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: H2 O, 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 transitions 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 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 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 measurements of the real and imaginary refractive index spectra of several liquid hydrocarbons 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/extinction diagnostic for simultaneous measurement of temperature and vapor-phase mole
fraction in an evaporating hydrocarbon fuel aerosol (vapor and liquid droplets) is described. 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 simultaneous 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 application 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 “Hanson 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 understanding 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
1.4
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
Light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4.1
Mie theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.4.2
Droplet distributions . . . . . . . . . . . . . . . . . . . . . . .
23
2 Measurements of Absorption Spectra in Liquid Fuels
2.1
2.2
25
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Absorption measurements in liquids . . . . . . . . . . . . . . . . . . .
27
2.2.1
27
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
x
2.3
2.4
2.2.2
Measurement procedure . . . . . . . . . . . . . . . . . . . . .
27
2.2.3
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.3.1
Hydrocarbons measured . . . . . . . . . . . . . . . . . . . . .
35
2.3.2
Plots of absorption spectra for liquid and vapor . . . . . . . .
36
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
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
3.3
4 Liquid Fuel Films
4.1
4.2
67
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.1.1
Fuel films and measurement techniques . . . . . . . . . . . . .
67
Two-phase laser-absorption measurements . . . . . . . . . . . . . . .
69
4.2.1
69
Beer’s law: vapor and liquid . . . . . . . . . . . . . . . . . . .
xi
4.2.2
4.3
4.4
Laser wavelength selection . . . . . . . . . . . . . . . . . . . .
72
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
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
5.1
5.2
89
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.1.1
Absorption spectra of liquid fuels . . . . . . . . . . . . . . . .
89
5.1.2
Evaporating fuel aerosols . . . . . . . . . . . . . . . . . . . . .
90
5.1.3
Fuel films . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
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. . . . . . . . . . . . . .
2.2
Fractional composition of alkanes, aromatics, and olefins for the three
gasoline blends analyzed. . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
42
Dependency of calculated extinction ratios (R21 = 1.08 and R31 = 1.14)
on initial droplet size distribution (log-normal, see equation 3.7). . . .
4.1
41
56
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 absorbing material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Schematic of laser light passing through a gas cell filled with an absorbing gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
6
Infrared absorption spectrum of n-decane (Top figure from Sharpe et.
al. [2]).
1.4
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Schematic of n-decane molecule showing fundamental C-H stretch vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . .
2.2
30
Measured extinctance of liquid toluene for three path lengths. Two
anchor points were selected near the absorption band (2681 and 3290
cm−1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
31
Extrapolation of measured refractive index data [3] at visible wavelengths 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. . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
33
2.4
Flow chart showing the data analysis in converting the measured extinction 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 correction, from the shortest path length extinction spectrum. . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
2.7
2.8
34
35
Measured real refractive index for liquid toluene near 3000 cm−1 are
compared with published data. The residual is defined as nporter − nbertie . 36
Measured real refractive index for liquid n-decane and n-dodecane
(2000 − 4000 cm−1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Measured absorption cross-section for liquid and vapor toluene. . . .
38
Measured absorption cross-section for liquid and vapor n-decane. . . .
38
2.10 Measured absorption cross-section for liquid and vapor n-dodecane. .
39
2.9
2.11 Measured absorption cross-section for liquid and vapor gasoline sample1. 39
2.12 Measured absorption cross-section for liquid and vapor gasoline sample2. 40
2.13 Measured absorption cross-section for liquid and vapor gasoline sample3. 40
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. . . . . . .
3.1
43
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. . . . . . . . . . . . . .
xv
52
3.5
FTIR measurements (corrected to ensure a constant integrated cross
section for this band) of temperature-dependent n-decane vapor absorption. Wavelengths ν̄1 and ν̄3 were chosen to maximize temperature
sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Temperature dependence of vapor-phase n-decane absorption crosssections at chosen wavelengths. . . . . . . . . . . . . . . . . . . . . .
3.7
55
Calculated extinction ratios during evaporation for two initial droplet
size distributions. Ratios are largely constant during evaporation. . .
3.8
54
57
Sensitivity analysis of measured mole fraction and temperature using
a 2 percent uncertainty. The ratio of droplet extinction to vapor absorption at ν̄1 , 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). . . . . . . . . . . . . . . . . . . .
3.9
58
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 (reference 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 ndecane 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 decreases as the aerosol evaporates. . . . . . . . . . . . . . . . . . . . .
xvi
64
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. . . . . . . . . . . . . . . . . . . . . . . .
4.1
Schematic of laser light passing through a window, a fuel film of thickness δ, and fuel vapor. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
69
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. . . . . . . . . . . . . . . . . . . . . . . . .
4.3
66
70
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]. . . . . . . . . . . . . . . . . . . . . . . .
4.4
71
a: Calculated reciprocal condition number of matrix A for all wavelength pairs (ν̄1 , ν̄2 ) between 2800 − 3000cm−1 . Darker regions show
wavelength pairs yielding well-conditioned matrices. a & b: Three
wavelength pairs are highlighted: the selected wavelength pair (ν̄1 =
2854.7cm−1 and ν̄2 = 2864.7cm−1 ) within the DFG phase matching
range, and two optimal wavelength pairs (I & II) not attainable with
the current DFG system.
4.5
. . . . . . . . . . . . . . . . . . . . . . . .
73
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 > nf ilm > ngas ), then Io,f ilm > Io . . . . . . . . . . . . . .
4.6
Measured refractive index of n-dodecane (right) and calculated baseline
offset for liquid n-dodecane films on several window materials (left). .
4.7
75
77
FTIR measurements of liquid n-dodecane films injected onto the windows listed in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
78
4.8
a: Measured absorbance of free-standing liquid n-dodecane for several film thicknesses, where the measured transmission, I, is equal to
Io,f ilm − absorption, and Io is the same as in Fig. 4.5a. b: Comparison of inferred cross-sections from free-standing film measurements to
previously measured cross-section.
4.9
. . . . . . . . . . . . . . . . . . .
80
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 onenear-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 subsequently 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 absorbance from n-dodecane liquid and n-decane vapor at both wavelengths. b: Measured liquid n-dodecane film thickness and vapor ndecane mole fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
86
Potential laser frequencies for sensitive detection of vapor temperature,
vapor mole fraction, and liquid film thickness. . . . . . . . . . . . . .
C.1 Reflection and transmission of normally incident light.
xviii
92
. . . . . . . . 109
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 continuouswave 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 temperature [9]. A large number of species important to combustion have been successfully measured including: H2 O [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], ndodecane [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 midinfrared 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 hydrocarbons 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 diagnostics 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 literature. 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 laserabsorption 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 measurements 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 temperature 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 heavily 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
4
CHAPTER 1. INTRODUCTION
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
1.2.1
Laser-absorption spectroscopy
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 mathematical statement of an experimental observation: the attenuation of light through certain
material samples is dependent on the sample’s thickness and concentration. The dependence 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 BeerLambert-Bouguer’s law or Beer-Lambert’s law, but it will be referred to as Beer’s law
in this thesis [31, 32].
Iv
Iv + dIv
dz
Figure 1.1: Schematic of monochromatic light passing through a thin slab of absorbing
material.
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 ν̄, where ν̄
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
= αL = Xnσ(ν̄, T )L
(1.2)
ν̄
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 crosssection, 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 schematically in Fig. 1.2. Light passes through the optical ports (windows) of a gas cell of
1
In this thesis frequencies and wavelengths are used interchangeably as they are inversely related:
1
.
ν̄[cm−1 ] = λ[cm]
6
CHAPTER 1. INTRODUCTION
Thermocouple
Pressure
Gas cell
Laser light (v )
Io
X, Tg , P
L
I
Windows
Figure 1.2: Schematic of laser light passing through a gas cell filled with an absorbing
gas.
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 ν̄ 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 measured directly. The absorption cross-section of the species of interest is typically a
7
1.2. LASER-ABSORPTION SPECTROSCOPY
function of temperature, pressure, and wavelength, and can be measured in the laboratory. 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
n-decane vapor
T = 300 K
P = 1 atm
50
0
2000
2
( , T ) [m /mole]
100
150
6000
T = 325 K
T = 500 K
T = 625 K
T = 725 K
100
50
4000 -1
Frequency [cm ]
P = 1 atm
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
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
8
CHAPTER 1. INTRODUCTION
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 vibrational 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
2
In this research, a Nicolet 6700 FTIR was used, with a minimum resolution of 0.09 cm−1
(FWHM).
9
1.2. LASER-ABSORPTION SPECTROSCOPY
n-decane molecule (C10H22)
ω
C
H
Figure 1.4: Schematic of n-decane molecule showing fundamental C-H stretch vibration.
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 transition 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 mechanical selection rules. These transitions show up as discrete absorption lines in the
10
CHAPTER 1. INTRODUCTION
absorption spectra of small molecules (e.g. H2 O, CO2 , and CO). For larger hydrocarbon 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 complexity 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
11
1.2. LASER-ABSORPTION SPECTROSCOPY
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
2
( , T ) [m /mole]
− ln
I
Io
ν̄1 =
I
Io
σ(ν̄1 , T )
= f (T )
σ(ν̄2 , T )
n-decane
T = 325 K
T = 500 K
T = 625 K
T = 725 K
v2
100
50
(1.3)
ν̄2
v 1
150
P = 1 atm
0
2700
200
2800
2900
3000
-1
Frequency [cm ]
3100
= 2938 cm
-1
2 = 2952 cm
-1
3
)/ (
)
2
1
2
(
1
2
100
1
0
300
400
500
600
700
800
Temperature [K ]
0
900
Absorption ratio
( T ) [m /mole]
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, ν̄1 , has been chosen near peak absorption, where the absorption
decreases rapidly with increasing temperature, and the second laser wavelength, ν̄2 ,
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
12
CHAPTER 1. INTRODUCTION
absorbing species can be determined simultaneously. This is important for applications 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
1.3.1
The refractive index
Beer’s Law from Maxwell’s equations
The derivation of Beer’s law presented previously was based on experimental observation 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 definition 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 electromagnetic radiation as plane waves. The plane wave approximation is characterized by a
traveling electromagnetic wave with harmonic oscillations in its electric and magnetic
13
1.3. THE REFRACTIVE INDEX
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).
