**2. Methodology**

## **2.1 Materials and methods**

Fresh apple (*Malus domestica* L.) of var. Granny Smith was locally purchased (San Andrés Cholula, Puebla, Mexico). Apples were washed, sanitized by immersion into peracetic acid solution (100 mg/L) for 10 min, peeled, and cut into cubes (1.2 cm 1.2 cm 1.2 cm) with an industrial vegetable cutter. Osmotic solutions were prepared with food-grade granulated refined sucrose and distillated water.

*Modeling of Drying Kinetics of Fresh and Osmodehydrated Apples during Convective Drying DOI: http://dx.doi.org/10.5772/intechopen.105162*

#### **2.2 Osmodehydration experiments**

Apple cubes (*w*0) were immersed in the osmotic solution (40, 50 and 60°Brix sucrose) at a mass ratio of 1:10 (to avoid excessive dilution), and at a temperature of 40°C [25] until an equilibrium state of mass transfer (without significative differences between water activity of osmotic solution and product) was reached. The mass transfer equilibrium state was determinate by measuring water activity in a digital hydrometer (Aqua Lab, 4TEV, USA) [17]. Then, the osmodehydrated apple cubes were removed from the osmotic solution. The excess liquid was removed with absorbent paper, weighed (*wOD*), and the moisture content (*Y*<sup>0</sup> and *YDO*, fresh and osmodehydrated product, respectively) was determined to calculate the water loss (*W*), Eq. (1), and solids gain (*G*) of the product, Eq. (2) [26]. All the osmodehydration experiments were carried out in triplication.

$$W = \frac{w\_0 Y\_0 - w\_{OD} Y\_{OD}}{w\_0} \tag{1}$$

$$G = \frac{w\_{OD}(\mathbf{1} - Y\_{OD}) - w\_0(\mathbf{1} - Y\_0)}{w\_0} \tag{2}$$

#### **2.3 Drying experiments**

#### *2.3.1 Drying curves*

Fresh and osmodehydrated apple cubes were subjected to convective drying in a dryer designed and built at the Universidad de las Américas Puebla. The dryer includes a fan (of axial flow), an electric device with two 1000-W electric resistances, and a drying chamber with three tray perforations (28 � 18 cm). The perforations allow the free passage of air, thus drying both sides of the samples. The samples were dried at 50°C and air velocity of 3.5 m/s [4, 5] until constant weight (�0.05 g). A digital balance was used to record the evolution of sample weight during the process. Weight data during the process was used to estimate the kinetics of free moisture fraction (Ψ) using the Eq. (3). Experimental kinetics of drying were carried out in triplicated.

$$
\Psi = \frac{mc\_{pt} - mc\_{p\infty}}{mc\_{p0} - mc\_{p\infty}} \tag{3}
$$

Where *mcp*<sup>0</sup> is initial moisture content, *mcpt* is moisture content at time *t* based on recoded weight, and *mcp*<sup>∞</sup> is equilibrium moisture content.

#### *2.3.2 Evolution of dimensionless volume*

Dimensionless volume was analyzed using image processing developed by González-Pérez et al. [4]. Briefly, a digital image of slices (1 mm-tick) of a group of 5 samples corresponding to certain free moisture fractions (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9) was acquired. The 4608 � 3456 pixels digital image was taken with a camara (Coolpix L810, Nikon Corp, Japan) using natural light and blue background (like contrast). ImageJ® (ver. 1.49 h, Dresden, Germany) was used to analyze the

bidimensional images. The measure of contour and surface (*S*) gave information about product shape [27]. Finally, the evolution of dimensionless estimated with Eq. (4) was fitted with Eq. (5), and characteristic length (*L*) was estimated with Eq. (6). Eq. (6) will be used in the next section to estimate the evolution of effective diffusion.

$$\frac{V}{V\_0} = \frac{\mathbb{S}^{3/2}}{\mathbb{S}\_0^{3/2}}\tag{4}$$

$$\frac{V}{V\_0} = m + (1 - m)\Psi^n\tag{5}$$

$$L = \left(V\_0(m + (1 - m)\Psi^n)\right)^{1/3} \tag{6}$$

Where *V* and *S* are the volume (m) and sample surface at time t, *V*<sup>0</sup> and *S*<sup>0</sup> are the volume (m) and sample surface at initial time, *m* and *n* are dimensionless parameters of the volume model.

