interface is at thermodynamic equilibrium. The basic idea is to calculate heat and masstransfer between gas and particles (i.e., the drying process) in suspension phase as for a packed bed ofparticles, where the drying rate (evaporated moisture flux) of the specific product is calculated by apply-ing the concept of a ‘‘drying coefficient’’. Mixing of the particles (i.e., the impact onto the heat and masstransfer coefficients) in the case of fluid bed is taken into account by means of the diffusion term in thedifferential equations, using an effective particle diffusion coefficient. Model validation was done on thebasis of the experimental data obtained with narrow fraction of poppy seeds characterized by meanequivalent particle diameter (dS,d = 0.75 mm), re-wetted with required (calculated) amount of water upto the initial moisture content (X0 = 0.54) for all experiments. Comparison of the drying kinetics, bothexperimental and numerical, has shown that higher gas (drying agent) temperatures, as well as velocities(flow-rates), induce faster drying. This effect is more pronounced for deeper beds, because of the largeramount of wet material to be dried using the same drying agent capacity. Bed temperature differencesalong the bed height are significant inside the packed bed, while in the fluid bed, for the same drying con-ditions, are almost negligible due to mixing of particles. Residence time is shorter in the case of a fluid beddrying compared to a packed bed drying.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Drying is an energy-intensive operation. Additionally, conven-tional dryers often operate at low thermal efficiency, typically be-tween 25% and 50%, but it may be as low as 10% [1,2]. The increasein energy costs, as well as the adoption of more strict safety and envi-ronmental regulations, initiated an increasing interest in designingenergy-saving systems all over the conventional chemical industry.Furthermore, in the case of industrial dryers, the wide variety ofproducts increases the concern to meet high quality specifications.Therefore, the need for optimal management of energy during dry-ing, with the demand for high quality products, leads to the develop-ment of control strategies for the drying plants studied.

Fluid bed processing of biological materials and food involvesdrying, cooling, agglomeration, granulation, and coating of particu-late materials. Fluid bed dryer is used because of the shorter dryingtime required and simple maintenance and operation. It is ideal for

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uclear Sciences Vinca, P.O.1 11 340 836.efan@vinca.rs (P. Stefanovic),kobalj).

a wide range of both heat sensitive and non-heat sensitive prod-ucts. Uniform processing conditions are achieved by passing agas (usually air) through a product layer under controlled velocityconditions to create a fluidized state. Heat is supplied by the fluid-ization gas, but it can be also effectively introduced by heating sur-faces (panels or tubes) immersed in the fluidized layer. Fluid beddrying offers important advantages over other methods of dryingparticulate materials such as easy material transport, high ratesof heat exchange at high thermal efficiency while preventing over-heating of individual solids.

The properties of a given product are determined from dryingkinetics data, i.e., moisture content and temperature changes withtime under controlled conditions. Other important properties aredimension, shape and density of the solids, fluidization gas veloc-ity, fluidization point (minimum fluidization velocity), equilibriummoisture content (desorption isotherms), and heat and mass trans-fer coefficients. These and other data are applied in a computa-tional model of fluid bed processing, thus enabling dimensioningof industrial drying systems.

Much work has been done to model and analyze both continuousand batch fluid bed dryers [3–12]. Each of the models has its own

Nomenclature

a, b exponents of Eq. (12) [–]ab bed specific surface area [m�1]AK coefficient with dimensions as defined by Eq. (14),B0 coefficient with dimensions as defined by Eq. (12)c specific heat capacity [J kg�1 K�1]d diameter [m]f volume fraction [–]h bed height [m](HBC)B volumetric heat transfer coefficient between bubble and

cloud-wake region [W m�3 K�1](HBE)B overall volumetric heat transfer coefficient between

bubble and emulsion based on volume of bubbles,[W m�3 K�1]

(HCE)B volumetric heat transfer coefficient between cloud-wake region and emulsion phase [W m�3 K�1]

ki internal mass drying coefficient [s�1](KBC)B coefficient of gas interchange between bubble and

cloud-wake region [s�1](KBE)B overall coefficient of gas interchange between bubble

and emulsion based on volume of bubbles [s�1](KCE)B coefficient of gas interchange between cloud-wake re-

gion and emulsion phase [s�1]M mass [kg]_M mass flow-rate [kg s�1]eM molecular mass [kg kmol�1]

nD, nT exponents of Eq. (14) [–]p pressure [Pa]r heat of evaporation [J kg�1]S surface [m2]T temperature, [�C,K]u velocity [m s�1]V volume [m3]_V volume flow-rate [m3 s�1]X material moisture content (dry basis) [–]Y gas moisture content (dry basis) [–]z axial coordinate [m]

