Moisture Diffusivity of Seedless Grape undergoing convective drying

2.1 Experimental protocol

2.1.2 Equilibrium moisture content

Desorption isotherms were obtained by keeping the sample at a constant partial pressure of water vapor at a given temperature until reach the hygroscopic equilibrium. This equilibrium was reached when we has not noticed a significant change in the sample weight. Desorption isotherms of seedless grape were determined at several temperatures by a static gravimetric method [12], for placing the sample in a enclosure (Figure 1) by maintaining with fixed temperature and where moisture control is provided by saturated solutions of salts (KOH, MgCl₂, K₂CO₃, NaBr, NaNO₂, NaCl, KCl and BaCl₂) which were selected to provide different water activities in the range of 0.1–0.9 (Table 1). The salt solutions and their water activities are taken from literature [13, 14]. The grape berries are suspended in hermetic jars half filled by the various salt solutions, without contact with its solutions. Then the loaded jars were placed in an oven at controlled temperature. The initial mass of the grape berry was about 3.040 ± 0.050 g. Four repetitions were performed (four samples per salt). The glass sorption jars were placed in a vacuum oven at the controlled temperatures of 40, 50, 60 and 70 °C ± 1 °C. Each sample was carefully weighed every week by using an analytical balance (Mettler, precision: 10^−4 g) until no significant change in weight was observed between two consecutive measurements [15]. Once equilibrium was reached, the moisture content of the sample was determined by drying 24 h in a vacuum oven at 70 °C [11]. The hygroscopic equilibrium was reached after 2 weeks.

Table 1:

Saturated salt solutions and corresponding water activities.

Solution	Temperature (°C)
	40°C	50°C	60°C	70 °C
KOH	0.063	0.057	0.055	0.053
MgCl2	0.316	0.304	0.295	0.288
K2CO3	0.433	0.427	0.421	0.416
NaBr	0.532	0.512	0.491	0.473
NaNO2	0.608	0.588	0.565	0.544
NaCl	0.747	0.718	0.703	0.689
KCl	0.823	0.777	0.751	0.727
BaCl2	0.891	0.882	0.872	0.863

Figure 1:

Experimental apparatus for determination of desorption isotherms.

Several mathematics models have been developed in literature to describe desorption isotherms for various food products [14]. Our experimental data were best fitted with the Guggenheim, Anderson and de Boer (GAB) model, expressed by the following equation:

(1)Xeq=XmCKaw1−Kaw1−Kaw+CKaw

The moisture content equilibrium value (X_eq) was used in the boundary conditions for the moisture diffusivity estimation model.

The adequacy of the model was evaluated by the correlation coefficient (r) and the standard error coefficients (s) calculated by the software Curve Expert 5.1. These parameters can be calculated as follows:

(2)s=∑i=1nexp.dataXeq−Xeq.cali2nexp.data−nparam

(3)r=1−∑i=1nexp.dataXeqi−Xeq.cali2∑i=1nexp.dataXˉeq−Xeqi2

The arithmetic average value of the experimental equilibrium moisture content (Xˉeq) was calculated by using this relation [15]:

(4)Xˉeq=1nexp.data∑i=1nexp.dataXeqi

2.1.4 Drying kinetics

The experimental drying kinetics were carried out in a climatic blower: convective vertical downward flow with controlled temperature (T), relative humidity (RH) and velocity (u), existing in the laboratory of Energy and Heat and Mass Transfers (LETTM) (Figure 2).

Figure 2:

Overall layout of the experimental laboratory dryer.

The grapes berries were placed on a perforated tray which was suspended to an electronic balance with a precision of 0.001 g. The drying experiments were established at four temperatures 40, 50, 60 and 70 °C and regulated to ± 1 °C and air velocity at 2.5 m/s. The mass of the product was continuously measured with constant time interval (2 min) in each drying temperature and recorded by a microcomputer. The experiments were replicated two times in each temperature in order to check the reproducibility of the drying curves. Drying was stopped when the weight of the test sample remained constant. The dry solid weight of the product was determined after each experiment by the vacuum oven-drying method at 70 °C for 48 h [11].

The dry basis equilibrium moisture content X (kg water/kg dry matter) is given by the following expression:

(5)X=Mf−MdMd

2.2 Numerical identification of the moisture diffusivity coefficient

The numerical procedure is essential to predict and simulate the drying behavior and it consists on solving the transfer conservation equations of both solid matter and water to take account for the material’s shrinkage. The equations are coupled by the solid phase velocity.

In order to facilitate the numerical resolution of the model equations many simplifying assumptions are formulated:

• The transfers of heat and mass are one-dimensional.

• The product is biphasic water-solid.

• Evaporation is carried out on the surface.

• The shrinkage is ideal and isotropic.

The conservation equations of the solid and water phases are written respectively:

with:

The liquid mass flux is given by Bird et al. [17]:

In convective drying of the grape the liquid migrates by diffusion and is transported by the shrinkage.

For an ideal shrinkage model, we can use the followings equations [18]:

By combining previous equations, we obtain finally the conservation equations of solid and liquid phases in spherical coordinates:

Equations 13 and 14 were numerically solved for us and X using the initial and boundary conditions at the surface (r = R) and at the center of a grain (r = 0):

The saturation vapor pressure was derived from the literature [14]:

The mass transfer coefficient to the level of the grape surface is evaluated starting from the heat transfer coefficient according to the following equation [19]:

Reynolds and Schmidt numbers were respectively calculated with the air experimental conditions:

The control volume spatial discretization with entirely implicit time integration which we were used was developed by Patankar [20]. The total number of calculation nodes over the sample thickness was kept fixed but the control volumes thicknesses were changed at each time step, as the sample thickness changed during drying. The thickness of each control volume was changed independently, according to the local moisture content variation and considering a constant dry solid mass within each control volume [7].

