Experimental and Theoretical Investigation of Low

Experimental and Theoretical Investigation of Low-Temperature Ignition in a Laminar Flow Reactor

Von der Fakultät für Maschinenwesen der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften genehmigte Dissertation

vorgelegt von

Tomoya Wada

Berichter:

Univ.-Prof. Dr.-Ing. Heinz Günter Pitsch Univ.-Prof. Dr.rer.nat. Katharina Kohse-Höinghaus

Tag der mündlichen Prüfung: 18. Juli 2011

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Acknowledgment This work summarizes my research activity at the Institute for Combustion Technology (ITV) at the Rheinisch-Westfälischen Technischen Hochschule (RWTH) Aachen University since June 2006. During this period, I was involved in three projects, namely, “NoNOx-project”, which was led by BMW Group Research and Technology, “Model Based Control of Homogenized Low-temperature Combustion” in Sonderforschungsbereich (SFB) 686, and “Taylor-Made Fuel from Biomass (TMFB).” In the first project, I collaborated with the following institutes: Institute of Reciprocating Engines (IFKM) at the Karlsruhe Institute of Technology, the Aerothermochemistry and Combustion Systems Laboratory (LAV) at the Swiss Federal Institute of Technology Zurich, and the Institute for Combustion and Gasdynamics (IVG) at the University of Duisburg-Essen. During the SFB project, the following institutes were involved: the Institute of Automatic Control (IRT) at the RWTH Aachen University and the Physical Chemistry 1 (PC1) at the University Bielefeld. In the third project, I was privileged to work at the Shock Wave Laboratory (SWL), the chair of Technical Thermodynamics (LTT), and the Institute of Combustion Engines (VKA) at the RWTH Aachen University. I would like to thank to Prof. em. Dr.-Ing. Dr.h.c. mult. Norbert Peters, who is my supervisor and the former institute chair. His supervision has contributed to my scientific curiosity remaining continuously at a high level, and he has encouraged me to face new problems with confidence. My thanks go to Prof. Dr.-Ing. Heinz Pitsch, who is the current institute chair. His advice based on his rich experience has led me to understand my work in further details. My appreciation goes to Prof. Dr. Katharina Kohse-Höinghaus for her kind revision of my dissertation and accepting to serve as a co-adviser of my thesis. I also would like to thank Prof. Dr.-Ing. (USA) Stefan Pischinger for accepting to chair my dissertation committee. I am indebted to Prof. Mohy Mansour (Beni-Suef University), Mr. Günter Paczko (ITV), Mr. Peter Spiekermann (Delphi), and Dr. Juan Pedro Mellado (Max Plank i

ii Institute of Meteorology). During my research at the ITV, their advice significantly improve the quality of my studies and will help my research carrier in the future. I would like to thank Mr. Walter Hübner (BMW), Dr. Udo Gerke (BMW), and Dr. Olaf Röhl (Daimler) for valuable discussions that helped to significantly improve the experimental apparatus. I am grateful to Mr. Fabian Jaromolowitz (IRT), Prof. Dr. rer. nat. Ravi Xavier Fernandez (SWL), and Dr. Changyoul Lee (SWL) for the collaborations we had together. To Mr. Leo Kuck and Mr. Dirk Grüterich, I am indebted for their work in helping to build my experimental apparatus. Many thanks go to Mr. Roman Silber and Mr. Siddharth Shah, who worked with me as student assistants. In the cause of my work I had valuable discussions with the following people: Prof. Konstantinos Boulouchos, Prof. Dr. Christof Schulz, Prof. Ulrich Spicher, Dr. Amin Velji, Dr. Christian Lämmle, Dr. Ulf Struckmeier, Mr. James Willie, Mr. Andreas Schmid, Mr. Dennis Bensing, and Mr. Christopher Gessenhardt. I want to thank all of my colleges at the ITV for the good time we had working together. It made my stay at the ITV valuable and enjoyable. My thanks go to my family members, father (Ruichi), mother (Fumiko), brother (Naoya), uncle (Yoshindo), aunt (Kimiko), and grandmother (Aya). They encouraged me to study abroad and were supportive of my stay in Germany. I would like to thank my friend Maria. Her support always helped me to concentrate on my research and her presence made life for me in Aachen worth living.

Aachen, September 2011 Tomoya Wada

Summary The main purpose of this dissertation is to investigate low-temperature ignition in detail. In the previous findings with conventional fuels such as diesel, gasoline, or jet fuels, first- and second-stage ignitions are observed in the negative temperature coefficient regime due to high- and low-temperature chemistry (LTC and HTC). As results from these investigations have shown, it is well-known that this temperature regime has high potential to reduce emission in applications. In general, these ignitions are observed sequentially (i.e., two ignition processes are observed as the overall-ignition). On the other hand, the importance of first-stage ignition due to low-temperature chemistry has been noted as one of the dominant phenomena in the overall-ignition process. In order to observe the ignition at low temperatures, a laminar flow reactor (LFR) has been developed. This LFR allows the experimental observations of only low-temperature ignition (i.e., first-stage ignition (FSI)). In other words, LFR extracts only FSI from the overall-ignition process. N-heptane (nC7 H16 ), dimethyl ether (DME) and their mixtures are chosen as fuels. The entire LFR remains at isothermal conditions. Temperature increase due to the FSI in LFR and the products from FSI are experimentally measured. Numerical calculations are conducted to simulate the experiments. Based on the experimentally and numerically determined results theoretical models are established. Obtained results are summarized in following in 8 chapters. In chapter 1, industrial requirements and related research in the combustion society are mentioned. Chapter 2 describes the mathematical model used in the numerical simulations. Theoretical investigations are performed based on this model. The applied assumptions are described in this chapter in details. Chapter 3 describes the specification of the LFR. Control devices, experimental methods, and experimental parameters are also given. In chapter 4, the evaluation of the LFR is conducted using 3-D calculations. The homogeneity of the mixture is mainly considered, and flow and the mixing fields are investigated in detail. The numerically determined data compensates for the restrictions in the experimental investigations and contributes to the establishment of a theoretical model. In chapter 5, the methodology is developed in detail using nC7 H16 as fuel. The heat loss iii

iv due to the walls is estimated by comparing the experimental data with simulation results. By using this heat loss, detailed investigations in reaction pathways are conducted. Based on the experimental and the numerical findings, a theoretical model is established and an analytical expression of first-stage ignition delay time (τ1st ) is obtained. In chapter 6, the developed methodology is tested with DME. The detailed analyses of reaction pathway at low temperatures are discussed. The theoretical model leads to the analytical expression of τ1st . The analytical expression explicitly describes the importance of specific elementary reactions in LTC and the criteria for validation of detailed reaction mechanism. In chapter 7, as a further development of the methodology, blended fuel that consists of DME and nC7 H16 is tested. Based on experiments, a blended mechanism is established using existing reaction mechanisms to predict τ1st with blended fuels. These experiments and simulations allow the derivation of an analytical expression for τ1st with blended fuels. τ1st explicitly demonstrates the rate-determining step in the oxidation process and non-linear mixing rule. The obtained results are summarized and an overview of future work is presented in chapter 8.

Zusammenfassung Diese Dissertation hat das Hauptziel, Niedertemperaturzündungen detailliert zu untersuchen. Bisherige Untersuchungen befassten sich mit der ersten und zweiten Zündstufe herkömmlicher Kraftstoffe wie Diesel, Benzin oder Flugzeugkraftstoffe. Die Beobachtung erfolgte in Regimen mit negativem Temperaturkoeffzienten bedingt durch Nieder- und Hoch-Temperatur Chemie (LTC und HTC). Diese Untersuchungen ergaben, dass Niedertemperaturregime ein großes Potenzial haben, um die Emissionen in technischen Anwendungen zu reduzieren. Generell werden die Zündungen sequentiell beobachtet. Das heißt, dass erste und zweite Zündphase getrennt analysiert werden und Gesamtzündprozess als Zusammensetzung aus erster und zweiter Zündphase betrachtet wird. Gleichzeitig ist die erste Zündstufe bedingt durch Niedertemperatur-Chemie bekannt als eines der dominierenden Phänomene des gesamten Zündungsprozess. Um die Zündung bei niedrigen Temperaturen zu beobachten, wurde ein Reaktor für laminare Strömungen (LFR) entwickelt. Dieser Reaktor ermöglicht es, ausschließlich Niedertemperaturzündungen experimentell zu untersuchen (hier: die erste Zündstufe). In anderen Worten: Der Reaktor extrahiert nur die erste Zündstufe aus dem gesamten Zündungsprozess. N-Heptan (nC7 H16 ), Dimethyl-Ether (DME) und ihre Mischung wurden als Kraftstoff verwendet. Der gesamte Reaktor arbeitet unter isothermen Bedingungen. Der Temperaturanstieg durch die erste Zündstufe und die Produkte aus dieser Stufe werden experimentell gemessen. Um die Experimente zu simulieren, werden numerische Rechnungen durchgeführt. Basierend auf den Ergebnissen dieser experimentellen und numerischen Untersuchungen werden theoretische Modelle entwickelt. Die gewonnenen Resultate sind in den folgenden acht Kapiteln zusammengefasst. Kapitel 1 gibt einen Überblick über die Anforderungen der Industrie und den Forschungsstand. Kapitel 2 beschreibt die mathematischen Modelle, die für die numerische Simulation benutzt werden. Theoretische Untersuchungen werden durchgeführt, basierend auf den numerischen Simulationen. Die hiefür angewandten Annahmen werden in diesem Kapitel detailliert beschrieben. Kapitel 3 enthält die Spezifiktionen des Reaktors für laminare Strömungen. Steuergeräte, v

vi experimentelle Methoden und Parameter werden hier ebenfalls beschrieben. In Kapitel 4 wird der Reaktor mit Hilfe von 3-D Berechnungen evaluiert. Die Homogenität der Mischung wird betrachtet. Darüber hinaus werden Strömungs- und Mischungsfelder eingehend untersucht. Die numerisch bestimmten Daten gleichen die Einschränkungen der Exprimente aus und tragen dazu bei, ein theoretisches Modell aufzustellen. In Kapitel 5 wird die Entwicklung der Methode anhand von n-Heptan beschrieben. Der Wärmeverlust durch die Wände wird berechnet, indem die experimentell gewonnen Daten mit den Simulationsergebnissen verglichen werden. Der Wärmeverlust wird dazu benutzt, genaue Untersuchungen des Reaktionsverlaufs durchzuführen. Basierend auf den Ergebnissen der Experimente und numerischen Simulationen wird ein theoretisches Modell entwickelt und eine analytische Lösung zur Darstellung der Zündverzögerung der ersten Zündstufe gewonnen. In Kapitel 6 erfolgt die Überprüfung der Methode mit DME. Die Analyse des Reaktionsverlaufs unter Niedertemperatur-Bedingungen wird besprochen. Das theoretische Modell führt zur analytischen Darstellung der Zündverzögerung. Die analytische Lösung beschreibt explizit die Bedeutung der Elementarreaktionen im Strömungsreaktor und die Kriterien für die Validierung des detaillierten Reaktionsmechanismus. Kapitel 7 beschreibt den Test einer Kraftstoffmischung aus DME und nC7 H16 als Weiterentwicklung der Methode. Basierend auf den Experimenten wird ein Mischungsmechanismus aus den bestehenden Reaktionsmechanismen entwickelt, um die Zündverzögerung für Kraftstoffmischungen vorherzusagen. Aus diesen Experimenten und Simulationen wird eine analytische Darstellung der Zündverzögerung für Kraftstoffmischungen hergeleitet. Die Zündverzögerung demonstriert explizit den geschwindigkeitsbestimmenden Schritt für die Oxidation und die nicht-lineare Mischungsregel. Die gewonnen Ergebnisse werden in Kapitel 8 zusammengefasst. Das Kapitel enthält außerdem einen Überblick über zukünftige Forschungsvorhaben.

Contents 1

Introduction

1

2

Model description

9

2.1

Molar- and mass-based reactions . . . . . . . . . . . . . . . . . .

9

2.2

Governing equations . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.4

Pipe flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.5

Plug flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.6

Mechanism reduction . . . . . . . . . . . . . . . . . . . . . . . .

15

2.6.1

Carbon Atom Screening (CAS) . . . . . . . . . . . . . .

15

2.6.2

Production Rate Screening (PRS) and Rate-of-Progress Screening (RPS) . . . . . . . . . . . . . . . . . . . . . .

16

2.7

Definition of the first-stage ignition delay time . . . . . . . . . . .

17

2.8

Calculation flow . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3

Experimental apparatus

21

3.1

Laminar flow reactor . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2

Gas supply system . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3

Power supply system . . . . . . . . . . . . . . . . . . . . . . . .

28

3.4

Data acquisition system . . . . . . . . . . . . . . . . . . . . . . .

30

3.5

Gas sampling system . . . . . . . . . . . . . . . . . . . . . . . .

31

3.6

Experimental parameters and methods . . . . . . . . . . . . . . .

31

vii

CONTENTS

viii 4

5

6

7

Evaluation of the laminar flow reactor

35

4.1

Experimental visualization . . . . . . . . . . . . . . . . . . . . .

35

4.2

Influence of the fuel injector . . . . . . . . . . . . . . . . . . . .

36

4.3

Mixing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Low temperature ignition of nC7 H16

47

5.1

Determination of temperature from experimental conditions . . .

47

5.2

Heat loss estimation . . . . . . . . . . . . . . . . . . . . . . . . .

48

5.3

Calculation of τ1st from Lign . . . . . . . . . . . . . . . . . . . .

51

5.4

Mechanism screening . . . . . . . . . . . . . . . . . . . . . . . .

52

5.5

The first-stage ignition of nC7 H16 . . . . . . . . . . . . . . . . . .

53

5.6

Influence of equivalence ratio and dilution . . . . . . . . . . . . .

64

5.7

Further reduction . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.8

Steady state assumption . . . . . . . . . . . . . . . . . . . . . . .

72

5.9

Simplified eigenvalues . . . . . . . . . . . . . . . . . . . . . . .

74

5.10 Analytical expression of τ1st . . . . . . . . . . . . . . . . . . . .

81

Low temperature ignition of DME

85

6.1

Temperature range and heat loss . . . . . . . . . . . . . . . . . .

85

6.2

The first-stage ignition of DME

. . . . . . . . . . . . . . . . . .

88

6.3

Differences of the mechanisms in LTC for DME . . . . . . . . . .

91

6.4

Reduction for the induction period . . . . . . . . . . . . . . . . . 100

6.5

Steady-state assumption . . . . . . . . . . . . . . . . . . . . . . . 103

6.6

Theoretical analysis of τ1st . . . . . . . . . . . . . . . . . . . . . 104

Low temperature ignition of the blended fuels

109

7.1

Experimental investigations with blended fuels . . . . . . . . . . 110

7.2

Comparison of τ1st for different blended fuels . . . . . . . . . . . 115

7.3

Numerical investigations with blended fuels . . . . . . . . . . . . 116

7.4

Model reduction at the induction period . . . . . . . . . . . . . . 119

7.5

Eigenvalue determination . . . . . . . . . . . . . . . . . . . . . . 124

7.6

Analytical solution of τ1st with blended fuel . . . . . . . . . . . . 129

CONTENTS 8

Conclusion and future work

ix 135

References

137

A Chem. mech. for lean DME/nC7 H16 mixture

155

B Ordinary differential equations

159

B.1 Homogeneous system . . . . . . . . . . . . . . . . . . . . . . . . 159 B.2 Inhomogeneous system . . . . . . . . . . . . . . . . . . . . . . . 161 B.3 Steady-state assumption . . . . . . . . . . . . . . . . . . . . . . . 162 B.4 Two intermediates model for nC7 H16 . . . . . . . . . . . . . . . . 165 B.5 Mechanism reduction . . . . . . . . . . . . . . . . . . . . . . . . 169 C Time evolution of functions

173

C.1 Quadratic (2nd-degree) polynomial . . . . . . . . . . . . . . . . . 173 C.2 nth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 C.3 Influence of zim on the time evolution of zcomp . . . . . . . . . . . 178 D Solution of higher polynomial

181

D.1 Quartic (4th-degree) polynomial . . . . . . . . . . . . . . . . . . 181 D.2 Cubic (3rd-order) polynomial . . . . . . . . . . . . . . . . . . . . 183 E Detailed information of the LFR

187

E.1 Fuel injector and fuel supplying pipe . . . . . . . . . . . . . . . . 187 E.2 Mesh for the CFD calculations . . . . . . . . . . . . . . . . . . . 188 E.3 2-D analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 E.4 Analysis of the mixing field . . . . . . . . . . . . . . . . . . . . . 196 E.5 Spontaneous 2nd-stage ignition in the LFR . . . . . . . . . . . . . 198

Nomenclature Roman Symbols A f rec Frequency factor

[depending on reactions] [m2 ]

ALFR Cross-section area of LFR

[kmol/m3 ]

C

Molar-based concentration

cp

Mass-based specific heat

Cvir

Virtual concentration

Ea

Activation energy

h

Mass-based enthalpy

I

Identity matrix

A

Jacobian matrix

Jdi f f

Diffusion mass flux

kf

Rate constant in forward direction

[J/(kgK)] [kmol/m3 ] [kcal/kmol] [J/kg]

[kg/(m2 s)] [depending on reaction] [m−1 ]

Khetero Heterogeneous reaction rate kr

Rate constant in reverse direction

kt

Turbulent energy

[depending on reaction] [m2 /s2 ]

ktherm Thermal conductivity

[W/(Km)]

Lbound Entrance length

[m]

Ldecay Length of temperature decay

[m]

LFI

[m]

Length of the fuel injector (= 60 mm) xi

NOMENCLATURE

xii Lign

Ignition length in LFR

LLFR Length of LFR (= 5.8 m) Lrz

Length of recirculation zone

[m] [m] [mm]

Mthird Third body nC

Number of cell

nR

Number of reactions

nS

Number of species

P

Modal matrix

Pr

Prandtl number

Q˙ loss Heat loss

[-] [J/s]

q˙ f

Forward rate-of-progress

[kmol/(m3 s)]

q˙r

Reverse rate-of-progress

[kmol/(m3 s)]

R

Universal gas constant (=1.9858775 cal/(K mol))

Re

Reynolds number

[-]

R pipe Radius of the pipe

[m]

Sc

Schmidt number fuel

Sstick Sticking coefficient

[cal/K mol]

[-] [-]

Toven Oven tempearture

[K]

Tpeak Peak temperature during FSI

[K]

Tre f

[K]

Temperature at the reference condition

vcross Cross-sectional averaged velocity V˙mix

Volumetric flow rate of air and nitrogen

vin j

Injection velocity from the fuel injector

[m/s] [m3 /s] [m/s]

vz

Axial component of velocity

[m/s]

NOMENCLATURE W

Molecular weight

Z

Mixture fraction

zcomp Complex number zim

Imaginary part of zcomp

zre

Real part of zcomp

Abbreviations CAD Computer Aided design CAS Carbon atom screening CFD Computational fluid dynamics CSP

Computational singular pertubation

DAE Differential algebraic equation DRG Directed relation graphs DRGEP Directed relation graphs with error propagation FID

Flame ionization detector

FI

Fuel injector

FR

Flow reactor

FSI

First-stage ignition

FV

Fuel vaporizer

GA

Gas analyzer

GC

Gas chromatography

HiCOT High temperature air combustion technology HTC High temperature chemistry ILDM Intrinsic low-dimensional manifolds ISO

Internatioinal standard organization

JSR

Jet-stirred reactor

xiii [kg/kmol] [-]

NOMENCLATURE

xiv LFR

Laminar flow reactor

LMFC Liquid mass flow controller LTC

Low temperature chemistry

MFC Mass flow contorller MILD Moderate and intensive low oxigen dilution MON Motor octane number NTC Negative temperature coefficient ODE Ordinary differential equation PRF

Primary reference fuel

PRS

Production rate screening

QSS

Quasi-steady state

RANS Reynolds averaged navier storks RCM Rapid compression machine RIF

Representative interactive flamelet

RMG Reaction mechanims generator RON Research octane number RPS

Rate-of-progress screening

SSI

Second-stage ignition

ST

Shock tube

TCD Thermal conductivity detector Greek Symbols β

Collision factor

[-]

δ Tˆ

Normalized δ T

[-]

λ

Eigenvalue

μ

Dynamic viscosity of the fluid

[kg/(ms)]

NOMENCLATURE

xv

νst

Stoichiometric mass fraction ratio YO2 /YO2

[-]

φ

Equivalence ratio

[-]

τ1st

First-stage ignition delay time

[s]

τ2nd

Second-stage ignition delay time

[s]

τchem Characteristic time scale of chemical reactions

[ms]

θ

Argument of zcomp

ϒˆ

Variation ratio

[-]

ν j,i

Net-stoichiometric coefficient (ν j,i = ν j,i − ν j,i )

[-]

ω˙ p

Production rate

[kmol/(m3 s)]

ω˙ d

Destruction rate

[kmol/(m3 s)]

ω˙ c

Creation rate

[kmol/(m3 s)]

ξ

Volumetric dilution ratio

[degree]

[-]

ΞRPS Threshold value for the RPS

[kmol/(m3 s)]

ΞPRS Threshold value for the PRS

[kmol/(m3 s)]

Other cl

Value on the centerline in the pipe



Cross-section averaged value

f

Fuel

i

Index for reaction

j

Index for species

laminar

Laminar condition

()

Mean value

 ()

Operator for the variance

trunc Truncation

Chapter 1 Introduction “Though the river’s current never fails, even the water flowing at every instance is never the same. Where the current pools, bubbles forming on the surface burst and disappear as others rise to replace them and none of them last long.” (Kamono-Chomei, Hojoki, 1212) The combustion phenomena, i.e., “chemical-reaction flow,” at low temperatures can be explained in a similar manner to the “river’s current” in the above sentence. During the oxidation process of fuels at low temperatures, intermediates are formed and are consumed (i.e., further oxidization). In other words, reaction flow is preserved if unburned fuel remains and the consumption of the intermediates is greater than their formation. Although intermediates are accumulated due to interrupted flow, other products may come into existence because of recombination reactions. This phenomenon can be explained by the dynamic behavior of the bubble in the sentence above. Combustion technology based on chemical reactions has a rich history, and can be found in a wide range of applications such as power generation and heating, and the evolution of this technology has contributed to advancements in various fields worldwide [1]. However, the negative impact of emissions on the environment is inevitable and is being investigated with the highest priority. The development of methods for the minimization of pollution [2, 3, 4, 5] and greenhouse gases, which absorb infrared radiation from the Earth’s surface and atmosphere [6, 7, 8], are the key topics in combustion research. Since the 1970s, when carbon dioxide (CO2 ) was first identified as a greenhouse gas, three major technologies have been developed to reduce CO2 emissions. The first technology, which is still under discussion, is the capture of CO2 from exhausts and its storage underground [9, 10, 11]; and the second technology still being investigated is the use of high-efficiency combustion systems for industrial applications, such as high 1

2

CHAPTER 1. INTRODUCTION

temperature air combustion technology (HiCOT) or moderate and intensive low oxygen dilution (MILD) combustion [12, 13], and the third method is the use of low-carbon, plant-based fuels or surrogate fuels with lighter hydrocarbon species [14, 15, 16]. Recently, methane (CH4 ) has also been recognized as a greenhouse gas. As CH4 is a major component of natural gas, which is widely used in industry, methods to improve its combustion efficiency have been intensively studied [17, 18]. Amongst atmospheric pollutants, nitrogen oxides (NOx ) including nitric oxide (NO) and nitrogen dioxide (NO2 ) have been found to have a strong impact on human health, and their forming mechanisms from fuel nitrogen (N) [19] and atmospheric nitrogen (N2 ) [20] have been investigated. These evaluations were applied to industrial applications such as the prediction of NOx formation [21, 22, 23, 24, 25]. Since the 1980s, investigations regarding the strong influence of nitrous oxide (N2 O) on the atmosphere as a greenhouse gas have been conducted [26, 27]. In spite of the critical consequences, our lives strongly depend on the energy generated using combustion technology. It is necessary to reduce these negative impacts on the environment by improving technology and exploring alternative energy sources. For example, since Japan became highly dependent on nuclear power for its industrial activities and transportation, its civil life faces critical problems at times of disaster, as experienced following the Fukushima earthquake on 11th March 2011. Therefore, we need to accelerate the use of alternative sources such as solar, wind, and geothermal energies. Although these energies are considered a wise substitute, their technologies still need to be improved. Hence, the efficiency of conventional energy sources, with the exception of nuclear, has to be enhanced until they can be referred to as “practical energy sources.” To satisfy such requirements, lean combustion at low temperatures is found to be a possible solution. In general, this method decreases flame temperatures and combustion emissions. However, combustion phenomena become unstable at low temperatures, and such instabilities retard the application of these technologies in many areas such as homogeneous charge compression ignition (HCCI) engines [28]. In order to predict these phenomena in such applications, combustion chemistry has been intensively studied. Combustion chemistry, comprising many elementary reactions (the reaction mechanism), has a long history, and has been investigated both experimentally and numerically. Experimentally, it has been investigated using various experimental apparatus such as shock tubes (STs), rapid compression machines (RCMs), jetstirred reactors (JSRs), and flow reactors (FRs). The ST is mainly employed for investigations at high temperatures and high pressures. Due to the pressure-driven shock wave, boundary conditions can be precisely measured and can be compared with numerical simulations. Studies of combustion chemistry using STs until the 1960s were summarized by Bauer [29] and Nettleton [30], and clearly

3 pointed to the potential of this experimental apparatus. A number of research groups (e.g., those of Hanson (Stanford, USA) [31], Petersen (Texas, USA) [32], Roth (Duisburg-Essen, Germany) [33], Adomeit (Aachen, Germany) [34, 35], and Simmie (Galway, Ireland) [36]) have developed STs, and have contributed to the understanding of the fuel oxidation process under a wide range of temperatures and pressures. Recently, experimental data has been obtained not only for single component fuel, but also for binary component fuels that represent natural gas [17, 37, 38, 39, 40]. RCMs were originally developed to study enginerelated phenomenon such as engine knock. During the 1960s, the main focus of RCMs was towards their usage in the automotive industries, as summarized by Levedahl (NIST, USA) [41]. However, the focus has since shifted from industrial to academic purposes because of their reusable experimental capability1 , and also because experiments can be conducted at relatively lower temperatures than those in STs. Recently, several types of RCMs (e.g., single piston, dual piston, air driven piston) have been developed by the research groups of Sung (CWRU, USA) [42], Simmie (Galway, Ireland) [43], Keck (MIT, USA) [44], Boulouchos (ETH, Switzerland) [45], Wooldridge (Michigan, USA) [46], and Sochet (CNRS, France) [47, 48], producing valuable data for detailed reaction mechanism validations. Although STs and RCMs have been used for about half a century to provide an insight into experimental investigations of combustion chemistry, auto-ignition phenomena in these apparatus were spontaneous. In principle, the auto-ignition delay time (i.e., overall ignition delay time) can be measured using these apparatus, but species concentration as a function of time cannot be measured. To overcome this issue, Hanson’s group reported hydroxide (OH)-time history measurements using a ST, thus demonstrating its potential [49, 50]. However, the available species in these time-history measurements are limited by the restrictions of current laser-diagnostics techniques. Therefore, in order to cover the requirements of reaction mechanism development, experimental investigations at not only high temperatures but also intermediate time histories have to be conducted. A JSR is widely used to measure intermediate species at low temperatures during auto-ignition processes (i.e., low-temperature chemistry (LTC)). JSRs were first built by Longwell [51] to study intermediate concentrations during the fuel oxidation process. By controlling the residence time of the mixture in the reactor, intermediates species can be measured using a JSR. Due to the longer ignition delay time at low temperatures, they have been used for investigations in LTC. Two types of JSR have been developed by Dagaut’s [52, 53, 54, 55, 56, 57, 58] and Ciajolo’s [59, 60] groups. Experimental data from both of these JSRs have been used for detailed reaction mechanism development. Dagaut’s JSR has been used for the study of various fuels (or specific stable intermediates) in their ox1 ST

needs to replace the diaphragm for each experiment.

CHAPTER 1. INTRODUCTION

4

idation process, and quantitative species concentrations have been measured by Battin-Leclerc’s group [61, 62, 63]. Ciajolo’s JSR has been used to study oscillatory phenomena in the JSR itself, and results were reported by Cavaliere’s group [64, 65]. Although JSRs allow the observation of intermediate concentrations during the fuel oxidation process, the ignition delay time and intermediate concentration measurements cannot be directly related because of the change in spontaneous residence time of the mixture. Recently, sequential ignition and extinction phenomena in the JSR were reported [66], but the interdependence between ignition delay time and measured species concentration has not been well investigated. The other investigations to measure intermediate species during the ignition process have used FRs. An FR is mainly designed to measure the intermediate species at similar conditions as a function of time, as in a JSR. By using a probe and gasand/or mass-chromatography, detailed species concentrations over time can be measured during the auto-ignition process. FRs used in the field of chemical engineering since the 1960s [67, 68, 69] have been adapted to aid the understanding of combustion phenomena. For example, several FRs have been developed by the groups of Dryer (Princeton, USA) [70], Koert (Drexel, USA) [71, 72], Mantzaras (PSI, Switzerland) [73, 74], Bowman (Stanford, USA) [75], Maas (KIT, Germany) [76], Beerer (UCI, USA) [77], and Frenklach (PSU, USA) [78]. Under steady conditions, optical measurements examined by Ghermay et al. [74] realized a number of advantages in comparison to JSRs. In addition, by measuring chemiluminescence, ignition delay times at low temperatures (i.e., a temperature range not available for STs or RCMs) were measured using FRs [77, 79]. However, the measured species concentrations and ignition delay times (i.e., first-stage ignition delay time, τ1st ) were not obtained simultaneously. Note that experimental data demonstrate only the results of possible elementary reactions. Precisely, the balanced concentrations occurring due to the production and consumption of related reactions are measured by the species concentration2 . Although each experimental apparatus has its advantages and disadvantages, together they provide a wide range of temperatures and pressures for reaction mechanism development. Based on these experiments, numerical simulations of the fuel oxidation process have been established. The experimental database contributes to the detailed reaction mechanism development. In general, there are two philosophies relating to the detailed reaction mechanisms development. According to the first philosophy, detailed and generalized reaction mechanisms are developed. By involving oxidation reaction pathways from not only one but several fuels, the detailed reaction mechanism can be used for another fuel under 2 Few

elementary reactions can be directly measured [80, 81].

5 various conditions. This philosophy is based on the principle that oxidation pathways of hydrocarbon species have a hierarchical structure. Based on this idea, for example, three of many reaction mechanisms have been established by the research groups of Ranzi (Milan, Italy) [82, 83, 84, 85, 86], Warnatz (Heidelberg, Germany) [87, 88], and Williams (UCS, USA) [89]. Remarkably, the detailed reaction mechanism from Ranzi’s research group involves the largest variety of fuels within a reaction mechanism, and it is therefore used for this study. Recently, the concept of a hierarchical structure has also been applied to blended fuels such as gasoline, diesel, and jet fuels. Gasoline and diesel fuels have been investigated using surrogate fuels (e.g., primary reference fuel (PRF) [90, 91] or a mixture characterized by using research octane number (RON) and motor octane number (MON) [92]). Another research angle is the investigation of suitable components for the surrogate, as was done by Pitsch and collaborators [93]. Jet fuel was also studied by investigating appropriate surrogate fuels [94, 95]. Recently, a surrogate fuel with a biomass-driven component has been focused upon, and its detailed investigation is in progress [91]. By using a general mechanism for the surrogate fuel that denotes the oxidation process of selected species, the combustion process in applications such as the internal combustion engine can be closely predicted. According to the second philosophy, a fuel-specific mechanism is developed. By collecting all of the existing experimental data, prediction using the specific mechanism in the experimental investigations is optimized. An example of this philosophy is summarized in the gas research institute (GRI) mechanism [96]. It is well known that this mechanism is specialized for CH4 oxidation processes, but there are cases in which propane (C3 H8 ) and formaldehyde (CH2 O) were also chosen as a reference. Another example comes from a research group led by Westbrook at the Lawrence Livermore National Laboratory (LLNL). They have established reaction mechanisms for a variety of fuels (e.g., hydrocarbon, oxygenated, and organophosphorus compounds). These mechanisms have been used as a reference for specific fuels. Two mechanisms from the LLNL research group, for dimethyl ether (DME) and n-heptane (nC7 H16 ), are used in this study. Besides these philosophies, Cˆome’s and Green’s groups have developed computational programs (i.e., EXGAS [97] and RMG [98]) to generate the reaction mechanism. These programs allow the generation of mechanisms for specific conditions (i.e., fuel concentration, temperature, and/or pressure). Although such a detailed reaction mechanism predicts combustion phenomenon accurately, it involves a large number of species and reactions, which may prove to be disadvantageous. As an example, in computational fluid dynamics (CFD), due to limited computational resources, the nonlinearity of a system has to be minimized. Thus, in CFD, it is difficult to consider all species in the detailed reaction mechanisms. Hence, mechanism reduction has been considered as one of

6

CHAPTER 1. INTRODUCTION

the main topics in combustion chemistry since the 1970s. In order to reduce the detailed reaction mechanism, i.e., reactions and species, Ranzi’s group has developed a lumping method that considers their chemical structures (especially isomers) [84, 99, 100]. Even after the lumping process, the detailed reaction mechanism still involves significantly more species and reactions than the CFD calculations are capable of handling. To overcome this issue (to convey detailed information from the reaction mechanism to the CFD calculations), several mechanism reduction methods have been developed based on the time scale in the chemical reaction. Based on the findings of Dixon-Lewis [101], the influence of the reaction time scale has been the focus by removing unimportant reactions and species from the detailed reaction mechanism, thus establishing a reduced mechanism. There are two major mechanism reduction strategies. The aim of the first strategy is to reduce the mechanism to a manageable level for the CFD calculations by partially neglecting the chemical kinetics. The computational singular perturbation (CSP) method was then initiated by Lam and Goussi [102]. By using this concept, the groups of Maas and Law proposed intrinsic low-dimensional manifolds (ILDM) [103] and directed relation graphs (DRG) [104, 105], respectively. These methods locally resolve the stiffness of a reaction mechanism based on the time scale of the chemical reaction. Fuel oxidation processes can be predicted from their initiation by time dissolution considerations. Further methods proposed from 2008 to 2011 include, the directed relation graphs with error propagation (DRGEP) method established by Pepiot-Desjardins and Pitsch [106], principle component analysis by Esposito and Chelliah [107], level of importance analysis by Løvås [108], and path flux analysis by Ju et al. [109]. These methods reproduce the results obtained using detailed reaction mechanisms, and successfully reduce the number of species and reactions. In addition, the reduced mechanisms can be applied to predict flame propagation and its structure (e.g., 1-D flame calculations). In other words, the fuel oxidation process and mass diffusion can be predicted. However, due to locally resolved reaction mechanisms by the above mentioned methods, the reduced mechanism still surpasses the capabilities of current simulation techniques. Therefore, for CFD calculations, further modeling such as representative interactive flamelets (RIFs) [110], multi-zone [111], or tabulated models [112] are required to convey the necessary information. The aim of the second mechanism reduction strategy is to express the combustion (i.e., oxidation process) analytically from the systematic reduction of nonlinearity in combustion chemistry. In terms of time scale considerations, this strategy is similar to that mentioned above. However, by considering the order of magnitude of the reaction mechanisms, only time scales of leading order are considered. In particular, the slow time scale (i.e., that corresponding to the slow manifold in the ILDM method) is extracted as the leading order time scale. Since corresponding

7 time scales are neglected, this method does not allow the prediction of oxidation processes at the beginning of the reaction. However, instead of inaccurate predictions, the stiffness of the reaction mechanisms is further reduced as compared to the first strategy. The oxidation process can thus be analyzed analytically by this method. Especially, this reduction considers obvious difference in terms of the flame propagation. This strategy was initiated by Peters and Williams, and it explicitly demonstrates the oxidation process of CH4 by global reactions based on quasi-steady state and partial equilibrium assumptions with asymptotic theory [113, 114]. This procedure has been applied to laminar premixed flames [115], partially premixed flames [116], and diffusion flames [117] to demonstrate the oxidation and diffusion processes. In addition, this procedure has been successfully applied to turbulent flames [118]. As mentioned, due to the simplifications of the analytical expressions, slightly less accurate results may be achieved than from the detailed reaction mechanism [119]. However, as shown in a recent publication from Williams’ group, this analytical expression can be applied with reasonable accuracy in the numerical studies of gas-turbines [120]. As described above, better insight into combustion chemistry is obtained by reducing the detailed reaction mechanism. Although the detailed reaction mechanism is well-validated and accurately predicts the experiments, most of the elementary reactions involved in the mechanism are estimated (i.e., estimated reaction pathway and Arrhenius parameters). A lack of experimental data concerning minor intermediates leads to insufficient feedback from the reduced to detailed reaction mechanism level, and also from experiments to a specific intermediate. Fortunately, due to the development of measurement techniques, a wide range of intermediates are now measurable to a certain extent. For instance, KohseHöinghaus’ group demonstrated measurements of a burner-stabilized flame on a low-temperature burner [121, 122]. The research groups of Qi, Battin-Leclerc, and Taatjes measured intermediates in LTC, and realized further possible improvements to the detailed reaction mechanism using state-of-the-art measurement techniques [61, 123, 124, 125]. It can be inferred from these experiments that, once the important species in the oxidation process are determined by the reduced model, one can exclusively measure the concentration of specific species and improve the accuracy of a specific elementary reaction. Also, the specific target allows the quantum mechanics calculations to conduct further detailed validations, such as the thermal properties of the intermediates involved and the pressure dependency of the reaction. Once the detailed reaction mechanism has been improved, it is subsequently reflected in the accuracy of the reduced model. Therefore, obtaining an appropriate reduced model and theoretical understanding of the oxidation process are important steps for advancing combustion chemistry.

