1 NUMERICAL SIMULATIONS OF TEMPERATURE MAPPING IN INDUSTRIAL COMBUSTION ENVIRONMENTS Michael P. Wood 2013 School of Electrical and Electronic Engineering A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences.
ThesisSchool of Electrical and Electronic Engineering
A thesis submitted to the University of Manchester for the degree
of
Doctor of Philosophy in the Faculty of Engineering and
Physical
Sciences.
2
1. Introduction
..............................................................................................................
18
1.1 Motivation
....................................................................................................
18
1.3 Current methods for temperature sensing
.................................................... 22
1.3.1 Invasive measurement
..................................................................................
22
1.3.2 Laser-induced fluorescence
..........................................................................
22
1.3.6 Optical pyrometry: soot
................................................................................
24
1.4 Laser absorption spectroscopy
.....................................................................
25
1.4.1 Line-of-sight thermometry by direct absorption spectroscopy
.................... 25
1.4.2 Modulation spectroscopy
.............................................................................
28
1.5 Aims and objectives
.....................................................................................
30
1.5.1 Objective
......................................................................................................
30
1.5.2 Aims
.............................................................................................................
30
1.6 Overview
......................................................................................................
31
2. Tomography
.............................................................................................................
32
2.1 Introduction
..................................................................................................
32
2.3 Filtered backprojection
.................................................................................
35
2.6.1 Existence
......................................................................................................
42
2.6.2 Uniqueness
...................................................................................................
42
2.6.3 Stability
........................................................................................................
43
2.7 Algorithms
....................................................................................................
43
3.1 Introduction
..................................................................................................
49
3.3 Infrared-active
species..................................................................................
52
3.5 The two-state transition model
.....................................................................
55
3.6 Boltzmann
statistics......................................................................................
58
3.7.1 Natural broadening
.......................................................................................
62
3.7.2 Doppler broadening
......................................................................................
62
3.7.3 Pressure broadening
.....................................................................................
63
3.7.5 Pressure shifting
...........................................................................................
65
3.9 Line-of-sight thermometry
...........................................................................
66
3.10 Temperature tomography
.............................................................................
72
4.4.1 Voigt calculation
..........................................................................................
83
4.5 Radon transform
...........................................................................................
85
5.2.3 Genetic algorithm
.........................................................................................
97
absorption methods
.......................................................................................................
113
6.2.1 Introduction
................................................................................................
113
6.4 Detailed simulations
...................................................................................
131
6.4.2 Results
........................................................................................................
133
6.4.3 Conclusions
................................................................................................
136
6.5 Annular reconstructions
.............................................................................
139
List of figures
Figure 1.1. Simple schematic of a turbofan engine. The cold,
atmospheric intake air is
sucked in, split, compressed, mixed with fuel, combusted, expanded,
and then funnelled
out of the rear nozzle imparting a forward thrust on the engine.
.................................... 20
Figure 1.2. Variation in temperature sensetivity between two
absorption transitions. ... 26
Figure 1.3. Relative similarity in mole fraction sensetivity
between two absorption lines.
.........................................................................................................................................
26
Figure 1.4. Effect of pressure increase on a set of near-infrared
absorption lines. ......... 28
Figure 2.1. A general point in the domain of , and line in the
domain of . 34
Figure 2.2. Graphical representation of the Fourier slice theorem.
The one-dimensional
Fourier transform of the Radon transform of at an angle is equal to
the
two-dimensional Fourier transform of along the radial slice .
.......................... 36
Figure 2.3. A conceptual illustration of the meaning of as the
length of line in
pixel . One method of reducing the number of unknowns is to
pixelate the image space.
The kernel of the Fredholm equation is represented by a matrix
operator. .................... 38
Figure 3.1. Direct absorption measurement over a single beam within
a tomography
system.
.............................................................................................................................
51
Figure 3.2. Geometric orientation of the water molecule. The
equilibrium bond angle
and bond length values are themselves calculated from spectroscopic
measurement. ... 53
Figure 3.3. Near-infrared spectrum of 1% water vapour at 300 K and
1 bar. On the far
left are the and bands, in the middle are ,
and bands, and on the right are , and
bands. The fundamental bands are found in the mid-infrared spectrum
below 5000
.
............................................................................................................................
55
Figure 3.4. Rotation-vibration partition function of water vapour
based on equation
3.14. The lower graph represents the fractional change where
: above , ’s estimate of is slightly
greater than that of Harris et al.
.......................................................................................
60
Figure 3.5. Doppler, Pressure and Voigt lineshapes for using
linear
and logarithmic vertical scales.
.......................................................................................
65
Figure 3.6. Contours of in the near-infrared at 1000 and 2000 K.
The
dotted line is the contour of .
..............................................................
69
Figure 4.1. Five steps to generate a random temperature phantom of
variable
smoothness.
.....................................................................................................................
78
Figure 4.2. Control of the smoothness is achieved by widening an
axisymmetric
Gaussian filter. 20 is used in all our simulations.
...........................................................
79
Figure 4.3. Temperature and concentration phantoms using Gaussian
peaks. ............... 80
Figure 4.4. Sample spectral absorption coefficients at a single
(central) pixel in the
phantom. Twenty data points split evenly between a low-temperature
(left) and high-
temperature (right) absorption line.
................................................................................
85
Figure 4.5. Left: temperature phantom; middle: mole fraction
phantom; right: spectral
attenuation coefficient at 5263.2 .
.......................................................................
85
7
Figure 4.6. Synthetic transmittance data over a single beam (#1)
through the
measurement zone.
..........................................................................................................
86
Figure 4.7. Synthetic absorption data set for 60 beams and 20
wavenumbers (divided
between two absorption transitions). Six peaks can be seen along
the ‘beam number’
axis because the beam configuration is par6, a parallel-beam
configuration with 6 angles
and 10 beams per angle. The peaks correspond to central beams with
the longest paths
through the measurement region.
....................................................................................
87
Figure 4.8. Sample reconstruction of the spectral attenuation
coefficient from the
transmittance data (Figure 4.6) generated from a high-resolution
phantom (Figure 4.5).
.........................................................................................................................................
88
Figure 4.9. Result of spectral fitting for a central pixel in the
image using data points
from a projected Landweber algorithm. The top-right figures show
the best-fit
results.
.............................................................................................................................
90
Figure 4.10. Result of spectral fitting for a central pixel in the
image using data points
from a Tikhonov algorithm. The top-right figures show the best-fit
results. ...... 91
Figure 4.11. Sample temperature reconstructions. Left: phantom;
centre and right:
reconstructions.
...............................................................................................................
91
Figure 4.12. Sample mole fraction reconstructions. Left: phantom;
centre and right:
reconstructions.
...............................................................................................................
92
Figure 5.1. Flow diagram for a genetic algorithm to optimise beam
configurations. ... 100
Figure 5.2. Progress of the genetic algorithm towards minimising ,
with the
optimised beam configuration in the top-right. The experiment was
performed twice.
The vertical dotted lines show when the perturbation noise was
reduced, and the
horizontal dotted line shows the overall minimum.
...................................................... 103
Figure 5.3. Left: beam configurations, optimised using a genetic
algorithm for varying
annular thicknesses: . Right: a sinogam representation, where
every
cross corresponds to a beam. The red borders represent the
boundaries of the
recontruction region.
.....................................................................................................
106
Figure 5.4. Distribution of for one million randomly selected beam
configurations.
1 in roughly 21,000 samples have below 5, which demonstrates how
rare good
beam configurations are.
...............................................................................................
107
Figure 5.5. Twelve candidate lines with relatively good spectral
isolation at 30 bar. .. 110
Figure 5.6. Left: linestrength ratio for the two transitions over
the
temperature range of interest; right: stimulated emission
ratio
for the line pair (red cross).
...........................................................................................
111
Figure 6.1. Left: temperature phantom; middle: mole fraction
phantom; right: beam
configuration.
................................................................................................................
114
Figure 6.2. Reconstruction errors with increasing uniform gas
pressure for three
competing reconstruction methods.
..............................................................................
116
Figure 6.3. Reconstructions of the three methods; left to right: SF
method, IA method,
PA method.
....................................................................................................................
