Received: 9 April 2017
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Revised: 21 July 2017
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Accepted: 11 September 2017
DOI: 10.1111/jfpe.12633
ORIGINAL ARTICLE
Fitting semi-empirical drying models using a tool based on wavelet neural networks: Modeling a maize drying process Carlos Alberto Claumann1 | Adriano Cancelier2 | Adriano da Silva1 | Wu € st Zibetti3 | Toni Jefferson Lopes4 Andre 1 Departamento de Engenharia Química e Engenharia de Alimentos, Universidade Federal de Santa Catarina – UFSC, Campus polis, Santa Catarina, Universitario, Floriano Brasil
Departamento de Engenharia Química – DEQ, Universidade Federal de Santa Maria – UFSM, Santa Maria, Rio Grande do Sul, Brasil
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^ nio Francisco Machado1 Ricardo Anto
Abstract Maize drying is an important process, especially for storage and conservation. For this study, the experimental stage was carried out using a forced convection dryer with air heated at different temperature conditions (306.05–441.85 K) and flow (0.13–0.256 m3/hr), totalizing 15 drying
2
Departamento de Informatica e Estatística – INE, Universidade Federal de Santa Catarina – UFSC, Campus Universit ario, polis, Santa Catarina, Brasil Floriano 3
s-Graduaç~ Programa de Po ao em Engenharia Química – PPGEQ, Universidade Federal do Rio Grande – FURG – Cidade ^nio da Patrulha, Rio Grande Alta, Santo Anto do Sul, Brasil 4
curves. Then the performances of the classic drying kinetics methodology and the approach proposed in this paper, in which the increase in moisture content of the product with time was represented combining exponential models and neural networks based on wavelets, were compared. Good performance was obtained in predictions using the proposed approach. One of the main differentials of the methodology adopted was the obtainment of a model that has a global predictive capacity, within the range of tested operating conditions, which can be used in predicting drying curves for different operating conditions.
Practical applications The drying process is also one of the most widely used methods for preserving food, and has the advantage of reducing the costs of storage and transport because of the low volume and weight of the end product. During the last years, this topic has attracted a broad industrial interest, resulting in
Correspondence W€ Andre ust Zibetti, Departamento de Informatica e Estatística – INE, Universidade Federal de Santa Catarina – polis, Santa Catarina, Brasil. UFSC, Floriano Emails:
[email protected];
[email protected]
many research studies investigating the drying process. Usually, with regard to the classic approach for modeling of the drying process, the kinetics of drying curves obtained in different operating conditions is affected separately, that is, the parameters are estimated independently, resulting in different regression problems. With the classical approach, in general, it is not possible to obtain a comprehensive prediction model with regards to operating conditions. We have proposed an alternative modeling method. Aiming to obtain a modeling tool with an overall predictive ability, an approach for drying kinetics prediction that combines exponential models and neural networks was proposed. The proposed modeling method was able to predict drying curves for different operating conditions.
~ oz, grants safe preservation (13–14.5%) (George, 2011; Martinello, Mun
1 | INTRODUCTION
& Giner, 2013). At the time of harvest, which occurs when the maize Maize (Zea mys L.) is a crop culture that originates from Central Amer-
grain reaches its physiological maturation, the grain moisture content
ica, and currently it is the most largely produced cereal in the world.
can reach 30% (Johnson, 2000).
According to the U.S. Department of Agriculture, the world production
Mechanical drying consists of subjecting the product to the action
from 2014 and 2015 totaled 1009.7 million tons, of which Brazil
of a hot air stream that passes through the mass of grains, which can
accounted for 85 million. (U.S. Department of Agriculture, 2016). Maize
also be understood as the activity intended to artificially lower the
is of great importance for global economics, representing 65% of the
humidity of the grains; this process can occur in various forms and in
food that is produced for animal consumption, and 15% of the food
complex ways until it reaches the desired limit (Hossain, Bala, & Satter,
produced for human consumption. Also, maize is used as raw material
2003; Maier & Saksena, 1997; Martinello et al., 2013). The drying pro-
for fuel production (Cunha & Fernandes, 2011; Facco et al., 2015; Sun
cess is also one of the most widely used methods for preserving food,
et al., 2016). The process of maize drying is used to guarantee that the
and has the advantage of reducing the costs of storage and transport
appropriate level of moisture for grain storage be obtained, a level that
because of the low volume and weight of the end product.
J Food Process Eng. 2017;e12633. https://doi.org/10.1111/jfpe.12633
wileyonlinelibrary.com/journal/jfpe
C 2017 Wiley Periodicals, Inc. V
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F I G U R E 1 Schematic representation of the experimental apparatus, consisted of a blower (1), an electric heater (5), an orifice flow meter (4) connected to a inclined manometer (3), copper–constantan thermocouples and a measuring cell (6)
Various phenomena related to mass and heat transfer are involved in
(Brazil), consisted of a blower (1), an electric heater (5), an Unochapeco
dehydration processes. Mass transfer kinetics is dependent on control
orifice flow meter (4) connected to an inclined manometer (3), copper–
factors such as temperature and drying air flow rate (Mariani, Perussello,
constantan thermocouples and a measuring cell (6). The cell consisted
rul & Pehlivan, 2004). MathematiCancelier, Lopes, & da Silva, 2014; Tog
of a square of 12 3 1022 m and height 2.0 3 1022 m, and metallic
cal models can aid in the optimization of the conditions for the drying
screens at the two extremities. The system was thermally isolated. The
process in order to minimize operating costs and thermal degradation of
thermocouples used for measuring the drying air temperature, and the
the product (Hossain et al., 2003; Kumar, Karim, & Joardder, 2014). While
dry and the wet bulb temperatures were calibrated by means of a ther-
there are theoretical models representing the phenomena associated
mostatic bath and a mercury thermometer with precision of 0.1 K. The
with food drying processes (Karim & Hawlader, 2005; Perussello, Kumar,
thermocouples were positioned in the air feeding tube.
