TABLE OF CONTENTS

1.INTRODUCTION _________________________________________________________________ 1

1.1. Dielectric Heating ___________________________________________________________ 2

1.2.Fundamentals of Radio Frequency and Microwave Heating __________________________ 3

1.2.1.Electromagnetic waves ____________________________________________________ 3

1.2.2.Dielectric properties ________________________________________________________ 5

1.2.3.Heating mechanism _______________________________________________________ 7

1.2.3.a) Ionic conduction _____________________________________________________ 7

1.2.3. b) Dipolar rotation _____________________________________________________ 8

1.2.4.Interaction of electromagnetic fields with materials _____________________________ 9

1.2.5.Parameters affecting the heating mechanism _________________________________ 10

1.3.Radio frequency heating characteristics _________________________________________ 14

1.3.1.Equipment: ____________________________________________________________ 14

1.3.2.Industrial Applications: ___________________________________________________ 14

1.3.2.a) Food Processing ____________________________________________________ 14

1.3.2.b) Various Material Processing ___________________________________________ 15

1.3.3.Advantages of Radio Frequency Heating _____________________________________ 15

1.4.Microwave heating characteristics _____________________________________________ 17

1.4.1.Equipment: ____________________________________________________________ 17

1.4.2.Industrial Applications: ___________________________________________________ 17

1.4.2.a) Food Processing ____________________________________________________ 17

1.4.2.b) Various Materials Processing __________________________________________ 19

1.4.3.Advantages of microwave heating __________________________________________ 19

1.4.4.Disadvantages of Microwave Heating ________________________________________ 20

1.5.Studies on Modeling of Microwave Heating _____________________________________ 21

1.6.Studies on Modeling of Radio Frequency Heating _________________________________ 27

2.MATERIALS AND METHODS ______________________________________________________ 31

2.1.Materials and Methods For Microwave Heating __________________________________ 31

2.2.Materials and Methods for Radio Frequency Heating ______________________________ 34

2.3.Analytical Solution __________________________________________________________ 37

2.3.1.Analytical Solution for Microwave Heating____________________________________ 37

2.3.2.Analytical Solution for Radio Frequency Heating _______________________________ 38

2.4.Numerical Solution _________________________________________________________ 40

2.4.1.Numerical Solution for Microwave Heating ___________________________________ 40

2.4.2.Numerical Solution for Radio Frequency Heating _______________________________ 43

3.RESULTS AND DISCUSSIONS ______________________________________________________ 46

4. REFERENCES __________________________________________________________________ 51

5. APPENDIXES __________________________________________________________________ 52

5.1. Microwave model computer program codes_____________________________________ 52

5.2. Radio frequency model computer program codes ________________________________ 57

5.3.List of Symbols _____________________________________________________________ 60

1.INTRODUCTION

Radio frequency and microwave heating appear to many engineers that these are new forms of

heating however in fact, practical applications began during World War II and the home

microwave oven was invented shortly after the war in 9 October 1945 (Schiffmann, 1995). The

first oven prototypes were placed in laboratories and kitchens throughout the United States to

develop microwave cooking technology. Consequently, microwave processing began on a

commercial scale in the early 1960s (Decareau and Peterson, 1986). However, these remained in

small industries. Generally, radio frequency heating is used in many industries, including plastics,

wood, ceramics, furniture, textiles, food and paper, where microwave heating is used both in

industrial and home applications.

In the past few years, there has been an increase of interest in the applications of microwave

and radio frequency heating for industrial purposes. This is due to the unique heating mechanisms

of microwaves and radio frequency, which permit energy savings in many instances, as well as

providing other benefits such as uniform heating, reduced production time, selective heating, etc.

1.1. Dielectric Heating

The term dielectric heating covers all electromagnetic frequencies, however, it is generally

accepted that radio frequency

heating is done at frequencies

between 1 and 100 MHz,

whereas microwave heating

occurs between 300 MHz and 300

GHz. The electromagnetic

spectrum is illustrated in Figure-1.

The microwave region

extends from the UHF band into

the sub-millimeter range

(Decareau and Peterson, 1986).

Frequency bands and ranges

allocated for Industrial, Scientific,

and Medical (ISM) and the

permitted countries are listed in

Table-1.

Figure-1 Electromagnetic spectrum

These frequency allocations are made by the International Telecommunication Union (ITU) and

some frequencies are specific to certain countries. Practical heating applications are generally

done at 13.56, 27.12, and 40.68 MHz in radio frequency heating, whereas 896, 915, and 2450

MHz in microwave heating.

Table-1 Frequencies Designated by the International Telecommunication Union for Use as

Fundamental Industrial, Scientific, and Medical Frequencies (Decareau and Peterson, 1986; Metaxas,

1991; Schiffmann, 1995; Datta and Davidson, 2000)

Frequency Generally

Used

(MHz)

Frequency range (MHz)

Permitted Countries

0,07 0,06-0,08 Russia

6.780

6.765-6.795

13.560

13.553-13.567

United Kingdom

27.120

26.957-27

Worldwide

40.680

40.66-40.70

Worldwide

433.920

433.05-434.79

Austria, Holland, Portugal, Yugoslavia

896.000 886-906 United Kingdom

915.000

902-928

North and South America

2375 2325-2425 Albania, Bulgaria, Hungary,

Romania, Czech Republic, Russia

2450

2400-2500

Worldwide except where 2375 MHz is used

3390 3369,66-3410,34 Holland

5800

5725-5875

Worldwide

6780 6739,32-6820,68 Holland

24150

24,025-24,275

Worldwide

40680 United Kingdom

61250

61,000-61,500

Although the basic principles of heating with radio and microwave frequencies are similar, the

methods of heat generation and the equipments are different.

1.2.Fundamentals of Radio Frequency and Microwave Heating

1.2.1.Electromagnetic waves

Light, x-irradiation, TV, AM and FM radio waves, ultraviolet, infrared, and microwaves are

some of the common waves. All bodies in the universe, above absolute zero temperature, emit

electromagnetic waves. The relationship of these waves is found in the electromagnetic

spectrum (Figure-1). All electromagnetic waves are characterized by their wavelength and

frequency, and an illustration of a plane monochromatic electromagnetic wave is shown in

Figure-2. It can be seen that an electromagnetic wave is a blend of an electric component E and

a magnetic component H. E and H are perpendicular to each other and both are perpendicular

to the direction of travel. This is what makes this a "plane" wave. The field strength at any

point may be represented by a sine or cosine function, which is what makes it

"monochromatic." Further, it is "linearly polarized" since the electric and magnetic field vectors

E and H lie in one direction only. The plane of polarization is YX for the E vector and ZX for the H

vector. (Schiffmann, 1995)

Figure-2: Diagrammatic illustration of a plane electromagnetic wave. E and H represent the electrical and magnetic

components of the wave; E0 and H0 are their respective amplitudes. (Schiffmann, 1995)

The wave is traveling in the X direction at the velocity C, which is equivalent to the speed of light

in air or vacuum, but slows as it passes through another medium. The distance λ, which is the

wavelength, has a relationship with the frequency of the wave and the velocity of propagation,

where f is how many times the wave goes through a full cycle per second;

λ

pVf = (Hz) (Schiffmann, 1995) (1)

As an electromagnetic wave passes through a material, its frequency remains the same;

therefore, its wavelength changes and this affects the depth of penetration. Penetration

depth is defined as the distance from the surface of a dielectric material at which the

incident power drops to 1/e. (Decareau and Peterson, 1986; Metaxas, 1991; Datta, 2001)

An electromagnetic wave is an energy wave that changes its energy content and amplitude as it

travels through a medium, as seen by a change in the amplitude of the wave. If the E component

is traced it is seen that at some point it is zero; then it builds up to a maximum value, decays

to zero, and again builds up to a maximum value with the opposite polarity before again

decaying to zero. The same thing happens to the H component. The amplitude of the wave at

any point along the X-axis represents the electrical (E) or magnetic (H) field strength, which

are measured as volts or amperes per unit distance, respectively. It is this periodic flip-flopping

of the wave's polarity and its decay through zero that causes the stress upon ions, atoms, and

molecules, which is converted to heat, and the greater the field strength, the greater will be the

whole effect. (Schiffmann, 1995)

1.2.2.Dielectric properties

Dielectric properties of materials are explained in similar ways with same equations by different

researchers. (Tinga and Nelson, 1973; Metaxas and Driscoll, 1974; Decareau and Peterson, 1986;

Schiffmann, 1995)

A dielectric material is defined by real and imaginary components; (Decareau and Peterson, 1986)

'''

* εεε j−= (F/m) (2)

where ε* is the complex permittivity (F/m), ε’ is the real part, directly influences the

amount of energy that can be stored in a material in the form of electric fields, whereas ε’’,

the imaginary part, also called the loss factor is a direct measure of how much energy a

material can dissipate in the form of heat. 1−=j indicates a 90º phase shift between the

real (ε’) and imaginary (ε’’) parts of the complex dielectric constant.

When ε is normalized with respect to the dielectric permittivity of free space ε0=10-9

/36π

farads/meter.

The complex relative permittivity;

'''0

jkkk −==ε

ε (3)

where, k’ is the relative permittivity or relative dielectric constant generally referred to as

the dielectric constant in the literature and tables.

The loss tangent tanδ, also called the dissipation factor, D represents the energy loss

characteristic of the material by the following equation;

'

''tan

k

kD ==δ (4)

where, k’’ is the relative loss factor also referred to simply as the loss factor and is the

product of the dielectric constant and the loss tangent. Eqn. (4) can be written in another

form; (Copson, 1962)

δtan'" kk = (5)

1.2.3.Heating mechanism

Radio frequencies and microwaves are not forms of heat, but forms of energy that are exhibited

as heat through their interaction with materials. It is as if they cause materials to heat

themselves. There are many mechanisms for this energy conversion; some of them are listed in

Table-2.

Table-2 List of Energy Conversion Mechanisms (Schiffmann, 1995)

Ionic Conduction (Ionic Polarization)

Dipole Rotation

Interface Polarization

Dipole Stretching

Ferroelectric Hysteresis

Electric Domain Wall Resonance

Electrostriction

Piezoelectricity

Nuclear Magnetic Resonance

Ferromagnetic Resonance

Ferrimagnetic Resonance

The ionic conduction and dipole rotation are primarily interested in dielectric heating

phenomena. The other mechanisms are not significant when heating is considered.

