a method of particle sizing using crossed laser beams
TRANSCRIPT
A METHOD OF PARTICLE SIZING USING
CROSSED LASER BEAMS
A thesis submitted for the degree of
Doctor of Philosophy
of the University of London
by
Nyi-Seng Hong, M.Sc., DIC.
Department of Chemical Engineering
and Chemical Technology,
Imperial College,
Prince Consort Road,
London, S.W.7 Feb.1977
ABSTRACT
Particle sizing using Laser Doppler Anemometry is based on
bringing two laser beams to intersect. An oscillating signal
is produced as a result of a particle traversing this intersection.
The visibility, Vsca , which is defined as the ratio of signal
A.C. to D.C. components, is a function of particle size, shape
and refractive index.
Theoretical calculations have been carried out to investigate
the behaviour of Vsca versus particle size parameter for different
values of angle of intersection of the beams ', angle of viewing
of the scattered signal and the size of the detector apertUre.
It is shown that the results can be used to determine particle
size distribution directly.
In order that the scattered signals should carry reliable
information of the particle size, it is shown that they
should be generated from a test-volume in which the interference
fringe contrast is near to unity. The properties of this volume
are examined in detail, including the effects of geometrical
mis-alignment of the incident beams.
Experimental studies of the signal shape characteristics
using a thin quartz fibre have shown that it is possible to distinguish signals that are generated from the test-volume from
others.
Finally, an optical system has been designed and constructed
to determine size distributions of glass ballotini in the range
of 1--10 stunseeded in a gas stream or thin flame. Analysis
of the scattered signals based on Mie theory was shown to be
generally in good agreement with independent optical microscopic
measurements. Limitations and the possibility of extending the
method for determining particle refractive index are also outlined.
Acknowledgement
I would like to thank Dr. A.R.Jones for his introduc-
tion to this field of research, his consistent guidance
and encouragement at all time.
My thanks also go to Professor F.J.Weinberg for many
useful discussions.
Without the technical assistance of R.Herris, B. mucus,
E. Barnes, the staff of the electronic workshop, C. Smith
and K. Grose, this work would not have been possible.
My appreciation is also due to the Procurement Executive
Ministry of Defence for their support.
Last, but not the least,' express my gratitude to my
wife for her endurance and support in every form possible,
and also Mr. and Mrs. C.M.Chai and Mr. A.H.Poi for their
assistance in the course of typing.
ii
Index
Chapter 1 Optical methods for particle sizing---a review
1-1 Introduction
1-2 Light scattering techniques
(A) Techniques using the ratio of signals
scattered at two angles in the forward
diffraction lobe.
(B) Angular position of the maxima and minima
in the scattered light.
(C) Complete scattering polar diagram.
(D) Polarization measurement.
1-3 Photographic technique.
1-4 Holographic technique.
1-5 Conclusion.
Chapter 2 Fringe anemometric method for particle sizing.
2-1 Introduction.
2-2 Formulation of the problem using Mie theory.
2-3 Modes of collecting the scattered light.
2-4 The properties of the scattered signals in the
z-y plane.
2-5 Effects on visibility of more ,than one particles
in the test space.
2-6 Noise in the detector.
Chapter 3 Optical system and its requirements.
3-1 Introduction,
3-2 The lay-out of the optical system.
3-3 The importance of the test space.
3-4 Definition of the test volume.
Chapter 4 A study of some particle samples.
4-1 Introduction.
4-2 Sample physical requirements and the method of
seeding.
4-3 Particle injection system and burner.
4-4 Study of samples with and without a flame.
Chapter 5 The choice of experimental parameters.
5-1 Introduction.
5-2 Choice of parameters.
5-3 Working parameters.
5-4 Other possible sources of Doppler signal.
5-5 Effect on visibility of the position of the
particle in the test volume
5-6 Alignment.
5-7 Calibration of oscilloscope.
5-8 Measurement of angle of interference.
5-9 Out-put signal shape as a function of particles
trajectories through the test space.
5-10 Experimental procedure.
Chapter 6 Experimental results and discussion.
6-1 Introduction.
6-2 Visibility measurements using quartz fibres.
6-3 Experimental results for glass ballotini with
and without seeding into a methane air flame.
iV
6-4 Experimental studies with particles of irregular
shape.
6-5 Sources of experimental errors.
6-6 The extension of LDA particle sizing to large
scale turbulent flames.
6-7 Possible extension of the method to particle
refractive index measurements.
6-8 Comparisons and discussions.
6-9 Limitations of the method and suggestions.
Chapter 7 Conclusions.
References.
Appendix A Theoretical background to light scattering
Mie theory.
Appendix B A diffraction theory for LDA.
Appendix C Detailed mathematical calculation of visibility.
Appendix D Derivation of fringe contrastjc in the
test space.
Appendix E
Notations:
a Particle radius.
D Particle diameter (=2a).
dp The diameter of the pinhole of the photomultiplier.
ElE0,EitEs Electric fields. Eois its amplitude while Ei and
Es are respectively incident and scattered fields.
I, Iii ,I1 Light intensities. Ii, represents polarization
parallel to the scattering plane while 1j is
perpendicular to it.
Isca Light intensity scattered from the common intersection
of two crossed laser beams.
k Wave number ( 2ff/7).
kf Defined as 211Y)1. .
Magnification of an optical system.
Refractive index ( m = n1- in2 , where 1. ). P(x) Particle size distribution function.
Psca Power of the scattered light from two crossed beams.
q Angle between two polarizations.
Ra Two beams intensity ratio.
Rs The radius of the test volume of fringe contrast
greater than 0.95.
(r,0,0) The orientation of the scattered ray, where r is
the distance from the particle to the point of
observation.
S(0) Mie amplitude function.
The velocity of the particle.
Vc Fringe contrast in the test space.
vi
K----- ACE. —31
vii
Vsca
x
Y 9 YO
p
Visibility of the scattered signal.
Particle size parameter ( x = 21-Ta/X).
The distance of the particle from the origin of
the test space along Y-axis.
Wavelength of the incident beam.
Fringe spacing.
Angle of sizing or viewing. In the special case
where sizing is done in the plane of incident
beams 0= Tr/2 and o(= 0 ,f3=0°.
Polarization ratio.
Half-width of the laser beam, defined at 1de2 point.
The angle of intersection of both laser beams.
11 The solid angle of the light collecting optics.
dOl ,d02,ea,415Size of the collecting aperture. The following
diagrams make the distinction:
Chapter 1
Methods for particle sizing---- a review.
1-1 Introduction.
A knowledge of particle size distribution is required
for studies on emission of radiation by particle laden flames.
This work was initiated to aid studies in which metallic
particles are used as additives in solid rocket propellants.
Upon burning they vaporize and react with oxygen to form
molten oxides, which play a prominent role in modifying
flame properties. These oxides can exist in different physical
forms being spherical when molten and perhaps irregular when
solid. Their refractive index is expected to vary depending
on the amount of imperity present, their history and
temperature.
Our aim therefore is to devise a means of measuring
size distributions in the range 1 pm -- 10 jam and refractive
index. As the first step it was decided to develop a sizing
method which could be relatively independent of refractive
index, so that eventually alternativemethods could independently
provide this information. Optical methods were preferred
because they do not disturb the system.
The method chosen had to take account of the dilute nature
of such particle systems, the presence of motion, the need
2
for spatial resolution and had to simply deal with particles
greater than 1 fain in size but which were too small for photo-
graphy or holography. The two main contenders were the forward
diffraction method ( Hodkinson, 1966 )and the fringe anemometer
method (Farmer, 1972; Fristrom et. al. 1973 ). In view of the
dilution and the spatial resolution requirement, the latter
was chosen. This was developed into a technique which would
simply provide particle size distribution directly with only
slight dependence on refractive index, and this forms the main
contribution of this thesis.
.1-2 Light Scattering Techniques.
When a beam of electromagnetic radiation falls on a
small particle, it is deflected in all directions with vary-
ing intensities and states of polarization. This phenomenon
is known as scattering. It is most effective for particles
of the order of size as the wavelength. Through a knowledge
of the scattered radiation, information concerning size,
shape, concentration and refractive index of the particle
cloud can be deduced.
Light scattering techniques are attractive because they
are useful for systems which are not readily accessible as
in the fields of astronomy, atmospheric pollution and
particle laden flames where the enviroAn ment is hostile.
Moreover, the sample required is small and measurements can
be carried out rapidly with automated electronics.
3
The important measurable parameters of the scattered
field are:-
a) Intensity,
b) State of polarization,
and c) Extinction efficiency.
Details of these and their comparison are described
by Hodkinson ( See Davies 1966 ) or more recently by
Kerker ( 1969 ). Other functions, for example the ratio
of scattered intensities at two different angles, may also
be defined depending on the problems at hand. These quan-
tities are in general functions of the following variables:
a) Particle size paraMeter x=2TraA
b) Particle shape,
c) Its refractive index relative to the surrounding
medium m=n1-in2
A theory that is able to describe the scattered field
due to an arbitrary particle necessarily has to incorporate
all these variables. This will be mathematically very
complicated. A special case of scattering by spheres was
developed by Mie in 1908. Even though Mie theory is appli- size
cable for spheres of arbitraryAand refractive index, the
solutions in the form of infinite series usually are difficult
to interpret physically and are mathematically complex. Van
De Hulst ( 1957 ) has mapped the x-m domain into different.
4
boundary regions in some of which simple formular can be
written down. These scattering regions are given briefly
in table T-1-2a and T-1-2b. It should be noted that all
the scattering intensity functions are deducible from Mie
theory.
Since our main concern was with physical systems
involving certain size distributions, we need only review
those techniques which are applicable to particulate Clouds
with a size distribution. They are given as follows:-
(A) Techniques using the ratio of signals scattered
at two angles in the forward diffraction lobe.
This method was first suggested by Hodkinson R. ( 1966 ).
It required measuring R(x) defined by
I(91,x,m) R(x) (1-2-1)
I(e,x,m)
The angles Gi and G2 are chosen within the forward scat-
tering lobe, since within this region angular intensities
change rapidly with size while remaining fairly insensitive
to refractive index. For particles within 0.3430,
Mie theory must be used to calculate I. For particles
having x-...301 a good approximation can be obtained using
Fraunhofer diffraction theory with an obliquity correctiont
An instrument for this purpose had been developed by
* Large error will still be introduced when nt-.=1 even for
large x ( Jones, 1976 )
Gravatt ( 1974 ) to dete -mine the size distribution of par- ;rt
ticulate matter„ouspension through individual particle counting.
The advantages of the method are that sizing can be done
in real time and the quanti±y R(x) is relative so that
complications like source intensity variations and non-uniform
particle velocities do not arise. However several problems
are also obvious with this method ( Self, 1976 ). First
the ratio R(x) is not monotonic for particles greater than
a certain size and second the signals to be compared are
proportional to which requires a comparator of very large
dynamic range. The practical upper limit is set by the
difficulty of measuring intensities at angles close to the
forward direction as the lobe becomes narrower with increasing
size.
An elegant apparatus has been designed by Swithenbank
et.al. ( 1976 ) to determine the particle size distribution.
This method involves the measurement of scattered light in
the far field by a cloud of particles. From Fraunhofer dif-
fraction theory, the intensity distribution, I(E) x ) formed
by a ingle particle is
I(e,x)-- 2 J1(x Sine)
x Sine
2 + Cos20) (1-2-2)
2
suppose the cloud has a distribution of size given by P(x),
00 2 JI (x Sine) + Cos29)
I(0) /-N-, i P(x)
0 dx (1-2-3)
x Sine 2
It can be seen that the contributions to the intensity
are from individual particle of all sizes. The size is
then obtained from 1(0) using an integral transform.
The far field condition can be simulated in practice
by illuminating the particle cloud with a parallel beam
and placing a converging lens in the beam which focuses the
cloud at a screen.
This method has several advantages. First the inten-
sity pattern on the screen remains stationary independent of
particle velocity in the beam and since only forward light
beam is measured the method remains fairly insensitive to
refractive index. This makes the method useful for sizing
particles in sprays. Secondly, a wide range of particle size
can be covered by changing the focal length of the focussing
lens. Good agreement has been obtained for droplets in the
range of 5-500jim using visible wavelengths. The lower
limit of applicability is expected to be about 1,Lun for a
He-Ne laser source. Thirdly, it can be automated and coupled
directly to a computer for fast analysis. On the other
hand, it integrates along the beam and thus lacks spatial
resolution within the test space. Also it would probably
not be so convenient at very low particle densities.
Recently, Jones ( 2_976 ) has carried out theoretical
comparision using rigorous Mie theory and diffraction
theory, --Shat is Eq.( 1-2-2 ).
This is shown in Fig.(1-2-a) where size parameter ka is
plotted against the real
part of refractive
index n1 and the numbers
shown in the figure are
percentage errors. As
can be seen the error
is an oscillating 30
function of both x and
m. This means that exact
prediction of the error
in using diffraction
theory required the
know-ledge of these
parameters. However, as 15-
a general guide for
n1 >1.2 and ka>25, a 10
maximum error of less
than 20% is encountered
when diffraction theory
is used.
1,2 0 1.4 1,5 1,6 n1
1.7
Fig.(1-2-a)
16
0
1
2X(m-1)
4 31.
Table (T-1-2=a) Scattering domains in terms of x and m.
Table T-1-2-b No. Scattering Boundaries Scattered Intensities I Comments
Regions
6 Rayleigh scattering
x<<1
xim-11<1
In =(I0014Cos28/r2
It =(I0k4A2)/r2
0(=Polarizability of the
medium per unit volume.
I= (111 +Ii. )/2
1
Rayleigh- Gans Scattering
xlm-ll<K1
j m=1 I<1
Ia=(Iok4 V2 /r2 )m-1 2 ROJO
2
1 R(8,0)=-,-„-.1Exp(iS)dV r v V
where o is the relative
phase of the ray scatter-ed by dV.
(2n,
Cos20
, 1 2 IJAI0k4 /r2)(4"-TT' R(et0)
2
3
Geometrical
otics(or 1p Diffraction
theory)
x>> X
I.(G2I0/)
2r2) D(8,0)2
where D(8,0)=(1/0 Exp(-0-
kxCos0+ikySin0)Sin0 dx dy
G=Geometrical crossection.
Diffraction pattern is
independent of states of
polarization,m and -oath-cle surface texture;.
5 Optical resonance
x<<1 m-1>1
x (m-1)=arbi-trary.
Scattered intensity is very sensitive to variat-ion of x and mx.
2 Anomalous diffraction
x)1 (m-1)<<1
I(9).(k2a4 A(F,z)2Io/r2
AQ,z)411(1-Exp(-i?Sint)) 0
Jo(zOoseCostSintdt
r=2x(m-1) z=x8
f5X1 Rayleigh-Gans theory.
SW Frauhofer diff. theory.
4 Total
reflector m —pop 6 1 os8)2 In =x (1-V
II =x6(Cos0-1/2)2 m=n1-n2 m.---co n1-->, 00 or n2-400
10
(B) Angular positions of the maxima and minima in
the scattered light.
That knowledge of these extrema could yield particle
size was first shown by Sloan ( 1955 ). Using Fraunhofer
diffraction theory, the value 9max at which the maximum
occured in the plot of i(e)e2 versus A.was inversely
proportional to radius, a, of the particle and can be cal-
culated from the equation
a Amax = 9.2 pm-degree
(1-2-4)
This rule was reinvestigated and shown to be correct for
particle with x-2 58 by Meehan et.al. ( 1973 ) using Mie
theory. For smaller particles the value of refractive index
affects the location of Gmax to some extent.
A more general form of this method is to determine
ith
minimum or maximum from the calculation of the per-
pendicular scattered intensity I./ . The size could be
calculated from
ki for minima (2Tra/X) Sin(ei/2)
(1-2-5) Ki for maxima
where ki and K1 were constants that could be determined
from Mie tabulation of This method in its simplest
11
form was suggested by Dandliker ( 1950 ) and extended by
later workers ( Nakagiki et.al. 1960; Maron et.al. 1963 ).
The range of applicability of the above equation was pre-
sented by Kerker et.al. (1964) in term of m-x domain.
The positions of the extrema were functions of refractive
index and size of the particle, therefore a main difficulty
of this technique is how to determine these extrema quickly
and accurately. One of the simplest methods was suggested
recently by Patitsas ( 1973 ). A precise determination
of the sensitivity of his method to change in m is very
difficult. It varies from 2% to 20% for 443[424 and
1.14 mz-2.10. Errors tend to be larger in the first and
second minima( or maxima ) and smaller for other extrema
in the size and refractive index ranges discussed here.
(C) Complete scattering polar diagram
A direct but tedious method is to measure a 360
degrees scattering diagram and compare it with theoretical
plots for assumed m and x in order to find the best fit.
Instruments designed for this purpose are described, for
example, by Gucker et.al. ( 1973 ) and Carabine et.al.
(1973 )® Size distribution can be determined using the
method in two ways. The first is counting a large number
of individual particles so that a size distribution can
be built up. Marshall et.al. (1974; 1975 ) employed
Gucker's instrument to determine aerosol size distributions
in this way. The method was obviously tedious since each
In Carabine's case particle size distribution has to be assumed..
12
polar diagram had to be fitted for m and x. The advantage
is that no size distribution function needS to be assumed.
The second way is via a single measurement of polar
diagram generated from a cloud of particles. A procedure
is to assume a size distribution function P(x) for the
cloud and further suppose that I(9,m,x) is the scattered
function at angleG for a fixed refractive index and size
parameter. The over all measured intensity II(9,m,x )
can be expressed as
It(8,m,x) = JP(x) I(e,m,x) dx (1-2-6)
This equation is, general and is applicable to all particu-
late clouds provided that
a) No multiple scattering occurp.
b) The particle number is large and they are randomly
positioned,
and c) No interference between particles. ( This
requires particle separation to be at least 3
radii,see eg. Van De Hulst, 1957 ).
