Model for microwave absorption and heat transfer in a combination washer dryer

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Model for microwave absorption and heat transfer

in a combination washer dryer

Dissertation submitted in fulfilment of the requirements for the degree Magister Ingeneriae at the Potchefstroom

campus of the North-West University

by

J.P. Smit

Supervisors Prof. G. van Schoor

Dr. K. Uren

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Abstract

The work presented within this dissertation focusses on the development of a finite element method (FEM) model for the microwave absorption and heat transfer within a microwave combination washer dryer (MCWD). FEM will be used to aid in the implementation of more advanced fluid dynamics such as laminar or turbulent flow, that may be present within the system. The intended use of the model is to aid a South African based company in the development of a control system for the MCWD. The model development presented focusses on the washing cycle of the MCWD and will therefore not take into account the drying cycle of the system. The target of the microwave heating within the model will be distilled water as the dielectric constant of water is a know quantity.

Various literature sources on microwave absorption and heat transfer models can be found, but none specific to the topic of the combination washer dryer. By reviewing literature from various sources, the finite element method was selected as the modelling technique and the COMSOL® software package was selected as the tool for developing the model.

A model for the MCWD will be developed within the COMSOL® environment which in turn implements FEM as a technique to solve the model. The model development is broken into nine stages. Stage one start by modelling the heat transfer within the washing drum. Each consecutive stage expands the model by adding features or model domains. Model verification takes place in parallel to the development by verifying each stage before moving to the next stage. The stage eight and nine models, which represent a full three dimensional model of the system, are selected to be validated as the final models. Stage eight models the system without an enclosure and makes use of convective cooling boundary conditions on the boundary of the air enclosed within the system enclosure. Stage stage nine models the system with the aluminium enclosure of the system and also implements convective cooling boundary conditions on the outer boundary of the aluminium. The boundary between the enclosed air and aluminium enclosure is implemented as a normal convective heat transfer boundary between a gas and solid.

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capturing platform is interfaced to the sensors by an in-house developed signal conditioning board.

Model validation is completed by comparing the response of the model to the practical system. Numerous simulations are completed to select the optimal configuration of the model that provides the optimal response.

The stage eight model was found to be more accurate then the stage nine model with respect to the difference between the simulated and expected response over the whole domain of the transient temperature response. A further method implemented to easily compare the results of various simulations is by comparing the average absolute temperature of the response over the whole domain of the transient response. The average absolute temperature is calculated by taking absolute difference between the expected results and the model response at each time step within the response domain and then to average the absolute difference. This enables the comparison of two responses using two values. Needles to say this method should not be used alone and should be used in conjunction with a comparison over the full response domain. Use of the average absolute temperature difference is aimed at filtering the results from a selection of results which warrants a more in depth investigation. Using a comparison of the average absolute temperature difference of the target in the 500 W model, it was found that their respective values are 2.92◦C and 11.36◦C. The stage eight model computation time was far less than the stage nine model and is therefore recommended for further development.

The final conclusion was made that the stage eight model represents the system fairly accurately at this stage and warrants further development by expanding the model to account for the drying cycle of the MCWD. The term fairy accurate is used to describe the results as further improvement of the model is definitely possible with regards to the accuracy of the transient response of the system. Further improvement of the model response may be possible by implementing a smaller mesh size or launching an in depth study on the effect of the various material thermal properties on the response of the system during various stages. For instance below a certain temperature the response closely represents the expected response and above that temperature the response various greatly from the expected response.

Future work on the model include, to change the target from distilled water to an actual representation of the textiles intended to be washed within the MCWD. This will require a study into how the various parameters such as the density and dielectric constant, of the heterogeneous mixtures of textiles and water, can be combined for use into the model. As a next step in the expansion of the model, the model can be configured to account for the drying cycle of the system which will require the model to account for the phase changes that the water will undergo.

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Foreword

First and foremost I would thank the Lord my saviour and creator for His guidance provided during the completion of my dissertation. Thank you Lord for your grace and love each day in my life and for providing me with the opportunity to expand my knowledge of your amazing creation we call the world.

I would secondly like to extend my never ending gratitude and love to the love of my life Christine Strauss. Thank you for your support and understanding during each day of my Masters’ study.

Thirdly I would like to thank my family for their support and love during the study period to complete this dissertation.

I would also like to extend my gratitude for the support provided to me by both my supervisors Prof. G. van Schoor and Dr. K. Uren. Thank you for guidance in the days that I felt I would not be able to solve certain problems presented to me.

Finally I would like to thank the National Research Foundation and the Technology and Human Resources for Industry Programme for providing financial support for the research.

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3.4 Stage two . . . 48 3.4.1 Model alterations . . . 48 3.5 Stage three . . . 50 3.5.1 Model alterations . . . 50 3.6 Stage four . . . 51 3.6.1 Model alterations . . . 51 3.7 Stage five . . . 54 3.7.1 Model alterations . . . 54 3.8 Stage six . . . 55 3.8.1 Model alterations . . . 55 3.9 Stage seven . . . 57 3.9.1 Model alterations . . . 57 3.10 Stage eight . . . 57

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3.11.1 Model alterations . . . 59 4 Model verification 60 4.1 Stage one . . . 60 4.2 Stage two . . . 64 4.3 Stage three . . . 66 4.4 Stage four . . . 68 4.5 Stage five . . . 70 4.6 Stage six . . . 71 4.7 Stage seven . . . 71 4.8 Stage eight . . . 72 4.9 Stage nine . . . 74 5 Practical implementation 77 5.1 Introduction . . . 77

5.2 Prototype configuration and testing . . . 77

5.3 Data capturing platform . . . 78

5.4 Sensor package selection and placement . . . 78

5.4.1 Sensor selection . . . 78 5.4.2 Sensor placement . . . 81 5.4.3 Sensor interface . . . 86 5.5 Data logging . . . 90 5.6 System evaluation . . . 92 5.7 Conclusion . . . 94 6 Model validation 95 6.1 Introduction . . . 95

6.2 Microwave absorption model validation . . . 95

6.3 Heat transfer model validation . . . 98

6.3.1 Stage eight model validation . . . 101

6.3.2 Stage nine model validation . . . 111

6.4 Validation conclusion . . . 111

7 Conclusion and Recommendations 116 7.1 Conclusion . . . 116 7.2 Recommendations . . . 118 7.2.1 Model . . . 118 7.2.2 Implementation . . . 119 7.3 Future work . . . 119 7.4 Closure . . . 120

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Bibliography 121

A COMSOL® development overview 125

B Model validation settings 128

C Practical measurements 140 D Data disc 147 D.1 Project proposal . . . 147 D.2 Dissertation . . . 147 D.3 dSpace® ControlDesk® . . . 147 D.4 Data processing . . . 148 D.5 Models . . . 148 D.6 Photos . . . 148 D.7 References . . . 148

