Introduction

Atemoya is a fruit of a plant from the *Annonaceae* family; it is a hybrid resulting from the cross between sugar-apple (*A. squamosa* L.) and cherimoya (*A. cherimola* Mill.) (^{Medeiros et al., 2009}). It has intense metabolic activity and is very perishable due to its high water content. Other unfavorable factors of atemoya are the rapid softening of the pulp and the darkening of its skin, as the main problems that affect its commercialization (^{Yamashita et al., 2002}).

Drying is a commercial process widely used to preserve the quality of agricultural products with the objective of promoting long storage periods without significant losses during the process (^{Martinazzo et al., 2010}). Foam-mat drying is a process that helps to preserve the nutrients of the pulp for prolonged period and immediately after dehydration; in addition, the powder can be reconstituted as juice and/or used as ingredient for the preparation of beverages and food (^{Kadam et al., 2010})

Although there is a great interest in its comprehension, drying is still one of the least understood operations, due to the complexity of the phenomena involved in the simultaneous transfer of heat, mass and amount of movement in the solid during the process (^{Kingsly et al., 2007}). Researchers study the drying process based on the external conditions of air, relative humidity and temperature. Thus, it requires mathematical models that represent the reduction in water content ratio of the products during the process (^{Alves, 2010}). The use of mathematical models to represent the process is important, because the generated information is utilized in the development of devices and prediction of drying times (^{Silva et al., 2009}).

Thus, this study aimed to dehydrate a suspension of atemoya through the foam-mat drying, for different values of layer thickness (0.5, 1.0 and 1.5 cm) and temperatures (60, 70 and 80 ºC), and study the adjustment of drying mathematical models to the experimental data.

Material and Methods

The raw material consisted of whole atemoya pulp, from the municipality of Petrolina, PE, and Emustab^{®} and Super Liga Neutra^{®}, both obtained from the market, in the municipality of Campina Grande-PB, Brazil.

Initially, Emustab^{®} and Super Liga Neutra^{®}, both at the concentration of 2%, were added to the whole atemoya pulp. This mixture was taken to a planetary mixer for homogenization for 20 min, in order to form a stable suspension to be dehydrated in layer for the evaluation of its drying kinetics.

The suspension was spread on stainless-steel trays forming a foam layer with different thicknesses (0.5, 1.0 and 1.5 cm), measured with a caliper. Then, they were placed in a forced-air oven at temperatures of 60, 70 and 80 ºC, defined by preliminary tests. The drying kinetics curves of the suspension were obtained by weighing the trays until constant mass, in regular intervals. Water content ratios (Eq. 1) were calculated based on the experimental data.

where:

RX - water content ratio, dimensionless;

X - water content, dry basis;

Xe - equilibrium water content, dry basis; and,

Xo - initial water content, dry basis.

The semi-theoretical models of Henderson & Pabis (Eq. 2), Page (Eq. 3) and Midilli (Eq. 4) were applied for the determination of the drying curves.

- Henderson & Pabis

where:

RX - water content ratio, dimensionless;

a - dimensionless constant of the equation;

k - constant of the equation, L h^{-1}; and,

t - time, min.

- Page

where:

RX - water content ratio, dimensionless;

k - constant of the equation, L h^{-1};

n - constant of the equation; and,

t - time, min.

- Midilli

where:

RX - water content ratio, dimensionless;

a - constant of the equation, dimensionless;

k - constant of the equation, L h^{-1};

b - constant of the equation, dimensionless;

n - constant of the equation; and,

t - time, min.

The model that best adjusted to the experimental data was evaluated through the parameters coefficient of determination (R^{2}) and mean quadratic deviation (Eq. 5).

where:

MQD - mean quadratic deviation;

RX_{pred} - water content ratio predicted by the model;

RX_{exp} - experimental water content ratio; and,

n - number of observations.

Results and Discussion

Figure 1 shows the kinetics of foam-mat drying for thicknesses of 0.5, 1.0 and 1.5 cm, at temperatures of 60, 70 and 80 ºC, in the dimensionless form of the water content (water content ratio) as a function of the drying time; the foams showed initial water content of 72.13% (d.b.).

The drying process of all samples was faster for smaller thicknesses (0.5 cm) at the three studied temperatures (60, 70 and 80 °C), and the drying times were equal to 420, 380 and 310 min, with final water contents of 13.23, 8.96 and 7.37% (w.b.), respectively. For the thickness of 1.0 cm, at the three temperatures, the drying times were equal to 1,000, 800 and 550 min, with final water contents of 12.48, 7.21 and 5.82% (w.b.), while for the thickness of 1.5 cm the drying times were 1,850, 1,100 and 850 min with final water contents of 12.26, 5.55 and 4.25% (w.b.), at the temperatures of 60, 70 and 80 °C, respectively.