117
8
Figure 6.4. Six examples of randomly generated temperature phantoms
used for the
simulations. The method of generation is documented in §4.2.1.3.
Concentration
phantoms are generated in the same way.
.....................................................................
121
Figure 6.5. Six beam configurations used in the comparative study.
........................... 122
Figure 6.6. Relative RMSEs of temperature reconstructions. Each
data point is an
average of 12 reconstructions from data generated from different
(randomised)
temperature and mole fraction phantoms.
.....................................................................
123
Figure 6.7. Relative RMSEs of individual temperature
reconstructions for par6, par10,
and fan3 beam configurations. Vertically aligned crosses of a
certain colour are relative
RMSEs using data from different randomised phantoms. The solid
lines connect
average values at each pressure for each method of reconstruction.
............................ 124
Figure 6.8. Relative RMSEs of individual temperature
reconstructions for fan5,
irreg001, and irreg002 beam configurations. Vertically aligned
crosses of a certain
colour are relative RMSEs using data from different randomised
phantoms. The solid
lines connect average values at each pressure for each method of
reconstruction. ...... 125
Figure 6.9. Left: temperature (top) and concentration (bottom)
phantoms; middle: par6
beam configuration; right: temperature (top) and concentration
(bottom)
reconstructions. This was the best reconstruction, occurring at 25
bar. ....................... 126
Figure 6.10. Left: temperature (top) and concentration (bottom)
phantoms; middle: fan3
beam configuration; right: temperature (top) and concentration
(bottom)
reconstructions. This was the worst reconstruction, occurring at 37
bar. ..................... 127
Figure 6.11. Images taken from [151] demonstrating a specific
dependence on the gas
pressure of reconstruction errors in both temperature and mole
fraction (concentration).
.......................................................................................................................................
128
Figure 6.13. Phantoms, beam configuration and time-averaged
reconstructions of
temperature distribution using 100 transmittance datasets.
Reconstructions using each
method are shown at the bottom of the image.
.............................................................
134
Figure 6.14. Comparative difference in errors between methods A, B
and C. The scatter
graph contains 36 vertically-aligned triplets of red, green and
blue data points which
each represent reconstructions of a single phantom using methods A,
B, and C
respectively. The horizontal position of a triplet is the average
of the relative RMSE of
the reconstruction for all three methods, and the vertical position
of each data point
within the triplet is the difference between the relative RMSE of
the reconstruction
(obtained using the corresponding method) and the average for all
three methods. The
top histogram represents the average relative RMSEs of all the
reconstructions for each
method, and the three histograms to the right are histograms of the
relative RMSEs of
the reconstructions using each method; they can be used to visually
interpret which
method is statistically advantageous, e.g. it appears that method C
performs marginally
better because the majority of the histogram volume is below the
dotted line. ............ 136
Figure 6.15. Percentage improvement in reconstruction accuracy as a
function of the
number of datasets used in the combination. Data points of
different colours are
reconstructions from data from different phantoms.
..................................................... 138
9
Figure 6.16. Sample beam configurations for an annulus .
.............................. 141
Figure 6.17. Reconstruction accuracies for different beam
arrangements for decreasing
annular thicknesses of . Each result is an average 24 randomised
phantoms.
.......................................................................................................................................
143
annular thicknesses of . Each result is an average 24
randomised
phantoms.
......................................................................................................................
144
reconstructions for r = 0.
...............................................................................................
145
Figure 6.20. Sample phantom, irregular beam configurations and
corresponding
reconstructions for r = 0.25.
..........................................................................................
146
Figure 6.21. Sample phantom, irregular beam configurations and
corresponding
reconstructions for r = 0.5. “irreg051” and “irreg052” are the
examples of the genetic
algorithm shown in Figure 5.2.
.....................................................................................
147
Figure 6.22. Sample phantom, irregular beam configurations and
corresponding
reconstructions for r = 0.7.
............................................................................................
148
Figure 6.23. Optimised configuration of 32 beams in an annulus .
.......... 151
Figure 6.24. Comparison of reconstruction errors for 24 randomised
phantoms using
Landweber and Tikhonov algorithms. Top: temperature errors; middle:
concentration
errors; bottom: pressure errors.
.....................................................................................
153
Figure 6.25. Projections of the data points for each
reconstruction onto - , - and -P graphs. Black lines connect
Landweber and
Tikhonov reconstructions from the same phantom, and hollow squares
represent mean
values.............................................................................................................................
154
Figure 6.26. Sample reconstruction 1 using non-uniform pressure
phantoms. ............. 155
Figure 6.27. Sample reconstruction 2 using non-uniform pressure
phantoms. ............. 156
Figure 6.28. Sample reconstruction 3 using non-uniform pressure
phantoms. ............. 157
Figure 8.1. Screenshot of graphic and inline output from beam
optimisation algorithm
for the first 177 generations. Filled data points represent
generation averages of and
circle data points represent generation best values of . In this
example, the mutation
noise is reduced by 25% if there is no improvement to the best beam
configuration after
25 successive generations. The beam configuration after 1000
generations was used in
§6.6.
...............................................................................................................................
174
Table 1.1. Symbols list for tomography and beam optimisation.
................................... 15
Table 1.2. Symbols list for spectroscopy.
.......................................................................
16
Table 3.1. Documented coefficients used in the approximation of in
equation 3.14.
.........................................................................................................................................
60
tomography algorithms).
...............................................................................................
129
reconstructions.
.............................................................................................................
130
The comparison between Tikhonov and projected Landweber methods is
difficult to
justify; both are performed with a somewhat ad hoc implementation
of the prior
assumption, and it is possible that any differences between these
two methods is due to
the implementation of the smoothing prior as opposed to any
fundamental advantage of
either method. As shown in table 6.3, the projected Landweber
iteration is slightly
faster on a desktop PC: it scales considerably better with
increasing reconstruction
resolution (any more than 1000 pixels leads to rapidly diminishing
returns for the
Tikhonov inversion), and the memory overhead is much smaller so it
is more
advantageous to independently reconstruct spectral absorption
coefficient images using
parallel threads without causing memory bottlenecks. It is for
these practical reasons
that the Landweber iteration is chosen for use in future
reconstructions. ..................... 130
Table 6.4. Comparison between three combination methods, averaged
over 36 different
phantoms. The difference between each method and the mean is taken
to show the
relative difference between methods.
............................................................................
135
Table 6.5. Computation times of the three methods with a dataset of
100. .................. 137
Table 6.6. Relative RMSEs of reconstructions
.............................................................
138
Table 6.7. Reconstruction resolutions for different annular
thicknesses ...................... 142
11
Abstract
This thesis presents the results from a set of numerical
experiments of two- dimensional gas temperature imaging using laser
absorption spectroscopy inside a turbofan engine. This measurement
environment is characterised by temperatures of 2000 K, pressures
of 45 bar, and extremely limited access for the installation of
measurement hardware, which renders invasive measurement
(thermocouple arrays) or direct imaging (PLIF or pyrometry) methods
unviable.
An alternative approach is indirect imaging of the temperature,
whereby the transmittance of a near-infrared laser light through
the gas is measured and used to make inferences about the
properties of the gas along the beam; specifically, its
temperature, pressure, and molecular constitution. The frequency of
the light is chosen to interrogate particular molecular transitions
of a target species—water— in such a way that the fraction of light
measured at the detector depends on the temperature of the gas
through which it has passed. This is an established measurement
technique known as tuneable diode laser absorption spectroscopy
(TDLAS), but it is possible to extend this method to two dimensions
if the transmittance measurements are made over set of coplanar
beams that transect the measurement region. Using the principles of
tomographic inversion, it becomes possible to image not only the
two-dimensional temperature distribution within a gas, but also the
pressure and molecular species concentration distributions.
In this thesis, extensive numerical simulations are used to
critically evaluate this approach when applied to the particular
case of the turbine engine, and a new methodology is developed for
use in this environment which opens up—for the first time, to the
best of the author’s knowledge—the possibility of tomographic
reconstruction of a gas pressure. This is challenging because the
gas pressure has a strong influence on not only the width of
absorption lines, but of their positions on the spectrum, with each
line being affected in a different way. To overcome and eventually
exploit this dependence, a robust approach which the author terms
the spectral fitting approach is developed and tested against the
two main existing methods found in the literature: integrated
absorbance and peak absorption reconstructions. The spectral
fitting approach was found to outperform both methods not only in
the high-pressure regime, but throughout the tested pressure range
( ).
