Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.11, n.2, p.117-121, 2009
ISSN 1517-8595
117
MATHEMATICAL MODELING Of THE DRYING KINETICS OF SUGARCANE
SLICES
Nahia Agote Goyalde1, Evandro de Castro Melo2, Ronicely Pereira Rocha3,
André Luis Duarte Goneli4, Fabiana Lana Araújo5
ABSTRACT
Sugarcane crop is most important economically, socially and environmentally. Brazil is
the largest sugarcane producer in the world. With the objective of contributing towards
precision agriculture, the air drying characteristics of sliced sugarcane (Saccharum spp)
were investigated and made to fit into semi-theoretical models used to describe drying
behavior. The drying tests were performed in an experimental fixed-bed dryer with
upward air flow. The drying was carried out at two air temperatures: 50 and 60ºc with
air relative humidity of 17.9 and 11.1, respectively. The time required for sugarcane
drying, from an initial moisture content of 70% w.b. to the final moisture content of 6%
w.b., was 7.5 and 3.5 h for drying temperatures of 50 and 60ºc, respectively.
Experimental data were adjusted to 4 traditional mathematical models in order to
represent the drying process of agricultural products. The Midilli model was the one that
best described the sliced sugarcane drying process.
Keywords: Saccharum spp, drying, mathematical modeling.
MODELAGEM MATEMATICA DACINETICA DE SECAGEM DA CANA DE
ACUCAR CULTIVADA
RESUMO
Notoriamente, a cultura da cana-de-açúcar tem grande importância econômica, social e
ambiental, fazendo do Brasil o maior produtor mundial de cana. Com o propósito de contribuir
com a agricultura de precisão, foi avaliada a obtenção das curvas de secagem da cana-de-açúcar
cultivada (Saccharum spp) e ajustada a diferentes modelos matemáticos. A secagem foi
realizada em um secador experimental a gás de leito fixo com fluxo de ar ascendente. Os testes
foram realizados utilizando-se duas temperaturas de secagem, 50 e 60 ºC, com umidade relativa
de 17,9 e 11,1, respectivamente. O tempo requerido para secar a amostra com teor de água de 70
para 6% b.u. foi de 7,5 e 3,5h para as temperaturas de secagem de 50 e 60ºC, respectivamente.
Aos dados experimentais, foram ajustados 4 modelos matemáticos tradicionais para a
representação do processo de secagem de produtos agrícolas. O modelo de Midilli foi o que
melhor se ajustou aos dados de secagem da cana-de-açúcar picada.
Palavras-chave: Saccharum spp, secagem, modelos matemáticos.
Protocolo 1102 de 02/06/2009
1
Agricultural Engineer, Undergraduate student in Agricultural Engineering, UPNA, Pamplona-Spain, e-mail: [email protected].
2
Agricultural Engineer, Dr. Professor in Agricultural Engineering, UFV, Viçosa – Brazil, Corresponding author. Tel.: +55 31 38991873, email address: [email protected]
3
Agricultural Engineer, Doctorate student in Agricultural Engineering, UFV, Viçosa – Brazil, email: [email protected]
4
Agricultural Engineer, Doctorate student in Agricultural Engineering, UFV, Viçosa - Brazil.
5
Animal Scientist, Máster student in Animal Science, UFV, Viçosa - Brazil.
Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.11, n.2, p.117-121, 2009
118
Mathematical modeling of drying kinetics of sugarcane slices
Goyalde et al.
the objective of this work is to fit mathematical
models to describe the drying process.
INTRODUCTION
Drying is one of the most widely used
primary methods of food preservation. The
objective of drying is the removal of water to
the level at which microbial spoilage and
deterioration reactions are greatly minimized
(Akpinar & Bicer, 2004).
In the development and improvement
of equipment used for drying, the simulation
and attainment of theoretical information about
the behavior of each product during water
removal is important. For the simulation,
which is based on the principle of successive
drying of thin layers of the product, a
mathematical
model
that
represents,
satisfactorily, their water loss during the drying
process is used (Berbet et al., 1995).
In recent years, various investigators
have undertaken studies covering mathematical
modeling and kinetics of the vegetable drying
process. For example, wheat (Sun & Woods,
1994), bean (Afonso Júnior & Corrêa, 1999),
rough rice (Basunia & Abe, 2001), red pepper
(Kaymak-ertekin, 2002; Akpinar et al., 2003),
pear fruit (Lahsasni et al., 2004), parbolized
wheat (Mohapatra & Rao, 2005), and tomato
(Sacilik et al., 2006).
Considering the fact that Brazil is the
principal producer of sugarcane and its
importance as an animal food and oil producer,
MATERIAL AND METHODS
The present work was carried out in
the Agricultural Engineering Department,
located at the Universidade Federal de Viçosa
(UFV), Viçosa-MG, Brasil.
The sugarcane used for the experiment
was taken from Usina Jatiboca, in Rio Casca
village, with traditional harvest techniques
used for the sugarcane. After harvesting, the
sugarcane was sliced with a mechanical
machine (Nogueira EM-6400) to 6 mm
approximately, and samples were taken for
drying tests and initial moisture content. The
sugarcane was sliced with moisture content of
70% w.b., measured by applying the
gravimetric method at 105 ± 1 ºC for a period
of 48 hours in triplicate.
The drying tests were performed in an
experimental fixed-bed dryer with an upward
air flow. The drying was carried out under two
air temperatures of 50 and 60ºC controlled by
an automatic controller with ± 2 ºC of
variation, as Jesus et al. (2001) describes
(Figure 1).
Figure 1 – Dryer view
Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.11, n.2, p.117-121, 2009
Mathematical modelling of drying kinetics of sugarcane slices
Air relative humidity was 17.9% and
11.1% at temperatures of 50 and 60ºC,
respectively, measured with a thermo-hydro
clock from Datamed company.
The samples used in the experiment
were placed in perforated plastic trays inside
the dryer. The weight of the original sample
was 140 g. A digital scale was used to weigh
the samples, model Mark 420LC. The drying
was completed when the samples weights were
found constant. Then three samples were
placed in the oven, model MA033, at 105 ± 1
°C for 48 hours to calculate the water content
final.
The moisture ratio (MR) was determined using
the following expression:
MR =
M θ − Me
Mi − M e
119
Goyalde et al
where:
MR: moisture ratio, dimensionless;
Mθ: moisture content at time θ (kg water/kg
dry mater);
Mi: initial moisture content, (kg water/kg
dry mater);
Me: equilibrium moisture content, (kg
water/kg dry mater).
The equilibrium moisture content for
sugarcane was taken from the work done by
Rao (2006). The drying curves were adjusted
from the experimental data using empiric and
semi-empiric models reported in the literature,
presented in Table 1 (Madamba et al, 1996;
Doymaz, 2004; Mohapatra & Rao, 2005;
Akpinar, 2006).
(1)
Table 1. Mathematical models used to describe the drying kinetics.
Model designation
Model
Henderson and Pabis
MR = a exp ( -k t )
(1)
Logarithmic
MR = a exp ( -k t ) + c
(2)
Midilli
MR = a exp -k t
Page
MR = exp -k t
(
(
n
n
)+bt
(3)
)
(4)
where:
t: Drying time (h);
k: empirical coefficients in the drying model (
h-1 );
a, b, c, n: empirical constants in drying model.
A regression analysis was performed
using the drying mathematical models and the
experimental data. The experimental data was
interpreted by means of a non-linear regression
analysis using the Quasi-Newton method
executed with the Statistica 6.0® computer
program. The drying models were selected
based on the mean relative error (MRE), the
standard error of estimate (SEE) and the
determination coefficient (R2). The MRE and
SEE were calculated for each model by the
following expressions (Madamba et al., 1996,
Mohapatra & Rao, 2005):
2
n
∑ M exp - M pre
SEE = i =1
Df
(
MRE =
)

