Convective and Microwave Assisted Drying of Wet Porous Materials with Prolate Spheroidal Shape: A Finite-Volume Approach

Convective heating is a traditional method used for the drying of wet porous materials. Currently, microwave drying has been employed for this purpose, due to its excellent characteristics of uniform moisture removal and heating inside the material, higher drying rate, and low energy demand. This paper focuses on the study of the combined convective and microwave drying of porous solids with prolate spheroidal shape. An advanced mathematical modeling based on the diffusion theory (mass and energy conservation equations) written in prolate spheroidal coordinates was derived and the numerical solution using the finite-volume method is presented. Here, we evaluated the effect of the heat and mass transport coefficients and microwave power intensity on the moisture removal and heating of the solid. Results of the drying and heating kinetics and the moisture and temperature distribution inside the solid are presented and discussed. It was verified that the higher the convective heat and mass transfer coefficients and microwave power intensity, the faster the solid will dry and heat up.


Introduction
Porous materials are a class of materials composed by a solid matrix with interconnected and disconnected pores of different shapes and sizes, and randomly distributed. In general, these void spaces are filled with one or more fluids. Several materials can be classified as porous materials such as fruits, vegetables and grains, which are rich in vitamins and minerals. Unfortunately, these post-harvested biological materials have high moisture content and are highly perishable.
Due to the great nutritional importance for human diet and nutrition, these materials must be dehydrated (moisture removal) in order to avoid nutritional, biological, physical or chemical

Mathematical Modeling
For an appropriate formulation, the following considerations are given: (a) The porous body is assumed to be isotropic; (b) The mass transport occurred only by diffusion inside the body and convection at the surface; (c) The heat transfer occurred only by conduction and volumetric heating inside the solid, and convection at the surface; (d) Thermo-physical properties are considered to be constant; (e) Dimension variations were neglected.
Then, for mass transfer and based on the considerations cited earlier, Fick's second law of diffusion (two-dimensional case) was written in the prolate spheroidal coordinate system, as follows: where D is the mass diffusion coefficient and M the moisture content (dry basis).

Mathematical Modeling
For an appropriate formulation, the following considerations are given: (a) The porous body is assumed to be isotropic; (b) The mass transport occurred only by diffusion inside the body and convection at the surface; (c) The heat transfer occurred only by conduction and volumetric heating inside the solid, and convection at the surface; (d) Thermo-physical properties are considered to be constant; (e) Dimension variations were neglected.
Then, for mass transfer and based on the considerations cited earlier, Fick's second law of diffusion (two-dimensional case) was written in the prolate spheroidal coordinate system, as follows: where D is the mass diffusion coefficient and M the moisture content (dry basis).

Mathematical Modeling
For an appropriate formulation, the following considerations are given: (a) The porous body is assumed to be isotropic; (b) The mass transport occurred only by diffusion inside the body and convection at the surface; (c) The heat transfer occurred only by conduction and volumetric heating inside the solid, and convection at the surface; (d) Thermo-physical properties are considered to be constant; (e) Dimension variations were neglected.
Then, for mass transfer and based on the considerations cited earlier, Fick's second law of diffusion (two-dimensional case) was written in the prolate spheroidal coordinate system, as follows: where D is the mass diffusion coefficient and M the moisture content (dry basis). For a well-posed formulation, the following initial and boundary conditions are used: For heat transfer, we use the heat conduction equation (two-dimensional case) as follows: In this equation, ρ is the density of the dry solid, c p is the specific heat, K T is the thermal conductivity, T is the temperature and . Q is the volumetric heat generation. For this governing equation, the initial and boundary conditions are as follows: For a better understanding, it can be stated the relationships between Cartesian and prolate spheroidal coordinate systems are follows [12]: where ξ = cosh µ; η = cos φ; ζ = cos ω and L = (L 2 2 − L 1 2 ) 1/2 ( Figure 2).
According to Lima et al. [4], an equation that has been reported in the literature for power absorbed and converted in heat during microwave irradiation can be expressed as: where P 0 is the power incident on the surface of the product (parameter to be determined experimentally), M → r , t is the moisture content of the material, M 0 is the initial moisture content, → r is the position vector of the microwave within the material, → n is the unit vector normal to the surface, ε r represents the dielectric loss factor relative to air and Ψ is the attenuation factor. The depth of penetration (D p ) of the microwave and the attenuation factor are given by: Agriculture 2020, 10, 507 5 of 17 where ε r represents the dielectric constant of the material. After obtaining the solution of Equations (1) and (4), the moisture content and temperature fields inside the material can be obtained throughout the process. Thus, the average moisture content and average temperature for the porous body during the drying process can be found using the following equations [31]:

