Simulation of cement clinker process by of microwaveheating

José Pedro Machado Mendes

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. José Carlos Fernandes PereiraDr. Duarte Manuel Salvador Freire Silva de Albuquerque

Examination Committee

Chairperson: Prof. Carlos Frederico Neves Bettencourt da SilvaSupervisor: Dr. Duarte Manuel Salvador Freire Silva de AlbuquerqueMember of the Committee: Prof. Viriato Sérgio de Almeida Semião

June 2017

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Dedicated to my grandfather, who didn’t make it to see his grandson graduate.

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Acknowledgments

I would like to start by thanking to my supervisors Prof. Jose Carlos Fernandes Pereira and Dr.

Duarte Manuel Salvador Freire Silva de Albuquerque for their guidance and orientation. And specially

to Dr. Duarte for his friendship, dedication, sacrifice and availability.

I would like to thank my colleagues at Laboratory of Simulation in Energy and Fluids (LASEF) for

their companionship.

Finally i would like to give a special thank to my beloved family and friends for their unconditional

comprehension and support during the preparation of this document.

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Resumo

A producao convencional de clınquer de cimento e um processo que requer bastante energia, ne-

cessitando de 3600 kJ para obter um unico quilograma. Alem disso, uma vez que e um dos materiais

mais utilizados no mundo, a sua producao representa ate 6% do total de CO2 emitido pela atividade

humana. A tecnologia de aquecimento por via de micro-ondas tem ganho interesse como uma alter-

nativa devido as suas vantagens. E uma energia limpa e sem emissoes diretas de CO2, ao contrario

do aquecimento convencional. Aquecimento por micro-ondas ocorre quando o campo eletrico interage

com o material, resultando num aquecimento volumetrico no interior do material, o que se traduz em

maiores eficiencias.

Esta Dissertacao aborda o tema do aquecimento por micro-ondas na producao de clınquer de ci-

mento, mais especificamente, na calcinacao de calcario. O objetivo inicial e validar as capacidades do

COMSOL no tratamento dos fenomenos fısicos envolvidos. Este programa provou ser uma ferramenta

adequada para prever os processos quımicos e o comportamento de um plasma induzido por micro-

ondas. Em seguida, propoe-se um novo algoritmo capaz de, autonomamente, simular uma unidade de

processamento de calcario com uma elevada eficiencia global. Este algoritmo e capaz de fazer os ajus-

tamentos necessarios na potencia imposta e no estado de ressonancia da cavidade de micro-ondas,

conseguindo assim, obter uma total conversao de calcario e evitando problemas relacionados com a

temperatura. Por fim, esta metodologia apresentada mostra melhoramentos significativos, face a outros

modelos numericos do genero existentes na literatura.

Palavras-chave: Modelo numerico de aquecimento por micro-ondas, Correspondencia au-

tomatica de impedancia, Otimizacao de potencia imposta, Producao de clınquer de cimento, Processa-

mento contınuo de calcario por micro-ondas.

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Abstract

Conventional cement clinker production is a very energy demanding process, that requires up to

3600 kJ for a single kilogram of cement clinker. Moreover, since it is one of the most used materials in

the world, its production accounts for up to 6% of the total CO2 emitted by the human activity. Microwave

heating technology has been gaining interest as an alternative for its advantages. It is a clean energy

with no direct CO2 emissions, and, unlike conventional heating, microwave heating happens when the

electric field interacts with the material, resulting in a volumetric heating inside the material, returning

higher efficiencies than the conventional counterpart.

This Thesis addresses the subject of microwave heating in the production of cement clinker, specifi-

cally in the limestone calcination. The initial goal is to validate capabilities of COMSOL in handling all the

physical phenomena involved. This software has been proven to be a suitable tool to predict chemical

processes and the behavior of a microwave induced plasma. Afterwards, a new method is proposed

and a code developed that can autonomously simulate a limestone processing unit with overall high

efficiency, using microwave energy. The code is capable of making the necessary adjustments to the

input power and resonant state of the microwave system, and was able to achieve total conversion of

limestone, while maintaining high efficiency and avoiding temperature related problems. As such, the

presented methodology shows a significant improvement over other available numerical models in the

literature for microwave heating.

Keywords: Microwave heating numerical model, Automatic impedance matching, Input power

optimization, Cement clinker production, Continuous microwave limestone processing, Autonomous sim-

ulation control.

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Conventional rotary kiln and cement chemistry . . . . . . . . . . . . . . . . . . . . 4

1.4.2 Microwave in clinker formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.3 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.4 Computational numerical models applied to microwave heating . . . . . . . . . . . 7

2 Background 9

2.1 Maxwell’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Microwave Rectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 TE Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 TM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Heat Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Numerical Simulations with COMSOL Multyphysics R© . . . . . . . . . . . . . . . . . . . . 16

2.5.1 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Transport of Concentrated Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.1 Mass transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.2 Arrhenius Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7.1 Electron Transport and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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2.7.2 Heavy Species Transport and Plasma Properties . . . . . . . . . . . . . . . . . . . 22

3 Verification and Validation 25

3.1 Chemical kinetics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.2 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Plasma modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Results of the plasma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Microwave Limestone Processing 37

4.1 Cavity Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Chemical model and material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Numerical model: domain description, interfaces and boundary conditions . . . . . . . . . 40

4.3.1 Domain description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.2 Interfaces and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Computational routine and process optimization . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.1 Optimum initial plunger position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.2 Microwave power input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.3 Matlab control routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Initial energy testing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6.1 Limestone processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6.2 Mass flow parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Conclusions 65

5.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 67

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List of Tables

3.1 Reactions, kinetics and heat of reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Chemical reactions’ writing simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Dimensions of the rotary kiln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Species’ molar mass and inlet mass fractions . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Species’ molar mass and inlet mass fractions . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Numerical and experimental results of the mass fraction at kiln’s exit . . . . . . . . . . . . 29

3.7 Chemical mechanism considered for the plasma model. . . . . . . . . . . . . . . . . . . . 32

3.8 Plasma processes for each correspondent reaction . . . . . . . . . . . . . . . . . . . . . . 33

3.9 Boundary conditions for the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Limestone calcination chemical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Limestone thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Microwave efficiency for different plunger positions with a 40% fill rate . . . . . . . . . . . 47

4.4 Microwave efficiency and optimal plunger position for various material fill rates . . . . . . 48

4.5 Energy balance of the initial model tested . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.6 Operational parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 Limestone processing figure’s content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 Steady state data of the limestone processing unit . . . . . . . . . . . . . . . . . . . . . . 58

4.9 Parametric study results by variation of the mass flow rate . . . . . . . . . . . . . . . . . . 61

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List of Figures

2.1 Frequency dependence of permittivity for a hypothetical dielectric material . . . . . . . . . 12

2.2 Schematic of a rectangular waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Schematic of the the multi-physical coupling. . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Temperature dependency of the reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Bed temperature input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Graphical representation of the rotary kiln . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Mass fractions’ evolution along the kiln’s length. . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Graphical representation of the plasma model . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Electron number density and temperature, time = 1× 10−2 s. . . . . . . . . . . . . . . . . 34

3.6 Electric field norm, time = 1× 10−2 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Maximum electron density, for a time of 1×10−2 s, versus microwave power and gas velocity 35

4.1 Geometry of the cavity for limestone processing . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Perspective of the computational grid with 2,354,801 tetrahedral elements. . . . . . . . . 38

4.3 Limestone’s permittivity and loss factor with temperature . . . . . . . . . . . . . . . . . . . 40

4.4 Domains of the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Wave equation domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Species’ transport domain for the chemical physics modeling . . . . . . . . . . . . . . . . 42

4.7 Boundary division used for each heat convection coefficient . . . . . . . . . . . . . . . . . 43

4.8 Flow diagram of the computational algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.9 Temperature profile used for the frequency domain study . . . . . . . . . . . . . . . . . . 45

4.10 Electric field norm distribution | ~E| in a XY plane . . . . . . . . . . . . . . . . . . . . . . . . 46

4.11 Microwave efficiency dependency on the plunger position for the initial temperature distri-

bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.12 2D cut on the ZX plane of the temperature field of the testing model . . . . . . . . . . . . 51

4.13 2D cut on the ZX plane of the CaCO3 mass fraction of the testing model . . . . . . . . . . 51

4.14 Stored heat, power input, microwave efficiency and plunger position evolution during the

simulation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.15 Bed power absorption, maximum bed temperature and outlet bed temperature versus

simulation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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4.16 Evolution of the thermal field over the course of the limestone processing simulation. . . . 55

4.17 Limestone mass fraction, maximum bed temperature and reaction heat source versus

simulation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.18 Bed power absorption and power losses/usage versus simulation time . . . . . . . . . . . 56

4.19 Energy balance of the limestone processing unit at steady state for a mass flow of 0.25 kg/h 57

4.20 Energy balance of the limestone processing unit at steady state for a mass flow of 0.25

kg/h considering the Pheat as useful power . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.21 Limestone processing steady state distributions (side view) of the temperature, loss factor,

microwave power dissipation and the chemical reaction heat source. . . . . . . . . . . . . 59

4.22 Limestone processing steady state distributions (top view). . . . . . . . . . . . . . . . . . 59

4.23 Converged solutions of the parametric study. 2D plots of the microwave power dissipation,

temperature field and chemical endothermic heat source. . . . . . . . . . . . . . . . . . . 63

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Nomenclature

Acronyms

CFD Computational Fluid Dynamics

EM Electromagnetic

EEDF Electron Energy Distribution Function

FDTD Finite Difference Time Domain

FEM Finite Element Method

MW Microwave

TE Transverse Electric

TEM Transverse Electromagnetic

TM Transverse Magnetic

Constants

ε0 Permittivity in vacuum, 8.85× 10−12 F/m

kB Boltzmann constant, 1.386488× 10−23 J/K

e electron charge, −1.602× 10−19 C

me electron mass, 9.109× 10−31 kg

R Gas constant, 8.314 kgm2mol−1K−1s−2

Greek symbols

α Thermal diffusivity (m2/s)

β Propagation mode

~Γ Flux

ε Permittivity (F/m)

εr Relative permittivity (F/m)

ε′r Relative dielectric Constant

ε′′r Relative loss factor

εp Electron energy (eV)

η Efficiency (%)

κ Thermal conductivity (Wm−1K−1)

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λ Wavelength (m)

µ Permeability (H/m)

νeN Collision frequency between electrons and neutral species (Hz)

ρ Density (kg/m3)

ρq Charge density (C/m3)

σ Electrical conductivity (S/m)

σk Collision cross section

φ Electric potential

ϕ Wave phase

ω Angular frequency (rad/s)

ωi Mass fraction of species i

Math operators

∆ Difference operator∂∂t Derivative with respect to time

∇· Divergence operator

∇× Rotational operator

∇ Gradient operator

j Imaginary number∫Integral operator

Roman Symbols

A Area(m3)

~A Magnetic vector potential (Vm−1s)

~B Magnetic flux density (Wbm−2)

Cp Heat capacity (Jkg−1K−1

)

~D Electric flux density (Cm−2)

~E Electric field (V/m)

Ea Activation energy (J)

e Nepper number

f Frequency (Hz)

~H Magnetic field (A/m)

~J Electric current density (Am−2)

k Arrhenius rate term

k0 Arrhenius pre-exponential factor

kn Wave number (m−1)

M Molar mass (kg/mol)

mflow Mass flow (kg/h)

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ni Number density of species i (K)~P Polarization (Cm−2)

P Power (W)

p Pressure (Pa)

Q Heat generation (Wm−3)

Ri Reaction rate of species i (K)

T Temperature (K)

Te Electron temperature (eV)

t Time (s)

~u Velocity (m/s)

V Volume (V)

x1 Mole fraction of species i

Z Wave impedance (Ω)

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Chapter 1

Introduction

In this chapter, the aim is to present the motivation for this Thesis and followed by a brief description of

the objectives. Then an outline to the thesis is presented and finally this chapter ends with the literature

review where the contributions to the topic are reported.

1.1 Motivation

Microwave technology for processing materials has been growing, finding use in a wide range of

applications. Opposing conventional heating, microwave-based applications offer a capability of gener-

ating heating in a way that avoids some constraints related to the first heating technology [1]. Whereas

in conventional heating, the material is gradually warmed from the outside inwards, through convection

or radiation, in microwave heating, heat is only produced where and when the energy is absorbed. As

microwave can penetrate materials easily (except metals), the full volume of the material can uniformly

absorb its energy leading to a volumetric heating [2]. This effect translates into higher heating rates,

increased efficiency and shorter processing times [3]. The relatively new technology can be applied to

drying calcining, curing or other manufacturing processes, which requires some sort of thermal process-

ing [2]. The food and rubber industries were the first ones to use this technology [4], where in the rubber

case, it was used for drying and vulcanization processes. Other industries are gradually seeking for the

introduction of microwave heating, such as, the wood, textile, and cement industries.

Cement is complex mixture of calcareous-, silica-, and alumina-based minerals. Its production pro-

cess consists of grinding and mixing the feedstock and then heating it to temperatures around 1450C

giving origin to a granular clinker composed of tricalcium silicate, dicalcium silicate, tricalcium aluminate

and tetracalcium aluminoferrite. This mixture forms a well known man-made construction material used

widely around the world, which is of great importance to civilization being the second most used material

in the world, only surpassed by water [5]. The main issues regarding the cement industry are the energy

required and the CO2 emissions resultant from the processes. A single kilogram of Portland cement

requires a theoretical value between 1674 and 1799 kJ. However,in a conventional cement production,

if one accounts for the heat losses due to the exhaust gas and cooling processes, the energy required

1

rises to values around 3100 to 3600 kJ. On the other hand, the cement formation emits CO2, during the

calcination process, where limestone (CaCO3) turns into lime (CaO) and CO2 (see the first reaction of

table 3.1 of chapter 3). Also in a conventional plant, a flame is used as the heat source, which also

produces CO2 contributing to the emissions of this gas at the plant location. Consider finally the huge

amount of cement produced worldwide, and then, taking everything into account, we have a cement

industry responsible for 5 − 6% of the carbon dioxide greenhouse gases generated by human activities

[6].

As it is impossible to change the cement chemistry, it is way more feasible to tackle the heat source,

in order to reduce the CO2 emissions and improve the industry’s sustainability. This is where microwave

heating technology comes in handy. It is a ”greener” technology if the electricity comes from a renewable

source, it is a clean heating process that does not produce any secondary waste and has the benefit of

being electronically controlled with precision. Further, this heating method is particularly efficient when

a dielectric material is being heated. The materials that compose cement excellent dielectric properties,

and so, they must be able to absorb microwave energy with high efficient levels [3] resulting in a more

efficient heating process.

Despite the clear advantages of using microwave radiation over conventional heating in thermal pro-

cesses, some issues may appear due to the use of this heating technology. Some of the materials have

the particularity to have an imaginary part of the permittivity that increases with temperature, cement

is a good example of such a material [7]. This creates a sort of loop mechanism where temperature

causes the increase of this characteristic which, in turn, enables the material to absorb more energy,

increasing the temperature. This can lead to high thermal gradients within the material as in certain

spots temperature will increase abruptly. The phenomenon depicted is known as thermal runaway [8]

and it can be catastrophic for a material’s structure when it occurs, thus presenting a major challenge

to the development of microwave heating [7]. When working with cement, other particular problem may

arise. Due to the CO2 emission during the calcination process, a plasma can be formed when the gas is

bombarded with microwave radiation [9]. The formation of the plasma may, due to its dielectric proper-

ties, divert the energy from the targetted cement, leading to low efficiencies. It also poses a threat to the

process apparatus as the concentrated energy can propagate and damage the surrounding equipment.

Finally, from an operational standpoint, the use of microwave heating technology pose some obsta-

cles inherent to the physical phenomena. The electric field distribution in a microwave cavity is very

sensitive to the changes of the dielectric and thermal fields of a material. Moreover, the geometry of the

cavity itself will impact the heating capacity of the whole system [10]. On top of that, many industries use

materials that only have an high loss factor for high temperatures (e.g. cement). As such, a method is

required to adjust the cavity geometry and input power continuously in order to optimize the microwave

absorption efficiency and maintain high temperatures at the materials[11].

