Received: 22 June 2017
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Revised: 25 September 2017
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Accepted: 11 October 2017
DOI: 10.1111/jfpe.12641
ORIGINAL ARTICLE
Generalization of a lumped parameters model using fractional derivatives applied to rice hydration Douglas Junior Nicolin1
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Thaisa Carvalho Volpe Balbinoti2 |
Regina Maria Matos Jorge2,3 | Luiz Mario de Matos Jorge3 1 Chemical Engineering Department, Federal Technological University of Parana (UTFPR), Linha Santa Barbara, PO Box 135, Francisco Beltr~ao, 85601-970, Parana, Brazil 2
Abstract This study presents a generalization of a lumped parameters model through the use of fractional derivatives of arbitrary order. The conventional model has an analytical solution in terms of expo-
Chemical Engineering Department, Food Engineering Graduate Program, Federal University of Parana (UFPR), Cel. Francisco H. dos Santos Avenue, 210, Curitiba, 81531-970, Parana, Brazil
nential function. However, when applied to kinetic data on moisture absorption in rice grains, its
3
tion of moisture values in relation to time and process temperature.
Chemical Engineering Graduate Program (PEQ), State University of Maring a (UEM), Colombo Avenue, 5790, Bl. E-46, Maring a, 87020-900, Brazil Correspondence Douglas Junior Nicolin, Chemical Engineering Department, Federal Technological University of Parana (UTFPR), Linha Santa Barbara, PO Box 135, Francisco Beltr~ao, 85601-970, PR, Brazil. Email:
[email protected] and Luiz Mario M. Jorge, Chemical Engineering Graduate Program (PEQ), State University of Maringa (UEM), Colombo Avenue, 5790, Bl. E-46, Maringa, 87020-900, Brazil. Email:
[email protected]
performance was worse when compared to the proposed generalized model. The results were analyzed for four hydration temperatures. Despite the similarity of the mass transfer coefficients for both models, the proposed generalization provided a statistically significant improvement in predic-
Practical applications The model proposed in this paper is applicable in describing the hydration kinetics of rice which is an important step in parboiling process, for instance. The simplicity of the analytical solution of the model and the improvement of its prediction capability makes the presented approach useful to project of equipment intended to process rice.
1 | INTRODUCTION
2005), minimization of grain breakage (Igathinathane, Chattopadhyay, & Pordesimo, 2005), as well as an increase in shelf life (Dutta & Mahanta,
Rice is considered one of the principal cereals being produced and con-
2014).
sumed around the world. It is estimated that half of the world’s popula-
The parboiling process usually consists of three stages: soaking the
tion consumes rice on a daily basis (Mir, Bosco, Shah, & Mir, 2016; Oli,
grain until reaching saturation, gelatinization of the starch of the grain
Ward, Adhikari, & Torley, 2014). Given the great economic and nutri-
by heating of the hydrated grains, and drying of the product until an
tional importance of rice, studies dedicated to its various processing
appropriate moisture level for grinding or storage is reached. According
methods are relevant to the development of the final product quality.
to Pascual et al. (2013) and Thammapat, Meeso, and Siriamornpun
Parboiling is one of the processes in which the main stage is the
(2016), hydration has the aim of providing the rice with the necessary
absorption of moisture by rice grains. Numerous studies have confirmed
moisture to induce starch gelatinization. Gelatinization provides mor-
the beneficial changes that the grain undergoes when parboiled, such as
phological transformations in the physical structure of the food (Ali &
alteration of texture (Buggenhout, Brijs, Celus, & Delcour, 2013), altera-
Pandya, 1974; Sittipod & Shi, 2016) involving the elimination of preex-
tion of color (Lamberts, Brijs, Mohamed, Verhelst, & Delcour, 2006),
isting fissures in the grain (Chen & Chen, 1986).
alteration in nutritional content (Paiva et al., 2016; Setyaningsih, Saputro,
Given the importance of the soaking stage in the processing of
Barbero, Palma, & García Barroso, 2015; Storck, Silva, & Alves Fagundes,
parboiled rice, study of the water absorption kinetics over time is of
J Food Process Eng. 2018;41:e12641. https://doi.org/10.1111/jfpe.12641
wileyonlinelibrary.com/journal/jfpe
C 2017 Wiley Periodicals, Inc. V
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particular interest, so as to enable modeling and optimization of such
within 3 hr and subsequently at intervals of 1 hr to complete the tests.
process.