Numerical Solution
To obtain the numerical solution of the heat and mass transfer equations (Equations (1) and (4)), the finite-volume method was used [32][33][34]. In this technique, the partial differential equation is integrated in volume and time. As a result, we obtain discretized linear algebraic equations for mass and heat transfer, as follows: The coefficients A K with K P represent the contributions of the diffusive transport of the variable Φ coming from the neighboring nodal points in the direction of node P. The coefficient A 0 P reflects the influence of the variable Φ in the previous time on its value in the present time. For mass transfer, we have Φ = M and B = 0, while for heat transfer, we have Φ = T and B = . Q 0 To obtain the simulated results, a computational code was implemented in the Mathematica ® software, using a uniform mesh with 20 × 20 nodal points ( Figure 3) and time step ∆t = 1.0 s, each obtained after refining the grid and time step. For the solution of the set of algebraic linear equations (Equation (15)), we used the Gauss-Seidel interactive method with a convergence criterion of 10 −8 kg/kg for moisture content and 10 −8 K for temperature.

Numerical Solution
To obtain the numerical solution of the heat and mass transfer equations (Equations (1) and (4)), the finite-volume method was used [32][33][34]. In this technique, the partial differential equation is integrated in volume and time. As a result, we obtain discretized linear algebraic equations for mass and heat transfer, as follows: The coefficients AK with K ≠ P represent the contributions of the diffusive transport of the variable Φ coming from the neighboring nodal points in the direction of node P. The coefficient A P 0 reflects the influence of the variable Φ in the previous time on its value in the present time. For mass transfer, we have Φ = M and B = 0, while for heat transfer, we have Φ = T and B = Q ≠ 0.
To obtain the simulated results, a computational code was implemented in the Mathematica ® software, using a uniform mesh with 20 × 20 nodal points ( Figure 3) and time step ∆t = 1.0 s, each obtained after refining the grid and time step. For the solution of the set of algebraic linear equations (Equation (15)), we used the Gauss-Seidel interactive method with a convergence criterion of 10 −8 kg/kg for moisture content and 10 −8 K for temperature.

Validation
For validation of the proposed model, two different cases were considered: (a) Mass transfer process in a prolate spheroidal body (Q = 0), without heat transfer and convective condition at the surface.

Validation
For validation of the proposed model, two different cases were considered: (a) Mass transfer process in a prolate spheroidal body . Q = 0, without heat transfer and convective condition at the surface.
(b) Heating process of a spherical body with a constant volumetric heat source, without mass transfer and equilibrium condition at the surface.
According to Carslaw and Jaeger [35], the analytical solution of the appropriated energy equation has the following form: where,

Simulated Cases
Microwave drying and heating simulations were performed for one prolate spheroid with aspect ratio L 2 /L 1 = 2.0, considering constant thermo-physical properties given in terms of the dimensionless parameters, mass and heat transfer Biot numbers (Bi m = h m L 1 /D and Bi c = h c L 1 /K T ), mass transfer Fourier number (Fo m = Dt/L 2 1 ) and the heat generation source term ( . Q). Table 1 presents a summary of the simulated cases. The following arbitrary data were considered in this research: initial moisture content (M 0 = 0.20 kg/kg dry basis), initial temperature (T 0 = 298.16 K), drying-air temperature (T eq = 373.16 K), mass diffusion coefficient (D = 1.22 × 10 −9 m 2 /s), specific heat (c p = 3395 J/kg·K), density (ρ = 1000 kg/m 3 ) and thermal conductivity (K T = 0.3395 W/m·K). Table 1. Parameters used in the simulations for the microwave drying and heating. Case

