© The Authors, 2024, Published by the Universidad del Zulia*Corresponding author:c27313@utp.edu.pe
Keywords:
Onion dehydration
Mathematical models
Kinetic parameters
Drying eciency
Environmental sustainability
Mathematical modeling of the onion drying process: Kinetic and Thermodynamic Parameters
Modelación matemática del proceso de secado en cebolla: Parámetros Cinéticos y Termodinámicos
Modelagem matemática do processo de secagem da cebola: Parâmetros Cinéticos e Termodinâmicos
Bruno César Giménez-López
1*
Cristhian Ronceros Morales
1
Alejandro Alfredo Quispe Mayuri
2
Rosalio Cusi Palomino
3
Manuel Antonio Giménez-Medina
2
Carmen Luz Cuba Cornejo
3
Rev. Fac. Agron. (LUZ). 2024, 41(3): e244127
ISSN 2477-9407
DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v41.n3.07
Food technology
Associate editor: Dra. Gretty R. Ettiene Rojas
University of Zulia, Faculty of Agronomy
Bolivarian Republic of Venezuela.
1
Universidad Tecnológica del Perú.
2
Laboratorio de Química Analítica Instrumental, Escuela
Profesional de Ingeniería Agroindustrial, Facultad de
Ingenierías, Universidad Privada San Juan Bautista, Ica Perú.
3
Universidad Nacional San Luis Gonzaga de Ica, Perú.
Received: 06-05-2024
Accepted: 13-07-2024
Published: 13-08-2024
Abstract
The dehydration processes of onions are governed by a series
of kinetic and thermodynamic parameters that, when controlled,
facilitate the identication of a mathematical model that will
improve the eciency and quality of the drying process in terms
of reducing production costs, environmental sustainability, and
the development of innovative products. In this study, various
mathematical models were validated to accurately describe the
drying process, and from them, the kinetic and thermodynamic
parameters governing the dehydration processes were determined.
For the experimental development, onions grown in the Ica region
of Peru were peeled, cut into pieces, and dehydrated (60, 70, and
80 ºC), and ve mathematical models were applied to model the
drying kinetics of the process. The Midilli model was the best t
for the experimental curves. Increasing the temperature reduced the
enthalpy and increased the entropy, Gibbs free energy, and eective
diusion coecient in both varieties of onions. Determining
the drying kinetics has been essential for establishing operating
conditions by understanding how temperature, relative humidity,
and other parameters aect the moisture removal rate, allowing for
the design of optimal equipment and predicting product behavior
during the drying process.
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2024, 41(3): e244127 July-September. ISSN 2477-9407.
2-6 |
Resumen
Los procesos de deshidratación de la cebolla están regidos por una
serie de parámetros cinéticos y termodinámicos que al ser controlados,
facilitan la identicación de un modelo matemático, que permitirá
mejorar la eciencia y calidad del proceso de secado en términos
de reducción de costos de producción, sostenibilidad ambiental y
desarrollo de productos innovadores. En este trabajo se validaron
diferentes modelos matemáticos que describan con precisión el proceso
de secado y a partir de ellos determinar los parámetros cinéticos y
termodinámicos que gobiernan los procesos de deshidratación. Para el
desarrollo experimental se utilizaron cebollas cultivadas en la región
de Ica, Perú, peladas, cortadas en trozos y deshidratadas (60, 70, y
80 ºC) y se aplicaron cinco modelos matemáticos para modelar la
cinética del secado del proceso. El modelo de Midilli fue el que mejor
se adaptó a las curvas experimentales. El aumento de la temperatura
redujo la entalpía e incrementó la entropía, la energía libre de Gibbs,
y el coeciente de difusión efectivo en ambas variedades de cebolla.
La determinación de la cinética de secado ha sido fundamental
para establecer las condiciones de operación, al comprender cómo
la temperatura, la humedad relativa y otros parámetros afectan la
velocidad de eliminación de la humedad, permitiendo el diseño de
equipos óptimos y prever el comportamiento del producto durante el
proceso de secado.
Palabras clave: deshidratación de cebolla, modelos matemáticos,
parámetros cinéticos, eciencia de secado, sostenibilidad ambiental.
Resumo
Os processos de desidratação da cebola são regidos por uma série
de parâmetros cinéticos e termodinâmicos que, quando controlados,
facilitam a identicação de um modelo matemático que permitirá
melhorar a eciência e a qualidade do processo de secagem em
termos de redução de custos de produção, sustentabilidade ambiental
e desenvolvimento de produtos inovadores. Neste trabalho, vários
modelos matemáticos foram validados para descrever com precisão o
