Mathematical modelling of the drying of jatropha fruit: an empirical comparison

Modelagem matemática da secagem dos frutos de pinhão-manso: uma comparação empírica

ENGENHARIA AGRÍCOLA

ARTIGO CIENTÍFICO

Mathematical modelling of the drying of jatropha fruit: an empirical comparison¹

Modelagem matemática da secagem dos frutos de pinhão-manso: uma comparação empírica

Valdiney Cambuy Siqueira^I,*; Osvaldo Resende^II; Tarcísio Honório Chaves^II

^IPrograma de Pós-Graduação em Engenharia Agrícola, Universidade Federal de Lavras, Lavras-MG, Brasil, [e-mail]

^IIInstituto Federal de Educação, Ciência e Tecnologia Goiano, Campus Rio Verde, Rio Verde-GO, Brasil, [e-mail]; [e-mail]

ABSTRACT

The jatropha has emerged on the world scene as a promising plant for the production of biodiesel. However, this crop still lacks the development of specific equipment for its processing, particularly post-harvest. Therefore, the aim of this work was to fit different mathematical models to experimental data obtained while drying jatropha fruits, and to recommend the one that best represents the facts. Jatropha fruit with a moisture content of 4.4 (kg moisture kg^-1 dry matter) were subjected to drying in a greenhouse with forced air-ventilation at five temperature conditions: 45; 60; 75; 90 and 10 ºC and relative humidities of 14.5; 7.4; 3.8; 2.2 and 1.4%, respectively, until reaching a moisture content of 0.10 ± 0.005 (kg moisture kg^-1 dry matter),with three replications. Ten mathematical models, used to represent the drying of agricultural products, were adjusted to fit the experimental drying data. The models were analyzed using the coefficient of determination, chi-square, mean relative error, mean estimated error and residual distribution. It can be concluded that the Page model satisfactorily describes the kinetics of jatropha-fruit drying, at temperatures of 60; 75; 90 and 105 ºC. However, drying at a temperature of 45 ºC produced different behavior, making necessary the adjustment of a new model to describe the phenomenon.

Key words: Biodiesel. Drying temperatures. Mathematical models.

RESUMO

O pinhão-manso tem se destacado no cenário mundial como planta promissora para a produção de biodiesel. No entanto, esta cultura ainda carece do desenvolvimento de equipamentos específicos para o seu processamento, principalmente na fase pós-colheita. Sendo assim, objetivou-se com o presente trabalho ajustar diferentes modelos matemáticos aos dados experimentais obtidos na secagem dos frutos de pinhão-manso e recomendar aquele que melhor representa o fenômeno. Os frutos de pinhão-manso com o teor de água de 4,40 (kg de água kg^-1 de matéria seca), foram submetidos à secagem em estufa com ventilação de ar forçada em cinco condições de temperatura: 45; 60; 75; 90 e 105 ºC e umidades relativas de 14,5; 7,4; 3,8; 2,2 e 1,4%, respectivamente, até atingirem o teor de água de 0,10 ± 0,005 (kg de água kg^-1 de matéria seca) em três repetições. Aos dados experimentais da secagem foram ajustados dez modelos matemáticos utilizados para representação da secagem dos produtos agrícolas. Os modelos foram analisados por meio do coeficiente de determinação, do qui-quadrado, do erro médio relativo, do erro médio estimado e da distribuição de resíduos. Conclui-se que, o modelo de Page descreve satisfatoriamente a cinética de secagem dos frutos de pinhão-manso nas temperaturas de 60; 75; 90 e 105 ºC. No entanto, para a secagem na temperatura de 45 ºC ocorreu um comportamento diferenciado sendo necessário um ajuste de um novo modelo para a descrição do fenômeno.

Palavras-chave: Biodiesel. Temperaturas de secagem. Modelos matemáticos.