All drying curves were influenced by the thickness of the layer. This behavior was also observed by ^{Raharitsifa & Ratti (2010)}, who applied lyophilization on apple juice foam prepared with 3% of egg albumin in different thicknesses (1, 4 and 6 cm) and observed that thickness has a significant impact on foam drying, i.e., the smaller the thickness, the shorter is the drying time.

During the drying process, besides the influence of foam layer thickness, there was the interference of drying air temperature, since the temperature of 80 ºC led to the shortest drying times for the thicknesses of 0.5, 1.0 and 1.5 cm, which were equal to 310, 550 and 850 min, respectively. At the temperature of 70 ºC, the drying times were equal to 380, 800 and 1,100 min, for the three thicknesses, while at 60 ºC the drying times were 420, 1,000 and 1,850 min.

The influence of temperature increment on the drying time is expected and reported by ^{Krasaekoopt & Bhatia (2012)}, who also observed reduction in the drying time with the increase in temperature, when drying yogurt through foam-mat drying. These differences in drying times occur mainly because of the different temperatures applied and the characteristics of each product. M^{elo et al. (2013)} observed the influence of drying temperature and foam layer thickness in foam-mat drying of cactus (*Cereus jamacaru*) fruit, and the drying occurred in a shorter time at the highest temperature for the smallest thickness.

For all studied temperatures and thicknesses, there was greater loss of water at the beginning of the process, with subsequent decrease in the rate of migration of water from the inside to the surface. As a result, lower drying rates were observed in the final drying stage. ^{Melo et al. (2013)}, studying the drying kinetics of cactus (*Cereus jamacaru*) fruit pulp, observed that the water ratio reduced rapidly in the beginning of the process and slowly as the drying time increased, for the studied temperatures.

Table 1 shows the coefficients of the models of Page, Henderson & Pabis and Midilli for the drying temperatures of 60, 70 and 80 °C and foam layer thicknesses of 0.5, 1.0 and 1.5 cm, as well as the coefficients of determination (R^{2}) and the mean quadratic deviations (MQD). For the three thicknesses, all models can be used to represent the foam-mat drying of atemoya pulp, because they showed coefficients of determination (R^{2}) higher than 0.99 and mean quadratic deviations below 0.05.