de Castilhos, & Karim, 2014; Zhu & Shen, 2014), empirical boundary layer
The experimental procedure began with the selection of maize
models based on analysis of experimental data—such as Page’s exponen-
(Zea mays L.) grains of similar sizes. The maize grains were subjected to
tial models (1949) and Henderson and Pabis (1961) (Henderson & Pabis,
a slow artificial remoistening process, through contact with nearly satu-
1961; Page, 1949)—are also used to predict the evolution of the product’s
rated ambient air. Initially the system was adjusted to the desired
moisture content over time; such models were used by several authors
operational conditions (speed and air temperature into the drying
(Babalis, Papanicolaou, Kyriakis, & Belessiotis, 2006; Goyal, Kingsly, Mani-
chamber). Next, the dry and wet bulb temperatures were measured. As
kantan, & Ilyas, 2007; Madhiyanon, Phila, & Soponronnarit, 2009), who
soon as the desired conditions were reached, the measuring cell was
obtained good results in their studies, eliminating the complex analysis of
inserted into the equipment, initiating the experiment (time zero). The
the interactions between the components of the food matrix and the
initial temperature of the maize grains (withdrawn from the humidifier)
changes in their thermo-physical properties during drying (Mariani et al.,
was equal to that of the drying air. The mass of the measuring cell was
2014). There are also studies that apply predictive models for heat and
periodically determined (every 1 min) using an analytical balance with a
mass transfer using artificial neural networks, where the objective is to
precision of 0.01 g. After the experiment, the grains were crushed and
obtain moisture from the change in kinetic predictions about the tempera-
left in an oven at 343.15 K for at least 24 hr or until they were com-
ture during drying, such as the drying of cassava and mangoes (Hern andez-
pletely dry, in order to obtain the dry matter (Mariani et al., 2014).
rez, García-Alvarado, Trystram, & Heyd, 2004), carrots (Erenturk & Pe
Drying curves were obtained for the operating conditions
Erenturk, 2007), tomatoes (Movagharnejad & Nikzad, 2007), organic
described in Table 1, being that the operating temperature (T) varied in
solid waste (Perazzini, Freire, & Freire, 2013), among others.
the range of 306.05 to 441.85 K and the air flow (Q) between 0.13 and
Usually, with regards to the classic approach for modeling of the drying process, the kinetics of drying curves obtained in different operating conditions is affected separately; that is, the parameters are esti-
0256 m3/h, amounting to 15 drying curves. Tables A1–A4 in Appendix show the drying curves under operating conditions described and indexed in Table 1.
mated independently, resulting in different regression problems. With the classical approach, in general, it is not possible to obtain a compre-
3 | CLASSIC DRYING MODELING PROCESS
hensive prediction model with regards to operating conditions. In this study, with the aim of obtaining a modeling tool with an
The mathematical description of the drying process is based on physical
overall predictive ability, an approach for drying kinetics prediction that
mechanisms of heat transfer and mass transfer. Waananen et al. (1993)
combines exponential models and neural networks was proposed.
presented an extensive review of models applied to the drying of porous solids (Waananen, Litchfield, & Okos, 1993). Such models are
2 | MATERIALS AND METHODS
useful to describing the processes of drying for the purposes of framing, analysis, and optimization.
The schematic diagram of the apparatus used for the convective hot air
Fick’s second law has been applied in classic modeling processes
drying experiment is shown in Figure 1. The experimental setup,
of food drying to describe the moisture diffusion inside an isotropic
located at the Laboratory of Chemical Engineering from the University
pez, Co rdova, Rodríguez-Jimenes, & homogeneous material (Ruiz-Lo
CLAUMANN
TA BL E 1
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ET AL.
Operational conditions used in drying experiments
T AB LE 2
3 of 12
Drying kinetics equation
Essay
Air flow (m3/hr)
Air temperature (K)
Name
Model
1
0.1300
311.85
Lewis
MR5exp ð2a1 tÞ
(1)
2
0.1300
369.35
Page
MR5exp ð2a1 t Þ
(2)
3
0.1300
396.55
Logarithmic
MR5u1 exp ð2a1 tÞ1d
(3)
4
0.1300
441.85
Two terms
MR5u1 exp ð2a1 tÞ1u2 exp ð2a2 tÞ
(4)
5
0.1824
306.05
6
0.1824
352.75
Henderson & MR5u1 exp ð2a1 tÞ1u2 exp ð2a2 tÞ1u3 exp ð2a3 tÞ (5) Pabis modified
7
0.1824
368.65
8
0.1824
439.95
9
0.2220
307.22
10
0.2220
348.65
2. The model must present monotonically decreasing behavior due
11
0.2220
389.45
to time, which is physically consistent with the particles mass loss.
12
0.2560
308.65
13
0.2560
341.55
14
0.2560
356.35
15
0.2560
381.15
b1
MR5u1 exp ð2a1 tb1 Þ1u2 exp ð2a2 tb2 Þ
This work
(6)
1. The relative humidity should be unitary at the beginning of drying (initial condition of the model);
Mathematically, the coefficients a1, a1, and a3 present in models described by Equations 1–6 must be positive; 3. For sufficiently long times the moisture content predicted by the model must tend to small values, consistent with the residual moisture; 4. In general, the more moisture is removed from the product to be dried, the harder it is to subsequently remove further moisture.
