Modeling the Electrohydrodynamic (EHD) Drying of Banana Slices

1.1 Theoretical considerations of EHD drying

A detailed mathematical description of EHD drying is complex. The drying rate in EHD depends on the strength of the electrical wind, which impinges on the wet material being dried and produces turbulent, vortex-like motions, enhancing the mass transfer rates. Thus, EHD drying is indeed an example of converting electrical to mechanical energy [4]. The ionic wind velocity (υ) induced by the EHD setup can be calculated from the following equation derived from the conservation of energy and Gaussian laws [9].

2.1 Empirical modeling

The moisture ratio (MR) of the banana slices was calculated using the following equation:

where M_e values are so smaller than M₀ and M that they can be neglected [11].

The experimental drying curves were fitted to the eight different empirical models of moisture ratio in Table 1. The models were then fitted to the experimental data to determine model constants. Nonlinear regression analyses were performed in the Matlab 7.6 software. The terms used to evaluate the goodness of fit were the coefficient of determination (R²) and RMSE between the experimental and predicated values of moisture ratio. The higher value of R² and the lower value of RMSE, as obtained by using the equations below, were interpreted as goodness of fit:

2.2 Theoretical modeling

A theoretical model was developed based on Fick’s second law and used to determine the moisture content inside the banana slices during drying. Preliminary studies showed a linear drying curve which was assumed to be externally controlled by EHD. EHD drying is the impingement of the corona wind (air ions) on wet sample that produces an impact in the form of a wind pressure, which decreased external resistance and enhances the mass transfer rate of water. The corona wind caused the moisture to move at the surface of the samples because of the lower external resistance against mass transfer than the internal resistance inside the nodes. The falling rate period showed that the internal transfer was dominant. A linear behavior has been observed in the drying curves for an average moisture content when the internal resistance is increased to twice as much as the external one [20]. However, we assumed that by applying EHD, moisture movement takes place only by the diffusion mechanism inside the slices in the two falling rate periods. In addition, the following assumptions were made:

1. Each banana slice is simulated using an infinite plate geometry (i.e., a one-dimensional moisture transfer);

2. A homogeneous initial moisture prevails inside the banana slices;

3. Non-shrinkage for banana slices [21];

4. Moisture evaporation occurs only on the upper surface; and

5. Moisture diffusivity is constant.

The governing one-dimensional transitional equation of the moisture transfer, based on Fick’s second law under the above assumptions, is written in Cartesian coordinates as follows:

where ∂M/∂t is the rate of change of moisture content dry basis at time t and point x with respect to time; and D_eff is the effective moisture diffusivity (m² s⁻¹) as described in eq. 10. The model is treated with the following boundary conditions (in eq. 6, drying occurs only in the upper surface of the slices, x = L, and t is the time passed) and the initial condition stating that moisture content is uniform in the solid (eq. 7).

2.3 Constant parameters of the theoretical model

The experimental drying data for calculating effective moisture diffusivity were interpreted using Fick’s diffusion model. Assuming a diffusive moisture migration, a negligible shrinkage, a uniform initial moisture distribution, infinite slab geometry, and constant moisture diffusivity, the average value of the infinite series on the space will be as in eq. 8. In other words, if a plane sheet of material at an initial moisture content is placed in an environment for which its equilibrium moisture content is M_e, the solution to eq. 5 will be as follows [22].

Thus, the MR gradients over the space at different times and points, x, can be calculated using eq. 8. The average moisture content at any time is given by eq. 9 [22]. Moisture gradients exist in the slice due to heat and mass transfers. However, average moisture content at any time on space which is only time-dependent is commonly used as these gradients cannot be experimentally measured in the product.

where D_eff is effective moisture diffusivity (m² s⁻¹), L is sample thickness (m), t is time (s), and n is a positive integer.

For long drying times, the first term of the series of solutions for eq. 9 is good enough for estimating the average moisture content in the infinite slab [23]. Equation 9 can be, therefore, simplified as in eq. (10):

Effective moisture diffusivity (D_eff) is also typically calculated using eq. 10. The appropriate value for D_eff is found by using D_eff as a fitting parameter and adjusting it in order to fit the numerical results to the experimental ones [24].

The moisture transfer equation (eq. 5) under relevant initial and boundary conditions and by applying the mass conservation in the small volume of the material was solved using the finite difference method. The implicit alternating direction method was also used in the solution. Applying the approximations for the first-order time derivation and second-order space derivation in the second Fick’s law, the relations between moisture content and time were obtained in the first, middle, and external nodes as in Figure 2 as well as in eqs (11), (12), and (13), respectively. Moisture content for each node (M_i,j) was estimated by such model constant parameters as average convective mass transfer coefficient and effective moisture diffusivity.

Mass transfer coefficient, k_m, was obtained by calculating the heat transfer coefficient, h_m. The dimensionless Nusselt number Nu was, therefore, used to relate a laminar flow for the convection heat transfer coefficient, h_m, to thermal conductivity, k, and product thickness, L.

The convective heat transfer is determined via the Nusselt–Reynolds–Prandtl correlation [25].

Corona wind velocity (eq. (1)) was used to calculate the Reynolds number, Re, for the flow over the product, which is related to EHD. The mass transfer coefficient, k_m, is determined via the Schmidt–Prandtl correlation [25].

Schmidt number is the ratio of kinematic viscosity to molecular diffusivity:

where Da is the diffusion coefficient of water vapor in air, ρ is air density, and cp is specific heat (Table 2). The model constants for theoretical modeling of drying behavior consisting of convective moisture transfer coefficient (k_m) and effective moisture diffusivity (D_eff) were used in Matlab to obtain the moisture content distribution inside the slices during drying.