100 n M exp - M pre
∑ 
n i =1 
M exp

(5)




(6)
where:
n:
Df:
Mexp:
Mpre:
number of observations;
degrees of freedom of the model;
experimental observed values;
estimated values by the model.
Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.11, n.2, p.117-121, 2009
120
Mathematical modeling of drying kinetics of sugarcane slices
Goyalde et al.
Madamba et al. (1996), for 50 and 60ºC drying
temperatures. The R2, SEE and MRE values of
Midilli model are 0.999, between 0.007 and
0.010, and between 2.677 and 7.615,
respectively. Therefore, the Midilli model may
be assumed to represent the drying behavior of
sugarcane slices.
RESULTS AND DISCUSSION
The
calculated
determination
coefficient (R2), standard error of estimate
(SEE) and mean relative error (MRE) for the
models presented in Table 1 for 50 and 60ºC
are presented in Table 2.
All models achieve a R2 greater than
0.99, which is acceptable according to
Table 2. Statistical analysis for the models using experimental data thin-layer drying of wheat.
50°C
Model
R
2
60°C
2
SEE
MRE (%)
R
SEE
MRE (%)
Henderson and Pabis
0.999
0.010
11.156
0.996
0.026
21.979
Logarithmic
0.999
0.010
13.206
0.999
0.015
8.807
Midilli
0.999
0.007
7.615
0.999
0.010
2.677
Page
0.999
0.008
12.029
0.999
0.009
6.775
The estimated parameters of the
Midilli model are represented in Table 3. The
observed and estimated data are represented in
Figure 1 as a function of the drying time for
each temperature. Figure 2 shows good
adjustment for the Midilli model to the
experimental data.
1,0
50°C
60°C
Estimated Midilli
Moisture ratio
0,8
0,6
0,4
0,2
0,0
0
1
2
3
4
5
6
7
8
Time, h
Figure 2. Experimental and predicted moisture ratios obtained using the Midilli model.
Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.11, n.2, p.117-121, 2009
Mathematical modelling of drying kinetics of sugarcane slices
The time required for sugarcane drying
from an initial moisture content of 70% w.b. to
the final moisture content of 6% w.b. was 7.5
and 3.5 hours, at temperatures of 50 and 60 oC.
Figure 1 verifies that higher temperatures
allow for faster evaporation rates.
Goyalde et al
121
slices. The time required for the sugarcane
drying from an initial moisture content of 70%
w.b. to the final moisture content of 6% w.b.
was 7.5 and 3.5 hours, at temperatures of 50
and 60ºC
ACKNOWLEDGMENTS
CONCLUSIONS
The Midilli model may be assumed to
represent the drying behavior of sugarcane
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