1.2.3.a) Ionic conduction

Since ions are charged units, they are accelerated by electric fields. In a solution, all of the ions

move in the direction opposite to their own polarity by the electric field. In so doing they collide

with un-ionized water molecules, giving up kinetic energy and causing them to accelerate and

collide with other water molecules. When the polarity changes, the ions accelerate in the

opposite manner. Since this occurs many millions of times per second, large numbers of

collisions and transfers of energy occur. Therefore, there is a two-step energy conversion: electric

field energy is converted to induced ordered kinetic energy, which in turn is converted to

disordered kinetic energy, at which point it may be regarded as heat. This type of heating is

not dependent largely upon either temperature or frequency (Schiffmann, 1995).

The power developed per unit volume (Pv) through ionic conduction is shown as;

µqnEPv

2= (W/m3) (Schiffmann, 1995) (6)

where,

E= electric field (V/m)

q= amount of electrical charge on each of the ions (Coulombs).

n= ion density (number of ions per unit volume).

μ= level of mobility of the ions.

1.2.3. b) Dipolar rotation

Many molecules, such as water, are dipolar in nature; that is, they possess an asymmetric charge

center. Water is typical of such a molecule. Other molecules may become "induced dipoles"

because of the stresses caused by the electric field. Dipoles are influenced by the rapidly

changing polarity of the electric field. Although they are normally randomly oriented, the

electric field attempts to pull them into alignment. However, as the field decays to zero

(relaxes), the dipoles return to their random orientation only to be pulled toward alignment

again as the electric field builds up to its opposite polarity. This buildup and decay of the field,

occurring at a frequency of many millions of times per second, causes the dipoles similarly to

align and relax millions of times per second. This causes an energy conversion from electric field

energy to stored potential energy in the material and then to stored random kinetic or thermal

energy in the material. This temperature and molecular size dependent time for buildup and

decay, defines a frequency known as the relaxation frequency. For small molecules, such as

water and monomers, the relaxation frequency is already higher than the microwave frequency

and rises further as the temperature increases, causing a slowing of energy conversion. On the

other hand, large molecules, such as polymers, have a relaxation frequency at room temperature

that is much lower than the microwave frequency but that increases and approaches it as the

temperature rises, resulting in better energy conversion into heat. This may lead to runaway

heating in materials that at room temperature are very transparent to the microwave field. This

must be superimposed upon the fact that such liquids as water and monomers are better

absorbers of microwave energy than polymers. The power formula for dipolar rotation is shown

as;

"2 εfkEPv = (W/m

3) (Schiffmann, 1995) (7)

where

k=constant dependent upon the units of measurement used

E=electric field strength (V/m)

f=frequency (Hz)

ε”=loss factor (dimensionless)

As frequency is changed it is necessary to increase the electric field strength E in order to

maintain a particular power level of Pv. Since radio frequency heating frequencies are much

lower than microwave frequencies, hence the field strengths should be much higher for

comparable power output in a radio frequency system (Schiffmann, 1995).

1.2.4.Interaction of electromagnetic fields with materials

The materials are divided by the way that they interact with electromagnetic fields into four

categories:

Conductors: Materials with free electrons, such as metals, reflect electromagnetic waves.

These materials are used to direct electromagnetic waves in the form of applicators and

waveguides (Schiffmann, 1995).

Insulators: Electrically nonconductive materials, such as glass, ceramics, and air, act as insulators,

which reflect and absorb electromagnetic waves to a negligible extent and primarily transmit

them (that is, they are transparent to the waves). They are therefore useful to support or

contain materials to be heated by the electromagnetic field and may take the form of conveyor

belts, support trays, dishes, or others. These materials may also be considered "nonlossy

dielectrics" (Schiffmann, 1995).

Dielectrics: These are materials with properties that range from conductors to insulators. There

is within this broad class of materials a group referred to as "lossy dielectrics," and it is this

group that absorbs electromagnetic energy and converts it to heat. Examples of lossy dielectrics

are water, oils, wood, food, and other materials containing moisture, and the like (Schiffmann,

1995).

Magnetic compounds: These are materials, such as ferrites, that interact with the magnetic

component of the electromagnetic wave and as such will heat. They are often used as

shielding or choking devices that prevent leakage of electromagnetic energy. They may also be

used for heating in special devices (Schiffmann, 1995).

1.2.5.Parameters affecting the heating mechanism

Heating mechanism is affected by various parameters, such as;

Moisture content: The amount of free moisture in a substance greatly affects its dielectric

constant since water has a high dielectric constant. Thus, with a larger percentage of water

the dielectric constant generally increases, usually proportionally. It should be emphasized that

very complex phenomena occur when different dielectrics are mixed. Various materials,

including alcohols and some organic solvents, also exhibit dielectric properties that make them

suiTable for heating with microwave and radio frequency energy and, so, behave similarly to

water (Schiffmann, 1995).

Density: The dielectric constant of air is 1.0, and it is, for all practical purposes, transparent to

electromagnetic waves at industrial frequencies. Therefore, its inclusion in materials reduces the

dielectric constants, and as density decreases so do the dielectric properties, and heating is

reduced (Schiffmann, 1995).

Temperature: The temperature dependence of a dielectric constant is quite complex, and it may

increase or decrease with temperature depending upon the material. In general, the values of

ε’ and ε” increase significantly with temperature as the food materials thaws, but as the

temperature rises further they tend to show a gradual decrease. However, some low moisture

content materials exhibit increases in ε’ and ε” with temperature due to dominant bound

water relaxation (Calay et al., 1995).

Frequency: Dielectric properties are affected by the frequency of the applied electromagnetic

field. The value of ε” for pure water decreases with falling frequency at the microwave

frequencies of interest in food processing, but for food materials ε” tends to increase with

decreasing frequency. Higher conduction losses at lower frequencies have a more pronounced

effect on the loss factor in foods with high salt contents (Calay et. al, 1995).

Conductivity: Conductivity refers to the ability of a material to conduct electric currents by the

displacement of electrons and ions. These charged units can have a major effect in heating,

and in a moisture loss situation in which the ion concentration increases as the water is

removed, this effect can be very complex (Schiffmann, 1995).

Thermal conductivity: Thermal conductivity often plays a lesser role in microwave and radio

frequency heating than in conventional heating because of the great speed with which the

former heat thus reducing the time in which thermal conductivity can be effective. There are

cases, however, in which it has a major role. For example, when penetration depth of the

electromagnetic energy is small in comparison with the volume being heated, thermal

conductivity may be depended upon to transfer the heat to the interior. Another important

case is to even out the nonuniformities of heating that may occur with electromagnetic fields.

Sometimes the microwave or radio frequency power is pulsed on and off to allow for this

evening out of temperature, as in microwave thawing (Schiffmann, 1995).

Specific heat: The specific heat parameter is often neglected while dealing with

electromagnetic heating and paying attention only upon the dielectric properties. However,

specific heat can have profound effects and may, in fact, be the overriding parameter, causing

materials to heat much faster than considering only dielectric properties (Schiffmann, 1995).

Penetration depth: Although not a property of a material but rather a result of its various

properties, penetration depth is of greatest importance. Since electromagnetic heating is bulk

heating, it is important that the energy penetrates as deeply as possible. If it does not, then the

heating is limited to the surface. The formula of attenuation factor is shown as;

( ) 21

2

0

1tan12

'2

−+= δ

λ

πα

k (1/m) (Datta, 2001) (8)

where α is the attenuation factor, 'k is the relative dielectric constant, 0λ is the wavelength

in free space.

The formula for the attenuation of the power is shown as;

dePP

α2

0

−= (W/m3) (Zhou et al., 1994) (9)

where, P is the power density at concerned point (W/m3), P0 is the surface power density and

d is the distance from the slab surface.

As P is proportional to E2, the power dissipated per unit volume decays as the energy traverses

the semi-infinite dielectric slab. (Metaxas, 1991).

Penetration Depth formula is shown as;

21

2

0

1tan1'

2

2

−+=

δπ

λ

kD (m) (Datta, 2001) (10)

where

α=attenuation factor (1/m), (1/α=D)

D= penetration depth at which the available power in the material has dropped to 1/e of its

value at the surface. (m)

λ0=free space wavelength (m)

k’=relative dielectric constant

tanδ=loss factor

This equation is reasonably accurate for most foods even though many have relatively high

k” values (Schiffmann, 1995).

Fat and sugar content: The absorption of electromagnetic energy by fats, in the absence of

moisture, is quite different from, presence of free water and ionic activity. Although the

actual mechanism that happens in such materials is not yet well understood, the basis for

the interactions is thought to result from the rotational modes of the molecules and is

related to the permanent and induced dipole moments. In general, ε’ and ε” values decrease

as the fat content increases (Desai, 1991).

Some data are available for sugars, such as glucose and sucrose, in solutions of various

concentrations (Kent, 1987). Because sugar molecules are relatively large and non-polar,

they do not dissociate on dissolving, and thereby inhibit orientation polarization. Thus, an

increase in the sugar content of a food decreases the dielectric constant, whereas the loss

factor increases due to the reduced free water relaxation frequency (Benson, 1963).

1.3.Radio frequency heating characteristics

1.3.1.Equipment:

In Radio frequency heating, the

material to be heated is often

placed between two parallel plates

(electrodes) where it becomes the

dielectric of a capacitor as shown in

Figure-3.

In general, the Radio frequency is

Figure-3: A simple radio frequency Applicator established in a tank circuit, which

is a resonant circuit predominantly made up of the distributive capacitance and inductance

of a metallic enclosure, although lumped elements are also used. The energy is then

transferred to an applicator, which contains the material to be processed.

1.3.2.Industrial Applications:

1.3.2.a) Food Processing

The unique processing benefits achievable with radio frequency technology have been used

in many food applications for many years, of which are listed below:

• Softening: There are number of food ingredients and products which can benefit

from softening either as a part of the main process or for recovery of trim or other

in-process waste. The ingredients include a range of fats such as shortening for

pastry making, cocoa butter and its substitutes.

• Cooking and blanching: Radio frequency heating is useful for a range of continuous

cooking operations. Their volumetric heating results in improved product quality.

The range of materials, which can be cooked, include meats and reformed meats,

pies and pizzas, vegeTables and potato products.

• Pasteurization and sterilization: Radio frequency heating offer significant

opportunities in extending the period between production runs of particular dishes

without compromising the shelf life.

• Defrosting: By using radio frequency heating to defrost frozen foods lowers the

possibility of bacterial contamination.

• Baking and post Baking: The process time decreases and separate control of surface

is provided.