B
x
E
z
y
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:
~c = E
~ 0 exp(iκz − i2πc0 ν̄t)k̂;
E
~c =B
~ 0 exp(iκz − i2πc0 ν̄t)k̂
B
(1.4)
~ 0 and B
~ 0 are constant vectors equal to the magnitude of the waves, κ is the
where E
propagation number3 , ν̄ is the optical frequency of the wave, t is time, k̂ 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′′ = 2πν̄co ǫµ ≡ 2πν̄(n + ik)
(1.5)
where co is the speed of light in vacuum, ǫ is the complex permittivity, µ is the permeability, 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 (independent of the position), and isotropic (independent of the direction). Substituting
3
Wavenumber is also used, but this would create confusion with frequency units.
14
CHAPTER 1. INTRODUCTION
the complex refractive index into Eq. 1.4 results in a plane wave of the form:
~c = E
~ 0 exp(−2πν̄kz)exp(i2πν̄nz − i2πc0 ν̄t)
E
(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:
o
n
~S = 1 Re E
~c ×B
~∗
c
2
(1.7)
where the complex conjugate of the magnetic field vector has been taken. Performing
the calculus one then obtains:
~S = 1 Re
2
r ǫ ~ 2
E0 exp(−4πν̄kz)k̂
µ
(1.8)
The magnitude of ~S is the irradiance
or intensity
of the light, which is given the
q
~ 2
ǫ
symbol I [W/m2 ]. Defining I0 ≡ 21 Re
E0 , results in Beer’s law derived from
µ
Maxwell’s equations:
I = I0 exp(−αz)
(1.9)
where z is the distance and the absorption coefficient α is defined as:
α ≡ 4πν̄k
(1.10)
1.3. THE REFRACTIVE INDEX
15
Thus, the imaginary refractive index is directly proportional to the absorption coefficient 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 component 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 physical 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 relative 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 measure of this phase shift, or equivalently, a measure of the speed of light in the medium
relative to vacuum.
4
This discussion follows that of Hecht [36].
16
CHAPTER 1. INTRODUCTION
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:
qe E(t) − me ω02 r − me γ
dr
d2 r
= me 2
dt
dt
(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) = E0 cosωt), 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 = qe rN .
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 +
N qe 2 X
fj
2
2πǫ0 me j 2π(ω0j − ω 2 ) + iγj ω
(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 summaP
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).
17
1.3. THE REFRACTIVE INDEX
0.3
n
k
1.4
1.3
1.2
2000
0.2
T = 300 K
P = 1 atm
2500
3000
3500
-1
Frequency [cm ]
0.1
Refractive Index, k
Refractive Index, n
1.5
0.0
4000
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
2
any one wavelength. Second, at frequencies away from absorption transitions, (ω0j
−
ω 2 ) >> iγj ω, and iγj ω 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
5
Spectral regions with dn/dω < 0, are referred to as exhibiting anomalous dispersion.
18
CHAPTER 1. INTRODUCTION
refractive index for n-decane liquid as a function of frequency, ν̄.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, allowing a simplified dispersion equation to be used that relates the real refractive index
to the optical frequency [37]:
n(ν̄)2 = A1 + A2 ν̄ 2 + A3 ν̄ 4
(1.13)
where A1 , A2 , and A3 are constant functions of the oscillator strengths and fundamental 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 required 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 refractive 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 covered here as it has little physical intuition, but the result is the Kramers-Kronig or
dispersion relations [34]:
2
n(ν̄i ) − n(∞) = P
π
Z∞
0
6
ν̄k(ν̄)
dν̄
ν̄ 2 − ν̄i2
The angular frequency is converted to wavenumbers by
ω
c0
= 2πν̄.
(1.14)
19
1.4. LIGHT SCATTERING
−2ν̄i
P
k(ν̄i ) =
π
Z∞
0
n(ν̄)
dν̄
− ν̄i2
ν̄ 2
(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 ν̄ = ν̄i .
The Kramers-Kronig relations state that the value of the real or imaginary component 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 spectrum, 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 wavelength 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 combustion 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
20
CHAPTER 1. INTRODUCTION
Chapter 3, but first a brief introduction to Mie scattering is needed.
1.4.1
Mie theory
Es , Bs
Eint ,Bint
Ei , Bi
D
m2
Et , Bt
m1
Figure 1.8: Schematic of light being scattered and absorbed by a liquid droplet.
Mie scattering has been extensively studied and analytical expressions for scattering efficiencies and directions can be found in many texts (Bohren and Huffman’s
book [34] is recommended). Mie theory considers light scattering by spherical particles, 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, corresponding 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
21
1.4. LIGHT SCATTERING
absorption within the droplet and the scattering in all directions other than forward,
by conservation of energy:
Wt (ν̄) = Wi (ν̄) − Ws (ν̄) − Wa (ν̄)
(1.16)
where Wt is the power of the transmitted light in the forward direction, Wi = Ii A
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:
∇2 ψ + κ2 ψ = 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
22
CHAPTER 1. INTRODUCTION
~ = ∇ × (~rψ)).
set of complete vector functions: the vector spherical harmonics (i.e. M
The vector spherical harmonics are in turn used to generate solutions to the vector
~ + κ2 E
~ = 0). After applying the appropriate boundary
wave equations (i.e. ∇2 E
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
2π X
=
(2n + 1)Re {an + bn }
= 2
Ii
κ n=1
[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 m̄:
x=
πm2 D
λ
m̄ =
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 m̄ = 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 ν̄ 4 ). 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
23
1.4. LIGHT SCATTERING
6
5
Qext
4
Liquid toluene droplet
D = 6 µm
T = 25°C
P = 1atm
D = 1 µm
D = 3 µm
3
2
1
0
0.5
1.0
1.5
-1
Inverse wavelength [µm ]
2.0
2.5
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
24
CHAPTER 1. INTRODUCTION
absorption coefficient presented earlier (Eq. 1.1):
τ=
X
Nj Cext,j
(1.20)
j
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:
τν̄ =
Z
Dmax
Dmin
N · f (D)
πD2
Qext,ν̄ (D, m)dD
4
(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++ routines developed for this research. The combined computer code is listed in Appendix
B, and calculates the extinction cross section, τν̄ , for user defined refractive index
spectra and droplet size distributions.
Chapter 2
Measurements of Absorption
Spectra in Liquid Fuels
2.1
2.1.1
Introduction
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 measurements 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 internal combustion engines [43, 44, 45, 46, 47] and pool fires [48], and laser light scattering 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 techniques [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 measurement 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
2.2
2.2.1
27
Absorption measurements in liquids
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 Michelson 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 software (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
= σ(ν̄, T )na L
(2.1)
ν̄
where I and I0 are the incident (baseline) and transmitted intensities, respectively,
σ(ν̄, T ) [cm2 /mole] is the temperature- and wavelength-dependent absorption crosssection, na [mole/cm3 ] is the concentration of the absorbing species, L [cm] is the
path length, and ν̄ [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 simultaneous determination of the real and imaginary refractive indices of a liquid sample
28CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
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 captures 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 simultaneous 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 positioning, 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 correction 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 resulting 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(ν̄) = n(ν̄) + ik(ν̄)
(2.2)
2.2. ABSORPTION MEASUREMENTS IN LIQUIDS
29
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(ν̄) =
4πν̄k(ν̄)
na
(2.3)
where σ(ν̄) [cm2 /mole] is the absorption cross-section and na [mole/cm3 ] is the concentration of the absorbing species in the gas, liquid or solid phase. The liquid concentrations 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(ν̄) ≡ − ln
I
I0
(2.4)
ν̄
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(ν̄), 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.
30CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
KBr window
Teflon spacer
drilled KBr window
32mm
3mm
liquid fill ports
Figure 2.1: Optical cell used in liquid absorption measurements. Teflon spacers with
thicknesses between 0.5 mm and 15 µm were used.
Extremely short path lengths were required to ensure sensitive absorption measurements at wavelengths near 3000 cm−1 where there is strong hydrocarbon absorption. 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(ν̄) 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(ν̄1 − ν̄2 )
(2.5)
where M is the number of intensity peaks between optical frequencies ν̄1 and ν̄2 .
The measured path lengths agreed within 10 percent with the stated Teflon insert
31
2.2. ABSORPTION MEASUREMENTS IN LIQUIDS
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 (ν̄) − Ext0,2
L1
Ext1 (ν̄) − Ext0,1
(2.6)
where L1 is the measured path length (≥ 15 µm), L2 is the unknown path length,
and Ext1 (ν̄) and Ext2 (ν̄) 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 introduced 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.
Extinctance
2.0
1.5
500 µm
25 µm
15 µm
Toluene
T = 27 °C
P = 1 atm
1.0
0.5
anchor
point #1
anchor
point #2
0.0
2600
2800
3000
3200
-1
Frequency [cm ]
3400
Figure 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 ).
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
32CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
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(ν̄) [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(ν̄) 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(ν̄), at the anchor points to approximate the imaginary refractive index using Eq. (2.3). A first approximation to the real refractive index was
then calculated using the Kramers-Kronig relation:
2
n(ν̄i ) − n(∞) = P
π
Z∞
0
ν̄k(ν̄)
dν̄
ν̄ 2 − ν̄i2
(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 ν̄ = ν̄i .