The numerical tests were also applied to more realistic measurement
environments, including annular measurement regions (modelling the
opaque central driveshaft of a turbine engine) with non-uniform
molecular species concentrations and gas pressures. In these
investigations, the temperature was reconstructed with a relative
root-mean-squared error of 2.47%. This demonstrates the theoretical
feasibility of tomographic reconstructions of gas temperature in
the turbine environment.
Numerical optimisation of the methodology is also addressed. The
geometric arrangement of beams through the measurement region is
investigated with a view to maximise the quality of the
reconstructed image, and a new design rule is analytically derived
and then applied to generate a set of viable beam arrangements that
perform competitively when compared to more conventional regular
arrangements. The selection of laser frequencies is also optimised
in the specific case of high-pressure spectroscopy, and two
near-infrared transitions are suggested as a possible candidate
pair for experimental verification.
12
Declaration
No portion of the work referred to in this thesis has been
submitted in support of an
application for another degree of qualification of this or any
other university or other
institute of learning.
Copyright statement
The author of this thesis (including any appendices and/or
schedules to this thesis) owns
certain copyright or related rights in it (the “Copyright”) and
s/he has given The
University of Manchester certain rights to use such Copyright,
including for
administrative purposes.
Copies of this thesis, either in full or in extracts and whether in
hard or electronic copy,
may be made only in accordance with the Copyright, Designs and
Patents Act 1988 (as
amended) and regulations issued under it or, where appropriate, in
accordance with
licensing agreements which the University has from time to time.
This page must form
part of any such copies made.
The ownership of certain Copyright, patents, designs, trademarks
and other
intellectual property (the “Intellectual Property”) and any
reproduction of copyright
works in the thesis, for example graphs and tables
(“Reproductions”), which may be
described in this thesis, may not be owned by the author and may be
owned by third
parties. Such Intellectual Property and Reproductions cannot and
must not be made
available for use without the prior written permission of the
owner(s) of the relevant
Intellectual Property and/or Reproductions.
Further information on the conditions under which disclosure,
publication and
commercialisation of the thesis, the Copyright and any Intellectual
Property and/or
Reproductions described in it may take place is available in the
University IP Policy
(see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in
any relevant
Thesis restriction declarations deposited in the University
Library, The University
Library’s regulations (see
http://www.manchester.ac.uk/library/aboutus/regulations) and
I would like to thank Prof Krikor Ozanyan for his continued support
throughout my
time at Manchester, along with Hugh McCann, Paul Wright, Ed
Cheadle, and Nataša
Terzija in the Industrial Process Tomography group at the
University of Manchester. In
addition, I thank the support of Rolls-Royce, plc. and in
particular the help given to me
by Dr John Black.
List of publications and conference presentations
Wood, M.P., Cheadle, E., Wright, P., Ireland, P., Black, J.,
McCann, H., and Ozanyan,
K.B., “Temperature tomography by NIR molecular absorption”, Proc.
Optics and
Photonics Conference (Photon10), p. 55, Southampton, UK,
2010.
Wood, M.P., Cheadle, E., Wright, P., Ireland, P., Black, J.,
McCann, H., and Ozanyan,
K.B., “Modelling the Performance of Temperature Tomography systems
with IR Laser
Sources”, Proc. 6th World Congress on Industrial Process
Tomography, Beijing, China.
2010. p. 772-777.
Wood, M.P. and Ozanyan, K.B., “Temperature Mapping from Molecular
Absorption
Tomography”, Proc. IEEE Sensors 2011 Conference, Limerick, Ireland,
2011. p. 865-
869; (10.1109/ICSENS.2011.6127014).
Wood, M.P. and Ozanyan, K.B., “Fan Beam Tomography in Annular
Geometry”, Proc.
6th International Symposium on Process Tomography, p.23, Cape Town,
South Africa,
2012.
Wood, M.P. and Ozanyan, K.B., “Concentration and Temperature
Tomography at
Elevated Pressures”, Sensors Journal, IEEE , vol.13, no.8, pp.3060,
3066, Aug. 2013;
(10.1109/JSEN.2013.2260535).
Wood, M.P. and Ozanyan, K.B., “Performance Simulation of a
Tomography Sensor for
Imaging of Temperature in a Gas Turbine Engine”, IEEE Sensors 2013
(submitted).
Symbol Description
( ) Cartesian coordinates
( ) Beam coordinates
( ) Objective function
( ) Kernel of the Fredholm integral equation
Beam index
Vector of
Relative root-mean-squared error in variable (RMSE)
16
Symbol Description
( ) Temperature / K
( ) Mole fraction
( ) Pressure / bar
Volumetric number density of molecule
Radiant energy density /
Natural broadening half width at half maximum (HWHM) /
Doppler broadening line shape function
Doppler broadening half width at half maximum (HWHM) /
Pressure broadening line shape function
Pressure broadening half width at half maximum (HWHM) /
Molecular mass / AMU
Vector of discretised phantom mole fractions
Vector of discretised phantom pressures /
Vector of reconstructed temperatures /
Vector of reconsructed pressures /
1. INTRODUCTION
“Turbojets are like people; if anything goes wrong, the temperature
rises”
– Sir Stanley Hooker
This thesis is a presentation of results which are intended to
demonstrate the feasibility
of remote gas temperature imaging using laser absorption
spectroscopy inside an
operational gas turbine engine. The work was undertaken at the
University of
Manchester from October March , and with the support of
Rolls-Royce
plc.
1.1 Motivation
The objective in this work is to develop a theoretical framework
for high-resolution gas
temperature imaging in a combustion environment where pressures may
reach ,
temperatures range from , invasive measurement is impossible
and
optical access is very limited. For reasons that are detailed in
the remainder of this
chapter, these constraints have led to the exploration of laser
absorption tomography as
a possible solution to this particular challenge. To elaborate on
the motivation for this
work, a description of the fundamental mode of operation of the
turbofan engine is
required.
1.2 The gas turbine engine
The gas turbine engine of an aeroplane is a machine designed to
convert the chemical
potential energy in jet fuel (typically kerosene) into forward
thrust by continually
imparting a rearward impulse on atmospheric air. This can be
modelled by a Brayton
cycle: air at the intake is adiabatically compressed, combusted (an
isobaric addition of
heat), adiabatically expanded, and then expelled into the
atmosphere [1]. In modern
designs the compression is achieved by rotating fan-shaped axial
compressors, which
19
sequentially increase the dynamic pressure, interspersed with fixed
stator vanes, which
redirect the swirling air flow and increase the static pressure.
After passing through a
series of these compressor stages the high-pressure (HP) gas passes
through a diffuser
before entering a combustion chamber where it is mixed with
vaporised jet fuel and
burnt in an exothermic reaction that significantly raises the
temperature of the gas. The
hot gas then expands through a set of turbines, which it does work
to rotate, as it flows
towards the rear of the engine. These turbines are mechanically
coupled to the
compressors by an axial driveshaft so that the work done by the
expanding gas is used
to compress more unburnt gas at the intake. There is a still a
significant quantity of
kinetic energy remaining in the hot gas as it exits the turbines,
and this is funnelled into
a jet using a nozzle. The rearward expulsion of this gas through
the nozzle provides the
forward thrust of the turbojet engine, which is the earliest form
of an aeroplane gas
turbine.
Development of the turbojet engine has since led to a variation: if
a large fan were
attached to the front of the driveshaft so that it overhangs the
front of the engine then a
large fraction of the airflow through it would bypass the
combustion chamber and be
propelled rearwards without being burnt. By adding a turbine to the
back of the engine
and connecting it to this front fan (two or more coaxial
driveshafts can be used in the
same engine by hollowing the outer one(s)), part of the kinetic
energy of the gas in the
jet can be diverted to kinetic energy in the bypass air; since this
air is travelling slower
than the jet air, this diversion can be used to increase the rate
of impulse acting on the
air at the intake (momentum is proportional to velocity, whereas
kinetic energy is
proportional to squared velocity). This configuration is more
fuel-efficient and quieter,
and so-called “turbofan” engines constitute the majority of engines
used in medium to
large-sized commercial aircraft (figure 1). For a more detailed
description and analysis,
see [2, 3].
