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Modelling of intermittent microwave convective drying: parameter sensitivity

  • Zhijun Zhang EMAIL logo , Wenchao Qin , Bin Shi , Jingxin Gao and Shiwei Zhang
Published/Copyright: June 14, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

The reliability of the predictions of a mathematical model is a prerequisite to its utilization. A multiphase porous media model of intermittent microwave convective drying is developed based on the literature. The model considers the liquid water, gas and solid matrix inside of food. The model is simulated by COMSOL software. Its sensitivity parameter is analysed by changing the parameter values by ±20%, with the exception of several parameters. The sensitivity analysis of the process of the microwave power level shows that each parameter: ambient temperature, effective gas diffusivity, and evaporation rate constant, has significant effects on the process. However, the surface mass, heat transfer coefficient, relative and intrinsic permeability of the gas, and capillary diffusivity of water do not have a considerable effect. The evaporation rate constant has minimal parameter sensitivity with a ±20% value change, until it is changed 10-fold. In all results, the temperature and vapour pressure curves show the same trends as the moisture content curve. However, the water saturation at the medium surface and in the centre show different results. Vapour transfer is the major mass transfer phenomenon that affects the drying process.

Keywords: Intermittent microwave convective drying (IMCD); modelling; multiphase porous media; COMSOL
PACS: 44.30.+v; 41.20.-q; 4.35.+c; 44.05.+e

1 Introduction

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 [3]. 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 [4]. There are some empirical models available in the literature for IMCD [5]; however, the empirical models are not helpful for optimization and are only applicable for specific experimental conditions [6]. 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 [4]. 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 [11]. 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.

Figure 1 1D model of the porous medium
Figure 1

1D model of the porous medium

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 [16].

The gaseous phase ensures that

ρa=MaPaRT(1)
ρv=MvPvRT(2)
Pg=Pa+Pv(3)
ρg=ρa+ρv(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

Ts=Tg=Tw=T(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:

∂ρs∂t=0(6)

The averaged mass conservation of the gas phase [11] is as follows

∂(ε⋅Sgρg)∂t+∇⋅(ρgkgkr,gμg∇Pg)=I˙(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 vapour [11, 14],

∂(ε⋅Sgρv)∂t+∇⋅(ρvVv)=I˙(8)

where Vv is the vapour velocity (m/s).

For free water [12],

∂(ε⋅Swρw)∂t+∇⋅(ρwVw)=−I˙(9)

where Vw is the water velocity (m/s).

For the Darcy flow of the vapour [14],

ρvVv=ρvVg−ρgDeff⋅∇ωv(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 [14],

ρaVa=ρaVg+ρgDeff⋅∇ωv(11)

The vapour fraction in the mixed gas is given by,

ωv=ρvρg(12)

The saturation of free the water and gas is

Sg+Sw=1(13)

The gas and free water Darcy velocity is given by [15],

Vg=−ki,g⋅kr,gμg⋅∇Pg(14)
vw=−ki,w⋅kr,wμw⋅∇Pw(15)

The total water velocity is given by [15],

Vw=−kin,w⋅kr,wμw⋅∇Pg−Dwρwε⋅∇Sw(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 [15]:

ρeCe∂T∂t+∇⋅(ρaVaCaT+ρvVvCvT+ρwVwCwT)=∇(ke⋅∇T)−hfg⋅I˙+Qmic(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 [15],

ρe=ρss+ε⋅Sgρa+ε⋅Sgρv+ε⋅Swρw(18)
ke=(1−ε)ks+ε(Swkw+Sg(ωkv+(1−ω)ka))(19)
Ce=(1−ε)ρsCs+ε(ρwSwCw+ρgSg(ωCv+(1−ω)Ca))(1−ε)ρs+ε(ρwSw+ρgSg)(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,

I˙=KevapMv(awPsat−Pv)SgεRT(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

Pmic=P0exp−2α(hs−z)(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 attenuation constant, α, is given by [17-21],

α=2πλε′1+ε″ε′2−12(23)

where λ is the wavelength of the microwave in free space and ε′ and ε″ are the dielectric constant and the dielectric loss, respectively [4].

