Thermodynamic properties and drying kinetics of Bauhinia forficata Link leaves

Propriedades termodinâmicas e cinética de secagem de folhas de Bauhinia forficata Link R E S U M O Objetivou-se, neste trabalho, determinar o coeficiente de difusão efetivo e as propriedades termodinâmicas das folhas de Bauhinia forficata Link, considerando-se duas formas de medição de espessura, bem como descrever o processo de secagem por meio do ajuste de modelos matemáticos. As folhas foram coletadas, levadas ao laboratório e preparadas para dar início ao processo de secagem em quatro temperaturas (40, 50, 60 e 70 °C). Após a secagem determinou-se o coeficiente de difusão efetivo por meio da teoria da difusão líquida permitindo a obtenção dos valores da energia de ativação, entalpia, entropia e energia livre de Gibbs. Já a descrição do processo de secagem foi realizada por meio do ajuste de treze modelos matemáticos constantemente utilizados para representação de secagem de produtos agrícolas. O modelo de Valcam foi selecionado para representar a cinética de secagem de folhas de B. forficata Link. O aumento da temperatura promove: decréscimo de entalpia e entropia; aumento da energia livre de Gibbs e do coeficiente de difusão efetivo. O coeficiente de difusão efetivo é maior quando se considera a espessura da nervura, recomendando-se a padronização e/ou especificação dos pontos de medição da espessura da folha.


Introduction
Medicinal plants have been used in the treatment of diseases since the past generations, which characterizes the millennial use of these products, combined with the popular knowledge and experience (Feijó et al., 2012).Among the more than 300 known species from the Bauhinia genus (Lusa & Bona, 2009), the species Bauhinia forficata Link, popularly known in Brazil as 'pata-de-vaca' , is constantly used in popular medicine, standing out in the treatment of diabetes mellitus (Rodrigues et al., 2012).
As most agricultural products, some medicinal plants need to undergo a drying process, but each agricultural product has a different behavior during this process, since, besides the characteristics of the drying air, its physical properties and chemical composition also have great influence.Because of that, various authors (Martinazzo et al., 2007;Prates et al., 2012;Rocha et al., 2012) use the technique of statistical modeling to predict such behavior.
Along the drying process, it is interesting not only to describe the drying kinetics, but also to observe the thermodynamic properties.They provide important information on the water properties and also on the energy necessary in the process (Corrêa et al., 2010).
This study aimed to determine the effective diffusion coefficient and the thermodynamic properties of leaves of 'pata-de-vaca' (Bauhinia forficata Link) considering two thicknesses, as well as describe the drying process through the fit of mathematical models.

Material and Methods
The leaves of 'pata-de-vaca' (B.forficata Link) were collected in October 2015 in the Medicinal Plants Garden of the Faculty of Agricultural Sciences -FCA, of the Federal University of Grande Dourados.
The initial and equilibrium moisture contents of the samples were determined using the gravimetric method proposed by ASABE (2010) and forced-air oven at 103 ± 1 ºC, for 24 h, in four replicates.
The initial and equilibrium moisture contents, for each temperature, were 1.61 ± 0.08 and 0.05 ± 0.008; 1.86 ± 0.09 and 0.04 ± 0.002; 1.81 ± 0.07 and 0.04 ± 0.008; 1.78 ± 0.07 and 0.04 ± 0.002, for the temperatures of 40, 50, 60 and 70 °C, respectively.The equilibrium moisture content was considered when there was no variation in the mass of the product in three consecutive weighings in intervals of 2 h.
The experiment was conducted using an experimental dryer with four trays equipped with a system that precisely controls the air flow and drying air temperature.The experimental dryer has, as heating source, a set of electrical resistances and a Sirocco fan, with 1-hp motor.Temperature is controlled through a universal process controller working with Proportional-Integral-Derivative (PID) control, while the air flow is selected by a frequency inverter connected to the fan motor.Drying air speed was monitored with the aid of a rotating vane anemometer and maintained around 0.4 m s -1 .
The moisture content ratio of B. forficata Link leaves at all temperatures was determined through Eq. 1.The data of moisture content ratio of B. forficata Link leaves were fitted to the thirteen mathematical models presented in Eqs. 2 to 14: -Diffusion approximation (2) (5) (10)

