LASER-BASED DIAGNOSTICS FOR HYDROCARBON FUELS IN …

LASER-BASED DIAGNOSTICS FOR

HYDROCARBON FUELS IN THE LIQUID AND VAPOR PHASES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Jason Morgan Porter

November 2009

iv

Abstract

Infrared laser-absorption diagnostics are widely used in combustion research for fast,

sensitive, and non-intrusive measurements of species concentration, temperature, and

pressure. A large number of species important to combustion have been successfully

measured including: H2O, CO2, CO, NO2, NO, and vapor-phase hydrocarbon fuels

and fuel blends. These measurements help designers, operators, and researchers of

engines and burners to reduce pollution, increase efficiency, and study combustion

chemistry.

Laser-absorption measurements of fuel concentration are often made at mid-infrared

wavelengths near 3.4 µm, which overlap with the strong C-H stretch vibrational tran-

sitions of hydrocarbons, ensuring sensitive detection even for short measurement path

lengths. Recent advances in laser technology have produced tunable mid-infrared

laser sources near 3.4 µm, using difference-frequency-generation (DFG). Previous re-

searchers in this laboratory have used DFG lasers operating at two wavelengths in

shock tube studies to simultaneously measure temperature and species concentration

for several gaseous fuels and fuel blends.

This thesis presents new applications of DFG lasers for multiphase (vapor and

liquid) measurements of hydrocarbon fuels in harsh environments. Quantitative mea-

surements of the real and imaginary refractive index spectra of several liquid hydro-

carbons are used to develop two novel multi-phase diagnostics: a three-wavelength

diagnostic for fuel-vapor mole fraction and temperature in evaporating aerosols, and a

two-wavelength diagnostic for fuel-vapor mole fraction and liquid fuel film thickness.

Quantitative absorption spectra for several hydrocarbon fuels in the liquid phase

at 25 ◦C are presented. Measurements of toluene, n-dodecane, n-decane, and three

v

samples of gasoline were made over the spectral region 2700− 3200 cm−1 to support

development of mid-infrared laser-absorption diagnostics for measurements of fuel

vapor in the presence of liquid films and aerosols. A procedure for quantitative FTIR

absorption measurements of strongly absorbing liquids is described and the resulting

absorption spectra are compared with previously measured absorption spectra in the

vapor phase. The measured absorption spectra for liquid gasoline are shown to scale

with the volume percent of olefin, alkane, and aromatic hydrocarbons in each sample.

Finally, the observed frequency shift of ∼8 cm−1 in the spectra of vapor and liquid

hydrocarbons near 3.4µm is discussed, which is important for liquid film thickness

measurements.

The development of a 3-wavelength mid-infrared laser-based absorption/extinc-

tion diagnostic for simultaneous measurement of temperature and vapor-phase mole

fraction in an evaporating hydrocarbon fuel aerosol (vapor and liquid droplets) is de-

scribed. The measurement technique was demonstrated for an n-decane aerosol with

D50 ∼ 3 µm in steady and shock-heated flows with a measurement bandwidth of 125

kHz. Laser wavelengths were selected from FTIR measurements of the C-H stretching

band of vapor and liquid n-decane near 3.4 µm (3000 cm−1), and from modeled light

scattering from droplets. Measurements were made for vapor mole fractions below 2.7

percent with errors less than 4 percent, and simultaneous temperature measurements

over the range 300 K < T < 900 K were made with errors less than 3 percent. The

measurement technique is designed to provide accurate values of temperature and

vapor mole fraction in evaporating polydispersed aerosols with small mean diameters

(D50 < 10 µm), where near-infrared laser-based scattering corrections are prone to

error.

Finally, a 2-wavelength mid-infrared laser-based absorption diagnostic for simul-

taneous measurements of vapor-phase fuel mole fraction and liquid fuel film thickness

is presented. The measurement technique was demonstrated for transient n-dodecane

liquid films in the absence and presence of n-decane vapor. Laser wavelengths were

selected from FTIR measurements of the C-H stretching band of vapor n-decane and

liquid n-dodecane near 3.4 µm (3000 cm−1). n-Dodecane film thicknesses < 20 µm

were accurately measured in the absence of vapor, and simultaneous measurements

vi

of n-dodecane liquid film thickness and n-decane vapor mole fraction (300 ppm) were

measured with < 10 % uncertainty for film thicknesses < 10 µm. A potential ap-

plication of the measurement technique is to provide accurate values of fuel-vapor

mole fraction in combustion environments where strong absorption by liquid fuel or

oil films on windows make conventional direct absorption measurements unfeasable.

vii

Acknowledgments

This thesis represents the culmination of over 11 years of study. In that time, I have

attended two community colleges: Butte College in Oroville, CA, and Utah Valley

State College in Orem, UT, and three universities: Brigham Young University in

Provo, UT, the University of Texas at Austin, and Stanford University in Palo Alto,

CA. In that time, I also married my wife Marilyn and we have had three children:

Dallin (7) while at BYU; Sarah (5) while an intern at Sandia National Laboratories

in Albuquerque, NM; and Ashley (2) while at Stanford. This has been a long journey,

with many advances and setbacks, but overall I have loved my time as a student.

Academically, I have been fortunate to study under some fantastic mentors of

which I name a few. Larisa Call (from my hometown near Corning, CA) tutored

me in math as a teenager and gave me a love for math and the confidence to learn

independently. Paul Mills and Gary Carlson at UVSC (now UVU) taught me to love

physics and calculus, respectively. Matt Jones at BYU taught me thermodynamics

and let me assist him with consulting in energy. Brent Adams at BYU was the first

to encourage me to pursue a PhD. Jack Howell, at UT Austin, advised my masters

work, taught me radiation heat transfer, provided me with a valuable internship at

Sandia, Albuquerque, and helped me obtain the NDSEG fellowship, which funded

much of my PhD work. His guidance and help were invaluable. The many “Han-

son students” I have worked with over the past four years have taught me so much

and have made working a joy. Jay Jeffries at Stanford provided invaluable guidance

and countless hours reviewing papers and presentations. Dave Davidson at Stanford

helped with all things experimental and gave me needed encouragement. My thesis

advisor at Stanford, Ron Hanson, taught me how to perform quality research and

viii

produce professional presentations and papers. I am especially grateful for his un-

derstanding of my unique work-life balance issues. He always pushed me to do my

best, but never pushed too hard. Ultimately, he made studying at Stanford possible,

fulfilling a childhood dream.

I save the most important thank you’s for last: my family. I would like to thank

my entire extended family for their encouragement; they always thought of me as

their “smart” brother, son-in-law, etc. at Stanford, even when I felt inadequate.

My parents provided ample opportunities for me to work and learn, and instilled

self confidence. My older brother, Mike, kept me grounded through weekly phone

conversations. My children, Dallin, Sarah, and Ashley, gave me their enthusiasm and

love. Most importantly, I want to thank my wife Marilyn for her perfect devotion

and patience. Neither she nor I knew at the start how long this educational journey

would be, but through it all she has been an outstanding companion and friend.

Above anyone else, she has made this possible.

ix

Contents

Abstract v

Acknowledgments viii

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Laser-absorption spectroscopy . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Beer’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 The absorption cross-section . . . . . . . . . . . . . . . . . . . 6

1.2.3 Measuring temperature . . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 Modifications to Beer’s law . . . . . . . . . . . . . . . . . . . . 12

1.3 The refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Beer’s Law from Maxwell’s equations . . . . . . . . . . . . . . 12

1.3.2 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Kramers-Kronig relation . . . . . . . . . . . . . . . . . . . . . 18

1.4 Light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.1 Mie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Droplet distributions . . . . . . . . . . . . . . . . . . . . . . . 23

2 Measurements of Absorption Spectra in Liquid Fuels 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Absorption measurements in liquids . . . . . . . . . . . . . . . . . . . 27

2.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 27

x

2.2.2 Measurement procedure . . . . . . . . . . . . . . . . . . . . . 27

2.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 Hydrocarbons measured . . . . . . . . . . . . . . . . . . . . . 35

2.3.2 Plots of absorption spectra for liquid and vapor . . . . . . . . 36

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Band strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2 Spectral shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4.3 Gasoline composition . . . . . . . . . . . . . . . . . . . . . . . 42

3 Evaporating Fuel Aerosols 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Laser diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Laser absorption in gases . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Laser extinction in absorbing aerosols . . . . . . . . . . . . . . 47

3.2.3 Droplet extinction model . . . . . . . . . . . . . . . . . . . . . 48

3.2.4 Liquid optical constants . . . . . . . . . . . . . . . . . . . . . 50

3.2.5 Droplet extinction calculation . . . . . . . . . . . . . . . . . . 50

3.2.6 Wavelength selection . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.7 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Validation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Optical arrangement . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.2 Aerosol flow-cell experiment . . . . . . . . . . . . . . . . . . . 60

3.3.3 Flow cell results . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.4 Shock tube experiment . . . . . . . . . . . . . . . . . . . . . . 63

3.3.5 Shock tube results . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Liquid Fuel Films 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Fuel films and measurement techniques . . . . . . . . . . . . . 67

4.2 Two-phase laser-absorption measurements . . . . . . . . . . . . . . . 69

4.2.1 Beer’s law: vapor and liquid . . . . . . . . . . . . . . . . . . . 69

xi

4.2.2 Laser wavelength selection . . . . . . . . . . . . . . . . . . . . 72

4.3 Refractive index matching . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Measuring I0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.2 Modeling I0 in the presence of a film . . . . . . . . . . . . . . 76

4.3.3 FTIR measurements of n-dodecane films . . . . . . . . . . . . 78

4.4 Demonstration experiments . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.2 Demonstration of liquid film measurement . . . . . . . . . . . 83

4.4.3 Demonstration of fuel vapor and liquid film measurement . . . 85

4.4.4 Films and vapor composed of same fuel . . . . . . . . . . . . . 87

5 Summary and future work 89

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 Absorption spectra of liquid fuels . . . . . . . . . . . . . . . . 89

5.1.2 Evaporating fuel aerosols . . . . . . . . . . . . . . . . . . . . . 90

5.1.3 Fuel films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Extension of film technique for vapor and film temperature . . 91

5.2.2 FTIR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 93

A Derivation of X and T in an aerosol 94

B C++ code for Mie scattering calculations 96

C Transmittance and reflectance for absorbing media 108

Bibliography 111

xii

List of Tables

2.1 Integrated band intensities of liquid (Al) and vapor (Av) absorption

spectra. The integration limits were from 2600cm−1 to 3400cm−1.

Temperatures are listed in the plotted spectra. . . . . . . . . . . . . . 41

2.2 Fractional composition of alkanes, aromatics, and olefins for the three

gasoline blends analyzed. . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Dependency of calculated extinction ratios (R21 = 1.08 and R31 = 1.14)

on initial droplet size distribution (log-normal, see equation 3.7). . . . 56

4.1 Refractive indices of common infrared window materials near 3.4 µm [1]. 76

xiii

List of Figures

1.1 Schematic of monochromatic light passing through a thin slab of ab-

sorbing material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Schematic of laser light passing through a gas cell filled with an ab-

sorbing gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Infrared absorption spectrum of n-decane (Top figure from Sharpe et.

al. [2]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Schematic of n-decane molecule showing fundamental C-H stretch vi-

bration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Temperature dependent absorption of n-decane. . . . . . . . . . . . . 11

1.6 Electromagnetic plane wave propagating in the z-direction. . . . . . . 13

1.7 Measured complex refractive index of liquid n-decane in the infrared. 17

1.8 Schematic of light being scattered and absorbed by a liquid droplet. . 20

1.9 Calculated extinction efficiency for liquid toluene droplets. . . . . . . 23

2.1 Optical cell used in liquid absorption measurements. Teflon spacers

with thicknesses between 0.5 mm and 15 µm were used. . . . . . . . . 30

2.2 Measured extinctance of liquid toluene for three path lengths. Two

anchor points were selected near the absorption band (2681 and 3290

cm−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Extrapolation of measured refractive index data [3] at visible wave-

lengths to 8000 cm−1. The extrapolated refractive indices at 8000

cm−1 for liquid toluene, n-decane and n-dodecane are 1.477, 1.404,

and 1.413, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xiv

2.4 Flow chart showing the data analysis in converting the measured ex-

tinction spectra to real and imaginary refractive index spectra. The

computer codes used [4] are shown in brackets (All computer codes

assume log base ten extinctance.). The initial n spectrum needed for

the ANCHORPT program was calculated, without anchor point cor-

rection, from the shortest path length extinction spectrum. . . . . . . 34

2.5 Measured absorption cross-sections for liquid toluene near 3000 cm−1

are compared with published data. The residual is defined as σporter −σbertie. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Measured real refractive index for liquid toluene near 3000 cm−1 are

compared with published data. The residual is defined as nporter −nbertie. 36

2.7 Measured real refractive index for liquid n-decane and n-dodecane

(2000 − 4000 cm−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.8 Measured absorption cross-section for liquid and vapor toluene. . . . 38

2.9 Measured absorption cross-section for liquid and vapor n-decane. . . . 38

2.10 Measured absorption cross-section for liquid and vapor n-dodecane. . 39

2.11 Measured absorption cross-section for liquid and vapor gasoline sample1. 39



2.14 Comparison of measured absorption near 3000 cm−1 for three liquid

gasoline samples. Absorption is proportional to alkane, aromatic and

olefin content of gasoline in the identified spectral regions. . . . . . . 43

3.1 Calculated evolution of an initially log-normal droplet size distribution

(D50 = 3.3 µm, q = 1.3 µm) in an evaporating n-decane aerosol. The

median diameter, D50, was calculated for each distribution. . . . . . . 49

3.2 Measured complex refractive index of liquid n-decane in the infrared. 51

3.3 Liquid n-decane droplet extinctance (no vapor absorption) calculated

from modeled droplet size distribution and measured optical constants. 51

3.4 Normalized extinction curves (at 2938 cm−1) showing wavelengths with

constant extinction ratios during evaporation. . . . . . . . . . . . . . 52

xv

3.5 FTIR measurements (corrected to ensure a constant integrated cross

section for this band) of temperature-dependent n-decane vapor ab-

sorption. Wavelengths ν1 and ν3 were chosen to maximize temperature

sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 Temperature dependence of vapor-phase n-decane absorption cross-

sections at chosen wavelengths. . . . . . . . . . . . . . . . . . . . . . 55

3.7 Calculated extinction ratios during evaporation for two initial droplet

size distributions. Ratios are largely constant during evaporation. . . 57

3.8 Sensitivity analysis of measured mole fraction and temperature using

a 2 percent uncertainty. The ratio of droplet extinction to vapor ab-

sorption at ν1, γ, was varied to show sensitivity to droplet loading (a

and b). The magnitude of extinction at 2938 cm−1 was also varied to

show sensitivity to extinctance (c). . . . . . . . . . . . . . . . . . . . 58

3.9 Time-division multiplexing used with two DFG lasers to generate three

mid-infrared wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . 59

3.10 Schematic showing the optical setup with common mode rejection (ref-

erence detector) and the combination of four laser beams with bandpass

optical filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.11 Flow-cell experiment (viewed from above). Ambient air flows through

the aerosol sweeping it into the laser path. . . . . . . . . . . . . . . . 61

3.12 Schematic of the aerosol shock-tube. The shock-tube is filled with n-

decane aerosol through the endwall. Endwall valves are closed and

a shock wave travels down the tube shock-heating the aerosol and

starting evaporation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.13 Measured vapor mole-fraction and temperature behind the incident

shock in the aerosol shock-tube. The measured mole fraction increases

as the liquid-phase n-decane evaporates. Measured temperature de-

creases as the aerosol evaporates. . . . . . . . . . . . . . . . . . . . . 64

xvi

3.14 Comparison between predicted and measured n-decane mole fraction

and temperature for 16 different shocks. Mole fraction measurements

showed agreement within 4 percent. Temperature measurements showed

agreement within 3 percent. . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Schematic of laser light passing through a window, a fuel film of thick-

ness δ, and fuel vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Measured absorption cross-section of n-decane liquid and vapor at 25◦C. A wavelength shift of ∼8 cm−1 is observed in the absorption spectra

of vapor and liquid phases. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 a: Comparison of the infrared absorption spectra of liquid n-decane

and liquid n-dodecane. b: Comparison of the absorption cross-section

of n-dodecane liquid and n-decane vapor at 25 ◦C , with ∼8 cm−1

wavelength shift evident [5]. . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 a: Calculated reciprocal condition number of matrix A for all wave-

length pairs (ν1, ν2) between 2800 − 3000cm−1. Darker regions show

wavelength pairs yielding well-conditioned matrices. a & b: Three

wavelength pairs are highlighted: the selected wavelength pair (ν1 =

2854.7cm−1 and ν2 = 2864.7cm−1) within the DFG phase matching

range, and two optimal wavelength pairs (I & II) not attainable with

the current DFG system. . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Schematic of light transmission through a window without a liquid

film (a) and a window with a non-absorbing liquid film (b). When the

refractive index of a film lies between that of the window and the gas

(i.e. nwindow > nfilm > ngas), then Io,film > Io. . . . . . . . . . . . . . 75

4.6 Measured refractive index of n-dodecane (right) and calculated baseline

offset for liquid n-dodecane films on several window materials (left). . 77

4.7 FTIR measurements of liquid n-dodecane films injected onto the win-

dows listed in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . 78

xvii

4.8 a: Measured absorbance of free-standing liquid n-dodecane for sev-

eral film thicknesses, where the measured transmission, I, is equal to

Io,film− absorption, and Io is the same as in Fig. 4.5a. b: Compari-

son of inferred cross-sections from free-standing film measurements to

previously measured cross-section. . . . . . . . . . . . . . . . . . . . 80

4.9 Schematic of DFG system. Two near-IR signal lasers are modulated

at 1 kHz to provide time-multiplexed mid-IR light at two wavelengths. 81

4.10 Optical setup for demonstrating the diagnostic. Two mid-IR and one-

near-IR beams used; mid-IR to measure absorption, near-IR to monitor

beam steering and other losses. . . . . . . . . . . . . . . . . . . . . . 82

4.11 A liquid n-dodecane film is injected onto a CaF2 window and subse-

quently removed by air flow. . . . . . . . . . . . . . . . . . . . . . . 83

4.12 n-Dodecane film measurement. a: Measured n-dodecane absorbance

at both wavelengths. b: Measured liquid film thickness. . . . . . . . . 84

4.13 A vapor cell with CaF2 windows containing 300 ppm n-decane vapor

in air at 1 atm and 24 ◦C. n-Dodecane films were injected onto and

removed from the CaF2 window. . . . . . . . . . . . . . . . . . . . . 85

4.14 n-Dodecane film and n-decane vapor measurement. a: Combined ab-

sorbance from n-dodecane liquid and n-decane vapor at both wave-

lengths. b: Measured liquid n-dodecane film thickness and vapor n-

decane mole fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1 Potential laser frequencies for sensitive detection of vapor temperature,

vapor mole fraction, and liquid film thickness. . . . . . . . . . . . . . 92

C.1 Reflection and transmission of normally incident light. . . . . . . . . 109

xviii

Chapter 1

Introduction

1.1 Overview

The advent of the laser (light amplification by stimulated emission of radiation) in

1960 ushered in an era of laser-based research that has contributed to nearly all

branches of science and engineering, with applications from nuclear fusion to digital

communication [6]. Lasers can provide a powerful light source with a narrow spectral

bandwidth (i.e. light at only one color) that is spatially coherent (easily collimated).

These attributes of lasers make them an excellent spectroscopic tool, as the narrow

bandwidth allows probing of individual absorption transitions in atoms and molecules,

and the spatial coherence allows for large light intensities over long path-lengths with

good spatial resolution ( < 1 mm) [7]. In addition, the time response of continuous-

wave laser-based sensors is often only limited by the detection electronics, resulting in

measurement bandwidths that can exceed 10 MHz. This high bandwidth allows for

near instantaneous measurements in transient environments (e.g. flames, high-speed

flows, and combustors).

Infrared laser-absorption diagnostics are widely used in combustion research for

fast, sensitive, and non-intrusive measurements of species concentration [8] and tem-

perature [9]. A large number of species important to combustion have been suc-

cessfully measured including: H2O [10], CO2 [11, 12], CO [13], NO2 [14], NO [8],

and vapor-phase hydrocarbon fuels and fuel blends [15, 16]. Fuel measurements are

1

2 CHAPTER 1. INTRODUCTION

often made using a helium-neon (HeNe) laser, as its fixed wavelength at 3.39 µm

overlaps with the strong C-H stretch vibrational transitions of hydrocarbons near 3.4

µm, ensuring sensitive detection even for short measurement path lengths. For many

fuels, absorption at the HeNe wavelength is largely independent of temperature and

pressure, allowing for accurate measurements over a large range of temperatures and

pressures [17]. Some examples of fuels and fuel blends successfully measured using

HeNe laser-based diagnostics include: methane [18], n-dodecane [19], diesel [20], and

iso-octane and gasoline in IC engines [16, 21, 22, 23].

Recent advances in laser technology have produced tunable mid-infrared laser

sources near 3.4 µm, using difference frequency generation (DFG) [24, 25]. These

tunable laser sources provide two key advantages over HeNe lasers: first, the laser

can be tuned to wavelengths for which the absorption by the vapor has the desired

temperature dependence and total absorption needed for the application. Second,

multiple wavelengths are possible, allowing for measurements of multiple species or

temperature. Previous researchers in this laboratory have used DFG lasers operating

at two wavelengths in shock tube studies to simultaneously measure temperature and

species concentration for several fuels and fuel blends, including: n-heptane [26], n-

dodecane [27], JP-10 (jet fuel) [28], and iso-octane and gasoline [29]. n-Dodecane

concentration has also been measured in the presence of interfering species and liquid

droplets [30].

This thesis will focus on the application of lasers as fuel sensors, as motivated

by combustion diagnostic measurements. In particular, applications of tunable mid-

infrared lasers for multiphase (vapor and liquid) laser-absorption measurements of

hydrocarbon fuels will be presented. This work is divided into three main parts:

1. Measurements of the real and imaginary refractive index spectra of liquid hy-

drocarbons in the mid-IR region of the C-H stretching vibrations

2. Development of a diagnostic for vapor-phase fuel mole fraction and temperature

in evaporating aerosols

3. Development of a diagnostic for vapor-phase fuel mole fraction and liquid fuel

film thickness

1.1. OVERVIEW 3

The order of the above topics is important; before developing laser-absorption di-

agnostics for multiphase environments, the wavelength-dependent refractive indices,

often referred to as the optical constants, must be measured quantitatively. There is

very little quantitative optical constant spectra for liquid hydrocarbons in the litera-

ture. Thus, it was necessary to make these measurements in the laboratory. As will be

shown in Chapter 2, these measurements are not straightforward, and a significant

portion of this thesis is dedicated to methods for accurate measurements of liquid

hydrocarbons and characterization of the measured spectra. This characterization

revealed a fortuitous wavelength shift in the liquid phase spectra relative to vapor

phase spectra that made the liquid film diagnostic (discussed in Chapter 4) possible.