20
Figure 1.1. Simple schematic of a turbofan engine. The cold,
atmospheric intake air is sucked in, split, compressed, mixed with
fuel, combusted, expanded, and then funnelled out of the rear
nozzle imparting a forward thrust on the engine.
The design of a turbofan engine involves many compromises between
conflicting
objectives: fuel consumption, thrust, range, safety, longevity,
noise, cost, emissions, and
weight, to name a few. For example, a conflict exists between fuel
consumption and
longevity/safety in the following way: the thermal efficiency of an
engine increases with
higher combustion temperature and pressure ratio, and an engine
which produces more
thrust for the same amount of fuel is extremely desirable
(especially if the engine weight
and size are also unchanged), but the higher operating temperatures
come at a price: the
turbines and nozzle guide vanes are both immediately downstream of
the combustor and
there are strict limitations to the temperature they can operate
at; it has been reported
that an increase of beyond a limiting operating temperature can
halve the
lifespan of a turbine blade [4].
Decades of work have been dedicated to pushing back this
temperature limitation:
better aerofoils have been manufactured using improving casting
methods from Nickel
superalloys with ever increasing melting points and creep strengths
[5], but such
metallurgical progress cannot be expected to continue indefinitely.
In addition to
improvements in the blade material itself, their surfaces are
treated with thermal barrier
coats (TBCs) [6] to limit the conductive heat flux between the
burnt gas and the metal
itself, and modern blades are hollowed and perforated to enable
cold HP air — up to
20% is diverted from the compressor yield [7] via air ducts at
significant cost to the
21
turbine pressure ratio and thermal efficiency — to be fed through
the blades and out of
the surface, creating an additional boundary layer between the gas
and the coating [8].
A different way to improving thermal efficiency lies in the
combustor design. A
typical gas turbine combustor is a perforated sheet metal parabolic
or hemispherical-
shaped bowl with the apex positioned upstream. High-pressure air
enters through the
perforations and mixes with vaporised kerosene that is injected via
a fuel nozzle. The
geometry is designed so that the chamber contains the flame, and is
suspended in the
turbine with cold, unburnt air flowing around the outside.
Downstream of the
combustion chamber the hot, burnt gas mixes with the cold, unburnt
gas before entering
the turbine section. If the mixing is sufficient then the
temperature of the gas, by the
time it reaches the turbines (turbine inlet temperature), will have
dropped sufficiently
that the turbine components are not endangered. Typically, the gas
temperature is
around 2200 K during combustion, 2000 K on entry to the HP
turbines, and 1200 K at
the low-pressure (LP) turbines. The design of the combustion
chamber itself therefore
affects the severity of the necessary trade-off between engine
performance and integrity,
and a well-designed combustion chamber whose exhaust gas mixes well
with the cold
external flow will cause the gas impacting on the turbine section
to be cooler which
may, for example, require fewer cooling ducts, or allow for the
turbine to be made from
a lighter alloy or using a cheaper manufacturing process. These
considerations are all
relevant when optimising blade and combustion chamber design.
As with any product, the research and design process is an
iterative exercise in
trial and error; a design is suggested, tested, and the results of
the test are analysed and
used to motivate modifications to the original design. In the case
of a turbine engine this
process may be repeated a large number of times as an initial
concept is developed into
a working prototype. The value of a testing phase lies in the data
that can be measured
whilst the engine is running on a test bed, and if more data can be
measured then it is
possible that the design of an engine will yield a better outcome
in fewer iterations. In
particular, useful data includes (but is not limited to)
information about the gas
temperature profile, species concentration profiles (fuel, , , and
), and flow
velocity profiles from the combustor to the turbine inlet [9,
10].
This provides the motivation for knowing temperature distributions
inside the turbine
engine. However, measurements of the gas temperature in this
hostile environment are
22
difficult to obtain due to the limited optical access, high
operating temperatures and
large amounts of engine vibration.
1.3 Current methods for temperature sensing
There are number of different methods of measuring temperature that
are currently used
for combustion diagnostics, and it is often the case that two or
more methods are used
simultaneously, often for the purposes of independent verification
or on-line calibration
of measurements [11, 12]. A brief summary of the methods are given
in this section.
1.3.1 Invasive measurement
One approach is to install one or more measurement probes into the
flow. Fine-wire
thermocouples are inexpensive and offer the capability of remote
sensing of in
combustion environments [13, 14]: Tungsten/rhenium-alloy
thermocouples are capable
of measuring temperatures up to and beyond 2500 K [15, 16] but in
oxidising conditions
the elevated temperatures cause rapid deterioration of the
thermocouple elements which
necessitates the use of protective sheathing. This sheathing
increases the thermal mass
of the device which reduces its temporal response, and further
limits the operational
temperature range of the device (e.g. platinum/rhodium, 1920 K).
Because of these
limitations gas turbine thermocouples are placed downstream of the
HP turbines,
usually in front of the LP turbine [17] where temperatures are far
lower. Another
limitation to this invasive approach is that the presence of the
probe and its connecting
and support wire(s) inside in the flow can cause local velocity and
temperature
perturbations which undermine the measured values, and the extent
of the perturbations
in turbulent flows is unpredictable [18]. Significant errors
associated with these
perturbations have been observed in relatively benign combustion
environments [19,
20]. Finally, thermocouples only give localised point measurements
of the temperature
rather than continuous distributions.
closely related remote sensing technique for imaging species
concentration and
temperature [11, 21-24] and pressure [22, 25] along one-dimensional
lines or one or
more two-dimensional planes within the imaging space
(three-dimensional imaging is
23
possible with multiple sheets [26]). This is normally achieved
using a pulsed laser
beam/sheet (although cw lasers have also been used [27]) with a
frequency that targets
an absorption transition of a particular species in the flow. This
causes a temporary
excitation of the targeted species. An off-beam/out-of-plane camera
is then used to
directly image the fluorescent radiation that is isotropically
emitted as the excited
molecules (or radicals, e.g. ) return to their lower-energy states;
the signal is
dependent on the concentration of excited species which in turn
depends on the
concentration of the species in the ground state. If the target
species has electronic
transitions then the resulting signal lies in the optical or
ultra-violet (UV) part of the
spectrum [28-30], but the advent of infrared fixed-plane array
cameras has led to the
development of infrared PLIF (IR-PLIF) which targets
vibration-rotation transitions of
small molecules (e.g. , and ) instead [31-33]. This part of the
spectrum is
often favourable because these molecules are natural combustion
products and there is
no requirement for any upstream doping of the flow with a UV-active
(but inert)
species, e.g. , which may not diffuse evenly in the flow.
LIF/PLIF are non-invasive remote sensing techniques but they can
suffer from an
effect known as radiative trapping whereby an amount of the
flourescent radiation is re-
absorbed by more target molecules en-route to the detector [22];
this attenuation
depends on out-of-plane gas properties which are often unknown.
Furthermore, the two-
line ratiometric approach does not warrant total cancellation of
this source of error if the
gas temperature along the emission-detection line-of-sight is not
isothermal.
In the context of imaging in restrictive geometries, PLIF’s primary
limitation is its
requirement of an out-of-plane detector with a full view of the
flourescent sheet. This
difficulty, along with the issues of radiative trapping, renders
the deployment inside a
turbine engine very challenging in practice.
1.3.3 Spontaneous Raman scattering
Another remote sensing technique is spontaneous Raman scattering.
This process
involves a molecular transition from one energy level, via a
virtual energy level, to a
different energy level. In this case a laser is tuned to a “pump”
frequency
(corresponding to the energy difference between the virtual level
and the original level),
and the molecule re-emits incoherent radiation isotropically at a
different “Stokes”
frequency (corresponding to the energy difference between the
virtual level and the new
24
level). This scattering effect is very weak, however, and the
signal is often dwarfed by
fluorescence or incandescence phenomenon in many conditions
[34].