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(ν̄)2 = A1 + A2 ν̄ 2 + A3 ν̄ 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
2.2. ABSORPTION MEASUREMENTS IN LIQUIDS
33
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.
Real Index, n
1.55
1.50
toluene fit
Wohlfarth
n-dodecane fit
Wohlfarth
n-decane fit
Wohlfarth
1.45
1.40
10000
15000
20000
-1
Frequency [cm ]
Figure 2.3: Extrapolation of measured refractive index data [3] at visible wavelengths
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.
The approximate reflection losses in the extinction measurements were calculated
from the Fresnel relations using the linear absorption coefficient, K(ν̄), 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
34CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
(from reference [4]), and to calculate the final σ(ν̄) 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.
Exncon spectra
(all path lengths)
Initial n spectrum
Path lengths
KBr index, nKBr (ν )
K (ν ) at
Anchor Points
(ANCHORPT)
K (ν
n∞
Path lengths
KBr index, nKBr (ν )
)
Baseline
Correcon
(EXTABS2K)
k spectra (all path lengths)
Build Composite
k Spectrum
Final k spectrum
Final n spectrum
Kramers-Kronig
Integraon
(KKTRANS)
Figure 2.4: Flow chart showing the data analysis in converting the measured extinction 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 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 technique is within 3 percent for the fundamental C-H stretch vibrational band. These
35
2.3. RESULTS
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(ν̄)
spectrum, which is likely due to the use of different anchor points than used by Bertie
et. al.
Residual
0.5
0.0
-0.5
Bertie et.al.
Porter et.al.
10
5
(
2
, T) [m /mole]
15
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 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 .
2.3
2.3.1
Results
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
36CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
Residual
2
0
-2x10
-3
Real Index, n
1.49
Toluene
T = 27 °C
P = 1 atm
Bertie et.al.
Porter et.al.
1.48
1.47
1.46
2000
2500
3000
3500
-1
Frequency [cm ]
4000
Figure 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 .
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 spectra 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 absorption 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
37
2.4. DISCUSSION
1.6
n-dodecane
n-decane
Real Index, n
T = 27 °C
P = 1 atm
1.5
1.4
1.3
2000
2500
3000
3500
-1
Frequency [cm ]
4000
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
2.4.1
Discussion
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 observations 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 increase 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
38CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
toluene
vapor (T = 27°C)
liquid (T = 27°C)
15
2
( , T ) [m /mole]
20
10
spectral shift
5
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.8: Measured absorption cross-section for liquid and vapor toluene.
2
( , T ) [m /mole]
120
n-decane
vapor (T = 25°C)
liquid (T = 27°C)
100
80
60
40
20
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.9: Measured absorption cross-section for liquid and vapor n-decane.
39
2
( , T ) [m /mole]
2.4. DISCUSSION
160
n-dodecane
140
vapor (T = 51°C)
liquid (T = 27°C)
120
100
80
60
40
20
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.10: Measured absorption cross-section for liquid and vapor n-dodecane.
50
gasoline sample 1
vapor (T = 53°C)
liquid (T = 27°C)
2
( , T ) [m /mole]
40
30
20
10
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.11: Measured absorption cross-section for liquid and vapor gasoline sample1.
40CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
60
40
2
( , T ) [m /mole]
gasoline sample 2
vapor (T = 53°C)
liquid (T = 27°C)
20
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.12: Measured absorption cross-section for liquid and vapor gasoline sample2.
50
gasoline sample 3
vapor (T = 53°C)
liquid (T = 27°C)
2
( , T ) [m /mole]
40
30
20
10
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 2.13: Measured absorption cross-section for liquid and vapor gasoline sample3.
41
2.4. DISCUSSION
the measured intensities are < 10 percent [64, 63]. The integrated fundamental vibrational 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 spectra. The integration limits were from 2600cm−1 to 3400cm−1 . Temperatures are
listed in the plotted spectra.
Name
Liquid
Vapor
−1
−1
2
Al [m mole cm ] Av [m mole−1 cm−1 ]
toluene
1180
1320
n-dodecane
9950
8330
n-decane
6260
7310
gasoline sample 1
3320
3470
gasoline sample 2
3500
3880
gasoline sample 3
3060
3230
2
Ratio
( AAvl )
0.89
1.19
0.86
0.96
0.90
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
1
=
Av
n̄
n̄2 + 2
3
2
(2.9)
where Al and Av are the band intensities in the liquid and vapor phases, respectively,
and n̄ 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
42CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
by the Polo and Wilson relation, whereas the integrated band intensities in liquid ndecane, 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 three
gasoline blends analyzed.
Name
Aromatics
Olefins
Alkanes
Volume % Volume % Volume %
gasoline sample 1
26.7
17.1
56.2
gasoline 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
43
2.4. DISCUSSION
60
liquid gasoline
sample 1
sample 2
alkanes
40
2
( , T ) [m /mole]
sample 3
olefins
20
aromatics
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
Figure 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.
the fractional composition of these hydrocarbon classes is listed in Table 2.2. Gasoline 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 composition 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 mixtures, 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
44CHAPTER 2. MEASUREMENTS OF ABSORPTION SPECTRA IN LIQUID FUELS
[69]. Therefore, 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 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, especially 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 spectroscopy (CARS) for point temperature measurements [72, 73], laser-induced fluorescence (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 hydrocarbon polydispersed (distribution of droplet diameters) aerosol with a measurement
bandwidth of 125 kHz. Measurements of light transmission at three wavelengths enable 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
3.2.1
Laser diagnostic
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:
Absν̄ ≡ − ln
I
I0
= σ(ν̄, T )na L
(3.1)
ν̄
where Absν̄ refers to the absorbance. For relatively large polyatomic hydrocarbons
like n-decane, the individual absorption transitions overlap, forming a broad spectrum
47
3.2. LASER DIAGNOSTIC
with the cross section, σ(ν̄, 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(ν̄, 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.
σ(ν̄1 , T )na L
σ(ν̄1 , T )
Absν̄1
=
=
= f (T )
Absν̄2
σ(ν̄2 , T )na L
σ(ν̄2 , 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 temperature sensing has been previously published for demonstration measurements of
n-heptane temperature and vapor mole fraction in shock-heated gases over a temperature 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]:
Extν̄ ≡ − ln
I
I0
=
ν̄
P Xa L
R̂T
σ(ν̄, T ) + τν̄ L
(3.3)
where τν̄ [cm−1 ] is the droplet extinction coefficient, the extinctance, Extν̄ , is defined
here as the combined light extinction from the wavelength-dependent vapor absorption (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(ν̄, T ) and τν̄ are known at both wavelengths.
However, for an evaporating aerosol, τν̄ will vary in time, making any static correction
prone to error. Instead, a three-wavelength technique is used that also infers τν̄ . Here
it is helpful to introduce an extinction ratio, which is defined as the ratio of droplet
48
CHAPTER 3. EVAPORATING FUEL AEROSOLS
extinction at wavelengths i and j :
Rij ≡
τi
τj
(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
=
P Xa L
i
R̂T
σi (T ) + Ri1 τ1 L 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, τν̄ 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,ν̄ (D, m), which is a measure of a particle’s ability to scatter and absorb light
at different wavelengths. The extinction coefficient,τν̄ , is determined by integrating
these parameters over the full range of droplet diameters, D, present in the aerosol.
τν̄ =
Z
Dmax
Dmin
N · f (D)
πD2
Qext,ν̄ (D, m)dD
4
(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
49
3.2. LASER DIAGNOSTIC
µm [77].
"
2 #
1
1 ln(D) − ln(D50 )
f (D) = √
exp −
2
ln(q)
2πln(q)D
(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 extinction calculations. (This analysis was repeated with a more sophisticated evaporation
model [78] with no significant change in the wavelengths chosen.)
D02 − 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 lognormal as the mean size decreases and the distribution widens, as illustrated in Fig.
3.1.
0.6
D50 = 3.3 µm
D50 = 2.5 µm
D50 = 1.7 µm
D50 = 1.3 µm
-1
f (D) [µm ]
0.5
0.4
0.3
0.2
0.1
0.0
0
2
4
6
Diameter [µm]
8
10
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.
50
CHAPTER 3. EVAPORATING FUEL AEROSOLS
3.2.4
Liquid optical constants
To calculate the extinction efficiency, Qext,ν̄ (D, m), the wavelength-dependent refractive index of the liquid was needed, which for an absorbing medium can be written
as a complex function.
m(ν̄) = n(ν̄) + ik(ν̄)
(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(ν̄) =
4πν̄k(ν̄)
na
(3.10)
where σ(ν̄) [cm2 /mole] is the absorption cross-section and na [mole/cm3 ] is the concentration 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,ν̄ (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 surrounding 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)
51
3.2. LASER DIAGNOSTIC
0.3
n
k
1.4
0.2
1.3
0.1
T = 300 K
P = 1 atm
1.2
2000
2500
3000
3500
-1
Frequency [cm ]
Refractive Index, k
Refractive Index, n
1.5
0.0
4000
Figure 3.2: Measured complex refractive index of liquid n-decane in the infrared.
Extinctance
2.0
1.5
N = 400,000 [cm-3]
L = 10 [cm]
D50 = 3.3
D50 = 2.5
D50 = 1.7
D50 = 1.3
D50 = 1.2
µm
µm
µm
µm
µm
1.0
0.5
0.0
3000
4000
5000
6000
-1
Frequency [cm ]
7000
Figure 3.3: Liquid n-decane droplet extinctance (no vapor absorption) calculated
from modeled droplet size distribution and measured optical constants.