processo de secagem e, a partir deles, determinaram-se os parâmetros
cinéticos e termodinâmicos que regem os processos de desidratação.
Para o desenvolvimento experimental, foram utilizadas cebolas
cultivadas na região de Ica, Peru, descascadas, cortadas em pedaços
e desidratadas (60, 70 e 80 ºC), e aplicaram-se cinco modelos
matemáticos para modelar a cinética de secagem do processo. O
modelo de Midilli foi o que melhor se ajustou às curvas experimentais.
O aumento da temperatura reduziu a entalpia e aumentou a entropia,
a energia livre de Gibbs e o coeciente de difusão efetivo em ambas
as variedades de cebola. A determinação da cinética de secagem
tem sido fundamental para estabelecer as condições de operação,
compreendendo como a temperatura, a umidade relativa e outros
parâmetros afetam a velocidade de remoção da umidade, permitindo
o design de equipamentos ótimos e prevendo o comportamento do
produto durante o processo de secagem.
Palavras-chave: desidratação da cebola, modelos matemáticos,
parâmetros cinéticos, eciência de secagem, sustentabilidade
ambiental.
Introducción
The drying of onion (Allium cepa L.) drying represents a crucial
aspect in the agri-food processing chain, with signicant implications
for conservation, quality, and production eciency (Attkan et al.,
2021). Inecient drying can lead to signicant quality losses of the
nal product, aecting its organoleptic and nutritional characteristics;
while at the production level it generates a signicant increase in
energy consumption, directly impacting production costs and the
environment (Babu et al., 2018; Braga da Silva et al., 2019).
It is essential to use mathematical modeling and simulation of
drying curves to ensure optimal process control and production of the
highest-quality products. Tools that improve operational eciency
and process sustainability by reducing energy consumption,
preventing premature wear of drying equipment, and preserving
the quality of the nal product (Fernando and Amarasinghe, 2016;
Lemus-Mondaca et al., 2015).
Prediction and control of the drying process make it possible
to guarantee maximum quality and energy eciency by accurately
describing the behavior of onion drying through mathematical
models, which allow optimization of the kinetic and thermodynamic
parameters of the process (Chakraborty et al., 2023; Silveira Dorneles
et al., 2019). Some authors such as Compaoré et al. (2019) indicate
that the mathematical model that best describes the drying kinetics
in various foods is the one proposed by Midilli, while Attkan et al.
(2021) and Revaskar et al. (2014) consider that Page’s model best ts
the experimental curves.
However, the mathematical model can be determined by the
type of drying process used for dehydration, such as convection
(Przeor et al., 2019) and osmodehydrofreezing (Bosco et al., 2018)
or microwave methods (Khodja et al., 2020; Süfer et al., 2017) and
uidized bed equipped with a heat pump dehumidier developed by
Jafari et al. (2016), among others.
Therefore, the objective of this research was to determine the
mathematical modeling, kinetic, and thermodynamic parameters that
describe the drying process of Allium cepa L., Noam, and Centrum
varieties. This allows the process to be optimized from an energetic
perspective and improve the quality of the product, enriching the
body of knowledge in the engineering of agri-food processes. It also
establishes the foundations for future research in this eld, opening
new opportunities for technological progress and innovation in food
processing.
Materials and Methods
Raw Material
The onions, Allium cepa L., Centrum (yellow) and Noam (red),
were acquired at the Arenales local market, in the city of Ica, Peru,
24 hours after harvest, guaranteeing their freshness and quality.
Subsequently, the bulb was peeled to remove the outermost layer
and cut into small uniform slices with a stainless steel chopper and
refrigerated at 6 ° C until the drying process began.
Drying process
The drying process of the two varieties of Allium cepa L. was
carried out in a tray dryer (Proctor and Schwartz, SCM Corporation),
with electrical resistances to generate heat. This equipment achieved a
drying speed of 11 m . s
-1
and maintained a constant dry air ow of 2.0
± 0.2 m . s
-1
, with a total load capacity of 5 kg. The drying temperatures
used were 60, 70 and 80 ° C and the dimensionless moisture ratio (MR)
was calculated according to the following equation:
Where M
t
indicates the moisture content over time (t) until the
constant weight is reached, Me is the equilibrium moisture content,
and M
0
is the initial moisture content.
=
 
