1 Introduction
The Brazilian Cerrado covers almost two million square kilometers of Brazil, and includes the states of Goias, Tocantins, Mato Grosso do Sul, Minas Gerais, Bahia, Mato Grosso, Maranhão, and Piauí. It contains a rich variety in fruits, some of which have medicinal properties, high nutritional potential, and unique flavors not found in other fruits (^{Klink & Machado, 2005}).
The cagaita is an indigenous of the Brazilian Cerrado fruit, which can be consumed fresh or processed (as jellies, ice creams, liqueurs, and juices). Because it is very perishable, it needs to be preserved by various techniques to increase its shelf life (^{Costa et al., 2017}).
One technique that preserves food products is foamlayer drying. Liquid or semiliquid foods are transformed into powder by agitation and incorporation of a foaming agent, followed by dehydration (^{Ng & Sulaiman, 2018}).
This drying process is recommended for products that are sensitive to heat or contain sugars, mainly because they require a shorter exposure time and lower temperatures than used in other techniques (^{Fernandes et al., 2014}).
To simulate and obtain data on product behavior during water removal, different mathematical models are used, which are based on external variables such as temperature and relative humidity of the air during the drying process (^{Resende et al., 2008}).
A mathematical modeling in the drying process is able to predict moisture removal behavior, estimate drying time, energy expenditure and equipment sizing (^{Keneni et al., 2019}).
Diffusion of water during drying is a complex process that involves different mechanisms, such as molecular diffusion, capillary diffusion, surface diffusion, hydrodynamic flow, vapor diffusion, and diffusion activation energy (^{Goneli et al., 2009}).
The objective of this work was to propose and adjust mathematical models of the foamlayer drying process of cagaita at different temperatures, and to determine the effective diffusion coefficient and activation energy for this process.
2 Material
Cagaita fruits were collected in the region of Montes Claros, Goiás, Brazil (16°06'20 “S and 51º17'11” W), packed in 30 × 40 cm polyethylene bags, placed in thermal boxes and transported to the Laboratory of Phytochemistry of IF Goiano–Campus Rio Verde. The albumina was purchased in powder by the Naturovos trademark (Salvador do Sul, RS, Brazil) at a store specializing in raw materials for the Food Industry in Rio Verde Goiás, Brazil.
3 Methods
The Cagaita were then selected for size, absence of mechanical injury, and maturation stage and the chosen fruits were then sanitized in chlorinated water for 15 min and dried on paper towels.
The fruits were homogenized in a Toturgan® electric pulper, and the pulp was packed in 25 × 35 cm polyethylene bags and stored in a freezer at 18 °C until analysis. For foaming, commercial, unflavored albumin (8% w/w) was added to the pulp, and the mixture was then shaken with a domestic shaker brand Arno, model SX 15 for 15 min.
Approximately 100 g of the formed foam was added onto unperforated aluminum trays and dried in an oven, using forced air circulation at 40, 50, 60, and 70 °C, to determine drying kinetics.
To determine drying curves and to adjust the models, the pulp was dried to a constant mass. The water contents of the product were determined in a greenhouse at 105 ± 3 °C, in three replicates, until constant mass was reached.
The experimental data obtained from the drying of the cagaita pulp with an emulsifier were used to adjust the mathematical models frequently used to describe drying processes of vegetable products, as presented in Table 1.
Equation  Model  


Wang and Sing  (1) 

Verma  (2) 

Thompson  (3) 

Page  (4) 

Newton  (5) 

Midilli  (6) 

Logarithmic  (7) 

Henderson and Pabis  (8) 

Henderson and Pabis, modified  (9) 

Twoterm Exponential  (10) 

Twoterm  (11) 

