Journal of Food Engineering 66 (2005) 455–461
www.elsevier.com/locate/jfoodeng
Modelling the drying of a twig of ‘‘yerba mate’’
considering as a composite material
Part I: shrinkage, apparent density and equilibrium moisture content
Miguel E. Schmalko
a
a,*
, Stella M. Alzamora
b
College of Exact, Chemical and Life Sciences, National University of Misiones, 1552 Felix de Azara St., 3300 Posadas, Misiones, Argentina
b
Departamento de Industrias, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,
Ciudad Universitaria, 1428 Buenos Aires, Argentina
Received 23 July 2003; accepted 19 April 2004
Abstract
Physical properties for bark and xylem of twigs of yerba mate were analyzed to use them in a mathematical model for describing
the drying of composite materials. Shrinkage coefficient and apparent density depended on moisture content and were found to be
different for both regions. Equilibrium moisture contents for different water activity values were measured at temperatures between
30 and 90 C. Desorption moisture isotherms varied with the type of material and with temperature. Between 30 and 60 C, the GAB
model exhibited the better fit to the sorption data, while in the range 70–90 C, experimental data were better described by the
Halsey’s model. Equilibrium moisture content between xylem and bark could be described through a potential function, which did
not depend on temperature.
2004 Elsevier Ltd. All rights reserved.
Keywords: Yerba mate; Xylem; Bark; Physical properties
1. Introduction
When drying of an isotropic material is modelling,
thermal and structural properties, moisture isotherms
and mass and heat transfer coefficients are required as
inputs. Generally, these properties depend on temperature and moisture content. On the other hand, some
materials are not isotropic and contain regions with
different structures, polarities and densities. This situation takes place, for example, in corn, rough rice and
some roots, where the pericarp has properties very
different from the other parts of the vegetable (AbudArchila, Courtois, Bonazzi, & Bimbenet, 2000; Mourad,
Hemati, & Laguerie, 1996; Sokhansanj, Bailey, & Van
Dalfsen, 1999; Tolaba, Suarez, & Viollaz, 1990). When
anisotropic products are drying, other supplementary
difficulties can appear: (1) each region has different
transport and physical properties; (2) the shrinkage for
each material is different, and (3) at the equilibrium, the
*
Corresponding author.
E-mail address: [email protected] (M.E. Schmalko).
0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfoodeng.2004.04.016
chemical potential of each component is uniform
throughout the system and therefore, in the interface,
the water activity (aw ) values but usually not the concentrations are equal.
Yerba mate or Ilex paraguariensis Saint Hilaire is
industrially processed as whole branches. The following operations are involved: blanching, drying, grinding, classification and seasoning (Ramallo, Pokolenko,
Balmaceda, & Schmalko, 2001). Along processing, the
material is heated to high temperatures (approximately
120 C) and looses water. The twigs of yerba mate can
be considered as a composite material with the xylem in
the centre and the bark at the periphery. Due to their
dimensions and the temperatures involved during
industrial processing, mass transfer is expected to control the drying rate. In this case, important properties in
mass transfer phenomena, like shrinkage coefficient,
apparent density, sorption isotherms, bark–xylem equilibrium and moisture diffusion coefficient, must be
known to model dehydration kinetics.
During most of the dehydration procedures, biological materials generally suffer a shrinkage that depends on its internal structure, and a variation in
456
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
Nomenclature
Ai ,Bi
a
aw
C
C0
d
H1
HL
Hm
k
k0
N
Qs
r
Sb
constants of the regression equations
constant of Halsey model
water activity
constant of GAB model
pre-exponential factor of GAB model
twig diameter (m)
heat of sorption of the first moisture layer
(kJ/mol)
heat of condensation of pure water (kJ/mol)
heat of sorption of the moisture multilayer
(kJ/mol)
constant of GAB model
pre-exponential factor of GAB model
number of experimental data
excess heat of sorption (kJ/mol)
constant of Halsey model
shrinkage coefficient
its density takes place. Lozano, Rotstein, and Urbicaın
(1983) have defined the shrinkage coefficient as the
relation between the present and the initial volume.