Theoretically, P(x) could be deterwined from equation
( 1-2-6 ) by inversion, provided that I( 9,m,x ) could be
calculated from the appropriate theory. For example, when
one is dealing with systems of spherical particles, then
1(9 ,m,x ) can be calculated from Mie theory.
13
In practical calculations, P(x) is usually assumed,
and then integration proceedsto find the best fit to the
experimental data. If P(x) is a Zero-order-log-normal
distribution, then it is described by only two parameters---
the modal value of the size parameter ( m, and the standard
deviation 6.. Otherwise higher moments are required to
describe P(x) fully. With this size distribution, the
problem now is of integration for different assumed values
of 4and 6-0 .
By using this method, an instrument has been constru-
cted capable of measuring aerosols with varying particle
size distribution ( Carabine, et.al. 1973 ).
Another parameter P(G,m,x) proposed by Kerker ( 1964 ) 1Rie
defined as (PisApolarization ratio for single incident beam)
.(G,m,x) dx P(e) =
P(x)I1 (O,m,x) dx (1-2-7)
has the advantage that only relative readings are required
rather than absolute intensities.
The limitation to these methods involving size distr-
ibution is that when the size distribution broadens,
information in the signals is washed out and makes inversion
of the light scattering data increasingly difficult. For
example, inverting light scattering data for size distri-
butions becomes multi-valued when Cio>0.20 ( Carabine, 1973 ).
Another technique of inverting light scattering data
has been proposed by Waterston ( 1976 ). The idea is the
use of a finite summation instead of integration. We
1.4
recast equation ( 1-2-6 ) as
(1-2-8) 1=1
where Ni is the number of particles of size xi . The
best values of Ni are determined from a least-square fit
of the form*
aNA I'(G. mi,x) - Ni I(Gpm,x)i l2
' 0 (1-2-9)
1=1
j = 1,2,...,n, where n is the
number of measuring angles.
-(D) Polarization measurement
When an unpolarized beam is incident on a particle,
the scattered light will in general be partially polarized
and can be resolved into a vertical component V and
horizontal component H with respect to the plane of scatt-
ering. The polarization ratio Rp defined as
= H/V (1-2-10)
at any point is a function of the relative refractive index
of the particle, angle of observation, and particle size.
Therefore a measurement of R with known refractive index
at a particular angle will give the size.
Eiden ( 1971 ) proposed that refractive index and
size distribution of the particle can be obtained from
measuring the states of the scattered light, i.e. degree of
*For more information on light scattering inverting technique see J. C. Vardan, 1973.
15
polarization, the ellipticity and the orientation of the
ellipse. So far its feasibility has not been seriously
assessed experimentally.
1-3 PhotoRraphic Technique.
Sizing.of particle clouds can be performed simply by
conventional photography provided certain conditions hold.
The essentials of the method are imaging onto a photographic
plate, processing and then measuring the size of individual
particles on projection. In order to obtain clear images
of moving particles ( eg. in liquid sprays ), a very short
exposure time is needed to 'freeze' their motion. This
requires a very bright source of shot duration, for instance,
pulsed laser and spark sources. An optical system for this
purpose using spark sources has been described by Beer and
Chigier ( 1972 ). Other parameters that govern the photo-
graphic process are focal length f, magnification M, and
resolution. Magnification is inversely proportional to
focal length, while resolution is given by WM), where AG
is the aperture size. For sizing small particles one needs
an optical system with short focal length as well as large
aperture. This makes the system havit” short depth of field.
Another disadvantage arethe lens abberations which affect
directly the quality of the images. In summary, the use-
fullness of the method in practice depends on
1.6
(a) the size of the particle,
(b) the speed it is travelling,
(c) the intensity and duration of the source,
and (d) the sensitivity of the film.
At present, the lower limit is for particles around
10pm in size.
Other useful versions of photography are shadowgraphy
and schlieren photography ( Weinberg, 1962 ).
1-4 Holographic Technique.
This methOd involves the use of two incident beams.
One beam carrying information about the particles is mixed
with another called the reference beam on the surface of
a photographic emulsion. The reference beam is necessary
for preserving phase information which would otherwise be
lost due to the square-law behaviour of the emulsion. Since
the final form of the image is recorded as interference
fringes on the plate, this necessitates that both beams be
highly coherent. After exposure, the plate is processed
and images can be reconstructed by illumination with the
reference beam. This method has been used by Bexon ( 1973 )
and others for particle sizes in the 4,um range. He also
attempted to achieve magnification by changing the wavelength
between exposure and reconstruction. In order to obtain
1.7
real linear magnification, not only should wavelength be
changed but the wavefront shape as well ( De Velis et.al.
1967 ). Another advantage of wavelength change, however,
is that resolution is governed only by the exposure wave-
length. Thus by taking the hologram, say, in the ultra-
violet and reconstructing in the visible, one has the
advantage of being able to see the image with the resolution
of ultra-violet wavelength. Apart from these, other advan-
tages and disadvantages of the method may be summarized
( For review article, see Thompson, 1974 ) as follows:-
Advantages: (a) The experimental set-up and optical
components are extremely simple.
(b) It allows us to see the particles on
reconstruction. Therefore, shape of the
particles can be recognized and measure-
ment is straight forward.( eg. by proje-
cting the image onto a TV screen, or by
optical microscope ).
One disadvantage is when applied to particles moving
at high velocity, the images suffer from loss of phase
information. Also, for irregular particles spinning at
high speed, no exact shape information will be obtained.
In both cases, the time of exposure must be made negligibly
small.
18
1-6 Conclusion.
The various techniques surveyed above have been applied
in practice for sizing particles with various degrees of
success. It is not possible at present to agree on which
one is the best because all of them are still developing
rapidly. Moreover, direct comparision of accuracy among
them is difficult, since there is no proven technique to
use as a basis.
The table given below summarizes roughly the size range
of applicability. They are classified into A and B whose
differences are given below:-
Table ( )
Class Techniques Size range (NF. 0.5 1-11r1 )
A
Light scattering
Using diffraction
theory
.42,am
2-10 ,um*
B Photographic method
Holographic method
--..10,um
4.. 2 ,um
(a) Class B is basically an imaging process so that
size and shape can be directly recognized and measured through
reconstructed images.
(b) In Class A , an appropriate theory is required to
calculate scattering function 1(9). If it is dependent on
refractive index, size can be deduced if m is known and vice
versa. Otherwise matching process is needed. This will be
convenient if I(9) is a monotonic function of both m and size.
In practice, I(9) is measured under two different conditions.
One is measurement of T(9)
19
from a large number of particles simultaneously and the
second one is to count and size individual particle one at
a time. In general, the method is mathematically complex.
Further improvements of techniques in class B will
depend upon the development of brighter and shorter
duration sources ( of the order of nano-seconds).
The main difficulty on class A is to devise an appro-
priate scattering function I(G). We know that its general
form would be a function of particle size, refractive index,
particle shape and anisotropy. Numerous' theories and
approximations are being proposed with the aim of incorp-
orating as many of these variables as possible. Some of
them are:-
(a) The point matching method,
( eg. Bates,et.al. 1973; Wilton, et.al., 1972 )
(b) The perturbation technique ( Yeh, 1964 )
(c) Integral equation formulation (eg. Barber and
Yeh, 1975 )
Another approach proposed by Chylek et.al. ( 1976 )
is to adapt Mie solutions with modifications to accept
irregular shaped particles:
* From pg. 18, The diffraction theory in itself does not
have upper size limit, rather the limit is on ability to measure forward scattered light close to the incident beam.
20
Chapter 2
Fringe anemometric method for particle sizing.
2-1 Introduction.
In the previous chapter different techniques for sizing
particulate clouds were reviewed. Some of them are already
quite sophisticated and have been applied very successfully
while others are still developing. An ideal method should be
accurate, fast, economic and have a wide dynamic range.
In this chapter, another feasible method called Laser
Doppler Anemometry LDA ) is presented. Theories based on
diffraction and that of Mie are also described.
Measuring particle size using LDA as described by
Farmer ( 1972 ), Fristrom et.al. ( 1973 ) and Eliasson et.alc
( 1973 ) is a direct extension of velocity measurement used
for local flow measurement ( Yeh and Cumming 1964; Durst
and Whitelaw, 1971 ). The method involves the observation
of the A.C. signal generated by a particle moving across a
region where two laser beams cross. We call this region
the test-space or control volume,
A fundamental notion behind the operation of LDA is that
of optical mixing ( Forrester 1961 ); that is mixing of two
coherent beams of slightly different frequencies so that the
21
resultant beat frequency is observed. The explanation
based on this point of view is called the Doppler shift
model. An alternative proposed by Rudd ( 1969 ) is the
fringe model, which describes the generation of the fre-
quency in terms of the variation in the light scattered by
a moving particle crossing the interference fringes. These
two models have been shown to be mathematically equivalent
and give the same beat frequency when c>> u, ( Fristrom
et.al. 1973 ;Lading 1971 ). Here u is the velocity of the
particle while cis the speed of light.
For particle sizing, measurements can be made either
by recording the doppler frequency, the signal envelope or
the signal visibility ( also called the modulation depth ).
The basic idea behind frequency measurement( Lading,
1971 ) is that the retarding force experienced by particles
in a fluid is a function of particle size. This method is k
best applied in flows involving droplets or solid particles
in a fluid where an accurate relationship between lag velocity
and particle size exists. Yanta ( 1974 ) has applied this
concept for measuring aerosol size distribution. A similar
method has also been applied by Ben-Yosef et.al. ( 1975 )
to determine the size distribution of rising bubbles. A
difficulty of the method is to find the correct flow regime
so that an appropriate form of drag coefficient can be chosen.
22
2-2 Formulation of the problem using Mie theory.
The diffraction theory as used, for example, by Robinson
and Chu ( 1975 ) is outlined in Appendix B. The theory
can take into account the dependence of visibility on the
particle size and the aperture of the detecting optics, whereas
it is inherently independent of refractive index. A compari-
sion of this method with Mie theory has been made by Hong
and Jones ( 1976 ).
Other workers ( Eliasson et.al. 1973; Jones, 1974 ) have
applied the Mie theory to relate the observed signal
particle size. The, main procedure is described below:
Figure ( 2-2-a ) shows the actual scattering situation
where we need to distinguish three coordinates systems
( X1, Y1 , Z1 ), ( X2, Y2, Z2 ) and ( X, Y, Z There are two
incident plane waves El and E2 both polarized to the
X-axis* with respective propagation vectors k1 and r2
( 1k1 I = 1r21 = 211r/ 2\ ). Vectors and and V.2 are along
the ZI axis of the ( X1 , Y1, Z1 )- system and Z2 axis
of ( X2, Y2, Z2 ) - system respectively. Both beams are in
the Y-Z-plane and cross at 0 , the origin of the ( X,Y,Z
* Mie theory is valid for any polarization. Perpendicular
polarization is chosen here because it results in simpler
formulae and is adequate practically for particle sizing.
+ Mie theory is used by Farmer (1972) as well. Recently another approach has been proposed by Chou(1976).
23
Fig.(2-2-a) The scattering coordinates systems.
24
system, making angles ± r symmetrically about Z-axis.
The region around 0 is called the test-space. Mathematically,
the systems ( X1 , Y1, Z1 ) and ( X2, Y2, Z2 ) arranged
in this way can each be regarded as pure rotation of angles
+ W and - about X-axis respectively.
A particle at the point A = ( Xn , yn Zn )
travels at a velocity j in the test-space. It is the
scattered light signals from this particle ( or particles )
that are detected.
Assuming the validity of linear superposition theory,
we can break down the scattering process by considering each
beam seperately in terms of scattered amplitudes, adding and
squaring* over the detector surface. This procedure is
described below.
Let Ul and U2 be the optical disturbances at the
point ( Xn Yn , Zn ) produced by incident beams 1 and
2 respectively. At any instant, the particle can be con-
sidered at rest. By applying Mie theory, the scattered
fields ( El° , Elo ) and ( E28 E20 ) generated
from beams 1 and 2 respectively, at the point P(rn, en,
Squaring procedure is valid for Square-law detectors.
Let E be any complex quantity, squaring then means taking
the real part of BE , where E is its complex conjugate.
25
and a distance 1: from the scattering center can be
written as:-
Ele = (-1/ikrn) Exp(-ikrn) Cos n S2(e1n) 1/11. (2-2-1)
410 = ( 1/ikrn) Exp(-ikrn) SinOin Si (e1n)
E20 = (-1/ikr )Exp(-ikrn) Cos02n S (02n) U2
E2 - ( l/ikrn) Exp(-ikr ) Sin02n S1(92n) U2 (2-2-4) 0 -
Where S1 and S2 are the respective scattering amplitude
functions ( Eg. Van de Hulst, 1957 ) for perpendicular and
parallel polarization with respect to the plane of
scattering and ( 0 - int 01n ) and ( 92n P02n ) are the
scattering angles with respect to the incident beam coordi-
nate systems. These angles are defined as shown in
Fig.( 2-2-b ).
The remaining steps are to supply the detailed form
of 111 and U2 and carry out the coordinate transformations.
Eliasson.et.al. ( 1973 ) have presented a detailed mathema-
26
tical analysis for IDA system where both incident beams are
focused by the same converging lens, and at the same time
have taken into account the Gaussian nature of the incident
beams.
If more than one particle is present simultaneously
at random in the test space, equations ( 2-2-1 )-- ( 2-2-4 )
can be summed to obtain the total scattered field. Here,
however, we follow the simpler approach given by Jones ( 1974 )
for the case of..only one scattering-centre in the test space
at a time. The single particle case will avoid much physical
and mathematical complexities..Moreover it is easily satisfied
experimentally by using a dilute particle cloud. For
simplicity, it is further assumed that the particle is cons-
trained to travel along Y-axis* and that the two incident
beams are of equal intensity and are infinite plane waves.
In practice when a non-focussed Gaussian profile laser beam
is used, the latter assumption necessitates that the ampli-
tude due to the profile should not vary significantly over
distances of the order of the particle size ( otherwise
* This restriction is un-necessary. It. is introduced to
simplify discussion. A particle with velocity ii. has components
ux , uy and uz . However, only that component across the
fringe ( ie. Y-direction ) generates a signal. Therefore one
measure u irrespective of u .
27
Scattered ray
• xn,Y,,Zn
cattering particle
Fig.(2-2-b) Notations used to specify the scattered
ray.
zo
0
Fig.(2-2-c) The phase relationship between the
particle and the origin 0.
28
particles would experience uneven illumination ). For
particles sizes less than 10,Alm , this assumption agrees
well with beam widths typical for unfocussed visible lasers.
e.g. for the Argon ion laser used in this work the beam width
was 0.65 mm ( 650.ium ).
With these assumptions,we suppose at some instant the
particle in the test space is at a distance Yo ( ie.
Pn = ( 0, Yo ,0 ) ) from the origin. This introduces phase
differences ± kY0 Sin ?5- with respect to the origin 0 .
Thus the incident waves can be written as ( Fig. 2-2-c )
El = E0 Exp(ikYoSin/) Exp(-ikZ1) (2-2-5)
E2 = E0 Exp(-ikYoSini) Exp(-11a2)
(2-2-6).
Therefore, we can substitude 14 and Ul in this case
by E0 exp (, ikY0 sin I' ) and Eo exp( -in.° sin X )
respectively into equations ( 2-2-1 ) to ( 2-2-4 ).
The resulting equations then are given by (omitting the sub- vs f n \ script n ) '10kr,°1,P1)
ES 2G LIn2'Wd
21
Eo Exp(-ikr) Exp(± ikYoSin)1 ikr
—00402(91 )
(2-2-7)
and s E10(1',01901)]
40(r,e2,02)
IExp(-ikr) Exp(ItikYoSinI)
cos02s2(92 )
S (01)
(2-2-8)
sin02s1 (02 )
29
Where 01 , 02, el and 02 are related to 0 and 0 by (from
Eq.(C-2a) and Eq.(C-2b) of appendix C)
• 011 Sine Sin0 Cost(±Cos0 Sind tan- (2-2-9)
Sine Cos0
02-
and
el] = Cos-1( ±Sine Sin0 Sin7S+ Cose Cos Y)
(;) (2-2-10)
Equations (2-2-7) and Eq.(2-2-8) are the required results.
In general, the total field E at the point P over the
detector surface is in fact the sum of scattered fields
and incident fields. Except along any one of the incident
beams, where scattered and incident waves have the same
propagation direction, the two incident waves El and E2
can practically be excluded from reaching a detector by
suitably designed optics. In this situation only scattered
waves are observed and the intensity 'sea at P is given by
Isca = 1 E0 r 1 E012
(2-2-11)
Ee and E0 are related to (E0 ,E0 and (Ee2 ,E02 )
through the matrix transformation:
30
Ee
E0
401 °601
(A01 d001
01
E01
c4002 0/002
°/002 E02
(2-2-12)
where 0( are matrix elements given by
h.Ou Q.1 uv hvav (2-2-13)
hu and h7 being the metric coefficients (Morse and
Feshbach,1953). The power received by a detector, at P,
sub-tending a solid angle SZ from the scattering centre
is
Psca = SIsca dlx = S SIsca r2 SinG de d$ (2-2-14)
60 a
Then the visibility Vsca is defineSfrom equation (2-2-14)
as (see appendix C )
sca)max (Psca)min
Vsca
(2-2-15)
(Psca)max + (Psca)min
31
Incident beams
in YZ -plane
Y A
Fig.(2-2-d) The diagram illustrating various angular
relationship. The scattering plane is
OABC.
32
Inspection of equation ( 2-2-14 ) reveals that Vsca is
in general a function of the following parameters:-
(a) The complex refractive index m = ni in2
(b) The size of the particles D ( = 2a ),
(c) Fringe spacing, ?f , of the test space,
(d) Angle of sizing ( 0 , 0 ),
(e) The size of the, collecting aperture AG
and b0.