In Chapter 3, the influence of optical scattering by liquid fuel aerosols on laser-

absorption measurements will be presented. This work was motivated by combustion

applications where liquid fuel is directly injected into the combustion chamber (e.g.

direct gasoline injection engines, diesel engines, and gas turbines). Making mea-

surements of vapor in the presence of droplets has been a topic studied by many

researchers. This work builds upon previous work to provide measurements of tem-

perature and vapor mole fraction by simultaneously measuring the absorption at

multiple mid-infrared wavelengths. This technique also allows for droplet scattering

corrections for mean droplet sizes < 10 µm, including applications in evaporating

aerosols, both of which are not possible with other techniques. This work relies heav-

ily on the optical constants measured in Chapter 2.

Finally, a method for measuring fuel vapor in the presence of absorbing liquid

films is presented. This work makes extensive use of the measured optical constants in

Chapter 2, and the multiple wavelength strategies developed in Chapter 3. Similarly,

this method was motivated by liquid fuel injection in combustion systems, this time

focusing on fuel and oil films that are known to exist within the combustion chamber.

This diagnostic not only makes it possible to measure the fuel vapor in the presence

of films, but also allows direct and simultaneous measurement of the film thickness.

Before presenting these three research topics, the fundamentals of light absorption

and scattering, including a derivation of Beer’s law and a discussion of its importance

to quantitative absorption measurements will be presented. The origin of the real


and imaginary refractive indices will then be explained, including dispersion and

the Kramers-Kronig relations. Finally, the Mie theory of light scattering by small

particles will be summarized along with applications to hydrocarbon liquid aerosols.

1.2 Laser-absorption spectroscopy

1.2.1 Beer’s Law

The fundamental physical principle behind laser absorption is the absorption of light

by atoms and molecules. Laser absorption can be converted to material properties of

interest (e.g. concentration and temperature) using Beer’s law, which is a mathemati-

cal statement of an experimental observation: the attenuation of light through certain

material samples is dependent on the sample’s thickness and concentration. The de-

pendence on sample thickness was first reported by Pierre Bouguer (1698-1758) in

1729. Johann Lambert (1728-1777) independently discovered the same phenomenon

in 1760 and for the first time expressed his observation in mathematical form. August

Beer’s (1825-1863) contribution was the addition of the concentration dependence in

1852. Because of multiple contributors, this relation is sometimes referred to as Beer-

Lambert-Bouguer’s law or Beer-Lambert’s law, but it will be referred to as Beer’s law

in this thesis [31, 32].

dz

vI vI v+ dI

Figure 1.1: Schematic of monochromatic light passing through a thin slab of absorbingmaterial.

The derivation of Beer’s law can be found in many texts (see for example [33, 34]),

and is repeated here for convenience. Imagine light of optical frequency ν, where ν

1.2. LASER-ABSORPTION SPECTROSCOPY 5

represents the optical frequency in wavenumbers1, with intensity, I, incident on a

slab of absorbing material with thickness dz. It was experimentally shown, that the

attenuation of light, dI, was proportional to the local intensity, I, and the path length,

dz, according to the relation:

dI = −αIdz (1.1)

where α is a coefficient of proportionality called the absorption coefficient and has

units of inverse length. Integrating Eq. 1.1 from z = 0 to z = L, where L (called the

path length) is the distance light has traversed through the material, assumed here

to be uniform, yields Beer’s law:

− ln

(

I

Io

)

ν

= αL = Xnσ(ν, T )L (1.2)

In Eq. 1.2, the term on the left is called the absorbance, if light attenuation is only

due to absorption, where I and Io are the light intensities with and without absorption.

Expressing the absorption coefficient in terms of the mole fraction of the absorbing

species, X, the total molar concentration, n [mole/cm3], and the absorption cross-

section, σ [cm2/mole], where T is the material temperature, gives a direct relation

between the concentration of the absorbing species in the medium and the absorbance.

Intuitively, one can think of the absorption cross-section as the effective size of an

absorbing molecule if it simply blocked light from passing through a volume. Imagine

molecules being replaced by opaque spheres of cross sectional area Am, where Am is

not the molecules actual size, but rather a measure of how much light it absorbs. The

absorption cross-section then expresses the sum over all molecular cross-sections, Am,

per mole of molecules. The product nσL is dimensionless and is referred to as the

optical thickness or opacity of a path length through a given material, where large

optical thicknesses correspond to significant radiation attenuation.

The basic arrangement for quantitative measurements of gases is shown schemat-

ically in Fig. 1.2. Light passes through the optical ports (windows) of a gas cell of

1In this thesis frequencies and wavelengths are used interchangeably as they are inversely related:ν[cm−1] = 1

λ[cm] .


Pressure

Thermocouple

Laser light (v )

Gas cell

Windows

X, Tg , P

L

Io I

Figure 1.2: Schematic of laser light passing through a gas cell filled with an absorbinggas.

length, L, and the incident light, I0, and transmitted light, I, are measured with a

photo detector. The amount of light attenuation at a laser optical frequency ν can be

related to the mole fraction, X, of the absorbing species in the medium using Beer’s

law (Eq.1.2) if the pressure and temperature are measured separately (e.g. with a

thermocouple and pressure gage). There are, however, several restrictions on the

applicability of Beer’s law:

• The light source must be monochromatic and spectrally narrow compared to

the probed absorption transition.

• The medium must be homogeneous in concentration and temperature along the

path length (or line of sight).

• Light attenuation not due to absorption (e.g. reflections, window fouling, and

scattering) must be included in I0.

Each of these items will be addressed later in the thesis, but it is important to note

that ignoring these restrictions can lead to significant measurement errors.

1.2.2 The absorption cross-section

For quantitative measurements, the absorption cross-section must be known or mea-

sured directly. The absorption cross-section of the species of interest is typically a


function of temperature, pressure, and wavelength, and can be measured in the lab-

oratory. The absorption cross-section at a single wavelength can be measured by

passing a laser beam at that wavelength through a gas cell as in Fig. 1.2, where the

exact path length, species mole fraction, temperature, and pressure are known. The

cross-section is then found directly from the measured absorbance at that wavelength.

150

100

50

0

600040002000Frequency [cm

-1]

T = 300 KP = 1 atm

n-decane vapor

150

100

50

0

(, T

)

[m2/m

ole

]

31003000290028002700Frequency [cm

-1]

T = 325 K T = 500 K T = 625 K T = 725 K

P = 1 atm

Figure 1.3: Infrared absorption spectrum of n-decane (Top figure from Sharpe et. al.[2]).

If the absorption cross-section spectrum is desired, a Fourier transform infrared

spectrometer can be used. The FTIR uses a broadband light source (in this case, a

black body which emits light from 600−10, 000 cm−1) which is intensity modulated by

a Michelson interferometer and passed through the gas cell. The transmitted light is

then detected and the Fourier transform is taken to recover the frequency-dependent

absorption spectrum. Commercial FTIRs are now available with internal software


that performs the post-processing of the data automatically, and provides the desired

absorbance spectrum of a particular gas sample directly2.

For hydrocarbons, which are the molecules of interest in this thesis, the strongest

absorption in the infrared is near 3.4 µm (3000 cm−1) corresponding to C-H stretching

vibrations within the molecule. Weaker absorption occurs near 1470 cm−1 and 1390

cm−1 due to bending vibrations [35]. Thus, the most sensitive diagnostics use lasers

near this wavelength, as the strong absorption will increase the minimum detectable

species concentration. FTIR measurements of the absorption spectrum of n-decane,

which is representative of most hydrocarbon fuels, is shown in Fig. 1.3 along with

the temperature dependent cross-section near 3.4 µm.

The strong absorption seen in Fig. 1.3 is due to fundamental C-H stretch vibra-

tional transitions near 3.4 µm. The terminology used in spectroscopy comes from the

classical mechanics notion of the harmonic oscillator, often represented by a mass on

a spring. When the mass is displaced from equilibrium and released, it will begin

to oscillate at a frequency which depends on the spring constant and the mass. A

harmonic oscillator may also be driven at some oscillation frequency by an external

force. When the driving frequency approaches the resonant frequency of the system,

the oscillation grows in magnitude. This situation is similar to what a hydrocarbon

molecule experiences when exposed to infrared light (see the schematic of an n-decane

molecule in Fig. 1.4). The individual atoms vibrate about an equilibrium inter-atomic

separation dictated by the charges of the electron cloud and nuclei. Collisions with

neighboring molecules cause these vibrations to increase or decrease and also cause

changes in rotation about the molecule’s center of mass. Unlike the mass-spring

system, a molecule’s vibrational and rotational energies are quantized, as stipulated

by quantum mechanics. Deriving the quantum mechanics solution for hydrocarbon

molecules is well beyond the scope of this thesis, but the basic principles are outlined

below.

In discussing the quantum mechanical solutions for molecular absorption, it is

more convenient to use the photon description of light instead of the electromagnetic

2In this research, a Nicolet 6700 FTIR was used, with a minimum resolution of 0.09 cm−1

(FWHM).


n-decane molecule (C10H22)

C H

ω

Figure 1.4: Schematic of n-decane molecule showing fundamental C-H stretch vibra-tion.

wave description. For photons of light incident on a molecule, when the photon

energy corresponds to one of the quantum mechanically allowed energy transitions

of the molecule, a photon is absorbed. Absorption is thus the conversion of the

photon’s energy to internal molecular energy (e.g. vibrations, rotations, or excitation

of electrons). In a vacuum, the molecule would subsequently re-emit a photon of the

same energy in a random direction and return to its previous energy state. In a gas

at moderate pressures, however, the excited molecule may loose the absorbed energy

by a collision with a neighboring molecule before re-emission of a photon can occur.

Thus, in a laser-absorption measurement, a large number of photons are absorbed

and subsequently converted to thermal energy within the gas. However, since the

powers of the lasers used in this research is small (< 1 mW), this thermal energy

increase is negligible.

When a hydrocarbon molecule absorbs an infrared photon, it undergoes a tran-

sition from its current vibrational energy level and a particular rotational level to

the next available vibrational level and a different rotational level; the difference in

energy being equal to that of the absorbed photon. Only certain frequencies of light

are absorbed, as only quantized energy levels are allowed according to quantum me-

chanical selection rules. These transitions show up as discrete absorption lines in the


absorption spectra of small molecules (e.g. H2O, CO2, and CO). For larger hydrocar-

bon molecules like n-decane, there are so many vibrational transitions available (e.g.

many C-H bonds) that the transitions are close together and no individual transitions

are visible in the spectrum (see Fig. 1.3). This overlapping of the transition lines

results in the spectra being largely independent of pressure, which reduces the com-

plexity of laser-absorption measurements for hydrocarbons. Only the temperature

dependence of the absorption spectrum in needed, and the pressure dependence does

not need to be characterized.

A number of broad peaks are observed in the absorption spectra of n-decane in Fig.

1.3. These peaks are all due to the C-H stretching vibrational mode of the molecule.

However, the number of hydrogen atoms bonded to each carbon atom affects the C-H

stretching transition wavelengths. For the alkanes studied in this thesis, two atomic

groupings dominate the absorption spectrum: the CH2 group and the CH3 group.

These groups are illustrated in Fig. 1.4 where the CH2 group corresponds to internal

carbon atoms each bonded to two hydrogen atoms, and the CH3 group, also referred to

as the methyl group, corresponds to the two carbon atoms at each end of the molecule

each bonded to three hydrogen atoms. The CH3 group absorbs at frequencies near

2960 cm−1 and 2870 cm−1, while the CH2 group absorbs at frequencies near 2925 cm−1

and 2855 cm−1 [35]. Different vibrational modes and spectral shapes are present in

other hydrocarbon molecules such as alkenes (double-bonded carbon molecules, also

referred to as olefins) and aromatics (carbon-ringed molecules).

1.2.3 Measuring temperature

The temperature dependence seen in the absorption feature shown in Fig. 1.3 is

largely due to the probability distribution of the rotational energy states. For a gas

at moderate pressure and temperature, there is a broad distribution of rotational

energy levels. As the temperature changes, this distribution changes, resulting in a

broadening effect on the spectra (i.e. lower peak cross-sections and a wider absorption

band).

Due to the temperature dependence of the absorption spectra, it is possible to


measure the temperature of a gas from the ratio of the measured absorbance at two

wavelengths. The ratio of absorbance is only a function of temperature as the mole

fraction, concentration, and path length all cancel in the ratio:

− ln(

IIo

)

ν1

− ln(

IIo

)

ν2

=σ(ν1, T )

σ(ν2, T )= f(T ) (1.3)

v1

v2

200

100

0

900800700600500400300Temperature [K ]

3

2

1

0

Ab

so

rptio

n ra

tio

1 = 2938 cm-1

2 = 2952 cm-1

( 1 ) / (2 )

n-decane150

100

50

0

31003000290028002700Frequency [cm

-1]

T = 325 K T = 500 K T = 625 K T = 725 K

P = 1 atm

(, T

)

[m2/m

ole

](

T )

[m

2/m

ole

]

Figure 1.5: Temperature dependent absorption of n-decane.

By careful selection of wavelengths it is possible to obtain a strong temperature

dependence in the absorption ratio, as demonstrated in Fig. 1.5 for n-decane. The

first laser wavelength, ν1, has been chosen near peak absorption, where the absorption

decreases rapidly with increasing temperature, and the second laser wavelength, ν2,

is chosen where the absorption is less sensitive to temperature. The resulting ratio

is temperature dependent, with decreasing temperature sensitivity as temperature

increases. Using two wavelengths, the gas temperature and the concentration of the


absorbing species can be determined simultaneously. This is important for applica-

tions where the temperature is needed either for the application, or to determine the

appropriate value of the temperature-dependent absorption cross-section.

1.2.4 Modifications to Beer’s law

The form of Beer’s law stated in Eq. 1.2 only accounts for changes in transmitted light

intensity due to absorption. Other common factors that can change the transmitted

light intensity include: light emission from heated gases, light scattering from small

particles, and changes in the real refractive index due to phase change (gas to liquid),

non-uniformity along the path length (e.g. due to temperature change), or density

changes (boundary layer effects). Interference from light emission is often corrected for

by momentarily interrupting the light beam and collecting the emitted radiation to be

subtracted in real time from the collected light signal. Alternatively, the laser intensity

can be modulated at some frequency and passed through a lock-in amplifier, effectively

rejecting the slowly varying emission signal. The effects of transverse gradients in

density, which steer the beam away from the normal path, can be minimized by

careful alignment of the laser beams and collection optics. The other two effects:

multi-phase absorption and light scattering, are more involved and will be addressed

next.

1.3 The refractive index

1.3.1 Beer’s Law from Maxwell’s equations

The derivation of Beer’s law presented previously was based on experimental obser-

vation of the attenuation of light as it passed through certain materials. It is also

possible to derive Beer’s law from Maxwell’s equations, and as this reveals the defi-

nition of the refractive index in an intuitive way, this derivation is summarized here.

For laser light, which is highly collimated, it is common to represent the electromag-

netic radiation as plane waves. The plane wave approximation is characterized by a

traveling electromagnetic wave with harmonic oscillations in its electric and magnetic

1.3. THE REFRACTIVE INDEX 13

fields. If an infinite plane is placed orthogonal to the direction of propagation, the

electric field (and accompanying magnetic field) is in-phase at all points on this plane

(e.g. the magnitude of the electric field and magnetic field are zero everywhere on

the plane in Fig. 1.6).

E

B

z

y

x

Figure 1.6: Electromagnetic plane wave propagating in the z-direction.

Using phaser notation, the electric field and magnetic field of an electromagnetic

plane wave traveling in the z-direction can be represented mathematically as:

~Ec = ~E0exp(iκz − i2πc0νt)k; ~Bc = ~B0exp(iκz − i2πc0νt)k (1.4)

where ~E0 and ~B0 are constant vectors equal to the magnitude of the waves, κ is the

propagation number3, ν is the optical frequency of the wave, t is time, k is the unit

vector in the positive z-direction, and i denotes imaginary numbers. The propagation

number, κ can be complex, in which case κ = κ′ + iκ′′. It can be shown that the

plane waves in Eq. 1.4 satisfy the Maxwell equations if the following relation is met:

κ = κ′ + iκ′′ = 2πνco√ǫµ ≡ 2πν(n+ ik) (1.5)

where co is the speed of light in vacuum, ǫ is the complex permittivity, µ is the per-

meability, and m = n + ik is introduced as the complex refractive index. This holds

for a medium that is linear (independent of the field strength), homogeneous (inde-

pendent of the position), and isotropic (independent of the direction). Substituting

3Wavenumber is also used, but this would create confusion with frequency units.


the complex refractive index into Eq. 1.4 results in a plane wave of the form:

~Ec = ~E0exp(−2πνkz)exp(i2πνnz − i2πc0νt) (1.6)

This equation represents a plane wave propagating in the z-direction, but with an

exponential decay that is a function of the imaginary refractive index. Thus, the

imaginary refractive index determines the attenuation of the light as it propagates.

The real refractive index is a measure of the relative wave speed in the material:

n = c/co, where c is the speed of light in the material and co is the speed of light in

vacuum. For anisotropic media, n is a function of position, causing the light beam to

bend or even reflect.

Equation 1.6 can be related to the light intensity using the time-averaged Poynting

vector, ~S, which is the rate of transfer of electromagnetic energy at all points in space

and is found by taking the cross product of the electric and magnetic field vectors:

~S =1

2Re

{

~Ec × ~B∗c

}

(1.7)

where the complex conjugate of the magnetic field vector has been taken. Performing

the calculus one then obtains:

~S =1

2Re

{√

ǫ

µ

}

∣

∣

∣

~E0

∣

∣

∣

2

exp(−4πνkz)k (1.8)

The magnitude of ~S is the irradiance or intensity of the light, which is given the

symbol I [W/m2]. Defining I0 ≡ 12Re

{

√

ǫµ

}

∣

∣

∣

~E0

∣

∣

∣

2

, results in Beer’s law derived from

Maxwell’s equations:

I = I0exp(−αz) (1.9)

where z is the distance and the absorption coefficient α is defined as:

α ≡ 4πνk (1.10)


Thus, the imaginary refractive index is directly proportional to the absorption coeffi-

cient derived from observation, and is a measure of the attenuation of electromagnetic

energy propagating through an absorbing medium. Equation 1.10 also shows that

measuring the absorption coefficient gives a direct measure of the imaginary compo-

nent of the refractive index. Before discussing experimental methods for measuring

the complex refractive index in Chapter 2, it is useful to first develop a deeper phys-

ical intuition for the real and imaginary refractive indices, including how they can be

derived from each other using dispersion relations.

1.3.2 Dispersion relation

The wavelength dependence of the refractive index (or alternatively, ǫ or µ) is referred

to as dispersion, dispersion relations being expressions of this wavelength dependence.

As just stated, the real index of refraction is the ratio of the speed of light in a medium

divided by the speed of light in vacuum: n = c/co. While this is true, it does not

describe the physical processes leading to the change in the speed of light. For a more

fundamental understanding, it is helpful to start at the atomic level and individual

photons4.

Atoms, either isolated or bound in molecules, are composed of nuclei surrounded

by an electron cloud. Electromagnetic radiation can perturb this electron cloud rel-

ative to the nucleus causing an oscillating dipole moment (i.e. positive and negative

charge separation). Once perturbed, this oscillating dipole then emits radiation at

the same wavelength as the incident photon, with the direction of the emitted photon

dependent on the polarization of the incident light. Dispersion arises from the fact

that the excitation of the electron cloud is not instantaneous, but depends on the

particular molecule and on the wavelength of the light. The resulting delay results in

a phase shift of the emitted light relative to the incident light, and a corresponding

reduction in the phase velocity of the light. Thus the real refractive index is a mea-

sure of this phase shift, or equivalently, a measure of the speed of light in the medium

relative to vacuum.

4This discussion follows that of Hecht [36].


Some further physical understanding can be achieved by modeling the nucleus

electron-cloud dipole as a forced harmonic oscillator with damping. The oscillating

electromagnetic wave acts as the forcing function driving the electron cloud of the

atom. The damping force is due mainly to photon emission, but also to collisions

with other molecules, which reduce the energy of the oscillation. The details of this

are not presented here, but can be found in many optics texts [36, 37]. The governing

equation is Newton’s Second Law:

qeE(t) −meω20r −meγ

dr

dt= me

d2r

dt2(1.11)

where r is the displacement of the oscillator, qe is the electron charge, me is the

electron mass, ω0 is the resonant frequency of the oscillator in radians per second, the

electric field, E(t), is harmonic (i.e. E(t) = E0cosωt), and the three force terms on

the left are due to the driving force, restoring force, and a damping force proportional

to the speed of the oscillation, respectively.

For N electrons per unit volume, the electric polarization is given by P = qerN .

Substituting the solution of Eq. 1.11 into the polarization and combining this with

the relations: P (t) = E(t)(ǫ− ǫ0) and m2 = ǫ/ǫ0, results in the dispersion equation:

m2(ω) = 1 +Nqe

2

2πǫ0me

∑

j

fj

2π(ω20j − ω2) + iγjω

(1.12)

where m is the complex refractive index (m = n + ik) and the solution has been

expanded to include multiple characteristic frequencies j, at which the molecule can

absorb or emit light. This relation holds for gases, where inter-molecular forces are

neglected, but can be applied to liquids by adding a factor of 1/3 to the summa-

tion [36]. The weighting factor, fj, where∑

fj = 1, corresponds to the number of

molecules per unit volume with natural frequencies ω0,j. This same relation can be

arrived at from the quantum mechanical approach, but with ω0,j being the quantum

mechanically allowed transitions and fj being the oscillator strength or transition

probability, which determines the likelihood of a transition occurring (i.e. absorption

or emission of a photon).


1.5

1.4

1.3

1.2

Refr

active Index,

n

40003500300025002000

Frequency [cm-1

]

0.3

0.2

0.1

0.0

Refra

ctiv

e In

dex, k

nk

T = 300 KP = 1 atm

Figure 1.7: Measured complex refractive index of liquid n-decane in the infrared.

The dispersion relation provides some valuable information about the relationship

between absorption and the refractive index. First, in order to calculate m(ω) at any

one frequency, a sum over all absorption transitions for that medium is required.

Thus, the full absorption spectrum is required for an accurate calculation of m at

any one wavelength. Second, at frequencies away from absorption transitions, (ω20j −

ω2) >> iγjω, and iγjω may be neglected. Thus, in the absence of damping, which is

largely due to absorption, the complex term in Eq. 1.12 vanishes and the refractive

index is real (k → 0 and m = n). At frequencies just below absorption transition

frequencies, as ω increases toward ω0, the summation in Eq. 1.12 is positive and n(ω)

increases above 1, corresponding to c < c0. At absorption transition frequencies, k

is non-zero, dn/dω is negative5, and n decreases with increasing wavelength and can

actually reach values where n < 1 (c > c0). At frequencies just above absorption

transition frequencies, ω increases away from ω0, and n(ω) increases with increasing

frequency. These trends are easily seen in Fig. 1.7, which shows the measured complex

5Spectral regions with dn/dω < 0, are referred to as exhibiting anomalous dispersion.


refractive index for n-decane liquid as a function of frequency, ν.6 Away from the

absorption feature, the real index increases with frequency as expected, increasing

more rapidly with frequency as it approaches the absorption feature, then rapidly

decreasing through each absorption peak. As expected, k is zero at non-absorbing

wavelengths, but scales with the absorption cross-section at absorbing wavelengths.

