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

Taras Holubets

doi:10.20944/preprints202505.0265.v1

Submitted:

05 May 2025

Posted:

06 May 2025

Read the latest preprint version here

Abstract

in this work the methods of applying the spatial averaging method to the description of the diffusion, electrodynamics and mechanical properties of a porous multiphase (composite) wetted medium are considered. the set of physical phenomena and processes in porous wetted materials is described in detail, as well as mathematical or theoretical models are proposed to describe the interaction of moisture, electromagnetic and mechanical fields. for a porous humidified medium, a closed system of equations was obtained, which connects the processes of heat and mass transfer with internal heat sources induced by external microwave electromagnetic radiation and mechanical stresses within the framework of the model of a continuous continuous medium.

Keywords:

microwave irradiation

;

porous media

;

heat and mass transfer

;

mechanical stress

Subject:

Physical Sciences - Theoretical Physics

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 [1,2], finite elements [3] or the line transfer matrix method [4] have been used to simulate microwave heating with varying degrees of success. In fact, when performing these calculations, it is necessary to know [5,6,7,8,9,10] the constant dielectric or thermal properties of the material.

A lot of research has been done in the modeling of microwave heating of materials [11], but we have little progress in modeling the transfer of heat and mass during microwave drying. 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 [4], 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. It is also assumed that the power of the external irradiation in the body decays in accordance [12] with the exponential law. Such approach was happy applied to modeling of heat and mass transport of dielectric the heated product of food [13,14]. A simple reflection was expressed according to investigation [15], where heat sources under dielectric losses of microwave irradiation in the range of solid product is assumed uniform. 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 [16] is applied and assumed [17,18], that heat gradient is the main division force into processes of mass transport or transfer. 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 [19] in the more cases.

Same researcher development of the models [20] into approach, when diffusion of water vapor is main mechanism of humidity transport during of drying processes. For too refined methods for describing of drying (microwave heating) the two area transport models [21,22,23] are treated. 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 [24] 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. The basic formulation was development for interconnected processes of heat and mass transfer into works [25,26,27]. 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 [28], the author is aware of only one scientific work [29], where the maximum values of thermal stresses caused by internal microwave heat sources were calculated in the specified material. Some attempts at computer simulation of the stressed state of the specified material under microwave irradiation were carried out in works [30,31], where only the energy (thermal) balance equation was used to calculate the thermal and elastic properties of the material.

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

ε = ε^{'} - і ε^{″}

(1)

here ε - complex dielectric constant, ε′- relative dielectric constant, ε″- dielectric loss factor, also

t g δ = ε^{″} / ε^{'}

, (2)

is the tangent of the dielectric loss angle.

Figure 1. The electromagnetically spectrum.

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

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

Where are any other types of dielectric relaxation (see Figure 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 its 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 mentioned physical properties of wetted or humidified porous material under influence of 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

L

(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.

Figure 1. Modeling of microwave treatment of porous sample under symmetric conditions.

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 [32,33] averaged heat and mass transport equations written in general form are used

where

and

, here

are the effective thermal characteristics of porous material,

and

are the heat and mass surfaces intensity correspondingly,

and

are the specific capacity and thermo conduction of σ-phase accordingly and

a is the latent heat of vaporization.

There are the expressions for concentrations

{⟨с_{L}⟩}^{L} = ⟨φ⟩ η_{L} {⟨ρ_{L}⟩}^{L}

,

{⟨с_{ν}⟩}^{G} = ⟨φ⟩ (1 - η_{L}) {⟨ρ_{ν}⟩}^{G}

,

{⟨с_{a}⟩}^{G} = ⟨φ⟩ (1 - η_{L}) {⟨ρ_{a}⟩}^{G}

and flows of phases (components) in the porous media

⟨J_{L}^{x}⟩ = - {⟨ρ_{L}⟩}^{L} \frac{k_{in} k_{rL}}{μ_{L}} \partial_{x} P_{L}

,

⟨J_{α}^{x}⟩ = - {⟨ρ_{α}⟩}^{G} \frac{k_{in} k_{rG}}{μ_{G}} \partial_{x} P_{G} - {⟨ρ_{G}⟩}^{G} D_{eff} \partial [{⟨ρ_{ν}⟩}^{G} / ρ_{G}]

,

here

ψ_{L} = [k_{in} k_{rL}] / μ_{L}

and

ψ_{G} = [k_{in} k_{rG}] / μ_{G}

, where

k_{in}

and

k_{r α}

,

α = \{L, G\}

a the intrinsic permeability of skeleton and liquid phase correspodingly,

{⟨ρ_{L}⟩}^{L} = ρ_{L}

,

{⟨ρ_{α}⟩}^{G} = p_{α} M_{α} / RT

, where

β = \{ν, a\}

a is the corresponding densities of liquid phase and components of gas mixture,

μ_{L}

and

μ_{G}

a is dynamical viscosity,

D_{eff} = [(M_{ν} M_{a}) / M_{G}^{2}] D_{G}^{eff}

is model diffusion coefficient,

P_{L}

and

P_{G}

a is the valueas of pressures in liquid and gas phases,

P_{С}

is a capillary pressure,

M_{ν}

and

M_{a}

a is the molar mass of water vapor and dry air respectively.