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List of Tables

Page

2.1 Results for plane wave propagation in various mediums . . . 21

2.2 Model parameters important in the context of this dissertation . . . 34

3.1 Model domain definitions . . . 36

3.2 Modelling stages overview . . . 37

3.3 Stage one Global Definitions Parameter node . . . 41

3.4 Stage one definitions sub-node - Variables as defined in COMSOL® using plain text format. . . 42

3.5 Stage six Global Definitions Parameter node . . . 55

3.6 Stage six material properties manually defined . . . 56

3.7 Stage eight Global Definitions Parameter node . . . 58

6.1 Wave guide power flow and target power absorption comparison . . . 96

6.2 Comparison of microwave power absorption in target . . . 97

6.3 Material properties for target and washing drum . . . 97

6.4 Parametric study of water dielectric constant . . . 98

6.5 Parametric study of polyethylene dielectric constant . . . 99

6.6 Microwave power absorption in target, new material parameters . . . 99

6.7 Validated material properties for target and washing drum . . . 100

6.8 Model parameters for stage eight 1000 W model validation . . . 101

6.9 Validated material properties for target and washing drum . . . 101

6.10 Stage eight model validation - Average absolute temperature difference between simulation and practical system. . . 107

6.11 Stage nine model validation - Average absolute temperature difference between simulation and practical system. . . 115

B.1 Stage eight 1000 W validation simulations - Model configurations . . . 128

B.2 Stage eight 500 W validation simulations - Model configurations . . . 135

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List of Figures

Page

1.1 Electromagnetic spectrum . . . 2

1.2 Laboratory prototype microwave combination washer dryer. . . 3

1.3 Schematic diagram of MCWD system. . . 4

2.1 (a) Closed or control mass system. (b) Open or control volume system . . . 12

2.2 Hull’s investigation into non-oscillating diodes . . . 23

2.3 Sinusoidal source with a capacitive load . . . 26

2.4 RLC circuit . . . 29

2.5 Iterative modelling process . . . 30

2.6 Model classification tree . . . 31

2.7 FEM Example . . . 32

3.1 Flow diagram of model development process . . . 38

3.2 Stage one model breakdown structure . . . 39

3.3 Domain used for stages one . . . 40

3.4 (a) Thermal conductivity, (b) Prandtl number and, (c) kinematic viscosity of air at one atmosphere . . . 43

3.5 Stage one pressure point constraints . . . 46

3.6 Stage one mesh configuration . . . 47

3.7 Domain used for stage two . . . 49

3.8 Stage two pressure point constraints . . . 49

3.9 Domain used for stage three . . . 50

3.10 Domains used for stages 4 & 6 & 8 . . . 51

3.11 Stage four convective cooling boundaries . . . 52

3.12 Stage four pressure point constraints . . . 53

3.13 Stage four mesh configuration . . . 53

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4.2 Stage one temperature distribution after heating at 100 W at time steps (a) 0 s,

(b) 50 s, (c) 100 s and (d) 200 s. . . 62

4.3 Stage one temperature distribution after 3600 s of heating at 100 W, both (a) with and (b) without laminar and convective heat flow. . . 63

4.4 Stage two transient average temperature response. . . 64

4.5 Stage two temperature distribution after 3600 s of heating at 100 W . . . 65

4.6 Stage three transient average temperature response. . . 66

4.7 Stage three fluid flow velocity. . . 67

4.8 Stage three Reynolds number. . . 67

4.9 Stage three temperature distribution after 3600 s of heating at 100 W . . . 68

4.10 Stage four transient average temperature response. . . 69

4.11 Stage four temperature distribution after 600 s of heating at 100 W . . . 69

4.12 Stage five transient average temperature response. . . 70

4.13 Stage five temperature distribution after 600 s of heating at 100 W . . . 70

4.14 Stage six electrical field with 100 W input power . . . 71

4.15 Stage seven electrical field with 100 W input power . . . 72

4.16 Stage eight electrical field with 100 W input power . . . 73

4.17 Stage eight transient average temperature response. . . 73

4.18 Stage eight temperature distribution after 600 s of heating at 100 W . . . 74

4.19 Stage nine electrical field with 100 W input power . . . 75

4.20 Stage nine transient average temperature response. . . 75

4.21 Stage nine temperature distribution after 600 s of heating at 100 W . . . 76

5.1 Wiring diagram of MCWD system. . . 79

5.2 Prototype commissioning test procedure. . . 80

5.3 3D view of two 2D temperature distribution surface plots within the system with 100 W microwave power at 600 s. . . 82

5.4 Front view of temperature distribution within system with 100 W microwave power at (a) 30 s, (b) 60 s, (c) 120 s and (d) 240 s. . . 83

5.5 Front view of temperature distribution within system with 100 W microwave power at (a) 360 s and (b) 600 s. . . 84

5.6 Right view of temperature distribution within system with 100 W microwave power at (a) 30 s and (b) 60 s. . . 84

5.7 Right view of temperature distribution within system with 100 W microwave power at (a) 120 s, (b) 240 s, (c) 360 s and (d) 600 s. . . 85

5.8 dSpace interface circuit . . . 87

5.9 Thermocouple amplifier circuit . . . 88

5.10 Temperature compensation circuit . . . 88

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5.12 dSpace® interface PCB . . . 89

5.13 System configuration in the laboratory . . . 90

5.14 Simulink® diagram for DS1104 program . . . 91

5.15 ControlDesk® layout to display realtime data . . . 92

5.16 Evaluation procedure. . . 93

6.1 Stage eight, 1000 W, Simulation 1 - Target temperature comparison . . . 102

6.2 Stage eight at 1000 W with 6 mm wall thickness; Simulation 1, 2, and 3 - Target temperature comparison . . . 102

6.3 Stage eight at 1000 W with 10 mm wall thickness; Simulation 4, 5, 15, and 16 -Target temperature comparison . . . 104

6.4 Stage eight at 1000 W with 8 mm wall thickness; Simulation 6, 7, 8, 17, 18, 19, 20, 21, 22, 23, and 24 - Target temperature comparison . . . 104

6.5 Stage eight at 1000 W with 7 mm wall thickness; Simulation 9, 10, 11, 12, 13, and 14 - Target temperature comparison . . . 105

6.6 Stage eight, 500 W, Simulation 1, 2, 3, 4, 5, and 6 - Target temperature comparison106 6.7 Stage eight, 100 W, Simulation 1, 2, 3, 4, 5, 6, 7, 8, and 9 - Target temperature comparison . . . 106

6.8 Temperature comparison for stage 8, simulation 19, 1000 W model; (a) Target (b) Above drum (c) Below drum (d) Left of drum (e) Behind drum . . . 108

A.1 Comsol development environment . . . 126

A.2 Comsol development tree and properties panel . . . 127

C.1 100 W practical microwave power measurements; (a) Input power (b) Reflected power (c) Net power . . . 141

C.2 100 W practical temperature measurements; (a) Target (b) Above drum (c) Below drum (d) Left of drum (e) Behind drum (f) Ambient . . . 142

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C.4 500 W practical temperature measurements; (a) Target (b) Above drum (c) Below drum (d) Left of drum (e) Behind drum (f) Ambient . . . 144 C.5 1000 W practical microwave power measurements; (a) Input power (b) Reflected

power (c) Net power . . . 145 C.6 1000 W practical temperature measurements; (a) Target (b) Above drum (c)

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Acronyms

AC alternating current

ADC analogue to digital converter CAD computer aided design EM electromagnetic FEM finite element method HT heat transfer

I/O input and output

MAL microwave assisted laundry

MCWD microwave combination washer dryer MW microwave

PCB printed circuit board PDE partial differential equation RF radio frequency

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Chapter 1

Introduction

This chapter firstly presents the background of the MCWD system. This is followed by the problem statement and the operational range for which the model shall be verified and validated. Next are the issues to be addressed, presenting the key concepts of the system model to be developed, followed by the proposed research methodology. The last section presents an overview of the dissertation.

