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Drying kinetics of crushed mass of 'jambu': Effective diffusivity and activation energy/Cinetica de secagem da massa triturada de jambu: Difusividade efetiva e energia de ativacao.

Introduction

'Jambu' (Acmella oleracea) is an annual ground herb, with cylindrical stem and height between 0.2 and 0.3 m. It is popularly known as 'jambu', 'agriao bravo' or 'agriao do Para', and its leaves and flowers cause a slight tingling and numbing sensation on the tongue (Nascimento et al., 2013; Aguiar et al., 2014).

This vegetable is mainly cultivated and consumed in the Northern region of Brazil, being widely used as seasoning in local foods. 'Jambu' has high water content and is perishable. Thus, it requires the application of postharvest technologies to preserve its quality for long periods, allowing for transport and commercialization in markets of other regions (Nascimento et al., 2013; Barbosa et al., 2016).

One technological alternative for preservation is drying, which partially removes the water, causing a reduction in water activity, microbial growth and in enzymatic, physical and chemical reactions (Correa et al., 2007). The drying process of a certain product can be described by mathematical models, which represent experimental data of water loss by the material and provide important information for equipment designing, dimensioning, optimization and determination of commercial application feasibility (Costa et al., 2015).

To fit mathematical models to the drying data of plant products, various criteria can be used, such as the magnitudes of coefficient of determination, mean relative error and mean estimated error, chi-square test and residual distribution. Nonetheless, some of these parameters have limitations, thus requiring the adoption of additional criteria in the selection of models to reinforce and endorse decision-taking. In this context, the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) consist in evaluating the models according to the principle of parsimony, since the number of parameters in the models varies.

This study aimed to evaluate the drying kinetics, test AIC and BIC model selection criteria, and determine effective diffusivity and activation energy of crushed mass of 'jambu' leaves for different conditions of temperature and thickness.

Material and Methods

'Jambu' plants were purchased in the municipality of Macapa, AP, Brazil. The species was botanically identified as Acmella oleracea by the Scientific and Technological Research Institute of Amapa, and the exsiccate was deposited at the HAMAB collection as Acmella oleracea F. P. Gomes 01.

The experiment was carried out in February 2017 at the Food Laboratory of the Brazilian Agricultural Research Corporation--Embrapa (Macapa, AP) (0[degrees] 00' 44" S; 51[degrees] 04' 46" W; ~460 m).

'Jambu' leaves (20 kg) were ground (without addition of water) in a food mixer to obtain a solid homogeneous mass formed by pieces of leaves. Drying was performed in a forced-air oven at temperatures of 60, 70 and 80[degrees]C, at relative air humidity 13.09, 8.14 and 5.45%, respectively, estimated by monitoring the relative humidity of the environment using a thermohygrometer, and with layer thicknesses of 0.005 and 0.010 m, measured with a caliper. Air speed of 1.0 m [s.sup.-1] was measured with an anemometer. The mass of the material was uniformly spread on rectangular stainless-steel trays (0.278 x 0.178 m) in thin layer.

Mathematical models were fit to the drying kinetics experimental data (Eqs. 1 to 11), according to a nonlinear regression analysis by the Gauss-Newton method.

--Page (Page, 1949)

RX = exp (-k [t.sup.n]) (1)

--Midilli (Midilli et al., 2002)

RX = a exp (-k [t.sup.n]) + b t (2)

--Henderson & Pabis (Henderson & Pabis, 1961)

RX = a exp (-k t) (3)

--Approximation of Diffusion (Sharaf-Elden et al., 1980)

RX = a exp (-k t) + (1 - a) exp (-k b t) (4)

--Two Terms (Henderson, 1974)

RX = a exp (-[k.sub.o] t) + b exp (-[k.sub.1] t) (5)

--Two-term Exponential (Sharaf-Eldeen et al., 1980)

RX = a exp (-k t) + (1 - a) exp (-k a t) (6)

--Logarithmic (Yagcioglu et al., 1999)

RX = a exp (-k t) + c (7)

--Thompson (Thompson et al., 1968)

RX = exp {[-a -[([a.sup.2] + 4 b t).sup.0.5]]/2b} (8)

--Newton (Lewis, 1921)

RX = exp (-k t) (9)

--Verma (Verma et al., 1985)

RX = a exp (-k t) + (1 - a)exp (-[k.sub.1] t) (10)

--Wang & Singh (Verma et al., 1985)

RX = 1 + a t + b [t.sup.2] (11)

where:

RX--moisture content ratio of the product, dimensionless;

k, [k.sub.0], [k.sub.1]--drying constants;

[h.sup.-1]; a, b, c, n--coefficients of the models; and,

t--drying time, h.