Practically, it is convenient to measure the angles
d and /3 instead of 0 and 0 as shown in Fig.( 2-2-d ). They are related through ( Appendix E )
tan0 = Sind/ tan8
Cosa = Cos/3 Cosd.
2-3 Mode's of collecting the scattered light.
For collecting scattered light, we may distinguish
between the following two cases:-
Case I :
A lens L1 is used to collect only parallel light
.............■•••■••••■••■•••■•■•■■••11
From here onwards the angle of sizing or viewing and the
size of signal collecting aperture are denoted respectively
as (0? , /3 ) and (4104 „zip )
Aperture
ze
33
x Fig.(2-3-a) Scattering angle 0 independent of
particle positions 1 and 2.
x
A
•
0
Fig.(2-3-b) Scattering angle changes with particle
position in the test space.
34
from the scattering centre ( or centres ) and focuses it to
a point P which is the pinhole of the photomultiplier.
The situation is shown in Fig.( 2-3-a ). This mode approaches
closest to the theoretical conditions described previously.
The scattering angle A is then independent of the position
of the particles in the test-space. The fact that only
parallel light is intercepted implies the pin-hole of the
photomultiplier is infinitesimally small. Therefore the
signal is likely to be weak in this case.
Case II :
Fig.( 2-3-b ) shows the optical system where
lens Lz focus a fraction of test-space so that its image
is formed at P . A cone of scattered light from the par-
ticle in the test-space is collected by the lens. The signal
will then be controlled by the size of entrance aperture.
However angle G in this mode will be dependent on the posi-
tion of the particle in the test-space. The error due to
this effect will be explored in 5-5.
2-4 The proper
A special situation where most of the qualitative
behaviour of the scattered field can be deduced is to sot.
-Eo =
Exp(-ikr) Exp(ikY0 Sini) ikr
Let f = ?/(2 Sin T)
35
0 7r/2. This situation corresponds to scattering in the
plane of the two incident beams. From equations ( 2-2-9 )
and ( 2-2-10 ), we obtain
951 = 932 =7/2 01 = 0 +
92 = 9
Using these relations, equations ( 2-2-7
reduce to
-E E10 =iTEP-Exp(-ikr) Exp(-ikY Sini)
and 1o Eo
( 2-4-1 )
) and ( 2-2-8 )
S (G-W) (2-4-2)
Si(G+W) (2-4-3)
(2-4-4)
denoting the fringe spacing in the test space and the inci-
dent beam intensity respectively.
Since S1 is in general a complex quantity, it can be repre-
sented in the form
S 1 ( ) = 6 Exp(iA )
S1 ( 0 -I- ) = 0-2 Exp(iA2)
wherec,2are amplitudes and e6) 2 are their respective phases.
( 2-4-5 )
36
With these notations Isca is
sca = (E10 40)(44 E20)
Ic
k2r2 + 0-2 6- + 2 62 + A1 - A2 ).]
1 Xf
(2-4-6)
Equation (2-4-6) can be used to define visibility Vsca
Envelope Env , Pedestal voltage Pev , and D.C. current Idc
respectively in the following ways:-
Vsca(Isca)max (Isca)min (Isca)max (Isca)min
Env 7 (Crl 02)2
Pev - 0-2)2
- and Idc (Pev Env)/2 = 0712+ 62
2O C12+ °-22
(2-4-7)
(2-4-8)
(2-4-9)
(2-4-10)
These quantities are shown in the following representative
signal(Fig.2-4-a). It can be seen that Vsca is also given
by
Vsca = (Env - Pev )/(2 Idc)
(2-4-11)
Apart from being an oscillating function having a frequency
proportional to velocity of the particle through the test
space, we notice also that the frequency for a given particle
is dependent only on Xf and particle velocity
•••••••■•
iy7110,
,1411-Ti ITN B
37
Vs ( Fey 'r Env)/ ( 2. c
C
Fig.(2-4-a) A typical signal shape indicating
various physical parameters.
38
and is independent of the angle of detection A.
It has been suggested ( Chigier, 1976 ) that the Envelope
measurement as-given in equation ( 2-4-8 ) might provide an
alternative for sizing particles. Theoretical calculations
for different fringe spacings are shown in Fig.( 2-4-b ),
Fig.( 2-4-c ). These suggest that owing to their undulatory
nature, size deduced from a single measurement would be ambi-
gious.
Another characteristic that can be studied using equation
( 2-4-6 ) is the intensity distribution of the scattered
light Isca as a function of y, 8 and D, because it is
advantageous to detect the scattered signal at an angle where
the intensity is the maximum. Theoretical calculations reveal
that the maximum of the scattered intensity is dependent on
the parameter D/ ?f . This is summarized as follows :-
(a) For D , the maximum is towards the bisector
of the incident beams, that is the Z-axis.
(b) As D/Af increases two maxima appear and they shift
in opposite directions so that when D >Af they
are along the incident beams.
These are shown in Fig.( 2-4-d ) to Fig.( 2-4-9 ).
Recently, Yule et.a1.(1977) has shown theoretically that oscillations actually damp out for particles of size
larger than 100:um.
D C INTENSTY(x) 7 ci
C INTENSTY(2) 4 ENVELOPE( x)
ENVELOPE(2) —env
Enic I , 111=1.6
0 )sAr = .9 76
3
2
5 6 7 °/1, 8
39
0-4 = M = 1.6
A Idc•I0 =.I22
B Erw10, C E,loo, ).41),. .0813 /
-.... ...,, / D Env 1 0 0, ).h.f = . 0 0 8 I 3
1 I
I.% / % /"N
i I /
/ ,,B
1 t I 1 I
A it 1 I %
%
N 1. I
It /
/ I i 4,
N...../ I I I % I I % II I / % ,
4. I I I
% I. S.... % I I
.. I I/
\,./ V
3
Fig.(2-4-b)
i 1 4 5 6 7 aA
Signals envelope and D.C.level versus size parameter.
Fig.(2-4-c) Scattered signal envelope against size parameter.
0 90
0 a/A=5.10 m=1.60 'Anti=0.093
Wy0.95 1=2.67°
00
-12
-24
-36
I I 0° 180
Fig;. (2-4-e)
10
-20
IMO
-40
-60;
a/A=5.10 m=1.60 ?/A 0.0098 40
DA4=0.10 1 =0,28°
( 0 90 0° 180
Fig.(2-4-d) For DWI the maximum scattered Intensity is along Z-axis.
8 AA f=0 .1 0 78 aA=5.10 m=1.60 D/\f=1.1 1.3°
,Incident beam 41
0
-16 f
-32
...481 I 11 1 1 1 1 1 1 1
. 0 90 0° 180
Fig.(2-4-f)
Incident beam
16
,\ -16 fiAiryv„,q
-32 0 90 Go
180
a/\=5.10. m=1.60. DAf=2. 1=5.6 °
XAf .0.1961
(2-4-,7) At DAr> 1 , maxima are al ong the incident beams.
42
2-5 Effect on visibility of more than one particle in the
test-space.
An exact mathematical approach towards this problem can
always be started by extending Eq.( 2-2-1 ) to ( 2-2-4 )
taking into account the position of each individual particle
in the test-space with respect to an arbitrary chosen origin.
The resulting signal follows by squaring and coordinate trans-
formation. The effect on visibility will be obtained by
carrying out the exact calculation of Eq.( 2-4-6 ). This
would be a very complicated and time consuming problem.
Since the exact effects of more than one particle in the
test-space is not relevant at the present stage of the work
only semi-qualitative approach of Fristrom et.al. ( 1972 )
on identical point particles is given.
Starting from Eq.( 2-4-6), we have that Isca for a Panic to
singlein the test-space is given by.
aca 0 [ -2+ (,--2 + 2 0-
2Cos(k*Y0 4. Al - A2)] ' k2r2 '1 '2 1
(2 5 1) where k* = TrAf •
For point particles we have A= A.2= const. and A l-A2 0.
The effect of more than one particle can be considered by
adding further particle in the test-space and using incoherent
addition. This is valid when the aperture is large enough
43
for incoherent mixing of the scattered signals. In general
for a particle cloud of thickness 2w , containing N randomly
distributed point particles in the test-space, the combined
visibility is given by
Vsca = (1/N) + (N-1)/N (2-5-2)
Thus, it is seen that, in general, more than one particle
placed randomly in the test-space leads to a decrease in
visibility.
2-6 Noises in the detector.
The quality of the output signal is proportional to the
total power received by the photomultiplier. Since we demand
that the signal should be resolvable for every cycle of the
beat frequency, it is necessary that within this time enough
photons reach the photomultiplier. It is expected that the
number of photons per cycle decreases as particle velocity
increases for a given fringe spacing, so that signal quality
decreases at high beat frequencies.
Ultimately a photomultiplier is limited by noise. A
useful quantity commonly used to describe the performance of
a detecting system is the signal-to-noise ratio denoted by S/N,
44
With only one particle at a time in the test-volume.
S/N ratio is affected by
(a) Photon noise,
(b) Lost of contrast of the fringes in the test-volume
( Durst and Whitelaw, 1971 ),
(c) Stray light entering the pupil of the photomulti-
plier,
(d) Dark current from the photomultiplier,
(e) The size of the light collecting solid angle.
Photon noise arises from the statistical nature of the
light because the number of quanta received by a detector
in a given time interval is subject to statistical fluctuation.
Photon and electronic noise were not serious in our case
provided that the photomultiplier was not overloaded.
Low fringe contrast reduces the S/N ratio, and at the
same time directly affects the measured visibility. Therefore
its maximization is essential.
Noise due to stray light can be controlled by employing
a suitable stop or a series of stops. Difficulty arises when
one is working near one of the incident beams.
Increasing the size of the collecting solid angle increases
the S/N ratio ( Cummins, 1970 ). This is because the amount
of light intercepted increases with aperture. The practical
45
limitation on the size of the aperture can be determined
from the following consideration :
If the source size in the test-volume has a linear dimen-
sion D, then using the Van-Cittert-Zernike
area formed at a distance dsaaway from the
2 Area 0.024 dsa
X2 / D2
theorem, the coherent
source is
( 2-6-1 )
( 2-6-2 )
or in terms of the linear dimension
2 da 0.32 dsa 7\/ D
According to Wang ( 1972 ), D can be taken as the
particle size in the test-volume and increasing the aperture
beyond the limit given in Eq.( 2-6-2 ) will not improve the
S/N ratio further. For example in one of our cases
d= 200 mm, = 0.488 um and average diameter of the particle
is D = 5.,gm, this gives ( 2d4) 6.25' mm implying
ACk 'AP = 0.9°.
46
Chapter
0•tical s stem and its reauirements
3-1 Introduction.
In this chapter the design of the optics for the laser
anemometer is described and emphasis is given to methods of
optimizing the system by maximizing the fringe contrast in
the test-space. Definition of test-volume is also given and
the effect of geometrical mismatching of the beams on the
test-volume is discussed in detail.
3-2 o ticaltem.
Extensive theoretical and experimental investigations
of various types of optical arrangements for LDV have been
carried out by Durst et.al. ( 1972 ). Depending on how the
scattered field and reference field are mixed together, they
can be classified as follows ( Wang, 1972 )
(a) Local-oscillator heterodyne,
(b) Differential heterodyne,
and (c) Symmetric heterodyne.
In our experiments the differential heterodyne arrangement
was used. This is shown diagr4tically in Fig.( 3-2--a )
47
Fig.(3-2-a) The lay-out of the optical system.
48
The system has the following main components :-
(a) An argon-ion laser light source ( LS ),
(b) A beam splitting system ( BSS ),
(c) A reflecting mirror M1 and two front aluminized
prisms 1%11 and 11,2 , and (d) A detecting system ( EMI photomultiplier type 9635B ).
The output from LS has its direction of polarization
perpendicular to the plane containing the interfering beams.
It is reflected by the mirror M1 , and intercepted by the
beam splitter which provids two beams of equal intensity.
These components are brought together by reflection from the
two prisms 11'10 and Pm2 at the centre of a circular
graduated track. The prisms are mounted on two separate tables which can be independently rotated and adjusted
horizontally (see photograph P-3-2-b )
The detecting system consists of a convex lens L an
aperture stop A which controls the signal collecting solid
angle, a photomultiplier and an oscilloscope. It is sometimes
convenient to use a series of stops on the other side of the
lens which effectively cuts off stray light. However, care
must be taken that effective size of the entrance aperture
is still given by that of A. The lens, apertures and the
photomultiplier are all mountalon an arm which can be moved
around the circular track so that sizing can be performed
at any angle. The centre of the track coincides with the test-
space.
Horizontall adjustable
table
Beam splitter
49
P(3-2-b) The beam splitting system.
50
3-3 The importance of the test-space.
In laser Doppler interferometry, the test space is
generally regarded as the volume where the signals are
generated by the presence of particles. The size, shape and
spatial characteristics of the light intensity distribution
are therefore important parameters to differentiate whether
signals are originatedin the test space or elsewtere. At
the same time, these parameters have to be properly defined
depending on the information required. Otherwise erroneous
results may be obtained. For example, if one is concerned be
with particle density, a wrong result wouldAdeduced if
incorrect size of the test-space was used in the calculation.
Farmer ( 1976 ) has shown that the definition is meaningful
only when it is related to particle size and the response
characteristics of the detecting system.
3-4 Definition of test-volume in our experiments.
For laser beams in the TEMooq mode, the test space can be
defined by adopting the Io/e2 modulation contour of the inten-
sity distribution ( Brayton 1974), here Io is the intensity
distribution in the common region where two laser beams cross.
This region is called probe volume. The shape of the volume
is then an ellipsoid of revolution with its major axis collinear
with the bisector between the beams. This definition is useful
51
for visualizing the focal characteristics of the collecting
lens used in the detecting optics, for yielding qualitative
predictions of signal shape and assessing the out-put signal-
to-noise ratio ( Mayers, 1971 ).
Another way of defining the test space commonly used
in LDV for local fluid velocity measurement ( Whitelaw, 1975),
is simply by adopting the common region where two laser beams
cross. This can be estimated geometrically as shown below:-
From Fig.( 3-4-a), the length D1 and width Dw are given by
Dw = 201( sin If cos If')
20/ cos ( 3-4-1)
where is the beam half-width at Io /e2 point. For example,
by taking 2a= 0.65 mm, and 1° and = 10°,we obtain
D1 = 0.65 mm, Dw = 37.2 mm and D1 = 0.66 mm Dw = 3.8 mm
respectively.
The above two definitions are insufficient in our case.
Here the following test space requirements apply:-
a) It must have sufficient intensity to give detectable
scattered signals. This means that the test space
will be a function of intensity of the incident beam,
particle density, particle size, y, responce threshold of the detecting system, aperture size
and angle of viewing of the detector.
•
52
Fig.(3-4-a) A simple way for estimating the dimensions
of the test space.
w
52b
b) The fringe contrast, Vc in the test space must be
greater than 0.95.
Vc is defined according to Michelson ( Jenkins and White, 1957 )
as
'max - 'min )/( Imax 'min )
where Imax and 'min are the maximum and minimum intensities
in the test space. We call the test space that satisfies
these requirements the test volume. The first condition is
a complicated one, because it is in itself a whole light
scattering problem. This is overcome experimentally by
ensuring that the light detecting system responds to the
smallest particles in the size range of interest. The second
requirement is based on the experimental observation that a
loss of fringe contrast in the test volume will artificially
reduce the scattered signal visibility ( see P-5-9-c on pg.
It is therefore important to discuss this aspect in detail.
Taking into account the individual intensity characteristics
of the laser beams, the combined intensity distribution, and
hence the fringe contrast, in the test space may vary from
point to point. The foctors that affect the contrast are given
below:
53
For two laser, beams, the fringe constrast in the test-
space is affected by:-
(a) The degree of coherence of the light source.
(b) The directions and degrees of polarization of the
two beams.
(c) The relative intensities of the beams.
(d) Mechanical vibration of the system.
(e) The Gaussian amplitude variation of the beams
across their wavefronts.
(f) Any mismatch'in geometry when they are brought
together.
Since the two coherent plane polarized beams are made
to interfere after several reflections and transmissions,
it is important in the adjustment that the reflection or
transmission surfaces should be parallel to the direction
of polarization. Then only unimportant reductions in beam
intensity and changes of phase are introduced, which can be
easily calculated using Fresnelt equations ( E. Stratton,.
1941 ). Any deviation from this condition may introduce
rotation of the polarization for each individual beam component.
The result of changes of polarization direction is a loss
of fringe contrast. Assuming that both beams are plane and
have unifo an intensity distribution across their wavefronts,
the change in fringe contrast can be calculated using formula
54
below ( Collier et.al. 1971 )
. 2 pa cos q Ra + 1) ( 3-4-2 )
Where Ra is the ratio between the beam intensities and
q is the angle between their respective directions of
polarization. ( Fig. 3-4-b )(1).
FIG. (3-4 b)(i)Interference of two beams with different directions of polarization.
We see that Vc = 1 only when Ra = 1, and q = 0. Closer
examination of Equation ( 3-4-2 ) is shown in Fig.( 3-4-b )00 where Vc is ploted versus q for different values of Ra.
It can be seen that V 0.95, is not difficult to obtain
• s
10 20 30 40 Angle
Fig.(3-4-bA The fringe contrast as functions of beam ratio and
differencein polarization angle.
56
even taking into their combined effect. Therefore, these
two effects are unlikely to give any problem here.
When taking the Gaussian nature of the interfering beams
into consideration, the fringe contrast in the test space
is given by ( Appendix D ),
Ve = 1 / cosh ( 4ZY sine cos y / 62 ) ( 3-4-3 )
A plot of Vc against Z and Y for certain values of
W is shown in Fig.( 3-4-c )(n.
In using LDA as a means of particle sizing a loss of
fringe contrast in the test space would reduce the visibility
of the scattered signal. This would result in over-estimation
of the particle size. From our calculations as shown in
Fig A( 3-4-c ), it was obvious the highest contrast of the
test-space is at the centre. We, therefore, define the test
volume as the maximum sphere of radius Rs which enclose
all fringes of contrast 0.95. Strictly speaking, the
test volume is in general cusp-shaped with its size a function
of 1 ( Fig.(3-4-c)(ii)).