Greek symbolsa heat transfer coefficient [Wm�2 K�1]

b mass transfer coefficient [m s�1]d diffusion coefficient [m2 s�1]g dynamic viscosity [Pa s]k thermal conductivity [W m�1 K�1]q density [kg m�3]u relative air humidity [–]/ sphericity of a particle [–]U general dependent variable [–]w bed porosity [–]s time [s]

Dimensionless numbersArS Archimedes number, d3

SqSðqS � qGÞg=g2G [–]

NuS Nusselt number, aS;GdS=kG [–]Pr Prandtl number, cGgG=kG [–]ReS Reynolds number, dSuGqG/gG [–]Sc Schmidt number, gG/(qGdV) [–]ShS Sherwood number, bS,GdS/dV [–]

Indicesat atmosphericb bedB bubbled dryE suspensioneq equilibriumG gasi internalL liquidin inletm moisturemf minimum fluidizationS solidssf surfacesat saturatedV vapor0 initial, superficial

M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74 67

specificity, nevertheless with common character of utilizing the heatand mass transport coefficients specific for the fluidized state.

This paper, however, presents a mathematical model to de-scribe the heat and mass transfer between solids, gas phase andbubble phase, based on two-phase model after Kunii and Leven-spiel [13], with the basic idea to calculate heat and mass transferbetween gas and particles (i.e., the drying process) in suspensionphase as for a packed bed of particles, using new approach in dry-ing equation definition. The influence of particle mixing, takingplace in the case of mobile beds (fluid bed, vibrated fluid bed, ro-tary bed, etc.), induced by the bubble flow in the case of fluidbed, is related to the diffusion term in the differential equations,using reported particle diffusion coefficients.

This model is successfully validated on the basis of originalexperimental data and can be utilized in order to improve theoperational conditions of the fluid bed dryers.

2. Materials and methods

2.1. Drying equipment and procedure

Experimental batch device (Fig. 1), consisting of a radial fan (1),a long horizontal pipe (2) with one flow-measuring orifice plate (3)

and an electrical heater (4), an air chamber (5) used to produceuniform air distribution through the bed of particles, and a sam-ple-vessel (6) with internal diameter of 150 mm, was equippedwith all necessary measuring and control instruments connectedto the Hewlett–Packard HP38528 data acquisition system andHP1000 computer system (7), thus making sampling and process-ing of experimental data fully automated. Air temperature after theorifice plate (T0), after the heater (Th), at the inlet of the sample-vessel (TG,in), and bed temperatures at different bed heights: T1at 0 mm, T2 at 5 mm, T3 at 15 mm, and T4 at 35 mm, were mea-sured using the pre-calibrated thermocouples (K-type with accu-racy of ±0.2 �C).

After turning on the fan and electric heater, the experimentalinstallation was preheated to the set temperature. The sample-ves-sel was preliminary weighted and connected to the experimentalsetup, while desired air flow and temperature were imposed andcontrolled. After obtaining preferred steady air parameters, mea-surement was started when the weighted mass of wet material,belonging to the prepared homogeneous sample mass, was in-serted into the vessel. From that moment all temperatureswere measured in equal time intervals, differential pressure onflow-measuring orifice plate as well as static pressures were con-trolled and measured. Dry and wet bulb temperature as well as

Fig. 1. Experimental setup; 1 – fan, 2 – long horizontal pipe (di = 100 mm), 3 – flow-measuring orifice plate (do = 33 mm), 4 – electric heater (2 kW), 5 – flow-equalizationchamber, 6 – sample-vessel (di = 150 mm), and 7 – acquisition system.

0.10

0.15

0.20

0.25T = 20.0oCT = 34.9oCT = 49.8oC

re C

onte

nt (k

g/kg

, d.b

.)

