Journal of Food Engineering 116 (2013) 109–117

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Journal of Food Engineering

journal homepage: www.elsevier.com/locate/jfoodeng

Coupled 3D heat and mass transfer model for numerical analysis of drying process
in papaya slices
Roberto A. Lemus-Mondaca a,⇑, Carlos E. Zambra b,c, Antonio Vega-Gálvez a, Nelson O. Moraga d
a
Departamento de Ingeniería en Alimentos, Universidad de La Serena, Av. Raúl Bitrán s/n, La Serena, Chile
b
Universidad Arturo Prat, Av. Arturo Prat 2120, Iquique, Chile
c
Centro de Investigación Avanzada en Recursos Hídricos y Sistemas Acuosos (CIDERH) CONICYT-Regional GORE, Tarapacá R90I1001, Iquique, Chile
d
Departamento de Ingeniería Mecánica, Universidad de La Serena, Av. Benavente 980, La Serena, Chile

a r t i c l e i n f o a b s t r a c t

Article history: An experimental and numerical study for the drying process of a solid food, Chilean papaya slices, was
Received 16 April 2012 carried out in a range of air temperatures from 40 to 80 °C. The unsteady temperature and moisture dis-
Received in revised form 17 October 2012 tributions results inside the sample were predicted by using an unsteady tri-dimensional coupled heat
Accepted 25 October 2012
conduction and mass diffusion mathematical model. The validation procedure includes a comparison
Available online 16 November 2012
with experimental and numerical temperature and moisture content results obtained from experimental
data. The samples thermophysical properties as density, specific heat, and thermal conductivity are
Keywords:
assumed to vary non-linearly with temperature. The convective heat and mass transfer coefficients were
Drying simulation
Heat conduction
found by the analytical model. The water effective diffusion coefficient, the drying curves and the center
Mass diffusion temperature were measured by physical experiments. It was found from the experimental results that
Convective coefficients slices of papaya present an isotropic behavior with an uncertainty between 6.0% and 9.0%. According
Physical properties to statistical test results (RE%), the finite volume method based calculations gave a very good fit quality.
Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction which are involved in the design of food storage and refrigeration
and drying equipment (Wang and Sun, 2003). The estimation of
The Chilean papaya (Carica pubescens), unlike tropical papaya process times for refrigerating, freezing, heating or drying of foods
(Carica papaya L.) cultivated in Brazil (var. Formosa), Colombia also requires information based on the thermal properties. Due to
(var. Maradol), and Caribbean countries (var. Hawaiana), grows the multitude of food items available, it is nearly impossible to
in colder climates (Moya-León et al., 2004). Further, it is smaller experimentally determine and tabulate the thermal properties of
and firm with a yellow pulp and its sensory attractiveness is the foods for all possible conditions and compositions. Because the
strong and characteristic aroma. It has an edible yield of 46%, a su- thermal properties of foods are strongly dependent upon chemical
gar content of 5%, and a high content of papain (Moya-León et al., composition and temperature, the most viable option is to predict
2004; Moreno et al., 2004). The papaya is an important fruit in the thermophysical properties of foods using mathematical models
Chile and attracts great technological interest, because it is widely which account for the effects of chemical composition and temper-
used in the development of different food products such as candied ature (Choi and Okos, 1986; Dincer, 1997). Thermophysical proper-
papaya, canned papaya, juice, syrup and marmalade. The papaya ties of foods which are often required for heat transfer calculations
fruit is a rich source of carbohydrates, vitamins C and A, b-carotene include ice fraction, density, specific heat and thermal conductiv-
and dietary fiber. In addition, it is a climacteric fruit and exhibits a ity. This paper provides a summary of prediction methods for esti-
characteristic rise in ethylene production during ripening, along mating these thermophysical properties. In addition, the
with softening, changes in color and a prominent development of performance of the various thermophysical property models is
aroma (Moyano et al., 2002). The dehydrated papaya has a high evaluated by comparing their calculated results with experimen-
commercial value and is commonly demanded by the gastronomy tally determined thermophysical property data available from
sector, because it is used as ingredient in different dishes mixed the literature.
with yogurt, ice cream, cocktails, salads and sweet courses. During the state of the art review it was found that the drying
Knowledge of the thermophysical properties of foods is re- kinetics process of bioproduct and food, has been studied consider-
quired to perform the various heat and mass transfer calculations ing the mass transfer by using the diffusion equation or Fick’s
Second Law in one-dimensional (thin-layer equation) (Goyal
⇑ Corresponding author. et al., 2006; Corzo et al., 2008; Lemus-Mondaca et al., 2009a). Fick’s
E-mail address: rlemus@userena.cl (R.A. Lemus-Mondaca). Second Law allows obtaining the drying parameters such as the