Diffusion Approximation  (12) 
t – time (h); k, k_{o}, k_{1} – constant of the equation (h^{1}); a, b, c, n – parameters of the equations; RX – moisture ratio, nondimensional.
For the adjustment of the mathematical models, nonlinear regression analysis (Equation 13) was performed by the Gauss Newton method.
Where:
RX – moisture ratio, nondimensional;
X* – moisture content, dry basis;
X*_{e} – moisture content equilibrium, dry basis;
X*_{i} – initial moisture content, dry basis.
The models were selected based on the magnitude of the coefficient of determination (R^{2}), the chisquared test (χ^{2}), the mean relative error (P), and the estimated standard deviation (SE). Additionally, an average relative error below 5% was considered one of the criteria for the selection of models, as per (^{Mohapatra & Srinivasa Rao, 2005}).
To evaluate net diffusion, we used the model of the flat plate geometry with an approximation of eight terms (Equation 14), which was adjusted to the experimental data of foamlayer drying of the cagaita pulp, and we considered the surface area and volume.
Where:
RX – moisture ratio, nondimensional;
X* – moisture content, dry basis;
X*_{e} – moisture content equilibrium, dry basis;
X*_{i} – initial moisture content, dry basis;
nt – number of terms;
S – surface area, m^{2};
V – volume, m^{3}.
The dimensions of length, width, and thickness were measured with a digital caliper.
The relationship between the diffusion coefficient and the drying air temperature is described by Equation 15.
Where:
D_{0} – Preexponential factor;
E_{a} – Activation energy, kJ mol ^{1};
R – Universal Gas Constant, 8.134 kJ kmol^{1} K ^{1};
T_{abs} – Absolute temperature, K.
The Arrhenius expression coefficients were loglinearized according to Equation 16.
In addition to the previous parameters, the Akaike information criterion (AIC) and the Bayesian Schwarz information criterion (BIC) were used. The AIC allows us to use the principle of parsimony in choosing the best model, that is, according to this criterion, the most parameterized model is not always the best (^{Burnham & Anderson, 2004}).
AIC is used to compare nonnested models or to compare three or more models. Lower AIC values reflect a better fit (^{Akaike, 1973}). BIC also considers the degree of parameterization of the model, and therefore, the smaller the BIC value is ^{Schwarz (1978)}, the better the model adjustment is.
3 Results and discussion
Drying curves of cagaita foam at 40, 50, 60, and 70 °C. The values were estimated from Midilli model are shown in Figure 1.
It was observed that the drying time decreases with the increase in the temperature. ^{Ferreira et al. (2012)} previously explained that due to the increase in the drying rate, the heat transfer potential between the air and the layer of the product is increased, resulting in greater water reduction in a shorter period compared with lower drying rates.
The curves have a steeper slope due to the increase in the amount of heat transferred from the air to the dried material. Similar behaviors were reported by ^{Baptestini et al. (2015)} in a study of foam drying of graviola.
As seen in Table 2, the Page model fitted better at 50, 60, and 70 °C than the other models; the Twoterm Exponential fitted better at 50 °C and 70 °C than the others ; and the Twoterm fitted better at 70 °C than the others. In order to adjust the mathematical models, P must be less than 10% (^{Mohapatra & Srinivasa Rao, 2005}), R^{2} must be next to the unit, and SE must be close to zero (^{Resende et al., 2006}).
Model  40 °C  50 °C  

