SIMULATION OF THE DRYING PROCESS OF SUNFLOWER SEEDS THROUGH ORDINARY DIFFERENTIAL EQUATIONS ABSTRACT

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1 59 ISSN: SIMULATION OF THE DRYING PROCESS OF SUNFLOWER SEEDS THROUGH ORDINARY DIFFERENTIAL EQUATIONS Camila N. Boeri Di Domenico 1*, Elen P. R. Leite 2, Maurício Bavaresco 3, Laís F. Serafini 4, Adriano Leite 5, Maiquiel S. Oliveira 6 ABSTRACT Drying is the most widely used process to ensure the quality and stability of a seed. From data collected from drying samples of 70 g of sunflower seeds, selected manually, with controlled temperatures of 75, 80 and 90 [ C], this work was carried out in order to provide, through an ordinary differential equation, the drying time of sunflower seeds until reaching constant mass. For the simulations, we used an ordinary differential equation of first order and separable variables, as well as the use of the Ordinary Least Squares method, to obtain coefficients and of the equation that best fit the curve, with the help of MATLAB. The simulations performed represented the experimental data in a satisfactory manner, which showed that the drying temperature directly influences the end of the process time. Keywords: Sunflower, differential equations, drying, modeling, seeds. SIMULAÇÃO DO PROCESSO DE SECAGEM DAS SEMENTES DE GIRASSOL ATRAVÉS DE EQUAÇÕES DIFERENCIAIS ORDINÁRIAS RESUMO A secagem é o processo mais utilizado para assegurar a qualidade e estabilidade de uma semente. A partir de dados coletados com a secagem de amostras de 70 g de sementes de girassol, escolhidas manualmente, com temperaturas controladas de 75, 80 e 90 [ºC], o presente trabalho foi realizado com o objetivo de prever, através de uma equação diferencial ordinária, o tempo de secagem de sementes de girassol até atingirem massa constante. Para as simulações, foi utilizada uma equação diferencial ordinária de primeira ordem, de variáveis separáveis, bem como também foi utilizado o Método dos Mínimos Quadrados, para serem obtidos os coeficientes K e c da equação que melhor se ajustaram à curva, com o auxílio do MATLAB. As simulações realizadas representaram de forma satisfatória os dados experimentais, os quais mostraram que a temperatura de secagem influencia diretamente no tempo final do processo. Palavras-chave: Girassol, equações diferenciais, mínimos quadrados, modelagem, sementes. 1* Departamento Acadêmico de Física, Estatística e Matemática Universidade Tecnológica Federal do Paraná Câmpus Francisco Beltrão PR. camiladomenico@utfpr.edu.br Autor para correspondência. 2 Curso de Engenharia Ambiental Universidade Tecnológica Federal do Paraná Câmpus Francisco Beltrão - PR. 3 Curso de Engenharia Ambiental Universidade Tecnológica Federal do Paraná Câmpus Francisco Beltrão - PR. 4 Curso de Engenharia Ambiental Universidade Tecnológica Federal do Paraná Câmpus Francisco Beltrão - PR. 5 Universidade Federal de Itajuba - Itajuba, MG 6 Departamento Acadêmico de Física, Estatística e Matemática Universidade Tecnológica Federal do Paraná Câmpus Francisco Beltrão - PR.

2 60 Simulation of the drying process of sunflower seeds through ordinary differential equations Domenico et al. INTRODUCTION The seeds from the field have generally inadequate water content for safe storage. The high water content of the seeds, during the period between harvesting and drying, contributes to speed up the deteriorating process due to the high metabolic activity. Moreover, there is the consumption of reserve substances and the release of energy and water, favoring the development of microrganisms and insects (PESKE et al., 2003). Drying is the most widely used process to ensure its quality and stability whereas the reduced amount of material water reduces biological activity and chemical and physical changes occurring in the grains during storage (Almeida et al., 2009). Drying seeds, aside from contributing to the preservation of physiological quality during storage, allows the early harvest avoiding several losses during the production process (Garcia et al., 2014). Mathematics consists of transforming real problems into mathematical problems and solve them, and thus interpreting the solutions in real-world language (Bassanezi, 2002). In this sense, this work aims to provide, through the use of ordinary differential equation, the sunflower seed drying time until constant mass. MATERIAL AND METHOD In this section, it will be detailed: (1) procedure for mass collection of each seed sample at every ins, (2) organization of the collected data, (3) choice of the mathematical model, (4) development of the system of equations based on the model from the tables of contents, and (5) mathematical method to calculate the best model parameters. Data collection The experiment was conducted at Universidade Tecnológica Federal do Paraná (UTFPR), on the campus of Francisco Beltrão (FB) in the year Sunflower seeds were received and selected manually, with an average of 13% (b.s) of water content and then were sent for drying, with controlled temperatures of 75, 80 and 90 [ C]. Drying was performed in convection oven with forced air. 70 g grain were used for each of the two samples. During drying, the samples were weighed periodically until no further mass variation of the sample. Organization of the data collected For each experiment, tables with the data collected from each seeds sample were prepared. In addition to the data collected for each sample, a field was added in order to identify the number of the sample, the average mass and the mass derived from the average mass, according to equation 1: where, d dt m[k] m[k + 1] m[k] t[k + 1] t[k] d dt (1) m[k] is the rate of mass variation at the time t[k]; m[k + 1] is the value of the seed sample mass at the time t[k + 1]; m[k] is the value of the seed sample mass at the time t[k]; t[k + 1] is the time at which the sample k + 1 was acquired; t[k] is the time at which the samplek was acquired; k is the sample number. The tables are designed to show more clearly how the seed samples behave according to time and temperature. Subsequently, the content of each table was used to elaborate an equation system according to the chosen model. The calculated averages are intended to improve the data collected and the derivatives were used in the solution according to the desired model. Mathematical model With the aid of a graphical tool, it is possible to observe that the behavior of the mass of each sunflower seed sample approaches the behavior of a decreasing exponential function. Thus, it was decided to represent the model that describes the relation of the sunflower seed mass over time by an ordinary differential equation (ODE) of first order, according to Equation 2:

3 dm dt = K. m + c (2) The following are detailed steps to get the mathematical expression which relates the mass of the sunflower seed sample over time during the drying process from the ordinary differential equation of Equation 2 using the method of Separable Variable equations of a first ordinary differential equation of first order. Starting from equation 2, the variables were separated and after integrated to the initial mass until any mass and at the initial time until any time, as shown in Equation 3. m m 0 1 K. m + c t dm = dt (3) t 0 To facilitate the full resolution of the integral in equation 3, the following simple replacement was used. u 0 = K. m 0 + c u = K. m + c du dm = K (4) dm = du K Replacing the data of equation 4 in equation 3. [ln u] u u0 u 1 u 0 u t du = dt (5) t 0 t = [ K. t] t0 (6) Where the start time is equal to 0: t 0 = 0 ln u ln u 0 = K. t (7) ln u u 0 = K. t (8) u u 0 = e K.t (9) Isolating u and e u 0 in equation 4 and replacing in equation 9: K. m + c K. m 0 + c = ek.t (10) Finally, the relation of mass over time during the sunflower seed drying procedure was obtained according to equation 11. m(t) = (m 0 + c K ) ek.t c K System of equations 11) With the data from the tables arranged in arrays, it is possible to apply the method of least squares and describe the data from each table, algebraically in matrix form: dm(t 1 ) dt m(t 1 ) 1 dm(t 2 ) m(t 2 ) 1 dt.. = [ K.. c ] (12).. dm(t n) [ m(t n ) 1] [ dt ] Parameter Estimator Due to the relation between number of equations and number of unknowns in the system established, there is a need to make use of a parameter estimator for the unknown coefficients of the model (K e c). In order to obtain the parameters that best fit the sampled data, the method of least squares was chosen. The method of least squares (MLS) is a procedure that solves invariant linear systems in time (ZILL & CULLEN, 2009). In other words, a system is time invariant when its parameters are constant, i.e., it does not change over time. In contrast, a system is said to be linear if it has no nonlinearities such as sine, exponential and logarithmic components, etc. Linearization is allowed. A time-invariant linear system given by: Where: Y = ψ θ

4 Y is the vector of derivatives of order nx1; ψ is the matrix of regressors of order nxm; θ is the vector of coefficients to be estimated of ordermx1; n is the number of equations; m is the number of unknowns. It is possible to obtain vector of the coefficients using equation (13): θ = (ψ T ψ) 1 ψ T Y (13) RESULTS AND DISCUSSION Here are presented the collected data, the code implemented to estimate the differential model, the estimated parameters of the model and the graphical representation of the resulting model on the data collected in this section for each working temperature: 75, 80 and 90 [ C]. Table 1: Drying at 75 [ºC] Sample Time [min] Mass A1 Mass A2 Average Mass Rate of Change [g/min] ,00 70,00 70,00-0, ,22 66,65 66,94-0, ,17 64,78 64,98-0, ,10 63,82 63,96-0, ,92 62,80 62,86-0, ,48 62,44 62,46-0, ,27 62,27 62,27-0, ,14 62,12 62,13-0, ,00 61,97 61,98-0, ,94 61,92 61,93-0, ,93 61,9 61,92-0, ,92 61,87 61,90 -