2

1.2 Objectives

The main objective of this Thesis is to address the process of cement clinker formation by the use

of microwave energy as an heating source. Great emphasis will be given to the conversion process

of limestone into lime, as one of the major steps of cement clinker formation. Limestone calcination

(conversion of limestone into lime) is subject to heavy interest, being extendedly studied in order to

develop calciner kilns [12] and pre-calciner kilns [13] then used in the cement industry.

COMSOL Multiphysics is the software chosen to conduct this work, due to its reported good capa-

bilities of handling several physics and coupled them together with great success. As an initial goal,

two cases will be studied. One targeting the governing chemical mechanisms of cement clinker and the

other aiming at understanding the formation of microwave induced plasma, the latter being an undesir-

able occurrence reported on cement processing by microwave heating. The goal of these studies is to

serve as an introduction to the software, better understanding of the all the phenomena involved and

assess the softwares capabilities by comparing the obtained results with data available in the literature.

After the initial cases, a numerical model for limestone processing by the means of microwave heating

will be built. The main goal underlined in this task is to obtain a steady state solution, where limestone is

converted into lime with maximum efficiency and using as little power as possible. Thus, a coupled way

to increase the efficiency and to manage the power input has to be formulated and executed. A careful

analysis will be made of the obtained results. The purpose is to observe the system’s behavior through

the simulation process, and then, evaluate the efficiency and power usage for various operational con-

ditions. The ultimate objective of the discussed task is to come up with an optimized numerical model

capable of processing limestone using microwave energy, and to assess its feasibility and capabilities

from an energy standpoint.

1.3 Thesis Outline

The present work is divided in five chapters. Chapter one is dedicated to the introduction of the

subject in discussion and to get acquaintance with the goals of this Thesis, with the contributions given

so far through works of other researchers. In chapter two, a brief insight is given to the fundamental con-

cepts and equations that rule all the physical phenomena involved in microwave heating. With chapter

three, the objective is to present the implementation of two models: one of a conventional cement kiln

and another one of a microwave plasma model, resorting to COMSOL in order evaluate the software’s

capabilities to preform chemical and plasma related studies. In chapter four a 3D model of a limestone

processing unit, that uses microwave energy, is depicted through its various stages. Insight is given

on the model building, assumptions and a priori tests. The results of the simulations carried out with

the model are also presented along with the respective analysis. At the end, final remarks and conclu-

sions can be found in chapter five, as well as future indications for continuing the development of the

addressed research field in this Thesis.

3

1.4 Literature Review

One can acknowledge that simulation of cement clinker formation by the use of microwave energy is

a multiphysics problem. This means that the subject under study in this work is not constricted to only

one matter. As such, this chapter aims to give a brief insight into the work already done in the various

domains, beyond the accomplishments regarding the subject of the work itself, which contribute to a

better understanding of all the phenomena involved.

1.4.1 Conventional rotary kiln and cement chemistry

Since the early 70’s several attempts have been made to characterize and predict the evolution of

cement clinker formation on conventional rotary kilns [14]. These attempts were based on a premise of

a rotary kiln that uses a flame as an energy source. With Spang [14] it is possible to witness one of the

early efforts to lay the first foundations of a coupled model of a rotary kiln. It involved flame modeling,

heat transfer and a simplified description of Portland cement chemistry with only five chemical equations

and twelve species. This work focused on the development of a one dimensional partial differential

model. And it was able to get temperatures profiles and mass fraction of each species in the bed and

freeboard regions, along the axial direction with accuracy, when compared to experimental data. In this

case the energy and material balance equations were solved for both phases (solid and gaseous).

Later, and with the development of more advanced computational tools, computational fluid dynamics

(CFD) was brought into the subject with Mastorakos et al. [15]. A 3-D CFD model (using FLOW-3D) of

a coal-fired rotary kiln, that included a Monte Carlo method for thermal radiation, was used for the

freeboard region . This model was then coupled to a 1-D model where a finite-volume heat conduction

code was used to compute kiln walls temperature and where participating species and energy equations

were solved. It had also the contribution of a more accurate description of the heat transfer phenomena

in a kiln, that accounted for heat transfer at the refractories and the bed, in addiction to the heat resulted

form incident radiation onto the exposed areas [16, 17]. Moreover, it can be seen that roughly 40% of

the energy input is absorbed by the bed, while 10% is lost to the environment through the refractory wall.

The study conducted in [15] included a clinker coating on the kilns walls, although it was considered to

be uniform along the kiln’s inner wall. Furthermore, the rate constants for the given chemical equations

were tuned (namely the pre-exponential factors and activation energy of the Arrhenius equation) by trial

and error in order to get the expected clinker chemical composition at the end of the kiln.

Meanwhile, efforts were made to better understand clinker’s chemistry. With the main focus lying one

the first and most important reaction [15], the calcination reaction where calcium carbonate is converted

into calcium oxide and carbon dioxide (first equation of table (3.1)) [18, 19, 20, 21, 22]. These studies

aim at understanding and describing the rate constants of calcium carbonate decomposition assuming

different models and using distinct techniques. The data was processed assuming a first-order reaction

and Arrhenius temperature dependence. Some experiments were made [19] using a structural model

and using a thin slab type pellets under isothermal conditions, resulting in good reported values. Others

opted for a thermogravimetric analysis technique [20, 21, 22]. In particular, Lee et al. [22] investigated

4

the decomposition of calcium carbonate spheres of various sizes under different atmospheric conditions

(pure N2 and various CO2-N mixtures) where the obtained results were in agreement with the literature

values and it was concluded that the activation energy and the pre-exponential factor of the Arrhenius

equation reduce as particle size increases. On the other hand Ar [21] tested 10 limestones samples

from different locations in Turkey and found that a shrinking core model with surface reaction controlling

mechanism gives the best fit to the experimental data. Finally Rao [20] conducted his analysis under

isothermal and non-isothermal conditions. It was found that both methods have given consistent data

between themselves and those in the literature. Moreover, and from the resulting analysis, it established

an isokinetic temperature of about 737oC to the calcium carbonate decomposition.

Taking this significant contribution to the characterization of the calcination reaction into account,

studies had been conducted to improve the model and to better portray the reality [23, 24, 25]. In

particular, all works referred used an updated data for the calcination reaction, although maintaining

Mastorakos et al. [15] data for the remaining reactions due to lack of new data regarding them. A one

dimensional model was built [23] to simulate major processes that occur in the solid bed of a cement

kiln. The mathematical model was first applied to a rotary calciner, where only the calcination reaction

occurs (the first reaction of table), being verified and validated with published data. The model was

then used to simulate the performance of actual industrial kilns. This time, however, it accounted for a

variation of bed height along the kiln, melting of the bed and for the formation of a clinker coating on the

kiln walls, near the end of the kiln, due to the increase in temperature in that region. This features were

later accounted for as well in a CFD model built by the same authors [24] with good results. It was found

that predictions achieved with the model were not sensitive to volume fraction of solids in the freeboard

region and that the results obtained are in a good agreement with the experimental data from an actual

industrial kiln. Based on this studies further work was developed aiming at reducing energy consumption

[13] and evaluating the existence of a pre-heater, calciner and a cooler on top of the original rotary kiln

apparatus [26].

1.4.2 Microwave in clinker formation

Despite the widespread penetration of the microwave heating technology in the most different indus-

tries, little has been done regarding the use of microwave energy to produce cement clinker from its raw

material.

The studies presented by Quemeneur et al. [27, 9] during the 1980’s are the earliest and still the most

effective work regarding this subject. The studies are quite similar. Quemeneur et al. [9] investigated the

clinkerability of different clinker raw mixes using microwaves with a 2.45 GHz frequency. It was reported

the formation of a plasma and its extension to the whole cavity. This work was able to conclude that at

1450C the dielectric losses of the cement consituents were able to maintain the material’s heating lead-

ing to a product with properties similar to the industrial counterpart. Quemeneur et al. [27] attempted the

clinkering of cement raw material under a microwave field using a grooved cavity (0− 3 KW, 2.45GHz).

While avoiding the formation of plasma and using only microwave, clinker similar to the industrial clinker

5

was formed, and a comparative analysis was done between the microwave prepared clinker and the

industrial one. This work was able to identify three different process domains by taking measures of

the dielectric properties along the processing time. The first correspond to the dehydration and de-

carbonation of the feedstock, the second to the formation of the dicalcium silicate (C2S) and and the

alumnimoferrite (C4AF), and the last one corresponds to the the formation of tricalcium silicate (C3S).

Fang et al. [28] conducted a study where regular and colored cement clinkers were prepared using

microwave energy. A multimode microwave cavity with a turntable to horizontally rotate a sample (like

a regular microwave oven) was used at 900 W and 2.45 GHz. A single-pellet sintering method was

applied and a comparison was made between microwave and conventional heating, here portrayed

through an electric furnace. It was possible to conclude that cement can be prepared by microwave

processing and it was also reported that by doing so, the clinkering temperature can be diminished by

100 C when compared to conventional heating. This microwave enhancement effect was reported to

be the result of the presence of Fe2O3, which held the responsibility for the temperature reduction due

to its good capacity to absorb microwave energy and to dissipate the heat through convection to the

surroundings. On the other hand Li et al. [29], who undertook similar endeavor, were able to find the

same enhancement effect but pointed out the increase in ion diffusion, caused by the interaction of them

with the electric field, as the responsible mechanism for the effect.

1.4.3 Plasma

During experiments of the clinkering of cement raw material under a microwave field there are reports

of a plasma appearing [27]. One of the sub-products of cement clinker production is the carbon dioxide,

more specifically, CO2 is a product of the calcination reaction [14]. The CO2 then flows through the

bed and into the gaseous phase of the kiln (the freeboard) mixing with the existent air. CO2 plasma is

then formed when appropriate conditions such as power, pressure and gas flow rate are met [30]. The

presence of the plasma involves additional energy losses [27] and, as such, it becomes important to

understand how microwave interacts with this gas and the consequences of its presence in the process.

Hence, this section aims at giving an insight into what has been done to understand and predict the

evolution of a plasma.

The most recent studies show a lot of interest in the CO2 plasma, mainly as a way to dissociate

it. The reasons that justify the interest are numerous. However, the most important seems to be the

environmental awareness with the contribution of CO2 to the greenhouse effect, [30, 31, 32, 33, 34].

Examples of experimental works [30, 31, 32] and computational studies [33, 34] are available. Regard-

ing the computational works, these present a zero-dimensional kinetic model of CO2 where the state

specific relaxation reactions and the effect of vibrational excitation on other chemical reactions are taken

into account. Both simulate non-thermal CO2 plasma generated by the microwave radiation, with the

first conducting a comparative study between a microwave discharge and a dielectric barrier discharge

regarding CO2 conversion and efficiency, while the second aims at understanding the effects of internal

plasma parameters on the CO2 conversion and efficiency. The kinetic model used in these studies was

6

the first to include all the relevant processes that characterize the CO2 dissociation process. However,

this zero-dimensional model, although with a great level of detail, is, due to its complexity, very hard to

implement in multidimensional models [35]. Hence, it is possible to notice an effort to come up with a

reduced kinetic model that is able to be implemented when a multidimensional model is required [35].

To the best of the author knowledge, no study was released where a multidimensional model of a CO2

kinetic model has been implemented, much less resorting to a commercial software such as COMSOL.

Hence, attention was focused on numerical studies regarding microwave sustained plasmas, but with a

simpler gas. After a brief search, Argon was found to be a less complex gas, with several works already

conducted in order to study plasma in Argon by the use of microwave energy [36, 37, 38, 39]. Several

of the previous works used COMSOL as a tool for their simulations [37, 38, 39], which turns to be rather

useful for the present work. With the analysis brought by Kabouzi and Graves [36] we have one of the

first two-dimensional fluid-plasma models for a microwave sustained discharge in Argon at atmospheric

pressure. The carried numerical simulation provided the full axial and radial structure of the plasma

and the distribution of the electromagnetic fields for a given set of operating conditions. The contraction

phenomenon [40], typical of atmospheric plasmas, was also studied.

With the latter three above referenced works, we can find two [37, 39] and three [38] dimensional

models of a microwave induced plasma in Argon, also at atmospheric pressure, resorting to the com-

mercial software COMSOL. The geometries are varied (but are similar) and the operating conditions

(such as power, gas flow...) vary accordingly, so that similar temperature ranges are obtained across

the studies. The frequency of operation is fixed to the typical 2.45 GHz and the Argon chemistry and

kinetics model used is almost the same in all three studies with seemly minor differences. Parametric

studies are carried out where gas flow [37, 38], power input [38, 39] and initial gas temperature [38] of

the discharge are varied. Regarding the output, all models are able to provide the radial and axial distri-

butions of the gas temperature, electron temperature and electron density. Further plasma parameters

are obtained such as ion and excited state number densities as the power deposited into the plasma

[37]. It was also observed in [37] that under 1200K a contraction of the discharge appears.

1.4.4 Computational numerical models applied to microwave heating

Numerical simulations and computational modeling are very powerful tools to predict electromagnetic

field distribution in the cavity and to anticipate the heating behavior of a sample material. Such tools can

then be used to optimize and develop more efficient microwave heating systems [10]. Several works on

microwave heating will be addressed below, namely a few that used COMSOL Multiphysics R© as the

numerical environment.

Alpert and Jerby [41] created a one dimensional model to study microwave heating of temperature

dependent dielectric materials though the two-way coupling of the thermal and electromagnetic fields.

The models resorted to FEM and two different time scales were used to reduce computational time.

Two separate diffusion functions were also derived in order to obtain more accurate temperature and

electromagnetic distributions. The developed model was successfully benchmarked against other works,

7

and the predicted temperature profiles allow to avoid thermal runaway. Zhu et al. [42] proposed a

mathematical model for microwave heating of liquids with a particularity of considering a continuous flow

rather than a stationary sample. Apple sauce, skim milk and tomato sauce flowed through a circular duct

which was bombarded with microwave radiation. The objective was to investigate the effect of dielectric

properties of the liquids, the dimensions and location of the circular duct as well as investigate the cavity

design. A finite difference time domain method was chosen to compute the electromagnetic field and,

through the obtained results, Zhu et al. [42] observed how the heating is strongly dependent on the

material’s dielectric properties and on the geometry of the microwave cavity.

Among others, authors such as Salema and Afzal [10] and Mimoso et al. [11] resorted to COM-

SOL to tackle their studies. Salema and Afzal [10] simulated microwave heating of stationary samples

of biomass, in bed and pellet form, in a multimode microwave oven and validated against experimen-

tal studies. It was showed that temperature and heating behavior of the samples are affected by the

biomass loading height and specific heat capacity. The obtained results allowed for the evaluation of the

optimal biomass loading height for a particular set of microwave power and frequency, where maximum

microwave energy absorption could be attained. Moreover, the model can be successfully used to iden-

tify hot and cold spots in the samples during the heating process, enabling the possibility of optimizing

the design of the microwave heating systems regarding uniformity. The work developed by Mimoso et al.

[11] consisted of a continuous glass melting in a single mode microwave cavity at 2,45 GHz. The main

objective of the conducted study was to optimize the microwave processing of glass while maintaining

high levels of energy efficiency and avoiding thermal related problems. Phase change from solid to liquid

in the glass region was included, as well as surface to surface radiation. A MATLAB code was devel-

oped to microwave power input and to maximize material’s energy absorption during the course of a 3D

transient simulation. The developed methodology was able to achieve high energetic efficiency while

maintaining specific power as low as possible. Through the obtained results, Mimoso et al. [11] were

able to conclude that, in order to maintain and high efficiency inside the cavity, a plunger adjustment to a

more efficient position is continuously required at specific times. Moreover, it was seen that pre-heating

the material at the inlet reduced global efficiency as it increased thermal losses due to radiation and

convection. Hence, when steady state is achieved pre-heating is turned out to be undesirable, however

pre-heating the whole material at the beginning of the process helps starting the heating process faster

and with more efficiency.

8

Chapter 2

Background

This chapter is dedicated to introduce the reader to the governing equations and important concepts

of the various phenomena discussed in this Thesis.