1.3.2.b) Various Material Processing

There are many applications for radio frequency heating. Recent advances in radio

frequency heating equipment have resulted in the development of innovative radio

frequency applications. Some of the many application areas for radio frequency heating are

listed below:

• Textiles: Drying of yarn packages, webs and fabrics

• Ceramic manufacturing: Drying ceramic parts or molds prior to firing.

• Pharmaceutical: Moisture removal in Tablet and capsule production processes

• Woodworking: Drying, gluing, restoration of wood

• Polymers: Welding, thermoforming and curing

• Glass Fiber Drying: Moisture removal in roving and bale with no over heating

• Drying of materials: Paper, book, tobacco and paint drying.

1.3.3.Advantages of Radio Frequency Heating

The radio frequency technology offers substantial benefits over conventional technologies

such as:

• Reduced production time: The speed of the heating is often substantially reduced

from that using other processes and the checking time is reduced.

• Competitive energy costs: In situations where small amounts of moisture are needed

to be removed radio frequency technology is extremely cost-effective.

• Reduced labor costs: Radio frequency heating can enable a streamlining of the

manufacturing process that may eliminate cumbersome batch heating methods. A

heating process that involved several people can often be accomplished with radio

frequency heating that is simply part of an inline continuous feed process.

• Floor space savings: Efficient heat transfer results in faster product transfer and

reduced oven length.

• Uniformity of product and product quality: Radio frequency is able to bring moisture

uniformity to many products, including food products, where it can provide added

crispness or make the moisture content consistent throughout the food. In drying

processes, it prevents overheating of dryer, exterior portions of bulk materials that

results in inconsistencies (variations in color or durability) in the overall product.

• Moisture elimination for heat sensitive products: Because radio frequency only

causes heating in areas where moisture exists, none of the product is excessively

overheated. This is especially important in dealing with heat sensitive compounds,

such as drug compounds, which can suffer degradation, or conversion when over

heating occurs.

• Preservation of surrounding materials: The material that needs to be cured or dried

in composition products can be targeted by using radio frequency, while surrounding

pieces or layers remain unaffected.

• Prevention of pollution: In radio frequency processing any by products or

combustion are formed.

1.4.Microwave heating characteristics

1.4.1.Equipment:

In a microwave heating equipment,

microwaves are generated at the required

frequency band in the power unit (Metaxas,

1991). Magnetron or klystron tubes are

used primarily to generate the microwave

power. Power is normally launched from the

microwave tube into a transmission line,

where it travels along the line as guided

waves

to a load or a microwave heating applicator.

Several types of transmission Figure-4: A typical microwave heating system lines are used for

microwaves, including coaxial line, hollow metallic waveguide, dielectric waveguide and strip

line. A typical microwave heating system is shown in Figure-4. There are three types of

transmission line modes: the principle transverse electromagnetic mode (TEM), the

transverse electric mode (TE), and the transverse magnetic mode (TE) (Decareau and

Peterson, 1986). The material is subjected to intense microwave fields in applicator and any

additional secondary process equipment such as pumps for operation under moderate

vacuum conditions, steam or hot air injection, must be connected to applicator. Applicators

are used to transfer energy from the microwave source to the load to be heated. Mode

stirrers are used to distribute the microwave energy among the different resonant modes of

the cavity, ensuring a homogeneous heating. Applicators may be classified into four

categories: the resonant cavity, the traveling wave applicators, the slow wave applicators

and free space applicators. Finally, a control circuitry is used to optimize and regulate the

overall performance of the microwave equipment.

1.4.2.Industrial Applications:

1.4.2.a) Food Processing

Food processes are successful applications of microwaves and are very widely used in many

areas such as;

• Blanching: Microwave heating is used to inactivate enzymes in fresh vegeTables and

fruit that lead to premature food spoilage at freezing temperatures.

• Cooking: Microwave heating is used for precooking bacon, meat patties and poultry

parts. Microwave heating is combined in sequence with resistance heating to brown

the surface of meat and poultry products where microwave energy is used to

complete the cooking process

• Baking: Microwave heating is often combined with a conventional baking process

either simultaneously or consecutively to yield effective baking and browning of the

surface of bread and doughnuts.

• Drying: Usually, drying is applied in combination with conventional heating and

microwave heating, to dry products like pasta, onions, rice cakes, seaweed, snack

foods and egg yolk. Combination of vacuum in order to dry heat sensitive materials

at lower temperatures are applied to products like citrus juices, grains and seeds.

• Tempering: The process of the raising the temperature of solidly frozen foods to a

temperature just below the freezing point of water is often done instead of

complete thawing. The microwave tempering process minimizes microbial growth

and spoilage in regions closer to the surface of products like meat, fish, butter and

berries.

• Freeze-drying: Microwave energy is applied directly to the ice core of the product

and is potentially used to reduce the process time of freeze-drying of foods like

meat, vegeTables and fruits.

• Pasteurization and sterilization: Microwaves are used to improve quality by means

of a substantial reduction in process time of products like bread, yogurt, milk and

prepared foods.

• Roasting: Microwaves are used to reduce the process time in roasting cocoa beans,

nuts and coffee beans. It results a slight gain in yield, much less smoke and cooler

equipment.

• Rendering: Microwaves are used in rendering of lard or tallow, as the product

comes out of the system, it requires little or no finishing before shipping.

1.4.2.b) Various Materials Processing

Microwaves are not only generally used in food processing; also, they are used successfully

in various materials processing such as;

• Woodworking: Reducing the drying time.

• Laboratory analysis: Several microwave systems have been developed to determine

solid and moisture content.

• Pharmaceutical: Microwave vacuum drying is used for the manufacture of Tablet

granulations.

• Industrial Coating: Microwave drying systems are used for drying coatings on

plastics and paper.

• Ceramics: Microwaves are used to reduce the initial and final drying time in the

mold after which the piece is glazed and fired.

• Foundry industry: Microwaves are used in drying and polymerizing the sand molds

used for casting.

1.4.3.Advantages of microwave heating

Microwave energy is a very convenient and versatile source of heat, which has many

advantages over conventional heating systems, such as;

• Uniform heating: Bulk heating effect produces uniform heating, avoiding large

temperature gradients that occur in conventional heating systems.

• Efficiency of energy conversion: The energy couples directly to the material being

heated. It is not expended in heating the air, walls of the oven, conveyor or other

parts. This leads significant energy savings.

• Better and rapid process control: The instantaneous on-off nature of the heating

and the ability to change the degree of heating by controlling the output power of

the generator leads fast, efficient and accurate control of heating.

• Floor space requirements: Due to the rapid heating, fewer floors are required.

• Selective heating: The electromagnetic field generally couples into the solvent, not the

substrate. Hence, it is the moisture that is heated and removed, whereas the carrier or

substrate is heated primarily by conduction. This also avoids heating of the air, oven

walls, conveyor, or other parts.

• Improved Product quality: Over heating of the surface and case hardening which are

common with conventional heating methods are eliminated. This leads to less rejected

products.

• Desirable chemical and physical effects: Many chemical and physical reactions are

promoted by the heat generated by microwaves, which leads to puffing, drying, melting,

protein denaturation and starch gelatinization.

1.4.4.Disadvantages of Microwave Heating

The advantages mentioned are to some extent compromised by several disadvantages,

which should never be overlooked, such as;

• High capital costs: Magnetrons and other electron tubes are more expensive than

devices used in conventional heating

• Inability to grill surfaces: As the microwave heating affects the whole volume of the

product, grilled surfaces, which are accomplished with conventional heating, is not

achieved.

• Durability: It is difficult to repair microwave devices or skilled personals are needed

for this purpose.

• Shielding: The equipment is needed to be shielded very carefully, in particular the

applicators, to avoid any hazard to their operators. Doors should be equipped with

chokes to reduce any leakage to outside.

1.5.Studies on Modeling of Microwave Heating

A concerted effort has recently been made to determine the temperature and moisture

distribution theoretically during microwave processing.

There are many ways of mathematical modeling of microwave heating of foods however, it

can generally be detached to three degrees of modeling possible based on the complexity of

the process or the desired accuracy.

1) Coupled model, where electromagnetics, heat transfer and moisture transfer are

included, and the temperature or moisture variation of properties require repeated

solutions of Maxwell’s equations (Appendix-1) during heating. This procedure is shown

schematically in Figure-5. This provides the most accurate description and as expected, is

the most complex. (Datta, 2001)

Start point

End point

Figure-5: Schematic diagram of the coupling of the electromagnetic and thermal calculations.

As the material heats non-uniformly, its properties change. Changes in dielectric properties,

ε’ and ε” cause the electric fields inside food to change, thus changing the energy

deposition.

Calculation of heat source from

electric field

Solve energy equation to get

temperature profile

Calculate properties at new

temperatures

Solve electro-magnetics to get electric field

In coupled models of microwave heating, dielectric properties of materials, electric field

distribution in materials, power dissipated within the material and temperature distribution

within the material should be considered. (Metaxas, 1991)

Dielectric properties of materials and variations of dielectric properties with temperature

and moisture content are defined by experimental measurement of dielectric properties

with varying temperatures and moisture contents. Then, regression analysis is done in order

to define the behavior of material due to varying temperature and moisture content by

several researchers such as Tinga and Nelson, (1973), Sun et al., (1995), Kim et al., (1998),

Ndife et al., (1998), Nelson and Bartley, (2000).

Dielectric properties of several materials are available in tabulated forms for several

materials at different temperatures and moisture content. (Bengtsson and Risman, 1971;

Ohlsson and Bengtsson 1975; Wang and Goldblith, 1976; Mudgett, 1979; Metaxas and

Meredith, 1983; Nelson, 1988)

In coupled models, electric field can be calculated by the equation below;

0.22 =+∇+

∇∇ EkEE

ε

ε (11)

where, k= wave number and shown by Eqn. (12).

*

00

22 εεµω=k (12)

where, ω is the angular frequency (rad/s), µ is the magnetic permeability.

These equations are results, obtained by solving Maxwell’s equations to calculate the

electromagnetic fields inside the microwave oven that are responsible for the heating of the

material. (Zhang and Datta, 2000)

Electric field is calculated in a different manner by the Eqn (13), (Fleischman, 1985). In this

study, electric field and volumetric heat generation term is based on a full solution of

Maxwell’s equation. However this analysis is based on steady-state microwave heating

resulting in either steady temperatures or temperature profiles and is applicable only to

very long term heating, where food is concerned, low power heating as well.

( ) ( )zii

food

zii

foodfoodx

foodfoodfoodfood eBeAEβαβα +−+

+=, (13)

Many other studies have been done by many researchers by the same manner where

calculations of electric field in multimode cavities are done by time domain or frequency

domain finite element method such as Smyth, (1989), Fu and Metaxas, (1994), Dibben and

Metaxas, (1997).

