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A generalized conjugate model for forced convection drying basedon an evaporative kinetics

Maria Valeria De Bonis, Gianpaolo Ruocco *

DITEC, Universit degli studi della Basilicata, Campus Macchia Romana, Potenza 85100, Italy

a r t i c l e i n f o

Article history:Received 6 August 2007Received in revised form 28 February 2008Accepted 4 May 2008Available online 15 May 2008

Keywords:Forced convectionFood dehydrationConjugate modelTemperature and moisture evolutionLocal heat and mass transfer

a b s t r a c t

A model describing the heat and mass transfer involved in food drying is presented. The aim is to deter-mine the effect of air temperature on the performance of the drying process applied to fresh-cut vegeta-ble slices, but other effects can be easily incorporated in the model. The model allows to disregard one of the most limiting parameters in such modeling, i.e. the average heat and mass transfer coefcients at thefood/drying substrate interface, which are generally taken from the literature. Such assumptions are lim-iting in the sense that they are referred to average transfer conditions and general geometries. The pre-sented model relies upon a nite-element solution of time-dependent differential equations forsimultaneous and conjugate heat and moisture transfer in a two-dimensional domain, without any infer-ence in such empiricism.

A special formulation for drying kinetic of the substrate is also exploited, and a treatment of thedependence of the properties upon the residual moisture content is included. After proper validation withthe available experimental measurements, the numerical solution is discussed by presenting eachinvolved eld variables, emphasizing on the conjugate nature of the drying process. Due to its exibilityand generality, the model can be used in common industrial driers optimization, even in the assumptionof a laminar ow eld.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

Drying, or dehydration, is one of the most common methods of preserving food, and involves a complex combination of transportphenomena such as the application of heat and the removal of moisture from a food substrate ( Barbosa-Canovas and Vega-Merca-do, 1996; Fellows, 2000 ). Drying systems optimization is stillsought nowadays and therefore full understanding of these phe-nomena can help to improve process parameters and hence prod-

uct quality, emphasizing on the external and internal processparameters that inuence drying behavior. The former includetemperature, velocity and relative humidity of the drying medium(air), while the latter include density, permeability, sorptiondesorption characteristics and physical substrate properties.

Starting from the seminal works by Luikov and Whitaker a vastnumber of contributions has been reported in the last decades onporous and multi-phase media drying by air convection ( Chenand Pei, 1989; Barbosa-Canovas and Vega-Mercado, 1996 ). But inthe past few years the multi-dimensional (distributed, transient)analysis has gained importance, specially for lumped moist prod-ucts, as a considerable computing power became more available,therefore many such studies could be conducted and nalized.

Shapes and detailed congurations were explored through a vari-ety of approach, though always appealing to empirical, average(i.e. independent on surface locations) relationships for interfacetransfer calculations.

These limitations affected many of the available works on dry-ing modeling in the last decade, as briey recalled in the following.

Wang and Chen (1999) presented a thorough diffusive model of heat and mass transfer in moist media, yet limited to a one-dimen-sional geometry. Chen et al. (1999) developed a nite element

model for coupled heat and mass transfer, to implement the ther-mal processing of chicken patties in a small convection oven withcooking condition and empirical, nonlinear thermal properties.Dincer and his co-workers have presented a great deal of workson the subject in the past and recently ( Kaya et al., 2006 ), wherethe simultaneous heat and mass transfer have been studied forthe spatial variations of the heat and mass transfer coefcientsalong the treated surface. Pasta drying was studied by Miglioriet al. (2005) on an axisymmetric geometry, followed by De Temm-erman et al. (2008) who added a radiation driving force. A nite-element approach was employed by Aversa et al. (2007) in orderto optimize the drying process, by accounting for local temperaturevariation of both air and food physical properties.

As drying is eminently a conjugate phenomenon (which means,the transfer of mass and heat is solved simultaneously in both solidand uid phases, and are strongly coupled through evaporation

0260-8774/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jfoodeng.2008.05.008

*

Corresponding author. Tel.: +393293606237.E-mail address: [email protected] (G. Ruocco).