García-Alvarado, 2004). The obtainment resulting from using these
This is due to the decrease of water activity over time during dry-
models assumed that there is no shrinkage, that the water diffusivity of
ing, which causes the solid to present a growing tendency to retain
the material is constant, and that the drying process is controlled by
residual water. Mathematically, it follows that the second order
diffusion (Crank, 1975; Pakowski & Mujumdar, 1995).
derivative of the drying curve must be positive, that is, it should
The analytic solution consists of series expansion of the exponen-
display concave behavior.
tials that depends on the diffusion coefficient. Based on such a solution, several semi-empirical models have been proposed in the literature
In the case of the models given by Equations 1–6, the properties described
pez, Montero, & (Table 2), according to Equations 1–5 (Celma, Rojas, Lo
will be met if the coefficients ai, bi, and ui are positive. If not, such proper-
Miranda, 2007; Erenturk, Gulaboglu, & Gultekin, 2004; Madhiyanon
ties can still be met depending on the combinations of these parameters.
et al., 2009; Panchariya, Popovic, & Sharma, 2002), depend on the parameters that are to be estimated from experimental data. Addition-
3.2 | Parameter estimation
ally, the model described by Equation 6 was proposed in this paper.
The identification of the drying process using the models described in
The models concerning Equations 1–6 are all based on series expansion of exponentials; however, they differ in the number of terms and parameters. As the moisture content ratio is unitary for the initial time, the
Section 3 consists in estimating these parameters optimizing an appropriate n performance index. Usually, the minimization of a criterion based on the approach of squared error is assumed, as described by Equation 8.
sum of coefficients ui preceding the exponential is also unitary; that is,
n X 2 MRExp;i 2MRMod;i ði51; . . . :nÞ F0 5
there is a restriction to the values of the coefficients u in a way that the
(8)
i51
number of free parameters is inferior to the total number of parameters. As an example, although the model given by Equation 5 presents a total number of six parameters, only five are free. In this case, there is a restriction (Equation 7) where, u1 1u2 1u3 51
(7)
therefore, one of the ui can be obtained from the other two. Table 3 shows the total number of free parameters of each model, as well as the total
T AB LE 3
Drying model characteristics
Model
Total number of parameters
Number of free parameters
Number of terms
Equation 1
1
1
1
Equation 2
2
2
1
Equation 3
3
2
2
Equation 4
4
3
2
Equation 5
6
5
3
Equation 6
6
5
2
number of terms.
3.1 | Desirable properties of drying models A suitable model to describing the drying process must attend to a series of properties, and the following can be cited:
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described in Section 3; nonetheless, by using a neural network, there is no guarantee that such properties are going to be met. What happens is that the parameters of the assessment stage (training) are performed using only the data, ignoring prior physical knowledge of the process. Instead of directly using a network trained with one output (MR) and three inputs (t, Q, and T), another approach was proposed, using a drying model (in principle any of the ones described in Section 3 could be chosen) and applying neural networks to describe each of these parameters. As an example, Figure 2 shows the application of the proposed approach for the case of the Equation 6. As it can be seen from Figure 2, the model depends directly on the drying time. Additionally, each of these parameters is described by a block that contains a (small) neural network due to the flow rate and the operating temperature. As mentioned, in principle any of the models described in Section 3 could be used to identifying the drying curves; however, this approach (semi-empirical model 1 neural networks), worked only with the model given by Equation 6, which performed best for the approximation of FIGURE 2
Proposed approach for general modeling of drying curves
drying curves separately (results will be presented in Section 5.2). Each block shown in Figure 2 contains, in addition to a neural net-
where MRExp,i and MRMod,i are the experimental moisture content ratio, and predicted by any model, respectively, for the ith data point; n is the number of experimental measurements for a given condition of Q and T.
work, operations necessary for the normalization and denormalization of the data. Additionally, it should be ensured that the value predicted for each parameter has physical meaning. Figure 3 shows, in detail, the components present in a block.
It is noteworthy that the sum of the approach square error is related to the coefficient of determination R2, so that minimizing the
In Figure 3, the block is split in smaller blocks; as follows:
first implies the maximization of the latter. Additionally, the parameter
1. Data Normalization. Each input to the neural network must be
estimation means that in non-linear optimization problems must be
standardized in a working range. In this case, the range [0, 1] was
solved numerically. The computational implementation for this study
adopted for operating conditions Q and T;
was made in MATLAB, and the function “fmincon,” present in the MATLAB optimization toolbox, was used. In terms of the resulting optimization problem, each curve is treated separately from the other, resulting in different regressions. Thus, an independent set of parameters for each curve is obtained.
4 | MODELING DRYING CURVES USING NEURAL NETWORKS Neural networks are characterized by being universal data approximators. In the context of identification of drying curves, several studies can be found in the literature (Çakmak & Yıldız, 2011; Ge & Chen, 2014; Huang & Chen, 2015; Karimi, Rafiee, Taheri-Garavand, & Karimi, 2012). If an approach similar to the one described in such references had been applied for the present case, a neural network should have been used to approach relative moisture due to drying, air flow and the operation temperature time. Such mapping can be described by a function f as shown in Equation 9. MR5fðt; Q; PÞ
(9)
Neural networks, in general, present a lot of free parameters, which grants them with high approximation power and adaptability; however, such flexibility can also bring disadvantages. For example, it would be desirable that a methodology for modeling respects the properties
Details of a block used to predict the value of a parameter of the model
FIGURE 3
CLAUMANN
ET AL.
