Numerical Simulation of Vacuum Drying by Luikov’s Equations F. Nadi, 1 G. H. Rahimi, 2 R. Younsi, 3 T. Tavakoli, 1 and Z. Hamidi-Esfahani 4 1 Agriculture Machinery Mechanics Department, Tarbiat Modares University, Tehran, Iran 2 Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran 3 Applied Sciences Department, University of Quebec at Chicoutimi, Chicoutimi, Quebec, Canada 4 Food Science and Technology Department, Tarbiat Modares University, Tehran, Iran A two-dimensional mathematical model was developed to simulate coupled heat and mass transfer in apple under vacuum dry- ing. Luikov’s equations are the governing equations in analyzing heat and mass diffusion problems for capillary-porous bodies. The model considers temperature- and moisture-dependent material properties. The aim of this study is to analyze the influence of some of the most important operating variables, in particular, pressure and temperature of drying air, on the drying of apple. The resulting system of unsteady-state partial differential equations has been solved by a commercial finite element method (FEM) package called FEMLAB (COMSOL AB, Stockholm, Sweden). Simula- tions, carried out in different drying conditions, showed that tem- perature is more effective than air pressure in determining the drying rate. A parametric study was also carried out to determine the effects of heat and mass transfer coefficients on temperature and moisture content distributions inside apple during vacuum dry- ing. A comparison between the theoretical predictions and a set of experimental results reported in the literature showed very good agreement, especially during the first 4,200 s, when experimental data and theoretical predictions overlapped and relative errors never exceeded 2%. Keywords 2D; Finite element; Luikov’s equations; Mathemat- ical modeling; Simultaneous heat and mass transfer INTRODUCTION The drying of fruit and vegetables is a subject of great importance. Dried fruit and vegetables have gained com- mercial importance, and their growth on a commercial scale has become an important sector of the agricultural industry. Lack of proper processing causes considerable damage and wastage of seasonal fruits in many countries, which is estimated to be 30–40% in developing countries. It is necessary to remove the moisture content of fruits to a certain level after harvest to prevent the growth of mold and bacterial action. [1] Apple is an important raw material for many food products, and apple plantations are cultivated in many countries all over the world. Thus, it is very important to define the conditions under which the characteristics of fresh apples can be preserved and to define optimal para- meters for their storage and reuse. Apples are the fourth most important tree fruit crop worldwide after citrus, grapes, and banana. The world production of apples is more than 71 million tons. China is the world’s greatest apple producer (29,851,163tons), followed by the United States in second place (4,358,710tons); Poland, Iran, Turkey, and Italy each produce more than 2 million tons annually. [2] High temperatures and long drying times required to remove the water from the fruit material in convection air drying may cause serious damage with regard to flavor, color, and nutrients and can reduce the bulk density and rehydration capacity of the dried product. [3] The main advantages of vacuum drying come from shorter drying times and lower operating temperatures. The lower drying temperatures result in less thermal darkening of the pro- duct, and the absence of oxygen prevents chemical staining. Drying of fruits and vegetables is a complicated process involving simultaneous heat and mass transfer, particularly under transient conditions. Understanding the heat and mass transfer in the product will help to improve drying process parameters and, hence, quality. Because of the complexity of the vacuum drying process, it takes a long time and considerable cost to optimize drying schedules directly in industrial kilns or in the lab- oratory. One way to achieve optimal control of products drying under vacuum is mathematical modeling based on a physical description of heat and mass transfer during drying. Many studies have been published on this subject. Moyne and Martin [4] presented a study on coupled heat and mass transfer during vacuum contact drying of wood based on Luikov’s approach. [5] Fohr et al. [6] estab- lished a model for continuous vacuum drying with hot plates, where the surfaces were kept at a constant tem- perature above the boiling point. Suvarankuta et al., [7] Correspondence: Dr. Goham-Hassien Rahimi, Mechanical Engineering Department, Tarbiat Modares University, P.O. Box 14155-4838, Gish Bridge, Tehran, Iran; E-mail: rahimi_gh@ modares.ac.ir 197 Drying Technology, 30: 197–206, 2012 Copyright # 2012 Taylor & Francis Group, LLC ISSN: 0737-3937 print=1532-2300 online DOI: 10.1080/07373937.2011.595860