1.3.4 Coherent anti-Stokes Raman scattering
Another method for remotely sensing temperature distributions is
coherent anti-Stokes
Raman scattering (CARS) [35]. As with the spontaneous case,
anti-Stokes Raman
transitions are targeted but the input laser source is a
combination of the pump
frequency and the Stokes frequency of a transition. This induces a
resonance in the
target species and causes the stimulated emission of coherent light
at the “anti-Stokes”
frequency (2*pump – Stokes), which is measurable along the same
path as the original
beam. CARS has been demonstrated for the purposes of concentration
imaging [36, 37]
and thermometry along a line of sight [11, 38], including at high
pressures [39].
Because the stimulated emission is unidirectional, a well-placed
detector can obtain a
far higher signal-to-noise ratio (SNR) than in the case of
spontaneous Raman scattering.
1.3.5 Optical pyrometry: turbine blades
Optical pyrometry is an established technology for the on-line
measurement of
turbine blade surface temperatures [17, 40, 41] via passive
measurement of thermal
emission. The temperature is inferred from an optical measurement
of the thermal
emission of a specific area (around ) of the blade using Planck’s
law. The fast
temporal response of the optical detector enables continuous
monitoring of all the
turbine blades, since multiple measurements of the same area can be
made for every
revolution of the turbine. This information is then used to
regulate the amount of fuel
burnt in the combustor to ensure that the turbine blades remain
within pre-defined
operating limits.
1.3.6 Optical pyrometry: soot
In many cases the flames themselves contain soot particles which,
as solids, emit
thermal radiation isotropically with spectrum approximated by a
blackbody curve. The
shape of this curve is temperature-dependent, and two measurements
of the emissivity
of a part of a flame at two different (but nearby) wavelengths
permits the two-
dimensional direct imaging of flame temperatures provided the
measurement
wavelengths do not suffer from interference by molecular or atomic
absorption bands.
25
This two-colour method has been known for a long time [42] but
recent advances in
both CCD and computer processing technology have brought about new
methods of
non-invasive, direct, and fast imaging of temperature distributions
of sooting flames
[43]. This method has been successfully employed in a coal-fired
reactor [44] and a
model turbine engine [45].
1.4 Laser absorption spectroscopy
Laser absorption spectroscopy is a diagnostic technique for the
measurement of gas
temperature, concentration, pressure and velocity along the
line-of-sight of a laser beam.
A single source-detector collimator pair either side of the
measurement region are used
to fire and collect a laser beam through the region, and the beam
frequency is tuned to
an absorption transition of a species within the flow. The
transmittance over the beam
can be measured as the ratio of the incident to transmitted power
to give a measurement
that depends on the spectral properties of the gas along the
beam.
1.4.1 Line-of-sight thermometry by direct absorption
spectroscopy
If the temperature at every point along the beam is constant then
its value can be
calculated from measurements of the beam absorption at two or more
absorption
transitions by direct absorption spectroscopy [46-53]. This
measurement is performed
experimentally using either wavelength-division multiplexing, where
light of different
frequencies is simultaneously sent through the same optical train,
or time-division
multiplexing where light of different frequencies is sent through
during alternating time
intervals. In certain cases (explained quantitatively in section
§3.9) it is possible to infer
the isothermal line-of-sight temperature by two absorption
measurements at the
linecentre frequencies of two absorption transitions of a target
molecule in the gas. The
absorption signals at each frequency are strongly dependent on the
molecular number
density when taken separately. Taking the ratio of these signals,
however, almost
entirely removes this dependency and produces a value that is
dependent on the relative
molecular quantum ground state populations of the two transitions
instead. These
populations are governed by temperature via the Boltzmann
distribution, and an
appropriate selection of absorption transitions can be used to
obtain sensitive
measurements of the gas temperature along the isothermal laser beam
[54]. These
different dependencies are shown in Figure 1.2 and Figure
1.3.
26
Figure 1.3. Relative similarity in mole fraction sensitivity
between two absorption lines.
The advent of relatively cheap, rugged, stable, and reliable laser
sources, detectors
and optical fibres in the near-infrared region over the last 20
years can be attributed to
the growth of commercial demand in the telecommunications and data
storage sectors
[55]. Diode lasers can operate at room temperature and produce
narrowband light whose
27
frequency can be tuned by controlling the diode temperature or
injection current. The
thermal mass of the diode limits the rate of tuning of the
frequency via temperature
control, but there is no such response time limitation on the
injection current and it is
possible to modulate diode laser sources at frequencies as high as
[56]. In
practice, direct absorption spectroscopy uses typical scanning
frequencies of
[57]; by sampling the transmitted light at the photodetector at a
higher frequency it is
possible to obtain a large ( ) number of transmittance measurements
for a single
sweep of the laser source, producing a continuous sample of the
absorption spectrum of
the species instead of a single peak-value. This technique leads to
a more general
method of temperature inference than the peak absorption method: if
the lineshapes of
the two transitions are different, then two lines with the same
linestrength will have
different peak heights; this introduces an error in the peak method
1 . The alternative is to
scan the laser source frequency over an entire transition and
integrate the detected
transmittance over every cycle. This gives a direct measurement of
the line strength
independently of the lineshape.
However, the integration of an absorption transition requires
well-defined limits, and
the existence and location of these limits becomes very difficult
at elevated gas
pressures because of the effects of collisional broadening on the
individual transition
lines [58], as shown in Figure 1.4. Absorption transitions have
narrow spectral widths
and are individually discernible at atmospheric pressure, but at
pressures above roughly
5 bar (depending on the specific spectral region of interest), the
lines blend together and
it becomes impossible to make a measurement of a transition
integrated absorbance
without contamination of the measurement by systematic error by
neighbouring
transitions. The positioning of the integration limits becomes a
matter of guesswork and
an alternative approach is necessary.
1 The rigorous definitions of a transition linestrength and
lineshape are given later in §3. For now
it is sufficient to know that the linestrength of a transition is a
measure of its absorbing strength, and the
lineshape is the shape of the absorption line but with an area
normalised to unity.
28
Figure 1.4. Effect of pressure increase on a set of near-infrared
absorption lines.
1.4.2 Modulation spectroscopy
One alternative to direct absorption measurements for high-pressure
gases is
wavelength-modulation spectroscopy (WMS) [59-63]. Using this
approach, the
frequency of the laser source is modulated at a much higher
frequency ( ) and
the detector is sampled at integer multiples of this frequency, for
example and .
This form of measurement is sensitive to the shape of the
absorption feature instead of
its absolute height, and the temperature and species mole fraction
can be calculated
from the measured values by decomposing the mathematical functions
which model the
expected absorption line shape (e.g. Lorentzian, Voigt, or Galatry
profiles) into
harmonic functions and calculating the and signal as a function of
these
harmonics. The complexity of this relationship also depends on the
modulation depth,
and the interference of the signal by a nonlinear amplitude
modulation in the laser
source (which results from a large modulation depth in the
injection current) can
introduce additional terms; these are discussed in detail in the
literature [62, 64, 65].
This approach is particularly suited to measurements over short
path lengths or for trace
species in the flow where the signal from direct absorption
spectroscopy is too small.
29
WMS is also beneficial in high-pressure environments where a
baseline absorption (the
measured valued in the case that there is no target species
present) is difficult to obtain
from high-pressure direct absorption measurements of transitions
with significant
amounts of interference. The drawback to this method lies in its
experimental and
mathematical complexity.
Laser absorption spectroscopy is notably non-invasive and, at
typical diode laser
powers ( ), the measurement causes virtually no alternation to the
measured
parameter(s). Furthermore, the geometric requirements are
particularly unrestrictive:
only hardware at either end is required to measure gas parameters
over the entire beam.
This is a highly desirable attribute for gasdynamic sensing and in
particular when
optical access comes at a premium, as would certainly be the case
inside a turbine
engine.
The concept of laser absorption spectroscopy can be expended to
higher-dimensional
reconstructions by recognising that the transmittance of light over
a beam can be related
to the Radon transform of a material property of the transected gas
that is called the
spectral absorption coefficient, which is a quantitative measure of
the local opacity of
the gas at a given frequency due to stimulated absorption. It is
then possible to use
tomographic inversion techniques to reconstruct images of this
quantity (or linear
functions of it, e.g. its integrals in the integrated absorbance
method) from line-of-sight
transmittance measurements. The theoretical basis of this was
published in 1975 [66]
and the numerical implementation was developed using a multitude of
mathematical
and spectroscopic approaches [67-70]. For example, in the
particular case of an
axisymmetric flame, 2-D temperature and species concentration
fields can be
reconstructed in a plane perpendicular to the axis of symmetry
using the onion-peeling
method [71, 72]. In, general, however, it is necessary to use
Fourier or algebraic
methods of reconstruction [73] depending on the number of available
measurements.