FIGURE 4
Scaling function and its wavelet (Citadin et al., 2016)
|
n5N Xf
2. Calculation of the output of the neural network (NN). This step
fðxÞ5
uses the neural network to provide the output given one or more
dn Un ðxÞ
5 of 12
(10)
n51
operating conditions of Q and T; where Nf: number of scaling functions used in the expansion. 3. Saturation. Saturates the value predicted by the neural network in a range of interest. The track between 21 to 1 was admitted;
The different U (indexed by n) represent shifts (translations) of the originally defined scaling function. As an example, Figure 5 shows two
4. Denormalization or conversion. A reverse step from the first one.
possible expansions in series scaling function (in this case, quadratic
At this point the obtained value becomes the parameter to be
splines) with different values of Nf. Figure 5a and b illustrate the case
used in the semi-empirical model drying. The 21 value of Step 3 is
with Nf 5 3 and 8, respectively.
mapped to the minimum physically acceptable value (LB) and the
As can be seen in Figure 5, with the increase in the Nf value, it
upper end 1 is mapped to the maximum value of the physically
follows that the functions of the expansion become more localized
acceptable parameter (UB)
and, therefore, the ability to approach the associated neural net-
It is noteworthy that the joint action of the stages of saturation and conversion ensures that the predicted value for the parameter has physical meaning, that is, it respects the desired properties for a drying model.
4.1 | Neural networks for predicting the parameters of the drying model
work increases. However, the risk with overtraining problems also increases. In principle, the Nf value is not known and must be determined in order to obtain a good identification performance. A good starting point is to begin with a small Nf (for example, 3, as shown in Figure 5a), and to verify that this expansion reaches the desired level of approximation. For this study, the quadratic spline function was used as a generator for the series expansion of functions. The quadratic spline was cho-
The most commonly used network topologies for modeling processes
sen because it presents soft behavior, and continued until the first
and time series are the feedforward type and variants (Fine, 1999; Hay-
derivative.
kin, 2008). These topologies can contain multiple hidden layers
The extended application to the multidimensional case can be eas-
between the inputs and outputs, as well as a mechanism with complex
ily made, and the most used technique is the product of the functions
connections between neurons.
of each coordinate (Alexandridis & Zapranis, 2013; Eslamimanesh,
Although feedforward networks could have been used for this
Gharagheizi, Mohammadi, & Richon, 2011). As an example, considering
study, we chose to implement networks based on wavelets (Alexandri-
a network whose input are Q and T and admitting three functions per
dis & Zapranis, 2013; Zhang & Benveniste, 1992). There are several
coordinate, the result is multidimensional scaling functions that would
types of neural networks based on wavelets, as described in (Strang &
be formed by the combination of all one-dimensional functions related
Nguyen, 1996). The one used for this study is described in (Citadin
to Q and T. In this case, the respective neural network can be written
et al., 2016), and it presents a simple topology that is equivalent to an
according to Equation 11.
series expansion functions. There are many types of W wavelets and for each of them there is an associated U scaling function (Alexandridis & Zapranis, 2013). For example, in the case of the quadratic spline wavelet, functions U and W are shown in Figure 4.
lðQ; TÞ5c1 U1 ðQÞU1 ðTÞ1c2 U1 ðQÞU2 ðTÞ1c3 U1 ðQÞU3 ðTÞ 1c4 U2 ðQÞU1 ðTÞ1c5 U2 ðQÞU2 ðTÞ 1c6 U2 ðQÞU3 ðTÞ1c7 U3 ðQÞU1 ðTÞ
(11)
1c8 U3 ðQÞU2 ðTÞ1c9 U3 ðQÞU3 ðTÞ
Series expansion with Wavelets, scale functions or both types of
As can be observed in Equation 11, such neural network has nine
functions can be used to approximate data (Citadin et al., 2016). For
parameters to be estimated by way of training (c1, c2,. . . c9). Functions
this study was used neural networks containing only scaling functions.
U1, U2, and U3 are the same as those shown in Figure 5a for the case
For the case with the single entry, it can be assumed that such a net-
of Nf 5 3, considering the application normalized area in the [0, 1]
work can be described in accordance with Equation 10:
range. This section presented a short description of networks based on
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(a) FIGURE 5
ET AL.
(b)
Illustration of two different series expansions in scaling functions
wavelets. Further details can be found in the work of Citadin (Citadin
necessary to adopt such an approach to get a model that respects the
et al., 2016).
properties described in Section 3.
4.2 | Training of the semi-empirical model 1 neural networks
Equation 6 model, in principle, it could be derived from a training
As described, the various neural networks contained in the model are
of the values of the coefficient of determination could be obtained
With the aim to training the neural networks associated with the method similar to backpropagation (Kim, Jung, Kim, & Park, 1996) to estimate weights; however, it was observed that good results in terms
postulated as functions of flow rate and operating temperature, which
directly from the application of the “fmincon” function contained in
results in two-dimensional neural networks. After several attempts, it
MATLAB. For this reason, in the same way as it was for the parameter
was determined that three scale functions for Q and T were sufficient
estimation problem described in Section 4.1, the “fmincon” function
to produce a model capable of identifying the drying curves, which
was used to minimize the quadratic error criterion approach.
means that each network associated with prediction of a parameter can be described by Equation 11.