Transcript
Numerical Simulation of Vacuum Drying by Luikov’sEquations
F. Nadi,1 G. H. Rahimi,2 R. Younsi,3 T. Tavakoli,1 and Z. Hamidi-Esfahani41Agriculture Machinery Mechanics Department, Tarbiat Modares University, Tehran, Iran2Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran3Applied Sciences Department, University of Quebec at Chicoutimi, Chicoutimi, Quebec, Canada4Food Science and Technology Department, Tarbiat Modares University, Tehran, Iran
A two-dimensional mathematical model was developed tosimulate coupled heat and mass transfer in apple under vacuum dry-ing. Luikov’s equations are the governing equations in analyzingheat and mass diffusion problems for capillary-porous bodies. Themodel considers temperature- and moisture-dependent materialproperties. The aim of this study is to analyze the influence of someof the most important operating variables, in particular, pressureand temperature of drying air, on the drying of apple. The resultingsystem of unsteady-state partial differential equations has beensolved by a commercial finite element method (FEM) packagecalled FEMLAB (COMSOL AB, Stockholm, Sweden). Simula-tions, carried out in different drying conditions, showed that tem-perature is more effective than air pressure in determining thedrying rate. A parametric study was also carried out to determinethe effects of heat and mass transfer coefficients on temperatureand moisture content distributions inside apple during vacuum dry-ing. A comparison between the theoretical predictions and a set ofexperimental results reported in the literature showed very goodagreement, especially during the first 4,200 s, when experimentaldata and theoretical predictions overlapped and relative errors neverexceeded 2%.
Keywords 2D; Finite element; Luikov’s equations; Mathemat-ical modeling; Simultaneous heat and mass transfer
INTRODUCTION
The drying of fruit and vegetables is a subject of greatimportance. Dried fruit and vegetables have gained com-mercial importance, and their growth on a commercialscale has become an important sector of the agriculturalindustry. Lack of proper processing causes considerabledamage and wastage of seasonal fruits in many countries,which is estimated to be 30–40% in developing countries.It is necessary to remove the moisture content of fruits toa certain level after harvest to prevent the growth of moldand bacterial action.[1]
Apple is an important raw material for many foodproducts, and apple plantations are cultivated in manycountries all over the world. Thus, it is very important todefine the conditions under which the characteristics offresh apples can be preserved and to define optimal para-meters for their storage and reuse. Apples are the fourthmost important tree fruit crop worldwide after citrus,grapes, and banana. The world production of apples is morethan 71 million tons. China is the world’s greatest appleproducer (29,851,163 tons), followed by the United Statesin second place (4,358,710 tons); Poland, Iran, Turkey, andItaly each produce more than 2 million tons annually.[2]
High temperatures and long drying times required toremove the water from the fruit material in convectionair drying may cause serious damage with regard to flavor,color, and nutrients and can reduce the bulk density andrehydration capacity of the dried product.[3] The mainadvantages of vacuum drying come from shorter dryingtimes and lower operating temperatures. The lower dryingtemperatures result in less thermal darkening of the pro-duct, and the absence of oxygen prevents chemical staining.
Drying of fruits and vegetables is a complicated processinvolving simultaneous heat and mass transfer, particularlyunder transient conditions. Understanding the heat andmass transfer in the product will help to improve dryingprocess parameters and, hence, quality.