0260-8774/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jfoodeng.2012.10.050
110 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117

Nomenclature

A solid surface area (m2) f fluid (air)

C moisture content (kg/m3) lv liquid–vapor transformation
T temperature (°C, K) max maximum
Cp specific heat (J/kg K) min minimum
D diffusion coefficient (m2/s) num numerical
g gravitational acceleration (m/s2) exp experimental
hlv vaporization latent heat (J/kg) o initial condition
k thermal conductivity (W/m K) ref reference state
h heat transfer coefficient (W/m2 K) s solid (food)
hm mass transfer coefficient (m/s) surf solid surface
n data number eq equilibrium
m_ rate mass (kg/s) i,j,k node position
r magnitude of a normal vector to the surface n-1, n iteration number
R gas constant (kJ/mol K)
t time (s) Greek symbol
V solid volume (m3) a thermal diffusivity (m2/s)
x,y,z coordinates (m) q density (kg/m3)
X mass fraction of each component / current variable (temperature and water content)

Subscripts
i components number

water effective diffusion coefficient, the activation energy and the developed a computer program for simulating the drying process
drying curve. Vega-Gálvez et al. (2009) studied and modeled the of rough rice in a deep bed batch dryer. The model consisted of four
drying kinetics of the blueberry at three temperatures (60, 70 non-linear partial differential equations as a result of the heat and
and 80 °C). Diffusional and empirical models were applied in the mass balances, together with an appropriate solution procedure
modeling of the drying kinetics of this fruit. The mass diffusion using the finite difference method. Validation of the computer sim-
coefficient, evaluated by an Arrhenius-type equation, and kinetic ulation was found in a good agreement with the measured values
parameters of each model showed dependence on temperature. along the depth of the dryer bed during the drying process.
The models obtained the well-fit quality for each drying curve, In the available literature there are important alternatives to
based on the statistical test. Doymaz (2009) dried spinach samples treat heat and mass convective transfer coefficients, especially
using a pilot-scale cabinet-type convective dryer at 50, 60, 70 and when they vary in space (Vitrac and Trystram, 2005; Haldera and
80 °C. The effective moisture diffusivity varied from 0.65– Datta, 2012). In addition, some approaches consider the use of
1.90  109 m2/s over the temperature range studied, with activa- the conjugate method that allows the solution of the velocity field
tion energy of 34.35 kJ/mol for spinach leaves. The researchers together with the temperature/concentration field (Nam and Song,
showed that the logarithmic model was found to be the best one 2007; De Bonis and Ruocco, 2008; Lamnatou et al., 2010; Defraeye
for explaining the drying characteristics of spinach leaves. et al., 2012). In this kind of model the exposed food surface is not
Some studies like Hussain and Dincer (2003a,b,c), Oztop and discontinuity and computed fields are quite realistic
Akpinar (2008) and Nilnont et al. (2011) have reported computa- (Lemus-Mondaca et al., 2011; Moraga et al., 2012).
tional simulations of drying fruits and vegetables considering the The aim of this study is to study and simulate the drying process
unsteady bidimensional thermal conduction equations and mass of a parallelepiped solid food using a coupled 3D unsteady heat
diffusion coupled. Janjai et al. (2008) developed a finite element conduction and moisture diffusion model with a computer code
model in two-dimensions to simulate moisture diffusion in mango (finite volume method), which can adequately predict the temper-
fruit during drying process. This theoretical study on drying mango ature and moisture content distributions. Experimental data is
only was based on pure mass transfer, neglecting the heat transfer used to validate the mathematical model and the simulation
and its effect on drying. The numerical results satisfactorily pre- procedure.
dicted the moisture diffusion coefficient during the drying process.
The authors predicted numerical moisture content profiles which
gave accurate descriptions of the movement of moisture inside 2. Physical layout
the fruit. In addition, this model could be used to provide informa-
tion on the dynamics of moisture movement without the need of 2.1. Temperature and moisture measurements
experimental measurements. Aversa et al. (2007) presented a the-
oretical model describing the transport phenomena involved in The drying experiment, performed in triplicate, was only carried
food drying by the use of finite elements method (commercial out at a drying air inlet temperature of 60 °C, employing a constant
FEMLAB package). The authors simulated the transient 2D heat air flow of 1.5 ± 0.2 m s1 (perpendicular direction to sample),
and moisture transfer accounting for the variation of both air and measured with an omnidirectional anemometer (451112, Extech
food physical properties as functions of local values of temperature Instrument Inc.). In addition, outlet relative humidity was
and moisture content. The results showed that air velocity and its 57.0 ± 3.8% measured by an ambient digital hygrothermometer
relative humidity were more effective than air temperature to (Extech Instrument Inc.). For each drying experience, a load density
determine the drying rate. A comparison between the theoretical of 10.5 ± 0.4 kg/m2 was used. The experimental design only evalu-
predictions and experimental results had a remarkable agreement, ated the effect of air drying temperature on the mass transfer
with relative error that never exceeded 5%. Zare et al. (2006) kinetics. Then, this design is based on a unifactorial design nk,
 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117 111

Athlon processor and 2 Gb RAM) were used to register the temper-

Food: T(x,y,z) ature with the MAC-14 v. 1.0 software incorporated into the sys-
C(x,y,z) tem data acquisition system (Vernon Hill, Illinois, USA).