The fit of the models to the experimental data was initially assessed by the standard error of the estimated coefficients and then by the magnitudes of the coefficient of determination ([R.sup.2]), mean relative error (P), mean estimated error (SE), and chi-square test ([chi square]) at 0.05 probability level, according to the following equations:

P = 100/n [summation] [absolute value of Y - [??]]/Y] (12)

SE = [square root of [summation] [(Y - [??]).sup.2]/DF] (13)

[chi square] = [summation] [(Y - [??]).sup.2]/DF] (14)

where:

Y--experimental RX value;

[??]--estimated RX value;

n--number of observations; and,

DF--degrees of freedom of the model.

In order to select a single model to describe the drying process under each condition, models that obtained the best fits were subjected to Akaike Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC). Lower AIC and BIC values indicate better fit of the model, and BIC is the most rigorous criterion (Wolfinger, 1993).

The information criteria were determined by the following equations:

AIC = -2 log L + 2p (15)

BIC = -2log L + p ln (N - r) (16)

where:

p--number of parameters of the model;

N--total number of observations;

r--rank of the matrix X (incidence matrix of fixed effects); and,

L--maximum likelihood.

Fick's diffusion model for a flat plate geometry (Brooker et al., 1992), with approximation of eight terms (Afonso Junior & Correa, 1999), was fitted to the experimental data of drying of 'jambu' leaves crushed mass, according to Eq. 17.

[mathematical expression not reproducible] (17)

where:

RX--moisture content ratio of the product, dimensionless;

D--effective diffusion coefficient, [m.sup.2] [s.sup.-1];

S--area of the equivalent plate, [m.sup.2];

V--volume of the equivalent plant, [m.sup.3];

n--number of terms in the equation; and,

t--time, s.

Arrhenius equation (Eq. 18) was used to correlate the dependence of effective diffusivity on temperature.

D = [D.sub.0] exp (-[E.sub.a]/R [T.sub.a]) (18)

where:

Do--pre-exponential factor, [m.sup.2] [s.sup.-1];

[E.sub.a]--activation energy, J [mol.sup.-1];

R--universal gas constant, 8.314 J [mol.sup.-1] [K.sup.-1]; and,

[T.sub.a]--absolute temperature, K.

Results and Discussion

Among the models fitted to the experimental data, Approximation of diffusion, Two Terms, Two-Term Exponential, Thompson and Verma reached convergence of fitness in the iterative process. However, they showed high standard error of the estimated coefficients, indicating lack of fit to the experimental data (Table 1). In the present study, standard errors of the estimates higher than 10 times the predicted value were assumed to demonstrate failure in predicting the coefficients. For the other models, the coefficients showed low standard errors of the estimates, indicating convergence of fitness.

The models Midilli, Logarithmic and Wang & Singh had coefficients of determination ([R.sup.2]) higher than 99%, mean estimated error (SE) below 0.0097 and chi-square test ([chi square]) lower than 9.4 x [10.sup.-5], for the drying conditions (Table 2).

Considering the mean relative error (P) lower than or equal to 10% as an adequate representation of the model (Mohapatra & Rao, 2005), it can be noted that Midilli, Logarithmic and Wang & Singh models were the ones that met this criterion and can adequately represent the drying of 'jambu' leaves crushed mass.

Along with the previous statistical parameters (Table 2), Akaike Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC) (Table 3) were also considered as additional parameters to select the best model.

Considering the lowest values of BIC and AIC, the Logarithmic model showed the best fit to the experimental data at temperature of 60[degrees]C and layer thickness of 0.005 m and for treatment of 80[degrees]C and thicknesses of 0.005 and 0.010 m. On the other hand, for the temperature of 60[degrees]C and thickness of 0.010 m and temperature of 70[degrees]C and thicknesses of 0.005 and 0.010 m, the Midilli model fitted best to the data.