2) Electromagnetics and moisture transfer are included; however, the electromagnetic

model is solved once and the resulting microwave pattern is used as a heat source term to

compute transient heat and/or moisture transfer. The complexities of electromagnetic

modeling are removed in this scenario. This solution can provide good description of the

process with short durations of heating.

This method is used by, Lian et al., (1997), Metaxas (1991).

3) Electromagnetics is simplified as Lambert’s law and used as a heat source in a heat or

moisture transfer model (Datta, 2001).

In simplified models of microwave heating, like conventional heating, the total power

absorbed increases with volume, eventually leveling off at a power primarily dependent on

the magnetron power level, and to a lesser extent on permittivity and geometry. Thus, this

relation is described by an empirical equation of the form;

( )bV

s eaP−−= 1 (14)

where, Ps is the power at material surface (W), V is the volume (m3), a and b are empirical

constants. (Zhang and Datta, 2001)

Due to the complexity and difficulty associated with determining the electromagnetic field

within an oven, Lambert’s Law, as shown below is used: (Datta, 1990; Metaxas, 1991; Zhou

et al., 1994; Calay et al., 1995; Khraisheh et al., 1997; Lu et al., 1998)

( )dePP

α2

0

−= (15)

where, P is the power density at concerned point, P0 is the incident power density, α is the

attenuation factor, which is a function of dielectric constant ε’ and loss factor ε” and d is the

distance from the surface.

Attenuation factor and loss factor are described by the Eqn. (16) and Eqn. (17), respectively.

( )

−+

=2

1tan1

'2

21

2 δε

λ

πα (Zhou, 1994) (16)

'

"tan

ε

εδ = (Zhou, 1994) (17)

where, λ is the wavelength of microwaves and tanδ is the loss factor.

The Volumetric heating rate or the power dissipation of the microwaves is related to the

electric field strength, E by Eqn. (18)

2

0 "2 rmsEfP εεπ= (18)

where, P is the power absorbed by the material (W/m3), f is the frequency, ε0 is the

permittivity of free space, ε” is the dielectric loss of the material, and Erms is the root-mean-

square average value of electric field at a location.

During initial or short periods of intense heating, and when thermal diffusion and surface

heat losses are assumed to be zero, Eqns. (18) and (25) can be combined as in Eqn. (19), to

estimate the electric field Erms from the measured temperature rise when experimental

results are available (Datta, 2001).

t

T

f

CE

eff

p

rms∂

∂=

,,

02 εεπ

ρ (19)

In other available studies, calculation and distribution of electric field is not explained clearly

or the electric field is assumed to make simplified solutions. (Datta and Liu, 1992; Calay et

al., 1995; Khraisheh et al., 1997; Chamchong, 1999; Clark et al., 2000)

Subsequent to calculation of electric field, power dissipation within the material is

calculated in the same manner.

2

0 "2

1foodEKP ωε= (Fleischman, 1985) (20)

where, ω is the angular frequency (rad/s), ε is the permittivity (F/m), K” is the relative

dielectric loss (dimensionless), Efood is the electric field developed in the slab.

2

0

2

0

2tan'2"2 EfEffEP ref δεεπεεπσ === (Clark et al., 1990) (21)

where,σ is the effective dielectric conductivity, ε” is the effective dielectric loss, f is the

frequency (MHz).

dxeETxf

PxTx

x

),(2

0

2

0

0 ),("2

2 αεεπ −

∫= (Calay, 1995) (22)

where, dielectric loss and power dissipation is dependent on position and temperature.

Temperature distribution within the material is calculated in available studies in the same

manner.

Metaxas (1991), Dibben and Metaxas (1994), Zhou et al., (1994), Lian et al., (1997),

Sundenberg (1998), Zhang and Datta (2000), were modeled three-dimensional heat transfer

either using Finite Element Method (FEM) or Finite Difference Method (FDT), numerically by

solving the general heat transfer equation shown in Eqn. (23).

PTkt

TC p +∇∇=

∂

∂)(ρ (23)

Calay et al., (1995) was modeled a one-dimensional model by solving the Eqn. (24) with FDT.

( ) ( )t

TCpTxP

x

TTk

xd

∂

∂=+

∂

∂

∂

∂ρ, (24)

Datta (1990), Metaxas (1991), Clark et al., (1995), were simplified the heat transfer equation

to Eqn. (25) and solved analytically.

Qt

TC p =

∂

∂ρ (25)

Moisture distribution during microwave heating or drying was studied by Lian et al., (1997),

Lu et al., (1998), Ni et al., (1998), Feng et al., (2001).

Microwave thawing models were studied by Virtanen et al., (1997), Chamchong and Datta,

(1999).

A study was made into the numerical modeling of wall losses for a microwave heating

application by Ehlers et al., (2000). It makes use of a surface integral term for both

frequency and time domain finite edge element formulation in order to model the wall

impedance of the enclosed microwave cavity.

1.6.Studies on Modeling of Radio Frequency Heating

Many studies are performed to determine the temperature and electric field distribution

theoretically during radio frequency heating. Generally the studies are performed on Radio

Frequency/Vacuum(RF/V) drying of wood. On the other hand, few studies are done in radio

frequency heating of food.

The studies are given in the chronological order :

1. A circuit model of a class C radio frequency industrial system is developed by Neophytou

and Metaxas, (1996). A novel method based on non-linear optimization is used to determine

the tank circuit parameters together with its mutual coupling to the applicator. The actual

parameters of a 13,56 MHz radio frequency system are measured and used in the electrical

model in order to enable their comparison. The model is then analyzed in the time-domain

using the Saber circuit simulator. The results obtained from the model show good

agreement with experiments. (Neophytou and Metaxas, 1996)

2. A study on commercial-scale RF/V drying of softwood lumber is performed in three parts

including basic kill design considerations, drying characteristics and quality and energy

consumption and economics.

The power dissipation term(P) is given as;

2

0

''2

rmsVCfP επ= (26)

where, f is the frequency (Hz), ε’ ‘ is the dielectric loss factor, C0 is the capacitance of the

empty chamber, Vrms is the voltage in root mean square term (Avramidis et al., 1996;

Avramidis et al., 1997)

3. Two specimens of two softwoods are dried in a laboratory RF/V kiln in order to investigate

the internal moisture flow patterns. The radio frequency generator operates at a fixed

frequency of 13,56 MHz, and has a maximum output of 10 kW at a maximum electrode

radio frequency voltage of 5 kV.

The experiment is performed under two different electrode voltages (0,4 kV and 0,8 kV).

Various measurements are done and shown graphically in the study. (Zhang et al., 1997)

4. A study on the loss factor of wood during radio frequency heating is performed with

different wood samples using the direct calorimetric method with a laboratory-scale RF/V

dryer at a frequency of 13.56 MHz, moisture content range between 10 and 80 %,

temperature range between 25 and 550C and root-mean-square electrode voltages of 0,8

and 1,1 kV, respectively.

( ) "10.56,5211 εfEPD −= (W/m

3) (27)

where E ( = V /d ) is the field strength, V/m; dis the thickness of the material between the

electrodes,m; and f is the frequency, Hz.

Some regression equations are written to calculate ε’’

ε’’ = a + b M

2 + cM + d T + e MT (28)

where a, b, c, d and e are the coefficients. (Avramidis and Zhou, 1997)

5. Finite element method for solution of the Laplace and the wave equations inside radio

frequency applicators are studied by Neophytou and Metaxas, (1998). The finite element

method is used to obtain the 3D electromagnetic fields inside various applicator

configurations.

A realistic drying problem is investigated using a combined circuit simulation and finite

element model. This demonstrated the ability of the model in providing detailed information

about the operation of a radio frequency heating system. (Neophytou and Metaxas, 1998)

6. In a study on the optimization of a RF/V kiln-drying schedule, for thick western hemlock

timbers, the drying temperature, vacuum pressure and drying rate are considered as

effective factors of the quality of products. Their relationships are obtained by regression

analysis and a drying schedule is established by an optimization method.

The results of the study have shown that the temperature, vacuum pressure and final

moisture content are the main factors that affect the RF/V drying quality of lumber

(Avramidis et al., 1998).

7. A new ‘’ boundary value ’’ approach to model the RF electric field strength during the

radio frequency assisted drying of particulate materials has been developed and validated

with experimental data in the study. Material dielectric data at varying moisture contents

and elevated temperatures are measured in a novel co-axial test cell and used in the

analysis.

The power dissipated in the load per unit volume(qrf) is given as;

2''

0 mrf Ewq εε= (W/m3) (29)

where, w is the angular frequency (rad/s),

The electric field in the material(Em) is determined as;

mba

m

ddd

VE

+++=

)(tan12' δε

(kV/m) (30)

where, da , db are the upper and lower gaps (m), dm :the depth of material within the basket

(m).

By using eqn.29 and 30, the radio frequency power density formula is given as;

2

2'

''

0

)(tan1

+++=

mba

rf

ddd

Vwq

δεεε (W/m

3) (31)

Presented experimental drying results shows how simultaneous application of radio

frequency heating complements heat pump drying, not only increasing drying rate, but also

increasing the heat pump coefficient of performance. (Marshall and Metaxas, 1998)

8. A research that is done by Marshall and Metaxas, (1998) describes an experimental heat

pump batch particulate dryer, which has been combined with radio frequency energy, the

latter being operated in a continuous pulsed mode. The results show several improvements

resulting from the combination drying process. A simplified mathematical model of the

dryer, including the radio frequency heating source has been developed using mass and

energy conservation, which show good agreement with experimental results.

The solution is accomplished by writing a program using the MATLAB version 5.0

computational package (Marshall and Metaxas, 1998).

9. The heating of liquid and particulate foods by a 40.68 MHz, 30 kW continuous flow radio

frequency unit is studied by Zhong and Sandeep, (2000). Temperatures at two locations at

the exit of the radio frequency unit are measured during continuous heating and are found

to be close to one another when water is used as the product. Temperature profiles of cut

sections of particulates (small whole carrots, potato cubes, and carrot cubes) are

determined using an Infrared camera. The results show that there is only a small

temperature gradient inside the particulates, which demonstrated the uniformity and the

advantage of radio frequency heating over microwave heating.

The power absorbed by the food products is calculated similar with eqn.27.(Zhong and

Sandeep, 2000)

10. In another paper the resonant modes of a radio frequency industrial heating applicator

system are determined numerically. This is carried out using a finite element eigenvalue

calculation of the electric field of the system. Both the complex linear and nonlinear forms

of the generalized large sparse eigenproblem are solved, the latter being obligatory when

material properties are frequency-dependent.