5 Table 2: Drying at 80ºC Sample Time [min] Mass A1 Mass A2 Average Mass Rate of Change [g/min] ,09 70,04 70,06-0, ,59 62,43 68,51-0, ,87 65,96 65,92-0, ,68 64,98 64,83-0, ,4 63,82 63,61-0, ,76 63,09 62,92-0, ,38 62,59 62,48-0, ,14 62,36 62,25-0, ,02 62,22 62,12-0, ,93 62,14 62,04-0, ,82 62,03 61,92-0, ,76 61,96 61,86-0, ,75 61,95 61,85 - Table 3: Drying à 90ºC Sample Time [min] Mass A1 Mass A2 Average Mass Rate of Change [g/min] ,04 70,03 70,035-0, ,99 67,42 67,2-0, ,24 64,64 64,44-0, ,4 63,42 63,41-0, ,55 62,63 62,59-0, ,24 62,19 62,22-0, ,04 62,03 62,04-0, ,91 61,87 61,89-0, ,83 61,77 61,8-0, ,75 61,75 61,75-0, ,73 61,7 61,72-0, ,69 61,69 61,69 -

6 64 Simulation of the drying process of sunflower seeds through ordinary differential equations Domenico et al. Accordingly, Tables 1, 2 and 3 above, obtain time constants ( τ) on which after 5 time constants (5τ) from the beginning of the experiment, the mass begins to have little variation and is already very close to its final value. Using the method of least squares in matrix form, it is possible to obtain the coefficients K and c that best fit the curve. To find these parameters, the software MATLAB was used. When the script drying at 75 C was executed, the following output was obtained: m f = c K = 61,9968 When the script drying at 90 C was executed, the following output was K = 0,03916 c = 2,4213 m 0 = m(t(0)) = 70,035 m f = c K = 61,8312 m (t) = K. m(t) + c Where the following results were obtained: K = 0,03852 c = 2,3926 To 75 ºC, solving the ordinary differential equation, it was obtained: where: m(t) = (m 0 m f )e K.t + m f m 0 = m(t(0)) = 70 m f = c K = 62, [ºC] = [ 75 [ºC] 75 [ºC] ] = [ 0, ,3926 ] To 80 ºC, the following results were obtained: K = 0, c = 1,666 m 0 = m(t(0)) = 70,06 From the data, it is possible to understand that the equation is decreasing because K is always negative in this case. With these data, graphical representations of the drying process for each temperature used in the experiments were made, to simulate the data collected with the model found. To understand the following charts, the transitional and permanent regime definitions were used. Transitional regime is the time interval in which the mass varies dramatically with time and permanent regime is the instant at which the mass variation is insignificant. It is possible to say that, from that moment, the drying process can be stopped because the seed has come close to its final mass. From this point, dissipating a great energy would be required in order to occur a small variation in the seed mass. In Figure 1, the transitional regime is between the start time until the first 100 minutes. After this time the mass begins to remain constant, i.e., permanent regime. In Figure 2, there was an increase in the transitional regime time from 100 to about 120 minutes. From this moment, the mass begins to vary very little and is already very close to its final value, ie the permanent regime. In Figure 3 the transitional regime fell, finishing in about 80 minutes. And so beginning the permanent regime, keeping constant mass.

7 65 Simulation of the drying process of sunflower seeds through ordinary differential equations Domenico et al. Figure 1: Sunflower seed drying at 75 C. Figure 2: Sunflower seed drying at 80 C.

8 Figure 3: Sunflower seed drying at 90 C. Thus, it is possible to rewrite equation 11 based solely on the initial (m 0 ) and final mass ( m f ) of the sunflower seed sample, according to equation 14. m(t) = (m 0 m f ) e t τ + m f (14) Where the final mass of the seed sample is given by: m f = c K And the time constant by: τ = 1 K CONCLUSIONS (15) (16) The present study examined the sunflower seed drying process at different temperatures and proposed the use of an ordinary differential equation of first order to simulate this process and predict the drying time necessary to achieve constant mass. The proposed model represented the experimental data obtained in our laboratory in a satisfactory manner, showing that the proposed equation can be used to perform projections for other drying temperatures. REFERENCES Almeida, P. D.; Resende, O.; Costa, L. M.; Mendes, U. C.; Sales, J. F. Cinética de secagem do feijão adzuki (Vigna angularis). Global Science and Technology, v. 2, n. 1, p , Bassanezi, R. C.; Ensino aprendizagem com modelagem matemática. São Paulo: Contexto, p.89, Chaia, Aline V.; Daibert, Maria R.; Mini Curso Introdução ao MATLAB. Juiz de Fora: UFJF, GET- Engenharia de Produção, Garcia, D. C.; Barros, A. C. S. A.; Peske, S. T.; Menezes, N. L. A secagem de sementes. Cienc. Rural, v. 34, n. 2, Santa Maria, Mar./Apr

9 Peske, S.T.; Rosenthal, M.D.; Rota, G.R.M. Sementes: Fundamentos científicos e tecnológicos. 3ª edição. Pelotas: Editora Rua, Pelotas, 415p, Zill, Dennis G.; Cullen, Michael R.; Matemática Avançada para Engenharia- Álgebra linear e cálculo vetorial. 3ª ed. Porto Alegre, Bookman, p159, 2009

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