Journal of Food Engineering 89 (2008) 232240

Contents lists available at ScienceDirect

Journal of Food Engineering

j ou rna l homepage : www.e l sev i e r. com/ loca t e / j foodeng

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and properties variation on moisture and temperature), an innova-tive, further approach is to solve a model in which the mass andenergy interface uxes vary seamlessly in space and time as thesolution of the eld variables. With this objective Oliveira and Hag-highi (1997) obtained the temperature and the moisture contoursfor the drying of wood, but their work was affected by the limita-tion given by considering a laminar boundary layer ow over thesubstrate. This approach was later complemented by Murugesanet al. (2001) for a timber block using a full Navier-Stokes formula-tion for the ow eld, allowing for the buoyancy term. While over-coming the limitations of the boundary layer assumptions, theirwork was focussed on Nusselt and Sherwood numbers on the ex-posed substrate surface. For the rst time a full conjugate modelof a drying food was presented by De Bonis and Ruocco (2007) ,yet for a specic exchange conguration (a thin baking productto be dried by an impinging turbulent jet draft), where the focuswas on residual local water activity.

A similar approach is carried out in the present work fora dryingvegetable substrate, by employing a nite-element approach.Residual water and temperature elds are computed locally withinthe substrate, when this interacts with a forced, laminar air ow.The later assumption allows to focus on the basic aspects of owtransport, focussing upon the vapor and liquid water production/depletion and transport, which is dealt with by an ad-hoc rst-or-der irreversible kinetics. Such kinetics is included to solve for tran-sient, two-dimensional ow, temperature and moisture elds.Realistic transfer exchanges are inherently considered that varywith process time and surface location, eliminating the need forempirical heatand mass transfer (averaged) coefcients evaluation.

2. Problem formulation

Convection and moisture removal by a bulk, hot air draft is as-sumed to a model substrate, in this case carrot slices, as reported inFig. 1. During processing, heat is transferred mainly by convectionfrom air to the products exposed surface, and by conduction fromthe surface toward the substrate interior. Meanwhile, moisture dif-fuses outward to the surface, where is vaporized. But if the sub-strate is water saturated, liquid can be converted into vapor even

within the substrate, depending on the heat perturbation front.The water transport mechanisms generally include the motion of liquid water (1) by diffusion caused by differences in the concen-tration of solutes at the surface and in the interior (Fickian diffu-

sion) and (2) by capillary forces, while the motion of water vaporis (3) by diffusion in air spaces within the substrate caused by va-por pressure gradients ( Fellows, 2000 ).

2.1. Assumptions

The following assumptions are considered in this work:

(1) the ow is laminar; the dryer is two-dimensional, and asmall portion in the vicinity of the product is studied only,for sake of simplicity;

(2) due to the adopted ow regime, no body force is accountedfor;

(3) the thermophysical food properties are moisture-dependent,as reported by Ruiz-Lpez et al. (2004) , and reported in Table1, while the air and water properties are temperature-dependent and are taken from Perry et al. (1997) : for sakeof simplicity their dependency are not reported in theformulation;

(4) the effect of capillary forces is included in liquid waterdiffusivity;

(5) the diffusivity of vapor in the substrate is the same than thediffusivity of liquid water, as implied for example by Braudet al. (2001) .

The following simplifying assumptions are adopted:

(1) the viscous heat dissipation in the drying medium and theheat generation within the moist substrate are neglected;

Fig. 1. Geometry and nomenclature.