Constitutive or material equations: 1. Pressure balance into liguid and gass phase:

P_{L} = P_{G} - P_{С}

, where

P_{С} (η_{L}) = σ \sqrt{k_{in} / ⟨φ⟩} J (η_{L})

a is capillary pressure,

J (η_{L})

is dimensionless characteristic of porous material named as J-Laverett [34] function; 2. The Daltone’s law:

p_{ν} + p_{a} = P_{G}

, where

p_{β}

is the partial pressures of gas mixture components (

P_{G} = ρ_{G} RT / M_{G}

, here

M_{G} = 1 / ([{⟨ρ_{ν}⟩}^{G} / {⟨ρ_{G}⟩}^{G}] / M_{ν} + [{⟨ρ_{a}⟩}^{G} / {⟨ρ_{G}⟩}^{G}] / M_{a})

a is the molar mass of mixture); 3. Equation of sorption equilibrium

P_{c} = - ρ_{L} RT Ln ϕ

, where

ϕ = p_{ν} / p_{ν s}

is the relative humidity and

p_{ν s}

is the partial pressure of water vapor into the gas phase.

Modification of heat and mass equations: 1. A transition from variables has been made from

\{η_{L}, T, P_{G}\}

to

\{η_{L}, T, x_{ν}\}

according to Rault’s law [32]

x_{α} = p_{α} / P_{G}

(

x_{ν} + x_{a} = 1

), where

x_{α}

is molar fraction of components, the intensity of evaporation sources in the internal volume and at the surface of the plate is determined in the different ways

2. An expression for the effective Burger[35] diffusion coefficient of the gas mixture in the pores of the material is proposed

D_{G}^{eff} = \frac{1}{τ} \frac{⟨φ⟩ (1 - η_{L})}{x_{ν} - ⟨φ⟩ (1 - η_{L}) (x_{ν} - 1)} D_{ν}^{a}

, (5)

where

D_{ν}^{a}

a is the diffusion coefficient of water vapor into dry air,

τ

a is the tortosity factor.

3.For the internal volume

\partial_{t} [{⟨c_{L}⟩}^{L} + {⟨c_{ν}⟩}^{G}] = - \partial_{x} [⟨J_{L}^{x}⟩ + ⟨J_{ν}^{x}⟩] \partial_{t} {⟨c_{a}⟩}^{G} = - \partial_{x} ⟨J_{a}^{x}⟩

(6)

and surface of plate

\partial_{t} {⟨c_{a}⟩}^{G} = - \partial_{x} ⟨J_{a}^{x}⟩

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

S_{1}

and

S_{2}

respectively.at the initial moment of time. The near-surface zones of wetting and heating of the material are considered, delimited by the surfaces

S_{1}

and

S_{2}

, 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

⟨J_{L}^{x}⟩ = ψ_{L} ρ_{L} (\frac{\partial p_{c}}{\partial η_{L}} - \frac{1}{x_{ν}} \frac{\partial p_{ν}}{\partial η_{L}}) \partial_{x} η_{L} - ψ_{L} ρ_{L} (\frac{\partial p_{c}}{\partial T} - \frac{1}{x_{ν}} \frac{\partial p_{ν}}{\partial T}) \partial_{x} T + ψ_{L} ρ_{L} \frac{P_{G}}{x_{ν}} \partial_{x} x_{ν} ⟨J_{ν}^{x}⟩ = - ψ_{G} \frac{1}{x_{ν}} {⟨ρ_{ν}⟩}^{G} \frac{\partial p_{ν}}{\partial η_{L}} \partial_{x} η_{L} - ψ_{G} \frac{1}{x_{ν}} {⟨ρ_{ν}⟩}^{G} \frac{\partial p_{ν}}{\partial T} \partial_{x} T + (ψ_{G} \frac{P_{G}}{x_{ν}} {⟨ρ_{ν}⟩}^{G} - D_{eff} {⟨ρ_{G}⟩}^{G}) \partial_{x} x_{ν} (8) ⟨J_{a}^{x}⟩ = - ψ_{G} \frac{1}{x_{ν}} {⟨ρ_{a}⟩}^{G} \frac{\partial p_{ν}}{\partial η_{L}} \partial_{x} η_{L} - ψ_{G} \frac{1}{x_{ν}} {⟨ρ_{a}⟩}^{G} \frac{\partial p_{ν}}{\partial T} \partial_{x} T + (ψ_{G} \frac{P_{G}}{x_{ν}} {⟨ρ_{a}⟩}^{G} + D_{eff} {⟨ρ_{G}⟩}^{G}) \partial_{x} x_{ν}

The thicknesses of the surface zones are determined by the ratios

{Δ h}_{1} = {\frac{1}{h_{m}} ψ_{G} \frac{p_{ν}}{x_{ν}}}_{{| S}_{0}}

{Δ h}_{2} = \frac{1}{h_{T}} {λ_{eff}}_{{| S}_{0}}

,

where

h_{m}

and

h_{T}

are the constant surface mass and heat exchange coefficients under convective heating conditions,

S_{0}

is a surface of symmetry of the porous plate.