Because of the complexity of the vacuum drying process,it takes a long time and considerable cost to optimizedrying schedules directly in industrial kilns or in the lab-oratory. One way to achieve optimal control of productsdrying under vacuum is mathematical modeling based ona physical description of heat and mass transfer duringdrying. Many studies have been published on this subject.Moyne and Martin[4] presented a study on coupled heatand mass transfer during vacuum contact drying ofwood based on Luikov’s approach.[5] Fohr et al.[6] estab-lished a model for continuous vacuum drying with hotplates, where the surfaces were kept at a constant tem-perature above the boiling point. Suvarankuta et al.,[7]
Kittiworrawatt and Devahastin,[8] and Swasdisevi et al.[9]
developed a liquid diffusion–based model to simulate thetransport of heat and mass during low-pressure drying.Ressing et al.[10] developed a two-dimensional finiteelement (2D FE) model in order to simulate the puffingof a dough ball, being dehydrated under vacuum, on thebasis of Fourier’s equations of heat conduction. Li et al.[11]
developed a one-dimensional mathematical model todescribe the process of wood microwave vacuum drying,based on the mechanism of moisture and heat transfer.Erriguible et al.[12] proposed an approach that includesthe solution of conservation equations for each mediumand linking the domains by fixing the appropriate bound-ary conditions. Koumoutsakos et al.[13] developed a 1Dmathematical model to describe the transport phenomenaduring continuous radio frequency–vacuum (RF=V) dryingof thick lumber, based on conservation of heat, mass,and momentum equations. Jaya and Das[14] developed amodel with regard to moisture diffusivity, and it wasfound that the model could provide close prediction ofthe moisture content of the pulp at different drying timesfor pure mango pulp and pulp components. Audebertet al.[15] proposed a simplified analysis of heat and masstransfer. This analysis is based on the existence of anevaporation front, determining two zones in the longi-tudinal direction.
One of the main difficulties in vacuum drying simulationis determination of boundary conditions. As reported bySebastian and Turner,[16] definition of boundary conditionsof the vacuum drying process remains the main difficultythat prevents any accurate analysis and advancement ofthis technology. Fohr et al.[6] assumed that there is noboundary layer. The boundary condition used by the latterauthors reflects the equilibrium state between the vapor atthe surface of the wood and in the chamber. This was alsothe case for the models used by Jomaa and Baixeras,[17]
Guilmain et al.,[18] Sebastian and Turner,[16] and Sebastianet al.[19] Defo et al.[20,21] estimated the convective masstransfer coefficient from the average drying rate deter-mined during the experiment. Even for conventional dry-ing, the use of boundary layer theory to calculate theheat and mass transfer coefficients is questioned.
Most previous studies on vacuum drying conducted onagriculture products have neglected heat transfer effectsduring drying and assumed the movement of moistureonly as diffusion. Kittiworrawatt and Devahastin,[8]
Suvarnakuta et al.,[7] and Swasdisevi et al.[9] simulated heatand mass transfer in an uncoupled manner. To the best ofour knowledge, no prior study has used coupled heat andmass transfer equations to model vacuum drying.
In this article, we used Luikov’s equation for simulationof heat and mass transfer simultaneously. We used convec-tive transfer coefficients calculated by the correlations offorced convection for different vacuum conditions. The
vacuum drying process was analyzed by Luikov’s equation,and then a numerical model was solved by means of acommercial package, FEMLAB,[22] to simulate heat andmass transfer in a 2D square apple. Also, the influence ofsome process parameters, especially pressure and tem-perature of drying air, on apple drying processes wasstudied. To validate the predicted temperature andmoisture distributions inside the object, we compared theprediction results with experimental data available in theliterature.
MATHEMATICAL AND NUMERICAL MODELING
The process of vacuum drying of apple involves simul-taneous heat and mass transfer. In this study we presentthe finite element formulation of a two-dimensional,unsteady-state model during vacuum drying. The transientLuikov’s formulation was used for mathematical formu-lation of the problem. All boundaries are in contact withthe surrounding hot air. In the model, these boundariesare represented using convection boundary conditions forthe temperature and moisture content.
Model Assumptions
Some assumptions were made to develop the two-dimensional, unsteady heat and mass equations, some ofwhich are as follows[23–25]:
� Thermophysical properties of apple are dependenton temperature and moisture content.
� All of the thermophysical properties of air areassumed constant except hm and hq.
� Apple was assumed to be an isotropic andhomogeneous material.
� Two-dimensional variation of temperature andmoisture is considered in square object (i.e., xand y).