By eliminating the magnetic field from Maxwell’s equations the following expression for the

electric field is obtained;

01

2

2'

0

0

=∂

∂+

∂

∂+×∇×∇

t

E

t

EE e

r

εεσµµ

(32)

where, σe is the effective conductivity, µr is the relative permeability, µ0 is the permeability

of free space.

Through some stages of formulations the following generalized eigenvalue is obtained;

[ ] [ ] [ ]xTxTkxS λ== 2 (33)

where, [S] and [T] are the stiffness and mass matrices. (Nephytou and Metaxas, 2000).

11. The potential of Radio Frequency/Vacuum(RF/V) drying to rapidly dry round wood is

described in the first part of the study. The investigation is carried out in a laboratory RF/V

dryer at a stabilized frequency of 13,65 MHz. During drying, temperature variance with time,

within each round wood section, is continuously monitored.

In the second part of the study the use of radio frequency heating at atmospheric pressure

to accelerate the fixation of chromated copper arsenate (CCA) in round wood is

investigated. (Avramidis, Fang, Ruddick, 2001)

12. A one-dimensional mathematical model to describe the transport phenomena during

continuous radio frequency vacuum (RF/V) drying of thick lumber is developed from general

conservation equations of heat, mass, and momentum.

Experiments were done at different voltages like 200 V, 250 V, and 300 V. The distance

between electrode plates is 250 mm. (Avramidis et al., 2001)

The power density (PD) is expressed as in eqn.27.

Electric field distribution is assumed constant and power absorption was measure

calorimetrically. (Avramidis et al., 2001)

2.MATERIALS AND METHODS

2.1.Materials and Methods For Microwave Heating

“Finite Element Modeling of Heat and Mass transfer in Food Materials During Microwave

Heating - Zhou, et al., 1994” is preferred as reference study that it has experimental results

to verify model results. In addition, power absorption equation is obtained by regressing the

experimental data like considered by Eqn. (14). By this way, electromagnetics is simplified by

using data of the study, as electric field calculation is suggested to be a study itself by Datta

(1990). Lambert’s law is applied to calculate the dissipation of heat within the material Eqn.

(36).

The microwave power absorption in food materials is mainly due to the presence of water

molecules. By regression analysis of experimental results, a and b constants of Eqn. (14) are

calculated in reference study, and power dissipation is formulated as;

( )[ ]4572,06826,4

1650−−−= W

e eP (W) (Zhou, et al., 1994) (34)

where, W is the weight of water (kg), Pe is the power absorbed by the material (W).

The surface power P0 is approximately calculated by dividing the power absorbed by the

total volume (Zhou, et al., 1994). The power dissipated per unit volume decays as the energy

traverses the dielectric slab. (Metaxas, 1991)

[ ]V

eP

W )4572,06826,4(

0

1650−−−

= (W/m3) (35)

Combining Eqn. (15) and Eqn. (35) yields,

[ ] )2()4572,06826,4(

1650 dW

eV

eP

α−−−−

= (Datta, 1990; Metaxas, 1991; Zhou, 1994) (36)

where, P is the power density at concerned point (W/m3), P0 is the surface power density

(W/m3), V is the volume of the slab (m

3), α is the attenuation factor, which is a function of

dielectric constant 'ε and loss factor "ε and d is the distance from the surface.

( )

−+

=2

1tan1

'2

21

2 δε

λ

πα (Zhou, 1994) (37)

'

"tan

ε

εδ = (Zhou, 1994) (38)

where, λ is the wavelength of microwaves, tanδ is the loss factor.

General heat transfer equation in three-dimensions can be shown as;

QTKt

TC p +∇∇=

∂

∂).(ρ (Zhou, 1994) (39)

where, ρ is the density, Cp is the specific heat, ∇ is the Laplacian operator, K is the thermal

conductivity and Q is the heat generation.

Heat loss at the boundaries due to the convection is shown as;

)(. TsTahnTK −−=∇ (Zhou, 1994) (40)

where, n is the unit outside the vector of the surface, h is the convective heat transfer

coefficient.

In the reference study, microwave-heating experiments were conducted in a GE 700W

microwave oven. Potato sample was cut in to slab shape with dimensions of 64(length) x48

(width) x30 (height) mm and sample was placed at the center of the microwave cavity and

on top of a box made from overhead transparencies. Fluoptric probes were used to measure

temperatures at different locations within the test sample. Air temperature in microwave

cavity is assumed to be 25°C. Physical and dielectric properties of potato are shown in Table-

3.

Table-3: Physical and dielectric properties of potato (Zhou et al., 1994)

ρ 1067 kg/m3

Cp 3630 J/kg.K

K 0,648 W/m.K

h 17,85 W/m2.K

To 23ºC

ε” 13

ε’ 58

Water content 85%

In present study, it is considered that, during initial or short periods of intense heating,

thermal diffusion and surface heat losses can be minimal in comparison to volumetric

microwave absorption. In such situations, the heat conduction in food is generally very small

compared to the rate of volumetric heating. Power absorption in the food may be uniform

or might be vary spatially. The energy equation given by Eqn. (39) becomes the following if

the convection and diffusion terms are dropped. Also, exact analytical solutions can only be

obtained for most simple cases, in which it is still necessary to assume constant ε”, ρ and Cp

parameters (Metaxas, 1991).

Pt

TC p =

∂

∂ρ (W/m

3) (Datta, 2001) (41)

For a given location, if the absorbed microwave power density, P does not vary with time,

the rate of temperature rise at the location is constant, giving rise to a linear temperature

raise with time. Such linear rise of temperature with time, finally reaching the boiling

temperature of water, has been observed in heating of moist food (Datta, 2001).

The numerical method of solution is used extensively in practical applications to determine

the temperature distribution and heat flow in solids having complicated geometries,

boundary conditions, and temperature-dependent thermal properties. To develop a

numerical solution, finite difference formulation for unsteady conduction heat transfer in a

semi-infinite slab with internal heat generation is applied.

2.2.Materials and Methods for Radio Frequency Heating

“Moisture flow characteristics during radio frequency vacuum drying of thick lumber

Avramidis,et al.,1996” is preferred as reference study that it has experimental results to

verify model results.

A schematic of the Radio Frequency/Vacuum(RF/V) kiln used in this work is illustrated in

Figure-6.The contact surfaces between the cylinder and the caps are fitted with rubber O-

rings to avoid air leakage.Two 30 by 224 by 1.27 cm in thickness aluminum electrodes (E)

supported by polyethylene bolts (S), are horizontally fixed in the center of the cylinder. The

RF generator operates at a fixed frequency of 13,56 MHz, and has a maximum output of 10

kW at a maximum electrode RF voltage of 5 kV. (Avramidis,et al.,1996)

Figure-6 Schematic design of RF/V dryer

In the experiment hemlock specimen with cross sections of 9,1 by 9,1 cm and 224 cm in

length were heated. The specimen was cut from 360 cm long, all-heartwood, green lumber

pieces. The moisture content of specimen was obtained by oven drying 3 cm thick slabs cut

from both ends. The constant electrode RF voltage level 0.8 kV was used for heating of

wood specimen. The absolute pressure in the chamber was maintained at about 24 torr in

all runs. (Avramidis, et al.,1996)

Wood under an RF electric field reveals its dielectric properties, which are characterized by

three parameters, namely, dielectric constant (ε’), loss tangent

(tan δ) and loss factor (ε’’). In order to determine the power dissipated in a dielectric

material that is under the influence of a high frequency electric field, knowledge of its ε’’ is

needed. The power loss in unit volume of a dielectric material such as wood under the

influence of an external high frequency electric field is known as power density ( PD,W/m3 ),

and is calculated as :

( ) "10.56,5211 εfEPD −= (Avramidis and Zhou, 1997) (42)

where E ( = V /d ) is the field strength, V/m; d is the thickness of the material between the

electrodes,m; and f is the frequency, Hz. (Avramidis and Zhou, 1997)

If heat loss due to the moisture evaporation is not taken into account, and if it is assumed

that there are no chemical reactions inside the wood, the equation to calculate the time rate

of temperature change (t

T∂

∂ )inside the wood, resulting from the conversion of high

frequency energy from the electric field to heat, is expressed as ;

pc

PD

t

T

ρ=

∂

∂ (Avramidis and Zhou, 1997) (43)

where ρ is the wood density kg/m3 ;and cp is the specific heat, J/kg

°C.

ε’’ is directly affected by moisture content and temperature, so a regression equation is

developed as :

ε’’= - 4,188M2+7,249M+0,006T+0,064MT (Avramidis and Zhou,1997) (44)

The specific heat of wood required for the determination of ε’’ was calculated by the

following equation (Avramidis and Zhou,1997) in the study .

( )

a

aa

pm

mTc

+

++=

1

0003,02393,0.4187 (45)

where cp is the specific heat of wood (J/kg°C),Ta is the average temperature range (°C) and

ma is the fractional moisture content.

For the accuracy of the calculations, cp that is used in the model is obtained with the values

of Ta=34°C and ma=0,65 where Ta is taken as initial temperature of wood and ma is assumed

to be 0,65.

The density (ρ) was calculated based on the measured green volume and calculated oven-

dry weight of the hemlock sapwood and given as 439,71 kg/m3 in the reference study.

Using the assumed moisture content, density of wood with 65% moisture content is

calculated as 1256,314 kg/m3.

As thermal conductivity (k) is not given in the reference study, the value of k for pine in

Geankoplis(1993) is used.

The heat transfer coefficient (h) is assumed to be 17,85 W/m2.K

Table-4: Physical and dielectric properties of hemlock sapwood

ρ 1256,314 kg/m3

Cp 2282,54 J/kg.K

K 0,151 W/m.K

h 17,85 W/m2.K

To 34ºC

ε” - 4,188M2+7,249M+0,006T+0,064MT

Water content 65%

2.3.Analytical Solution

2.3.1.Analytical Solution for Microwave Heating

Combining Eqn. (41) and Eqn. (15) yields the following equation;

d

p ePt

TC

αρ 2

0

−=∂

∂ (46)

Combining Eqn. (46) and Eqn. (36) gives;

[ ] dW

p eV

e

t

TC

αρ 2

)4572,06826,4(1650 −

−−−=

∂

∂ (47)

Combining Eqn. (47) and Eqn. (37) integrating and rearranging yields the final analytical

equation for the system;

( )[ ]te

VC

eTT

d

p

W

..1650

2

1'

"tan1

'2

2

4572,06826,4

12

21

2

−+=

−

+

−

−−

ε

ε

ελ

π

ρ

(48)

Assumptions:

1) Electric field distribution is constant in the slab during heating.