1.1 Background

This section firstly presents a short history of microwaves and microwave ovens. The connection between microwaves and the cleaning of laundry is then presented. This is followed by an overview of the prototype MCWD system and lastly an overview of the model development is given.

1.1.1 Microwaves and the microwave oven

As illustrated in figure 1.1 microwaves refer to electromagnetic (EM) waves in a portion of the EM spectrum with frequencies ranging from 300 MHz up to 300 GHz and with the respective wave lengths from 1 m down to 1 cm [1]. Microwaves have numerous applications in industry ranging from radar to telecommunications [2]. Other interesting applications include the drying of paper, textiles, photographic film and leather. In heavy industrial applications microwaves are used to extract oil from tar sands, cross link polymers and for vulcanisation and casting [3].

A common and probably the largest application of microwaves is the heating of foodstuffs in microwave ovens. Microwave ovens use a magnetron to generate microwaves, as the magnetron is a very common, cheap and easily manufactured microwave generator. The development of the magnetron and the first industrial or military application of microwaves date back

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Chapter 1. Introduction 1.1. Background

Figure 1.1: Electromagnetic spectrum [1]

to the second world war, where microwaves were used in the first 10 cm wavelength radars [1, 4, 5, 6]. Engineers searched for an alternative application for the magnetron knowing that after the war the demand for magnetrons would decrease. This search lead to the development of the microwave oven [7, 8], to be used in the food industry to heat foodstuffs. From that time onwards various new uses and applications of microwaves were developed as previously mentioned. One of these new applications is the use of microwaves to assist in the cleaning of textiles. This process, named microwave assisted laundry (MAL) revealed that when microwaves are introduced into the chamber during the wash cycle, the microwaves aid in the cleaning of the textiles. Using the microwaves reduces the amount of water needed to clean textiles as well as reducing and in some cases eliminating the need for detergent[2].

1.1.2 Cleaning with microwaves

The MAL process holds a few advantages over the normal washing process, which include using less water and detergent during the washing of textiles, and less time needed to clean certain stains [2]. These advantages that MAL present and the need for a small effective washing machine formed the basis of the motivation for a Pretoria based South African company Delphius Commercial and Industrial Technologies (Pty) Ltd. to develop a MCWD prototype, seen in figure 1.2. Preliminary studies completed by Delphius using the prototype MCWD indicated that the concept of MAL is sound. Further preliminary tests adhering to the standard testing procedure of a washing machine as defined by the South African Bureau of Standards (SABS) provided promising results.

The scope of the work conducted in this dissertation is limited to the model development that will aid in the development of a control system. Delphius remains responsible for the

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Chapter 1. Introduction 1.1. Background

Figure 1.2: Laboratory prototype microwave combination washer dryer.

providing the prototype system and expertise in the field of microwaves, microwave heating and cleaning of textiles with microwaves.

1.1.3 Prototype MCWD system

The first prototype of the system was manually controlled during the investigation to determine the feasibility of the MCWD development. The automation of the prototype and the need to study the MCWD system and various aspects thereof motivated the development of a numerica model. The model would enable the conduction of a parametric study on system parameters. By varying certain parameters, such as the microwave power level, the response of the MCWD can be studied. Results from the parametric study would enable a neural network model to be trained, with the advantage of allowing faster simulated responses compared to the model. Using a neural network model enables the selection of a control scheme best suited to the MCWD system.

The laboratory prototype MCWD makes use of a standard 1 kW microwave oven magnetron. Furthermore the first prototype seen in figure 1.2 is constructed using a normal square application chamber, with the washing drum placed within the chamber. The drum is actuated from outside the chamber and can be rotated to mimic the operational cycle of a normal washing machine. The second prototype system developed by Delphius has the form factor and feel of a small washing machine, the design of which is the patented property of Delphius and is not illustrated in this dissertation. Here after when referring to the system, this referral is aimed at the first prototype system as used for this study.

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Chapter 1. Introduction 1.2. Problem statement

Application chamber

Washing dum

Magnetron Isolator and

water load Transformer

Variac

Distillation tower

Figure 1.3: Schematic diagram of MCWD system.

The schematic diagram of the first prototype (Figure 1.3) presents the main components of the system. The variac is connected to the mains power, and is used to adjust the voltage supplied to a high voltage transformer. By adjusting the input voltage, the amount of microwave power generated by the magnetron is adjusted. The magnetron is connected to an isolator and water load via a standard wave guide for 2.45 GHz waves. The isolator isolates the magnetron from the application chamber, and diverts all microwaves reflected from the application chamber into a water load. Therefore the connection from the magnetron to the isolator is effectively one way, whereas the waveguide between the chamber and the isolator allows bidirectional microwave propagation. The water load requires water to be circulated to aid in the dissipation of heat generated by the reflected microwaves.

The system is manually operated and no automation is available for the first prototype. To adjust the load within the washing drum, the system needs to be powered down before the drum is removed from the system to open the drum hatch. Microwave power levels are manually adjusted during the operation of the system. To prevent any accidental exposure to microwaves, the application chamber of the system is fitted with an interlock relay to disable power to the magnetron.

1.2 Problem statement

The purpose of this project is to develop a model for the microwave absorption and heat transfer in a MCWD. The model should integrate the microwave absorption and thermal domains to the extent that accurate prediction of the temperature in the washing drum is achieved. Furthermore the thermal model should also account for advanced fluid dynamics

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Chapter 1. Introduction 1.3. Issues to be addressed

The model of the system will be validated for a microwave power range from 0 W to 1 kW and an average temperature ranging from 10◦C to 70◦C for the wash load. The microwave power range is limited by the magnetron in the system, whereas the temperature range is selected to exclude water undergoing a phase change as the temperature approaches freezing or the boiling point of water. The model is limited to the wash cycle of the system and will not account for the drying phase of the system as the model does not account for the phase change of water too gas. The model will also not account for the rotation of the washing drum at this stage and will be left for future work on the model. Finally the target load will be distilled water as the material properties of water is a know quantity and it is not in the scope of the research to investigate the thermal and electrical properties of various mixtures of materials, water and detergent.

1.3 Issues to be addressed

This section presents a few issues to be addressed during the model development process. In the following section a methodology to solve these issues is presented. The first issue to be addressed is the development of the system model, followed by the laboratory implementation of the MCWD system. The laboratory implementation will address the issue of gathering experimental data to validate the model. The last issue to be addressed is to make use of experimental data to evaluate the model.

1.3.1 System model development

The two main physics phenomena simulated by the model are the heat transfer and microwave propagation within the system. Each of these phenomena have their own issues to be addressed, which will be presented in this section. The electromagnetic phenomenon part of the model will implement the following aspects and features:

• Microwave model aspects:

– The generation of microwaves at the wave guide port; – Propagation of microwaves within the system;

– Microwave power dissipation in various parts of the system. • Microwave model features:

– Microwave power flow through the wave guide; – Power dissipation density within model domains;

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Chapter 1. Introduction 1.3. Issues to be addressed

– Electrical field distribution throughout the system.

