Drying is a method of removing moisture for the purpose of preserving food that prevents microbial growth, increases shelf-life, and significantly reduces product weight [1, 2]. Conventional convective drying combined with microwave heating could be a good method to reduce the drying time and drying energy costs . To overcome the overheating problem, intermittent microwave convective drying (IMCD) with various intermittent times is practiced. Because it is a relatively new technique for food drying, modelling studies of IMCD are very limited . There are some empirical models available in the literature for IMCD ; however, the empirical models are not helpful for optimization and are only applicable for specific experimental conditions . Apart from empirical modelling, several diffusion-based theoretical models exist that consider the intermittency of microwave power [1, 7-9]. However, the mass transfer has been neglected in these models. A multiphase porous media model considering the liquid water, gases and the solid matrices inside food during drying can provide an in-depth understanding of IMCD . In fact, the parameters used for modelling are very important because the simulation results are decided by them. The mass transfer coefficient is a boundary condition that is very difficult to retrieve. However, most modelling is conducted using previous references [10-13]. The impact of the model parameters on the model predictions must be clarified . In fact, the parameter sensitivity of IMCD has not been reported. In this paper, the heat and the mass transfer within porous media during the IMCD process are characterized by using a non-equilibrium method. Then, the parameter sensitivity analyses for the microwave power level, ambient temperature, effective gas diffusivity, evaporation rate constant, surface mass, heat transfer coefficient, relative and intrinsic gas permeability, capillary diffusivity of water and evaporation rate constant are examined. Because most measurement errors are less than 20%, the parameters were only changed by 20%, except for special cases.
2 Mathematical model
2.1 Problem description and assumptions
Because the model is for a multiphase and multiphasic process, the computation is a time-cost process. To simplify the calculation, a physical one-dimensional (1D) model that explains the drying process was used, as shown in Figure 1. The heat and mass transfer was considered only in the y direction. The total height of the porous medium is 1 cm. The bottom and top surfaces are the heat and mass transfer boundaries.
The porous medium consists of a continuous rigid solid phase, an incompressible liquid phase (free water), and a continuous gas phase. The gas is assumed to be a perfect mixture of vapour and dry air. It is considered an ideal gas. The moisture flow in the inner porous medium is liquid water and vapour flow. The liquid water can become vapour. The vapour and liquid water are moved by the pressure gradient. The heat and mass transfer theory can be found elsewhere [11-15], or in our other study .
The gaseous phase ensures that (1) (2) (3) (4) where ρs, ρa, ρv and ρg are the densities of the solid, air, vapour and gas (kg/m3), respectively, Pa, Pv and Pg are the pressures of the air, vapour and gas (Pa), respectively, Ma and Mv are molar masses of the air and vapour, respectively, (kg/mol), R is the universal gas constant (J/mol/K), and T is the temperature (K).
The assumption of the local thermal equilibrium between the solid, gas, and liquid phases involves (5) where Ts, Tg, and Tw are the temperatures of the solid medium, gas and water, respectively (K).
2.2 Governing equations
2.2.1 Mass and momentum transfer
Mass conservation equations are written for each component in each phase. Given that the solid phase is rigid, the following is given: (6)
The averaged mass conservation of the gas phase  is as follows (7) where ε is the porosity, Sg is the gas saturation, kg is the intrinsic permeability of the gas (m2), kr,g is the relative permeability of the gas, μg is the viscosity of the gas (Pa s), and İ is the phase change rate of water to vapour.
For free water , (9) where Vw is the water velocity (m/s).
For the Darcy flow of the vapour , (10) where Vg is the gas velocity (m/s), Deff is the effective diffusivity of the vapour and air (m2/s), and ω is the mass fraction of the vapour.
For the Darcy flow of the air , (11)
The vapour fraction in the mixed gas is given by, (12)
The saturation of free the water and gas is (13)
The gas and free water Darcy velocity is given by , (14) (15)
The total water velocity is given by , (16) where Dw is the capillary diffusivity (m2/s).
2.2.2 Energy transfer
By considering the hypothesis of the local thermal equilibrium, the energy conservation is reduced to a unique equation : (17) where ρe is the effective density (kg/m3), Ce is the effective specific heat (J/kg/K), ke is the effective thermal conductivity (W/m/K), Ca, Cv and Cw are the specific heats of the air, vapour and water, respectively, (J/kg/K), and λ is the phase change heat of water to vapour (J/kg).