Hydrocarbons have few absorption transitions through the visible to near-IR, al-

lowing a simplified dispersion equation to be used that relates the real refractive index

to the optical frequency [37]:

n(ν)2 = A1 + A2ν2 + A3ν

4 (1.13)

where A1, A2, and A3 are constant functions of the oscillator strengths and funda-

mental frequencies of individual transitions in the ultra-violet, which are dominant

transitions for all atoms and molecules. The utility of Eq. 1.13 is for extrapolating

refractive index data measured at visible wavelengths to the infrared, as will be re-

quired for the refractive index measurements of liquid hydrocarbons discussed in the

next chapter.

1.3.3 Kramers-Kronig relation

We have shown from classical mechanical arguments that the real and imaginary re-

fractive indices are dependent upon each other. A similar but more useful relation can

be deduced mathematically. It can be shown that the real and imaginary refractive

indices belong to a family of functions for which the real and imaginary components

are related through integral equations. The derivation of this relationship is not cov-

ered here as it has little physical intuition, but the result is the Kramers-Kronig or

dispersion relations [34]:

n(νi) − n(∞) =2

πP

∞∫

0

νk(ν)

ν2 − ν2i

dν (1.14)

6The angular frequency is converted to wavenumbers by ω

c0

= 2πν.

1.4. LIGHT SCATTERING 19

k(νi) =−2νi

πP

∞∫

0

n(ν)

ν2 − ν2i

dν (1.15)

where n(∞) is the real refractive index at the highest frequency analyzed, and the P

indicates that the principal value of the improper integral must be taken because of

the singularity at ν = νi.

The Kramers-Kronig relations state that the value of the real or imaginary com-

ponent of the complex refractive index is dependent upon the value of the other

component at all wavelengths. Using the convolution theorem, it can be shown that

the finite response time of molecules to electromagnetic radiation leads to this result.

In practice, the Kramers-Kronig relations can be used to calculate the real refractive

index spectrum by integrating over the measured imaginary refractive index spec-

trum, and visa-versa. This relation will later prove very useful when measuring the

optical constants of liquid fuels.

Thus far, the fundamental processes of light absorption and the origins of the

refractive index have been presented. With this fundamental background established,

the first application of these principles can be presented: light scattering by small

particles.

1.4 Light scattering

Light scattering by small particles can be divided roughly into two domains: particles

with diameters much smaller than the wavelength of light (i.e. D << λ), referred to

here as Rayleigh scattering, and particles with diameters on the order of the wave-

length of light. This latter form of scattering is referred to as Mie scattering when

the particles can be approximated as spheres. Rayleigh found that the scattering of

light intensity by particles much smaller than the wavelength of light was inversely

proportional to λ4. As this work is performed in the infrared (long wavelengths), this

effect is small and not addressed here. However, many liquid aerosols used in com-

bustion have particle sizes on the order of microns, which leads to larger scattering

of laser light. Methods for correcting for this effect will be addressed at length in


Chapter 3, but first a brief introduction to Mie scattering is needed.

1.4.1 Mie theory

Ei , B

i D

Es , B

s

m1m

2

Eint

,Bint

Et , B

t

Figure 1.8: Schematic of light being scattered and absorbed by a liquid droplet.

Mie scattering has been extensively studied and analytical expressions for scat-

tering efficiencies and directions can be found in many texts (Bohren and Huffman’s

book [34] is recommended). Mie theory considers light scattering by spherical parti-

cles, which is typical of liquid droplets in the absence of strong pressure fluctuations.

A schematic showing light scattering by a sphere of diameter D is shown in Fig. 1.8.

The refractive index of the sphere, m1, and the surrounding gas, m2, are sufficiently

different to cause reflections at the boundaries of the two media. For liquid fuel

droplets, m1 is complex in the mid-infrared due to light absorption by the liquid.

When an incident electromagnetic field interacts with the droplet, an internal field, a

scattered field, and a transmitted field are generated. Solving the Maxwell equations

for the incident, internal, and scattered fields results in the scattered intensity as a

function of wavelength and scattering direction.

For line-of-sight measurements, only the transmitted light is measured, corre-

sponding to the total extinction of light in the forward direction. Extinction of light

in the forward direction is due to the interaction of the incident light field and the

forward scattered light field. Alternatively, the transmitted field can be related to the


absorption within the droplet and the scattering in all directions other than forward,

by conservation of energy:

Wt(ν) = Wi(ν) −Ws(ν) −Wa(ν) (1.16)

where Wt is the power of the transmitted light in the forward direction, Wi = IiA

is the incident light power, Ws is the light power lost to scattering in directions

other than forward, and Wa is the power absorbed within the particle. Thus, a

detector placed in the path of the incident beam, with a sufficiently small solid angle

of detection, would measure the extinction of light due to absorption and scattering.

A small solid angle of detection is required to minimize detection of scattered light at

small angles to the incident direction, as these rays are not included in the extinction

calculation. In the scattering measurements presented in Chapter 3, the solid angle is

reduced by placing the detector far from the measurement location, and by passing the

transmitted beam through an optical iris. The measured extinction can be related

to an extinction cross section per particle, Cext [cm2], which is analogous to the

absorption cross section in Beer’s law. Gustav Mie (1869-1957) developed analytical

expressions for the scattering and extinction cross sections by solving the plane wave

version of Maxwell’s equations for spherical particles. A derivation of Mie’s solutions

would be beyond the scope of this thesis, but the basic solution proceeds as follows.

Maxwell’s equations for plane electromagnetic waves can be manipulated to yield

the scalar wave equation:

∇2ψ + κ2ψ = 0 (1.17)

where ψ is a scalar function corresponding to the amplitude of the wave. The wave

equation can be converted to spherical polar coordinates and solved by separation of

variables (i.e. ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ)) yielding two infinite series solutions: ψomn

and ψomn (caution: do not confuse the integers n and m in these solutions with the

refractive index m and real refractive index n). These solutions, also called generating

functions, are complete, meaning they can be expanded to generate any function that

satisfies the scalar wave equation in spherical polar coordinates. Although not shown

here, these solutions to the scalar wave equations can also be used to generate another


set of complete vector functions: the vector spherical harmonics (i.e. ~M = ∇× (~rψ)).

The vector spherical harmonics are in turn used to generate solutions to the vector

wave equations (i.e. ∇2~E + κ2~E = 0). After applying the appropriate boundary

conditions, these solutions can be combined to determine rates of energy loss due to

scattering and absorption, which by conservation of energy, equals the total extinction

(Ws +Wa = Wext). The ratio of the rate of energy loss to the incident light intensity

has units of area and is given by:

Cext =Wext

Ii=

2π

κ2

∞∑

n=1

(2n+ 1)Re {an + bn} [cm2] (1.18)

which is the desired extinction cross section . The terms an and bn are coefficients in

the series solutions to the vector wave equations and are functions of two variables,

x and m:

x =πm2D

λm =

m1

m2

(1.19)

where D is the particle diameter, λ is the wavelength of the incident light, and m1 and

m2 are the refractive indices of the particle and the surrounding medium, respectively.

For the simulations and experiments in this work, the medium surrounding the droplet

is a gas with m2 ∼ 1, therefore m = m1 = m. The extinction cross section is

often reported as an extinction efficiency, Qext, which is the extinction cross section

normalized by the cross sectional area of the particle (Qext = 4Cext/πD2). The term

efficiency can be misleading however, as Qext > 1 is common due to edge diffraction

effects that remove more light than given by the particle’s cross sectional area.

It is informative to plot the extinction efficiency versus optical frequency, which

is done in Fig. 1.9 for three liquid toluene droplet sizes [38] (toluene is used here

because its optical constants were known over the large spectral range needed to

generate this plot). For λ >> D (i.e. low frequency), Rayleigh scattering dominates

and Qext varies as 1/λ4 (or as ν4). This is observed as the near-monotonic increase

in Qext at low frequencies. In the Rayleigh scattering region, extinction is dominated

by bulk absorption within the droplet, which can be observed as two large spikes in

Qext below 0.1 µm−1. These spikes correspond to large absorption features in the


6

5

4

3

2

1

0

Qext

2.52.01.51.00.5Inverse wavelength [µm

-1]

Liquid toluene droplet

D = 6 µm

D = 3 µm D = 1 µm

T = 25°CP = 1atm

Figure 1.9: Calculated extinction efficiency for liquid toluene droplets.

absorption spectrum of liquid toluene. This analysis only considers droplets down to

tens of nanometers. At the nanometer scale the bulk absorption may be influenced by

inter-molecular surface effects, which are not addressed here. The large oscillations in

Qext are referred to as interference structure and are caused by interference between

the incident and forward scattered light. The finer oscillations are referred to as ripple

structure and will not be discussed here. As measurements were only made in aerosols

with many thousands of droplets with different sizes, this ripple structure was always

averaged out. For λ << D (i.e. high frequency), Qext asymptotes to a value of 2.

Thus, for large particles relative to the wavelength, Qext may be considered constant

at 2 and the scattering calculations are greatly simplified.

1.4.2 Droplet distributions

The extinction cross section calculated from Mie theory can be thought of as the area

of the detector surface that is blocked by one particle. In practice, many particles

of many different sizes are present along the optical path of the laser. Adding up

over many particles, it is possible to define an extinction coefficient similar to the


absorption coefficient presented earlier (Eq. 1.1):

τ =∑

j

NjCext,j (1.20)

where τ [cm−1] is the extinction coefficient, and Nj [cm−3] is the number of particles

per unit volume of type j. For example, if there were ten particles per unit volume

with four of diameter D1, four of diameter D2, and two of diameter D3, Eq. 1.20

would yield: τ = 4Cext,1 + 4Cext,2 + 2Cext,3. The number of particles of each diameter

is referred to as the droplet distribution. This droplet distribution can be discrete,

as in the above example, or it can be continuous (i.e. f(D)), in which case Eq. 1.20

takes an integral form:

τν =

∫ Dmax

Dmin

N · f(D)πD2

4Qext,ν(D,m)dD (1.21)

where the limits of integration are from the smallest diameter to the largest. In Eq.

1.21, N [cm−3] is the total number of droplets of all sizes per volume, also called the

droplet loading, and f(D) [cm−1] is the size distribution function, or the number of

particles per differential diameter dD.

Calculating the series solution to Cext (Eq. 1.18) is best done using a computer,

as an and bn require solving recursive relations for Bessel functions. Many computer

codes have been written to perform these calculations, and the SCATMECH C++

class library developed at NIST [39] was used for this work as it provided the most

flexible open source computer code. As a C++ library, SCATMECH provided Mie

scattering functions (e.g. for calculating Qext) which could be called in the C++ rou-

tines developed for this research. The combined computer code is listed in Appendix

B, and calculates the extinction cross section, τν , for user defined refractive index

spectra and droplet size distributions.

Chapter 2

Measurements of Absorption

Spectra in Liquid Fuels

2.1 Introduction

2.1.1 Motivation

In practical combustion environments, fuels are often injected as a liquid spray which

quickly evaporates at elevated temperatures. Direct measurement of vapor-phase

properties (e.g. vapor-phase mole fraction and temperature) in the presence of liquid

fuels is valuable for studying combustion in gas turbine engines and direct-injection

gasoline and diesel engines [23, 40, 21, 41]. Laser-absorption measurements at these

conditions are difficult, however, because light scattering by liquid droplets and strong

absorption by liquid films interfere with measurements of the vapor. Liquid films are

also common in liquid fuel injection combustion systems, with film thicknesses up to

5 µm having been observed in direct injection IC engines [42]. Moreover, liquid films

present a challenge for optical measurements as strongly absorbing liquid films on

optical access ports interfere with vapor-phase measurements. Tunable mid-infrared

lasers are promising for direct measurements of liquid hydrocarbon films, as strong

liquid absorption in this spectral region allows for sensitive film thickness measure-

ments down to sub-micron thicknesses. However, in order to develop diagnostics

25

26CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS

for liquid hydrocarbon films and aerosols, quantitative absorption spectra of liquid

hydrocarbon fuels in the mid-infrared spectral region are required.

The absorption spectra of liquid hydrocarbon fuels are used for many engineering

modeling and diagnostic applications, including radiative heat-transfer modeling in in-

ternal combustion engines [43, 44, 45, 46, 47] and pool fires [48], and laser light scatter-

ing corrections for optical diagnostics in absorbing liquid aerosols [49, 50, 51, 40, 23].

Many different techniques for measuring absorption in liquids have been reported in

the literature using both reflection techniques [52, 44, 45, 53] and transmission tech-

niques [47, 54, 55, 56]. For a summary of available methods see the review by Bertie

[57]. Despite this body of previous work, there remains little quantitative absorption

data for liquid hydrocarbons with the spectral resolution needed for optical diagnostic

development. Further, there is virtually no quantitative data available for fuels in the

spectral region of the fundamental C-H stretch vibrational transition, where strong

liquid absorption requires short transmission measurement path-lengths (< 15µm).

In this chapter a Fourier Transform Infrared (FTIR) spectrometer transmission mea-

surement technique for absorbing liquid hydrocarbons, originally developed within

the chemistry community [58], is described. More specifically, details of this method

as applied to quantitative absorption measurements over the fundamental C-H stretch

absorption band are presented.

Mid-infrared FTIR transmission measurements of the absorption spectra of three

liquid hydrocarbons (toluene, n-dodecane, and n-decane) and three hydrocarbon fuel

blends (gasoline) were made at 27 ◦C in the spectral region 2700− 3200 cm−1. These

are the only known quantitative absorption measurements of n-decane, n-dodecane,

and gasoline in this important spectral region. The toluene measurements were made

for comparison with the published data of Bertie et. al. [38]. The influence of

gasoline composition on the liquid absorption spectrum is presented. The measured

spectra are also compared with similar measurements in the vapor phase, revealing

a shift in the location of the C-H stretch vibrational transition between liquid and

vapor hydrocarbons. The potential to use the observed spectral shift for laser-based

absorption measurements of fuel vapor in the presence of liquid films is also discussed.

2.2. ABSORPTION MEASUREMENTS IN LIQUIDS 27

2.2 Absorption measurements in liquids

2.2.1 Experimental setup

Measurements were made using a Nicolet 6700 FTIR spectrometer with a spectral

range from 650 to 10,000 cm−1 (wavelengths from 1 to 15 µm). Light from the FTIR’s

internal light source was focused through an iris, collimated, and passed into a Michel-

son interferometer that caused each wavelength to be modulated. This modulated

beam was then passed through a short path length liquid cell and focused onto a

HgCdTe detector. The detector signal was analyzed with the manufacturer’s soft-

ware (OMNIC Ver. 7.3), which uses Mertz phase correction and boxcar apodization

without zero filling to provide spectrally resolved transmitted intensity.

For each measurement, the transmitted intensity as a function of wavelength was

related to absorption using Beer’s law:

− ln

(

I

I0

)

ν

= σ(ν, T )naL (2.1)

where I and I0 are the incident (baseline) and transmitted intensities, respectively,

σ(ν, T ) [cm2/mole] is the temperature- and wavelength-dependent absorption cross-

section, na [mole/cm3] is the concentration of the absorbing species, L [cm] is the

path length, and ν [cm−1] is the optical frequency in wavenumbers. The incident

intensity, or baseline, was measured with the optical cell removed from the beam’s

path and was corrected for reflection losses through the cell windows, as discussed in

detail below. The transmitted intensity was measured with the filled cell placed in

the optical path. Measurements were made with a resolution of 1 cm−1 at a pressure

of 1 atm and a temperature of 27 ◦C.

2.2.2 Measurement procedure

The absorption spectra of the liquid samples were measured using an experimental

method first developed by Jones and coworkers [58]. This method allows for simul-

taneous determination of the real and imaginary refractive indices of a liquid sample


from FTIR transmission measurements. The greatest challenge in using this technique

is determining the baseline accurately. In gas phase measurements, the baseline is

measured with the optical cell evacuated. Thus, the measurement of the transmitted

light includes transmission losses from cell window absorption and reflection. As the

index change from vacuum to vapor is typically small, this baseline measurement cap-

tures all light transmission losses except for absorption by the gas. In contrast, when

measuring liquid samples, the index change from vacuum to liquid is large, causing

significant variation between the baseline measured with the cell evacuated and the

true baseline of the filled cell. Jones and coworkers developed a technique for simul-

taneous measurement of the absorption and real refractive index of the sample liquid

by using the measured refractive index to correct the baseline for reflection losses.

In addition to providing the measurement procedure, Jones and coworkers published

several Fortran codes for performing the needed calculations.

In calculating the reflection losses, Jones and coworkers assumed ideal cell posi-

tioning, where light passing through the cell was strictly orthogonal (90 ◦ incidence)

to the window surface. Bertie and coworkers [59] later showed that this is often not

the case, and supplemented the technique by adding a second-order baseline correc-

tion using anchor points. The anchor point correction provided a direct measure of

the transmission losses, due to non-ideal cell positioning, at select wavelengths. In

this work, the methods of Jones and coworkers are used with the Bertie et. al. anchor

point correction as implemented in updated C++ computer codes [4]. The computer

codes used in this work, along with software documentation, are freely available on

the world wide web [60]. In addition to the many references listed here, numerous

other publications can be found in the literature describing this technique and re-

sulting measured spectra. Therefore, this description is limited to a brief summary,

providing the details of implementation that are useful to others interested in making

similar measurements.

In this chapter, the real and imaginary refractive indices of the liquid hydrocarbons

studied are referred to as optical constants.

m(ν) = n(ν) + ik(ν) (2.2)


The real part of the refractive index (Eq. (2.2)), n, is related, through the Fresnel

relations, to the reflection and transmission of light at the interface of disparate

media. The imaginary part of the refractive index, k, also called the absorption

index, is related to optical absorption (for gases, liquids or solids) by the following

relation [37]:

σ(ν) =4πνk(ν)

na

(2.3)

where σ(ν) [cm2/mole] is the absorption cross-section and na [mole/cm3] is the con-

centration of the absorbing species in the gas, liquid or solid phase. The liquid con-

centrations used in this work were obtained from online tables [61]. In this technique

the fractional transmission of light through the cell includes both liquid absorption

and reflection losses. For clarity, the extinctance is defined as the negative natural

logarithm of the uncorrected fractional transmission, which includes both absorption

and reflection losses:

Ext(ν) ≡ − ln

(

I

I0

)

ν

(2.4)

where I is the transmitted intensity through the filled cell including reflection losses,

and I0 is the transmitted intensity with the cell removed from the beam’s path.

To determine the true absorption spectrum of the liquid, the extinction spectrum

must be corrected for reflection losses. Under ideal conditions (e.g. orthogonal light

transmission) the transmission losses can be calculated using the Fresnel relations

if the real refractive index of the liquid and windows are known. Thus, in order to

make accurate transmission measurements of σ(ν), both n and k must be determined

simultaneously.

The liquid samples were placed in a short path-length optical cell (Fig.2.1) and

the fractional transmission of light was measured over the full spectral range of the

FTIR (650 to 10,000 cm−1). A commercial (ThermoFisher) cell with KBr windows

was used with exchangeable Teflon spacers to vary the cell path length from 0.5 mm

down to < 15 µm. A syringe was used to inject each liquid sample into the cell

through holes in one of the KBr windows.


Te�on spacer

drilled KBr window

KBr window

liquid !ll ports

32mm 3mm

Figure 2.1: Optical cell used in liquid absorption measurements. Teflon spacers withthicknesses between 0.5 mm and 15 µm were used.

Extremely short path lengths were required to ensure sensitive absorption mea-

surements at wavelengths near 3000 cm−1 where there is strong hydrocarbon absorp-

tion. The desired range of measured extinctance values to ensure good signal-to-noise

ratios is limited to a lower bound of ∼ 0.02 by intensity noise and an upper bound

of ∼ 3 by insufficient light transmission and reduced exponential sensitivity of Beer’s

law. To ensure accurate extinction data at each wavelength, FTIR measurements of

each sample were performed using several different path lengths ranging from 0.5 mm

to less than 15 µm.

Cell path lengths ≥ 15 µm were determined by measuring Ext(ν) in the evacuated

cell, from 650 to 10,000 cm−1, and counting the interference fringes. For the path

length measurements, I was measured in the evacuated cell and I0 was measured with

the cell removed from the beam’s path. The cell path length can be related to the

separation of the etalon fringes by:

L[cm] =M

2(ν1 − ν2)(2.5)

where M is the number of intensity peaks between optical frequencies ν1 and ν2.

The measured path lengths agreed within 10 percent with the stated Teflon insert


thickness. Teflon spacers were not available for path lengths < 15 µm. For these

measurements, the liquid sample was inserted between the two KBr windows without

a spacer. Path lengths in these cases were determined from linear extrapolation of

the measured extinction spectra at path lengths ≥ 15 µm using the following relation:

L2[cm] =Ext2(ν) − Ext0,2

Ext1(ν) − Ext0,1

L1 (2.6)

where L1 is the measured path length (≥ 15 µm), L2 is the unknown path length,

and Ext1(ν) and Ext2(ν) are corresponding measured extinctance values at a chosen

frequency, before correction for reflection losses. The constants Ext0,1 and Ext0,2 are

subtracted from the extinctance at each path length, forcing the extinctance to zero

at non absorbing frequencies. This subtraction serves as a first-order correction for

reflection losses, and is only used for determining the path length. The uncertainty in-

troduced by using extrapolation to find the path length is < 10 percent, as determined

by applying this technique and the fringe technique to absorption measurements with

path lengths ≥ 15 µm.

2.0

1.5

1.0

0.5

0.0

Extincta

nce

34003200300028002600

Frequency [cm-1

]

500 µm

25 µm

15 µmTolueneT = 27 °CP = 1 atm

anchor

point #1

anchor

point #2

Figure 2.2: Measured extinctance of liquid toluene for three path lengths. Two anchorpoints were selected near the absorption band (2681 and 3290 cm−1).

Measurements at 0.5 mm were used to calculate the baseline correction at selected

anchor points. Wavelengths on both sides of the absorption band were chosen as


anchor points, as shown in Fig. 2.2 for the toluene measurements. The measured

extinction at the anchor points in a long path length cell (e.g. 0.5 mm) can be

approximated as the sum of the ideal transmission losses and absorption by the liquid

sample, as non-ideal losses (e.g. due to cell positioning) are small compared to the

absorption. The linear absorption coefficient, K(ν) [cm−1], at the anchor points is

determined by subtracting the calculated ideal transmission losses from the measured

extinctance at the anchor points and dividing by the path length. The baseline

correction for non-ideal losses at shorter path lengths can then be determined by

subtracting the calculated ideal transmission losses from the measured extinctance at

non-absorbing anchor points. The baseline for the absorption band was found by a

linear fit to the baseline correction at the anchor points. The program ANCHORPT

(from reference [4]) was used to calculate the K(ν) values (See [59] for a more detailed

discussion of anchor points.).