Flow balances: Is defined by equalities for liquid

{⟨J_{L}^{x}⟩}_{{| S}_{1^{-}}} = {J_{L}^{*}}_{{| S}_{1}^{+}}

and components

{⟨J_{ν}^{x}⟩}_{{| S}_{1^{-}}} = {J_{ν}^{*}}_{{| S}_{1}^{+}}

and

{⟨J_{a}^{x}⟩}_{{| S}_{1}^{-}} = {J_{a}^{*}}_{{| S}_{1}^{+}}

of gas mixture .

Boundary conditions: Are defined relative to transport flows of mass on the surface

S_{0}

under additional conditions: 1. There is no liquid flow (

{P_{L}}_{{| S}_{1}} = 0

)

{P_{C |}}_{S_{1}} = {P_{G}}_{{| S}_{1}}

and the transformation

{⟨ρ_{L}⟩}^{L}_{{| S}_{1}} = {⟨ρ_{ν}⟩}^{G}_{{| S}_{1}}

of the liquid into water vapor on the boundary surfaces

S_{1}

,

x = \{(- L {+ Δ h}_{1}) / 2, (L - {Δ h}_{1}) / 2\}

of constant wetting occurs instantly; 2. Heat sources in the near surface areas

x \in [- L / 2, - L / 2 + {Δ h}_{2}] \cup [L / 2 - {Δ h}_{2}, L / 2]

of plate heating are uniform.

Than

J_{L}^{*} = - \frac{h_{T}^{*}}{⟨r⟩} (T_{{| S}_{1}} - T_{| amb}) J_{ν}^{*} = - ⟨φ⟩ (1 - η_{L}) \frac{{⟨ρ_{a}⟩}^{G}}{{⟨ρ_{ν}⟩}^{G}} h_{m}^{*} ({⟨ρ_{ν}⟩}^{G}_{{| S}_{1}} - {⟨ρ_{ν}⟩}^{G}_{| amb}) - \frac{1}{T} {⟨ρ_{a}⟩}^{G} \frac{D_{ν}^{a}}{λ_{G}} \frac{h_{T}^{*}}{⟨r⟩} (T_{{| S}_{1}} - T_{| amb}) + + ⟨φ⟩ (1 - η_{L}) \frac{1}{x_{ν}} {⟨ρ_{a}⟩}^{G} (h_{m}^{*} + \frac{x_{ν}}{1 - x_{ν}} \frac{M_{ν}}{M_{g}} D_{S}^{a}) ({x_{ν}}_{{| S}_{1}} - {x_{ν}}_{| amb}) J_{a}^{*} = - ⟨φ⟩ (1 - η_{s L}) h_{m}^{*} ({⟨ρ_{ν}⟩}^{G}_{{| S}_{1}} - {⟨ρ_{ν}⟩}^{G}_{| amb}) - - ({⟨ρ_{ν}⟩}^{G} \frac{D_{ν}^{a}}{λ_{G}} \frac{1}{T} + 1) \frac{h_{T}^{*}}{⟨r⟩} (T_{| S_{1}} - T_{| amb}) + ⟨φ⟩ (1 - η_{L}) \frac{1}{x_{ν}} {⟨ρ_{ν}⟩}^{G} (h_{m}^{*} - \frac{M_{a}}{M_{g}} D_{S}^{a}) ({x_{ν}}_{{| S}_{1}} - {x_{ν}}_{| amb})

,

here

h_{m}^{*} = [{ψ_{G} \frac{p_{ν}}{x_{ν}}}_{{| S}_{1}} / ψ_{G} {\frac{p_{ν}}{x_{ν}}}_{{| S}_{0}}] h_{m}

and

h_{T}^{*} = [{λ_{eff}}_{{| S}_{1}} / {λ_{eff}}_{{| S}_{0}}] h_{T}

are the variable with time heat and mass transfer coefficients,

is the known Millington-Quirk [36] coefficient of diffusion for dry air component in the pores of material at surface

S_{2}

of evaporation.

Electromagnetic interaction As was described in the paper [37] for a flat plate into the electric field strengths, the wave equation will have the form

here

is the complex refractive index,

{\bar{k}}_{ω}^{'} (\vec{x}, t)

is the wave vector into the porous (inhomogenius) media,

k_{0} = ω \sqrt{μ_{0} ε_{0}} = ω / c_{0}

(where

c_{0} = 1 / \sqrt{μ_{0} ε_{0}}

is the light velocity in vacuum) is the wave vector of electromagnetic irradiation in the vacuum,

ω = 2 π f

a is the corner frequency electromagnetic field (

f

is the linial frequency),

μ_{0}

and

ε_{0}

are the magnetic and electric constants in the vacuun correspondengly.