2.2. Physical properties and convective coefficients

Composition data for papaya samples are readily available by

c a Lemus-Mondaca et al. (2009b). This data consists of the mass frac-
tions of the major components found in food items. Such compo-
nents include water, protein, fat, carbohydrate, fiber and ash.
b Papaya samples thermophysical properties can be predicted by
using this composition data in conjunction with temperature
Air: Ta , h, hlv dependent mathematical models of the thermal properties of the
Ca , hm individual components. Choi and Okos (1986) have developed
mathematical models for predicting the thermophysical properties
Fig. 1. Schematic view of the geometry of the food and drying air flow. of food components as functions of temperature in the range of
40 to 150 °C (Eq. (1)).

where n is the number of levels and k is the number of factors, 1 X

n Xn
X =q
where temperature is the one factor under study (k = 1) with only q¼X Cp ¼ Cpi X i k¼ ki n i i ð1Þ
n X
i¼1 i¼1
one level (n = 1), i.e. 60 °C. Fig. 1 presents the schematic illustration X i =qi X i =qi
i¼1 i¼1
for unsteady 3D coupled heat and mass transfer during convective
drying process of a papaya parallelepiped sample. The dimension The convective mass transfer coefficient (hm) in the surface of the
of each sample was a = 30 mm, b = 20 mm and c = 10 mm. Several sample was obtained according to the procedure described by Kaya
authors have used experimental data to validate the mathematical et al. (2007). The convective mass transfer coefficient as a function
models through analytical and numerical methods (Hussain and of the drying curve (MR = (C  Ceq)/(Co  Ceq)), is shown in Eq. (2):
Dincer, 2003a,b; Kaya et al., 2007; Villa-Corrales et al., 2010). The  
drying process was carried out in a convective tray dryer designed V C  C eq
hm ¼ ln : ð2Þ
and built by the Food Engineering Department of Universidad de La At C o  C eq
Serena. The dryer has a control unit to set the velocity and temper-
The convective heat transfer coefficient (h) in the surface of the slice
ature of air, which is heated through electrical resistances (Fig. 2).
was obtained carrying out an energy balance in the boundary. The
Moisture content (in triplicate) was determinated using of pa-
procedure used resembles those reported by Karim and Hawlader
paya slabs samples (100.0 ± 1.2 g, i.e. 25 samples, approximately)
(2005) and Rahman and Kumar (2007). The convective heat transfer
placed as a thin layer in a stainless steel basket, which hangs on
coefficient was calculated as a function of the measured tempera-
a balance (SP402 Scout-Pro, Ohaus, Shanghai, China) with an accu-
ture, as following:
racy of ±0.01 g, communicated with an interface system (SP232
Scout-Pro, Ohaus) connected to a personal computer, which re- Cp m DT  hAlv dC
dt
cords and stores the weight changes in real time by means of the h¼ : ð3Þ
T surf  T f
Microsoft Hyperterminal software (Redmond, WA). The weight of
samples was recorded at 1 min intervals until reaching a constant A desorption isotherm was determined at 45 °C in order to estimate
weight (equilibrium condition). Temperature data was measured the equilibrium moisture content (Ceq). The standard gravimetric
with T-type thermocouples (0.3 mm in diameter) located in the in- method recommended by the European Cooperative Project COST
ner-center of the papaya slabs (5 samples by triplicate). A Cole 90 (Spiess and Wolf, 1983) was used. Experimental data were fitted
Parmer Scanner (model LA-AI-48-RTG-SC-220, Vernon Hill, Illinois, using the Guggenheim, Anderson and de Boer (GAB) equation,
USA) and a personal computer (Dual Core 2X 64, 2.4 GHz AMD frequently used in food studies because it is considered to have

Fig. 2. Schematic diagram of drying equipment.

