Mathematical modeling of drying the pulped coffee ( Coffea arabica l . ) at different air conditions

O objetivo do estudo foi descrever a cinética de secagem do café despolpado (Coffea arabica L.) e avaliar o melhor modelo matemático para ajuste dos dados experimentais de secagem, realizada com diferentes umidades relativas do ar (40, 50 e 60 %) e temperaturas (23, 40 e 60 °C). Os frutos de cafés foram padronizadas na lavagem, separação e seleção manual, dos cafés verdes, passa cana, verde e boia. Em seguida, cerca de 150L de café cereja foi despolpado e levado diretamente para o terreiro. A secagem do café despolpado foi completada em um secador mecânico e no terreiro. Os resultados obtidos mostraram que as diferentes condições do ar ambiente influenciou significativamente nos processos de secagem do café despolpado. O teor de água de equilíbrio higroscópico do café despolpado é proporcional à atividade de água e a umidade relativa, diminuindo com o aumento da temperatura, para o mesmo valor de umidade de equilíbrio higroscópico. O modelo Oswin foi o que melhor representou a higroscopicidade do café despolpado, enquanto o modelo Midilli apresentou o melhor ajuste das curvas de secagem do café. O coeficiente de difusão eficaz aumentou com o aumento da temperatura do ar de secagem e redução da umidade relativa, sendo descrito pela equação de Arrhenius.

The aim of the study was to describe the drying kinetics of washed coffee (Coffea arabica L.) and evaluate the best mathematical model to fit the experimental drying data conducted with different air humidity (40 %, 50 %, and 60 %) and temperatures (23 °C, 40 °C, and 60 °C).The fruit shakes were standardized washing, separation, and manual selection of green coffees, pass cane, and green buoy.Then, approx.150 L of coffee cherries were pulped and taken directly to the yard.Drying the washed coffee was completed in a mechanical dryer and yard.The obtained results showed that the different conditions of the ambient air significantly influenced the processes of drying pulped coffee.The water content of the hygroscopic equilibrium of pulped coffee is directly proportional to the water activity and relative humidity, decreasing with increasing temperature, for the same value of equilibrium relative humidity.The Oswin model was best represented by the hygroscopicity of the pulped coffee, while the Midilli model shows the best fit to describe the drying curves of the washed coffee.The effective diffusion coefficient increases with increasing temperature of the drying air and reducing of relative humidity, being described by the Arrhenius equation PALAVRAS-CHAVE: Drying.Mathematical Modeling.Pulped Coffee.

INTRODUCTION
There are several factors that influence the final quality of the coffee, as soil and climate characteristics, cultivars, driving and crop management, harvesting, processing, drying, and storage.There are various forms of processing that result in major differences in the sensory attributes and there are common reports of superiority to coffee peeled and pulped and in relation to natural coffee.Drying is one of the most important stages in the processing of coffee, both from the standpoint of energy consumption and the influence this has on the operation quality of the final product.
Given these problems, we seek greater control of the drying parameters (temperature of the drying air temperature of the grain mass, relative humidity, and air flow) in order to minimize adverse situations to the product.On the other hand, if the best drying techniques are not used, the quality may be impaired as a result of physical, chemical, and sensory (BORÉM et al., 2008;SAATH et al., 2010).The drying of agricultural products, thin layer, has the purpose of determining the rates of drying of the product using for data collection recording the mass loss occurred in a sample during water removal (RESENDE et al., 2009).Thus, the drying curves, thin layer, vary with species, variety, environmental conditions, methods staging post-harvest, among other factors.Accordingly, various mathematical models have been used to describe the drying of agricultural produce, although in most cases, the semi-empirical relationships and empirical have been shown to predict the best options for the drying of grains and seeds, although its validity is restricted to the conditions under which the experimental data were obtained (RESENDE et al., 2009;CORADI et al., 2014).These models generally are based on variables external to the product, such as the temperature and relative humidity of the drying air.The semi-empirical equations are based on Newton's law of cooling heat transfer by convection, assuming that during the drying conditions are isothermal and that the water transfer is restricted to the surface of the product.Thus, the aim of the study was to describe the drying kinetics of washed coffee (Coffea arabica L.) and evaluate the best mathematical model to fit the experimental drying data conducted with different air humidity (40 %, 50 %, and 60 %) and temperatures (23 °C, 40 °C, and Página | 2399

This work was conducted at the Department of Engineering and Technology
Center of Post-Harvest Coffee, Federal University of Lavras.The coffee was harvested manually and selectively removing only the cherry fruit from the plant.For each repetition, 800 liters of the coffee variety Topazio were collected.All the raw materials were standardized by the washing, separation, and manual selection of green coffees; green cane passes, and buoy (Figure 1).Then, about 150 liters of coffee cherries were pulped and taken directly to the yard.The pulped coffee was divided into distinct segments in the yard, remaining for two days, so that the beans were taken for mechanical drying (40 and 60 + 2 °C to 40, 50, and 60 + 5% RH) and complete drying in the yard (23 + 2 °C to 40, 50, and 60 + 5% RH) (Figure 2).
During the time that the coffee remained in the yard, turnings were made every half hour and monitoring the temperature and relative humidity of the ambient air using a term hygrograph.Mechanical drying was conducted on two prototypes of fixed layer.To obtain the air flow diaphragm used a graduated opening in the fan inlet.The determination of the water content was performed by standard oven at 105 + 3 °C for 24 hours (BRASIL, 2009).The drying curves were fitted to the experimental data using thirteen different semi-empirical and empirical equations discriminated below.