8

CHAPTER 1. INTRODUCTION

In this dissertation, low-temperature ignition is studied experimentally, numerically, and theoretically using a laminar flow reactor (LFR). This LFR is developed to obtain appropriate boundary conditions for numerical and theoretical investigations. The remainder of this dissertation is arranged as follows: chapter 2 describes the theory used in the numerical simulations and the theoretical investigations. The governing equations (species and energy equations) and the assumptions applied are described in this chapter. Chapter 3 describes the LFR and the measurement procedure in detail. This apparatus is capable of using liquid and/or gaseous fuel at atmospheric pressure. The available temperature range is 300–1000 K. The LFR is significantly different from conventional flow reactors. In chapter 4, the LFR is evaluated to examine its ability. Due to the restrictions of the LFR, 3-D simulations are used for these investigations. The homogeneity of the mixture in the LFR (i.e., flow and the mixing fields) is thoroughly investigated, and appropriate assumptions for numerical and theoretical investigations are discussed. In chapter 5, the methodology is developed to analytically express the low-temperature ignition of nC7 H16 in detail. nC7 H16 has been extensively studied in the combustion community, and is a suitable reference fuel. Heat loss due to the reactor walls is estimated using experimental data as a reference. By using this heat loss, detailed investigations of reaction pathways are conducted. Based on the experimental and numerical findings, a theoretical model is established and analytical solutions representing low-temperature ignitions are obtained. In chapter 6, the developed methodology is applied to DME. DME is used as an alternative fuel from biomass. The validity of the estimated heat loss is examined for different fuels. Three mechanisms (the LLNL and Ranzi mechanisms, as well as one generated by RMG) are compared, and their reaction pathways at low temperatures are discussed in detail. The theoretical model leads to an analytical expression for low-temperature ignition with DME. This analytical solution explicitly describes the importance of specific elementary reactions in LTC and provides criteria for the validation of the detailed reaction mechanism. In chapter 7, as a further development of the methodology, blended fuels consisting of DME and nC7 H16 are tested. These blended fuels represent the blending of conventional and biomass-driven fuels. In addition, the fuels demonstrate the interaction between hydrocarbon and oxygenated species in their oxidation processes. Based on inferences from the experiments, a blended mechanism (a combination of the LLNL and Ranzi mechanisms) is established to predict low-temperature ignition with blended fuels. These experiments and simulations aid the derivation of an analytical expression for low-temperature ignition of blended fuels. This low-temperature ignition explicitly demonstrates the rate-determining step in the oxidation process, and analytically expresses the nonlinear interaction between fuels. The obtained results are summarized in chapter 8, where an overview of future works is also presented.

Chapter 2 Model description Investigations of combustion chemistry involve analysis of detailed reaction mechanism through computational resources. For accurate analysis of reaction mechanism one has to compromise on limited computational resources. Although the mechanisms well predict combustion phenomena such as auto-ignition using large number of species and reactions, they further need to be verified with other references such as flame propagation, flame extinction, or flame structure. For consideration of these references, influence of diffusion and flow field should be taken into account. However, due to limitation in computational resources, considering diffusion and flow field with detailed reaction mechanism (i.e., direct numerical simulation of reacting flow) in areas like internal combustion engines and gas turbines have not been carried out. To overcome this issue, improvements in combustion modeling have been carried out since the 1970s. A number of modeled burners (reactor) with proper simplification of the governing equations have been proposed. However, as described in references [126, 127], the derived model strongly depends on the assumptions concerning its configuration (i.e., closed, opened, tube or sphere). Hence, it is necessary to elucidate the assumptions made for the governing equations to correctly interpret the experiments. In this chapter, the low-dimensional governing equations (species and energy equations) are derived from the general form to explain the assumptions made in this study.

2.1

Molar- and mass-based reactions

The general form of elementary reaction with forward and reverse rate constants (denoted by k f , kr ) is described as   k f nS     ν ∑ j,i X j  ∑ ν j,i X j nS j

kr

j

9

(i = 1, 2, 3, ..., nR)

(2.1)

CHAPTER 2. MODEL DESCRIPTION

10

  Here, X j , nR, and nS denotes jth species, number of species and number of reactions. ν j,i denotes a stoichiometric coefficient of jth species in the forward direction of the ith reaction, respectively. Contrarily, ν j,i denotes a stoichiometric coefficient of jth species in the reverse direction of the ith reaction. The rate constant (k) of these reactions is referred as the Arrhenius expression and can be expressed as   Ea B k = A f req T exp − (2.2) RT where A f rec , Ea , and R denote frequency factor, activation energy, and universal gas constant, respectively. In general, to determine k for specific reaction, gas composition is experimentally measured in unit of ppm (i.e., molar-based concentrations (C) ) and is converted to molar fraction (X)1 . It satisfies the following relationship. Xj =

Cj ∑j Cj

(2.3)

For most applications in combustion technology, not only chemical reaction but also diffusion and convection play a major role. Hence, it is favorable to consider mass-based reactions instead of molar-based reactions. Mass-based continuity (i.e., mass fraction (Y )) is used by the following conversion from X to Y and to C. Yj =

Wj ρC j Xj = Wj W

(2.4)

where W and () denote molecular weight and its mean value. W is defined as W ≡ ∑ j X jW j . For homogeneous systems such as premixed combustion, its composition is expressed using equivalence ratio (φ ) with mass fractions of fuel and oxidizer, denoted by subscripts f and O2 . ν Y Y f /YO2  = st f φ≡ YO2 Y f /YO2 st

(2.5)

νst Y f −YO2 +YO2 ,2 νst Y f ,1 +YO2 ,2

(2.6)

  Here, νst denotes stoichiometric coefficient and is defined as νst ≡ YO2 /Y f st . For inhomogeneous systems such as non-premixed combustion, fuel distribution caused by two streams is expressed using mixture fraction (Z). Z= 1X

= [ppm] / 1 × 106

2.2. GOVERNING EQUATIONS

11

where subscript 1 and 2 denotes feed streams. For stoichiometric mixtures, by definition from equation 2.5 (νst Y f = YO2 ), the stoichiometric mixture fraction Z can be expressed as

νst Y f ,1 Zst = 1 + YO2 ,2

−1 (2.7)

If two feed streams are identical, the mixture is a homogeneous mixture. Under this condition, φ and Z are related by the following equation which can be expressed with local equivalence ratio φ in the mixing cold flow field as φ=

2.2

Z (1 − Zst ) 1 − Z Zst

(2.8)

Governing equations

The jth species equation is based on the species conservation in a control volume and is expressed as DY j −−−→ = −∇ · Jdi f f , j + ω˙ jW j (2.9) Dt where Jdi f f and ω˙ denote mass flux and production rate. By assuming no spatial distribution of mean molecular mass (W ), Jdi f f can be written as ρ

−−−→ Jdi f f , j = −ρD j ∇Y j

(2.10)

and equation 2.9 can be expressed with mixture averaged diffusion coefficient of jth species (D j ) as ρ

DY j = ρD j ∇2Y j + ω˙ jW j Dt

(2.11)

In cylindrical coordinates, the equation can be written as  ρ

∂Y j ∂ vrY j 1 ∂ vθ Y j ∂ vzY j + + + ∂t ∂r r ∂θ ∂z



 = ρD j

  ∂Y j 1 ∂ 1 ∂ 2Y j ∂ 2Y j r + 2 + 2 + ω˙ jW j r ∂r ∂r r ∂θ2 ∂z (2.12)

For a perfect gas, energy equation without dissipation can be expressed with the low Mach number assumption [128, 129] as

CHAPTER 2. MODEL DESCRIPTION

12 ρc p

nS nS DT −−−→ = ∇ · (λ ∇T ) − ∑ c p, j Jdi f f , j · ∇T − ∑ h j ω˙ jW j − Q˙ loss Dt j j

(2.13)

Here, c p , h, and Q˙ loss denote mass-based specific heat, mass-based enthalpy, and heat loss, respectively. Further, by neglecting conduction and applying equation 2.13 the energy equation can be written as ρc p

nS nS DT = ρ ∑ c p, j D j ∇Y j · ∇T − ∑ h j ω˙ jW j − Q˙ loss Dt j j

(2.14)

In cylindrical coordinates, equation 2.14 is written as  ρc p

∂ T ∂ vr T 1 ∂ vθ T ∂ vz T + + + ∂t ∂r r ∂θ ∂z



nS

= ρ ∑ c p, j D j j



∂Y j ∂ T 1 ∂Y j ∂ T ∂Y j ∂ T + + ∂ r ∂ r r2 ∂ θ ∂ θ ∂ z ∂ z



nS

+ ∑ h j ω˙ jW j − Q˙ loss

(2.15)

j

2.3

Chemical reactions

For these elementary reactions, the production rate of the jth species (i.e., the change of jth species molar-concentration (C) due to the chemical reaction) can be written as   nR dC j = ∑ ν j,i q˙nj (2.16) ω˙ j = dt chem i Here, ν j,i and q˙nj denote net stoichiometric coefficient (cf. ν j,i = ν j,i − ν j,i ) and net rate-of-progress of the jth species. q˙nj is defined as the balance between contribution from forward and reverse reactions and is expressed as q˙nj

f

= q˙i − q˙ri

nS ν j,i = k f ,i C j − kr,i j j nS

∏

ν 

∏ C j j,i

Hence, the production rate of the jth species can be written as

(2.17)

2.4. PIPE FLOW

13 

ω˙ j =

dC j dt

 chem

nR

= ∑ ν j,i i

nS ν k f ,i C j j,i − kr,i j j nS

∏

∏

ν  C j j,i

 (2.18)

For some of the elementary reactions (e.g., recombination or dissociation reactions), the third body (Mthird ) is used for considering the collision efficiency of each species. For such reactions, equation 2.19 is modified as

    nR nS nS ν  nS ν  dC j k f ,i ∏ C j j,i − kr,i ∏ C j j,i (2.19) ω˙ j = = ∑ ν j,i ∑ β j,iC j dt chem j j i j where β is the collision factor.

2.4

Pipe flow

Velocity profile on surface had been studied largely as one of the major topics in fluid dynamics since the 18th century. From a number of experimental, numerical, and theoretical findings, the velocity profile in a pipe of diameter R pipe with laminar flow, i.e., fully developed flow, is referred to as the Hagen-Poiseulle flow and is expressed as 2  vz r = 1− (2.20) vcl R pipe where vz and subscript cl denotes velocity and value on the centerline, respectively. In this flow, the cross-sectional averaged velocity (vcross ) is 0.5 vcl , and ρv 2R the Reynolds number (Re) in the pipe can be determined from Re = crossμ pipe , where μ denotes dynamic viscosity of the fluid. In a pipe with laminar flow, the jth species and energy equations can be simplified further. By the definition of the Hagen-Poiseulle flow, the flow is in a steady ∂ condition and has only axial direction (i.e., ∂t = 0 and vr = vθ = 0). In addition, negligible diffusion in circumferential and axial directions can be neglected (i.e., diffusion occurs only in the radial direction) [130]. This assumption  leads to ∂ 2Y

∂Y

∂ 2Y

∂Y

D j r12 ∂ θ 2j = D j ∂ z2j = 0 in the species equation and D j r12 ∂ θj ∂∂ Tθ + ∂ zj ∂∂Tz in the energy equation, respectively. Together with these assumptions, the jth species and energy equations can be simplified as ∂Y j = Dj vz ∂z nS ∂T c p vz = ∑ c p, j D j ∂z j

 

  ∂Y j ω˙ jW j 1 ∂ r + r ∂r ∂r ρ ∂Y j ∂ T ∂r ∂r



˙ jW j Q˙ loss ∑nS j h jω + − ρ ρ

(2.21)

(2.22)

CHAPTER 2. MODEL DESCRIPTION

14

Further, due to the steady velocity profile, z and t are related by t = z/vcl . Therefore, equations 2.21 and 2.22 can be written as partial differential equations with respect to t and r as 



1− 



cp 1 −

2.5

r

2 

r R pipe

2 

R pipe

∂Y j = Dj ∂t

nS ∂T = ∑ c p, j D j ∂t j





  ∂Y j ω˙ jW j 1 ∂ r + r ∂r ∂r ρ

∂Y j ∂ T ∂r ∂r



(2.23)

˙ jW j Q˙ loss ∑nS j h jω + − (2.24) ρ ρ

Plug flow

A multi-dimensional governing equation is obtained for the simplest flow such as the Hagen-Poiseulle flow. To solve these multi-dimensional equations with a detailed reaction mechanism, large computational resource is required. Availability of such computational resource is still restricted, only some limited cases (i.e., small hydrocarbon species) have been studied. Hence, a lower dimensional model referred as Plug flow model is developed for further simplification. The plug flow model simplifies the governing equations into one-dimension equations by neglecting velocity and concentration distribution in the radial direction. By assuming a unique velocity profile (i.e., vz is constant) and negligible diffusion in the radial direction, equations 2.21 and 2.22 can be written as vz

∂Y j ω˙ jW j = ∂z ρ

˙ jW j Q˙ loss ∑nS ∂T j h jω = − c p vz ∂z ρ ρ

(2.25)

(2.26)

Furthermore, due to the steady and unique velocity profile, z and t can be related as t = z/vz . Therefore, equations 2.25 and 2.26 can be written as partial differentiation equations with respect to t and r as ∂Y j ω˙ jW j = ∂t ρ cp

˙ jW j Q˙ loss ∑nS ∂T j h jω = − ∂t ρ ρ

(2.27)

(2.28)

2.6. MECHANISM REDUCTION

15

In this study, the plug flow model is used and the influence of diffusion in detail is discussed in the appendix.

2.6

Mechanism reduction

Chemical mechanisms from Ranzi’s group (Ranzi mechanism, Milan, Italy)[82] and Lawrence Livermore National Laboratory (LLNL mechanism, CA, USA) were used as the reference reaction mechanisms. Ranzi’s group concentrates on the prediction of the oxidation processes by the use of one detailed reaction mechanism for various fuels. The detail reaction mechanism consists of large number of species and reactions. The number of species and reactions are then reduced further using the lumping method [84]. LLNL’s group has investigated various fuels by the use of individual detailed reaction mechanisms. Detailed reaction mechanisms from both group involves large number of species and reactions. For instance, to predict the oxidation process of nC7 H16 , 341 and 561 species has to be considered using Ranzi and LLNL mechanisms, respectively. Although these mechanisms accurately predicts the oxidation process and have been validated under a wide temperature and pressure ranges, these number of species and reaction hinders the deep understanding of oxidation processes. To overcome this, a number of reduction methods have been proposed [106, 88, 131, 132, 108, 133, 115]. Although these methods can be applied for Ranzi and LLNL mechanisms, insignificant species can be removed from the mechanism in advance by specifying the tested conditions (i.e., lean mixture at low temperatures). There are two criteria for a reduction process. First, the calculated peak temperature (Tpeak ) has to remain constant. A lower or higher Tpeak caused by the reduction process indicates a significant change of reaction pathways. Second, the ignition delay time (i.e., τ1st at low temperatures) has to remain constant. Reductions consist of step-by-step processes including the above mentioned restrictions [66] and they include: 1) Carbon atom screening (CAS), 2) Production rate screening (PRS), and 3) Rate-of-progress screening (RPS). These screening methods focus on the importance of the species and the reactions one after the other by using a specific threshold value.

2.6.1

Carbon Atom Screening (CAS)

CAS is a reduction process to remove insignificant hydrocarbon (or oxygenated) species from a reaction mechanism. A reaction mechanism consists of a hierar-

CHAPTER 2. MODEL DESCRIPTION

16

chical structure based on the number of carbon atoms within the species [88]. The highest class of the hierarchy describes H2 oxidation and has to be considered for all combustion phenomena. On the other hand, lower classes which describe the other reactions with CO, methanol (CH3 OH), or heavier hydrocarbons (or oxygenated) species, have to be considered, depending on the tested fuel. These classes can be removed from the reaction mechanism if it has no influence on the entire combustion process. In this study, this screening removes species which belongs to the lower classes with more than two carbon atoms in their molecular structure.

2.6.2

Production Rate Screening (PRS) and Rate-of-Progress Screening (RPS)

Further screenings also consider the same restrictions (i.e., Tpeak and τ1st ). PRS and RPS are the screening processes to remove negligible species and elementary reactions from the reaction mechanism, respectively. For these processes, the production rates are regarded. The production rate in equations 2.24, 2.26, 2.28, 2.22, and 2.28 can be written as  ω˙ j =

dC j dt

 chem

nR

= ∑ ν j,i i

nS ν k f ,i C j j,i − kr,i j j nS

∏

  = ∑ ν j,i q˙i f − q˙ri

∏

ν  C j j,i

 (2.29) (2.30)

i

= ω˙ cj − ω˙ dj

(2.31)

Here, ω˙ c , ω˙ d , q˙ f , and q˙r denote creation rate, destruction rate, forward and reverse rate-of-progress, respectively. The net production rate of the jth species corresponds to equation 2.31 (i.e., ω˙ nj = ω˙ j ), and the net rate-of-progress of the ith reaction (q˙ni ) is expressed as q˙ni ≡ q˙i f − q˙ri . For the PRS, equation 2.31 is used to evaluate the important species. Species with extremely smaller ω˙ c and ω˙ d than others are removed from the mechanism. By connecting the remaining important species, the process specifies a reaction network. This network identifies the reaction pathway (i.e., oxidation process) from the fuel to the product which are to be studied. Note that the number of species should be constrained after this screening. For the RPS, equation 2.30 is used to examine the importance of the reactions at the individual reaction pathway. Negligible reactions are removed from the mechanism with the same concept as in the PRS. During these

2.7. DEFINITION OF THE FIRST-STAGE IGNITION DELAY TIME

17

screenings, Tpeak and τ1st are monitored to sustain the consistency from the original mechanism. After these screenings, a skeletal mechanism is established (see details in section 5.4).

2.7

Definition of the first-stage ignition delay time

Figure 3.5 shows an experimentally determined normalized temperature profile (δ Tˆ ) throughout the reactor with φ = 0.8 (based on nC7 H16 as fuel). V˙mix = 63 ×10−3 m3 /min at 620 K. δ T denotes the temperature difference between the reference temperature (Tre f ) and the measured temperature (see details in section 3.6). Note that this maximum value of δ T may not demonstrate the temperature difference between the peak temperature (i.e., equilibrium temperature) and the oven temperature. Due to the specified distance between the thermocouples (see chapter 3), it is not possible to measure the temperature changes in between them. Hence, the temperature peaks occurring in the region between the thermocouple locations are not distinguished. In addition, the temperature distribution of Tre f may cause difficulty to detect the temperature increase caused by the FSI (see chapter 3). To measure the low-temperature ignition delay time (i.e., first-stage ignition delay time, τ1st ), the temperature increase caused by the low-temperature chemistry (LTC) should be measured. Therefore, in this study, normalized value of δ T by difference between the minimum and maximum values of δ T are used and are denoted by δ Tˆ . This temperature profile shown in figure 2.1 clearly demonstrates a temperature peak caused by low temperature ignition in the middle of the LFR. This peak is caused by the first-stage ignition (FSI, see the discussion in chapter 5). In order to compare this with the numerical simulations, τ1st must be obtained using the temperature profile. By applying Lagrangian concept to the mixture in the LFR, τ1st can be obtained by using equation 2.32 given below.

Lign τ1st = 0

1 dz vz

(2.32)

Here Lign denotes the length to reach the first δ Tˆ = 0.7 from the inlet (i.e., δ Tˆ = 0 at L = 0 corresponding FI location). In the shown figure, Lign was found to be at 2.3 m. In this concept the control volume is transported by the flow, and chemical reaction occurs within the control volume (i.e, Plug flow model). Therefore, τ1st can be determined by comparing the time with a 0-D flame calculation and the

CHAPTER 2. MODEL DESCRIPTION

18

time computed from Lign . In order to experimentally determine τ1st , the density variation is neglected because the variation due to temperature increase is relatively small2 . However, τ1st strongly depends on the velocity profile. By assuming the velocity profile of laminar flow, τ1st can be written as τ1st, laminar =

Lign ALFR Lign ρ = 2 vcross 2 m˙

(2.33)

Here, ALFR denotes cross-sectional area of LFR. From the experiments, τ1st cannot be measured exactly due to the restriction of the experimental apparatus. In order to overcome this hurdle, CFD calculations are used to investigate the influence of Re on FSI (see detail in chapter 5). By doing this, it is concluded that τ1st,laminar is the proper value for the experiments, and is determined experimentally τ1st (see detail in chapter 5).

1

δ Tˆ [−]

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

L [m]

Figure 2.1: Normalized temperature profile with φ = 0.8, V˙mix = 63 ×10−3 m3 /min, and ξ = 10% at 620 K.

2

Temperature increase caused by FSI is below 300 K.

2.8. CALCULATION FLOW

2.8

19

Calculation flow

Figure 2.2 shows the calculation flow with Cantera for this study. A subroutine of Cantera in Matlab was used for solving the governing equations [134]. Information on gas-phase chemistry and thermodynamic data are combined using Cantera. The governing equations with the combined data are solved using an ordinary differential equation (ODE) solver (ode-15s) in Matlab with input parameters such as the composition of gas, T ∗ , and the calculation time. In this solver, the governing equations are considered as a system that consists of (nS + 1)th ODEs (all species and energy equations). The calculation is carried out using adaptive time stepping with an absolute and a relative tolerance of 1×10−20 and 1×10−8 . In order to store the Jacobian matrix at every time step, the solver is modified [66]. For theoretical analysis (see details in chapters 5 and 6), the same solver is used with the same tolerances. The mechanism screening is conducted until a skeletal mechanism is established.

Gas-phase chemistry

Thermodynamic Data Cantera Input parameters

Mechanism screening

Matlab ode15s

Postprocess

Figure 2.2: Calculation flow with Cantera.

Chapter 3 Experimental apparatus Auto-ignition has been studied using shock tubes (ST) [38, 135, 49, 136, 137, 138, 39] and rapid compression machines (RCM) [139, 140, 44] to investigate the ignition delay time. They measure the second-stage ignition delay time (τ2nd ) of an overall ignition process (second-stage ignition (SSI)). SSI sequentially occurs within a negative temperature coefficient (NTC) regime after the first-stage ignition (FSI). Hence, the first-stage ignition delay time (τ1st ) has a significant influence on ignition characteristics of fuels involving active chemical reactions within the NTC regime. In order to develop a chemical kinetic mechanism for future fuels like biodiesel, better insight of the FSI is required. This section describes the laminar flow reactor (LFR), which is developed to decouple the FSI from the overall ignition.

3.1

Laminar flow reactor

An overview of the LFR is shown in figure 3.1. The LFR consists of a number of flanged segments as shown in the figure 3.2a and 3.2b. All the segments are made of stainless steel1 and can withstand pressures up to 5 atm. The inner diameter and the wall thickness of the segments are 20 mm and 2.5 mm, respectively. For ensuring the flexibility in the configuration of the LFR, a number of segments with a wide variation in length are used. The minimum and maximum lengths are 50 to 300 mm, respectively. At each connection, a gasket made of graphite is used. In this study, the total length is set to 5.8 m (LLFR ). The oxidizer flows into the LFR from the open end positioned at the bottom. The gentle curved section does 1 DIN-1.4301,

SUS304 equivalent

21

CHAPTER 3. EXPERIMENTAL APPARATUS

22

not create a recirculation zone, and it was confirmed by CFD calculations. Under typical test conditions of this study, Re of the LFR is approximately 1250 at 600 K with air2 . This Re proves that the size of the reactor in conjunction with the overall mass flow results in a lower Re value and hence ensuring laminar flows at all times.

Figure 3.1: Overview of the LFR.

Figure 3.3a shows the fuel injector (FI), and figure 3.3b shows its schematic (its technical drawing can be found in the appendix). This FI is designed to rapidly form a homogeneous mixture. It is a 60 mm long two component assembly (LFI ) located 170 mm downstream of the oxidizer inlet. The downstream component with 12 circumferentially-arranged holes (1.5 mm diameter) is assembled upstream with another component comprising of three mixing-enhancing rings (the difference between inner and outer diameters is 2.5 mm and the interval between adjacent rings is 7.5 mm). The gaseous fuel3 flows in a cavity inside the FI and L for air in the LFR can be determined as follows. Re = ρvcross = 192 vcross . vcross under the μ typical tested conditions is 6.5 m/s at 600 K. 3 The gasification of the liquid fuel is described in section 3.2. 2 Re

3.1. LAMINAR FLOW REACTOR

23

(a) Flanged joints.

(b) Segment.

Figure 3.2: Components of the LFR. flows out through the small holes. The fuel mixes with the oxidizer inside the LFR. In order to conduct experiments with a premixed mixture at close to its auto-ignition temperature, the mixing of fuel and oxidizer has to occur in a very short time. Mixing-enhancing rings form a local turbulent flow field around the FI and exert a strong shear force on it. This local turbulent flow field enhances the mixing and prevents the flame attachment on the FI. The fuel immediately starts to react with the oxidizer, and a partially premixed flame may be formed over the holes [141, 142] without the local turbulent flow field (this is also experimentally investigated with diffusion flame [143, 144]). On the other hand, due to the presence of only molecular diffusion in the laminar flow, moderate mixing leads to large spatial distribution in the mixture (i.e., fuel distribution at re-laminarized regime hardly decreases). It may lead to random auto-ignition although a flame does not form over the holes. This mixing time (i.e., the time required to moderate the distribution of fuel) has strong influence on the ignition delay time [145]. A

CHAPTER 3. EXPERIMENTAL APPARATUS

24

detailed analysis is performed using CFD calculations and is discussed in section 4 and in the appendix.

20

Oxidizer

(a) Photo.

Fuel (b) Schematic

Figure 3.3: Fuel injector. In order to measure the gas temperature in the LFR, the thermocouple-integrated flanges and segments are used as shown in figure 3.4. 27 k-type thermocouples with the sheath diameter of 1 mm are used. The thermocouples are placed at non-equidistant interval ranging between 60 to 320 mm because of the flanged joints (see figure 3.5). They are inserted into the LFR such that about 1 mm of their tips is located inside the LFR. This inserted length is selected to compensate for the opposing effects, absolute temperature measurements and bluff-body effects for the flow field. Previously, the gas temperatures were measured using a thermocouple with 20 mm inserted length along the upstream on the center axis. However, the thermocouple acts as a bluff-body, and a flame is stabilized at the thermocouple even when the experimental conditions are significantly changed. The inserted thermocouples are strongly affected by the boundary layer: Around the boundary layer, the thermocouples exert a strong shear force and damping influence on the flow field. In addition, this length is close to the quenching distance from the walls, as learnt from previous experimental findings [146]. Although the

3.2. GAS SUPPLY SYSTEM

25

short inserted thermocouple does not show an absolute gas temperature, the temperature increase caused by the heat release from the chemical reactions is still detectable. By monitoring the temperature increase throughout the LFR, it is possible to detect the location of the auto-ignition (see details in section 2.7).

(a) Flange.

(b) Segment.

Figure 3.4: Thermocouple-integrated flange and flanged segment. The entire flow reactor is located in the oven and is isothermally heated up to a preset temperature. The oven consists of two half-cylindrical heaters (Kantahl: Fibrotah HAS, 9 kW). In order to avoid buoyancy effect in the oven, an additional air tube is used to form a tumbling flow field inside it. Figure 3.5 shows the time-averaged temperature profile throughout the LFR. The temperature profile obtained from the time-averaged temperature shows small temperature distribution throughout the LFR, which has a distribution of ±5% from set temperature. This ensures that the entire reactor is isothermally maintained at a set temperature.

3.2

Gas supply system

A schematic diagram of the LFR with its gas supply system is shown in figure 3.7. N-heptane (nC7 H16 ) and dimethyl ether (DME) are used as fuels, and air is used as oxidizer. Nitrogen (N2 ) is used both as a carrier gas of nC7 H16 and as a dilution gas. Liquid nC7 H16 is measured by a liquid mass flow controller (LMFC, Brooks

CHAPTER 3. EXPERIMENTAL APPARATUS

26 800

Tempearture [K]

700

Set temperature: 560K

600

500

400

300

Fuel injector 0

1

2

3

4

5

L [m]

Figure 3.5: Temperature profile. instruments: FLOMEGA). Air, N2 , and DME are measured by gas mass flow controllers (MFCs, Brooks instruments: TR5850 and SLA5851 series). These LMFC and MFC are controlled by the use of a control unit (Brooks: Control unit 0154). In order to improve the accuracy of the experiments, all the MFCs are calibrated by a gas flow calibrator (Bios: Definer 220). Expected errors from the MFCs are ±1%. The fuel vaporizer (FV) is used to continuously supply the gaseous fuel to the LFR. Figures 3.6a and 3.6b show the FV. The FV is located in an isolated container, which is kept at a significantly higher temperature than the boiling temperature of the liquid fuel (e.g., temperature set to 473 K for nC7 H16 ) using a heating tape (Tyco: IT-H, 1.09 kW). The measured liquid nC7 H16 enters the FV, and a mixture of vaporized nC7 H16 and carrier gas flows out of it. A liquid-supply pipe is connected to the FV, where its temperature is maintained far below the boiling temperature of the liquid fuel, and the fuel is supplied as droplets at the inlet of the FV. Due to the set temperature (i.e., 473 K), supplied liquid fuel is completely evaporated in the FV. Therefore, the droplets can be continuously supplied, and the fuel concentration in the outgoing mixture can be maintained without fluctuation. In addition, to avoid local condensation of the fuel, the carrier gas forms a tumbling flow field in the FV (see figure 3.6b). This flow field increases the residence time of the mixture in the FV and assist evaporization. Previously, experiments were carried out without using the FV (i.e., liquid fuel was directly supplied to the FI). As a result, large fluctuation in the flow rate of the supplied liquid fuel was observed. This was caused by rapid evaporation of the liquid fuel. The supplied fuel came into contact with the hot gas (i.e., liquid fuel and gas at significantly

3.2. GAS SUPPLY SYSTEM

27

higher temperature than the boiling temperature) and was evaporated. This evaporation continued until the entire heat was transferred from the gas to the liquid and the heat of vaporization of the liquid fuel was balanced. After the evaporation, additional liquid fuel was supplied to replace the evaporated fuel. This evaporation and subsequent supply of the liquid fuel occurs periodically in a short time. On the other hand, the response time of the LMFC was relatively long. Therefore, the fluctuation caused by the periodic evaporation could not be compensated and this led to a large uncertainty of the fuel concentration in the unburned mixture.

nC7H16 (Liquid)

Carrier gas (N2)

Gaseous fuel

(a) FV in the isolated container.

(b) Schematic of the FV.

Figure 3.6: Fuel vaporizer. The measured air and dilute gas is heated up to the preset temperature of the oven, depending on the set temperature and the flow rate of the oxidizer using an air heater (Omega: AHPF-122, 1.2 kW) or a homemade heater having a heating element (Thermocoax: Inconel 600, 2 kW). In order to minimize the temperature fluctuations in the heated oxidizer caused by switching the air heater, a heating tape (Tyco: IT-H, 1.09 kW) is used between the air heater and the oven over a length of 1500 mm. The same type of heating tape is used to avoid the condensation of nC7 H16 between the FV and the oven and is kept at the same temperature as the FV. Before the fuel enters the FI, it is heated up to the same temperature as that of the oven. A portion of the fuel supplying pipe (1.5 m out of 4 m in length) connected to the FI is located close to the heating element of the oven. In this portion, gaseous fuel is heated to the same temperature as the oven (see detail in the appendix). The oxidizer, the fuel, and the oven (i.e., the entire LFR) are maintained at the same temperature, and all experiments are carried out at isothermal conditions. In order to blend the DME into the nC7 H16 /air mixture, a DME

CHAPTER 3. EXPERIMENTAL APPARATUS

28

supply line along with N2 supply pipe is connected to the FV. DME does not interrupt the vaporization of nC7 H16 because it is gaseous under atmospheric pressure and the temperature of the FV is sufficiently above the boiling temperature of the nC7 H16 . The measured DME is mixed with N2 and the mixture flows into the FV. A mixture of the DME, nC7 H16 , and N2 flows into the LFR through the FI.

MFC: LMFC: FV: FI: GS: H:

Mass flow controller Liquid mass flow controller Fuel vaporizer Fuel injector Gas sampling Heaters

DME

MFC

nC7H16

LMFC

N2 Air

MFC

FV

MFC MFC

GS Oven

H1

H2, H3

FI

Figure 3.7: Schematic diagram of the LFR.