68 M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74

atmospheric pressure of the surrounding air, were also measured.After a given period of time (5 min), measuring was stopped. Sam-ple-vessel filled with material was disconnected and weighedquickly on digital scale with accuracy of 0.1 g. For the same steadyair parameters, described procedure was repeated with equal massof wet material from the same sample, but during a longer period(10 min). In that way, keeping constant chosen particular dryingregime, several measurements (each of them 5–10 min longer thatthe previous one) were necessary for the single drying kinetics ofthe prepared material sample.

2.2. Drying process parameters

Experimental investigations of drying kinetics in a fluid bedwere carried out using previously described procedure. Bed heightsof 15 and 30 mm, drying agent temperatures of 49.8, 70.1 and90.8 �C, and drying air velocities 0.573–0.642 m/s were used. Nar-row fraction of dry poppy seeds from 0.6 to 1.0 mm was used, char-acterized by the mean equivalent particle diameter dS,d = 0.75 mm.The other particle properties (qS,d, qb, w, /S, umf) are summarized inTable 1. Poppy seeds have been re-wetted with required (calcu-lated) amount of water up to the initial moisture content(X0 = 0.54) for all experiments. Note that this is approximately65% of the maximum moisture content for poppy seeds(Xmax = 0.84). After wetting, material was kept in a hermetic spacefor one day before the drying experiment was carried out, in orderto get the uniform moisture content of the solids (X0), verified bydrying the several samples in a laboratory dryer at 105 �C, andby measuring the sample mass before and after drying using ana-lytical scale with accuracy of 0.0002 g. Drying was terminatedwhen no mass change was attained.

2.3. Sorption data

Equilibrium moisture content was determined experimentallyusing a pre-calibrated Thermoconstanter TH2 (NOVASINA), basedon a static method for determination of the state of equilibrium

Table 1Particle (poppy seeds) properties.

dS,d qS,d qb w /S umf

mm kg/m3 kg/m3 – – m/s

0.75 1140 617.5 0.458 0.9 0.283

in a very small volume at constant temperature (0–50 �C). Afterreaching the equilibrium at the certain temperature, air relativehumidity (with accuracy of ±2%), as well as temperature, weremeasured. Moisture content of the material sample (poppy seeds)was determined using the standard method of drying the materialin a laboratory dryer at 105 �C. Experiments were carried out atthree temperatures (20.0, 34.9, and 49.8 �C). Moisture (desorption)isotherms are presented in Fig. 2.

3. Modeling of convective drying process

The conventional evaporative convective drying process in-volves complex transport phenomena and it can be considered asmade up of three consecutive steps. The first step is moisture (li-quid) movement in solids from the wet interior to the gas–solidinterface (internal pore, particle surface, etc.). This step is slowerfor larger solids and/or low moisture content in the material. Thesecond step is evaporation due to heat (energy) supplied to trans-form liquid into vapor. The last step is vapor movement to the sur-rounding gas by diffusion and convection. The slowest of the stepswill be rate-determining. Prediction of falling-rate drying kineticsby theory alone is very difficult, and accurate small-scale experi-ments are required instead. It is possible to estimate drying ratesunder different conditions by applying concepts such as the ‘‘char-acteristic drying curve’’ ([5,14,15] etc.) or the ‘‘drying coefficient’’([16,17] etc.).

0.00

0.05

0 20 40 60 80 100Relative Humidity (%)

Moi

stu

Fig. 2. Moisture isotherms for poppy seeds.

M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74 69

A reliable model of a drying process can substitute lengthy andexpensive experimental investigations of the process by fastnumerical simulations. The main objective of simulating the dryingprocess is to accurately predict the temperature and moisture con-tent of the dried material as functions of time and/or position inthe dryer. Similarly, one can estimate the residence time or dimen-sions of a dryer required for drying a particular material to the tar-get moisture.

3.1. General equation

The differential equations describing the conservation of a gen-eral dependent variable, U, in the case of an unsteady problem canbe written in a generalized form after Patankar [18]:

@

@sðqUÞ þ divð~uqUÞ ¼ divðCU gradUÞ þ SU: ð1Þ

The terms in Eq. (1) are called: the ‘‘unsteady term’’, the ‘‘convec-tion term’’, the ‘‘diffusion term’’, and the ‘‘source term’’. Interphasetransport laws must be incorporated in the ‘‘source term’’, repre-senting generation and dissipation of the variable U. Expressionsfor CU and SU depend on the physical meaning of the variable U.In convective drying, moisture content and enthalpy (temperature)of the material being dried, as well as humidity and enthalpy (tem-perature) of the used drying agent, are particular cases of the gen-eral dependent variable to be determined.