2.1 Maxwell’s equation

Maxwell’s equations depict the interactions between the electric and magnetic field at a macroscopic

level. These equations are based on other’s empirical and theoretical knowledge, namely Faraday,

Gauss and Ampere. The differential form of these equations are presented next. A more in-depth

explanation of the Maxwell’s equations and the equivalent integral form can be found in many textbooks

such as in Popovic and Popovic [43].

∇× ~E = −∂~B

∂t(2.1)

∇× ~H =∂ ~D

∂t+ ~J (2.2)

∇ · ~D = ρq (2.3)

∇ · ~B = 0 (2.4)

Where:~E is the electric field, in volts per meter (V/m);~B is the magnetic flux density, in webers per squared meter(Wb/m2);~H is the magnetic field, in amperes per meter (A/m);~D is the electric flux density, in coulombs per squared meter (C/m2);~J is the electric current density, in amperes per squared meter (A/m2);

ρq is the electric charge density, in coulombs per cubic meter (C/m3);

t is time, in seconds (s).

Equation 2.1 is Faraday’s law of induction. It is the cornerstone of electromagnetism and constitutes

an example of the profound connection between ~E and ~B and it states that time variation in the magnetic

9

field has a correspondent spatial variation in the electric field. Equation 2.2 is the Ampere’s law. From it

one knows that a variation in the electric field causes a change in the magnetic field. The electric current

density term ( ~J) is the responsible for the creation of a magnetic field that circles the electric current.

Equation 2.3 is the Gauss law. It expresses how the electric field behaves around electric charges and

it states that the divergence of the electric flux density, over any region of space, is equal to the charge

density. Finally, equation 2.4 is Gauss law for magnetism. Looking to equation 2.3 it is possible to

see the similarity, being the difference the field in the divergence operator. In this case we have the

divergence of the magnetic flux density and it is expected to be equal to the magnetic charge (the same

way eq. 2.3 is equal to the electric charge). However, so far, there is no evidence that magnetic charges

do exist and thus the divergence of ~B equals zero, and so, no second term is visible in equation 2.4.

Through some algebraic manipulation it is possible to deduce another relation from Maxwell’s equa-

tions. Knowing that ∇ · (∇ × ~H) = 0 (divergence of the curl of any vector field is equal to zero) one

can substitute equation 2.2 into it. Then, using the relation established in equation 2.3 the continuity

equation of energy can be derived:

∇ · ~J +∂ρq∂t

= 0 (2.5)

To fully describe the fields involved in electromagnetism additional relations need to be added to

Maxwell’s equations. This are called constitutive equations and establish the relations between the vec-

tor entities seen in the above Maxwell’s equations. These equations depict the macroscopic properties

of the material medium where the electromagnetic field exists.

~D = ε ~E (2.6)

~B = µ ~H (2.7)

~J = σ ~E (2.8)

Where ε, µ and σ are the constitutive parameters of the material medium: ε is the permittivity (F/m), µ is

the permeability (H/m) and σ is the electrical conductivity (S/m). The equations are written for a linear

isotropic and homogeneous medium. Equation (2.8) is the vectorial form of te Ohm’s law.

Sometimes it is useful to form problems in terms of the magnetic and electrical potential, respectively~A and φ. As such, it is necessary to establish the respective general relations between ~B and ~E and the

respective potentials introduced above:

~B = ∇× ~A (2.9)

~E = −∇φ− ∂ ~A

∂t(2.10)

As stated above, the Maxwell’s equations (eqs. 2.1 to 2.4) are partial differential equations having

time and the spatial coordinates as independent variables. However it is frequent to have a sinusoidal

time variation of the sources. Plus if the medium is linear, we can conclude that all quantities have

10

a sinusoidal variation in time. So, taking it all into account, time can be removed form the equations

simplifying them in the process. Sinusoidal time dependent quantities can have their time dependence

written in the form:

cos(ωt+ ϕ) (2.11)

With ω = 2ϕf being the angular frequency (in rad/s), f the frequency (in Hz), and ϕ the initial phase.

So considering ~X a time-harmonic field, it can be expressed as follows:

~X = ~X0 cos(ωt+ ϕ) (2.12)

Resorting to Euler’s formula it is possible to write the cosine as a sum of complex exponentials with

j =√−1:

cos(ωt+ ϕ) =exp(j(ωt+ ϕ)) + exp(−j(ωt+ ϕ))

2(2.13)

The variables presented in Maxwell’s equation can all be represented resorting to this relation, and

so it is common to replace the cosine for exp(jωt). Hence, to represent the derivative of a sinusoidal

quantity in respect to time it just needed to replace the time derivative with jω [43].

According to this formulation it is now possible to write Maxwell’s equations in the complex (or phasor)

form resulting in fields possessing both real and complex part as can be seen below.

∇× ~E = −jω ~B (2.14)

∇× ~H = ~J + jω ~D (2.15)

∇ · ~D = ρq (2.16)

∇ · ~B = 0 (2.17)

2.2 Complex Permittivity

The microscopic behavior of a material subjected to an external field is generally dependent on

the field’s frequency. This dependence is a reflection of the non-instantaneous response to an applied

field by the material’s polarization. The response gives rise to a phase shift difference between the

polarization (~P , in (C/m2)) and the electric field ( ~E). This behavior can be analytically represented by

means of a complex permittivity, ε, as follows:

ε = ε′ − jε′′ (2.18)

The permittivity of dielectric materials can then be normalized to that of free space:

ε′ = ε0ε′r ; ε′′ = ε0ε

′′r (2.19)

Where ε0 = 8.85× 10−12 (F/m) is the free space permittivity, and so the complex relative permittivity is

11

represented as follows:

εr = ε′r − jε′′r (2.20)

Permittivity describes the interaction of the material with the electric field [44] and these interactions

can happen in one of two distinct ways: energy storage and energy dissipation. Energy storage de-

picts the lossless energy exchange between the electric field and the material, while energy dissipation

happens when the energy is absorbed by the material. Regarding equation (2.20) the capacity of the

material to store energy is translated by the real part of the complex permittivty (ε′r) which it is known

as dielectric constant and is responsible for phase shift of the electric field and storing its energy. On

the other hand, the capability of the material to dissipate energy, one of the most important properties

for microwave heating, is represented by the imaginary part of the same equation (ε′′r ) which is called

the loss factor and is responsible for the electric energy losses that are transformed into heat. In a

qualitative form, the time-varying field induces time-varying dipoles in the dielectric, as such, it starts to

vibrate more vigorously due to these variations which result in heat [43]. A similar demonstration to the

one presented above can be made for permeability.

The complex permittivity depends upon the frequency and temperature. Regarding the frequency,

for different ranges, the material will have distinct responses which correspond to various phenomena.

The major mechanism that contribute to the permittivity of a certain dielectric material are the ionic

conduction, dipolar relaxation, atomic polarization and electronic polarization as can been seen in Fig.

2.1. It is noteworthy to highlight that dipolar relaxation is the major cause for permtittivity variation in the

microwave range of frequencies.

Figure 2.1: Frequency dependence of permittivity for a hypothetical dielectric material [45].

In respect to temperature dependence, it is usually subject of more focus as frequency of the source

is fixed to one value in most applications [8]. From the literature, it is known that there is an increase

of the loss factor when increasing the temperature [8], however in some materials this increment is

greater than others [46], which can lead to some unwanted problems. What happens is that the initial

12

microwave energy will be absorbed and temperature will increase causing ε′′ to increase, and as a result

temperature will rise again and so on.

2.3 Microwave Rectangular Waveguide

A brief description of a rectangular waveguide is the topic of this section as it is used in all of the

studies conducted in this work. A waveguide is a structure that is able to conduct electromagnetic energy

along a determined path. Although waveguides can be of various shapes and forms, we will be focused

in studying waveguides in the form of hollow metallic tubes, and in particular the rectangular shaped

ones, as the one seen in Figure 2.2 where a is the largest dimension of the waveguide (a > b).

Figure 2.2: Schematic of a rectangular waveguide.

The rectangular waveguide can propagate Transverse Magnetic (TM) and Transverse Electric (TE)

modes, but not Transverse Electromagnetic (TEM) modes as only one conductor is present [47]. Elec-

tromagnetic waves are transversal waves. In Transverse Electric modes there is no electric field in the

direction of propagation, and it is characterized by Ez = 0. The Transverse Magnetic mode is charac-

terized by Hz = 0, so there is no magnetic field in the direction of propagation. In the case of the TEM

mode, there is no electric nor magnetic fields in the direction of propagation, as they are both transversal

to it.

2.3.1 TE Mode

According to Pozar [47], assuming time-harmonic fields and using the separation of variable meth-

ods, it is possible to obtain the equations for the electromagnetic field of a TEmn mode propagating in a

rectangular waveguide, filled with a material of permittivity ε and permeability µ, as follows:

13

Hx =jβmπ

k2caAmnsin

mπx

acos

nπy

be−jβz (2.21)

Hy =jβnπ

k2cbAmncos

mπx

asin

nπy

be−jβz (2.22)

Hz = Amncosmπx

acos

nπy

be−jβz (2.23)

Ex =jωµnπ

k2cbAmncos

mπx

asin

nπy

be−jβz (2.24)

Ey =−jωµmπk2cb

Amnsinmπx

acos

nπy

be−jβz (2.25)

Ez = 0 (2.26)

In the previous equations Amn is an arbitrary amplitude constant that depends on the level of excitation

of the wave [47] and β is the propagation mode which is expressed as follows:

β =√k2n − k2c =

√k2n −

(mπa

)2−(nπb

)2, (2.27)

with kc being the cutoff wave number and kn = ω√µε the wave number. The parameter β corresponds

to a propagating mode when it is real, and that happens when kn > kc.

For each TE mode there will be a correspondent cutoff frequency as expressed by (2.28), which is

the lowest frequency for which a mode will propagate in a waveguide with section axb. A wave with a

frequency under this value will see its fields decay as it travels along the waveguide.

fcmn=

kc2π√µε

=1

2π√µε

√k2n −

(mπa

)2−(nπb

)2(2.28)

The TE mode for which we have the lowest cutoff frequency is called fundamental mode (or dominant

mode). This mode (assuming a > b) will be the TE10 mode (with m = 1 and n = 0) and its cutoff

frequency is:

fc10 =1

2a√µε

(2.29)

Usually, in the majority of waveguide applications, the dimensions and frequency of operation are

chosen in order to only allow the TE10 mode to propagate. Hence, and due to their importance, equa-

tions (2.30) to (2.33) describe the electromagnetic field of this mode inside de waveguide (with m = 1

and n = 0).

Hz = A10cosπx

ae−jβz (2.30)

Ey =−jωµaπ

A10sinπx

ae−jβz (2.31)

Hx =jβa

πA10sin

πx

ae−jβz (2.32)

Ex = Hy = Ez = 0 (2.33)

The cutoff wave number and the propagation constant for this mode can then be expressed respec-

14

tively as:

kc =π

a; β =

√k2n − (π/a)2 (2.34)

Finally, the wave impedance (ZTE) and the guide wavelength (λg) can also be introduced [47][43]:

ZTE =ExHy

=knη

β=

√µ/ε√

1− f2c10/f2(2.35)

λg =2π

β=

λ0√1− f2c10/f2

(2.36)

Where λ0 is the wavelength of a plane wave with the same frequency and in the same considered

medium [43] and η =√µ/ε is the intrinsic impedance of the material that fills the waveguide.

2.3.2 TM Mode

Taking an identical approach as in the TE mode and recalling that TM modes are characterized by

having Hz = 0, the field components for a TMmn mode can be presented:

Hx =jωεnπ

k2cbBmnsin

mπx

acos

nπy

be−jβz (2.37)

Hy =−jωεmπk2ca

Bmncosmπx

asin

nπy

be−jβz (2.38)

Hz = 0 (2.39)

Ex =−jβmπk2ca

Bmncosmπx

asin

nπy

be−jβz (2.40)

Ey =−jβnπk2cb

Bmnsinmπx

acos

nπy

be−jβz (2.41)

Ez = Bmnsinmπx

asin

nπy

be−jβz (2.42)

In the previous equation Bmn is an arbitrary amplitude constant like Amn was for the the TE mode.

For the TM modes the propagation constant and cutoff frequencies assume the same formulas as the

ones seen for the TE mode (equations 2.27 and 2.28 respectively). The waveguide length (λg) is also

the same as it was for the TE mode (equation 2.36), while the impedance assumes the following form:

ZTM =ExHy

=βη

kn(2.43)

2.4 Heat Transport Equation

The following equation (2.44) describes the heat transfer phenomenon, ignoring the pressure work

and the viscous heating. Its solution will give the temperature field.

ρcp∂T

∂t+ ρcp~u · ∇T = ∇ · κ∇T +Q (2.44)

where ρ is the density (kg/m3), cp is the specific heat capacity (J/(kg ·K)), u is the velocity vector

15

(m/s), κ is the thermal conductivity (W/(m · K)), T is the temperature (K) and Q (W/m3) is the heat

generation term (W/m3). In microwave heating this last term Q represents the coupling of Maxwell’s

equations with the heat equation and is the sum of the power dissipated by Joule effect and the dissipa-

tion of the electromagnetic power (due to the presence of a material with the dielectric loss effect) and

can be represented by the following equations:

Qj = σ|E|2 (2.45)

Qdiss = ε0ε′′rω|E|2 (2.46)

Regarding equation (2.46), recall that it expresses the energy deposition phenomenon that results

from the alternating electric field in a medium with dielectric loss properties [46]. Further, note that it

includes the dielectric loss factor ε′′r which is responsible for the electromagnetic energy dissipation that

is transformed into heat, as discussed in section 2.2.

Furthermore, it is now necessary to attend the cement clinker and plasma domains regarding this

matter. To what the cement clinker is concerned, the heat generated and absorbed by the chemical

reactions that take place need to be added to the microwave heat generation term, and so Q turns out

to be the sum of two heat generation terms: one related to the heat coming from the chemical reactions

(one for each reaction), as seen in equation (2.47); and another to the microwave heating. Finally, in

the plasma domain, to the heat coming form the microwave heating generation term, it is necessary to

take into account and add the heat that is transfered from the free electrons of the plasma to the heavy

species present in it, translated through equation (2.48) [37]. Hence, the term Q of the heat equation

will be the added contributions of all these mechanisms depicted.

Qreact = ∆H Rreact (2.47)

Qel = 3me

MkBνeN (Te − T ) (2.48)

In equation (2.47), ∆H is the enthalpy change for a given reaction and Rreact is the reaction rate for

a given reaction. The meaning for the variables of equation(2.48) can be found on section 2.7.

2.5 Numerical Simulations with COMSOL Multyphysics R©

The numerical simulations carried in this work were performed resorting to COMSOL. It is a finite

element software that is a powerful resource to solve partial differential equations in domains with com-

plex geometries. The finite element method (FEM) is a computational method where an object is divided

into small but finite-size elements. Each element is treated individually, and to each one an array of

equations is assigned that depict physical properties, boundary conditions and imposed forces. These

equations are then solved simultaneously in order to predict the behavior of the object under study.

16

2.5.1 Coupling

The intention of this section is to address the coupling between the multiple physics used in the

simulations conducted throughout this Thesis. The main focus will lie on the strong coupling between

the electromagnetic and thermal fields as it is the mechanism that makes microwave heating rather

unique and an advantageous technology.

It is possible to obtain an equation that is able to compute the electromagnetic field using harmonic

sources. This equations results form manipulation of the Maxwell’s equations and is known as the

Helmholtz equation [38].

∇× (µ−1r ∇× ~E)− k2fs(εr −

j~σ

ωε0

)~E = 0 (2.49)

In order to describe microwave heating alone, it is necessary to couple the heat equation (equation

2.44), from the previous section (section 2.4), to the Helmholtz equation (2.49). This interaction can

be seen in further detail in the work of Zhao et al. [48] and is represented by the ”Electromagnetic

field” and ”Temperature field” balloons of figure 2.3. The electromagnetic field is obtained through the

Helmholtz equation and so, the heat source provided by microwave dissipated power can be calculated

and the heat equation solved. The temperature field can be computed and all temperature dependent

properties calculated, namely the complex permittivity, which will have an impact in the EM field. After,

the Helmholtz equation is solved again and this process continues until a solution is obtained.

