Mathematical Modeling of Mechanical Thermo-Diffusion Processes Under Influence of Microwave Irradiation

Introduction

Microwave technologies are widely used in industrial processes, such as drying (textiles, wood, paper and ceramics), heat treatment of metals (hardening), disposal (recycling or disposal) of radioactive waste. In medicine, they are used to restore frozen tissues, warm blood, and treat tumors. The greatest consumer interest in microwave technologies arises in the food preparation industry, namely the processes of baking, pasteurization, dehydration and sterilization. Volumetric heating of the material by microwave irradiation (internal dielectric heating) refers to accelerated methods of heat exchange and removal of moisture from the material. For various methods of dielectric heating, the prediction (forecasting) of heat and mass transfer (transport) processes is extremely important in terms of equipment development, optimization processes, and product quality improvement. Direct experimental measurement of temperature and moisture content in porous material is a complex and troublesome process. Therefore, a significant amount of scientific research was carried out to model the transport (diffusion) processes of heat and mass transfer for such heterogeneous materials. Several different techniques or methods of numerical calculations, which are based on the use of the finite difference method, finite elements or the line transfer matrix method have been used to simulate microwave heating with varying degrees of success [1-4]. In fact, when performing these calculations, it is necessary to know the constant dielectric or thermal properties of the material [5-10].

A lot of research has been done in the modeling of microwave heating of materials, but we have little progress in modeling the transfer of heat and mass during microwave drying [11]. Since under influence of dielectric heating or drying in an electromagnetic field, the material with dielectric loss of power of microwave irradiation has no permanent electro-physical properties, for most materials the dielectric properties change as a function of moisture content and temperature. Therefore, during microwave heating (drying), the distribution of the electromagnetic field in the body (material) is strongly related to the processes of heat and mass transfer. Changes in the local moisture content and temperature affect to the dielectric properties of the material, including the distribution of the electromagnetic field. The size of the load (material) relative to the waveguide or the drying cavity, the effect of the wave resistance (impedance) of the cavity, the amount of the radiation power that is reflected from the inner walls of the cavity (resonator) in the direction of the on magnetron determine the quality of the microwave equipment.

Since the detailed distribution of the power of the electromagnetic field (internal heat sources) within the material loading is quite difficult to model, most studies assume that the lines of force of the electromagnetic field during microwave irradiation on the surface of the material are uniform and normal to the surface [4]. It is also assumed that the power of the external irradiation in the body decays in accordance with the exponential law. Such approach was happy applied to modeling of heat and mass transport of dielectric the heated product of food [12-14]. A simple reflection was expressed according to investigation, where heat sources under dielectric losses of microwave irradiation in the range of solid product is assumed uniform [15]. In reality into the mentioned theoretical works the understanding of drying processes is limited, because under description of humidity transfer in porous media at isothermal conditions is used in general. For more developed models a law of molecular diffusion is applied and assumed, that heat gradient is the main division force into processes of mass transport or transfer [16- 18]. Its need to pointed, that results of numerical modeling according to mentioned above models for humidity diffusion as rule do not agree with experimental measured in the more cases [19].

Same researcher development of the models into approach, when diffusion of water vapor is main mechanism of humidity transport during of drying processes [20]. For too refined methods for describing of drying (microwave heating) the two area transport models are treated [21-23]. In this models is assumed, that two different area (region) for describing of humidity transport during drying (wetted and dried area) is existed. In the wetted area moisture content is grate as maximal sorption value for water vapor and basic mechanism for transport (transfer) of humidity is movement of liquid. In sorption region the movement of adsorbed (bounded) water and water vapor is basic reason of mass transport. The applying of this model is limited, because during the long (limited) process of drying the division (boundary) between wetted and sorption areas (regions) is conventional.

A reference review is showed that main progress was made into development of transport (diffusion) models under investigation of physics of ground, where main interest was concentrated on the utilization of nuclear remains and management of water resources [24]. The basic formulation was development for interconnected processes of heat and mass transfer into works [25-27]. The main approaches are based on assumption, that general transport potential consist on the two components: temperature and capillary potential.

During microwave drying an induces by thermal (microwave) heating increment in density of water vapor and humidity potential is calling of movement for humidity from more heated into cooled area. A general restriction or deficiency of mentioned isothermal models is absence of the thermal induced changing of humidity. First of all, this is due to a significant difference in the distribution of moisture in the porous material, which is caused by the processes of evaporation or condensation.

In fact, in the context of drying processes, reformulation of the problem in terms of moisture content and temperature is desirable for a fundamental understanding of drying processes, especially when modeling diffusion (transport) processes under conditions of intense internal microwave heating. First of all, this is due to a significant difference in the distribution of moisture in the porous material, which is caused by the processes of evaporation or condensation.

The nature of stresses in porous materials that arise during microwave heating, as a result of changes in temperature and moisture distribution, also remains insufficiently researched. For example, despite the fact that the hydrothermal behavior of concrete under thermal (radiation) heating is described in detail in the work, the author is aware of only one scientific work, where the maximum values of thermal stresses caused by internal microwave heat sources were calculated in the specified material [28,29]. Some attempts at computer simulation of the stressed state of the specified material under microwave irradiation were carried out in works where only the energy (thermal) balance equation was used to calculate the thermal and elastic properties of the material [30,31].

Special attention should be paid to the study of the properties of thermal and moisture stresses during microwave irradiation under conditions of phase transformation (evaporation or condensation) and changes in the pressure of the gas medium into the pores of the moistened material. Theoretical models for describing the dependence of microwave heating sources on the distribution of moisture and temperature in a porous medium also need improvement and development, especially in view of modeling effective (measured) characteristics due to the corresponding properties of phases or components in the studied wetted porous sample.