The uncorrected extinction spectra were used along with the measured linear

absorption coefficient, K(ν), at the anchor points to approximate the imaginary re-

fractive index using Eq. (2.3). A first approximation to the real refractive index was

then calculated using the Kramers-Kronig relation:

n(νi) − n(∞) =2

πP

∞∫

0

νk(ν)

ν2 − ν2i

dν (2.7)

where n(∞) is the real refractive index at the highest frequency analyzed, and the P

indicates that the principal value of the improper integral must be taken because of

the singularity at ν = νi.

The real refractive index of each liquid at 8000 cm−1 was used for n(∞). The

n(∞) values for toluene, n-dodecane, and n-decane were determined by extrapolation

from measured values [3] at visible wavelengths using the relation:

n(ν)2 = A1 + A2ν2 + A3ν

4 (2.8)

where A1, A2, and A3 are constants. The extrapolation was calculated by least squares

regression using the program NFIT (from reference [4]). This relation holds for the


refractive index of clear liquids at non-resonant wavelengths only [37]. The resulting

extrapolation and visible refractive index data are shown in Fig. 2.3.

1.55

1.50

1.45

1.40

Re

al In

de

x,

n

200001500010000

Frequency [cm-1

]

toluene fit Wohlfarth n-dodecane fit Wohlfarth n-decane fit Wohlfarth

Figure 2.3: Extrapolation of measured refractive index data [3] at visible wavelengthsto 8000 cm−1. The extrapolated refractive indices at 8000 cm−1 for liquid toluene,n-decane and n-dodecane are 1.477, 1.404, and 1.413, respectively.

The approximate reflection losses in the extinction measurements were calculated

from the Fresnel relations using the linear absorption coefficient, K(ν), at the anchor

points, and the initial estimate for n calculated from the uncorrected k spectrum.

These calculations also required the known wavelength-dependent refractive index of

the KBr windows [62]. A new n spectrum was then calculated from the corrected

k spectra, and this process was repeated until the solutions converged. Iterative

calculations of the n and k spectra were performed using the program EXTABS2K

(from reference [4]).

The resulting k spectra (2700 − 3200 cm−1) from all path lengths were combined

into one composite spectrum, which was composed of the averaged k values calcu-

lated from spectral regions where the measured extinctance was between 0.02 and 3.

Variations in the k spectra were greatest near peak absorption where measurements

with path lengths > 10 µm were optically thick. The final composite k spectrum was

used to calculate the final n spectrum from Eq. (2.7) using the program KKTRANS


(from reference [4]), and to calculate the final σ(ν) spectrum from Eq. (2.3). A flow

chart summarizing the steps needed to convert the measured extinction spectra into

real and imaginary refractive index spectra is given in Fig. 2.4.

at

Anchor Points

(ANCHORPT)

Ex�nc�on spectra

(all path lengths)

( )νK

Baseline

Correc�on

(EXTABS2K)

Initial n spectrum

Path lengths

KBr index, ( )νKBr

n

n∞

Path lengths

KBr index, ( )νKBr

n

Build Composite

k Spectrum

k spectra (all path lengths)

Kramers-Kronig

Integra�on

(KKTRANS)

Final k spectrum

Final n spectrum

( )νK

Figure 2.4: Flow chart showing the data analysis in converting the measured ex-tinction spectra to real and imaginary refractive index spectra. The computer codesused [4] are shown in brackets (All computer codes assume log base ten extinctance.).The initial n spectrum needed for the ANCHORPT program was calculated, withoutanchor point correction, from the shortest path length extinction spectrum.

2.2.3 Validation

To validate the measured optical constants, measurements made of liquid toluene in

this laboratory were compared with previously published data [38] (Figs. 2.5 and

2.6). The reference data (Bertie et. al.) was measured independently by multiple

researchers in different laboratories, and showed that the reproducibility of this tech-

nique is within 3 percent for the fundamental C-H stretch vibrational band. These

2.3. RESULTS 35

measurements are in excellent agreement with the published data. Small variations

in the real index are due to uncertainties in n(∞) and the small offset in the σ(ν)

spectrum, which is likely due to the use of different anchor points than used by Bertie

et. al.

15

10

5

0

(, T

) [m

2/m

ole

]

320031003000290028002700

Frequency [cm-1

]

Bertie et.al. Porter et.al.

-0.5

0.0

0.5

Resid

ual

Figure 2.5: Measured absorption cross-sections for liquid toluene near 3000 cm−1 arecompared with published data. The residual is defined as σporter − σbertie.

2.3 Results

2.3.1 Hydrocarbons measured

Measurements were made for toluene, n-decane, n-dodecane (Sigma-Aldrich > 99

percent purity) and three gasoline samples that were chosen with varying composition

of aromatic, olefin and alkane hydrocarbons. The measured n spectra of n-decane and

n-dodecane are shown in Fig. 2.7. Values of n(∞) were not available for the gasoline

samples, so an n(∞) value of 1.4 at 8000 cm−1 was used. The measured k spectrum

is likely not significantly affected by this small uncertainty in n(∞), as the anchor


1.49

1.48

1.47

1.46

Real In

dex,

n

40003500300025002000

Frequency [cm-1

]

Bertie et.al. Porter et.al.

TolueneT = 27 °CP = 1 atm

-2x10-3

0

2

Resid

ual

Figure 2.6: Measured real refractive index for liquid toluene near 3000 cm−1 arecompared with published data. The residual is defined as nporter − nbertie.

point baseline correction would largely correct for uncertainty in this value. Because

of the uncertainty in n(∞) and the compositional variation of gasoline, which varies

from one refinery lot to another, the measured n spectra of the gasoline samples are

not reported.

2.3.2 Plots of absorption spectra for liquid and vapor

The measured absorption spectra of the liquids were compared with absorption spec-

tra from the same liquids in the vapor phase, also measured at 1 atm. The liquid

densities of the gasoline samples, needed to convert the measured k spectra to ab-

sorption cross section, were reported previously (see gasoline samples 3, 7, and 9 in

[63]). Where available, vapor spectra measured at the same temperature as the liquid

measurements were used. However, vapor data for n-dodecane and the three gasoline

samples were only available at 53 ◦C due to their low vapor pressure [64, 63]. The

n-decane vapor absorption spectrum was obtained from the online spectral library

maintained by Pacific Northwest National Laboratory (PNNL) [2]. All other vapor

2.4. DISCUSSION 37

1.6

1.5

1.4

1.3

Real In

dex,

n

40003500300025002000

Frequency [cm-1

]

n-dodecane n-decane

T = 27 °CP = 1 atm

Figure 2.7: Measured real refractive index for liquid n-decane and n-dodecane (2000−4000 cm−1).

spectra were previously measured in this laboratory and have been published [64, 63].

The resulting absorption spectra of the vapor and liquid have been overlaid in Figs.

2.8-2.13 to facilitate direct comparison of absorption in the two phases. In all cases

the absorption spectra of both phases have similar structure, with the spectra of the

liquids shifted by ∼ 8 cm−1 to lower frequencies relative to the vapor.

2.4 Discussion

2.4.1 Band strength

The absorption spectra of liquid and vapor hydrocarbons have been compared by

several investigators to study intermolecular forces in solutions [65, 66]. Two obser-

vations of these studies are relevant to this analysis: first, the vibrational energies

within excited molecules are redistributed among vibrational and rotational energy

modes upon condensation [65]. Depending upon the molecule, this leads to an in-

crease or decrease of as much as 50 percent in the integrated band intensity of the

fundamental C-H stretch vibrational transition [65]. The integrated band intensities

of all 12 liquid and vapor spectra were computed (Table 2.1). The uncertainties in


20

15

10

5

0

(, T

)

[m2 /m

ole

]

320031003000290028002700

Frequency [cm-1

]

vapor (T = 27°C) liquid (T = 27°C)

spectral shift

toluene

Figure 2.8: Measured absorption cross-section for liquid and vapor toluene.

120

100

80

60

40

20

0

(, T

)

[m2/m

ole

]

320031003000290028002700

Frequency [cm-1

]


n-decane

Figure 2.9: Measured absorption cross-section for liquid and vapor n-decane.

2.4. DISCUSSION 39

160

140

120

100

80

60

40

20

0

(, T

)

[m2 /m

ole

]

320031003000290028002700

Frequency [cm-1

]


n-dodecane

Figure 2.10: Measured absorption cross-section for liquid and vapor n-dodecane.

50

40

30

20

10

0

(, T

)

[m2 /m

ole

]

320031003000290028002700

Frequency [cm-1

]


gasoline sample 1

Figure 2.11: Measured absorption cross-section for liquid and vapor gasoline sample1.


60

40

20

0

(, T

)

[m2/m

ole

]

320031003000290028002700

Frequency [cm-1

]


gasoline sample 2


50

40

30

20

10

0

(, T

)

[m2/m

ole

]

320031003000290028002700

Frequency [cm-1

]


gasoline sample 3


2.4. DISCUSSION 41

the measured intensities are < 10 percent [64, 63]. The integrated fundamental vi-

brational band intensity of vapor hydrocarbons has been found to be only weakly

dependent on temperature [64, 67], therefore, intensities from vapor measurements

made near 50 ◦C can be correctly related to liquid band intensities measured at 27◦C.

Table 2.1: Integrated band intensities of liquid (Al) and vapor (Av) absorption spec-tra. The integration limits were from 2600cm−1 to 3400cm−1. Temperatures arelisted in the plotted spectra.

Name Liquid Vapor RatioAl[m

2mole−1cm−1] Av[m2mole−1cm−1] ( Al

Av)

toluene 1180 1320 0.89n-dodecane 9950 8330 1.19n-decane 6260 7310 0.86

gasoline sample 1 3320 3470 0.96gasoline sample 2 3500 3880 0.90gasoline sample 3 3060 3230 0.95

Comparing the ratio of integrated band intensities in liquid and vapor phases

reveals differences of up to 19 percent. In comparing liquid and vapor band intensities

it is customary [57] to employ a relation proposed by Polo and Wilson [68]:

Al

Av

=1

n

(

n2 + 2

3

)2

(2.9)

where Al and Av are the band intensities in the liquid and vapor phases, respectively,

and n is the real index at band center if no absorption band were present. This

equation assumes an unperturbed molecule in the liquid phase. Thus, deviation from

this relation likely indicates a change in the shape of molecules upon condensation.

Most hydrocarbon fuels have n ≈ 1.4 at 2950 cm−1, which gives a value of about 1.25

for the right side of equation (2.9). Thus, a roughly 25 percent increase in the band

intensity is expected in going from the vapor to the liquid phase. Comparing this

value to the measured ratios in Table 2.1, only n-dodecane has a ratio > 1 as predicted


by the Polo and Wilson relation, whereas the integrated band intensities in liquid n-

decane, toluene, and gasoline are less than in the vapor phase. Other researchers have

also measured lower integrated intensities for the C-H stretching vibrational band (see

Bertie et. al. for similar results in liquid benzene [66]). Understanding these changes

in integrated intensity upon condensation is still an active area of research within the

chemistry community.

2.4.2 Spectral shift

The second conclusion from earlier studies is that a spectral shift occurs as vapors

condense and as the pressure of liquids increases [69]. The magnitude of this shift

depends upon the molecule, however, most hydrocarbons experience a spectral shift

of ∼ 10 cm−1 at 3000 cm−1 to lower frequencies upon condensation [69]. This is due

to an increase in intermolecular attractive forces in the liquid phase, which decrease

the intramolecular vibrational transition energies [70, 71]. Measurements revealed an

∼ 8 cm−1 shift to lower frequencies for all 6 liquids measured in the region of the C-H

stretching vibrational transitions, which is consistent with the literature. Moreover,

the shape of the liquid and absorption spectra is relatively unchanged, indicating that

there is similar absorption temperature-dependence in both phases.

2.4.3 Gasoline composition

Table 2.2: Fractional composition of alkanes, aromatics, and olefins for the threegasoline blends analyzed.

Name Aromatics Olefins AlkanesVolume % Volume % Volume %

gasoline sample 1 26.7 17.1 56.2gasoline sample 2 13.6 11.9 74.5

gasoline sample 3 39.0 8.5 52.5

The three gasoline samples were chosen with varied composition to investigate the

influence of the aromatic, olefin and alkane content on the absorption spectrum, and

2.4. DISCUSSION 43

60

40

20

0

(, T

)

[m2/m

ole

]

320031003000290028002700

Frequency [cm-1

]

liquid gasoline sample 1

sample 2

sample 3

ole�ns

alkanes

aromatics

Figure 2.14: Comparison of measured absorption near 3000 cm−1 for three liquidgasoline samples. Absorption is proportional to alkane, aromatic and olefin contentof gasoline in the identified spectral regions.

the fractional composition of these hydrocarbon classes is listed in Table 2.2. Gaso-

line sample 1 had a large olefin content, sample 2 a large alkane content, and sample

3 a large aromatic content (These three gasoline samples do not contain ethanol.).

The liquid absorption spectra for the three gasoline samples are overlaid in Fig. 2.14

showing the contribution of the gasoline composition to the absorption band shape.

Researchers in this laboratory have previously studied the influence of gasoline com-

position on the absorption spectrum in the vapor phase [63] and demonstrated that

the cross section for the C-H stretch vibrational band could be estimated, for a given

gasoline sample, if the fractional composition of alkanes, olefins, and aromatics in the

gasoline was known. The liquid measurements reported here indicate that similar

trends exist in the liquid phase, as evidenced by the absorption being proportional to

the composition in the identified spectral regions. However, in mixtures of gases, the

absorption by the different constituents simply adds. For liquid hydrocarbon mix-

tures, this may not be the case as solvation effects have been observed [69] to cause a

redistribution of the energy modes of the constituent molecules. Such redistribution

has been estimated to change the integrated band strength by as much as 20 percent


[69]. Therefore, further measurements of absorption in liquid hydrocarbon mixtures

are needed before it can be concluded that the absorption spectrum of a liquid mix-

ture (e.g. gasoline) can be accurately estimated from the sum of spectra of its pure

liquid components (This summation worked well for vapor phase gasoline [63].).

Chapter 3

Evaporating Fuel Aerosols

3.1 Introduction

In practical combustion environments, fuels are often injected as a liquid spray which

quickly evaporates at elevated temperatures. Direct measurement of vapor-phase

properties (e.g. vapor-phase mole fraction and temperature) in the presence of

droplets is valuable for studying the combustion of liquid sprays and aerosols, es-

pecially in practical devices such as gas turbine engines and direct-injection gasoline

and diesel engines. Laser-absorption measurements at these conditions are difficult,

however, because light scattering by liquid droplets interferes with measurements of

the vapor.

Several optical techniques to monitor fuel vapor have been applied to combustion

environments with hydrocarbon aerosols, including coherent anti-Stokes Raman spec-

troscopy (CARS) for point temperature measurements [72, 73], laser-induced fluores-

cence (LIF) for two-dimensional temperature measurements [74, 75], and laser-based

direct absorption for line-of-sight measurements of vapor concentration [49, 30, 50]. In

this paper a three-wavelength direct-absorption diagnostic is presented that provides

near-simultaneous measurement of temperature and vapor mole-fraction in a hydro-

carbon polydispersed (distribution of droplet diameters) aerosol with a measurement

bandwidth of 125 kHz. Measurements of light transmission at three wavelengths en-

able solution of a modified version of Beer’s law for the unknown temperature, vapor

45

46 CHAPTER 3. EVAPORATING FUEL AEROSOLS

mole fraction, and droplet extinction. This technique is demonstrated for an n-decane

aerosol in steady and shock-heated flows. N-decane was chosen due to its relevance

as a fuel surrogate and because the vapor pressure of n-decane allowed for sensitive

absorption measurements in the 10 cm path across the shock tube. These are the first

known laser-absorption measurements of both vapor mole-fraction and temperature

in the presence of an evaporating polydispersed hydrocarbon aerosol.

Recently, tunable diode-laser sources using difference-frequency-generation (DFG)

have been developed [24, 25] for use in the spectral region from 2850-3050 cm−1(3.3-3.5

µm), thereby enabling detection of hydrocarbon fuels via the fundamental C-H stretch

vibration. The absorption strengths of these transitions are over 100 times stronger

than those in the overtone bands in the near-infrared where tunable laser sources have

long been available. Researchers in this laboratory have previously demonstrated the

use of DFG lasers to measure vapor concentration in the presence of shock-heated

fuel aerosols [30] and to simultaneously measure concentration and temperature in

gaseous hydrocarbon fuels [26]. These techniques are now extended to simultaneous

measurements of fuel concentration and temperature in evaporating liquid aerosols.

Before describing the measurement technique and demonstration experiments with

an n-decane aerosol, the relevant fundamental spectroscopy will be reviewed.

3.2 Laser diagnostic

3.2.1 Laser absorption in gases

As stated before, in direct absorption spectroscopy, narrowband laser light is passed

through a gas sample and the fractional transmission, I/I0, of light is related to the

concentration of the absorbing species by Beer’s law:

Absν ≡ − ln

(

I

I0

)

ν

= σ(ν, T )naL (3.1)

where Absν refers to the absorbance. For relatively large polyatomic hydrocarbons

like n-decane, the individual absorption transitions overlap, forming a broad spectrum

3.2. LASER DIAGNOSTIC 47

with the cross section, σ(ν, T ), nearly independent of pressure. The concentration

of the absorbing species (e.g. n-decane), na, can be determined by measuring the

fractional transmission at one wavelength, if σ(ν, T ) and the temperature are known.

To measure the gas temperature, an additional wavelength is needed as the ratio

of absorbance at two wavelengths can be reduced to a function of temperature.

Absν1

Absν2

=σ(ν1, T )naL

σ(ν2, T )naL=σ(ν1, T )

σ(ν2, T )= f(T ) (3.2)

For accurate temperature measurements, the ratio of absorption at these wavelengths

should vary strongly over the temperature range of interest. This approach to tem-

perature sensing has been previously published for demonstration measurements of

n-heptane temperature and vapor mole fraction in shock-heated gases over a temper-

ature range from 300 K to 1300 K [26].

3.2.2 Laser extinction in absorbing aerosols

When an aerosol is present in the beam path, in addition to vapor-phase absorption,

laser light is attenuated by droplet absorption and scattering (together called droplet

extinction). This adds an additional term to Beer’s law (Eq. 3.1) which becomes [34]:

Extν ≡ − ln

(

I

I0

)

ν

=PXaL

RTσ(ν, T ) + τνL (3.3)

where τν [cm−1] is the droplet extinction coefficient, the extinctance, Extν , is defined

here as the combined light extinction from the wavelength-dependent vapor absorp-

tion (first term) and droplet extinction (second term), and the concentration has been

replaced by temperature, pressure and vapor mole fraction using the ideal gas law.

From this expression, the vapor mole fraction and temperature can be found from

the measured fractional transmission at two wavelengths using equations (3.1) and

(3.2) if P and L are measured, and σ(ν, T ) and τν are known at both wavelengths.

However, for an evaporating aerosol, τν will vary in time, making any static correction

prone to error. Instead, a three-wavelength technique is used that also infers τν . Here

it is helpful to introduce an extinction ratio, which is defined as the ratio of droplet


extinction at wavelengths i and j :

Rij ≡τiτj

(3.4)

Substituting this ratio into Eq. (3.3), results in a form of Beer’s law that is a function

of the vapor mole fraction, Xa, vapor temperature, T, and the droplet extinction

coefficient at a chosen wavelength, τ1.

− ln

(

I

I0

)

i

=PXaL

RTσi(T ) +Ri1τ1L i = 1, 2, 3 (3.5)

This expression can be solved for Xa, T, and τ1 from the fractional transmission at

three wavelengths (three equations in three unknowns), if the extinction ratios R21

and R31 are known and constant during the experiment. To identify three wavelengths

for which Ri1 is constant during evaporation of the absorbing n-decane polydispersed

aerosol used in the demonstration experiments, τν is calculated for all wavelengths in

the C-H stretch vibrational band using Mie theory [34, 76].

3.2.3 Droplet extinction model

In a polydispersed aerosol, the droplet extinction coefficient is dependent on several

parameters [34]: the total number density of droplets (droplet loading), N [cm−3],

the droplet size distribution function, f(D) [cm−1], and the extinction efficiency,

Qext,ν(D,m), which is a measure of a particle’s ability to scatter and absorb light

at different wavelengths. The extinction coefficient,τν , is determined by integrating

these parameters over the full range of droplet diameters, D, present in the aerosol.

τν =

∫ Dmax

Dmin

N · f(D)πD2

4Qext,ν(D,m)dD (3.6)

The initial n-decane aerosol droplet size distribution produced by the nebulizer used

in these measurements was previously measured to be approximately log-normal with

a median diameter, D50, of about 3.3 µm, and a distribution width, q, of about 1.3


µm [77].

f(D) =1√

2πln(q)Dexp

[

−1

2

(

ln(D) − ln(D50)

ln(q)

)2]

(3.7)

However, in an evaporating aerosol f(D) changes in time. A D2-law evaporation

model (Eq. (3.8)) was used for evaporating n-decane aerosols in the droplet extinc-

tion calculations. (This analysis was repeated with a more sophisticated evaporation

model [78] with no significant change in the wavelengths chosen.)

D20 −D(t)2 = κt (3.8)

In Eq. (3.8), t is time after evaporation begins, D0 is the initial droplet diameter,

D(t) is the droplet diameter at time t, and κ is the evaporation rate constant. The

value used for κ in this analysis was arbitrary as only the evolution of the shape of

f(D) during evaporation was needed and not the rate. The modeled variation of

f(D), for the initially log-normal evaporating n-decane aerosol, deviates from log-

normal as the mean size decreases and the distribution widens, as illustrated in Fig.

3.1.

0.6

0.5

0.4

0.3

0.2

0.1

0.0

f ( D

) [µ

m-1

]

1086420

Diameter [µm]

D50 = 3.3 µm

D50 = 2.5 µm

D50 = 1.7 µm

D50 = 1.3 µm

Figure 3.1: Calculated evolution of an initially log-normal droplet size distribution(D50 = 3.3 µm, q = 1.3 µm) in an evaporating n-decane aerosol. The median diameter,D50, was calculated for each distribution.


3.2.4 Liquid optical constants

To calculate the extinction efficiency, Qext,ν(D,m), the wavelength-dependent refrac-

tive index of the liquid was needed, which for an absorbing medium can be written

as a complex function.

m(ν) = n(ν) + ik(ν) (3.9)

The real part of the refractive index, n, is related, through the Fresnel equations,

to the reflection and transmission of light at the interface of disparate media. The

imaginary part of the refractive index, k, is related to optical absorption (for gases,

liquids, or solids) by the following relation [37]:

σ(ν) =4πνk(ν)

na

(3.10)

where σ(ν) [cm2/mole] is the absorption cross-section and na [mole/cm3] is the con-

centration of the absorbing species in the gas, liquid, or solid phase.