While the iterative process of the EM field coupled with the thermal field progresses, some non-

uniformity in the heated material may occur. This can lead to dramatic temperature changes over small

distances within it. As temperature changes, the complex permittivity ε′′ may increase abruptly and

will enhance the microwave energy absorption on hotter areas, even if the electric field is unchanged,

leading to very rapid heating which can result in the occurrence of an unwanted phenomenon known as

thermal runaway [8]. This phenomenon, due to its unstable nature, can cause a local sharp increase of

temperature in the process material, called ”hot spots” which can lead to its catastrophic degradation.

In the case of the present work, where there are heat generation terms from sources other than the

microwave dissipated power alone, the relation discussed above turns to be a little more complex as can

be observed in figure 2.3. As the temperature distribution profile is computed from the heat equation

(provided with the microwave heat source) some processes will be triggered, like the chemical reactions

of the cement clinker and of the Plasma. On the other hand, electromagnetic energy will be absorbed by

the electrons through Joule heating [40], increasing their temperature and triggering electron collision

(excitation and ionization) and chemical reactions (some depend on the gas and electron temperature)

in the plasma region. This processes will, on their own, generate heat sources and sinks that need to

be added to the term Q in the heat equation. With this contribution a new temperature profile can be

calculated as can the temperature dependent properties, and only then the process can continue with

the re-calculation of the Helmholtz equation.

17

Figure 2.3: Schematic of the multi-physical coupling. Adapted from Zhao et al. [48].

2.6 Transport of Concentrated Species

This section aims to give an insight to the transport equations of the chemical species, as well as the

mechanisms behind the chemical reactions that take place to form the cement clinker.

2.6.1 Mass transport Equation

The mass transport equation for an individual species, part of a mixture with i species can be formu-

lated in the following way [49]:

∂

∂t(ρωi) +∇ · (ρωi~u) = −∇ · ~ji + ~Ri (2.50)

The subscripted terms are specific for an individual species, i while the remaining concern the mix-

ture as a whole. From left to right, ρ stands for the density of the mixture (in kg/m3), ωi represents the

mass fraction of species i (non dimensional), ~u is the average velocity (m/s), ~ji is the mass flux, of a

species i, relative to the average velocity (kg/(m2 · s)). The ~ji term can include added contributions of

molecular an thermal diffusion and mass flux resulting from the migration in an electric field. Finally the

last term, ~Ri is the rate expression and depicts the production or consumption of a species i.

The sum of all transport equations, one for each species, will give the mass conservation equation

as follows:∂ρ

∂t+∇ · (ρ~u) = 0 (2.51)

To get this relation, it was assumed that the sum of all species mass fractions (ωi) equals one, that

the sum of the mass fluxes (~ji) returns zero and that the sum of the rate expressions ( ~Ri) of all the

species must be zero, as the rates of the produced species must match the rates of the consumed ones.

18

Thus, with this result, the species transport for an individual species can be presented:

ρ∂

∂t(ωi) + ρ(~u · ∇)ωi = ~Ri (2.52)

With this relation i − 1 of the species are independent and possible to solve. The mass fraction of

the remaining species can be computed knowing that the sum of all mass fractions must be one. As it

might been noticed the term ”−∇ · ~ji” is missing, this happens as no mass diffusion is accounted for in

the studies conducted in this Thesis resulting in ~ji = 0.

2.6.2 Arrhenius Law

For many chemical reactions, the rate expression can be expressed as the product between a tem-

perature dependent term and a composition dependent term [50]. For the majority of reactions, the

reaction rate constant is well represented by the Arrhenius’ equation (2.53),

k = k0e−Ea/RT , (2.53)

where k0 is the frequency factor (or pre-exponential factor), Ea is the a activation energy (usually comes

in Joule per mole), R is the universal gas constant (in J/(K ·mole)) and T is the temperature in Kelvin.

This expression is able to be a good fit of the experiment over a broad range of temperatures, and

is considered to be a good approximation to the real temperature dependency of the reactions rates.

The energy activation term Ea and the temperature level of the reaction will determine the temperature

dependency. As can be seen in figure 2.4, reactions with high energy activation are very temperature-

sensitive while reactions with low Ea are less sensitive to temperature [50]. It can also be observed

that at lower temperature it is required less temperature input (∆T ) to double the rate of reaction, which

means reactions at lower temperature are more temperature sensitive.

2.7 Plasma

Plasma is an ionized gas and is usually considered as the fourth state of the matter [35, 40]. It can be

produced in a gas when a constant or alternating electric field is applied. As temperature increases, the

atoms and molecules in a certain state of matter will see their energy increase and thus increasing their

vibration and movement. Eventually the transition from one state of matter to another more energetic

happens, form solid to liquid, from liquid to gas and, if the temperature keeps raising, from gas to plasma

phase.

In a plasma, two major events occur: molecules dissociate into atoms and radicals, and the elec-

trons of molecules and atoms lose their bonds, resulting in the formation of ions. Thus, plasma can

be considered as a mixture of electrons, ions and neutral species. Further it is also considered electri-

cally neutral or quasi-neutral, however its electrical properties are affected by the significant number of

charged particles [51].

19

Figure 2.4: Temperature dependency of the reaction rate (adapted from Levenspiel [50]).

It is also possible to categorize plasmas in thermal and non-thermal plasmas ones. Thermal plasmas

is a quasi-equilibrium plasma that is in chemical equilibrium where local thermodynamic conditions are

met. It is described by only one temperature (the gas temperature), which then determines the ionization

and chemical processes occurring in the plasma [40].

The non-thermal plasma, also known as cold plasma, is a plasma that is not in thermodynamic equi-

librium. This kind of plasma has a multitude of temperatures that describe it and that are associated to

different species in the plasma. In this case, and in opposition to what happens with the thermal plas-

mas, there are other temperatures, besides the gas one, able to determine the ionization and chemical

processes and it is possible to present them as follows:

Te > Tv > Tr ≈ Ti ≈ Tg (2.54)

In equation (2.54) Te is the electron temperature and is the highest temperature, then we have the

vibrational temperature Tv and the rotational temperature Tr. It is common for Tr to be very similar to

the ion temperature Ti and the gas temperature Tg. The gas temperature is also known as translational

temperature and is the lowest of all temperatures presented [40]. Non-thermal plasmas are usually

generated at low power inputs or at low pressure.

2.7.1 Electron Transport and Energy

In general, electron transport is depicted by the Boltzmann equation [38]. The Boltzmann equation

is a complicated integrodifferential equation and so far it is not possible to solve it in an efficient manner

[52]. The equation is approximated by two fluid equations that after some mathematical treatment turn

20

out to be equations (2.55) and (2.57) that describe the electron number density ne and electron energy

density nεp respectively.

∂

∂t(ne) +∇ · ~Γe = Re − (~u · ∇)ne (2.55)

~Γe = −(µe · ~E)ne −De · ∇ne (2.56)

Here, ne denotes electron number density (1/m3), Re is the electron rate expression (1/(m3 · s)) and

~u is the neutral fluid velocity vector (m/s) and comes from Navier-Stokes equations. ~Γe is the electron

flux as seen in equation (2.56), ~µe is the electron mobility (m2/(V · s)), ~E is the electric field (V/m) and

finally ~De is the electron diffusivity (m2/s).

∂

∂t(nεp) +∇ · ~Γεp + ~E · ~Γe = Rεp − (~u · ∇)nεp (2.57)

~Γεp = −(µεp · ~E)nεp −Dεp · ∇nεp (2.58)

In the electron energy density equation (2.57), nεp stands for the electron energy density (V/m3) and

Rεp represents the energy loss/gain due to inelastic collisions (V/(m3 · s)). Regarding equation (2.58),~Γεp is the electron energy flux, µεp is the electron energy mobility (m2/(V · s)) and Dεp is the electron

energy diffusivity (m2/s).

Due to their high mobility and low mass, electrons are the first to receive energy from the electric

fields. This energy is then transfered to the other plasma components giving the required energy for

the plasma-chemical processes to occur (ionization, excitation and dissociation) [40]. The rate of those

processes depend on the amount of electrons that have the energy to start them. This energy can be

depicted by resourcing to the electron energy distribution function (EEDF) f(εp), which is the probability

density for an electron to have its value of energy equal to εp. The electron diffusivity, energy mobility

and energy diffusivity can be calculated by using Einstein’s relation for a Maxwellian EEDF (there are

other distributions besides the Maxwellian but only this one is used in studies conducted) such that:

De =kBTee

µe (2.59)

µεp =

(5

3

)µe (2.60)

Dεp =kBTee

µεp (2.61)

µe =e

meνeN(2.62)

In the previous equation kB is the Boltzmann constant (1.386488(13)× 10−23J/K), e the electron charge

(−1.602× 10−19C), me is the electron mass (9.109× 10−31kg) and with νeN being the collision frequency

between electrons and Neutrals. The EEDF can then be used to calculate reaction rates coefficients

of plasma-chemical processes involving electrons (electron impact reactions and electron attachment

21

reactions):

kk =

(2e

me

)1/2 ∫ ∞0

εpσk(εp)f(εp) dεp (2.63)

In equation 2.63 σk is the collision cross section. Collisions cross sections for several reactions are

available in many references of the literature [53] and databases [54]. However these reactions rates

can also be defined by inputing a specific forward rate constant coefficient.

The mean electron energy ε (eV ) and the electron temperature Te (eV ) can be calculated from:

εp =nεpne

(2.64)

Te =

(2

3

)εp (2.65)

2.7.2 Heavy Species Transport and Plasma Properties

The heavy species in a plasma are the neutral, excited and ionized components that take part in

it. The transport equation is equation (2.52) seen in the section (2.6.1). As in section (2.6.1) i − 1

species are possible to solve, and the mass fraction of the remaining species is computed from the

mass constraint. To calculate the source coefficients (Ri) it is possible to just specify a forward rate

constant coefficient directly or, again, use the Arrhenius Law as described in (2.6.2) to compute the

reaction rate. It is however common to use the modified version of the Arrhenius equation as it explicits

the temperature dependence of the pre-exponential factor.

k = k0Tne−Ea/(RT ) (2.66)

In equation 2.66 n typically ranges between −1 < n < 1. it is convenient to also present a way to

determine the number density of a certain species i as it will be useful in the studies regarding plasmas,

ni =

(p

kBT

)xi, (2.67)

where p stands for pressure and xi for the mole fraction of the species i.

Finally and regarding the plasma mixture itself, there are still some variables of great importance yet

to be introduced. Namely the electrical conductivity, σ, the relative electrical permittivity, εr [37] and the

gas heat capacity, Cp [39], all presented in equations (2.68) to (2.70) in the same order while the Plasma

density is computed through the ideal gas law as in equation (2.71).

σ =nee

2

me

1

νeN(2.68)

εr = 1− nee2

ε0me

1

(ωf − jνeN )(2.69)

Cp =5kB2M

(2.70)

ρ =pM

RT(2.71)

22

With M being the mole averaged molecular weight (kg/mole) and ω the angular frequency (ω = 2πf ).

23

24

Chapter 3

Verification and Validation

3.1 Chemical kinetics model

In this chapter, a verification and validation of the numerical models with COMSOL Multiphysics R©

for the chemical kinetics model for the cement clinker was made.

The study conducted by Mujumdar and Ranade [23] is chosen for this comparison, as it describes a

very consistent kinetics model. Such model was developed with experimental data obtained from actual

industrial kilns yielding results with high accuracy. Thus, the purpose of this sections is to conduct a

verification and validation process by comparing the results from COMSOL against both the numerical

and experimental ones presented by Mujumdar and Ranade [23].

Recall that the main objective of the numerical simulation is to verify and evaluate the validity of the

chemical kinetics model of the cement clinker built in a COMSOL environment. Hence the focus is on the

chemical reactions, their inputs and outputs, leaving all other aspects aside, such as energy balances,

energy losses and kiln design specifics.

Starting form this premise, the reactions set, chemical kinetics and heat of reactions can be es-

tablished, being taken and adapted (as explained below) from Mujumdar and Ranade [23], and are

presented in the table (3.1).

Reaction k0 Ea (kJ/mol) ∆H (kJ/mol)

1. CaCO3 → CaO + CO2 4.07× 106(s−1) 185 179.4

2. 2CaO + SiO2 → C2S 1.0× 107(m3 kg−1 s−1) 240 −127.6

3. C2S + CaO → C3S 1.0× 109(m3 kg−1 s−1) 420 16.0

4. 3CaO +Al2O3 → C3A 1.0× 108(m3 kg−1 s−1) 310 21.8

5. 4CaO +Al2O3 + Fe2O3 → C4AF 1.0× 108(m6 kg−1 s−1) 330 −41.3

Table 3.1: Reactions, kinetics and heat of reaction.

Regarding table (3.1) there are some assumptions made to simplify the writing of the equations [14]

25

C2S => 2CaO · SiO2 (dicalcium silicate)C3S => 3CaO · SiO2 (tricalium silicate)C3A => 3CaO ·Al2O3 (tricalcium aluminate)C4AF => 4CaO ·Al2O3 · Fe2O3 (tetracalcium aluminoferrite)

Table 3.2: Chemical reactions’ writing simplification.

as can been seen in the table (3.2).

The adaptation made in table (3.1) refers to the frequency factor of the first reacton. It was converted

from the value presented in Mujumdar and Ranade [23], 1.18×103 (kmol m−2 s−1), to a more convenient

form, resorting to Rao [18]. By dividing 1.18× 103 by the molar density of the solid and the particle size

of the limestone, assumed to be 27.1 (kmol/m3) and 1.07× 10−5 (m) respectively, the new value for the

frequency factor is found and is the one presented for the first reaction of table (3.1).

3.1.1 Numerical Model

In order to fulfill the targeted goal, the approach to the problem consisted on building a three di-

mensional model of the first industrial kiln presented in Mujumdar and Ranade [23]. The majority of

the geometric constraints were respected, as was the mass inflow of the raw material and the relative

percentage of all the inflow species (i.e. the inlet mass factions). Further, the resulted bed temperature

profile obtained from the study conducted , as seen in figure 3.1, was taken as an input to the numerical

simulation carried. This approach was followed due to the far greater complexity of building and mod-

eling a fully coupled three dimensional conventional kiln. As an example, that would require to model

the combustion process of the conventional heat source. That would be a tricky task and would divert

the subject away from the main study in this Thesis, microwave as an heat source, and so, it would not

be beneficial to this work. Hence, and as the temperature profile in the bed reflects, ultimately, all the

energy changes in it, it was considered a reasonable approach to take this profile as an input to the

numerical model. Finally, it was also assumed a constant bed height along the kilns, that there is no

formation of the melted phase and that no coating forms along the kiln walls.

With the approach clarified, it is now possible to start building the model in COMSOL. The transport

of concentrated species interface was chosen as it is able to solve the mass fraction of the all partici-

pating species of a mixture (in this case a pseudo-liquid one), where species concentrations are of the

same order and none can be identified as the solver, which is the case. Transport through convection

was included. The geometry of the model includes the bed region, the freeboard domain, where air and

gaseous chemical products are supposed to exist, i.e., CO2, and the refractory shell, that enclosures

the other two domains along the kiln. The dimensions can be found on table 3.3, and on figure 3.2 it is

possible to see a graphical representation of it.

26

Figure 3.1: Bed temperature input obtained from Mujumdar and Ranade [23]..

Length (m) 50

Inner refractory diameter (m) 3.4

Outer refractory diameter (m) 3.8

Height at solids entry (m) 0.46

Table 3.3: Dimensions of the rotary kiln.

Figure 3.2: Graphical representation of the rotary kiln..

With the height of the solids entry the inlet area can be calculated. And knowing the solids inlet flow

and the mixture density, 38.88Kg/s and 1200Kg/m3 respectively, it is possible to compute the velocity

of the bed mixture (u) through the following equation:

˙mflow = ρAu (3.1)

27

The inlet mass fractions and the respective molar masses of the participant species are presented in

table (3.4). These values will be placed in the referred interface as required. Namely, a inflow boundary

conditions will be established with the inflow mass fractions at the entrance of the kiln. While and outflow

boundary condition will be placed at the opposite end.