Figure 4 illustrates a comparison between the predicted moisture ratio [M
] in a prolate spheroidal solid with the analytical results reported in the literature by [36][37][38]. The results are purely related to the convective drying phenomenon. From the analysis of this figure, it can be seen that there is an excellent concordance between the results. Thus, validating the model and methodology presented herein.  [36][37][38]. The results are purely related to the convective drying phenomenon. From the analysis of this figure, it can be seen that there is an excellent concordance between the results. Thus, validating the model and methodology presented herein.  [35] within a sphere of radius L1 = 1.0 cm for different Fourier numbers ( Fo = αt/L = 0.06, 0.10, 0.20 and 0.60). This physical problem corresponds to an intense heat convection condition at the surface of the sphere in conjunction with an intense microwave heating with constant heating power density P 0 = 10 6 W/m 3 , and considering the attenuation factor Ψ = 0.0 m -1 . Again, from the analysis of this figure, it can be seen that there is an excellent concordance between the results.   Table 1) present the average moisture content and average temperature in a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0 for different mass transfer Biot numbers (Bi m = Bi c ). In these figures, Bi c = 1.0 corresponds to a weak heat convection phenomenon at the surface of the solid and Bi c →∞ corresponds to a strong physical interaction of the thermal convection, i.e., instantaneously the surface temperature equals the temperature of the drying-air and the moisture content at the surface reaches its hygroscopic equilibrium condition. From the analyses of these figures, it is observed that an increase in the Biot number (mass or heat) provides an increase in the drying and heating rates of the solid, that is, the body reaches hygroscopic and thermal equilibrium in a shorter process time. A small value of mass transfer Biot number causes a decrease in the drying rate of the material and therefore a longer exposure time to the microwave radiation, which can cause damage to the product being dried. Meanwhile, with a higher heat transfer Biot number, the material   Table 1) present the average moisture content and average temperature in a prolate spheroidal solid with aspect ratio L2/L1 = 2.0 for different mass transfer Biot numbers (Bim = Bic). In these figures, Bic = 1.0 corresponds to a weak heat convection phenomenon at the surface of the solid and Bic→∞ corresponds to a strong physical interaction of the thermal convection, i.e., instantaneously the surface temperature equals the temperature of the drying-air and the moisture content at the surface reaches its hygroscopic equilibrium condition. From the analyses of these figures, it is observed that an increase in the Biot number (mass or heat) provides an increase in the drying and heating rates of the solid, that is, the body reaches hygroscopic and thermal equilibrium in a shorter process time. A small value of mass transfer Biot number causes a decrease in the drying rate of the material and therefore a longer exposure time to the microwave radiation, which can cause damage to the product being dried. Meanwhile, with a higher heat transfer Biot number, the material reaches the thermal equilibrium in a shorter heating time. Further, it can be stated that the heating process of the solid is faster than the drying process.        Table 1) present the moisture content profiles on the y-axis (z = 0) inside a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0 for different mass transfer Biot numbers at different moments of drying. Analyzing these figures, it is verified that an increase in the Biot number (mass) corresponds to an increase in the drying rate, which leads to higher moisture gradients inside the solid, especially in the initial drying moments and near the surface of the solid.      Table 1) show the temperature profiles on the y-axis (z = 0) for a prolate spheroidal solid with aspect ratio 2.0 and different heat transfer Biot numbers at the instants of time 300, 1000 and 4000 s, respectively. By analyzing these figures, it is noticed that an increase in the heat transfer Biot number implies a higher heating rate of the solid, thus reaching its equilibrium temperature at the surface more quickly. In the initial heating times, the heat flux occurs from the surface to the center of the solid, i.e., the heating process is governed by heat convection. After a   Table 1) show the temperature profiles on the y-axis (z = 0) for a prolate spheroidal solid with aspect ratio 2.0 and different heat transfer Biot numbers at the instants of time 300, 1000 and 4000 s, respectively. By analyzing these figures, it is noticed that an increase in the heat transfer Biot number implies a higher heating rate of the solid, thus reaching its equilibrium temperature at the surface more quickly. In the initial heating times, the heat flux occurs from the surface to the center of the solid, i.e., the heating process is governed by heat convection. After a certain warm-up period, this behavior is reversed, and heat flux occurs from the center to the surface of the solid. Now, the heating process is governed by microwave heating, and the lower the heat transfer Biot number, the higher the temperature inside the prolate spheroidal solid, for a longer drying time. From the physical point of view, when the value of the heat transfer Biot number is small, one has a physical situation with low heat transfer by convection and the microwave heating dominates the phenomenon. Figures 13 and 14 (cases six to nine of Table 1) illustrate the temperature distribution within a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0 for different microwave power densities and different moments of the drying process. By analyzing these figures, it can be observed that for the microwave power densities ranging from 10 4 to 10 6 W/m 3 , the temperature profiles showed similar behaviors; however, for the higher microwave power density of 10 7 W/m 3 , there was a substantial increase in temperature at the beginning of the microwave heating process, which may cause damage to the physical structure of the product being heated or even fully incinerated. Furthermore, it is known that the higher the applied microwave power density, the greater the amount of heat generated internally in the product, providing greater rigidity and less shrinkage. Obviously, these phenomena are dependent on the nature of the product, for example, biological and non-biological materials, especially for heat sensitive materials such as fruit, vegetables, grains, and many others. certain warm-up period, this behavior is reversed, and heat flux occurs from the center to the surface of the solid. Now, the heating process is governed by microwave heating, and the lower the heat transfer Biot number, the higher the temperature inside the prolate spheroidal solid, for a longer drying time. From the physical point of view, when the value of the heat transfer Biot number is small, one has a physical situation with low heat transfer by convection and the microwave heating dominates the phenomenon.   