Martins et al. (2015), working with 'timbo' (Serjania marginata Casar) leaves, observed that the Logarithmic and Midilli models showed satisfactory fit to the experimental data for the temperatures of 40 and 50[degrees]C, but only Midilli showed good fit for the temperatures of 60 and 70[degrees]C. These authors recommend the Midilli model to predict thin-layer drying of 'timbo' leaves, because of its good fit to all drying conditions studied. The same behavior was found by Goneli et al. (2014a) in the drying of aroeira leaves at temperatures of 40, 50, 60 and 70[degrees]C, and by Martinazzo et al. (2007) in the drying of lemongrass leaves at temperatures of 30, 40, 50 and 60[degrees]C.

According to Goneli et al. (2014b), the best fit of the Midilli model to the experimental data of drying of medicinal plants is probably associated with the fast loss of water in the initial stages of the process in these materials, generating a drying curve that is sharper and best characterized mathematically by this model.

Only the parameter k tended to vary its magnitude with the variation in the drying layer thickness (Table 4). As drying air temperature increased, no trend was observed. Goneli et al. (2014a), studying the drying kinetics of 'erva baleeira' (Cordia verbenacea DC.) leaves, observed no trend in the parameters (a) and (n) for the Midilli model. Sousa et al. (2017) did not find a defined trend for the constant (n) of the Midilli model with the increment in temperature and thickness, observing values from 0.949 to 1.148 in the pequi pulp drying. The values found in the present study are within the range described by Sousa et al. (2017).

The drying constant k of the Midilli and Logarithmic models tended to decrease with the increment in layer thickness for the same temperature, except at 80[degrees]C for the Midilli model. According to Sousa et al. (2017), increments of thickness reduce the drying rate and also the constant k.

The curves fitted by the Logarithmic model and behavior of the effective diffusion coefficients for the drying kinetics of 'jambu' leaves crushed mass are shown in Figure 1.

Increments in drying air temperature directly reduce the time required to dry the product (Figure 1A). This phenomenon has also been observed by different researchers in various agricultural products (Martinazzo et al., 2007; Goneli et al., 2014a, b; Nascimento et al., 2015; Smaniotto et al., 2017). The drying of 'jambu' leaves crushed mass was influenced by layer thickness along the drying time, and the values increased over time as thickness increased. Similar behavior was observed by Sousa et al. (2017) studying pequi pulp drying kinetics in convective drying under different conditions of temperature (50, 60, 70 and 80[degrees]C) and thickness (0.005; 0.010 and 0.015 m).

As demonstrated in Figure 1B, the effective diffusion coefficient tended to increase with the elevation of drying air temperature. In addition, the 0.010 m thickness of the material led to higher effective diffusion coefficients in comparison to 0.005 m for the three temperatures studied. This fact may have occurred because of the water mass flow generated when a homogeneous mass of material is subjected to drying. Sousa et al. (2017) observed an increasing trend in effective diffusivity with the increment in pequi pulp layer thickness and with the elevation of temperature. Diffusivity represents the speed with which the water moves from the inside to the surface of the material, thus being vaporized (Menezes et al., 2013). Therefore, the higher the temperature, the faster the water movement from the food to the environment.

Effective diffusion coefficients between 0.66 x [10.sup.-11] and 12.07 x [10.sup.-11] [m.sup.2] [s.sup.-1] were reported by Martins et al. (2015) for the drying of 'timbo' leaves within the temperature range from 40 to 70[degrees]C, respectively. Lower values were obtained for the effective diffusion coefficients of 'jambu' leaves crushed mass, which ranged between 5.79 x [10.sup.-10] and 2.03 x [10.sup.-9] [m.sup.2] [s.sup.-1] for the thicknesses of 0.005 and 0.010 m, respectively. These values reinforce the higher speed of exit of water from the product when the material is homogeneous.