They found that shrinkage coefficient was mainly
dependent on material structure and moisture content
and developed different models for mathematically
describing this coefficient as a function of moisture
content. A linear dependence between apparent density
and moisture content was reported by these authors in
pears, carrots, potatoes, sweet potatoes and garlic.
Arnosti, Freire, and Sartori (2000), working with Brachiaria prizanlha seeds, found that shrinkage coefficient
depended on an adimensional moisture content (present moisture content/initial moisture content), while
apparent density did not.
Ramallo et al. (2001) reported that shrinkage coefficient and apparent density of leaves of ‘‘yerba mate’’
linearly depended on moisture content but were independent on drying temperature. Sokhansanj et al.
(1999), working with ginseng roots, developed a
shrinkage model that considered two different regions
(two tank shrinkage model). This model fitted better the
experimental data than the model considering homogeneous material.
Sorption isotherms are needed to determine the
equilibrium moisture content in the interface air–solid
during drying. Because of the complexity of solid
matrices, it is not possible to theoretically predict sorption characteristics based on knowledge of composition
and structure and experimental data are required for
each material. A number of mathematical equations
having two or more parameters have been developed to
model isotherm data. Chirife and Iglesias (1978) made a
review of the different models appearing in the literature,
T
x
X
Xm
Xm0
DH 0
q
temperature (K)
moisture content, wet basis (kg water/kg wet
solid)
moisture content, dry basis (kg water/kg dry
solid)
monolayer moisture content of GAB’s model
(kg water/kg dry solid)
pre-exponential factor of GAB model
Arrhenius type energy factor (kJ/mol)
apparent density (kg/m3 )
Subscripts
A
air
B
bark
M
mean value of a property considering bark
and xylem
X
xylem
classifying them in linear and non-linear ones. In recent
years, the GAB (Guggenhein, Anderson and de Boer)
three parameter model, applicable up to water activities
of about 0.90, has became one of the most used (van den
Berg & Bruin, 1981). It offers many advantages: (1) it
has available theoretical background; (2) it describes
sorption behaviour between 0 and 0.9 aw ; (3) its
parameters have a physical meaning and (4) it is able to
describe temperature effects.
The aims of the first part of this research were the
determination of some physical properties (shrinkage
coefficient, apparent density, desorption isotherms and
bark–xylem equilibrium moisture content) for bark and
xylem of twigs of ‘‘yerba mate’’ and their fitting to different models.
2. Materials and methods
2.1. Material
To study shrinkage during drying of branches of
‘‘yerba mate’’, three twig representative diameters were
assayed: 2.5 · 103 m; 5.0 · 103 m and 7.5 · 103 m.
Twigs for each diameter were selected and cut into 5 cm
length cylinders. To study xylem properties, twigs were
hand peeled with a knife.
For determining equilibrium moisture content, bark
was separated from xylem, and the last one was ground
in a laboratory knife miller. Both materials were sprayed
with a 1% w/w potassium sorbate aqueous solution and
exposed to UV light during 30 min to avoid microbial
growth during the experiments.
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
2.2. Moisture content
Moisture content was determined in an oven at
103 ± 2 C until constant mass was reached (approximately 6 h) (IRAM, 1995).
Table 1
Equations obtained for shrinkage coefficient (Sb) and apparent density
(q) for twigs with (M) and without bark (X) as a function of moisture
content (X ), P -value and mean percentile error (MPE)
Equation
2.4. Desorption isotherms and bark–xylem equilibrium
Equilibrium data were obtained by the isopiestic
technique. Bark and xylem were put in close flasks
containing saturated salt solutions which generated
atmospheres with different water activities. They were
maintained in an oven at constant temperature (30, 40,
50, 60, 70, 80 or 90 C) until the equilibrium was reached
(approximately 15 days). Seven different saturated salt
solutions were used (LiCl, MgCl2 , CoCl2 , NaBr,
P -value
MPE
SbX ¼
0:706 þ 0:038XX
0:706 þ 0:038XX0
0.009
6.7
SbM ¼
0:613 þ 0:192XM
0:613 þ 0:192XM0
0.0001
11.3
0.0001
0.0001
12.4
9.8
2.3. Shrinkage coefficient and apparent density
To determine shrinkage coefficient (Sb) and apparent
density of twigs with (qM ) and without bark (qX ) and its
dependence on moisture content, the method of weight
and volume determination was used. At the beginning,
samples were saturated by immersion in water during 5
days. Then, weight, length and diameter in three points
were measured using an optical microscope Nicon
model 104. It was previously calibrated with an optical
micrometer with an absolute scale of calibration.