3.3

Power supply system

For the experiments, approximately a power of 22 kW is supplied, and a three phase electric system is used. Figure 3.8a shows the schematic of the power supply diagram. In this figure, the same nomenclature as in figure 3.7 is used. Five power controllers (PMA: ECO24) are used in this system. These controllers supply power to the heating devices with an on-off control. To minimize the temperature fluctuation during the experiments, threshold temperatures (low- and high-limit values) and the duration of the power added are optimized for each heating device. As a result, the fluctuation of the oven temperature is kept negligibly small (temperature amplitude of ±2 K and frequency of 5 × 10−4 Hz). A

3.3. POWER SUPPLY SYSTEM

29

controller is installed in the main control box as shown in figure 3.8b. The others are installed in the four individual boxes also shown in figure 3.8c. The oven is regulated by a controller in the main box with a voltage of 400 V due to its high power consumption. The heating tapes and the air heater are controlled by the rest of the controllers with a voltage of 230 V. The temperature is monitored using one thermocouple for each device. An additional thermocouple is used as security for the oven.

Fuel

H1

Oven

FV H2 H3

Oxidizer

CB1 CB2 CB3 CB4 CB: MCB: : :

MCB

Control box Main control box Power Power + TC

(a) Schematic diagram of the LFR

(b) Main control box

(c) Control box

CHAPTER 3. EXPERIMENTAL APPARATUS

30

3.4

Data acquisition system

In order to monitor the temperature throughout the LFR, 27 k-type thermocouples are used with a data logger system. Figure 3.8 shows the data acquisition system. Analog signals from the thermocouples are converted to digital and are corrected by the input modules (National instruments: TC-2209, SCXI-1102, SCXI-1000, SCXI-1349, SHC58-68, and SH96-96). The corrected signals are stored by an acquisition board (National instruments: PCI-6259) with a data acquisition program, VI-logger. The entire measuring systems is regulated by using the VI-logger with input parameters such as the number of thermocouples and the sampling rate. In this study, only steady conditions are considered, and the sampling rate is set to 2 Hz. The temperature data is recorded individually starting from thermocouple one at short time intervals. However, the time difference between two thermocouples is approximately 80 μs4 , and it is significantly smaller than the sampling rate. Therefore, in this study, temperature measurement is considered to be done simultaneously using 27 thermocouples.

Chassis PC PCI

SC1

IM

TCC SC2

TC

PCI: PCI-data acquasition board (PCI-6259) Chassis: SCXI chassis (SCXI-1000 & -1349) IM: Input module (SCXI-1102) TCC: Thermocouple connector (TC-2095) TC: Thermocouples (k-type) SC1: Shielded cable (SHC68-68) SC2: Shielded cable (SH96-96) Figure 3.8: Schematic of the data acquisition system.

4 The

maximum sampling rate of the system is 333 kHz due to the SCXI chassis.

3.5. GAS SAMPLING SYSTEM

3.5

31

Gas sampling system

The mixture is sampled at the end of the reactor (i.e. 5.7 m on the downstream side of the FI). Figure 3.5 shows a schematic of the gas sampling system. The sampling system consists of a gasmaus (gas collection tube), a sampling cylinder, and a vacuum pump. In order to avoid the residue from the previous experiment, the system is purged. The vacuum pump is then used to create a negative pressure in the system. By opening the valve (i.e., BV1) which is connected to the LFR, burned gas flows into the system. This continues until the pressure in the system returns to the normal atmospheric value (approximately 3 min). Sampling is performed when the temperature profile reaches a steady condition; the sampling on the FSI has no influence. In order to measure oxygen (O2 ), carbon monoxide (CO), and carbon dioxide (CO2 ), a gas analyzer is used (GA, Testo: 350 XL) with a sampling rate of 1 Hz. In the GA, O2 and CO are measured from an electromotive force using a galvanic cell and controlled-potential electrolysis, respectively. CO2 is measured as absorption of infrared radiation (4.27 μm). For the measurement of the other species, a gas chromatography system (GC, Agilent: 6890N) is used. The GC has three columns: HP-PLOT Q (30 m), HP-PLOT Al2 O3 (48 m), and HP-PlOT MolesSieve (5 m). The detectors used are a flame ionization detector (FID) and a thermal conductivity detector (TCD). Helium (He) and N2 gases are used as the carrier gases for the FID and the TCD, respectively. By adjusting the retention time (i.e., residence time of the sampling gas in each column), species are separated and their concentrations can be measured. The retention time is specified for the columns, and conversion factors are determined using a certificated reference gas. Sample gas for the GC is taken from the gasmaus and is manually injected to GC using a syringe.

3.6

Experimental parameters and methods

The experimental parameters are mass flow rate (m), ˙ equivalence ratio (φ ) of the mixtures, and temperatures. The m˙ is related to the convection speed of mixture (i.e., transport speed of the control volume in the LFR) and consists of m˙ of air, nC7 H16 , DME, and N2 . In order to investigate the influence of the convection speed on FSI, the volumetric flow rate of air and N2 (V˙mix ) and φ are specified. The V˙mix is determined as the sum of the volumetric flows of air and N2 (i.e., dilution and carrier gas). The ratio between the air and the carrier gas is fixed to 0.1. The m˙ of nC7 H16 and DME are determined based on the YO2 in the oxidizer

CHAPTER 3. EXPERIMENTAL APPARATUS

32

BV2

BV3

GC

P

BV1 LFR

GM

SC

BV: Ball valve GA: Gas analyzer (350 XL) GC: Gas chromatography (6890N) GM: Gasmaus LFR: Laminar flow reactor SC: Sample cylinder (2L) VC: Vacuum pump

VP BV4

GA

Figure 3.9: Schematic of the gas sampling system. and specified φ . Volumetric flow rate of nC7 H16 (in liquid) and DME (gaseous) are converted to mass flow rates using their densities. Note that the carrier gas is not used in the experiments with DME because it is a gaseous fuel at atmospheric pressure. In this case, the missing N2 which is not added to the DME as the carrier gas, is added to the oxidizer in order to keep same mass flow of the mixture. With blended fuels, the specified ratio of its mass is replaced by DME (see chapter 6). The dilution of the oxidizer using N2 (in volume percent) is denoted by ξ and the the volume is considered within the V˙mix . The temperatures of the oxidizer, fuel, and oven (T ∗ ) are set to be equal. The experimental procedures are as follows: Firstly, all gases (i.e., air, dilution gas, and carrier gas) flow into the reactor. The FV is heated and maintained at 473 K. The air heater and the oven are heated to a set value. When the fluctuation in the temperatures becomes less than ±1%, the temperatures are recorded as Tre f . The time series of the temperatures throughout the LFR is used to obtain the time-averaged temperature profile without fuel injection (see figure 3.5). Then the fuel (liquid nC7 H16 or gaseous DME) is injected into the FV. In order to avoid random auto-ignition caused by the fluctuation in φ , the flow rate of the liquid fuel is increased gradually (i.e., the time required to attain the set value is about 30 min). In the case with blended fuels, after the volumetric flow rate of nC7 H16 reaches the required value, DME is gradually added in the same manner as nC7 H16 to the fuel supplying pipe. In this process, the temperature profile is monitored to exclude randomness. When the temperatures attain a steady state, a time-averaged temperature profile throughout the LFR is obtained (see details in chapter 5). The profile of temperature difference (δ T ) throughout the LFR is obtained by subtracting corresponding Tre f from the measured temperature at each

3.6. EXPERIMENTAL PARAMETERS AND METHODS

33

location. As mentioned, the measured temperatures do not show the absolute gas temperatures and δ T shows small difference from the set temperature. However, by subtracting the Tre f from the temperature compensates for the error caused by the individual thermocouples. Further, the normalized temperature (δ Tˆ ) profile is obtained by the use of minimum and maximum values of temperature profile. Through this profile of δ Tˆ the effect of the fuel injection and the influence of the temperature increase caused by FSI can be observed. Lign is determined from the δ Tˆ profile (see detail in section 2.7). τ1st is obtained by the use of the residence time of the mixture in the LFR and the Lign . During the recording of the temperature at steady state, exhaust gas is sampled three to five times at the end of the LFR and analyzed using the GA and the GC. By purging the gas sampling system (see section 3.5), no influence of the previous sample gas was observed. Experiments with the same parameters (i.e., m, ˙ φ , and T ∗ ) are conducted three to five times to ensure their repeatability and the associated hysteresis caused by increasing and decreasing T ∗ . There are variations in the value of temperature distribution throughout the LFR, the Lign , and the flow rate measurements. Maximum allowable errors are 4% in the temperature distribution, 2% in the Lign , and 1% in the flow rate measurements. So, a total maximum experimental error of 7% is expected in this study.

Chapter 4 Evaluation of the laminar flow reactor The mixing field downstream of the fuel injector (FI), formed by separately supplying the oxidizer and the fuel injection through the FI, plays a major role in the investigation of the laminar flow reactor (LFR). The mixtures formed by the FI should be highly homogeneous throughout the cross-section. The residence time of a homogeneous mixture in the LFR is regarded as the first-stage ignition delay time (τ1st ). To accurately obtain and theoretically model τ1st , rapid mixing of the oxidizer and the fuel should be ensured. This chapter deals with a detailed discussion of the flow field and the mixing field around the FI using 3-D calculations from Reynolds averaged Navier Stokes (RANS) models.

4.1

Experimental visualization

In the laminar flow of the premixed gas, a steady premixed flame is formed along the velocity profile of the unburned mixture. The flame is formed in the region where its laminar burning velocity is balanced with the flow velocity normal to the flame surface of the unburned mixture. An example of this is a wrinkling flame formed on a slot burner [147]. It was shown that the wrinkling flame shape (i.e., the location of the flame) with the same velocity profile has a significant dependency on an equivalence ratio (φ ). Hence, the premixed flame formed may demonstrate the flow field and homogeneity of the unburned mixture. By closely observing the influence of the cross-sectional averaged velocity (i.e., bulk velocity, vcross ) and φ on the premixed flame, qualitative investigations of the flow field and mixing field can be conducted. Although a different fuel and temperature is 35

36

CHAPTER 4. EVALUATION OF THE LAMINAR FLOW REACTOR

used for the investigation, qualitative results can be obtained by considering the μ Reynolds number (Re) and the Schmidt number (Sc ≡ ρD ) of the mixtures.CH4 /air mixtures are used at atmospheric temperature and pressure in the measurements. Figures 4.1a and 4.1b compare the flames formed downstream of the FI. These pictures were taken at an angle slightly above the end of the pipe. The tip of the FI is located 10 mm upstream from the end of the pipe as shown in the figures. For these observations, a digital camera (Canon: Power Shot A70) was used with the following settings: exposure time of 0.3 s, F-number of 8, and International Standard Organization (ISO) sensitivity of 200. Figure 4.1a shows the flame variation for 1 < φ < 1.5 at a constant vcross = 1.0 m/s, and figure 4.1b shows the flame variations for 0.7 m/s < vcross < 1.0 m/s at a constant φ = 1.2. The corresponding values of Re are 881, 1133, and 1260 for vcross = 0.7, 0.9, and 1.0, respectively. Figure 4.1a shows that the flames formed behave similarly to a Bunsen flame. As φ is increased from its stoichiometric ratio, the inner flame elongates and merges with the outer flame close to its flammability limit of φ = 1.5. In contrast, the inner flames shown in figure 4.1b demonstrate slight variations when vcross is varied with a constant φ . By increasing vcross , the inner flame is linearly elongated. These qualitative results show that the FI rapidly forms a premixed mixture at the downstream point. In addition, these results clearly show that there is a structural flow field formed by the FI. In the following sections, the mixing field and the structural flow field will be analyzed using numerical simulations.

4.2

Influence of the fuel injector

Around the FI, chemical reactions do not play an important role due to the induction period of the fuel oxidation. In this period, the characteristic time of the chemistry occurring is significantly longer than that of the fluid. In the experiments, this time difference is observed. The first-stage ignitions (FSI) are always observed away from the FI (i.e., the distance at which FSI occurs is significantly longer than the length of the FI, see details in chapter 5). Therefore, the heat release from the chemical reaction does not need to be considered. Simulations under isothermal conditions with frozen chemistry can predict the flow field and the mixing field around the FI. Since the flow around the FI is expected to be turbulent, the kt − εt model with low Re and re-laminarization is used for the RANS calculations in StarCD [148], and Paraview is used for visualization of the simulated results [149]. Figure 4.2 shows a computer-aided design (CAD) model of the FI (length LFI = 60 mm). A hybrid or tetrahedral mesh is generated from 20 mm upstream of the FI tip to 130 mm downstream of the other tip (see details of the

4.2. INFLUENCE OF THE FUEL INJECTOR

37

vcross = 1.0 m/s

20 mm F = 1.0

F = 1.2

F = 1.5

(a) Influence of φ on flame structure at vcross = 1.0 m/s F = 1.2

20 mm vcross = 0.7 m/s

vcross = 0.9 m/s

vcross = 1.0 m/s

(b) Influence of vcross on flame structure with φ = 1.2

Figure 4.1: Direct flame observations at the end of the pipe. FI is located 10 mm upstream of the end. All images are taken with an exposure time of 0.3 s, F-number of 8, and ISO sensitivity of 200.

38

CHAPTER 4. EVALUATION OF THE LAMINAR FLOW REACTOR

r

z

Figure 4.2: CAD model of the FI. technical drawing and the mesh in the appendix). As mentioned in section 2.7, the inlet of the fuel supply line is considered as the origin and a cylindrical coordinate system is used. Previously, the boundary condition at the 12 circumferentiallyarranged holes were studied, and the pressure distribution over the holes and their outflows was investigated. It was found that the 12 holes worked perfectly as a diffuser, and the inflow of air into the cavity in the FI flows equally out to the LFR from the 12 holes. Therefore, an inlet set at a direction normal to the holes is used for this study. Note that only the time-averaged value is considered, because the results are obtained using a RANS model. The scalar repetition of the time-averaged value is considered in the appendix (not described in this chapter). Two injection velocities (vin j = 0.1 and 1.0 m/s) are tested with vcross = 10 m/s to study the influence of the injected flow from the FI. For simplification, air is used at ambient temperature as the fluid. As mentioned above, qualitative results can be obtained even at different temperatures. Figure 4.3 shows the pressure distribution for the entire calculation domain. It can be observed that the upstream pressure is relatively high due to the presence of the FI in the tube. However, the difference in pressure upstream and downstream of the FI is negligibly small compared to the ambient pressure. In addition, there is no pressure distribution downstream of the FI. A slightly higher pressure is observed with vin j = 1.0 m/s, but the pressure is significantly lower than the ambient and is still negligible. Figure 4.4 shows velocity vectors of the vertical and the horizontal cross-sections of the flow (cf. corresponding plane shown in figure 4.3) downstream of the FI. The upper two planes of figure 4.4 show results at vin j = 0.1 m/s. In plane 1 (vertical cross-section), a non-symmetric recirculation zone is observed. In contrast, plane 2 (horizontal cross-section) displays a symmetric recirculation zone. These results show that two shorter and one longer recirculation zones are formed downstream of the FI. This is caused by the differences in the upstream leg thickness (see figure4.2 and the technical drawing in the appendix). One leg has a thickness

4.2. INFLUENCE OF THE FUEL INJECTOR Pressure difference from the ambient [Pa]

/s

m 10

900 795 690 585 480 375

39

e1

n Pla

vinj

270 165 60 -45 -15

2 ane

Pl

60

mm

Diameter 20 mm

Figure 4.3: Influence of the injected velocity on the pressure distribution around the FI. of 4 mm instead of 2 mm as it encloses a fuel supply pipe. Plane 1 refers to the cross-section across the thicker leg. This leg separates the flow more widely than the other two legs, and decreases the local velocity magnitude downstream of it. This supports the findings shown in figures 4.1a and 4.1b. Figure 4.5 shows the axial velocity profile along the centerline with vcross = 10 m/s and vin j = 10 m/s. The horizontal axis is the distance from the FI tip normalized by LFI (LFI = 60 mm). As the distance increases, the velocity increases from negative to positive values, and reaching zero at a normalized distance of approximately 0.25 (i.e., corresponding distance from the FI is 15 mm). This distance characterizes the length of the recirculation zone and is denoted as Lrz . Figure 4.6 shows the influence on Lrz of varying vin j from 0.1 to 10 m/s with a fixed value of vcross = 10 m/s. This result supports the above explanation of the weak influence of vin j on the flow field with the constant Lrz /LFI under the tested conditions. Hence, vcross has a strong influence on the downstream flow field, whereas the injected flow has a weak influence on it. To observe the influence of the mixing-enhancing rings, a downstream flow field (i.e., velocity profile at a cross-section) is studied. Figure 4.7 shows the velocity magnitude distribution at the same cross-section as plane 2 in figure 4.3 and the axial velocity profiles at different locations with vcross = 1.0 m/s. The intervals

40

CHAPTER 4. EVALUATION OF THE LAMINAR FLOW REACTOR

Plane 1

Plane 2 Velocity [m/s] 35 31.5 28 24.5 21 17.5 14 10.5 7 3.5 0

Top row: vinj = 0.1 m/s Bottom row: vinj = 10 m/s

vcross = 10 m/s

Figure 4.4: Influence of the injected velocity on the flow structure downstream of the FI. Planes 1 and 2 are shown in figure 4.3.

15

vcl [m/s]

10

5

0

−5

−10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized distance from the FI tip [−]

Figure 4.5: Velocity profile downstream of the FI.

4.3. MIXING FIELD

41

0.5

Lrz / LFI [−]

0.4

0.3

0.2

0.1

0

−1

0

10

10

v

inj

1

10

[m/s]

Figure 4.6: Influence of vin j on Lrz with vcross = 10 m/s. between the observation locations are 30 mm (i.e., LFI /2). To reduce the computational load, vin j is not considered in this simulation. Figure 4.8 shows the influence of vcross on Lrz by varying vcross from 1.0 to 10 m/s. The corresponding Re varies between 1260 and 12600. Although the velocity magnitude at the FI tip is proportional to vcross (i.e, the axial velocity of the permitting gas between the walls and the mixing-enhancing ring), the same length of the recirculation zone is formed downstream of the FI. In other words, although the local velocity magnitude of the permitting gas differs due to variations in vcross , the downstream flow field is almost identical. This result supports the experimental investigations shown in figure 4.1b. In these experiments, the slight influence of vcross on Lrz can be seen.

4.3

Mixing field

If one assumes that the Sc is smaller than unity, then it can be expected that the fuel distribution is homogenized before re-laminarization of the flow field. In that case, the concentration boundary layer becomes thicker than the velocity boundary layer, and the concentration boundary layer develops fully within a shorter length in the pipe than that of the velocity. In general, the value of Sc with a lean

CHAPTER 4. EVALUATION OF THE LAMINAR FLOW REACTOR

2

3

0

3.5

vcross = 1 m/s

0.9

0

−1

1

2

Normalized radius [-]

Velocity magnitude [m/s]

1

3

1

0.5

1.5

1

0.5

1.5

0.5

1

1.5

1

0.5

1.5

0

42

Axial velocity [m/s]

Figure 4.7: Velocity magnitude and axial velocity profiles at different locations with vcross = 1.0 m/s. Intervals of the velocity profiles are 30 mm (LFI /2).

v

cross

[m/s]

0.794 0.5

7.94

0.3

rz

FI

L / L [−]

0.4

0.2

0.1

0 3 10

4

Re [−]

10

Figure 4.8: Influence of vcross on Lrz .

4.3. MIXING FIELD

43

CH4 /air mixture is lower than unity, but that of a lean nC7 H16 /air mixture is higher than unity, and a lean DME/air mixture has a value of Sc close to unity. Hence, once the homogeneity of the nC7 H16 mixture is ensured, the homogeneity of DME or CH4 mixture can be ensured. Thus, in this section, to observe the mixing field downstream of the FI, the φ distribution is investigated. Note that, in the consideration of the induction period, frozen chemistry is assumed. The test conditions are volumetric flow rates of the oxidizer, consisting of the air and the N2 (denoted by V˙mix ), with values of 16 and 63 l/min (corresponding to vcross = 1.8 and 7.0 m/s, respectively) with dilution (ξ ) of 10%, φ = 0.8, and T ∗ = 620 K. These conditions correspond to those discussed in chapter 5. Figure 4.10 shows a comparison of the mixing fields with nC7 H16 /air/N2 mixtures. The cross-sections considered (CS1 to CS3) are at LFI , 1.5 LFI , and 2 LFI from the FI tip (i.e., corresponding distances from the FI tip are 60, 90, and 120 mm, respectively). The densities of both mixtures (i.e., main and injected flows) is obtained at 620 K. Figure 4.9 shows a comparison of the variance in φ at each cross-section for both of the flow rates shown in figure 4.10. The variance (σφ ) is expressed as 2

∑ (φ − φ ) (4.1) = nC nC where nC denotes the number of cells. The cross-section averaged value is denoted using an operator  and can be written as σφ2

φ  =



φ dS

ALFR

(4.2)

Here, ALFR denotes a cross-section area of LFR (ALFR = 1 × 10−4 π m2 ). These results show that the mixing field is weakly influenced by vcross . Due to the recirculation zone at the FI and local turbulent flow because of gas acceleration between the walls and the mixing-enhancing ring, the injected fuel is rapidly mixed with the oxidizer. As a result, a small distribution can be seen at CS1 in figure 4.10 (i.e., 60 mm downstream of the FI tip). However, the distribution decreases as the distance from the FI tip increases (CS2 and CS3, corresponding 90 and 120 mm from the FI tip). Therefore, no influence of vcross on the mixing field can be assumed under these conditions. A theoretical analysis of the mixing can be found in the appendix. The experimentally observed structural flow fields formed by the FI are numerically verified using CFD. Although the length considered for the calculation is significantly shorter than LLFR (i.e., total domain length and LLFR are 2.1 and 580 cm, respectively), a significantly developed boundary layer can be observed. In addition, a quasi-homogeneous mixture with a parabolic velocity profile is evident 6

44

CHAPTER 4. EVALUATION OF THE LAMINAR FLOW REACTOR 0

Variance of φ [−]

10

16 l/min 63 l/min −1

10

−2

10

0

0.5

1

1.5

2

Normalized distance from the FI tip [−]

Figure 4.9: Comparison of the variance of φ downstream of the FI tip under the same conditions as those of figure 4.10. cm downstream of the FI tip. In order to study further details of the flow field (i.e., re-laminarization) and the mixing field, advanced turbulent models are required. However, experimentally and numerically determined results show a significant mixing within LFI , which is significantly shorter than Lign (i.e., Lign /LFI > 20). Furthermore, the residence time of the mixture at the recirculation zone is significantly shorter than τ1st (i.e., Lign /Lrz > 100). Therefore, in the following chapters, the formation of a homogeneous mixture at the FI tip will be assumed.

4.3. MIXING FIELD

45

z

CS3

CS2

CS1

. Vmix = 16 l/min

CS1

Top: Bottom: . Vmix = 63 l/min

. V. mix = 16 l/min Vmix = 63 l/min

f 0.9

0.8

CS2 0.7

CS3

Figure 4.10: Comparison of the mixing field with nC7 H16 /air/N2 mixtures. Conditions are V˙mix = 16 and 63 l/min, ξ = 10%, φ = 0.8, and T ∗ = 620 K. Crosssections (CS1-CS3) are located at 60, 90, and 120 mm from the FI tip, respectively.

Chapter 5 Low temperature ignition of nC7H16 In order to study the first-stage ignition (FSI) including the first-stage ignition delay time (τ1st ) and its theoretical analysis, a laminar flow reactor (LFR) is developed and examined at atmospheric pressure. A well-known alkane fuel, n-heptane (nC7 H16 ), is used as a reference fuel and a detailed reaction mechanism from the Ranzi’s research group [83, 82, 84, 85, 86] is used for the simulations. Firstly, the available temperature range for the experiments and the heat loss due to the walls are examined. The temperature range to detect the FSI is obtained by the use of the molar fraction of CO (XCO ) as a benchmark, and the heat loss is estimated by comparing experimental and numerical obtained temperature decays after the FSI (i.e., thermal relaxation). Secondly, the screening processes (Carbon atom screening (CAS), Production rate screening (PRS), and Rate-of-progress screening (RPS)) are applied to the detailed reaction mechanism in order to obtain a skeletal reaction mechanism and to simulate the auto-ignition in the LFR. By applying the screening processes, better insight of the FSI is obtained. Based on this understanding, a reduced mechanism can be obtained. Thirdly, theoretical τ1st is derived from the mechanism. In order to reduce the nonlinearity of the system, additional assumptions are applied and their validity is discussed in detail.

5.1

Determination of temperature from experimental conditions

The FSI occurs only if the oxidation process of fuel involves low-temperature chemistry (LTC). The premixed mixture has to be at a significantly low tempera-

47

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

48

ture which is below 1300 K1 . Although the temperature is significantly lower than that of a flame, most of the fuel is consumed and intermediates are formed during the FSI. If the second-stage ignition (SSI) does not occur, the formed intermediates are preserved in the mixture (i.e., the products from the FSI). To distinguish the auto-ignition in the LFR as FSI or SSI, XCO can be used as an indicator. CO is one of the most stable intermediates and is relatively easily measured. In addition, incomplete combustion of most of the hydrocarbon (or oxygenated) fuel is caused by terminating the CO2 -forming reaction from CO [151]. Figure 5.1 shows XCO in a temperature range from 550 to 800 K with equivalence ratio (φ ) of 0.8, volumetric flow rate of air and N2 (V˙mix ) of 10 × 10−3 m3 /min, and dilution (ξ ) of 0%. This result clearly shows that only the FSI occurs under the tested conditions. Below 550 K, CO is not produced because the temperature is too low to oxidize nC7 H16 . Above 550 K, the influence of the FSI appears, and significant amount of CO is produced. Over 800 K, the temperature exceeds the upper limit of the LTC [114], and less CO is produced. CO2 has significantly low values than CO, and it indicates that the CO2 -forming reaction does not play an important role under the tested conditions. The nC7 H16 cracking reaction (i.e., pyrolysis of nC7 H16 ) is observed only above 830 K [152]. In addition, residence time of the gaseous nC7 H16 in the fuel supply pipe is about 0.5 s (see details in the appendix). Therefore, this result indicates that the effects of the FSI are maximized at about 600 K, and the pyrolysis can be neglected under the tested conditions. This result also suggests that the temperature range around 600 K is a suitable experimental condition with nC7 H16 using the LFR.

5.2

Heat loss estimation

The heat loss due to the walls has a significant influence on not only the flame structures but also on the auto-ignition delay times. Figure 5.2 shows the temperature profile with φ = 0.8, ξ = 10%, 620 K inlet temperature (T ∗ ), and with varying heat transfer coefficient (α). In the case of α = 0 cal/(cm2 Ks) (i.e., adiabatic condition), typical FSI and SSI can be observed. The first temperature rise to about 800 K because the FSI occurs approximately in 0.19 s. The temperature reaches the adiabatic equilibrium temperature by the SSI in about 0.53 s. In the case of α = 1 × 10−5 cal/(cm2 Ks), the time of SSI is increased with respect

1 This

temperature was proposed as the inner-layer temperature for the premixed combustion of methane. It denotes the minimum temperature to sustain chain-breaking reaction [150]. In the case with H2 /O2 mixtures, the inner-layer temperature is 1000 K.

5.2. HEAT LOSS ESTIMATION

49

−3

8

x 10

CO CO

7

2

6

X [−]

5 4 3 2 1 0

550

600

650

700

750

800

*

T [K]

Figure 5.1: Influence of temperature on XCO with φ = 0.8, V˙mix = 10×10−3 m3 /min, and ξ = 0%. to the previous one, and a temperature decay after the SSI can be observed. For the FSI, there is no significant difference from the adiabatic case. In the case of α = 1 × 10−4 and 1 × 10−3 cal/(cm2 Ks), the SSI disappears and only moderate temperature decay after the FSI can be observed. The SSIs are strongly affected by the heat loss because of the large temperature difference between the burned gas and the walls. Adversely, the FSI is weakly affected by the heat loss. It is shown that the heat loss due to the walls freezes chemical reactions after the FSI and does not allow the SSI to occur. Therefore, heat loss leads to the observation of the FSI from the overall ignition processes (i.e., sequential occurrence of FSI and SSI). In addition, the heat loss must be correctly estimated to carry out a comparison between the experimental and simulated results. Figure 5.3 shows a comparison between the normalized temperature (δ Tˆ ) decay after the FSI with φ = 0.8, ξ = 10%, and T ∗ = 620 K for different values of α. In figure 5.3, the circles and lines denote the experimental and numerical results, respectively. The horizontal axis represents the length from the beginning of the temperature decay (Ldecay ). Note that this origin does not correspond to the location of the fuel injector (i.e., Ldecay = 0 is not L = 0). The position at the peak δ Tˆ is defined as the beginning of the temperature decay for experimentally determined values. On the other hand, the inflection point in the temperature profile after the

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

50

Temperature [K]

2000

α=0 −5

1800

α = 1×10

1600

α = 1×10

−4 −3

α = 1×10

1400 1200 1000 800 600 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Figure 5.2: Profiles obtained with the same values of φ , ξ , and T ∗ , but different values of α ([cal/(cm2 Ks)) : φ = 0.8, ξ = 10%, and T ∗ = 620 K.

FSI (i.e., inflection point during temperature decay) is defined as the beginning for the numerically determined values. In addition, the numerically determined values of inlet temperature and the temperature at the beginning are used to find out the normalized temperature. The inflection point may differ in its time for the chosen experimental peak δ Tˆ . Comparing the experimental and numerically determined profiles having exponential decaying nature, different time of inflection point and the peak δ Tˆ has a negligible influence. This result shows that the simulations with α = 1 × 10−3 cal/(cm2 Ks) reproduce the experiments under the tested conditions. In the following section, this α will be used for calculations. To check experimentally the veracity of the above arguments for the extraction of the FSI from the overall ignition process, the experiments are conducted to observe the SSI after the FSI. By controlling the heat loss due to the walls after the FSI (i.e., by increase temperature at the corresponding segments of the LFR), the secondary temperature peak appears in the LFR. Due to the rapid thermal expansion caused by the SSI, the steady state condition cannot be achieved and only spontaneous temperature peak is observed (see details in the appendix).

5.3. CALCULATION OF τ1ST FROM LIGN

51

1

δ Tˆ [−]

0.8 −4

α = 1×10

0.6

−3

α = 1×10

α = 5×10−3 Exp.

0.4

0.2

0 0

0.5

1

1.5

L

decay

2

2.5

[m]

Figure 5.3: Comparison of temperature decays after the FSI with φ = 0.8, T ∗ = 620 K, and ξ = 10% for different values of α.

5.3

Calculation of τ1st from Lign

Figure 5.4 shows the influence of V˙mix (i.e., corresponding vcross ) on the τ1st . The definition described in section 2.7 is used to calculate τ1st using Ling . This result shows that τ1st is constant over the tested range of V˙mix and a maximum Re of 1260. For a laminar flow, the entrance length (i.e., the length to reach fully-developed flow in a tube) and Re are linearly proportional (see details in the appendix). Within the entrance length, the velocity at the centerline (vcl ) exponentially increases and asymptotically reaches to 2 vcross . In other words, the boundary layer significantly develops within a short length, and the parabolic velocity profile appears (see figure 4.7). Hypothetically, τ1st and V˙mix should be proportional if the influence of the entrance length plays an important role in the LFR. The accelerating flow increases the local transport speed of the mixture at the centerline in comparison with the fully developed flow. This reflects in the decrease of the experimentally determined τ1st . However, figure 5.4 does not show their proportional relationship and demonstrate a negligible influence of the acceleration. This implies that the velocity profile in the LFR can be assumed to be parabolic all the time, although the velocity profile may deform while passing the curved sections of the LFR2 . In addition, it is verified that the influence of diffusion caused by 2 This

is confirmed by the CFD calculations.

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

52

0.6 0.5

τ1st [s]

0.4 0.3 0.2 0.1 0 10

20

30

40

V˙ mix [l/min]

50

60

70

Figure 5.4: Influence of V˙mix on the τ1st with φ = 0.8, ξ = 10%, and T ∗ = 620 K. the parabolic velocity profile on the τ1st is also negligible (see details in the appendix). Therefore, under tested conditions, the experimental τ1st was calculated using equation 2.33 (assuming the plug flow model) and this value was adapted in the measurements.

5.4

Mechanism screening

Ignition process is an irreversible phenomenon and has less complexity than a reversible phenomenon like oscillations. It implies that ignition can be numerically reproduced by the use of smaller reaction mechanism (i.e., with smaller number of species and reactions) than that of a reversible phenomenon. To reproduce only the FSI, the reaction within the high-temperature chemistry (HTC) that are activated after the peak temperature caused by the FSI (Tpeak ), are insignificant. The screening processes for nC7 H16 begins with the C7 mechanism (234 species and 5487 reactions). This mechanism is obtained using Carbon atom screening (CAS) and involves all species consisting of carbon atoms below 7 in the molecular structure, but nitrogen species are removed due to the small influence of nitrogen chemistry (see details in section 5.5). The reference conditions are φ = 0.8, ξ = 10%, and T ∗ = 620 K. Figures 5.5a and 5.5b show the influence of screening process

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

53

on the τ1st and Tpeak during Production rate screening (PRS) and Rate-of-progress screening (RPS), respectively. To sustain the consistency of the reaction mechanisms, deviation of τ1st and Tpeak from their reference values (i.e., with the C7 mechanism) should be negligible. From these figures, the threshold values of PRS (ΞPRS ) and RPS (ΞRPS ), namely, 6 × 10−4 kmol/(m3 s) and 2 × 10−9 kmol/(m3 s), respectively are found as the optimal values. The number of species decrease from 234 to 48 using the PRS, and the number of reactions decrease from 5487 to 366 using RPS (see the reduction in the numbers of species and reaction by varying ΞPRS and ΞRPS in the appendix). The obtained mechanism is used as the skeletal mechanism for the FSI simulations in the following section. Figure 5.6 shows a comparison of XCO , XCO2 , and XO2 at t = 5 s. This chosen time is significantly far from the temperature peak caused by the FSI. No temperature rise is observed after the FSI due to large heat loss at the walls (see figure 2.1). Although a number of species and reactions are removed from the mechanism, the skeletal mechanism shows a good agreement with the detailed reaction mechanism. This result ensures that the irreversible phenomenon requires less number of species and reactions than those of reversible phenomenon, and the HTC does not play an important role during the FSI.

5.5

The first-stage ignition of nC7H16

Figure 5.7 shows a comparison of τ1st between previous findings measured by different research groups and that obtained in this study with φ = 0.8 and ξ = 10%. The symbols used in the figure are described as follows: the circle denotes the experimental data of this work, and the inverted triangle, diamond, square, and asterisk denote the measured values from the research groups of Silke [153], Griffith [154], Minetti [47], and Adomeit [34, 35], respectively. The solid line denotes the numerically determined values using 0-D flame calculations. In order to emphasize the relationship between the tested condition and the negative temperature coefficient (NTC) regime, the numerically determined second-stage ignition delay time (τ2nd ) with adiabatic condition (i.e., α = 0 cal/(cm2 Ks)) using the C7 mechanism is also plotted with a dash line. As shown in the figure, the tested conditions are close to the lower limit of the NTC regime. Moving towards the high temperature regions from the tested conditions, the difference of τ1st and τ2nd becomes larger. In this figure, the consistency of the mechanism is also shown by

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

54

0.205

τ1st [s]

0.2

0.195

0.19 −4

−3

10

10 3

TVPRS [kmol/(m s)] (a) PRS

800 700

T

peak

[K]

600 500 400 300 200 100 0 −10 10

10

Ξ

−9

3

RPS

[kmol/(m s)]

(b) RPS

Figure 5.5: Screening result of Production rate screening (PRS) and Rate-ofprogress screening (RPS) in τ1st and peak temperature.