Early tomographic reconstructions of temperature and OH
concentration were achieved
using a continuous wave dye laser [74]. Successful reconstructions
of 2-D temperature
profiles using electronic ( ) transitions of were produced [75,
76], with
another target molecule using a He-Ne laser [77, 78]. This was
followed by chemical
species tomographic reconstructions using near-infrared transitions
of hydrocarbon
30
molecules in chemical reactor environments [79, 80]. A similar
method was used by the
same group to obtain time-resolved species reconstructions inside
an automotive engine
cylinder [81-83].
Very recent experiments in temperature tomography using laser
absorption
spectroscopy have focused on reconstructing two [84, 85] or more
than two [86] images
of integrated absorbances. These approaches demonstrate the
feasibility of temperature
tomography using laser absorption spectroscopy at atmospheric
pressures, where target
lines are well-isolated
1.5.1 Objective
The objective is to develop a theoretically viable solution to
permit temperature imaging
inside a turbofan environment characterised by high temperature and
pressure and
limited optical access, and to demonstrate the viability of this
solution using numerical
simulations.
synthetic data predicted at turbine operating conditions.
To evaluate the method over a wide range of possible
conditions.
To compare the method to existing methods of temperature
tomography
found in the literature
To investigate how best to use post-processing to aggregate the
large amount
of time-series data that can be captured from fast modern
photodiode
detectors.
To develop a methodology to optimise the beam configurations for a
variety
of differently-shaped annular geometries.
To identify a set of candidate absorption lines that are suitable
for high-
pressure temperature measurement.
1.6 Overview
The second and third chapters are dedicated to the description of
the fields of
tomography and spectroscopy, tailored to this particular
application. These chapters
form an introduction to the methods of temperature tomography,
which are introduced
at the end of the third chapter.
The fourth chapter contains a description and illustrated example
of the processes
involved in (1) generating synthetic data and (2) reconstructing
temperature fields from
that data.
The fifth chapter is dedicated to numerical analysis with the aim
of optimising the
measurement system by selecting a good beam configuration and good
water vapour
absorption lines.
The feasibility of the developed method is analysed by computer
modelling, and
chapter five is used to explain the modelling process that is used.
Chapter six contains
the results of a large number of numerical simulations of
temperature tomography. The
data are divided into sub-sections which focus on different aims.
For each sub-section,
the methodology is given, the results are presented as
reconstructions and data
overviews, and a set of conclusions are recorded.
Chapter seven summarises the conclusions of the work, the
assumptions made in the
numerical investigations and a list of possible future work in the
field, and chapter 8
contains truncated MATLAB code.
2. TOMOGRAPHY
2.1 Introduction
Tomography is the mathematical study of image reconstruction from
data that is
measured only at the periphery of the imaged object. In the case of
a flat 2D image the
measurement hardware is restricted to the edge of the imaging
plane, and not out of the
plane (e.g. in the case of direct imaging using a camera) or in the
imaging space itself
(e.g. invasive imaging using a detector array). Tomographic imaging
is a specialist
technique which is often exploited in cases where the desired
information is, using
alternative imaging methods, either inaccessible or accessible at a
prohibitive cost. X-
ray CT scans, PET scans, and NMR imaging are all forms of
tomography which allow
physicians to view distributions of physical quantities inside the
body without the need
for invasive surgery, and the advent of reliable imaging machines
in hospitals has
revolutionised the diagnostic process.
However, the issue of limited measurement access is not unique to
the medical
profession. Although tomographic reconstructions differ on a
case-by-case basis, there
are mathematical concepts and approaches that are common to many
applications and
the same ideas which were first used to image human bone are also
applicable to the
imaging of gas parameters in a combustion environment. The history
of tomographic
reconstruction is a story of researchers in many different branches
of science working in
parallel on many different imaging problems, and it is neither
uncommon for a
particularly useful concept to be independently discovered multiple
times, nor for an
33
idea in one branch of science to be adopted with great success in
an entirely separate
branch by interdisciplinary communication.
The relationship between the imaged quantity and the measurement
dictates the
method of inversion, and each relationship can be classified. In
many tomographic
applications, a single measurement at the boundary will be
dependent on every value of
the object function in the imaging space. This general case is true
in electrical
capacitance tomography and electrical impedance tomography.
However, there are
certain special cases in which a single measurement will depend
only on a readily
identifiable subset of the imaging space; for example, in laser
absorption tomography,
the absorption measurement is attributed to the spectral absorption
coefficient of the gas
along the beam only. This is known as the hard-field
approximation.
The Radon transform is the key to characterising the link between
the imaged object
and the measured data in this approximation. In the following
section, the imaged object
is treated as a generic scalar function over two-dimensional space,
and the two-
dimensional Radon transform is defined accordingly. The finite and
approximate nature
of the resulting measurements is used to derive a practical link
between the desired
quantity and the known information. To make progress, a
discretisation scheme is
employed to reduce the scale of the image to a finite dimension,
and recast the
relationship as a discrete linear inverse problem. The issues with
such a problem are
discussed and then addressed using the two competing reconstruction
algorithms:
projected Landweber iteration and Tikhonov inversion. Finally, the
issue of
measurement optimisation via beam placement is discussed in the
specific context of
limited-data tomography.
2.2 Defining the Radon transform
Consider a scalar quantity which varies over the two-dimensional
unit disc
{( ) }; this defines the nondimensionalised
measurement region, which can later be generalised to a
two-dimensional annulus
{ } for a given parameter . and are Cartesian coordinates,
and
( ) . The two-dimensional Radon transform [ ]( ) is a mapping
from
onto the integrals of over the set of straight lines that pass
through the disk:
34
The coordinates ( ) [ ) [ ) represent the position and orientation
of a
straight line in the following way: is the minimum distance between
the origin and the
line, and is the angle between the -axis and the line connecting
the origin to the
closest point on the line (Figure 2.1); the equation of this line
is given by:
, 2.2
and is the delta function which is used to select only the points
that reside on the line.
This transform is named after Johann Radon who published an
expression for the
inverse transform in 1917 [87, 88].
Figure 2.1. A general point ( ) in the domain of , and line ( ) in
the domain of .
By writing ( ) and substituting into equation 2.1:
[ ]( ) ∫ ∫ ( ) ( )
2.3
Equation 2.1 can be interpreted as a Fredholm integral equation of
the first kind with a
kernel . The objective is to find the ‘object’ given the
‘measurements’ . In
35
practice, it is only possible to measure a discrete number of line
integrals, and the task is
to find from a small number of samples of . Let these discrete
measurements over
the line ( ) be represented by , indexed by , with measurement
errors
. Then equation 2.3 is:
2.3 Filtered backprojection
With enough measurements of this line integral at different angles,
a fast solution of
2.4 is possible using the Fourier slice theorem, which states that
the one-dimensional
Fourier transform of a single projection of the object (i.e. the
function ( ) for a fixed
) is equal to a “slice” of the two-dimensional Fourier transform of
the object over the
∫ [ ]( )
( ( ) ( )
) ( )
2.5
and are coordinates in the frequency domain and parameterises the
“slice” in the
frequency domain. is used to select values of the Fourier transform
along the slice
(the phase shift of exists because the slice is perpendicular to
the direction of the
parallel beams used to form the projection); this is shown
graphically in Figure 2.2.
36
Figure 2.2. Graphical representation of the Fourier slice theorem.
The one-dimensional Fourier transform of the Radon transform of ( )
at an angle is equal to the two-dimensional
Fourier transform of ( ) along the radial slice .
Multiple projection angles can be used to generate multiple slices
of the Fourier
transform of , but the discrete Fourier transform must be used
because ( ) is a
discrete function. According to the Nyquist sampling theorem, some
high-frequency
components of will be lost due to the finite spacing between the
lines within a single
projection. This can be remedied by applying a high-pass filter to
Fourier transform. A
solution is then found by taking the two-dimensional discrete
inverse Fourier transform
and interpolating the result to obtain a solution. This is the
filtered backprojection
algorithm; it was first derived by Bracewell [89] in radio
astronomy and later
independently by Cormack [90] who shared a Nobel prize with
Hounsfield for their
work towards the development of the first x-ray computerised
tomography imager.