The activation energy ([E.sub.a]) for the drying of 'jambu' leaves crushed mass was equal to 16.61 kJ [mol.sup.-1] for the thickness of 0.005 m and to 16.97 kJ [mol.sup.-1] for the thickness of 0.010 m. Goneli et al. (2014a), studying the drying of 'erva baleeira' leaves, reported activation energy of 62.89 kJ [mol.sup.-1]. Rocha et al. (2012) found Ea of 77.16 kJ [mol.sup.-1] in the drying of thyme, whereas Goneli et al. (2014b) reported 74.96 kJ [mol.sup.-1] for the drying of aroeira leaves and Martins et al. (2015) obtained 81.39 kJ [mol.sup.-1] in the drying of 'timbo' leaves. Lower Ea values in the 'jambu' leaves crushed mass evidenced the need for lower energy to trigger the water diffusion process, in comparison to the leaves of 'erva baleeira', thyme, aroeira and 'timbo'.

It is important to highlight that 'jambu' leaves were crushed before drying, and such alteration in its original structure may have favored the exit of water from the material, resulting in lower activation energy.

Such different values of activation energy for agricultural products can be attributed to their physical and biological characteristics (Martins et al., 2015). Correa et al. (2007) described the activation energy as the difficulty with which water molecules break the energy barrier during the movement inside the product, and the lower the activation energy, the higher the water diffusivity.

Conclusions

1. Midilli and Logarithmic models showed the best fit to the experimental data of drying of 'jambu' leaves crushed mass.

2. Effective diffusion coefficient tends to increase with the increment in layer thickness and elevation of temperature.

3. The dependence of diffusivity on temperature was described by the Arrhenius equation, with activation energy of 16.611 kJ [mol.sup.-1] for the thickness of 0.005 m and 16.975 kJ [mol.sup.-1] for the thickness of 0.010 m.

4. The Akaike Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC) can be additionally included to select models of drying.

DOI: http://dx.doi.org/10.1590/1807-1929/agriambi.v22n7p499-505

Acknowledgments

To the Federal Institute of Education, Science and Technology Goiano for financial support, to the Federal Institute of Amapa for the release to conduct the study and to EMBRAPA/Amapa for providing the facilities. To CAPES, FAPEG, FINEP and CNPq for the financial support.

Literature Cited

Afonso Junior, P. C.; Correa, P. C. Comparacao de modelos matematicos para descricao da cinetica de secagem em camada fina de sementes de feijao. Revista Brasileira de Engenharia Agricola e Ambiental, v.3, p.349-353, 1999. https://doi. org/10.1590/1807-1929/agriambi.v3n3p349-353

Aguiar, J. P. L.; Yuyama, L. K. O.; Souza, F. das C. do A.; Pessoa, A.; Biodisponibilidade do ferro do jambu (Spilanthes oleracea L.): Estudo em murinos. Revista Pan-Amazonica de Saude, v.5, p.1924. 2014. https://doi.org/10.5123/S2176-62232014000100002

Barbosa, A. F.; Sabaa-Srur, D. F.; Maia, J. G. S.; Sabaa-Srur, A. U. O. Microbiological and sensory evaluation of jambu (Acmella oleracea L.) dried by cold air circulation. Food Science Technology, v.36, p.24-29, 2016. https://doi.org/10.1590/1678-457X.6827

Brooker, D. B.; Bakker-Arkema, F. W.; Hall, C. W. Drying and storage of grains and oilseeds. Westport: The Avi Publishing Company, 1992. 450p.

Correa, P. C.; Resende, O.; Martinazzo, A. P.; Goneli, A. L. D.; Botelho, F. M. Modelagem matematica para a descricao do processo de secagem do feijao (Phaseolus vulgaris L.) em camadas delgadas. Engenharia Agricola, v.27, p.501-510, 2007. https://doi. org/10.1590/S0100-69162007000300020

Costa, L. M.; Resende, O.; Goncalves, D. N.; Oliveira, D. E. C. de. Modelagem matematica da secagem de frutos de crambe em camada delgada. Bioscience Journal, v.31, p.392-403, 2015. https:// doi.org/10.14393/BJ-v31n2a2015-22340

Goneli, A. L. D.; Nasu, A. K.; Gancedo, R.; Araujo, W. D.; Sarath, K. L. L. Cinetica de secagem de folhas de erva baleeira (Cordia verbenacea DC.). Revista Brasileira de Plantas Medicinais, v.16, p.434-443, 2014a. https://doi.org/10.1590/1983-084X/13_041 Goneli, A. L. D.; Vieira, M. do C.; Vilhasanti, H. da C. B.; Goncalves,