112 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117

parameters which prove to be useful to the physicochemical descrip- 4. Numerical procedure

tion, such as the monolayer moisture content (Cm), Co and Ko (Eq. (4)).
C m C o K o aw The mathematical model based on a non-linear partial differen-
C eq ¼ ð4Þ tial equation is solved by finite volume method (FVM) (Versteeg
ð1  K o aw Þ ð1 þ ðC o  1ÞK o aw Þ
and Malalasekera, 1995). The development of the mathematical
model and numerical simulation is based on a non-commercial
3. Mathematical model computational program implemented in FORTRAN language. The
FVM is used in an implicit formulation procedure to obtain the dis-
The mathematical model used to predict the drying process of cretized system of algebraic equations (Patankar, 1980). Previ-
papaya samples is based on Fourier’s Law and Fick’s Second Law ously, several calculations changing the mesh and time step,
to calculate the unsteady three-dimensional temperature and from 403, 603 to 803 nodes, from 0.001 to 0.1 s, were performed.
moisture fields inside the food in rectangular coordinates (x,y,z) In each case the variations in the isotherms were evaluated in
with temperature-dependent thermophysical food properties. terms of precision and computational time (Moraga and Barraza,
The model is also based on certain assumptions which are convec- 2003). Due to the previous analysis, it was found that the three-
tive drying with constant air drying temperature, homogeneous dimensional staggered for the spatial discretization (Table 1), a
initial fields of temperature and moisture content inside the sam- uniform grid of 603 nodes in the three directions x, y and z
ple, negligible shrinkage and deformation of food during drying (Fig. 3). The time discretization was based on a constant time step
and thermal radiation around the food, as well as absence of heat of 0.01 s (Table 1). In addition, Table 2 shows a comparison be-
generation inside the food. Several researches have used different tween experimental and numerical results for temperature and
infinite geometry assumptions to simplify the heat and mass trans- moisture ratio, in which the simulations were accomplished by
fer problems without any restriction (Califano and Zaritzky, 1997; using three meshes and three time steps, in order to select a suit-
Zhao et al., 1998; Aversa et al., 2007; Bakalis et al., 2009). However, able time step and the appropriate number of nodes needed to ob-
the weight loss by water evaporation from the surface of the food tain accurate solutions in reasonable CPU times. The nodes were
was introduced in the heat diffusion equation (vaporization latent located in specific spatial locations in order to compare the results
heat, hlv). Under these assumptions, the mathematical model for given by FVM with the experimental data measured. The con ver-
the unsteady tri-dimensional drying process is (Wu et al., 2004): gence criterion applied to stop the temperature calculations at
Heat transfer: each time step, with a 99% convergence of nodes, was based in a
      maximum value for the difference in the calculated value at two
@T @ @T @ @T @ @T
qðTÞCpðTÞ ¼ kðTÞ þ kðTÞ þ kðTÞ ð5Þ successive iterations (Eq. (13)). Computations were accomplished
@t @x @x @y @y @z @z in an IntelÒ CoreÒ 2 Duo T5750/2.0 GHz personal computer with
The initial temperature is uniform and equal to: 3.0 GB of RAM.

t ¼ 0 Tðx; y; z; 0Þ ¼ T o ð6Þ jT ni;j;k  T n1

i;j;k j  10
4
jC ni;j;k  C n1
i;j;k j  10
4
ð13Þ
The temperature-dependent boundary conditions at the surface The numerical strategy used to solve the coupling between en-
and in the center-line take into account the thermal convection to ergy and moisture was the following. First, a reference temperature
the surrounding air: of 10 °C was used to calculate the physical properties of food. Then,
@T the diffusion coefficient was calculated for the food with the given
t>0: kðTÞ ¼ h ðT s  T f Þ þ hlv  hm ðC s  C f Þ ð7Þ temperature, and the 3D mass diffusion equation (Eq. (8)) was
@r
solved. Next, the value of water content on the food surfaces was
Mass transfer: calculated in order to apply the boundary conditions at the food
     
@C @ @C @ @C @ @C surface (Eq. (10) and Eq. (7)). Finally, the transient diffusion heat
¼ DðTÞ þ DðTÞ þ DðTÞ ð8Þ conduction was solved and a first approach for the temperature
@t @x @x @y @y @z @z
distribution was obtained (Eq. (5)). With these temperature values
With the following initial and boundary conditions: the physical properties were updated. The calculation procedure
t¼0: Cðx; y; z; t ¼ 0Þ ¼ C o ð9Þ was implicit. Iterations for water content (C) and temperature (T)
were done until a maximum error between previous and updated
The moisture content-dependent boundary conditions at the / values were less or equal than 104 for each control volume and
surface and in the center-line take into account the mass convec- at each time step.
tion to the surrounding air: The fit quality was evaluated by means of the Relative Error (Eq.
@C (14)), which compares the numerical values calculated by the finite
t>0: DðTÞ ¼ hm ðC s  C f Þ ð10Þ volume method and the experimental data.
@r
The latent heat of vaporization equation (Eq. (11)) had been re- ðT; CÞexp  ðT; CÞnum
RE ð%Þ ¼  100 ð14Þ
ported by Youcef-Ali et al. (2001). The evaporation at the food sur- ðT; CÞmax  ðT; CÞmin
face, included in the last term of Eq. (5), allows the coupling
between the heat and mass transfer unsteady diffusion equations:
hlv ðTÞ ¼ 4186:8ð597  056  ðT þ 273:15ÞÞ ð11Þ
The moisture effective diffusivity (Eq. (12)) in papaya samples Table 1
had been reported by Lemus-Mondaca et al. (2009a) by an Arrhe- Influence of time step and nodes number on the computational time (h) employed.
nius-type equation with dependence of the drying temperature
Time step (s) Nodes number
from 40 to 80 °C. This property included in the Eq. (8) also contrib-
utes to the coupling between the heat and mass transfer unsteady 403 603 803