The effective density, effective specific heat and effective thermal conductivity are given by , (18) (19) (20) where ks, kw, ka, and kv are the thermal conductivities of solid medium, water, air and vapour, respectively, (W/m/K).
2.2.3 Phase change
The phase change modelling was provided in two forms: equilibrium and non-equilibrium. Recent studies have shown that evaporation is not instantaneous and that non-equilibrium conditions exist during rapid evaporation between water vapour in the gas phase and water in the solid phase [14, 15]. The more general expression for the non-equilibrium evaporation rate used for modelling the phase change in porous media, which is consistent with the studies on pure water mentioned above, is given by, (21) where aw is the coefficient and Psat is the saturated vapour pressure of water (Pa).
2.2.4 Microwave power absorption
Lambert’s Law [4, 17-21] was used to calculate the microwave energy absorption inside the food samples. This law considers the exponential attenuation of microwave absorption within the product and is given by (22) where P0 is the incident power at the top surface, αis the attenuation constant, hs is the thickness of the media, and (hs − z) is the distance from the top surface.
The volumetric heat generation, Qmic (W/m3), is given by, (24) where V is the volume of the medium (m3).
The microwave power is intermittent input. A duration of 20 s of microwave power is given followed by 80 s without microwave power until the drying process is ended.
2.3 Initial conditions
2.4 Boundary conditions
The bottom and top surfaces are the heat and mass transfer boundaries. The other boundaries of the model are insulated and impermeable. The boundary conditions are given as:
3 Numerical solution
A one-dimensional (1D) grid was used to solve the equations using COMSOL. The mesh consisted of 200 elements (1D), and variable time stepping was used. Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results were grid-independent. The relative tolerance was set to 1e-4, whereas the absolute tolerance was set to 1e-6. The simulations were performed using a ThinkPad with an Intel i7 Duo processor with a 2.3 GHz processing speed and 16 GB of RAM running Windows 7.
4 Results and discussion
4.1 Model validation
The microwave power input was intermittent with a 100 s interval. The microwave power input occurred during the first 20 s. The temperature and moisture curve is shown in Figure 2. The results agree well with the reference results  (not shown in the present paper), except that the temperature is slightly higher than in the present study. The reason of that may be due to the 2D and 1D differences. Therefore, the model in the present study was used to determine the sensitivity parameter. The temperature increased during the first 20 s and decreased during the later 80 s. It was cycled every 100 s until the drying ended. However, the average temperature increased over time.
4.2 Microwave power level
The microwave power is the major input parameter. In the present study, the power level was changed to -20% and +20%P0. The results are shown in Figure 3 forP1 =80%P0, P2 =100%P0, and P3 =120%P0. Lower moisture content was obtained with higher power levels. In addition, higher moisture content was obtained with lower power levels. The moisture content changed to −8.2% and 7.5%, respectively. With an increase in the power level, more microwaves are absorbed, resulting in more moisture loss through higher temperatures, vapour pressures and evaporating rates . The vapour pressure at the surface was approximately the same under the three power levels because it is controlled by the vapour pressure in the ambient. It is different from the vapour pressure in the centre, which had a larger pressure difference. The temperature of the surface barely changed, but the temperature in the centre was much different under the three power levels. The water saturation at the surface was obviously different under the three power levels. The water saturation at the surface and in the centre of the medium decreased with increasing power levels. The results agree well with the moisture content curve. The water saturation was near zero at the end of the drying stage on the sample surface for the 120% P0 power level. This is because the water transfer rate was lower than the vapour transfer rate. The vapour transfer rate decreased during the drying stage. In fact, the drying temperature had a large effect on the product quality. Usually, a lower temperature was good for drying the product; although, most of time, it prolonged the drying time. The increasing power level needs more energy input. Therefore, the drying process should be carefully considered .