INTRODUCTION

According to Abreu et al. (2009), among the oilseed crops that have been presented as another means of diversification, and one that can be planted in order to increase the production of biodiesel, the jatropha (Jatropha curcas L.) stands out. Laviola and Day (2008) claim that depending on the spacing adopted when planting jatropha it is possible to achieve a productivity of over 2,000 kg oil ha^-1, and that with genetic improvements and changes to the production system, it is believed that the jatropha is capable of producing up to 4,000 kg oil ha^-1. This crop therefore becomes an extremely viable alternative for small, medium and large producers.

There are other attributes related to jatropha oil, as it is not edible and therefore cannot be diverted for human consumption (SATURNINO et al., 2005), unlike soybean oil, which is considered as the reference for biodiesel production on an industrial scale, and is derived from raw material which is abundant in Brazil, with the technology for its production being readily available (SANTOS et al. 2009).

Given the characteristics of jatropha oil, this crop has been prominent on the world stage as one of the most promising for the production of biodiesel. However, the steps for processing the grains and fruit need to be studied in order to achieve more efficiency. Among the postharvest steps, drying is one of the most important as it is directly related to the end quality of the product, and as oil is the main product of the jatropha, it is necessary to develop drying techniques, which will give better yield and oil quality.

Mathematical modeling of the drying process of agricultural products allows predicting their behavior during the removal of water, reducing the time and costs involved in practical drying methods which are aimed at the development of suitable equipment. Among the models of thin-layer drying, the most used are: Exponential, Henderson and Pabis, Two-term, Lewis, Page, Thompson, and Wang and Sing (MOHAPATRA; RAO, 2005).

The process of thin-layer drying of agricultural products aims at determining the drying rate of the product by recording the weight losses which occur in any one sample during the removal of water (HILL et al. 2008). The drying curves of thin layers vary with the species, type, environmental conditions, and postharvest preparation methods, among other factors. In this regard, various mathematical models have been used to describe the drying process of agricultural products (RESENDE et al. 2008).

Empirical models are usually based on variables external to the product, such as temperature and the relative humidity of the air used for drying. However, they give no information about the phenomena of energy and water transport inside the grains, and consider that the entire drying process occurs only in the falling-rate period (RESENDE et al., 2009).

Given the above, the aim of the present work was to fit different mathematical models to experimental data obtained when drying jatropha fruit, and recommend the one that best represents the phenomenon.

MATERIAL AND METHODS

The experiment was developed in the Laboratory for the Postharvest of Vegetable Products at the Federal Institute of Education, Science and Technology of Goiás, Rio Verde Campus (IF Goiano). After manual harvesting, the jatropha fruit was left out in a laboratory environment for 30 hours in order to reduce and homogenise its moisture content to 4.40 (kg water kg^-1 dry matter), and then subjected to drying in a hothouse with forced air ventilation at five temperatures: 45; 60; 75; 90 and 105 ºC, promoting the respective relative humidities: 14.5; 7.4; 3.8; 2.2 and 1.4%. The fruit was kept in the hothouse at 105 ± 1 ºC for 24 hours until reaching a moisture content of 0.005 ± 0.10 (kg water kg^-1 dry matter). There were three replications (BRAZIL, 2009).

For the drying process, three aluminum trays each containing 250 g of jatropha fruit distributed in a single layer to an approximate height of 2.7 cm were left inside the hothouse until they reached the desired moisture content. During drying, in between weighings, the positions of the trays were rotated and the fruit turned.

The temperature of the air used for drying was monitored by a thermometer installed inside the hothouse. The relative humidity was obtained by employing the basic principles of psychrometrics, using the GRAPSI software (MELO; LOPES; CORREA, 2004). Equation 1 was used to determine the moisture-content ratios of jatropha fruit during drying.

Where, RX: dimensionless moisture-content ratio of the product; X moisture content of the product (kg water kg^-1 dry matter); Xi: initial moisture content of the product (kg water kg^-1 dry matter); Xe: equilibrium moisture content of the product (kg water kg^-1dry matter).

The equilibrium moisture content of jatropha fruit at each temperature was obtained experimentally. The trays with the samples stayed in the hothouse until the weight of the product had remained unchanged for three consecutive weighings.

The mathematical models frequently used to represent product drying were adjusted to the data from the drying process of the jatropha fruit, as shown in Table 1.