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

Full Skeletal

−1

10

X [−]

55

−3

10

−5

10

CO

CO2

O2

Figure 5.6: Comparison of gas composition using the C7 and the skeletal mechanism. the perfect overlapping of τ1st , which are determined using the C7 and the skeletal mechanisms. These results show that the LFR can be used to obtain the FSI from the overall ignition, and that the FSI is also reproducible from the skeletal mechanism. This result also shows that the LFR is capable of measuring τ1st at lower temperature than the other experimental apparatus. During the experiments using the rapid compression machine (RCM) and the shock tube (ST), it was difficult to synchronize the compressed duration (i.e., the time when the mixture is at a desired temperature) and auto-ignition as T ∗ decreases. On the other hand, for the experiments using the LFR, all experimental parameters such as V˙mix , φ , and T ∗ can be explicitly specified. Also experiments can be conducted using the LFR at steady conditions. Therefore, the LFR has a higher potential to measure the FSI at lower temperatures than the RCM and the ST. The result also shows an exponential increase of τ1st under the tested condition (i.e., linearly increase in logarithmic scaled diagram) as an extension of the previous findings even when T ∗ is below 600 K. The previous experimental findings are observed under different conditions in pressure, φ , and ξ . Hence, this indicates a weak influence of pressure, fuel concentration including dilution on τ1st (this is discussed in section 5.6).

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

56

T* [K] 714

667

625

588

556

3

2

10

τ

1st

or τ

2nd

[ms]

10

This work ( φ=0.8, 1 bar) Silke et al. ( φ=1, 10 bar) Griffith et al. ( φ=1, 10 bar) Minetti et al. (φ =1, 3.3 bar) Ciezki & Adomeit (φ=1, 13.5 bar) Numerical (1st stage) Numerical (2nd stage)

1

10

0

10 1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1000/T* [K−1]

Figure 5.7: τign s with equivalence ratio φ of 0.8 and dilution ξ of 10%. ˆ in the X valFigures 5.8a and 5.8b show a comparison of the variation ratio (ϒ) ues of ethene (C2 H4 ), propene (C3 H6 ), O2 , and CO at the end of the LFR as T ∗ increases under the corresponding conditions for figure 5.7. The variation ratio of the jth species is determined as X j − X j,re f ϒˆ j = X j,re f

(5.1)

Here, X j,re f denotes the molar concentration of the jth species at a reference temperature (i.e., 600 K). As shown in the figures, the experimental and numerical results show a good trend. As T ∗ increases, the effect of the LTC, the CO production and the O2 consumption consequently decreases. Contrarily, the production of intermediates such as C2 H4 and C3 H6 are almost constant in a temperature range from 600 to 630 K. This implies that the production of these intermediates and CO occurs through different reaction pathways. Figures 5.9a and 5.9b show the numerically determined temperature and major species profiles with φ = 0.8 and ξ = 10% at T ∗ = 620 K. As the temperature increases from T ∗ , nC7 H16 is consumed and H2 O and CO are produced in large amounts. The large amounts of products remain in the exhaust gas, and their molar fraction X show plateau profiles. They clearly show the effect of “freezing” the chemical reactions. The heat loss due to the wall prevents the reactions that should

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

57

0.1

ˆ j [−] Υ

0 −0.1 −0.2 C2H4 −0.3 −0.4

CH

3 6

O

2

CO −0.5 600

610

620

630

640

650

640

650

*

T [K] (a) Experimental

0.1

ˆ j [−] Υ

0 −0.1 −0.2 C2H4 −0.3

CH

−0.4

O2

3 6

CO −0.5 600

610

620

630 *

T [K] (b) Numerical

Figure 5.8: Comparison of variation ratio ϒˆ based on molar fraction X.

58

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

follow the FSI and this does not allow the SSI to occur. As a by-product, it allows measuring the formed intermediates at the end of the LFR because the reactions are interrupted. This figure also shows a significantly small amounts of CO2 in the exhaust gas and also small change of XO2 from unburned to burned condition. Therefore, it is possible to assume that during the FSI, no CO2 production occurs and XO2 remains constant. This assumption will be used in section 5.7. In order to analyze this in detail, reaction path analysis is conducted wherein the temperature increase is considered to be 1% from T ∗ (i.e. 626.3 K). Figures 5.10a and 5.10b show the reaction pathway diagrams starting from nC7 H16 and ending with the C1 species. Figure 5.10a shows the reaction pathways of nC7 H16 to the C5 species. Most of nC7 H16 reacts with OH and their oxidation process start. They pass through a reaction pathway to sequentially form the following intermediates: n-heptyl radical (C6 H13CH2 , Rhep ), n-heptyl-peroxy radical (C6 H13CH2 O2 , Rhep −OO), n-heptane-hydroperoxyalkyl radical (CH2C5 H10 OOH, Qhep OOH), n-heptane-hydroperoxyalkylperoxy radical (O2CH2C5 H10 OOH , OOQhep OOH), and n-heptane-ketohydroperoxyde (OCHC5 H10CH2 OOH, OQhep OOH). Apart from this main pathway, there are minor reaction pathways to produce n-heptene (nC7 H14 ) and oxocane (nC7 H14 O). These intermediates, OQhep OOH, nC7 H14 , and nC7 H14 O are oxidized and form C4 or smaller hydrocarbon species. The succession is shown using I to IV in the figure. It should be noted that n-pentene (nC5 H10 ) is also formed from Qhep OOH and OQhep OOH. However, the skeletal mechanism does not involve any nC5 H10 -consuming reactions. Although they are involved in the C7 mechanism, due to small contribution on the oxidation process, they are removed from the mechanism during screening processes (see details in section 5.4). Hence, the C5 reactions do not play an important role on the oxidation process under tested conditions. Figure 5.10b shows the reaction pathways below the C4 reactions and the succeeding reactions are shown in figure 5.10a. The colored arrows indicate the continuation from species with 5 or more carbon atoms. For example, red arrows demonstrate connection from nC7 H14 O, and the pointed species are the formed products of its decomposing reaction. It shows that there are two pathways to form the C1 species: One pathway starts with propanal (propionaldehyde, C2 H5CHO) and ends with methylcarbonyl radical (CH3CO) through propanal radical (C2 H4CHO), propenal (acrolein, C2 H3CHO), and ethanal (acetaldehyde, CH3CHO). The other pathway starts with the ethyl radical (C2 H5 ) and ends with ethenone (ketene, CH2CO) through ethene (ethylene, C2 H4 ) and vinyl radical (C2 H3 ). These two pathways involve interactions between C2 H4CHO, C2 H5 , and CH2CO. The latter is merged with other reaction pathways from 1-methylvinoxy (acetonyl, CH3COCH2 ) and allyl radical (CH2CHCH2 ). Besides them, the CH2CHCH2 forms n-butene (nC4 H8 ) and propene (C3 H6 ).

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

59

800 780

Temperature [K]

760 740 720 700 680 660 640 620 0.17

0.18

0.19

0.2

0.21

0.22

0.21

0.22

Time [s]

(a) Temperature

0.02

X [−]

0.015

0.01

O2 × 0.1 nC H

7 16

0.005

HO 2

CO CO

2

0 0.17

0.18

0.19

0.2

Time [s]

(b) Major species

Figure 5.9: Temperature and major species profiles during the FSI.

60

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

The skeletal mechanism does not involve any reactions that significantly consume n-butane (nC4 H10 ) and nC5 H12 . In addition, the C2 H4 -consuming reaction leading to C2 H3 takes place at a significantly slower rate. Therefore, all alkenes formed during the FSI are accumulated. This implies that the stable intermediates such as alkanes or alkenes remain as the products and can be experimentally measured at the end of the LFR. By measuring these concentrations as the products of the FSI, their creation and destruction rate can be directly validated. These results also demonstrate that the screening processes discuss in the section 5.4 successfully capture characteristic reaction pathway from the detailed reaction mechanism and reduce the number of species and reactions. In order to study the relation between C7 and the reactions below C4 (i.e., the relation between figures 5.10a and 5.10b), the OQhep OOH-consuming reactions are further investigated. OQhep OOH is mainly consumed by the following three reactions: R.5.1:

OQhep OOH → OH +C2 H5CHO +CH3CO + 0.4C2 H4 + 0.4C3 H6

R.5.2:

OQhep OOH → OH +CH3CHO + 0.84C2 H4CHO +0.16CH3COCH2 + 16 nC5 H10 + 16 nC7 H14

R.5.3:

OQhep OOH → OH +CH2 O +CH3CO + nC4 H8

It should be noted that these reactions from the detailed mechanism and their noninteger coefficient is obtained using lumping approach [84]. In order to evaluate the influence of each of these three reactions on the FSI, each reaction is removed from the skeletal mechanism, and the resulting mechanisms are named as RM1, RM2, and RM3, respectively (i.e., RM1 denotes a mechanism obtained by removing R.5.1 from the skeletal model). Figure 5.11 shows a comparison of τ1st for the various mechanisms with φ = 0.8 and ξ = 10% at T ∗ = 620 K. This result shows that the influence of R.5.1 and R.5.3 on the FSI is weak. The difference in τ1st from the skeletal mechanism is approximately 10%. On the other hand, the influence of R.5.2 on the FSI is strong. τ1st is approximately doubled from its value obtained using the skeletal mechanism. This result clearly shows that OQhep OOH is consumed through R.5.2. The main products of R.5.2 are CH3CHO and C2 H4CHO. The most contributed consuming reactions are as follows:

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

61

nC7H16 Rhep

nC7H14

Rhep-OO

Rhep: C6H13CH2 Qhep: CH2C5H10CH2 Q’hep: CHC5H10CH2

QhepOOH nC5H10 OOQhepOOH @ OOH OQhep

I

II nC7H14 nC7H14O

I III

nC7H14

I

nC5H10 IV C1 (a) For species with 5 or more carbon atoms (C5 to C7 species).

: I (nC7H14) : II (QhepOOH)

C2H5CHO

: III (nC7H14O)

C2H4CHO

C2H5

C2H3CHO

C2H4

: IV (OQ’hepOOH)

C1 nC4H8

CH2CHCH2 C1

C1 CH3CHO

C2H3

C1

C1

C1 CH3CO C1

C3H6

CH2CHO

CH2CO

C1

CH3COCH2

C1

(b) For species with 4 or less carbon atoms (below C4 species).

Figure 5.10: Reaction pathway diagrams at 626.3 K with φ = 0.8 and ξ = 10%.

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

62 0.4

τ1st [s]

0.3

0.2

0.1

0

Skeletal

RM1

RM2

RM3

Figure 5.11: Comparison of ignition delay times obtained using the skeletal model and the RM1, RM2, and RM3 models (see text above). R.5.4:

O2 +C2 H4CHO → 14 C2 H4 + 14 CO2 + 14 OH + 34 QC2 OOH + 34 CO

R.5.5:

OH +CH3CHO → H2 O +CH3CO

where QC2 stands for CH3CH2 (C2 H5 ). Figure 5.12 shows the net rate-of-progress (q˙n ) of R.5.4 and R.5.5. Note that q˙ni is the net value of the forward and the reverse rate-of-progress of the ith reaction. If no reverse reaction is considered, the net value corresponds to that of the forward reaction. In order to compare it with their forming reaction, q˙n of R.5.2 is also plotted in the figure. The results show that C2 H4CHO formed through R.5.2 immediately leads to the product of R.5.4. (i.e., QC2 OOH and CO). The CH3CHO reacts with OH, but q˙n of R.5.5 is significantly smaller than others, and CH3CHO is accumulated. It shows that the oxidation through C2 H4CHO is a major reaction pathway for Qhep OOH oxidation. In addition, it can be seen that the accumulated acetaldehyde has high potential as a good indicator for the FSI in the measurements and can be used for the validation of the detailed reaction mechanism. As a result, the overall reaction pathway for the FSI with nC7 H16 can be summarized in figures 5.13 and 5.14. R.5.6 and R.5.7 denote the most contributing reactions of nC7 H16 and CO.

5.5. THE FIRST-STAGE IGNITION OF NC7 H16

63

0.035

q˙in [kmol/(m3 s)]

0.03 0.025

R.5.2 R.5.4 R.5.5

0.02

0.015 0.01

0.005 0 0.196

0.198

0.2

0.202

0.204

Time [s]

Figure 5.12: Net rate-of-progress q˙n of the CH3CHO- and C2 H4CHO-related reactions. R.5.6:

OH + nC7 H16 → H2 O + Rhep

R.5.7: CO + OH = CO2 + H Figure 5.13 shows q˙n of nC7 H16 -, OQhep OOH-, and CO-consuming reactions as a function of temperature (i.e., R.5.2, R.5.6, and R.5.7). The included figure shows a magnified view of the plot at around 780 K. This result shows that nC7 H16 is consumed even at low temperatures and forms intermediates. Once the gas temperature increases and R.5.2 is activated, the accumulated intermediates are consumed with CO formation. When R.5.2 proceeds faster than R.5.4, thermal runaway occurs and the temperature increase rapidly (up to Tpeak ). However, this rapid increase in the temperature terminates when the R.5.4 proceeds faster than R.5.2. This indicates that the production of OQhep OOH is greater than its consumption and interrupts the subsequent reactions. Over the entire temperature range, q˙n of the CO-consumption is negligible. Figure 5.14 shows a schematic diagram of the major reaction pathway of nC7 H16 for the FSI. nC7 H16 forms Rhep radical and decomposes to OQhep OOH through the main nC7 H16 reaction pathways. OQhep OOH leads to the formation of CH3CHO and CO. When the consumption of OQhep OOH becomes larger than that of nC7 H16 , the tempera-

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

64

0.04

x 10

−3

R.5.2 R.5.6 R.5.7

6

q˙in [kmol/ (m 3 s)]

4

0.03

2 0

778

780

782

0.02

0.01

0

650

700

750

Temperature [K] Figure 5.13: Net rate-of-progress q˙n of the nC7 H16 -, OQhep OOH-, and COconsuming reactions. ture increases rapidly. Once the nC7 H16 -consuming reaction becomes faster than the OQhep OOH-consuming reaction, the intermediates are accumulated. If their amount is sufficiently large, the entire oxidation process is interrupted and the accumulated intermediates are preserved in the mixture. The LFR is capable of measuring these accumulated intermediates under fixed conditions (i.e., φ , V˙mix , and ξ ). Hence, with this advantage, the LFR is a more suitable experimental apparatus to study the FSI compared to RCM or ST.

5.6

Influence of equivalence ratio and dilution

In order to study the influence of the equivalence ratio φ and the dilution ξ on the FSI, experiments and simulations are conducted. Experiments are carried out over a temperature range of 590 to 650 K with four different mixture combinations varying φ with ξ such as φ = 0.8 with ξ = 20%, φ = 0.6 with ξ = 20%, φ = 0.72 with ξ = 9%, and φ = 0.8 with ξ = 10%. Figure 5.15 shows the experimentally determined τ1st at the tested conditions. This result indicates that the obtained τ1st is inversely proportional and decreases monotonically with increase of T ∗ . This indicates a weak influence of φ and ξ on the FSI. In order to verify this, individual influence of φ and ξ on the FSI are studied.

5.6. INFLUENCE OF EQUIVALENCE RATIO AND DILUTION

65

nC7H16 ’ OOH OQhep

C5 & C7

C2H4CHO CH3CHO CH3CO CO

: Non radical species : Fast reaction : Slow reaction

CO2 Figure 5.14: Reaction flow.

Firstly, the influence of varying equivalence ratio at φ = 0.6 and 0.8 on the FSI is experimentally studied with ξ = 20% at T ∗ = 600, 620, and 640 K . Figure 5.16 shows its influence on XCO . The variation ratio ϒˆ is calculated using the equation 5.1 with reference molar fraction XCO,re f at φ = 0.6 and T ∗ = 600 K. ϒˆ with φ = 0.6 and 0.8 monotonically decrease, and values of ϒˆ at φ = 0.6 is higher than at φ = 0.8 for all temperatures. As mentioned, nC7 H16 is consumed by R.5.2 and R.5.6 and forms CO. However, relatively small amount of nC7 H16 still remains in the burned gas because of R.5.2 (see figures 5.9b and 5.13). Therefore, under tested conditions, the influence of φ on FSI (i.e., varying amount of nC7 H16 in the unburned mixture) is compensated by the remaining fuel and the accumulated intermediates in the exhaust gas. In order to verify this result, the influence of equivalence ratio φ is numerically examined in the range 0.4 < φ < 1.0. Figure 5.17 shows its influence on Tpeak , XCO , and τ1st with ξ = 10% at T ∗ = 620 K. ϒˆ can also be calculated by replacing XCO with Tpeak or τ1st , and XCO,re f with Tpeak,re f or τ1st,re f in the equation 5.1 at φ = 0.4. These results demonstrate that only XCO is influenced by varying φ .

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

66

T* [K]

τ1st [ms]

649

633

617

602

φ = 0.80, ξ = 20% φ = 0.60, ξ = 20% φ = 0.72, ξ = 9% φ = 0.80, ξ = 10%

2

10

1.52

1.54

1.56

1.58

1.6 *

1.62 −1

1.64

1.66

1.68

1000/T [K ]

Figure 5.15: Influence of φ and ξ on the τ1st . On the other hand, because of the varying φ both Tpeak and τ1st do not have a significant change on the FSI. Numerical results obtained are in good agreement with the measurements. Hence, it is experimentally and numerically shown that φ has a weak influence on the FSI. Secondly, the influence of the dilution ξ on the FSI is experimentally investigated with φ = 0.8. Figure 5.19 shows the influence of ξ on XCO at T ∗ = 600, 620, and 640 K. The variation ratio ϒˆ is determined using equation 5.1 with reference molar fraction XCO,re f at ξ = 10% and T ∗ = 600 K. Variation ratios at ξ = 10% and 20% monotonically decrease. This decrease is also caused by the larger amounts of nC7 H16 in the unburned mixture, as mentioned above. Hence, it has the same effect as the influence of φ . To emphasize the influence of ξ , further numerical investigations are conducted with φ = 0.8. Figure 5.17 shows the influence of ξ with φ = 0.8 at T ∗ = 620 K on Tpeak , XCO , and τ1st . ϒˆ can also be calculated by replacing XCO with Tpeak or τ1st , and XCO,re f with Tpeak,re f or τ1st,re f in the equation 5.1 at ξ = 0% and T ∗ = 600 K. Below ξ = 30%, the influence of ξ is negligible because all variation ratios display values within ±20%. On the other hand, the influence becomes significant over ξ > 30%. The variation ratio at ξ = 50% exceeds 0.2 for τ1st and −0.35 for XCO . Therefore, the influence of dilution can be neglected in this study due to ξ < 30%.

5.6. INFLUENCE OF EQUIVALENCE RATIO AND DILUTION

1.2

67

φ = 0.6 φ = 0.8

1.1

ˆ j [−] Υ

1 0.9 0.8 0.7 0.6 0.5 0.4

600

610

620 *

630

640

T [K]

Figure 5.16: Influence of the equivalence ratio φ on the molar fraction of CO XCO in the exhaust gas with a dilution ξ of 20% (experimental).

0.8 0.6

ˆ [−] Υ

0.4 0.2 0 −0.2 T −0.4 −0.6 −0.8 0.4

peak

XCO τ

ign

0.5

0.6

0.7

φ [−]

0.8

0.9

1

Figure 5.17: Influence of the equivalence ratio φ on the peak temperature Tpeak , the molar fraction of CO XCO , and the first-stage ignition delay time τ1st (numerical).

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

68 1.2

10% 20%

1.1

ˆ j [−] Υ

1 0.9 0.8 0.7 0.6 0.5 0.4

600

610

620 *

630

640

T [K]

Figure 5.18: Influence of the dilution ξ on the molar fraction of CO XCO at an equivalence ratio φ of 0.8 (experimental).

For a detailed study a comparison is made using numerical results for two different mixture combinations φ = 0.4 with ξ = 10%; and the other φ = 0.8 with ξ = 50% at T ∗ = 620 K. Under these conditions, the amount of XnC7 H16 is approximately equal in the unburned mixture, and there are significant differences in the amounts of only XN2 and XO2 . In other words, a large amount of O2 is replaced by N2 for the latter mixture. Figure 5.20 shows the net rate-of-progress q˙n (i.e., f corresponding forward and reverse rate-of-progress, q˙i and q˙ri ) of R.5.2 as a function of temperature. A mixture condition with φ = 0.4 and ξ = 10% has a higher value of q˙n and Tpeak (i.e., the maximum value in temperature on the horizontal axis) in comparison to the mixture condition with φ = 0.8 and ξ = 50%. R.5.2 is a C2 H4CHO-consuming reaction with reaction partner as O2 . As the amount of O2 increases, R.5.2 is enhanced, and the difference of q˙n between the two mixtures increases. Therefore, it is shown that R.5.2 plays an important role during the FSI with nC7 H16 even when the unburned mixture is highly diluted.

5.6. INFLUENCE OF EQUIVALENCE RATIO AND DILUTION

69

0.8 0.6

ˆ [−] Υ

0.4 0.2 0 −0.2 Tpeak

−0.4

X

CO

−0.6

τign

−0.8 0

10

20

ξ [%]

30

40

50

Figure 5.19: Influence of the dilution ξ on the peak temperature Tpeak , the molar fraction of CO XCO , and the first-stage ignition delay time τ1st (numerical).

−3

x 10

q˙n [kmol/(m3 s)]

4 φ = 0.8 ξ = 50 % φ = 0.4 ξ = 10%

3

2

1

0

620

640

660

680

700

Temperature [K]

Figure 5.20: Influence of dilution ξ on the R.5.2.

720

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

70

5.7

Further reduction

In the previous sections, it was experimentally and numerically shown that OQhep OOH plays an important role, and equivalence ratio and dilution have a weak influence on the FSI. These results lead to an assumption that only part of the reaction pathway from the nC7 H16 -consuming to the OQhep OOH-consuming reactions play an important role on the FSI. In addition, it can be assumed that the radicals which are formed only at high temperatures such as H and O, have a weak influence on the FSI3 . By removing the reactions associated with these extra species from the skeletal mechanism, the number of species and reactions dramatically decrease. After the removal, three reversible reactions such as Rhep  Rhep − OO, Rhep − OO  Qhep OOH, and Qhep OOH  OOQhep OOH exist. If these reverse reactions have a weak influence on the FSI, they can be excluded from the mechanism. By using CHEMClean and CHEMRev [155], an elementary reaction with the Arrhenius parameters for the equilibrium are separated as forward and reverse reactions. Figure 5.21 shows comparisons between the forward and reverse reactions, and they are compared with their net rate-of-progress q˙ni . There is a large difference between forward and reverse reactions of Rhep  Rhep − OO and Rhep − OO  Qhep OOH. At 750 K, q˙ni of the forward reactions show significantly larger value than those of the reverse reactions. On the other hand, Qhep OOH  OOQhep OOH do not show such large differences. These results indicate that the Qhep OOH-forming reaction from OOQhep OOH plays an important role in the FSI, and the rest of the reverse reactions can be excluded from the skeletal reaction mechanism. By applying these considerations, the number of species and reactions are further decrease, and 16 species and 8 reactions remain. This mechanism is referred as the reduced mechanism in following sections. Figure 5.22 shows species profiles mentioned in the previous sections with the same conditions as that shown in figure 5.9b. According to the temperature profile as shown in figure 5.9a, a temperature of 630 K (i.e., 10 K increase from the T ∗ ) is reached approximately in 0.19 s and peak temperature (Tpeak ) is reached approximately in 0.205 s. As shown in figure 5.22, all intermediates are rapidly formed at the beginning (Time < 0.01 s) and increase exponentially (i.e., linearly increase in logarithmic scaled diagram) until around 0.19 s when the temperature recognizably increases from T ∗ . On the other hand, molar fraction of nC7 H16 does not change with the exponential the increase of intermediates, because the value of nC7 H16 is significantly larger than other intermediates. These results clearly demonstrate that the time duration characterizes the FSI, when the all intermediates increase exponentially. In other words, the other durations can be neglected 3 This

was also pointed out by Peters et al. [114] using another mechanism (see details in the appendix).

5.7. FURTHER REDUCTION

71

1 R

hep

q˙in [kmol/(m3 s)]

0.8

→R

−OO

hep

Reverse

0.6 0.4 0.2 0

620

640

660

680

700

720

740

760

740

760

740

760

Temperature [K]

(a) Rhep  Rhep − OO

1.4 Rhep−OO → QhepOOH

q˙in [kmol/(m3 s)]

1.2

Reverse

1

0.8 0.6 0.4 0.2 0

620

640

660

680

700

720

Temperature [K]

(b) Rhep − OO  Qhep OOH

10 Q

OOH → OOQ

q˙in [kmol/(m3 s)]

hep

8

OOH

hep

Reverse

6 4 2 0

620

640

660

680

700

720

Temperature [K]

(c) Qhep OOH  OOQhep OOH

Figure 5.21: Comparisons between the forward and the reverse reactions in the net rate-of-progress q˙ni .

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

72 0

10

−5

X [−]

10

−10

10

nC H

7 16 ′ OOH hep

−15

OQ

10

CH CHO 3

OH

−20

10

0

0.05

0.1

0.15

0.2

Time [s]

Figure 5.22: Species profile on the logarithmic scale. for detailed investigation of the FSI. By considering this limited time duration, temperature can be assumed as constant due to its smaller change as compare to the maximum value (i.e., Tpeak ). This time duration is often referred to as the induction period. Since the temperature remains constant, excluding the energy equation from the governing equations, FSI can be analyzed using only the species equations. Furthermore, by assuming constant temperature, the rate constants of the elementary reactions can be considered as constant values with respect to T ∗ .

5.8

Steady state assumption

The steady state assumption is one of the major strategies used to reduce nonlinearity in the governing equations. The assumption is called a quasi-steady state (QSS) assumption based on the fact that the creation rate of the species is significantly faster than its destruction rate, and the species concentration remains at a lower value. In a system consisting of the governing equations, this QSS assumption is expressed by setting a time derivative of the species in QSS to zero. To find out possible species for the QSS assumption from the governing equations, two criteria are considered (see details in the appendix), these include: significantly small eigenvalues (λ ), and the corresponding element in the inverse modal matrix (P−1 ) has to have limited entries. For example, λ of the Jacobian matrix (A) of

5.8. STEADY STATE ASSUMPTION

73

the governing equations which consists of nC7 H16 , Rhep , Rhep − OO, Qhep OOH, OOQhep OOH, OQhep OOH, and OH at condition discussed in the section 5.7 with consideration as lean mixture (i.e., CO2 can be assumed as constant) can be written as

diag (Λ) = =

 

−6.6 × 106 ,



λ 1 , λ 2 , · · · , λ7

−6.6 × 106 ,

−2.8 × 106

;

−5.7 × 103 − 71 1.1 × 10−7 − 77



(5.2) (5.3) (5.4)

In these λ s, there are one positive λ (λ7 ), one significantly small λ (λ6 ), two negative λ s (λ3, 4 ), and three significantly large negative λ s (λ1, 2 ). By comparing λ6 with other λ s, it can be assumed to be zero. By analyzing P−1 , it is suggested that Rhep , Qhep OOH, and OH are considered as QSS species (see details in the appendix). By applying the QSS assumption for these species, the following equation can be obtained. dCQhep OOH dCRhep dCOH = = =0 dt dt dt

(5.5)

This equation leads to the following algebraic equations.

q˙hep,1 + q˙hep,2 − q˙hep,3 = 0 q˙hep,4 − q˙hep,5 f + q˙hep,5r = 0 −q˙hep,1 + q˙hep,6 + q˙hep,7 = 0

(5.6) (5.7) (5.8)

By applying the above equations to the governing equations, one obtains the ordinary differential equations (ODE)-system below. dCnC7 H16 = −q˙hep,2 − q˙hep,6 − q˙hep,7 dt dCRhep −OO = q˙hep,2 + q˙hep,6 + q˙hep,7 − q˙hep,4 dt dCOOQhep OOH = dt dCOQhep OOH = dt

(5.9) (5.10)

q˙hep,4 − q˙hep,6

(5.11)

q˙hep,6 − q˙hep,7

(5.12)

74

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

As already mentioned above, CnC7 H16 can be assumed constant during the induction period. Hence, dynamic behavior of this system depends only on the intermediates. By using a simplified notation, I, II, III, and IV for nC7 H16 , Rhep − OO, OOQhep OOH, and OQhep OOH, one can obtain the above system as ⎞ ⎞ ⎛ ⎞ ⎛ CII CII εhep d ⎝ CIII ⎠ = Ahep ⎝ CIII ⎠ + ⎝ 0 ⎠ dt 0 CIV CIV ⎛

(5.13)

where ⎛

⎞ −khep,4 khep,6 khep,7 ⎠ 0 Ahep = ⎝ khep,4 −khep,6 0 khep,6 −khep,7

(5.14)

Here, εhep is a constant term during the induction period and is defined as εhep ≡ khep,2CICO2 .

5.9

Simplified eigenvalues

The λ of the system can be easily determined numerically (e.g., by using Matlab). This numerically determined λ is useful to investigate a large system (i.e., governing equations with large number of species). However, numerically determined λ does not explicitly demonstrate a role of each elementary reaction. To demonstrate their roles, sensitivity analysis has been widely used. By using the analysis, the influence of the elementary reactions on the system can be shown as an “accelerating” or “decelerating” system. However, this analysis shows only relative relationship of the considered elementary reaction with the system and does not describe the quantitative relationship. In order to describe the influence quantitatively, an analytical solution of the system is needed. For Ahep   , one can obtain the corresponding characteristic equation using det Ahep − λ I where I denotes the identity matrix (see detail in the appendix) as λ 3 + a3h,2 λ 2 + a3h,1 λ + a3h,0 = 0

(5.15)

a3h,2 = khep,4 + khep,6 + khep,7   a3h,1 = khep,7 khep,4 + khep,6 a3h,0 = −khep,4 khep,6 khep,7

(5.16) (5.17) (5.18)

where

5.9. SIMPLIFIED EIGENVALUES

75

Here, a3h denotes a coefficient in the characteristic equation with 3rd-degree polynomial for nC7 H16 and the following numbers in the subscript from zero to three denotes corresponding order with respect to λ . For rate constant k, subscript hep and the following number denote the value for nC7 H16 and the number of reactions, respectively. By solving equation 5.15 and applying the obtained λ s, the system can be mathematically solved. However, due to a number of terms in the solutions caused by nonlinearity of the combustion chemistry, quantitative insight of the influence of individual elementary reactions on the system is hardly obtained from mathematically determined solutions. As described in the appendix,  could be a complex number with larger imaginary numbers the solution (i.e., C) than λ itself. Hence, it is needed to simplify λ to derive the analytical solutions. Precisely, less root terms in the expression of λ are preferred to avoid mathematical complexity. To solve equation 5.15, a depressed cubic polynomial needs to be considered (see details in the appendix). It can be written as ψ 3 + b3h,1 ψ + b3h,0 = 0

(5.19)

where

λ =ψ− b3h,1 = − b3h,0 =

a3h,2 3

a23h,2 3

2 a33h,2 27

(5.20)

+ a3h,1 +

a3h,1 a3h,2 + a3h,0 3

(5.21) (5.22)

Here, ψ and b denote converted λ and coefficients in the depressed characteristic equation. A solution of the depressed cubic polynomial can be written as         3 2 3 3 b3h,0 b3h,1  b23h,0 b33h,1  b3h,0 b3h,0 + + + − − + ψ1 = − 2 4 27 2 4 27 b2

b3

b

(5.23)

3h,0 3h,1 In this equation, 3h,0 4 + 27 < 0 and − 2 > 0. Therefore, the term with the cubic root is a complex number, and the first and second terms are in symmetry 1/3 with respect to the real axis. The first term in the equation 5.23 (i.e., zcomp , see details in the appendix) can be written in the complex plane with rcomp and θcomp .

76

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

3

zcomp zcomp,1 z im

comp,2

Z

zcomp,3

Zre

Figure 5.23: Complex number plot.

 3 2 b3h,0 2 b3h,0 b3h,1 + = − + 2 4 27  ⎞ ⎛ 3 2 81 b + 12 b 3h,1 3h,0 1 ⎠ θcomp = arctan ⎝− 2 9 b3h,0 

3 rcomp

(5.24)

(5.25)

Figure 5.23 shows the three possible roots of this complex number. Out of this, one of the solutions lies close to the real axis (i.e., θ =178.98 degree) and it can be assumed as θ ≈ π. In addition, the corresponding imaginary part has a weak influence on the time evolution of the function (see the appendix for the behavior of this function). By applying this assumption, one can simplify the ψ1 as 

b3h,0 ψ1 ≈ ψ1,trunc = −2 2

1/3 (5.26)

Here, the subscript trunc denotes a truncated function. Figure 5.24 shows the comparison between the simplified and numerical ψ1 . They show a good agreement under the entire range of temperatures (e.g., ψ1,trunc and ψ at T ∗ = 620 K are −3.8132 × 103 and −3.8190 × 103 , respectively). The error caused by this simplification is 0.15% and can be neglected.

5.9. SIMPLIFIED EIGENVALUES

77

2

−10

ψ1,trunc or ψ1

ψ1,trunc ψ1 3

−10

4

−10

5

−10 560

580

600

620

640

660

680

700

*

T [K]

Figure 5.24: Influence of the simplification on ψ1 . Together with the simplified ψ1 , one can obtain λ with a simple expression as λ1 ≈ −khep,4 − khep,6 − khep,7

(5.27)

By applying this λ1 , the rest of the λ s can be obtained by following equation. a2h,2 λ 2 + a2h,1 λ + a2h,0 = 0

(5.28)

The coefficients in this equation can be determined by comparing them with equation 5.15, and can be written as a3h,2 a3h,1 a3h,0 − 2 − 3 λ1 λ1 λ1 a3h,1 a3h,0 − 2 a2h,1 = − λ1 λ1 a3h,0 a2h,0 = − λ1

a2h,2 = −

(5.29) (5.30) (5.31)

Figure 5.25 shows the coefficients of the equation 5.28. This figure suggests that a2h,2 ≈ 1. Therefore, equation 5.28 can be simplified as

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

|a2h,1 |, |a2h,2 |, or |a2h,3 |

78

0

10

−2

10

−4

a2h,1 a2h,2 a2h,3

10

−6

10

560

580

600

620 *

640

660

680

700

T [K]

Figure 5.25: Coefficients of equation 5.28.  a3h,0 a3h,1 a3h,0 λ − + 2 λ− =0 λ1 λ1 λ1 

2

(5.32)

By solving equation 5.32, one can obtain the rest of the λ s as    a3h,1 a3h,0 1 a3h,1 a3h,0 2 a3h,0 λ2,3 = + ± + 2 − 2 λ1 4 λ1 λ1 2 λ12 λ1

(5.33)

Figure 5.26 shows the comparison between the cubic and the simplified quadratic polynomials. The λ1 is shown with squared symbol in the figure. The included figure shows a magnified view of the plot around the origin. This result clearly shows that the roots of the polynomials show good agreement with the original function, and applied simplifications do not modify the behavior of the function itself.