We are finding the solution of the wave equation as it is specified into the work [37] according to the method of W.K.B. (Wentzel-Kramers-Brillouin) [38] in the form

⟨E_{* y}^{t}⟩ (x) = \frac{1}{\sqrt{{\bar{n}}_{ω}^{eff} (x, t)}} \begin{matrix} (A (t) \sqrt{{\bar{n}}_{ω}^{L} (t)} Exp [- {ik}_{0} \int_{- L / 2}^{x} {\bar{n}}_{ω}^{eff} (x, t) dx] + B (t) \sqrt{{\bar{n}}_{ω}^{R} (t)} Exp [{ik}_{0} \int_{L / 2}^{x} {\bar{n}}_{ω}^{eff} (x, t) dx]) \end{matrix}

where

\bar{n_{ω}^{α}} (t) = \bar{n_{ω}^{'}} (\pm L / 2, t), α = \{L, R\}

are the constant values of the refractive index on the edges of porous plate,

A (t)

and

B (t)

are the unknown functions of time.

Under applying of boundary conditions [37] to the components and derivatives of EMF at the edges of the plate we get the time-varying coefficients

where

T_{α}^{t} = 2 / [1 + {\bar{n}}_{ω}^{α} (t)]

and

R_{α}^{t} = [1 - {\bar{n}}_{ω}^{α} (t)] / [1 + {\bar{n}}_{ω}^{α} (t)]

(here

α = \{L, R\}

is the index, which corresponds to

L

and

R

edged of plate), are respectively the coefficients of transmission and reflection of a plane wave at the boundaries of the plate in the accepted notations.

In the approximation of large plate

L

thicknesses acording to [37] for the power of microwave irradiation into the porous plate we have

⟨P_{x}^{t}⟩ (x) = P_{in} |{\bar{n}}_{ω} (t)| {|T (t)|}^{2} Re [\sqrt{\frac{{\bar{\tilde{n}}}_{ω}^{eff} (x, t)}{{\bar{n}}_{ω}^{eff} (x, t)}}] \times (Exp [- 2 \int_{- L / 2}^{x} α (x, t) dx] + Exp [- 2 \int_{x}^{L / 2} α (x, t) dx])

where

P_{in} = E_{0^{2}} / 2 η_{0}

is the power of external microwave irradiation,

{\bar{n}}_{ω} (t) = {\bar{n}}_{ω}^{eff} (\pm L / 2, t)

and

T (t) = 2 / [1 + {\bar{n}}_{ω} (t)]

are the boundary values of refractive index and transmission coefficient for electromagnetic wave correspondingly.

The poser of heat sources of electromagnetic (microwave) heating is obtained by the relation

{\dot{q}}^{t} (x) = - \partial ⟨P_{x}^{t}⟩

and is gets the following view

\begin{matrix} {\dot{q}}^{t} (x) = 2 P_{in} |{\bar{n}}_{ω} (t)| {|T (t)|}^{2} α (x, t) \sqrt{\frac{1}{2} (1 + \frac{1}{\sqrt{1 + {tg}^{2} δ_{ω} (x, t)}})} \times \\ \times (Exp [- 2 \int_{- L / 2}^{x} α (x, t) dx] + Exp [- 2 \int_{x}^{L / 2} α (x, t) dx]) \end{matrix}

,

where

Re [\sqrt{\frac{{\bar{\tilde{n}}}_{ω}^{eff} (x, t)}{{\bar{n}}_{ω}^{eff} (x, t)}}] = \sqrt{\frac{1}{2} (1 + \frac{1}{\sqrt{1 + {tg}^{2} δ_{ω} (x, t)}})}

is the same equal notation.

Given the ratio

k_{0} {\bar{n}}_{ω}^{eff} (x, t) = β (x, t) - i α (x, t)

, where

α (x, t) = - k_{0} Im [\sqrt{{\bar{n}}_{ω}^{eff} (x, t)}]

and

β (x, t) = k_{0} Re [\sqrt{{\bar{n}}_{ω}^{eff} (x, t)}]

are the indicators of adsorbtion and transmission of plane electromagnetic wave, we obtain the effective quantities of wave length

λ_{ω}^{eff} (x, t)

and penetration depth

D_{p}^{eff} (x, t)

for such wave process

λ_{ω}^{eff} (x, t) = λ_{0} \frac{\sqrt{2}}{\sqrt{{\bar{ε}}_{ω}^{eff (1)}}} \frac{1}{\sqrt{\sqrt{1 + {tg}^{2} δ_{ω} (x, t)} + 1}}

,

D_{p}^{eff} (x, t) = \frac{с_{0}}{ω} \frac{\sqrt{2}}{\sqrt{{\bar{ε}}_{ω}^{eff (2)}}} \frac{1}{\sqrt{\sqrt{1 + {tg}^{2} δ_{ω} (x, t)} - 1}}

under conditions of executing of this

2 α β = k_{0}^{2} \sqrt[3]{{tg}^{2} δ_{ω}} Re [{\bar{ε}}_{ω}^{eff}]

rationed relation, where

tg δ_{ω} (x, t) = {\bar{ε}}_{ω}^{eff (2)} / {\bar{ε}}_{ω}^{eff (1)}

is tangent of dielecric loss angle.