Filtered backprojection is a fast solution method that is
well-suited for medical
applications because, as long as a patient remains still, it is
possible to measure a very
large number of line integrals and the resulting filtering and
interpolation errors are
small. However, when measurement data are limited, the quality of
the reconstructed
image is heavily degraded [73] and alternative approaches must be
sought.
37
2.4 Discretisation
Given the incomplete nature of the measurements, the problem can be
better approached
via the expansion of in a finite series: ( ) is approximated using
the weighted sum
of a finite number of predetermined basis functions ( ):
( ) ( ) ( ) ∑ ( )
( ) 2.6
( ) {
⁄ ⁄
. 2.7
Where is the size of the pixel along the coordinate axes and is the
total number of
pixels. are the pixel values, and they completely specify .
Substitution of into 2.4
yields:
[ ]( ) 2.8
The order of the double integral and the sum in the first term on
the second line can be
changed and, because it is not a function of or , can be taken
outside of the double
integral:
[ ]( ) 2.9
and are both known so the double integral can be precalculated for
each pixel and
line . These values can be stored in a coefficient (or sensitivity)
matrix :
38
2.10
In physical terms, is the length of the segment of beam that
resides inside pixel
(Figure 2.3).
Figure 2.3. A conceptual illustration of the meaning of as the
length of line in pixel . One
method of reducing the number of unknowns is to pixelate the image
space. The kernel of the Fredholm equation is represented by a
matrix operator.
This discretisation procedure is a form of numerical quadrature
where a curve is
approximated by a series of flat steps: in this case, the curve is
a one-dimensional slice
of , and the steps are defined between the intersection points of
the line and the pixel
edges. The result is a linear system of equations:
∑
[ ]( )
2.11
The second and third error terms can be combined into a single new
term for brevity:
[ ]( ) 2.12
39
. 2.14
is an ( ) vector of the measured values, is an ( ) matrix of the
discretised
kernel, is an ( ) vector of pixel values, and is an ( ) vector of
the
combined modelling and data errors. Solving equation 2.14 for is a
discrete linear
inverse problem.
2.5 Ill-posedness
It is clear from equation 2.14 that it is easier to find given than
to find given .
These two problems satisfy a commonly accepted definition of a
forward-inverse
problem pair offered by Keller [93] as one in which “the
formulation of each involves
all or part of the solution of the other”. The naming of each
problem as direct or inverse
is, by convention, chosen so that the direct problem involves the
acquisition or
prediction of the measurement data from the state of a physical
system, whereas the
inverse problem involves estimating the state of a physical system
from such
measurement data. Because the objective is to find from , this
represents an inverse
problem. A common feature of inverse problems is ill-posedness
which is characterised
by a failure to satisfy any of Hadamard’s three criteria:
1. The problem has a solution,
2. The solution to the problem is unique, and
3. The solution depends continuously on the data.
The problem of reconstructing a discrete image from a limited set
of its line integrals is
an ill-posed problem because it will typically fail all three of
these conditions. The
following discussion will demonstrate why.
If is naturally pixellated in the same way as then the modelling
errors will be
zero, and if the measurement errors are also zero then and .
Additionally,
if the number of measurements equals the number of pixels then is
square, and if none
of these measurements are redundant (e.g. if no two beams intersect
exactly the same
40
pixels by the same amounts) then is non-singular and an exact
solution can be found
via . This is an ideal case where the problem is well-posed because
the solution
exists, it is unique, and it is stable; Hadamard’s criteria are
satisfied.
This ideal case will not be realised in tomography of real objects
for three reasons:
(1) unless physical distributions of the temperature, mole
fraction, pressure, and local
attenuation are naturally pixelated, and there will be inherent
modelling errors.
(2) All measurement data should be assumed to contain random (if
not also systematic)
errors. Either of these reasons is sufficient to ensure that . The
true (unknown)
line integral values are said to be in the column space of because
there exists
an image such that . If the error vector is in the column space of
then
there exists a second image such that , so the measurement
( ) corresponds to the image . However, since contains a
stochastic
contribution from the random measurement errors, it is extremely
unlikely that is in
the column space of and, by implication, extremely unlikely that
exists. It is
therefore reasonable to believe that there exists no image that
would reconcile
with the measured data . In short, there is no solution to , and
direct inversion
via is impossible. This is in violation of Hadamard’s first
criterion (of existence).
(3) There are a relatively small number of measurements in
limited-data tomography: it
might be possible to place as many as 60 beams inside a combustion
environment [79,
82, 83], and if the primary goal is to ensure that (so that is
square), the image
resolution is limited to 60 pixels or an grid from which it is
difficult to discern
structures of interest in . A better approach involves choosing a
sufficiently large set of
basis functions to make it possible for these structures to appear
in the image, and work
towards remedying the under-constrained problem which results from
. This
problem can be explained using the concept of the matrix
rank.
The rank of a matrix is equal to the number of linearly independent
rows or columns
it has, and cannot be greater than either the number of rows or
columns. If the
measurements are independent then ( ) , but for an
under-constrained
problem and is called rank-deficient. By the rank-nullity theorem,
if a matrix is rank-
deficient then there must exist least one vector in the nullspace
of that matrix, i.e. there
exists a vector such that . This vector is troublesome because it
introduces a
degree of freedom in the image space which cannot be detected in
the measurement
space: no measurement will be able to help distinguish between an
image or ,
41
because ( ). Any solution to equation 2.14 will be one
amongst
infinitely many because , and Hadamard’s second criterion of
uniqueness is
violated. So far, the data neither matches the model nor is capable
of fully constraining
it—this is certainly an inverse problem.
A fourth problem may exist as a result of the discretization of the
Fredholm integral
equation. This process may have reduced the number of unknown
quantities to , but it
does not do much to address the problems that are often inherent to
integral equations.
The Radon transform, being a form of integration, is a natural
low-pass filter. High-
frequency (local) noise in the image is often smoothed when the
measurements are
taken, because integration over a line is effectively an averaging
process: the
measurements are relatively insensitive to localised variations in
the image. Any
inversion attempt can be expected to exhibit the opposite effect
because local variations
in the image become relatively sensitive to the measurements. Many
naïve numerical
solutions of Fredholm integral equations with noisy data will
generate nonsensical
results, due to the amplification of the noise. This feature is
true in the continuous case,
and is also true in the discretised case, where it can be partially
quantified using the
concept of a matrix condition number. The condition number of a
matrix is defined as
the ratio of its first to its last singular values: ( ) , where
are
found via the process of singular value decomposition. If this
number is large then is
called an ill-conditioned matrix and there are often problems with
the stability of the
solution of equation 2.14.
A problem with the ill-conditioning of is that two or more of the
rows of may be
nearly linearly dependent. These rows might then correspond to a
pair of integrals of
over nearly identical lines. Any discrepancy between the two
measurements will be
attributed to the small fraction of pixels that one line transects
but the other does not.
The length of one of the lines through this small number of pixels
is much smaller than
the total length of the line, so a large difference in the values
of these pixels must result
from a small discrepancy in the measurement. A linear system of
equations which
exhibits this problem is ill-conditioned, and violates Hadamard’s
third condition of
stability.
42
2.6.1 Existence
Data and modelling errors will render the measurements incompatible
with any possible
image because for any . It is instead reasonable to seek a solution
that
minimises the error in lieu of an exact match. A scalar functional
can be defined as
the square of the length of this error vector:
( )
2.15
Any image which minimises is a least-squares fit to the data. Note
that if there are
no errors then , and is the exact solution. The minimum point can
be
calculated by taking the gradient vector of :
( ) ( ) 2.16
and setting it equal to . The resulting image is a global minimum
because is a
positive quadratic function. This generates the corresponding Euler
equation:
( ) . 2.17
This formulation ensures the existence of a solution, but it does
not mean that this
solution is unique.