2) Conduction and convection heat transfer is very small to be negligible compared to the

rate of volumetric heating.

3) Dielectric properties of sample are constant.

4) There is no phase change and no evaporative heat loss.

5) Thermophysical properties of potato are constant during process.

6) There are no heat losses to environment.

2.3.2.Analytical Solution for Radio Frequency Heating

Combining Eqn. (42) and Eqn. (43) yields the following equation;

=

∂

∂ −"...10.56,5

2

11 ερ fd

V

t

Tc p (49)

Combining Eqn. (49) and Eqn. (44) yields the following equation;

( )

+++−

=

∂

∂ −mTTmmf

d

V

t

Tc p 064,0006,0249,7188,4...10.56,5

2

2

11ρ (50) By

integrating and rearranging Eqn.(49) yields the final analytical equation for the system;

[ ]( ) ( )

M

eMMT

CptfE

MMTTMM

064.0006.0

249,7188.4.

...10.56,5064.0006.0064,0006,0249,7188,4ln

2

2

211

112

+

+−=

+++++−

−

ρ

(51)

1) Electric field distribution is uniform and constant in the slab.

2) Conduction and convection heat transfer is very small to be negligible compared to the

rate of volumetric heating.

3) There is no phase change and no evaporative heat loss.

4) Thermophysical properties of hemlock sapwood are constant during process.

5) There are no heat losses to environment

6) Dielectric properties change only with temperature

2.4.Numerical Solution

2.4.1.Numerical Solution for Microwave Heating

1, −nm yq∆

y∆

1, +nm yyq ∆+∆

Figure-7: Representative interior node.

Using energy balance method, the first law of thermodynamics is applied to each interior

and exterior node at an instant of time t. The development of such energy balance on an

interior subvolume such as the one shown in Figure-7 gives;

t

TVCqVPq pyyy

∆

∆∆+∆=∆+∆ ∆+ ρ (Thomas, 1992) (52)

where, yq is the heat transfer in y direction (W), V is the volume (m3), ρ is the density

(kg/m3), Cp is the specific heat (W/m

2.K), t is the time.

Second-order finite-difference approximation for the Fourier law of conduction is

substituted into Eqn (52), to obtain;

(53)

dividing by y∆ ,

( ) ( )t

TACTT

y

kA

y

VPTT

y

kAp

t

n

t

n

t

n

t

n∆

∆+−

∆−=

∆

∆+−

∆− +− ρ1212

(54)

t

TVC

y

TTxkVP

y

TTxk p

t

n

t

n

t

n

t

n

∆

∆∆+

∆

−∆−=∆+

∆

−∆− +− ρδδ 11

m,n

After rearranging, this equation is put into the form;

( ) ( )t

TTT

yC

PTT

y

t

n

t

n

p

t

n

t

n∆

∆+−

∆−=+−

∆− +− 1212

α

ρ

α (55)

where, α is the thermal diffusivity (m2/s)

To complete the formulation, t

T

∆

∆is usually expressed in terms of forward time difference

that the equation becomes explicit where nodal temperature distribution is known at some

instant of time, that unknown nodal temperatures at the next instant of time can be

calculated directly.

t

TT

t

Tt

n

tt

n

∆

−=

∆

∆ ∆+

(Thomas, 1992) (56)

This is a first-order approximation.

Utilizing forward-time difference, Eqn (55) takes the form;

( ) ( )t

TTTT

yC

PTT

y

t

n

tt

nt

n

t

n

p

t

n

t

n∆

−+−

∆−=+−

∆−

∆+

+− 1212

α

ρ

α (57)

The explicit nodal equation for an interior node is shown as;

( ) [ ]FoTyk

PTTFoT

t

n

t

n

t

n

tt

n 212

11

1 −+

∆++= +−

∆+ (58)

Finally, combining with Eqn. (58 and Eqn. (36) yields;

[ ]

( ) [ ]FoTyk

V

ee

TTFoTt

n

dW

t

n

t

n

tt

n 21

1650

2

2)4572,06826,4(

11

1 −+

∆

−

++=

−−−

+−∆+

α

(59)

where, Fo is the Fourier number (dimensionless)

yq∆ 1, −nm convection )(" 10 a

t

s TThq −= at node 0,

insulated Surface 0" =sq at node 10.

sq∆

Figure-8, Representative exterior node

Applying convection boundary for node 0 as shown in Figure-8, the explicit nodal equation

for a regular exterior node, with convection boundary condition, is given by;

( )

∆

−∆=

∆+−

∆+−

∆+

t

TTyAC

yPATT

y

kATThA

ttt

p

tt

a

00

01022

)( ρ (60)

Rearranging Eqn. (60), gives;

( ) [ ]BiFoFoTyk

PBiTTFoT

t

a

ttt221

22 0

2

10 −−+

∆++=∆+ (61)

where, Bi is the Biot number (dimensionless).

Finally, combining with Eqn. (61) and Eqn. (36) yields;

[ ]

( ) [ ]BiFoFoTyk

V

ee

BiTTFoTt

dW

a

ttt221

2

1650

2 0

2

2)4572,06826,4(

10 −−+

∆

−

++=

−−−

∆+

α

(62)

Applying insulated surface boundary for node 10,

( )

∆

−∆=

∆+−

∆

∆+

t

TTyAC

yPATT

y

kAttt

p

tt 1010

10922

ρ (63)

Rearranging Eqn. (63), gives;

nm,

( ) [ ]FoTyk

PTFoT

tttt21

22 10

2

910 −+

∆+=∆+ (64)

Finally, combining with Eqn. (64) and Eqn. (36) yields;

[ ]

( ) [ ]FoTyk

V

ee

TFoT t

dW

ttt21

2

1650

2 10

2

2)4572,06826,4(

910 −+

∆

−

+=

−−−

∆+

α

(65)

To avoid the resulting finite-difference become unstable and blow up after a number of time

steps have been taken, stability criterion which is shown in Eqn. (66) is provided for all

nodes,

2

1)1(0 ≤+ BiF (Thomas, 1991; Incropera,2001) (66)

2.4.2.Numerical Solution for Radio Frequency Heating

1, −nm yq∆

y∆

1, +nm yyq ∆+∆

Figure-9: Representative interior node.

Using energy balance method, the first law of thermodynamics is applied to each interior

and exterior node at an instant of time t. The development of such energy balance on an

interior subvolume such as the one shown in Figure-9 gives;

m,n

t

TVCqVPq pyyy

∆

∆∆+∆=∆+∆ ∆+ ρ (Thomas, 1992) (67)

where, yq is the heat transfer in y direction (W), V is the volume (m3), ρ is the density

(kg/m3), Cp is the specific heat (W/m

2.K), t is the time.

Second-order finite-difference approximation for the Fourier law of conduction is

substituted into Eqn (67), to obtain;

(68)

Dividing by y∆ ,

( ) ( )t

TACTT

y

kA

y

VqTT

y

kAp

t

n

t

nrf

t

n

t

n∆

∆+−

∆−=

∆

∆+−

∆− +− ρ1212

(69)

After rearranging, this equation is put into the form;

( ) ( )t

TTT

yC

qTT

y

t

n

t

n

p

rft

n

t

n∆

∆+−

∆−=+−

∆− +− 1212

α

ρ

α (70)

where, α is the thermal diffusivity (m2/s)

To complete the formulation, t

T

∆

∆is usually expressed in terms of forward time difference

that the equation becomes explicit where nodal temperature distribution is known at some

instant of time, that unknown nodal temperatures at the next instant of time can be

calculated directly.

t

TT

t

Tt

n

tt

n

∆

−=

∆

∆ ∆+

(Thomas, 1992) (71)

This is a first-order approximation.

t

TVC

y

TTxkVq

y

TTxk p

t

n

t

n

rf

t

n

t

n

∆

∆∆+

∆

−∆−=∆+

∆

−∆− +− ρδδ 11

Utilizing forward-time difference, Eqn (71) takes the form;

( ) ( )t

TTTT

yC

qTT

y

t

n

tt

nt

n

t

n

p

rft

n

t

n∆

−+−

∆−=+−

∆−

∆+

+− 1212

α

ρ

α (72)

The explicit nodal equation for an interior node is shown as;

( ) [ ]FoTyk

qTTFoT

t

n

rft

n

t

n

tt

n 212

11

1 −+

∆++= +−

∆+ (73)

Finally, combining with Eqn. (73) and Eqn. (50) yields;

( )( ) [ ]FoTy

k

mTTmmfd

V

TTFoTt

n

t

n

t

n

tt

n 21

064,0006,0249,7188,4...10.56,5

2

2

2

11

11

1 −+

∆

+++−

++=

−

+−∆+

(74)

where, Fo is the Fourier number (dimensionless)

yq∆ 1, −nm Insulated Surface 0" =sq

sq∆

Figure-10 Representative exterior node

Applying same procedure for exterior nodes (node 5), as shown in Figure-10, the explicit

nodal equation for a regular exterior node, with insulated surface boundary condition, is

given by;

( )

∆

−∆=

∆+−

∆

∆+

t

TTyAC

yAqTT

y

kAttt

prf

tt 55

5422

ρ (75)

Rearranging Eqn. (75), gives;

nm,

( ) [ ]FoTyk

qTFoT

trfttt21

22 5

2

45 −+

∆+=∆+

(76)

Finally, combining with Eqn. (76) and Eqn. (50) yields;

( )( ) [ ]FoTy

k

mTTmmfd

V

TFoTtttt

212

064,0006,0249,7188,4...10.56,5

2 5

2

2

2

11

45 −+

∆

+++−

+=

−

∆+ (77)

To avoid the resulting finite-difference become unstable and blow up after a number of time

steps have been taken, stability criterion which is shown in Eqn. (78) is provided for all

nodes,

2

1)1(0 ≤+ BiF (Thomas, 1991; Incropera,2001) (78)

3.RESULTS AND DISCUSSIONS

In the reference study, a potato sample was used as the test material with dimensions of

64mm (x direction), 48mm (y direction), 30mm (z direction). The sample was placed at the

center of the microwave cavity and on top of a box made from overhead transparencies.