Measurements of the liquid n-decane absorption spectrum in the infrared have

long been available [56]. However, these measurements are not quantitative because

the path length is not reported. Quantitative transmission measurements of liquid

n-decane have been published from 650 cm−1 to 4000 cm−1 by Tuntomo et. al. [47]

and by Anderson et. al [79]. However, the data spacing (> 4 cm−1) and path lengths

used (> 15 µm) are inadequate for quantitative analysis near 3000 cm−1. Therefore,

the optical constants of liquid n-decane were measured in this laboratory as described

in Chapter 2, and the resulting optical constant spectra (2000 - 4000 cm−1) are shown

in Fig. 3.2.

3.2.5 Droplet extinction calculation

The extinction efficiency, Qext,ν(D,m), was calculated using the measured optical

constant spectra (2000 - 4000 cm−1) for D < 20 µm using the Mie scattering program

in the SCATMECH C++ class library developed at NIST [39] with the surround-

ing gas having a real refractive index of 1. The expected droplet extinction for an

evaporating n-decane aerosol was calculated by numerically integrating Eq. (3.6)


1.5

1.4

1.3

1.2

Refr

active Index,

n

40003500300025002000

Frequency [cm-1

]

0.3

0.2

0.1

0.0

Refra

ctiv

e In

dex, k

nk

T = 300 KP = 1 atm

Figure 3.2: Measured complex refractive index of liquid n-decane in the infrared.

2.0

1.5

1.0

0.5

0.0

Extincta

nce

70006000500040003000

Frequency [cm-1

]

D50 = 3.3 µm

D50 = 2.5 µm

D50 = 1.7 µm

D50 = 1.3 µm

D50 = 1.2 µm

N = 400,000 [cm-3]

L = 10 [cm]

Figure 3.3: Liquid n-decane droplet extinctance (no vapor absorption) calculatedfrom modeled droplet size distribution and measured optical constants.


using the calculated extinction efficiency, Qext,ν(D,m), and the time-varying f(D)

from the D2 evaporation model (Fig. 3.3). Several trends are evident when compar-

ing the calculated extinction curves: first, as expected, extinction decreases as the

droplets evaporate; second, at wavelengths with strong absorption there is significant

variation with wavelength; and third, there is poor correlation between extinction at

non-resonant wavelengths (e.g. in the near infrared around 6500 cm−1 or 1.5 µm)

and extinction at resonant wavelengths (e.g. near 3000 cm−1). Further calculations

3.0

2.5

2.0

1.5

1.0

No

rma

lize

d E

xtin

ctio

n

70006000500040003000Frequency [cm

-1]

D50 = 3.3 µm

D50 = 2.5 µm

D50 = 1.7 µm

D50 = 1.3 µm

D50 = 1.2 µm

1.4

1.2

1.0

3100300029002800

v3 v1

v2

Figure 3.4: Normalized extinction curves (at 2938 cm−1) showing wavelengths withconstant extinction ratios during evaporation.

showed that for initial f(D)s with D50 > 10 µm the wavelength dependence decreased

markedly and the droplet extinction spectra varied less during evaporation, allowing

for a simple droplet-extinction correction using a single non-resonant wavelength.

Indeed, others have successfully used a single non-resonant beam to correct vapor-

phase measurements for droplet scattering in experiments with large mean-diameter

aerosols [23, 49, 50]. However, for strongly absorbing aerosols with small mean di-

ameters (D50 < 10 µm), large measurement uncertainties would result from ignoring

the wavelength dependence of the droplet extinction [50].


3.2.6 Wavelength selection

The calculated extinction spectra were normalized at 2938 cm−1, and three wave-

lengths were identified with extinction ratios that are independent of droplet evap-

oration, as shown in Fig. (3.4). These wavelengths are listed below with the corre-

sponding calculated droplet extinction ratios, R21 and R31.

ν1 = 2938 cm−1, R11 ≡ 1.00

ν2 = 2952 cm−1, R21 = 1.08 (calculated)

ν3 = 2862 cm−1, R31 = 1.14 (calculated)

The vapor-phase absorption cross-section, σ(ν, T ), of n-decane for 325 K < T

< 725 K was measured in a heated cell using the FTIR spectrometer (Fig. 3.5).

The experimental setup and procedure for similar measurements have been reported

previously [64]. Measurements at 325 K were compared with the spectra measured

by Sharpe et. al. [2] with good agreement. However, a leak at elevated temperatures

led to errors < 20% in the measured cross sections at temperatures above 500 K. This

discrepancy was found by comparing the integrated band intensities (Ψ in Eq. 3.11)

at 300 K (see Table 2.1) and 325 K to those at 500 K, 625 K, and 725 K.

Ψ =

∫

band

σ(ν, T )dν 6= f(T ) (3.11)

It has been shown theoretically [67, 80] and verified experimentally for 26 different

hydrocarbons [64] that the integrated band intensities of the C-H stretching band

are independent of temperature for the moderate temperatures measured here (i.e.

< 725K). At higher temperatures (e.g. > 2000K) combination bands (e.g. due

to bending vibrational modes near 1400 cm−1) and hot bands begin to contribute,

causing increased temperature dependence in the integrated band intensity [67]. The

measured cross sections at 500 K, 625 K, and 700 K were corrected by scaling the

spectra so that the integrated band intensities were independent of temperature (Eq.

3.12).

σscaled(ν, Thigh) =Ψ(T300K)

Ψ(Thigh)σ(ν, Thigh) (3.12)

These scaled spectra agreed with previous measurements of n-decane cross sections


at 2950 cm−1 for 300 K < T < 725 K [17]. The temperature dependence of the scaled

cross-sections at the three chosen wavelengths was fit with second-order polynomials

(Fig. 3.6):

σ(ν, T ) = aiT2 + biT + ci i = 1, 2, 3 (3.13)

where ai, bi, ci, are wavelength-dependent constants. The strong temperature depen-

dence at ν1 and ν3 enables sensitive vapor phase temperature measurements. The

final governing equation for the measurement of extinction in an evaporating aerosol

was obtained by inserting Eq. (3.13) into Eq. (3.5).

− ln

(

I

I0

)

i

=PXaL

RT(aiT

2 + biT + ci) +Ri1τ1L i = 1, 2, 3 (3.14)

150

100

50

0

(, T

)

[m2/m

ole

]

305030002950290028502800

Frequency [cm-1

]

T=300 K (Sharpe et. al.)

T=325 K (Sharpe et. al.)

T=500 K T=625 K T=725 K

n-decane (vapor)v1

v2

v3

P = 1 atm

Figure 3.5: FTIR measurements (corrected to ensure a constant integrated crosssection for this band) of temperature-dependent n-decane vapor absorption. Wave-lengths ν1 and ν3 were chosen to maximize temperature sensitivity.

With the wavelength-dependent absorption and extinction ratios known, temper-

ature and vapor n-decane mole fraction were determined from the measured extinc-

tance at the three chosen wavelengths by solving this system of equations. An explicit

analytical solution was found that enabled fast post-processing of the data, and the


120

100

80

60

40

20

(, T

)

[m2/m

ole

]

900800700600500400300

Temperature [K]

1 = 2938 cm-1

2 = 2952 cm-1

3 = 2862 cm-1

n-decane

Figure 3.6: Temperature dependence of vapor-phase n-decane absorption cross-sections at chosen wavelengths.

solution details are given in the appendix. The solutions for T, Xa, and τ1 are given

below, where A, B, C and x, y, z are constants defined in the appendix.

T =−y +

√

y2 − 4xz

2x(3.15)

Xa =(R21Ext1 − Ext2)RT

PL(AT 2 +BT + C)(3.16)

τ1 =Ext1L

− PXa

RT

(

a1T2 + b1T + c1

)

(3.17)

The sensitivity of this solution to uncertainties in the measured values of Exti, L, P,

and Ri1 was analyzed in detail with a brute force sensitivity analysis. This analysis

first assessed the sensitivity of R21 and R31 to variations in the initial f(D). The

sensitivity of T and Xa was then analyzed for increasing levels of droplet extinction

relative to vapor absorption and for low light transmission (Ext > 2).


3.2.7 Uncertainty analysis

Table 3.1: Dependency of calculated extinction ratios (R21 = 1.08 and R31 = 1.14)on initial droplet size distribution (log-normal, see equation 3.7).

D50 = 2µm D50 = 3.3µm D50 = 5µm∆R21[%] ∆R31[%] ∆R21[%] ∆R31[%] ∆R21[%] ∆R31[%]

q = 1.5µm -0.5 0.0 -2.4 -4.1 -5.8 -9.5q = 1.4µm -0.3 0.9 -1.2 -2.1 -5.1 -8.6q = 1.3µm -0.5 1.0 0.0 0.0 -4.0 -7.2q = 1.2µm -1.1 0.5 0.8 1.6 -2.6 -5.0q = 1.1µm -1.6 0.0 1.2 2.6 -1.3 -3.1

Precise measurements of temperature and vapor mole fraction require accurate

extinction ratios R21 and R31, which are dependent on the droplet f(D). To study

the dependence of R21 and R31 on the initial f(D), both ratios were computed for a

range of initial size distributions (Table 3.1). The calculations show that changing the

distribution width, q, from 1.3 µm to 1.5 µm changed the extinction ratios by only 4

percent, while increasing the initial mean diameter from 3.3 µm to 5 µm (a substantial

increase) changed the ratios by only 10 percent. Reducing the droplet diameter had

a much smaller effect on extinction ratio. Moreover, there was little variation of the

ratios during evaporation (Fig. 3.7). A brute force sensitivity analysis was performed

by increasing each parameter listed in Fig. 3.8 by 2 percent and recording the resulting

change in the temperature and mole fraction. An uncertainty of 2 percent is not an

estimate of the actual uncertainty in each measurable quantity, but was chosen in

order to compare relative sensitivity of the solution for each measurable quantity.

This was repeated for increasing droplet loading using the parameter γ, which was

defined as the ratio of droplet extinction to vapor absorption at 2938 cm−1 (Fig. 3.8

a and b). Values of γ in practical fuel sprays and aerosols depend upon the vapor

pressure of the fuel and the measurement location. Values of γ < 1 were typical for

the n-decane aerosol studied here for most flow conditions. This analysis revealed

that mole fraction measurements were most sensitive to uncertainty in the measured

extinctance at ν3 (Ext3), and R31. The temperature was most sensitive to R31,


1.20

1.15

1.10

1.05

1.00

Extinctio

n R

atio

7654321Mean Diameter, D50 [µm]

= 3.3 µm

R21

R31

= 5 µm

R21

R31

Figure 3.7: Calculated extinction ratios during evaporation for two initial droplet sizedistributions. Ratios are largely constant during evaporation.

and all uncertainties increased with droplet loading (increasing γ). The sensitivity

of the measurement to measured extinctance was also analyzed, and showed that

measurement uncertainty increases rapidly for Extν > 2 (Fig. 3.8 c). Similar trends

were observed in the demonstration experiments discussed below.

The influence of the temperature dependence of the measured optical constants

on the extinction ratios was also considered. The optical cell used for the liquid

measurements did not allow for measurements at elevated temperatures. However,

based on the boiling point of n-decane (447 K) and assuming similar temperature

dependence for liquid and vapor absorption, the measured optical constants vary

by less than 20 percent at peak absorption. Given the accuracy of the measured

temperature (< 5 percent), uncertainties in the optical constants due to temperature

fluctuations are likely much less than 20 percent.


γ = 1

γ = 1

γγγ

γγγ

a cb

Figure 3.8: Sensitivity analysis of measured mole fraction and temperature using a 2percent uncertainty. The ratio of droplet extinction to vapor absorption at ν1, γ, wasvaried to show sensitivity to droplet loading (a and b). The magnitude of extinctionat 2938 cm−1 was also varied to show sensitivity to extinctance (c).

3.3 Validation experiments

3.3.1 Optical arrangement

Light at the three selected sensor wavelengths was generated using two Novawave

Technologies difference-frequency-generation (DFG) laser systems (Fig. 3.9). This

commercial laser generates mid-infrared light at the difference frequency of a near-

infrared signal laser and a pump laser using a periodically-poled lithium niobate

crystal, and is similar to designs in the literature [25]. A recent review by Tittel and

coworkers [16] and references therein demonstrates the utility of these laser sources

for sensitive absorption measurements of polyatomic molecules in the atmosphere [2].

Near-simultaneous measurements at the three selected wavelengths were made using

time-division multiplexing, which allows a single detector to monitor extinction at

multiple wavelengths. Two DFG systems were used to generate light at the three

wavelengths needed, as illustrated in Fig. 3.9. The DFB signal lasers were modulated

at a 125 kHz repetition rate, to time-division-multiplex the three lasers. Light output

from the DFG lasers with power ∼120 µW, was combined on a bandpass filter (2926

- 3046 cm−1) and passed through a ZnSe wedge, splitting the beam (Fig. 3.10). The

reference beam was collected onto a cryogenically-cooled indium-antimonide (InSb)

3.3. VALIDATION EXPERIMENTS 59

DFG #1

DFG #2

PPLN DFG

Crystal

PPLN DFG

Crystal

Near-IR Pump Laser #1 (1064 nm)

Near -IR Pump Laser #2 (1074 nm)

Yb/Er Fiber Amplifier

Yb/Er Fiber Amplifier

Fiber Combiner

Fiber Combiner

Near-IRDFB #1

1547.9 nm

Near- IRDFB #2

1551.1 nm

DFB Signal Lasers

Near-IRDFB #3

1550.6 nm

I

I

I

Time

Pulse Generator

2 µs pulse

Mid-IR Output

All Off

Laser Intensity

v1

Laser Intensity

v2

v3

Figure 3.9: Time-division multiplexing used with two DFG lasers to generate threemid-infrared wavelengths.

Near-IR

λ4

InGaAs

Detector

Mid-IR

InSb

Detector

Mid-IR

ZnSe

Wedge

InSb

Reference

Detector

Sapphire

Windows

BP filter #3

BP filter #2

BP filter #1

L

Vapor &

Aerosol

Iris

v1,v2

v3

Figure 3.10: Schematic showing the optical setup with common mode rejection (refer-ence detector) and the combination of four laser beams with bandpass optical filters.


detector (2.0 mm diameter, 30◦ field of view, 800 kHz bandwidth). The transmitted

mid-infrared beam was passed through the test section and focused onto a second

InSb detector. The solid angle of transmitted light collected by the InSb detector

was reduced by placing the detector far from the measurement location (∼1 m away),

and by passing the transmitted beam through an optical iris (∼1 cm diameter, 20 cm

from the measurement location). The resulting collection half-angle of 1.4◦ resulted

in < 5% of the scattered light being incident on the detector 1. Each laser was on

for 2 µs of the 8 µs period, and the extinctance was averaged over each 2 µs time

window. The detector background was collected in the remaining 2 µs of the 8 µs

period.

To independently verify the presence of droplets and determine complete evapora-

tion, a fourth near-infrared laser at 1.5 µm was combined with the three mid-infrared

lasers to monitor the droplet scattering without interference from vapor and liquid

absorption. The near-infrared laser beam was combined with the mid-infrared beams

on a second bandpass filter (2630 - 3570 cm−1) and passed through the test section.

It was separated from the mid-infrared beams with a third bandpass filter (2820 -

3035 cm−1), and focused onto an InGaAs detector (Fig. 3.10).

Measurements were first made in an aerosol flow cell to measure the extinction

ratios and demonstrate the technique at low temperature. Higher temperature, tran-

sient measurements were then performed in an aerosol shock-tube [30, 19].

3.3.2 Aerosol flow-cell experiment

A series of measurements were made at three temperatures (300 K, 315 K, and 330

K) in an aerosol flow cell (Fig. 3.11). Multiple measurements were made at each

temperature set point as the droplet extinctance was incrementally increased from

0 to 2 by changing the gas flow rate through the cell. These measurements served

two purposes. First, measurements of saturated vapor mole fraction at 300 K, at all

droplet loading conditions, were used to precisely determine the extinction ratios R21

1Angular dependent scattering was calculated using Mieplothttp://www.philiplaven.com/mieplot.htm


and R31. Second, measurements as a function of droplet loading at the three tempera-

tures verified the diagnostic’s ability to recover the saturated vapor mole-fraction and

temperature. The n-decane aerosol was produced by an ultrasonic nebulizer [77] and

(Ambient Air)

Flow Control Valve

Thermocouple

To

Vacuum

Heated

Windows

5.8 cm

Heated Section

Nebulizer

Valve

Pressure

z

x

y

Liquid n-Decane

Figure 3.11: Flow-cell experiment (viewed from above). Ambient air flows throughthe aerosol sweeping it into the laser path.

entrained in an ambient air flow which passed through the 5.4 cm path length cell.

The flow channel upstream of the test section was heated to increase the aerosol tem-

perature to ∼330 K. Measurements at temperatures above 330 K were not possible

as the saturation vapor pressure reached a level that is optically thick (extinctance

> 3) for the path length across the cell. Both optical-access ports were heated to

∼ 10 K above the gas temperature to avoid condensation of n-decane on the 5 cm

sapphire windows. Pressure and temperature were independently monitored by a

pressure transducer and thermocouple, with the pressure in the cell maintained at

approximately 700 torr. The aerosol loading in the cell was varied by changing the

air flow rate over the nebulizer. Higher flow rates decreased the aerosol loading in


the cell due to the fixed aerosol production rate of the nebulizer.

3.3.3 Flow cell results

To determine the extinction ratios experimentally, seven experiments were performed

at 300 K as the aerosol extinctance was varied between 0 and 2. The extinction ratios

R21 and R31 were then varied until the saturated vapor mole fraction and temperature

were recovered for all aerosol loadings. The resulting ratios were R21 = 1.02 and

R31 = 1.10, corresponding to a 6 percent and 4 percent difference from the values

calculated from the FTIR measurements. This is considered excellent agreement given

the uncertainties in the measured refractive index, the laser wavelength (± 0.25 cm−1

due to laser switching), and the measured aerosol f(D). These measured extinction

ratios were subsequently used to improve measurement accuracy, as even a 4 percent

uncertainty in R31 would have a significant impact on measured temperature and

mole fraction.

The calibrated extinction ratios were then used to make measurements at 300

K, 315 K, and 330 K as the aerosol extinctance was again incrementally increased

from 0 to 2 at each temperature set-point. As expected, when the aerosol flow rate

was sufficiently high, the aerosol completely evaporated before reaching the cell. In

this case, the measured mole fraction was below the saturated value; however, the

correct vapor temperature was still recovered. When the droplet-extinction to vapor

absorption ratio was less than unity (γ < 1) and the flow rate was low enough to

saturate the vapor, the vapor mole fraction and temperature determined from the

optical measurements agreed (5 percent) with the saturated vapor mole fraction and

thermocouple temperature. It was found that for high aerosol loadings (i.e. as γ

was increased beyond 1) the temperature determined from the optical measurements

increasingly deviated from the thermocouple values, as predicted in the uncertainty

analysis. However, the optical determination of the vapor mole fraction continued to

agree with the saturated vapor mole fraction.


Laser Path

Figure 3.12: Schematic of the aerosol shock-tube. The shock-tube is filled with n-decane aerosol through the endwall. Endwall valves are closed and a shock wavetravels down the tube shock-heating the aerosol and starting evaporation.

3.3.4 Shock tube experiment

An aerosol shock-tube, which was designed to load fuel aerosol for the study of com-

bustion of low vapor-pressure hydrocarbons, was used to demonstrate simultaneous

temperature and vapor measurements at higher temperatures and in transient condi-

tions [19]. Shock tubes are widely used in combustion research to measure combustion

parameters at short time scales. The shock tube used in this study has a 3 m driver

section and a 9.6 m driven section separated by a polycarbonate diaphragm (Fig.

3.12). Shock waves are formed by over-pressurizing the driver section with helium

to rupture the diaphragm. The incident shockwave travels the length of the driven

section, reflects off the endwall and the reflected shock wave continues back toward

the driver section. In the aerosol shock tube, the driven section is filled with an

aerosol/gas mixture which is shock-heated when the incident wave arrives. Immedi-

ately behind the incident wave the aerosol begins to evaporate, and evaporation was

shown to be complete within 200 µs. Measurements across the 10 cm wide tube were

performed in three shock regions: pre-shock (region 1), behind the incident shock

wave (region 2), and behind the reflected shock wave (region 3).


3.3.5 Shock tube results

2.0

1.6

1.2

0.8Mo

le F

ractio

n [

%]

2001000-100

Time [µs]

5

4

3

2

1

0

NIR

Extin

cta

nce

Mole Fraction NIR extinction

n-Decane/ArgonP1 = 0.27 atm

T1 = 302 K

P2 = 1.02 atm

T2 = 475 K

700

600

500

400

300

Tem

pera

ture

[K

]

2001000-100

Time [µs]

5

4

3

2

1

0

NIR

Extin

cta

nce

Temperature NIR extinction

n-Decane/ArgonP1 = 0.27 atm

T1 = 302 K

P2 = 1.02 atm

T2 = 475 K

Predicted increase

due to evaporation

Xsat(T1)

T2 prediction without

evaporation

T2 prediction with

evaporation

Figure 3.13: Measured vapor mole-fraction and temperature behind the incident shockin the aerosol shock-tube. The measured mole fraction increases as the liquid-phasen-decane evaporates. Measured temperature decreases as the aerosol evaporates.

Measurements were made with initial driven section pressures from 40 torr to

220 torr. The gas temperature and pressure after the incident shock passed were

calculated using the shock jump equations from the shock velocity and the measured

vapor mole fraction after complete evaporation [81]. In an aerosol shock, the post-

shock temperatures are lower than in a vapor-only shock due to evaporative cooling of

the aerosol, as demonstrated in [19]. Before arrival of the shock, the n-decane vapor


was at the saturated pressure at 300 K. Once the shock arrived, the vapor pressure

increased until all of the liquid n-decane was evaporated (Fig. 3.13).

The amount of liquid aerosol present before the shock was calculated from the

near-infrared extinctance using the initial f(D). Converting this liquid to vapor in-

creased the mole fraction to the level indicated by the dashed line in Fig. 3.13. In

this measurement, the measured mole fraction increased from the saturated state

to within 5 percent of the predicted post-shock vapor mole fraction. Similar mea-

surements were made with 16 different shocks (Fig. 3.14). Agreement was within 4

percent for vapor-phase mole fractions below 2.7 percent.

The measured temperature for a single shock is shown in Fig. 3.13 with the pre-

shock temperature and post-shock temperature limits represented by dashed lines.

The upper temperature limit was the post-shock temperature without evaporative

cooling and the lower limit was the post-shock temperature with evaporative cooling.