Figure 2:

Schematic of meshing for numerical modeling of moisture values in different spaces of banana slice.

3.1 Empirical modeling of drying behavior

The values of R² and RMSE used for fitting the experimental moisture ratios to the 8 thin layer models ranged from 0.9612 to 0.9994, and 0.00702 to 0.05360, respectively (Table 3). The diffusion model [17] had a higher average value of R² but a lower average value of RMSE than those obtained by other models. Therefore, the diffusion model was considered to be the best for predicting the thin layer drying behavior of EHD-treated banana slices. It should be noted that our solution is identical to the third model in Table 1. Previous studies have shown that the diffusion model predicts a better fit to the experimental thin layer solar drying than the other models do [28]. Similar results have been reported for apricot and fig [29]. In addition, the Page and modified Page models as well as the Wang & Singh model have been found to be the best for describing high temperature drying at 50 and 70°C of two kinds of banana varieties [30]. The quadratic model with a nonlinear regression analysis has been found appropriate for describing the drying rate of fish with drying voltage, ambient temperature, and drying time as the parameters involved [31].The variations in experimental and predicted moisture ratios obtained by the diffusion model for EHD at an electrical field strength of 6 kV cm⁻¹ are presented in Figure 5. A good agreement was observed between the experimental moisture ratio and that predicted by the diffusion model. This reveals the fitness of the diffusion model for describing the drying behavior of slices treated by EHD. Multiple regressions were undertaken to determine the effects of the electric field strength on the coefficients (a, b, and k) of the diffusion model, which were obtained as follows:

The coefficients of the empirical models are not phenomenologically constant; however, they contain meaningful parameters such as D_eff in analytical solutions.

Figure 5:

Experimental and predicated values of moisture ratios by the empirical diffusion model for EHD-treated banana slices at electric field strength of 6 kV cm⁻¹.

3.2 Theoretical modeling of drying behavior

Determining the moisture content distribution inside each banana slice during drying depends on the values of convective moisture transfer coefficient (k_m), which were obtained in this study to be 3.08 × 10⁻⁷, 4.62 × 10⁻⁷, and 5.65 × 10⁻⁷ m s⁻¹ for electric field strengths of 6, 8, and 10 kV cm⁻¹, respectively. These results corresponded to values within the ranges previously obtained for drying banana slices using a hot air dryer [32].

The values of effective diffusivity of banana slices, as the second parameter in the theoretical models, slightly increased with increasing electric field strength (Table 4). The corona wind enhanced moisture movement at the surface of the samples by applying the EHD field. Hence, external resistance to moisture transfer decreased compared to the internal resistance due to moisture gradients. As already mentioned, moisture gradients exist in slices due to the heat and mass transfers; however, the average equation on space which is only time-dependent is generally used as these gradients cannot be experimentally measured in the product. Previous research has revealed that drying banana slices is controlled by the internal moisture transfer. This is deduced from the best fit of the diffusion equation and a Biot number greater than 10 [33]. If Bi >10, the drying process is controlled by the internal resistance; if Bi <0.1, the surface moisture transfer is dominant and the drying process will be controlled by the boundary resistance. The values for D_eff for agricultural products have been found to lie within the range of 10⁻¹¹−10⁻⁹ m² s⁻¹ [26]. The diffusivity for bananas was best described as a function of its moisture content [1]. Previously reported values of D_eff for drying banana were in the ranges of 1.3 × 10⁻¹⁰ to 7.8 × 10⁻¹⁰ m² s⁻¹ [34], 2.94 × 10⁻⁹ to 5.26 × 10⁻⁹ m² s⁻¹ [1], and 4.31 × 10⁻⁹ to 4.44 × 10⁻⁹ m² s⁻¹ [35]. The differences between these values could be due to differences in the drying methods used, banana varieties, thicknesses of banana slices used, air and ambient conditions, and perhaps some other uncontrolled constant parameters.

Figure 6 presents the results of theoretical modeling of moisture ratio versus time for banana slices treated by EHD at 6 kV cm⁻¹. When the constant rate period is considered, it is assumed to be an externally controlled stage which only depends on air conditions and product geometry, but not influenced by product internal characteristics. Variations in moisture ratio were obtained by averaging the moisture distribution in each time step. Moisture ratio decreased because of moisture gradients, indicating an external control without moisture gradients inside the food [20]. The values predicted by the theoretical model were compared with experimental measurements and reported in Figure 6 under EHD with an electric field strength of 6 kV cm⁻¹. The results for 8 and 10 kV cm⁻¹ are not reported. The predicted values are adequately consistent with experimental values, with maximum differences observed in moisture ratio being equal to 0.16, 0.15, and 0.10 g g⁻¹ for 6, 8, and 10 kV cm⁻¹, respectively. The differences may be due to some of the assumptions made during model development including constant moisture diffusivity, negligible shrinkage of the sample (constant thickness and density), etc. Figure 7 illustrates a numerical solution dependent on both space and time. It is clear that moisture gradients exist in slice thickness, but they cannot be experimentally measured in the thin layers; hence, average moisture content is commonly used over the space. In other words, Figure 7 shows the slice internal moisture ratio as a function of space and time obtained from the finite difference method. Slice moisture content dropped to about 0.2 g g⁻¹ in a short period of time at the initial stage of drying. The average moisture content on the space is considered in empirical methods but predicting moisture ratio versus various positions (spaces) can be numerically shown as in Figure 7.

Figure 6:

Experimental and predicated moisture ratios by the numerical modeling for EHD-treated banana slices at electric field strength of 6 kV cm⁻¹.

Figure 7:

Numerical modeling depends on both space and time of moisture ratio versus time at various spaces of a slice with constant thickness for banana slice treated by EHD at 6 kV cm⁻¹.