� The temperature and moisture content are initiallyuniform inside the apple.
� The shrinkage is neglected and no degradation ofthe solid occurs.
� There is no heat generation inside the apple.� Evaporation only takes place at the surface of theapple slab.
� Air distributes throughout the dryer uniformly.� The air assumed ideal gas.
Governing Equations
Many foods can be considered as solids consisting ofseveral pores and capillaries in which water is bound andvapor can move around. Heat and mass transfer in porousmedia subjected to convective boundary conditions can bemodeled by means of Luikov’s coupled system of partialdifferential equations. Using the listed assumptions, the
198 NADI ET AL.
equations describing two-dimensional heat and mass trans-fer in porous materials (apple) are given as follows[26,27]:
� Heat transfer equation:
qcq@T
@t¼ r � ðkq þ ekkmdÞrT þ ekkmrU
� �ð1Þ
� Mass transfer equation:
qcm@U
@t¼ r � kmdrT þ kmrU½ � ð2Þ
where T is the temperature; U is the moisture potential; t istime; q is the dry-body density; cq and cm are the heat andmoisture capacities, respectively; kq and km are the thermaland moisture conductivity coefficients, respectively; e is theratio of the vapor diffusion coefficient to the coefficient oftotal moisture diffusion; k is the latent heat of water; and dis the thermal gradient coefficient.
Luikov used the principles of irreversible thermody-namics to derive his equations and introduced the conceptof moisture potential to describe the equivalence betweenheat and potential U not due to a gradient in moisture con-tent C.[5] When several bodies are in thermodynamic equi-librium with each other, the moisture potentials of thesebodies are equal to one another, though their specificmoisture contents are not necessarily equal. A comparisoncan be made with heat transfer in the sense that the drivingforce is not a gradient in enthalpy but a temperature gradi-ent. Bodies of the same temperature usually have a differ-ent enthalpy, which depends on the heat capacity andphase. By convention, moisture transfer from one bodyto another takes place from the body with higher potentialto the body with lower potential.
By analogy with the specific heat capacity cq, the con-cept of isothermal specific moisture capacity was developedto connect the moisture content of a body with the moist-ure transfer potential[5]:
cm ¼ @C
@U
� �ð3Þ
where C is the moisture content. For a constant specificmoisture capacity, C¼ cmU, the specific moisture capacityis obtained by finding the equilibrium moisture contentof a body in contact with moist air, which is characterizedby its relative humidity and, hence, its moisture transferpotential. Because the model parameters in the literatureavailable on the validity of Luikov’s coupled heat and masstransfer equations are expressed in terms of moisturepotential, in this article the same unit for moisture poten-tial will be used so the numerical parameter values donot need to be converted.[5]
Moisture content in the heat and mass transfer equa-tions is nondimensionalized using the following equation:
M:R ¼ C � Cd
Co � Cdð4Þ
Boundary Conditions
In drying surfaces, it is assumed that both heat andmoisture are lost through convection. Thus, the boundaryconditions associated with this system of equations werewritten in a generalized form as:
kq@T
@nþ hqðT �TaÞþ ð1� eÞkh�mðU �UaÞ ¼ 0 at the surface
ð5Þ
km@U
@nþ kmd
@T
@nþ h�mðU �UaÞ ¼ 0 at the surface ð6Þ
where n is the normal to the surface; Ta and Ua are thetemperature and moisture potential of the drying, respect-ively; and hq and h�m are convective heat and mass transfercoefficients, respectively.
It assumed that air follows the ideal gas law; if the airpressure decreases, air density will decrease according tothe ideal gas law:
qa ¼paRTa
ð7Þ
where qa is the air density, R is the air constant, and Pa isthe air pressure.
A decrease in the air density influences the heat andmass transfer coefficients, which in turn influence theboundary conditions.