It should be noted that equation ( 3-4-3 ) is derived
under the assumption that both beams are completely matched
at the test-space. That is the central lines of both beams
crossed at a point.
Fig.(3-4-c)(i) Test volume as a function of I
/ = Vc>0,95
Fig.(3-4-c)(ii) The structure of the test volume.
58
Owing to the nature of the two Gaussian beams, geome-
trical matching becomes very important in practice. Suppose
the central line of one beam crossea a point in the test-
space, while the other has its central line slightly displaced
from this point by amounts DX , DY , and DZ. Then the
visibility distribution around the point due to mismatching
can be written down as ( Appendix D ),
V6 = 2 BiB2 / ( Bi + B22 ) ( 3-4-4 )
where B1 = exp 2ZY cos sin T ) /6 2]
B2 = exp [( 2ZDZ + ( DZ )2 sin2)+ (2YDY + (DY)2cos26
- 2 ( ZY + ZDY + YDZ + DZDY ) cos Ysin
( 2 XDX + ( DX )2 /
Using Eq.( 3-4-4 ), we examine the effects of mismatching
- on the test-space in terms of two quantities - change of test
volume size and fringe contrast. It is possible to break
down the mismatching into two directions, one along X-axis
and another in the Y-Z plane Fig.( 3-4-d ). Generally, any
case can be viewed as a combination of these.
Case I :
The mismatching corresponds to a parallel displacement
59
Case I: Displacement of one of the laser
beams along the X-axis.
Case II: Horizontal displacement of one
the laser beams from the origin,
Fig.(3-4-d) The cross sections of both laser beams
• in XY -plane.
60
of one of the beams along the X-axis. That is DX = rx while DY = DZ = O. The situation is shown in Fig.( 3-4-d )
Detailed study of the beam mismatching is rather complicated
because the test volume doe'S not vary symmetrically.
Observations from calculations using equation ( 3-4-4 )
are described below :-
In the plane X = 0, when DX is increasing, the test
volume expands horizontally outwards and at the same time
contracts vertically. The visibility at the point ( 0,0 )
begins to drop from unity. This is shown in Fig.( 3-4-e )
and Fi ° As DX>0.20 mm the visibility
of the origin drops below 0.95[Fig.( 3-4-g )] and the test-
space in the Y-Z -plane has the shape shown in Fig.( 3-4-h ).
Obviously, the test volume with respect to the origin ( 0,0 )
does not exist in this case. The behaviour of the boundaries
of the test-volume at other planes of X = conts. are very
complicated. They have to be obtained graphically whenever
this information is need. For DX = 0.10 mm, two situations
for X = 0.12 mm and X = -0.18 mm are shown. in Fig.( 3-4-i )
and Fig.( 3-4-j ) respectively. Note that in both cases, the
visibilities at Y = 0.0 are all> 0.95.
Since mismatching involves defolfflation of the test
volume, there arises a question as to how much mismatching
can be allowed so that a minimum size of the test volume
will still exist. The answer to this question depends on
•
61
X=10°
,2 ,3 Y(mm) .5
Fig.(3-4-e) Variation of fringe contrast in the plane X=0.
Fig.(3-4-f) The test volume shrinks with increasing Z.
Notice that at (0,0) Vc is still greater
than 0.95.
x=0 LZ=tY=0 AX= Q2 , T=le
0,4 Y(rnm) 0.6
62
Fig.(3-4-g) As DX= 0.2 mm. the visibility at the origin
drops below 0.95.
X =0 1=100 /IX= 0,20 AY =AZ = 0
Vc < 0,95
0,2 0,4 mm
Fig.(3-4-h) The shape of the test volume at the plane X=0 as DX = 0.20 mm.
X =-0,18mm AX= 0,10 mm AY =AZ = 0
Y mm .6
63
X =0.12 mm AX= 0.10 mm AY=AZ = 0 r=10°
0
Fig.(3-4-i) Fringe contrast at the plane X = -0.18 mm.
The difference in behaviour of these curves from Fig.(3-4-f) to (3-4-j) is because the Y-axis is dissymmetrically placed.
Fig.(3-4-j) Fringe contrast at various values of Z at the plane X = 0.12 mm.
a
64
the definition of the minimum size required, and this will
depend on the number of fringes required in the test volume.
That it is necessary to have more than two fringes in the
test volume for measurement is obvious. Experience showed
that a convenient number was of the order 50-100 fringes
( also, Whitelaw 1975 ). Therefore, for a fixed number of
fringes, the minimum size of the test volume is a function
of . For example, in the present case, at I( = 10° and
A = 0.488 hum, a test volume containing 100 fringes would
need a sphere of diameter 0.14 mm. Calculations show as
in Figs.( 3-4-i ) and Fig.( 3-4-j ) that for mismatching
of DX = 0.10 mm, we have a test volume of size around 0.3 mm.
Therefore, this is not going to cause any theoretical problem
except that practical control of particle trajectories and
optical alignment become difficult. On the other hand, for
a smaller angle of interference, a larger test volume is
required to accomodate the same number of fringes. Then
mismatching becomes critical.
Case II :
This case corresponds to a horizontal displacement of
one of the beams from the origin. Without loss of generality,
we only consider displacement of the form DX = 0, DZ=DY= ro. This is depicted in Fig.( 3-4-d ) and Fig.( 3-4-k ).
The fringe contrast of the test volume in this case is
exactly the same as for perfect alignment save for the fact
displaced beam
= New test volume.
Un-displaced test volume.
N
I
65
6
Fig.(3-4-k) Geometrical mis-matching in YZ-plane.
66
0,2 0,1 I I I
O
O
0,1
0,2 Y
O
\ O
O
Y =100
(a)= AX=AY=A2 =0
( b)=AX =Oa X =0,
AY = AZ=0,2 mm.
N
Fig.(3-4-.) The origin of the test volume is displaced
horizontally towards the first quardrant.
a
67
that the volume is displaced horizontally ( in the Y-Z-plane )
from the origin. The amount of displacement depends on the
magnitude of DZ and DY. As can be seen from Fig.( 3-4-1 ),
a spherical volume of radius, Rs' 0.36 mm can still be
obtained at ( 0,0 ) with DX = 0 and DZ = DY = 0.2 mm.
As this is compared to Rs 2'0.385 mm for perfect matching,
we conclude that mismatching due to vertical displacement
is more critical than for horizontal displacements.
68
Chapter 4
A Study Of Some Particle Samples
4-1 Introduction
The visibility parameter formulated using Mie theory
in the previous chapter is best applied to a single particle
of spherical shape. Verification of the theory using a
particle sample necessitates an understanding of how
individual particle behave when they are dispersed.
Here, particles such as glass ballotini, Titanium
dioxide (rutile), Magnesium oxide and Aluminium oxide were
investigated. A design used for injecting particles from
a fluidized bed into the test space is described and a
study of particle behaviour, using this system, both in
cold gas and a flame is presented.
4-2 Physical requirements of the sample and the method
of seeding.
Theoretical studies using Mie theory (chapter 2)
for spherical particles lead to certain conclusions which
must be tested experimentally. To do this we need particles
in the form of uniform spheres. Although liquid aerosols
generated by blast or pressure atomisers give perfect
spherical particles,they were not used in these experiments
because they vaporized easily in flames. Of course; flames
are not necessary for verifying the theory, however, our aim
is trying to size particles in the presence of flames.
69
Known sources of solid spheres include Dow latex and
glass ballotini. The former, being polystyrene cannot be
added to flames. The smallest commercially available glass
ballotini has sizes ranging from044-60um. It was found
that seeding of particles into the gas stream was most
conveniently done by boiling particles from a fluidized
bed. Different ranges of size could be obtained by chan-
ging the flow rate. For example, intermediate sizes could
be obtained by first boiling away smaller particles using
lower flow rates and then increasing the flow rate succes-
sively to get larger particles. The flow rate and maximum
particle size can be related by the following consideration:-
As shown in Fig.( 4-2-a ), for ease of investigation,
assume that the flow is laminar and thus the velocity
profile in the tube is parabolic at a certain height above
the bed of powder. The profile is given by ( Terence,1968 )
u AP(12.— rc2 )
(4-2-1) 411 L
where AP = pressure drop at the length L across the
tube, i.e. p1 - p
2
= fluid viscosity,
and / = radius of the tube.
The total volume flow rate Q can be found by integration,
that is
1 Q= 2Trrcud4,— Avir)4
0 89,L (4-2-2)
a
70
The maximum velocity umax occurs when 0 and is equal to
umax = 2um , where um = QA1 12) = Cdpi2 BILI,) is the
mean velocity. We have then
umax = 2Q/u12
(4-2-3)
Suppose Dmax is the diameter of the spherical particles
that can be supported by air of density4o at the velocity
umaxil We obtain Eq.(4-2-4) from Stokes law
3nDmaxl umax = (Ps - Pf ) n 0max/6 (4-2-4)
Substituting this equation into (4-2-3) gives
1/2 36 11 Q
Dmax = (4-2-5) g (i? 4 )111
where g = 9.81 m/sec2,
= the density of the solid,
4? = the density of the fluid.
Equation (4-2-5) predicts that particleS with diameter
less than Dmax will be removed from the bed. As a
numerical example for air, we havellair = 0.064 Kg/hr-m
4; = 2.5 • 103Kg/m3 ,Pair = 1.3 Kg/m3 and A!. 0.0328m.
From these data, we obtain Dmax = 10,Am as Q = 13.10-3m3/sec
Details of the relationship of Eq.(4-2-5) are plotted in
Fig.(4-2-b) for samples of rutile Ti02, MgO andAl203.
.. •
UnCI"
J~-- -_. ,,-' p. ..... / II',
I I. \ . I
I I ~
. I " L I I I I I D"
71
l'2-- ---.---__ ).Ao Particles
- -- -1-- ---
Fig.{4-2-E!) The velocity profile.
.. -f.~. _ . - .. ..... -........ ~ . .. '-- .
Injection nozzle
~~ .. \. '--'-~~~~--'---!--' ------- --~ll I II I I~
J==r=-=;:.=,;::..;:...:;.:t;tt--ll/ll/l I~
\-Nozzle hold er
Carbon cooted grid
Fig. (4-3-b) Particle injection and sampling system.
2.5.103 Kg/m3
3.97:103Kg/m3
4.26.103Kg/m3
3.58-103Kg/m3
3.28.1072 m
Pglass =
PA1203
PTiO2 =
PMg0
Atz0,
10 x10-6 Q( ISec )
72
s. Fig. (4-2.-b)k plot of maximum particle diameter
versus volume flow rate of air.
0
73
4-3 Particle injection system and Burner
We require a method of seeding a flame with particles.
Since the flame would be illuminated by the fringe pattern
0 formed by the two crossed laser beams, some experimental
requirements of the system have to be borne in mind. First,
at any flow rate, the particle density should be readily
variable. Secondly, the flame must be premixed with
sufficient air to prevent soot formation, since this would
both scatter and emit light. Thirdly, the flame should be
as small as possible, at least initially, to avoid undue
disturbance of the laser beams. The system used is shown
in Fig.(4-3-a). R1 and R2 are filtered dry air supplies.
R1 passes through the fluidized bed carrying particles
with it while R2 acts as a diluent. R1 and R2 can be
independently varied to keep total flow rate R1 + R2 constant.
The size of the flame can be adjusted by diverting part of
Rl + R2 .
The particles boiled off the fluidized bed have to pass
through a certain length of tubing before entering the test
space. The size distribution is expected to vary along the
tubing. Therefore, any sampling is best done near the in-
jection nozzle.
The nozzles are designed to be removable so that differ-
ent diameters can be fitted if required and, at the same
time, the sampling procedure is made easier. Each nozzle
was about 0.025m long and the inner diameters ranged from
0.0002-0.0005m.
•
R3+010( RI + R2
flame trap
Clean dried air (Ri + R2) cc( + R2)
flame
1,,(R1 +R2)(1 -00
filter
R9
Air outlet
Fig,(4-3 -a ) Particle. injection and
burner system.
Mixing chamber
a,
TR1
dried air
• t
Methane' supply
75
Samples of particles were obtained using carbon coated
electron microscope grids: The grid was fitted at one end
of the nozzle, half covering the tube as shown in Fig.(4-3-b).
4-4 Study of samples with and without a flame
Three samples of metal oxides, namely Ti02, Mg0, A12031
and commercially available glass ballotini were studied.
Table (4-4-a) summarized their sources and physical appear-
ance.
Table (4-4-a)
Samples Sources Size range Physical appearance
MgO- 4031
(light)
BDH Chemic- als, Ltd., Poole, England.
Thin film
0.3 gm
white, irregular shaped light , scale-like. See Photo P-4-4-c.
Al23 0 -101 (light) calcined
Same as
above
Thin film
1.5 gm white, irregular shaped
appears as lumps. See Photo P-4-4-d.
TiO2 rutile
Tioxide In-ternational Ltd.,Stock- ton, Eng.
o4.--40,um white irregular shaped heavy. See P-4-4-b.
Glass
ballotini Jencon's
Ltd.,Yark
Hemel Hem- pstead,He-rtfordshire, England.
o.01.--60 ,um hard, transparent,
smooth,spherical
See Photo P-4-4-a-
For our purpose, we need the particle cloud to be well
dispersed without agglomeration. This requires the
•
76
particles and gas to be very dry. Other factors are the
design of the fluidized bed and physical properties (such
as shape and hardness ) of. the particles. The
design and optimization of fluidized beds is a matter of
experience. However, in general larger beds are better
for irregular particles, and constant mechanical vibration
is necessary to prevent slugging which gives non-uniform
bed density and consequently erratic variations in particle
concentration.
All samples were studied with and without seeding into
a methane-air premixed flame. To collect samples of glass
particles seeded into a flame, a cleanAslide attached to a
constant speed motor was passed just over the reaction
zone of the flame a number of times at constant intervals.
In this way the glass slide was kept cool. In each case
these samples were taken first with a flame, second with-
out and finally with a flame again. This was to ensure
that any difference observed in the particles was not due
to changes in the bed condition.
In the presence of a flame a sufficient sample could
be obtained in 15 minutes, but when the same method was
applied without a flame, only a very small number of
particles stuck on the slide even after a few hours.
Presumably this was due to the particles striking the
glass surface with a lower momentum. A way round this
difficulty was to invert the nozzle with the glass slide
placed underneath and surrounded by a suitable sized glass
cylinder to prevent draughts. Samples without a flame
•
77
P-4-4-a Original sample of glass ballotini.
P-4-4-b Original sample of TiO2 x1000.
•
78
P-4-470 Original sample of T1g0 x1000
P-4-4-d Original sample of A1203 x500.
•
79
•
P-4-4-e Mg0 without seeding into a flame.
P-4-4-f MgO in the presence of a flame.
-•
80
,74171.7",
P-4-4-g A1903 without a flame.
Ilp..!IrtsgsPAtmerwcro7Nr riammurrorogree”mrgrtr7
:1
I
P-4-4-h Al203 in the presence of a flame.
81
L.
f
TiO2 without a flame.
1-4-4-j TiO2 in the presence of a flame. •
- •
82
•
P-4-4-k Glass ballotini without a flame.
P-4-4-1 Glass ballotini in the presence
of a flame.
83
taken using carbon coated electron microscope grids for
TiO2 and ballotini using method described in 4-3 were
compared and found to be little different. Eight pictures
of these samples each with and without flames are shown in
P-4-4-e to P-4-4-t. It is clear from these observation
that:-
(a) TiO2 and ballotini show no difference in size
with or without a flame. Furthermore, no aggl-
omerates are found in ballotini and very few
are found with Ti02.
(b) Mg0 tends to form agglomerates with and without
a flame and generally the size of the particles
is smaller in a flame than otherwise. The same
situation is found in A1203 as well. These
effects may be due to a number of reasons,
including,(i) the particles or agglomerates
break up upon hitting the glass slide due to
their higher momenta in a flame, (ii) they break
up inside the flame due to heating or electric
charging( Waterston, 1975 ).
84
Chapter5
Choice of experimental parameters
5-1 Introduction
In this chapter we first present some theoretical curves
calculated using Eq.(2-4-6). A study of the behaviour of
these curves suggest a practical range of experimental
parameters to be used. These parameters are size parameter
x, angle of sizing e, angle of interference of the laser
beams 1 and the aperture size of the signal collecting opt-
ics.
Secondly, the Rayleigh qu4;ter wavelength criterion is
used to relate the size of the collecting aperture and
diameter of the entrance pupil of the photo-multiplier in
such a way that the photomultiplier will 'see' a well
defined space totally within the test-volume of diameter
Rs. However, these values only offer a general guide-line
in practice. The photomultiplier will actually see a
greater volume than that given by Rs. This volume is called
the extended test volume.
A study of the effect of trajectory on the charact-
eristic shape of the out-put signals was made using a
thin quartz fibre. It was found experimentally that it was
possible to differentiate signals that originated in the
test volume and those that did not. This knowledge provided
useful information on the sampling of signals generated
from particles seeded into an air stream the trajectory of
• which are partially controllable.
85
5-2 Choice of parameter
As was shown in chapter 2, the parameter immediately
measurable in this experiment was the visibility, Vsca, of
the scattered A.C. signal. Among other factors (eg. 49, m,
Ac(, a), this is a function of the angle if and some prior
knowledge of the appropriate maximum particle size was
required in order to establish the best interference angle
to use.