Nomenclature

a temperature factor in Eq. (8) (dimensionless)c concentration (mol/m 3)

c p constant pressure specic heat (J/kg K)D diffusivity (m 2/s)D hvap latent heat of evaporation (kJ/kg)E a activation energy (kJ/mol)k thermal conductivity (W/mK)K rate of production of water vapor mass (1/s)K 0 reference constant (1/s) in Eq. (8)K 1 temperature factor in Eq. (8) (dimensionless)H height (m)L; L

0

; L00 lengths (m)

M molecular weight (g/mol)l dynamic viscosity (Pa s)x air absolute humidity (kg water vapor/kg dry air) p pressure (Pa)_q cooling rate due to evaporation (W/m 3)

R universal gas constant (kJ/mol K)

q air density (kg/m 3)t time (s)

T temperature (K)u ; v horizontal and vertical component of air velocity (m/s)u air velocity vector (m/s)U moisture content, wet basis (kg liquid water/kg sub-

strate) x,y horizontal and vertical coordinate (m) X moisture content, dry basis (kg liquid water/kg dry sub-

strate)

Subscripts0 initiala airl liquid waters substrate, bulkv water vapor

M.V. De Bonis, G. Ruocco / Journal of Food Engineering 89 (2008) 232240 233

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(2) due to the nature of the interacting species, no diffusionuxes are accounted for in the energy equation;

(3) neither shrinkage nor deformation of drying substrate areaccounted for.

2.2. Governing equations

With reference to the previous assumptions, the governing con-servation equations in vector form are enforced to yield for concen-tration of vapor and liquid water, pressure, velocity andtemperature ( Bird et al., 2002 ) in two distinct air and substratesub-domains:

In the substrate: Continuity, liquid water o c lo t

r Dlsr c l Kc l 1

Continuity, water vapor

o c vo t r Dvsr c v Kc v 2

Energy

q sc pso T o t r ksr T

_q 3

In the drying air: Continuity, water vapor o c vo t

r Dva r c v u r c v 4

Momentum

q ao uo t

u r u r p l r 2 u 5Energy

q ac pao T o t r kar T q ac pa u r T 6

2.3. Evaporation cooling and vapor production rates

The cooling rate due to evaporation _q can be computed as_q D hvap M lKc l 7

The concept of vapor rate of production Kc is adopted in this pa-per: a negative source term Kc l (K being the rate of production of water vapor mass per unit volume) is included in Eq. (1), to ac-count for the depletion of liquid water; symmetrically, a positivesource term Kc v is included in Eq. (2) , to account for the productionof water vapor. The motivation of the kinetic-like approach forevaporation will be now briey addressed. The rst notion usedto describe a drying process incorporating a single constant K forthe combined effect of the various existing transport phenomenawas suggested by W.K. Lewis in 1921, as recently recalled by

Babalis et al. (2006) while reviewing the leading kinetic formula-tions. Since then, many variations of the basic equations have beenreported in the open literature, based on purely empirical models,directly relating moisture ratio with drying time, and incorporating

various parameters that describe both the inherent water phaseconversion and interface conditions.

In this paper a modied exponential model of evaporation hasbeen adopted, based on an Arrhenius rst-order irreversible kinet-ics formulation. Several works have been presenting such an ap-

proach, as Panagiotou et al. (1999), Azzouz et al. (2002) andRoberts and Tong (2003) . It is seen here that the inherent (volu-metric) evaporation physics ( Roberts and Tong, 2003 ) must be joined to interface conditions ( Panagiotou et al., 1999; Azzouz etal., 2002 ), such that the thermal, uid dynamic and concentrationregimes could be all be represented in the mass source term.

The present work is focussed upon the additional dependenceon process temperature variation, so that the basic Arrhenius-typerelationship can be modied as follows:

K K 0e E a =RT K a1 8

where

K 0 is a reference constant, to be found empirically by matching a

parametric numerical analysis with the available experimental/numerical data for each conguration, meaning that for a givenconguration (air velocity/humidity, and geometry) K 0 is heldconstant in the present model;

the activation energy E a is taken as 48.7 kJ/mol ( Roberts andTong, 2003 );

T is the local substrate temperature; K 1 is the ratio of the process temperature to the referencetemperature;

a is a dimensionless temperature factor varying with each dif-ferent process temperature T a .