0
 
 (1) 1
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Giménez-López et al. Rev. Fac. Agron. (LUZ). 2024 41(3): e244127
3-6 |
Mathematical Models
Table 1 shows ve simplied mathematical models explaining the
evolution of the moisture content during drying of the two varieties
of Allium cepa L.
Table 1. Mathematical models describing drying processes.
Model name Model equation Equation
Page

)
(2)
Midilli


(3)
Lewis

(4)
Logarítmico

(5)
Henderson y Pabis

(6)
Here, a, b, c, k, and n represent the kinetic constants of the models, while MR is the
moisture ratio and t is the drying time.
Diusion coecient, activation energy, and thermodynamic
properties
The eective moisture diusion coecient (D
e
), expressed
in square meters per second (m².s
-1
), was calculated using Fick’s
second law for diusion, applied at dierent drying temperatures.
This determination is based on the assumption that the temperature
remains constant during the drying process, without undergoing
signicant changes. For this calculation, the equation that models the
drying process was used:
By analyzing the relationship between drying time (t) and
moisture ratio (MR) of Allium cepa L., and taking into account that
its geometry resembles a sheet, where L
0
represents half the thickness
of the material and D
e
as the measure of the capacity of moisture to
diuse through the material, from the slope of the straight line (m)
obtained from the linear graph of equation 7, D
e
can be calculated.
Considering that D
e
in foods presents a markedly dependent
relationship with the drying temperature and that its behavior is
aligned with the Arrhenius model, it is established that the relationship
between D
e
and the drying air temperature (Ta) is well described by
the Arrhenius model.
Where, R is the universal gas constant 8.314 J.mol
-1
.K
-1
, Ea is
the activation energy (J.K
-1
.mol
-1
), D0 is the preexponential factor or
initial diusion constant (m
2
.s
-1
) and Ta is the absolute temperature
(K). From the linearized Arrhenius equation, as shown in equation
10, it is observed how D
e
varies with the absolute temperature (Ta)
measured in kelvin.
Ea and D
0
are calculated from the slope (m) and intercept (b),
respectively, of the graph of Equation 10. Furthermore, once Ea is
known, it is possible to calculate the dierentials of enthalpy (ΔH),
entropy (ΔS) and Gibbs free energy (ΔG) dierentials using the
following equations:
Where: K
B
represents Boltzmann’s constant (1.38 x 10
−23
J.K
−1
),
hp is Planck’s constant (6,626 x 10
−34
J.s) and T is the absolute
temperature (Ta) measured in kelvin.
Statistical analysis
The suitability of the proposed models for the drying kinetics of
the two onion varieties was evaluated by statistical tests involving
the sum of squares error (SSE) using equation 14 and employing the
Excel Solver that allows maximization of an objective function that is
subject to certain restrictions. The lowest values of SSE (≈0.0) were
used as a criterion to choose the model that best ts the experimental
curve.
In the equation, MR
e,i
represents the experimental moisture
content, MR
c,i
is the calculated moisture content, i is the number of
terms, and N is the number of data.
The values of De, ΔH, ΔS and ΔG for the same variety of Allium
cepa L. were compared using the analysis of variance test (ANOVA)
with a condence level of 95 %. On the other hand, when comparing
the values of De, Ea, ΔH, ΔS and ΔG between the two varieties of
Allium cepa L., Student’s t test for independent samples was used,
also with a condence level of 95%.
Results and discussion
Drying kinetic curves
Figure 1 shows the drying curves obtained experimentally for
Allium cepa L., Centrum, and Noam varieties that were subjected to
dierent drying temperatures (60, 70, and 80 ºC).
Figure 1 shows that as the air drying temperature increases, the
relative humidity decreases signicantly in both onion varieties
analyzed, observing that the drying process reaches equilibrium faster
at 80 °C, with equilibrium times of 120 minutes for the Noam variety
and 150 minutes for the Centrum. This dierence in equilibrium times
is statistically signicant (p-value = 0.000) at a condence level of 95
%. According to Pasechny et al. (2023) this may be due to a higher
phenolic content in the Noam variety, a red onion, compared to the
white or yellow varieties.
The study did not detect signicant dierences in initial
moisture levels between the two onion varieties, with mean values
of approximately 88.6 % for Centrum and 86.5 % for Noam, with
a signicance level of 5 %. This indicates a uniformity in the initial
moisture content, which facilitates a fair comparison of their behavior
during the drying process. The trend of a faster reduction in drying
time with increasing temperature aligns with the results of research
such as those developed by Siqueira et al. (2015) on timbo leaves,
Silva-Paz et al. (2023) on muña, Silva et al. (2015) on jenipapo and
Gasparin et al. (2017) with mentha piperita leaves. This phenomenon
is mainly attributed to variations in vapor pressure between the
product and the surrounding air, a fundamental principle in drying
dynamics.The mathematical model that showed the best t based on
the obtained value of the sum of squares error (SSE) for Allium cepa
L., the Noam variety and the Centrum variety was the Midilli model
as can be seen in table 2.
=
 