The heat transfer phenomena part of the model shall implement the following: • Heat transfer model aspects:

– The generation of heat within the target using the calculated power dissipation from the microwave model;

– Heat transfer between domains, both for solids and fluids;

– Laminar or turbulent fluid dynamics for the air within the system; – Convective cooling of the system enclosure.

• Heat transfer model features:

– Temperature of model domains at each time step; – Velocity fields of the fluid flow;

– Configuration of different initial system temperatures for different model domains.

1.3.2 Gathering experimental data

The next issue of the dissertation is to gather experimental data to use during the evaluation of the model. The secondary issues include the selection of the appropriate temperature sensors, installation of these sensors and the interfacing of all sensors with a platform that would enable the capturing of the measured sensor data. Keeping the model evaluation in mind, the implementation of the system should allow for the system to be operated with various configurations or conditions e.g. different microwave power levels or different loads within the washing drum.

1.3.3 Model evaluation

The final issue to be addressed is the evaluation of the model. Model verification and validation and the understanding of this terminology vary from research topic to research topic. Clarification on what these terms mean in the context of this dissertation is presented in this section. Verification of the model is regarded as the process to check whether the model responds as it is expected to respond based on literature. Validation of the model is seen as the step in which the results of the model are compared to measurements taken from the prototype to determine how accurately the model can simulate the prototype system.

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Chapter 1. Introduction 1.4. Research methodology

1.4 Research methodology

1.4.1 System model development

The development of the model for the MCWD system requires the knowledge of the physics phenomena that may be present within the system. The phenomena will therefore be identified and the relevant literature on these will be studied. The FEM is selected to model the system. By using the multiphysics software package COMSOL®that implements FEM, it would enable the modelling of a full three dimensional (3D) representation of the system. Before a FEM model can be developed a basic understanding of the system needs to be formed. This is done by studying models developed from first principles for microwave absorption and heat transfer in various systems.

Model development starts with a two dimensional model of the system simulating the basic heat transfer within the washing drum and continues to gain complexity with each model stage building on the preceding stage of development. This stage based development concludes with the development of a full 3D model that is based on the geometry of the laboratory prototype of the MCWD. The final full 3D model accounts for both the microwave propagation and heat transfer within the system.

The study attempts to develop a model that would accurately represent the system, while at the same time is simplistic enough to save on computation time. Various system parameters are therefore investigated to determine whether the parameter needs to be represented accurately, or whether an approximation of the parameter could be implemented. For instance, the complex dielectric constant of materials is a function of temperature and would therefore affect the amount of microwave power absorbed. Another example of such an investigation is to determine if the model needs to account for laminar or turbulent flow of fluids.

1.4.2 Gathering experimental data

The task of gathering experimental data requires that temperature sensors be installed at various positions within the system. Since the voltages induced by the sensors are small and need to be amplified, an interface or signal conditioning printed circuit board (PCB) will be designed and manufactured. This PCB will enable the data capturing hardware to interface with the sensors of the system.

The data capturing platform selected for the logging of sensor data is the dSpace® platform which provides the hardware and software needed to perform the task. The hardware is controlled by developing a Matlab® Simulink® diagram to graphically represent the connection between the analog to digital converters and the software data processing

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Chapter 1. Introduction 1.5. Dissertation overview

implemented in real time before the data is exported to a file containing the logged data. The dSpace® environment enables the realtime presentation of data while the data is being logged by making use of the software front end ControlDesk® . The ControlDesk® environment enables emulations of instrumentation to display data captured from the sensors. A ControlDesk® layout shall be developed to display the voltages measured by the hardware and to display the realtime calculated microwave power levels alongside the temperatures. During the capturing of data, the system is manually operated at the required microwave power level by actuating a variac, since the prototype system does not have a controller.

1.4.3 Model evaluation

Model development was broken into stages aiding in verification of the model. Stage one of the model will start with a basic model taking into account the basic physics phenomena within the system. With each progressive stage the model will be refined by taking into account an additional aspect or phenomena. For instance, stage one simulates the heat exchange within the washing drum, and stage two additionally models the air surrounding the drum to the model. This incremental modelling will enable each stage to be compared to literature to verify how the respective stage of the model reacts as it is expected before the model is expanded by the next model stage.

The model results are validated by comparing these results to measurements from the prototype system. Multiple tests will be conducted using the prototype system, by varying the microwave power levels. Using the information from the sensors, the average ambient temperature over all the sample points collected is calculated and used to define the ambient temperature within the model. Secondly, the average of temperature at time step zero using all five sensors within the system is calculated and used to define the initial temperature of the model. Lastly the average microwave input power is calculated from all the collected sample pints and used to define the input power for the model. After the model parameters are defined, and the simulation is completed, the results from the model are compared to the data from the system prototype to judge how accurately the model represents the system.

1.5 Dissertation overview

Chapter two of the dissertation presents the relevant literature. The literature provides an essential background to the development of a multiphysics model for the MCWD system. Topics discussed include first principle concepts of heat transfer, thermodynamics and electromagnetic waves. Further topics include the generation of microwaves using the

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Chapter 1. Introduction 1.5. Dissertation overview

with the concepts of computer modelling, specifically FEM modelling, and a critical review of the literature presented in the chapter.

Following the literature overview, chapter three discusses the model development. It firstly discusses an analytical model derived from first principles to form an understanding of the modelling problem before discussing the FEM model. The FEM model discussion in chapter three presents the whole development procedure describing the methodology and assumptions made during the development.

Chapter four of the dissertation presents the results obtained from the simulation and discusses the verification of the model data. The verification stage is to test whether the model responds to the parameters as expected as well as to verify if the results obtained agree with the literature.

Chapter five of the dissertation presents the work completed to enable the practical implementation of the prototype system. Work conducted during the implementation phase includes the setup and calibration of sensors and the configuration of the data logging system. Chapter six discusses the model validation where the results from the model are compared to the measurements taken from the prototype system. The validation of the model is to ensure the model results are correct and that the model can be used to approximate the system for various input parameters.

The dissertation concludes with chapter seven presenting the conclusions and recommendations for this dissertation. Issues that are encountered during the research period are discussed. Lastly possible future work is discussed.

Following the background to the dissertation as presented in this chapter the second chapter of the dissertation will present the literature needed to form a solid basis for the dissertation.

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Chapter 2

Literature study

This chapter presents an overview of the literature considered to complete this dissertation. It starts with an introduction on work completed and available literature on the numerous models developed for microwave absorption. Following the introduction are the concepts of heat transfer and thermodynamics, since a large part of the model makes use of these concepts to model the heat transfer. This chapter next presents the literature on EM waves, used by the model to simulate the microwave propagation and absorption. An overview of magnetron microwave generation is then presented. Following the literature on heat transfer, EM waves and microwave generation, the next section will discuss heating with microwaves. Second to last is a section on computer modelling and the common methodology followed to solve modelling problems. Finally a critical review of the literature in the context of the dissertation is presented.