52
CHAPTER 3. EVAPORATING FUEL AEROSOLS
using the calculated extinction efficiency, Qext,ν̄ (D, m), and the time-varying f (D)
from the D2 evaporation model (Fig. 3.3). Several trends are evident when comparing 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
Normalized Extinction
3.0
2.5
D50 = 3.3 µm
D50 = 2.5 µm
D50 = 1.7 µm
D50 = 1.3 µm
D50 = 1.2 µm
2.0
1.4
v3
v1
v2
1.2
1.0
2800
2900
3000
3100
1.5
1.0
3000
4000
5000 -1 6000
Frequency [cm ]
7000
Figure 3.4: Normalized extinction curves (at 2938 cm−1 ) showing wavelengths with
constant 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 vaporphase measurements for droplet scattering in experiments with large mean-diameter
aerosols [23, 49, 50]. However, for strongly absorbing aerosols with small mean diameters (D50 < 10 µm), large measurement uncertainties would result from ignoring
the wavelength dependence of the droplet extinction [50].
53
3.2. LASER DIAGNOSTIC
3.2.6
Wavelength selection
The calculated extinction spectra were normalized at 2938 cm−1 , and three wavelengths were identified with extinction ratios that are independent of droplet evaporation, as shown in Fig. (3.4). These wavelengths are listed below with the corresponding calculated droplet extinction ratios, R21 and R31 .
ν̄1 = 2938 cm−1 , R11 ≡ 1.00
ν̄2 = 2952 cm−1 , R21 = 1.08 (calculated)
ν̄3 = 2862 cm−1 , R31 = 1.14 (calculated)
The vapor-phase absorption cross-section, σ(ν̄, 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=
Z
band
σ(ν̄, T )dν̄ 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 (ν̄, Thigh ) =
Ψ(T300K )
σ(ν̄, Thigh )
Ψ(Thigh )
(3.12)
These scaled spectra agreed with previous measurements of n-decane cross sections
54
CHAPTER 3. EVAPORATING FUEL AEROSOLS
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(ν̄, T ) = ai T 2 + bi T + ci
i = 1, 2, 3
(3.13)
where ai , bi , ci , are wavelength-dependent constants. The strong temperature dependence at ν̄1 and ν̄3 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
=
P Xa L
R̂T
2
( , T ) [m /mole]
v2
v3
(3.14)
n-decane (vapor)
v1
150
100
(ai T 2 + bi T + ci ) + Ri1 τ1 L i = 1, 2, 3
T=300 K (Sharpe et. al.)
T=325 K (Sharpe et. al.)
T=500 K
T=625 K
T=725 K
P = 1 atm
50
0
2800
2850
2900
2950
3000
-1
Frequency [cm ]
3050
Figure 3.5: FTIR measurements (corrected to ensure a constant integrated cross
section for this band) of temperature-dependent n-decane vapor absorption. Wavelengths ν̄1 and ν̄3 were chosen to maximize temperature sensitivity.
With the wavelength-dependent absorption and extinction ratios known, temperature and vapor n-decane mole fraction were determined from the measured extinctance 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
55
3.2. LASER DIAGNOSTIC
n-decane
100
1
= 2938 cm
2
= 2952 cm
3
= 2862 cm
-1
-1
-1
2
, T ) [m /mole]
120
80
(
60
40
20
300
400
500
600
700
Temperature [K]
800
900
Figure 3.6: Temperature dependence of vapor-phase n-decane absorption crosssections 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 =
Xa =
−y +
p
y 2 − 4xz
2x
(R21 Ext1 − Ext2 )R̂T
P L(AT 2 + BT + C)
(3.15)
(3.16)
Ext1 P Xa
−
a1 T 2 + b 1 T + c 1
(3.17)
L
R̂T
The sensitivity of this solution to uncertainties in the measured values of Exti , L, P,
τ1 =
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).
56
CHAPTER 3. EVAPORATING FUEL AEROSOLS
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).
q
q
q
q
q
= 1.5µm
= 1.4µm
= 1.3µm
= 1.2µm
= 1.1µm
D50 = 2µm
D50 = 3.3µm
D50 = 5µm
∆R21 [%] ∆R31 [%] ∆R21 [%] ∆R31 [%] ∆R21 [%] ∆R31 [%]
-0.5
0.0
-2.4
-4.1
-5.8
-9.5
-0.3
0.9
-1.2
-2.1
-5.1
-8.6
-0.5
1.0
0.0
0.0
-4.0
-7.2
-1.1
0.5
0.8
1.6
-2.6
-5.0
-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 ν̄3 (Ext3 ), and R31 . The temperature was most sensitive to R31 ,
57
3.2. LASER DIAGNOSTIC
Extinction Ratio
1.20
= 3.3 µm
R21
R31
1.15
1.10
= 5 µm
R21
R31
1.05
1.00
1
2
3
4
5
6
Mean Diameter, D50 [µm]
7
Figure 3.7: Calculated extinction ratios during evaporation for two initial droplet size
distributions. 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 Extν̄ > 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.
58
CHAPTER 3. EVAPORATING FUEL AEROSOLS
γ
γ
γ
γ
γ
γ
γ=1
γ=1
a
b
c
Figure 3.8: Sensitivity analysis of measured mole fraction and temperature using a 2
percent uncertainty. The ratio of droplet extinction to vapor absorption at ν̄1 , 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).
3.3
3.3.1
Validation experiments
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 nearinfrared 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)
59
3.3. VALIDATION EXPERIMENTS
DFB Signal
Pulse
Generator Lasers
2µs pulse
Near- IR
DFB #1
I
1547.9 nm
Mid-IR Output
DFG #1
Near-IR Pump
Laser #1
(1064 nm)
Yb/Er Fiber
Amplifier
Fiber
Combiner
Near- IR
DFB #2
1551.1 nm
I
Laser
Intensity
PPLN
DFG
Crystal
v1v2
DFG #2
Near -IR Pump
Laser #2
(1074 nm)
Near- IR
DFB #3
1550.6 nm
I
Yb/Er Fiber
Amplifier
Fiber
Combiner
PPLN
DFG
Crystal
Time
Laser
Intensity
v3
All Off
Figure 3.9: Time-division multiplexing used with two DFG lasers to generate three
mid-infrared wavelengths.
Mid-IR
 v3
ZnSe
Wedge
Near-IR
λ4
Vapor &
Aerosol
BP filter #1
InGaAs
Detector
Iris
BP filter #3
Mid-IR
v1,v2
BP filter #2
Sapphire
Windows
InSb
Reference
Detector
InSb
Detector
L
Figure 3.10: Schematic showing the optical setup with common mode rejection (reference detector) and the combination of four laser beams with bandpass optical filters.
60
CHAPTER 3. EVAPORATING FUEL AEROSOLS
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 evaporation, 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, transient 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
1
Angular
dependent
scattering
http://www.philiplaven.com/mieplot.htm
was
calculated
using
Mieplot
61
3.3. VALIDATION EXPERIMENTS
and R31 . Second, measurements as a function of droplet loading at the three temperatures 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
Liquid n-Decane
Valve
(Ambient Air)
Heated Section
Nebulizer
Pressure
Heated
Windows
Thermocouple
5.8 cm
x
y
z
Flow Control Valve
To
Vacuum
Figure 3.11: Flow-cell experiment (viewed from above). Ambient air flows through
the 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 temperature 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
62
CHAPTER 3. EVAPORATING FUEL AEROSOLS
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.
63
3.3. VALIDATION EXPERIMENTS
Laser Path
Figure 3.12: Schematic of the aerosol shock-tube. The shock-tube is filled with ndecane aerosol through the endwall. Endwall valves are closed and a shock wave
travels 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 combustion of low vapor-pressure hydrocarbons, was used to demonstrate simultaneous
temperature and vapor measurements at higher temperatures and in transient conditions [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. Immediately 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).
64
CHAPTER 3. EVAPORATING FUEL AEROSOLS
3.3.5
Shock tube results
1.6
1.2
5
Mole Fraction
NIR extinction
4
n-Decane/Argon
P1 = 0.27 atm
T1 = 302 K
P2 = 1.02 atm
T2 = 475 K
3
Predicted increase
due to evaporation
0.8
2
1
Xsat(T1)
NIR Extinctance
Mole Fraction [%]
2.0
0
600
500
400
0
Temperature
Time
NIR extinction
n-Decane/Argon
P1 = 0.27 atm
T1 = 302 K
P2 = 1.02 atm
T2 = 475 K
100
[µs]
200
T2 prediction without
evaporation
5
4
3
T2 prediction with
evaporation
2
1
300
NIR Extinctance
Temperature [K]
700-100
0
-100
0
100
Time [µs]
200
Figure 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 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 postshock 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
3.3. VALIDATION EXPERIMENTS
65
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 increased 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 measurements 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 preshock 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).
66
Measured Mole Fraction [%]
CHAPTER 3. EVAPORATING FUEL AEROSOLS
3
I
2
1
Post Incident Shock
Calculated Mole Fraction
0
Measured Temperature [K]
0
1000
1
2
3
Calculated Mole Fraction [%]
I
800
600
400
Pre Shock
Post Incident Shock
Post Reflected Shock
Calculated Temperature
400
600
800
1000
Calculated Temperature [K]
Figure 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.
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 directgasoline-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 overlaps 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 detect 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 requires 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, quantitative 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 components 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.
69
4.2. TWO-PHASE LASER-ABSORPTION MEASUREMENTS
4.2
4.2.1
Two-phase laser-absorption measurements
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.
gas
window
δ
laser
light
liquid film
Figure 4.1: Schematic of laser light passing through a window, a fuel film of thickness
δ, and fuel vapor.
− ln
I
Io
= Xng σg (ν̄i , Tg )L + nL σL (ν̄i , TL )δ
i = 1, 2
(4.1)
ν̄i
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 crosssections 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 ν̄i , which represents both laser
frequencies in wavenumbers. If the pressure and temperature are known, X and δ
70
CHAPTER 4. LIQUID FUEL FILMS
can be determined from the absorbance measured at two wavelengths (2 equations in
2 unknowns).