Fig.(5-2-a) shows three groups of curves (a), (b),
and (c) which are theoretical plots of Vscaversus a/N for
various values of I . A main feature to be observed is
the variation in sensitivity to changes in a/A . For
example, in curve (c) particles with sizes below a/A 0.4
cannot be distinguished because they all give rise to
visibilities nearly equal to unity. We call this the
insensitive region. Similary, those having sizasgreater
than a/A".. 0.8 give more than one possible size for each
visibility. This is the ambiguous region. Therefore, the
only region that is sensitive to size variation and gives
unique results is 0.44a/7 .4: 0.8. By changing the value
of , this unique region can be shifted to other values
of a/A Fig.(5-2-b). For a fixed particle size distri-
bution boiling off a fluidized bed containing glass
ballotini, a qualitative behaviour of the distribution of
visibility is shown in Fig.(5-2-c) for three different
angles of interference. As can be seen for 1,,0.44owhich
a
0, ■ .„. Ifi %N.
tv7........ .
1 1 1 1 i 1
2
86
Fig.(5-2-a) Visibility as a function of size parameter. Curves c and d have three regions.
a)Ap .0.59°,4a .2.16°, 71A. =0.05 tot= t3 =0°, m=1.60 ;
/\f =0.5, /8..--0? na=1.50-i0.1; b) a=0.5°, c)c) =5°,c00(.14.480.
a. d: i
■ •
\
02 / _---
, .% . . ..
Fig. (5-2-b).Vscc, versus a/A for different angles of viewing. (19=o( for 4 =IA, ).
0,4 XiAt=0,5176 ■ \ •
X/Xf =0,2437 , i% ■ , , ,.
•,.....„-
„
-
...
,P.......s%
b,c: • ,7., • •
\•,.,i m =160-i0,10
.,....... ....".
alx 3
0I.
-0
0
z0
1,10 f171 OZIS p T.
aoj ‘y qq_Tm c:11.0,VilogsT,4 14TTTEiTsTAOL 1,101.1S s8anlITJ7 GSaU,W
(3-Z- Sr5U (!!)
el 1. D3sA L' RI 6' I
if 1-1
CH!)
0 Z' 911
(!)
1. 6' L. 0 '
•
88
gives Xf = 31,um almost all particles have visibilities
greater than 0.95. As the interfering angle becomes larger, lower
particles with slightlyxvisibility start to appear and when
is increased still further to 4.2°, the visibility
distribution appears as in Fig.(5-2-c) (iii). Actually
this curve needs a slight correction owing to the fact that
the oscilloscope response is in the non-linear region ( see
Fig.5-7-a ). This means the curve as a whole shifts
slightly towards higher visibility.
Another variable which can be used is 8, the sizing
angle. The plots of Vsca versus aA for different values
of 6 are shown in Fig.(5-2-d) and Fig.(5-2-e). They reveal
similar characteristics as variable 1-
Visibility with respect to changes in refractive in-
dex m is shown in Fig.(5-2-f) and Fig.(5-2-g). A comparision
between visibility with and without integration over an
aperture is also shown in Fig.(5-2-h). Some obvious conclu-
sions follow:-
(a) Visibility curves are more complicated for least
absorbing particles and have higher sensitivity
to changes in refractive index at secondary
maxima.
(b) For a given refractive index, and angles of sizing
and interference, the general effect of increasing
the size of the collecting aperture is to decrease
visibility when the particle sizes are within the
first minimum. However, the variation becomes
unpredictable for larger sizes.
•
• .2
m=1.60-i0:10
X/Af =0.3473
"r=10. I • • %. •. • `•
0 1 1.5 • 2 25 a/X 3
. 9
.8
.7
.6
.5
.4
.3
.1
I I I i J r I r t l r 1 t I
89
1 3 0 2
)4)f=0.5176
I =15°
Fig.(5-2-d) Theoretical curves show how the sensitive region shifting with sizing angle 0; a:0=1° , b:0=3°, *c:0=5°, d:9=100 and e:0=20°.
Fig.(5-2-e) Another example for y.15°. a: 0=1°, b: 0=3°, c: 0=5°, d: 9=10°,and e: 0=20°.
•
90
8=100 , X/Xt = 0,3473
(a) m=1,60-10,01 (b) m=1,6-10,05
(c) m=1,60-10;10 (d) m=1,5-10,10
• •
• 1 •
\ • 1
0 cn 0,8
0,6
0,4
0,2
a/ N 3 2
• .‘
Fig.(.5-2-f) Theoretical curves showing Vsca as a
X function of a at various values of m.
0,8
(a) m =1,60-10,01
0 4
(b) m =1,60-10,05
9=5° , N/Ai = 0,3473 IN" t ■
• I i
i t 1,
I 1 1t. 1
n \a I % l 4 I %
, 1 v. i i ■% It A f %
I t -. I ..•••• .....
• I 11 il I t \ I / ........ I • • 1
1\
• I I
I 1 i
bI :
•
t \ , I I
•t '
•t ‘ 1 I I
r
1
9 .'
a
\
I • tI ‘,.V"
,\ 1 /
.. 11
i r • \k‘ t
I I i
% ■ , If 1
I • i . i I V I /
V %
N..•il ' ,
I / % , '5'
I 1
2 a/x 3
(e) m =1,60-10,10
(f) m =1,50-10,10 0,
Fig.(5-2-g) Theoretical plots of Vsca versus a/N
at different values of refractive indices.
0 2 a/X
91a
U
0,
m = 1.60 -i0.10
0°
2,5 a/x
Fig.(5-2-h)(ii).The effect of aperture size on visibility, with = 50 ando= 2° .
1
0,5
Fig.(5-2-h)(iii). The effect of aperture on with different and o( .
2,0 a/A 3.0 1,
m =1.60-i0.10 r=10° , (X= 5°0
0.6 0 =O.
Without integration a =
Aot 0.040
0,4 b = Aar. 2° , /43 =1°
c = =12° , Ap = 6°
0.2
1.0 a U
0
91
Fig.(5-2-h) The effect of aperture size on visibility.
Observation from Fig.(5-2-h) (ii) and (iii) shows that for the same aperture size Vsca is affected differently for different values of 'y and ok .
92
5-3 Working parameters
Given the physical properties of the interfering beams
and the angle of interference , the radius Rs of the test
volume containing fringes of contrast greater than 0.95
can be determined graphically using Eq.(3-4-3), or Eq.(3-4-4)
when geometrical mis-matching is to be taken into account.
The focal depth, A of the lens L, which forms part of the
light collecting optics as in Fig.(5-3-a) is given by
(Jenkins and Wh.ite,1957)
A /2mua2 (5-3-1)
where X = wavelength of the incident beam,
m = the refractive index of the test volume,
and Ua = the maximum angle sub-tended by the entrance
pupil of the collecting optics.
In order that the pupil of the photomultiplier accepts no
other scattered light than that generated from this volume,
Aperture
Fig.(5-5-a)
•
93
A should be 4 Rs . If the distance da>> ds , then (Fig.5-3-b)
Ua=daids , and by putting m.1, we obtain
ds2 2d2 - Rs a
or tan2Ua / (2R )
(5-3-2)
The transverse size of the test volume is determined by the
,size of the pin-hole of the photomultiplier. If this has
radius' dp /2 /2 then
d e 2MR d R Ids, (5-3-3)
where M is the magnification.
Equations (5-3-2) and (5-3-3) together precisely determine
the test volume. If the pinhole is bigger than the limit
given by Eq.(5-3-3), then it will 'see' regions of test
volume of low contrast so that measured visibilities will be
artificially low( see 5-9 and P-5-9-d). As a numerical example
for 67. 0.325 mm and Y= 1.44° , graphical calculation using
Eq.(3-4-3) give Rs = 0.82 mm. Substituting this Rs into
Eq.(5-3-2) and Eq.(5-3-3), we obtain for ? = 0.488 pm ,
Ua 0.99°
and dp -40.82 mm
if magnification of M = 1 is used.
A lower limit to dp is set by the requirement that
* Precisely means the photomultiplier only Iseeethe sphere having fringe contrast greater than 0.95.
Aperture Test Volume
Lens
d., ds,
Fig.(5-3-b) The various working parameters of the signal collecting
system.
95
there must be more than two fringes in the test volume for
measurement. The fringe spacing Xf of the interference pattern
at the angle is given by( see 2-4 )
= A / (5-3-4)
so that for two fringes in the test volume of transverse
linear size 2R5, dp is set by the limit
MW(2Sini d /2 MRs (5-3-5)
From this, it is obvious that there is also a lower limit for T •
5-4 Other possible sources of Doppler sig
Theoretically, the size of the test volume can be limited
using Eq.(5-3-2) and (5-3-3). However, the practical volume
is much larger in length. The reason is that Eq.(5-3-2) is
based on quarter-wave criterion. This extended test space
is depicted in Fig.(5-9-b). This volume covers regions of
lower fringe contrast depending on angle e and at the same time can see other possible sources of Doppler signals. These
sources were discussed and verified experimentally by Durst
(1972), and are:-
(a) Laser Doppler signals from two particles illuminated
by one beam.
96
Aperture
Fig.(5-4-a) Laser Doppler signals generated
from two particles illuminated by
one beam. V represents particle
velocity.
Fig.(5-4-b) Laser Doppler signals from two
particles illuminated by different
beams. •
w
97
Fig.(5-4-a) shows graphically two particles simultaneously
in the same beam and travelling at different velocities. It
is not known how the overall shape of the signal looks like in
this case. It is expected that the chance of them giving a
reasonable Gaussian shaped signal will be very small. However,
in practice a simple check was performed before the start of
particle counting to ensure that this case did not arise.
This was done by blocking off each incident beam alternately
for a few minutes to see how the signals behaved. We observed
in both cases that only Gaussian pulse without A.C. components
were present. One representative signal is given in (p-5-4-a)
on page 115.
(b) Laser Doppler signals from two particles illuminated
by different light beams.
This physical situation is given in Fig.(5-4--b).
It is very difficult to carry out an experimental
check in this case, since both beams are needed to
generate the Doppler signal. Therefore one would
by no means be able to differentiate whether the
signal arose from a single particle traversing the
interference pattern or from two particles in
different beams. However, one would expect the
probability of having both particles to be in the
right position to generate a symmetrical Gaussian
signal to be minute, especially when operating at
low particle concentrations.
98
5-5 Effect on signal visibility of the position of the
particle in the test volume.
As has been cited in 2-3, the effects on the visibility
of the signals arise solely from the collecting optics adopted.
The variation is due to the fact that scattered light reaching
the collecting lens varies with the position of the particle
in the test volume. This situation is shown in Fig.(5-5-a).
Theoretical calculations were carried out for two specific
examples. Initially , we assumed a one dimensional case
to see how visibility would be affected.
In the system shown in Fig.(5-5-a), the photomultiplier
is aligned to 'see' the centre of the test space at (0,0).
The correct angle of sizing would be eo, that is when the
particle is at (0,0). However, 00 becomes el when the
particle is at distance Y1 away. If ropYi , the following
approximation is valid,
and eo1 82
Combining these two equations gives,
= eo (4-- Y1 / ro ) (5-5-3)
In one of our experimental cases we had eo = 3.2°,
ro = 20 cm = 1.44° and the maximum value of Y1 = 0.82 mm.
99
›-
Fig.(5-5-a) Particle position dependence of scattering
angle in the test space.
100
Details of calculations for this case are given in Table 5 5-a).
Table (5-5-a)
an = 8.00 (D/Xf = 0.804 ), m = 1.60.
Y1 mm 0 00 Vsca
0.82 3.1959 . 0.2103
0.70 3.1965 0.2104
0.50 3.1975 0.2106
0.30 3.1985 0.2108
0.10 3.1995 0.2110
0.00 3.2000 0.2111
-0.10 3.2005 0.2112
-0.30 3.2015 0.2114
-0.50 3.2025 0.2115
-0.70 3.2035 0.2117
-0.82 3.2041. 0.2118
Another example for A/Xf = 0.3473, r0 = 20 mm, Yl =1 0.82 mm
and 80 = 10ois shown in Table (5-5-b). The effect is expected
to decrease with increasing angle of sizing and is greatest
at Go = 00 ( i.e. along the Z-axis ).
For a three dimensional case, the maximum distance, 4max
= ±(x2+ y2+ z2)1/2 f.rom the origin ( 0,0 ) is pmax . For
X=Y=Z= 0.85 mm , we obtain ,Amax = ±1.44 mm. This, means that
for G0= 3.2° , ro = 20 cm and 1= 1.44°, the sizing angle varies in the range 5.195°“0..;;5.207° . The corresponding
limit of visibilities are found from calculation to vary
2111ax e8 Vsca '
-1.44 mm 3.1929 0.2173
0 3.2000 0.2186
1.44 mm 3.2071 0.2199 Aperture size
as = =
o)
100b
within 0.2098 4 Vsca < 0.2124 with a mean of 0.2111 at %=3.2°. This is shown in table 5-5-a )(i). Table ( 5-5-a )(ii)
shows the same situation integrated over an aperture size
one degree square at 04= 3.2° ,p. o . We conclude that
in general the effect is very small in cases of interest
here and well within the accuracy of measurement.
Table ( 5-5-a )(i)
a/A = 8.00, m = 1.60, Y. 1.44°.
&max eo Vsca
-1.44 mm 3.1929 0.2098
0 3.2000 0.2111
1.44 mm 3.2071 0.2124
Table ( 5-5-a)(ii)
a/A = 8.00, m = 1.60, /5= 1.44°.
101
Table (5-5-b)
a/x = 8.00 (D/xf = 0.804 ), m = 1.60
Y1 mm o
Go Vsca
0.82 9.9959 0.1238
0.62 9.9969 . 0.1238
0.42 9.9979 0.1238
0.22 9.9989 0.1238
0.00 10.0000 0.1238
-0.22 10.0011 0.1238
-0.42 10.0021 0.1237
-0.62 10.0031 0.1237
-0.82 10.0041 0.1237
5-6 Alignment
In these experiments laser output at the 0.488dum wave-
length was employed. The output power was 10 mw in the
TEMooq mode.
Measurement of the scattered signals was made only very
near to the plane containing the two beams.
The two components from the beam splitter were adjusted to
be as equal in intensity as possible with the help of grey
filters. Their intensities could be checked by traversing
the photomultiplier through both beams as seen in P-5-6-a.
102
Measurement showed that this method enables one to balance the
beams within 99 %.
The fringe spacing, which is a critical value, was
measured using a photographic plate and an ordinary optical
microscope with a calibrated filar microscopic eye-piece.
12
Plate(P-5-6-a). Intensities of laser beams.
The size of the entrance pin-hole, d of the photo-
multiplier was chosen so that it satisfied inequality (5-3-3).
The same was true for the aperture size, A , which was based
on equation (5-3-2). In all of our experiments, to be
presented in chapter 6 , dp 0.48 mm, M Ua = l°.
The position of the test volume was adjusted so that
its position coincided with a pin proViding out of the centre
of the track. The pinhole of the photomultiplier was then
placed at the position where the image of the pin was sharp.
The lens,L, was then displaced to by-pass the laser beam.
103
If the beam arrived exactly at the pin-hole, it meant that
the pin, the lens axis and the pin-hole of the photomultiplier
were all in line. The lens was then returned to its original
position. Usually, one found that the backward reflected
beams from the two lens surfaces were very helpful in the
alignment of the lens. The arm containing the collecting
optics was then moved to check the other incident beam using
the same procedure as before.
When the alignment was complete, the pin was removed
and the particle injecting nozzle put in its place, together
with the suction nozzle.
When a flame was required, a mixture of methane and
air was used to boil off the particles.
5-7 Calibration of Oscilloscope
To determine whether the beat frequencies generated by
particles traversing the test volume were well within its
linear voltage response, the oscilloscope needs to be calib-
rated. Using a reference frequency generator whose out-put
voltage is known to be independent of frequency, the oscil-
loscope was fed with reference signals through a load
resistor and a length of co-axial cable corresponding to
that used in the experiments. The result is shown in
Fig•(5-7-a} for two voltage sensitivities; namely 10 mv/cm
and 1 mv/cm. The ordinate denotes values of peak-to-peak
response voltages while the abscissa refers to frequency.
• As can be seen 30 KHz is the upper limit in both cases.
0 rn 0 05
4
2
10 102 I t I l I I I 1 It!!!
Frequency KHz 103 0 t 1 t 1 1 1 t 1 t 1 I 1
Fig.(5 -7 -a) Frequency response curve of Oscilloscope DM 53A (Telequipment).
. 1 my/cm 10 my/pm
♦ • \
♦
1 0
105
5-8 Measurement of angle of interference "r
Direct measurement of y from the interfering beams is
inaccurate due to the finite width of the laser beam. A more
accurate way is to measure the fringe spacing from from which)"
is deduced using Eq.(2-4-4), that is
Xf = A/2Sinl (5-8-1)
To estimate the error, we differentiate on both sides of
Eq.(5-8-1) and obtain
ktan YiNf )1 AAf (5-8-2)
It can be seen that the error committed in measuring XI" that
is becomes, important for larger interfering angles:
Fig.(5-8-a) shows a plot of Eq.(5-8-2) for several values of
. For example, when = 1.440, an error in X = 0.368 hum
would introduce error in -?! arround 0.0010 .
In our experiments, a Kodak photographic plate was used
to record the fringe patterns. Measurement was then made
with an ordinary optical microscope. Emulsion shrinkage has
very little effect on measured fringe spacing ( Williams, 1973 )
especially when the emulsion is coated on a rigid substrate
as in our case. Shrinkage in this case was mainly alteration
of emulsion thickness which affects the reconstructed image
quality.
S
0 1
Tr. 0,5°
2,0 =0,84°
I ; 1.44°
106
0,8
0,4
1
3 5 AY° 7.10-2
1,6 0 (*)
(5-8-a) A plot of AAf versus A y for
different values of 1 .
•
107
5-9 Out-put signal shape as a function of particle trajectory
in the test space.
The extended test volume discussed in 5-4 makes possible
out put signals having a variety of shapes depending on the
particle trajectory. A qualitative study of these signals is
useful to assist in sampling signals passing only through
the test volume. The signals were studied here using a thin
fibre ( preferably less than a fringe spacing in diameter )
in order to obtain good output signals in the sense of high
A.C. component and offer controllable trajectory.