Figure 2. Schematic depiction of porous plate under microwave irradiation.

Thermal stress under moisture loads For a spatially isotropic homogeneous solid phase (framework or skeleton) in a porous medium under the condition of the absence of external force loads, the quasi-static problem of linear thermo-elasticity is formulated in the form of the ratios

{\hat{T}}_{α β, β} + \hat{I} F_{α} = 0

,

F_{α} = - ⟨φ⟩ [\frac{\partial P_{G}}{\partial x_{α}} - η_{L} \frac{\partial P_{C}}{\partial x_{α}}] + ⟨φ⟩ P_{C} \frac{\partial η_{L}}{\partial x_{α}}

,

where

{\hat{T}}^{*} = (1 - ⟨φ⟩) ({\hat{t}}_{S}^{'} - ⟨K⟩ ⟨α_{S}^{T}⟩ (T - T_{0}) \hat{I})

is the leading stress tensor,

F_{α}

is the components of internal (pondermotore) forces due to gas pressure

P_{G}

, capillary pressure

P_{C}

and liquid pore saturation

η_{L}

under averaged constant porosity of body

⟨φ⟩

and

\hat{I}

is diagonal tensor.

In the case of a one-dimensional plate (Figure 2), which is under influence of internal microwave heating according to conditions of compatibility of Saint-Venant deformations [39], the system of equations for the stress component of the tensor has the following form

\frac{\partial^{2} T_{xx}^{*}}{\partial x^{2}} + \frac{\partial F}{\partial x} = 0 \frac{\partial^{2} T_{yy}^{*}}{\partial x^{2}} + \frac{1}{1 - ⟨ν⟩} (⟨E⟩ \frac{\partial^{2} L}{\partial x^{2}} + ⟨ν⟩ \frac{\partial F}{\partial x}) = 0 \frac{\partial^{2} T_{zz}^{*}}{\partial x^{2}} + \frac{1}{1 - ⟨ν⟩} (⟨E⟩ \frac{\partial^{2} L}{\partial x^{2}} + ⟨ν⟩ \frac{\partial F}{\partial x}) = 0

Force loads in the internal volume of the plate are determined as

Φ^{*} (x) = - \int_{0}^{x} F (x) dx = - [Φ (x) - Φ (0)], L^{*} (x) = L (x) - L (0)

, (9)

where

Φ (x) = - \int F (x) dx = ⟨φ⟩ [P_{G} - η_{L} P_{C}]

and

L (x) \equiv - (1 - ⟨φ⟩) ⟨K⟩ ⟨α_{S}^{T}⟩ (T - T_{0})

are the quantity of humidity and thermal stress and their corresponding constant values

Φ (0)

and

L (0)

on the internal surface of symmetry (see Figure 2 is the surface

YZ

at

x = 0

) for the porous plate.

According to the symmetry conditions, the boundary conditions for a plate of finite

L

thickness are applied: 1. There are no internal forces

Φ^{*} (0) = L^{*} (0) = 0

along the plane of symmetry of a plate; 2. The boundary surfaces of the plate are free from the external loads

\int_{- L / 2}^{L / 2} T_{xx}^{*} dx = \int_{- L / 2}^{L / 2} T_{yy}^{*} dx = \int_{- L / 2}^{L / 2} T_{zz}^{*} dx = 0

; 3. The derivative of stresses along a plane of symmetry

\frac{\partial T_{xx}^{*} (0)}{\partial x} = \frac{\partial T_{yy}^{*} (0)}{\partial x} = \frac{\partial T_{zz}^{*} (0)}{\partial x} = 0

is equal to zero.