Heat and mass transfer coefficients for different vacuumconditions were calculated on the basis of the well-knownsemi-empirical correlations expressing the dependence ofNusselt number upon Reynolds and Prandtl numbers andof Sherwood number on Reynolds and Schmidt numbers,respectively. The Chilton-Colburn analogy holds[28]:
Sh ¼ 0:664 Re0:5Sc0:33 ð8Þ
Nu ¼ 0:664 Re0:5Pr0:33 ð9Þ
with
Nu ¼ hqL
kqaSh ¼ h �
mL
DRe ¼ LqV
la
Pr ¼ cqalakqa
Sc ¼ laDqa
ð10Þ
NUMERICAL SIMULATION OF VACUUM DRYING 199
The mass transfer coefficient hm used in the literature isrelated to h�m by the equation[29]
h�m ¼ hmqacm
where Nu, Re, Pr, Sh, and Sc are the Nusselt, Reynolds,Prandtl, Sherwood, and Schmidt numbers, respectively. Lis the length of the sample, V is the air velocity, and D isthe diffusion coefficient of vapor in air. kqa, la, and cqaare the thermal conductivity, viscosity, and specific heatof the air, respectively. Because the effects of temperatureand pressure are insignificant on kqa, la, cqa, those thermalproperties of the air were assumed constant.
The diffusion coefficient, D,[15] was calculated by thefollowing expression:
D ¼ 2:17� 10�5 101; 325
Pa
� �Ta
273:16
� �1:88
ð11Þ
A useful and simple analogy relating heat and mass trans-fer simultaneously is the Chilton-Colburn model, which iswritten as:
hm ¼ hqPr23
qaCqaSc23
¼ hq
qaCqaLe23
ð12Þ
where Le is the Lewis number, defined as:
Le ¼ aD
ð13Þ
where a is thermal diffusivity and is equal to
a ¼ kqaqacqa
ð14Þ
Initial Conditions
The sample is initially assumed to be at uniformtemperature and moisture content. Therefore, the initialconditions can be expressed as:
Tðx; y; tÞ ¼ T0ðx; yÞ at t ¼ 0 ð15Þ
Uðx; y; tÞ ¼ U0ðx; yÞ at t ¼ 0 ð16Þ
Equations (1) and (2) are nonlinear. Therefore, theequations were solved numerically together with relevantboundary conditions.
Model Implementation
The coupled partial differential equations were solvedusing commercial software FEMLAB,[22] which operatesin the MATLAB environment.[30] The packages usednumerical algorithms, based on the finite element method.The direct (UMFPACK) linear system solver was used.The initialization step created the elements’ geometriesand assigned material properties for all elements. Fromthe geometrical viewpoint, apple was assumed to be a2� 2 cm square object. The geometry of the sample wasdrawn using FEMLAB, version 3.1, as shown in Fig. 1.
The thermophysical properties of apple are required tosolve the simultaneous heat and mass transfer equations.The accuracy of the mathematical model depends on theaccuracy of the values of thermophysical constants. Thethermophysical properties of apple were obtained fromthe literature[23,31,32] and are given in Table 1. The coeffi-cients for the boundary condition stated in Eqs. (5) and(6) are given in Table 2.[23,31,32] The load step applied theinternal overpressure of trapped air relative to the chambervacuum. Different mesh densities were tested to obtainmesh-independent solutions. The mesh-independent solu-tions were obtained by using 2,196 triangular elements,generated automatically in FEMLAB.[22]
FIG. 1. Schematic representation of the geometry and mesh.
200 NADI ET AL.
RESULTS AND DISCUSSION
In the present study, Luikov’s equations were solvedand the resultant model was validated. Furthermore, theeffect of drying temperature and pressure as well as theeffect of mass transfer and heat transfer coefficients hmand hq on the drying process were studied. The resultsare shown in Figs. 2 to 6.
Model Validation
To validate the model, the predicted center temperatureand moisture evolutions inside the square object (apple)were compared with experimental data available in theliterature and are shown in Figs. 2a and 2b. Significantgood agreement was found between the numericalresults and the experimental data taken from Chiang andPeterson.[32] Hussain and Dincer[33] presented an analyticalmodel for the experimental results of Chiang and Peterson.The maximum difference between numerical results andexperimental data was less than �1.00% for temperatureevolution and less than �0.82% for moisture evolution,whereas the difference between the analytical results ofHussain and Dincer and measured values was higher thanthe difference between numerical and measured ones.Therefore, it can be stated that the present methodologyis a good tool for estimating temperature and moistureevolutions of solid objects=products subjected to vacuumdrying.