The establishment of moisture diffusivity was carried by the minimization of the difference between the experimental drying kinetics (X = f (t)) and the simulated one.

Moisture diffusivity depends on the variation of the moisture content during drying, the physical structure of the products and the experimental drying conditions [21]. Generally the variation of the moisture diffusivity with temperature is represented by Arrhenius equation 8, [22–24]. However the influence of the moisture content has not been performed into a general accepted model. In the present study, the effect of the moisture content is suitable presented by an exponential function [7]:

with:

3.1 Desorption isotherms

The results of the experimental measurement of the desorption isotherms of grapes at 40, 50, 60 and 70 °C are shown in Figure 3. The shape of the isotherms is sigmoid corresponding to type II curves according to BET classification [25], similar to the majority of food products.

Figure 3:

Desorption isotherms for prickly seedless grape at various temperatures.

An acceptable agreement between experimental and calculated fit can be observed. Commonly, at each temperature, all experimental data of desorption isotherms, show an increase of the equilibrium moisture data with increasing of the water activity. In addition, these curves show that the equilibrium moisture content increases when the temperature decreases, which is consistent with the literature [26]. At higher temperature and at a given water activity, the equilibrium moisture content is lower, signifying less hygroscopic capacity due to modifications induced by temperatures in the product and the higher energy levels of the water molecules [27].

The three parameters of the GAB model (X_m, C and K) depend on the characteristics of the product and the equilibrium temperature. The values ​​of these parameters at various temperatures and the statistical analysis are given in Table 2.

3.2 Shrinkage curves

Figure 4 presents the curves of the dimensionless relative diameters characteristics (a) (d_v/d_v0) and (b) (d_h/d_h0) of a grape versus the moisture content, for various temperatures of drying. We note that, the shrinkage following vertical diameter d_v varies linearly lasting all the period of drying. On the other hand, the shrinkage following the horizontal diameter d_h is constant. This particular evolution of the shrinkage, at the end of drying is explained by the fact why the transfer of water is dominating according to the vertical diameter of the grape. For low moisture content air penetration and development of air-filled porosity in the material at the end of drying.

Figure 4:

Curves of the dimensionless relative diameters characteristics (a) (d_v/d_v0) and (b) (d_h/d_h0) of seedless grape.

The predictive correlation of the shrinkage is given by the expressions bellow:

3.3 Drying kinetics

The evolution of the drying kinetics as a function of drying time and the variation of the drying rate versus moisture content of Sultana grape at various air conditions, were both shown in Figure 5 and Figure 6. These curves illustrate the absence of constant drying period, they show only the falling-rate period. This result designate that diffusion is the physical process governing moisture movement in berries grape. We note a reduction in the drying time when the drying air temperature increase. These results are the same for other food products [28].

Figure 5:

Variation of moisture content of Sultana grape versus drying time at different air temperatures.

Figure 6:

Drying rate of Sultana grape versus moisture content at different air temperatures.

3.4 Identification of the moisture diffusion

Figure 7 presents the evolution of the moisture diffusivity as function as moisture content of the sultana grape. at different temperatures and an air velocity equal to 1.5 m/s. these curves shown a significant increasing of the moisture diffusivity with the air drying temperature and a moderate increasing with the moisture content. Similar results have been reported by Kiranoudis et al. [29], Garau et al. [30],and Guiné et al. [31].

Figure 7:

Moisture diffusivity of the grape estimated by numerical method at various temperatures.

The reduction of the moisture diffusivity with the reduction of the moisture content is generally considered as a normal behavior in biological products [22, 32, 33].

The variation of (D_x1) versus (1/T) for Sultana grape is shown in Figure 7. By the analysis of the experimental data, it was revealed the existence of exponential relationship between these parameters.

Figure 8:

Variation of predicted diffusivity coefficient with (1/T) for seedless grape.

The variation of (D_X2) with temperature for sultana grape was shown in Figure 9. We revealed a linear relationship between these parameters. Correlation coefficient (r) of the fitted lines with experimental data was obtained as 0.9972.

Figure 9:

Variation of (D_X2) with temperature for seedless grape.

Hence, taking into account relations from previous curves (Figures 8 and 9) over the range of water content from 3.8 to 0.3 (kg /kg dry matter) and temperature values ranging from 40 up to 70 °C, our findings are quite similar to those reported by Doymaz [23],who found that the water diffusivity of grape varies between 3.5610⁻¹⁰ (m²/s) and 12.610⁻¹⁰ (m²/s). The water diffusivity of Sultana grapes is expressed by the following equation:

In order to compare our semi-empirical diffusivity correlation with those reported in literature, the moisture diffusivity versus moisture content obtained with eq. 27 and with two others correlations selected from Saravacos [34] and Raghavan et al.[35] for seedless grapes were compared (Table 3). It can be noted that our results is very close to that found in literature for the cited references. The slight difference between the compared data can be explained mainly by the estimating methods of moisture diffusivity, what stresses the need for standardization and benchmarking of moisture diffusivity measurements. Otherwise, it should be noted that our activation energy value (57.56 kJ/mol) is far from to that found by Azzouz et al. [8],using a numerical approach to estimate the moisture diffusivity of grape and identifies a value of activation energy equal to 34,87 kJ/mol. This difference can be attributed mainly to the grapes varieties and the measurements errors.