Then the solution of the equations of the quasi linear thermo-elastic problem under the action of internal (moisture and thermal) loads has the form of stresses

T_{xx}^{*} = Φ^{*} (x) - ⟨Φ^{*} (x)⟩

T_{yy}^{*} (x) = T_{zz}^{*} (x) = \frac{⟨E⟩}{1 - ⟨ν⟩} [M^{*} (x) - ⟨M^{*} (x)⟩]

and deformation

\begin{matrix} E_{yy}^{*} (x) = E_{zz}^{*} (x) = \frac{1}{⟨E⟩} [(1 - ⟨ν⟩) T_{yy}^{*} (x) - ⟨ν⟩ T_{xx}^{*} (x)] + L (x) - L (0), \\ E_{xx}^{*} = \frac{1}{⟨E⟩} [T_{zz}^{*} - 2 ⟨ν⟩ T_{yy}^{*}] + L (x) - L (0), \end{matrix}

correspondingly, where

M^{*} (x) = L^{*} (x) - \frac{⟨ν⟩}{⟨E⟩} Φ^{*} (x)

is an expression for the total internal forces, and

⟨M^{*}⟩ = \frac{1}{L} \int_{- L / 2}^{L / 2} M^{*} (x) dx

and

⟨Φ^{*}⟩ = \frac{1}{L} \int_{- L / 2}^{L / 2} Φ^{*} (x) dx

are the corresponding averaged effort values.

The main mechanical thermo-diffusion equations In such way, we obtain the system of closed equations of heat and moisture diffusion

({ρ C}_{p})_{eff} \partial_{t} T (x, t) - \frac{1}{2} ⟨r⟩ \partial_{t} [{⟨c_{L}⟩}^{L} - \partial_{t} {⟨c_{ν}⟩}^{G}] = \partial_{x} [λ_{eff} \partial_{x} T (x, t)] + + \frac{1}{2} ⟨r⟩ \vec{\nabla} \cdot [⟨J_{L}^{x}⟩ - ⟨J_{ν}^{x}⟩] + {\dot{q}}^{t} (x)

,

\partial_{t} {⟨c_{L}⟩}^{G} + \partial_{x} ⟨{\vec{J}}_{L}⟩ = \dot{I,}

\partial_{t} {⟨c_{ν}⟩}^{G} + \partial_{x} ⟨{\vec{J}}_{ν}⟩ = - \dot{I}

,

\partial_{t} {⟨c_{a}⟩}^{G} = - \partial_{x} ⟨J_{a}^{x}⟩

with the corresponding boundary conditions described above, the solutions of which can be found according to the obtained expression for the heat sources

\begin{matrix} {\dot{q}}^{t} (x) = 2 P_{in} |{\bar{n}}_{ω} (t)| {|T (t)|}^{2} α (x, t) \sqrt{\frac{1}{2} (1 + \frac{1}{\sqrt{1 + {tg}^{2} δ_{ω} (x, t)}})} \times \\ \times (Exp [- 2 \int_{- L / 2}^{x} α (x, t) dx] + Exp [- 2 \int_{x}^{L / 2} α (x, t) dx]) \end{matrix}

and may determine the stress-strain state of the body with values the calculated relative to the plane of symmetry

Φ (x) = - \int F (x) dx = ⟨φ⟩ [P_{G} (x, t) - {η_{L} P}_{C} (x, t)] L (x) \equiv - (1 - ⟨φ⟩) ⟨K⟩ ⟨α_{S}^{T}⟩ [T (x, t) - T_{0}]

humidity and thermal forces loadings.

The closed system of equations is obtained by fulfilling the conditions of weak variability of volume (phase), dielectric (wave) properties of an investigated three-phase porous wetted body

\frac{1}{θ_{σ} (x, t)} \frac{\partial θ_{σ} (x, t)}{\partial t} < < ω_{0}

and

\frac{1}{{\bar{n}}_{ω}^{eff} (x, t)} \frac{\partial {\bar{n}}_{ω}^{eff} (x, t)}{\partial x} < < k_{ω}^{eff} (x, t)

as well as conditions

λ_{ω}^{eff} (x, t) = \frac{2 {π υ}_{ω}^{eff} (x, t)}{ω_{0}} > > l

,

which are determines the possibility of applying the approximation of the effective macroscopic field in the study (determination) of the effective electro-physical properties of the porous body according to the method of local spatial averaging.

Here

k_{ω}^{eff} (x, t) = 2 π / λ_{ω}^{eff} (x, t)

and

υ_{ω}^{eff} (x, t) = c_{0} / {\bar{n}}_{ω}^{eff} (x, t)

wave vector and phase velocity of electromagnetic (TEM) wave propagation in the simulated environment,

{\bar{n}}_{ω}^{eff} (x, t)

is the effective value of refractive index,

θ_{σ} (x, t)

is the volume fraction of

σ

- phase (here

ω_{0}

a is the angle frequencyand 1 is the characteristic (REV) length of averaged volume).