Species Molar mass (kg/mole) Inlet mass fraction

CaCO3 0.1 0.340

CaO 0.056 0.396

SiO2 0.06 0.179

Al2O3 0.1 0.0425

Fe2O3 0.16 0.0425

CO2 0.044 −C2S 0.172 −C3S 0.228 −C3A 0.27 −C4AF 0.486 −

Table 3.4: Species’ molar mass and inlet mass fractions

Regarding the chemical reactions, the Arrhenius law is used, through equation (2.53), to compute

the rate constant of formation or decompositions of the species. With those set it is possible to compute

the rate of each reaction as follows:

RCaCO3= kCaCO3

CCaCO3(3.2)

RC2S = kC2S(CCaO)2CSiO2(3.3)

RC3S = kC3SCCaOCC2S (3.4)

RC3A = kC3A(CCaO)3CAl2O3(3.5)

RC4AF = kC4AF (CCaO)4CAl2O3CFe2O3 (3.6)

Finally, with equations 3.2 to 3.6, one can compute the chemical conversions of each species [55].

To solve the numerical problem a stationary solver was used, as such, in the mass transport equation

(2.50), the time dependent term drops out resulting in equation (3.7).

∇ · (ρωiu) = Ri (3.7)

Furthermore, a mesh needs to generated by the software. The procedure followed to obtain the mesh

consists in generating a free tetrahedral mesh for the domains of the geometry and set a maximum size

for each element. The imposition will block the maximum size of the elements but not the minimum, as

so, some will be of that size while others, due to geometric constraints (like curvature) will be smaller.

A mesh independence study was carried by comparing the mass fraction of C3S at the end of the kiln

after conducting several simulations where the maximum element size was decreased for each one.The

objective is to monitor C3S mass fraction until it becomes independent of the mesh. The study results

28

are summarized in table 3.5 where it is possible to see the mass fraction for a correspondent element

size used, divided by the length of the kiln (50 m), giving a characteristic length of an element in the

mesh.

Element characteristic length (m) C3S Mass Fraction at kiln’s exit

6.53× 10−3 0.52686

5.15× 10−3 0.52652

3.15× 10−3 0.52625

3× 10−3 0.52618

2.30× 10−3 0.52618

1.68× 10−3 0.52618

Table 3.5: Results for the mesh independence study

As it is possible to see, from table 3.5, the mass fraction of C3S changes with the mesh refinement,

although only with small variations. Considering the mass fraction of C3S it is possible to conclude that

the minimum characteristic element length that should be used is 3 × 10−3, so that one gets accurate

results.

3.1.2 Verification and Validation

With the model built and the study conducted, it is possible to compare the results and accomplish

the verification and validation process. The table 3.6 presents a comparison between the numerical data

obtained from Mujumdar and Ranade [23], the experimental data presented in the same study and the

results obtained from the numerical studied carried in COMSOL. Regarding the data itself, it shows the

mass fraction of product species, namely: CaO, C2S, C3S, C3A, C4AF.

Species Experimental data [23] Numerical data [23] Numerical data from COMSOL

CaO 0.084 0.075 0.0771

C3S 0.483 0.503 0.5262

C2S 0.239 0.222 0.2159

C3A 0.051 0.051 0.0829

C4AF 0.143 0.149 0.0988

Table 3.6: Numerical and experimental results of the mass fraction at kiln’s exit.

As it is possible to see on table 3.6 the numerical data obtained from the COMSOL model are close

to the numerical results presented in Mujumdar and Ranade [23], with the mass fractions of the first

three species presenting similar values. In the last two species, the difference is significant, probably

due to an approximation made or a mistake in the model construction. Regarding the experimental

data, the results are not far apart, however it is possible to notice significant deviations from the values

presented for the mass fractions, namely in the last two species, where again, the difference is steeper

when compared to the other species.

Although the presented comparison holds its value, it fails to give an insight into the evolution of the

29

mass fractions along the kiln’s length. This happens due to lack of experimental data along the kiln as

mass fractions are only known in the beginning (the inlet) and at the end (the outlet) of the rotary kiln.

However, Mujumdar and Ranade [23], in its numerical study, presents a typical evolution of the mass

fractions of the relevant species for the case in study. As such, and with the objective to assess the

performance of the built model, a comparison (see figure 3.3) was made where the mass fractions along

the kiln obtained in Mujumdar and Ranade [23] will be presented against the ones obtained from the

numerical simulation carried in COMSOL.

(a) Mass fractions from Mujumdar and Ranade [23]. (b) Mass fractions from COMSOL.

Figure 3.3: Mass fractions’ evolution along the kiln’s length.

From the observation of the figure 3.3, a very reasonable agreement in the behavior of the mass

fraction along the kiln length between the two models can be seen. It is possible to verify from a qual-

itative point of view that the curves presented follow a close evolution. Except, as referred, for the last

two species, C3A and C4AF, that despite appearing at similar time when compared to Mujumdar and

Ranade [23] results and sharing a correct relative position, their mass fractions appear to be apart from

the compared numerical model [23].

Recall that attempting to model a fully coupled conventional kiln is complex and challenging task

where any assumption and simplification will make room for uncertainty, and so, slightly different results.

Moreover, it would require the demanding undertaking of modeling the necessary combustion process,

which in its own would be rather unnecessary as the main heat source used in this work is microwave

energy. Hence, from the study conducted, it may be reasonable to look at the numerical model, built in

COMSOL as a fair tool to model the chemical reactions and respective kinetics of the cement clinker.

The results show that the data obtained at the kiln’s exit is not very far from the compared data in the

literature, furthermore, the evolution of the mass fractions along the kiln length shows a fair concordance

with the compared results.

30

3.2 Plasma modeling

In this section an effort is made to study and understand major operational features of a numerical

model for plasma driven by microwave energy, as it is an important application of this type energy and a

reported phenomena in clinker formation with a microwave applicator [27].

In the production of cement clinker, according to Spang [14], CO2 will be formed as a reaction product

of the first reaction (see table 3.1). The CO2 will then flow from the bed region to the freeboard. This

gas, when bombarded with the microwave energy can turn into a plasma, posing new challenges to the

cement clinker formation when using microwave energy. It involves an additional loss of energy and it

makes it difficult to evaluate the effects of the microwave and the plasma in the overall process [27].

The initial objective was to study the required conditions to create a microwave induced plasma in

CO2, resorting to a numerical model built in COMSOL. And then, compare it to similar works in the

literature with the purpose to verify and possibly validate the current model. However, after extended

search, no study was found with a suitable numerical model of a CO2 microwave induced plasma.

Nonetheless several works were found where a COMSOL numerical model was used to simulate a

microwave induced plasma in an Argon atmosphere.

Although Argon is not the required gas, it is a simpler species to model, and despite stepping aside of

the initial motivation, it seemed to be a good first step to understand the plasma modeling in a COMSOL

environment. Hence the focus lied down on recreating the numerical model of a Argon microwave

induced plasma in order to study and evaluate the concepts behind it. Efforts were made to tackle this

objective and models were built in COMSOL resorting to several studies, namely Baeva et al. [37], Yang

et al. [38] and Nowakowska et al. [39].

Unfortunately, no attempts were completed with success. Modeling a microwave induced plasma (in

this case using Argon as the working gas) has proven to be a much harder endeavor than expected.

The reasons behind this failure may be due to the complex and rather large subjects involved in the

process as can be attested by Yang et al. [38]. The objective was to add plasma, later, as a feature of

a microwave heating model, however, given the faced challenge, it was considered that pursuing such

goal any longer would mean going way beyond the scope of a this master thesis. Ultimately, the author

acknowledges that a lack of sufficient background regarding this subject may have also compromised

the proposed work.

It was then considered to adapt a test model course available in COMSOL, and build a paramet-

ric study of a microwave induced plasma in Argon. It can be considered to be a reasonable tool to

understand how operating factors ,like the power input or gas velocity, can affect an ionized gas.

3.2.1 Numerical Model

The numerical model was based on the test model available in the commercial software used in

this Thesis. It consists on a simple two dimensional model where a flow of Argon is bombarded with

microwave in order to generate a plasma. A schematic of the geometry of the model can be seen in

figure 3.4 and represents the computational domain used to conduct this work. The yellow domains

31

(both 0.05 m wide and 0.1 m long) represent the waveguide through which the electromagnetic wave

propagates. While the blue domain (0.05 m wide and 0.25 m long) is the 2-D representation of the tube

used to convey the flow of gas that enters as indicated, intercepts with the waveguide in the middle and

leaves the domain through boundary ”C”.

Figure 3.4: Graphical representation of the plasma model.

The chemical mechanism for the plasma consists of 4 species and 14 reactions taken from the

work conducted by Baeva et al. [37], with the rate coefficient of the last set being taken from available

databases [54] and they can be found on table 3.7. Surface reactions, that describe the neutralization

transition of ionic species back to their ground state, were also included.

Reaction j Rate coefficientKj ∆εj (eV )

1. e+Ar → e+Ar∗ 4.9× 10−15Te[eV ]0.5exp(−11.65/Te[eV ]),m3/s 11.52. e+Ar∗ → e+Ar 4.8× 10−16Te[eV ]0.5,m3/s -11.53. e+Ar → 2e+Ar+ 1.27× 10−14Te[eV ]0.5exp(−15.76/Te[eV ]),m3/s 15.764. e+Ar∗ → 2e+Ar+ 1.37× 10−13Te[eV ]0.5exp(−4.11/Te[eV ]),m3/s 4.115. e+ e+Ar+ → e+Ar 8.75× 10−39Te[eV ]−4.5,m3/s 1.5Te6. e+Ar+2 → Ar +Ar∗ 1.04× 10−12(Te[K]/300)−0.67 1−exp(−418/T [K])

1−0.31exp(−418/T [K]) ,m3/s 1.5Te

7. e+Ar+2 → e+Ar +Ar+ 1.11× 10−12exp[−(2.94− 3T [eV ]−0.026Te[eV ] )],m3/s 1.25

8. Ar+ + 2Ar → Ar+2 +Ar 2.25× 10−43(T [K]/300)−4

,m3/s9. Ar+2 +Ar → Ar+ + 2Ar 0.552× 10−15exp(−1.304/T [eV ])T−1,m3/s10. Ar∗ +Ar∗ → Ar+ +Ar + e 6.2× 10−16,m3/s -1.5Te11. Ar∗ +Ar → Ar +Ar 3.0× 10−21,m3/s12. e+M → e+M(M = Ar,Ar+, Ar+2 )

Table 3.7: Chemical mechanism considered for the plasma model.

Each reaction translates a mechanism in plasma formation. Those processes are presented for the

respective reactions (of table 3.7) in table 3.8 [38].

The temperature and pressure are fixed to 350 K and approximately 135 Pa respectively. The mag-

32

Reaction j Process

1. Ground state excitation2. Superelastic collisions3. Ground state ionization4. Step-wise ionization5. Three-body recombination6. Dissociative recombination7. Electron impact8. Atomic ions conversion9. Molecular ions dissociation Atom impact10. Metastable pooling11. Two-body quenching12. Elastic scattering

Table 3.8: Plasma processes for each correspondent reaction.

netic flux is set to zero, the density is computed with the ideal gas law, the reduced electron mobility

is set to 4 × 1024 and finally the initial electron density is set to 1 × 1017 to make calculations converge

faster. Flow speed and power input will be addressed later.

As for boundary conditions, regarding the species (heavy species, ions and electrons), the derivative

of the participant particles density number and of the electron temperature is set to zero in the plasma

domain (blue domain) boundaries. The surface reactions take place on the walls of the tube (blue plasma

domain) and the species outlet is made through ”C”.

The boundary conditions for the electromagnetic field can be found in table 3.9. At the port (boundary

”B”) is where power input is defined with TE10 mode.

Boundaries Condition type

B Excitation Port (TE10mode)Blue domain walls except wall C Ground (V = 0)all except B Perfect electric conductor (n× ~E = 0)

Table 3.9: Boundary conditions for the electromagnetic field.

3.2.2 Results of the plasma model

The aim of the study is to get a glimpse of microwave interaction with a plasma and its development

and behavior under different powers and flows to evaluate its influence. As such a frequency transient

simulation was conducted with a defined time interval of 1× 10−8 to 1× 10−2 seconds, a power of 500W

and a flow speed of 10 m/s. After, studies were conducted where power was varied from 50 W to 1000 W

and flow velocity was set range from 1 m/s to 30 m/s.

The results for the first study for a time of 1× 10−2 s can be found in figures 3.5 and 3.6. In figure 3.5

the electron number density (a) and temperature (b) are plotted. As it is possible to notice the electron

number density has its maximum of 6, 82×1017 m−3 slightly after the crossing point between the plasma

domain and the waveguide. More important, electron density is found to be asymmetric, being closer to

the top. Such behavior occurs because the electromagnetic waves are also absorbed in a non symmetric

33

way. The electron temperature reaches its maximum right below the waveguide as the wave is absorbed

at this zone.

(a) Electron number density (m−3). (b) Electron temperature (eV).

Figure 3.5: Electron number density and temperature, time = 1× 10−2 s.

In figure 3.6 the electric field norm is plotted. As it can be seen the electric field is high inside

the waveguide and the electromagnetic wave propagates with no losses. When the wave reaches the

plasma, its energy is absorbed by the electrons. As the electron absorb the microwave energy it will

see its temperature raise and by that it will generate more electron through the process of ionization that

sustains the plasma.

Figure 3.6: Electric field norm (V/m), time = 1× 10−2 s.

The results of the parametric studied can be found in figure 3.7 where the maximum electron density

is plotted against a varied input power and gas velocity. The power parametric study was done with

a gas velocity of 10 m/s and while the varied velocity study was conducted with a microwave power of

500W . One can assess that the electron density increases directly with microwave power. This is fair

considering that the increase in power will result in more energy absorbed and hence more electrons

will be generated. Regarding the gas velocity it can be seen that it has almost no effect on the electron

number density. This happens because although the gas velocity may increase the discharge gas

34

volume remains about the same and so the electrons density remains somewhat constant.

Figure 3.7: Maximum electron density, for a time of 1×10−2 s, versus microwave power and gas velocity.

35

36

Chapter 4

Microwave Limestone Processing

From what was learned in the previous chapters, namely in chapter 3, it it is now time to develop

a numerical model where the microwave, thermal and chemical interfaces are coupled. Although the

objective was to model the cement clinker processing by means of microwave energy, some adjustments

and simplifications had to be made in order to conceive a more feasible goal. Due the complexity of the

cement clinker production and the non-existent of a continuous microwave cavity design proven to cope

with such process, the focus targeted on the prediction of the first and major equation of the cement

clinker formation: turn limestone into lime as in table 3.1. Plasma revealed himself to be a way more

challenging enterprise than initially though, and hence, it was dismissed, although the conducted study

helped understanding microwave interactions with the absorbing medium. As a result, the following

sections aim at presenting a 3D model of microwave limestone processing unit, using a simulation

method able to optimize efficiency and power usage during the continuous operation, while avoiding

temperature related problems.

4.1 Cavity Description

In this section a brief description of the cavity is made. It is a 86.36x43.19x500 mm rectangular

cavity with a single aluminum waveguide (standard WR-340) coupled with a short plate (or plunger) so

that microwave heating efficiency can be maximized. The 500 mm long waveguide allows to observe the

whole wavelength inside it and to evaluate the absorption efficiency as it will be described further down.

The cavity only has one resonance mode (monomode cavity), that must be maintained if maximum

efficiency is to be obtained.

The material flows horizontally (see figure 4.4) with a 40% fill rate through a 2 mm thick quartz tube,

320 mm long and with an inner radius of 25 mm. The tube is located at the center of the waveguide and

intercepts it through its widest surface (86.36 mm wide). The residence time should be enough for the

full endothermic reaction of limestone to take place. A graphical representation of the model’s geometry

can be found below in figure 4.1. More details on how the system works can be found on section 4.3.1

and in figure 4.4.

37

Figure 4.1: Geometry of the cavity for limestone processing.

Figure 4.2 presents the computational grid with 2,354,801 tetrahedrons of the cavity used created

in COMSOL. The general 3D perspective can be seen in on the left (a). On the right (b) a zoomed

perspective can be found to show how the local mesh was made smaller in the regions of interest and

in objects with a small dimension like the thickness of the quartz tube.