The calculation of the parameters of the model (transformed into a matrix multiplication) and the evaluation of the proper models consti-
The only difference between networks associated to diverse
tute a quick procedure, and it is noteworthy that the training of the
parameters is the values of the weights, since the activation functions
model based on neural networks can be made in seconds in a regular
are the same. An important feature is that the values of such activation
computer.
functions can be calculated prior to training, being stored as a matrix
It is worth noting that, had multilayer networks of the feedforward
for use during this process. For example, if a dataset for training with
type been used, the nonlinearities present would have to have been
12 drying curves are considered, the computer implementation of neu-
assessed during the training, which would significantly increase compu-
ral networks would be based on a single matrix with dimension 12
tational time.
(number of operating conditions) and 9 (number of activation functions). In this case, to obtain the value of a particular model parameter
5 | RESULTS AND DISCUSSION
(Equation 6), it suffices to multiply the referred matrix by its weight vector.
5.1 | Classic modeling
With respect to the resulting regression problem, it follows that
The models described in Section 3 were used to identify the drying
the total number of variables to be optimized is equal to the sum of
data; shown in Table 4 are the performance results (values of R2), solely
the number of each network weights. As function object, in the same
for the two best models, indicated by Equation 6, followed by the one
way as it was for Equation 8, the sum of the square error was used
associated with Equation 5. As expected, due to the similarity between
though, considering the overall approximation error for all the drying
Equations 1–6, models with better performance were those with the
curves simultaneously.
highest number of free parameters.
The optimization problem described in this section is considerably
According to Table 4, comparing the Equations 6 and 5 models, it
larger than that explained in Section 4.1. Considering the approach that
is possible to observe that a significantly superior performance was
uses neural networks, the number of variables is higher and, for an
obtained with the first, even if both models have the same number of
evaluation of the function object, all the drying curves considered in
free parameters (in this case, 5). The reason for this difference is associ-
the training dataset must be evaluated. As described above, it was
ated with significant differences of the regressor used.
CLAUMANN
TA BL E 4
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ET AL.
Comparison between the performances of the models
T AB LE 5
7 of 12
Results at training and validation
Essay
R2 for Equation 6
R2 for Equation 5
Essay
R2 at training
R2 at testing
1
0.99953
0.99560
1
0.99633
–
2
0.99958
0.99795
2
0.99675
–
3
0.99920
0.99243
3
0.99390
–
4
0.99904
0.99677
4
0.99481
–
5
0.99960
0.99563
5
0.99214
–
6
0.99928
0.99660
6
0.99235
–
7
0.99973
0.99883
7
–
0.99574
8
0.99914
0.99818
8
–
0.99788
9
0.99926
0.99798
9
0.99680
–
10
0.99959
0.99820
10
0.98532
–
11
0.99989
0.99863
11
0.99344
–
12
0.99768
0.99027
12
0.98857
–
13
0.99990
0.99911
13
0.99870
–
14
0.99994
0.99845
14
–
0.99892
15
0.99874
0.99435
15
0.99755
–
In the case of Equation 6 the regressors are [exp(2a1tb1), exp
As described, the parameters of each drying curve were estimated
(2a2tb2)] and for Equation 5 they are [exp(2a1t), exp(2a2t), exp
separately. In this case, it is interesting to analyze the behavior of a sin-
(2a3t)]. All the regressors are based on exponentials, and it is necessary
gle parameter; however, estimated at different operating conditions. As
to ensure linear independence, that the internal parameters of the
an example, Figure 6 shows the surface of parameter a1 (model refer-
exponential are significantly different. For example, in the case Equa-
ent to Equation 6), depending on the Q and T of each operation.
tion 5 if a1 5 a2 5 a3 then all terms would be linearly dependent (in this case, equal) and could be replaced by only one regressor.
In Figure 6, the values of a1 associated with the same flow were connected to ease understanding. It is possible to observe the lack of
In terms of the number of internal parameters contained in each
regularity between the values of a1 for different operating conditions,
exponential, Equation 6 has two (ai and bi) and Equation 5 only one
that is, the graph displays an aleatory behavior and not suitable for
(ai). This allows for greater independence between the regressor in the
interpolation. The same behavior due to Q and T was observed for all
first case, which explains the better performance obtained with Equa-
other estimated parameters.
tion 6 in terms of approaching capacity, when compared with the
This aleatory behavior can be attributed to the separate regression
model of Equation 5. The result is significant because Equation 6 has a
of the drying curves. In a given problem of estimation of various param-
lower number of terms than Equation 5.
eters, there may be an approximation error compensation, being that different sets of parameters can result in almost the same value of the approximation criteria. This results in a difficulty in relating what happens to certain parameter in different regressions, considering different sets of data. Many studies described in the literature typically treat parameter estimation of drying curves separately (Babalis et al., 2006; Goyal et al., rul & Pehlivan, 2004; Zhu & Shen, 2007; Madhiyanon et al., 2009; Tog 2014). However, by what was exposed above, such a procedure is not justified.
5.2 | Modeling using the approach based on neural networks By making the identification using neural networks, data should be split FIGURE 6
and T
Surface of the parameter a1 due to the operation Q
into groups, one for training and another for testing. The first serves the purposes of estimating the weights, and the latter, considering the
|
8 of 12
CLAUMANN
8 that the surface of the parameter a1 due to Q and T, which displays
1 7
8
14
0.9
smoother behavior, unlike what occurred with the same parameter
0.8
when it was estimated from separate curves, as shown in Figure 6. The same smooth behavior in terms of Q and T was observed for
0.7
other parameters predicted by the neural networks. Again, this vali-
M
0.6 UR
ET AL.
dates the methodology proposed, for a comprehensive model for pre0.5
dicting drying curves.