certain warm-up period, this behavior is reversed, and heat flux occurs from the center to the surface of the solid. Now, the heating process is governed by microwave heating, and the lower the heat transfer Biot number, the higher the temperature inside the prolate spheroidal solid, for a longer drying time. From the physical point of view, when the value of the heat transfer Biot number is small, one has a physical situation with low heat transfer by convection and the microwave heating dominates the phenomenon.      Table 1) illustrate the temperature distribution within a prolate spheroidal solid with aspect ratio L2/L1 = 2.0 for different microwave power densities and different moments of the drying process. By analyzing these figures, it can be observed that for the microwave power densities ranging from 10 4 to 10 6 W/m 3 , the temperature profiles showed similar behaviors; however, for the higher microwave power density of 10 7 W/m 3 , there was a substantial increase in temperature at the beginning of the microwave heating process, which may cause damage to the physical structure of the product being heated or even fully incinerated. Furthermore, it is known that the higher the applied microwave power density, the greater the amount of heat generated internally in the product, providing greater rigidity and less shrinkage. Obviously, these phenomena are dependent on the nature of the product, for example, biological and non-biological materials, especially for heat sensitive materials such as fruit, vegetables, grains, and many others.     Table 1) illustrate the temperature distribution within a prolate spheroidal solid with aspect ratio L2/L1 = 2.0 for different microwave power densities and different moments of the drying process. By analyzing these figures, it can be observed that for the microwave power densities ranging from 10 4 to 10 6 W/m 3 , the temperature profiles showed similar behaviors; however, for the higher microwave power density of 10 7 W/m 3 , there was a substantial increase in temperature at the beginning of the microwave heating process, which may cause damage to the physical structure of the product being heated or even fully incinerated. Furthermore, it is known that the higher the applied microwave power density, the greater the amount of heat generated internally in the product, providing greater rigidity and less shrinkage. Obviously, these phenomena are dependent on the nature of the product, for example, biological and non-biological materials, especially for heat sensitive materials such as fruit, vegetables, grains, and many others.  Temperature inside a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0, different microwave power densities and fixed heat transfer Biot number (y-axis, z = 0, t = 300 s).   Table 1) illustrates the heating kinetics of a prolate spheroidal solid for different attenuation factors of the electromagnetic waves when subjected to a microwave power density of 10 6 W/m 3 and heat transfer Biot number equal to 5.0. Upon analyzing this figure, it can be seen that the temperature increases with time, and this phenomenon is more intense for lower attenuation factors. For this physical situation, we have a greater depth of penetration of the electromagnetic waves, providing faster heating or even incineration of the porous body.    Table 1) illustrates the heating kinetics of a prolate spheroidal solid for different attenuation factors of the electromagnetic waves when subjected to a microwave power density of 10 6 W/m 3 and heat transfer Biot number equal to 5.0. Upon analyzing this figure, it can be seen that the temperature increases with time, and this phenomenon is more intense for lower attenuation factors. For this physical situation, we have a greater depth of penetration of the electromagnetic waves, providing faster heating or even incineration of the porous body.   Table 1) illustrates the heating kinetics of a prolate spheroidal solid for different attenuation factors of the electromagnetic waves when subjected to a microwave power density of 10 6 W/m 3 and heat transfer Biot number equal to 5.0. Upon analyzing this figure, it can be seen that the temperature increases with time, and this phenomenon is more intense for lower attenuation factors. For this physical situation, we have a greater depth of penetration of the electromagnetic waves, providing faster heating or even incineration of the porous body.    Table 1) illustrates the temperature distribution inside a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0 for different attenuation factors and fixed microwave power density and heat transfer Biot number at 900 s. By analyzing the temperature profile on the y-axis (z = 0), it is observed that the highest temperature value, 461.28 K, occurred just for the lowest attenuation factor (Ψ = 0.0 m −1 ). This is due to the fact that the attenuation factor is inversely proportional to the penetration depth of the electromagnetic waves, which is mainly responsible for the rise in temperature during the microwave heating process for wet solids. The high temperature value inside the product can cause irreparable damage to the product and may even incinerate it.  Table 1) illustrates the temperature distribution inside a prolate spheroidal solid with aspect ratio L2/L1 = 2.0 for different attenuation factors and fixed microwave power density and heat transfer Biot number at 900 s. By analyzing the temperature profile on the yaxis (z = 0), it is observed that the highest temperature value, 461.28 K, occurred just for the lowest attenuation factor (Ψ = 0.0 m −1 ). This is due to the fact that the attenuation factor is inversely proportional to the penetration depth of the electromagnetic waves, which is mainly responsible for the rise in temperature during the microwave heating process for wet solids. The high temperature value inside the product can cause irreparable damage to the product and may even incinerate it.