3.2. Packed bed model

In order to analyze convective drying in a packed bed, an un-steady-state one dimensional mathematical model was developedand has already been validated for different biological materials(corn grains, soybean, potato cubes, poppy seeds, etc., see [19–22]).

The basic assumptions of this model that describes simulta-neous heat and mass transfer between the gas phase and the prod-uct during the convective drying in a packed bed are as follows:

– drying parameters vary in one dimension, namely in the direc-tion of gas flow (usually vertical), and only changes of theparameters in this direction will be discussed;

– all solids are of the same size, shape, and density at one momentof time;

– the gas–solid interface is at thermodynamic equilibrium;– the drying rate (evaporated moisture flux) of the specific product

is calculated by applying the concept of a ‘‘drying coefficient’’;– heat transfer inside the solids (temperature gradient) is neglected;– dispersion of mass or heat in the considered direction of gas

flow is neglected.

On the basis of Eq. (1) and with the assumptions mentionedabove, the corresponding system of partial differential equationscan be used to describe heat and mass balances. Mentioned equa-tion system for the case of packed bed is fully described and givenby Stakic and Tsotsas [23].

3.3. Fluid bed model

The assumptions of the model for a fluid bed, in addition tothose for a packed bed, are:

– bubbles are of the same size in one cross section of the bed anduniformly distributed,

– bubbles contain no solids,– every uprising bubble drugs the solids in its wake, while after-

wards the solids sink back through the suspension phase, thusobtaining mixing of the solids,

At the same time, the corresponding system of partial differentequations derived on the basis of Eq. (1) for the case of fluid bedcan be written as (with the symbols defined in the Nomenclaturesection):

Conservation of moisture in bubble phase (solids-free gas):

@

@sðMG;B;dYBÞ þ

@

@zðuG;BMG;B;dYBÞ ¼ MG;B;dðKBEÞBðYE � YBÞ: ð2Þ

where, after Kunii and Levenspiel [13]:

1ðKBEÞB

¼ 1ðKBCÞB

þ 1ðKCEÞB

;

ðKBCÞB ¼ 4:5 � uG;mf

dBþ 5:85 � D

0:5V g0:25

d1:25B

; ðKCEÞB

¼ 6:78 �wmf DV uG;B

d3B

!0:5

:

Conservation of moisture in suspension phase:Gas:

@

@sðMG;E;dYEÞ þ

@

@zðuG;EMG;E;dYEÞ ¼ MG;B;dðKBEÞBðYB � YEÞ þ _Mm: ð3Þ

Solids:

@

@sðMS;dXÞ þ @

@zðuSMS;dXÞ ¼ @

@zMS;dDeff

@uS

@z

� �� _Mm; ð4Þ

Conservation of enthalpy, bubble phase (solids-free gas):

@

@sðMG;B;dcBTBÞ þ

@

@zðuG;BMG;B;dcBTBÞ ¼ VBðHBEÞBðTE � TBÞ; ð5Þ

where: cB = cG,B,d + cm,VYB,and after Kunii and Levenspiel [13]:

1ðHBEÞB

¼ 1ðHBCÞB

þ 1ðHCEÞB

;

ðHBCÞB ¼ 4:5qBcB �uG;mf

dBþ 5:85 � ðkBqBcBÞ0:5g0:25

d1:25B

; ðHCEÞB

¼ 6:78 � ðqBkBcBÞ0:5 �wmf uG;B

d3B

!0:5

Conservation of enthalpy, suspension phase:Gas:

@

@sðMG;E;dcETEÞ þ

@

@zðuG;EMG;E;dcETEÞ

¼ VBðHBEÞBðTB � TEÞ þ aS;GabVðTS � TEÞ þ cV ðTE � TSÞ _Mm; ð6Þ

where: cE = cG,E,d + cm,VYE.Solids:

@

@sðMS;dcSTSÞ þ

@

@zðuSMS;dcSTSÞ ¼

@

@zVSkeff

@TS

@z

� �þ aS;GabVðTE

� TSÞ � r _Mm; ð7Þ

where: cS = cS,d + cm,LX.Equation of continuity for gas phase:

@

@sðMGÞ þ

@

@zðuGMGÞ ¼ _Mm: ð8Þ

Initial mass flow-rate of the gas phase can be divided, according tothe bubbling bed (two-phase/region) fluidization model, to one partflowing through the bubble phase and another flowing through thesuspension phase:

Table 2Parameters of drying kinetics (Eq. (14)) and sorption equilibria (Eq. (12)) for poppyseeds.