0.4 0.3
6 | CONCLUSION
0.2
In this study, the application of a modeling tool for predicting maize
0.1 0
0
10
20
30
40 sample
50
60
70
80
drying kinetics was evaluated, combining exponential models and neural networks based on wavelet.
Performance evaluation of the global model for the testing dataset FIGURE 7
The developed modeling tool has overall predictive ability, within the range of tested operating conditions of flow and the temperature of the drying air, and can be used in kinetic drying prediction in differ-
fixed weights, is used to assess the ability of generalization and verifi-
ent operating conditions from those used in the training of neural net-
cation of the occurrence of over-training problems.
works associated with the parameters of the model.
From the 15 curves described in Tables A1–A4 of Appendix, 12
Considering the use of neural networks based on wavelets, the cal-
were chosen for training (curves 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 15)
culation of the drying kinetics was a quick procedure, and it was
and three for the test (curves 7, 8, and 14).
observed that the training of the model based on neural networks can
Table 5 shows the performance results for training and testing
be accomplished in a matter of seconds in a standard computer.
data, in terms of the values of R2. As can be seen, the coefficients of determination for the testing data are similar to those observed for the training data; this shows that the proposed methodology resulted in
ORC ID Toni Jefferson Lopes
http://orcid.org/0000-0001-6210-4508
networks with good generalization capability. Comparing Table 4 (R2 column for Equations 6 and 5), it is notable that R2 values decreased substantially in the last case. However, the performance obtained with the overall model is sufficient for prediction purposes of drying curves. Figure 7 shows a comparison between the prediction from the semi-empirical model 1 neural network, in the case of the testing data set, which confirms the prediction capability of the proposed method. The parameters contained in the global model were predicted by small neural networks (only nine activation functions each) and, therefore, it is expected that there are no abrupt changes in parameter values for different operating conditions. As an example, it is shown in Figure
0.6 α1
Alexandridis, A. K., & Zapranis, A. D. (2013). Wavelet neural networks: a practical guide. Neural Networks : The Official Journal of the International Neural Network Society, 42, 1–27. Babalis, S. J., Papanicolaou, E., Kyriakis, N., & Belessiotis, V. G. (2006). Evaluation of thin-layer drying models for describing drying kinetics of figs (Ficus carica). Journal of Food Engineering, 75, 205–214. Çakmak, G., & Yıldız, C. (2011). The prediction of seedy grape drying rate using a neural network method. Computers and Electronics in Agriculture, 75, 132–138. pez, F., Montero, I., & Miranda, T. (2007). ThinCelma, A. R., Rojas, S., Lo layer drying behaviour of sludge of olive oil extraction. Journal of Food Engineering, 80, 1261–1271. €st Zibetti, A., Marangoni, A., Bolzan, Citadin, D. G., Claumann, C. A., Wu A., & Machado, R. A. F. (2016). Supercritical fluid extraction of Drimys angustifolia Miers: Experimental data and identification of the dynamic behavior of extraction curves using neural networks based on wavelets. The Journal of Supercritical Fluids, 112, 81–88.
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Crank, J. (1975). The mathematics of diffusion (2nd ed.). Ely House, London, UK: Oxford University Press.
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Cunha, S. C., & Fernandes, J. O. (2011). Multipesticide residue analysis in maize combining acetonitrile-based extraction with dispersive liquidliquid microextraction followed by gas chromatography-mass spectrometry. Journal of Chromatography A, 1218, 7748–7757.
450
400
350 T [K]
FIGURE 8
R EFE R ENC E S
300
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Q [m3/h]
Surface of parameter a1 due to the operation Q and T
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of temperature and velocity on kinetic parameters. Journal of Food Process Engineering, 37, 619–627. https://doi.org/10.1111/jfpe. 12118 ~oz, D. J., & Giner, S. A. (2013). Mathematical modMartinello, M. A., Mun elling of low temperature drying of maize: Comparison of numerical methods for solving the differential equations. Biosystems Engineering, 114, 187–194. Movagharnejad, K., & Nikzad, M. (2007). Modeling of tomato drying using artificial neural network. Computers and Electronics in Agriculture, 59, 78–85. Page, G. E. (1949). Factors influencing the maximum of air drying shelled corn in thin layer. Purdue University. Pakowski, Z., & Mujumdar, A. S. (1995). Basic process calculations in drying. In A. S. Mujundar (Ed.), Handbook of industrial drying (2nd ed.). Marcel Dekker: New York, US. Panchariya, P. C., Popovic, D., & Sharma, A. L. (2002). Thin-layer modelling of black tea drying process. Journal of Food Engineering, 52, 349–357. Perazzini, H., Freire, F. B., & Freire, J. T. (2013). Drying Kinetics prediction of solid waste using semi-empirical and artificial neural network models. Chemical Engineering & Technology, 36, 1193–1201. Perussello, C. A., Kumar, C., de Castilhos, F., & Karim, M. A. (2014). Heat and mass transfer modeling of the osmo-convective drying of yacon roots (Smallanthus sonchifolius). Applied Thermal Engineering, 63, 23–32. pez, I. I., Co rdova, A. V., Rodríguez-Jimenes, G. C., & GarcíaRuiz-Lo Alvarado, M. A., (2004). Moisture and temperature evolution during food drying: effect of variable properties. Journal of Food Engineering, 63, 117–124. Strang, G., & Nguyen, T. (1996). Wavelets and filter banks (2nd ed.). Wellesley-Cambridge Press. Sun, L., Liu, S., Wang, J., Wu, C., Li, Y., & Zhang, C. (2016). The effects of grain texture and phenotypic traits on the thin-layer drying rate in maize (Zea mays L.) inbred lines. Journal of Integrative Agriculture, 15, 317–325. _ & Pehlivan, D. (2004). Modelling of thin layer drying kinetics To grul, IT., of some fruits under open-air sun drying process. Journal of Food Engineering, 65, 413–425. USDA - United States Department of Agriculture. (2016) Grain: World Markets and Trade Report, Erie, KS. Waananen, K. M., Litchfield, J. B., & Okos, M. R. (1993). Classification of drying models for porous solids. Drying Technology, 11, 1–40. Zhang, Q., & Benveniste, A. (1992). Wavelet networks. IEEE Transactions on Neural Networks and Networks, 3, 889–898. Zhu, A., & Shen, X. (2014). The model and mass transfer characteristics of convection drying of peach slices. International Journal of Heat and Mass Transfer, 72, 345–351.