Effects of Drying Temperature and Pressure onVacuum Drying
The effects of drying temperature and pressure on thevacuum drying process are shown in Figs. 3a, 3b and 4a,4b, in which drying curves (moisture ratio vs. time) andtemperature evolution in the sample center under differentdrying conditions are plotted.
Figure 3a shows the moisture evolution inside an appleslab under vacuum drying for various temperatures. At thesame vacuum level, the rate of moisture reduction increaseswith an increase in drying temperature because of the tem-perature gradient between the drying product and the sur-rounding medium. Also, it can be seen from Fig. 3a thatthe rate of initial moisture loss was very high until 4 hand almost two thirds of the drying time was required toremoved the last one third of the moisture. In other words,at the beginning of drying process, the drying ratio is high
FIG. 2. Measured and predicted (a) center moisture content and (b)
temperature for a square object (apple).
NUMERICAL SIMULATION OF VACUUM DRYING 201
and decreases as time progresses. This phenomenon couldbe attributed to the very low moisture diffusivity duringthe later part of the drying process.
Figure 3b shows the temperature evolution within anapple slab. At the beginning of the drying process, the tem-perature reduces suddenly because water immediately eva-porates from the apple due to abrupt pressure drop. Theevaporated water removes heat from the slab in the formof the heat of vaporization; hence, the temperature drops.After this period, the temperature of the sample increasesrapidly until it reaches a constant value around the pre-determinate medium temperature as expected. Then, itremains unchanged at this level until the surface of thesample starts to dry.
Figure 4a illustrates that the vacuum level affectsmoisture evolution of the apple slab but not as stronglyas drying temperatures do. This may be due to the fact that
temperature is the main factor influencing the air thermalproperties within the tested operations. A similar type ofdrying behavior has been reported by Nimmol et al.[34]
for drying of banana slices. For the present vacuum dryingconditions, the effect of drying chamber pressure on thedrying process is significant. This observation is in goodagreement with that reported by Arevalo-Pinedo andMurr[35,36] for carrot and pumpkin, Cui et al.[37] for carrot,Giri and Prasad[38] for mushroom, and Methakhup et al.[39]
for Indian gooseberry, where drying pressure had a certaineffect on the drying process where the drying time wasreduced by decreasing drying pressure.
Figure 4b presents the temperature evolution of anapple slab under vacuum drying at various vacuum levels.The phenomena obtained for various temperature levels issimilar to those obtained for various pressure levels.However, the effects of various vacuum levels are less
FIG. 3. Predicted (a) moisture ratio and (b) temperature at the apple
center for various temperatures; pressure was kept at 40 kPa.
FIG. 4. Predicted (a) moisture ratio and (b) temperature at the apple
center for various pressures; temperature was kept at 60�C.
202 NADI ET AL.
pronounced than the effect of temperature. According tothe reports of Swasdisevi et al.[9] for the effect of vacuumlevel on the temperature evolution of carrot, the differencesbetween the temperature curves are not very significant.Swasdisevi et al.[9] used the diffusion model to simulateheat and mass transfer. The differences between the resultsof this study and those of Swasdisevi et al.[9] could beattributed to the fact that their model was uncoupled andignored capillary flow. The approach to be taken here isto work from the more general theory of heat and masstransport in porous capillary media. Luikov[5] consideredthe distribution of moisture in capillary-porous bodies tobe either in the form of noncondensable airs, vapor, liquid,or possibly solid. The major difference between Luikov’smodel and the diffusion model is that in the former modelthe capillary forces are considered, and a differentiationbetween air, vapor, liquid, and solid is made. Luikovdescribed the two phenomena associated with the transport
of air=vapor and liquids through the porous media asmolecular transport and molar transport.