5.9. SIMPLIFIED EIGENVALUES

Polynomial (cubic and quadric)

4

x 10

79

10 4

x 10 3

3

2 1 0

2

−1 −200

−100

0

100

200

1 0 −1

Cubic Trunc. Quad. L 1

−2 −6000 −5000 −4000 −3000 −2000 −1000

L

0

1000

2000

Figure 5.26: Comparison between the cubic and the simplified quadratic polynomials. In order simplify the expression for the λ2 and λ3 , additional assumption is applied. That is, khep,4 +khep,6 khep,7 (i.e., khep,4 +khep,6 +khep,7 ≈ khep,4 +khep,6 ). By applying this assumption, the coefficients of equation 5.32 can be truncated a a 6 k7 with − λ3h,1 ≈ khep,7 and − λ3h,0 ≈ kk44k+k . Figure 5.27 shows the influence of the 6 1 1 truncation on the behavior of the functions. Both coefficients indicate no difference caused by this truncation. Therefore, it is demonstrated that this truncation also does not change the behavior of the function. By applying the truncated λ s, one can obtain them in simple expression as ⎛ 1 λ2,3 = − ⎝khep,7 ± 2

 khep,7 −

a3h,0 λ12

⎞

2 +

4 khep,4 khep,6 khep,7 ⎠ − a3h,0 (5.34) khep,4 + khep,6 2 λ12

Figures 5.28 shows the analytically and the numerically determined λ s. Because of the symmetry of λ3 with λ2 , only λ2 is shown. This result clearly demonstrates that the analytical expression perfectly reproduces the numerical values. For example, the analytical and numerical λ1 at T ∗ = 620 K are 70.9991 and 71.4345, respectively. Therefore, the simplification in the complex number and truncation successfully simplified their analytical expressions.

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

80

3

8

10

10

2

6

2h,0

10

1

10

a

a

2h,1

10

0

10

2

10

Trunc. Full

Trunc. Full

−1

10

4

10

0

560

580

600

620

640

660

680

700

10 560

580

600

*

620

640

660

680

700

*

T [K]

T [K]

(a) 1st order

(b) 0th order

Figure 5.27: Influence of truncation on the coefficients of the 1st and the 0th order terms in the equation 5.32.

3

Analytical Numerical

Analytical Numerical

1

λ [1/s]

−10

2

2

10

1

λ [1/s]

10

2

−10

1

10

3

−10 580

600

620

640 *

T [K]

(a) λ1

660

680

700

580

600

620

640

660

*

T [K]

(b) Influence of truncated

Figure 5.28: Comparison of the analytical and the numerical λ s

680

700

5.10. ANALYTICAL EXPRESSION OF τ1ST

81

The difference between the current model and that from Peters et al. (see details in the appendix) lies in the consideration of intermediates. In Peters’ model, only two intermediates (Rhep and OQhep OOH) are considered, and reaction 6 (the OOQhep OOH-consuming reaction) is assumed as infinitely fast. By manipulating khep,6 → ∞, one can obtain the asymptotic limit of λ2,3 as   1 2 lim λ2,3 = − khep,7 ± khep,7 + 4khep,4 khep,7 khep,6 →∞ 2

(5.35)

This is the same expression in Peters’ model [114]. This also indicates that the procedure to obtain the simple expression does not modify the system and the obtained expression is an improved model of Peters’.

5.10

Analytical expression of τ1st

By substituting the obtained λ s, one can obtain the modal matrix (P) and the particular solution (veig,p ) (see details in the appendix) ⎛ ⎜ ⎜ P=⎜ ⎝

(khep,6 +λ1 )(k7 +λ1 )

(khep,6 −λ2 )(khep,7 −λ2 )

khep,4 khep,6 khep,7 +λ1 khep,6

khep,4 khep,6 khep,7 −λ2 khep,6

1

1

⎞ 1 −1 khep,6 khep,4 +khep,6 −khep,7

⎟ ⎟ ⎟ ⎠

(5.36)

and   εhep εhep εhep T ,− ,− veig,p = − khep,4 khep,6 khep,7

(5.37)

Figure 5.29 shows comparison between the analytically and numerically determined values of an element of eigenvector (i.e.,veig,11 ). Due to their overlapping profile, the result shows a good agreement between the analytically and the numerically determined values and confirms the negligible influence of the simplification on the FSI. Together with P and veig,p , the equation 5.13 can be solve as follows  = c1veig,1 exp(λ1t) + c2veig,2 exp(λ2t) + c3veig,13 exp(λ3t) + c4veig,p C

(5.38)

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

82 0.04

Analytical Numerical

0.035

P11

0.03 0.025 0.02 0.015 580

600

620

640

660

680

700

*

T [K]

Figure 5.29: Influence of the simplification on P11 (i.e., veig,11 ) Due to the significantly small value of veig,p , one can neglect the inhomogeneous term (see details in the appendix). By neglecting it, one can rewrite equation 5.38 as  = c1veig,1 exp(λ1t) + c2veig,2 exp(λ2t) + c3veig,13 exp(λ3t) C

(5.39)

where   ελ2 khep,4 + khep,6   c1 = khep,7 (λ2 − λ1 ) khep,4 + khep,6 + λ1   ελ1 khep,4 + khep,6    c2 = khep,7 (λ1 − λ2 ) khep,4 + khep,6 + λ2   ελ1 λ2 khep,4 + khep,6 − khep,7       c3 = khep,4 + khep,6 + λ2 khep,6 khep,7 khep,4 + khep,6 + λ1

(5.40) (5.41) (5.42)

By considering only the growing term at the induction period with the lean mixture, one can express τ1st as τ1st

1 = log λ1

   khep,7 λ1 (λ2 − λ1 ) khep,4 + khep,6 + λ1    khep,2CO∗ 2 λ2 khep,4 + khep,6 2khep,7 + λ1

(5.43)

5.10. ANALYTICAL EXPRESSION OF τ1ST

83

Figure 5.30 shows the comparison between the analytically and the experimentally determined τ1st . The experimental values shown in the figure 5.7 are plotted as a reference. The analytically determined values show slightly longer τ1st than the experimental values. However, the analytically determined τ1st is based on the detailed reaction mechanism used and shows perfect overlap to the numerically determined τ1st . In order to minimize the difference between the experiments and the analytical values, the detailed reaction mechanism may need further validation. The current detailed reaction mechanism can also describe the FSI. For example, the mechanism predicts the importance of OH in LTC and the negligible influence of the pressure and the dilution on the τ1st . This is also pointed out by Peters et al. [114] by using their reaction mechanism. Although the qualitative results are approximately the same using different reaction mechanisms (e.g., Ranzi mechanism and one used in Peters’ study), there is a significant quantitative difference between these reaction mechanisms. For example, the same QSS assumptions cannot be applied for different mechanisms without analyzing the Jacobian matrix as described in section 5.8. The validation of elementary reactions within the LTC can be done using quantitative mechanism calculations. However, such validation cannot be conducted for entire reactions because of the lack of experimental data. On the other hand, equation 5.43 suggests that the Rhep − OO-, OOQhep OOH-, and OQhep OOHconsuming reactions may need to be validated with higher priority. In addition, equation 5.43 suggests that, the revised rate constants of the elementary reactions have to satisfy the experimentally determined values. This developed procedure demonstrates how to experimentally, numerically, and theoretically investigate low-temperature ignition. It suggests that experimentally determined values (i.e., τ1st ) can be directly used to point out important elementary reactions within the LTC and their constraints (i.e., rate constants). In the following chapters, the same procedure will be applied to the DME and the DME/nC7 H16 blended fuels.

CHAPTER 5. LOW TEMPERATURE IGNITION OF NC7 H16

84

*

T [K] 667

τ1st [ms]

10

10

10

10

3

625

588

This work ( φ=0.8, 1 bar) Silke et al. ( φ=1, 10 bar) Griffith et al. ( φ=1, 10 bar) Minetti et al. (φ =1, 3.3 bar) Ciezki & Adomeit (φ=1, 13.5 bar) Analytical

2

1

0

1.45

1.5

1.55 *

1.6 −1

1.65

1.7

1000/T [K ]

Figure 5.30: Comparison between the analytically and experimentally determined τ1st s.

Chapter 6 Low temperature ignition of DME Extensive research has been carried out to investigate the low temperature chemistry (LTC) of dimethyl ether (DME). DME as a biofuel has high potential in applications such as internal combustion engines or gas-turbines. Although DME has a low boiling point, it can be used as liquid fuel at high pressures (over 5 bar). Shock tubes (STs) are mainly used to investigate its ignition delay times (i.e., firstand second-stage ignition delay times (τ1st and τ2nd )) [17, 18, 156, 157]. Flow reactors and Jet stirred reactors (JSRs) are used to measure the species concentration as a function of temperature [53, 58]. Based on the above mentioned studies, the negative temperature coefficient (NTC) regime of DME has been qualitatively well determined. On the other hand, because of a strong dependence between first- and second-stage ignition (FSI and SSI), FSI has not been well investigated. FSI has a significant influence on the overall ignition process and needs separate investigation. In this chapter, experiments with lean DME/air/N2 mixture at low temperatures and their detailed analyses in numerical simulations and theoretical modeling are conducted. The results describe the low temperature ignition of DME and examines the capability of the theoretical model as described in the previous chapter.

6.1

Temperature range and heat loss

In order to observe the measurable temperature range of DME using the laminar flow reactor (LFR), molar fraction of CO and CO2 (XCO and XCO2 ) are experimentally measured in the manner described in the section 5.1. The influence of the heat transfer coefficient (α) on the flame structure is numerically investigated. The obtained value from section 5.2 is then used to predict the temperature profile of the FSI. Figure 6.1 shows XCO and XCO2 in the temperature range of 550 to 575 K (i.e., varied inlet and oven temperature (T ∗ )) with equivalence ratio (φ ) of 0.8, 85

86

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

volumetric flow rate of air and N2 (V˙mix ) of 40×10−3 m3 /min, and dilution (ξ ) of 10%. Although significantly small amount of CO2 is observed under the entire temperature range, large amount of CO is observed above 570 K. Similar result with n-heptane (nC7 H16 ) can be found in chapter 5, and hence demonstrates the significant role of FSI under these conditions. The CO2 -forming reaction from CO does not play an important role in this temperature range, and the formed CO remains as product of the FSI in the exhaust gas (see detail in chapter 5). Thus, this sharp increase in XCO with increase of T ∗ (with significantly small amount of CO2 ) gives the lower-limit of the measurable temperature using the LFR. Balance of the ignition delay time (i.e., τ1st ) and the residence time of the mixture in the LFR does not have an influence on the lower limit of the measurable temperature. During the experiments, no temperature peak such as the one shown in figure 2.1 is observed throughout the LFR. FSI may occur downstream of the LFR just next to the fuel injector (FI). The location lies between the FI and the first thermocouple downstream of the FI, and temperature decays before the thermocouple, and this may lead to the thermocouple not detecting the temperature. If the FSI occurs at a lower temperature, temperature peak and significant amount of CO should be observed. Therefore, this lower-limiting temperature is independent of the apparatus and denotes the minimum temperature of its LTC. Hence, the LFR can also experimentally distinguish whether the testing fuel (liquid and/or gaseous) involves LTC. Figure 6.2 shows the influence of the heat loss on the ignition process with φ = 0.8, ξ = 10%, and T ∗ = 620 K by changing the heat transfer coefficient α = 0 and 1×10−3 cal/(cm2 Ks). In these simulations, the Lawrence Livermore National Laboratory (LLNL) mechanism is used1 . This α is estimated from section 5.2 for nC7 H16 (see details in section 5.2). In the adiabatic case (i.e., α = 0 cal/(cm2 Ks)), the FSI and the SSI can be observed at around 0.07 s and 0.5 s, respectively. A small temperature increase is caused by the FSI (up to around 800 K) and a large increase by the SSI (up to the equilibrium temperature). When considering the heat loss, SSI disappears and only the FSI is observed. FSI does not show any influence of heat loss. Hence, this demonstrates that the estimated α with nC7 H16 is valid for the LFR and is independent of the kind of fuel. In addition, this also proves the validity of the estimating procedure of the heat transfer coefficient α.

6.1. TEMPERATURE RANGE AND HEAT LOSS

87

−3

x 10 5

CO CO

2

X [−]

4 3 2 1 0

550

555

560

565

570

575

*

T [K]

Figure 6.1: Influence of temperature on XCO with φ = 0.8, V˙mix = 40×10−3 m3 /min, and ξ = 10%.

Temperature [K]

2000 1800

α=0 α = 1×10−3

1600 1400 1200 1000 800 600 0

0.1

0.2

0.3

0.4

0.5

0.6

Time [s]

Figure 6.2: Temperature profiles obtained with the same values of φ , ξ , and T ∗ but different values of α ([cal/(cm2 Ks)) : φ = 0.8, ξ = 10%, and T ∗ = 620 K.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

88

6.2

The first-stage ignition of DME

Figure 6.3 shows the comparison of τ1st for the findings done by different research groups and the result of this study with φ = 0.8 and ξ = 10%. The symbols used in the figure are described as follows: the circle denotes the experimental data of this work, the inverted triangle and the diamond denote the measured values of Adomeit’s group [35], and the square denote the values of Fast’s group [76]. The solid and dashed lines denote the values obtained using numerical simulation with the LLNL and the Ranzi mechanisms, respectively. Due to the lack of experimental data of the FSI with DME, there is a significant difference between the experimentally and the numerically obtained τ1st s. In addition, the numerically determined τ1st s using different reaction mechanisms show significantly different values from each other. Firstly, experimentally determined τ1st s are considered. The experimental data from Fast et al. [76] demonstrates significantly longer τ1st than the usual trend from other experimental τ1st (i.e., from Pfahl et al. [35] by using the ST and this study). Fast et al. observed the FSI using a pressurized chamber with jet fuel injection (i.e., oxidizer and DME are separately injected). They neglected the mixing time of the fuel and the oxidizer because of the negligibly short τ1st at high pressures. However, this difference in the experimentally obtained data shown in figure 6.3 indicates that the mixing plays an important role on the τ1st . By separately injecting the fuel and the oxidizer, FSI may occur in a non-homogeneous mixture. Although the influence of the mixing at high temperatures is insignificant due to negligible ignition delay time (i.e., τ2nd ), but the influence at low temperatures cannot be neglected. At low temperatures, the ignition delay time is much longer (i.e., τ1st τ2nd ) and may be in the same order as the mixing time (i.e., chemical and fluid dynamical characteristic time scales are within the same order). Hence, the formed intermediates and the surrounding gas may be mixed (i.e., diffused by mixing) at low temperatures. Jet configuration may demonstrate longer τ1st due to this mixing effect. On the other hand, in the ST and the LFR, inhomogeneity does not play an important role. Therefore, linearly distributed values (in logarithmic scale) measured by Pfahl et al. using the ST and in this investigation using the LFR show results of a pure oxidation process of DME without mixing. They also indicate the weak influence of pressure and equivalence ratio on the LTC. Secondly, the difference between the numerically determined τ1st using LLNL and the Ranzi mechanisms are considered. τ1st using LLNL mechanism shows

1A

comparison between Ranzi and LLNL mechanisms is discussed in section 6.3

6.2. THE FIRST-STAGE IGNITION OF DME

89

T* [K] 679

645

606

571

2

τ1st [ms]

10

1

10

This work (φ=0.8, 1 bar) Pfahl et al. (φ=1, 13 bar) Pfahl et al. (φ=1, 40 bar) Fast et al. (Jet, 30 bar) Numerical (LLNL) Numerical (Ranzi)

0

10

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1000/T* [K−1]

Figure 6.3: τ1st with φ = 0.8 and ξ = 10%.

shorter time values than the Ranzi mechanism. In order to clearly differentiate the values of the two mechanisms, their oxidation pathways at low temperatures are analyzed. The reaction pathway analyses are conducted in the same manner as described in section 5.5. Figure 6.4 shows the overview of the LTC for both mechanisms. DME reacts with O2 and the entire oxidation starts (R.6.1). DME passes through a reaction pathway (i.e., R.6.2 to R.6.5) to sequentially form the following intermediates: methoxymethyl radical (CH3 OCH2 , Rdme ), methoxymethyl-peroxy radical (CH3 OCH2 O2 , Rdme −OO), methoxymethyl-hydroperoxyalkyl radical (CH2 OCH2 OOH, Qdme OOH), hydroperoxymethoxymethyl radical (O2CH2 OCH2 OOH, OOQdme OOH), and hydroperoxy-methyl-formate (OCHOCH2 OOH, OQdme OOH 2 ).

2 This

specie corresponds the ketohydroperoxide for nC7 H16

90

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME CH3OCH3 Rdme

CH2O + CH3

Rdme-OO QdmeOOH

2 CH2O + OH

OOQdmeOOH ’ OOH + OH OQdme

Rdme: CH3OCH2 Qdme: CH2OCH2 Q’dme: CHOCH2

Figure 6.4: Reaction pathways using DME at 626.3 K with φ = 0.8 and ξ = 10%. R.6.1:

DME + O2 → Rdme + HO2

R.6.2:

DME + OH → Rdme + H2 O

R.6.3:

Rdme + O2 → Rdme − OO

R.6.4:

Rdme − OO → Qdme OOH

R.6.5:

Qdme OOH + O2 → OOQdme OOH

R.6.6:

OOQdme OOH → OQdme OOH + OH

There are two reactions pathways to form CH2 O from LTC as shown in figure 6.4. They are as follows: R.6.7:

Rdme → CH2 O +CH3

R.6.8:

Qdme OOH → 2CH2 O + OH

The reactions R.6.1 to R.6.8 are involved in both mechanisms, but they are differently evaluated (i.e., they have different Arrhenius parameters). These differently evaluated elementary reactions may cause the difference in τ1st .

6.3. DIFFERENCES OF THE MECHANISMS IN LTC FOR DME

91

Thirdly, the qualitative trend of the numerically determined τ1st s shown in figure 6.3 is considered. The solid line i.e., trajectory of the τ1st using LLNL mechanism slightly bends at around 679 K toward the larger τ1st . The difference between the experimental and the numerical values increases because of the bend. On the other hand, the dashed line i.e., trajectory of the τ1st using the Ranzi mechanism is almost a straight line along the entire temperature range. This indicates that there is a significant difference between these mechanisms over 679 K and this qualitative difference can be a hint in improving the detailed mechanisms.

6.3

Differences of the mechanisms in LTC for DME

To investigate the difference between the numerically determined τ1st using the LLNL and the Ranzi mechanisms and the qualitative difference above 679 K, the two mechanisms are compared in detail. Firstly, equilibrium rate constants (Keq ≡ k f /kr ) of the CH2 O-forming reactions from LTC and the oxidation reaction pathways are compared. Figure 6.5 shows the comparison of the mechanisms in the CH2 O-forming reaction from Rdme and the Rdme -oxidizing reaction (i.e., R.6.3 and R.6.7). This figure clearly demonstrates that the preferred reaction pathway from Rdme is its oxidation reaction. Although the CH2 O-forming reaction indicates significant difference in temperature dependency (i.e., only Ranzi mechanism shows its temperature dependency), obtained values using both LLNL and Ranzi mechanisms are significantly smaller than those of the Rdme -oxidation reactions and do not play a major role. In contrast, the determined value of the Rdme -oxidation reaction shows significantly large values and demonstrate their significant role in the oxidation process. In addition, due to the larger value of Keg in the Rdme -oxidizing reaction, the LTC of DME is greatly preferred in the Ranzi mechanism. Figure 6.6 shows the comparison of the mechanisms in the CH2 O-forming reaction from Qdme OOH and the Qdme OOH-oxidizing reaction (R.6.5 and R.6.8). This figure clearly shows a significant difference in the preferred reaction pathways between the LLNL and the Ranzi mechanisms. Although the the Qdme OOHoxidizing reaction shows the same trend as shown in figure 6.5 (i.e., Reaction pathway of the LTC is more preferred than the LLNL one), a large difference in the values of Keq in the CH2 O-forming reaction can be seen. The values using the LLNL mechanism are several times higher in order of magnitude than those of the Ranzi mechanism. Furthermore, these large values are close to the Qdme OOH-oxidizing reaction. For example, at around 1000/T ∗ = 1.3 K−1 (i.e., T ∗ = 769 K), values using the LLNL mechanism of both the CH2 O-forming and the Qdme OOH-oxidizing reactions are approximately equal. This implies that relatively larger amount of CH2 O is formed than that of Ranzi. Contrarily, using the

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

92

*

T [K] 769

667

588

526

10

Keq [−]

10

Rdme = CH2O + CH3 (LLNL) Rdme = CH2O + CH3 (Ranzi)

5

10

Rdme + O2 = Rdme−OO (LLNL) Rdme + O2 = Rdme−OO (Ranzi)

0

10

1.3

1.4

1.5

1.6 *

1.7

1.8

1.9

−1

1000/T [K ]

Figure 6.5: Comparison of the equilibrium rate constant in the LLNL and the Ranzi mechanisms for the CH2 O- and Rdme − OO-forming reactions from Rdme . Ranzi mechanism, LTC is the preferred reaction pathway even at those temperatures. Secondly, temperature and species profiles using the LLNL and the Ranzi mechanisms are compared. Figure 6.7 shows corresponding temperature and species profiles. Note that the horizontal axes are adjusted for each FSI and shows different ranges. On the other hand, the vertical axes of the species profile are set to the same range of molar fraction. From these figures, one can see a significant difference between these mechanisms. The predicted peak temperatures (i.e., the equilibrium temperature of the FSI) using the LLNL and the Ranzi mechanisms are 778 K and 844 K, respectively. In addition, a significant difference can also be observed in the Xdme of the exhaust gas. Larger amount of DME is consumed and larger amount of XOQdme OOH is formed in the case using the Ranzi mechanism. In addition, although almost the same amount of H2 O is formed in both mechanisms, larger amount of CO and CO2 are formed from the LTC in the case with the Ranzi mechanism. These differences (i.e., 66 K difference in the peak temperature and in the amount of products) is caused by the different evaluation of the elementary reactions. In addition, this strongly affect the post FSI period, which is the SSI, due to the influence of the radical species formed during the FSI.

6.3. DIFFERENCES OF THE MECHANISMS IN LTC FOR DME

93

T* [K] 769

667

588

526

10

Keq [−]

10

5

10

0

10

QdmeOOH + O2 = OOQdmeOOH (LLNL) QdmeOOH + O2 = OOQdmeOOH (Ranzi) Q

OOH → 2 CH O + OH (LLNL)

Q

OOH → 2 CH O + OH (Ranzi)

dme

−5

10

2

dme

1.3

2

1.4

1.5

1.6 *

1.7

1.8

1.9

−1

1000/T [K ]

Figure 6.6: Comparison of the equilibrium rate constants in the LLNL and the Ranzi mechanisms for the CH2 O- and OOQdme OOH-forming reactions from Qdme OOH.

Thirdly, further oxidation pathways from OQdme OOH is compared in both mechanisms. Figure 6.8 shows a comparison of the mechanisms in the oxidation pathways beginning with OQdme OOH . In the Ranzi mechanism, OQdme OOH is decomposed to CH2 O, CO2 , OH, and H by R.6.7. R.6.9:

OQdme OOH → CH2 O +CO2 + OH + H

On the other hand, in the LLNL mechanism, there are several elementary reactions involved for the further oxidation of OQdme OOH. Several reaction pathways are involved in the LLNL mechanism to form different products of the FSI such as CH2 O, HCO2 H, or CO. One of the reaction pathway is to form the intermediates OQdme O, HOCH2 OCO, and HOCH2 O by R.6.10 to R.6.13. This reaction pathway leads to HOCH2 as the product.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

94

850

800

Temperature [K]

Temperature [K]

750

700

650

750

700

650

600 0.055

0.06

0.065

0.07

0.075

0.08

0.085

600 0.12

0.13

0.14

0.15

Time [s]

0.2

x 10 8 DME × 0.2 OQ′ OOH dme

5

CO H O × 0.5

4

CO

7 ×7

6

2

2

4

CO2

2

1

1 0.07

0.075

Time [s]

(c) Species-LLNL

0.08

0.085

×3

2

3

0.065

′ OOH dme

OQ

5

2

0.06

DME × 0.2 CO H O × 0.3

3

0 0.055

0.19

−3

−3

X [−]

X [−]

6

0.18

(b) Temperature-Ranzi

8 7

0.17

Time [s]

(a) Temperature-LLNL

x 10

0.16

0 0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

Time [s]

(d) Species-Ranzi

Figure 6.7: Temperature and species profiles using the LLNL and the Ranzi mechanisms at T ∗ = 620 K.

6.3. DIFFERENCES OF THE MECHANISMS IN LTC FOR DME ’ OOH OQdme

’ OOH OQdme

Q’dme: CHOCH2

95

Q’dme: CHOCH2

’ O + OH OQdme

HOCH2OCO CH2O + CO2 + OH + H

CHO2

CH2OH + CO2 HOCH2O + CO CH2O HCO2H + H

(a) Ranzi

(b) LLNL

Figure 6.8: Comparison of the oxidation pathways beginning with OQdme OOH. R.6.10:

OQdme OOH → OQdme O + OH

R.6.11:

OQdme O → HOCH2 OCO

R.6.12:

HOCH2 OCO → HOCH2 O +CO

R.6.13:

HOCH2 O → HCO2 H + H

Another reaction pathway to form the product is HCO2 H from OQdme O via an intermediate CHO2 by R.6.14 and R.6.15 is shown below. R.6.14:

OQdme O → CHO2 +CH2 O

R.6.15: CHO2 + HO2 → HCO2 H + O2 Figure 6.9 shows q˙ni of R.6.14 and R.6.15. This result shows that the isomarization and decomposition of OQdme O are equally important in LTC. In other words, the reaction pathway to form HCO2 H through CHO2 act as a bypass for the other reaction pathway mentioned above. The other reaction pathway is to form CH2 O from HOCH2 OCO through CH2 OH by R.6.16, R.6.17, and R.6.18.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

96 0.04

OQdme O → HOCH2 OCO

q˙in [kmol/(m3 s)]

OQdme O → CHO2 + CH2 O

0.03

0.02

0.01

0

600

650

700

750

Temperature [K]

Figure 6.9: Comparison of the oxidation and the decomposition of OQdme O at T ∗ = 620 K. R.6.16:

HOCH2 OCO → CH2 OH +CO2

R.6.17: CH2 OH + O2 → CH2 O + HO2 R.6.18: CH2 O + H (+M) ↔CH2 OH (+M) Figure 6.10 shows q˙ni of R.6.16, R.6.17, and R.6.18. CH2 O has a small influence on the formation of CH2 OH, but most of the CH2 OH leads to CH2 O. The formed CH2 O from 6.7, R.6.8, R.6.14, and R.6.17 is oxidized to CO through HCO by 6.19 and 6.20. R.6.19: CH2 O + OH → HCO + H2 O R.6.20:

HCO + O2 → CO + HO2

Figure 6.11 shows q˙ni of R.6.19 and R.6.20. This result shows that CO is produced not only by R.6.12, but also from the oxidation of CH2 O. From the above analyses of further oxidation pathways in the LLNL mechanism from OQdme OOH, it is shown that the stable products of the FSI are CO, H2 O,

6.3. DIFFERENCES OF THE MECHANISMS IN LTC FOR DME

−3

3.5

x 10

HOCH2 OCO → CH2 OH + CO2

q˙in [kmol/(m3 s)]

3

CH2 OH + O2 → CH2 O + HO2 CH2 O + H (+ M ) ↔ CH2 OH (+ M )

2.5 2

1.5 1

0.5 0

600

650

700

750

Temperature [K]

Figure 6.10: Reaction pathway from HOCH2 OCO to CH2 O at T ∗ = 620 K.

0.1 CH2 O + OH → HCO + H2 O

0.08

q˙in [kmol/(m3 s)]

HCO + O2 → CO + HO2

0.06 0.04 0.02 0

600

650

700

750

Temperature [K]

Figure 6.11: Reaction pathway from CH2 O to CO at T ∗ = 620 K.

97

98

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME ’ OOH OQdme

Q’dme: CHOCH2

’ O + OH OQdme

CH2O + CHO2 HCO CO

CO2

^ CHO2: O=CH-O

Figure 6.12: Reaction pathway from OQdme OOH using RMG for the LTC of ˆˆ DME. () denotes the location of the free electron. CO2 , and HCO2 H, whereas, in the Ranzi mechanism, CH2 O, CO2 , OH, and H are formed as the products due to the OQdme OOH-decomposing reaction. In the Ranzi mechanism, CH2 O forms CO and H2 O with R.6.19 and R.6.20 (i.e., with different Arrhenius coefficient than one in LLNL mechanism), and H reacts with DME. Therefore, the difference between these mechanisms is only involving HCO2 H related reactions. Fourthly, to understand the importance of HCO2 H on FSI, an oxidation process starting from OQdme OOH is investigated using the Reaction mechanism generator (RMG). RMG is a program to generate detailed reaction mechanisms based on additive theory and is described in the reference [98, 158]. The same set of parameters were previously tested for OQhep OOH and the result demonstrated the main reaction pathway of the OQhep OOH as OQhep OOH → OQhep O → OCHC5 H10 → OCHC4 H8CH3 → OCC4 H8CH3 → CH3C3 H6CH2 (C5 H11 ) with stable products of CO, CH2 O, and H2 O. Although this reaction pathway involves some additional species such as OCHC5 H10 , OCHC4 H8CH3 , and OCC4 H8CH3 , the reaction pathway from OQhep O to C5 H11 holds good agreement with the one proposed by Peters et al. [114]. By using the same set of parameters with DME, a reaction mechanism is obtained which involves only the OQdme OOH oxidation process. ˆˆ The obtained reaction pathway is shown in figure 6.12. In the figure, () denotes the location of the free electron. In the reaction pathways generated by using the RMG, HCO2 H is not involved. However, similar products are formed as those from the Ranzi and the LLNL

6.3. DIFFERENCES OF THE MECHANISMS IN LTC FOR DME

99

mechanisms. Therefore, HCO2 H has a small influence on the LTC, whereas species involved in the oxidation process of DME to OQdme OOH have a major influence. These results indicate that the difference is caused by the preferred reaction pathways between the oxidizing reaction in the LTC and the decomposing reaction (i.e., the CH2 O-forming reactions). One of the example is the Rdme -consuming reactions. Due to the decomposing reaction at the beginning of DME oxidation process, the time to reach to the equilibrium condition of the FSI is shortened. This shortened time directly leads to shorter τ1st . In the Ranzi mechanism, the entire oxidizing reaction pathway in the LTC (e.g., Rdme - and Qdme OOH-oxidizing reactions, see figures 6.5 and 6.6) are greatly preferred. This leads to less influence of the decomposing reaction as compare to the oxidizing reaction, and leads to larger τ1st than those obtained using the LLNL mechanism. Therefore, the difference in τ1st shown in figure 6.3 is caused by evaluating the oxidizing reactions in the LTC (R.6.5 and 6.8 in both mechanisms) differently. In addition, the bend of the τ1st profile around 679 K using the LLNL mechanism shown in the figure can also be described as mentioned above. As shown in figure 6.5, the difference in equilibrium rate constants between R.6.5 and R. 6.8 in the LLNL mechanism becomes smaller as T ∗ increases. This shows that the role of the reaction pathway from Qdme OOH to OOQdme OOH (i.e., oxidizing reaction , R.6.5) becomes less important as T ∗ increases, and large amount of CH2 O is formed. This change reflects in the overall oxidation rate of DME and τ1st . Hence, the bend of τ1st using the LLNL mechanism results from the influence of the oxidizing reaction on the LTC in the CH2 O-forming reaction. These results demonstrate that OH plays an important role on the FSI. In the induction period, it reacts with DME and its reaction flow continues until the OH production is terminated. On the other hand, after the FSI, OH reacts with CH2 O. To consider the regime between the induction period and the FSI, it is important to consider DME- and CH2 O−consuming reaction with OH. This is discussed in details in the next section. Unlike the LTC of DME (C2), the C1 chemistry also plays an important role. If the CH2 O-consuming reactions leading to CO (R.6.19 and 6.20) are not correctly evaluated, its formation from Rdme cannot predict the oxidation process. By using results of τ1st and species concentrations (i.e., the products from the FSI), the C1 chemistry can be validated at low temperatures. Together with the validation of the C1 chemistry at high temperatures, the oxidation process at wider-range of temperatures can be predicted. Therefore, the experimental observation of the FSI using the LFR has high potential to investigate not only the LTC of the tested fuel, but also to validate the C1 (or other related ) chemistry. Moreover, this

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

100

validation improves the prediction of the FSI, species profiles before the secondstage ignition and after FSI, and the accuracy of prediction of the τ2nd .

6.4

Reduction for the induction period

In the previous section, the influence of CH2 O related reactions on the FSI was investigated by comparing the results using the LLNL and the Ranzi mechanisms. In this section, the specific influence of the LTC on the induction period is discussed by using the LLNL mechanism. The LLNL mechanism predicts XCO2 and τ1st to values closer to the experiments and uncertainty is observed in the relatively high temperature range (i.e., around 679 K, see figure 6.3). The OQdme OOHconsuming reaction (R.6.10) is an important reaction pathway in the LTC. The oxidation pathways further consist of mainly radicals3 and one can assume that they have significantly shorter life. Therefore, the OQdme OOH-consuming reaction is the rate-determining step in the LTC of DME. In other words, one can deal with OQdme O as a final product from the FSI (see the considered reactions in the appendix). Within the reaction pathways from the DME-consuming reaction to the OQdme OOH-consuming reaction, there are five reversible reactions involved, and they include: DME  Rdme , Rdme  Rdme − OO, Rdme − OO  Qdme OOH, Qdme OOH  OOQdme OOH, and OOQdme OOH  OQdme OOH. Figure 6.13 shows the influence of the reverse reactions as compared to the forward ones. These results show that the reverse reaction of Rdme  Rdme − OO has a relatively large value of rate-of-progress as compared to forward reaction. This implies that the FSI has a significant influence on this reverse reaction. In contrast, the FSI has a negligible influence on the other reverse reactions. The reduced mechanism involving the oxidation pathways until the OQdme OOHconsuming reaction and subtracting the insignificant reverse reactions are designated as RM1. In the RM1, two reaction pathways to form CH2 O (R.6.7 and R.6.8) remains. As mentioned in section 6.3, R.6.8 can be assumed to have a negligible influence on the FSI. In addition, by neglecting R.6.7, one can establish the smallest reduced mechanism, RM2. Figure 6.14 shows the influence of the reduction process with φ = 0.8, ξ = 10%, and T ∗ = 620 K. The temperature profiles using the RM1 and the RM2 mechanisms demonstrate good agreement with the LLNL mechanism. This result clarifies the assumption that the OQdme OOH-consuming reaction is the rate-determining step in the LTC, and the influence of CH2 O-forming reaction from the LTC can be neglected. On 3 Existence

of HCO2 H is neglected due to the comparison of the mechanisms (see details in the section 4.2)

6.4. REDUCTION FOR THE INDUCTION PERIOD

DME + OH → R

dme

+H O

Rdme + O2 → Rdme−OO

0.4

2

Reverse

Reverse

q˙in [kmol/(m3 s)]

q˙in [kmol/(m3 s)]

0.2

101

0.3

0.15

0.2

0.1

0.1

0.05

0 620

640

660

680

700

720

740

760

0

780

650

700

750

Temperature [K]

Temperature [K]

(a) DME + OH ↔ Rdme + H2 O

(b) Rdme + O2 ↔ Rdme − OO

0.3 R

−OO → Q

dme

OOH

0.04

dme

Reverse

0.2

Q

OOH + O → OOQ

dme

2

OOH

dme

Reverse

q˙in [kmol/(m3 s)]

q˙in [kmol/(m3 s)]

0.25

0.03

0.15

0.02

0.1

0.01

0.05 650

700

0

750

(c) Rdme − OO ↔ Qdme OOH

0.04

OOQ

650

700

(d) Qdme OOH + O2 ↔ OOQdme OOH

′ OOH dme

OOH → OQ

dme

+ OH

Reverse

0.03

0.02

0.01

0

650

750

Temperature [K]

Temperature [K]

q˙in [kmol/(m3 s)]

0

700

750

Temperature [K]

(e) OOQdme OOH ↔ OQdme OOH + OH

Figure 6.13: Influence of the reverse reactions at T ∗ = 620 K.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

102 1100

Full RM1 RM2

Temperature [K]

1000

900

800

700

600 0

0.1

0.2

0.3

0.4

0.5

Time [s]

Figure 6.14: Influence of model reduction in temperature profile with φ = 0.8, ξ = 10%, and T ∗ = 620 K using the LLNL mechanism. the other hand, the predicted peak temperature shows a significant difference and the one using the RM2 mechanism shows higher value. This high peak temperature can be explained by the heat release from the R.6.3 and R.6.7 reactions. The R.6.3 and the6.7 are endothermic and exothermic reactions, respectively.

kJ mol kJ Rdme + O2 → Rdme − OO + 149.05 mol Rdme → CH2 O +CH3 − 25.60

(6.1) (6.2)

By neglecting the endothermic reaction (R.6.3), larger mole of Rdme is oxidized through the exothermic reaction (R.6.7). By subtracting the influence of the endothermic reaction and adding the extra influence of the exothermic reaction, the peak temperature increases (i.e., significantly high peak temperature using the RM2 mechanism is obtained). This result clearly indicates that the RM2 mechanism can be used as a reduced model. On the other hand, the CH2 O related reactions should be considered to investigate the transition from FSI to SSI (i.e., the induction period). In the following section, the RM2 mechanism is used as the reduced mechanism for the LTC.