150
n-decane
vapor
liquid
T = 25°C
P = 1 atm
2
( , T ) [m /mole]
200
100
spectral shift
50
0
2750
2800
2850
2900
2950
-1
Frequency [cm ]
3000
3050
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 and
liquid phases.
The proposed 2-wavelength laser-absorption diagnostic requires quantitative absorption cross-sections at both chosen wavelengths in vapor and liquid fuel. A difference 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 under well-controlled conditions at 25 ◦ C: n-decane (Psat = 1.425 torr) and n-dodecane
71
4.2. TWO-PHASE LASER-ABSORPTION MEASUREMENTS
(Psat = 0.136 torr) [61], both common fuel surrogates (Sigma-Aldrich > 99 percent
purity). Fourier transform infrared (FTIR) spectrometer (Nicolet 6700) measurements 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
a
2
( , T ) [m /mole]
T = 25°C
P = 1 atm
n-dodecane liquid
100
n-decane liquid
0
200
b
n-decane vapor
n-dodecane liquid
100
0
2750
2800
2850 2900 2950
-1
Frequency [cm ]
3000
3050
Figure 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].
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).
72
CHAPTER 4. LIQUID FUEL FILMS
A comparison of the mid-IR absorption spectra of liquid n-decane and liquid ndodecane (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).
"
ng σg (ν̄1 )L
ng σg (ν̄2 )L
#" #
nL σL (ν̄1 ) X
nL σL (ν̄2 )
δ

=
− ln
− ln

I
Io ν̄1 
I
Io
(4.2)
ν̄2
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 absorption band (2800 − 3000cm−1 ) and its reciprocal is plotted in Fig. 4.4 with darker regions corresponding to wavelength pairs that yield a well-conditioned A matrix. The
73
4.2. TWO-PHASE LASER-ABSORPTION MEASUREMENTS
a
0
1
cond ( A)
0.1
0.2
0.3
-1
Frequency,v1 [cm ]
2980
2940
II
2900
ing
se
2860
G
DF
h
atc
m
a
ph
v1 ,v2
2820
2820
I
2860
2900
2980
-1
Frequency,v2 [cm ]
b
200
n-decane vapor
n-dodecane liquid
I
II
150
v1 ,v2
2
( , T ) [m /mole]
2940
100
50
T = 25°C
P = 1 atm
0
2750
2800
2850
2900
2950
-1
Frequency [cm ]
3000
3050
Figure 4.4: a: Calculated reciprocal condition number of matrix A for all wavelength
pairs (ν̄1 , ν̄2 ) between 2800−3000cm−1 . Darker regions show wavelength pairs yielding
well-conditioned matrices. a & b: Three wavelength pairs are highlighted: the selected
wavelength pair (ν̄1 = 2854.7cm−1 and ν̄2 = 2864.7cm−1 ) within the DFG phase
matching range, and two optimal wavelength pairs (I & II) not attainable with the
current DFG system.
74
CHAPTER 4. LIQUID FUEL FILMS
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. Comparing Fig. 4.4a and b, it becomes clear that wavelength pairs I (ν̄1 = 2845cm−1 and
ν̄2 = 2975cm−1 ) and II (ν̄1 = 2895cm−1 and ν̄2 = 2975cm−1 ) are optimal because ν̄1
is dominated by absorption from the liquid, and ν̄2 is dominated by absorption from
the vapor, approaching an independent measurement of vapor and liquid.
4.3
4.3.1
Refractive index matching
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,f ilm , 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,f ilm in these absorbance measurements, Io and Io,f ilm 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.
75
4.3. REFRACTIVE INDEX MATCHING
a
Incident light
Window
1
2
Ii
Transmitted light
Reflected light
b
1
Window
Io
2’
Film
3
Ii
Io,film
δ
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 of
a film lies between that of the window and the gas (i.e. nwindow > nf ilm > ngas ), then
Io,f ilm > Io .
76
CHAPTER 4. LIQUID FUEL FILMS
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 =
4ni nj
(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,f ilm , using Eq. 4.5 (see Fig. 4.5):
Io = Ii T1 T2
(4.4)
Io,f ilm = Ii T1 T2′ 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
CaF2
Fused Silica
KBr
Sapphire
ZnSe
n (3.4 µm)
1.41
1.41
1.54
1.70
2.44
The window materials analyzed in these calculations, including their respective
77
4.3. REFRACTIVE INDEX MATCHING
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 > nf ilm > ngas ), then
Io,f ilm > 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,f ilm /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
1.6
Liquid
0.10
Window materials
0.05
CaF2, Fused Silica
KBr
Sapphire
ZnSe
n-Dodecane
1.2
CaF2, FS
0.8
KBr
0.00
-0.05
-0.10
2400
ZnSe
2600
Sapphire
2800 3000 3200
-1
Frequency [cm ]
0.4
Refractive Index, n
-ln(Io, film/Io)
n-Dodecane
0.0
3400
Figure 4.6: Measured refractive index of n-dodecane (right) and calculated baseline
offset for liquid n-dodecane films on several window materials (left).
Calculating the light transmission, including internal reflections in both the window 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,f ilm 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,
78
CHAPTER 4. LIQUID FUEL FILMS
the offset is largely independent of wavelength. Because of this wavelength independence, 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 absorbing 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
FTIR light
δ
Window
Figure 4.7: FTIR measurements of liquid n-dodecane films injected onto the windows
listed 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,
4.3. REFRACTIVE INDEX MATCHING
79
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,f ilm 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 measurements 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 FTIRbased 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 absorbance (−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 wavelengthdependent difference between the film and cell measurements (i.e. σf ilm − σcell ) were
then plotted, where the discrepancy is largely due to variation of the refractive index of n-dodecane at absorbing wavelengths (again due to the inability to measure
Io,f ilm 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
80
CHAPTER 4. LIQUID FUEL FILMS
a
4
n-dodecane
-ln(I/Io)
3
2
CaF2 Wedge
T = 25 °C
P = 1 atm
δ = 7.9 µm
δ = 1.9 µm
δ = 1.0 µm
δ = 0.7 µm
Cell
1
0
2
( , T ) [m /mole]
film
-
cell
b
10
0
-10
200
150
100
50
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
Figure 4.8: a: Measured absorbance of free-standing liquid n-dodecane for several
film thicknesses, where the measured transmission, I, is equal to Io,f ilm − absorption,
and Io is the same as in Fig. 4.5a. b: Comparison of inferred cross-sections from
free-standing film measurements to previously measured cross-section.
81
4.4. DEMONSTRATION EXPERIMENTS
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
4.4.1
Demonstration experiments
Experimental setup
Current
Modulation
Channel 1
Pump laser
(1.0 µm)
Signal laser #1
(1.5 µm)
Fiber
combiner
Channel 2
Signal laser #2
(1.5 µm)
Fiber amplifier
(Yb/Er)
Fiber
combiner
ν1 ,ν2
PPLN crystal
mount
Figure 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.
The DFG system used for the diagnostic is shown in Fig. 4.9 [96]. Two polarizationmaintaining 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. ν̄1 = 2854.7cm−1 and ν̄2 = 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
82
CHAPTER 4. LIQUID FUEL FILMS
InGaAs
Detector
Near-IR
ZnSe
Wedge
v 3
BP filter #2
Mid-IR
v1,v2
BP filter #1
Iris
Window
InSb
Detector
InSb
Reference
Detector
Figure 4.10: Optical setup for demonstrating the diagnostic. Two mid-IR and onenear-IR beams used; mid-IR to measure absorption, near-IR to monitor beam steering
and other losses.
(Fig. 4.10). The reference beam was collected onto a cryogenically-cooled indiumantimonide (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 (ν̄3 , 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 indiumgalium-arsenide (InGaAs) detector (5.0 mm diameter, 3 MHz bandwidth). The timemultiplexed 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.
83
4.4. DEMONSTRATION EXPERIMENTS
4.4.2
Demonstration of liquid film measurement
δ
Laser (v1,v2,v3 )
Air flow
CaF2 Window
Figure 4.11: A liquid n-dodecane film is injected onto a CaF2 window and subsequently 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 thickness 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 ν̄3 (non-absorbing) is due to beam steering and scattering by the
84
CHAPTER 4. LIQUID FUEL FILMS
a
-ln(I/Io)
8
-1
ν1 = 2854.7 cm
-1
ν2 = 2864.7 cm
6
-1
ν3 = 6424.2 cm
4
2
Film Thickness, δ [µm]
b
T = 24 °C
P = 1 atm
ν1
ν3
ν2
0
30
ν2
20
ν1
10
0
0
2
4
6
8
Time [seconds]
10
12
Figure 4.12: n-Dodecane film measurement. a: Measured n-dodecane absorbance at
both wavelengths. b: Measured liquid film thickness.
85
4.4. DEMONSTRATION EXPERIMENTS
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 ν̄1 , 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 ν̄2 was
greater than ν̄1 , being inversely proportional to the cross section ( δmax (ν̄2 ) = 20µm
vs δmax (ν̄1 ) = 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 measurement
Thermocouple
Pressure
Test cell
δ
T = 24 °C
P = 1 atm
Laser (v1,v2,v3 )
Air flow
L
CaF2 windows
Figure 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.