Samples with different moisture contents were obtained by drying the material in an oven at 60 C over
silica gel in ten steps. After each step, samples were
maintained during four days in a tightly closed flask in
order to assure moisture content uniformity in the
material. Then, they were weighed and measured in the
same three points of each twig (Schmalko, Morawicki, &
Ramallo, 1997). Experiences were carried out with 20
twigs, with and without bark, and for initial diameters
equal to 2.5 · 103 m; 5.0 · 103 m and 7.5 · 103 m (initial average moisture content: 59 ± 1% w/w, wet basis).
457
qX ¼ 458 þ 1040XX
qM ¼ 701 þ 730XM
NaNO3 , NaCl and KCl) (Greenspan, 1977). Measurements were made in triplicate and the average was reported.
2.5. Sorption isotherms models
From models of sorption isotherms for foods exposed
by Chirife and Iglesias (1978), the Halsey and GAB
equations were selected. The Halsey model had been
found to be the linear model with the better fit for
sorption isotherms of twigs of ‘‘yerba mate’’ (K€
anzig,
Novo, & Schmalko, 1987) and has the following form:
a aw ¼ exp r ;
ð1Þ
X
where constants ‘‘a’’ and ‘‘r’’ are temperature dependent. This dependence can be expressed as an Arrhenius
type equation (Rao & Rizvi, 1995; Garcıa, Kobylanski,
& Pilosof, 2000).
For fitting the data to the Halsey’s model, a nonlinear regression technique was used. A basic program
was developed to estimate the parameters of the model.
Mean percentile error (MPE) was minimized using the
convergence method. MPE was calculated using the
following equation:
P jXexperimental Xcalculated j
MPE ¼
Xexperimental
N
100
ð2Þ
Fig. 1. Apparent density (qM and qX ) of twig with and without bark as a function of moisture content (xM and xX ). Experimental and predicted data
calculated with equations from Table 1.
458
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
where N is the number of observations in the experiment.
GAB model was selected because its good fit in many
foodstuffs and their advantages previously mentioned.
GAB’s model has the following form:
X ¼
Xm Ckaw
ð1 kaw Þð1 kaw þ Ckaw Þ
ð3Þ
where Xm is the monolayer moisture content and C and k
are constants. Xm , C and k show an Arrhenius type
dependence with temperature (Garcıa et al., 2000; Rao
& Rizvi, 1995):
Xm ¼ Xm0 expðDH 0 =RT Þ
ð4Þ
C ¼ C0 exp½ðH1 Hm Þ=ðRT Þ
ð5Þ
k ¼ k0 exp½ðHL Hm Þ=ðRT Þ
ð6Þ
The best fit was obtained minimizing MPE with the nonlinear regression technique previously described.
The excess heat of sorption was evaluated at each
moisture content using the equation (Labuza, Kaanane,
& Chen, 1985)
lnðaw Þ ¼ ðQs =RÞ 1=T þ constant
ð7Þ
2.6. Bark–xylem equilibrium models
Models for describing the moisture contents corresponding to bark–xylem equilibrium were obtained
from sorption isotherm ones. A function of water
activity versus equilibrium moisture content was obtained for bark and for xylem by applying each model.