It is emphasized here that present approach, that couples theheat and mass transfer through the use of the K and _q source terms,simplies the analysis with respect to the classical Luikovs ap-proach, employed by Oliveira and Haghighi (1997) and Murugesanet al. (2001) .

2.4. Initial conditions

For the substrate:

initially in thermal equilibrium ( T T 0) with the quiescent ambi-ent air the moisture content is such that

c l0 1000U 0 q s

M l9

It is also c v0 0; for the drying air:

with reference to Fig. 1, no-slip ( u 0) is enforced for the dryingair at every solid surface; air ows, with a fully-developed (par-abolic) horizontal component u a , through the left inlet at givenprocess temperature T a and absolute humidity x a (as usual, re-lated to the relative humidity) such that

c v0 1000 x a q aT ax a 1M l

10

It is also c l0 0.

2.5. Boundary conditions

Full continuity is assumed for vapor mass and temperature

through the substrates surface, to solve for concentrations andtemperature seamlessly across the interface. With reference toFig. 1, the mass, momentum and thermal boundary conditions(where applicable) are as follows:

Table 1

Food properties functions, dependent on the moisture content X (Ruiz-Lpez et al.,2004 )

Property Function

q s (kg/m3) 440 :001 90 X

c ps (J/kgK) 1750 2345 X 1 X ks (W/mK) 0 :49 0 :443exp 0:206 X

Dls , Dvs (m 2/s) 2 :8527 1010 exp 0 :2283369 X

234 M.V. De Bonis, G. Ruocco / Journal of Food Engineering 89 (2008) 232240

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Process inlet x 0; 0 < y < H a

c v c v0 ; u ua ; v 0; T T a 11

Bottom plate, air interface 0 < x < L0s and L0s L

00s < x < L p;

y 0

o c v;lo y

0; u v 0; oT

o y 0 12

Bottom plate, substrate interface L0s < x < L0s L

00s ; y 0

o c lo y

0; oT

o y 0 13

Upper open surface 0 < x < La ; y H p o c vo y

0; ou

o y o vo y

0; oT

o y 0 14

Process outlet x La ; 0 < y < H a o c vo x

0; ou

o x 0; v 0;

oT o x

0 15

Finally, continuity is ensured by enforcing the following positions:

Across the horizontal sub-domains interface L0s < x < L0s L

00s ;

y H s

c va c vs ; oc l

o y 0; u v 0; T a T s 16

Across the upwind x L0s ; 0 < y < H s and downwind x L00s ; 0 < y < H s (vertical) sub-domains interfaces

c va c vs ; oc l

o x 0; u v 0; T a T s 17

2.6. Numerical method and additional considerations

A nite-element commercial solver has been employed to inte-grate the partial differential equations system ( COMSOL Multi-physics Users Guide, 2007 ). A preliminary grid independencytest has been carried out with 3 different grids of approximately2000, 4000 and 6000 triangular elements, respectively, and thesecond grid was selected as the local heat ux across the interfacevary less than 2% in all locations with respect to the one computedwith the third grid. The mesh was distorted locally ( Fig. 2) to allow

for concentration, velocity, temperature and pressure gradientsresolution in the boundary layer and within the substrates ex-posed surface, induced by the heating and evaporation. Executiontime for t = 18000 s elapsed time has been approximately 20 minon a Pentium Xeon PC (WindowsXP Pro OS, 3.0 GHz, 2 GB RAM).

Specic underrelaxation factors have been employed to solve theNavier-Stokes equations in the start-up phase of drying.