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
86
CHAPTER 4. LIQUID FUEL FILMS
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.
a
6
-1
ν 1 = 2854.7 cm
-1
-ln(I/Io)
ν 2 = 2864.7 cm
4
-1
ν3 = 6424.2 cm
ν1
ν3
2
T = 24 °C
P = 1 atm
ν2
0
500
X (n-decane)
δ (n-dodecane)
400
25
20
X
300
15
200
10
δ
100
5
0
0
0
2
4
6
8
Time [seconds]
10
Film thickness, δ [µm]
Mole fraction, X [ppm]
b
12
Figure 4.14: n-Dodecane film and n-decane vapor measurement. a: Combined absorbance from n-dodecane liquid and n-decane vapor at both wavelengths. b: Measured 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
4.4. DEMONSTRATION EXPERIMENTS
87
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 ν̄3 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 impingement, 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 ν̄1 decreased below 4, the film thickness and
vapor mole fraction were recovered. The largest film thickness measurable was limited
by the absorbance at ν̄1 , 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 validation 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.
88
CHAPTER 4. LIQUID FUEL FILMS
For application to other fuels, new wavelengths should be selected from the absorption 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 (vapor and liquid) laser-absorption measurements of hydrocarbon fuels as motivated by
combustion applications (e.g. direct-gasoline-injection engines). This work was divided 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
5.1.1
Summary
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 measured 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 simulate 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 extinction 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 fraction 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
5.1.3
91
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 demonstrated with 300 ppm of n-decane measured with < 10 % uncertainty in the presence
of n-dodecane films < 10 µm.
5.2
5.2.1
Future work
Extension of film technique for vapor and film temperature
Most combustion applications have elevated gas temperatures, and fuel film temperatures 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 vapor temperature, vapor mole fraction, and liquid film thickness. In Fig. 5.1, the
absorption at ν̄1 is strongly dependent on the liquid, while absorption at ν̄2 is more
sensitive to the vapor absorption and the vapor temperature. The third frequency,
ν̄3 , 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.
v2
n-decane
vapor (T = 25°C)
liquid (T = 27°C)
100
v1
2
( , T ) [m /mole]
150
v3
50
0
2700
2800
2900
3000
-1
Frequency [cm ]
3100
3200
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 temperature. 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 liquid 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 reduce 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
5.2.2
93
FTIR spectroscopy
The uncertainty in liquid film temperature could be resolved by developing a diagnostic 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 fuels 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 provide explicit relations for vapor mole fraction, Xa , temperature T, and the droplet
extinction coefficient at ν̄1 , τ1 .
− ln
I
I0
i
=
P Xa L
R̂T
(ai T 2 + bi T + ci ) + Ri1 τ1 L i = 1, 2, 3
(A.1)
The following constants were defined for simplicity in the solution:
A = R21 a1 − a2
D = R31 a1 − a3
B = R21 b1 − b2
E = R31 b1 − b3
C = R21 c1 − c2
F = R31 c1 − c3
Z=
R21 Ext1 − Ext2
R31 Ext1 − Ext3
(A.2)
(A.3)
x = ZD − A
y = ZE − B
z = ZF − C
94
(A.4)
95
The solutions for T, Xa , and τ1 are
T =
Xa =
−y +
p
y 2 − 4xz
2x
(R21 Ext1 − Ext2 )R̂T
P L(AT 2 + BT + C)
(A.5)
(A.6)
Ext1 P Xa
a1 T 2 + b 1 T + c 1
(A.7)
−
L
R̂T
The extinction coefficients at ν̄2 and ν̄3 are found using the measured extinction ratios,
τ1 =
R21 and R31 :
τ2 = R21 τ1
(A.8)
τ3 = R31 τ1
(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, τν̄ [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 generate 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 wavelengths 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 distribution 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 example1.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
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
// ∗∗ F i l e : example1 . cpp
// ∗∗ Jason M. Porter , S t a n f o r d HTGL
// ∗∗ Phone : ( 6 5 0 ) 723 −0941 , Email : j a s o n p o r t e r @ s t a n f o r d . edu
// ∗∗ Purpose : Calc l i g h t e x t i n c t i o n by a b s o r b i n g l i q u i d d r o p l e t d i s t r i b u t i o n s
// ∗∗ This code u s e s a C++ l i b r a r y d e v e l o p e d a t NIST :
// ∗∗ SCATMECH: P o l a r i z e d L i g h t S c a t t e r i n g C++ C l a s s L i b r a r y
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
#include ” d r o p d i s t . h”
// d r o p l e t d i s t r i b u t i o n h e a d e r
#include <i o s t r e a m >
using namespace s t d ;
// u s e u n q u a l i f i e d names f o r Standard C++ l i b r a r y
using namespace SCATMECH;
// u s e u n q u a l i f i e d names f o r t h e SCATMECH l i b r a r y
i n t main ( )
{
double f v = 2 . 4 e −5;
// d r o p l e t l o a d i n g s t a t e d as volume f r a c t i o n
double rmax = 5 ;
// maximum d r o p l e t r a d i u s [ microns ]
double rmin = . 0 0 0 5 ;
// minimum d r o p l e t r a d i u s [ microns ]
double lmax = 4 ;
// maximum l i g h t w a v e l e n g t h [ microns ]
double lmin = 1 . 4 ;
// minimum l i g h t w a v e l e n g t h [ microns ]
double Dbar = 3 . 3 ;
// mean d r o p 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 w i d t h
int l t o t = 1000;
// t o t a l w a v e l e n g t h s ( d e t e r m i n e s s p e c t r a l r e s o l u t i o n )
int r t o t = 100;
// t o t a l number o f r a d i i b e t w e e n rmin & rmax
// i n s t a n t i a t e m i e s c a t t e r e r ,
Iinitialize
variables
DropDist Lnorm ( r t o t , l t o t , rmin , rmax , lmin , lmax ) ;
Lnorm . SetLambda ( ) ;
// c a l c u l a t e s w a v e l e n g t h s from i n p u t s
Lnorm . S e t R a d i u s ( ) ;
// c a l c u l a t e s r a d i i and d i a m e t e r s from i n p u t s
Lnorm . GetQ ( ) ;
// c a l c u l a t e Qext a r r a y ( r t o t x l t o t )
//Lnorm . GetQArray ( ) ;
// u p l o a d Qext a r r a y from f i l e
//Lnorm . GetDandN ( ) ;
// g e n e r a t e s drop d i s t from i n p u t f i l e s
Lnorm . LogNormal ( Dbar , q ) ; // g e n e r a t e l o g −normal d i s t
//Lnorm . Rosin Rammler ( Dbar , q ) ; / / g e n e r a t e Rosin−Rammler d i s t
//Lnorm . NTot ( f v ) ;
// c a l c u l a t e d r o p l e t l o a d i n g [ cm−3] from f v
Lnorm . SetNTot ( 1 0 0 0 0 0 ) ; // s e t d r o p l e t l o a d i n g 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] v s lambda
Lnorm . D 32 ( ) ;
// c a l c u l a t e s S u a t e r mean d i a m e t e r
Lnorm . PrintQ ( ) ;
// p r i n t s Qext a r r a y t o f i l e
//Lnorm . PrintLambda ( ) ;
// p r i n t s w a v e l e n g t h t o f i l e
//Lnorm . P r i n t D i a m e t e r ( ) ; / / p r i n t s d i a m e t e r t o f i l e
Lnorm . P r i n t E x t ( ) ;
// p r i n t s e x t i n c t i o n v s w a v e l e n g t h t o f i l e
//Lnorm . P r i n t D i s t ( ) ;
// p r i n t s f (D) and i n t e g r a l o f f (D) t o f i l e
//Lnorm . P r i n t I n d e x ( ) ;
// 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 i n d e x t o f i l e
//Lnorm . QvsRadius ( 1 . 5 5 ) ; / / p r i n t s Qext v s r a d i u s f o r one w a v e l e n g t h
//Lnorm . QvsLambda ( 1 . 5 ) ; // p r i n t s Qext v s w a v e l e n g t h f o r one r a d i u s
return 0 ;
}
98
APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
// ∗∗ SCATMECH: P o l a r i z e d L i g h t S c a t t e r i n g C++ C l a s s L i b r a r y
// ∗∗ F i l e : d r o p d i s t . h
// ∗∗ Jason M. Porter , S t a n f o r d HTGL