The volumetric heat generation, Qmic (W/m3), is given by,

Qmic=PmicV(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

The initial moisture of the porous medium is represented by the liquid water saturation; different initial water saturation values are used.

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:

  • B.C. for Eq. (7),

    Ps=Pamb(29)

  • B.C. for Eq. (8),

    n→v,s=−hmεSg(ρv−ρv,amb)(30)

  • B.C. for Eq. (9):

    n→w,s=−hmεSw(ρv−ρv,amb)(31)

  • B.C. for Eq. (11),

    qs=−hT(T−Tamb)+(λ+CwT)nw,s+CvTnv,s(32)
where Pamb is the ambient pressure (Pa), hm is the mass transfer coefficient (m/s), ρν,amb is the vapour density of the ambient (kg/m3), hT is the heat transfer coefficient (W/m2/K), and Tamb is the ambient temperature (K).

2.5 Input parameter

In the present study, the input parameters from the literature [4, 15] are listed in Table 1. The major focus of this work is the parameter sensitivity. In order to finish this work, the different values or adapted equations will be shown in the later sections.

Table 1

Input properties for the model

Parametervalue
Sample thickness, hs40 mm
Porosity, ε0.922
Initial saturation of water, Sw00.794
Initial saturation of gas Sg00.206
Initial mass fraction of the vapour, ων00.026
Initial temperature of the sample, Sw0303 K (30 °)
Universal gas constant, R8.314 J mol−1K−1
Molecular weight of water, Mw0.018016 kg mol−1
Molecular weight of vapour, Mν0.018016 kg mol−1
Molecular weight of air, Ma0.028966 kg mol−1
Latent heat of evaporation, hfg2.26e6 J kg−1
Ambient pressure, Pamb101325 Pa
Ambient vapour pressure, Pν,amb2992 Pa
Specific heat of the solid sample, Cs3734 J kg-1 K−1
Specific heat of water, Cw4183 J kg−1 K−1
Specific heat of vapour, Cν1900 J kg−1 K−1
Specific heat of air, Ca1005.68 J kg−1 K−1
Thermal conductivity of the solid sample, ks0.46 W m−1 K−1
Thermal conductivity of water, kw0.644 W m−1 K−1
Thermal conductivity of air or vapour, kνkν0.026 W m−1 K−1
Density of the solid apple, ρs1419 kg m−3
Density of water, ρw1000 kg m−3

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 [4] (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.

Figure 2 Temperature and moisture content curves (d. b.) vs time
Figure 2

Temperature and moisture content curves (d. b.) vs 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 [22]. 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 [22].

Figure 3 Moisture content, temperature, vapour pressure and water saturation using different microwave power levels
Figure 3

Moisture content, temperature, vapour pressure and water saturation using different microwave power levels

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.

Figure 4 Water and vapour flow rates using different power levels
Figure 4

Water and vapour flow rates using different power levels

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.

Figure 5 Moisture content, temperature, vapour pressure and water saturation under different ambient temperatures
Figure 5

Moisture content, temperature, vapour pressure and water saturation under different ambient temperatures

Figure 6 Surface heat transfer under different ambient temperatures (a) and different heat transfer coefficients (b)
Figure 6

Surface heat transfer under different ambient temperatures (a) and different heat transfer coefficients (b)

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.

Figure 7 Moisture content curves with different heat transfer coefficients
Figure 7

Moisture content curves with different heat transfer coefficients

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.

Figure 8 Moisture content, temperature, vapour pressure and water saturation using different mass transfer coefficients
Figure 8

Moisture content, temperature, vapour pressure and water saturation using different mass transfer coefficients

The original value was 0.067904 m s−1 [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.

Figure 9 Water transfer rates and vapour transfer rates using different mass transfer coefficients
Figure 9

Water transfer rates and vapour transfer rates using different mass transfer coefficients

4.6 Gas permeability

The relative permeability values for free water and gas in the present study were obtained from [14] using Eq. (33) and Eq. (34). However, different gas relative permeability values were calculated using Eq. (35). Their effects are shown in Figure 10.

kr,w=Sw−Sr1−Sr30Sw>SrSw≤Sr(33)
Figure 10 Moisture content, temperature, vapour pressure and water saturation under different relative permeability values
Figure 10

Moisture content, temperature, vapour pressure and water saturation under different relative permeability values

The gas relative permeability is given by [14],

kr,g1=Sg(34)

Another gas relative permeability is given by [21],

kr,g2=Sg2(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 [23]. 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.