Then, an equation to correlate equilibrium moisture
contents of xylem and bark was developed by equalizing
water activity values. Two kinds of models were assayed
(Chirife & Iglesias, 1978):
(1) Logarithmic models (obtained from Halsey, Henderson, Caurie, Day and Nelson and Oswin models)
ln XX ¼ A1 þ B1 ln XB
ð8Þ
(2) Linear models (obtained from Bradley, Chen,
Chung and Pfost, Kuhn, Mizrahi and Smith equations)
XX ¼ A2 þ B 2 X B
ð9Þ
3. Results and discussion
3.1. Shrinkage and apparent density
When bark and xylem thicknesses of raw ‘‘yerba
mate’’ were measured, different volume proportions between them were found for the different twig diameters.
Mean values of xylem volume/bark volume, according to
Table 2
Water activities (aw ) and equilibrium moisture contents of bark (XB )
and xylem (XX ) at 30, 40, 50, 60, 70, 80 and 90 C
aw
XB
XX
30 C
0.113
0.324
0.560
0.614
0.731
0.751
0.836
0.0486
0.0729
0.1006
0.1071
0.1560
0.1890
0.2483
0.0455
0.0730
0.1174
0.1185
0.1735
0.2415
0.3618
40 C
0.112
0.316
0.532
0.555
0.710
0.747
0.823
0.0313
0.0696
0.0943
0.0992
0.1330
0.1696
0.2309
0.0276
0.0803
0.1033
0.1067
0.1607
0.2386
0.3766
50 C
0.111
0.305
0.500
0.509
0.690
0.744
0.812
0.0288
0.0665
0.0721
0.0807
0.1141
0.1699
0.2195
0.0296
0.0745
0.0883
0.0977
0.1375
0.2322
0.3269
60 C
0.110
0.293
0.467
0.497
0.674
0.745
0.802
0.0251
0.0601
0.0700
0.0840
0.1103
0.1622
0.1819
0.0247
0.0683
0.0867
0.0969
0.1377
0.2130
0.2781
70 C
0.108
0.278
0.470
0.497
0.660
0.751
0.795
0.0220
0.0591
0.0606
0.0702
0.1045
0.1701
0.1751
0.0193
0.0656
0.0616
0.0785
0.1196
0.1799
0.2294
80 C
0.105
0.260
0.514
0.652
0.763
0.789
0.0174
0.0471
0.0532
0.0674
0.1160
0.1223
0.0182
0.0525
0.0568
0.0740
0.1142
0.1210
90 C
0.102
0.241
0.650
0.785
0.0198
0.0350
0.0603
0.1114
0.0236
0.0402
0.0619
0.1067
their diameters, were 0.53/0.47 (d ¼ 2:5 103 m),
0.60/0.40 (d ¼ 5:0 103 m) and 0.65/0.35 (d ¼ 7:5
103 m).
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
459
Table 3
GAB’s model constants for bark and xylem between 30 C and 60 C
Bark
Xm
C
k
MPE
Xylem
30 C
40 C
50 C
60 C
30 C
40 C
50 C
0.0509
20.15
0.954
11.11
0.0486
16.84
0.959
6.23
0.0463
11.81
0.972
9.69
0.0403
10.68
0.989
8.65
0.0548
14.56
0.990
9.68
0.0523
12.52
1.004
14.11
0.0510
10.91
1.019
11.72
0.4
60 C
0.0458
9.29
1.048
9.79
0.4
30°C
30°C
40°C
40°C
50°C
0.3
0.3
50°C
60°C
60°C
X B 0.2
X X 0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
aw
Bark
0
0.2
0.4
Xylem
0.6
0.8
1
aw
Fig. 2. Sorption isotherms for bark and xylem between 30 and 60 C. Experimental data and the fit of the GAB’s model.
Table 4
GAB constants of Eqs. (4)–(6) for bark and xylem between 30 C and 60 C (P < 0:05)
Bark
Xylem
Xm0
DH 0 (kJ/mol)
C0
H1 Hm (kJ/mol)
k0
HL Hm (kJ/mol)
4.37 · 103
8.60 · 103
6.24
4.70
1.08 · 102
10.49 · 102
19.00
12.46
1.57
1.83
)1.28
)1.55
Experimental data of shrinkage coefficient and
apparent density for twigs with (M) and without (X)
bark for each diameter were adjusted to a linear relationship with moisture content. No significant differences were found between constants of the model for the
three diameters assayed, and accordingly all data were
fitted to only one model. The best model was obtained
by fitting the experimental data of shrinkage coefficient
with moisture content in dry basis and the apparent
density with moisture content in wet basis. Equations
obtained are shown in Table 1.