A higher vapour pressure will accelerate the mass transfer of the water and vapour. The mass flow rates of the water and vapour on the surface were calculated by Eq. (30) and (31), respectively. The mass flow rates of the water and vapour in the centre were ρwVw and ρνVν respectively. The results are shown in Figure 4. The mass transfer of water on the surface was larger than the vapour transfer in the present study, especially before 400 s when compared with Figures 4(a) and 4(c). However, as the drying continued, the mass transfer rate of vapour on the surface increased until it was higher than the mass transfer of water. The reason is that the water saturation is larger during the beginning of the drying process. The vapour mass transfer is difficult because most gaps in the porous medium surface are filled by water. During the water mass transfer and evaporation, more and more gaps were filled by vapour, and the mass transfer of vapour increased. The water mass transfer in the centre was always greater than the vapour mass transfer in the centre. The water transfer was the major parameter inside the porous medium.
4.3 Ambient temperature
During IMCD, convective air is used to move the vapour from the microwave drying chamber. It can lower the vapour concentration around the surface of the medium. In addition, it can accelerate the drying process. The surface temperature of the medium was lower than the centre temperature during evaporation. Therefore, the ambient air temperature increased the surface temperature. It also increased the vapour pressure and vapour concentration. The mass transfer from the surface to the ambient could also increase. The effect of the ambient temperature is shown in Figure 5. The temperature was 60°C, and it also changed by ±20%. The ambient temperatures were Tamb1 = 48°C, Tamb2 = 60°C and Tamb3 = 72°C. The moisture content changed to −2.4% and 2.3%, respectively. The temperature and vapour pressure in the medium barely changed. However, the water saturation on the surface incurred large differences, but in the centre, it barely changed. A lower moisture content was obtained with higher ambient gas temperatures for the same drying time. The major water reduction was due to the surface water evaporating. The ambient temperature effect was calculated with q = −hT(T − Tamb), as shown in Figure 6(a). The increase in temperature noticeably increased the strength of the heat transfer.
4.4 Heat transfer coefficient
The heat transfer coefficient effect is shown in Figure 7. The heat transfer coefficients also changed ±20% from their initial values. They were hT1 =13.411, hT2 =16.764, and hT3 =20.117 W m2 K−1. Compared with Figure 5, the heat transfer coefficients had a minimal effect. The temperature, vapour pressure and water saturation were approximately constant for different heat transfer coefficients. The reason is that the heat transfer coefficients were large enough to strengthen the heat transfer process. The results can also be found in Figure 6(b). In comparison of Figures 6(a) and 6(b), the ambient temperature shows little sensitivity.
4.5 Mass transfer coefficient
The water inside the medium transfers to the surface of the porous medium and evaporates. The vapour exchanges mass with the ambient medium. The mass transfer coefficient is the key factor because it is the last step of the drying process to transfer mass outside of the porous medium. In the present study, the mass transfer coefficient was varied. The results are in shown in Figure 8.
The original value was 0.067904 m s−1 , and it was changed by ±20%. The values were hm1 =0.054323, hm2 =0.067904 and hm3 =0.08148 m s−1. The moisture content curve showed minimal changes. It is similar to the temperature, vapour pressure and water saturation curves. The centre water saturation curve, especially, showed no difference. The water transfer rate and vapour transfer were calculated by Eq. (30) and Eq. (31). The results of the water transfer and vapour transfer rates showed minimal differences in Figure 9.
4.6 Gas permeability
The relative permeability values for free water and gas in the present study were obtained from  using Eq. (33) and Eq. (34). However, different gas relative permeability values were calculated using Eq. (35). Their effects are shown in Figure 10. (33)
The gas relative permeability is given by , (34)
Another gas relative permeability is given by , (35)
The different values for the gas relative permeability did not substantially change the results of the model, as shown in Figure 10. The results were similar to those in reference . The temperature and vapour pressure were approximately the same. However, the water saturation at the surface and in the centre exhibited different results. The water saturation at the surface was higher, while the water saturation in the centre was lower using the gas relative permeability given by Eq. (35). The water saturation in the surface was lower, but the water saturation in the centre was higher using the gas relative permeability given by Eq. (34). The vapour flow rate in the centre was ρνVg from Eq. (10). The results of the different gas relative permeability values on vapour flow rate showed minimal differences, as shown in Figure 11. The reason is that the relative permeability barely changed with relative to other parameters in Eq. (14). It was of the order of 10−12.
The intrinsic permeability was studied by three groups of values:
kin,g1 = 4.0×10−12 m2 and kin,w1 = 4.0×10−12 m2,
kin,g2 = 2.7×10−11 m2 andkin,w2 = 4.0×10−12 m2, and
kin,g3 = 2.7×10−11 m2 and kin,w3 = 2.7×10−11 m2.