3. Results and discussion

3.1. Model validation

The available literature data are rather limited in order to vali-date the model and its numerical treatment, as geometry and owregimes were always left unspecied and transfer coefcients wereassumed from empirical correlations, except in Murugesan et al.(2001) (who dealt with a non-food substrate). However, the exper-imental average residual moisture reported by Ruiz-Lpez et al.(2004) has been rst compared with the present numerical solu-tion and reported in Fig. 3. A 4 h baking process of a thin carrotslice with Ls 0:06 m and H s 0:0050 or 0.0075 m (for Data Set1 and 2, respectively) was congured with the following drivingparameters: T a 343 or 323 K (for Data Set 1 and 2, respectively),T 0 = 298 K, U 0 0:87, and inlet air relative humidity of 45%. Carewas exercised to adapt the present model so that the inlet airvelocity ua was 2.0 m/s, but still in the laminar regime, withLa 0:20 m and H a 0:10 m. The reference constant K 0 for the gi-ven conguration was 7 10 3 , while the temperature factor a wasfound to be 0 and -10 for Data Set 1 and 2, respectively.

For Data Set 1 there is a good agreement at the beginning of treatment, while a maximum difference of approximately 15% isdetected after 2 h. At the end of drying the measured and com-puted moisture are again very similar. In Data Set 2 the drying con-

dition are milder therefore the kinetic parameters (in absence of athickness adjustment) underestimate the measurements, the max-imum difference being less than 10% after 3 h.

A second such benchmark has been found in the numerical datafrom Aversa et al. (2007) and reported in Fig. 4. A similar process(baking of a carrot substrate) was congured, with Ls 0:06 mand H s 0:015, and with the following driving parameters:T a 353, 343 or 333 K (for Data Set 3 to 5, respectively),T 0 = 303 K, U 0 0:64, and inlet air relative humidity of 75%. Carewas exercised, as well, to adapt the present model so that the inlet

Fig. 2. Close-up of distorted grid at the substrate (dimensions in m).


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air velocity ua was 0.3 m/s, in the laminar regime, with La 0:20 mand H a 0:10 m. The reference constant K 0 for the given congu-

ration was 90, while the temperature factor a was found to be 0,17 and 10 for Data Set 3 to 5, respectively.

The same limitations with the earlier benchmark were found, asno information was available on employed conguration. In all

cases a good agreement is detected between the two differentmodels. Small discrepancies (less than 5%) are found after 1 hr of

treatment only, due to the condensation phenomenon reportedin the benchmark work, which remains unjustied for empiricaltransfer coefcients such as the ones reportedly employed in Aver-sa et al. (2007) .

Fig. 3. Average U evolution during process: comparison with Ruiz-Lpez et al. (2004) measurements for 2 different Data Sets.

Fig. 4. Average U evolution during process: comparison with Aversa et al. (2007) computations for 3 different Data Sets.


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3.2. Flow and temperature eld

The simulation results for Data Sets 3 conguration ( Aversaet al., 2007 ) are then briey presented in the form of velocity, tem-perature or moisture distributions. Fig. 5 shows rst the vector and

scalar distributions of velocity in the drying air. Due to the oweld contraction and speed-up, the action of the drying air is stron-gest on top of the substrate, while the front and back faces are subject to stagnation and recirculation ow regions, respectively. This justies the adoption of a fully conjugate model for a detaileddescription, as transfer properties vary considerably with exposedsurface location.

Depending on the ow eld, the temperature distribution in Fig.6 presents a related non-homogeneous behavior, due to the non-uniform heat transfer, which will then reect upon the residualmoisture distribution. On the three exposed substrate sides, dueto the conjugate nature of the model, the isotherms are obviouslyinclined. The substrate is found to be more than 3 K warmer on theleading edge, with respect to the trailing edge, and its left side is

being heated more effectively (as expected) than the right one.The lowest temperature of about 340 K is detected on substratebottom, by the adiabatic oor, with the slowest heating point beinglocated slightly in the ow direction.

In addition to the available Data Set 3, the present model hasbeen exercised by varying the nominal value of velocity. Fig. 7shows the new ow eld generated with ua = 0.3 m/s, 10 timeshigher. The velocity distribution is very similar to the previousone, but the velocity local values are much higher indeed. These

in turn reect on the higher thermal regime, reported in Fig. 8,where the product center temperature increase by 4 K with respectto Data Set 3 comparison. A more dynamic ow situation dictatesan overall more even side-to-side treatment, therefore the slowestheating point is almost perfectly centered this time.

