MEASUREMENT OF PARTICLE DENSITY AND VELOCITY
FROM AUTOCORRELATIONS IN AN OPTICAL CHANNEL
by
James Andrew Nolan
B.S., Illinois Institute of Technology, 1979
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fullfillment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
1993
This thesis for the Master of Science
degree by
James Andrew Nolan
has been approved for the
School of
Engineering
by
Arun K. Majumdar
Joseph Thomas
Marvin Anderson
L 2, S 9 2_
Date
This thesis for the Master of Science
degree by
James Andrew Nolan
has been approved for the
School of
Engineering
by
Arun K. Majumdar
Joseph Thomas
Marvin Anderson
Date
Nolan, James Andrew (M.S., Electrical Engineering)
Measurement of Particle Density and Velocity from
Autocorrelations in an Optical Channel
Thesis directed by Professor Arun K. Majumdar
An optical channel that has particles traversing the
field of view is modeled. The particles are assumed to
absorb an amount of the transmitted light proportional to
their crosssectional area. The particles traversing the
field of view are modeled as a periodic pulse trains at the
photodetector output. The pulse width is determined by the
field of view crosssection and particle velocity, the
pulse interval by the particle spacing and velocity.
The particle density is simulated by the number of
pulse trains summed at the photodetector output. The time
autocorrelation function is taken at the photodetector
output for several particle densities. The effects of the
particle density on the autocorrelation function are
studied and compared to experimental data.
The form and content of this abstract are approved.
I recommend its
Signed
run
ajumaar
CONTENTS
FIGURES ......................................... V
CHAPTER
1. INTRODUCTION .................................. 1
2. MODEL ......................................... 3
Description .................................. 3
Mathematics of the Model ..................... 4
Results of the Model.......................... 7
Discussion of the Results .................... 7
3. EXPERIMENTAL TESTING ......................... 11
Description.................................. 11
Results...................................... 12
Discussion of the Experiment................. 13
Suggestions for Improvements .......... 14
4. SUMMARY AND CONCLUSIONS....................... 16
APPENDIX
A. FIGURES....................................... 18
BIBLIOGRAPHY
28
FIGURES
Figure
2.1. Timesignal at density = 6.......................19
2.2. Timesignal at density = 10......................20
2.3. Timesignal at density = 16......................21
2.4. Tauzero for model at 3.0 seconds...............22
2.5. Tauzero for model at 7.0 seconds...............23
3.1. Block diagram of measurement system..............24
3.2. Photograph of measurement system .............. 25
3.3. Schematic of amplifier circuitry . ..............26
3.4. Tauzero for experiment..........................27
v
CHAPTER 1
INTRODUCTION
The effects of particle density on temporal
autocorrelation functions in an optical channel are
studied. A basic optical channel system consists of a
light source or transmitter, a path over which the light
propogates, and a photodetector or receiver. The optical
path, or channel, can alter the intensity, phase, freguency
or polarization of the transmitted light. These
alterations can distort information transmitted through the
channel, or if the transmitted signal is known, the
distortions could reveal properties of the channel.
For measurement applications such as determining
particulate density in exhaust plumes, water droplet or ice
crystal content in clouds, rainfall or snowfall rate and
particle content in a flowing fluid, the density of
;l
particles passing through a given volume is desired to be
known. One technique used to measure particle density is
to compare the distorted optical channel intensity to an
undistorted one. This technique of measurement has several
disadvantages. First an undistorted optical channel may be
difficult to obtain. If an undistorted optical channel is
available, then an optical switching network or two
photodetectors are required. The optical switching network
requires additional optical components that reduce
reliability, accuracy, with increased cost and complexity.
Two photodetectors may not have the same response or age
differently and this would require repeated calibration.
Hence, it would be desirable to develop a system that was
simpler and impervious to long term detector gain
variations.
A simpler system would use the information contained
in the time varying waveform of the distorted channel. The
absolute intensity of the waveform would not be necessary,
only the scintillations. A requirement on this type of
system is that the transmitted waveform be known. One way
this can be accomplished is by transmitting a signal of
constant intensity. The spatial correlation function of
the scintillations are taken to give the particle velocity
perpindicular to the propagation path. The temporal
autocorrelation function is taken to give the spectral
content of the signal. The particle density can be gotten
through a relationship of the velocity and the first zero
crossing, r0, of the temporal autocorrelation function.