In the experiment, the fibre was aligned using the fibre-
holder shown in p(5-9-a),see pg.115. A circular rotatable ring
having a quartz or carbon fibre fixed across it was mounted
j on a table which could be a d usted horizontally and could be A
operated manually to travel along two directions, i.e. the
Z and Y-axes . The fibre was adjusted so that it was normal
to the plane containing the two incident beams. ( Fig. 5-9-a ).
This could be checked by observing the back-ward scattered
light(in this case the scattered light has its highest inten-
sity in the plane of incident beams) and by projecting the
fibre and fringes simultaneously using a. microscope ( here
the fibre is adjusted to be parallel to the fringes). The
collecting optics used here were the same as described in
3-2. Other geometrical details are provided below:-
Focal length of the collecting lens f = 15 cm.
Objedt distance ds = 27 cm.
Image distance dB 1 =33.7 cm. •
Dialneter of the fibre
Fringe spacing
Size of pin-hole
Collecting aperture
D = 7.99 'urn • Xf = 11.68 ,um.
d = 0.38 mm.
6C{ = 1624°. 013=0.72°.
108
The characteristic shape of the output signal is a
function of the light intensity distribution along the parti-
cle trajectory. The intensity distribution in the extended
test space is to some extent/dependent on the sizing angle 8 .
As can be seen from Fig.(5-9-b) the volume of laser energy
intercepted by the extended test space decreases as 49 is
increased for a fixed )' and approaches a minimum when 0 = goo.
The manner in which it varies with y is fairly obvious.
Although only two angles, namely G = 0o and G = 8.4 ,
were studied here, the results are quite general qualitatively.
Details of the study are shown in Fig.(5-9-c) and Fig.(5-9-d).
The following sums up a few relevant observations:-
(a) When the fibre was within the test volume, at both
angles out-put signals always had a symmetrical
Gaussian shape(SGS). For a small angle of inter-
ference, SGS could in fact be obtained at a certain
distance on both sides outside the test volume along
the Z-axis. This distance varies with and 9 .
For instance, at 49 = 0° , = 1.19°, the distance
is around ± 1 cm. and 0 = 8.4°, = 1.19° it is
about ± 0.4 cm.
(b) Outside the test volume, the pedestal voltage has
double peaks at 0 = 0° while at 9 0°, the shape
2 — _
'Q ISca
109
back-scattered light
Fig.(5-9-a) The geometry of the quartz fibre scattering
experiment. The inclination of the fibre
in the test volume can be specified by
and T . The required alignment is 0 =cf. O. Whencis displaced from zero , the back
scattered light is shifted towardst X depen-
ding on the direction of '' . By displacing
4 from zero, the first observed effect is
the tilting of scattered light about the Z-
axis. A further increased in (1). makes the
scattered light from each beam visible ( i.e.
they are seperated ).
0
Y 0
a)
CD
e -z
Extended Test Space
Fig.(5-9-b) The extended test volumes as a function of angles
of sizing.
•
Text Volume
Extended Text Volume
,-___----,,___----_,------_---------z_-__-__ ---...---- __-_. ----- -.„3-_-_---7,--_----- ------- --7----,-t-- N-7------,./. ■
<0,95
Direction of
sizing e
-V, ,95
U3
U1
Fig.( 5-9 -c ) Signal shapes as function of particle trajectories at 8 =0° .
• •
Extended test volume
Ui
Fig.(5-9-d ) Signal shapes a function of particle trajectories at 8 =-8.4°.
11 3
of the signal usually becomes asymmetrical and
irregular depending on various factors such as G ,
geometrical mis-matching and intensity distribution.
These charactristic shapes are shown in the figures.
(c) One would expect that the signals from the test
volume would be unpredictable when geometrical
mis-matching is serious.
These qualitative conclusions drawn from the fibre can
be extended directly to particles though there are more
. complicated trajectories due to an additional degree of
freedom compared to the fibre. Few possible trajectories
are depicted as ui to till. in Fig.(5-9-e). If they are conf-
ined in the test volume, the signals could only suffer from
a reduced number of cycles without affecting the visibility
(see P-5-9-b).
In the above study on the scattered signal shape as a
function of particle trajectory, both laser beam intensities
were well balanced. We observed in this case that. at 9 = 0°,
almost unit visibility was obtained. Other situations for
unbalanced beam intensities were given in P-5-9-c and P-5-9-d.
They represent signals for II 12 and Il = 0 12 resp-
ectively. This observation proved that a loss.of fringe
contrast in the test volume would artificially reduce signal
visibility.
Test volume
Fig.(5-9-e) Various possible trajectories for a
free particle.
114
P -5 -b P -5 -9 -c. Scattered signal
when incident beam
intensities are hi
hly Unbalanced.
Compare with oscillosoore
trace u1 on pg. 111.
tt
115
P-5-9-d Scattered signal
P-5-4-a Oscilloscope trace when one of the incident
for a particle illuminated
beam is blocked (1=0). by a single beam.
P-5-9-a The fibre holder.
116.
5-10 Experimental procedure
Initially, spherical glass ballotini particles were used.
A few special experimental situations can give rise to signals
of unit visibility, and can be conveniently used to check
whether the test volume conditions are satisfied. These are
(a) For particles of size in the Rayleigh scattering
regime or particles with diameter very much smaller •
than fringe spacing.
(b) For a collection angle Ok= 0 P= 0° . In both cases, signals of visibility greater than 0.95
should be observed. For Rayleigh scattering the result can
be explained easily using Eq.(2-4-7) where substituting (
Van de Hulst, 1975)
sl(e+w) = s2(e-w = iotk3
(5-10-1)
where p( is polarization per unit volume.
gives
Vsca = (2k6 d 2 )/(k
6 0(2 + k6 2 ) = 1
(5;-10-2)
01 - 02 = 7/2
independent of angle of sizing.
A particle very much smaller than the fringe spacing
follows the intensity variation of the test space exactly,
since < sca I0 at each point and I0 is approximately
constant across the particle surface. Vsca of the scattered
signal will be a copy of Vc of the test space. At 0 = CP
117
Fig. (5-10-a) Visibility curves as a function of size
parameter. The interference angle is , 1.44° and
signal is collected at o( = 0°withdp =0.Y:
dE:192=1.08?
•
a 118
and 180°we have a symmetrical situation and hence balanced
amplitudes of the scattered light. For collecting aperture
-.-2°square, a computer calculation for ci.,=e,13.0° is shown
in Fig.(5-10-a).
The fringe spacing and angle of sizing were chosen so
that the whole particle size distribution lay within the
sensitive region of the visibility-size curve.
In taking measurements, only highly symmetrical traces
having Gaussian shapes were chosen ( see 5-9 ), and visibi.r
lities were measured at the middle of each trace(See 6-5). this
usually coincided with the extrema of the envelopes. From
each trace, values A, B and C were measured (see Fig.2-4-a )
and Vsca was then calculated using the relation (See Fig.2-4-a
on pg. 37 )
Vsca = ( B - C )/( 2A - B - C ) (5-10-3)
a 119
•
Chapter
Experimental result and discussion
6-1 Introduction.
This chapter summarises the experimental results on
quartz fibres2glass ballotini and some metal oxides with
and without seeding into a flame. The metal oxide particles
are all irregular in shape. Owing to the fact that signals
from these irregular particles are apparently erratic in
shape, measurement of their size distribution in flames
has not been successful. In all the measurements results
were compared to independent optical microscopic measurement
of samples taken using carbon coated electron microscope
grids described in 4-3. A small thin flame was used so
as not to cause unnecessary disturbance to the light beams.
Finally, we discuss some of the difficulties encountered
when applying the sizing technique to large scale turbulent
flames based on the geometrical mismatching of the beams.
A possible technique for refractive index measurements is
also outlined.
•
120
6-2 Visibility measurements using quartz fibres.
Visibility measurements of some quartz fibres are
presented here. Theoretical-calculations are done using
the theory of Jones ( 1973 ).
i 00 Isca = (210/ffkr 2] an Exp(ikYSinl) Cosn(e-T) +
2 Exp(-ikYSinW) Cosn(e4)1 (6-2-1)
By letting
00 an Cos n(O-') Exp(ikY Sind) = 6aExp(i0a)
no (6-2-2)
and 5 an Cos n(04) Exp(-ikY Sin') =c51 Exp(1010) n=o
where T b are amplitudes while 0a bare their a, respective phases.
Eq.(6-2-1) can be recasted as
Isca = (2I0/7kr)K + 1613 + 2CF Crb Re fExp(i0a - 1041 a
The visibility is then given by
(Isca)max (Isca)min V SCQ (Isca)max (Isca)min
2O 61 (6-2-4)
6a +Vb
410
121
Preparation of the fibre:
Quartz fibres in the size range of 1-10 pm were
prepared by blowing a high temperature propane-oxygen flame
at a very high velocity towards a thin quartz rod 1 mm in
diameter. Molten quartz was torn off and solidified in the
air. The fibre produced in this manner vary in thickness
along their length. Observation by optical microscope
showed that variation of 0.5-1,um in a distance of 1 cm is
not uncommon. If we assume that the thickness varies in a
uniform way, then over a length of 0.1 mm we would encounter
a size difference of > 0.05 ).un. This variation will be
important at certain angles where visibility is very sensitive
to size.
Experimental results:
Owing to the inaccuracy of our apparatus when measuring
a full polar diagram, visibility polar diagrams over a limited
range of angles for fibres of size parameters x = 2.499 and
x = 4.248 only are presented here. A computer is used to
search for the best fit of m and x by comparing Vsca •
obtained from theory with those from experiment. A count
is registered when the following inequality holds
I v sca;tii. (gi) - Vsca,exp. (0i) < 0.05 ( 6-2-5 )
at any Gi
122
Here sca th.(gi) Theoretical value of V sca at (g).
andVsca,exp (Q-) is the corresponding experimental value.
Let Nbf denote the total number of counts for all Qi at a particular value of m and Plots of Nbf versus
an and m are shown in Fig.( 6-2-a ) to ( 6-2-f ) on pages
fgO* to 185. The _search starts from the value ofm ( =1.463 )
taken from Handbook of Chemistry and Physics ( West, 1969 )
for pure quartz and a/7 from the measured value. The resulting
theoretical and experimental curves are shown in Fig.( 6-2-g )
and ( 6-2-h )•
Values of a/) obtained from the best fit are compared
with values from the optical microscope below :
Refractive Optical D index microscope
(1) 2.318 m = 1.460 2.439 0.25 ;um
(2)3.972 m = 1.465 4.146 4- 0.25 ,um
We see then that except for a slight differences in refractive
index the theoretical sizes agree well with measurement.
Fig.( 6-2-i ) shows how the fit varies with refractive index
for the case of D.2.406/trl.
Observation of Fig.( 6-2-g ) and ( 6-2-h ) shows that
theoretical and experimental curves fit better in the forward
direction. Slight discrepency at larger angles may either
be due to some misalimment of the system or inhomogeneity
I
•
,8
/ • /‘ ....•
i1 ■ %. / / /
/ i / /1"- l
I -. I / / / I N i 1 / 0 / /
`\\ /1 N
/ / /
1 \ i 1 1 / \
I 1 / ...0
1 0 0 \ • 1
/
\ O/ I
\• 1 0\
O 1
/ \ i
% ... / •
•0
,6
,4
I i I 1 1 I L I l / I 1 1 , t I I f I u L • i___J 1 1 i L 1 1 1
5 10 15 20 25 9 30°
Fig.(6-3-g) Comparision of theory with experiment. = Theory ;
o = Experiment. 1) = 3.972)am , m = 1.465 ,/= 1.97g.
- - - - 0 = 4,021,u m
124
.. I'S / • .... / ! \ ■ •-■ -
\ --... N. .■ %
o o.∎ 0,..
■ 0 ‘. e" i 0 .521 / 0....
No, 0 i Ot /
■_......
\ ! ! . r .■ ./ .,, i ■ /
r ■ ./ \ .....
\ i
— D.2.406 _Am m=1.460 D=2.289 pm m=1.460. o Experiment. - D.2.318 Jam m=1.460.
5 10 15 20
Fig.(6-2.7h) Theoretical curves compared with experimental.
1
0 0 Q.-
,8
0 0 ■ 041 \
\\ •
1./ \
o 0
I L I I i 1 l I I 0 5 10 15
90 20
Fig.(6-2-i) shows how the fit varies with refractive indice
at D=2.406. o experimental; m=1.460; ---- m=1.462; m=1.465; ---m=2.720.
•
125
(hence the variation in refractive index ) of quartz fibres
prepared in this way. However, as far as particle sizing at
a fixed angle is Concerned , this will not be a problem since
most of our measurement is made near the forward direction
and the accuracy of the angle can be checked before measure-
ment.
It should be noted that the limit ( i.e. 0.05 ) of the
inequality in Eq. (6-2-5) depends upon the accuracy of
measurement and the number of points measured. When highly
accurate measurement is possible and more data points are
available, one can shrink the limit of the inequality to
obtain a more accurate and unique fit. However, it has
no advantage in setting the limit lower than experimental
accuracy.
One disavantage of having too little data is that
the fit is not unique. For example, detailed calculations
have shown that for the case of a quartz fibre with
D = 2.318,um, several ' best fits ' exist. The one shown
in Fig. ( 6-2-h ) is that which agrees most closely with
the optical microscope measurement. For the case of
D = 3.927Am, we did not find any other fits in the ranges
investigated:
that is
1.35 < m 1.50
and
2 xun < D < 6 ,um.
126
6-3 Experimental results for glass ballotini with and without seeding into a methane air flame.
Signal visibilities of more than 300 particles were
measured from oscilloscope traces. A histogram was then
constructed of the fractional number of particles against
visibility ( Fig.6-3-a ) and this was converted into a size
distribution using theoretical curves of the type shown in
Fig.( 6-3-C ), The same figure also shows the effect of
varying refractive index. It was found that the uncertainty
due to not knowing this parameter and also due to oscillation
of the curve could be estimated readily since all the variation
can be emcompassed within the two envelopes shown. The
resulting size distribution is shown in Fig.( 6-3-10 ) together
with the histograi of particle size distribution measured
from samples using an optical microscope. The bars express
the uncertainty within the envelopes. Further results are
shown in ( 6-3-d ) to ( 6-3-h ). They are summarized in
Table T6-3-a.
• •
O
20
10
,3 ,5 ,7 ,8 ,9 Vsca
Fig.(6-3-a) ViSibility histogram from measurement. A=4/4= 0.540
1= 1.03°, c= 34 . 0(;' dp 0.930.
2 3 4 5 6 D (urn)
Fig.(6-3-b) Theoretical and experimental result.
Histogram : Microscope. bars : Visibility
measurement.
Q-4..2 0....°...9 ,I. t 1 040 0 4 +•„4.- --c.,:1 -. — 0 -.. ...- ., co. c 00. - 0 ) 0, 4 • '-- . :4.0 0 .a 0
o 0 7;4: 4' :00 ...4...: I : 4 . - . 4 .• : 0,„. '—' .,:e. ■ ■ ......
• 00 0 •:oo , "•••■
`s. 0 0 0° 4•• , .• .......
oo 0 •• 0 0 \.
0 ,°•40e• 4
• * • • ■ ••• 0 •
..• • O ‘‘. ...... . \• ..
: 000 N .
ee
\ \ 0t;),++ . 0„
\ 0 • \ o a
4. 6. \
\ 0 • 0°• .4. •
• • .\
is. • 0 • + . \
\ •'• • 0 4 • + 0 . ai° O\
\• : +•0 •• 0 0
\ 40. 4,
• . \ 0.0 0 -
• • 0
• \
0° \
.
\0* . 0 0
\ 00 •' ••
' \
.e. 0 •
• 0
oo 0 O
O
3
4
5
6 a/A
Fig.(6:-.3-c) Theoretical calculations of ca versus a/A using
Mie theory. cle1=d92=0.54°,75.=1.03°, ()=3.6°, p=00,
dp , o m=1.60, v m=1.65, + m=1.55.
,6
2
1 -23°°1ro° 65- ' --- •---- 006 ---- --- -...... 0,...0 Cil 0 0 090■.+44.847.s. .......
-.',. 0 ++ ■,.
NO +
''... + + + .(13- +4. ‘...", Ns.. 4' 4. + 0 0000 +++++,,,‘
N \
9) O cfPNa, oto _O N
sta,°_,0 0 0.\
0%r0u +-I- °O N
\ 4- + +
▪
?t 0
0
++
4- ++\
\ + + \ 0 4- 99 cb \ 0 N
\ °CI) 0 0+ % 0 6) N
0° 4. N N 0 + 0
\ . +-1- °2)
0+
+ +\
+ \
\ o
\ 4- + . \ \ ++ 4. 0 o (5Po
+
o°
0 0+ o \
Fig.(6 -3-d) Theoretical plot. 0o
00 + eP \
o \ y =1.44°, 0=5.20 , 13 =0°, \ oo
000 \ + + 4. +0
÷ \ °° \
dB, =d9z =1.08°, di3=0.59°. + \
4. + 00 0 o
+ ni= 1 • 5 5 g 0 m=1.60. 4, 4- \ + 4' 0
3 1
5
++ 4. 0 0 dP +
0 0 _s.
VA •
6 an 7
s)
O
20
10
0 2 4 6 8 1 0 D(um)
Fig.(6-3-e) Light microscopic measurement
compares with visibility measurement
using theoretical plot of Fib, (6-3-d).
Histograms: From microscope.
Ears: From visibility measurement.
S
• •
,4
..,
ox 0 c . 'N.
x0 90)/"--
N 000 )c? x xoxx .....`
N x 0 x N...„. o x "N.
N. x 0° N
0N. \o
\ xn_i x 5 oo x x N. N. x x
o \ occ'' \ x x 0 x N.