When the air pressure decreases, the air densitydecreases (Eq. (7)). Decreasing the air density will lead toan increase in thermal diffusivity (Eq. (14)). This in turncause the apple slab to reach air temperature faster.
Effect of Mass Transfer and Heat Transfer Coefficientshm and hq
The proposed model is a tool for calculation of theaverage moisture and temperature to investigate the effectof heat and mass transfer coefficients. The calculationresults are given in Figs. 5. and 6 for the effect of masstransfer coefficient and heat transfer coefficient, respect-ively. Here, the heat and mass transfer coefficients are var-ied separately; that is, only one is varied while the other iskept constant.
FIG. 6. Effect of the heat transfer coefficient on surface and center tem-
peratures and moisture ratio at temperature¼ 80�C and pressure¼ 1 bar
(color figure available online).
FIG. 5. Effect of the mass transfer coefficient on surface and center tem-
peratures and moisture ratio at temperature¼ 80�C and pressure¼ 1 bar
(color figure available online).
NUMERICAL SIMULATION OF VACUUM DRYING 203
The effect of convective mass transfer coefficients on theapple moisture ratio and the temperature of apple is pre-sented in Figs. 5a and 5b, respectively. Once the moistureis transferred to the surface of the apple, it has to beremoved from the surface and transferred to the air by con-vection. The convection mass transfer depends on the mag-nitude of the mass transfer coefficient and the moisturegradient. In order to remove the moisture, its gradientbetween the apple surface and the bulk air has to be largeenough to drive this removal process. If the moisture con-tent of the air is very high and close to saturation, then thegradient will be low, and consequently the moisture cannotbe removed. Similarly, if the magnitude of the mass trans-fer coefficient is low, the moisture removal from the surfacewill be very slow. If this transfer is slow, the drying willtake a longer time, which decreases the productioncapacity. If it is too fast, then the surface moisture contentis removed rapidly. However, moisture from the interior ofapple cannot quickly diffuse to the surface. The differencebetween the surface moisture content and that of the inte-rior parts becomes large, which can cause shrinkage andcase hardening of apple. Therefore, the moisture removalrate from the surface has significant effects on the process.Changing the sample size or the air velocity can influencethe magnitude of the mass transfer coefficient (seeEq. (10)). As can be seen from Fig. 5, the apple tempera-ture is not significantly affected by the magnitude of themass transfer coefficient. It can be seen in Fig. 5a, under thesame conditions of operation, that increasing mass transfercoefficient leads to different moisture contents. For ahigher mass transfer coefficient, heat propagation is fasterwithin the apple. In other words, the higher the masstransfer coefficient is, the higher the drying rate will be.
From Fig. 5a, it was also found that the mass transfercoefficient is a very small number but highly significantto describe the process and very dependent on the timeelapsed during the drying process.
Figure 6 shows the effect of the heat transfer coefficienton surface-average and center temperature profiles for thesquare object. The temperature rises in the early dryingprocess due to convective boundary conditions at the sur-face. This is more pronounced on the surface. A tempera-ture rise on the surface of the slab is due to the internalenergy gain in this region. In this case, convective heatingof the surface results in increased internal energy of theapple slab in the surface region. Moreover, energy gainfrom the convective surface dominates over the conductionlosses from the surface vicinity to the bulk of the appleslab. It should be noted that in the early heating periodthe temperature gradient in the surface region is not con-siderably high to enhance the conduction energy transportfrom this region. As the heating progresses, the tempera-ture gradient in the surface region becomes high, whichin turn accelerates conduction energy transport from the
surface region to the solid bulk of the apple slab. For ahigh heat transfer coefficient, the center and surface tem-peratures more quickly reach the equilibrium temperaturebecause more heat is transferred from the fluid to the applesurface. This results in a temperature gradient between thesurface and interior of this square object. In this case, heatpropagation is faster within the particle. As the heat trans-fer coefficient increases, both surface and center tempera-tures increase significantly, which also increases moistureloss. However, increasing the heat transfer coefficient leadsto a higher rate of moisture loss. Therefore, total moistureloss in the dried apple slab can be less for a higher heattransfer coefficient. At a given time, the temperature inc-reases with the heat transfer coefficient, and it has almostno effect on the moisture characteristics. According tothe reports of Younsi et al.[24] for the effect of heat transfercoefficient on the surface temperature of wood, the differ-ences between the temperature curves are not very signifi-cant, although the heat transfer coefficient has aneightfold increase. The difference between the results of thisstudy and the related literature could be attributed to thedifferent thermal conductivities of apple and wood. Wood’sthermal conductivity is very low and wood is considered avery good insulating material. Therefore, the resistance toheat transfer is considerably larger inside the apple thanthat on the air side. The model takes into account thiseffect.