Analysis and Discussion

Numerical simulation of the stress state of a flat unbounded plate under symmetric microwave irradiation was carried out for the selected porous material: historic ceramic brick. The sorption and capillary properties of the studied material are described in detail into the paper [40]. In particular, according to a comparative analysis of semi-empirical models of moisture release or retention, it was established, than at defined averaged porosity

⟨φ⟩ = 0 . 46

the internal permeability of porous skeleton is

k_{in} = 2 . 29 \cdot 10^{- 13} [m^{2}]

and empirical parameter in the model of relative permeabilities for fluid (liquid and gas) phase by van Genuchen [41]

m = 0 . 67

. Also into the work [40] the relative permeability of liguid

k_{rL}

and gas

k_{rG}

phases are defined. Its pointed, that capillary properties of porous material is identically determined by the use of dimensionless

J

- Leverett function , which takes a form

J (η_{L}) = {({[1 / Θ (η_{L})]}^{1 / m} - 1)}^{1 - m}

,

here

Θ = (η_{L} - η_{L}^{ir}) / (η_{L}^{cr} - η_{L}^{ir})

is effective pore saturation by liquid,

η_{L}^{ir} = 0 . 015

and

η_{L}^{cr} = 1

are residual and critical pore saturation by liquid for the mentioned material accordingly.

The beginning or initial equilibrium wetting

η_{L}^{eqv}

of porous material is defined by the know [40] relation

η_{L}^{eqv} = (η_{L}^{cr} - η_{L}^{ir}) / {[1 + {(\frac{P_{amb} \sqrt{k_{in} / ⟨φ⟩}}{σ (T_{amb})})}^{\frac{1}{1 - m}}]}^{m} + η_{L}^{ir}

, (10)

where

P_{amb} = 10325 [Pa]

and

T_{amb} = 293.15 [K]

are pressure and temperature of the vapor like mixture in surrounding equipment accordingly, and

σ

is coefficient of surface tension.

At calculation of matrix elements in the generalized form of heat and mass equation according to known expressions (8) for phase and component flows the following values of molar mass

M_{a} = 0 . 029 [к mol / kg]

for dry air and

M_{ν} = 0 . 018 [к mol / kg]

water vapor in two component gas mixture are used. Than the expression for density

{⟨ρ_{G}⟩}^{G}

of gas mixture in the approximation of light solution

P_{G} = p_{ν} / x_{ν}

has been view

where

p_{ν} = ϕ p_{ν s} (T)

a is partial pressure of unsuturatedwater vapor,

ϕ

a is relative humidity of air,

p_{ν s}

is the equilibrium pressure of saturated water vapor, for this value a temperature approximation [42] is satisfied

\begin{matrix} p_{ν s} (T) = E x p [\begin{matrix} - 5800 . 2206 / T + 1.3914993 - 4.8640239 \cdot 10^{- 2} T + + 4.1764768 \cdot 10^{- 5} T^{2} - \\ - 1.4452093 \cdot 10^{- 8} T^{3} + 6.5459673 L n (T) \end{matrix}], \end{matrix}

here

T

is termodynamic temperature.

Because relative humidity

ϕ

is defined according to the known for investigated material dependence of capilary pressure

p_{c} (η_{L}, T) = σ (T) \sqrt{⟨φ⟩ / k_{in}} J (η_{L})

from temperature

T

and pore saturation

η_{L}

by the liquid, in such case the eqvilibrium (initial) value of water vapor molar fraction

x_{ν}^{eqv}

in wetted porous material is

where

is the initial value of relative humidity,

η_{L}^{eqv}

is the corresponding equilibrium (beginning) value of wetting, which has been defined according to relation (10) above.

Effective thermal characteristics of the composite material have been calculated according to known [43,44] experimental propertied of phases, as it depicted in the Table 1.

For the defined effective

{\bar{ε}}_{ω}^{eff}

electro physical properties (11) in the investigated material, including dielectric loss tangent

tg δ_{ω}

, the numerical calculations were performed in the approximation of the effective medium (EMA) [45], according to the relation

\sum_{σ} θ_{σ} \frac{ε_{c}^{σ} (ω) - {\bar{ε}}_{ω}^{eff}}{ε_{c}^{σ} (ω) + 2 {\bar{ε}}_{ω}^{eff}} = 0

. (11)

The last cubical equation is solved relatively to effective dielectric constant of composite

{\bar{ε}}_{ω}^{eff}

, were coefficients are defined by known dielectric constant

ε_{c}^{σ} (ω)

and volume fraction

θ_{σ}

of porous media component with usage of experimental data, as it is depicted into Table 2.

At calculation it is taken into account, that density of solid skeleton

{⟨ρ_{S}⟩}^{S} = 1500 [к g / m^{3}]

and incompressible liquid

{⟨ρ_{L}⟩}^{L} = 1000 [к g / m^{3}]

.

A surface tension coefficient

σ

and latent heat (enthalpy) of the vaporization

⟨r⟩

[46] has been reviewed in the form of approximated temperature dependencies

σ (T) = 0 . 121978 - 0 . 0001683 T

and

where

T_{cr 1} = 647. 3 [K]

is the critical temperature (triple point).