Equation
Models Number It is usual to consider the value of the diffusion coefficient constant or linearly dependent on the temperature of the drying air.
The coefficients of the Arrhenius expression were linearized by applying the logarithm of the form: To obtain the water content of the hygroscopic equilibrium of coffee the dynamic-gravimetric method was used.A desorption thin layer of the product was performed for different controlled conditions of temperature (23, 40, and 60 °C) and relative humidity of the drying air 40, 50, and 60 % until the product reached the equilibrium moisture content with air condition specified.Temperature and relative humidity were monitored by means of a psychrometer installed next to trays containing the samples.During the drying, the trays with the product were weighed periodically and the hygroscopic equilibrium was reached when the mass change of the containers to remain unchanged for three consecutive weightings.The experimental data of the equilibrium water content was adjusted mathematical models are frequently used to represent the hygroscopic agricultural products, whose expressions are shown below.The experimental design was a completely randomized design (CRD) with three tests for each drying air velocity and drying temperatures.To adjust the mathematical models analysis were performed nonlinear regression, Quasi-Newton method, using the computer program Statistica 7.0®.To check the degree of fit of each model was considered the significance of the regression coefficient by t-test, adopting the 5% level of probability, the magnitude of the coefficient of determination (R 2 ), the mean relative error values (P) and the average estimated error (SE) and verified the behavior of distribution of residuals.The relative average error and the average error estimated for each model were calculated according to the following expressions, respectively:

RESULTS AND DISCUSSION
It can be seen in Table 1 that the mathematical models used to describe the fermented coffee hygroscopicity presented, for most of its coefficients, a regression significance level of 5% probability level by the t test and, in general, the models showed values of high coefficient of determination greater than 0.90 except for the models BET, GAB, Henderson, Modified Henderson, Chung, and Pfost that were below 80%.For further analysis, we used other statistical parameters to support the selection of the best model.Table 1 shows the summary of the mathematical models evaluated, with the parameters adjusted by nonlinear regression to the experimental data of the equilibrium moisture content of the washed coffee, obtained by desorption with the coefficients adjusted determination (R²) and average errors for (P) and estimated (SE).It is observed in Table 1 that the equations based on the models of Oswin, Sigma Copace, and Copace showed satisfactory adjustments to the experimental data of the equilibrium moisture content of the washed coffee, with better results for the Oswin model, since it had coefficients of determination set high and average relative errors and estimated very low, independent of the temperature and relative humidity of the drying air.Therefore, when comparing the values of the equilibrium moisture content of hygroscopic coffee that was not pulped, note that the values of equilibrium water content were higher for lower temperatures and higher relative humidity of the air.The equations based on the models of Oswin, Sigma Copace, and Copace showed satisfactory adjustments to the experimental data of equilibrium moisture content of the washed coffee, with better results for the Oswin model (Table 1), since it had coefficients of determination set high and average relative errors and estimated very low, independent of temperature and relative humidity of the drying air.Therefore, when comparing the values of equilibrium moisture content of hygroscopic coffee that was not pulped, note that the values of equilibrium water content were higher for lower temperatures and higher relative humidity of the air.Figure 3   increased speed of withdrawal of water at 60°C and 40% relative humidity.As expected, the drying time is affected by air temperature, noting a greater difference between temperatures of 60 °C and 23 °C.It is also observed in Figure 4 that with increasing temperature of the drying air, there is a higher rate of removal of product water, as noted by many researchers for many agricultural products (PENA et al., 2010;REIS et al., 2011).Table 2 shows the coefficients of the models adjusted for the coffee that was not pulped and that was analyzed during drying at different drying air temperatures and relative humidity conditions of the air.Among the models that gave good results, the Midilli model was selected to represent the phenomenon of drying coffee due to its simplicity compared to other models and selected to present a number of significant coefficients.It was observed that the magnitude of the drying constant (k) for the model Midilli, which represents the effect of external conditions drying increases linearly with the rise in temperature of the drying air (Table 2).The coefficient of determination was above 98% (Table 3), which according to MADAMBA et al. (1996), indicates a satisfactory representation of the phenomenon under study.According to this researcher, the use of the coefficient of determination as the only evaluation criterion for the selection of nonlinear models is not a good parameter to represent the drying phenomena.The models of Thompson and Diffusion of Eight Terms achieved, for modeling the drying of washed coffee, the biased distribution of waste, thus resulting in poor fits to the experimental data, while all other models corroborated with results verified by RESENDE et al. (2009) for the modeling of drying coffee clones.All the models that were evaluated attended satisfactory the P, SE, and distribution residue, but the Midilli model was better that other evaluated because presented higher R² for all.), due to different air temperatures drying of pulped coffee.