Kinetics Equilibria

AK nD nT B0 a b

After Eq. (14) – – After Eq. (12) – –2.8 � 10�4 0.98 3.19 14.822 �3.027 1.948

Fig. 3. State variables and discretization along the height of the bed.

70 M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74

_M0 ¼ SqGuG;0 ¼ _MB þ _ME ¼ SfBqGuG;B þ Sð1� fBÞqGuG;E; ð9Þ

where uG,0 is the gas velocity in a free cross section of the apparatus(superficial velocity). Then, gas velocity through the suspensionphase, uG,E, and gas velocity through the bubble phase, uG,B, can beobtained from: uG,E = uG,mf/wmf, uG,B = uG,0/fB � uG,E(1 � fB)/fB.

Note that the suspension phase is in the state of minimum flu-idization, and can be assumed as a packed bed having porosity[13]:

wmf ¼ 0:586 � /�0:72S � g2

G

qGðqS � qGÞgd3S

" #0:029

� qG

qS

� �0:021

:

As in the present case of a fluid bed (packed bed as well) there is nodirected movement of the solids, the solids velocity is set to uS = 0.

3.4. Source term

In order to define the mass source, _Mm, which is, for the case ofconvective drying, the evaporated moisture flow-rate, the systemof coupled partial differential Eqs. (2)–(8) has to be completed witha drying rate equation. To this purpose, the concept of a ‘‘dryingcoefficient’’ is used in the present work. In the scope of the ‘‘dryingcoefficient’’ concept, the transport phenomena inside and outsidethe solids are considered separately. The reason for this is the exis-tence of two types of mass transfer resistances during convectivedrying due to:

– moisture transport from the solids interior to the solids surface,– moisture convection from the solids surface to the surrounding

gas.

Internal moisture transport is the more complex problem, dueto a variety of involved mechanisms (capillarity, diffusion, thermaldiffusion, bulk and molecular flow, surface diffusion) that dependon the structure of the specific product.

The moisture flow-rate from the interior to the surface of thesolids is, therefore, expressed empirically as

_Mm ¼ MS;dki � ðXsf ;eq � XÞ; ð10Þ

where ki is the internal moisture transport coefficient (the alreadymentioned ‘‘drying coefficient’’).

On the other hand, all the moisture transported from the solidsinterior to its surface has to be subsequently transferred to the sur-rounding gas. It is assumed that the solids surface and the gas in itsimmediate vicinity are at hygroscopic equilibrium with each other.Therefore, the evaporated moisture flow-rate can also be expressedas:

_Mm ¼ MG;d bS;Gab � ðY � YeqÞ: ð11Þ

Since the Eqs. (10) and (11) are mutually coupled by equilibrium,they have to be solved simultaneously.

The unknown variables Yeq and Xsf,eq (equilibrium gas resp. sol-ids moisture content at the solid surface) must fulfill the conditionof hygroscopic equilibrium for the specific moisture on the specificproduct, in our present case for water on poppy seeds. The empir-ical drying coefficient, ki, must also be specified for the consideredproduct. On the other hand, the gas-side mass transfer coefficient,bS,G, may be extracted from general, non-product-specificequations.