2.1 Introduction

Extensive work has been done by various researchers to develop microwave absorption models for various applications and industries of which a few are mentioned here. Budd et.al. [9] presented a comparison of models for heating moist foodstuffs using a one dimensional approximation. Ratanadecho et.al. [10] investigated the impact that natural convection and the dielectric properties of liquids have when liquids are being heated with microwaves. Salvi et.al. [11] presented work done on the development of a COMSOL®model for continuous flow microwave heating of liquids. Pandit et.al. [12] conducted work into the development of a two dimensional FEM model to predict the transient microwave heating of solid foodstuffs. Chen et.al. [2] presents the initial work done on the cleaning of textiles with microwaves and defines the MAL term. Drying textiles with microwaves is also of interest with, Vrba et.al. [13] presenting such work. Interesting work has been done by Pengfei et.al. in the attempt to

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Chapter 2. Literature study 2.2. Thermodynamics and heat transfer

Literature on industrial microwave heating, addressing all the topics needed to grasp the subject of heating with microwaves is presented by Metaxas et.al. [15], and is complimented by Meredith [16]. These references discuss the basic concepts of heat transfer and focus more on the microwave aspect of the process, with advanced topics such as the design of microwave application chambers. To fully understand some of the more complex concepts of heat transfer and thermodynamics, the reference list is expanded with the literature from [17, 18, 19], that present in depth discussions on the concepts needed to grasp heat transfer and thermodynamics. Topics of interest include natural convection, heat conduction, heat transfer in fluids and solids as well as laminar and turbulent flow.

2.2 Thermodynamics and heat transfer

Thermodynamics is described by [18] as the science of heat. The name thermodynamics originates from the two Greek words therme (heat) and dynamis (power). The first and second laws of thermodynamics capture the concept of heat science. The first law formulates the principle of energy conservation with the second law defining that energy has quality and quantity. Thermodynamics is concerned with the amount of energy being transferred between two mediums, or a medium and its environment. The science of heat transfer is concerned with the transient aspect of this energy transfer, i.e. how long does the transfer process take until the mediums reach equilibrium [18].

Heat transfer or the flow of heat is described by [19] as being all-pervasive, i.e. it occurs everywhere to a greater or lesser extent. The temperature difference between two objects gives rise to a thermal gradient. This thermal gradient will level out when the heat is transferred. The levelling of the thermal gradient is described as the driving force behind heat transfer. The heat transfer process will stop when the thermal gradient is level, in other words the system reaches thermal equilibrium. Therefore the concept of heat transfer is that heat flows from the hotter object to the colder object. Since the levelling of the thermal gradient drives heat transfer, a temperature difference is required between objects if any heat transfer is to take place. From this it can be concluded that when the temperature difference between objects increases, the temperature gradient becomes steeper, and the rate of heat transfer will also increase [18, 17, 19].

The authors of [19] agree that thermodynamics alone is not enough to describe the energy transfer between two items. A combination of the first and second laws of thermodynamics and the transport laws, including Newton’s law of cooling, Fourier’s law and the Stefan-Boltzmann law are needed to completely describe the energy transfer process [19]. Presented in this section of the dissertation is the literature on both the concepts of thermodynamics and heat

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(a) (b)

Figure 2.1: (a) Control mass system. (b) Control volume system. Adapted from [18]

transfer. The literature presented here will only be an overview of the basic concepts considered important in the context of the dissertation.

2.2.1 Thermodynamics concepts

In terms of thermodynamics, a system is defined as a quantity of matter or a region in space. Everything outside the system is referred to as the surroundings. The separation between the system and surroundings is visualised as an imaginary wall or boundary to the system. Typically a system is classified as one of two types. Firstly, a closed system or control mass, refers to a system that does not allow mass to cross the system boundaries, but does allow energy in the form of heat to cross the boundary (Figure 2.1(a)). The second classification for a system is an open system or a control volume system, which allows for both mass and energy to cross the boundary of the system (Figure 2.1(b)). When energy is transferred from a system due to a temperature difference the energy is defined as heat, otherwise it is defined as work [18].

Thermodynamics is mainly concerned with the first and second laws of thermodynamics and the interaction of heat, and is used for calculations when materials are in equilibrium. For instance, thermodynamics can be used to calculate the amount of energy needed for a material to transition from one equilibrium state to another [20].

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2.2.1.1 First law of thermodynamics

The first law of thermodynamics, also known as the conservation of energy principle, states that, “energy can neither be created nor destroyed during a process; it can only change forms” [17, ?]. For a system undergoing any process the energy balance equation is defined as

Ein− Eout = ∆Esystem, (2.1)

where Einis the energy transfer into the system and Eoutthe energy transfer out of the system.

∆Esystemtherefore represents the net change in the internal energies of the system. The energy

transfer mechanisms are defined as heat transfer, work and mass flow [17, 18].

As previously defined, two classes of systems exist. For a control volume system there are sub-classifications. The first classification is the steady flow process, which experiences no change in the flow of mass through the control volume over a period of time. Therefore the mass and energy content of a control volume system will remain constant, since the flow rates of the inlet and outlet to the control volume remain constant. The second classification is the unsteady flow process, except that the control volume experiences a mass change during the process. For an unsteady flow system the inlet and outlet to the control volume are averaged and treated as constant during the entire process, and can therefore be modelled as a uniform flow process.

2.2.1.2 Second law of thermodynamics

The second law of thermodynamics, as defined by [18], states that “processes occur in a certain direction, not in any direction.” Furthermore, a process of heat exchange will not occur unless the process adheres to both the first and second law of thermodynamics. Commonly the second law is used to determine the theoretical limits of system performance, for example a heat exchanger. The second law also asserts that the energy of a system has quality and quantity [18].

A process is defined to be reversible only if both the system and the surroundings can be restored to their original conditions before the process was initialised. To verify that a process does not violate the second law of thermodynamics, a property defined as entropy is used. The concept of entropy is hard to describe, and it is commonly viewed as a measure to indicate the molecular disorder or molecular randomness of the medium. Entropy S can be calculated using [18],

dS = δQ T

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with δQ the rate of heat transfer and T represents temperature. Important aspects of entropy, include that a process must occur in the same direction of the entropy increase otherwise the process is impossible. Secondly, entropy can be used as a measure of a system’s irreversibility, since the more entropy generation takes place during a process, the greater the extent of the irreversibility. Thirdly, entropy is an extensive system property and lastly entropy is non conserved property, i.e. there is no principle for the conservation of entropy. The entropy of a system is related to the number of possible microscopic states of the system, called the thermodynamic probability, with Boltzmann’s relation [18],

S = k ln p, (2.3)

where S is the entropy, k = 1.3806 × 1023 J/K is the Boltzmann constant and p is defined as the thermodynamic probability [18].