Other sides rather than top surface are assumed to be insulated. As it was mentioned that, it

was not possible to derive generalized analytical solutions for that set of equations, which

contains a heat generation term, a heat conduction term and a heat accumulation term

(Zhou, 1994). However, for a given location, if the absorbed microwave power density does

not vary with time, the rate of temperature rise at the location stays constant and a linear

temperature raise with time is observed (Datta, 2001). To find out an analytical solution

conduction and convection terms were dropped from the formula as it was assumed that

conduction, convection and radiation heat transfer to be very small to be negligible

compared to the rate of volumetric heating. Constant ε”, ρ, and Cp parameters are assumed

according to Metaxas, (1991). A computer program was written to obtain analytical results.

After 50s of heating period, the difference between analytical results and the measured

temperature was 21,36 ºC (relative difference of 44,1% from the measured value) at the

geometric center of the slab and 5,71ºC (relative difference of 7,5% from the measured

value) at the surface of the slab. The reasons for these differences may be explained as

follows:

1) Insufficient probe location – it was difficult to locate a probe at the exact

geometric center or the surface (Zhou, 1994).

2) Insufficient accuracy of material properties, which were obtained from the

literature (Zhou, 1994).

3) Non-uniform electric field distribution thus non-uniform power distribution.

4) Presence of conduction, convection and radiation heat transfer.

5) Significant variations of dielectric properties of potato with temperature.

6) Variations of thermophysical properties of potato with time.

Comparison of measured and analytically predicted temperature at the geometric center

and the surface of the slab are shown in Figure-11 and Figure-12.

Figure 11: Microwave Heating Results

Comparison of Measured and Predicted Temperature at the Geometric Center of the Slab

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Heating Time (s)

Tem

pera

ture

(C

)

Measured Numerical Analytical

To obtain the solution of heat transfer equation, which includes conduction and convection

terms, a one-dimensional finite difference formulation was developed.

The slab was assumed to be insulated from all sides except top surface. Top surface is

assumed to be exposed to convection and bottom surface is assumed to be an insulated

surface as the sample was placed on top of a box made from overhead transparencies.

Figure 12: Microwave Heating Results

Comparison of Measured and Predicted Temperature at the Surface of the slab

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Heating Time (s)

Tem

pera

ture

(C

)

Measured Numerical Analytical

The slab was divided into ten nodes. According to Lambert’s Law, heat generation was

assumed to decay exponentially from surface node to bottom node. A visual basic program

was written to obtain numerical solutions.

After 50s of heating period, the difference between numerical results and the measured

temperature was 8,01 ºC (relative difference of 16,5% from the measured value) at the

geometric center of the slab and 6,46ºC (relative difference of 8,56% from the measured

value) at the surface of the slab. The reasons for these differences may be explained as

follows:

1) Insufficient probe location – it was difficult to locate a probe at the exact geometric

center or the surface (Zhou, 1994).

2) Insufficient accuracy of material properties, which were obtained from the literature

(Zhou, 1994).

3) Occurrence of heat transfer in all directions.

4) Significant variations of dielectric properties of potato with temperature.

5) Variations of thermophysical properties of potato with time.

6) Occurrence of evaporative loss.

In modeling of radio frequency heating, due to insufficient studies on radio frequency

heating of foods, temperature-time parameters of hemlock sapwood, which is a kind of pine

is taken from the reference study “ Moisture flow characteristics during radio frequency

vacuum drying of thick lumber” (Zhang, et al., 1996).

Heat capacity of hemlock sapwood is calculated with Eqn.45 and found to be 2282,54 J/kg°C,

density of wet hemlock sapwood is calculated, assuming 65% moisture content, which also

affects heat capacity, dielectric loss factor and temperature distribution. All sides are

assumed to be insulated. A computer program was written to calculate the for both

analytical and numerical temperature distribution.

As the experiment of the reference study contain evaporation, results of the model are

compared with the part of the measured results up to 100°C of the reference study,

assuming evaporative heat loss is negligible below 100°C.

Figure –13 compares the analytically and numerically calculated center temperature of the

sample with experimentally obtained data below 100°C.

Figure-13: Radio Frequency Heating Results

Comparison of Measured and Predicted Temperature at the Geometric Center of the Slab

05

101520253035404550556065707580859095

100105110

0 1 2 3 4 5 6 7 8 9 10

time (min)

Tem

pera

ture

(ºC

)

Measured Analytical Numerical

Both analytical and numerical model results showed agreement with experimental results.

After 9 minutes of heating period, the difference between analytical results and the

measured temperature was 3,058°C (relative difference of 3% from the measured value), at

the geometric center of the slab and the difference between numerical results and the

measured temperature was 2,82°C (relative difference of 2,81% from the measured value).

The reasons agreement may be explained as follows:

1) Temperature dependent dielectric properties of material are used.

2) Due to slab location there are no convection heat losses, which leads small

difference between analytical and numerical results.

3) In analytical solution conduction heat transfer is assumed to be negligible when

compared with volumetric heating.

4) Moisture content is assumed such to provide best results.

5) Thermophysical properties of the sample are assumed to be constant.

In modeling of microwave heating, the power term is obtained by regressing the

experimental data, which lead dependence of the solution to the conditions of the

reference model. Also, dielectric properties were taken as constant according to

reference study. While, in modeling of radio frequency heating, a general model, which

will be valid in all process conditions, was attempted to be developed.

4. REFERENCES

5. APPENDIXES

5.1. Microwave model computer program codes

Private Sub About_Click()

Form5.Show

End Sub

Private Sub Command1_Click()

L = Length.Text

W = Wid.Text

h = Heigth.Text

V = L * W * h

'VOLUME

Form1.Text1.Text = V

a = 2 * L * W + 2 * W * h + 2 * L * h

Form1.Text5 = a

'MASS

d = Density.Text

M = d * V

Form1.Text2 = M

'WATER CONTENT

Mo = Mo.Text

Mwater = Mo * M

Form1.Text3 = Mwater

'POWER ABSORBED

P = 650 * (1 - Exp(-4.6826 * Mwater - 0.4572))

Form1.Text4 = P

'PENETRATION DEPTH

E1 = E1.Text

E2 = E2.Text

d = E2 / E1

f = f.Text

L = 300 / f

pi = 3.14159265358979

delta = 2 * Atn(d) * 180 / pi

a = (2 * pi / L)

b = (1 - Cos(delta * pi / 180)) / (1 + Cos(delta * pi / 180))

c = (1 + b) ^ (0.5)

Pd = a * (E1 * (c - 1) / 2) ^ (0.5)

Form1.Text5 = Pd

'POWER DISSIPATION

Pdis = (1 - (2.718282 ^ (-2 * Pd * h))) * 100

Form1.Text6 = Pdis

End Sub

Private Sub Command1_Click()

Den = Form1.Density.Text

Cp = Form1.Cp.Text

dt = Form2.dt.Text

V = Form1.Text1.Text

T1 = Form1.To.Text

Pdis = Form1.Text6.Text

P = Form1.Text4.Text

T2 = T1 + (Pdis / 100) * P * dt / (Den * Cp * V)

While T2 < 100

Form2.AnaltyicResults.AddItem T2

T1 = T2

T2 = T1 + (Pdis / 100) * P * dt / (Den * Cp * V)

Wend

End Sub

Private Sub Command2_Click()

Form2.Show

End Sub

Private Sub Command3_Click()

Form3.Show

End Sub

Private Sub Command1_Click()

'DELTA X

Dx = Form1.Heigth.Text / 10

Form3.Text1 = Dx

'BIOT NUMBER

h = Form1.hc.Text

k = Form1.k.Text

Bi = h * Dx / k

Form3.Text2 = Bi

'THERMAL DIFFUSIVITY

d = Form1.Density.Text

c = Form1.Cp.Text

TD = k / (d * c)

Form3.Text4 = TD

'FOURIER NUMBER

'tin is calculated for convergence

tin = (Dx ^ 2) / (2 * TD * (1 + Bi))

If tin > 1 Then T = 0.1

If 1 > tin > 0.1 Then T = 0.01

If tin < 0.1 Then T = 0.01

Form4.Text2 = T

Form3.Text5 = T

Fo = (TD * T) / (Dx ^ 2)

Form3.Text3 = Fo

End Sub

Private Sub Command2_Click()

Dim T(11), q(11) As Variant

Dim Ttemp(11) As Variant

k = Form1.k.Text

Ta = Form1.To.Text

Tint = Form3.Text5

Tamb = Form1.Tamb.Text

Fo = Form3.Text3

Dx = Form1.Heigth.Text / 10

Bi = Form3.Text2

For i = 0 To 11

T(i) = Ta

Ttemp(i) = Ta

Next

'q(0)= surface List 11

q(0) = Form1.Text4.Text / Form1.Text1.Text

q(1) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 1 * Form1.Text5.Text) / Form1.Text1.Text

q(2) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 2 * Form1.Text5.Text) / Form1.Text1.Text

q(3) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 3 * Form1.Text5.Text) / Form1.Text1.Text

q(4) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 4 * Form1.Text5.Text) / Form1.Text1.Text

q(5) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 5 * Form1.Text5.Text) / Form1.Text1.Text

q(6) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 6 * Form1.Text5.Text) / Form1.Text1.Text

q(7) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 7 * Form1.Text5.Text) / Form1.Text1.Text

q(8) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 8 * Form1.Text5.Text) / Form1.Text1.Text

q(9) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 9 * Form1.Text5.Text) / Form1.Text1.Text

q(10) = Form1.Text4.Text * 2.718282 ^ (-2 * Dx * 10 * Form1.Text5.Text) /

Form1.Text1.Text

Form4.Show

T(0) = 2 * Fo * (Ttemp(1) + Bi * Tamb + ((q(0) * Dx ^ 2) / (2 * k))) + Ttemp(0) * (1 - 2 * Fo -

2 * Bi * Fo)

T(1) = Fo * (Ttemp(0) + Ttemp(2) + ((q(1) * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

T(2) = Fo * (Ttemp(1) + Ttemp(3) + ((q(2) * Dx ^ 2) / k)) + Ttemp(2) * (1 - 2 * Fo)

T(3) = Fo * (Ttemp(2) + Ttemp(4) + ((q(3) * Dx ^ 2) / k)) + Ttemp(3) * (1 - 2 * Fo)

T(4) = Fo * (Ttemp(3) + Ttemp(5) + ((q(4) * Dx ^ 2) / k)) + Ttemp(4) * (1 - 2 * Fo)

T(5) = Fo * (Ttemp(4) + Ttemp(6) + ((q(5) * Dx ^ 2) / k)) + Ttemp(5) * (1 - 2 * Fo)

T(6) = Fo * (Ttemp(5) + Ttemp(7) + ((q(6) * Dx ^ 2) / k)) + Ttemp(6) * (1 - 2 * Fo)