0
 
 (1) 1

(

)
=
2

4
0
2
 
8
 (7) 1
4
0
2
2
=

 (8) 1

=
0



 (9) 1
=
   (11
)
1
=  
0
  
    (12) 2
 =     (13)
3
=
1

,
 
,
2
=1
 (14) 1
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2024, 41(3): e244127 July-September. ISSN 2477-9407.
4-6 |
Figure 1. Dimensionless moisture ratio (MR) curves as a function of time, under three dierent drying temperatures (60, 70 and 80 ºC),
for Allium cepa L., varieties Noam (a) and Centrum (b).
Table 2. Empirical models and regressive statistical parameters for Centrum and Noam varieties of Allium cepa L.
Allium cepa L., var. Centrum
Model T (˚C) SSE K N A C B
Lewis
60
0.144 0.0077
Page 0.050 0.0430 0.6
Henderson y Pabis 0.029 0.0056 0.79
Logaritmo 0.024 0.0025 1.30 -0.543
Midilli 0.021 1.7 x 10
-5
2.5 x 10
-8
0.70 -2.5 x 10
-3
Lewis
70
0.051 0.0096
Page 0.035 0.0045 1.2
Henderson y Pabis 0.050 0.0099 1.03
Logaritmo 0.010 0.0061 1.17 -0.222
Midilli 0.008 0.0031 1.2 0.91 -3.6 x 10
-4
Lewis
80
0.073 0
.0204
Page 0.066 0.0337 0.9
Henderson y Pabis 0.046 0.0172 0.84
Logaritmo 0.019 0.0097 0.96 -0.208
Midilli 0.005 1.22 x 10
-4
2.1 0.62 -3.3 x 10
-5
Allium cepa L., var. Noam
Lewis
60
0.025 0.0095
Page 0.023 0.0116 1.0
Henderson y Pabis 0.022 0.0091 0.97
Logaritmo 0.022 0.0086 0.98 -0.025
Midilli 0.021 8.3 x 10
-3
2.5 x 10
-8
0.97 -2.5 x 10
-3
Lewis
70
0.085 0.0113
Page 0.022 0.0022 1.4
Henderson y Pabis
0.058 0.0127 1.12
Logaritmo 0.028 0.0089 1.20 -0.152
Midilli 0.015 1.8 x 10
-2
1.5 0.92 -2.9 x 10
-4
Lewis
80
0.057 0.0224
Page 0.023 0.0217 1.4
Henderson y Pabis 0.042 0.0196 0.87
Logaritmo 0.024 0.0136 0.92 -0.115
Midilli 0.008 4.2 x 10
-2
1.9 0.65 -2.8 x 10
-6
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Giménez-López et al. Rev. Fac. Agron. (LUZ). 2024 41(3): e244127
5-6 |
Table 3. Eective diusivity coecient for Allium cepa L., varieties
Noam and Centrum at dierent drying temperatures
(60, 70 and 80 ºC).
T (K)
Allium cepa L.
Noam
D
e
(m
2
.s
-1
)
Centrum
D
e
(m
2
.s
-1
)
333.15 8.75 x 10
-8
1.79 x 10
-10
343.15 1.87 x 10
-7
9.71 x 10
-10
353.15 4.47 x 10
-7
3.26 x 10
-9
Se observó que los valores de De del agua para la variedad
Noam de Allium cepa L. aumentaron conforme se incrementó
la temperatura de secado de 8.75 x 10
-8
hasta 4.47 x 10
-7
m
2
.s
-1
en
el rango de 60 – 80 ºC, y para Allium cepa L., variedad Centrum
aumentó desde 1.79 x 10
-10
hasta 3.26 x 10
-9
.
Los valores son muy
similares a los que han sido reportados previamente
por Süfer et al.
(2017) que obtuvo coecientes de difusión entre 1.962 × 10
−9
y 1.372
× 10
−8
m
2
.s
-1
cuando el secado fue por convección, entre 9.8 × 10
−9
y 1.7 × 10
−8
m
2
.s
-1
para el secado en vacío, ambos procesos llevados
a cabo en 50 y 70 ºC y entre 3.193 × 10
−8
y 9.139 × 10
−7
m
2
.s
-1
para
secado en microondas. Por otra parte, Attkan et al. (2021) reportaron