#### **2.4 Modeling of drying curves**

### *2.4.1 Mathematical models*

Mathematical models have been developed to optimize the dryers and/or predict the quality of the dried product [28, 29]. To find out the best mathematical model to predict drying curves some thin-layer drying models were used, including Newton [30], Henderson and Pabis [31], Page [32] and Weibull [33] model according to Eqs. (7)–(10), respectively. Non-linear regression was used to estimate model parameters. The correlation coefficient (R2 ) and root mean square error (RMSE) were estimated to evaluate the goodness of fit of selected models.

$$
\Psi = \mathfrak{e}^{(-k\_1 t)} \tag{7}
$$

$$
\Psi = a e^{(-k\_2 t)} \tag{8}
$$

$$
\Psi = \mathfrak{e}^{\left(-k\_{\beta}t^{b}\right)} \tag{9}
$$

$$
\Psi = \mathfrak{e}^{-\left(t/k\_4\right)^\varepsilon} \tag{10}
$$

Where *k1*, *k*<sup>2</sup> and *k*<sup>3</sup> are rate constants in 1/s for Newton, Henderson and Pabis, and Page model, respectively; *k*<sup>4</sup> is scale parameter in s; *a* and *b* are empirical parameters (dimensionless) of Henderson and Pabis model, respectively; *c* is shape-parameter of Weibull model.

#### *2.4.2 Effective water diffusion*

Effective water diffusion was determined with the Fick's second law for a cubic geometry according to Eqs. (11)–(12) [34, 35]. The solution of Eq. (11) considers an initial homogenous water distribution, a constant water diffusivity parameter, a solid sample with no volume change, and an isothermal process [34].

*Modeling of Drying Kinetics of Fresh and Osmodehydrated Apples during Convective Drying DOI: http://dx.doi.org/10.5772/intechopen.105162*

$$\Psi = \left(\frac{8^3}{\pi^2} \sum\_{i=1}^{\infty} \frac{1}{\left(2i+1\right)^2} e^{\left(\frac{-\left(2i+1\right)^2 \pi^2}{4}\right)}\right)^3 \tag{11}$$

$$
\sigma = \frac{Dt}{0.5L} \tag{12}
$$

Where τ is the Fourier number (dimensionless), *D* is the effective water diffusivity (m<sup>2</sup> /s), *t* is the process time (s), and *L* is the thickness of the sample (m).

The modified slope method was used to calculate the diffusivity of water in the product considering its shrinkage. Briefly, the τ term of Eq. (11) was replaced by a more general definition (*θ*), as shown in Eq. (13) [36].

$$\theta = \int\_0^t \frac{D(t)}{\left(0.5L(t)\right)^2} dt = \int\_0^t \frac{D\left(\Psi\right)}{\left(0.5L\left(\Psi\right)\right)^2} dt\tag{13}$$

Where *D*(*t*) and *L*(*t*) are the diffusivity and sample thickness as a function of time, respectively. *D*(*Ψ*) and *L*(*Ψ*) are the diffusivity and sample thickness as a function of free moisture fraction, respectively.

#### **2.5 Modeling of drying curves**

Fick's second law solutions were performed with the Matlab software and its Statistic Toolbox 7.3 (Matlab R2020b, MathWorks Inc., Natick, MA, USA). Experimental data were analyzed with an ANOVA using Minitab v.17 Statistical Software (Minitab Inc., State College, PA, USA) and the Tukey test was used to analyze the mean comparisons considering a 95% confidence.