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

0.5 | Page | 60 | 0.0065 | 1.0792 | - | 0.9980 | 0.0386 | |

70 | 0.0053 | 1.1674 | - | 0.9983 | 0.0290 | |||

80 | 0.0034 | 1.2630 | - | 0.9983 | 0.0251 | |||

Henderson & Pabis | 60 | 1.0073 | 0.0096 | - | - | 0.9974 | 0.0278 | |

70 | 1.0309 | 0.0116 | - | - | 0.9964 | 0.0352 | ||

80 | 1.0577 | 0.0125 | - | - | 0.9943 | 0.0416 | ||

Midilli | 60 | 0.9819 | 0.0062 | 1.0746 | -0.00008 | 0.9988 | 0.0017 | |

70 | 0.9842 | 0.0049 | 1.1713 | -0.00008 | 0.9989 | 0.0024 | ||

80 | 0.9935 | 0.0038 | 1.2303 | -0.00012 | 0.9989 | 0.0020 | ||

1.0 | Page | 60 | 0.0024 | 1.0407 | - | - | 0.9983 | 0.0266 |

70 | 0.0016 | 1.1815 | - | - | 0.9973 | 0.0381 | ||

80 | 0.0013 | 1.3015 | - | - | 0.9989 | 0.0293 | ||

Henderson & Pabis | 60 | 1.0020 | 0.0030 | - | - | 0.9981 | 0.0212 | |

70 | 1.0266 | 0.0046 | - | - | 0.9948 | 0.0386 | ||

80 | 1.0505 | 0.0066 | - | - | 0.9940 | 0.0446 | ||

Midilli | 60 | 1.0017 | 0.0038 | 0.9429 | -0.00009 | 0.9996 | 0.0007 | |

70 | 0.9853 | 0.0016 | 1.1652 | -0.00006 | 0.9983 | 0.0017 | ||

80 | 0.9868 | 0.0012 | 1.3070 | -0.00004 | 0.9994 | 0.0049 | ||

1.5 | Page | 60 | 0.0012 | 1.0802 | - | - | 0.9975 | 0.0880 |

70 | 0.0019 | 1.0679 | - | - | 0.9981 | 0.0364 | ||

80 | 0.0009 | 1.2865 | - | - | 0.9982 | 0.0404 | ||

Henderson & Pabis | 60 | 1.0135 | 0.0021 | - | - | 0.9971 | 0.0360 | |

70 | 1.0017 | 0.0029 | - | - | 0.9975 | 0.0249 | ||

80 | 1.0443 | 0.0046 | - | - | 0.9931 | 0.0460 | ||

Midilli | 60 | 0.9861 | 0.0009 | 1.1164 | -0.00002 | 0.9989 | 0.0018 | |

70 | 0.9890 | 0.0025 | 0.9997 | -0.00008 | 0.9997 | 0.0007 | ||

80 | 0.9871 | 0.0009 | 1.2802 | -0.00005 | 0.9989 | 0.0022 |

For the thickness of 0.5 cm, among the tested models, Midilli showed the best fit to the experimental data, with the highest coefficients of determination and the lowest mean quadratic deviations. ^{Furtado et al. (2010)} studied the drying kinetics of 'ceriguela' (*Spondias purpurea*) pulp, at temperatures of 60, 70 and 80 ºC, and observed that the Midilli model was satisfactory. According to ^{Azzouz et al. (2002)}, the parameter *n* in the Page model depends on the drying air speed and initial water content of the product and *k* depends on temperature and initial water content. The obtained *k* values decreased with the increase in drying temperature, while the parameter *n* increased as temperature increased.

^{Giraldo-Zuniga et al. (2010)} studied the drying kinetics of sliced cupuaçu (*Theobroma grandiflora*) pulp, at temperatures of 50, 60 and 70 ºC, and observed variation in the parameters *k* and *n* in relation to the studied temperatures in the model of Page. ^{Alexandre et al. (2009)} studied the drying of sliced pineapple (*Ananas comosus* L.), cv. 'Pérola', at temperatures of 50, 60, 70 and 80 ºC, and obtained R^{2} > 0.97 for the model of Page.

In the model of Henderson & Pabis, the values of the parameter *k* increased with the increment in temperature. This behavior can be observed with the parameter *a* and this fact occurred because the drying curves were very close to each other.

The values of *k* and *n* for the Midilli model decreased and increased with the increment in temperature, respectively. ^{Madureira et al. (2011)}, adjusting the drying data of the formulation of cactus pear (*Opuntia fícus-indica* Mill.) pulp mixed with 25% of modified starch, highlight that *n* values increased as temperature increased.

For the thickness of 1.0 cm, in the three models, the parameter *k* decreased with the increase in temperature for the models of Page and Midilli, while the opposite occurred for the model of Henderson & Pabis. For the parameter *n*, there was an increase with the increment in temperature. ^{Perez et al. (2013)} observed that the model of Page adjusted well to the experimental data of drying of 'cupuaçu' (*Theobroma grandiflorum*) pulp for the studied treatments, showing coefficient of determination (R^{2}) higher than 0.99 and mean percent errors lower than 8.5%; thus, it can be used in the prediction of cupuaçu pulp drying kinetics. Among the tested models, Midilli showed the best adjustments, which is confirmed by the coefficients of determination (R^{2}) above 0.99 and mean quadratic deviations below 0.05.

For the thickness of 1.5 cm, in the three models used, the parameters *k* and *n* did not show a defined behavior as temperature increased in the models of Page and Midilli, and the parameter *k* increased with the increment in temperature in the model of Henderson & Pabis. ^{Santos et al. (2010)} analyzed the drying kinetics of starfruit (*Averrhoa carambola* L.) at the temperatures of 50, 60 and 70 ºC and observed that the model of Page showed the best fit for the experimental data, with the highest R^{2} and the lowest relative mean error. Among the tested models, Midilli showed the best fits, with R^{2} > 0.98, and the lowest MQD, all below 0.09.

Figure 2 shows the drying kinetics curves for atemoya pulp foam at different temperatures and foam layer thicknesses of 0.5, 1.0 and 1.5 cm, fitted to the Midilli model.

As shown in Figure 2A, the experimental data are close to the curves established by the model and the drying temperature influences the kinetics, indicating the tendency that the higher the temperature, the shorter the drying time. There are small differences between the curves of 70 and 80 ºC, with greater distance from the curve of 60 ºC.

According to Figure 2B and 2C, the experimental data are close to the curves predicted by the model and the loss of water is more intense in the beginning of the drying. There was influence of temperature on the drying curves, which indicates that the higher the temperature, the faster is the drying. ^{Sousa et al. (2011)} also obtained excellent accuracy using the Midilli model in the prediction of drying data in a thin layer of oiti (*Opuntia fícus-indica* Mill.) pulp at the temperatures of 50, 60 and 70 ºC.