diffusion equations: 0.001 18 20 45

0.01 14 15 30
ð3159
DðTÞ ¼ 1:41  104  e RT
Þ
ð12Þ 0.1 12 13 19
 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117 113

5. Results and discussion

5.1. Thermal properties and convective coefficients

X
Z
0.03
The proximal analysis of fresh papaya (Lemus-Mondaca et al.,
0.01 2009b) showed a high moisture content of 10.20 ± 0.41 g/100 g
0.02 sample with a water activity of 0.996 ± 0.002. The protein content
0.005 showed a value of 5.35 ± 0.78 g/100 g sample, lipids showed a re-
0.01 sult of 1.20 ± 0.21 g/100 g sample, ashes revealed a content of
7.02 ± 1.08 g/100 g sample, fiber content was of 19.05 ± 1.22 g/
0 0
100 g sample and nitrogen-free extract content had a value of
0 0.005 0.01 0.015 0.02 Y 64.38 ± 3.30 g/100 g sample. From these values, the empirical
equations were obtained for each thermophysical property, density
Fig. 3. Discretized domain and coordinate system. (Eq. (15)), specific heat (Eq. (16)) and thermal conductivity (Eq.
(17)), of Chilean papaya, depending on the drying temperature
for a range from 10 to 90 °C.
Density as a function of temperature:
qðTÞ j10  C90  C ¼ 1:03  103  7:90  103  T  3:69  103  T 2
Table 2
Comparison of experimental temperature and moisture ratio versus numerical results ð15Þ
used to select suitable time step and the appropriate number of nodes (3000 s).
Specific heat as a function of temperature:
Temperature (°C) Moisture ratio (dimensionless)
CpðTÞj10  C90  C ¼ 3:93  103 þ 8:66  102  T þ 4:53
Nodes number 403 603 803 Exp. 403 603 803 Exp.
Dt = 0.01 s 56 55 55 53 0.77 0.78 0.78 0.79  103  T 2 ð16Þ
Time step 0.1 0.01 0.001 Exp. 0.1 0.01 0.001 Exp.
603 57 55 55 53 0.75 0.78 0.78 0.79 Thermal conductivity as a function of temperature:
kðTÞj10  C90  C ¼ 5:52  101 þ 1:73  103  T  6:48
 106  T 2 ð17Þ

(a) 4.00 The values of hm and h, obtained from Eq. (2), were in the range
80°C from 3.10  107 to 6.05  106 m/s and from 0.25 to 4.50 W/m2 K,
3.50 70°C respectively, for a drying temperature between 40 and 80 °C
(Fig. 4). These results were found in the range of those available
3.00 60°C
in the existing literature for different fruits and drying conditions
50°C (Verboven et al., 1997; Hussain and Dincer, 2003c; Kaya et al.,
2.50
h (W/m 2 K)

40°C 2006; Oztop and Kavak, 2008; Nilnont et al., 2011). The precise
2.00 knowledge of the heat and mass transfer coefficient has been found
to be the main error source in drying time prediction methods.
1.50 However, the small deviations of the heat and mass convective
coefficient were found, by the use of a finite volume method, to re-
1.00 sult in small deviations in the core temperature mass average of
food drying using air. The time variations of mass and heat convec-
0.50 tive coefficients as well as the procedure used to determine them
are two issues that affect the accuracy of the numerical prediction
0.00
0 10000 20000 30000 40000 of drying process. The mass and heat convective coefficients were
Time (s) allowed to change with time in the polynomial form hm(t) = A + B
t + C t2 and h(t) = A + B t + C t2 + D t3 + E t4, where the values of
(b) 7.0E-06 the coefficients are defined as:
80°C Mass transfer coefficient as a function of process time for each
6.0E-06 drying temperature:
70°C
60°C hm ðtÞ j40 C ¼ 3:52  107 þ 7:59  1012  t ð18Þ
5.0E-06
50°C
hm ðtÞ j50 C ¼ 5:41  107  7:36  1012  t  1:92
4.0E-06 40°C
hm (m/s)