SE  χ^{2}  P  R^{2}  SE  χ^{2}  P  R^{2}  
Wang and Sing  0.0013  0.9988  6.56  0.9975  0.0012  0.9989  8.6330  0.9978  
Verma  0.0026  0.9974  14.26  0.9949  0.0034  0.9967  18.6592  0.9934  
Thompson  0.0121  0.9881  38.88  0.9764  0.0089  0.9913  28.9043  0.9827  
Page  0.0020  0.9981  10.51  0.9961  0.0011  0.9989  5.7286  0.9979  
Newton  0.0121  0.9881  38.87  0.9764  0.0089  0.9913  28.8968  0.9827  
Midilli  0.0011  0.9989  6.95  0.9978  0.0010  0.9990  6.9005  0.9981  
Logarithmic  0.0025  0.9976  13.51  0.9952  0.0028  0.9973  16.8926  0.9945  
Henderson and Pabis  0.0088  0.9914  32.38  0.9828  0.0057  0.9944  22.1547  0.9889  
Twoterm Exponential  0.0023  0.9978  14.52  0.9956  0.0012  0.9988  6.5252  0.9977  
Twoterm  0.0022  0.9978  12.35  0.9957  0.0024  0.9953  15.3219  0.9953  
Diffusion Approximation  0.0026  0.9974  14.26  0.9949  0.0034  0.9967  18.6591  0.9934  
Henderson and Pabis, modified  0.0022  0.9978  12.37  0.9957  0.0024  0.9977  15.3223  0.9953  
60 °C  70 ºC  
SE  χ^{2}  P  R^{2}  SE  χ^{2}  P  R^{2}  
Wang and Sing  0.0031  0.9965  9.46  0.9930  0.0014  0.9982  4.9067  0.9964  
Verma  0.0038  0.9957  13.93  0.9914  0.0025  0.9969  9.7008  0.9938  
Thompson  0.0180  0.9792  47.07  0.9588  0.0098  0.9876  23.4831  0.9754  
Page  0.0010  0.9989  9.83  0.9978  0.0005  0.9994  2.8481  0.9988  
Newton  0.0180  0.9792  47.07  0.9588  0.0098  0.9876  23.4804  0.9754  
Midilli  0.0007  0.9992  4.98  0.9984  0.0004  0.9994  1.7747  0.9989  
Logarithmic  0.0027  0.9969  10.97  0.9939  0.0019  0.9976  8.1507  0.9953  
Henderson and Pabis  0.0025  0.9971  10.06  0.9942  0.0062  0.9922  18.0108  0.9844  
Twoterm Exponential  0.0016  0.9982  15.72  0.9964  0.0006  0.9992  5.0435  0.9985  
Twoterm  0.0025  0.9971  10.06  0.9942  0.0005  0.9994  3.5532  0.9988  
Diffusion Approximation  0.0038  0.9957  13.93  0.9914  0.0025  0.9969  9.7008  0.9938  
Henderson and Pabis, modified  0.0025  0.9971  10.06  0.9942  0.0017  0.9979  7.4712  0.9957 
The Diffusion Approximation model and the Henderson and Pabis Modified model were adapted to the experimental data obtained at drying at 70 °C. The Midilli model, followed by the Wang and Sing model, was the best model to represent the drying kinetics of powdered cagaita pulp at all temperatures.
The models for the same mathematical models in foamlayer drying were found by ^{Silva et al. (2008)} in the dehydration of the tamarind pulp, with temperatures between 50 and 80 ° C, where the best models that represent the behavior of the dehydration curves were of Midilli and Kucuk. ^{Alves & Rodovalho (2016)} with avocado pulp, with temperatures of 50 to 80 ° C, and the best model of Wang and Sing. In Table 3, the AIC and BIC confirmed that the fit of the Wang and Sing model was better at 40 and 50 ºC, whereas the Midilli model was better adjusted at 60 and 70 °C.
Wang and Sing  Midilli  

Temperature (°C)  AIC  BIC  AIC  BIC 
40  101.57  98.583  100.08  95.105 
50  97.819  94.981  96.744  92.022 
60  59.796  57.672  78.389  74.858 
70  65.79  63.872  78.883  65.79 
When analyzing the drying kinetics of the crushed jambu mass, evaluating the AIC and BIC parameters, it found the best Midilli model for temperatures of 60 and 70 ° C (^{Gomes et al., 2018}). The Midilli model is probably associated with the rapid loss of water at the beginning of the process steps, generating a drying curve that best characterizes mathematically by this model (^{Goneli et al., 2009}).
The values of the net diffusion obtained for the different drying temperatures are shown in Figure 2. The curve shows a linear, increasing behavior in which the values of the diffusion coefficient increased with the increase in the temperature.
According to ^{Baptestini et al. (2015)}, this behavior is predictable, because the temperature is inversely correlated to the viscosity, which facilitates the diffusion of water molecules in the capillaries of the product. The same behavior was observed by ^{Alves & Rodovalho (2016)}, in their study of foamlayer drying kinetics of avocado pulp. The variation in the effective diffusion coefficient as a function of drying temperature is described by the ratio of Arrhenius (Figure 3).
The activation energy for drying the foam of the powdered cassava pulp was 25.368 kJ mol^{1} at a temperature range of 40 to 70 °C. Similar values were found for bananas dried by the same method (^{Thuwapanichayanan et al., 2008}), which had an activation energy of 25.19 kJ mol^{1} at a temperature range of 60, 70 and 80 °C.
The lower the activation energy is, the greater the water diffusivity in the product is. Our reported values are within the energy activation range reported by ^{Zogzas et al. (1996)}, i.e., between 12 and 110 kJ mol^{1}.
4 Conclusion
The Wang and Sing models at 40 and 50 °C and the Midilli model at 60 and 70 °C were the ones that best fit the experimental data of the drying curves of the foamlayer drying of cagaita using AIC and BIC parameters. With increasing air temperature, a reduction in the drying time of the cagaita pulp foam occurred. The net effective diffusion coefficient increases with increasing drying temperature. The value of the activation energy is within the established parameters.