A. A. Modelagem matematica e difusividade efetiva de folhas de aroeira durante a secagem. Pesquisa Agropecuaria Tropical, v.44, p.56-64, 2014b. https://doi.org/10.1590/S198340632014000100005

Henderson, S. M. Progress in developing the thin layer drying equation. Transactions of the American Society of Agricultural Engineers, v.17, p.1167-1168. 1974. https://doi.org/10.13031/2013.37052

Henderson, S. M.; Pabis, S. Grain drying theory. II: Temperature effects on drying coefficients. Journal of Agricultural Engineering Research, v.6, p.169-174. 1961.

Lewis, W. K. The rate of drying of solid materials. The Journal of Industrial and Engineering Chemistry, v.13, p.427-432, 1921. https://doi.org/10.1021/ie50137a021

Martinazzo, A. P.; Correa, P. C.; Resende, O.; Melo, E. de C. Analise e descricao matematica da cinetica de secagem de folhas de capim-limao. Revista Brasileira de Engenharia Agricola e Ambiental, v. 11, p.301-306, 2007. https://doi.org/10.1590/S1415-43662007000300009

Martins, E. A. S.; Lage, E. Z.; Goneli, A. L. D.; Hartmann Filho, C. P.; Lopes, J. G. Cinetica de secagem de folhas de timbo (Serjania marginata Casar). Revista Brasileira de Engenharia Agricola e Ambiental, v.19, p.238-244, 2015. https://doi.org/10.1590/18071929/agriambi.v19n3p238-244

Menezes, M. L. de; Stroher, A. P.; Pereira, N. C.; Barros, S. T. D. de. Analise da cinetica e ajustes de modelos matematicos aos dados de secagem do bagaco do maracuja-amarelo. Engevista, v.15, p.176-186, 2013.

Midilli, A.; Kucuk, H.; Yapar, Z. A new model for single layer drying. Drying Technology, v.20, p.1503-1513. 2002. https://doi. org/10.1081/DRT-120005864

Mohapatra, D.; Rao, P. S. A thin layer drying model of parboiled wheat. Journal of Food Engineering, v.66, p.513-518, 2005. https://doi. org/10.1016/j.jfoodeng.2004.04.023

Nascimento, A. M.; Souza, L. M. de; Baggio, C. H.; Werner, M. F. de P.; Ferreira, D. M.; Silva, L. M. da; Sassaki, G. L.; Gorin, P. A. J.; Iacomini, M.; Cipriani, T. R. Gastroprotective effect and structure of a rhamnogalacturonan from Acmella oleracea. Phytochemistry, v.85, p.137-142, 2013. https://doi.org/10.1016/j. phytochem.2012.08.024

Nascimento, V. R. G.; Biagi, J. D.; Oliveira, R. A. de. Modelagem matematica da secagem convectiva com radiacao infravermelha de graos de Moringa oleifera. Revista Brasileira de Engenharia Agricola e Ambiental, v.19, p.686-692, 2015. https://doi. org/10.1590/1807-1929/agriambi.v19n7p686-692

Page, G. E. Factors influencing the maximum rates of air drying shelled corn in thin layers. West Lafayette: Purdue University, 1949. Thesis

Rocha, R. P. da; Melo, E. de C.; Corbin, J. B.; Berbert, P. A.; Donzeles, S. M. L.; Tabar, J. A. Cinetica del secado de tomillo. Revista Brasileira de Engenharia Agricola e Ambiental, v.16, p.675-683, 2012. https://doi.org/10.1590/S1415-43662012000600013

Sharaf-Eldeen, Y. I.; Blaisdell, J. L.; Hamdy, M. Y. A model for ear corn drying. Transactions of the American Society of Agricultural Engineers, v.23, p.1261-1265. 1980. https://doi. org/10.13031/2013.34757

Smaniotto, T. A. de S.; Resende, O.; Sousa, K. A. de; Oliveira, D. E. C. de; Campos, R. C. Drying kinetics of sunflower grains. Revista Brasileira de Engenharia Agricola e Ambiental, v.21, p.203-208, 2017. https://doi.org/10.1590/1807-1929/agriambi.v21n3p203-208