-Thompson
The thickness (L) of the B. forficata Link leaves was measured with a digital micrometer with resolution of 0.001 mm, using 40 fresh leaves, which were the replicates, 10 for each drying temperature.The thickness of the leaves was determined in two ways: four points on each side of the midrib, totaling eight points on the entire leaf area (Figure 1A) and fifteen points on the entire leaf area; eight outside the ribs and seven on the ribs (Figure 1B).In both cases, the contraction of the thickness was disregarded.
Then, the mean thickness of the B. forficata Link leaves was calculated considering the points outside the rib (OSR), whose mean value was 0.250 ± 0.07 mm, and those outside and on the ribs (ONR), whose mean value was 0.583 ± 0.12 mm.The higher standard deviation of this mean results from the large variation in the thickness of the ribs.
The Arrhenius equation, described in Eq. 16, was used to evaluate the behavior of the effective diffusion coefficient in relation to the different temperatures applied during the drying process for both situations: OSR and ONR.
where: t -time of drying, h; k, k 0 , k 1 -constants of drying, h -1 ; and, a, b, c, d, n -coefficients of the models.
The effective diffusion coefficient at the various drying temperatures was determined using Eq. 15, based on the theory of liquid diffusion, which considers the geometric form of the product as close to a flat plate with approximation of eight terms.where: D 0 -pre-exponential factor; E a -activation energy, KJ mol -1 ; R -universal gas constant, 8.314 kJ kmol -1 K -1 ; and, T a -absolute temperature, K.
After selecting the model to represent the effective diffusivity, an equality hypothesis test of the models was performed using the method of Regazzi (2003).This method allows the analysis of equivalence between the models and aims to establish a single equation to describe the studied phenomenon.
The parameters of the linear model used to describe the effect of the thickness of B. forficata Link leaves on the effective (11) (12) diffusivity (a i and b i ) were compared to verify their equality.The formulated hypotheses were: H o -a 1 = a 2 and b 1 = b 2 ; and, H a -there is at least one inequality between the parameters.
According to Regazzi (2003), to obtain the forms of the complete model of the equations with restrictions, dummy variables (D) were considered: D i = 1 if the observation y ij belongs to the group i, and D i = 0 in the opposite case.
The decision rule was based on the chi-square test (χ 2 ), according to Eq. 17.
error (P) and standard deviation of the estimate (SE), calculated according to Eqs. 22 and 23.
where: N -number of observations; RSS Ω -residual sum of squares of the complete model; and, RSS Wi -residual sum of squares of the restricted parameter space.
. The tabulated value of (χ 2 ) is a function of the significance level α and the number of degrees of freedom, described in Eq. 18.The diffusion models were compared using the two previously mentioned values of thickness (0.250 and 0.583 mm).
The thermodynamic properties associated with the drying process were determined according to the method proposed by Jideani & Mpotokwana (2009), presented in Eqs. 19, 20 and 21, respectively, specific enthalpy, specific entropy and Gibbs free energy.
The degree of fit of each model was analyzed using the magnitudes of the determination coefficient (R 2 ), relative mean The experimental data from the drying kinetics of B. forficata Link leaves were analyzed and subjected to nonlinear regression analysis, through the Gauss-Newton method, using the software Statistic 8.0.

Results and Discussion
Only the Wang-Singh model showed determination coefficients (R 2 ) lower than 0.95, which, according to Kashaninejad et al. (2007), is the minimum value to obtain a satisfactory representation of models of the drying process (Table 1).However, the determination coefficient is not a correct parameter for this type of characterization when it is individually analyzed (Madamba et al., 1996).According to Siqueira et al. (2012), the lower the SE values, the better the fit of the models to the experimental data.In this case, it is possible to claim that the capacity of a model to precisely describe certain physical process is inversely proportional to the value of the standard deviation of the mean (Draper & Smith, 1998).
The acceptable values of P must be lower than 10% (Aguerre et al., 1989;Mohapatra & Rao, 2005).Therefore, the models Modified Henderson-Pabis, Logarithmic, Valcam and Verma are the only ones to meet this requirement, besides exhibiting low SE and high R 2 values (Table 1), at all drying air temperatures.
The model selected to represent the B. forficata Link drying curves was the Valcam model, for presenting a simplified form and lower number of coefficients, thus being easily used in drying simulation processes.
The values of the effective diffusion coefficient calculated without considering the thickness of the leaf ribs (0.250 mm) and the effective diffusion coefficient calculated considering the points outside and on the leaf ribs (0.583 mm) increased as the drying air temperature increased (Table 2).Martinazzo et al. (2007) and Prates et al. (2012) observed the same behavior for lemon grass and 'fruta-de-lobo' leaves.
The effective diffusion coefficient serves as an indication of the speed of water outlet.With the increase in drying air temperature and consequent increment in the difference of partial pressure of water vapor between the drying air and    the product, the effective diffusion coefficient becomes higher.Such behavior can be related to the viscosity of the water, which decreases with the temperature; oscillations in the behavior of this property leads to alterations in water diffusion, favoring the movement of water through the capillaries of the leaves (Goneli et al., 2014).In addition, the coefficients "a", "b", "c" and "d" of the Valcam model showed high degree of significance for all drying conditions (Table 2).
The values of OSR effective diffusion coefficient varied from 6.4236 x 10 -12 to 3.9491 x 10 -11 m 2 s -1 , a behavior similar to that observed in lemon grass leaves (Martinazzo et al., 2007) and basil leaves (Reis et al., 2012).Both authors obtained effective diffusion coefficients ranging from 10 -12 to 10 -11 m 2 s -1 .On the other hand, the ONR effective diffusion coefficients remained in the range between 1.7829 x 10 -11 and 1.0961 x 10 -10 m 2 s - content ratio of the product, dimensionless; U -moisture content at a certain time, decimal (b.s); Ue -equilibrium moisture content, decimal (b.s); and, Ui -initial moisture content, decimal (b.s).
the product, m; θ -drying time, s; and, n -number of terms of the model.

Figure 1 .
Figure 1.Points of thickness measurements on B. forficata Link leaves outside the rib (OSR) (A), and outside and on the ribs (ONR) (B) freedom of the model; P W -number of parameters of the complete model; and, P Wi -number of parameters of the model with restriction.
moisture content ratio observed experimentally; RX est -moisture content ratio estimated by the model; and, v -degrees of freedom of the model.

Figure 2 .
Figure 2. Moisture content obtained experimentally and estimated by the Valcam model, at the different drying temperatures of B. forficata Link leaves

Table 2 .
Estimated values of the parameters of the Valcam model and effective diffusion coefficient considering (ONR) and disregarding (OSR) the thickness of the ribs of B. forficata Link leaves

Table 3 .
1. Thermodynamic properties of the drying process of B. forficata Link leaves: specific enthalpy (h), specific entropy (s) and Gibbs free energy (G)