2
CHAPTER 2
MODEL
Description
The optical channel is modeled as a clear, non
compressible fluid that is moving at right angles to the
path of optical propogation. The fluid is assumed not to
distort the transmitted light signal in any way. Suspended
in the moving fluid are particles. As each particle passes
through the receiver field of view, it collects the light
proportional to its crosssectional area illuminated by
multiple scattering and reflections. The particle is
assumed not to change its crosssectional area during its
translation through the field of view.
If the particle density is sufficient, they could be
modeled as several strings of particles. Each string would
have its own uniform particle spacing, velocity, and
particle size. This simplifying assumption allows for the
modeling of the photodetector output as a summation of
periodic rectangular pulse trains.
Each pulse train represents a string of particles.
The pulse width is the string velocity divided by the field
of view width, the pulse repetition interval is the string
velocity divided by the particle spacing, and the pulse
amplitude is proportional to the particle crosssectional
area divided by the field of view area.
A time signal at the photodetector output is generated
from the summation of the pulse trains. The time signal is
analyzed through its autocorrelation function for different
particle densities and particle velocities. The particle
density is determined by the number of pulse trains used in
the summation. All the particles are given the same
velocity for each time signal.
Mathematics of the Model
The generation and summations of the pulse trains were
performed using the mathematics modeling program MATHCAD.
The input variables are:
the number of pulse trains to be summed
the start time and time increment for which the pulse
trains are summed, timstart and timinc
the velocity of each train, vel
the particle spacing in the train, space
the particle diameter for each train, dia
the detector field of view side dimension, fieldvw
Note that certain variables are denoted by vectors
(bar), for there is a specific value for each pulse train.
4
The pulse width is given by:
width
fieldvw
vel
(2.1)
The pulse repetition interval is given by:
pri
space
vel
(2.2)
The amount of light collected by each particle is
proportional to:
fieldvw2
(2.3)
To determine if a given pulse train had a particle in the
field of view of the receiver, the following procedure was
used:
outputm n = if[mod[n* timinc+ timstart,prin^
(2.4)
5
The pulse train is a known function of time. The
measurement time, n timinc + timstart, is divided by the
period, priB, of the pulse train. If the remainder of the
division is less than the pulse width, then the particle is
in the field of view.
The photodetector output time function is then:
timeseriesn = Â£ outputB#a (2.5)
m
Where n is the time index, and m is the pulse train index.
The autocorrelation function of the time function is taken
as:
P(t) = <(Â£( fc + T ) < f> ) ( f ( t) ) ) (2.6)
Where the angled brackets (<,>) denote the ensemble average
or mean of the function and r is a time increment variable.
The autocorrelation function is a function of r and is
normalized to its value when r = 0.
Results of the Model
The computer model was run for six different densities
6
at two different time periods. The densities were 6, 8,
10, 12, 14, and 16 pulse waveform summations. 256 time
points were calculated in .001 second steps starting from
both 3.0 and 7.0 seconds. Two different starting times
were chosen to test if the model is time dependent. The
chosen field of view was a square of .004 meters on one
side. This closely approximates the field of view of the
experimental setup. The particle diameters were chosen to
be 50 x 106 meters, again approximately the size in the
experiment. The velocities for the particles were 0.5, 1,
1.5, 2.0, 2.5, and 3.0 meters/second.
The typical time waveforms for three different
velocities and densities are shown in the Figures 2.1, 2.2,
and 2.3. The first zero of the correlation function (r0)
of all the test cases are are plotted in the Figures 2.4
and 2.5.
Discussion of the Results
Note that the tq's for both time periods are close to
each other in value. This is an indication that the model
follows a random stationary process. A random stationary
process is the one that has its probability distribution
functions invariant under time translations [1].
r0 is a measure of the rate of change of the time
7
waveform. Small t0's indicate a rapidly changing waveform
and large rc/s indicate a more slowly changing waveform.
As the velocity of the particles increased the r0 decreased,
corresponding to a faster waveform. From the Figures 2.4
and 2.5 it can be seen that the rD curve had a steeper slope
at the lower velocities and began to flatten out at higher
velocities. This would indicate a saturation in rQ for
higher velocities.
For the r0 data taken at time = 3.0 seconds there is
a rise in r0 at velocity = 2.0 m/sec. This rise was also
observed in the model at time = 7.0 seconds. This anamoly
can be attributed to the nature of the model. The time
waveform is modeled as a sum of periodic pulse trains. For
each of these trains there exists Fourier Transform and for
the composite waveform there exists a Fourier Transform.