 1014  t 2 ð19Þ
3.0E-06
hm ðtÞ j60 C ¼ 5:25  107 þ 1:59  1010  t  3:52
2.0E-06  1014  t 2 ð20Þ

1.0E-06 hm ðtÞ j70 C ¼ 7:61  107 þ 2:68  1010  t  2:88

 1014  t 2 ð21Þ
0.0E+00
0 10000 20000 30000 40000
Time (s) hm ðtÞ j80 C ¼ 1:12  106 þ 3:70  1010  t  6:68  1014  t 2
ð22Þ
Fig. 4. Time evolution of the heat and mass transfer coefficients on the food surface.
114 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117

Heat transfer coefficient as a function of process time for each and Kavak, 2008). The unsteady values for heat and mass convec-
drying temperature: tive coefficients have been included in the boundary conditions
(Eqs. (7) and (10)) in terms of the drying temperature, for the inter-
h ðtÞ j40 C ¼ 6:38  101 þ 1:08  105  t  2:23  108  t 2 val between 40 and 80 °C. Even though the heat and mass convec-
þ 2:66  1012  t3 ð23Þ tive coefficients change also in space, our previous findings have
shown that the heat and mass transfer coefficients remain almost
constant in space for almost a 90% of each surface, with major
h ðtÞ j50 C ¼ 1:01  100 þ 2:27  104  t  1:36  107  t2
changes near the surface boundaries (Moraga and Medina, 2000;
þ 2:08  1011  t 3 ð24Þ Moraga and Barraza, 2003; Moraga and Lemus-Mondaca, 2011).

h ðtÞ j60 C ¼ 4:23  101 þ 2:21  103  t  1:19  106  t 2

5.2. Mathematical model validation
þ 2:25  1010  t3  2:57  1014  t 4 ð25Þ
In order to validate the mathematical model and the numerical
h ðtÞ j70 C ¼ 6:05  101 þ 3:32  103  t  1:47  106  t 2 solution procedure, the equilibrium moisture content (Ceq) to be
reached in the experimental drying process by the food must be
þ 2:48  1010  t3  1:84  1014  t 4 ð26Þ
known, because this moisture content depends on the relative
humidity based on the dry bulb temperature. The equilibrium
h ðtÞ j80 C ¼ 2:34  101 þ 6:36  103  t  3:07  106  t 2 moisture content was calculated by means of the desorption iso-
þ 5:84  1010  t3  4:99  1014  t 4 ð27Þ therm at 45 °C calculated by the GAB equation. The values for
the parameters used in the calculations for Cm, Co and Ko were of
The use of unsteady heat and mass coefficients is an improve- 0.11 g water/g d.b., 68.0 and 0.95, respectively. The equilibrium
ment with respect to the use of values that that remains constant moisture content value reached in the drying process by the prod-
in time and in space for each surface as in many references uct was 0.10 ± 0.01 g water/g d.m., with respect to the equilibrium
(Califano and Zaritzky, 1997; Hussain and Dincer, 2003a,b,c; Oztop relative humidity (7.5 ± 0.5%), based on the drying temperature
surrounding the food.

(a) 70
(a) 90 80°C
60 70°C
80 60°C
Temperature ( °C)

50°C
40°C
50 70
Temperature (°C)

60
40
50

Experimental
30 40
Numerical 3D
Numerical 2D 30
20
0 2000 4000 6000 8000 10000 20
Time (s) 0 1000 2000 3000 4000 5000
Time (s)
(b) 1.00
Experimental (b) 1.00
Moisture ratio (dimensionless)

Numerical 3D 40°C
0.80 50°C
Moisture ratio (dimensionless)

Numerical 2D
0.80 60°C
70°C
0.60 80°C
0.60

0.40
0.40

0.20 0.20

0.00 0.00
0 5000 10000 15000 20000 0 10000 20000 30000 40000
Time (s) Time (s)

Fig. 5. Experimental data and numerical 2D and 3D values for center-temperature Fig. 6. Temperature and moisture content numerical values in papaya sample
and average-moisture content inside papaya samples at 60 °C. center at selected temperatures.
 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117 115

Time (a) Time (b)

200 s 1800 s

1000 s 3200 s

2000 s
7500 s

5000 s 10500 s

T(K) MR

Fig. 7. Time evolution of (a) temperature and (b) moisture content distribution inside the rectangular sample at 60 °C.