6.5. STEADY-STATE ASSUMPTION

6.5

103

Steady-state assumption

Species equations of DME, Rdme , Rdme −OO, Qdme OOH, OOQdme OOH, OQdme OOH, and OH are considered. By using the concentration of these species, one can obtain an ODE-system with the Jacobian matrix (Adme ), eigenvalues (Λ), and inverse modal matrix (P−1 ) at a condition with φ = 0.8, ξ = 10%, and T ∗ = 620 K (their determination procedure is explained in the appendix). They are as follows: ⎛ ⎜ ⎜ ⎜ ⎜ Adme = ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 −6396524 266 0 0 0 6396524 −1732 804241 0 0 0 1466 −3043025 0 0 0 0 2238783 −11178 0 0 0 0 11178 0 0 0 0 11178

⎞ 0 −3974939 0 3974939 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ −104 0 ⎠ 104 −3974939

diag (Λ) = (λ1 , λ2 , · · · , λ7 )  = −6.4 × 106 , −3.0 × 106 , −1.2 × 104 , 1.9 × 10−8 ;  268, −380, −4.0 × 106

⎛ ⎜ ⎜ ⎜ ⎜ −1 P =⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 −1.0 0 0 0

1.0 0 0.1 −2.0 2.3 −1.6 0

0 0 0.1 −2.0 2.3 −1.6 0

0 −1.0 −0.7 −2.0 3.1 −1.2 0

0 0 −0.9 −2.0 3.3 −1.0 0

0 0 0 −1.0 0.7 0.6 0

−1.6 0 0.1 −1.0 2.6 −1.6 2.6

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

In Λ, λ4 can be considered as zero, and λ1 , λ2 , and λ7 demonstrate the fast decay modes of the corresponding variable. Therefore, quasi-steady state (QSS) should be applied for the first, second, and the 7th rows of the P−1 (refer to the details as in the appendix). This result clearly indicates that Rdme , Qdme OOH, and OH can be assumed as QSS species during the induction period.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

104

6.6

Theoretical analysis of τ1st

By applying QSS condition to the species Rdme , Qdme OOH, and OH, one can obtain following ODE-system for the LTC of the DME.

dCdme = −q˙dme,2 − q˙dme,6 − q˙dme,7 dt dCRdme = q˙dme,2 + q˙dme,6 + q˙dme,7 − q˙dme,4 dt dCOOQdme OOH = dt dCOQdme OOH = dt

(6.3) (6.4)

q˙dme,4 − q˙dme,6

(6.5)

q˙dme,6 − q˙dme,7

(6.6)

Additionally, by assuming the concentration of the O2 to be constant during the induction period (i.e., lean mixture, see detail in chapter 5), one can derive an ODE T  system for three-intermediates with C = CR −OO , COOQ OOH , COQ OOH dme

dme

dme

(see details in chapter 5). ⎞ ⎛ ⎞ ⎛ −kdme,4 kdme,6 εdme kdme,7  dC ⎝  +⎝ 0 ⎠ ⎠C kdme,4 −kdme,6 0 = dt 0 kdme,6 −kdme,7 0

(6.7)

∗ C∗ . By noticing the order of the rate constants (O (k )) where εdme = kdme,2Cdme i O2 such as O (k4 ) ≈ O (k6 ) and O (k4 ) O (k7 ), one can simplify the expression to



kdme,4 + kdme,6

3

3  − kdme,4 kdme,6 kdme,7 ≈ kdme,4 + kdme,6

(6.8)

and

2 kdme,7



kdme,4 + kdme,6

4

!  "# 2 + kdme,4 kdme,6 2 kdme,4 + kdme,6 − 3kdme,4 kdme,6  4 2 kdme,4 + kdme,6 ≈ kdme,7 (6.9)

6.6. THEORETICAL ANALYSIS OF τ1ST

105

By substituting equations 6.8 and 6.9for the eigenvalues (λ s) from the Jacobian matrix in equation 6.7, one obtains: λ1 = −kdme,4 − kdme,6 ⎛ 1 λ2,3 = − ⎝kdme,7 ± 2



a3d,0 kdme,7 − 2 λ1

2

(6.10)

⎞ 4 kdme,4 kdme,6 kdme,7 ⎠ a3d,0 + − kdme,4 + kdme,6 2 λ12 (6.11)

where

a3d,0 kdme,4 kdme,6 kdme,7 = 2 2 λ12 kdme,4 − kdme,6

(6.12)

Here, a3d,0 denotes the constant of the characteristic polynomial for the DME system. Note that a solution of the depressed cubic polynomial locates with θ = 177.1 degree in the complex plane and the imaginary part can be neglected (refer to section 5.9for details). Figure 6.15 shows the comparison between the analytical and the numerical λ s. All λ s along the entire temperature range show good agreement and hence, the nonlinearity is successfully extracted from the original system. Together with these λ s, one can obtain τ1st in the same manner as described in chapter 5. τ1st can be expressed as τ1st

1 = log λ1

   kdme,7 λ1 (λ2 − λ1 ) kdme,4 kdme,6 + λ1    kdme,2CO∗ 2 λ2 kdme,4 + kdme,6 2kdme,7 + λ1

(6.13)

Figure 6.16 shows the comparison between the analytically and the experimentally determined τ1st . The experimentally determined values are the same values that are represented in figure 6.3. Due to the perfect overlapping of the numerically and the analytically determined τ1st , the numerically determined values are not shown. The analytically determined τ1st demonstrate relatively larger τ1st than that of the experimentally determined values. τ1st decreases linearly on a logarithmic scale (i.e., exponentially decrease on the normal scale) with increase in T ∗ . Although the predicted values using the Ranzi mechanism are significantly large (see figure 5.30), both of the τ1st values show a linear decrease and are in good agreement trendwise. This linear decrease of τ1st indicates a weak influence of the pressure and the equivalence ratio on the FSI. In addition, by demonstrating the perfect overlap between the numerically and the analytically determined values, it demonstrates that OH is an important species in the LTC in the induction period.

CHAPTER 6. LOW TEMPERATURE IGNITION OF DME

3

Analytical Numerical

λ1 [1/s]

10

2

10

1

10 550

600

650

700

*

T [K]

(a) λ1

Analytical Numerical 2

λ2 [1/s]

−10

3

−10

4

−10 550

600

650

700

T* [K]

(b) λ2

Analytical Numerical

λ3 [1/s]

106

4

−10

550

600

650

700

T* [K]

(c) λ3

Figure 6.15: Comparison of analytical and numerical λ s.

6.6. THEORETICAL ANALYSIS OF τ1ST

107

Also the CH2 O-forming reaction from the Rdme does not have a strong influence on FSI within the tested temperature range. These results suggest that the LTC of DME has a similar oxidation process as nC7 H16 . The reaction of the fuel DME with OH takes place in the LTC. Similarly, theOQdme OOH-consuming reaction is the rate-determining step in the LTC. *

T [K] 667

625

588

2

τ1st [ms]

10

1

10

This work (φ=0.8, 1 bar) Pfahl et al. No. 1 (φ=1, 13 bar) Pfahl et al. No. 2 (φ=1, 40 bar) Fast et al. (Jet, 30 bar) Aanalytical

0

10

1.45

1.5

1.55

1.6 *

1.65 −1

1.7

1.75

1000/T [K ]

Figure 6.16: Comparison between the analytically and the experimentally determined τ1st . A methodology to investigate the low temperature ignition of nC7 H16 is applied for DME, and a criterion of the rate constants is successfully derived. Obtained results suggest that the methodology can be used for oxygenated species as well as for hydrocarbon species. By using this methodology, the important species for the FSI and the rate-determining step of the LTC can be pointed out. Comparison between the experimentally and the analytically determined data gives a deeper insight into the LTC such as pressure and equivalence ratio dependencies. This insight may support detailed reaction mechanism developments using quantum mechanics calculations. In the next section, blended fuel of DME/nC7 H16 are considered and the potential of this methodology will be further discussed.

Chapter 7 Low temperature ignition of the blended fuels Conventional fuels for internal combustion engines and gas turbines comprises of many constituents. In simulation of the mentioned applications, all constituents of fuels cannot be considered for investigations because of their highly nonlinear combustion chemistry. To overcome these difficulties, certain compositions are chosen as references and their mixtures are considered as fuel (i.e., surrogate fuel). For example, the mixture of nC7 H16 and 2,2,4-trimethylpentane (isooctane, i −C8 H18 ) has been intensively studied as primary reference fuel (PRF) for gasoline and diesel fuels [159]. More recently, the mixture of i −C8 H18 , nC7 H16 , ethanol, toluene, methylcyclohexane, 1-methylnaphtalene, and n-decane are also considered as a candidate composing these fuels [93]. As a research for natural gas, the effect of blended fuel consisting of CH4 and C3 H8 has been intensively studied [17, 18, 160]. Recently, the blended fuels comprising of CH4 and DME are being studied to reduce emissions from gas turbines [161, 162]. Although reaction mechanisms have been developed for these individual species and have been tested for blended fuels, they have mainly been validated only in terms of overall ignition phenomena at high temperatures. CH4 is one of the species which does not involve low temperature chemistry (LTC) in its oxidation process (i.e., around 600 K). Hence, a binary fuel with CH4 involves only single LTC in the chemical reactions at low temperatures. Therefore, these blended fuels cannot describe the interaction of two species at low temperatures (i.e., two LTCs). In order to have a concrete understanding of the influence of blending at low temperatures, well-defined experiments with simple fuels (e.g., a fuel involving less carbon atoms needs to be intensively studied) are needed. In this chapter, the effect of the blending fuels at low temperatures will be experimentally, numerically, and theoretically investigated with two fuels, namely, nC7 H16 and DME, as dis109

110CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS cussed in the previous chapters. By using two fuels having significantly different number of carbon atoms in their molecular structures, the individual influence of LTC on the overall oxidation process may be emphasized and their influence can be experimentally detected.

7.1

Experimental investigations with blended fuels

Blended fuels have been experimentally evaluated by using flow reactors, shock tubes (STs), or rapid compression machines (RCMs). These experiments are conducted at different conditions (temperature, fuel concentrations, and/or pressures), and the influence of blending has been described. On the other hand, the laminar flow reactor (LFR) can directly demonstrate the effect of blending in a single experiment. By adding DME to the fuel stream during experiments with nC7 H16 , the influence of DME additive can be directly shown as the change of ignition length in the LFR (Lign , see its definition in section 2.7) at constant inlet and oven temperatures (T ∗ ). As shown in figure 5.17, φ has a weak influence on the FSI with nC7 H16 . Hence, one can expect the same Lign at the same T ∗ if DME acts in the same manner as nC7 H16 on the first-stage ignition (FSI). This kind of active experiments can be carried out only if the FSI is observed under steady condition (i.e., not spontaneous conditions). On the other hand, they cannot be carried out using STs or RCMs due to their non-steady conditions. Figure 7.1 shows a comparison of temperature profiles at steady state conditions within a single experiment. These temperature profiles are obtained by adding DME. Firstly, the normalized temperature (δ Tˆ , see its definition in section 2.7) is measured with only nC7 H16 . Chosen conditions are dilution (ξ ) of 10%, sum of volumetric flow rates of air and N2 (V˙mix ) of 80 l/min, and T ∗ of 620 K. In this step of the experiment, only half mass of nC7 H16 with equivalence ratio (φ ) of 0.8 is injected (i.e., actual φ = 0.4). However, the first-stage ignition delay time (τ1st ) is 0.19 s and it shows good agreement with values shown in figure 5.17. Therefore, this also supports the arguments that φ and ξ have a weak influence on the FSI. Secondly, DME is injected so as to supply the other half of the mass into the fuel stream. With increasing flow rate of DME, small steps are chosen to avoid spontaneous change in φ and temperature profile, which is carefully monitored. Note that the same oxidizer and nC7 H16 continuously flow into the LFR as in the previous step. In addition, all temperatures are kept at the same value (i.e., T ∗ ) during the DME manipulation. The total m˙ (i.e., sum of air, N2 , nC7 H16 , and DME) after reaching the set manipulated values have values that are the same as those of nC7 H16 . This result clearly demonstrate that τ1st with blended fuel is smaller than that of nC7 H16 .

7.1. EXPERIMENTAL INVESTIGATIONS WITH BLENDED FUELS

111

1

δ Tˆ [−]

0.8

nC7H16 nC7H16:DME = 1:1

0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

L [m]

Figure 7.1: Effect on Lign of adding DME into nC7 H16 mixture.

Figure 7.2 shows τ1st with nC7 H16 , DME, and the blended fuel. The triangle, diamond, square, circle, and inverted triangle represent nC7 H16 in fractions of 100%, 75%, 50%, 25%, and 0% (i.e., DME) mixtures, respectively. The required mass of the fuel is calculated on the basis of nC7 H16 with φ = 0.8, and the proportion of mass obtained follows the substitution of DME. τ1st with the pure components are shown in figures 5.7 and 6.3. This result clearly shows that τ1st with DME is smaller than with nC7 H16 , and replacing nC7 H16 with DME decreases τ1st s monotonically. The obtained results imply that there is a significant difference in the oxidation processes between nC7 H16 and DME, although their oxidation processes in low temperatures are similar as discussed in chapters 5 and 6. To understand the oxidation processes of the blended fuel, firstly the amount of unburned nC7 H16 in the exhaust gas from the FSI at T ∗ = 620 K is compared as shown in figure 7.3. This result shows that the amount of unburned nC7 H16 in the exhaust gas is proportional to nC7 H16 in the unburned mixtures. In other words, this result also shows that the amount of nC7 H16 in the exhaust gas is inversely proportional to the amount of DME in the unburned mixture. Therefore, it implies

112CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS T* [K] 625

588

nC7H16 nC7H16 : DME = 3:1 nC H : DME = 1:1 7 16

τ1st [ms]

nC7H16 : DME = 1:3 DME

2

10

1.55

1.6

1.65

1.7

1.75

1000/T* [K−1]

Figure 7.2: Influence of the blending of DME on τ1st s with φ = 0.8 (based on nC7 H16 as fuel) and ξ = 10%. The shown ratio of mass is replaced by DME. that the influence of DME blending on nC7 H16 does not change the composition drastically. This is because analysis of the exhaust gas shows both fuels still exist in the same proportion, even though this proportion is lower than it was in the unburnt mixture. ˆ see its definition in Figure 7.4 shows the comparison of the variation ratio (ϒ, equation 5.1) in molar fraction (X) for C2 H4 , C3 H6 , 1-buten (CH2 = CHCH2CH3 , 1−C4 H8 ), O2 , CO, and CO2 with DME/nC7 H16 mixtures in the temperature range ˆ X for each species measured with only nC7 H16 of 600 to 650 K. To calculate ϒ, at T ∗ = 600 K is used as the reference value for the molar fraction (X j,re f ). The considered mixtures are the ones with nC7 H16 having a proportion of 100%, 75%, and 50%, with the rest of the contribution coming from DME. The measured ϒˆ indicate not only the influence of temperature (i.e., as compared to 600 K) for a mixture, but also the influence of the blending at a specific temperature. These results support the idea that hydrocarbon species (i.e., C2 H4 , C3 H6 , and 1 −C4 H8 ) are formed almost in the same manner as the one shown in figure 7.3. If the oxidation process of nC7 H16 is terminated by the existence of DME (i.e., the LTC of DME and nC7 H16 strongly interact), its influence should be observed with a different trend than those represented in the figures. Precisely, if the LTC of DME disturbs the oxidation process of nC7 H16 , at least the value of C2 H4 should show

7.1. EXPERIMENTAL INVESTIGATIONS WITH BLENDED FUELS

113

1

ˆ nC H [−] Υ 7 16

0.8 0.6 0.4 0.2 0

Hep100

Hep75

Hep50

Figure 7.3: Comparison of the experimentally determined variation rates of Xhep for DME/nC7 H16 mixtures with φ = 0.8 (based on nC7 H16 as fuel) and ξ = 10% at T ∗ = 620 K.

a different trend, since the amount of DME in the mixture increases due to both DME and C2 H4 in C2 chemistry. However, as seen from the figure 7.3, as the amount of nC7 H16 decreases, the corresponding amount of hydrocarbon species (i.e., C2 H4 , C3 H6 , and 1 − C4 H8 ) decrease in the tested temperature range. In addition, no difference can be found in the values of O2 , CO, and CO2 in terms of temperature and the amount of nC7 H16 . These trends clearly demonstrate that both LTCs (i.e, that of DME and nC7 H16 ) independently oxidizes the fuels. According to the experimental observations shown in figures 5.1 and 6.1, the LTC of both fuels produce mainly CO as product (see detail in sections 5.5 and 6.3). Hence, one can expect that the CO-forming reactions from both fuels interact with each other. On the other hand, the experimental results do not explicitly show this interaction. Therefore, it is important to investigate whether an interaction between the two LTCs has a significant influence on the overall oxidation process of the mixture.

1.2 1 0.8 0.6 0.4

nC7H16

0.2

nC H

: DME = 3 : 1

nC H

: DME = 1 : 1

7 16 7 16

0

600

610

620

630

640

Variation ratio of XC3 H6 [−]

Variation ratio of XC2 H4 [−]

114CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS

1.2 1 0.8 0.6 0.4

nC7H16

0.2 0

650

600

610

630

640

650

*

(b) C3 H6

1.2

1.2 1 0.8 0.6 0.4

nC7H16

0.2

nC H

: DME = 3 : 1

nC H

: DME = 1 : 1

7 16 7 16

600

610

620

630

640

Variation ratio of XO2 [−]

Variation ratio of XC4 H8 [−]

620

T [K]

(a) C2 H4

1 0.8 0.6 0.4

nC7H16

0.2

*

600

610

620

0.8 0.6 0.4

nC7H16

0.2

nC H

: DME = 3 : 1

nC H

: DME = 1 : 1

7 16 7 16

630

T* [K]

(e) CO

640

650

Variation ratio of XCO2 [−]

1

620

: DME = 1 : 1

640

650

(d) O2

1.2

610

630

T* [K]

(c) C4 H8

600

: DME = 3 : 1

nC H

7 16

0

650

nC H

7 16

T [K]

Variation ratio of XCO [−]

: DME = 1 : 1

7 16

*

0

: DME = 3 : 1

nC H

7 16

T [K]

0

nC H

1.2 1 0.8 0.6 0.4

nC7H16

0.2

nC H

: DME = 3 : 1

nC H

: DME = 1 : 1

7 16 7 16

0

600

610

620

630

640

*

T [K]

(f) CO2

ˆ of Figure 7.4: Comparison of the experimentally determined variation ratio (ϒ) XnC7 H16 for DME/nC7 H16 mixtures with φ = 0.8 (based on nC7 H16 as fuel) and ξ = 10%.

650

7.2. COMPARISON OF τ1ST FOR DIFFERENT BLENDED FUELS

7.2

115

Comparison of τ1st for different blended fuels

Experimentally and numerically determined values of τ for different blended fuels are compared. For this comparison, the Lawrence Livermore National Laboratory (LLNL) mechanism for nC7 H16 is also used [163, 164, 165] in addition with mechanisms for DME and Ranzi. Figure 7.5 shows the influence of blending fuels (i.e., replacing nC7 H16 with DME) on τ1st at T ∗ = 620 K. The LLNL mechanisms cannot be used for simulations with blended fuels because each mechanism is validated for individual species1 . On the other hand, the Ranzi mechanism is a general mechanism and can be used for simulations with DME and nC7 H16 . In case of nC7 H16 , the predicted τ1st based on both the Ranzi and the LLNL mechanisms have a good agreement with the experimental values. However, in the case with the DME/nC7 H16 mixture and only the DME, it shows a significant difference in the predicted and the experimental values. Firstly, in the case with DME/nC7 H16 mixture, the predicted τ1st using Ranzi’s mechanism increases as the ratio of nC7 H16 in the mixture decreases (i.e., the proportion of DME in the mixture increases). However, the experimentally determined value decreases under these conditions. In the case with 25% nC7 H16 in proportion, the predicted value shows one order of magnitude higher than that of the experiments. Secondly, in the case of DME, both predicted values using the LLNL and the Ranzi mechanisms show a significant difference as compare to the experiments (difference between the predicted and the experimentally obtained values are also shown in figure 6.3). Notable is the fact that the difference between the predicted value using the Ranzi mechanism and the experiments further increases. Therefore, the result shows that the investigation using the Ranzi mechanism may describe the difference between the predicted and the experimentally determined values. As shown in figure 7.4, the LTCs in the blended fuels are independent from each other until C2 chemistry. In other words, it implies that their LTCs oxidize the fuels parallely (i.e., from fuels-consuming reaction to OQdme OOH-consuming (or OQhep OOH-consuming) reactions) and can be added as a reaction mechanism for the blended fuels. Therefore, the LTC with nC7 H16 from the Ranzi mechanism and the one with the DME from the LLNL mechanism are combined as a blended mechanism. By using this mechanism the interaction of LTCs is further discussed in the following section. Note that the Ranzi mechanism is the only available mechanism which includes the LTC of DME and nC7 H16 into one reaction mechanism. This mechanism strictly follows the concept that combustion chemistry has a hierarchical structure 1 Oxidation

pathway of another fuel may involved, but the entire reaction mechanism is not validated for the fuel.

116CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS 250

Ranzi Exp. LLNL

τ

1st

[ms]

200 150 100 50 0

Hep100

Hep75

Hep50

Hep25

Hep0

Figure 7.5: Influence of replacing nC7 H16 based on mass with DME on τ1st s at T ∗ = 620 K. [88, 99]. The same elementary reactions (i.e., elementary reactions with the same Arrhenius parameters) appears for different fuel oxidation processes and allows comparing the specific part of their oxidation process. On the other hand, LLNL mechanisms results in better prediction with DME and nC7 H16 than the Ranzi mechanism, but 118 reactions of the same elementary reactions have different Arrhenius parameters [155]. This suggests that although the LLNL mechanism can reasonably predict τ1st with individual species, the oxidation process cannot be compared because of those different Arrhenius parameters.

7.3

Numerical investigations with blended fuels

Figure 7.6 shows the predicted τ1st using blended mechanism. As mentioned above, this blended mechanism consists of the LTC of DME from the LLNL mechanism and that of nC7 H16 from the Ranzi mechanism, respectively. The predicted τ1st becomes shorter at the same temperature as the amount of DME increases in the unburned mixture. This trend shows a good agreement with the experimentally observed trend as shown in figure 7.4 and correctly predicts that τ1st depends on the amount of DME rather than those shown in figure 7.5.

7.3. NUMERICAL INVESTIGATIONS WITH BLENDED FUELS

117

*

T [K] 625

3

10

588

nC H

7 16

nC7H16 : Dme = 3 : 1 nC H

7 16

: Dme = 1 : 1

τ1st [ms]

nC7H16 : Dme = 1 : 3 Dme

2

10

1.55

1.6

1.65

1.7

1.75

1000/T* [K−1]

Figure 7.6: Comparison of the numerically obtained τ1st using the blended mechanism with φ = 0.8 (based on nC7 H16 as fuel) and ξ = 10%. Figure 7.7 shows a comparison of profiles of molar fraction X with 50% in amount of nC7 H16 (mass ratio between nC7 H16 and DME is 1 : 1). Comparison are conducted between fuels (i.e., DME and nC7 H16 ) in figure 7.7a and between intermediates OQdme OOH and OQhep OOH in figure 7.7b. In figure 7.7a, the insert shows a magnified view of the plot at around 0.09 s. Note that the corresponding mass of nC7 H16 is replaced by DME as mentioned. Therefore, the X value of DME shows a larger mass fraction of nC7 H16 at the initial condition. The comparison of both fuels shows almost a flatter profile in the entire time range during the induction period. The profiles are supported by the assumption of the induction period and the assumption is well validated (see detail in section 5.7). Although both fuels are consumed during the induction period, their profiles show a difference at around 0.09 s (i.e., at the end of the induction period). nC7 H16 is consumed completely before DME depletes to zero (see the magnified view). In addition, although the molar fraction X of DME at initial condition is larger than that of nC7 H16 , OQhep OOH and OQdme OOH have almost the same value of X within the induction period. These results indicate that the consumed nC7 H16 is accumulated as OQhep OOH and the consumed DME does not remain in the exhaust gas as an intermediate, but it is consumed to form products. Figure 7.8 shows a comparison of q˙ni profiles with the same mixtures as shown in figure 7.7. These results show that no significant difference is found between the two fuels until the OOQdme OOH- and OOQhep OOH-consuming reactions. On the

118CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS

0.014 0.012

X [−]

0.01 0.008 0.006

−3

10

x 10

8

0.004

6

nC7H16

0.002

DME 0 0

0.01

4 2 0 0.0898

0.02

0.03

0.0899

0.04

0.05

0.06

0.07

0.08

0.07

0.08

Time [s] (a) DME and nC7 H16

0

10

′ OOH hep ′ OQ OOH dme

OQ −5

X [−]

10

−10

10

−15

10

0

0.01

0.02

0.03

0.04

0.05

0.06

Time [s] (b) OQdme OOH and OQhep OOH

Figure 7.7: Comparison of molar fraction X profiles with 50% in the amount of nC7 H16 at T ∗ = 620 K.

7.4. MODEL REDUCTION AT THE INDUCTION PERIOD

119

other hand, the OQdme OOH-consuming reaction shows a higher value than that of OQhep OOH. This difference indicates that a large amount of OQhep OOH is accumulated in the exhaust gas because they have the same production rates (i.e., the OQdme OOH- and OQhep OOH-forming reactions, see figure 7.8.b). These results clearly indicate that the rate-determining step in the oxidation process of the blended fuel is the OQhep OOH-consuming reaction. In addition, this is supported by the shown result in figure 7.7, and the difference in τ1st (i.e., shorter τ1st with DME than that of nC7 H16 ) can be explained as due to the effect of the blending. By considering the fact that DME and nC7 H16 mainly react with OH (i.e., DME + OH → Rdme + H2 O and nC7 H16 + OH → Rhep + H2 O), the oxidation process of the blended fuel simply interacts with each other with respect to the OH concentration. Note that, in case of a single fuel (DME or nC7 H16 ), this concentration is significantly small within the entire induction period and can be assumed to be a quasi-steady state (QSS) species (see details in chapters 5 and 6). Although the rate-determining step is the OQhep OOH-consuming reaction, which forms OH, τ1st can be correctly predicted unless the interaction between the two LTCs occurs for the QSS species. Independent oxidation of DME and nC7 H16 is also validated by the experimentally obtained results as shown in figure 7.4. This also leads to an important conclusion that a combined mechanism may correctly predict the LTC of a mixture if the interaction solely occurs with the QSS species. To verify this argument, a theoretical analysis is performed in the following sections.

7.4

Model reduction at the induction period

To establish a system for theoretical analysis with blended fuel, firstly, only the main species from both LTCs with the same assumptions for the induction period with lean mixtures are considered (see details in sections 5.8 and 6.6). The considered species are   C = Cdme , CRdme , CRdme −OO , COOQdme OOH , COQdme OOH ; CnC7 H16 , CRhep , CRhep −OO , COOQhep OOH , COQhep OOH , COH

T

(7.1)

By applying the same procedure as described in the sections 5.9 and B.1, one can obtain the eigenvalue (Λ) and inverse modal matrix (P−1 ) of the ordinary differential equation (ODE)-system as

120CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS

0

10

Rhep−OO → QhepOOH R

−OO → Q

OOH

0.02

0.04

q˙in [kmol/(m3 s)]

dme

dme

−5

10

−10

10

0

0.06

0.08

Time [s]

(a) R − OO → QOOH

0

10

′

OOQhepOOH → OQhepOOH ′

q˙in [kmol/(m3 s)]

OOQdmeOOH → OQdmeOOH −5

10

−10

10

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time [s]

(b) OOQOOH → OQ OOH

0

10

′

q˙in [kmol/(m3 s)]

OQhepOOH → P OQ′

OOH → P

dme

−5

10

−10

10

0

0.02

0.04

0.06

0.08

Time [s]

(c) OQ OOH → P

Figure 7.8: Comparison of q˙ni profiles with 50% in the amount of nC7 H16 at T ∗ = 620 K.

7.4. MODEL REDUCTION AT THE INDUCTION PERIOD

121

diag (Λ) = (λ1 , λ2 , · · · , λ9 )  = −3 × 106 , −1.1 × 104 , −2.7 × 103 , −1.5 × 103 ;  −409, 283, −9.9, −4.4 × 10−8 , 5.0 × 10−8 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ −1 P =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 0 0 0 −0.1 1.0 0 0 0 0 0 0 1.0 0 0 0 −2.5 0 0 0 1.8 0 0 0 −1.7 −1.0 0 0 1.5 −0.6 −0.6 0 1.9 0 0 0 −2.8 −0.7 −0.7 0 −2.6 0 0 0 2.1 1.1 1.1 0 0 −1.0 −0.9 −0.5 0.7 0.3 0.3 1.1 0.7 0.3 0.3 0.5 1.6 0.8 0.8 1.1 1.6 0.8 0.8

−1.0 −0.1 −0.9 −0.1 1.7 −2.7 1.0 0.3 0.8

(7.2) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (7.3)

This result suggests that OOQdme OOH and OH can be assumed as QSS2 (see details in section 5.8 and B.3). Secondly, by applying the QSS for OH, one can derive COH as COH =

γ   kdme,1 + khep,1

(7.4)

where γ = q˙dme,6 + q˙dme,7 + q˙hep,6 + q˙hep,7

(7.5)

and  kdme,1 = kdme,1Cdme

(7.6)

 khep,1

(7.7)

= khep,1CnC7 H16

By substituting the expressions 96, 97, and 98 into 95 and putting expression 95 into q˙dme,1 and q˙hep,1 , one obtains the following

2 From

the first and second rows of P−1

122CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS q˙dme,1 = Fdme γ

(7.8)

q˙hep,1 = (1 − Fdme ) γ

(7.9)

where

Fdme =

 kdme,1

  kdme,1 + khep,1

(7.10)

Thirdly, by substituting these expressions (99 and 100) and applying QSS for OOQdme OOH in the ODE systems in chapters 5 and 6 (for pure nC7 H16 and DME, respectively) one can derive the system as

dCdme = dt

−Fdme γ − q˙dme,2

dCRdme −OO =Fdme γ + q˙dme,2 − q˙dme,4 dt dCOQdme OOH = q˙dme,4 − q˙dme,7 dt dCnC7 H16 = −Fhep γ − q˙hep,2 dt dCRhep −OO = Fhep γ + q˙hep,2 − q˙hep,4 dt dCOOQhep OOH = q˙hep,4 − q˙hep,6 dt dCOQhep OOH = q˙hep,6 − q˙hep,7 dt

(7.11)

This ODE-system involves 5 intermediates (i.e., with the exception of the fuels). Due to the mathematical restriction, this system cannot be solved analytically. In order to overcome this issue, its Jacobian matrix (Ablend ) is considered. Ablend is expressed as

7.4. MODEL REDUCTION AT THE INDUCTION PERIOD ⎛

Ablend

⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

0 −686 −48 0 −781 48 0 1466 −104 0 −781 −55 0 781 55 0 0 0 0 0 0

0 −1965 0 0 1965 0 0 0 0 0 −2239 0 0 −1965 0 0 4204 −1520 0 0 1520

123 −2 2 0 −3 3 0 −5

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(7.12)

From Ablend , one can realized that OOQhep OOH is isolated from the reaction pathways (i.e., only two elements has non-zero values in the 6th row and the 6th column). Therefore, this suggest that OOQhep OOH also can be assumed as QSS species. By applying the QSS for OOQhep OOH, one can derive the system as

dCdme = dt

−Fdme γ − q˙dme,2

dCRdme −OO =Fdme γ + q˙dme,2 − q˙dme,4 dt dCOQdme OOH = q˙dme,4 − q˙dme,7 dt dCnC7 H16 = −Fhep γ − q˙hep,2 dt dCRhep −OO = Fhep γ + q˙hep,2 − q˙hep,4 dt dCOQhep OOH = q˙hep,4 − q˙hep,7 dt

(7.13)

By assuming a negligible influence of change in C of fuels (i.e., Cdme and CnC7 H16 ) and considering lean mixture (see details in B.4), one can derive the system for 4 intermediates as ⎛ d ⎜ ⎜ ⎜ dt ⎝

where

CRdme −OO COQdme OOH CRhep −OO COQhep OOH

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ = Ablend ⎜ ⎠ ⎝

CRdme −OO COQdme OOH CRhep −OO COQhep OOH

⎞

⎛

⎞ εdme ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎟ ⎟+⎝ εhep ⎠ ⎠ 0

(7.14)

124CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS ⎞ Fdme kdme,7 Fdme khep,4 Fdme khep,7 (Fdme − 1) kdme,4 ⎟ ⎜ kdme,4 −kdme,7 0 0 ⎟ Ablend = ⎜ ⎝ (1 − Fdme ) kdme,4 (1 − Fdme ) kdme,7 −Fdme khep,4 (1 − Fdme ) khep,7 ⎠ 0 0 khep,4 −khep,7 ⎛

and

εdme = kdme,2CO2 Cdme εhep = khep,2CO2 CnC7 H16 This is a 4×4 system that can be analytically solved. By considering not only P−1 but also Ablend , the system is simplified and analyzed analytically.