Sousa, E. P. de; Figueiredo, R. M. F. de; Gomes, J. P.; Queiroz, A. J. de M.; Castro, D. S. de; Lemos, D. M. Mathematical modeling of pequi pulp drying and effective diffusivity determination. Revista Brasileira de Engenharia Agricola e Ambiental, v.21, p.493-498, 2017. http:// dx.doi.org/10.1590/1807-1929/agriambi.v21n7p493-498

Thompson, T. L.; Peart, R. M.; Foster, G. H. Mathematical simulation of corn drying: A new model. Transactions of the American Society of Agricultural Engineers, v.11, p.582-586. 1968. https:// doi.org/10.13031/2013.39473

Verma, L. R.; Bucklin, R. A.; Endan, J. B.; Wratten, F. T. Effects of drying air parameters on rice drying models. Transactions of the American Society of Agricultural Engineers, v.28, p.296-301. 1985. https://doi.org/10.13031/2013.32245

Wolfinger, R. D. Covariance structure selection in general mixed models. Communications in Statistics, v.22, p.1079-1106, 1993. https://doi.org/10.1080/03610919308813143

Yagcioglu, A.; Degirmencioglu, A.; Cagatay, F. Drying characteristics of laurel leaves under different conditions. In: International Congress on Agricultural Mechanization and Energy, 7, 1999, Adana. Proceedings... Adana: Faculty of Agriculture, Cukurova University, 1999. p.565-569.

Francileni P. Gomes (1), Osvaldo Resende (2), Elisabete P. Sousa (1), Daneil E. C. de Oliveira (3) & Francisco R. de Araujo Neto (2)

(1) Instituto Federal do Amapa/Departamento de Alimentos, Tecnologia em Alimentos. Macapa, AP. E-mail: fpgleni@yahoo.com.br (Corresponding author)--ORCID: 0000-0003-3107-5374; elisabete_pianco@yahoo.com.br--ORCID: 0000-0003-2055-6674

(2) Instituto Federal Goiano/Campus Rio Verde/Diretoria de Pesquisa e Pos-Graduacao. Rio Verde, GO. E-mail: osvaldo.resende@ifgoiano.edu.br--ORCID: 0000-0001-5089-7846; francisco.neto@ifgoiano.edu.br--ORCID: 0000-0003-1064-5614

(3) Instituto Federal de Educacao, Ciencia e Tecnologia Goiano/Campus Ipora. Ipora, GO. E-mail: oliveira.d.e.c@gmail.com--ORCID: 0000-0002-3824-994X

Ref. 183817--Received 09 Aug, 2017 * Accepted 26 Jan, 2018 * Published 28 May, 2018

Caption: Figure 1. Drying kinetics data obtained experimentally and estimated by the Logarithmic model (A) and mean value of effective diffusion coefficient--D (B), obtained in the drying of 'jambu' leaves crushed mass for thicknesses of 0.005 and 0.010 m and temperatures of 60, 70 and 80[degrees]C
Table 1. Standard errors of the estimated coefficients for the models
Approximation of diffusion, Two Terms, Two-term Exponential, Thompson
and Verma evaluated in the drying kinetics of 'jambu' leaves crushed
mass

                                             0.005 m

                  Temp.                           Coefficients
Model           ([degrees]
                    C)               a                  b

Approximation       60        1.6 x [10.sup.4]   1.8 x [10.sup.2]
of diffusion        70        3.2 x [10.sup.4]   1.1 x [10.sup.2]
                    80        3.4 x [10.sup.4]   7.1 x [10.sup.4]
                    60        1.3 x [10.sup.6]   1.3 x [10.sup.6]
Two terms           70        9.7 x [10.sup.5]   9.7 x [10.sup.5]
                    80        1.3 x [10.sup.6]   1.3 x [10.sup.6]
Two-term            60        8.4 x [10.sup.3]         ---
Exponential         70        2.0 x [10.sup.3]         ---
                    80        5.1 x [10.sup.3]         ---
                    60        5.6 x [10.sup.5]   2.9 x [10.sup.3]
Thompson            70        3.8 x [10.sup.5]   2.9 x [10.sup.3]
                    80        7.6 x [10.sup.5]   6.1 x [10.sup.3]
                    60        1.8 x [10.sup.4]         ---
Verma               70        3.2 x [10.sup.4]         ---
                    80        2.8 x [10.sup.2]         ---