// ∗∗ Phone : ( 6 5 0 ) 723−0941
// ∗∗ Email : j a s o n p o r t e r @ s t a n f o r d . edu
// ∗∗ C l a s s m o d e l i n g e x t i n c t i o n by d r o p l e t d i s t r i b u t i o n s
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
#i f n d e f DROPDIST H
#define DROPDIST H
#include ” m i e s c a t . h”
#include <i o s t r e a m >
using namespace s t d ;
// Use u n q u a l i f i e d names f o r Standard C++ l i b r a r y
using namespace SCATMECH;
// Use u n q u a l i f i e d names f o r t h e SCATMECH l i b r a r y
c l a s s DropDist {
public :
DropDist ( int , int , double , double , double , double ) ;
void S e t R a d i u s ( ) ;
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 P r i n t E x t ( ) ;
void P r i n t D i s t ( ) ;
void P r i n t I n d e x ( ) ;
void GetQArray ( ) ;
void SetNTot ( double ) ;
void PrintLambda ( ) ;
void P r i n t D i a m e t e r ( ) ;
void GetDandN ( ) ;
˜ DropDist ( ) ;
private :
double ∗ l , ∗ r , ∗d , ∗∗ Qarray , ∗Ndens , ∗ Cumulative , ∗ E x t i n c t i o n ;
double ∗N1 , ∗N2 , ∗Qvr , ∗ Qvl ;
double lmax , lmin , rmax , rmin , d e l d , Ntot , D32 ;
int rtot , l t o t ;
s t a t i c i n t DEBUG;
M i e S c a t t e r e r mie ; // i n s t a n t i a t e M i e S c a t t e r e r
};
#endif
99
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
// ∗∗ SCATMECH: P o l a r i z e d L i g h t S c a t t e r i n g C++ C l a s s L i b r a r y
// ∗∗ F i l e : d r o p d i s t . cpp
// ∗∗ Jason M. P o r t e r
// ∗∗ S t a n f o r d HTGL
// ∗∗ Phone : ( 6 5 0 ) 723−0941
// ∗∗ Email : j a s o n p o r t e r @ s t a n f o r d . edu
// ∗∗ C l a s s m o d e l i n g e x t i n c t i o n by d r o p l e t d i s t r i b u t i o n s
// ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
#include ” m i e s c a t . h”
#include <i o s t r e a m >
#include ” d r o p d i s t . h”
#include <s t r i n g >
#include <f s t r e a m >
#include <iomanip>
#include <cmath>
#include <ctime>
#include <c s t d l i b >
using namespace s t d ;
// Use u n q u a l i f i e d names f o r Standard C++ l i b r a r y
using namespace SCATMECH;
// Use u n q u a l i f i e d names f o r t h e SCATMECH l i b r a r y
// I n s t a n t i a t e m i e s c a t t e r e r
i n t DropDist : : DEBUG = 0 ;
// i n i t i a l i z e
variables
DropDist : : DropDist ( i n t r t , i n t l t , double rmn , double rmx , double lmn , double lmx ) {
rtot = rt ;
ltot = lt ;
rmax = rmx ;
rmin = rmn ;
lmax = lmx ;
lmin = lmn ;
d e l d = 2 ∗ ( rmax − rmin ) / ( r t o t −1);
// dD i n e x t i n c t i o n e q u a t i o n
// d y n a m i c a l l y a l l o c a t e memory f o r a r r a y s
l = new double [ l t o t ] ;
// w a v e l e n g t h [ microns ]
E x t i n c t i o n = new double [ l t o t ] ;
// d r o p l e t e x t i n c t i o n , Qext
N1 = new double [ l t o t ] ;
// r e a l r e f r a c t i v e i n d e x , n
N2 = new double [ l t o t ] ;
// i m a g i n a r y r e f r a c t i v e i n d e x , k
r = new double [ r t o t ] ;
// d r o p l e t r a d i u s [ microns ]
d = new double [ r t o t ] ;
// d r o p l e t d i a m e t e r [ 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 f u n c t i o n , 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 v s r a d i u s
Qvl = new double [ l t o t ] ;
// Qext v s w a v e l e n g t h
Qarray = new double ∗ [ l t o t ] ;
// Qext a r r a y ( r t o t x l t o t )
f o r ( i n t i = 0 ; i <l t o t ; i ++){
Qarray [ i ] = new double [ r t o t ] ;
}
// Query u s e r f o r mie parameters , e . g . t h e r e f r a c t i v e i n d e x f i l e ( lambda , n , k )
100
APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS
// SCATMECH w i l l e x t r a p o l a t e , i n t e r p o l a t e as needed
mie . AskUser ( ) ;
// c o u t << ”
Qext = ” << mie . Qext()<< e n d l ;
}
DropDist : : ˜ DropDist ( ) {
// f r e e −up d y n a m i c a l l y a l l o c a t e d memory
delete [ ]
l;
delete [ ] r ;
delete [ ] d ;
delete [ ] Ndens ;
delete [ ] Cumulative ;
delete [ ]
Extinction ;
delete [ ] N1 ;
delete [ ] N2 ;
f o r ( i n t i = 0 ; i <l t o t ; i ++)
delete [ ] Qarray [ i ] ;
delete [ ] Qarray ;
delete [ ] Qvr ;
delete [ ] Qvl ;
}
// g e n e r a t e s drop d i s t from i n p u t 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 a m DIN( ” Din . t x t ” ) ;
// c o u t << ” r e a d i n g Din . t x t \n ” ;
i f (DIN . f a i l ( ) ) {
c e r r << ” Din . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
f o r ( i n t i =0; i <r t o t ; i ++){
DIN>>d [ i ] ;
r [ i ]=d [ i ] / 2 ;
}
i f s t r e a m NIN( ” Nin . t x t ” ) ;
// c o u t << ” r e a d i n g Nin . t x t \n ” ;
i f (NIN . f a i l ( ) ) {
c e r r << ” Nin . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
f o r ( i n t i =0; i <r t o t ; i ++){
NIN>>Ndens [ i ] ;
}
double sum = 0 ;
f o r ( i n t n=0;n<r t o t ; n++){
sum = sum + Ndens [ n ] ;
Cumulative [ n ] = ( sum ∗ ( d [ n+1]−d [ n ] ) ) ;
}
}
101
// l o a d s p r e v i o u s l y c a l c u l a t e d Qext a r r a y ( r t o t x l t o t ) from t e x t
void DropDist : : GetQArray ( ) {
i f s t r e a m QIN( ” Q Array . t x t ” ) ;
// c o u t << ” r e a d i n g Q Array . t x t \n ” ;
i f (QIN . f a i l ( ) ) {
c e r r << ” Q Array . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
f o r ( i n t i =0; i <l t o t ; i ++){
f o r ( i n t j =0; j <r t o t ; j ++){
QIN>>Qarray [ i ] [ j ] ;
}
}
}
// p r i n t s Qext v s lambda t o f i l e
void DropDist : : P r i n t E x t ( ) {
o f s t r e a m QDIST( ” E x t i n c t i o n . t x t ” ) ;
// c o u t << ” P r i n t i n g E x t i n c t i o n . t x t \n ” ;
i f (QDIST . f a i l ( ) ) {
c e r r << ” E x t i n c t i o n . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
QDIST<<”Lambda\ t E x t i n c t i o n \n” ;
f o r ( i n t i =0; i <l t o t ; i ++){
QDIST<<l [ i ]<< ’ \ t ’<<E x t i n c t i o n [ i ]<< ’ \n ’ ;
}
}
// p r i n t s r e f r a c t i v e i n d e x v s lambda t o f i l e
void DropDist : : P r i n t I n d e x ( ) {
o f s t r e a m INDEX( ” Index . t x t ” ) ;
// c o u t << ” P r i n t i n g I n d e x . t x t \n ” ;
i f (INDEX . f a i l ( ) ) {
c e r r << ” Index . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
INDEX<<”Lambda\ tn \ t k \n” ;
f o r ( i n t 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 o p l e t d i a m e t e r t o f i l e
void DropDist : : P r i n t D i a m e t e r ( ) {
o f s t r e a m DIAM( ” Diameter . t x t ” ) ;
// c o u t << ” P r i n t i n g Diameter . t x t \n ” ;
i f (DIAM. f a i l ( ) ) {
c e r r << ” Diameter . t x t c o u l d not be opened \n” ;
exit ( 1 );
file
102
APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS
}
f o r ( i n t i =0; i <r t o t ; i ++){
DIAM<<d [ i ]<< ’ \n ’ ;
}
}
// p r i n t s w a v e l e n g t h t o f i l e
void DropDist : : PrintLambda ( ) {
o f s t r e a m LAMBDA( ”Lambda . t x t ” ) ;
// c o u t << ” P r i n t i n g Lambda . t x t \n ” ;
i f (LAMBDA. f a i l ( ) ) {
c e r r << ”Lambda . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
f o r ( i n t i =0; i <l t o t ; i ++){
LAMBDA<<l [ i ]<< ’ \n ’ ;
}
}
// p r i n t s d r o p l e t s i z e d i s t r i b u t i o n t o f i l e
void DropDist : : P r i n t D i s t ( ) {
o f s t r e a m NDENS( ”NDens . t x t ” ) ;
// c o u t << ” P r i n t i n g NDens . t x t \n ” ;
i f (NDENS. f a i l ( ) ) {
c e r r << ”NDens . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
NDENS<<” d i a m e t e r \ tNdens \ tCumulative \n” ;
f o r ( i n t 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 a r r a y t o f i l e ( r t o t x l t o t )
void DropDist : : PrintQ ( ) {
o f s t r e a m QARRAY( ” Q Array . t x t ” ) ;
// c o u t << ” P r i n t i n g Q Array . t x t \n ” ;
i f (QARRAY. f a i l ( ) ) {
c e r r << ” Q Array . t x t c o u l d not be opened \n” ;
exit ( 1 );
}
f o r ( i n t i =0; i <l t o t ; i ++){
f o r ( i n t 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 d i a m e t e r s from i n p u t s