Model and Analysis

Interaction of the material with the electromagnetic field for a non-magnetic dielectric medium with conductive properties adsorbs the energy of the electromagnetic field and converts it into the heat.

In the microwave or dielectric ranges of frequency (Figure.1) area the properties of the material with dielectric losses are determined by the ratio

Figure 1: The electromagnetically spectrum

The main dielectric losses is linked with properties of free (not adsorbed or non bounded) water, as shown on Fig.2 below

Figure 2: The dielectric properties of free water in the specific spectral interval

Where are any other types of dielectric relaxation (see Fig.2) may to meets in the porous body. We are considering only dipole rotation under condition that scattering according to electric conductivity or joule heat release are neglibles.

Figure 3: Types of dielectric relaxation in the porous wetted material

Comparing to convective drying we can poit to advantage of microwave or dilectric drying: 1. High energy efficiency in the period of falling speed, which is a consequence of the concentration of the energy of heat release in places of liquid concentration; 2. A significant increase in fluid mobility with an increase in the internal pressure of water vapor due to the effect of liquid evaporation; 3. Significant improvement or preservation of the quality of the drying product due to the near- uniform distribution of heat sources, which corresponds to the insignificant distribution of stresses along the thickness of the body; 4. Significant improvement or preservation of the quality of the drying product due to the near-uniform distribution of heat sources, which corresponds to the insignificant distribution of stresses along the thickness of the body; 5. The duration of convective drying can be significantly reduced in accordance with the expediency of saving energy costs.

The acceleration of convective drying applying microwave is demonstrated on the Figure.4 below. There is the significant time saving in drying according to time point of applying microwave. In other word it's possible to combine convective and microwave drying for reduction of drying period.

Figure 4: Curves of Mixed Convective and Microwave Drying

There are also differences into nature of destruction for investigate porous material, as it is depicted on the Figure.5 below

Figure 5: Examples of destruction of ceramics under convective (A) and microwave (B) drying

The main goal of this work to divide of mechanical, electro- magnetical and diffusional processes to correct describes of the mentioned physical properties of wetted (humidified) porous material under influence of the symmetric microwave irradiation for the one-dimensional case into approach of continues media. In general, allow me to characterize and name these phenomena as mechanical thermo-diffusion in the proposed terminology.

The object of studies Let's define the object of studies. This is one dimensional porous wetted plate length of (Figure.1) which is under influence of symmetrical microwave irradiation of the fixed power. The constant convective flow by the hot air at the surfaces of plate is also additionally applied during all time of microwave treatment, so in general it is can be classified as the mixed microwave-convective drying.

We are modeling such drying system mathematically via introduction of dynamical boundary conditions. There is confirmed by other authors [21-23] that conditionally it is possible to highlight the two zones during of microwave treatment for porous sample: wetted and heated zone. The term zone is meaning as a one-dimensional area from the internal surface to the real hard boundaries of body into the direction of external environment. The moving that zones is interconnected due to dependence of diffusion coefficients in the internal volume of sample from humidity as well as the time dependences of surface heat and mass transfer coefficients on the real physical boundaries of mentioned porous sample.

To describe the diffusion processes in a porous plate, the well- known averaged heat and mass transport equations written in general form are used [32,33].

the systems of heat and mass transport equations are separated.

Modeling of microwave treatment processes: Let`s define the real body surfaces as S (see Figure.1), heating and evaporation surfaces as S1 and S2 respectively.at the initial moment of time. The near-surface zones of wetting and heating of the material are considered, delimited by the surfaces S1 and S2, within which the solutions of the system of equations and the power of the microwave heating sources are assumed to be constant.

Internal diffusion processes: The modified system of equations (6) and (7) with a defined effective diffusion coefficient (5) of the gas mixture in the pores of the moistened material with corresponding expanded expressions for the flows is considered

Conclusions

According to results of heat and mass process numerical simulation the time shift of near surface wetted and heated zones which are separated by the surfaces S1 and S2 is obtained due to distribution of dimensionless sickness Δh1 ⁄ L and Δh2 ⁄ L as it depicted on the (Figure.3) and (Figure.4) relatively to surface S0 of plate symmetry.

Conclusion III:

Decreasing of moisture content is caused by increasing of evaporation sources that as a result of inhomogeneous microwave heating of plate and change of thermodynamic state of two component mixture.

On the pictures below it depicted the distribution of flows by liquid (Figure.10), water vapor (Figure.11) and dry air (Figure.14) along half thickness of plate in the different time intervals.

Conclusion IV: In the neighborhood of time point tb, which corresponds to achievement of boiling temperature at normal conditions its fixed change of flows distribution for liquid and gas mixture components by thickness of plate (dashed line). This is due to the increase in the intensity (Figure. 13) of internal sources of evaporation i(int) along the thickness of the plate.

The distribution of specific stresses (Figure. 14) and (Figure. 15) and small deformations (Figure. 16) and (Figure. 17) in a solid matrix (frame or skeleton) over the half-thickness of the plate is shown by the following graphical dependencies.

The dynamics of thermal stress distribution (Figure. 19) along the thickness of the plate is also shown in the form of the corresponding graphical dependence:

Figure 18: Distribution of Thermal Stress In Porous Skeleton

Form the pictures below (Figure.14 and Figure.15) it is shown that longitudinal components of stress tension is joined first with distribution of the temperature field (see Figure.8) and nonlinear behavior of tangential component of stress tension is results of influence of nonlinear evaporation processes (see Fig.7) especially near surface of sample.

Acknowledgement

I am grateful to my scientific mentors Prof. O.R. Hachkevych and Doc. of Science R.F. Terletskiy for their patience in the development of these studies.

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