2.2.1.3 System and material properties

Properties or characteristics of a system that should be familiar to all, with basic science knowledge, is the concepts of pressure P , volume V , mass m and temperature T . As [18] describes some properties of systems are defined in terms of other properties, such as density being defined as the mass per unit volume:

ρ = m

V [kg/m

3_], _(2.4)

where the density is ρ, m represents the mass and the volume is presented by V . Density is a function of temperature and pressure, where for instance the density of gas under pressure will increase. Mostly the density of common solids and liquids is assumed to be constant. Water for instance at a temperature of 3.98 ◦C has a density of 1000 kg/m3 [21] vs. the density of 958.38 kg/m3 _{at 100} ◦_{C [21]. This change in density amounts to 4.162 %, which for most}

calculations is small enough so that the density can be assumed constant, where more advanced models will take these changes into account in density [18].

The density of a substance is sometimes defined relative to the density of a standard substance, typically water at 4◦C with ρH2O= 1000 kg/m

3_{. This method of definition is called the specific}

gravity (SG) or relative density, and is defined by [18]: SG = ρ

ρH2O

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reciprocal of the density [18]:

v = V m =

1 ρ [m

3_/kg]. _(2.6)

Properties of a system can be grouped into one of two categories, either intensive or extensive. Intensive properties are independent of the system size, where extensive properties vary based on the size of the system. Extensive properties are commonly referred to as specific properties, for example the specific volume of a material [18].

Specific heat C of a material or substance, is defined as “the energy required to raise the temperature of a unit mass of a substance by one degree” [18]. Specific heat is subdivided into two types, the specific heat at constant pressure Cp and the specific heat at constant volume

Cv [18]: Cp= ∂h ∂T p [J/kg · K], (2.7) Cv = ∂u ∂T v [J/kg · K], (2.8)

with Cp the specific heat at constant pressure, Cv the specific heat at constant volume, h the

specific enthalpy and u the specific internal energy. The specific heat at constant pressure and volume of ideal gasses can be related by the gas constant R with [18]

Cp= Cv+ R, (2.9)

or by the specific heat ratio k defined as [18] k = Cp

Cv

. (2.10)

2.2.2 Heat transfer concepts

The heat transfer process is classified into one of three methods of heat transfer: conduction, convection and radiation. The topic of heat transfer is concerned with the nonlinear or non-equilibrium part of the process, and what happens between two non-equilibrium states. Therefore the heat transfer equations of the three heat transfer processes are used to quantify the rate of heat transfer [20].

Heat conduction is defined as, “The transfer of energy from the more energetic particles of a substance to the adjacent, less energetic ones as a result of interactions between the particles” [17]. Fourier’s law of heat conduction is defined as [18]

˙

Qcond= −kA

dT

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with ˙Q the rate of heat transfer, k the thermal conductivity of the material, A the heat transfer area normal to the direction of conduction, T the temperature and x the distance of conduction. The thermal conductivity k of a material is defined as “the rate of heat transfer through a unit thickness of the material per unit area per unit temperature difference” [18].

Conduction can take place in all three phases of materials. In fluids and gasses the heat conduction is because of the collision and diffusion of the particles. In solids it is due to a combination between the vibrations of particles in the material lattice and the energy transferred by free electrons. The rate of conduction through a medium depends on the geometry, thickness and material of the medium [18].

Two properties commonly encountered during a transient heat transfer analysis is the heat capacity of a material and the thermal diffusivity of a material. The heat capacity of the material is defined as the product ρCp [J/m3· K] and the thermal diffusivity α is defined as

α = k ρCp

[m2/s], (2.12)

with k [W/m · K] the thermal conductivity of the material. The thermal diffusivity is the rate at which heat diffuses through a material and is also the ratio of heat conducted vs. the heat stored.

Heat convection is defined as, “The mode of transfer between a solid surface and the adjacent liquid or gas that is in motion, and it involves the combined effects of conduction and fluid motion” [17]. The rate of convection is proportional to the temperature difference and is defined by Newton’s law of cooling as

˙

Qconv= hAS(TS− T∞) [W], (2.13)

with h [W/m2· K)] the convection heat transfer coefficient, A_S[m2] the surface area over which convection takes place, TS[K] the surface temperature and T∞[K] the temperature of the fluid

adjacent to the solid surface at a sufficient distance from the surface. The convection heat transfer coefficient h is defined as the heat transfer between a solid surface and a fluid per unit surface area per unit temperature [18].

Heat radiation is defined as, “The energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules” [17]. For a heat transfer analysis, the main focus typically falls on thermal radiation, which is as a result of the temperature of a medium and no attention is paid to

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Chapter 2. Literature study 2.3. Electromagnetic waves

The rate of radiation from a surface is governed by the maximum rate of radiation possible, defined by the Stefan-Boltzmann law. The law is defined as

˙

Qemit,max= σASTS4 [W], (2.14)

with σ = 5.67 × 10−8[W/m2· K4_{] the Stefan-Boltzmann constant, A}

S[m2] the area of radiation

and TS [K] the surface temperature. A black body can be defined from this equation . A black

body is an ideal surface that can radiate at the theoretical maximum rate. The radiation from real surfaces is related to a black body by the emissivity of the surface, and is defined by as [18]:

˙

Qemit = σASTS4 [W]. (2.15)

The emissivity is defined as “the ratio of the radiation emitted by the surface at a given temperature to the radiation emitted by a blackbody at the same temperature” [18] and takes a value between 0 and 1, 0 ≤ ≤ 1. Similarly the absorptivity of a surface is defined as α with 0 ≤ α ≤ 1 [18].

2.3 Electromagnetic waves

Since the invention of the first radios the role that EM waves play in our lives has dramatically changed from sending simple Morse code, to transmitting any type of analogue or digital data wirelessly. Not only is EM waves used for data communications or radar, it is widely used in the food industry to heat foodstuffs. Industrial microwave heating of materials also plays a large role in various industrial processes.

When working with EM waves, the equations presented by Maxwell define the reaction and propagation of the waves. The four Maxwell equations come in various formulations for different applications. For instance, waves in a vacuum and waves in a conductor adhere to the same equations, but certain parts of the equation do not necessarily apply to vacuum as it does when a wave travels along a conductor.

2.3.1 Maxwell’s equations

2.3.1.1 Electromagnetic waves in a vacuum

Maxwell’s equations in a vacuum, or a region with no charges or currents present, are defined as [22],

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Chapter 2. Literature study 2.3. Electromagnetic waves (i) ∇ · E = 0, (iii) ∇ × E = −∂B ∂t, (ii) ∇ · B = 0, (iv) ∇ × B = µ00 ∂E ∂t,          (2.16)

with E [V/m] the electric field intensity, B [T/m] the magnetic field intensity, µ0 the

permeability of free space and 0 the permittivity of free space. The equations can be

represented in a decoupled form by taking the divergence of Maxwell’s equations [22],

∇ × (∇ × E) = ∇ (∇ · E) − ∇2E = ∇ × −∂B ∂t = −∂ ∂t(∇ × B) = −µ00 ∂2E ∂t2 , (2.17) ∇ × (∇ × B) = ∇ (∇ · B) − ∇2B = ∇ × µ00 ∂E ∂t = µ00 ∂ ∂t(∇ × E) = −µ00 ∂2B ∂t2 . (2.18)

The final decoupled second order equations take the form [22] of,

∇2E = µ00 ∂2E ∂t2 , (2.19) ∇2B = µ00 ∂2B ∂t2 . (2.20)

2.3.1.2 Electromagnetic waves in a linear medium

For EM waves travelling within a linear medium and in a region where no free charges or currents are present, Maxwell’s equations as defined in (2.16) hold true and are rewritten [22], (i) ∇ · D = 0, (iii) ∇ × E = −∂B ∂t, (ii) ∇ · B = 0, (iv) ∇ × H = ∂D ∂t .          . (2.21)

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D = E, H = 1

µB.