\ xx 0 x .\\.' x 00 Nx o o 0
o o° 0
Fig.(6-3-f) Theoretical plots for 00 xx 0 Zf
ox
=0.84°, .3.2°, p.o° , dp.o.39°, \
\ ci 0 , 0
d91' =d92 — m.1.67 - io.oi , x o 0
x m=1.55, o m=1.65. xx 0 0 0
0
0 \00
1 1 ci/X 8
1
,8
,6
X X
0
0
8 2 4 6 8 D(urn)
• 1
Fig.(6-3-g) Result for glass
ballotini seeded into a thin flame. Histogram : microscope.
Bars = visibility measurement.
Fig.(6-3-h) Result for glass ballo-
tini without seeding into a flame. Theoretical curves shown
in Fig.(6-3-f) are used for visibility calculation.
--- 4
-
111••■■••■
F- FT1
20
,; • 0
E z
10
133
Fig.( 6-3-g ) shows result obtained for ballotini
seeded into a premixed methane-air flame. The flame was
chosen to be small at this stage so that difficulties arising
from disturbances of the laser beams by the hot gases did
not arise.
The following table summarises the experimental results.
The angles and d were chosen using the following consi-
deration :-
(a) Particle size should lie in the region where it
gives unique visibility,
(b) Visibility must be sensitive to particles size,
(c) No stray light should enter the entrance aperture.
Material Shape
r
Flame Figure Result compare with Microscope(Ma::.frequency)
Glass Spherical 1.44 3.2 No 6-3—e good,, 5% Glass Spherical 1.44 0 No not shown showing high visibility
in accordance with theory Glass Spherical 1.03 3.6 No 6-3—b fairly good 15% TiO2 irregular 1.03 3.6 No 6-4.—b poor 20% Glass Spherical 0.84 3.2 No 6-3—h fairly good 15% Glass Spherical 0.84 3.2 Yes 6-3—g poor • 20%
Table T6-3-a.
134
The maximum error for spherical particles not in a flame
was less than 15%. This is well within experimental error if
we note that there is about 5% due to loss of contrast in the
test space, 5% due to angular measurement and 5% from the
measurement of the traces on the oscilloscope (For detailed
analysis of errors, see article 6-5 on pg. 142 ).
When particles are seeded into a flame the error tends
to reach 20%. One reason for this increased error is that
the expansion of the hot gas increases the velocity of the
particle through the test volume so that the Doppler frequency
of the signal is not in the linear response of the scope. From
P-6-3-a we observe that th Doppler frequency is about 35 KHz
which is 5 KHz outside the linear region of the scope. Another
cause.-is the slight increase in signal noise owing to the
presence of flames. This noise increases the 'random error
when reading the signal trace to about 7%. The effect of
particle refractive index(probably becoming absorbing) cannot
account for this increased deviation. As can be seen from
Fig.(6-3-f) the absorbing particles also have visibility lying
inside the two envelopes.
6-4 Experimental studies with particles of irreoular sha•e
Measurement of particle size distribution was attempted
for TiO2 , A1203 and Mg0 in a cold gas stream. In all
cases, the output Doppler bursts were similar to those for
spherical glass ballotini. However, irregularity presented
a difficult problem in size measurement through a microscope.
This was particulary true for A1203 and Mg0 which follaed
"clusters ". Therefore, comparision was only successfully
carried out for rutile Ti02. The results are shown in
Fig..( 6-4-a ) and Fig.( 6-4-b ). Here the " maximum " projected
S size when measuring particle samples under microscope was chosen.
134b
•
time scale 200 susec/cm
P-6-3-a Signal trace from glass ballotini seeded in
Methane-air premixed flame. Note that the
Doppler frequency is about 35 KHz( as compare
to 30 KHz in article 5-7 ) and the presence of
flame did cause some background noise as
indicated by the arrow.
•
0 U
L. ,8
,6
I •
1 6-OM + 0 +0 + + 0 o 0 0 t it 0 oZ
6:4 oc,. op 40+0
N +cbo 00 o N - o +°1)°0° x000 0 0 +00 o:P o0d4.
sr.)o 0 009 + +0 x + 0°00
+ n o - 0+ N 0 0 0 00 \ 0 00t40 000 0 0 N
00 c, \ 0000 p
0 + 0
0 \ 0 0 r, _ 0 0 + v 0 0 + +
0 cPb +00
N
0 00 o o
Fig. (6-4-a) Theoretical plot for ?f=1.03°, o(=-3.6°, 0 o 0 0
2 /3 .0°, dp=0.93°, dGi =d92 =0.54a.
. o m=2.65, + m=2,63.
0
0
\ 0 EQ o +
3,0 4,0 5,0 6,0 ci/X 7,0 2f)
\ 1 1 1 1 1 1 1 i 1
• •
•
0
2
4
6 D(iwm) 8
Fig.(6-4-b) Experimental result for rutile TiO2 in cold.
Histogram : Microscopic measurement.
Bars : Visibility measurement.
137
disagreement is around 20% indicating that some other size
parameter would be more suitable. In fact the scattering
method predicts a size of the order of 0.85 of the measured
maximum size. If one compares the area of an ellipse to
that of a sphere then a2 = a1a2 , where a is the sphere
radius and a1 and a2 are semi-minor and semi-major axes
of the ellipse. For the TiO2 the ratio of dimensions was
about 2:3 giving a * 0.82 a2 in close agreement with the
above number.
When similar measurements were carried out in a flame,
the signals appeared to be irregular, in sharp contrast to
signals from glass ballotini in a flame. Two representative
bursts are shown in P(6-4-a ) and P( 6-4-b ). These signals
made visibility measurement very difficult if not impossible,
Further tests have suggested that they might be due to spin
of the particles as a result of shear forces experienced in
the flame reaction zone and also the breaking up of the larger
lumps in a flame ( See 4-4 ). Both effects could occur
individually or together.
One possible -„ray to overcome this difficulty was by
making measurements at small intervals throughout the whole be
trace. The most probable visibility could thenAtaken as
representative. Fig.( 6-4) (c), (d) and (e) show how this
is done, A frequency spectrum of occurrence of visibility
for 34 particles in the presence of a flame was then plotted
138
P(6-4-a) A trace for A1203 seeded in a flame.
P(6-4-b) Signal originated from A1 203 seeded
in a flame.
139
.80
.70
.60
31" .•• x X X
. yx
• .
xxy
it 1'
YY .3( ..V
X
Visibility per min. along the trace.
a =.769 b =.76 3 are visibilities obtained by taking readings
c =.750 on the envelopes shown as dotted lines.
Fig.(6-4-c) A1203trace in the absence of flames.
•
Particle's trace.
Particle's trace. VSCa .8 •-•
.7
.6
.5
.4
.3
Visibility per mm. along the trace.
Visibility per mm. along the trace.
Fig.(6-4-d) A1203 in the presence of flame.
Fig.(6-4-e) A1203 in the presence of --A
--
-
flame.
.2
11111.111.
.1 .2 .3 .4 .5 .6
10•■••••■
■•■■••■■
30—
.7 .8 .9 I I Vsca
20
0
. I
0 .1 .2 .3 .4 .5 .6
304)
E 2:
20
141
Fig.(6-4-f) Visibility
distribution of A1203
with a flame.
I0
NI ca (most probable) Vs
Fig.(6-4-g) Visibility distribution of A1203without a flame,
142
as shown in Fig.( 6-4-f ). This was compared with the
visibility distribution Fig.( 6-4-g ) without a flame under
the sane, experimental conditions. Although the sample is
rather small for particles in a flame, it does indicate that
quite similar visibility distribution are obtained in
both cases.
6-5 Sources of experimental error.
We discuss here three main sources of error and show how
they are estimated. The first is due to loss of fringe contrast
in the test volume. This may be due to any one of the reasons
discussed on pg. 35 ( article 3-4 ). The results is the
reduction of scattered signal visibility ( see pg.1111.) which
then tends to overestimate particle size in the region where
size and visibility have a one-to-one correspondence. From
the definition of test volume, the maximum expected error is
less than 5% . Of course the error will be larger if the test
volume does not satisfy the definition.
Another main source of error arises from noise in the
signals, including photon noise, stray light and dark current
from the photomultiplier. This noise appears as irregularity
of the signal envelopes. This is made clear when we analyse
three Doppler signals from a) the quartz fibre wher one has
a
143
high scattered intensity (see the trace on pg. 37 ), b)
spherical glass ballotini and c) A1203 in the absence of
flames ( see pg. (36i ). The voltages corresponding to the
signal's pedestal Pevand Envelope Env are traced and analysed
for visibility at, for example, every millimeter along the
trace. These are shown in Fig. ( 6-5-a Vas A, B, and C.
We observe that the visibilities vary along the trace, whereas
theoretically they should be constant. This deviation is
partly due to the fact that as we go away from the extrema of
the Gaussian envelopes ( i.e. away from both sides of line AA )
the magnitudes of the voltages become so small that accurate
reading of the values becomes difficult, and partly due to
lower light intensity so that instrument noise becomes inport-
ant . For these reasons and those discussed on pg. 98,
measurements of visibility of a trace should be taken near the
peak of both envelopes (i.e. near the line AA ). Furthermore,
we observe in trace A that for the region in which visibilities
are calculated, the average Vsca is 0.824 and lies between
0.805 and 0.840 with a maximum difference of dV= 0.035 this
gives a maximum random error of about 4% if measurement of the
trace is done within this region. The same can be done for
traces B and C where we obtain
Trace B: Mean Vsca =0.912, dV = 0.05, Maximum error= 7% ,
Trace C: Mean Vsca =0.753. dV =0.120, Maximum error= 12% .
•
Fig. (6-5-aN)"Signal traces for A: quartz fibre, B: glass ballotini in cold stream and C: A1203
in the absence of flames.
Abscissa: Visibility per mm along the trace.
• • • • a• to
.••. • ..• •
• •
.80 t
*le • " .1, • • x ••
g .1(7 •% % .
X X
y v.
.70 X0.0
.60
Oa.
A
A
ENV
,95
0,9
"T
B
A
•
0
145
The smallest value of dV in the case of the quartz fibre is
because it has the highest scattered light intensity so that
noise is relatively less important. In practical measurements,
a dimension of a few millimeters, say 2 mm, on both sides of
the line AA of a signal trace allows sufficient space for
measurement, we would expect the error ( i.e. dV ) within that
space to be smaller. However, when for particle system having
a size distribution, noise varies from trace to trace because
of the differences in scattered intensity. By analysing several
small traces in this way, it was found that 5% seems to be a
reasonable upper limit in the case of glass ballotini,
7% for irregular particles in gas stream and glass ballotini
seeded in a flame.
The third class of error arises from angular measurement
of a t W and the size of signal collecting aperture &and .
These errors have been estimated from calculations such as
those shown in Fig. ( 6-5-a ) and Fig.( 6-5-b ). They are
summarized in Table ( T-6-5-a ) and ( T-6-5-b ). In table
( T-6-5-a ) Act , 4p and AT are the maximum difference
( i.e. deviation ) from a number of similar measurements.
These uncertainties arie from the finite width of the laser
beam and alignment of the collecting optics. In table ( T-6-5-b ),
the effect of the size of collecting aperture is investigated.
Here we assume that the value 0.54° is the correct one and
calculate how the visibilities are affected when the size of
aperture departs from that value.
•
44• 00A P e
4 ApX.-V
Xx - X A A 0 A 09
XA .>C X X° 0° x° X 0
>).1
9 X A •9 0° xxxds
4X X0
X A •
ex
e 0 a
AO A. A
0 4eX0 A 0 0X x x4, Ax x
tstt 4 4 44 A
491 944
A8221 4X A • 444 XVO
•
Xx•
m=1.60 -10.00 , dA1=d92=0.54°. 0:(3.00, dp.o. 93°,7=1.03°, d=3.5°.
0,6 x:p.o°, d1S.0.93°,1=1.07°, c4=3.6°. dp.0.93°,1.1.03°, t1.3.60.
0:3.o.f,dp.o .93°,1.1.03°, o1=3.6°.
1
U
,9
X
0 X
2,5 3,0 4,0
5,0
Fig.(6-5-a) The effect of parameters !3, W, and on visibility.
1
U
,s
0
• 0
0 4.
•
4
,6
0
0 • 0 •
A4 +
& .12 0
A
8
+ •
A A 0 •0 +• A+
A
A
0
• 0 0
A
4 + A
A • 0 , 0
5 6 7 8 • a/X
•
0 0
o 6. o
6 6
6
Af.13.611m , 1.103°,p=0°,
0! =3.6°, dp=0.93°, m=1.60,
o : (191=d99=0.34°. + c191=d92=0.74°.
d91=d92=0.54°.
A : d91 =d92=1.00°.
Fig.(6-5-b) The effect of slight variation in aperture
size on visibility.
148
One assumed correct value for theoretical calculations is del =dG2=0.54o
Table T6 -5 -a
Parameters Maximum error Maximum for error in visibility45%
1:1/X
ot La = + 0. 10 6 (upper limit)
iE3 +
I Li = + 0.04° 6.5 11
Table T6-5---b
d91=d02=d0 Magnitude of assumed error
Maximum for error in visibility < 5%
ctIX • -
o.34o Mae) = -0.2 8.2(upper limi
0.54° A(dg) = 0° OD II
0.74° L(d9) = +0.2° 7.1 11
1.00o
b(d9) = +0.7° 6.4
The deviation of visibility from the one calculated for this
aperture is shown in Fig.( 6-5-b ).
•
149
From these tables, we note that :
(a) For small particles ( diameter 41,-Xf/ 2 ) all errors
are less than 5%, for the cases calculated.
(b) The errors in visibility incurred due to uncertainty
in measurement tends to be smaller for a larger
fringe spacing at a particular particle size,
(0) In terms of relative magnitude, size of the aperture
gives rise to insignificant error compared to those
due to p , and /. Among those lc is the most important parameter,
(d) In our case, if Ad, Ap and Ocala be measured
to the order of 0.01° while Al. to the order of 0.001°, errors in visibility can be neglected
for sizes lying between 1 - 10jum.
6-6 The extension of LDA particle sizing to large scale turbulent flames.
When LDA is applied to size particles in a large scale
turbulent flame, the interactions of laser beams with flame
boundaries between cold and hot gases and the turbules become
major problems. These interactions give rise to spurious fre-
quencies depending on the velocity of approach of the turbules
towards the beams as well as their curvature. An experimental
test was carried out in which a thin fibre was held stationary
Flame
Stationary fibre (D =7, 49ium )
150
Fig.(6-6-a) Apparatus used to demonstrate the occurrence
Of Doppler signals owing to the interaction
between incident beams and flame boundaries.
V-- 1 --4 2 k— 3
lo msec/cm-----> Time
P-6-6-a A Doppler signal containing three bursts 1,2 an d 3.
•
151
in the test volume and a turbulent flame passed through
both laser beams at a small distance away from the fibre
as in Fig.( 6-6-a ). A characteristic signal is shown in
P( 6-6-a ). This signal is caused by : ( Hong et.a1.1977 )
(a) Moving fringes owing to the change in phase
difference without beam displacement,
(b) Moving fringes owing to deflection ( or wobbling )
of one or both beams.
Both'effects can occur individually but most likely act
together. Since these effects are integrated along the beams,
the longer the laser paths through the flame, the larger
will be the disturbance.
In connection with particle sizing, these effects play
different roles. The former one is of no importance for it
affects only the frequency without altering the fringe con-.
trast or spacing in the test volume. The latter is important
because it causes deflection of the laser beams. This
introduces geometrical mismatching ( See 3-4 ) which will
either reduce the size of the test volume or the.fringe
contrast to an unacceptable level.
152
6-7 Possible extension of the method to particle refractive index measurements.
Radiative transfer from flames containing particles can
only be fully understood if the optical properties of the
particles are known. An example is the flame from solid
propellant rockets containing metal oxide particles. Experi-
mental studies made by Carlson( 1965 ) revealed that these
properties have to be measured at the temperature and wave-
length of interest. In this hostile environment in situ
measurement of optical properties using electromagnetic waves
as a probe is the most promising. A general procedure, for
example Bhardwaja, et.al. 1974, to obtain refractive index
is given below :-
(a) Define a measurable parameter, say
R -
sat Q- (a)1Ta2 N(a) da
al (6-7-a) a2
a QbsP (a)-Tra2 N(a) da l
Where a1 and a2 is lower and upper limit of particle
size and
Qsp = Total scattering coefficient,
Qbsp= Back-scattering efficiency factor, from an appro-
priate theory.
•
153
s
•
(b) Assume a size distribution function N(a) ( E9, Junge's
power law or ZOLD ) or indepently determine using
a method which is insensitive to refractive index,
(c) Obtain computer plots of R against ranges of
refractive indices,
and (d) Search for the m which gives the best fit for
experimental and theoretical values of R.
The LDA technique described previously offers a possible
alternative for determining refractive index through visibility
measurement. We know that visibility is a function of the
size of particles, angle of sizing, refractive index of the
particle, the fringe spacing and the size of signal collecting
aperture. Although theory predicts high sensitivity of Vsca
on refractivity at secondary extrema, computer plots of Vsca
versus x reveal too great a complexity to be of any use with
a range of sizes.
Various techniques arise depending on the choice of
variables. They are described below :-
(a) Refractive index as a function of N/Xf at fixed
0,, a, X and aperture size. This method requires
a technique which is able to sample single particles
with different values of 1" A few theoretical plots
of Vsca angainst ( or X /Xf ) are shown in
Fig.( 6-7-a ) to Fig.( 6-7-d ). The strong dependence
Junge's power law see Junge(1951); ZOLD is zero order log-
normal distribution see eg. Clyde Orr(1966).
J 0.2 0,8 0,6 0,4
m =1,40
154
Fig. (6-7-a) Theoretical plots of Vscaagainst )\ A f •
.,, i/ i v,„. ....,, ,./ , ,.% Jr1 ■ / . , i ;I 1.1
l k / I
: /
11 li % a I it 1 ii ■ ‘ 11
V fri =1,60 - 10,001 m =1,50 - 10,1
— m =1,60 -10,0 --i0,00010
ZVX =5,01 9 =1°
0,2 0,4 0,6 0,8 Of 1,0
Fig. (6-7-b) Theoretical plots of Vsca versus yx for
aborbing particles.