The measured temperature agreed well before the incident shock, and reached the

upper limit after the shock before falling to within 2 percent of the lower limit. The

initial overshoot of the upper temperature limit was likely due to beam steering caused

by the passing of the incident shock, which was exaggerated by the time division

multiplexing technique used. The decreasing temperature in the post incident-shock

region was likely due to evaporative cooling. The temperatures measured in the three

shock regions were within 3 percent of the calculated temperatures (with evaporative

cooling) for all 16 recorded shocks (Fig. 3.14).


1000

800

600

400

Me

asu

red

Te

mp

era

ture

[K

]

1000800600400

Calculated Temperature [K]

Pre Shock Post Incident Shock Post Reflected Shock Calculated Temperature

3

2

1

0Measure

d M

ole

Fra

ction [%

]

3210

Calculated Mole Fraction [%]

Post Incident Shock Calculated Mole Fraction

I

I

Figure 3.14: Comparison between predicted and measured n-decane mole fraction andtemperature for 16 different shocks. Mole fraction measurements showed agreementwithin 4 percent. Temperature measurements showed agreement within 3 percent.

Chapter 4

Liquid Fuel Films

4.1 Introduction

In this chapter a laser-absorption diagnostic is presented for measuring vapor mole

fraction and liquid film thickness for potential application in combustion environments

where strong absorption by liquid fuel or oil films on windows make conventional direct

absorption measurements of the gas problematic. Before presenting this diagnostic,

a discussion of the impact of liquid films in IC engines and a survey of liquid film

thickness measurement techniques are presented.

4.1.1 Fuel films and measurement techniques

Fuel films are common in IC engines [82], and understanding how these films influence

unburned hydrocarbon pollution is an active area of research [83, 84]. Fuel films

have been observed in the intake port and on the cylinder walls of port-injection IC

engines under cold start conditions [85], and on the piston top of diesel and direct-

gasoline-injection (DGI) engines [82, 84]. Fuel film thicknesses of up to 10µm have

been measured on piston surfaces [86], and these films have been shown to increase

pollutant emissions. For example, Martin et. al. showed that fuel films and the

resultant pool fires within the piston bowl of a diesel engine increased unburned

hydrocarbon emissions and NOx formation [84].

67

68 CHAPTER 4. LIQUID FUEL FILMS

Fuel films on optical ports can also present significant challenges to laser-absorption

measurements of fuel vapor in IC engines, as the absorption band of liquid fuel over-

laps with the absorption band of the vapor fuel [5]. Single-wavelength absorption

diagnostics are unable to separate absorption from vapor, needed to quantify fuel

concentration, from absorption by liquid films. Thus, a technique is needed to de-

tect the presence of fuel films and correct vapor-phase absorption measurements for

absorption from liquids.

Techniques for measuring and/or imaging liquid films have long been available

[87], and the range of techniques is too broad to recount here. Limited to a discussion

of optical methods only, the approaches used include: Schlieren [83], laser absorption

[87, 88, 89], optical interference [90, 91, 92], and laser induced fluorescence [93]. All

these techniques, perhaps with the exception of Schlieren, have been demonstrated on

films < 10µm thick, as needed for this application. For line-of-sight measurements,

only absorption and LIF have been shown to be successful. Of these two, LIF re-

quires seeding the liquid [94], which impacts engine performance, and neither method

provides quantitative measurement of films and vapor.

In this chapter the development of a line-of-sight laser absorption diagnostic for

simultaneous measurement of fuel vapor mole fraction and liquid fuel film thickness is

described. Although practical measurements would likely be in environments where

the liquid films and vapor are composed of the same fuel, different fuels were used

for vapor and liquid in the demonstration experiments to ensure controlled, quan-

titative validation measurements. n-Dodecane was chosen for the liquid fuel due to

its low vapor pressure, which provided a liquid film with minimal evaporating vapor,

while n-decane was chosen for the vapor-phase fuel due to its higher vapor pressure,

providing sufficient absorption for vapor-phase measurements at room temperature.

Demonstrating the diagnostic on pure fuels instead of fuel blends (e.g. gasoline or

diesel), avoided preferential evaporation, meaning smaller molecular weight compo-

nents evaporate first, which changes the fuel composition and introduces uncertainty

in the absorption cross-sections. All measurements were performed at 25 ◦C and 1

atm.

4.2. TWO-PHASE LASER-ABSORPTION MEASUREMENTS 69

4.2 Two-phase laser-absorption measurements

4.2.1 Beer’s law: vapor and liquid

A schematic of a laser-absorption measurement of liquid film and vapor is shown in

Fig. 4.1. Laser light intensity decreases as it passes through absorbing liquid and

vapor, and Beer’s law (Eq. 4.1) relates the amount of light absorbed by the vapor

and liquid to the vapor mole fraction, X, and liquid film thickness, δ.

δ

gas window

liquid �lm

laser

light

Figure 4.1: Schematic of laser light passing through a window, a fuel film of thicknessδ, and fuel vapor.

− ln

(

I

Io

)

νi

= Xngσg(νi, Tg)L+ nLσL(νi, TL)δ i = 1, 2 (4.1)

In Eq. 4.1, the term on the left is called the absorbance, where I and Io are the

measured laser intensities at the detector, with and without absorption. This relation

requires that all transmission losses not due to absorption are included in Io (this is

addressed in detail below). The remaining quantities are ng and nL, which are the

molar densities of the gas and liquid, σg and σL, which are the absorption cross-

sections for the gas and liquid, Tg and TL, which are the temperatures of the gas

and liquid, L, which is the vapor path length, and νi, which represents both laser

frequencies in wavenumbers. If the pressure and temperature are known, X and δ


can be determined from the absorbance measured at two wavelengths (2 equations in

2 unknowns).

200

150

100

50

0

(, T

)

[m2/m

ole

]

3050300029502900285028002750

Frequency [cm-1

]

n-decane

vapor liquid

T = 25°CP = 1 atm

spectral shift

Figure 4.2: Measured absorption cross-section of n-decane liquid and vapor at 25 ◦C.A wavelength shift of ∼8 cm−1 is observed in the absorption spectra of vapor andliquid phases.

The proposed 2-wavelength laser-absorption diagnostic requires quantitative ab-

sorption cross-sections at both chosen wavelengths in vapor and liquid fuel. A differ-

ence in only the magnitude of absorption by the liquid relative to the vapor would lead

to an unsolvable linearly-dependent set of equations. To differentiate absorption from

vapor and liquid, there must be significant differences in the wavelength-dependent

absorption spectra of the two phases. Such differences in spectra are present when the

liquid film is of a different composition than the vapor (e.g. oil film and fuel vapor).

For vapor and film composed of the same fuel, however, the spectra of the two phases

are highly similar, but the absorption spectrum of the liquid is shifted in wavelength

due to intermolecular attractive forces in the higher density liquid (see Fig. 4.2 for

an example of this wavelength shift for n-decane). The wavelength shift due to phase

change makes it possible to distinguish between absorption by the vapor and liquid

phases of the same fuel.

We chose fuels with appropriate vapor pressures to demonstrate the diagnostic un-

der well-controlled conditions at 25 ◦C: n-decane (Psat = 1.425 torr) and n-dodecane


(Psat = 0.136 torr) [61], both common fuel surrogates (Sigma-Aldrich > 99 percent

purity). Fourier transform infrared (FTIR) spectrometer (Nicolet 6700) measure-

ments of the infrared absorption spectra of vapor and liquid n-decane (Fig. 4.2) show

the shift in the absorption band of vapor and liquid. A nearly identical wavelength

shift is seen for n-dodecane liquid and vapor, as a shift of ∼10 cm−1 is common for

most hydrocarbon fuels [5].

200

100

0

n-dodecane liquid

n-decane liquid

T = 25°CP = 1 atm

200

100

0

(, T

)

[m2/m

ole

]

3050300029502900285028002750

Frequency [cm-1

]

n-decane vapor n-dodecane liquid

a

b

Figure 4.3: a: Comparison of the infrared absorption spectra of liquid n-decane andliquid n-dodecane. b: Comparison of the absorption cross-section of n-dodecane liquidand n-decane vapor at 25 ◦C , with ∼8 cm−1 wavelength shift evident [5].

Demonstration experiments were performed under two controlled conditions: First,

the thickness of a free-standing liquid film was measured in the absence of vapor,

which required a low vapor pressure fuel (n-dodecane). Second, a known amount of

vapor was measured in the presence of a fuel film. This second measurement required

a fuel with sufficient vapor pressure at 25 ◦C (n-decane) and no additional vapor from

evaporating liquid (low vapor pressure liquid n-dodecane).


A comparison of the mid-IR absorption spectra of liquid n-decane and liquid n-

dodecane (Fig. 4.3a) reveals that the spectra are highly similar, with the integrated

band strength of n-dodecane being ∼30% greater than that of n-decane. The stronger

absorption by liquid n-dodecane means that for a given wavelength, thicker films are

measurable with n-decane than n-dodecane, due to reduced light transmission as

film thickness increases. The similarity in the liquid spectra indicated that the same

approach (i.e. same wavelengths) would be applicable to measurements with liquid

and vapor consisting purely of either n-decane or n-dodecane. Therefore, wavelengths

were selected using the absorption spectra of n-decane vapor [2] and n-dodecane liquid

[5] as shown in Fig. 4.3b.

4.2.2 Laser wavelength selection

The wavelengths were chosen to make the diagnostic as robust as possible. As seen

in Eq. 4.1, the vapor mole fraction, X, and the liquid film thickness, δ, can be found

by solving a system of two equations, which can be represented in matrix form as,

Ax = b (Eq. 4.2).

[

ngσg(ν1)L nLσL(ν1)

ngσg(ν2)L nLσL(ν2)

][

X

δ

]

=

− ln(

IIo

)

ν1

− ln(

IIo

)

ν2

(4.2)

The accuracy of the measurement is impacted by uncertainties in: L, I, Io, ng,

nL, σg, σL, and laser and detector noise. The coefficient matrix, A, determines

the sensitivity of the solution (x) to uncertainties in the measured absorbances (b).

Wavelengths must be chosen that ensure A is well-conditioned, so that x is not

overly sensitive to measurement uncertainties. The condition number is a measure

of how well-conditioned a matrix is, with small condition numbers corresponding to

well-conditioned matrices [95]. Therefore, the condition number of A was used to

evaluate potential laser wavelength pairs.

The condition number of A was calculated for all wavelength pairs in the absorp-

tion band (2800− 3000cm−1) and its reciprocal is plotted in Fig. 4.4 with darker re-

gions corresponding to wavelength pairs that yield a well-conditioned A matrix. The


200

150

100

50

0

(, T

)

[m2/m

ole

]

3050300029502900285028002750

Frequency [cm-1

]

n-decane vapor n-dodecane liquid

T = 25°CP = 1 atm

II

I

v1 ,v2

2820 2860 2900 2940 2980

2820

2860

2900

2940

2980

v1 ,v2

Frequency,v2 [cm-1]

Fre

qu

en

cy,v

1 [

cm

-1]

DFG phase

matchin

g

DFG phase

matchin

g

DFG phase

matchin

g

DFG phase

matchin

g

DFG phase

matchin

g

DFG phase

matchin

g

0 0.1 0.2 0.3

)(

1

Acond

I

II

a

b

Figure 4.4: a: Calculated reciprocal condition number of matrix A for all wavelengthpairs (ν1, ν2) between 2800−3000cm−1. Darker regions show wavelength pairs yieldingwell-conditioned matrices. a & b: Three wavelength pairs are highlighted: the selectedwavelength pair (ν1 = 2854.7cm−1 and ν2 = 2864.7cm−1) within the DFG phasematching range, and two optimal wavelength pairs (I & II) not attainable with thecurrent DFG system.


two chosen wavelengths are shown in Fig. 4.4 along with the phase-matching range

of the DFG laser used. The DFG tuning range was limited to 2820 − 2920cm−1 and

the maximum wavelength separation was 10 cm−1 to maintain quasi-phase-matching

in the DFG crystal [24, 26]. The two most promising candidate wavelength pairs (I

& II in Fig. 4.4) were not achievable with one DFG system due to phase-matching

constraints, but would be reachable using two independent laser systems. Compar-

ing Fig. 4.4a and b, it becomes clear that wavelength pairs I (ν1 = 2845cm−1 and

ν2 = 2975cm−1) and II (ν1 = 2895cm−1 and ν2 = 2975cm−1) are optimal because ν1

is dominated by absorption from the liquid, and ν2 is dominated by absorption from

the vapor, approaching an independent measurement of vapor and liquid.

4.3 Refractive index matching

4.3.1 Measuring I0

As seen in Eq. 4.1, an accurate value of Io (also referred to as the baseline) is required

for absorption measurements. For vapor phase measurements, Io is typically measured

with the gas evacuated, but the presence of a liquid film (even non-absorbing) on

the window surface can change the light transmission through the system because

the real refractive index of the liquid and window are typically not the same. This

situation is represented schematically in Fig. 4.5, where laser light is shown passing

through a window in vapor (Fig. 4.5a) and through a window with a non-absorbing

liquid film on one surface (Fig. 4.5b). The baseline needed for laser-absorption

measurements with liquid films is actually Io,film, which would require a liquid with

the same wavelength-dependent real refractive index as the liquid fuel, but without

absorption. Such a liquid does not exist, so only Io (Fig. 4.5a) is measurable. To

quantify the uncertainty introduced into this measurement by using Io instead of

Io,film in these absorbance measurements, Io and Io,film were calculated as a function

of wavelength for n-dodecane films on several common infrared window materials.

These calculations were then validated with FTIR measurements.

4.3. REFRACTIVE INDEX MATCHING 75

Window

δ

Window Film1 2’ 3

Incident light

Re!ected light

Transmitted light

Io

Io,film

1 2

a

b

Ii

Ii

Figure 4.5: Schematic of light transmission through a window without a liquid film(a) and a window with a non-absorbing liquid film (b). When the refractive index ofa film lies between that of the window and the gas (i.e. nwindow > nfilm > ngas), thenIo,film > Io.


4.3.2 Modeling I0 in the presence of a film

The transmission of laser light depicted in Fig. 4.5 can be calculated if the refractive

indices of the different media are known. The transmittance, Tij, at the interface of

two media (i & j) is given by:

Tij =4ninj

(ni + nj)2(4.3)

where ni and nj are the refractive indices of medium i and j, respectively [36]. This

relation assumes light is perpendicular to the window surface, but it is useful in

understanding the trends. Equation 4.3 is only valid for non-absorbing media (see

Appendix C for a full derivation). However, since the effect of absorption on Tij is

< 1% for the liquid hydrocarbons studied, this equation is retained for simplicity.

Neglecting multiple reflections within the window and film, the light transmission

through the window, Io, can be approximated using Eq. 4.4, and the transmission

through the window and film, Io,film, using Eq. 4.5 (see Fig. 4.5):

Io = IiT1T2 (4.4)

Io,film = IiT1T2′T3 (4.5)

where Ii is the incident light intensity, and the transmittance subscripts are shown in

Fig. 4.5.

Table 4.1: Refractive indices of common infrared window materials near 3.4 µm [1].

window material n (3.4 µm)CaF2 1.41

Fused Silica 1.41KBr 1.54

Sapphire 1.70ZnSe 2.44

The window materials analyzed in these calculations, including their respective


refractive indices at 3.4µm, are shown in Table 4.1. The refractive index of n-dodecane

is n = 1.4 at 3.4µm [5], representative of most hydrocarbons, which have refractive

indices between 1.4 and 1.5 near 3.4µm [3]. When the refractive index of the film

lies between that of the window and the gas (i.e. nwindow > nfilm > ngas), then

Io,film > Io. For example, calculating Eqs. 4.4 and 4.5 for an n-dodecane film on a

sapphire window in air gives an Io/Ii of 0.87, and an Io,film/Ii of 0.90. This change

in transmission can be reduced by matching the refractive indices of the window and

liquid fuel. Of the window materials in Table 4.1, CaF2 and fused silica appear to be

the best choices as their refractive indices are closest to 1.4.

0.15

0.10

0.05

0.00

-0.05

-0.10

-ln

(Io

, fi

lm/ I

o)

340032003000280026002400

Frequency [cm-1

]

1.6

1.2

0.8

0.4

0.0

Refra

ctiv

e In

dex, n

Liquid n-Dodecane

Sapphire ZnSe

CaF2, FS

KBr

Window materials

CaF2, Fused Silica

KBr Sapphire

ZnSe

n-Dodecane

Figure 4.6: Measured refractive index of n-dodecane (right) and calculated baselineoffset for liquid n-dodecane films on several window materials (left).

Calculating the light transmission, including internal reflections in both the win-

dow and liquid film, for all wavelengths in the absorption band confirms these choices.

Fig. 4.6 shows the results of the transmission calculations, where the negative natural

logarithm of the ratio of Io,film over Io has been taken to show the expected offset in

baseline for each window material. This offset decreases the measured absorbance in

the film measurement and should be minimized by index matching. The calculated

baseline offsets in Fig. 4.6 reveal that at wavelengths outside of the absorption band,


the offset is largely independent of wavelength. Because of this wavelength indepen-

dence, a third laser at a wavelength outside of the absorption band could be used to

correct for the index mismatch between the liquid and window material. This was

verified experimentally with the FTIR measurements reported below.

The calculations also reveal wavelength-dependence in the baseline offset at ab-

sorbing wavelengths. Even with index matching and baseline offset correction using

a third wavelength, there remains some unavoidable baseline variation due to the

strongly wavelength-dependent real refractive index of n-dodecane at wavelengths in

the absorption band. However, this variation is small (absorbance < 0.02) compared

to the absorbances measured in the demonstration experiments.

4.3.3 FTIR measurements of n-dodecane films

δ

Window

FTIR light

Figure 4.7: FTIR measurements of liquid n-dodecane films injected onto the windowslisted in Table 4.1.

To experimentally validate the calculations, an FTIR spectrometer was used to

measure the absorbance of liquid n-dodecane films on the window materials listed in

Table 4.1 from 2700− 3100cm−1. Each window was placed vertically in the FTIR at

the focal point of the internal light beam (Fig. 4.7). Liquid n-dodecane was injected

onto one side of each window and the absorbance spectra were collected (10 sample

averaging, 4cm−1 resolution) for a range of film thicknesses. For each window sample,


the light transmission in the absence of a liquid film was measured (this is Io in Fig.

4.5). The light transmission, I, was then measured with a film of unknown thickness,

δ, on one surface of the window.

The negative natural logarithm of the ratio of I over Io is shown in Fig. 4.8a for

n-dodecane films on a CaF2 window at four different film thicknesses (δ inferred from

absorbance, see below). The attenuation increased with film thickness as expected

and there was no attenuation at wavelengths outside of the absorption band (i.e. zero

baseline offset).

As discussed above, absorption measurements of free-standing films on window

surfaces include transmission changes from refractive index mismatch (e.g. Io,film vs

Io in Fig. 4.6). It was shown that with index matching this difference was small at

wavelengths outside of the absorption band (see Fig. 4.6 and Fig. 4.8a). However,

it is clear from Fig. 4.6 that there were wavelength-dependent film effects in the

absorption band. To quantify the uncertainty of free-standing film absorbance mea-

surements in the absorption band, a comparison was made between free-standing film

measurements and previously measured liquid n-dodecane absorption cross-sections.

These absorption cross-sections were previously measured using an established FTIR-

based measurement technique for liquid samples in short path length liquid cells

(δ < 200µm) as presented in Chapter 2.

Since the free-standing film thickness was not known a priori, the measured ab-

sorbance (−ln( IIo

)) was divided by the liquid density, nL, and a film thickness, δ,

which was chosen to ensure that the integrated areas of the spectra from the film

measurement and from the cell measurement were equal (Fig.4.8b). The wavelength-

dependent difference between the film and cell measurements (i.e. σfilm − σcell) were

then plotted, where the discrepancy is largely due to variation of the refractive in-

dex of n-dodecane at absorbing wavelengths (again due to the inability to measure

Io,film directly). The deviation from the measured cross-section was < 3% at the

chosen wavelengths, indicating that light transmission changes were minimal (similar

agreement was seen for fused silica).

This direct comparison was possible because there was no significant baseline

offset using CaF2 windows. Baseline offsets (i.e. negative absorbance) following


200

150

100

50

0

(, T

)

[m2/m

ole

]

31003000290028002700

Frequency [cm-1

]

n-dodecane δ = 7.9 µm

δ = 1.9 µm

δ = 1.0 µm

δ = 0.7 µm Cell

4

3

2

1

0

CaF2 Wedge

T = 25 °C P = 1 atm

-10

0

10

film

-

ce

ll

a

b

-ln

(I/I

o)

Figure 4.8: a: Measured absorbance of free-standing liquid n-dodecane for severalfilm thicknesses, where the measured transmission, I, is equal to Io,film− absorption,and Io is the same as in Fig. 4.5a. b: Comparison of inferred cross-sections fromfree-standing film measurements to previously measured cross-section.

4.4. DEMONSTRATION EXPERIMENTS 81

the trends calculated in Fig. 4.6 were observed for n-dodecane film measurements

on KBr, sapphire, and ZnSe windows. However, it was observed that a baseline

correction using a non-absorbing wavelength near 1.5µm was sufficient to bring the

measurements within ∼ 5% of the known cross section. For these window materials,

the baseline offset was first subtracted before inferring the film thickness. Thus, if

the index mismatch between the liquid and window material is large, a non-absorbing

beam may be used for baseline offset correction.

4.4 Demonstration experiments

4.4.1 Experimental setup

Signal laser #2

(1.5 µm)

Fiber ampli!er

(Yb/Er)

Fiber

combiner

PPLN crystal

mount

Signal laser #1

(1.5 µm)

Fiber

combiner

Current

Modulation

Channel 1

Channel 2

ν1

,ν2

Pump laser

(1.0 µm)

Figure 4.9: Schematic of DFG system. Two near-IR signal lasers are modulated at 1kHz to provide time-multiplexed mid-IR light at two wavelengths.

The DFG system used for the diagnostic is shown in Fig. 4.9 [96]. Two polarization-

maintaining near-IR lasers (NEL: 1550 & 1552 nm) were time-division-multiplexed

by switching them on and off at a 1 kHz repetition rate (i.e. each laser on for 0.5 ms).

These two lasers were then amplified and combined with a third near-IR pump laser

(1076 nm) and passed through the DFG’s PPLN crystal (periodically poled lithium

niobate), which emitted time-multiplexed light at the difference frequencies of the

input lasers relative to the pump laser (i.e. ν1 = 2854.7cm−1 and ν2 = 2864.7cm−1).

In the demonstration experiments, the light from the DFG laser, with average

output power of about 120 µW, passed through a ZnSe wedge, splitting the beam


Near-IR

v3

InGaAs

Detector

Mid-IR

InSb

Detector

ZnSe

Wedge

InSb

Reference

Detector

BP filter #2

BP filter #1v1,v2

WindowIris

Figure 4.10: Optical setup for demonstrating the diagnostic. Two mid-IR and one-near-IR beams used; mid-IR to measure absorption, near-IR to monitor beam steeringand other losses.