7.5

Eigenvalue determination

The characteristic polynomial of the Ablend is defined with the eigenvalue (λ ) and the Identity matrix (I) as det(Ablend − λ I) = 0 and can be written as λ 4 + a4m,3 λ 3 + a4m,2 λ 2 + a4m,1 λ + a4m,0 = 0

(7.15)

where   a4m,3 = Fdme khep,4 − kdme,4 + kdme,4 + kdme,7 + khep,7      a4m,2 = Fdme khep,4 kdme,7 + 2khep,7 − kdme,4 khep,7 + 2kdme,7     + kdme,4 kdme,7 + khep,7 + khep,7 kdme,7 − khep,4      a4m,1 = Fdme kdme,4 khep,4 khep,7 − kdme,7 + 2kdme,7 khep,7 khep,4 − kdme,4    + khep,7 kdme,4 kdme,7 − khep,4 kdme,4 + kdme,7 a4m,0 = −kdme,4 kdme,7 khep,4 khep,7 where a denotes coefficients of the polynomial, and their subscripts 4m denote quartic (4th-degree) polynomial with blended fuels. Figure 7.9 shows this polynomial with respect to λ .

7.5. EIGENVALUE DETERMINATION

125

12

8

x 10

6

det(A−λI)

4 2 0 −2 −4 −6 −3000 −2500 −2000 −1500 −1000 −500

λ

0

500

1000

Figure 7.9: Quartic (4th-degree) polynomial. As mentioned above, one of the λ is zero and one can realized that the quartic polynomial can be approximated as a linear polynomial around the origin. Figure 7.10 shows a 1st-degree polynomial with the original polynomial (quartic polynomial).

From this approximation, one can derive one of the eigenvalues (λm1 ) as λm1 ≈ −

khep,7 Fdme

(7.16)

Figure 7.11 shows a comparison of the approximated and the numerically determined λ s. This shows the validity of the approximation for λ . By substituting λm1 , the quaritic polynomial can be expressed as a cubic polynomial. λ 3 + a3m,2 λ 2 + a3m,1 λ + a3m,0 = 0 where

(7.17)

126CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS

11

1

x 10

Approx. Poly. Root 4th Poly.

det ( A−λI )

0.5

0

−0.5

−1

−400

−300

−200

−100

λ

0

100

200

300

Figure 7.10: Approximation for the quartic polynomial by 1st-degree polynomial.

−1

−10

Analytical Numerical 0

λ

m1

−10

1

−10

2

−10

3

−10

560

580

600

620

640

660

680

700

*

T [K]

Figure 7.11: Comparison of the analytically and the numerically determined λm1 .

7.5. EIGENVALUE DETERMINATION

127

2

−10

Analytical Numerical 3

λ

m2

−10

4

−10

5

−10 560

580

600

620

640

660

680

700

T* [K]

Figure 7.12: Comparison of the truncated and numerically determined λm2 .

a3m,2 = khep,4 Fdme + kdme,4 (1 − Fdme ) + kdme,7   a3m,1 = kdme,7 kdme,4 (1 − 2Fdme ) + khep,4 Fdme − khep,4 khep,7 a3m,0 = −kdme,4 kdme,7 khep,4 Fdme By applying the same argument as the one described in sections 5.9 and C.3, one can simplify one of the λ (i.e., λm2 ) as   λm2 = Fdme kdme,4 − khep,4 − kdme,4

(7.18)

Figure 7.12 shows a comparison between the numerically determined and this simplified value. There is a slight difference between the two values over 660 K due to the variation of θ (i.e., θ = 9.37 and 19.7 at 560 and 620 K, respectively). With an increase in θ a slight difference is seen in the profiles of λm2 because of the imaginary part of the solution (see details in section 5.7). However, the difference is acceptable because of the simplification of the system to reduce its complexity. By neglecting the difference in λm2 , the cubic (3rd-degree) polynomial can be expressed as a 2nd-degree polynomial

128CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS 4

10

Analytical Numerical 3

λm3

10

2

10

1

10 560

580

600

620

640

660

680

700

*

T [K]

Figure 7.13: Comparison of the analytically and the numerically determined λm3 .

λ 2 + a2m,1 λ + a2m,0 = 0

(7.19)

where

a2m,1 = khep,7 a2m,0 = −kdme,4 kdme,7 Fdme − khep,4 khep,7 The solution of this polynomial (λm3 and λm4 ) can be expressed as

λm3, m4 =

−kdme,7 ±



2 kdme,7 + 4kdme,4 kdme,7 Fdme + 4khep,4 khep,7

2

(7.20)

Figures 7.13 and 7.14 show the comparison of the analytically and the numerically determined λ s. These results also ensure that the applied assumptions and the truncation of the polynomials are correctly done.

7.6. ANALYTICAL SOLUTION OF τ1ST WITH BLENDED FUEL

129

1

−10

Analytical Numerical 2

λ

m4

−10

3

−10

4

−10 560

580

600

620

640

660

680

700

T* [K]

Figure 7.14: Comparison of the analytically and the numerically determined λm4 .

7.6

Analytical solution of τ1st with blended fuel

By applying the obtained λ s, the modal matrix of the blended fuel (Pm ) can be written as



Pm = Pm1 ,Pm2 , Pm3



⎛

⎞ P11 · · · P14 ⎜ ⎟ = ⎝ ... . . . ... ⎠ P41 · · · P44

(7.21)

The first-column is corresponding to λm1 and can be written as

Pm1 =

  λm1 khep,7 Fdme − khep,7 , 2 kdme,4 Fdme

λm1 , kdme,4

khep,7 − λm1 , khep,4

T 1

(7.22)

The second-, third-, and fourth-columns (corresponding to λm2 , λm3 , and λm4 , respectively) can be written as,

130CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS Pm2 = Pm3 = Pm4 =



2 kdme,4 λm2 − , ϕ1



kdme,4 λm2 − , ϕ1









ϕ2 kdme,7 + λm3 , 

ϕ3 kdme,7 + λm4 ,

λm2 , khep,4

T 1

kdme,4 ϕ2 ,

khep,7 + λm3 − , khep,4

ϕ3 kdme,4 ,

khep,7 + λm4 , khep,4

T −1 T 1

where,   ϕ1 = kdme,4 λm2 + kdme,7 kdme,4 + khep,4   Fdme 2khep,7 + λm3 ϕ2 = − ϕ2    ϕ2 = λm3 + kdme,4 λm3 + kdme,7 − 2kdme,4 kdme,7 Fdme   Fdme 2khep,7 + λm4 ϕ3 = ϕ3    ϕ3 = λm4 + kdme,4 λm4 + kdme,7 − 2kdme,4 kdme,7 Fdme By considering only the growing terms (see details in section 5.10), one can express the time evolution of the intermediates as ⎛ ⎜ ⎜ ⎜ ⎝

CRdme −OO COQdme OOH CRhep −OO COQhep OOH

⎞

⎛

⎞ P13 ⎟ ⎜ P23 ⎟ ⎟ ⎟ ⎟ ≈ C3 ⎜ ⎝ P33 ⎠ exp (λm3t) ⎠ P43

(7.23)

By applying these concentrations into the system, one can obtain the time derivative of the fuels as

d dt where,



Cdme CnC7 H16



 exp (λm3t) = −G

(7.24)

7.6. ANALYTICAL SOLUTION OF τ1ST WITH BLENDED FUEL = G



G1 G2





= C3 kdme,4 P13 + kdme,7 P23 + khep,4 P33 + khep,7 P43





131 Fdme 1 − Fdme



From the above equation and the same assumptions of the induction period as discussed in section 5.10 and B.4, first-stage ignition τ1st of the blended fuel can be derived with respect to DME and nC7 H16 . However, because of the complete consumption of nC7 H16 before the depletion of DME to zero (see figure 7.23), it can be clearly concluded that the τ1st of DME represents the LTC of the blended fuel. Together with these considerations, τ1st can be expressed as τ1st

  ∗ Cdme λm3 1 = log λm3 Fdme G1

(7.25)

Figure 7.15 shows a comparison of the analytically and the numerically determined values of τ1st . The solid and dashed lines represent the analytically and the experimentally determined values, respectively. Although the analytically determined τ1st is derived on the assumption of complete fuel consumption (see details in section 5.10), it demonstrates the same trend as the one of the numerically determined values and is in good agreement with the experimentally determined values. Therefore, the analytical expression of the τ1st correctly represent the LTC of the blended fuel and ensures correct extraction of nonlinearity. In addition, it verifies the validity of the assumptions for the induction period with blended fuels, and it confirms the weak influence of the pressure, the equivalence ratio φ , and the dilution ξ on the FSI due to a linear decrease of the τ1st in the logarithmic diagram with an increase of T ∗ . In this study, it is concluded that λ with a positive value has a significant influence on the FSI of blended fuels as well as single component fuels. This implies that the characteristic time scale of the chemical reactions (τchem ) at low temperatures can be expressed as τchem ∝

1 λm

(7.26)

Figure 7.16 shows the distribution of τchem in the Fdme -T ∗ plane and the figure 7.17 shows the distribution at T ∗ = 620 K. This clearly demonstrates a strong influence of the temperature on τchem and summarizes the entire findings in this study. Firstly, in the case with the pure components (i.e., Fdme = 0 and 1 for nC7 H16 and DME, respectively), τchem with Fdme = 1 shows higher value than the one with Fdme = 0 at the same temperature. This confirms the fact that DME has

132CHAPTER 7. LOW TEMPERATURE IGNITION OF THE BLENDED FUELS T* [K] 625

610

595

2

10

τ

1st

[ms]

Analytical Exp.

1.58

1.6

1.62

1.64

1.66

1.68

1000/T* [K−1]

Figure 7.15: Comparison of analytically and experimentally determined τ1st s with 50% in the amount of nC7 H16 . shorter τ1st than nC7 H16 . Secondly, τchem linearly decreases (on the logarithmic scale) as T ∗ increases to represent the same trend as that of τ1st s. This trend demonstrates a weak influence of the pressure, φ , and ξ on the FSI. Thirdly, τchem is influenced by the blending (i.e., Fdme ). As Fdme increases (i.e., the amount of DME increases), τchem becomes shorter and represents the same trend as shown in figures 7.2 and 7.6. These finding indicates that the observed trend (e.g., linearly decreasing) can be used as a criteria for preliminary investigations of the reaction mechanism. This τchem can be numerically obtained with the use of either detailed mechanism or a blended mechanism. By substituting the numerically determined positive λ , one can examine the mechanism without conducting detailed analysis in the elementary reactions or the reaction pathways. In addition, one can examine the possibility of blending the two mechanisms to predict the oxidation process of blended fuels at low temperatures. After these preliminary investigations, one can conduct detailed quantum mechanics calculations to determine the rate constant of the important elementary reactions for FSI which is pointed out by reaction path analysis. The validity of those rate constants can then be verified using experimentally determined τ1st with its analytical expression.

7.6. ANALYTICAL SOLUTION OF τ1ST WITH BLENDED FUEL

2

τ

chem

[ms]

10

0

10

550 1

600 0.5

650 700

0

Fdme [−]

T* [K]

Figure 7.16: Influence of Fdme and T ∗ on τchem distribution.

12

τchem [ms]

10

8

6

4 0

0.2

0.4

0.6

F

dme

0.8

[−]

Figure 7.17: τchem profile at T ∗ = 620 K.

1

133

Chapter 8 Conclusion and future work A sequential approach is established using experimental, numerical and theoretical study to examined and observe low-temperature ignition in a laminar flow reactor (LFR). The oxidation processes of the most candidate fuel from biomass occurs even at low temperatures (below 800 K) and forms significant number of intermediates involving reactive radicals due to its low-temperature chemistry (LTC). The formed intermediates lead to low-temperature ignition (i.e., first-stage ignition, FSI). In this study, to observe FSI, n-heptane (nC7 H16 ), dimethyl ether (DME), and their blended fuels are studied using a LFR at atmospheric pressure. The fuels (nC7 H16 , DME, and their mixtures) and the oxidizer (air) and N2 separately flow into LFR. The LFR and the fed streams are kept at the same temperature. The parameters of the tested conditions are the equivalence ratio (φ ), ratio of dilution in volume (ξ ), set temperature (T ∗ ), and volumetric flow rate of oxidizer (V˙mix with N2 if ξ > 0%). Their ranges are: φ from 0.6 to 0.8 (based on nC7 H16 as a fuel in the case with blended fuels), ξ from 9% to 20%, T ∗ from 550 to 700 K, and V˙mix from 10 to 90 l/min. The obtained conclusions are as follows: Firstly, the LFR is successfully developed to experimentally extract FSI from the overall ignition process (i.e., FSI with sequentially occurring second-stage ignition (SSI) caused by the further oxidization of species after FSI). In order to form a homogeneous mixture, a fuel injector (FI) is designed and installed at the beginning of the LFR. By conducting computational fluid dynamics (CFD) simulations, the flow and mixing fields around FI are investigated in detail. As a result of these simulations, the oxidizer (and N2 ) and the fuel is rapidly mixed, and the inhomogeneity of the mixture becomes negligibly small at the downstream of the LFR close to the FI. The temperature throughout the LFR is monitored by a number of thermocouples, and the temperature changes due to the fuel injection are 135

136

CHAPTER 8. CONCLUSION AND FUTURE WORK

measured. A single temperature peak is observed, and a large amount of carbon monoxide (CO) is detected in the exhaust gas. CO is one of the main products of the FSI. Therefore, this result experimentally demonstrates that only FSI occurs in the LFR. Secondly, by comparing the experimentally and numerically determined temperature profiles, an influence of the heat loss due to the walls on the ignition process in the negative temperature coefficient (NTC) regime is examined. Although FSI and SSI are numerically observed, it is seen that only FSI appears (SSI disappears) as the heat loss increases. FSI induces a small temperature increase, and this increase leads to a smaller influence on the FSI caused by the heat loss. For a range of heat loss (>1×10−4 ), only the FSI is obtainable. The heat loss in the LFR is estimated by comparing experimentally and numerically obtained temperature profiles during the thermal relaxation period. The estimated heat loss is examined with both fuels (i.e., nC7 H16 and DME) and show good agreement with the experiments. These results indicates that the FSI is successfully extracted from the overall ignition process using the LFR and allows the measurement of products only form FSI at the end of the LFR. Thirdly, the first-stage ignition delay time (τ1st ) is determined from the temperature profiles in the LFR. The calculation based on the temperature profile in the LFR for τ1st is discussed based on the results form CFD simulations and theoretical analysis. They are compared with the numerically determined values and shows good agreement with each other. This indicates that the LFR is capable of measuring τ1st at lower temperatures than other experimental apparatus (i.e., shock tubes or rapid compression machines). By using the LFR, one can clearly distinguish the FSI from the overall ignition in the NTC regime, and the obtained data at low temperatures can be used as a reference for reaction mechanism validations. Fourthly, the theoretical analysis in the induction period is conducted. For single component fuels (i.e., nC7 H16 or DME), reduced mechanism is established based on the experimental measurements of the intermediate composition in the exhaust gas, and the numerical simulations and analysis of the reaction pathways are conducted. From these results, it is found that nC7 H16 and DME are oxidized in a similar manner in their induction period. They sequentially form intermediates (or radicals) by H-abstraction, isomarization, and molecular-oxygen adding reactions. By applying the quasi-steady state (QSS) assumption to certain species, an analytical expressions for τ1st s are derived. These analytical expressions indicate that the Rhep − OO-, OOQhep OOH-, and OQhep OOH-consuming reactions (subscript is dme in case with DME) play an important role in both LTCs. In addition, it also demonstrates that both the decomposing reactions of OQhep OOH and OQdme OOH are the rate-determining step in their respective LTCs.

137 Finally, experimental, numerical, and theoretical analyses are conducted for DME/nC7 H16 blended fuels. Experimentally obtained τ1st with different compositions demonstrates that τ1st becomes shorter as the amount of DME in the unburned mixture increases. By analyzing the exhaust gas, no interaction between the two LTCs (i.e., from fuel-consuming to OQdme OOH- (OQhep OOH-) consuming reactions) is found. Based on these experimental results, two mechanisms representing the LTCs (DME and nC7 H16 from the LLNL and the Ranzi mechanisms, respectively) are combined as blended mechanism. This blended mechanism predicts the experimentally obtained results and indicates a significant role of OH in the oxidation process at low temperatures. In addition, it is found that the OQhep OOHconsuming reaction is the rate-determine step even for the blended fuels. By applying further QSS assumptions, analytical expression for τ1st with blended fuels are derived. From this expression, the experimentally and the numerically determined results are explicitly demonstrated. Furthermore, from this expression, a blending rule is derived and signifies not only the influence of one component on another, but also the basis of optimization for the desired fuel. In combustion chemistry, only few elementary reactions can be directly measured, and the rest of the reactions are estimated. On the other hand, the experimental data has been used as reference for the overall predicted results such as laminarburning velocity or the flame structure. However, this overall phenomena involves uncertainty caused by the use of the estimated elementary reactions and cannot be avoided. To include the experimentally determined data into combustion chemistry in detail, simplification (or modeling) of the governing equations is one of the most important steps. In this study, the combination of several simplification methods based on experiments are applied to study low-temperature ignitions. The procedure used shows how to incorporate experimentally determined values into the level of elementary reactions. In addition, the procedure also demonstrates how to evaluate the important elementary reactions and specifically determine their rate constants. These results would enhance the efficiency of research in the field of quantum mechanics calculations and would help to improve the accuracy of detailed reaction mechanism.

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Appendix A Chem. mech. for lean DME/nC7H16 mixture The elementary reactions for the blending fuel (DME and nC7 H16 ) are taken from the Ranzi mechanism [83, 82, 85, 84, 86] for nC7 H16 and the LLNL mechanism [166] for DME. Table A.1a shows the considered elementary reactions. In the table, Rhep , Qhep , and Qhep denote C6 H13CH2 (C7 H15 ), CH2C5 H10CH2 (C7 H14 ), and CHC5 H10CH2 (C7 H13 ), respectively. Rdme , Qdme , and Qdme denote CH3 OCH2 , CH2 OCH2 , and CHOCH2 , respectively. P denotes product of the first-stage ignition, for example, HO2 , H2 O, CH3CHO, C2 H4CHO, CH3COCH2 , nC5 H10 , nC7 H14 , or OCH2 OCHO. The change in molar-concentration C due to the chemical reactions is expressed with using Arrhenius parameters as 

where

dC j dt



nR

chem

= ∑ ν j,i i

nS ν k f ,i C j j,i − kr,i j j nS

∏

∏

ν  C j j,i



  Ea k = A f req T exp − RT B

By assuming constant temperature, the change in C of OOQhep OOH can be written with the rate-of-progress (q) ˙ and production and consumption rates (ω˙ p and c ω˙ ) as follows: 155

156 

APPENDIX A. CHEM. MECH. FOR LEAN DME/NC7 H16 MIXTURE

dCOOQhep OOH dt

 p c = ω˙ OOQ − ω˙ OOQ hep OOH hep OOH chem   = q˙hep,5 f − q˙hep,5r + q˙hep,6   = khep,5 f COOQhep OOH CO2 − khep,5r + khep,6 COOQhep OOH

157

Table A.1: Considered reactions from Ranzi and LLNL mechanism. (a) Reaction for heptane from Ranzi mechanism

Reactions

A f req

B

Ea

OH + nC7 H16

khep,1

→

Rhep + P

0.4793 × 107

2

-2259.83

O2 + nC7 H16

khep,2

→

Rhep + P

0.2045 × 108

2

40722.49

O2 + Rhep

khep,3

Rhep − OO

0.2 × 1013

0

0

Rhep − OO

khep,4

Qhep OOH

0.3 × 1013

0

25100

Qhep OOH + O2

khep,5 f

→

OOQhep OOH

0.2 × 1013

0

0

khep,5r

→

Qhep OOH + O2

0.2 × 1015

0

29000

khep,6

→

OQhep OOH + OH

0.1 × 1013

0

25000

khep,7

OH + P

0.63 × 1015

0

40000

OOQhep OOH OOQhep OOH OQhep OOH

→ →

→

(b) Reaction for DME from LLNL mechanism

A f req

B

Ea

Rdme + P

9.35 × 105

2.3

-780

O2 + DME

kdme,2

→

Rdme + P

4.1 × 1013

0

44910

O2 + Rdme

kdme,3 f

→

Rdme − OO

4.439 × 1019

-1.59

36240

Rdme − OO

kdme,3r

O2 + Rdme

0.2 × 1012

0

0

Rdme − OO

kdme,4

Qdme OOH

6.0 × 1010

0

21580

Qdme OOH + O2

kdme,5

→

OOQdme OOH

7.0 × 1011

0

0

kdme,6

OQdme OOH + OH

4.0 × 1010

0

18580

OH + P

2.0 × 1016

0

40500

Reactions OH + DME

kdme,1

→

OOQdme OOH OQdme OOH

→

→ →

kdme,7

→

Appendix B Ordinary differential equations Ordinary differential equations (ODEs) for scalar x1 and x2 can be written as dx1 = a11 x1 + a12 x2 + zinh,1 dt dx2 = a21 x1 + a22 x2 + zinh,2 dt

(B.1) (B.2)

where a11 , a12 , a21 , and a22 are arbitrary constants. zinh,1 and zinh,2 are inhomogeneous terms. Together with these equations (ODE-system), it can be written in matrix form as d dt

B.1



x1 x2



 =A

x1 x2



 +

zinh,1 zinh,2



 ,

A=

a11 a12 a21 a22

 (B.3)

Homogeneous system

In order to solve the ODE system, firstly, the homogeneous system is considered. The homogeneous system corresponds to the one that neglectszinh in the system. The considered system is d→ − − x = A→ x dt

(B.4)

where x = (x1 , x2 )T . Eigenvalue of the homogeneous system can be determined by det (A − λ I) = 0, where λ and I denote eigenvalue and identity matrix, respectively. This equation is referred to as characteristic polynomial, and λ can written as 159

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

160

λ 2 + λ (a11 + a22 ) + a12 a21 − a11 a22 = 0

λ1,2 =

− (a11 + a22 ) ±



(B.5)

(a11 + a22 )2 − 4 (a12 a21 − a11 a22 ) 2

(B.6)

By applying the λ s to Ax = 0, eigenvector (veig ) can be determined. Note that veig corresponds to each λ . For example, veig,1 can be determined as a result of x, which satisfies the following equation. 

a11 − λ1 a12 a21 a22 − λ1



x1 x2

 =0

(B.7)

By collecting veig s, modal matrix (P) can be obtained, and the inverse matrix of P can be written as 

P = veig,1 , veig,2

−1

P

=







 inv,2  inv,1 , M M

= 



=

veig,11 veig,12 veig,21 veig,22



Mivn,11 Minv,12 Minv,21 Minv,22



P, P−1 , and A are related by the diagonalization of A. −1

P AP =



λ1 0 0 λ2

 =Λ

(B.8)

where Λ denotes the eigenvalues. Here, transform x to y using the following equation  x = Py, y =

y1 y2

 (B.9)

By applying equation B.9 to equation B.4 and multiplying P−1 from the left on both sides of the equation1 , one obtains 1P

is independent from t

B.2. INHOMOGENEOUS SYSTEM

161

d→ −y = P−1 APy = dt



λ1 0 0 λ2



→ −y

(B.10)

In the equation B.16, each differential algebraic equation (DAE) can be independently solved, and they have solutions of the form y1 = c1 exp (λ1t) and y2 = c2 exp (λ2t), where c1 and c2 are constants. The solutions of y1 and y2 demonstrate a general behavior of each dependent variable By applying these solutions to equation B.9, the following ODE-system can be written as  x =

veig,11 veig,12 veig,21 veig,22 

= c1



y1 y2





veig,11 vdig,21

 exp (λ1t) + c2

veig,12 veig,22

 exp (λ2t)

(B.11)

By applying the initial conditions, c1 and c2 can be determined. By differentiating x with respect to t with the obtained c1 and c2 , the time derivative of x (i.e., the left-hand side of equation B.4) can be expressed as d c1 x = dt λ1

B.2



veig,11 vdig,21



c2 exp (λ1t) + λ2



veig,12 veig,22

 exp (λ2t)

(B.12)

Inhomogeneous system

If zinh in equation B.3 is of the form g exp (λ pt) where g and subscript p denote constant and particular solution, respectively, one can assume the form of the particular solution as veig,p exp (λ pt). By applying the assumed solution to equation B.3, one can obtain d x = λ pveig,p exp (λ pt) = Aveig,p exp (λ pt) +g exp (λ pt) dt By canceling exp (λ pt), one can obtain (λ p − A)veig,p = g that can be transformed to the same form as the one used above (A − λ p I)veig,p = −g

(B.13)

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

162

By solving equation B.13 (e.g., Gauss-Jordan elimination), one can obtain veig,p . The solution for x with the particular solution can be written as  x = c1

veig,11 vdig,21



 exp (λ1t) + c2

veig,12 veig,22



 exp (λ2t) +

veig,p1 veig,p2

 exp (λ pt) (B.14)

By applying the initial conditions, c1 and c2 are determined and the solution of equation B.3 can be obtained. Ifzinh is independent of time (i.e., λ p = 0), equation B.15 can be written as  x = c1

B.3

veig,11 vdig,21



 exp (λ1t) + c2

veig,12 veig,22



 exp (λ2t) +

veig,p1 veig,p2

 (B.15)

Steady-state assumption

In this section, an inhomogeneous system which is independent of time (t) is described. The quasi-steady state (QSS) assumption can be used for species in which the consumption rate is significantly faster than its production, and its concentration is significantly small within the entire time range. In order to demonstrate the role of the QSS assumption, the transformed ODE system (i.e., variable is y) with inhomogeneous terms which is independent of t (i.e., g exp (λ pt) where λ p = 0) is considered. By transforming x in equation B.3 to y with equation B.9, and multiplying P−1 from the left on both sides of the equation, the time derivative of y and its solution can be written as

d→ −y = dt



λ1 0 0 λ2



→ −y + P−1z inh

y =y∗ exp (λit) + P−1zinh

(B.16)

(B.17)

where superscript ∗ denotes initial condition. Equation B.17 shows that the inhomogeneous term is the driving term of the system if y∗ = 0 like a system with radicals (see details in section B.4). On the other hand, it also shows a strong influence of the inhomogeneous term ony only when t is significantly small. Therefore, the non-homogeneous term does not play an important role in considering

B.3. STEADY-STATE ASSUMPTION

163

the QSS. By multiplying P and P−1 from the left- and the right-hand sides of both terms in equation B.8, it can be written as A = PΛP−1

(B.18)

By applying this to equation B.4 and multiplying P−1 from the left on both sides of the equation, one can obtain d −1 → − P − x = Λ P−1 → x dt

(B.19)

This equation implies that there are two criteria for the QSS assumption. Firstly, if the corresponding λ has a significantly large and is negative value, the influence of the species on the system is negligible. This λ indicates that the corresponding y exponentially decays and it can be assumed to be zero for longer time scale. Secondly, if only small number of elements in the corresponding row of P−1 have non zero values, this species can be assumed as QSS species. This suggests that time evolutions of corresponding x are explicitly described. If both of the conditions are satisfied, the corresponding species can be assumed as QSS species. An example discussed in section 5.8 is described in detail. The conditions are φ = 0.8 (based on nC7 H16 as fuel), ξ = 10%, V˙mix = 63 l/min, and T ∗ = 620 K (see their notation in chapter 3). The considered species are nC7 H16 , Rhep , Rhep − OO, Qhep OOH,  (i.e., OOQhep OOH, OQhep OOH, and OH. The A and Λ of the system, and the C corresponding x in equation B.4) can be written as ⎛ ⎜ ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 −6604880 0 0 0 0 6604880 −4204 0 0 0 0 4204 −6604880 11801 0 0 0 6604880 −13321 0 0 0 0 1520 0 0 0 0 1520 ⎛

λ1 0 0 0 ⎜ 0 λ2 0 0 ⎜ Λ=⎜ ⎝ 0 0 ... 0 0 0 0 λ7 Continue to the next page.

⎞ ⎟ ⎟ ⎟ ⎠

⎞ 0 −2773526 0 2773526 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ −5 0 ⎠ 5 −2773525

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

164

Continue from the previous page. ⎛ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 −6.6 × 106 6 0 0 0 0 −6.6 × 10 6 0 0 −2.8 × 10 0 0 0 0 0 −5724 0 0 0 0 0 71 0 0 0 0 0 0 0 0 0 0

⎞

0 0 0 0 0 0 0 0 0 0 0 0 0 −77

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

 T  C = CnC7 H16 CRhep CRhep−OO CQhep OOH COOQhep OOH COQhep OOH COH From Λ2 , the following can be seen: first, λ6 demonstrates a constant mode because its value is 0, λ4 and λ7 demonstrate decay because their values are negative, and λ5 demonstrates growth because its value is positive. In addition, from Λ, it is also clearly indicated that λ1 , λ2 , and λ3 demonstrate extremely fast decay. Therefore, the fast decay λ s are considered as a candidates for QSS. The P−1 is shown below. ⎛ ⎜ ⎜ ⎜ ⎜ −1 P =⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 1 0

1.5 −0.5 0 1.1 −10.1 2 11.36

0 0 0 1.1 −10.1 2 11.3

0 −1.4 0 −0.4 −10 2 11.5

0 0 0 −0.4 −10 2 11.5

0 0 0 0 0.7 1 0.7

−0.1 0 1.8 1.1 −10.1 1 11.3

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

The corresponding rows for λ1 , λ2 , and λ3 are the top three rows. From the 1st row, it can be seen that only the 2nd and the 7th elements (i.e., Rhep and OH) play an important role for the ODE. The ODE can be written as    d  1.5CRhep − 0.1COH = λ1 1.5CRhep − 0.1COH dt

(B.20)

  ∗ 1.5CRhep − 0.1COH = 1.5CRhep − 0.1COH exp (λ1 t)

(B.21)

and solved as  2 The

diagonal element of Λ can be expressed by diag (Λ)

B.4. TWO INTERMEDIATES MODEL FOR NC7 H16

165

In a similar manner, the solution for the 2nd and the 3rd rows can be written as 

  ∗ −0.5CRhep − 1.4CQhep OOH = −0.5CRhep − 1.4CQhep OOH exp (λ2 t) (B.22) 1.8COH = (1.8COH )∗ exp (λ3 t)

(B.23)

The results show that CRhpe , CQhep OOH , and COH decay significantly faster than other species, their decay do not have an influence on the time evolution of other species. Therefore, these species can be assumed as QSS species.

B.4

Two intermediates model for nC7H16

Peters et al. [114] analyzed low temperature chemistry (LTC) of nC7 H16 with two intermediates during the induction period. By assuming partial equilibrium of isomer of Rhep and negligible influence of temperature, and using steady-state assumption for Rhep , Qhep OOH, OOQhep OOH, and OH, the ODE is analytically solved, and one can analytically express τ1st . In their paper, it was pointed out that reactions 11 and 15 (corresponding reactions are the Rhep − OO- and OQhep OOHconsuming reactions) play an important role on LTC of nC7 H16 during the induction period. Note that they used a mechanism different from the one used in this study and the Arrhenius parameters used are not same as shown in table A.1a. Figure B.1 shows a comparison of LTCs between their mechanism and the one used in this study. Isomer of Rhep and additional product from OQhep OOH are considered in Peters’ reaction mechanism. However, the same reaction pathways pointed out are considered in both mechanisms and essentially they are identical. Therefore, the reaction numbers from table A.1a is used to describe Peters’ finding in detail in the following section. Considering the ordinary differential equations (ODEs) for the induction period one get dCnC7 H16 =−q˙hep,2 − q˙hep,4 − q˙hep,7 dt dCRhep −OO = q˙hep,2 + q˙hep,7 dt dCOQhep OOH = q˙hep,4 − q˙hep,7 dt

(B.24) (B.25) (B.26)

166

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

nC7H16

nC7H16

Rhep

1-Rhep

2-Rhep

Rhep-OO

Rhep-OO Reaction 11

QhepOOH

QhepOOH

OOQhepOOH

OOQhepOOH HO2Q’hepOOH

OQ’hepOOH

OQ’hepOOH Reaction 15 ’ O OQhep

C1

Rhep: C6H13CH2 Qhep: CH2C5H10CH2 Q’hep: CHC5H10CH2

Left: Ranzi’s mechanism Right: Peters’ mechanism

C1

Figure B.1: Comparison of LTCs between Peters’ and Ranzi mechanisms.

B.4. TWO INTERMEDIATES MODEL FOR NC7 H16

167

where C and q˙ denote molar concentrations and rate-of-progress. The main innovation in this paper is the assumption of the negligible influence of temperature on the overall oxidation process during the induction period (see details in section 5.7). In addition, nC7 H16 and O2 are assumed to be constant (i.e., they remains at their initial values) because of the lean mixture during the induction period. Therefore, the considered system for the intermediates can be written as

d dt

CRhep −OO COQhep OOH



 =



khep,7 −khep,7

0 khep,4

CRhep −OO COQhep OOH



 +

ε 0

 (B.27)

where ε = khep,2CO2 CnC7 H16 . For simplification, nC7 H16 , Rhep −OO, and OQhep OOH  This is are denoted by I, II, and III, and the concentration vector is denoted by C. an inhomogeneous ODE system and it can be solved using the solution described in appendix B.1. In this system, the Jacobian matrix A is  A=

khep,7 −khep,7

0 khep,4

 (B.28)

By substituting equations B.5 and B.6 into the eigenvalue of the matrix above, λ can be determined as  khep,7 ± λ1,2 = − 2

2 khep,7

+ khep,4 khep,7

4

(B.29)

Together with these obtained λ s and equation B.7, one can obtain P.