The mathematical models were adjusted by nonlinear regression analysis using the Gauss-Newton method and the Statistica 7.0 software ^®. The models were selected taking into account the magnitude of the coefficient of determination (R²), the chi-square (χ²), the mean relative error (P) and the standard error of estimate (SE), in addition to checking the behavior of residual distribution. Because it is a numeric value that indicates how much of the total observed variation in the experimental values is explained by the model, the closer it is to one or to 100% of the value of R², the better the fit of the model. For evaluating the mean relative error, a value of less than 10% was considered as one of the criteria for selecting the models, according to Mohapatra and Rao (2005).

The mean relative error, the standard deviation of the estimate and chi-squared for each model were calculated according to the expressions 12; 13 and 14:

Where, Y: observed experimental value; Ŷ: value calculated by the model; N: number of experimental observations; GLR: degrees of freedom of the model (number of observations less the number of model parameters).

RESULTS AND DISCUSSION

Table 2 shows the average ratio values for the moisture content of jatropha fruit when subjected to drying under different air conditions. It can be noted that the jatropha fruit reached a moisture content of 0.007 ± 0.10 (kg water kg^-1 dry matter), with drying times of 10.28; 10.98; 15; 26; 21.83 and 38.62 hours at temperatures of 105; 90; 75; 60 and 45 ºC respectively. Therefore, an increase in temperature decreased the drying time of jatropha fruit, showing an increase in the drying rate, a fact noted by various researchers for many agricultural products such as red pepper (AKPINAR; BICER; YILDIZ, 2003), parboiled wheat (MOHAPTRA; RAO, 2005), beans (RESENDE et al., 2008), jatropha seeds (SIRISOMBOON; KITCHAIYA, 2009; ULLMANN et al., 2010).

The reason drying takes place faster at higher temperatures is related to a greater difference between the vapour pressure of the air used in drying and that of the product, resulting in the water being removed faster. Thus, the higher the temperature, the greater the energy efficiency for drying time: this is beneficial, providing there is no morphological, physiological and/or biochemical damage to the product.

It can be further noted in Table 2 that the ratio of the moisture content of jatropha fruit was 0.12 at a temperature of 45 ºC; 0.13 at 60 ºC; 0.18 at 75 ºC; 0.18 at 90 ºC and 0.19 at 105 ºC, there is therefore an increase in the moisture content ratio with an increase in the temperature of the air used in drying, due to the lower equilibrium-moisture content of the fruit under these drying conditions. Similar results were found for the turnip (SOUSA et al., 2011), the crambe (COSTA et al., 2011) and jatropha seeds (RESENDE et al., 2011; ULLMANN et al., 2010).

Table 3 shows some statistical parameters employed to compare the ten models used to describe the drying kinetics of jatropha fruit. Note that the models tested had coefficients of determination (R²) greater than 95%, which according Kashaninejad et al. (2007) indicates a satisfactory representation of the drying process. However, Madamba, Driscoll and Buckle (1996), claim that the coefficient of determination (R²) alone is not a good criterion for selecting nonlinear models. The authors state that assessments are needed of other parameters such as the estimated mean error and the relative mean error, distribution of the residual values and the value of chi-squared.

Of all those tested, the Page model (5) was that which most presented values of mean relative error (P) below 10%, the criterion adopted for using the proposed model. The values of the mean relative error (P) indicate the deviation of the observed values from the curve estimated by the model (KASHANINEJAD et al., 2007). Thus the lower the value of P, less the deviations between the experimental values and those estimated by the model. However, the Page model showed satisfactory performance for only four of the drying conditions (60; 75; 90 and 105 ºC). It is also noted in Table 3 that the Page model, as with the other models tested, did not satisfactorily represent the drying of jatropha fruit at a temperature of 45 ºC, indicating that under this condition the jatropha fruit have their moisture content reduced differentially.

Generally, the Page model and the Midilli model are those which present the lowest values of SE for most drying conditions. It is worth noting that the lower the value of SE, the better the quality of the adjustment of the model in relation to the experimental data (DRAPER; SMITH, 1998).