que los valores efectivos de difusividad de la humedad en las rodajas
de cebolla variaron entre 1.33 × 10
−8
m
2
.s
-1
y 2.49 × 10
−8
m
2
.s
-1
en un
secador solar híbrido asistido por aire de baja humedad.
The water De values for the Noam Allium cepa L. variety were
observed to increase as the drying temperature increased from 8.,75
x 10
-8
to 4.47 x 10
-7
m
2
.s
-1
in the range of 60 - 80 °C, and for Allium
cepa L., the Centrum variety increased from 1,79 x 10
-10
to 3.26 x 10
-
9
. The values are very similar to those previously reported by Süfer
et al. (2017) who obtained diusion coecients between 1.962 ×
10
−9
y 1.372 × 10
−8
m
2
.s
-1
when drying was by convection, between
9.8 × 10
−9
and 1.7 × 10
−8
m
2
.s
-1
for vacuum drying, both processes
carried out at 50 and 70 °C and between 3.193 × 10
−8
and 9.139 × 10
−7
m
2
.s
-1
for microwave drying. On the other hand, Attkan et al. (2021)
reported that the eective moisture diusivity values in onion slices
ranged from 3,193 × 10
−8
y 9,139 × 10
−7
m
2
.s
-1
in a low humidity air-
assisted hybrid solar dryer.
Thermodynamic properties of the Allium cepa L., Noam and
Centrum varieties
Knowledge of activation energy values allows selecting the
optimal temperature and operation time; very high Ea values lead
to slow and long drying processes with direct eect on production
costs; in contrast, low Ea lead to fast drying processes that can
cause degradation of the quality of product sheets (Fernando and
Amarasinghe, 2016; Padilla-Frias et al. 2018).
Based on the slope of the line described by the Arrhenius equation,
as shown in Figure 2, the activation energy (Ea) was calculated for
the two varieties of Allium cepa L., Noam, and Centrum. Statistically
signicant dierences in Ea (p-value = 0.000) were found for both
varieties, being (79.685 ± 0.012) kJmol-1 for the Noam variety and
(141.653 ± 0.014) kJmol-1 for the Centrum variety. The results
obtained are much higher than those reported by Süfer et al. (2017) in
Allium cepa L., in the range of 3,28 to 34,13 kJ.mol
-1
for convection
and vacuum drying processes, while 2.25 to 6.08 W/kg for microwave
drying. According to the above, the activation energy is independent
of the drying conditions as proposed by Compaoré et al. (2019).
No statistically signicant disparities with dierential enthalpy
(∆H) for Allium cepa L., variety Noam (p-value = 0.579) and Centrum
(p-value = 0.858) when varying drying temperatures (60, 70, and 80
°C), for a 95 % condence level. However, when comparing between
varieties, statistically signicant disparities (p-value = 0.000) were
found in ∆H, being 76.8 ± 0.2 kJ.mol
-1
for the Noam variety and 138.9
± 0,3 kJ.mol
-1
for the Centrum variety, as detailed in table 4.