Comparison of Figs. 3a and 6a shows that the effect oftemperature is much greater than the effect of heat transfercoefficient on drying.
The model developed in this study could be a useful toolfor prediction of the drying times; that is, the end point of adrying process. The optimum moisture content of the finalproduct is important for stability of the product duringstorage. Preservation involves the removal of enough waterto prevent microbiological deterioration or the growth oforganisms that could give rise to food poisoning. The pre-diction of moisture and temperature distribution could alsobe used in the study of chemical reactions that occur duringdrying processes; such reactions generally depend on bothtemperature and moisture content. Therefore, deteriora-tion data as a function of moisture content and tem-perature, as well as knowledge of the food’s moisturetemperature distribution as a function of drying time, areimportant in the prediction of deterioration.
One of the irreversible changes accompanying thedehydration of a food product is nonenzymatic browning,which can lead to a loss of protein biological value. Predict-ing the extent of nonenzymatic browning that occurs infood dehydration should be based upon moisture contentand temperature in the sample. Therefore, it is importantto know the moisture and temperature distribution in thesample during drying when predicting this browningextent.
204 NADI ET AL.
CONCLUSIONS
In this article, a numerical solution is presented for thetemperature and moisture evolutions inside a two-dimensional square object under vacuum drying at thecenter. Validation of the results obtained from the presentanalysis was performed with experimental data available inthe literature. The numerical model shows considerablyhigher agreement with the experimental values than withthe analytical model.
It was found that the temperature rises rapidly in theearly heating period. As the heating period progresses,the temperature rise reaches an almost permanent state.The moisture gradient is higher in the early heating periodand, as heating progresses, the moisture gradient remainsalmost steady. The effects of operating parameters, thatis, drying temperature and pressure, on the drying behaviorof apple slabs were investigated numerically. The resultshowed that the effect of operating pressure was less thanthat of the temperature. Higher air temperature and heattransfer coefficient reduce the drying time. The mass trans-fer coefficient has a pronounced effect on the averagemoisture content field, though it does not cause a signifi-cant change in temperature. It was found that althoughthe mass transfer coefficient hm is a small number, it con-trols the apple’s moisture during drying. Comparing theresults of this study with related literature indicates thatthe devised model can take into account the effect of ther-mal conductivity on the drying process and the effect ofpressure on the body temperature.
NOMENCLATURE
C Moisture contentcm Moisture capacity (kgmoisture=kgdrybody � �M)cq Heat capacity (J=kg �K)D Diffusion coefficient (m2=s)hm Mass transfer coefficienth�m Convective mass transfer coefficient (kg=
m � s � �M)kq Thermal conductivity coefficient (W=m �K)L Length of the sample (m)N Normal to the surfaceNu Nusselt numberP Pressure (kPa)Pr Prandtl numberR Air constantRe Reynolds numberSc Schmidt numberSh Sherwood numbert Time (s)T Temperature (�C)
U Moisture potential (�M)V Velocity (m=s)
Greek Letters
E Vapor diffusion coefficient to coefficient oftotal moisture diffusion
a Thermal diffusivityd Thermo gradient coefficient (�M=K)k Latent heat of water (J=kg)l Viscosityq Density (kg=m3)
Subscripts
a Surrounding mediumd Dry0 Initial
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