The dynamical viscosity of binary gas mixture has been modeled according to Wilke [47] theory, when for clear components of water vapor

μ_{ν}

and dry air

μ_{a}

it is selected the following [48] numerical approхimations

μ_{ν} (T) = 8 . 85 \cdot 10^{- 6} + 3 . 53 \cdot 10^{- 8} (T - T_{cr 2})

,

here

T_{cr 2} = 273.15 [K]

is the fixed (critical) temperature of water freezing, also

where

μ_{L}

a is the dynamical viscosity approximation according to data [49].

Diffusion properties of components of vapor-air mixture in the pores have been defined with usage of introduced by the extended through the author of research the diffusion coefficient

D_{G}^{eff} = \frac{1}{τ} \frac{⟨φ⟩ (1 - η_{L})}{x_{ν} - ⟨φ⟩ (1 - η_{L}) (x_{ν} - 1)} D_{ν}^{a}

,

where

D_{ν}^{a} = 2 . 29 \cdot 10^{- 13} [m / s]

is the diffusion coefficient of water vapor into the dry air.

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

S_{1}

and

S_{2}

is obtained due to distribution of dimensiomless sickness

{Δ h}_{1} / L

and

{Δ h}_{2} / L

as it depicted on the (Figure 3) and (Figure 4) relatively to surface

S_{0}

of plate symmetry.

Conclusion I: 1. Its fixed a critical value of time

t_{cr}

there is a point of the crossaver surfaces

S_{1}

and

S_{2}

about that the surface mass transfer coefficient

h_{m}^{*}

(Figure 5) increase nonlinially. It is joined with beginning of processes of intensity liquid evaporation at the near surface wetted zone;

2. It's also not significal nonlinear behaviour of surface heat transfer coefficient is delivered

h_{T}^{*}

(Figure 5) along the surface

S_{1}

, which is joined with deviation of averaged liquid pore saturation

η_{L}

from equvilibrium values at surface of symmetry

S_{0}

of plate.

Figure 5. The surface emission coefficient for heat and mass transfer.

Figure 6. The values of internal and surface diffusion coefficients at the evaporation surface.

Conclusion II: 1. In the neighborhood of time value

t_{eqv}

an equilibrium state of two component mixture of water vapor and dry air is fixed (

x_{ν} = x_{a} ≃ 1 / 2

), abouve it the dry air is gradually displaced through the surface

S_{1}

of wetted and molar fraction of water vapor

x_{v}

in critical point of time

t_{cr}

follows to maximum value

x_{ν} = 1

of saturation. 2. Diffusivity

D_{G}^{eff} / D_{ν}^{a}

of gas mixture in pores of material for the internal volume of plate and at the evaporation surface are defined in various way, but over time in the region

t \geq t_{cr}

the calculated values ( at

τ = 4

a this model tortosity factor) of difusion coefficiens on the surface

S_{1}

is aligned.

Solutions of modified system of heat and mass balance for material of historical ceramic brick are represented in the form of graphical dependencies as its showing below, here

L^{*} = n^{*} Δ L / L_{0} (n^{*} = Δ L / L_{0})

is dimensionless half thicknes of plate.

Figure 7. The distribution of the pore saturation by liquid.

Figure 8. The distribution of the temperature fiel.

Figure 9. The distribution of the molar fraction for water vapor.

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 its 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.

Figure 10. Distribution of the liquid flow.

Figure 11. Distribution of the water vapor flow.

Figure 12. Distribution of the dry air flow.

Figure 13. Distribution of an internal evaporation sources.

Conclusion IV: In the neighborhood of time point

t_{b}

, which corresponts to achievement of boiling temperature at normal conditions its fixed chane of flows distributionfor 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

{\dot{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.

Figure 14. Distribution of stress in longitudinal direction.

Figure 15. Distribution of stress in tangential direction.

Figure 16. Distribution of strains in longitudinal direction.

Figure 17. Distribution of strains in tangential direction.

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 19. 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 Figure 7) especially near surface of sample.

Acknowledgment

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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Table 1. The thermal properties of phases in the wetted porous material.

Phases	Solid ( $S$ )	Liquit ( $L$ )	Gas ( $G$ )
Thermal conductivity $[W m^{- 1} K^{- 1}]$	0.55	0.65	0.025
Thermal capacity $[J {kg}^{- 1} K^{- 1}]$	810	4180	1006

Table 2. Dielectric constants of components for investigation material according to EMA approximation (

ε_{S}

-solid phase,

ε_{L}

-water,

ε_{G}

-air.

Table 2. Dielectric constants of components for investigation material according to EMA approximation (

ε_{S}

-solid phase,

ε_{L}

-water,

ε_{G}

-air.

v, [Gz]	ε_S		ε_L		ε_G
v, [Gz]	ε_S′	ε_S″	ε_L′	ε_L″	ε_G′	ε_G″
2.45·10⁹	5,86	0,703	80	20	1	0

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

Abstract

Keywords:

Subject:

Introduction

Model and Analysis

Comparing to Convective Drying we can Poit to Advantage of Microwave or Dilectric Drying

Analysis and Discussion

Conclusions

Acknowledgment

References

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