How to cite this article: Claumann CA, Cancelier A, da Silva A, Zibetti AW, Lopes TJ, Machado RAF. Fitting semi-empirical drying models using a tool based on wavelet neural networks: Modeling a maize drying process. J Food Process Eng. 2017; e12633. https://doi.org/10.1111/jfpe.12633
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CLAUMANN
A P P E N DI X
T AB LE A1
(Continued)
Essay 1
Appendix contains drying data for the operating conditions described
ET AL.
Essay 2
Essay 3
t [min]
MR
t [min]
MR
160
0.2490
84
0.1580
It can be seen in Tables A1–A4 that the drying time decreases
170
0.2424
87
0.1478
with increasing flow rate and operating temperature, which was
180
0.2240
90
0.1367
190
0.2055
93
0.1265
200
0.1844
96
0.1178
allocated in a regular grid (equidistant points) in terms of the operating
210
0.1746
99
0.1091
temperature.
220
0.1594
102
0.1019
230
0.1410
105
0.0956
240
0.1331
115
0.0752
250
0.1192
125
0.0641
260
0.1054
135
0.0511
270
0.0935
145
0.0375
300
0.0547
155
0.0249
330
0.0323
165
0.0143
180
0.0081
t [min]
Essay 4 MR
t [min]
MR
in Table 1. The data were grouped in different Tables A1–A4 corresponding to different air flow rates used.
expected due to the increased drying rates. To avoid extreme drying rates, experiments for combinations of higher temperatures and flow were not conduced. For this reason, the experimental points were not
TA BL E A 1
Drying curves for experiments 1–4 (Flow rate 0.13 [m3/hr])
Essay 1
Essay 2
Essay 3
Essay 4
t [min]
MR
t [min]
MR
t [min]
MR
t [min]
MR
0
1.0000
0
1.0000
0
1.0000
0
1.0000
5
0.9401
3
0.9027
5
0.7641
8
0.6810
10
0.8841
6
0.7982
10
0.6583
13
0.5358
15
0.8300
9
0.7145
15
0.5883
18
0.4084
20
0.7938
12
0.6593
20
0.5268
23
0.3414
25
0.7523
15
0.6114
25
0.4673
28
0.2941
30
0.7154
18
0.5684
30
0.4129
33
0.2597
35
0.6910
21
0.5306
35
0.3824
38
0.2181
40
0.6502
24
0.4972
40
0.3405
43
0.1935
45
0.6317
27
0.4677
45
0.3105
48
0.1685
50
0.6041
30
0.4406
50
0.2867
53
0.1395
55
0.5784
33
0.4150
55
0.2538
58
0.1185
60
0.5534
36
0.3908
60
0.2266
63
0.0993
65
0.5277
39
0.3661
65
0.1847
68
0.0778
70
0.5092
42
0.3283
70
0.1680
73
0.0613
75
0.5007
45
0.3225
75
0.1475
78
0.0470
80
0.4888
48
0.3051
80
0.1294
83
0.0313
85
0.4743
51
0.2887
85
0.1194
88
0.0237
90
0.4519
54
0.2746
90
0.0861
93
0.0126
95
0.4295
57
0.2620
95
0.0761
98
0.0077
100
0.4104
60
0.2490
100
0.0708
103
0.0050
105
0.3979
63
0.2359
105
0.0675
118
0.0001
110
0.3801
66
0.2228
120
0.0532
115
0.3663
69
0.2107
135
0.0356
120
0.3511
72
0.1996
150
0.0284
130
0.3340
75
0.1885
180
0.0036
140
0.3109
78
0.1783
150
0.2833
81
0.1682
T AB LE A2
Drying curves for experiments 5–8 (Flow rate 0.1824
[m3/hr]) Essay 5
Essay 6
Essay 7
Essay 8
t [min]
MR
t (min)
MR
t (min)
MR
t (min)
MR
0
1.0000
0
1.0000
0
1.0000
0
1.0000
5
0.8703
5
0.7996
5
0.7964
5
0.7334
10
0.8159
10
0.7132
10
0.6771
10
0.5917
15
0.7655
15
0.6395
15
0.5919
15
0.4486
20
0.7163
20
0.5727
20
0.5196
20
0.4073
25
0.6779
25
0.5186
25
0.4650
25
0.3321
30
0.6370
30
0.4688
30
0.4136
30
0.2717
35
0.6076
35
0.4269
35
0.3696
35
0.2264
40
0.5731
40
0.3893
40
0.3328
40
0.1856
45
0.5475
45
0.3601
45
0.2933
45
0.1529
50
0.5175
50
0.3283
50
0.2645
50
0.1234
55
0.4932
55
0.3002
55
0.2374
55
0.1016
60
0.4696
60
0.2779
60
0.2139
60
0.0829
65
0.4447
65
0.2514
65
0.1886
65
0.0738
70
0.4216
70
0.2302
70
0.1700
70
0.0612
75
0.3993
75
0.2048
75
0.1567
75
0.0525
80
0.3756
80
0.1936
80
0.1381
80
0.0490
85
0.3552
85
0.1693
85
0.1327
85
0.0403 (Continues)
(Continues)
|
CLAUMANN
ET AL.