Fig. 5 shows the experimental center temperature (5 samples transfer coefficients could be deviation sources between experi-
by triplicate) and average moisture (25 samples by triplicate) con- mental and numerical profiles results. However, an acceptable
tent profiles of the sample along with the numerical results corre- agreement is found between the numerical results and the exper-
sponding to the drying curve at 60 °C. From the Fig. 5 it can be imental data.
observed that in a short time the slice warms up with a little mois- The results for the time evolution of the temperature and mois-
ture content decrement. Later on, the sample is still warming up ture content in Fig. 5 are consistent with the behavior reported by
and the moisture evaporates at a bigger drying rate. Afterwards, Doymaz (2009), Vega-Gálvez et al. (2009), Hussain and Dincer
the drying rate decreases when the slice temperature is close to (2003a,b,c), Wu et al. (2004), Kaya et al. (2007), Janjai et al.
the air drying temperature. The drying process keeps going more (2008) and Doymaz (2009). Fig. 5 shows a comparison between
slowly due to the temperature gradients and less considerable. Fi- experimental and numerical results for temperature and moisture
nally, the process ends up when the moisture steam pressure in- ratio in the center of the food calculated by using a 2D model, in
side the slice is the same that the one in the drying air. which a RE% of 15.7% for the moisture and 11.2% for the tempera-
Experimental drying conditions, experimental thermophysics ture was established. Results obtained with the general 3D mathe-
properties and the inclusion of experimental heat and mass matical model are seen to be in a better agreement with the
116 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117

Once the values of thermophysical properties and convective

(a) 60
heat and mass transfer coefficients were calculated, the temperature
and moisture content distributions were predicted for the drying
simulation at 60 °C and the results are shown in Fig. 7a and b,
50 respectively. Furthermore, the developed numerical code gives the
three-dimensional fields of temperature and moisture content dis-
Temperature (°C)

tributions inside the sample during the drying process. The moisture
40 diffusion phenomenon from the inside of the tridimensional sample
toward its boundaries while it is heating up from its boundaries to
the inside is determined and represented by Fig. 7b. The dimension-
Center less moisture gradients are higher during the first hour of drying in
30 the range from 0.5 to 2.5  101, this mainly occurs in its sample
Surface boundaries. The drying rate in the boundaries is faster than the dry-
ing rate inside the slice because the moisture gradients between the
20 air drying and the boundaries are higher (Hussain and Dincer,
2003a,b,c; Oztop and Kavak, 2008; Janjai et al., 2008; Aversa et al.,
0 1000 2000 3000 4000 5000
2007; Villa-Corrales et al., 2010; Nilnont et al., 2011).
Time (s) As the process advances in time, temperature gradients decrease
faster than moisture gradients, in such a way that temperature in the
(b) 1.00 food tends asymptotically towards the drying ambient temperature.
Longer times are required for the moisture in the food reaches the
Center same moisture of the drying ambient air (Zare et al., 2006; Oztop
Moisture ratio (dimensionless)