While ( ) ( ), there will be infinitely many least-squares
solutions that
satisfy 2.17. However, a large fraction of these possible solutions
are very different
from the sort of physical distributions that might be expected to
be found in nature; in
other words, very few of the possible solutions reconcile with
knowledge of the
physical processes which are expected to determine the distribution
of . When faced
with chosing from one amongst many solutions, it is sensible to
favour the one which
least contradicts the available information about from alternative
sources, and this
concept— the incorporation of prior knowledge — can be used to
adequately constrain
the problem.
43
This incorporation can be achieved in many different ways. If the
priors are known
for certain and are not likely to change then they can be encoded
into the basis functions
themselves, as was demonstrated by Verhoeven [91]: rather than use
square flat blocks
for , one can use pyramid-shaped functions to enforce a degree of
smoothness from
the outset of the discretisation; the prior knowledge is that is
small. If, on the other
hand, the priors might require adjustment, then another approach is
to incorporate them
when solving the discrete inverse problem.
2.6.3 Stability
Priors can be used to constrain the system of equations 2.17 but,
even so, the solution is
not trivial to solve. As was explained above, ill-conditioned but
full-rank matrices are
still tricky to deal with in the presence of measurement noise if
the condition number of
is high. The amplification of noise by the inversion process can
mean that two
measurements of a system in the same state can lead to two very
different reconstructed
images because the measurement noise is expected to vary between
results.
In hard-field tomography, particularly with limited-data, the main
problem is that
. However, the limited number of measurements and large number of
pixels
means that it is very unlikely for two measurements to be linearly
or even nearly
linearly dependent, and the condition number of a typical kernel
matrix is ( ) ;
other reconstruction applications face far more severely
ill-conditioned matrices; this
feature has also been reported by Twynstra & Daun in their
analysis of a similar
problem [95]. The main issue is the severe rank-deficiency of but,
in addressing this
by incorporating certain priors, the propagation of measurement
noise during inversion
is limited. In this case, the prior information is that the image
is smooth, and it is likely
that the enforcement of this prior has a secondary advantage of
suppressing the natural
amplification of high-frequency measurement noise by the inversion
process.
2.7 Algorithms
2.7.1 Landweber Iteration
Landweber’s method is a stable algorithm for finding a minimum by
iterative
means, which also permits simple incorporation of priors via a
projection operator that
is applied between the iterative steps. The final equation in 2.16
contains an expression
44
for the gradient of the functional that is to be minimised; by
definition the gradient
vector points in the direction of the steepest increase of the
function, so the deduction of
it from an initial guess of the solution is an effectively a
gradient descent step:
( ) 2.18
with step size . This is Landweber’s iterative scheme [96-98].
Re-arranging 2.18
gives:
( ) . 2.19
To ensure this does not diverge, the spectral norm of ( ) must be
between
and 1, which means should be chosen so that ( ( )) . Because
the eigenvalues of are equal to the squares of the singular values
of , an
equivalent condition is ⁄ . In practice, if is too small then the
iteration
requires many steps to reach a minimum value, if is too large then
the iterates tend to
oscillate between values of corresponding to poor solutions
[99-102], if it is zero then
the iteration achieves nothing, and if it is negative the iteration
represents a gradient
ascent and it should be expected that the new result is further
from the solution than
before.
As long as is chosen correctly the iterative scheme 2.18 will
exhibit semi-
convergence towards a minimum of [100, 103]. However, this solution
is rarely
meaningful in the case of limited-data tomography owing to the
severe rank-deficiency
of , which the iterative scheme does little to address: a
least-squares solution can be
found by standard Landweber iteration, but it will still be one of
an infinite set, and will
generally bear little resemblance to the imaged object.
A more general form of Landweber iteration [103, 104] applies an
operator
to the solution after each iteration:
( ( )) 2.20
This is the projected Landweber iteration and can be used to impose
prior
knowledge on the solution after each repetition. This step serves
to alleviate the
problems caused by the rank-deficiency of because the operator
enforces the
interdependency between adjacent pixels in the image. This creates
an additional set of
45
constraints on the solution. For example, a median filter with an
additional non-
negativity prior [81, 82] has been used to reconstruct species mole
fraction images. This
can be written analytically in terms of an operation on each
element in the image:
( ) { |( ) ( )| }. 2.21
is a raidal distance which is used to control the size of the
filter. The median filter
step updates every pixel in the image whether they are intersected
by beams or not, and
the non-negativity prevents unphysical results in the case that
measured data implies
negative mole fractions of a species. This filter has the effect of
smoothing the image
whilst preserving sharp edges.
( ) {
⁄
2.22
Where equals the number of active pixels sharing an edge with pixel
. This is
effectively a smoothing convolution which tends to lessen steep
gradients in the
reconstructed solution and ensure that every pixel’s value is
similar to the neighbouring
ones regardless of whether or not a beam transects it. It has been
found that no non-
negativity step is required with this projection, and the numerical
implementation is
extremely fast.
Projected Landweber iteration is a very fast method of
regularisation if the projection
operation can be computed at a similar speed as the iteration step.
Furthermore, the
iterative step does not require large-scale matrix manipulations,
and the method scales
well as becomes large. The projection operator enables a wide range
of priors to be
incorporated (some at the expense of computation time), including
nonlinear functions
of the image, but one disadvantage is that the projection and
iterative steps will tend to
conflict. As a result, the iteration will tend to converge to two
separate images
depending on whether it is terminated after the iterative step or
the projection step, and
it is tricky to control the balance between the competing
influences on the solution of
the data and the priors.
46
2.7.2 Tikhonov inversion
A different approach to solving this problem is to modify the
original least-squares
problem to include an additional penalty term. This regularises the
problem by
incorporating priors from the outset. Tikhonov regularisation
[105-107] achieves this by
defining a new functional:
2.23
is a scalar and is a square ( ) matrix. The minimiser of is no
longer a
simple least-squares best fit, but instead a solution fitted to the
data and some additional
condition that penalises large . This step can be regarded as a
form of regularisation
using prior information, which shall be encoded into .
By expanding 2.23 and setting its gradient to zero (as before) it
is possible to find the
equivalent minimiser:
( ) 2.24
An appropriate choice of can ensure that the augmented matrix is
full-
rank and permits direct matrix inversion. This method has been
independently proposed
in the field of statistical optimisation wherein it is called
“ridge regression” [108-110].
is sometimes absorbed into , but its value is important in
controlling the relative
adherence of the solution to fitting the data (small ) and
minimising (large ).
can take many forms depending on the problem; for example if
solution with a
small norm is considered favourable then , the identity matrix,
will serve to
penalise solutions with large norms. In ridge regression, and the
singular
matrix is augmented by adding a “ridge” of ones to the diagonal
components. In
this case, the least-squares solution with minimum norm is the
solution.
The prior knowledge does not involve any information about the norm
of the
solution, so ridge regression is not appropriate. Indeed, least
squares solutions of limited
data tomography problems with minimum norms do not give good
reconstructions
because the pixels which are not intersected by beams are
automatically set to zero
[111, 112]. Instead, the prior knowledge that the reconstructed
image must be smooth to
some degree is used, because it is based on physical quantities
which are assumed to be
continuous over space: the gas temperature is governed by the flow
of heat in the fluid
47
according to the Heat equation, individual molecular species
concentrations are
governed by the convection-diffusion equation, and the pressure is
governed by the
Navier-Stokes equation. In each of these three equations, the
specific gas parameters
evolve in a way that tends to lessens sharp spatial gradients, and
it is assumed that there
are no additional point-wise source terms that will counter the
smoothing effect of these
physical laws (e.g. a point-wise source of heat or water vapour
from a localised
chemical reaction).
{
⁄
2.25
Where equals the number of active pixels sharing an edge with pixel
. This is a
discrete version of the Laplace operator, and has been used in
previous reconstructions
of physical variables [95, 111, 113] because is a scalar measure of
the non-
smoothness of . If is relatively smooth, then its value at some
pixel will be almost
equal to its value at all its the neighbouring pixels (say, pixels
and ). The
corresponding row of is then equal to the sum ⁄ ⁄ , which is close
to
zero if . It is in this way that the penalty factor serves to
favour
smooth images.
A benefit of Tikhonov regularisation lies in the ease with which
priors can be
directly implemented via , and controlled via . Provided the
problem scale is not too
large ( was found to be a good rule of