(a) 3D perspective. (b) Zoomed perspective.

Figure 4.2: Perspective of the computational grid with 2,354,801 tetrahedral elements.

4.2 Chemical model and material properties

As explained above, the chemical model used will only include the limestone calcination reaction

where limestone turns into lime and carbon dioxide according equation (4.1).

CaCO3 → CaO + CO2 (4.1)

According to section 2.6, for the reactions model, a transport equation has to be solved for each

species. However the principle of mass conservation must be respected, so it serves as a constraint,

hence, the model only requires two equations to account for all three species. The transport equation for

each species mass fraction can be found on equation (2.52) from section 2.6. In order to compute the

consumption and production rate of limestone and lime, respectively, the following equations are used:

RCaCO3 = −ρκωCaCO3 (4.2)

RCaO = ρkωCaCO3

(MCaO

MCaCO3

), (4.3)

and the CO2 mass fraction is computed from the mass constraint. Recall that k corresponds to the

38

Arrhenius reaction rate constant defined in equation (2.53).

To finalize, the energy absorbed by the endothermic reaction must be added to the heat equation

(2.44). That is achieved resorting to equation (2.47) and adding it as an energy source term.

Qchem = ∆HRCaCO3= −∆HρkωCaCO3

(4.4)

The chemical constants used in this model are presented on table 4.1.

Chemical model constant Value

Ea 185 [kJ/mol]

k0 4.07× 106 [s−1]

∆H 1794 [kJ/Kg]

MCaCO30.1 [kg/mol]

MCaO 0.056 [kg/mol]

Table 4.1: Limestone calcination chemical constants.

Regarding the materials used, aluminum, air and glass (quartz) are from the software materials li-

brary. Regarding the properties of limestone, data is required from the literature. The density, thermal

conductivity and heat capacity are presented below on table 4.2.

Property Value

ρ 1680 [kgm−3]

Cp 800 [J kg−1K−1]

κ 0.69 [W m−1K−1]

Table 4.2: Limestone thermal properties from Mujumdar and Ranade [23].

Another property that need definition is the limestone complex permittivity. The complex permittivity

assumes a major role in the process. It enables the absorption of microwave energy, and thus, fills the

void between the electromagnetic and the thermal fields in the material heating. The constituents of

the complex permittivity, relative permittivity (or dielectric constant) and the loss factor were acquired

from the work developed by Behrend et al. [56] and are presented in figure 4.3 in order to illustrate their

evolution with temperature. They were added to the limestone’s material bed interface node according

to equation (2.20).

39

Figure 4.3: Limestone’s permittivity and loss factor with temperature [56].

4.3 Numerical model: domain description, interfaces and bound-

ary conditions

In this section the goal is to explain the domains decomposition, the imposed boundary conditions

and the interfaces used in the numerical model.

4.3.1 Domain description

A section plane normal to the y axis that crosses through the middle of the cavity was defined to get

the representation of the model seen in figure 4.4. There, it is possible to find the domain selection for

each component belonging to the model. This domain selection is particularly useful as it is then used

to assign each domain to the respective materials and to the interface nodes in the software. Interface

nodes are the physical models that COMSOL provides for each physical phenomenon, in this case there

will be three representing the microwave, heat transfer, and chemical process. Each has its own set of

equations, assumptions and setting defined according to the physics it wants to simulate.

In figure 4.4, besides the domain definition, its possible to examine the model’s general layout and

how it works. First notice that figure 4.4 presents the model horizontally as it fits better in the document,

in fact, gravity is considered to act along the x axis, in the negative way as indicated. Thus, material

flows horizontally (in relation to gravity) along positive z and it is bombarded and heated by microwave

energy that propagates through the cavity along the positive x axis. The quartz tube rotates at 5 rpm,

though the material bed is in a fixed position, so it will only affect the tube thermal field. Finally one

can easily find the plunger, represented in purple. Its function is to concentrate the electric field at the

material’s location, by moving along the x axis, maximizing the microwave absorption efficiency.

40

Figure 4.4: Domains of the numerical model.

4.3.2 Interfaces and boundary conditions

In order to fulfill the objective established, the adequate COMSOL interfaces must be selected so

that each physical and chemical process can be accurately depicted in the simulation. Therefore, as

the power source is microwave energy on a single frequency (2.45 GHz), the ”electromagnetic waves,

frequency domain” interface was designated to handle the microwave physics. The ”transport of con-

centrated species” interface was chosen to accommodate the chemical process in study, constricted to

the bed domain, and finally the ”heat transfer in solids” was selected so that the thermal field can be

computed. In all interfaces pressure was set to 1 atmosphere.

The first referred interface computes the electric field in the domains where the wave propagates

through. As can be attested on figure 4.5, the wave equation (Helmholtz equation (eq. 2.49)) is set to

the whole model except for the volume behind the plunger. The moving plunger will increase or diminish

the waveguide domain as to maximize microwave absorption efficiency.

Figure 4.5: Wave equation domain.

The waveguide port is a boundary condition imposed at the port with its location indicated on fig-

ure 4.5. A TE10 mode is applied at the input port and the frequency is fixed at 2.45 GHz, as already

mentioned, for all simulations, the resonance of the microwave model is accomplished by making ad-

justments to the plunger along the vertical (x axis) orientation.

For all the boundaries of the wave equation domain selected in figure 4.5, the perfect electric con-

41

ductor condition ( ~E × ~n = 0) is imposed, except for the input port boundary, because electromagnetic

energy is not supposed to propagate through metallic walls.

The transport interface is responsible for computing the mass fractions of each participant species

along the tube and is assigned only to the bed domain as highlighted in figure 4.6. In this domain the

chemical model described in 4.2 is established. The mass fraction of the reactant (CaCO3) is set to 1

as an initial value (the other two species have their mass fraction set to zero). An inflow, with the same

species proportions, is set at tube’s entrance (on the left of figure 4.6) to account for the entrance of new

material and an outflow boundary condition is applied to the opposite end. The velocity of the material

along the tube is set axially from the inflow to the outflow. Its value corresponds to a mass flow (mflow)

of 0.25kg/h which was established to be a suitable value as it allows for a low axial speed increasing the

simulation’s stability and helping to reach a converged steady state solution. Later higher mass flows

will be experimented on, for comparison with the results of the proposed mass flow value. The aim is to

increase to mass flows of the same magnitude of the ones found in similar studies [57] and preform a

parametric study. Finally, for the remaining boundaries a no flux condition (~n · (ρωi~u) = 0) is applied so

that no mass flows through these boundaries.

Figure 4.6: Species’ transport domain for the chemical physics modeling.

Finally, the last interface computes the thermal field by accounting the heat sources provided by the

other physical processes and the convective heat losses of the microwave cavity. It is set to all the

domains present in the model. A velocity consistent with the one defined in the chemical interface is

set to the heat generated in the material’s bed domain (highlighted in figure 4.6). For the quartz tube a

rotary motion of 5 rpm is imposed in order to the temperature field, so that its influence can be modeled

by this interface.

Concerning the heat sources of this model, two must be accounted (in the energy equation (2.44)) to

promote the coupling between the three used physics. One comes from the microwave power deposition

(equations 2.45 and 2.46) and its assigned to the wave domain (colored domain in figure 4.5). The

other from the endothermic reaction (equation 2.47) and is assigned only to the bed domain (the one

highlighted in figure 4.6).

The bed material domain has an initial temperature of 900 K so that it helps with the start of the

process simulation. It recreates heating the material, with an electric oven, to help the material to

reach a temperature high enough to ensure the microwave radiation absorption, facilitating the whole

42

operation. In the remaining domains, initial temperature is set to an ambient temperature of 293 K.

Regarding the boundaries, a fixed temperature of 293 K is imposed at the material’s inlet. In the

remaining a convective heat flux boundary condition described by equation (4.5) is set to account for

the power loss (Pconv [W/m2]) through convection losses at the walls. Radiation was not considered,

temperature of the walls are considerably low so external radiation can be considered negligible, as was

the internal radiation.

Pconv = h(Tamb − T ) (4.5)

In equation (4.5) Tamb is the ambient temperature of the medium surrounding the model and is

assumed to be 293 K. The convective heat transfer coefficient h is computed resorting to the empirical

correlations for external free convection flows found in Incropera et al. [58] and Churchill and Chu [59, 60]

and are presented in equations (4.6) to (4.8).

Ra =gβair(Ts − T∞)X3

ναair(4.6)

Nu =hX

κair=

0.825 +0.387Ra

16(

1 + ( 0.492Prair

)916

) 827

2

for 0.1 < Ra < 1012 (4.7)

Nu =hX

κair=

0.60 +0.387Ra

16(

1 + ( 0.559Prair

)916

) 827

2

for Ra ≤ 1012 (4.8)

In equations (4.6) to (4.8), βair is the expansion coefficient of air (K−1), νair is the kinematic viscosity

in (m2s−1), Ts is the surface temperature (K), Tinf is the environment temperature (K), X is the charac-

teristic dimension (m), αair is the thermal diffusivity (m2s−1), λair is the thermal conductivity and Prair

is the Prandtl number. Equation (4.6) gives the Rayleigh number that will be used to compute the Nus-

selt number for both cases, vertical wall (equation 4.7) and horizontal cylinder (equation 4.8), with the

correspondent characteristic dimension (X), to compute the heat convection coefficient.

The boundary surfaces are considered to have an isothermal temperature distribution and an average

temperature is computed in order to apply these correlations. To achieve a more accurate average

temperature, the boundary surfaces are divided so that different geometrically similar boundaries will

have different heat convection coefficients. Figure 4.7 displays the surface division, for each highlighted

surface there is a computed heat convection coefficient from the local average temperature.

Figure 4.7: Boundary division used for each heat convection coefficient.

43

4.4 Computational routine and process optimization

The objective of the following sections is to describe the computational procedure adopted to obtain

a converged solution. A brief explanation of an improved method of the one developed by Mimoso et al.

[11] is presented right below and a summary of the whole routine can be found on figure 4.8.

Figure 4.8: Flow diagram of the computational algorithm.

To start, it is necessary to build the CAD of the model and assign all the interfaces (physics, materials

and boundary conditions) to the correct domains and boundaries as prescribed in the previous sections.

As for the transient study, mass flow, inlet temperature and initial conditions are defined as found suitable

and the power input is set with the procedure in subsection 4.4.2.

Before starting the process, an initial temperature profile must be specified and it should be a good

guess of the temperature field obtained in the final solution. The established temperature distribution is

then used to compute all the temperature dependent properties, namely the dielectric properties. With

the referred conditions, a fixed frequency domain simulation is carried solving only for the electromag-

netic equations and for different plunger positions. By doing so, it is possible to obtain the optimum

plunger position that will grant the maximum microwave efficiency for the initial temperature distribution.

This procedure will be depicted in more detail in subsection 4.4.1.

Then, the optimum plunger position found above will be used in a transient simulation that couples

the three physics of the model (electromagnetic, energy/heat and chemical). During the course of the

simulation, temperature distribution is expected to change and hence dielectric properties may change

as well. The last field will, in turn, dictate how the microwave energy is absorbed. So, at the end of

the simulation, a new frequency domain study is necessary as it is possible that the optimum plunger

position has been changed. After a new fully coupled the transient simulation, with a new defined power

input (explained in subsection 4.4.2), starts and the process continues.

The briefly depicted process will last until the numerical solutions is found to be converged. Power

and plunger position will be adjusted as necessary resorting to a MATLAB control routine as described in

subsection 4.4.3 until the stopping criterion is achieved. The convergence criterion is selected through

the evaluation of two separate variables: the integration of the temporal term of the energy equation

(2.44), the stored energy, observed in equation (4.9), and the average limestone mass fraction at the

44

tube’s exit. When the term of equation (4.9) gets close to zero (≤ 1 W flexibility is considered), one is

able to conclude that the overall temperature distribution is suffering minor changes through time and

so, that steady state has been reached for the thermal field. On the other hand, it was considered that

total conversion of limestone was attained if the average mass faction of limestone at the tube’s exit

dropped below 1.5%. Therefore, the simulation is considered to be converged when this two criteria are

simultaneous present.

∫ρCp

∣∣∣∣∂T∂t∣∣∣∣ dV (4.9)

4.4.1 Optimum initial plunger position

The key to achieve the goals of this study is to maintain high power absorption so that the material to

be processed during the transient simulation gets totally converted. Hence an adequate plunger position

is required.

For the start of the process, a temperature distribution is set as already stated above. It is wise

to establish a distribution similar to the one obtained in the final solution. Otherwise more plunger

positions changes will be required to maintain a high power absorption by the material, and so, more

computational time will be needed. In figure 4.9 it is possible to see the initial temperature profile used

in this work.

Figure 4.9: Temperature profile used for the frequency domain study.

The temperature profile is then used to compute the dielectric properties inside the cavity. With the

current defined data, a frequency domain study is conducted with a constant frequency of 2.45 GHz and

a parametric sweep of the plunger position. The efficiency η is computed resorting to the S11 factor,

which is calculated automatically by the software, and is presented in the following on equation (4.10).

η = 1− |S11|2 (4.10)

The S parameter is a scattering parameter that describes the relationship between the input and output

wave [44]. So S11 represents the reflected power at port 1 (defined as the microwave port of the model)

and it is known as the reflection coefficient or return loss.

The main feature of the plunger is impose a zero electric field condition at its location, and as a

45

consequence, it will define the maximum of the electric field norm location throughout the waveguide. In

figure 4.10 the plunger depicted influence can be attested. It is possible to see the electric field norm

distribution. The higher value are colored red, while the closest to zero are colored blue.

Figure 4.10: Electric field norm distribution | ~E| in a XY plane.

Briefly, by adjusting the plunger, the objective is to place the zone with the highest electric field

norm right on top of the material that is required to be processed. The range of adjustments is set

taking into account the wavelength inside the waveguide. If one sweeps the plunger along the full

distance correspondent to the local wavelength, then two positions with maximum efficiency (two peaks)

are expected to be found. The distribution of the electric field norm is periodic each half wavelength

producing two peaks due to that. These peaks happen because there are also two peaks of the electric

field norm (| ~E|2) along the distance of a wavelength.

In order to find the waveguide wavelength, the following equation (2.36) is used which is manipulated

so that equation 4.11 is obtained [61]. Considering the WR340 waveguide and a frequency of 2.45 GHz

(giving λ0 = 122mm), the waveguide wavelength λg returns 173 mm. So the distance between the peaks

of electric field norm colored in red in figure 4.10 will be of 86.5 mm (λg/2) as it is known to repeat every

half wavelength [46].

λg =λ0

2

√1−

(λ0

2a

)2 (4.11)

It would be reasonable to assume then, that the plunger position could be set half wavelength away

from the material in order to place the high intensity node over it. However, as the electric field propa-

gates in the presence of the material it will be distorted by it [62]. The dielectric properties of the material

will cause a phase shift and an amplitude reduction of the electric field norm according to Mehdizdeh

[46]. This very effect can be observed in figure 4.10 as a distorted light blue manifestation of the elec-

tric field can be spotted, placed in the bed material domain as expected. As result, other means are

necessary to find the best plunger position.

The microwave dependency on the plunger position for the considered temperature profile can be

attested in figure 4.11. In this case the plunger range was equal to do the waveguide’s wavelength as

to observe the expected two high efficiency peaks. In this case, maximum efficiency was attained for a

plunger position of 144 mm from the center of the quartz tube and then again roughly 86.5 mm after, at

46

231 mm. So as can be seen, choosing a position half wavelength apart would be equally correct. An

observed particularity, presented in table 4.3, is that there is a very significant variation of the efficiency

for small plunger position shifts. The precision of the plunger position is of 1 mm, which represents only

0.578% of the wavelength. However the change in efficiency is rather considerable, therefore, it can be

verified the great importance of making small, refined adjustments in the plunger position. The observed

efficiency variation will translate into an added difficulty to achieve and maintain high efficiencies during

the transient simulations.

Other fill rates were previously tested and a summary of the results obtained can be found on table

4.4. Where it is possible to attest the suitability of the chosen 40% fill rate for the conducted stimulations.

Figure 4.11: Microwave efficiency dependency on the plunger position for the initial temperature distri-bution.