0.80 and Kavak, 2008; Kaya et al., 2007). After the 15,000 s, the size of
Surface the moisture gradients is 1.0  104, the drying rate slows down
and the process ends when those gradients are considered to be
0.60
insignificant. The drying rate in the boundaries is faster than the dry-
ing rate inside the slice due to the moisture gradients between the
0.40 air drying and the boundaries are higher. Also, symmetric distribu-
tions of the moisture content and temperature inside the slice are
observed from the numerical results. This it is attributable to the
0.20 boundary conditions used in the model that are symmetrical as
pointed out by Hussain and Dincer (2003a,b,c), Oztop and Akpinar,
(2008), Janjai et al. (2008), Aversa et al. (2007) and Villa-Corrales
0.00 et al. (2010). Therefore, we can state that the present methodology
0 10000 20000 30000 40000 is capable of estimating the temperature and moisture content dis-
Time (s) tributions of solid objects/products subjected to drying.
Fig. 8. Numerical results for time evolution of (a) temperature and (b) moisture The time variation of temperature profile along the sample for
ratio selected locations (center and surface). different positions (center and surface) is shown in Fig. 8a. The tem-
perature distribution increases all the time from the surface to the
centre of the sample. As the drying time progresses, the temperature
experimental results, with a relative error of 9.5% for the moisture gradient decreases until a steady stage is reached, in which heat gain
and 5.4% for the temperature. In addition, this error gives an indi- due to convective boundary condition balances the heat transfer
cation that the temperature dependent moisture diffusivity varies through conduction (Janjai et al., 2008; Villa-Corrales et al., 2010;
inside the object, which in turn affects the moisture diffusion rate Wu et al., 2004). Fig. 8b shows the moisture ratio at the centre and
in the sample. Consequently, this shows that the numerical meth- at the surface of the sample object subjected to drying.
odology presented above with a developed computer code is capa- The moisture content inside the sample reduces as the time period
ble of simulating 3D coupled heat and mass transfer inside the progresses. In the early heating period the moisture content reduces
papaya sample during drying process. rapidly and as the heating period progresses the rate of reduction of
moisture content becomes less; i.e., it reduces almost steady with pro-
5.3. Drying process simulation gressing heating period. This is more pronounced in the surface local-
ity. The rapid drop of moisture content in the early heating period is
Temperature and moisture content predicted by a 3D unsteady caused by the high moisture gradient in this region, which in turn de-
mathematical model and numerical solution procedure presented, rives considerable diffusion rates from centre of the sample to the sur-
with temperature-dependence thermophysical properties and face. Also, it is clear from the Fig. 8b that during the early heating
time-dependence heat and mass convective coefficients were vali- period, the drying rate is constant, thus exhibiting a constant rate per-
dated using experimental data. Fig. 5a and b depict food initial iod. As the time period progresses, the drying rate continuously de-
temperature was 40 °C and air temperature was 80 °C a numerical creases representing falling rate periods. It is also noted that the
study was done where the drying temperature was the one study moisture content is non-uniform in the sample and varies at each loca-
experimental factor. The drying process simulation through the tion (boundary and center) (Hussain and Dincer, 2003a,b,c; Kaya et al.,
initial and boundary conditions was considered from 40 to 80 °C. 2006; Oztop and Akpinar, 2008; Janjai et al., 2008; Aversa et al., 2007;
Fig. 6a and b shows the simulating center-temperature and aver- Villa-Corrales et al., 2010).
age-moisture content inside papaya samples, respectively, in
which the drying time has been reduced because the drying rate 6. Conclusions
increases due to the drying surface area has increased. The air dry-
ing temperature variation has effect in the inner-center tempera- A 3D unsteady mathematical model for drying process of
ture of the slice too, as it shows in Fig. 6a and b. papaya sample was used and a computer program in FORTRAN
 R.A. Lemus-Mondaca et al. / Journal of Food Engineering 116 (2013) 109–117 117

language was also developed to implement the model. The mois- Lemus-Mondaca, R., Betoret, N., Vega-Galvéz, A., Lara, E., 2009a. Dehydration
characteristics of papaya (Carica pubenscens): determination of equilibrium
ture diffusivity, thermophysics properties and heat and mass
moisture content and diffusion coefficient. Journal of Food Process Engineering
transfer coefficients of the papaya used in the simulation of drying 32, 645–663.
were determined experimentally. The relative error of the simu- Lemus-Mondaca, R., Miranda, M., Andres, A., Briones, V., Villalobos, R., Vega-Gálvez,
lated values on experimental data was less than 9.5% for tempera- A., 2009b. Effect of osmotic pretreatment on hot air drying kinetics and quality
of Chilean papaya (Carica pubescens). Drying Technology 27, 1105–1115.
ture and 5.4% for moisture content using the 3D mathematical Lemus-Mondaca, R., Vega-Gálvez, A., Moraga, N., 2011. Computational simulation
model. The quality of the predicted results illustrates that the 3D and developments applied to food thermal processing. Food Engineering
model of the coupled heat and liquid moisture transfer in solid Reviews 3 (3–4), 121–135.
Moraga, N., Barraza, H., 2003. Predicting heat conduction during solidification of a
food is satisfactory. In addition, this model provides a better under- food inside a freezer due to natural convection. Journal of Food Engineering 56,
standing of the heat and mass transport phenomenon inside the 17–26.
papaya samples. Moraga, N., Jauriat, L., Lemus-Mondaca, R., 2012. Heat and mass transfer in
conjugated food freezing/air natural convection. International Journal of
Refrigeration 35 (4), 880–889.
Acknowledgments Moraga, N., Lemus-Mondaca, R., 2011. Numerical conjugate air mixed convection/
non-Newtonian liquid solidification for various cavity configurations and
rheological models. International Journal of Heat and Mass Transfer 54, 5116–
The authors acknowledge the financial support of CONICYT– 5125.
CHILE through FONDECYT 1111067 and 11110097 projects. Roberto Moraga, N., Medina, E., 2000. Conjugate forced convection and heat conduction with
A. Lemus-Mondaca acknowledges the financial support given by freezing of water content in a plate shaped food. International Journal of Heat
and Mass Transfer 43, 53–67.
the National Doctoral Fellowship of the Advanced Human Capital Pro-
Moreno, J., Bugueño, G., Velasco, V., Petzold, G., Tabilo-Munizaga, G., 2004. Osmotic
gram CONICYT. dehydration and vacuum impregnation on physicochemical properties of
Chilean papaya (Carica candamarcensis). Journal of Food Science 69 (3), 102–
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