The moisture gain of the samples were evaluated according to AOAC
One way to mathematically model hydration kinetics is the proposi-
International (2000).
tion of lumped parameters models. In this methodology, moisture is considered to be distributed homogeneously inside the food, without the presence of moisture gradients inside the grains (Hangos & Cameron,
2.2 | Theory
2001). Such approach was efficiently used in hydration modeling of peas (Omoto et al., 2009) and soybean (Coutinho, Conceiç~ao, Omoto, Andrade, & Jorge, 2007; Coutinho, Omoto, Andrade, & Jorge, 2005; Nicolin, Neto, Paraíso, Jorge, and Jorge, 2015). On account of the simple nature of these models, the equation proposed by Omoto et al. (2009) enables easy practical application. However, given the fact that the analytical solution of these models has an exponential nature, it can fail in the description of kinetic curves that are nonexponential. An alternative for solving such situations would be the use of a fractional calculus, which generalizes the model and introduces improved performance. The fractional calculus has the characteristic of capturing behavior that traditional calculus often cannot, by considering operators such as
The generalized model of the present study was based on the model originally proposed by Omoto et al. (2009). In their study, the authors proposed a model in which the rate of moisture accumulation, inside spherical pea grains, was caused by the convective flow of moisture around the grains, which were submersed in water. The convective flow of water was given by NA 5KS Aðqeq 2qA Þ, where NA is the flow of water (g/cm2hr), A the surface area of the grain (cm2), qA water concentration in the grain (g/cm3), and qeq water concentration in the grain at the point of equilibrium (g/cm3). Therefore, the obtained mass balance generated a lumped parameters model (Equation 1), in which V designates the volume (cm3) and qA0 the initial moisture of the grain (g/cm3). dqA AKS 5 ðqeq 2qA Þ dt V
integrals and derivatives of arbitrary order (Oldham & Spanier, 1974; Podlubny, 1999). Depending on the fractional order, the parameters of
(1)
this kind of model may lose their physical reality, although generally
The initial condition used for the solution of the model considered
they supply a better representation of the hydration process. For
that the concentration of water in the grains was known at the begin-
kinetic models, for example, this mathematical technique can be carried
ning of the hydration process, such that qA ð0Þ5qA0 .
out considering a derivative of fractional order to represent the varia-
The original work considered spherical geometry for pea grains as
tion of the property to be described. In recent studies on the adsorp-
a hypothesis for the model. However, the grains of rice investigated in
tion kinetics of heavy metals, Friesen, Leitoles, Gonçalves, Lenzi, and
this study were considered cylindrical. Thus, the model for rice grains
jo, Lenzi, Silva, and Lenzi (2013) proLenzi (2015) and Gomes, Arau
becomes:
posed lumped parameters models of fractional order and obtained satisfactory results in the description of the experimental data analyzed. In this context, the present study has the objective of using fractional calculation to generalize the exponential model proposed by
dqA 5kðqeq 2qA Þ dt
(2)
qA ð0Þ5qA0
(3)
Omoto et al. (2009), and of evaluating the benefits of this generalization
ðh1RÞ , h being the length of the cylinder (cm) and R Where k5 2KS hR
on the performance of the original model, when applied to water
the radius of the cylinder (cm). The constant was calculated using the
absorption data of rice grains. Furthermore, it is the intention of this
equations for surface area and for cylinder volume, A52phR12pR2
study to analyze the behavior of the main parameters present in the
and V5pR2 h. A dimensional analysis of the constant k allowed to con-
conventional model and fractional model as a function of temperature at
clude that it has units of hour21 (hr21).
which the hydration process occurs. Omoto’s original equation and the proposed fractional order model were compared statistically in order to identify the best performance in describing the experimental data.
2 | MATERIALS AND METHODS
The model presented by Equations 2 and 3 has an analytical solution as follows: qA ðtÞ5qeq 1ðqA0 2qeq Þe2kt
(4)
The generalization proposed in the present study replaces the first derivative present in Equation 2 with a derivative of arbitrary order (or
2.1 | Obtaining experimental data In order to obtain moisture of rice as a function of time tests were performed with paddy rice variety SCS117CL (0.177 6 0.001 kg/kg dry basis; 1.207 6 0.063 mm radius; and 7.162 6 0.327 mm long) grown in
of fractional order) as presented below: d a qA 5kðqeq 2qA Þ dta
(5)
qA ð0Þ5qA0
(6)
Navegantes, SC, Brazil, belonging to the 2014/2015 harvest. For grain
Where a is the fractional order of the derivative (dimensionless).
moisturizing 1 kg of paddy rice was submitted in triplicate to the hydra-
For practical applications, the best definition for the fractional
tion process done with thermostatic bath (SOLAB/SL-143155/22/Bra-
derivative of order a is the Caputo derivative (Caputo, 1967). This defi-
zil) containing 4 L of distilled water. The test conditions were 35.5,
nition involves the initial values of the differential equation of fractional
45.5, 55.5, and 65.5 8C for 10 hr, in which samples of approximately
order, which makes it adequate for application to practical problems.
20 g (Shimadzu/Scale AY220/Japan) were collected every 30 min
Furthermore, the Caputo derivative has a known Laplace transform
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(Oldham & Spanier, 1974; Podlubny, 1999). The Caputo derivative has the following definition:
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Where / is the objective function (g2/cm6) and Nexp the number of experimental data (dimensionless).
ðt ðmÞ qA ðsÞ da qA 1 5 ds Cða2mÞ ðt2sÞa1m11 dta
The adjustment of the models to the experimental data was qualifor m21