P=

khep,7 λ1

khep,7 λ2

1

1

 (B.30)

In addition, by putting λinh = 0 into equation B.13, one can obtain veig,p  veig,p = −

ε khep,4

,−

ε khep,7

T (B.31)

Therefore, from equation B.15, the solution (i.e., concentration as a function of time) can be written as

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

168

 = c1 C

khep,7 λ1

1

 exp (λ1t) + c2

khep,7 λ2

1

 exp (λ2t) +

ε − khep,4 ε − khep,7

 (B.32)

Together with the initial condition, Chep,II = Chep,III = 0 at t = 0. khep,7 khep,7 ε + c2 − λ1 λ2 khep,4 ε = 0 = c 1 + c2 − khep,7

CII∗ = 0 = c1 ∗ CIII

(B.33) (B.34)

From these algebraic equations, c1 and c2 can be written as   khep,4 − λ2 ελ1 c1 = khep,4 khep,7 λ1 − λ2   khep,7 − λ1 ελ2 c2 = − khep,4 khep,7 λ1 − λ2

(B.35) (B.36)

From equations B.32, B.35, and B.36, a time evolution of each species in equation B.27 can be solved. Within the induction period, the intermediate growth plays an important role, and its decaying and constant terms are not important. For example, if λ1 > 0, CII can be written as khep,7 khep,7 ε exp (λ1t) + c2 exp (λ2t) − λ1 λ2 khep,4 khep,7 exp (λ1t) ≈ c1 λ1

CII = c1

(B.37) (B.38)

In the same manner for CIII and neglecting the influence of reaction nC7 H16 + O2 (it is also discussed in the section 5.5), time derivative of CI can be written with Chep,II and Chep,III as dCI = −q˙hep,2 − q˙hep,4 − q˙hep,7 dt ≈ −q˙hep,4 − q˙hep,7 = −khep,4CII − khep,7CIII   khep,4 + 1 exp (λ1t) = −c1 khep,7 λ1

(B.39) (B.40) (B.41) (B.42)

B.5. MECHANISM REDUCTION

169

This equation demonstrates that CI decrease as the intermediates are formed. From this, by replacing c1 , one can get    ε khep,4 − λ2 khep,4 + λ1 dCI = exp (λ1t) dt khep,4 (λ1 − λ2 )

(B.43)

By integrating from t = 0 to τ1st and assuming CI = 0 at t = τ1st , one can obtain following equation.    k − λ + λ ε k 2 1 hep,4 hep,4 CI∗ ≈ exp (λ1 τ1st ) λ1 khep,4 (λ1 − λ2 )

(B.44)

This equation implies that τ1st can be expressed by using only a value at the initial condition. By manipulating the equation, one can obtain τ1st as τ1st

1 = log λ1



C∗ λ k (λ − λ2 )   I 1 hep,4  1 ε khep,4 − λ2 khep,4 + λ1

 (B.45)

Because ε = khep,2CO2 CI and the assumption that Chep,I and CO2 are constant during the induction period, the equation can be rewritten as τ1st

1 = log λ1



λ1 khep,4 (λ1 − λ2 )    khep,2CO2 khep,4 − λ2 khep,4 + λ1



Furthermore, by assuming that λ1 = −λ2 = λ , khep,4 +λ ≈ khep,4 , and λ ≈ one can obtain 1

τ1st ≈ $ log khep,4 khep,7

B.5

2khep,7 khep,2CO∗ 2

(B.46) $ k4,0 k7,0

 (B.47)

Mechanism reduction

Production rate screening (PRS) and Rate-of-progress screening (RPS) are applied to reduce the number of species and reaction from a detailed reaction mechanism for nC7 H16 . The number of species and reactions were successfully reduced from 234 to 48 and from 5487 to 366, respectively. The optimal threshold values are used as the criteria for these screening processes (i.e., ΞPRS and ΞRPS , see details in

170

APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS

chapter 5). Figures B.2a and B.2b show the variation in the number of species and reactions with the change in their threshold values. Result of PRS (figure B.2a) shows a significant change when a value crosses the optimal point. On the other hand, the result from RPS (figure B.2b) does not show a notable change when a value crossing the optimal point. These results clearly imply that the monitoring multiple variables (i.e., peak temperature, ignition delay time, number of species, and number of reactions) should be carefully observed during the screening processes.

B.5. MECHANISM REDUCTION

171

130 120 110

nS

100 90 80 70 60 50 40 10

−4

−3

Ξ

PRS

10 3

[kmol/(m s)]

(a) Production rate screening (PRS)

400

390

nR

380

370

360

350 −10 10

10

Ξ

RPS

−9

3

[kmol/(m s)]

(b) Rate-of-progress screening (RPS)

Figure B.2: Influence of threshold values of screening process on the number of species and reactions.

Appendix C Time evolution of functions C.1

Quadratic (2nd-degree) polynomial

As described in section B.3, the eigenvalue (λ ) of the system describes the time evolution of a system. It (λ ) may have an imaginary part. This imaginary part plays an important role on the time evolution of a function, and it also plays a major role in simplifying the function. In order to describe the general behavior of the function, a quadratic (2nd-degree) polynomial as the characteristic equation is considered. λ 2 + a2,1 λ + a2,0 = 0

(C.1)

where a denotes coefficient in a polynomial with subscript 2 denoting the degree of the polynomial and the number that follows denotes the order of the term. This solution and its discriminant (D2 ) are expressed as   1 −a2,1 ± a22,1 − 4 a2,0 2 D2 = a22,1 − 4 a2,0 λ=

(C.2) (C.3)

when D2 = 0, λ has only real part. On the other hand, λ has positive or negative imaginary part when D2 = 0. To demonstrate the time evolutions of the quadratic 173

APPENDIX C. TIME EVOLUTION OF FUNCTIONS

174 2.5 2

D=0

1.5 1

a

2,1

0.5 0 −0.5 −1 −1.5 −2 −2.5 −1.5

−1

−0.5

0

0.5

1

1.5

a2,0

Figure C.1: Considered conditionsa2,1 − a2,0 phase diagram. polynomial, 12 conditions depending on a2,1 , a2,0 , and D2 are considered. The considered conditions are shown with symbols in the a2,1 − a2,0 phase diagram in figure C.1. By comparing equations B.5 and C.11 , and setting a11 = a22 and a21 = 1 for simplification, one can obtain a11 = a22 = − a12 =

a22,1 4

a2,1 2

− a2,0

(C.4) (C.5)

The equilibrium point of the system is (x1 , x2 ) = (0, 0). For simulations, the initial conditions is set to (x1 , x2 ) = (0, 1). Figure C.2 shows the time evolution of the variables at each condition (entries of the Jacobian matrix). The corresponding relationship between the two scalar variables are plotted in the phase diagrams in figure C.1. In these figures, the symbol denote the initial condition and lines denotes the trajectories of the functions.

1a

2,0

= a11 a22 − a12 a21 and a2,1 = −a11 − a22

C.1. QUADRATIC (2ND-DEGREE) POLYNOMIAL

1

175

1

1

0.8

0.8

0.5

x

0.6

x2

x2

2

0.6 0

0.4

0.4 0.2 −0.5

0.2 0

−1 −1

−0.5

0

x

0.5

0

−0.2 −1

1

−0.5

0

x

1

0.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

1

1

17

x 10 5

0.8

0.9

4

0.6

0.8

3

x

x2 0.4

2

1

x2

1

2

0.7

0.2

1

0.6

0

0 −0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.5 −0.1

0.8

0

0.1

0.2

x

1

0.4

0.5

0

0.6

1

2

3

104

x 10

5

6

7 17 x 10

86

x 10

x 10

3.5

12

4

x1

1

42

14

0.3

x

3.5

3

3

2.5

2.5

10

6

2

x2

x2

x2

8

1.5

4

2 1.5

1

1

0.5

0.5

2 0 0 0

5

10

x1

1

2

3

4

x1

x 10

−0.5

5 104 x 10

−1

0

1

2

3

4

x1

3

x 10

86

x 10

21

x 10 1.5

2.5

1

5

2

0.5

4

1.5

0

3

x2

x2

6

x2

0

43

73

7

0

−0.5

15 42 x 10

1

2

−0.5 −1

0.5

−1.5

1

−2

0 0

−2.5 −2

0

2

4

x1

−0.5

6 73

x 10

−2

−1

0

x

1

1

2 43 x 10

−1

0

1

2

x1

3

4 21

x 10

Figure C.2: Time evolutions of functions at the considered conditions in figure C.1 shown x1 − x2 phase diagram.

APPENDIX C. TIME EVOLUTION OF FUNCTIONS

176

From the figures, two types of trajectories can be found depending on whether they reach the equilibrium point. At a2,1 > 0 and a2,0 > 0 in figure C.1 (corresponding symbols: circle, inverted triangle, and diamond), the trajectories reach the equilibrium point. On the other hand, the trajectories move away from the initial points and diverge with the other conditions. These figures also show that the imaginary part of the function plays an important role in its trajectories (i.e., depending on D2 ). At D2 < 0 in figure C.1 (corresponding symbols: diamond and asterisk), the trajectory spiral into the equilibrium point or spiral away from the initial point. Under other conditions, the trajectories do not follow a spiral-like locus and move linearly away from the initial point. An exceptional condition is at a2,1 = 0 and a2,0 > 0 (correspond symbol: x). At these conditions, the function has only imaginary part. The absence of a real part does not allow the trajectory to reach or to leave from the equilibrium point. Therefore, the trajectory is around the equilibrium point in an arc. This is a typical Lyapunov stability.

C.2 nth roots A complex number (zcomp ) to the nth power with the Euler’s theorem can be written as zncomp = rn exp (θn ) = rn (conθn + i sinθn )

(C.6)

Here, rn , θn and i denote the radius of the circle and the argument to the nth power, and imaginary unit. By raising to the 1/nth power based on the de Moivre theorem, the equation can be rewritten as zcomp = r1/n (cosθ + i sinθ ) ,

kcomp = 0, 1, 2, . . . n − 1

(C.7)

where θ=

θn + 2kcomp π n

(C.8)

Here, θ denotes argument of zcomp . This equation indicates that the nth root is located on a circle with a radius of r1/n , and the interval between the roots in angle is 2nπ . For example, if z3comp with r = 1 and θn = 90 degree (n=3), zcomp is located at θ =30, 150, and 270 degree on the unit circle. In order to consider simplification of a function, the time evolution of each root is important. To observe the time evolution of the root, real and imaginary parts (zre and zim ) of zcomp are defined as

C.2. NTH ROOTS

177

1.5 1

z

im

0.5 0 −0.5 −1 −1.5 −1.5

−1

−0.5

0

0.5

1

1.5

z

re

Figure C.3: Considered conditions in the complex plane.

zcomp = zre + i zim

(C.9)

In a complex plane, 8 conditions depending on zre and zim are examined as shown in figure C.3. Equation C.9 denotes the eigenvalue of equation C.2. By applying zcomp as λ , one can obtain from zcomp the corresponding x1 and x2 in an ODE system. This conversion allows one to evaluate the time evolutions of zcomp . In other words, by converting a complex number to a2,1 and a2,0 , the corresponding condition can be plotted and the behavior of the function can be described. From equations C.2 and C.9, one can convert from zcomp to a2,1 and a2,0 .

a2,1 = −2 zre

(C.10)

a2,0 = z2re − zim

(C.11)

With the same simplification as above, the elements in equation B.4 can be obtained as

APPENDIX C. TIME EVOLUTION OF FUNCTIONS

178 3

D=0 2

a2h,1

1 0 −1 −2 −3 −1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

a

2h,0

Figure C.4: Focused conditions in the a2,1 − a2,0 phase diagram.

a11 = a22 = zre a12 = zim

(C.12) (C.13)

Figure C.4 shows the corresponding condition from figure C.3. The same symbols in figures C.1 and C.2are used. From these figures, the time evolution of the function in algebraic and complex expressions are visualized. By calculating zre and zim and finding a quadrant in the complex plane, one can obtain the corresponding symbol in the figure C.4. In addition, by finding the symbol in figure C.2, one can estimate the behavior of the function.

C.3

Influence of zim on the time evolution of zcomp

As mentioned, the oscillation is caused by the imaginary number of the eigenvalue. In general, when the complex number is in the fourth quadrant (i.e., zre > 0 and zim < 0), the function exponentially grows and eventually oscillates. On the other hand, for dynamic behavior, the frequency of the oscillations are also important. The expression of one of the roots of zcomp to the nth power with r and θ is

C.3. INFLUENCE OF ZIM ON THE TIME EVOLUTION OF ZCOMP 50

10

5000

40

10

179

1

3000 0.5

30

10

x2

2

x

x

2

1000

20

0

−1000

10

−0.5 10

−3000

10

0

10

0

20

40

60

80

100

−1

−5000 0

20

(a) θ = 0 degree

40

60

80

100

Time [s]

Time [s]

(b) θ = −85 degree

0

20

40

60

80

100

Time [s]

(c) θ = −90 degree

Figure C.5: Influence of θ on the time evolution of the function.

 (zre )2 + (zim )2   zim θ = arctan zre r=

(C.14) (C.15)

For convenience, the example with r = 1 is considered. Figure C.5 shows the time evolution of the function with varied θ . At θ = 0 (i.e., the root is on the real axis), the function exponentially diverges without oscillations2 . Once θ is close to −π like θ = −85 degree, the function exponentially diverges, but the oscillating behavior can be easily distinguished. However, θ = −π (i.e., on the imaginary axis), the function does not diverge because of the Lyapunov stability.

In order to simply the function, it is important to know the influence of the imaginary number on the solution. If the influence from the imaginary number is significantly small, the complex number can be treated as a real number (i.e., neglecting the zim ). Figure C.6 shows the value of x2 at a valid θ value. This result shows a strong influence of zim for a large value of t. At t = 4 s, the difference between θ = -8 and 0 degree is about one order of magnitude. On the other hand, at t = 1 s, the function can be treated as a real number within |θ | < 2 degree because the imaginary part has an extremely weak influence. It has to be noted that the neglect of zim is valid within a limited time duration. By considering the influence of zim and considering the time scale, different simplification of the function can be carried out. symbol is  in figure C.3 and it on the discriminant in the fourth quadrant in figure C.4. In addition, corresponding figure are the center at the bottom row in figure C.2. 2 Corresponding

APPENDIX C. TIME EVOLUTION OF FUNCTIONS

180

2

x

2

10

θ=0 θ = −2 θ = −4 θ = −6 θ = −8

1

10

0

10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [s]

Figure C.6: Influence of θ on the time evolution of the function.

Appendix D Solution of higher polynomial D.1

Quartic (4th-degree) polynomial

There are some solutions for the quartic polynomial since the 18th century. Here, Ferrari’s method is described. A quartic polynomial expressed as λ 4 + a4,3 λ 3 + a4,2 λ 2 + a4,1 λ + a4,0 = 0

(D.1)

where a denotes constants for the polynomial. Its subscripts denotes the degree of the polynomial and the order of coefficient. In order to cancel the third-order term, the equation become depressed quartic polynomial by changing the variable a λ = s − 4,3 4 . s4 + b4,2 s2 + b4,1 s + b4,0 = 0

(D.2)

where

a4,2 a4,3 3 1 b4,2 = a4,2 − a24,3 , b4,1 = a4,1 + a34,3 − 8 8 2 2 a4,2 a4,3 a4,1 a4,3 3 4 b4,0 = a4,0 − a4,3 + − 256 16 4 where b denotes constants for depressed polynomial. In order to make both sides have perfect squares in the equation, move terms that are below the 2nd-order 181

APPENDIX D. SOLUTION OF HIGHER POLYNOMIAL

182

2

term from the left-hand side to the other side and add us2 + u4 to both sides of the equation. v is the unknown variable at the current step and is defined later1 . 

  2 u 2 u 2 − b4,0 s + = (u − b4,2 ) s − b4,1 s + 2 4 2

(D.3)

Note that the right-hand side is the 2nd-order polynomial with respect to u. In order to form a perfect square on the right-hand side, its discriminant (D) has to be zero. This equation is a cubic polynomial with respect to u, and this has to be solved separately to obtain the solution of the quartic polynomial (see its solution in the section D.2). D is expressed as  D2 = b24,1 − 4 (u − b4,2 )

 u2 − b4,0 = 0 4

(D.4) 2

By manipulating this equation, one can obtain part of equation D.3 ( u4 − b4,0 = b24,1

4(u−b4,2 )

). Moreover, by substituting the equation and squaring both sides,  $ b4,1 u s + = ± u − b4,2 s − 2 2 (u − b4,2 ) 2

(D.5)

From this, one can obtain the 2nd-order polynomial with respect to u as

s2 ∓

$

 u − b4,2 s +

b4,1 u ± $ 2 2 u − b4,2

 =0

(D.6)

This equation is a quadratic polynomial in s. Therefore, from each equation, one can solve for the solution of the quartic polynomial in terms of u.  ⎧ ) *  ⎪ $ b ⎪ 1 ⎪ u − b4,2 ± (u − b4,2 ) − 4 u2 + √ 4,1 ⎪ ⎨2 2 u−b4,2  ) s= *  ⎪ $ ⎪ b 4,1 1 u ⎪ ⎪ ⎩ 2 − u − b4,2 ± (u − b4,2 ) − 4 2 − 2√u−b4,2 1 u3 − b

4,2 u

2 − 4b

4,0 u +

  4b4,2 b4,0 − b24,1 = 0

(D.7)

D.2. CUBIC (3RD-ORDER) POLYNOMIAL

D.2

183

Cubic (3rd-order) polynomial

A general cubic polynomial is expressed as λ 3 + a3,2 λ 2 + a3,1 λ + a3,0 = 0

(D.8)

In order to solve equation D.4, variables has to be transformed as λ = u, a3,2 = −b4,2 , a3,1 = −4b4,0 , and a3,0 = 4b4,2 b4,0 − b24,1 . By substituting the 2nd-order a term (u = w − 3,2 3 ), one can obtain the depressed cubic polynomial as ψ 3 + b3,1 ψ + b3,0 = 0

(D.9)

where

b3,1 = a3,1 −

a23,2

3 a4,2 a4,0 2 3 b3,0 = a3,0 + a4,2 − 27 3

(D.10) (D.11)

To solve this equation, additional variables, such as ψ = Φ + Ψ, should be introduced. By comparing D.9 and the sum ofψ raised to the third power and b3,1 ψ, one can obtain Φ3 + Ψ3 + (Φ + Ψ) (b3,1 + 3ΦΨ) + b3,0 = 0

(D.12)

To satisfy the equation, two conditions can be derived. Φ3 + Ψ3 + b3,0 = 0 (Φ + Ψ) (b3,1 + 3ΦΨ) = 0

(D.13) (D.14)

From equation D.14, one can obtain conditions as b3,1 + 3ΦΨ = 0. In addition, by moving b3,1 from the left- to the right-hand sides and cubing the equation, the conditions can be expressed with cubic Φ and Ψ as Φ3 + Ψ3 = −b3,0 Φ3 Ψ3 = −

b33,1 27

(D.15) (D.16)

APPENDIX D. SOLUTION OF HIGHER POLYNOMIAL

184

By using Viete’s formulas2 with an additional variable (Ω), which is defined later, one can obtain the following equation.

Ω + b3,0 Ω − 2

b33,1 27

=0

(D.17)

This equation is a quadratic polynomial with respect to Ω. Therefore, its solution is 1 Ω= 2

+ −b3,0 ±

4 b23,0 + b33,1 27

 (D.18)

   If Ω is defined to satisfy Ω − Φ3 Ω − Ψ3 = 0, one can obtain its solution as  Φ3 = −

b3,0 + 2

b3,0 Ψ =− − 2 3



b23,0 4 b23,0 4

+ +

b33,1 27 b33,1 27

(D.19) (D.20)

Because of the cubic Φ and Ψ, the solution can be obtained in the same manner as described above. For example, Φ has the following solutions. ⎧+  2 ⎪ 3 b33,1 b3,0 b3,0 ⎪ ⎪ − + + ⎪ 2 4 27 ⎪ ⎪ + ⎨  2 √  3 b3,0 b33,1 b3,0 3 1 Φ= − 2 + + 27 −2 + 2 i ⎪ 4 ⎪ + ⎪  2 ⎪  √  3 ⎪ b3,0 b33,1 ⎪ b3,0 ⎩ −1 − 3i − + + 2 2 2 4 27

(D.21)

From the three solutions of Φ and Ψ, one can obtain the solution which satisfies equation D.15 and D.16. Finally, one can obtain the solution of the cubic polynomial as 2 For

αβ =

c a

f (x) = ax2 + bx + c, assuming solution α and β . The roots satisfy α + β = − ba and

D.2. CUBIC (3RD-ORDER) POLYNOMIAL

185

+ ⎧+  2  2 3 ⎪ 3 3 b b b3,0 b33,1 b b ⎪ 3,0 3,0 3,0 3,1 ⎪ − + + + − − + ⎪ 2 4 27 2 4 27 ⎪ ⎪ + + ⎨  2  2  √  3 √  3 3 b b b3,0 b33,1 b b3,0 3,1 3,0 3,0 3 3 1 1 ψ= + i − + + + − − i − − + − ⎪ 2 2 4 27 2 2 4 27 ⎪ + 2 + 2 ⎪  2  2 ⎪     √ √ ⎪ 3 3 ⎪ ⎩ − 1 − 3 i 3 − b3,0 + b3,0 + b3,1 + − 1 + 3 i 3 − b3,0 − b3,0 + b3,1 2 2 2 4 27 2 2 2 4 27 (D.22)

Appendix E Detailed information of the LFR E.1

Fuel injector and fuel supplying pipe

Figures E.1 show the technical drawing of the fuel injector (FI). The FI consists of two parts with a cavity in them. After manufacturing both of the components, they are welded into one piece (see figure 3.3a). An additional connector is used at the inlet of the FI to connect the fuel supplying pipe. A gasket made of graphite is used to avoid leakages. The fuel supplying pipe is made of stainless steel1 , and its inner and outer diameters are of diameters 4 and 6 mm, respectively. The length between the FI and the fuel vaporizer (FV) is 4 m. As described in chapter 3, a portion of it (1.5 m) is in the oven and is heated to the same temperature as that cp μ of the oven. If the Prandtl number (Pr = ktherm where ktherm denotes the thermal conductivity) and Re are determined, the development of the thermal boundary layer can be estimated because Pr relates the analogy between momentum and thermal boundary layers. The entrance length is proportional to the Re by using the following equation [129]. Lbound = 0.06 Re d

(E.1)

Here, Lbound and d denote the entrance length and diameter of the pipe, respectively. Figure E.2 shows Pr of the supplying fuel (i.e., gaseous fuel and carrier gas) and Lbound . The considered condition is V˙mix = 63 l/min, ξ = 10%, and φ = 0.8 (based on nC7 H16 as the fuel, see their notation in chapter 3). This result shows that Pr can be assumed to be unity. Therefore, by analogy, Lbound can be assumed to be a thermal boundary layer. This result also shows that Lbound has 1 The

same material as that of the LFR.

187

188

APPENDIX E. DETAILED INFORMATION OF THE LFR

a significantly smaller value than that of the length of the pipe in the oven. The temperature distribution in radial direction can be neglected because of a fully developed thermal boundary layer. In other words, the gaseous fuel through the pipe is heated to the oven temperature. By supplying the fuel at the same temperature as that of the oven and oxidizer, it is shown that the adiabatic condition in the LFR is maintained.

E.2

Mesh for the CFD calculations

Figure E.3 shows the computational grid used. As described in section 4.2, the domain covers from 20 mm upstream of the FI tip to 130 mm downstream. The hybrid mesh (the hexahedral cells for core regime and the tetrahedral cells for the rest) with two layers on the wall are used. For the wall, normal wall assumption (hydrodynamically smooth) is used2 . The numbers of cells and boundaries are 627,129 and 62,396, respectively. In order to correctly reproduce the fuel injection from the FI, the normal inlet conditions are applied on the flat surface at the circumferentially-arranged 12 holes (see the enlarged image in figure E.3). For simulations, residual tolerances of momentum, turbulent energy, and turbulent energy dissipation are set to 5 × 10−3 , and those of pressure and temperature are set to 2.5 × 10−3 and 1 × 10−4 , respectively.

E.3

2-D analysis

In order to investigate the influence of the velocity profile on the τ1st , the HagenPoiseuille flow in the LFR is considered. As described in section 2.4, the jth species equation can be written as ∂C j vz = Dj ∂z



∂ 2C j 1 ∂C j + ∂ r2 r ∂r





∂C j + ∂t

 (E.2) chem

where  vz = vcl 1 − 2 The



r

2 

RLFR

pipe surface is characterized by the maximum height of the profile of 25.

(E.3)

E.3. 2-D ANALYSIS

189

I

II

Component of the injector I

Component of the injector II

Figure E.1: Detailed technical drawings of the fuel injector.

Pr [−]

0.94

0.17

0.92

0.16

0.9

0.15 Pr L

bound

0.88 0.86 0.84 500

0.14

Lbound [m]

APPENDIX E. DETAILED INFORMATION OF THE LFR

190

0.13

550

600

650

0.12 700

Temperature [K]

Figure E.2: Pr and Lbound of the supplying fuel in the pipe.

r 11 mm z

Figure E.3: Mesh of the model with the FI. Cross section at z = 11 mm and its enlarged image is shown.

E.3. 2-D ANALYSIS

191

Here, vz , vcl , D j , and RLFR denote the axial component of velocity, velocity at the centerline, mass diffusivity of jth specie, and the radius of the LFR, respectively. In general, the species profile is independent of velocity and has two limits. One of the limit is represented by radical type species with a sticking coefficient (Sstick ) of unity. Under this condition, within the mean free path thickness all molecules do not reflect from the walls (this layer is known as Knudsen layer [167]). These molecules stick on the wall and react with reaction partners at the same time (i.e., surface reaction). Due to this surface reaction, the species is completely consumed at the wall and its concentration reaches to zero. By defining the heterogeneous reaction rate (Khetero ) which denotes the derivative in the radial direction of the concentrations on the walls, this condition is expressed as Khetero → ∞ (i.e. C =0 at the walls). The other limit is represented by stable type species such as the fuel with Sstick = 0. Under this condition, all molecules perfectly reflect from the walls and no surface reaction occurs. Because of the absence of the surface reactions, the concentration near the wall remains unchanged. This condition is expressed as Khetero → 0 (i.e., ∂C ∂ r = 0 at the walls). Ju et al. theoretically investigated the velocity and species profiles within the Knudsen layer on the micro scale (i.e., micro flow reactor) [167]. On the other hand, experimentally obtaining Sstick in the LFR is impossible. The concentration profile near the wall cannot be measured due to experimental restrictions, and the quenching distance caused by the heat loss due to the walls has a strong influence on the species concentrations near the walls [146]. In order to overcome this problem, the concentration profile is assumed to have symmetric parabolic form as C f = a f ,2 r2 +C f ,cl

(E.4)

Here, a f ,2 and C f ,cl denotes the coefficient of the polynomial and concentration at the centerline. Note that the variables a f ,2 and C f ,cl are function of z to reproduce the time evolution of its concentration profiles. Figure E.4 shows a schematic of concentration profiles assumed. By defining Khetero (it will be specified later), the boundary condition at the walls (i.e., r = RLFR ) can be written as ∂C f = 2RLFR a f ,2 = Khetero, f C f ,cl ∂r

(E.5)

This concentration profile has a convex shape with respect to the upstream, and Khetero, f has a positive value3 . By differentiating equation E.4 with respect to r and z and applying E.5, equation E.2 can be rewritten as 3 If

considered species is intermediate, its profile has a convex shape respect to the downstream and Kheter has a negative value.

APPENDIX E. DETAILED INFORMATION OF THE LFR

192

C Ccl Cwall

-KheteroCcl

RLFR

r

RLFR- Dr

Dr

Figure E.4: Schematic of Khetero and the concentration profile.

  Kdme r2 ∂Cdme,cl ∂Cdme = 1+ ∂z 2R ∂z

(E.6)

By considering only the centerline (i.e., r = 0) and applying equation E.3 (i.e., vz = vcl ), it can be written as   2D f Khetero, f ∂C f ,cl ∂C f = vcl C f ,cl + ∂z RLFR ∂t chem

(E.7)

Due to the fully developed flow (z = vcl t), z can be converted to t as 1 ∂ ∂ = ∂ z vcl ∂t

(E.8)

Therefore, species equation with Khetero, f on the centerline with respect to t can be written   2D f Khetero, f ∂C f ,cl ∂C f = C f ,cl + ∂t RLFR ∂t chem

(E.9)

As an example, DME as a fuel and considering two intermediates model for simplification, DME, Rdme − OO, and OQdme OOH are denoted by I, II, and III,

E.3. 2-D ANALYSIS

193

respectively. Following the method of DME as described above, the species equation for Rdme − OO and OQdme OOH (i.e., II and III) can be written as ∂CI,cl ∂t ∂CII,cl ∂t ∂CIII,cl ∂t

  2DI Khetero,I ∂CI = CI,cl + RLFR ∂t chem   2DII Khetero,II ∂CII = − CII,cl + RLFR ∂t chem   2DIII Khetero,III ∂CIII = − CIII,cl + RLFR ∂t chem

(E.10) (E.11) (E.12)

As described in chapter 5, only intermediates should be considered within the induction period. In addition, the CO2 is assumed as a constant and the influence of temperature is neglected. Considering this system, the intermediates can be written as d dt



CII CIII



 = A2d

CII CIII



 +

ε2d 0

 (E.13)

and  A2d =

k7 −σII k4 −σIII − k7

 (E.14)

 C where ε2d = kdme,2CdmeCO2 = kdme dme and

2DII KII , RLFR

σII =

σIII =

2DIII KIII RLFR

(E.15)

The eigenvalue (λ ), modal matrix (P), and the particular solution (veig,p ) can be written as (see details in appendix)

λ2d,1, 2d,2 =

P= veig,p =



−

−γ1 ±



γ12 − 4γ0

(E.16)

2

kdme,7 σII +λ1

kdme,7 σIII +λ2

1

1

ε (kdme,7 +σIII ) γp

,−

 (E.17) kdme,4 ε γp

T (E.18)

APPENDIX E. DETAILED INFORMATION OF THE LFR

194

  where γ p = kdme,7 kdme,4 − σII − σII σIII . By applying the same assumptions described in chapter 5, one can obtain the time derivative of the DME (i.e., species indexed I) as   2DI Khetero,I ∂CI = CI − kdme,4CII + kdme,7CIII ∂t RLFR   = σdme,1Cdme − σdme,2 exp λ2d,1t , λ2d,1 > 0, λ2d,2 < 0

(E.19) (E.20)

where σdme,1 =

2DI Khetero,I RLFR

(E.21)

and

σdme,2 = kdme,4 kdme,7 σdme,3 − σdme,4

(E.22)

with ε σdme,3 = γp





kdme,4 +1 σII + λ1

(E.23)

and      ε λ1 σIII + kdme,7 − γ p kdme,4 (σIII + λ2 ) + (σII + λ1 ) (σIII + λ2 ) σdme,4 = γ p (σII + λ1 ) (σIII − σII + λ1 − λ2 ) (E.24) Then one can obtain the first-stage ignition delay time in the LFR (τ1st,2d ) as

τ1st,2d

   CI∗ λ1 − σdme,1 1 = log λ1 − σdme,1 σdme,2

(E.25)

Figure E.5 shows the general relationship between Khetero and concentration profiles of the fuel. As Khetero increases, the profile becomes a parabolic one. The

E.3. 2-D ANALYSIS

195 2RLFR (= dLFR)

v

C

vcl

Ccl

RLFR

r

Khetero

RLFR

r

Figure E.5: Influence of Khetero on the concentration profile. Khetero specifies not only the concentration gradient near the wall, but it also specifies the concentration at the wall (Cwall ). Considering the two limits, Khetero → ∞ and Khetero → 0, the Cwall is not zero. Virtual concentration (Cvir ) is assumed as the intermediate species type (i.e., Rdem − OO or OQdme OOH) profile with the same Khetero at the wall. In addition, by specifying Cvir = 0 at the wall, one can obtain the Cvir profile as follows. For convenience, the subscript, which denotes the species (i.e., II or III) will be omitted henceforth.  Cvir = Cvir,cl 1 −



1

2 

RLFR

(E.26)

Note that this concentration profile has the same curve as CII or CIII , but it is shifted to a zero concentration at the wall (i.e,. Cwall,vir = 0). By differentiating it with respect to r and considering it at the wall (r = RLFR ), Cvir,cl ∂Cvir = −2 ∂r RLFR

(E.27)

By comparing equations E.5 and E.27, one can obtain Cvir,cl =

KheteroCcl RLFR 2

(E.28)

APPENDIX E. DETAILED INFORMATION OF THE LFR

196

T* [K] 667

588

556

0−D 2−D

3

10

τ1st [ms]

625

2

10

1

10

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1000/T* [K−1]

Figure E.6: Comparison between the 0-D flame calculation and the 2-D model. The Cwall can be obtained by Ccl −Cvir,cl . Note that the Cwall has to be a positive value. By considering them, one can obtain a restriction for the Khetero as Khetero
2 m). Therefore, Tincrease should be subtracted to observed the temperature peaks in the LFR. Hence, δ T in the oven can be calculated by δ T = T − Tre f − Tincrease

(E.36)

Here, T denotes the measured temperature. For example, 30 K is subtracted from the measured values where z >2 m in the case for Toven = 880 K (i.e., temperature difference between Tre f and Toven is 30 K). This figure clearly demonstrates that there is one peak outside the oven which is at 600 K. On the other hand, a peak inside the oven appears only when Toven =880 K. Figure E.9b shows the time

200

APPENDIX E. DETAILED INFORMATION OF THE LFR

evolution of δ T (δ T  , δ T  is an instantaneous value and is the sum of the time averaged value (δ T ) and its fluctuation) where z = 2.57, 2.77, 3.07, and 3.37 m at Toven = 880 K. Their locations correspond to around the second peak shown in figure E.9a. At z = 2.57, δ T  is close to zero. This small value indicates that the difference between the set value and measured temperature at this location is kept small. On the other hand, at z = 2.77 and 3.07 m, δ T  shows several peaks. Each peak appears only for about 20 s. The small and large peaks at z = 2.77 and 3.07 m demonstrate that the peak temperature caused by the SSI is transported by convection, and it extinguishes before z = 3.37 m. δ T  at z = 3.37 shows the highest δ T  because the burned gas from the SSI do not show any peaks. Figure E.10 shows the influence of Toven on XCO and XCO2 in the exhaust gas with the same mixture as the one used in figure E.9a. As Toven increases, XCO and XCO2 show an opposite behavior. At the Toven = 850 K, a large amount of CO exists in the exhaust gas. Note that this measured value is significantly higher than the one shown in figure 5.1. This CO is mainly produced by FSI outside the oven which is kept at 600 K. Although the oven is kept at a higher temperature than that outside (i.e., T ∗ < Toven at z > 2 m of LFR), this temperature is still insignificant to activate the CO2 -forming reaction from CO. At Toven = 880 K, XCO and XCO2 showed closer values. As shown in figure E.9b, SSI spontaneously occurs and consumes large amount of CO (i.e., CO2 is produced). On the other hand, during this spontaneous phenomenon, the exhaust gas is continuously sampled at the end of the LFR. Therefore, at the end of the LFR, partially burned gas (i.e., mixture of spontaneously produced burned gas of SSI and continuously produced products from the FSI) is taken as the gas sample. Therefore, not only CO, but also a large amount of CO2 is detected in the exhaust gas. At Toven = 890 K, XCO decreases and XCO2 increases. They indicate that the influence of the FSI becomes smaller than the one in the SSI. These results experimentally demonstrate that the heat loss due to the walls plays an important role in the SSI. By controlling the heat loss, one can observe only the FSI due to freezing chemical reactions after the FSI. Furthermore, one can experimentally observe the FSI and the SSI in the LFR which can validate chemical kinetics in the negative temperature coefficient regime.

E.5. SPONTANEOUS 2ND-STAGE IGNITION IN THE LFR

201

60 T

= 850 K

T

= 880 K

oven

50

oven

δT [K]

40 30 20 10 0 0

100

200

300

400

500

600

700

L [m]

(a) Time averaged temperature profile at Toven = 850 and 880 K

80 z = 2.57 m z = 2.77 m z = 3.07 m z = 3.37 m

70

δT  [K]

60 50 40 30 20 10 0 0

20

40

60

80

100

120

140

160

Time [s]

(b) Time evolution of local temperatures at Toven = 880 K

Figure E.9: Temperature measurements with V˙mix = 10 l/min, φ = 0.8, ξ = 10%, and T ∗ = 600 K

APPENDIX E. DETAILED INFORMATION OF THE LFR

202

−3

x 10

CO CO

8

2

7

X [−]

6 5 4 3 2 1 0

850

860

870

T

oven

880

890

[K]

Figure E.10: Influence of Toven on XCO and XCO2 with the same mixture as the one used in figure E.9a.