Figure 11 Vapour flow rates in the centre of the sample under different gas relative permeability values
Figure 11

Vapour flow rates in the centre of the sample under different gas relative permeability values

The intrinsic permeability was studied by three groups of values:

  1. kin,g1 = 4.0×10−12 m2 and kin,w1 = 4.0×10−12 m2,

  2. kin,g2 = 2.7×10−11 m2 andkin,w2 = 4.0×10−12 m2, and

  3. kin,g3 = 2.7×10−11 m2 and kin,w3 = 2.7×10−11 m2.

The effect of the intrinsic permeability with different values is shown in Figure 12. There was no appreciable change in the final moisture content of the material with the changing permeability. The temperature and vapour pressure values were almost the same. However, as shown in Figure 12(d), the water saturation was noticeably different. The curves with group parameters (1) and (3) agreed well. However, the water saturation at the surface was lower, and the water saturation in the centre was higher for group parameter (2). This is also the reason why the moisture content curves were approximately the same. The water and vapour flow rates are shown in Figure 13. The water flow rates agreed well with the moisture content results. Group parameters (1) and (3) were the same, but group parameter (2) was different. However, the vapour flow rate was almost the same for the three group parameters. The high sensitivity parameter of the water flow rate was demonstrated. In fact, the mass transfer from the sample to the ambient occurred through evaporation, especially through water evaporation on the surface. Therefore, the varying water flow rate did not greatly affect the drying process, especially the moisture content.

Figure 12 Moisture content, temperature, vapour pressure and water saturation under different intrinsic permeability values
Figure 12

Moisture content, temperature, vapour pressure and water saturation under different intrinsic permeability values

Figure 13 Water flow rate and vapour flow rate in the sample centre under different intrinsic permeability values
Figure 13

Water flow rate and vapour flow rate in the sample centre under different intrinsic permeability values

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 [4], namely,

Deff1=Dva(Sgε)4/3(36)

It can also be calculated as [21],

Deff2=2.13PgT2731.8(Sgε)3−εε(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 [22]. As shown in Figure 14, the change was very large with different effective gas diffusivity values, which is similar to the results in reference [22]. 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.

qvD=ρgDeff⋅∇ωv(38)

The vapour diffusion rate showed obvious differences. In addition, it actually affected the moisture content.

Figure 14 Moisture content, temperature, vapour pressure and water saturation under different effective gas diffusivity values
Figure 14

Moisture content, temperature, vapour pressure and water saturation under different effective gas diffusivity values

Figure 15 Vapour diffusion rate under different effective gas diffusivity values
Figure 15

Vapour diffusion rate under different effective gas diffusivity values

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 [4]. 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

qwD=Dwρwε⋅∇Sw(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.

Figure 16 Moisture content, temperature, vapour pressure and water saturation under different capillary diffusivity values for water
Figure 16

Moisture content, temperature, vapour pressure and water saturation under different capillary diffusivity values for water

Figure 17 Water diffusion rate under different diffusivity values
Figure 17

Water diffusion rate under different diffusivity values

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 [15]. 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.

Figure 18 Moisture content, temperature, vapour pressure and water saturation using different evaporating rate constants
Figure 18

Moisture content, temperature, vapour pressure and water saturation using different evaporating rate constants

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.

Figure 19 Water evaporating rate using different evaporation rate constants
Figure 19

Water evaporating rate using different evaporation rate constants

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.

5 Conclusions

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.

Acknowledgement

This research was supported by the National Natural Science Foundation of China (Grant Nos. 31371873, 31000665, 51176027, and 31300408).

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Received: 2016-8-1
Accepted: 2017-3-9
Published Online: 2017-6-14

© 2017 Zhijun Zhang et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0045
Keywords for this article
Intermittent microwave convective drying (IMCD); modelling; multiphase porous media; COMSOL
Creative Commons
BY-NC-ND 3.0

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  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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