3.5. Parameter estimation

Because of the complex bond mechanisms of moisture to solids,the equilibrium between a certain material and moist air at a pre-scribed temperature (sorption isotherm) can only be specified

experimentally, and is usually correlated empirically. To this pur-pose, the empirical relationship after Milojevic and Stefanovic [17],

1�ueq ¼ eB; B ¼ �B0TaSXb

eq ð12Þ

with TS in K and

ueq ¼pat

psat� Yeq

Yeq þ ð1� YeqÞ �eMVeMG

ð13Þ

is used in the present work. The same authors proposed a relation-ship for the internal moisture transport coefficient (‘‘drying coeffi-cient’’) ki in Eq. (10) that accounts for the overall resistance tomoisture transport in the material, namely:

ki ¼ AK dnDS �

XX0

� �� TnT

S ð14Þ

with TS in �C and dS in m.All parameters of Eq. (14) (AK, nD, nT) were determined by using

the classical static method of equilibration of small specimen withair over saturated salt solutions for measuring the sorption iso-therms that have subsequently been correlated with Eq. (14), whilethe parameters of Eq. (12) (B0, a, b) were obtained using dryingkinetics experiments carried out in very thin beds (just one ortwo particle layers) of poppy seeds, using air at temperatures be-tween 50 and 100 �C, moisture contents of 0.53 kg/kg, and veloci-ties of 0.1 and 0.2 m/s. The air condition can be assumed toremain approximately constant during the flow through such thinlayers, so that the parameters of Eq. (14) can be derived immedi-ately (compare with Hirschmann et al. [24]). All product specificparameters characterizing hygroscopicity and particle-side dryingkinetics of poppy seeds are summarized in Table 2.

In contrary to the previously discussed coefficients, the heat andmass transfer coefficients between particle surface and gas in apacked bed (suspension phase of fluid bed as well), aS,G and bS,G,do not depend on the internal structure of the product. They havebeen investigated extensively in literature and generalized in formof suitable, non-dimensional correlations. The correlations of Kuniiand Levenspiel [13] are known to give good results in case of par-ticles smaller than about 1 mm in diameter:

0 10 20 30 40 50 600

10

20

30

40

50

T1, h = 0 mm T2, h = 5 mm T3, h = 15 mm T4, h = 35 mmBe

d Te

mpe

ratu

re (o C

)

Time (min)0 10 20 30 40 50 60

0

10

20

30

40

50

T1, h = 0 mm T2, h = 5 mm T3, h = 15 mm T4, h = 35 mmBe

d Te

mpe

ratu

re (o C

)

Time (min)(a) packed bed, G, niu = 0.19 m/s. (b) fluid bed, G, niu = 0.59 m/s.

Fig. 4. Experimentally determined bed temperature in the case of poppy seeds convective drying; dS,d = 0.75 mm, X0 = 0.53, TS,0 = 20.0 �C, Yin = 0.01, TG,in = 50.0 �C, andh0 = 30 mm.

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

TG,in = 50.0 oC

TG,in = 70.2 oC

TG,in = 90.4 oC

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)0 10 20 30 40

0102030405060708090

TG,in = 50.0 oC

TG,in = 70.2 oC

TG,in = 90.4 oCBed

Tem

pera

ture

(o C)

Time (min)(a) packed bed, G, niu = 0.19 m/s.

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

TG,in = 50.0 oC

TG,in = 70.1 oC

TG,in = 90.8 oC

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)0 10 20 30 40

0102030405060708090

TG,in = 50.0 oC

TG,in = 70.1 oC

TG,in = 90.8 oCBed

Tem

pera

ture

(o C)

Time (min)(b) fluid bed, G, niu = 0.59 m/s.

Fig. 5. Effect of gas inlet temperature on the experimentally determined height-averaged poppy seeds drying kinetics in a packed and a fluid bed; dS,d = 0.75 mm, X0 = 0.53,TS,0 = 20.0 �C, Yin = 0.01, and h0 = 15 mm.

M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74 71

NuS ¼ 0:977 � Pr0:33Re0:595S ; ReS > 300

NuS ¼ 1:83 � Pr0:33Re0:485S ; ReS 6 300

ð15Þ

ShS ¼ 0:977 � Sc0:33Re0:595S ; ReS > 300

ShS ¼ 1:83 � Sc0:33Re0:485S ; ReS 6 300

ð16Þ

and have been used in the present investigation.Minimum fluidization velocity, uG,mf, can be calculated after

Kunii and Levenspiel [13] as:

umf ¼ ReS;mf �gG

qG � dS; ReS;mf ¼ ðj2

1 þ j2ArSÞ0:5 ð17Þ

where j1 = 24, j2 = 0.049 [25].