                                                0.005 m

                  Temp.                       Coefficients
Model           ([degrees]
                    C)               k           n           g

Approximation       60              1.2                     ---
of diffusion        70              2.7                     ---
                    80              0.0                     ---
                    60        1.2 x [10.sup.3]        7.7 x [10.sup.2]
Two terms           70        2.3 x [10.sup.2]        6.5 x [10.sup.2]
                    80        2.3 x [10.sup.2]        2.1 x [10.sup.2]
Two-term            60              8.9                     ---
Exponential         70              5.6                     ---
                    80              11.5                    ---
                    60              ---                     ---
Thompson            70              ---                     ---
                    80              ---                     ---
                    60              1.2                     1.2
Verma               70              2.7                     2.8
                    80              0.1                     0.1

                                                     0.010m

                  Temp.                            Coefficients
Model           ([degrees]
                    C)               a                  b

Approximation       60        2.6 x [10.sup.4]   2.7 x [10.sup.2]
of diffusion        70        3.0 x [10.sup.4]   1.2 x [10.sup.3]
                    80        4.2 x [10.sup.3]   4.9 x [10.sup.3]
                    60        1.3 x [10.sup.6]   1.3 x [10.sup.6]
Two terms           70        1.9 x [10.sup.5]   1.9 x [10.sup.5]
                    80        1.9 x [10.sup.5]   1.9 x [10.sup.5]
Two-term            60            1.9E+03              ---
Exponential         70        3.1 x [10.sup.3]         ---
                    80        6.7 x [10.sup.3]         ---
                    60        5.4 x [10.sup.5]   2.3 x [10.sup.3]
Thompson            70        6.9 x [10.sup.5]   3.3 x [10.sup.3]
                    80            6.6E+05        3.7 x [10.sup.3]
                    60        3.0 x [10.sup.4]         ---
Verma               70        4.0 x [10.sup.4]         ---
                    80        5.2 x [10.sup.2]         ---
                                            0.010m

                                          Coefficients

                  Temp.
Model           ([degrees]
                    C)               k           N       g

Approximation       60              0.9                 ---
of diffusion        70              1.7                 ---
                    80              0.8                 ---
                    60        1.5 x [10.sup.2]        2.5E+02
Two terms           70              5.2                 5.4
                    80              7.5                 7.3
Two-term            60              2.6                 ---
Exponential         70              4.4                 ---
                    80              7.4                 ---
                    60              ---                 ---
Thompson            70              ---                 ---
                    80              ---                 ---
                    60              0.9                 1.0
Verma               70              1.8                 1.8
                    80              0.2                 0.2

Table 2. Mean estimated error (SE), mean relative error (P), chi-
square test ([chi square]) and coefficient of determination
([R.sup.2]) for the models evaluated in the drying of 'jambu' leaves
crushed mass for different temperatures and thicknesses of 0.005 and
0.010 m

                                60[degrees]C - 0.005 m

                                        [chi square]
Model                  SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.007      4.2        0.05          99.94
Page                  0.030     18.5        0.90          99.06
Newton                0.063     41.1        3.94          95.72
Midilli               0.007      3.5        0.05          99.96
Logarithmic           0.007      3.8        0.05          99.96
Henderson & Pabis     0.056     36.4        3.17          96.70

                                60[degrees]C - 0.010 m

                                        [chi square]
                       SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.009      7.3        0.09          99.9
Page                  0.028     24.1        0.78          99.16
Newton                0.061     52.2        3.71          95.88
Midilli               0.007      5.1        0.05          99.95
Logarithmic           0.007      6.2        0.05          99.94
Henderson & Pabis     0.054     46.5        2.91          96.89

                                70[degrees]C - 0.005 m

                                        [chi square]
Model                  SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.01       6.2        0.13          99.87
Page                  0.025     22.1        0.62          99.38
Newton                0.058     51.2        3.35          96.50
Midilli               0.008      4.1        0.07          99.94
Logarithmic           0.009      5.8        0.08          99.92
Henderson & Pabis     0.050     44.5        2.49          97.52