void DropDist : : S e t R a d i u s ( ) {
103
i f (DEBUG == 1 )
c o u t << ” r a d i i \n” ;
i f ( r t o t == 1 ) {
r [ 0 ] = rmin ;
d [ 0 ] = 2∗ rmin ;
}
else {
f o r ( i n t i =0; i <r t o t ; i ++){
r [ i ] = rmin + s t a t i c c a s t <double>( i ) ∗ ( rmax−rmin )
/ ( s t a t i c c a s t <double>( r t o t ) − 1 ) ;
d [ i ] = 2∗ r [ i ] ;
i f (DEBUG == 1 )
c o u t << r [ i ] << ’ \n ’ ;
}
}
}
// c a l c u l a t e s w a v e l e n g t h s from i n p u t s
void DropDist : : SetLambda ( ) {
i f (DEBUG == 1 )
c o u t << ” lambdas \n” ;
i f ( l t o t == 1 )
l [ 0 ] = lmin ;
else {
f o r ( i n t i =0; i <l t o t ; i ++){ // 0−99
l [ i ] = lmin + s t a t i c c a s t <double>( i ) ∗ ( lmax−lmin )
/ ( s t a t i c c a s t <double>( l t o t ) − 1 ) ;
//
l [ i ] = 1/ l [ i ] ; // temporary c o n v e r s i o n f o r QVL c a l c
i f (DEBUG == 1 )
c o u t << 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 f u n c t i o n s
void DropDist : : GetQ ( ) {
int i , j ;
i f (DEBUG == 1 )
c o u t << ” Qarray [ lambda ] [ r a d i u s ] \ n” ;
i f (DEBUG == 1 ) {
c o u t << ’ \ t ’ ;
f o r ( i =0; i <r t o t ; i ++)
c o u t << r [ i ] << ”
”;
c o u t << ’ \n ’ ;
}
f o r ( j =0; j <r t o t ; j ++){
mie . s e t r a d i u s ( r [ j ] ) ;
f o r ( i n t k =0;k<l t o t ; k++){
mie . s e t l a m b d a ( l [ k ] ) ;
104
APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS
Qarray [ k ] [ j ] = mie . Qext ( ) ;
// c a l c u l a t e s f o r w a r d s c a t t e r i n g w/o a b s o r p t i o n
// Qarray [ k ] [ j ] = mie . Qsca ( ) ;
}
}
}
// c a l c u l a t e s l o g −normal d r o p 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 S i g ) {
i f (DEBUG == 1 ) {
c o u t << ” diams \n” ;
f o r ( i n t n=0; n<r t o t ; n++){
c o u t << d [ n ] << ’ \n ’ ;
}
}
i f (DEBUG == 1 )
c o u t << ” Ndens \n” ;
double term1 , term2 ;
f o r ( i n t n=0; n<r t o t ; n++){
term1 = ( l o g ( d [ n]) − l o g ( DistD ) ) / ( l o g ( S i g ) ) ;
term2 = ( 1 / s q r t ( 2 ∗ p i ) / l o g ( S i g ) / d [ n ] ) ;
Ndens [ n ] = term2 ∗ exp ( ( − 0 . 5 ) ∗ ( term1 ∗ term1 ) ) ;
i f (DEBUG == 1 )
c o u t << Ndens [ n ] << ’ \n ’ ;
}
double sum = 0 ;
f o r ( i n t n=0;n<r t o t ; n++){
sum = sum + Ndens [ n ] ;
Cumulative [ n ] = ( sum∗ d e l d ) ;
}
}
// c a l c u l a t e s Rosin−Rammler d r o p 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 ) {
c o u t << ” diams \n” ;
f o r ( i n t n=0; n<r t o t ; n++){
c o u t << d [ n ] << ’ \n ’ ;
}
}
i f (DEBUG == 1 )
c o u t << ” Ndens \n” ;
f o r ( i n t n=0; n<r t o t ; 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 )
c o u t << 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 u s e r i n p u t d r o p l e t l o a d i n g , Ntot
void DropDist : : SetNTot ( double Nt ) {
Ntot = Nt ;
c o u t << Ntot << ’ \n ’ ;
double Vtot = 0 ;
f o r ( i n t n=1;n<r t o t ; n++){
i f ( ( n = = 0 ) | | ( n==(r t o t −1)))
Vtot += ( p i / 6 ) ∗ pow ( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ;
else
Vtot += ( p i / 6 ) ∗ pow ( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ;
}
c o u t << Vtot << ’ \n ’ ;
i f (DEBUG == 2 )
c o u t << ” Ntot = ” << Ntot << ’ \n ’ ;
}
// c a l c u l a t e s d r o p l e t l o a d i n g from l i q u i d volume f r a c t i o n
double DropDist : : NTot ( double f v ) {
double Vtot = 0 ;
f o r ( i n t n=1;n<r t o t ; n++){
i f ( ( n = = 0 ) | | ( n==(r t o t −1)))
Vtot += ( p i / 6 ) ∗ pow ( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ;
else
Vtot += ( p i / 6 ) ∗ pow ( ( d [ n ] ∗ 0 . 0 0 0 1 ) , 3 ) ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ;
}
Ntot = f l o o r ( f v / ( d e l d ∗ Vtot ) ) ;
i f (DEBUG == 2 )
c o u t << ” Ntot = ” << Ntot << ’ \n ’ ;
return Ntot ; // [ cm−3]
c o u t << Vtot << ’ \n ’ ;
}
// c a l c u l a t e s S u a t e r mean d i a m e t e r D32 d ˆ3/ d ˆ2
double DropDist : : D 32 ( ) {
double Num = 0 ;
double Den = 0 ;
f o r ( i n t n=1;n<r t o t ; n++){
i f ( ( n = = 1 ) | | ( ( n==r t o t −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 ; // d e l d c a n c e l s
106
APPENDIX B. C++ CODE FOR MIE SCATTERING CALCULATIONS
i f (DEBUG == 2 )
c o u t << ”D32 = ” << D32 << ’ \n ’ ;
return D32 ;
// c o u t << ”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] v s lambda , r e f r a c t i v e i n d e x v s lambda
void DropDist : : QDist ( ) {
COMPLEX i n d e x ;
i f ( (DEBUG == 1 ) | | (DEBUG == 2 ) )
c o u t << ” E x t i n c t i o n \n” ;
double h o l d =0;
f o r ( i n t i =0; i <l t o t ; i ++){
hold = 0 ;
f o r ( i n t n=1;n<r t o t ; n++){
i f ( ( n = = 1 ) | | ( n==(r t o t −1)))
h o l d +=0.5∗( Ntot ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ) ∗ Qarray [ i ] [ n ]
∗pow ( ( ( d [ n]+d [ n − 1 ] ) / 2 ∗ 0 . 0 0 0 1 ) , 2 ) ∗ ( d [ n]−d [ n − 1 ] ) ;
else
h o l d+=(Ntot ∗ 0 . 5 ∗ ( Ndens [ n]+ Ndens [ n − 1 ] ) ) ∗ Qarray [ i ] [ n ]
∗pow ( ( ( d [ n]+d [ n − 1 ] ) / 2 ∗ 0 . 0 0 0 1 ) , 2 ) ∗ ( d [ n]−d [ n − 1 ] ) ;
}
// E x t i n c t i o n [ i ]= h o l d ∗ ( d e l d ) ∗ p i / 4 ;
E x t i n c t i o n [ i ]= h o l d ∗ p i / 4 ; // d i a m e t e r b i n s i z e c h a n g e s
i n d e x = mie . g e t s p h e r e ( ) . i n d e x ( l [ i ] ) ;
N1 [ i ] =
r e a l ( index ) ;
N2 [ i ] =
imag ( i n d e x ) ;
}
}
// c a l c u l a t e s Qext v s d r o p l e t r a d i u s f o r one w a v e l e n g t h
void DropDist : : QvsRadius ( double lambda ) {
i f (DEBUG == 3 ) {
// c o u t << ” r a d i u s \n ” ;
// f o r ( i n t i =0; i <r t o t ; i ++)
// c o u t << r [ i ] << ’ \ n ’ ;
}
i f (DEBUG == 3 )
c o u t << ”Qvr [ r a d i u s ] \ n” ;
mie . s e t l a m b d a ( lambda ) ;
// c o u t << mie . g e t l a m b d a ()<< ’ \ n ’ ;
f o r ( i n t k =0;k<r t o t ; k++){
mie . s e t r a d i u s ( r [ k ] ) ;
// c o u t << mie . g e t r a d i u s ()<< ’ \ n ’ ;
Qvr [ k ] = mie . Qext ( ) ;
i f (DEBUG == 3 )
c o u t << Qvr [ k ] << ” \n” ;
107
}
}
// c a l c u l a t e s Qext v s lambda f o r one d r o p l e t d i a m e t e r and p r i n t s t o f i l e
void DropDist : : QvsLambda ( double r a d i u s ) {
SetLambda ( ) ;
i f (DEBUG == 3 )
c o u t << ” Qvl [ lambdas ] \ n” ;
i f (DEBUG == 3 ) {
c o u t << ’ \ t ’ ;
f o r ( i n t i =0; i <l t o t ; i ++)
c o u t << l [ i ] << ”
”;
c o u t << ’ \n ’ ;
}
mie . s e t p a r a m e t e r ( ” r a d i u s ” , r a d i u s ) ;
f o r ( i n t j =0; j <l t o t ; j ++){
mie . s e t l a m b d a ( l [ j ] ) ;
Qvl [ j ] = mie . Qext ( ) ;
i f (DEBUG == 3 )
c o u t << Qvl [ j ] << ” \n” ;
}
o f s t r e a m QVL( ”QVL. t x t ” ) ;
// c o u t << ” P r i n t i n g QVL. t x t \n ” ;
i f (QVL. f a i l ( ) ) {
c e r r << ”QVL. t x t c o u l d not be opened \n” ;
exit ( 1 );
}
QVL<<”Lambda\tQ\n” ;
f o r ( i n t 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 =
4ni nj
(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:
1
r̃ =
1−
m1
m2
m1
m2
(C.2)
t̃ =
2
m1
1+ m
2
(C.3)
1+
This derivation follows that of Born and Wolf [37] and Bohren and Huffman [34]
108
109
m2 = n 2
m1 = n1+ik1
Er
Et
Ei
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 = r̃E0,i and E0,t = t̃E0,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):
1
Si = Re
2
r
ǫ2
µ2
|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:
1
Sr = Re
2
r
ǫ2
µ2
|r̃|2 |E0,i |2
(C.5)
1
St = Re
2
r
ǫ1
µ1
2
t̃ |E0,i |2
(C.6)
110APPENDIX C. TRANSMITTANCE AND REFLECTANCE FOR ABSORBING MEDIA
The reflectance and transmittance are defined as:
R=
Sr
= |r̃|2
Si
St
T =
= Re
Si
r
ǫ1 µ 2
µ 1 ǫ2
(C.7)
2
t̃
(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 r̃ and t̃ from Eqs. C.2 and C.3:
R = |r̃|2 =
T =
(n2 − n1 )2 + k12
(n2 + n1 )2 + k12
m1 2
4n1 n2
t̃ =
m2
(n2 + n1 )2 + k12
(C.9)
(C.10)
Equations C.9 and C.10 are applicable for plane waves in non-conducting homogeneous 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 transmittance 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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