3.6. Numerical procedure

A numerical procedure based on Patankar [18] has been used tosolve the partial differential equations. Eqs. (2)–(8) are discretizedby means of the control-volume method, i.e., the packed bed is di-vided by means of a vertical grid into a finite number of control

0 10 20 30 40 50 600.0

0.1

0.2

0.3

0.4

0.5

0.6

h0 = 15 mmh0 = 30 mm

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)0 10 20 30 40 50 60

0

10

20

30

40

50

h0 = 15 mmh0 = 30 mm

Bed

Tem

pera

ture

(oC

)

Time (min)(a) packed bed; G, niu = 0.19 m/s.

0 10 20 30 40 50 600.0

0.1

0.2

0.3

0.4

0.5

0.6

h0 = 15 mmh0 = 30 mm

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)0 10 20 30 40 50 60

0

10

20

30

40

50

h0 = 15 mmh0 = 30 mm

Bed

Tem

pera

ture

(oC

)

Time (min)

(b) fluid bed; G, niu = 0.59 m/s.

Fig. 6. Effect of bed height on the experimentally determined height-averaged poppy seeds drying kinetics in a packed and a fluid bed; dS,d = 0.75 mm, X0 = 0.53, TS,0 = 20.0 �C,Yin = 0.01, and TG,in = 50.0 �C.

72 M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74

volumes, as shown in Fig. 3. The iterative line-by-line method isused for solving the obtained linearized algebraic equations, apply-ing a recurrence formula during the calculation of the variable’svalues for every line, and following the same procedure for allthe lines in one direction. This method, called the Thomas algo-rithm or the TDMA (TriDiagonal-Matrix Algorithm), is describedin detail in the book of Patankar [18].

The calculation starts for the first control volume, and continuesfor the next one in the direction of gas flow after the balances pre-sented by Eqs. (2)–(8) are fulfilled. The fact that there is no directedmovement of the solids inside the fluid bed (uS = 0), i.e., the ab-sence of ‘‘convection terms’’ in Eqs. (4) and (7), does not influencethe possibility of calculating the height-dependent change of solidsmoisture content and enthalpy, but creates some problems ofnumerical stability. In spite of this, convergent solutions could al-ways be obtained by appropriate selection of the under-relaxationfactors [18].

As Fig. 3 shows, the developed mathematical model allows thecalculation of the local values of all state variables along the bed atany time. For the sake of presentation, time averages of the localvalues, as well as height-averaged values of the state variables ata specific time can be calculated.

4. Results and discussion

4.1. Experimental

In order to highlight the difference between parameters in apacked and a fluid bed, bed temperatures measured at differentheights in a packed bed for the case of initial bed heighth0 = 30 mm are shown in Fig. 4a. Note that thermocouple T4,regarding its position (5 mm above the bed), has measured the

drying agent temperature above the bed. It can also be seen thatthe differences between the temperatures inside the bed(T1,T2,T3) are significant (up to 18 �C). In the case of a fluid bed,for the same drying conditions (Fig. 4b), the differences betweenthe temperatures inside the bed (T1,T2,T3) are almost negligible(±1.0 �C) due to mixing of particles.

Experimental drying kinetics for poppy seeds drying in a packedbed, as well as in a fluid bed, at different air temperatures are givenin Fig. 5. It is obvious that higher drying agent temperature induceshigher drying rates. It can also be seen that the drying process isfaster in the case of fluid bed compared to the same process in apacked bed.

Comparison of the drying kinetics of poppy seeds drying at dif-ferent initial bed heights (Fig. 6) shows the results as expectedfrom the drying theory. More material to be dried (deeper bed) in-duces slower drying because of the larger amount of wet materialto be dried using the same drying agent capacity. In the specificcases, when insufficient air flow-rates are applied, air saturationcan additionally induce lower drying rates. However, dissimilaritybetween different bed types (packed or fluid) is significant whenequal amount of wet material is dried.

4.2. Model

The packed bed model was validated in the past by measuringequilibria and single-particle – or equivalent (thin layer) – dryingkinetics, fitting the product-dependent model parameters, simulat-ing the operation of deep-bed drying of the considered material,and comparing the calculated results with respective experiments.This procedure has been conducted for corn grains, corn on the cob,wheat, coal etc., see [19,20,22,23]. In the frame of the present work,the same type of validation has been carried out for the convective

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)

experiment num. simul.