Fig. 1 shows the agreement between experimental
data for apparent density and predicted ones using
equations from Table 1.
3.2. Desorption isotherms
Mean values of equilibrium moisture content of bark
and xylem at temperatures ranging from 20 to 90 C are
shown in Table 2. In all cases, the standard deviation
was less than 10%.
Fig. 3. Excess heat of sorption (Qs ) for bark and xylem at different
moisture content.
460
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
Table 5
Halsey’s model constants for bark and xylem at 70 C, 80 C and 90 C
Bark
a
r
MPE
Xylem
70 C
80 C
90 C
70 C
80 C
90 C
0.0326
1.162
11.88
0.0151
1.311
13.55
0.0095
1.429
11.23
0.0502
1.025
12.92
0.0135
1.372
13.55
0.0053
1.658
11.10
0.25
0.25
70°C
70°C
80°C
90°C
0.2
80°C
0.2
0.15
90°C
0.15
XB
XX
0.1
0.1
0.05
0.05
0
0
0
0.2
Bark
0.4
0.6
0.8
aw
1
0
0.2
Xylem
0.4
0.6
0.8
1
aw
Fig. 4. Sorption isotherms for bark and xylem at 70, 80 and 90 C. Experimental data and the fit of Halsey’s model.
Fig. 5. Equilibrium moisture content between xylem (XX ) and bark
(XB ). Experimental data and predicted values calculated with Eq. (10).
Halsey and GAB’s models fitted the data well
(P < 0:01), but with an MPE relatively high. This was
probably due to the low values of moisture content
obtained and the low uniformity of the material. The fit
of GAB’s model was good until 60 C. Above this
temperature, MPE values were too high, fitting better
the Halsey’s model. Table 3 shows the values of the
GAB equation constants, Xm , C and k, for each temperature. These values were in general different for bark
and xylem and exhibited a typical behaviour with tem-
perature: Xm and C decreased and k increased when
temperature increased. Fig. 2 shows experimental
desorption isotherms and the fit of GAB equation for
bark and xylem between 30 and 60 C.
Table 4 shows the constant values of Equations (4)–
(6) for the GAB’s model. Different values of the constants were observed for bark and xylem. The excess
heat of sorption (Qs ) for bark and xylem versus moisture
content was represented in Fig. 3. The values obtained
are similar to the ones reported for others foodstuffs
(Garcıa et al., 2000; Kim, Kim, Kim, Shin, & Chang,
1999).
Table 5 shows the constants of the Halsey’s model for
bark and xylem at 70, 80 and 90. In both materials, ‘‘a’’
value decreased and ‘‘r’’ value increased with temperature. Fig. 4 shows the agreement between experimental
data and the fit of Halsey’s model at these temperatures.
3.3. Bark–xylem equilibrium
Experimental data for bark and xylem equilibrium
moisture content were fitted to sorption models (Eqs. (8)
and (9)), and a good fit was obtained in both cases with
P < 104 . MPE values were minor for the logarithmic
model (4.10%) than for the linear one (9.96%). When
the slope and the intercept were compared for different temperatures, no significant differences were found
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 66 (2005) 455–461
between them. So, all data were fit to only one equation
(Eq. (8)) with an MPE ¼ 8:16%
XX ¼ 1:61XB1:14
ð10Þ
The agreement between experimental data and the values
calculated using Eq. (10) is observed in Fig. 5. Appreciable differences were found between bark and xylem
equilibrium moisture contents. This behaviour was also
found by Tolaba et al. (1990), when considering the
moisture content of the pericarp and the dehulled corn.
4. Conclusions
Xylem and bark of twigs of ‘‘yerba mate’’ exhibited
different shrinkage coefficient, apparent density and
equilibrium sorption moisture contents. Shrinkage
coefficient and apparent density were found to depend
on moisture content in a linear way.