Conclusions
In this paper, an advanced study to predict coupled microwave heating and drying processes of a wet porous material with complex geometric shape (ranging from sphere and prolate spheroid to infinite cylinder) was developed. All formalism of the mathematical model and numerical solution of the governing equations were presented. From the obtained results, it can be concluded that: (a) For a fixed microwave power density, a drying process with a higher Biot number for heat and mass transfer will result in the solid drying and heating up faster. The effect is more evident in moisture ratio than in average temperature; (b) In hybrid processes, for a shorter drying time, moisture migration occurs from the center to the surface and heat flux occurs in the opposite direction, generating high thermal and hydric gradients inside the solid; (c) In hybrid processes, for a longer dying time, moisture migration and heat flux occurs in the same direction, from the center to the surface, generating low hydric gradients and high thermal gradients inside the solid; (d) The higher the microwave power density, the greater the amount of heat generated, and the higher the temperature inside the solid. For the drying of prolate spheroidal solids, a microwave power density less than 10 6 W/m 3 provoked temperatures less than 450 K inside the solid. Temperature inside a prolate spheroidal solid with aspect ratio L 2 /L 1 = 2.0, different attenuation factors and fixed microwave power density and heat transfer Biot number (y-axis, z = 0, t = 900 s).

Conclusions
In this paper, an advanced study to predict coupled microwave heating and drying processes of a wet porous material with complex geometric shape (ranging from sphere and prolate spheroid to infinite cylinder) was developed. All formalism of the mathematical model and numerical solution of the governing equations were presented. From the obtained results, it can be concluded that: (a) For a fixed microwave power density, a drying process with a higher Biot number for heat and mass transfer will result in the solid drying and heating up faster. The effect is more evident in moisture ratio than in average temperature; (b) In hybrid processes, for a shorter drying time, moisture migration occurs from the center to the surface and heat flux occurs in the opposite direction, generating high thermal and hydric gradients inside the solid; (c) In hybrid processes, for a longer dying time, moisture migration and heat flux occurs in the same direction, from the center to the surface, generating low hydric gradients and high thermal gradients inside the solid; (d) The higher the microwave power density, the greater the amount of heat generated, and the higher the temperature inside the solid. For the drying of prolate spheroidal solids, a microwave power density less than 10 6 W/m 3 provoked temperatures less than 450 K inside the solid. (e) The lower the attenuation factor (higher penetration depth), the higher the temperature inside the solid. For the drying of prolate spheroidal solids, an attenuation factor less than 50 m −1 provoked an average temperature greater than 410 K inside the solid.