3.3. Moisture and vapor removal

Based on the above ow eld and temperature maps, it is ex-pected that (1) the evaporation occurs non-homogeneously withinthe substrate, and (2) the vapor mass transfer across the uid-sub-strate interface also occurs non-uniformly. Consequently, themoisture will be non-homogeneously removed within thesubstrate.

Residual moisture distribution after the treatment is reported inFig. 9 for Data Set 3. The evaporation and depletion of water ismore effective where the temperature is the highest ( Fig. 6) atthe leading edge (a triangular chunk, one-fth of the entire prod-

Fig. 5. Close-up of ow eld (vector eld and streamlines) in the vicinity of substrate for Data Sets 3 to 5 uid dynamic conguration, after a 5 h drying. ju j values range from0 to 0.21 m/s.

Fig. 6. Close-up of temperature eld (isotherms) in the substrate and its vicinity for Data Sets 3 conguration, after a 5 h drying. T values range from 341 to 352 K.


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uct), but the trailing edge is dried more than the average too, dueto the favorable momentum transport in its vicinity.

Fig. 10 describes the effect of ten-fold velocity increment onresidual moisture. The drying process is stronger, so the humidity

Fig. 7. Close-up of ow eld (vector eld and streamlines) in the vicinity of substrate for a higher air velocity, after a 5 h drying. ju j values range from 0 to 2.7 m/s.

Fig. 8. Close-up of temperature eld (isotherms) in the substrate and its vicinity for a higher air velocity, after a 5 h drying. T values range from 345 to 352 K.

Fig. 9. Close-up of residual moisture concentration eld (isolines) in the substrate for Data Set 3 conguration, after a 5 h drying. c l values range approximately from 3.19 to3.33 10 4 mol/m 3.


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distribution decrease accordingly when compared with the DataSets 3. The side-to-side treatment is slightly more homogeneous,but a larger concentration is detected y-wise in turn.

Finally, Fig. 11 shows the removal of water vapor from the sub-strate in the uid phase. The much higher vapor concentration inthe substrate is not reported in the Figure, for sake of clarity. Thevapor is non-uniformly blown away by the air ow. It is interestingto note that such irregular eld, together with the associated mois-ture, that was presented by one of the benchmark adopted here(Aversa et al., 2007 ), cannot be predicted under the assumptionof average heat and mass transfer coefcients as claimed in theirwork. On a general basis, an uneven distribution can be computedby adopting local transfer values, as in Kaya et al. (2006) , but againthis work does not consist in a conjugate model, as the transfercoefcient are empirically implied.

4. Conclusions

In this work a generalized conjugate model of forced convectiondrying has been proposed. A modied exponential model for dry-ing kinetics has been adopted, based on an Arrhenius rst-orderirreversible formulation.

The basic Arrhenius-type relationship has been modied withan appropriate temperature factor, to deal with the process tem-perature variations, and validated against the available experimen-tal or numerical literature data. Such an approach is independenton empirical heat and mass transfer coefcients.

The proposed model can be complemented by additional air

velocity and humidity variation factors, also taking into accountof different multi-physics effects such as microwave or ultrasoundexposure, and can be readily extended to allow for full three-dimensional geometries.

Acknowledgement

This work was funded by MIUR Italian Ministry of Scientic Re-search, Grant No. 2006093719002 entitled Transport phenomenaof heat and mass from plates and modied surfaces by air imping-ing jets.

References

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Fig. 10. Close-up of residual moisture concentration eld (isolines) in the substrate for a higher air velocity, after a 5 h drying. c l values range approximately from 3.10 to3.23 10 4 mol/m 3 .

Fig. 11. Close-up of vapor excess eld (contours) in the vicinity of substrate for Data Set 3 conguration, after a 5 h drying. Representation in 10 levels of gray, for a c v rangefrom approximately 6.05 to 6.20 mol/m 3 .


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