Figure 1 -
Figure 1 -Process of coffee stripping (left) and process of coffee pulping (right)

Figure 2 -
Figure 2 -Drying of natural and washed coffees in yard (left) and fixed bed dryers prototypes (right) water content of product (% d.b.); Ui*:initial water content of the product (% d.b.); Ue*: equilibrium water content of the product (% d.b.).
activity, decimal; T: temperature, °C; a, b, c: coefficients that depend on the product.
, mean relative error (P), standard deviation of the estimate (SE), coefficient of determination (R2) and residual distribution for mathematical models of drying relative humidity average of 40, 50 and 60% of washed coffee (Coffea arabica L.)-c cont.Parameter values estimated, mean relative error (P), standard deviation of the estimate (SE), coefficient of determination (R2) and residual distribution for mathematical models of drying relative humidity average of 40, 50 and 60% of washed coffee (Coffea arabica L.)-c cont.estimated coefficients were significant at 5% probability by t test.A -aleatory distribution Fonte: Authors elaboration(2014) shows the experimental values of equilibrium water content of the fermented coffee obtained by desorption isotherms as well as estimated by the model Oswin.The constant water activity values of equilibrium water content of hygroscopic fermented coffee decreased with increasing temperature and with decreasing relative humidity.

Figure 3 -
Figure 3 -Observed and predicted values by Oswin model of water content equilibrium moisture content of the natural coffee obtained by desorption for different conditions of temperature and water activity.*Significant at 5% probability by the t test

Figure 4 -
Figure 4 -Curves of drying coffee cherries processed naturally However, analyzing the estimated average error (SE), which describes the value of the standard deviation of the estimate, it was found that the models Wang and Singh, Page, and Logarithmic Approximation of Diffusion, Midilli, Exponential for Two terms showed lower values for drying performed in different temperatures and relative humidity of the air.It is noteworthy that the lower the value of the standard deviation of the estimate (SE) is the better the quality of fit of the model will be relative to the observed data.RESENDE et al. (2009) also observed that the models Page, Diffusion Approximation and Midilli showed a low average error estimated during the modeling of drying coffee clones of Coffea canephora.It appears that most of the models presented values mean relative error less than 10%, which according to MOHAPATRA and RAO (2005) indicates an adequate representation of the phenomenon, except for models Thompson, Newton, Henderson, and two terms and Pabis.

Figure 5
Figure5shows the values of the effective diffusion coefficient for the pulped coffee beans during different drying conditions.It appears that during the drying the effective diffusion coefficient increased significantly (p<0.05), with the rise of temperature and increase in relative humidity.ALMEIDA et al. (2009) found that during the drying of adzuki effective diffusion coefficients had magnitudes between 0.51x10 -10 and 2.23x10 -10 m 2 s -1 for the temperature range from 30 °C to 70 °C.Their dependence on the temperature of the drying air is described by the Arrhenius equation as shown in Figure6.

Figure 5 -
Figure 5 -Values for the effective diffusion coefficient (m 2 s -1

Fonte:
Figure 6 -Representation Arrhenius relationship for the effective diffusivity and air temperature drying of natural coffee.

Table 1 -
Parameter values estimated, mean relative error (P), standard deviation of the estimate (SE), coefficient of determination (R2) and residual distribution for mathematical models of drying relative humidity average of 40, 50 and 60% of washed coffee (Coffea arabica L.)

Table 1 -
Parameter values estimated

Table 2 .
Parameters obtained from models fitted to the data for drying of washed coffee processing in the different temperatures of air drying and relative humidity of 40, 50 and 60%

Table 2 .
Parameters obtained from models fitted to the data for drying of washed coffee processing in the different temperatures of air drying and relative humidity of 40, 50 and 60%

Table 2 .
Parameters obtained from models fitted to the data for drying of washed coffee processing in the different temperatures of air drying and relative humidity of 40, 50 and 60%

Table 2 .
Parameters obtained from models fitted to the data for drying of washed coffee processing in the different temperatures of air drying and relative humidity of 40, 50 and 60%

Table 3 .
Coefficients of determination (R

Table 3 .
Coefficients of determination (R 2 ), mean relative errors (P) and mean estimated errors (SE) for the models analyzed during drying of the washed coffee under various temperature conditions and relative humidity of 40, 50 and 60%

Table 3 .
Coefficients of determination (R 2 ), mean relative errors (P) and mean estimated errors (SE) for the models analyzed during drying of the washed coffee under various temperature conditions and relative humidity of 40, 50 and 60%

Table 3 .
Coefficients of determination (R 2 ), mean relative errors (P) and mean estimated errors (SE) for the models analyzed during drying of the washed coffee under various temperature conditions and relative humidity of 40, 50 and 60%