Plunger position [mm] Microwave efficiency (%)

140 57.61

141 71.31

142 86.09

143 96.88

144 99.89

145 96.27

146 89.5

147 82.3

148 75.83

Table 4.3: Microwave efficiency for different plunger positions with a 40% fill rate.

47

Fill rate [%] Optimum plunger position [mm] Microwave efficiency (%)

20 150 62.11

30 148 73.62

40 144 99.89

50 140 95.07

Table 4.4: Microwave efficiency and optimal plunger position for various material fill rates.

4.4.2 Microwave power input

Defining a suitable microwave power input either for the initial and the subsequent studies is a key

aspect that will dictate the success of the whole converging process. The main objective is to find the

required power input capable of converting all the material from limestone to lime. And, it is of the utmost

importance to specify a power input that avoids the occurrence of temperature related problems such as

thermal runaways.

For the transient simulation the power input will have two stages: one for the initial times step and

other for the subsequent ones. The power input of the initial study comes from the energy required to

convert all limestone that passes through the model. This power is computed by multiplying the mass

flow for the enthalpy change of the reaction and then divide this value for the computed efficiency found

in subsection 4.4.1. In this stage losses are not yet considered as they are not yet known. It was possible

to take a guess by running an a priori study. However it was decided to continue as described so that

it was possible to see the MATLAB controller react to this conditions. An equation for the initial power

input can be found in equation (4.12) considering no losses and an outlet temperature equal to the inlet

one.

The second stage of power input regards all the subsequent studies until steady state is achieved.

In this stage power input is the result of the final conditions of the previous iteration. In this case losses

are taken into account as well as the energy carried by the material when it leaves the tube. The power

input (Pinput) in this stage is set according to equation (4.12):

Pinput =Pchem +mflowCp(Texit − Tin) + Plosses

ηparam(4.12)

where:

• Pchem is the theoretical power required to convert a determined mass flow of limestone into lime,

mflow∆H, (W );

• mflow is the mass flow of limestone material (kg/h);

• Cp is the heat capacity of limestone ([J kg−1K−1] );

• Texit is the average temperature of the bed material at the tube’s exit at the end of the previous

study (K);

• Tin being the imposed bed temperature at the tube’s inlet (K);

48

• Plosses is the computed convective losses of the model at the end of the previous study (W );

• ηparam is the maximum computed efficiency of the parametric study, carried after the transient

study, used find the new optimal plunger position.

Two operational procedures were implemented regarding power between two transient studies. When

the computed power for the new study is found to be higher than the one from the previous study, the

new power is set linearly over the course of the simulation. Although this approach extends the overall

simulation time it prevents the instabilities created by a sudden increase in power. On the other hand,

when the new computed power was lower than the previous one, then the new power will be set immedi-

ately at the beginning of the new iteration. If the newly found power input is lower than the previous one,

it means the system requires less power than before. Therefore, by reducing it immediately, temperature

related problems can be avoided.

4.4.3 Matlab control routine

In order to construct and obtain a high efficiency study, a method was developed, based the work

undertaken by Mimoso et al. [11], that comprised all the major operational steps considered necessary

in a multiphysics microwave heating problem.

First it is necessary to define the different operational parameters such as the axial velocity of the

bed, the initial power input and the inlet temperature of the bed material. Then, the initial temperature

and electromagnetic conditions for the model are defined; additionally an initial plunger position is also

set according to a study conducted as described in subsection 4.4.1. Finally the simulation time is

defined and is set to be a quarter of the residence time, which is the time for the material to go from the

inlet all the way to the outlet of the quartz tube.

With the model parameters defined the first transient study can be carried out. Once finished, the

frequency domain parametric study is used to obtain the new optimum plunger position. Then, a new

power input is computed from equation (4.12). The newly computed power and plunger positions are

automatically inserted into the model by the MATLAB algorithm and a new transient study can start. The

steps presented in this paragraph are then repeated until the transient term of the energy equation (4.9)

is negligible and the average limestone mass fraction goes below 1.5%.

Through initial learning experiments with the model, it was found that a coupled study that included

microwave, heat and chemical interfaces is very time consuming. Furthermore, given the nature of the

following method, without the MATLAB control algorithm, as described above, it would require constant

supervision so that no time is lost in idle between studies. This controller’s main task is to automate the

proposed methodology, given only the initial conditions and parameters in need of definition in the initial

study.

The built MATLAB code consists on the steps presented below:

1. Define initial variables: mass flow, initial power input, simulation time, inlet material temperature

and plunger position;

49

2. Load the COMSOL model;

3. Start Loop 1: Running and converging the sequence of simulations;

(a) Set variables defined at ”1” in the COMSOL model opened at ”2”;

(b) Run the first transient simulation;

(c) Save the new model and extract all the relevant data;

(d) Carry a frequency domain parametric study. To save time, run it for the current plunger

position and for its adjacent ones;

(e) Read the computed efficiencies for the three plunger positions;

(f) Start loop 2: Finding optimal plunger position;

i. If the maximum efficiency is found for the current position leave loop 2;

ii. If not, run a new frequency domain simulation for the plunger position that returned the

highest efficiency and for the its two neighboring positions;

(g) Compute the new power input required from the data extracted at (c) using equation (4.12);

(h) If transient term of the energy balance equation and the average percentage of reactant at

the quartz tube exit are ≤ 1 and < 1.5% respectively, leave loop 1;

(i) If the new power input is greater than the power input at the end of the previous simulation,

then the new power is linearly increased during the course of the new transient simulation;

(j) if the new power input is lower than the previous onde, then the new power input is established

in its full magnitude at the beginning of the new transient simulation;

4. End of the MATLAB code and the last solution is the converged one for the set of parameters of

velocity and inlet temperature.

4.5 Initial energy testing model

Before initiating a fully coupled simulation between the three physics involved a test was made to

assess if the model was operating as expected from a chemical standpoint and its interaction with the

thermal interface. An energy balance has to be made in order to check if the energy input is being

conveyed to the right places. This study was conducted considering only the chemical and heat transfer

interfaces so that simulation time is reduced when compared to a fully coupled one. To simulate the

microwave power deposition, an artificial heat source is imposed in the material bed domain.

The mass flow of limestone is set to 0.75 kg/h. A transient study is conducted until the integral of the

temporal term of the energy equation is converged and the limestone is converted at the exit. Meaning

little temperature variations over the whole model and that the simulation has reached steady state. A

steady state situation is achieved for a power of 640 W and a residual mass fraction of 0.5% of CaCO3 is

still present at the tube’s exit so a complete conversion of the reactant can be considered.

50

Through figures 4.12 and 4.13 it can be attested the interconnection between the temperature and

the CaCO3 (limestone) conversion. Regarding the energy balance, table 4.5 refers to the extracted data

used to compute the balance and summarizes the power usage and losses of the model.

Figure 4.12: 2D cut on the ZX plane of the temperature field of the testing model.

Figure 4.13: 2D cut on the ZX plane of the CaCO3 mass fraction of the testing model.

Convective losses 233.35W

Material’s exit energy 45.5W

Endothermic reaction heat source ) + 372.65W

Total used energy = 651.5W

Power input 640W

Table 4.5: Energy balance of the initial model tested.

In table 4.5, the convective losses are computed using the equation (4.5). The energy lost through

the material exiting the tube is known resorting to the second term of equation (4.12). The endothermic

reaction heat source is computed by integrating equation (4.4) over the entire volume of the bed domain.

As can be seen the total amount of power lost to the various processes sums up to 651.5W . Facing

the computed value with the input power of 640W referred above, an 1.76% difference can be spotted

which is fairly reasonable. Other important value to take into account is the endothermic reaction heat

source imposed at the bed material domain. It presents a value of 372.65W which is roughly the same

as the theoretical one of 372.05 W computed by multiplying the converted mass flow by the energy of

the reaction (1795 [kJ/kg]). Hence it is possible to conclude that the power absorbed by the chemical

reaction is being well computed.

51

4.6 Results

4.6.1 Limestone processing

In the present section the aim is to expose and discuss the results obtained from the simulations car-

ried out. It is the culmination of the process that has been presented so far. The operational parameters

have already been discussed, however, for convenience they are recalled in table 4.6.

Operational parameter Value

Initial bed temperature 900K

Remaining model initial temperature 293K

Material’s inlet temperature 293K

Mass flow rate 0.25 kg/h

Table 4.6: Operational parameters.

From the presented mass flow, power input for the first simulation step is computed according to

equation 4.12 giving approximately 125 W of power to be delivered by the microwave source. Plunger

will be set at a distance of 144 mm from the tube’s center as previously addressed and the simulation

time is set to be 1/4 of the residence time, and so it is adjusted to 1512 seconds.

From figures 4.14, 4.15, 4.17 and 4.18 it is possible to observe the major variables obtained through

the control algorithm implemented in MATLAB. They enable the evaluation of the conditions on which

the model is operating and allow for the MATLAB controller to intervene and adjust power and plunger

position as suitable until a steady state status is attained. Figure 4.16 presents the thermal distribution

evolution through the simulation at key time points marked in 4.15 respectively. Table 4.7 summarizes

the following set of five figures in order to give an initial brief understanding about the content of each

one, due to the interconnection of the data presented between them. The whole simulation process

required 9 transient studies in order to achieve a steady state for a total simulated time of 13 608

seconds corresponding to roughly 3 hours and 46 minutes of computing time.

Figure Content

Figure 4.14 Controlling variables of the MATLAB algorithmFigure 4.15 Temperature evolutionFigure 4.17 Chemical evolutionFigure 4.18 Energy balanceFigure 4.16 Thermal field distribution evolution at key time points

Table 4.7: Limestone processing figure’s content.

In figure 4.14, access is given to the evolution of the microwave efficiency, plunger position, power

input and the stored heat rate which is the transient term from the energy equation 2.44. It is possible

to notice that in the first time interval (0s-1512s) the efficiency drops dramatically although the plunger

positions is spot on (because initially efficiency is high). This happened mainly due to the low input

power of around 125 W set in this stage. As power is not enough to overcome the convection losses

52

Figure 4.14: Stored heat, power input, microwave efficiency and plunger position evolution during thesimulation time.

throughout the bed domain and hence to maintain the material’s temperature, as can be seen in figure

4.15, where temperature falls till point (a) with the temperature distribution presented in 4.16 (a), the

dielectric properties will change due to this temperature drop. As a result, with a lower dielectric loss

distribution the capacity of the material to absorb the microwave radiation is also diminished, and so, the

microwave efficiency is reduced. The stored heat rate represents the state of cooling or heating of the

model, in this case this variable drops from zero to a negative value, as can be spotted, indicating that

the materials’ temperature is dropping.

For the second time interval (1512s-3024s) a new plunger position is obtained and the input power

is update according to 4.12. As can be noticed, the new power input is not established right away,

instead it is linearly increased from the beginning to the end where it reaches the computed power.

Although this increases the whole simulation time, it helps with the stability and diminishes the chances

of temperature related problems, from sudden increases in energy absorption (increased efficiency)

before the controller intervention. At the end of this time interval an example of such event is present,

as it can be attested in figure 4.15 in the maximum temperature line. However, due to lower power input

in the next time interval this effect was mitigated. For the remaining time intervals the model suffers less

changes, and minor adjustments in power are done until steady state and total conversion (see figure

4.17) are obtained. Steady state is obtained when the stored heat rate reaches zero, this means that

the transient term of equation (2.44) is negligible and that no overall temperature changes are occurring

in the model.

As the bed material is the only capable of absorbing significant amounts of microwave energy, it is

53

safe to say that the power input multiplied by the microwave efficiency returns the power absorbed by the

bed which is represented in figure 4.15. In this figure the maximum temperature recorded on the material

bed and the mean material temperature at the outlet are also under display. From careful observation

it is possible to notice the strong coupling between the thermal field and the microwave interface. With

the increase in power absorption (due to the increase in input power) a higher maximum temperature is

obtained, as a consequence, the dielectric loss factor increases, and so, it will allow for a better power

absorption and increased efficiency. This feedback loop will develop a steeper slope of the maximum

bed temperature and absorption power through points (a), (b) and (c) in figure 4.15. This phenomenon,

although useful cause it accelerates the process, if not contained, can lead to the already discussed

problems of hot spots or thermal runaway.

Figure 4.15: Bed power absorption, maximum bed temperature and outlet bed temperature versussimulation time.

In figure 4.16 the thermal field distribution corresponding to the time instances marked in figure 4.15

can be found. With these distributions it is possible to discuss how the coupling affects the present

results, specially the feedback mechanism between the temperature and electric fields. From (a), (b)

and (c) one can see the consequence of power deposition to the warmer areas, making those areas

hotter and smaller. If no measures were taken at moment (c) the hot spot would only get hotter and

more concentrated absorbing the majority of the input power. Figure (d) is displayed to give an insight to

the temperature distribution at steady state. As it is possible to observe, the concentrated hot area of the

previous time mark has dissipated, becoming wider and better distributed. The mean outlet temperature

of the material is presented in figure 4.15 as to show its relation to the maximum temperature. It behaves

54

(a) Time=1512s; Max. Temperature:487 K. (b) Time=2268s; Max. Temperature:553 K.

(c) Time=3024s; Max. Temperature:1154 K. (d) Steady state (13608s); Max. Temperature:1123 K.

Figure 4.16: Evolution of the thermal field over the course of the limestone processing simulation.

just as expected being a key variable when computing the power input for each time interval as verified

by equation (4.12).

With the temperature increase, temperature dependent chemical processes are expected to occur.

The interconnection between the thermal field and the chemical interface is briefly presented in figure

4.17. It is possible to verify that with temperature increase the endothermic reaction begins to convert

limestone into lime and the respective heat source starts to increase. The endothermic reaction is

considered to hold according to Arrhenius law, which as previously seen, is temperature dependent. The

Arrhenius term ”K” will increase resulting in a higher reaction rate. As conversion proceeds, CaCO3

will decrease which results in a reduce in pace of the reaction rate until it reaches total conversion,

explaining the behavior of the limestone mass fraction. The reaction heat source will absorb energy

from the system (endothermic) and its evolution depends on temperature due to the connection with the

reaction rate (equation 2.47). As it is possible to see it gets triggered when the reaction begins to take

place. Later, at steady state, the heat source stabilizes to a fixed value corresponding to the energy

required to convert the limestone that flows through the tube.

The objective in the displayed information of figure 4.18 is to present the evolution of the energy

consumption and losses. Energy consumption happens though the present chemical reaction, on the

other hand, energy losses are portrayed by convection losses and enthalpy loss trough the tube’s outlet.

However, not all losses can be technically considered as losses. In order to start the reaction, a certain

temperature must be met (around Treact=1073K), thus, it is required energy to heat the material up to

that temperature according to equation 4.13. If the system were to be adiabatic the outlet temperature

would be the reaction temperature. Yet, it is not. That means, that some of the heating enthalpy was

lost through convection and did not leave the system by the tube’s outlet. That means, the convection

losses will be the value obtained from COMSOL subtracted by the difference between the heating power

(Pheat) and the enthalpy of the material at the tube’s outlet

Pheat = mflowCp(Treact − Tin) (4.13)

55

Figure 4.17: Limestone mass fraction, maximum bed temperature and reaction heat source versussimulation time.

Figure 4.18: Bed power absorption and power losses/usage versus simulation time.

As it is possible to notice, all the power usage and losses are conveniently presented in absolute values

and their summation without the stored heat rate is also given as to compare it to the power absorption.

56

It can be seen that the difference between sum of the power usage line and the bed power absorption

line is the stored heat energy. The bigger the difference, the farthest away from steady state the system

is. As efficiency increases so does power absorption, and after a while, power losses start to increase

and catch up. At steady state it is possible to observe a balance between the power that is absorbed,

the one being used and the one that is lost, while the stored heat is negligible. This helps to conclude

that the majority of the energy of the model is being accounted for.