(Fig. 4.10). The reference beam was collected onto a cryogenically-cooled indium-

antimonide (InSb) detector (2.0 mm diameter, 30◦ field of view, 800 kHz bandwidth)

to monitor intensity fluctuations in the mid-IR lasers. The remaining mid-IR light

was combined with a third near-IR wavelength (ν3, NEL: 1556 nm) using a band-pass

filter (2600− 3600 cm−1) before passing through an iris to reduce the beam diameter

to ∼ 1mm. Air flow, used to accelerate removal of the film from the window, created

transient spatial variation of the film thickness over the window surface. The iris was

used to reduce the beam diameter and ensure collinear beams so that both beams

(wavelengths) passed through the same film thickness.

After light at all three wavelengths passed through the window (or cell in the case

of vapor and liquid detection), the near-IR beam was separated from the mid-infrared

beams with a second bandpass filter (2820 - 3035 cm−1) and focused onto an indium-

galium-arsenide (InGaAs) detector (5.0 mm diameter, 3 MHz bandwidth). The time-

multiplexed mid-IR light was collected onto a second InSb detector and separated

into two data traces using time-demultiplexing. Due to the long test times required

(20 seconds), detectors were sampled at 100 kHz, and low pass RC filters (corner

frequencies of 15 kHz) were used to reduce aliasing and detector noise. Averaging 10

points from each wavelength’s 0.5 ms “on” period provided data with an effective 1

kHz bandwidth.


4.4.2 Demonstration of liquid film measurement

δ

CaF2 WindowAir !ow

Laser (v1,v2,v3 )

Figure 4.11: A liquid n-dodecane film is injected onto a CaF2 window and subse-quently removed by air flow.

For film measurements in the absence of vapor, each of the mid-IR wavelengths

provides an independent measure of the liquid film thickness. For accurate film thick-

ness measurements at two wavelengths, three things are required: First, accurate

values for the absorption cross-section of the liquid at both wavelengths, second,

minimal transmission changes due to the wavelength-dependent real refractive index

of the liquid, and third, no baseline offset. To verify that these three requirements

were satisfied, the diagnostic was demonstrated on a free-standing n-dodecane film on

a CaF2 window in the absence of vapor absorption. Liquid films were formed using

a syringe to quickly inject ∼ 50 µl of liquid n-dodecane onto the window surface. A

steady air flow removed the liquid film from the laser path within 20 seconds (Fig.

4.11), with the low vapor pressure of n-dodecane ensuring negligible interference from

evaporated vapor.

Results from the time-resolved liquid n-dodecane film thickness measurement at

both wavelengths are shown in Fig. 4.12. The attenuation of the two mid-IR lasers

and the third near-IR laser is plotted in Fig. 4.12a, while the film thickness measured

at each wavelength is plotted in Fig. 4.12b. Initially, there is no attenuation at any of

the three wavelengths as no liquid is present. At about 2 seconds, attenuation at all

three wavelengths dramatically increases as the fuel spray impinges on the window.

The attenuation at ν3 (non-absorbing) is due to beam steering and scattering by the


30

20

10

0Film

Thic

kness,

δ [µ

m]

121086420

Time [seconds]

8

6

4

2

0

-ln(I

/Io)

ν1 = 2854.7 cm-1

ν2 = 2864.7 cm-1

ν3 = 6424.2 cm-1

T = 24 °CP = 1 atm

a

b

ν1

ν2

ν3

ν2

ν1

Figure 4.12: n-Dodecane film measurement. a: Measured n-dodecane absorbance atboth wavelengths. b: Measured liquid film thickness.


liquid fuel impinging on the window. This attenuation lasts for ∼200 ms, after which

no beam steering or scattering is seen for the remainder of the experiment.

The two mid-IR beams are initially attenuated by scattering, beam steering, and

absorption. After the initial beam steering and scattering subsides, the mid-IR beams

remain highly attenuated due to the initial thickness of the film (in fact at ν1, the

liquid film is opaque) . After ∼ 500ms, the air stream reduced the film thickness

to < 20µm allowing sufficient light transmission for quantitative film thickness mea-

surements at both wavelengths. For absorbances < 4, the film thickness at both

wavelengths showed excellent agreement. However, the dynamic range of ν2 was

greater than ν1, being inversely proportional to the cross section ( δmax(ν2) = 20µm

vs δmax(ν1) = 10µm). By tuning the DFG laser to wavelengths with stronger or

weaker absorption, film thicknesses from < 1µm to over 100 µm could be accurately

measured.

4.4.3 Demonstration of fuel vapor and liquid film measure-

ment

Pressure

Thermocouple

Laser (v1,v2,v3 )δ

Test cell

CaF2 windows

T = 24 °C

Air "ow

P = 1 atm

L

Figure 4.13: A vapor cell with CaF2 windows containing 300 ppm n-decane vapor inair at 1 atm and 24 ◦C. n-Dodecane films were injected onto and removed from theCaF2 window.

To demonstrate simultaneous measurement of vapor mole fraction and liquid film

thickness, an n-dodecane film was injected onto the CaF2 window of a 34.7 cm path

length cell filled with 300 ppm of n-decane vapor in air (Fig. 4.13). The cell was fitted

with a pressure transducer and thermocouple to monitor temperature and pressure


inside the cell. Air flow was again used to remove the liquid film from the window

within 20 seconds. Injecting liquid n-dodecane on the outside of the cell avoided the

fluid dynamics of liquid injection in the quiescent n-decane vapor, and the resulting

potential for beam steering and fluctuations in vapor concentration. Thus, this setup

provided a controlled vapor concentration and a variable liquid film thickness for

demonstration of the diagnostic.

25

20

15

10

5

0

Film

thic

kn

ess, δ [µ

m]

121086420Time [seconds]

500

400

300

200

100

0 Mole

fra

ction, X

[ppm

] X (n-decane) δ (n-dodecane)

6

4

2

0

-ln(I

/Io)

ν1 = 2854.7 cm-1

2 = 2864.7 cm-1

3 = 6424.2 cm-1

T = 24 °CP = 1 atm

ν

ν

a

b

ν1

ν2

ν3

δ

X

Figure 4.14: n-Dodecane film and n-decane vapor measurement. a: Combined ab-sorbance from n-dodecane liquid and n-decane vapor at both wavelengths. b: Mea-sured liquid n-dodecane film thickness and vapor n-decane mole fraction.

The results of the vapor and film measurement are shown in Fig. 4.14 where

both vapor mole fraction and liquid film thickness are plotted. The attenuation of

the two mid-IR lasers and the third near-IR laser is plotted in Fig. 4.14a, while the


measured film thickness and vapor mole fraction are plotted in Fig. 4.14b. Initially,

there is no attenuation of the near-IR beam, but there is small attenuation of the two

mid-IR wavelengths due to absorption by the vapor. At about 2 seconds, attenuation

at all three wavelengths sharply increases as the fuel spray impinges on the window.

The attenuation at ν3 is again due to beam steering and scattering as the liquid fuel

impinges on the window. This attenuation again lasts for ∼200 ms, after which no

beam steering or scattering is seen for the remainder of the experiment.

After fuel injection on the window, the two mid-IR beams are initially attenuated

by scattering, beam steering, and absorption by vapor and liquid. After fuel impinge-

ment, the mid-IR beams are again highly attenuated due to the initial thickness of the

film. After ∼ 500ms, the air stream reduced the film thickness to < 10µm allowing

sufficient light transmission for quantitative film thickness and vapor mole fraction

measurements. After the absorbance at ν1 decreased below 4, the film thickness and

vapor mole fraction were recovered. The largest film thickness measurable was limited

by the absorbance at ν1, as the two Beer’s law equations must be solved together (Eq.

4.2). Due to the much larger absorption from liquid compared to the vapor, there

was as much as 10% uncertainty in the measured vapor mole fraction. This quickly

decreased to less than 5% as the total absorbance fell below 2. An uncertainty < 10

% is considered excellent agreement given that the ratio of liquid absorption to vapor

absorption was > 30:1 at a total absorbance of 4.

4.4.4 Films and vapor composed of same fuel

Different fuels were used for the film and vapor demonstration experiments for val-

idation purposes only. The same technique can be applied, without alteration, for

measuring films and vapor of the same fuel. In fact, the similarity between the spectra

of n-decane and n-dodecane means that the same laser wavelengths could be used for

film and vapor measurements in either fuel. Although measurements of n-decane fuel

films and vapor were made, these results are not presented as the evaporation and

transport rates produced an uncertain amount of vapor in the laser path.


For application to other fuels, new wavelengths should be selected from the absorp-

tion spectra of the liquid and vapor phase fuel, and the condition number approach of

Fig. 4.4 provides a quantitative selection method. For applications with fuel blends

such as gasoline or diesel, care must be taken to account for preferential evaporation

of the fuel films, especially for elevated film temperatures.

Chapter 5

Summary and future work

This thesis presented applications of tunable mid-infrared lasers for multiphase (va-

por and liquid) laser-absorption measurements of hydrocarbon fuels as motivated by

combustion applications (e.g. direct-gasoline-injection engines). This work was di-

vided into three sections: measurements of the real and imaginary refractive index

spectra of liquid hydrocarbons; development of a diagnostic for fuel mole fraction

and temperature in evaporating aerosols, and the development of a diagnostic for

fuel mole fraction and liquid fuel film thickness. Each of these efforts is summarized

below.

5.1 Summary

5.1.1 Absorption spectra of liquid fuels

Measurements of the absorption spectra of three liquid-phase hydrocarbons (toluene,

n-decane, and n-dodecane) and three gasoline blends were made over the spectral

region 2700 − 3200 cm−1. An FTIR transmission technique provided quantitative

absorption and refractive index spectra of the fundamental C-H stretch vibration

band. Analysis of the gasoline samples showed dependence of the absorption spec-

trum on gasoline composition similar to previous studies in vapor gasoline, in that

the measured liquid gasoline absorption spectra are shown to scale with the volume

89

90 CHAPTER 5. SUMMARY AND FUTURE WORK

percentage of olefin, alkane and aromatic hydrocarbons in each sample. The mea-

sured spectra were also compared with vapor-phase spectra of the same hydrocarbons

revealing a shift of 8 cm−1 in the location of the liquid’s C-H stretching absorption

band relative to the vapor.

5.1.2 Evaporating fuel aerosols

A novel 3-wavelength mid-infrared laser-based absorption diagnostic, which provides

simultaneous measurement of temperature and vapor-phase mole-fraction in a gas

flow with evaporating aerosol, has been designed and tested. FTIR measurements

of the optical constants of liquid n-decane were combined with Mie theory to sim-

ulate the wavelength-dependent droplet-extinction for an evaporating polydispersed

n-decane aerosol. These simulations, combined with FTIR measurements of n-decane

vapor, guided the selection of three wavelengths with temperature-dependent vapor

absorption and constant droplet-extinction ratios.

Fast time-multiplexed measurements at these colors provided a droplet extinc-

tion correction, gas temperature, and fuel-vapor mole-fraction. This technique was

demonstrated using n-decane vapor and aerosol in a flow cell and aerosol shock-tube.

Flow cell measurements at room temperature were used to calibrate the technique.

Results indicate that accurate measurements were possible when vapor absorption

was less then droplet extinction. Measurements were made for vapor mole fractions

below 2.7 percent with errors less than 4 percent. Temperature measurements were

made over the range 300 K < T < 900 K, with errors less than 3 percent.

These are the first known laser-absorption measurements of both vapor mole frac-

tion and temperature in the presence of an evaporating polydispersed hydrocarbon

aerosol. This three-wavelength technique can be applied to any aerosol-laden flow in

gases with broad unresolved absorption spectra (e.g. hydrocarbons with C# > 4), but

is especially well suited for measurements in aerosols with small mean diameters (D50

< 10 µm) where increased wavelength dependence of the droplet extinction causes

large uncertainties to corrections derived using a single non-resonant (e.g. near-IR)

beam.

5.2. FUTURE WORK 91

5.1.3 Fuel films

A 2-wavelength absorption diagnostic, which uses tunable mid-IR light from a DFG

laser for simultaneous measurements of fuel vapor mole-fraction and liquid fuel film

thickness, is reported. Optimal wavelengths were chosen from FTIR measurements of

the C-H stretching band of vapor n-decane and liquid n-dodecane near 3.4 µm (3000

cm−1), as constrained by the wavelength tuning range of the DFG laser. Modeling the

light transmission through liquid films on windows revealed that CaF2 and fused silica

reduced baseline offsets for the fuels measured, due to refractive index matching of

the window and liquid. These predictions were validated using FTIR measurements

of n-dodecane films on several window materials.

The diagnostic was demonstrated for n-dodecane films on CaF2 windows in the

absence and presence of n-decane vapor near 25 ◦C. n-Dodecane films < 20 µm were

accurately measured in the absence of n-decane vapor as limited by an absorbance

of ∼4 for the thickest films. Thicker films could be measured by choosing wave-

lengths with smaller absorption cross-sections. By adjusting the wavelengths used,

this technique is suitable for fuel films with thicknesses from < 1µm up to > 100µm.

Simultaneous measurements of n-dodecane films and n-decane vapor were demon-

strated with 300 ppm of n-decane measured with < 10 % uncertainty in the presence

of n-dodecane films < 10 µm.

5.2 Future work

5.2.1 Extension of film technique for vapor and film temper-

ature

Most combustion applications have elevated gas temperatures, and fuel film tem-

peratures up to the fuel’s boiling point. In combustion environments, the gas and

liquid temperatures are typically unknown, and a diagnostic that recovers the gas

and liquid temperatures in addition to the film thickness and vapor mole fraction

would be preferred. Fortunately, the temperature-dependent absorption spectra of

many vapor-phase fuels have been measured [64] and an extension of this technique

92 CHAPTER 5. SUMMARY AND FUTURE WORK

to three mid-IR wavelengths to recover the gas temperature is currently underway.

The technique is similar to the current 2-wavelength approach as illustrated for

n-decane in Fig. 5.1, which shows possible frequencies for sensitive detection of va-

por temperature, vapor mole fraction, and liquid film thickness. In Fig. 5.1, the

absorption at ν1 is strongly dependent on the liquid, while absorption at ν2 is more

sensitive to the vapor absorption and the vapor temperature. The third frequency,

ν3, is chosen to be insensitive to vapor temperature, with near equal dependence on

liquid and vapor absorption. If the temperature dependence of the vapor is known,

it would be possible to systematically choose wavelengths that maximized sensitivity

to vapor temperature, mole fraction, and liquid film thickness.

150

100

50

0

(, T

)

[m2/m

ole

]

320031003000290028002700

Frequency [cm-1

]


v1

v2

v3

n-decane

Figure 5.1: Potential laser frequencies for sensitive detection of vapor temperature,vapor mole fraction, and liquid film thickness.

A potential complication of this technique is uncertainty in the liquid-film tem-

perature. Hydrocarbon fuels have boiling points from 60 ◦C to over 200 ◦C, which

could lead to changes of up to 20 percent in the absorption cross-section of the liq-

uid film, assuming a similar temperature dependence in liquid and vapor. However,

liquid films typically result from cold liquid fuel impinging on cold surfaces, leading

to average liquid temperatures significantly below the boiling point, which would re-

duce uncertainties in the absorption cross-section of the liquid. Measurements of the

temperature-dependent absorption spectra of liquids might lead to multi-wavelength

strategies to determine liquid film temperature.

5.2. FUTURE WORK 93

5.2.2 FTIR spectroscopy

The uncertainty in liquid film temperature could be resolved by developing a di-

agnostic that measured this temperature directly. Such a diagnostic would require

quantitative absorption cross sections of the liquid fuel over a range of temperatures.

Therefore, studying the temperature-dependent absorption of liquid hydrocarbon fu-

els is an important future research direction, as this data is currently unavailable for

most fuels.

It was stated earlier that hydrocarbon absorption spectra are largely pressure

independent. This has been verified for pressures up to 5 atm. However, many

combustion applications (e.g. diesel engines) have pressures well above 30 atm. FTIR

measurements of hydrocarbon absorption spectra at these high pressures may be

needed to assess the feasibility of fuel diagnostics in these high pressure environments.

It was stated in Chapter 2 that the absorption spectrum of a vapor fuel blend

could be constructed by the absorption spectra of the constituent hydrocarbons. The

applicability of this technique for liquid fuel blends has not been verified. Further

measurements of absorption in liquid hydrocarbon mixtures are needed before it can

be concluded that the absorption spectrum of a liquid mixture (e.g. gasoline) can

be accurately estimated from the sum of spectra from its pure liquid components.

An initial study of binary liquid-mixtures (e.g. toluene and n-decane) in underway.

Developing this understanding could lead to models of absorption in liquid fuel blends

useful for fuel processing and combustion.

Appendix A

Derivation of X and T in an aerosol

The modified form of Beer’s law at three wavelengths (Eq. 3.3) was solved to pro-

vide explicit relations for vapor mole fraction, Xa, temperature T, and the droplet

extinction coefficient at ν1, τ1.

− ln

(

I

I0

)

i

=PXaL

RT(aiT

2 + biT + ci) +Ri1τ1L i = 1, 2, 3 (A.1)

The following constants were defined for simplicity in the solution:

A = R21a1 − a2 D = R31a1 − a3

B = R21b1 − b2 E = R31b1 − b3 (A.2)

C = R21c1 − c2 F = R31c1 − c3

Z =R21Ext1 − Ext2R31Ext1 − Ext3

(A.3)

x = ZD − A

y = ZE −B (A.4)

z = ZF − C

94

95

The solutions for T, Xa, and τ1 are

T =−y +

√

y2 − 4xz

2x(A.5)

Xa =(R21Ext1 − Ext2)RT

PL(AT 2 +BT + C)(A.6)

τ1 =Ext1L

− PXa

RT

(

a1T2 + b1T + c1

)

(A.7)

The extinction coefficients at ν2 and ν3 are found using the measured extinction ratios,

R21 and R31:

τ2 = R21τ1 (A.8)

τ3 = R31τ1 (A.9)

The sensitivity analysis for this solution is given in Fig. 3.8.

Appendix B

C++ code for Mie scattering

calculations

To determine the extinction coefficient, τν [cm−1], for a distribution of droplets, Eq.

1.21 needed to be integrated. This required a droplet distribution function, f(D),

and the Mie extinction efficiency, Qext. A C++ computer code was written to gen-

erate f(D) and integrate Eq. 1.21, which uses the SCATMECH C++ class library

developed at NIST to calculate Qext from the measured complex refractive index of

the liquid droplets [39]. The only input file required by the computer code is the

complex refractive index of the liquid. The computer code includes the option to

generate f(D) in three ways: load the distribution from file, or generate a log-normal

or Rosin-Rammler distribution from a user defined mean diameter and distribution

width. The output of the computer code is τ [cm−1] vs λ over the range of wave-

lengths desired. Other options include printing Qext vs λ for a single droplet diameter

(this option was used to generate Fig. 1.9) or Qext vs D for a single wavelength.

The computer code is composed in object oriented form, with a droplet distri-

bution header and class file called dropdist.h and dropdist.cpp, respectively. Input

variables such as droplet size range and wavelength range are input into file ex-

ample1.cpp, which calls functions defined in the dropdist.cpp and the SCATMECH

library. The computer code is included here with extensive commenting to guide the

reader.

96

97

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

//∗∗ Fi l e : example1 . cpp

//∗∗ Jason M. Porter , Stanford HTGL

//∗∗ Phone : (650) 723−0941, Email : jasonpor ter@stanford . edu

//∗∗ Purpose : Calc l i g h t e x t i n c t i o n by absorb ing l i q u i d d rop l e t d i s t r i b u t i o n s

//∗∗ This code uses a C++ l i b r a r y deve loped at NIST:

//∗∗ SCATMECH: Po lar i z ed Ligh t Sca t t e r i n g C++ Class Library

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

#include ” dropd i s t . h” // d rop l e t d i s t r i b u t i o n header

#include <iostream>

using namespace std ; // use unqua l i f i e d names f o r Standard C++ l i b r a r y

using namespace SCATMECH; // use unqua l i f i e d names f o r the SCATMECH l i b r a r y

int main ( )

{

double fv = 2 .4 e−5; // d rop l e t l oad ing s t a t e d as volume f r a c t i o n

double rmax = 5 ; // maximum drop l e t rad ius [ microns ]

double rmin = . 0 005 ; // minimum drop l e t rad ius [ microns ]

double lmax = 4 ; // maximum l i g h t wave length [ microns ]

double lmin = 1 . 4 ; // minimum l i g h t wave length [ microns ]

double Dbar = 3 . 3 ; // mean d rop l e t s i z e (D50)

double q = 1 . 2 7 ; // s i z e d i s t r i b u t i o n width

int l t o t = 1000 ; // t o t a l wave lengths ( determines s p e c t r a l r e s o l u t i on )

int r t o t = 100 ; // t o t a l number o f r a d i i between rmin & rmax

// i n s t a n t i a t e miesca t terer , I i n i t i a l i z e v a r i a b l e s

DropDist Lnorm( rtot , l t o t , rmin , rmax , lmin , lmax ) ;

Lnorm . SetLambda ( ) ; // c a l c u l a t e s wave lengths from inpu t s

Lnorm . SetRadius ( ) ; // c a l c u l a t e s r a d i i and diameters from inpu t s

Lnorm .GetQ ( ) ; // c a l c u l a t e Qext array ( r t o t x l t o t )

//Lnorm . GetQArray ( ) ; // upload Qext array from f i l e

//Lnorm .GetDandN ( ) ; // genera te s drop d i s t from input f i l e s

Lnorm . LogNormal (Dbar , q ) ; // generate log−normal d i s t

//Lnorm . Rosin Rammler (Dbar , q ) ;// generate Rosin−Rammler d i s t

//Lnorm .NTot( f v ) ; // c a l c u l a t e d r op l e t l oad ing [cm−3] from f v

Lnorm . SetNTot (100000 ) ; // s e t d r op l e t l oad ing d i r e c t l y [cm−3]

Lnorm . QDist ( ) ; // c a l c u l a t e s e x t i n c t i o n [cm−1] vs lambda

Lnorm . D 32 ( ) ; // c a l c u l a t e s Suater mean diameter

Lnorm . PrintQ ( ) ; // p r i n t s Qext array to f i l e

//Lnorm . PrintLambda ( ) ; // p r i n t s wave length to f i l e

//Lnorm . PrintDiameter ( ) ; // p r i n t s diameter to f i l e

Lnorm . PrintExt ( ) ; // p r i n t s e x t i n c t i o n vs wave length to f i l e

//Lnorm . Pr in tDis t ( ) ; // p r i n t s f (D) and i n t e g r a l o f f (D) to f i l e

//Lnorm . PrintIndex ( ) ; // p r i n t s i n t e r p o l a t e d r e f r a c t i v e index to f i l e

//Lnorm . QvsRadius (1 . 55 ) ; // p r i n t s Qext vs rad ius f o r one wave length