Table 4 shows the chi-squared values obtained for the different models adjusted to the drying curve of the jatropha fruit. All the models analysed were within the confidence interval of 99%. It can be noted that the Page model and the Midilli model were those with the lowest values for chi-squared. The lower the value of chi-squared, the better the adjustment of the model (AKPINAR; BICER; YILDIZ, 2003; GÜNHAN et al., 2005; MIDILLI; KUCUK, 2002). Therefore, even if the models show a satisfactory adjustment (for this parameter), the numerical value must be taken into account when choosing the model which will best fit the experimental values.

Table 5 shows the behavior of residual distribution for those models studied. It appears that the models of Page (5) and Midilli (7) were the two which more often represented the random distribution, confirming the analyses presented above, this parameter therefore, would be a good indicator for choosing the model. According to Goneli et al. (2011), a model is considered random when the residual values fall near the horizontal and are around zero, but do not form defined figures, indicating no bias in the results. If it presents a biased distribution, the model is considered inappropriate to represent the phenomenon in question. For this reason most of the tested models could not be used to represent the drying process of jatropha fruit, as they all presented a biased distribution.

The traditional Page model has been consistently shown to represent the phenomenon of drying of various agricultural products: amaranth seeds (ABALONE et al., 2006), tomato (DOYMAZ, 2007), apple pulp (WANG et al., 2007), arabica (GONELI et al., 2009), tarragon (ARABHOSSEINI et al., 2009), coffee (RESENDE et al., 2009), lemon grass (MARTINAZZO et al., 2010), among others.

Figure 1 shows illustrations of the residual distribution: Random for the Page model and Biased for the Two-term model, when drying jatropha fruit at a temperature of 105 ºC.

Considering that none of the models used, satisfactorily represented the drying of jatropha fruit at 45 ºC, a computational analysis was carried out to adjust an equation that would meet all the statistical requirements necessary to validate any one model.

The process of drying the jatropha fruit at a temperature of 45ºC was described by equation 15, which for purposes of identification was named Valcam.

The statistical parameters used for selecting equation 15 are shown in Table 6. Note that the model presents a mean relative error of less than 10%, low chi-squared values and low standard deviation of the estimate, besides having a high coefficient of determination and random residual distribution (Figure 2).

Figure 2

Table 7 shows the values of the coefficient values of the Page model at temperatures of 60; 75; 90 and 105 ºC, and the Valcam model at 45 ºC, to represent the drying of jatropha fruit.

It can be seen in Table 7 that the values of the drying-constant "k", which according to Goneli et al. (2009) represents the effect of external drying-conditions, increase with a rise in temperature of the air used in drying. According to Madamba, Driscoll and Buckle (1996) and Babalis and Belessiotis (2004), the drying constant "k" can be used as an approximation to characterize the effect of temperature, and is related to the effective diffusivity of the drying process for the falling-rate period, when liquid diffusion controls the process.

Normally, as the temperature of the air used in drying increases, the lower the value of "n" since there is a greater difference between the vapor pressure of the air and of the seed, which facilitates the removal of water. However, just as in the results obtained by Goneli et al. (2009), for the drying kinetics of husked coffee beans in a thin layer, it was not possible to observe any clear behavior of the coefficient "n" with increasing temperature of the air.

In Figure 3, the drying curves are shown for the jatropha fruit as estimated by the Page model, at temperatures of 60; 75; 90 and 105 ºC, and by the Valcam model for drying at 45 ºC. Note that the models used, satisfactorily fit the phenomenon of drying the jatropha fruit.

CONCLUSION

Given the above it is concluded that the Page model showed the best adjustment when describing the drying kinetics of jatropha fruit at the higher temperatures (60; 75; 90 and 105 ºC). When drying at 45 ºC the behavior was different, and a new representative model (Valcam) was proposed.

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Recebido para publicação em 04/10/2011; aprovado em 01/07/2012

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Autor para correspondência

1

Trabalho financiado pelo CNPq e CAPES

Publication Dates

• Publication in this collection
  14 Feb 2013
• Date of issue
  June 2013

History

• Received
  04 Oct 2011
• Accepted
  01 July 2012