(2.22)

Finally because the media is homogeneous Maxwell’s equations reduce to [22],

(i) ∇ · E = 0, (iii) ∇ × E = −∂B ∂t, (ii) ∇ · B = 0, (iv) ∇ × B = µ∂E

∂t.          (2.23)

Noted from the equations the only difference between Maxwell’s equations for a vacuum and within a medium is the replacement of the vacuum parameters (µ0 & 0) with the medium

properties (µ & ).

2.3.1.3 Electromagnetic waves in conductors

Ohm’s law defines the current density Jf to be proportional to the electric field [22],

Jf = σE, (2.24)

with σ the surface charge density. Taking into account the definition of the free current density, Maxwell’s equations can be rewritten as [22],

(i) ∇ · E = 1

ρf, (iii) ∇ × E = − ∂B

∂t, (ii) ∇ · B = 0, (iv) ∇ × B = µσE + µ∂E

∂t,          (2.25)

with ρf the free charge density. By simultaneously making use of the continuity equation for

free charge, Ohm’s law and Gauss’s law [22],

∇J_f = −∂ρf ∂t [A/m 2_], _(2.26) and, ∂ρf ∂t = −σ(∇ · E) = − σ ρf. (2.27) It follows that [22],

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ρf(t) = e−(σ/)tρf(0). (2.28)

Any free charge density initially on the conductor will therefore dissipate in the characteristic time of τ ≡ /σ. As in [22], it is assumed that there will be a time laps and the accumulated free charge will have time to disappear. With ρf = 0, it follows that [1, 22],

(i) ∇ · E = 0, (iii) ∇ × E = −∂B ∂t, (ii) ∇ · B = 0, (iv) ∇ × B = µσE + µ∂E

∂t.          (2.29)

When the curl is applied to (iii) and (iv) the resulting wave equations for E and B are defined [22], ∇2E = µ∂ 2_E ∂t2 + µσ ∂E ∂t, (2.30) ∇2_{B = µ}∂2B ∂t2 + µσ ∂B ∂t. (2.31)

2.3.2 Plane wave propagation

Electromagnetic waves can propagate inside and/or through various mediums. The types of mediums are typically divided into three groups of materials; lossless mediums, general lossy mediums and good conductors. The way EM waves propagate inside a medium is determined by a few factors, such as the propagation constant, phase constant and skin depth [1]. With a linear and homogeneous medium free from any sources Maxwell’s curl equations can be presented in phasor form [1],

∇ × E = −jωµH, (2.32)

∇ × H = jωE, (2.33)

with ω the radian frequency. By taking the curl of (2.32) and substitution of (2.33), it follows that [1]

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Table 2.1: Results for plane wave propagation in various mediums [1] Type of medium

Quantity Lossless (00= σ = 0)

General lossy Good conductor 00 0 _{or σ ω}0 Complex propagation constant γ = jω√µ γ = jω√µ = jω√µ0 r 1 − j σ ω0 γ = (1 + j)pωµσ/2 Phase constant (wave number) β = k = ω√µ β = Im(γ) β = Im(γ) = pωµσ/2 Attenuation constant α = 0 α = Re(γ) α = Re(γ) = pωµσ/2 Impedance η =pµ/ = ωµ/k η = jωµ/γ η = (1 + j)pωµ/2σ Skin depth δS = ∞ δs= 1/α δs =p2/ωµσ Wavelength λ = 2π/β λ = 2π/β λ = 2π/β Phase velocity (wave velocity) vp = ω/β vp= ω/β vp= ω/β

When making use of the vector identity ∇×∇×A = ∇(∇·A)−∇2A, (2.34) can be rewritten to the form [1]

∇2E + ω2µE = 0. (2.35)

Equation (2.35) is the resulting wave equation for E. Similarly, an equation for H can be derived in the form of [1],

∇2H + ω2µH = 0. (2.36)

The wave number is defined as [1],

k = ω√µ [1/m]. (2.37)

An understanding of the wave equations is necessary and therefore the propagation of plane waves in all three groups are studied. Table 2.1 is a comparison between the three groups. Refer to [1] to view how this information is derived.

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Chapter 2. Literature study 2.4. Microwave generation

2.4 Microwave generation

With the advent of microwaves in a wide range of applications, specifically in the field of microwave heating, a basic understanding of microwave generation is important when designing a model for microwave absorption. For this dissertation the focus of microwave generation will be magnetrons. The amount of literature that can be presented on the topic of microwave generation using magnetrons is far too great for this dissertation. A summary of the most relevant literature will therefore be presented. For more information refer to [23, 24].

2.4.1 Magnetron as microwave generator

Various methods for generating microwaves are available, including Klystrons, Magnetrons, Travelling-wave thermionic devices, Gyrotrons, Magnicons, Ubitrons and Peniotrons. The first magnetrons date back to 1940 during World War II, with a power output of several hundred watts with a wave length of 9.8 cm [6, 23, 24]. The magnetron, because of its efficiency and versatility is widely used for the generation of microwaves in the industry of microwave heating.

2.4.1.1 The development of the magnetron

The most basic form of the magnetron is a vacuum diode placed in a magnetic field parallel to the axis of the diode. Authors of Microwave Magnetrons [24] are in agreement that Hull’s investigation [25] into non-oscillating diodes within magnetic fields provided much of the theoretical understanding of how magnetrons operate. Hull’s work focused on how electrons behaved in a cylindrical diode with the strong magnetic field with an orientation parallel to the axis of the diode. Seen in figure 2.2(a) is a representation of such a diode used in the experimental work of Hull.

The construction of the diode of figure 2.2(a) can be described as follows. The cathode is centrally placed with respect to the surrounding cylindrical anode, which in turn is placed in a nearly uniform magnetic field parallel to the axis of the diode. Electrons are released from the cathode when the cathode is heated. When the electrons are exposed to the electrical and magnetic fields the electron follows a quasi-cycloidal orbit seen in figure 2.2(b)

By connecting the non-oscillating diode to a resonant circuit, the diode can be made to oscillate at very high frequencies. The study of these diodes connected to a resonant circuit was the groundwork for the development of the first magnetrons [24]. As this section of the dissertation will not cover the whole development history of the magnetron, refer to [24] for a more detailed

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Chapter 2. Literature study 2.4. Microwave generation

2.4.1.2 Magnetron efficiency and frequency stability

The typical applications of magnetrons require both high efficiency and frequency stability. To obtain both is not possible and most magnetrons are designed to achieve a compromise between the efficiency and stability. Depending on the application of the magnetron various designs of oscillators can be used at a certain frequency, where each of these designs will have a different impedance. The design of the oscillators depend on the impedance the magnetrons is required to have for a certain application [24].

Typically the losses within a magnetron are described by two types of losses, and to help distinguish between the two, the total magnetron efficiency is broken into two components. One is the electronic efficiency ηe and the second is the circuit efficiency ηc with η = ηeηc.