Figure 2. Curves of the eective diusivity coecient as a function of the inverse of temperature (60, 70 and 80 ºC), for Allium cepa L.,
varieties Noam and Centrum.
Noam
Centrum
1
Noam
Noam
Centrum
1
Centrum
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2024, 41(3): e244127 July-September. ISSN 2477-9407.
6-6 |
Table 4. Thermodynamic parameters for red and yellow onion.
T
(°C)
ΔH
(kJ mol
-1
)
Allium cepa L.
ΔS
(kJ mol
-1
K
-1
)
Allium cepa L.
ΔG
(kJ mol
-1
)
Allium cepa L.
Noam Centrum Noam Centrum Noam Centrum
60 76,9
138,9
-48,3
-486,1
93,0
30,1
70 76,8
138,8
-48,5
-486,3
93,5
30,6
80 76,7
138,7
-48,8
-486,6
94,0
31,1
However, no statistically signicant disparities were found in
the values of the entropy dierential (∆S) for Allium cepa L., variety
Noam (p-value = 0.832) and Centrum (p-value = 0.828) when varying
the drying temperatures (60, 70, and 80) ºC, for a condence level
of 95 %. However, when comparing between varieties, statistically
signicant disparities (p-value = 0.000) were found in the ∆S, being
−48.4 ± 0.3 kJmol
-1
K
-1
for the Noam variety and −486.4 ± 0.2 kJ.mol
-
1.
K
-1
for the Centrum variety, as detailed in Table 4. Therefore, it can
be observed that the values obtained for ∆H, ∆S are lower than those
recorded by Braga da Silva et al., (2019).
In the Gibbs free energy dierential, no statistically signicant
disparities were found for Allium cepa L., variety Noam (p-value
= 0.923) and Centrum (p-value = 0.885) when varying the drying
temperatures (60, 70 and 80) °C, for a condence level of 95 %.
However, when comparing between varieties, statistically signicant
disparities (p-value = 0.000) were found in the Gibbs free energy
dierential, being 94.0 ± 0,1 kJ.mol
-1
for the Noam variety and 30.1
± 0,2 kJ.mol
-1
for the Centrum variety, as detailed in Table 4. These
results are lower than those published by Braga da Silva et al. (2019)
on pretreated Piper aduncum leaves (108.955 y 113.889 kJ.mol
-1
), and
higher than those reported by Quequeto et al. (2019) on laurel leaves
(53.038 kJ.mol
-1
).
Conclusions
The Midilli model was determined to be the most appropriate
model to represent the experimental data on the drying process of
Allium cepa L., applicable to the Noam and Centrum varieties
regardless of variations in drying temperature. Furthermore, increasing
the temperature during the drying process signicantly reduced the
time required to remove water from both varieties of Allium cepa L.
and caused an increase in the eective water diusion coecient.
However, this increase in temperature did not signicantly aect
the values of enthalpy dierential, entropy, and Gibbs free energy,
demonstrating that these thermodynamic properties remain relatively
stable under the drying conditions studied.
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