In figure 4.19 an energy balance for the steady state is presented in order to complement the previous

information. The absorbed power accounts for 84.6 % (301 W) of the input power (356 W) while 15.4 %

of it is reflected back. The thermal losses are constituted by the convection losses, and, accounting the

energy that leaves the tube through the outlet material, it adds to roughly half of the total input power

in the model. The power used to heat the material up to the reaction temperature is computed using

equation (4.13), returning 43.4 W. This means that the energy truly lost was of 133.6 W or 37.5% of the

input power. On the other hand, the chemical power used to convert limestone into lime was of almost

35 % of total power, adding the heating power as in equation (4.14), a total power of 167.4 W is used to

heat and convert the limestone, 47% of the input power, giving a thermal efficiency of 55.1 % (obtained

dividing the Pconversion by the absorbed power). In equation (4.14), the first term represents the integral

of the chemical heat source over the bed domain, it will be close to the theoretical value Pchem. A

revised energy balance is displayed in figure 4.20 where the energy used to raise the temperature is not

considered a loss.

Pconvertion =

∫bed

QChem dV + Pheat (4.14)

Figure 4.19: Energy balance of the limestone processing unit at steady state for a mass flow of 0.25kg/h.

The steady state data obtained with the given operational conditions (table 4.6) are given below in

table 4.8 where a summary of the major operational variables can be consulted. In figure 4.21 the

steady state distribution in a isoplane align with the zz axis for the temperature (a), loss factor (b),

57

Figure 4.20: Energy balance of the limestone processing unit at steady state for a mass flow of 0.25kg/h considering the Pheat as useful power.

microwave power deposition (c) and reaction heat source (d) are presented. From the displayed figures,

it is possible to visualize the interdependence of the four variables presented. For a higher temperature

there is a correspondent stronger loss factor, which will result in a hn higher microwave power deposition

at the same zone. Finally through figure 4.21 (d), one can attest how the reaction starts as soon as

temperature increases. It can be observed its evolution along the zz axis. Afterwards, the reaction rate

starts to decrease due to the lack of reactant.

Power input 356W

Material’s inlet temperature 293K

Max. bed temperature 1124K

Outlet mean bed temperature 583K

Outlet limestone mass fraction 0.7 %

plunger position 144mm

absorption efficiency 84.6 %

Bed power absorbed 301W

Convection losses 161W

Material’s outlet enthalpy 16W

Heating power to raise the material’s temperature 43.4W

Reaction power consumption 124W

Simulation time 13608 seconds

Table 4.8: Steady state data of the limestone processing unit.

Through figure 4.22 it is possible to visualize the electric field distribution (a), loss factor (b) and

microwave power deposition (c) from a top perspective (by defining a horizontal planar cut at roughly half

bed height normal to the xx axis). By analyzing the three images it is possible to notice the influence of

the electric field and the loss factor, two key variables of the Helmholtz equation (2.49), on the microwave

power deposition. It can be noticed how the power deposition manifests itself, in the both presence of

58

(a) Temperature distribution (K). (b) Loss factor.

(c) Total microwave power dissipation (W/m3) . (d) Heat source from the endothermic chemical re-action (W/m3).

Figure 4.21: Limestone processing steady state distributions (side view) of the temperature, loss factor,microwave power dissipation and the chemical reaction heat source.

a significant high electric field and high loss factor, as expected through equation (2.46). Hence it is

possible to attest that the microwave power dissipation field results from the intersection of the electric

field peaks with the loss factor ones (meaning high temperatures). Although the electric field plays an

important role in triggering the microwave heat source, its the loss factor that is the most crucial factor,

since its only peak is localized in a small zone. Further downstream the power deposition starts do

decay due to the absence of a strong electric field. A curious fact can be seen in the electromagnetic

field as two intense peaks are present.

(a) Electric field distribution(V/m).

(b) Loss factor. (c) Total microwave power dissi-pation (W/m3) .

Figure 4.22: Limestone processing steady state distributions (top view).

Regarding the Matlab controller, a remark should be made. The whole simulation discussed above

was conducted only with the autonomous intervention of the controller, once the initial conditions were

set. From the obtained results, it might be reasonable to consider that the approach taken to develop the

controller was a success. It was verified that the controller, by making an energy balance to the model

and accounting for the microwave efficiency, could cope with the evolution of the simulation and be able

to achieve a steady state condition. The particular moment corresponding to the first time interval (see

59

figure 4.14), can be considered note worthy. The controller was able to recover from a great efficiency

loss, due to an intentional low initial power input (that did not take into account the convective losses).

Regarding efficiency, it was noted that a 900 K initial temperature returned very high efficiency values

of roughly 97%. particularly when compared to the lowest efficiency obtained in the first time interval

(around 29%). This means that an high initial temperature, a sort of material pre-heating, can in fact

”kickstart” the process. And, when coupled with an adequate power, can reduce the required time to get

a steady state solution while always maintaining an high microwave efficiency.

Finally, it should be noticed the different variables used to develop de controller, that were defined

as probes in the COMSOL environment in order to return the required variables evolution. So that, the

power input and the optimum plunger position could be computed by equation 4.12 and the frequency

domain study. The particularity of the variables obtained through probes rely on the fact that they can

be related with the experimental apparatus, which means that the developed algorithm is not limited to

the numerical environment. For example, both the temperature at the metallic walls and of the material

at the outlet can be can be measured. Therefore, the convective power loss through the walls and

the enthalpy variation at the tube’s outlet can be estimated. So, by measuring the mass flow, one can

determine the amount of power required to convert all the material. Finally by measuring the reflected

microwave power, microwave efficiency can be found.

4.6.2 Mass flow parametric study

With the sole objective of understanding the impact of certain operational conditions on the developed

model, several simulations were conducted for different material’s mass flow while maintaining the same

material’s inlet temperature.

The steady state results for the limestone processing for a variety of different mass flows are dis-

played in table 4.9. The initial state for each mass flow simulation set was defined to be the steady state

of the previous. So, for a mass flow of 0.5 kg/h initial conditions were taken from the 0.25 kg/h mass flow

steady condition, for the 0.625 kg/h were taken from the 0.5 kg/h and so on. The aim of this approach

was to ensure a reduced simulation time, as it will strongly depend on the quality of the initial solution.

The last three entries of table 4.9 give information about the physical time required for each mass flow.

The time interval for each transient study was defined as approximately one quarter of the residence

time of each mass flow. In the entries ”g.” to ”m.” it is possible to find the power losses/usage the

respective impact in the energy balance. Entry ”k.” reflects the the ratio between the energy losses and

the absorbed power according to equation 4.15. It is possible to see that the true losses are computed

subtracting the heating enthalpy that was lost trough convection and did not leave at the tube’s outlet.

Losses% =Convection losses− (Pheat −Material′s outlet energy)

Asborbed power× 100 (4.15)

Entry ”m.” is the thermal efficiency for each mass flow. It is the ratio between the useful energy and

the absorbed power according to equation 4.16. The useful energy will be the energy consumed by the

reaction and the energy spent to heat the material to a suitable temperature (the considered 1073 K).

60

mass flow (Kg/h) 0.25 0.5 0.625 0.75 1

a. Power input (W) 356 573 684 793 1005b. Max. bed temperature (K) 1124 1156 1167 1171 1183c. Outlet mean bed temperature (K) 583 758 815 852 911d. Outlet limestone mass fraction (%) 0.7 0.4 0.7 1.2 1.1e. Plunger position (mm) 144 144 144 144 144f. Microwave absorption efficiency (%) 84.6 84.3 83.8 83.3 82.9g. Bed power absorbed (W) 301 483 573 661 833h. Convection losses (W) 161 185 193 198 208i. Material’s outlet enthalpy (W) 16 52 73 93 138j. Heating power to raise the material’s temperature (W) 43.4 86.78 108.47 130.16 173.55k. Losses (%) (eq: 4.15) 44.3 31.10 27.49 24.22 20.70l. Reaction power consumption (W) 124 247 309 370 489m. Thermal efficiency (%) (eq: 4.16) 55.6 69.72 72.86 75.67 79.54n. Total efficiency (%) 47.03 58.77 61.05 63.03 65.94o. Specific power (W/(Kg/h)) 1434 1151 1102 1070 1016p. Time interval for each study (s) 1512 756 600 504 372q. Number of required transient studies 9 7 7 7 7r. Total physical time of the simulations (s) 13608 5292 4200 3528 2604

Table 4.9: Parametric study results by variation of the mass flow rate.

Thermal efficiency% =Pheat +

∫bed

QChem dV

Asborbed power× 100 (4.16)

The specific power found in entry ”o.” is the power that was required to convert the tested mass flow

at the given conditions. It translates the ratio between the power input (entry ”a.”) and the converted

limestone mass flow (the converted mass flow was computed by subtracting the outlet mass fraction ”d.”

with the one from inlet that is equal to one).

From the gathered data of the steady state solutions, several remarks can be made. Regarding

the obtained temperatures, maximum bed temperature increases slightly over every mass flow and, as

expected, outlet temperature increases significantly, from the lowest (0.25 kg/h) to the highest (1 kg/h).

This behavior results from the increased axial speed, resulting in an hotter surface area and higher

convective losses. Microwave efficiency is showed to be quite similar over the range of simulated mass

flow rates. The entries ”k.” and ”m.” are the percentage of energy conveyed to the convective losses

and to convert the limestone, respectively. As can be noticed, there is a clear trend over the range of

mass flows. For a higher mass flow, the convection losses increase, however their weight decreases,

representing a lower percentage of the absorbed power. The behavior of useful power, used to heat

and convert the material, increases with the mass flow, but, this time, the thermal efficiency (eq: 4.16)

increases as well. Entry ”n.” displays the total efficiency for each mass flow, computed by multiplying

the microwave absorbed efficiency (entry f.) with the thermal efficiency (entry m.). The total efficiency

increases with mass flow, which was to be expected due to the increase in thermal efficiency for roughly

the same microwave absorption one. The discussed acknowledgment is reflected in the power input

61

required to convert one kilogram of limestone (entry ”o.”). As can be seen, for a higher mass flow, a

lower power input is enough to convert the same amount of mass when compared to a lower mass flow.

From the analysis to the parametric study it is possible to see that for a higher mass flow the total

efficiency increases while the specific power decreases. To obtain better conclusions about the impact

of mass flow in a limestone processing unit, it would be necessary to increase the mass flow. That would

have to be made until reach a point where it would be impossible to obtain a steady state solution, the

total efficiency would start to decrease or a temperature limit of the materials of the apparatus would be

violated.

In figure 4.23 the space distribution of the microwave power dissipation, temperature and chemical

heat source are displayed. The purpose of this figure is to serve as a supplement of the analysis of table

4.9 through a visual assist. The color range for the mass flows of 0.5 Kg/h and on are the same, so that,

the variations of each field could be easily noticed from one mass flow to another. As expected all three

field grow throughout the mass flow range. The microwave power dissipation increases due to the input

power due to the higher mass flow rate, which in turn, will produce a stronger electric field for the same

loss factor distribution. Although the loss factor is not represented, it peaks in the high temperature

zones, as so, it will be higher in the zones colored in dark red of the temperature field. As can be noticed

the power dissipation’s peak starts to develop in hotter areas, due to the higher loss factor.

Regarding the temperature, the dissipation is obtained when increasing the mass flow rate. With the

material being heated to the roughly the same temperature, combined with the faster travel time (less

residence time, so less convection time), temperatures will increase downstream, until the tube’s outlet.

The chemical reaction heat source increment happens because a higher mass flow means that more

material is required to be converted, and so, for each increasing mass flow an increasing part of the

power input is dedicated to this process. As can also be notice the temperature is the main responsible

for triggering the chemical heat source and, therefore, the chemical reaction.

62

(a) Microwave power dissipation(W/m3) for a mass flow of 0.25 kg/h.

(b) Temperature field (K) for a massflow of 0.25 kg/h.

(c) Chemical heat source (W/m3) fora mass flow of 0.25 kg/h.

(d) Microwave power dissipation(W/m3) for a mass flow of 0.5 kg/h.

(e) Temperature field (K) for a massflow of 0.5 kg/h.

(f) Chemical heat source (W/m3) for amass flow of 0.5 kg/h.

(g) Microwave power dissipation(W/m3) for a mass flow of 0.625 kg/h.

(h) Temperature field (K) for a massflow of 0.625 kg/h.

(i) Chemical heat source (W/m3) for amass flow of 0.625 kg/h.

(j) Microwave power dissipation(W/m3) for a mass flow of 0.75 kg/h.

(k) Temperature field (K) for a massflow of 0.75 kg/h.

(l) Chemical heat source (W/m3) for amass flow of 0.75 kg/h.

(m) Microwave power dissipation(W/m3) for a mass flow of 1 kg/h.

(n) Temperature field (K) for a massflow of 1 kg/h.

(o) Chemical heat source (W/m3) fora mass flow of 1 kg/h.

Figure 4.23: Converged solutions of the parametric study. 2D plots of the microwave power dissipation,temperature field and chemical endothermic heat source.

63

64

Chapter 5

Conclusions

This thesis is divided into two parts and presents an insight into the use of microwave energy as

an heating source in the formation of cement clinker. It enabled the understanding of the impact of key

operational parameters in microwave heating and the development of an efficient microwave limestone

processing unit.

In the first part the chemical mechanism and the behavior of microwave induced plasma were ad-

dressed. Two models were built, respectively in order to reenact related literature data. The results

obtained turn out to be a mild success. The chemical model was able to predict the evolution of the

complete cement clinker chemical process with reasonable results, being validated with the available

data in the literature. Regarding plasma, the developed model strafed way from the intended purpose,

nevertheless, it was able to capture the typical behavior of a microwave induced plasma. In all, both mod-

els contributed to the general understanding of the involved processes and were reasonable enough to

prove that the software used, COMSOL, was able to handle with such challenges.

On the second part, a 3D microwave heating model was developed for converting a key component

in cement clinker, limestone. 3D transient simulations were carried resorting to COMSOL and controlled

by a code developed in a MATLAB environment. This simulation coupled the three involved physics. The

chemical and thermal fields are coupled through the Arrhenius equation used to compute the reaction

rate. The electromagnetic and thermal fields are coupled through the dielectric properties of temperature

and the resulting feedback dependency of the heating source.

By controlling the cavity geometry, and hence, the resonance of the electromagnetic field, and con-

trolling the microwave power input, the model was able to attain a steady state with an optimized mi-

crowave efficiency and power usage. It is rather clear how the interception of the the electromagnetic

field and the loss factor spatial distribution results in the much needed microwave power dissipation,

which triggers both the heating and chemical processes. Therefore, it was showed the importance of

a correct electric field placing (through varying the plunger position) and maintaining a high material

temperature in the cavity (matching an high loss factor) in order to achieve optimum conditions for lime-

stone processing. From the conducted parametric study, it was possible to observe a tendency where an

higher mass flow returns a lower specific power input required to convert the same amount of limestone.

65

5.1 Achievements

Considering all the carried simulations it is quite safe to conclude that COMSOL is a suitable tool to

tackle a variety of multiphysics problems. The MATLAB controller was a proven success, as it conducted

the transient simulation with the necessary adjustments without any user intervention during the con-

verging process. Thus, enabling to achieve an optimized and efficient microwave limestone processing

model, while avoiding temperature related problems, by adjusting operational parameters such as the

plunger position and input microwave power. Moreover, and unlike other models published in the litera-

ture, the proposed controller is also able to guarantee total conversion of the chemical reactant. Finally,

it was showed that the developed controller could be applied outside of the numerical simulation, as all

the data used by it can be also be measured in an experimental apparatus.

5.2 Future Work

Considering the continuation of this work, there is plenty of ways to improve the accuracy and better

portray the reality of limestone or cement clinker processing. Although increasing the complexity, some

steps/mechanisms can be added to the model. Variations in bed height and density shall be accounted

for due to the gaseous chemical products (CO2 for limestone calcination) that flow out of the bulk ma-

terial. A diffusion model can be coupled to better depict the gaseous phase. The pseudo-fluid behavior

assumed for the bed material can be changed for particles and small bulks of material, which, through

the use of a discrete element method, can better portray the movement of the chunks of material along

the rotating tube, describing the real behavior of the material inside the cavity. This would increase the

importance of the cavity geometry adjustments, as the temperature field would be far more dynamic.

Finally the controller can be used to study the design of new cavities and controllers.

66

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