Electronic efficiency ηe is defined as the ratio of input power to the radio frequency (RF)

power generated within the magnetron, where the circuit efficiency ηcis defined as the fraction

of RF power transferred to the load [24].

The most efficient magnetron design is then one where there is a compromise between a high impedance oscillator with a high circuit efficiency and a low impedance oscillator with a high electric efficiency. Under normal operational conditions if the load on the magnetron is increased, the efficiency of the magnetron will rise. The opposite is also true:too much load does not necessarily result in an efficiency drop. An overload on the magnetron can result in a drop in efficiency in some cases[24].

(a) Cylindrical diode

(b) Quasi-cycloidal orbit of electron

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Chapter 2. Literature study 2.5. Heating with microwaves

2.5 Heating with microwaves

Microwave heating works on the principle of converting the electromagnetic energy carried by the microwaves into heat within the work load or target being heated. The heat generated is because of dielectric losses in the materials exposed to the microwaves. This section of the dissertation will present the literature on the aspects of microwave heating applicable inside the application chamber, and how the microwaves interact with the target being heated. Refer to [15] for more information on the different types of dielectric losses and the exact role these losses play in microwave heating.

2.5.1 Target power dissipation

One of two methods is commonly used to derive the equation to determine the power dissipated within the target. The first method is to derive the equation from first principles using Maxwell’s equations. A second approximation of a parallel capacitor with a dielectric between the two plates is used to derive the equation. Once the power dissipated within the material is calculated, it can be used to determine the temperature rise within the material during a defined time period [15].

2.5.1.1 Power dissipation using first principles

It is argued by [15] that the amount of power flowing through a closed surface can be calculated by integrating the Poynting vector of the electromagnetic wave passing through the surface,

ρ = E × H [W/m2]. (2.38)

By first making use of Maxwell’s current law [15],

∇ × H = J + jω0∗E, (2.39)

and by substituting J = σE along with ∗ = 0− j00, (2.39) it can be shown that [15],

∇ × H = σE +ω0 00 + jω0 0 E = ω0 00 ef fE + jω0 0 E, (2.40)

with 00_{ef f} = 00+ σ/ω0, and where

00

is the permeability, representing all of the losses present except for the conductivity losses. The dot product of (2.40) using E follows as [15],

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Chapter 2. Literature study 2.5. Heating with microwaves (∇ × H) · E = ω0 00 ef fE · E∗− jω0 0 E∗· E. (2.41)

By using Maxwell’s third law (∇ × E) and calculating the dot product using H∗ it presents [15],

(∇ × E) · H∗ = −jωµ0µ

0

H · H∗ (2.42)

If one subtracts (2.41) from (2.42) it yields [15],

(∇ × E) · H∗− (∇ × H∗) · E = − jωµ0µ 0 H · H∗ + jω0 0 E∗· E − ω₀00_{ef f}E · E∗ (2.43) By integrating (2.43) over a volume V and by using the divergence theorem, it can be calculated that [15], Z V ∇ · (E × H∗) dV = Z S (E × H∗) · dS0 = −jω Z V µ0µ 0 H∗· H − 0 0 E · E∗dV − Z V ω0 00 ef fE · E ∗ dV. (2.44)

The definition for average power Pav, used by [15], is presented as,

Pav= − 1 2 Z S Re (E × H∗) · dS0, (2.45) from which it follows that [15],

Pav = − 1 2ω0 00 ef f Z V (E∗· E) dV. (2.46)

During practical application of microwave heating it follows that the electric field is not uniform and varies in intensity from one point in space to another. In the special case that the electric field can be assumed to be uniform and the average power (2.46) can be simplified to be [15],

Pav = ω0

00

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In the event that the material being heated by microwaves exhibits magnetic losses in addition to the electrical losses, the average power dissipated is calculated by using [15],

Pav = ω0

00

ef fErms2 V + ωµ0µ

00

ef fHrms2 V. (2.48)

2.5.1.2 Power dissipation derived using approximation

Both [15] and [16] implement a second method to determine the average power dissipated in a material. The method makes use of a lossy capacitor within an alternating current (AC) circuit, such as seen in figure 2.3.

The voltage applied across the capacitor is [15],

V = ˆV ejωt, (2.49)

with ˆV being the peak voltage value. Total current flowing through the capacitor is calculated using [15], i = iR+ iC = V R0 + C dV dt . (2.50)

Differentiating equation (2.49) results in [15, 16], dV

dt = jωV. (2.51)

From the definition of a capacitor it follows that [15, 16],

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C = 0

0

A

d , (2.52)

where Ac is the capacitor plates area and d the distance between them [15, 16].

Substituting (2.51) and (2.52) into (2.50) yields [15, 16],

i = jω0 Ac d 0 − j 0 R0ωC ! V. (2.53) Following that [15, 16], Q = ωCR0 = 0 00_{ef f}, (2.54)

the maximum current density follows as [15, 16],

ˆ J = jω0∗E,ˆ (2.55) with [15, 16], ˆ E = ˆ V d. (2.56)

By making use of the power dissipated per unit volume [15, 16], 1

2Re( ˆJ

∗_{· ˆ}

E), (2.57)

and (2.55), the power per unit volume follows as [15, 16],

Pav/V = ω0 00 ef f E_rms2 2 = ω0 00 ef fErms2 . (2.58)

2.5.2 Rate of temperature change

Multiple parameters play a role to determine the rate of temperature increase of a target being heated with high frequency electrical waves. The power required to heat a selected material to a set temperature within a set time period can be calculated using [15],

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P = Qh t =

MaCp(T − T0)

t , (2.59)

where Ma[kg] is the mass, T0 mathsf [K] is the initial temperature and T mathsf [K] is the

target temperature of the material. By substituting (2.47) into (2.59), it follows that [15],

(T − T0) t = ω0 00 ef fErms2 ρCp [◦Cs−1]. (2.60)

It is noted from this equation that the rate of temperature increase depends on 00_{ef f}E_rms2 , where 00_{ef f} determines the amount of power dissipated and E_rms2 indicates the electrical field strength [15].

2.5.3 Electric field strength

The previous section indicated that rate of temperature increase is dependent on the electrical field strength. In actual fact, the electrical field strength is the most important factor when heating with microwaves or for that matter any high frequency waves [15]. To determine the electrical field strength within the target is not as straight forward as one may think. Typical mathematical methods to estimate the field is used, but care must be taken when implementing these methods to ensure the theory holds true. Another factor that complicates the calculation of the electrical field in the target is the dielectric properties of materials that play a role in the field calculations [15].

It is suggested by [15] that one can make use of calorimetry and use

Erms = ρCp(T − T0)/t 0.556 × 10−10_f00 ef f !1/2 [V/m], (2.61)

to estimate the value of the effective electrical field within the target. A second equation presented by [15] allows for the calculation of Emax using a measure of temperature increase.

This equation takes the form

P = 0.556 × 10−10f E_max2 00_{ef f}V (J₀2(KRw) + J

0₂

0 (KRw)), (2.62)

with Rwthe radius of the dielectric and K a constant. Substituting P into the equation enables

Model for microwave absorption and heat transfer in a combination washer dryer