Hence the model is not truly stochastic. The rise in r0 at
2.0 m/sec is a small bit of periodicity remaining in the
composite waveform from the periodicity of the constituent
waveforms.
It can also be surmised that the entire composite time
waveform is itself periodic at the least common multiple of
all the periods of the constituent waveforms. As the
number of waveforms summed are increased, the period of the
composite waveform is also increased.
As the density is increased for any given velocity,
8
the t0 become less for any given velocity. This would
indicate a saturation in r0 with increasing density. It
seems counterintuitive that as the density increased, tq
would increase. To explain this phenomena one should
examine the mechanism of the process. For a single
particle traversing the field rQ would be equal to the pulse
width of the waveform. For two consecutive particles t0
would be greater, equal to two pulse widths. For two
almost simultaneous particles r0 would be greater than one
pulse width, but less than two. As the number of particles
in the field of view increases, then it could be argued,
that t0 would increase. However as the density increases,
T0 cannot increase unbounded. As r0 * o this would indicate
a direct current waveform, or almost no scintillations.
This is similar to the Rytov solution for the
intensity fluctuations of a plane wave in weak turbulence
and observed saturation effects. The Rytov solution is:
o2lnJ = 1.23 CD2 k gL 6 1 '
Where a2Un r) is the variance of the natural log of the
intensity fluctuations, C2n is the changes in refractive
index, k is a constant, L is the path length. The density
in the model could be viewed as changes in L or C2n in the
Rytov equation. The Rytov equation states that the
9
variance of the intensity fluctuations increases without
bound on Cn or L. From experimental and theoretical
investigations [2] saturation effects on a2(lnI) have been
found.
The computer model shows that as the velocity of the
particles and the density increases, the r0 decreases. The
model also demonstrates the saturation effect on t0 as
either velocity or density increases. Also the computer
model indicates that within a certain range of densities
and velocities, that if the particle velocity is known,
relative particle density can be determined.
10
CHAPTER 3
EXPERIMENTAL TESTING
Description
An experimental apparatus was built to test the model.
A block diagram of the functions of the apparatus is shown
in the Figure 3.1. A photograph of the experimental setup
is reproduced in the Figure 3.2. The apparatus had to
provide an optical channel in which particles could be
suspended. The density and velocity of the particles were
controlled over the measurement period.
A clear plastic tube was used for the optical channel.
The optical path was at right angles to the tube. Clear
water was pumped through the tube by a small voltage
controlled pump. The flowing water had negligible
distortion on the optical signal. A cylindrical plastic
collar was slipped over the clear plastic tubing. On the
collar were the photodetectors and their apertures. Light
entered the aperture, passed through the flowing water and
then arrived at the photodetector.
Finely ground pepper was used for the suspended
particles. Pepper was used because it did not dissolve in
water, and it was black which made it absorptive of all
light frequencies. Two disadvantages of pepper is that its
size varies and the crosssection changes with rotation.
The photodetector output was fed to an amplifier, then
to an A/D converter and an oscilloscope. The amplifier had
adjustable gain so the output could accomodate an A/D
converter. A schematic of the amplifier is shown in the
Figure 3.3.
The density of the particles was controlled by the
amount of pepper mixed in the water. First a small amount
of pepper was used, then increased. The pump was run at
four voltage settings for four different speeds. The
velocity was measured by timing a group of particles over
a given path length. Care was taken to avoid air bubbles
in the optical channel. The light source used was
household flashlight. Alignment of the light source and
photodetector were critical because of the lens effect of
the clear plastic tubing and the photodetector lens.
Results
Experimental data was taken for three different
densities at four different velocities. The results for r0
are graphed in the Figure 3.4. The velocity could be
determined by timing the fluid flow over a known distance.
The particle density could not be determined, so it is
taken on gualitative terms high density, medium density
12
and low density.
Discussion of the Experiment
The results of the experiment show that r0 decreases
for increasing particle density, which is contrary to the
results of the mathematical model. It is in agreement with
the model in that the r0 decreases with velocity. The model
assumed that the particles all translated with uniform
velocity and spacing. This was not observed in the
experiment. The dispersion of the particle velocity caused
the experiment to disagree with the model. Particles close
to the walls of the tube moved more slowly than the ones in
the center. As the velocity of the fluid increased, it
could be observed that the velocity distribution became
greater.