T(7) = Fo * (Ttemp(6) + Ttemp(8) + ((q(7) * Dx ^ 2) / k)) + Ttemp(7) * (1 - 2 * Fo)

T(8) = Fo * (Ttemp(7) + Ttemp(9) + ((q(8) * Dx ^ 2) / k)) + Ttemp(8) * (1 - 2 * Fo)

T(9) = Fo * (Ttemp(8) + Ttemp(10) + ((q(9) * Dx ^ 2) / k)) + Ttemp(9) * (1 - 2 * Fo)

T(10) = 2 * Fo * (Ttemp(9) + Bi * Tamb + ((q(10) * Dx ^ 2) / (2 * k))) + Ttemp(10) * (1 - 2 *

Fo - 2 * Bi * Fo)

While T(0) < 95

Form4.List11.AddItem T(0)

Form4.List1.AddItem T(1)

Form4.List2.AddItem T(2)

Form4.List3.AddItem T(3)

Form4.List4.AddItem T(4)

Form4.List5.AddItem T(5)

Form4.List6.AddItem T(6)

Form4.List7.AddItem T(7)

Form4.List8.AddItem T(8)

Form4.List9.AddItem T(9)

Form4.List10.AddItem T(10)

For i = 0 To 11

Ttemp(i) = T(i)

Next

T(0) = 2 * Fo * (T(1) + Bi * Tamb + ((q(0) * Dx ^ 2) / (2 * k))) + T(0) * (1 - 2 * Fo - 2 * Bi * Fo)

T(1) = Fo * (Ttemp(0) + Ttemp(2) + ((q(1) * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

T(2) = Fo * (Ttemp(1) + Ttemp(3) + ((q(2) * Dx ^ 2) / k)) + Ttemp(2) * (1 - 2 * Fo)

T(3) = Fo * (Ttemp(2) + Ttemp(4) + ((q(3) * Dx ^ 2) / k)) + Ttemp(3) * (1 - 2 * Fo)

T(4) = Fo * (Ttemp(3) + Ttemp(5) + ((q(4) * Dx ^ 2) / k)) + Ttemp(4) * (1 - 2 * Fo)

T(5) = Fo * (Ttemp(4) + Ttemp(6) + ((q(5) * Dx ^ 2) / k)) + Ttemp(5) * (1 - 2 * Fo)

T(6) = Fo * (Ttemp(5) + Ttemp(7) + ((q(6) * Dx ^ 2) / k)) + Ttemp(6) * (1 - 2 * Fo)

T(7) = Fo * (Ttemp(6) + Ttemp(8) + ((q(7) * Dx ^ 2) / k)) + Ttemp(7) * (1 - 2 * Fo)

T(8) = Fo * (Ttemp(7) + Ttemp(9) + ((q(8) * Dx ^ 2) / k)) + Ttemp(8) * (1 - 2 * Fo)

T(9) = Fo * (Ttemp(8) + Ttemp(10) + ((q(9) * Dx ^ 2) / k)) + Ttemp(9) * (1 - 2 * Fo)

T(10) = 2 * Fo * (T(9) + Bi * Tamb + ((q(10) * Dx ^ 2) / (2 * k))) + T(10) * (1 - 2 * Fo - 2 * Bi *

Fo)

Wend

End Sub

Private Sub Form_Load()

End Sub

5.2. Radio frequency model computer program codes

Dim Tini, T2 As Variant

Dim num, lineno As Integer

Private Sub Command1_Click()

'DEFINITIONS

Dim Tnu(6) As Variant

Dim Ttemp(6) As Variant

f = Form1.Text4 * 1000000

M = Form1.Text1

pi = 3.14159265358979

ro = Form1.Text8

Cp = Form1.Text9

k = Form1.Text10

h = Form1.Text11

d = Form1.Text2

V = Form1.Text5

Tamb = Form1.Text12

Tini = Form1.Text13

wa = 2 * pi * f

tinc = 1

E = V / d

es = 2.718281828

Form2.List9.AddItem tinc

'ANALYTICAL SOLUTION

Form2.Show

T1 = Form1.Text13.Text

T2 = ((4.188 * M ^ 2 - 7.249 * M) + Exp((Log(-4.188 * M ^ 2 + 7.249 * M + 0.006 * T1 + 0.064

* M * T1)) + ((0.006 + 0.064 * M) * (0.0000000000556 * E ^ 2 * f * tinc) / (ro * Cp)))) /

((0.006 + 0.064 * M))

While T2 < 104

Form2.List1.AddItem T2

T1 = T2

T2 = ((4.188 * M ^ 2 - 7.249 * M) + Exp((Log(-4.188 * M ^ 2 + 7.249 * M + 0.006 * T1 +

0.064 * M * T1)) + ((0.006 + 0.064 * M) * (0.0000000000556 * E ^ 2 * f * tinc) / (ro * Cp)))) /

((0.006 + 0.064 * M))

Wend

'NUMERICAL SOLUTION

'Position increment

Dx = Form1.Text2.Text / 10

'Biot Number

h = Form1.Text11

k = Form1.Text10

Bi = h * Dx / k

'Thermal Diffusivity

td = k / (ro * Cp)

'Fourier Number

'tara is calculated for the convergence

'(tn represents time increment for numerical calculations)

tara = (Dx ^ 2) / (2 * td * (1 + Bi))

If tara > 1 Then Tn = 1

If 1 > tara > 0.1 Then Tn = 0.1

If tara < 0.1 Then Tn = 0.1

Form2.List8.AddItem Tn

Fo = (td * Tn) / (Dx ^ 2)

'Solution

Form2.Show

For i = 0 To 5

Tnu(i) = Tini

Ttemp(i) = Tini

Next

Pnu0 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(0) +

0.064 * M * Tnu(0))

Pnu1 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(1) +

0.064 * M * Tnu(1))

Pnu2 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(2) +

0.064 * M * Tnu(2))

Pnu3 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(3) +

0.064 * M * Tnu(3))

Pnu4 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(4) +

0.064 * M * Tnu(4))

Pnu5 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Tnu(5) +

0.064 * M * Tnu(5))

Tnu(0) = Fo * (Ttemp(1) + Ttemp(1) + ((Pnu1 * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

Tnu(1) = Fo * (Ttemp(0) + Ttemp(2) + ((Pnu1 * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

Tnu(2) = Fo * (Ttemp(1) + Ttemp(3) + ((Pnu2 * Dx ^ 2) / k)) + Ttemp(2) * (1 - 2 * Fo)

Tnu(3) = Fo * (Ttemp(2) + Ttemp(4) + ((Pnu3 * Dx ^ 2) / k)) + Ttemp(3) * (1 - 2 * Fo)

Tnu(4) = Fo * (Ttemp(3) + Ttemp(5) + ((Pnu4 * Dx ^ 2) / k)) + Ttemp(4) * (1 - 2 * Fo)

Tnu(5) = 2 * Fo * (Ttemp(5) + ((Pnu5 * Dx ^ 2) / (2 * k))) + Ttemp(5) * (1 - 2 * Fo)

While Tnu(1) < 104

Form2.List2.AddItem Tnu(0)

Form2.List3.AddItem Tnu(1)

Form2.List4.AddItem Tnu(2)

Form2.List5.AddItem Tnu(3)

Form2.List6.AddItem Tnu(4)

Form2.List7.AddItem Tnu(5)

For i = 0 To 5

Ttemp(i) = Tnu(i)

Next

Pnu0 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(0) +

0.064 * M * Ttemp(0))

Pnu1 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(1) +

0.064 * M * Ttemp(1))

Pnu2 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(2) +

0.064 * M * Ttemp(2))

Pnu3 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(3) +

0.064 * M * Ttemp(3))

Pnu4 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(4) +

0.064 * M * Ttemp(4))

Pnu5 = 0.0000000000556 * (E) ^ 2 * f * (-4.188 * M ^ 2 + 7.249 * M + 0.006 * Ttemp(5) +

0.064 * M * Ttemp(5))

Tnu(0) = Fo * (Ttemp(1) + Ttemp(1) + ((Pnu1 * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

Tnu(1) = Fo * (Ttemp(0) + Ttemp(2) + ((Pnu1 * Dx ^ 2) / k)) + Ttemp(1) * (1 - 2 * Fo)

Tnu(2) = Fo * (Ttemp(1) + Ttemp(3) + ((Pnu2 * Dx ^ 2) / k)) + Ttemp(2) * (1 - 2 * Fo)

Tnu(3) = Fo * (Ttemp(2) + Ttemp(4) + ((Pnu3 * Dx ^ 2) / k)) + Ttemp(3) * (1 - 2 * Fo)

Tnu(4) = Fo * (Ttemp(3) + Ttemp(5) + ((Pnu4 * Dx ^ 2) / k)) + Ttemp(4) * (1 - 2 * Fo)

Tnu(5) = 2 * Fo * (Ttemp(5) + ((Pnu5 * Dx ^ 2) / (2 * k))) + Ttemp(5) * (1 - 2 * Fo)

Wend

End Sub

5.3.List of Symbols

a, b empirical constants

Bi Biot number, dimensionless

Cp specific heat, kJ/kg.K

D penetration depth, m

d distance, m

δ width of the slab

E electric field, V/m

Fo Fourier number, dimensionless

f frequency, Hz

H Magnetic field, H/m

h convective heat transfer coefficient, W/m2.K

K thermal conductivity

k wave number

k’ relative permittivity, dimensionless

k’’ relative loss factor, dimensionless

n ion density, number of ions per unit volume

n unit outside the vector of the surface

Pe power calculated experimentally

P power density at concerned point, W/m3

P0 incident power density, W/m3

Ps power at material surface, W

Q heat generation, W/m3

q amount of electrical charge on each of the ions, Coulombs

xq Heat transfer in x direction, W

t time, s

T temperature, ºC

Ta ambient temperature, ºC

T0 initial temperature, ºC

tanδ loss factor, dimensionless

V the volume of the slab, m3

Vp velocity of propagation, m/s

W weight of water, kg

x, position

α Thermal diffusivity, m2/s

α Attenuation factor, 1/m

Δ increment

ε*

complex permittivity, F/m

ε’ dielectric constant, dimensionless

ε’’ dielectric loss factor, dimensionless

ε0 permittivity of free space, F/m

∇ Laplacian operator

λ Wavelength, m

λ0 free space wavelength, m

μ level of mobility of the ions.

µ Magnetic permeability, H/m

ω angular frequency, rad/s

ρ Density of material, kg/m3

σ Effective dielectric conductivity