The fit of the GAB and Halsey’s models to sorption
properties had relatively high mean percentual errors,
probably be due to the low uniformity of the material
and the low moisture contents resulting. Desorption
isotherms for both materials varied with temperature.
A potential equation had the best fit when bark and
xylem equilibrium moisture contents were related, this
relationship not depending on temperature.
References
Abud-Archila, M., Courtois, F., Bonazzi, C., & Bimbenet, J. J. (2000).
A compartmental model of thin-layer drying kinetics of rough rice.
Drying Technology, 18, 1389–1414.
Arnosti, S., Freire, J. T., & Sartori, D. J. M. (2000). Analysis of
shrinkage phenomenon in Bachiaria brizantha seeds. Drying
Technology, 18, 1339–1348.
Chirife, J., & Iglesias, H. A. (1978). Equations for fitting water
sorption isotherms of foods: part 1. A review. Journal of Food
Technology, 13, 159–174.
461
Garcıa, L. A., Kobylanski, J. R., & Pilosof, A. M. R. (2000).
Modelling water sorption in okara soy milk. Drying Technology,
18, 2091–2103.
Greenspan, L. (1977). Humidity fixed points of binary saturated
aqueous solutions. Journal of Research of the National Bureau of
Standards, 81A(1), 89–96.
IRAM (1995). Yerba mate: determinaci
on de la perdida de masa a 103
C. Instituto de Racionalizaci
on de Materiales, No 20503, Argentina.
K€anzig, R. G., Novo, M. A., & Schmalko, M. E. (1987). Comparaci
on
estadıstica de las isotermas de sorci
on de la Yerba Mate. Revista de
la Secretarıa de Ciencia y Tecnologıa de la Universidad Nacional de
Misiones, 5, 13–24.
Kim, S. S., Kim, S. Y., Kim, D. W., Shin, S. G., & Chang, K. S.
(1999). Moisture sorption characteristics of composite foods filled
with chocolate. Journal of Food Science, 64, 300–302.
Labuza, T. P., Kaanane, A., & Chen, J. Y. (1985). Effect of
temperature on the moisture sorption isotherms and water activity
shift of two dehydrated foods. Journal of Food Science, 50, 385–
391.
Lozano, J. E., Rotstein, E., & Urbicaın, M. J. (1983). Shrinkage,
porosity and bulk density of foodstuffs at changing moisture
contents. Journal of Food Science, 48, 1497–1502.
Mourad, M., Hemati, M., & Laguerie, C. (1996). A new correlation for
the estimation of moisture diffusivity of corn kernels from drying
kinetics. Drying Technology, 14, 876–894.
Ramallo, L. A., Pokolenko, J. J., Balmaceda, G. Z., & Schmalko, M.
E. (2001). Moisture diffusivity, shrinkage and apparent density
variation during drying of leaves at high temperatures. International Journal of Food Properties, 4, 163–170.
Rao, M. A., & Rizvi, S. S. H. (1995). Engineering properties of foods p.
252 (2nd ed.). New York: Marcel Dekker Inc.
Schmalko, M. E., Morawicki, R. O., & Ramallo, L. A. (1997).
Simultaneous determination of specific heat capacity and thermal
conductivity using the finite-difference method. Journal of Food
Engineering, 31, 531–540.
Sokhansanj, S., Bailey, W. G., & Van Dalfsen, K. B. (1999). Drying of
north american ginseng roots (Panax quinquefolius L.). Drying
Technology, 17, 2293–2308.
Tolaba, M. P., Suarez, C., & Viollaz, P. E. (1990). The use of a
diffusional model in determining the permeability of corn pericarp.
Journal of Food Engineering, 12, 56–66.
van den Berg, C., & Bruin, S. (1981). Water activity and its estimation
in food systems: theoretical aspects. In L. B. Rockland & G. F.
Stewart (Eds.), Water Activity: Influences on Food Quality (pp. 1–
61). New York: Academic Press.