                                70[degrees]C - 0.010 m

                                        [chi square]
                       SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.013      4.2        0.18          99.81
Page                  0.023      8.6        0.54          99.45
Newton                0.067     26.7        4.52          95.19
Midilli               0.007      1.6        0.05          99.95
Logarithmic           0.010      3.1        0.10          99.90
Henderson & Pabis     0.058     22.6        3.35          96.58

                                80[degrees]C - 0.005 m

                                        [chi square]
Model                  SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.01       2.3        0.13          99.84
Page                  0.026      6.9        0.65          99.22
Newton                0.065     18.8        4.21          94.71
Midilli               0.008      1.5        0.06          99.93
Logarithmic           0.008      1.7        0.06          99.93
Henderson & Pabis     0.055     15.4        3.04          96.36

                                80[degrees]C - 0.010 m

                                        [chi square]
                       SE         P       (decimal)     [R.sup.2]
                    (decimal)    (%)    x [10.sup.-3]      (%)

Wang & Singh          0.008      1.8        0.06          99.93
Page                  0.026      8.1        0.70          99.19
Newton                0.061     20.4        3.78          95.42
Midilli               0.006      1.3        0.04          99.96
Logarithmic           0.006      1.3        0.04          99.96
Henderson & Pabis     0.054     17.5        2.89          96.65

Table 3. Schwarz's Bayesian Information Criterion (BIC) and Akaike
Information Criterion (AIC) for the models that fitted best to the
drying data of the 'jambu' leaves crushed mass under different
conditions of temperature and layer thickness

                               Wang & Singh             Midilli
Model/
Treatments                   BIC        AIC        BIC        AIC

60[degrees]C -  0.005 m    -166.91    -170.57    -166.66    -172.75
60[degrees]C -  0.010 m    -159.29    -163.39    -182.00     188.84
70[degrees]C -  0.005 m    -133.46    -136.87    -144.68    -150.35
70[degrees]C -  0.010 m    -148.28    -152.17    -177.95    -184.43
80[degrees]C -  0.005 m    -121.09    -124.22    -132.93    -138.16
80[degrees]C -  0.010 m    -162.98    -166.63    -172.85    -178.94

                               Logarithmic
Model/
Treatments                   BIC        AIC

60[degrees]C -  0.005 m    -169.44    -174.32
60[degrees]C -  0.010 m    -167.35    -172.81
70[degrees]C -  0.005 m    -141.36    -145.90
70[degrees]C -  0.010 m    -161.70    -166.88
80[degrees]C -  0.005 m    -134.09    -138.26
80[degrees]C -  0.010 m    -175.79    -180.67

Table 4. Coefficients of the models that fitted best to the drying
data of 'jambu' leaves crushed mass under different conditions of
temperature and thickness

                                              0.005 m
             Temperature
Model        ([degrees]C)                    Coefficients

                               a           b           k          n

             60            1.003416    -0.044528    0.073914   1.061702
Midilli      70            1.005617    -0.032338    0.139091   1.123543
             80            1.010381    -0.090987    0.075519   1.129995
             60            3.286140    -2.279470    0.037430      --
Logarithmic  70            1.951988    -0.936192    0.095699      --
             80            12.695900   -11.680600   0.013600      --

                                              0.010 m
             Temperature
Model        ([degrees]C)                   Coefficients

                              a           b          k          n

             60            1.004232   -0.025116   0.043710   1.115153
Midilli      70            1.001743   -0.024922   0.051771   1.249553
             80            1.007867   -0.055776   0.048866   1.048359
             60            3.378600   -2.368020   0.022860      --
Logarithmic  70            4.302200   -3.286370   0.022420      --
             80            6.177150   -5.167520   0.017370      --
COPYRIGHT 2018 ATECEL--Associacao Tecnico Cientifica Ernesto Luiz de Oliveira Junior
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Author:Gomes, Francileni P.; Resende, Osvaldo; Sousa, Elisabete P.; de Oliveira, Daneil E.C.; de Araujo Net
Publication:Revista Brasileira de Engenharia Agricola e Ambiental
Date:Jul 1, 2018
Words:5097
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