0 10 20 30 400

102030405060708090

Bed

Tem

pera

ture

(o C)

Time (min)

experimentnum. simul.

(a) G, niT = 50.0 oC, inY = 0.01, G, niu = 0.59 m/s, = 20.0 oC.

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)

experiment num. simul.

0 10 20 30 400

102030405060708090

Bed

Tem

pera

ture

(o C)

Time (min)

experiment num. simul.

(b) G, niT = 70.1 oC, inY = 0.01, G, niu = 0.59 m/s, = 20.0 oC.

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

0.6

Moi

stur

e C

onte

nt (k

g/kg

, d.b

.)

Time (min)

experiment num. simul.

0 10 20 30 400

102030405060708090

Bed

Tem

pera

ture

(o C)

Time (min)

experiment num. simul.

(c) G, niT = 90.8 oC, inY = 0.01, G, niu = 0.59 m/s, 0S,T

0S,T

0S,T

= 20.0 oC.

Fig. 7. Effect of gas inlet temperature on the simulated and experimentally determined height-averaged poppy seeds drying kinetics in a fluid bed; dS,d = 0.75 mm, X0 = 0.53,and h0 = 15 mm.

M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74 73

drying of poppy seeds in a fluid bed. The product-dependentparameters have been determined (Table 2) based on the discussedmeasurements. The respective data have been systematically com-pared with model results in the present work, with very goodagreement. Several representative comparisons are presented inFig. 7. To this purpose, and as previously explained, the local valuesof solids moisture content and temperature have been averagedover the bed height and are plotted against time. The agreementshows that the model can reliably describe transition from the thinimmobile layer to the fluid bed without additional fitting oradaptation.

The model has a lot of potential in predicting the relevantparameters of the fluid bed convective drying. The values oftime-averaged and height-averaged parameters can be calculatedreliably.

The values of the solids parameters, such as temperature andmoisture content, over the bed height (i.e., along the flow-direc-tion) are not inconsistent for one moment of time (for examplesee Fig. 4b) due to the fact that mixing of the particles exists in afluid bed. On the other hand, the existence of the temperature gra-dient as well as the moisture gradient along the flow-direction in-side the bed is well known. The higher the bed height, the morepronounced gradients are.

5. Conclusions

1. Based on experiments, it was shown that drying process is fas-ter in the case of a fluid bed than in the packed bed. Lower bedtemperatures during the drying process as well as at the end ofthe process are in the case of a packed bed.

74 M. Stakic et al. / International Journal of Heat and Mass Transfer 59 (2013) 66–74

2. The comparisons of the corresponding parameters for the dry-ing process obtained by the calculations on the basis of a devel-oped model, and by the experimental investigations (for appliedparameter range), have shown a good agreement. Therefore thismodel can be used for calculating the drying processes for thecase of fine-grained materials (dS = 0.1–1.0 mm) in a fluid bedup to 0.1 m high, with data accuracy greater than 5%. Yet, theequilibrium gas and solids moisture content at the solid surfacemust fulfill the condition of hygroscopic equilibrium for thespecific moisture on the specific product. The empirical dryingcoefficient, ki, must also be specified for the considered product,as well as the equivalent particle diameter and void fraction ofthe bed.

3. The results obtained can be summarized as following:– Drying time becomes shorter with the increase of the inlet

drying agent temperature (Figs. 5 and 7). The longest oneis for air having an inlet temperature of app. 50 �C. Thepoppy seeds temperature at the end of the drying increaseswith an increase of the inlet drying agent temperature.

– The influence of the bed height at the drying rate is clearlynoticeable (Fig. 6). It is obvious that higher drying agentvelocity induces lower drying rates. In certain cases the dry-ing process is slower mostly because of the low drying agentflow-rate, i.e., because of the insufficient drying agent capac-ity (saturation of the drying agent).

4. Finally, it is very important to point out that utilizing theparameters and correlations proposed for the packed bed, theconvective drying process in a fluid bed can be calculated withdeveloped computer program. The basic significance of thedescribed mathematical model and the computer program isits universality. It is feasible, by changing a very few of the datain the input files, to calculate the convective drying process in afluid and/or in a packed bed.

Acknowledgments

The authors would like to acknowledge the Ministry of Scienceand Technological Development of the Republic of Serbia for its in-put along the course of preparing this publication.

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