TA BL E A 2
(Continued)
Essay 5
T AB LE A3
Essay 6
Essay 7
Essay 8
11 of 12
(Continued)
Essay 9
Essay 10
Essay 11
t [min]
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
90
0.3373
90
0.1560
90
0.1150
90
0.0303
70
0.4082
70
0.2054
70
0.0335
95
0.3175
95
0.1380
95
0.1070
95
0.0247
75
0.3852
75
0.1851
75
0.0250
100
0.2951
100
0.1258
100
0.0986
100
0.0186
80
0.3686
80
0.1671
80
0.0179
105
0.2779
105
0.1104
105
0.0888
105
0.0151
85
0.3469
85
0.1496
85
0.0127
110
0.2619
110
0.0956
110
0.0760
115
0.0047
90
0.3284
90
0.1316
90
0.0086
115
0.2465
115
0.0813
115
0.0711
95
0.3125
95
0.1164
100
0.0001
120
0.2299
120
0.0733
120
0.0653
100
0.2952
100
0.1030
125
0.2165
125
0.0595
125
0.0547
105
0.2748
105
0.0947
130
0.2037
130
0.0447
130
0.0494
110
0.2607
110
0.0841
135
0.1897
135
0.0394
135
0.0427
115
0.2448
115
0.0768
140
0.1762
140
0.0335
140
0.0378
120
0.2314
120
0.0662
145
0.1622
145
0.0192
150
0.0329
125
0.2148
130
0.0472
150
0.1494
150
0.0134
160
0.0227
130
0.2014
140
0.0339
160
0.1258
155
0.0097
170
0.0179
135
0.1924
150
0.0122
170
0.1047
165
0.0001
200
0.0014
140
0.1816
160
0.0007
180
0.0849
145
0.1682
190
0.0689
150
0.1554
200
0.0535
155
0.1465
210
0.0420
160
0.1350
240
0.0037
165
0.1299
170
0.1190
175
0.1114
180
0.1050
190
0.0884
TA BL E A 3
Drying curves for experiments 9–11 (Flow rate 0.2220
[m3/hr]) Essay 9
Essay 10
Essay 11
t (min)
MR
t (min)
MR
t (min)
MR
200
0.0750
0
1.0000
0
1.0000
0
1.0000
210
0.0552
5
0.9176
5
0.7773
5
0.7921
240
0.0118
10
0.8551
10
0.6790
10
0.6455
250
0.0003
15
0.7925
15
0.5928
15
0.5264
20
0.7466
20
0.5305
20
0.4223
25
0.7006
25
0.4826
25
0.3297
30
0.6559
30
0.4411
30
0.2534
Essay 12
Essay 13
Essay 14
Essay 15
35
0.6202
35
0.4019
35
0.1957
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
40
0.5850
40
0.3659
40
0.1500
0
1.0000
0
1.0000
0
1.0000
0
1.0000
45
0.5525
45
0.3369
45
0.1176
5
0.9227
5
0.8550
5
0.8181
5
0.6939
50
0.5295
50
0.3060
50
0.0916
10
0.8626
10
0.7483
10
0.6714
10
0.5051
55
0.4989
55
0.2778
55
0.0719
15
0.8007
15
0.6564
15
0.5466
15
0.2826
60
0.4708
60
0.2515
60
0.0551
20
0.7088
20
0.5775
20
0.4485
20
0.1714
65
0.4510
65
0.2271
65
0.0440
25
0.6389
25
0.5131
25
0.3609
25
T AB LE A4
(Continues)
Drying curves for experiments 12–15 (Flow rate 0.2560
[m3/hr])
0.1158 (Continues)
12 of 12
|
TA BL E A 4
CLAUMANN
(Continued)
T AB LE A4
(Continued)
Essay 12
Essay 13
Essay 14
Essay 15
Essay 12
Essay 13
Essay 14
Essay 15
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
MR
t (min)
t (min)
30
0.5950
30
0.4547
30
0.2895
30
0.0689
100
0.2192
100
0.0714
35
0.5503
35
0.4024
35
0.2271
35
0.0365
105
0.1965
105
0.0623
40
0.5145
40
0.3593
40
0.1833
40
0.0277
110
0.1859
110
0.0540
45
0.4779
45
0.3153
45
0.1404
45
0.0172
115
0.1591
115
0.0462
50
0.4470
50
0.2822
50
0.1123
50
0.0089
120
0.1273
120
0.0361
55
0.4266
55
0.2500
55
0.0857
55
0.0019
125
0.1111
130
0.0183
60
0.3876
60
0.2221
60
0.0709
60
0.0001
130
0.0924
140
0.0104
65
0.3624
65
0.1951
65
0.0542
135
0.0761
150
0.0026
70
0.3420
70
0.1716
70
0.0400
140
0.0631
75
0.3193
75
0.1503
75
0.0314
145
0.0517
80
0.2924
80
0.1307
80
0.0219
150
0.0387
85
0.2664
85
0.1141
85
0.0152
155
0.0135
90
0.2453
90
0.0976
90
0.0090
160
0.0021
95
0.2363
95
0.0854
95
0.0038 (Continues)
ET AL.
MR
MR