//Lnorm .QvsLambda ( 1 . 5 ) ; // p r i n t s Qext vs wave length f o r one rad ius

return 0 ;

}

98 APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗


//∗∗ Fi l e : d ropd i s t . h

//∗∗ Jason M. Porter , Stanford HTGL

//∗∗ Phone : (650) 723−0941

//∗∗ Email : jasonpor ter@stanford . edu

//∗∗ Class modeling e x t i n c t i o n by d rop l e t d i s t r i b u t i o n s

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

#ifndef DROPDIST H

#define DROPDIST H

#include ”miescat . h”

#include <iostream>

using namespace std ; // Use unqua l i f i e d names f o r Standard C++ l i b r a r y

using namespace SCATMECH; // Use unqua l i f i e d names f o r the SCATMECH l i b r a r y

class DropDist{

public :

DropDist ( int , int , double , double , double , double ) ;

void SetRadius ( ) ;

void SetLambda ( ) ;

void GetQ ( ) ;

void LogNormal (double , double ) ;

void Rosin Rammler (double , double ) ;

double NTot(double ) ;

double D 32 ( ) ;

void QDist ( ) ;

void QvsRadius (double ) ;

void QvsLambda(double ) ;

void PrintQ ( ) ;

void PrintExt ( ) ;

void Pr intDi s t ( ) ;

void PrintIndex ( ) ;

void GetQArray ( ) ;

void SetNTot (double ) ;

void PrintLambda ( ) ;

void PrintDiameter ( ) ;

void GetDandN ( ) ;

˜DropDist ( ) ;

private :

double ∗ l , ∗ r , ∗d , ∗∗Qarray , ∗Ndens , ∗Cumulative , ∗Ext inc t ion ;

double ∗N1 , ∗N2 , ∗Qvr ,∗Qvl ;

double lmax , lmin , rmax , rmin , deld , Ntot , D32 ;

int r tot , l t o t ;

stat ic int DEBUG;

MieScat te re r mie ; // i n s t a n t i a t e MieScat terer

} ;

#endif

99

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗


//∗∗ Fi l e : d ropd i s t . cpp

//∗∗ Jason M. Porter

//∗∗ Stanford HTGL

//∗∗ Phone : (650) 723−0941

//∗∗ Email : jasonpor ter@stanford . edu

//∗∗ Class modeling e x t i n c t i o n by d rop l e t d i s t r i b u t i o n s

// ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

#include ”miescat . h”

#include <iostream>

#include ” dropd i s t . h”

#include <s t r i ng >

#include <f stream>

#include <iomanip>

#include <cmath>

#include <ctime>

#include <c s t d l i b >

using namespace std ; // Use unqua l i f i e d names f o r Standard C++ l i b r a r y

using namespace SCATMECH; // Use unqua l i f i e d names f o r the SCATMECH l i b r a r y

// I n s t a n t i a t e mie sca t t e r e r

int DropDist : :DEBUG = 0 ;

// i n i t i a l i z e v a r i a b l e s

DropDist : : DropDist ( int rt , int l t , double rmn , double rmx , double lmn , double lmx ){

r t o t = r t ;

l t o t = l t ;

rmax = rmx ;

rmin = rmn ;

lmax = lmx ;

lmin = lmn ;

de ld = 2∗( rmax − rmin )/ ( r tot −1); // dD in e x t i n c t i o n equat ion

// dynamica l ly a l l o c a t e memory fo r arrays

l = new double [ l t o t ] ; // wave length [ microns ]

Ext inc t ion = new double [ l t o t ] ; // d rop l e t e x t i nc t i on , Qext

N1 = new double [ l t o t ] ; // r ea l r e f r a c t i v e index , n

N2 = new double [ l t o t ] ; // imaginary r e f r a c t i v e index , k

r = new double [ r t o t ] ; // d rop l e t rad ius [ microns ]

d = new double [ r t o t ] ; // d rop l e t diameter [ microns ]

Ndens = new double [ r t o t ] ; // drop s i z e d i s t r i b u t i o n funct ion , f (D)

Cumulative = new double [ r t o t ] ; // i n t e g r a l o f f (D)

Qvr = new double [ r t o t ] ; // Qext vs rad ius

Qvl = new double [ l t o t ] ; // Qext vs wave length

Qarray = new double ∗ [ l t o t ] ; // Qext array ( r t o t x l t o t )

for ( int i = 0 ; i<l t o t ; i++){

Qarray [ i ] = new double [ r t o t ] ;

}

// Query user f o r mie parameters , e . g . the r e f r a c t i v e index f i l e ( lambda , n , k )


// SCATMECH w i l l e x t rapo l a t e , i n t e r p o l a t e as needed

mie . AskUser ( ) ;

// cout << ” Qext = ” << mie . Qext()<< end l ;

}

DropDist : : ˜ DropDist ( ){

// free−up dynamica l ly a l l o c a t e d memory

delete [ ] l ;

delete [ ] r ;

delete [ ] d ;

delete [ ] Ndens ;

delete [ ] Cumulative ;

delete [ ] Ext inc t ion ;

delete [ ] N1 ;

delete [ ] N2 ;

for ( int i = 0 ; i<l t o t ; i++)

delete [ ] Qarray [ i ] ;

delete [ ] Qarray ;

delete [ ] Qvr ;

delete [ ] Qvl ;

}

// genera te s drop d i s t from input f i l e s (#rows = r t o t ) : Nin . t x t and Din . t x t

void DropDist : : GetDandN(){

i f s t r e am DIN( ”Din . txt ” ) ;

// cout << ” reading Din . t x t \n”;

i f (DIN . f a i l ( ) ){

c e r r << ”Din . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

for ( int i =0; i<r t o t ; i++){

DIN>>d [ i ] ;

r [ i ]=d [ i ] / 2 ;

}

i f s t r e am NIN( ”Nin . txt ” ) ;

// cout << ” reading Nin . t x t \n”;

i f (NIN . f a i l ( ) ){

c e r r << ”Nin . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

for ( int i =0; i<r t o t ; i++){

NIN>>Ndens [ i ] ;

}

double sum = 0 ;

for ( int n=0;n<r t o t ; n++){

sum = sum + Ndens [ n ] ;

Cumulative [ n ] = (sum∗(d [ n+1]−d [ n ] ) ) ;

}

}

101

// loads p r e v i ou s l y c a l c u l a t e d Qext array ( r t o t x l t o t ) from t e x t f i l e

void DropDist : : GetQArray ( ){

i f s t r e am QIN( ”Q Array . txt ” ) ;

// cout << ” reading Q Array . t x t \n”;

i f (QIN . f a i l ( ) ){

c e r r << ”Q Array . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

for ( int i =0; i<l t o t ; i++){

for ( int j =0; j<r t o t ; j++){

QIN>>Qarray [ i ] [ j ] ;

}

}

}

// p r i n t s Qext vs lambda to f i l e

void DropDist : : PrintExt ( ){

ofstream QDIST( ” Ext inc t ion . txt ” ) ;

// cout << ”Pr in t ing Ex t inc t i on . t x t \n”;

i f (QDIST. f a i l ( ) ){

c e r r << ” Ext inc t ion . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

QDIST<<”Lambda\ tExt inc t i on \n” ;

for ( int i =0; i<l t o t ; i++){

QDIST<<l [ i ]<< ’ \ t ’<<Ext inc t ion [ i ]<< ’ \n ’ ;

}

}

// p r i n t s r e f r a c t i v e index vs lambda to f i l e

void DropDist : : Pr intIndex ( ){

ofstream INDEX(” Index . txt ” ) ;

// cout << ”Pr in t ing Index . t x t \n”;

i f (INDEX. f a i l ( ) ){

c e r r << ” Index . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

INDEX<<”Lambda\ tn\ tk\n” ;

for ( int i =0; i<l t o t ; i++){

INDEX<<l [ i ]<< ’ \ t ’<<N1 [ i ]<< ’ \ t ’<<N2 [ i ]<< ’ \n ’ ;

}

}

// p r i n t s d r op l e t diameter to f i l e

void DropDist : : PrintDiameter ( ){

ofstream DIAM(”Diameter . txt ” ) ;

// cout << ”Pr in t ing Diameter . t x t \n”;

i f (DIAM. f a i l ( ) ){

c e r r << ”Diameter . txt could not be opened \n” ;

e x i t ( 1 ) ;


}

for ( int i =0; i<r t o t ; i++){

DIAM<<d [ i ]<< ’ \n ’ ;

}

}

// p r i n t s wave length to f i l e

void DropDist : : PrintLambda ( ){

ofstream LAMBDA(”Lambda . txt ” ) ;

// cout << ”Pr in t ing Lambda . t x t \n”;

i f (LAMBDA. f a i l ( ) ){

c e r r << ”Lambda . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

for ( int i =0; i<l t o t ; i++){

LAMBDA<<l [ i ]<< ’ \n ’ ;

}

}

// p r i n t s d r op l e t s i z e d i s t r i b u t i o n to f i l e

void DropDist : : Pr in tDi s t ( ){

ofstream NDENS( ”NDens . txt ” ) ;

// cout << ”Pr in t ing NDens . t x t \n”;

i f (NDENS. f a i l ( ) ){

c e r r << ”NDens . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

NDENS<<”diameter \ tNdens\ tCumulative \n” ;

for ( int i =0; i<r t o t ; i++){

NDENS<<d [ i ]<< ’ \ t ’<<Ndens [ i ]<< ’ \ t ’<<Cumulative [ i ]<< ’ \n ’ ;

}

}

// p r i n t s c a l c u l a t e d Qext array to f i l e ( r t o t x l t o t )

void DropDist : : PrintQ (){

ofstream QARRAY(”Q Array . txt ” ) ;

// cout << ”Pr in t ing Q Array . t x t \n”;

i f (QARRAY. f a i l ( ) ){

c e r r << ”Q Array . txt could not be opened \n” ;

e x i t ( 1 ) ;

}

for ( int i =0; i<l t o t ; i++){

for ( int j =0; j<r t o t ; j++){

QARRAY<<Qarray [ i ] [ j ]<< ’ \ t ’ ;

}

QARRAY<< ’ \n ’ ;

}

}

// c a l c u l a t e s r a d i i and diameters from inpu t s

void DropDist : : SetRadius ( ){

103

i f (DEBUG == 1)

cout << ” r a d i i \n” ;

i f ( r t o t == 1){

r [ 0 ] = rmin ;

d [ 0 ] = 2∗ rmin ;

}

else {

for ( int i =0; i<r t o t ; i++){

r [ i ] = rmin + static cast<double>( i )∗ ( rmax−rmin )

/( static cast<double>( r t o t ) − 1 ) ;

d [ i ] = 2∗ r [ i ] ;

i f (DEBUG == 1)

cout << r [ i ] << ’ \n ’ ;

}

}

}

// c a l c u l a t e s wave lengths from inpu t s

void DropDist : : SetLambda ( ){

i f (DEBUG == 1)

cout << ” lambdas\n” ;

i f ( l t o t == 1)

l [ 0 ] = lmin ;

else {

for ( int i =0; i<l t o t ; i++){ // 0−99

l [ i ] = lmin + static cast<double>( i )∗ ( lmax−lmin )

/( static cast<double>( l t o t ) − 1 ) ;

// l [ i ] = 1/ l [ i ] ; // temporary convers ion fo r QVL ca l c

i f (DEBUG == 1)

cout << l [ i ] << ’ \n ’ ;

}

}

}

// c a l c u l a t e Qext by c a l l i n g SCATMECH l i b r a r y func t i ons

void DropDist : : GetQ(){

int i , j ;

i f (DEBUG == 1)

cout << ”Qarray [ lambda ] [ r ad iu s ]\n” ;

i f (DEBUG == 1){

cout << ’ \ t ’ ;

for ( i =0; i<r t o t ; i++)

cout << r [ i ] << ” ” ;

cout << ’ \n ’ ;

}

for ( j =0; j<r t o t ; j++){

mie . s e t r a d i u s ( r [ j ] ) ;

for ( int k=0;k<l t o t ; k++){

mie . set lambda ( l [ k ] ) ;


Qarray [ k ] [ j ] = mie . Qext ( ) ;

// c a l c u l a t e s forward s c a t t e r i n g w/o absorp t ion

//Qarray [ k ] [ j ] = mie . Qsca ( ) ;

}

}

}

// c a l c u l a t e s log−normal d r op l e t s i z e d i s t r i b u t i o n from Dbar and q

void DropDist : : LogNormal (double DistD , double Sig ){

i f (DEBUG == 1){

cout << ”diams\n” ;

for ( int n=0; n<r t o t ; n++){

cout << d [ n ] << ’ \n ’ ;

}

}

i f (DEBUG == 1)

cout << ”Ndens\n” ;

double term1 , term2 ;


term1 = ( log (d [ n])− l og ( DistD ) ) / ( l og ( S ig ) ) ;

term2 = (1/ sq r t (2∗ pi )/ l og ( S ig )/d [ n ] ) ;

Ndens [ n ] = term2∗exp (( −0.5)∗ ( term1∗ term1 ) ) ;

i f (DEBUG == 1)

cout << Ndens [ n ] << ’ \n ’ ;

}

double sum = 0 ;


sum = sum + Ndens [ n ] ;

Cumulative [ n ] = (sum∗deld ) ;

}

}

// c a l c u l a t e s Rosin−Rammler d rop l e t s i z e d i s t r i b u t i o n from Dbar and q

void DropDist : : Rosin Rammler (double Dbar , double q ){

i f (DEBUG == 1){

cout << ”diams\n” ;


cout << d [ n ] << ’ \n ’ ;

}

}

i f (DEBUG == 1)

cout << ”Ndens\n” ;


Cumulative [ n ] = 1−exp(−pow( ( d [ n ] / Dbar ) , q ) ) ;

Ndens [ n ] =(q/Dbar )∗pow( ( d [ n ] / Dbar ) , q−1)∗exp(−pow( ( d [ n ] / Dbar ) , q ) ) ;

i f (DEBUG == 1)

cout << Ndens [ n ] << ’ \n ’ ;

}

}

105

// c a l c u l a t e s t o t a l l i q u i d volume from user input d r op l e t loading , Ntot

void DropDist : : SetNTot (double Nt){

Ntot = Nt ;

cout << Ntot << ’ \n ’ ;

double Vtot = 0 ;


i f ( ( n==0) | |(n==(rtot −1)))

Vtot += ( pi /6)∗pow( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ 0 . 5 ∗ ( Ndens [ n]+Ndens [ n−1 ] ) ;

else

Vtot += ( pi /6)∗pow( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ ( Ndens [ n]+Ndens [ n−1 ] ) ;

}

cout << Vtot << ’ \n ’ ;

i f (DEBUG == 2)

cout << ”Ntot = ” << Ntot << ’ \n ’ ;

}

// c a l c u l a t e s d r op l e t l oad ing from l i q u i d volume f r a c t i o n

double DropDist : : NTot(double fv ){

double Vtot = 0 ;


i f ( ( n==0) | |(n==(rtot −1)))

Vtot += ( pi /6)∗pow( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ 0 . 5 ∗ ( Ndens [ n]+Ndens [ n−1 ] ) ;

else

Vtot += ( pi /6)∗pow( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ ( Ndens [ n]+Ndens [ n−1 ] ) ;

}

Ntot = f l o o r ( fv /( de ld ∗Vtot ) ) ;

i f (DEBUG == 2)

cout << ”Ntot = ” << Ntot << ’ \n ’ ;

return Ntot ; // [cm−3]

cout << Vtot << ’ \n ’ ;

}

// c a l c u l a t e s Suater mean diameter D32 dˆ3/dˆ2

double DropDist : : D 32 ( ){

double Num = 0 ;

double Den = 0 ;


i f ( ( n==1) | |((n==rtot −1))){

Num+=0.5∗(0.5∗(Ndens [ n]+Ndens [ n−1]))∗pow(d [ n ] , 3 ) ;

Den+=0.5∗(0.5∗(Ndens [ n]+Ndens [ n−1]))∗pow(d [ n ] , 2 ) ;

}

else {

Num+=(0.5∗(Ndens [ n]+Ndens [ n−1]))∗pow(d [ n ] , 3 ) ;

Den+=(0.5∗(Ndens [ n]+Ndens [ n−1]))∗pow(d [ n ] , 2 ) ;

}

}

D32 = Num/Den ; // de ld cance l s


i f (DEBUG == 2)

cout << ”D32 = ” << D32 << ’ \n ’ ;

return D32 ;

// cout << ”D32 = ” << D32 << ’\n ’ ;

}

// c a l c u l a t e s e x t i n c t i o n [cm−1] vs lambda , r e f r a c t i v e index vs lambda

void DropDist : : QDist ( ){

COMPLEX index ;

i f ( (DEBUG == 1 ) | | (DEBUG == 2))

cout << ” Ext inc t ion \n” ;

double hold=0;

for ( int i =0; i<l t o t ; i++){

hold = 0 ;


i f ( ( n==1) | |(n==(rtot −1)))

hold+=0.5∗(Ntot ∗0 .5∗ ( Ndens [ n]+Ndens [ n−1]))∗Qarray [ i ] [ n ]

∗pow ( ( ( d [ n]+d [ n−1 ] )/2∗0 .0001) , 2 )∗ ( d [ n]−d [ n−1 ] ) ;

else

hold+=(Ntot ∗0 .5∗ ( Ndens [ n]+Ndens [ n−1]))∗Qarray [ i ] [ n ]

∗pow ( ( ( d [ n]+d [ n−1 ] )/2∗0 .0001) , 2 )∗ ( d [ n]−d [ n−1 ] ) ;

}

// Ex t inc t i on [ i ]=ho ld ∗( de ld )∗ p i /4;

Ext inc t ion [ i ]=hold ∗ pi /4 ; // diameter b in s i z e changes

index = mie . g e t sphe r e ( ) . index ( l [ i ] ) ;

N1 [ i ] = r e a l ( index ) ;

N2 [ i ] = imag ( index ) ;

}

}

// c a l c u l a t e s Qext vs d r op l e t rad ius f o r one wave length

void DropDist : : QvsRadius (double lambda ){

i f (DEBUG == 3){

// cout << ” rad ius \n”;

// fo r ( i n t i =0; i<r t o t ; i++)

// cout << r [ i ] << ’\n ’ ;

}

i f (DEBUG == 3)

cout << ”Qvr [ r ad iu s ]\n” ;

mie . set lambda ( lambda ) ;

// cout << mie . get lambda()<< ’\n ’ ;

for ( int k=0;k<r t o t ; k++){

mie . s e t r a d i u s ( r [ k ] ) ;

// cout << mie . g e t r a d i u s ()<< ’\n ’ ;

Qvr [ k ] = mie . Qext ( ) ;

i f (DEBUG == 3)

cout << Qvr [ k ] << ”\n” ;

107

}

}

// c a l c u l a t e s Qext vs lambda fo r one d rop l e t diameter and p r in t s to f i l e

void DropDist : : QvsLambda(double rad iu s ){

SetLambda ( ) ;

i f (DEBUG == 3)

cout << ”Qvl [ lambdas ]\n” ;

i f (DEBUG == 3){

cout << ’ \ t ’ ;

for ( int i =0; i<l t o t ; i++)

cout << l [ i ] << ” ” ;

cout << ’ \n ’ ;

}

mie . s e t parameter ( ” rad iu s ” , r ad iu s ) ;

for ( int j =0; j<l t o t ; j++){

mie . set lambda ( l [ j ] ) ;

Qvl [ j ] = mie . Qext ( ) ;

i f (DEBUG == 3)

cout << Qvl [ j ] << ”\n” ;

}

ofstream QVL( ”QVL. txt ” ) ;

// cout << ”Pr in t ing QVL. t x t \n”;

i f (QVL. f a i l ( ) ){

c e r r << ”QVL. txt could not be opened \n” ;

e x i t ( 1 ) ;

}

QVL<<”Lambda\tQ\n” ;

for ( int i =0; i<l t o t ; i++){

QVL<<l [ i ]<< ’ \ t ’<<Qvl [ i ]<< ’ \n ’ ;

}

}

Appendix C

Transmittance and reflectance for

absorbing media

As stated in Chapter 4, the equation commonly used for normal transmittance at the

interface of two media i and j:

Tij =4ninj

(ni + nj)2(C.1)

where ni and nj are the refractive indices of medium i and j, respectively, is only

valid if neither medium i nor j are absorbing. A derivation1 of the transmittance and

reflectance for light incident from a non-absorbing medium, with refractive index m2,

to an absorbing medium, with complex refractive index m1 = n1 + ik1 is given below.

Figure C.1 depicts a plane wave electric field incident normal to the interface of two

media. Conservation of the tangential electric and magnetic fields at the boundary

results in expressions for the reflection and transmission coefficients:

r =1 − m1

m2

1 + m1

m2

(C.2)

t =2

1 + m1

m2

(C.3)

1This derivation follows that of Born and Wolf [37] and Bohren and Huffman [34]

108

109

m1 = n1+ik1

Er

Ei

Et

m2 = n2

Figure C.1: Reflection and transmission of normally incident light.

The amplitudes of the transmitted and reflected electric fields are then given by:

E0,r = rE0,i and E0,t = tE0,i. Relations of transmitted and reflected light intensity

are what is needed in practice, which can be found using the Poynting vector for the

incident field evaluated at z = 0 (refer to Eq. 1.8):

Si =1

2Re

{√

ǫ2µ2

}

|E0,i|2 (C.4)

Substituting E0,r and E0,t into Eq. C.4 results in expressions for the transmitted and

reflected light intensities:

Sr =1

2Re

{√

ǫ2µ2

}

|r|2 |E0,i|2 (C.5)

St =1

2Re

{√

ǫ1µ1

}

∣

∣t∣

∣

2 |E0,i|2 (C.6)

110APPENDIX C. TRANSMITTANCE AND REFLECTANCE FOR ABSORBING MEDIA

The reflectance and transmittance are defined as:

R =Sr

Si

= |r|2 (C.7)

T =St

Si

= Re

{√

ǫ1µ2

µ1ǫ2

}

∣

∣t∣

∣

2(C.8)

From the definition of the refractive index: m = c0√ǫµ, and from the fact that for

non-conducting media the approximation µ1 = µ2∼= 1 can be used, Eqs. C.7 and C.8

can be solved by substituting r and t from Eqs. C.2 and C.3:

R = |r|2 =(n2 − n1)

2 + k21

(n2 + n1)2 + k21

(C.9)

T =m1

m2

∣

∣t∣

∣

2=

4n1n2

(n2 + n1)2 + k21

(C.10)

Equations C.9 and C.10 are applicable for plane waves in non-conducting homoge-

neous media, normally incident on the interface of two media, where one medium is

absorbing and the other is not. For the case studied here, light incident on a liquid

film either from gas or from a window material, the difference between the transmit-

tance calculated using Eq. C.10 and Eq. C.1 is < 1% for k < 0.2, typical of the liquid

hydrocarbons studied. Hence, in this analysis, Eq. C.1 is used for simplicity.

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LASER-BASED DIAGNOSTICS FOR HYDROCARBON FUELS IN …