4.7 Effective gas diffusivity
The effective gas diffusivity can be calculated as a function of the gas saturation and porosity according to the Bruggeman correction , namely, (36) It can also be calculated as , (37) where Dνa is the binary diffusivity between the air and water vapour (m2s−1).
Because several parameters affect the effective diffusivity of water vapour in porous media (molecular diffusivity, porosity, and tortuosity), this coefficient is variable from one kind of material to another . As shown in Figure 14, the change was very large with different effective gas diffusivity values, which is similar to the results in reference . The resistance of the porous medium to gaseous migration was very important, and this showed that their model was sensitive to the changes in the effective diffusivity of the water vapour. The temperatures with different effective gas diffusivity values were obviously different. They were higher using Eq. (36) and lower using Eq. (37). The water saturation was the same as that in Figure 10(d). With Eq. (36), the water saturation at the surface was higher, but the water saturation in the centre was lower. With Eq. (37), the water saturation at the surface was lower, but the water saturation in the centre was higher. The vapour diffusion rate was calculated from Eq. (38); the results of the calculation are shown in Figure 15. (38) The vapour diffusion rate showed obvious differences. In addition, it actually affected the moisture content.
4.8 Capillary diffusivity of liquid water
The capillary diffusivity of liquid water is various for different materials. In the present study, the capillary diffusivity of liquid water is the same as in the previous reference . The parameter sensitivity of the capillary diffusivity is shown in Figure 16. The effect on moisture curve is minimal. It is also the same with the temperature and vapour pressure. However, the water saturation exhibits different results compared to those in Figure 7(d). The biggest difference occurred at the surface for the water saturation. A relatively small difference was observed in the centre. The water diffusion rate was calculated as (39) The results under different capillary diffusivity values for liquid water are shown in Figure 17. The water diffusion rate showed obvious differences. However, compared with Figure 16, an effect on the moisture curve was not observed. As shown in Figure 13(a), the water flow rate was not a major factor on the drying rate because the water changed to vapour, especially on the surface.
4.9 Evaporation rate constant
Another important model parameter is the non-equilibrium evaporation rate constant, Kevap. Figure 18 shows its effect on the moisture profiles. The evaporation rate constant required a 10-fold change, not just by ±20%, because of its lower parameter sensitivity. As seen in Figure 18, for low values of K, the moisture content is higher. However, with larger Kevap values, the difference between them decreases. Therefore, the model can simulate an equilibrium problem . The temperature increased with a lower evaporation rate constant because less energy input was required for evaporation. The vapour pressure was the same for the temperature curve. The water saturation was almost the same in the centre, but the difference was huge at the surface. This shows that the water evaporating near the surface increased with the higher evaporation rate constant.
The evaporating rate results under different evaporating rate constants are shown in Figure 19. The evaporating rate increased as the evaporating rate constant increased. During the non-equilibrium simulation, the larger evaporating rate constants brought the system closer to an equilibrium state. The Kevap. =10000 was sufficient for the simulation.
In a comparison of the results of the capillary diffusivity of the liquid water, evaporating rate constant, effective gas diffusivity, gas permeability, and effective gas diffusivity, vapour transfer is the deciding factor of the drying process. For parameters that have an effect on the vapour transfer, the drying process will be obviously different.
A multiphase porous media model of intermittent microwave convective drying was developed based on reference values. The model considered liquid water, gas and the solid matrix inside of food. The model was simulated by COMSOL software. Its parameter sensitivity was analysed by changing the parameter values ±20%, with exceptions for several parameters. Sensitivity analyses of the process to the microwave power level, ambient temperature, effective gas diffusivity, and evaporation rate constant show that they all have significant effects on the process. However, the surface mass, heat transfer coefficient, gas relative and intrinsic permeability, and capillary diffusivity of water did not have a considerable effect. The evaporation rate constant exhibited minimal parameter sensitivity with a ±20% value change, until it was changed 10-fold. In all results, the temperature and vapour pressure showed the same trends as the moisture content. However, the water saturation at the medium surface and in the centre showed different results. Vapour transfer was the major mass transfer phenomenon that affected the drying process.
This research was supported by the National Natural Science Foundation of China (Grant Nos. 31371873, 31000665, 51176027, and 31300408).
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Published Online: 2017-06-14