As the density of the particles increased it could be
observed that there was a higher proportion of particles
clinging to the tube walls. This gave an increasing
velocity distribution with density. As the density and
velocity increased it appeared that a smaller portion of
particles travelled down the center of the tube at high
velocity.
13
This problem could be remedied by using a larger
diameter tube. The ratio of the crosssectional area to
perimeter would be greater, thus reducing the drag effects
of the walls.
Another problem encountered was in controlling the
density of the particles. The particles dispersed unevenly
in the reservoir and the density of the fluid passiing
through the detector varied with time.
Gravitational effects on the particles were apparent
at low velocities. A portion of the particles tended to
drag along the bottom of the tube at low velocities,
further effecting density.
The contribution of a noncoherent light source in the
experiment was not as large a factor in the outcome error
as was the particle flow. The particles are much larger
than a light wavelength so the absorption is not light
wavelength dependent. Since the particles are black in
color, they absorb all wavelengths of visible light. This
results in a temperature rise in the particle, rather than
a reradiation, therefore scattering, coherency and
polarization have less of an effect on received intensity.
Suggestions for Improvements
The problems with the particle flow was that it was
not uniform in velocity and density. A larger diameter
14
tube would alleviate some of the problem. A more
controllable solution would be to disperse particles
between two clear plastic or glass sheets. The sheets
would hold the particles in place. In the case of long
rectangular sheets, the sheet could translate at a uniform
velocity. In the case of circular sheets, they could be
made to rotate.
Clamping the particles between plastic sheets would
eliminate the nonuniformities in velocity, and density.
Particles or dots could be etched or drawn on the sheets,
allowing a much more controllable particle size and
particle rotation would not be a problem.
15
CHAPTER 4
SUMMARY AND CONCLUSIONS
The flow of particles traversing an optical channel
are modeled. The model assumes a frozen translation of
uniform absorptive particles. Because of this assumption,
a summation of rectangular pulse trains is used to model
the photodetector output. The model correctly predicts
saturation effect of density and velocity as is observed in
atmospheric measurements. The t0 approached a limit as
velocity or density increased.
The computer model also showed that relative particle
densities could be determined if the velocity of the fluid
is known. The first zero crossing of the autocorrelation
increased for either velocity or density increases. Hence,
if a series of particles passed through an optical channel
and velocity or density was constant, the relative value of
nonconstant quantity could be determined through rQ.
The flow of particles is also simulated with a
laboratory apparatus. The apparatus suspended particles in
water flowing in a tube. This technique did not represent
the model due to the viscosity effects between boundary of
the tube walls and the fluid. The shear forces between the
fluid and the wall did not allow for a uniform translation
of the particles. At low fluid velocities gravitational
effects further increased the velocity distribution.
The suspension of particles in a moving fluid are
suitable for frozen translation models if viscosity and
gravitational effects can be minimized. This can be
accomplished by using smaller particles and larger diameter
tubing. For the range of particle sizes and velocities
used in the model, clear plastic or glass sheets with the
particles imprinted on them would serve as a better
simulation of frozen translation.
17
APPENDIX A
FIGURES
18
\
Figure 2.1. Timesignal at density = 6
19
Figure 2.2. Timesignal at density = 10
20
Figure 2.3. Timesignal at density = 16
21
Figure 2.4. Tauzero for model at 3.0 seconds
22
Figure 2.5. Tauzero for model at 7.0 seconds
23
w
p
iQ
0
I
U)
H
ro
H
O
0
w
01
p
P>
iQ
P>
O
H>
3
(D
P)
w
0
>1
0)
3
fl>
3
r+
in
co
r+
ELUID PUMP
*
OPTICAL LIGHT
CHANNEL SOURCE
A/D
CONVERTOR
COMPUTER
25
v+
Figure 33. Schematic of amplifier circuitry
26
velocity, m/sec
Figure 3.4. Tauzero for experiment
27
TauZero
velocity, m/sec
Figure 3.4. Tauzero for experiment
27
BIBLIOGRAPHY
[1] Tatarskii, V. I., The Effects of the Turbulent
Atmosphere on Wave Propagation, p. 4, Israel Program
for Scientific Translations Ltd., Jerusalem (1971).
[2] Ishimaru, Akira, Wave Propagation and Scattering in
Random Media. Volume 2, p. 445, Academic Press, San
Diego (1978).
28