155
0,4 -
a/X=5,01 1 1
I Iii 0 = 25°
1 1 1
A I ri
1 Iii
1 1 1 Ii Il 1
1 ‘ =1,62 i .
1 \ / l 1
, l m
/ till m=2,72 1
i 11 % Si i 1
i. i I
1 A. '1 , 1! \ I
li 1 •
Ii 1 1 m =1.60
\../ % \ % iiiiit i ‘I % 1 I
1 1 1 1 \ m=1.40 \ \ti ‘ i 1. %
/ ■,..... , 1 0 1 % 1
,0 if ..k-
.
Iv\
0,8 Of 1,0 0 0,2
0,4 I
0,6
Fig.(6-7-c) Theoretical curves of Vsca versus N)4.
for non-absorbing particles at 9=25°.
1,0 ill 1 li-c\ it i / it 1 I i% • t
It •1 I I .!I i
it !, 1 i I t
,1 !I I *1 I 1! 1 !I 1 I i 11
!I 1 I • I II ! I I 1 I 1 11. !I . j,
1 1 % 1 I. • 1 s ' 1 ! 1 . 11 !I 1 1 ! II 1 . 1 !, ! / I 1 1 ! 1 ‘ i I I I i! I V I 1 I 1 I i It I I. 1 11 i 1 I i
I 1 i 1 i
/ i ! I I 1 ! 'S iI 1 ! I I 1 ! 1 I 1 ! I i
I
• .1
t`• 0. / 1
j I I'
4 I 0,8 I
I
0,6 I
I
- I
I
-
I
II
- 0,4
II
0,2
I
1 I i 1 i I i i 1
I .i 1 i 1 iii
1 11 I i I i i i 1III 1 !
I!
I I 1 ■• IVI I \ i/
Ili '1 i / L 1 —.... '... .i // i, I,/ .`., . . . %. .ii
.., --......._ .s.,.....,../.,
I I 1--- 1 0,4 0,6 0,8 1,0 XIXf
rl it II
FIG.
= 5,01 8 = 25°
m =1,60-10,010
m =1,60-10,0 —10,00010
m =1,60-i010
0 0,2
Fig.(6-7-d) Same as above figure for absorbing particles.
• 156-
of the curves on refractive index is obvious.
(b) Visibility as a function of aperture size, and
refractive index. •
It has been shown by Hong and Jones ( 1976 ) through
theoretical calculation using I4ie theory that visi-
bility can vary quite significantly at different
aperture size with refractive index. In practice
one way is to use different aperture size at different
angles to record scattered light from a particle so
that sizing is done by smaller aperture while refractive
index is measured from larger aperture.
(c) Variation of visibility with refractive index.
This proposed method use two laser beams of different 50 cue
wavelength twoAsets of fringes,superimposed at the
test-space. Each particle that traverses the test-
space generates signals at the different wavelengths.
These can be recorded using two photomultipliers,
each with an appropriate filter to block off the other
wavelength. Provided that refractive index does not
change much with wavelength or one laser measured size
independently of refractive index, otherwise compli-
cations will arise. Alternatively, the single laser
could be used and measurements made at two different
angles.
•
•
ta9
• • •
a/N =1.89
157
•
•
.
,6 • • •
• • . • .
.r̀ '-a/N =4.58 ,4
a/N =7.17 •
. • •
••••• ..•
• • • •
• • • •
•
• • • •
0 2,70
2,75 2,80 2,85 2,90 n1
Fig. (6-7-e) Theoretical plots of Vsca versus real part of
refractive index for Y=3.1°, 0.2°, d81.de2=1°, o dp =0.25°.
0 8
X x X X X X X st . x X X X X X x x x x x X x
N a/A =1.89
X
.8
• •
za/A =4.58 • •
•
• •
• •
•
4
• •
z a/A =7.17
• • • • • • • .•
e
• • •
• •
• •
•
0 1,40.
• L 1,45 1,50 1,55 1,60 1,65 n1
Fig. (6-7-f) 1r =3.1° 0=2° =d92=1° 3 =00 dp.0 . 25° .
O 0 0.0 0 O
0
• • x * x t -51\ K OOOOOOOOOOOO
158
3:0 n 10
b:e = 9°
°:8 = 17°
4:0 =25°
e , 0
1:0 =9° g:e =170
b:e =25°
1:8 = 14°
,
10-5 1073 • ,1 1 10
Fig.(6-7—g)
I i 1 I
: hb
xx
m=1.60-im.
),A1= 0.9756 DAf= 16
ci/X=8.20
X/Xf=04 8 78
°Ai= 8
NNINItig
0,4
0,2 0
-g
0,8
0,6
n2 103 0 io-5
g i e = 3°
11:0 = 5°
1.0 = 4°
=14389
a, 0 = 30
b:e =
c:O=.23° d,O= 17°
e, = 22°
1:0 =30°
m Cl/2 =6.15
X/Xf =0.0407
Fig.(6-7-h) Theoretical calculations of Vsca versus n2 •
159
One signal would determine size, and the other refrac-
tive index using that knowledge of size. This is explored
theoretically by calculating Vsca versus refractive index
m = ni - in2 using several possible combinations of
( D /Xf Q ). Examples are shown in Fig.( 6-7-e ) to Fig.
( 6-7-h ). A look at these plots shows that the visibility
is oscillating within a narrow limits throughout the whole
range of n1 investigated, while it seems the most
sensitive regions all occur at about the same value of
n2 Attempts to establish a ( D /Xf Q )
pair that will give visibility sensitive to m outside the
region n2 = 0.001 to 10 have not succeeded. There is no
rule that can be used to determine which ( D /Xf Q ) is the
most suitable, but in general sensitive pairs are towards
larger D /Nf and Q away from the forward direction.
It is interesting to compare the variation of absorption
efficiency Qabs versus n2 . ( Q bs is given in Appendix A ).
This is shown in Fig.( 6-7-i ) and Fig.( 6-7-j ) for
m = 1.60 - in2 and m = 2.71 - in2 respectively. It can
be seen that significant variations in absorption efficiency
also occur only for imaginary refractive index in the same
region. It may be, therefore, that the visibility method is
capable of measuring the complex refractive index precisely
in the region of most interest.
2
0
Qabs0.60-i105)=Clox IC4
or,0
3
a=6.46, go =3.80I
b=554, go= 4.14 I
, C10 = 4.522
d=3.0I qo =1.555
165 to-" 163 WI .1
160
Fig.(6-7-i) Absorption efficiency versus imaginary refractive index.
The values a,b,c,and d are size parameter a/ A..
4
tki 0
• m .7-- 2,71 -in2
Qabs(2.71-i10) =Clox 104
Clo= 1'778 1)2.7.07
q3=3.624.
c=646
q3=5.343
d= 5.54
q3=5.774
c■4
a
2
.1■1-11t te 111,111
10 10-4 163 10 z 1 10.1 nA
161 \ •
4
Fig.(6-7-j) Absorption efficiency versus
imaginary part of refractive index.
The values given in a, b, c, and d
are size parameter a/X .
162
6-8 Comparisons and discussion.
Before making comparisons with other workers we introduce
two physical 'terms not mentioned previously. They are
the visibility of extinction, Vext, and V. Vext is defined
as (Jones,1974)
Vext = Re(S1(2X))/Re(S1(0))
(6-8-1)
where Re stands for real part of the quantity in bracket.
Physically this means the visibility of the signal due to all
light lost in the scattering-absorbing process. For non-absorbing
particles it would correspond to collecting all the scattered
light.
The term, V, is given by
V = ( 2J1(kfa))/( a) (6-8-2)
where kf = 211*Af , "Xf = X/(2SinX) and J1 is the Bessel
function of the first kind and first order. It represents
Vext as calculated using geometrical optics (Fristrom et.al.,1973)
or diffraction (Farmer, 1972).
Comparisons between Vext and V with respect to the
parameter a/ ).f has been carried out by Jones(1974) and revealed
excellent agreement for large absorbing particles (a/WO ).
Slight disagreement was observed for small (aA::,..1 ) and
non-absorbing particles.
163
Comparisons of Vscain the forward direction (ck.e = 00)
with Vext and V against particle size parameter a/X. are shown
in Fig.(6-8-a) to Fig.(6-8-d). The following sums up the main
observations:-
a) Vext is dependent on Xf , a and refractive index and
independent of aperture size because infinite aperture
is assumed in its derivation. It is only meaningful
when all the scattered light is collected for non-
absorbing particles or the total extinction is measured.
V is a function of f and a and is independent of the
refractive index of the particle. It applies when all
the forward scattered light is measured. Indeed, it
has been indicated by Robinson and Chu(1975) using
diffraction theory that only by collecting all the
light scattered forward by spherical particles ( i.e.
integrating over 2Trsteradian ) should the visibility
of the scattered signal reduce to Eq.(6-8-2). moreover,
V applies only to the paraxial direction of viewing.
Vsca is a measurable parameter depending on Xf, a, m,
angle of sizing and size of the aperture of the
* detecting optics.
b) For highly absorbing particles, the curves of Vsca and
Vext are smooth (Fig.6-8-a). However for non-absorbing
particles, the curves show a lot of ripples (Fig.6-8-b).
A detailed comparison of both graphs reveals that the
ripples oscillate along the smooth lines of the same
n1 and other physical parameters (i.e.)i, oI ,3 ,4c( andzip)
a cn
0,5
164
--- :\ • ---• • -•-.
% ■ ' a a \ . •--..
‘`\ s • N •••• s■ a . . • \ . • • \ \ (.0{,(3) ,(0.0)
..\ • ....S. \ \ \\ . \
A . . bl' \ -/ P \ \
\ • \ \\ ...
A \
v \ \ , .....
\ \ \
vext —\\■ I. \ \ ∎ .. ,\\ . \
. .
\ \ \
% \.. , \ V ' . \ \ *.
\ V d \ •
\ \ ■ . \
\ , \
..
\
% \ ‘
V. \
% • • \
% Os \
\ \s• I\ % le % \.
\ I
‘ % t ` N \ : \ .
\ ...I .‘,.,,,, \.\ I\ I. u \ •
N \ A 1\ \ 1 \ i..\:. \ S.\.\ \ -
I I I , I I I I ......, , 1 . 1 6 a/N
a) isd ...-Ap=0.4° b) AC( =13=40 .
C) Ad =4=80 . d) ac( = Ap=i 2° .
It
Fi. (6-8-a). Visibility as a function of aperture size. m= 1.50 -10.10, l= 3o t a= oo , p= 0°.
165
0,5
..--.. ..
...\,.. , ,\ .1 6-1,•-•.,,,. ._ - -- ^ .....
■ ••• \ • • : *,N4 \ ea is
\ ‘ \ . .. ...•\
., V ; ‘-' \ \ , I. .. • \ A., 9 . : • \ 1 % i \ . • •I , • I I 1 r\ . • • ,
■ \ ‘ / 1
1
V I . .
1 1 • i I I
\ \. •' 1.1 t 7, c
% . . . • I
.......___s l';',
. : .0
t • ,. % . 1 / II
A II • 1 I
II •
• 6 I ( 1 0 •
V k II 10 1 \ I i• t \ .. . I. 1
A •II • g ..\ 1
■
A \ ■ •°9 i
1
'\
• % •
A % I / A 6 d i
■ ,N) v i.\ 1 V ‘' I. I. i \ . •• : ., . . . , • • 1
1 0 : : 1 1\st : I . I t . I \..1 li \ 1,: : I ."‘ s
I
1" \ ' ' s : - I. " 1 a) ii(4=Api=0.4° . I ‘ i \ \ b) zok=zsp= 4°. ,/
1 • i 1
%. :. a 'N Si c) Ati,..elf3= 8°. - I: d) AoL--..413= 120. \ e \ :
1. I\ v : \ ‘•
i I I I 1 I i I I I \\' 1 \ 1 t 4 6 a/X 0 2
Fig.(6-8-b). Visibility as a function of aperture size
• for m= 1.50 y =
3° ,
= 00 6 AO
•
\ •••••-• %-/ \
\ VNt 11 • ••••• I
r \
\16::„..N
PO\
CT.". %.* 'S.. •
. 4 41e-
.• •• \ • • • .0.
%. ‘••-• •,% •
'"k44,N.■
Ve-xt
a
v y
•
4 It\ . . 4 ■ . ,
I . .."N■ ■ e
a . . .. 4
N, 1,
d Z..: •• v.i, N\ . • , s.`. . A
• • • S.
O. .0
a) Lot =ap= 0.40 b) AC( = 4f3= 40.
c) 4.d. =A13. 8°. d) 401\--.41= 12°.
Fig.(6-8-c). Visibility as a function of aperture size for m= 1.50, 1c =1.2°,04= oo , p= 00.
6 8 aiN 10 J
0,6
0,4
0,2
WNW
0,6
a) Ad=t1f3= 6°.
Fig.(6-8-d) Visibility as a function of aperture size for m = 1.20, f=
0.50 0(= oo to= oo.
I
0 (f)
0,8
2 a /X 1 0
168
c) Vsca varies strongly with aperture size. As this
increases while other parameters are held constant,
the value of Vsca drops. One cause is that as the
size is increased larger phase variations are introduced
across the aperture. Another interesting point about
Vsca against size of aperture is that at small aperture
and at of .f3= 0° , Vsav̂ A for all sizes of particle.
Indeed as iiot =4/3 vsel, 1 , for a/x >> 1 .
d) Other conclusions which may be deduced include:
(i) Vext...V when Xf > a and Zr\-0 (Fig.6-8-c). However, it should be noted that outside the
forward lobe, large disagreements may exist
especially at large angle of intersection (Eg.
Chou,1976 for X = 200.).
(ii) For transparent particles of low refractive index
(n1 ), Vext agrees very well with V. Reasonable
agreement with Vsca also occurs for particles
of diameter smaller than the fringe spacing.
From these observations, it seems that diffraction
theory for dual crossed laser beams agrees well with
rigorous wave theory in the forward direction when
a >> X •
(A) Comparison with Farmer(1974).
Farmer uses Mie theory to obtain the scattered light field.
By using the approximations ir and 2a much smaller than the
1.69
incident laser beam waist at the point of intersection, he
arrived at the visibility of the scattered signal, V, given
by. Va-2J1(kfa)/(kfa) (6-8-3)
This is independent of the Mie amplitude functions S1 or S2.
Farmer showed that the visibility curves versus particle size
off axis behave differently from the forward direction case
in that:
(i) The first diffraction lobe drops more quickly to
its minimum value for the off axis observation.
(ii) In the forward direction the secondary maxima
( i.e. for large particles) can approach unity
as well.
These observations agree qualitatively with us in terms of Vsca
( see Fig.5-2-b and Fig.5-2-e on pg. 89 ). Quantitative
comparison is not possible because the angle of intersection
is not given in his paper.
(B) Comparison with Robinson and Chu (1975).
This comparison is carried out in terms of Vscaagainst
detector aperture size. Here, a square aperture of sides
2L is placed perpendicular to the Z-axis at a distance Zff
from the origin (0,0) and passing through the centre of
the aperture. The visibility is calculated against the
parameter R=2aL/(XZff ). R is thus a measure of aperture
size. The results are shown in Fig.(6-8-e) and (6-8-f).
It is found that visibility is strongly dependent on
refractive index and size of aperture for particles in the
b
IA a
Fig.(6-8-e)
o Experimental points from Robinson and Chu.
Scattered visibility vs aperture size as a function of refractive index. (1) D/X = 2.06, a1 = 6.5 pm, D/X1 = 0.20; (a) Robinson and Chu, (h) n = 1.4, (c) n = 1.6. (11) /)/X = 9.56, Xi = 6.5 pm, DA, .4 0.93; (1) n - 1.46, (2) n = 1,49, (3) n t. 1.60, (4) 1.60 — i 0.10 (5) n = 2.79,'
(6) Robinson and Chu.
0.9
0.8
0.7
0.6
0.5
0.4
6 0.3 0
02
MN.
0.1
5'
• ■•
a
b_
1.0
0.9
0.8
0.7
Fig.(6 —8 —f)
Scattered visibility vs aperture size as a function of particle size. AA/ = 0.0974, Al = 6.5 mm (a) n = 1.4, DA = 1.80; (b) Robinson and Chu, DA = 2.06;(c) n = 1.4, D/X = 2.40; (d) n = 1.6, DA = 7.02; (e)n = 1.6 —1 0.1,1)/X = 7.02; 4) n = 1.6, D/X = 8.60; (g) -7-- 1.6 — i 0.1, D/A = 8.60; (h) = 1.6, D/A = 9.56; (i) = 1.6 — 0.1, DA = 0.56;
(j) ry = 1.6, D/X = 10.52; (k) a = 1.6 — 10.1,1)/A = 10.52.
0.6
0.5
04
0.3
0.2
0.1 *4, ••■ ••••.•
— — — - - - 0 2 3
172
size range considered by them (i.e. 2a410,um) for aperture
R71. In Fig.(6-8-e) experimental points from Robinson and
Chu are also plotted. As can be seen more experimental points
are really needed for rigorous verification.
(C) Comparison with ChIgier et.al.(1977).
In this paper, Chegier et.al. investigated visibility as
a function of particle size in the range 30 .um to 250 jun.
They found that at T=3.406° ,0t=§ =0° (i.e. along Z-axis),
visibility can be much higher than predicted by diffraction
theory, and that visibility drops as the aperture size
is enlarged.
These phenomena are explained qualitatively by using Vsca.
We have already observed that Vsca-•.1 at p(= (3 = 0°
with small aperture for very large at-A, and as the aperture
is increased Vsca is reduced in such a way that larger
particles have relatively lower V - sca They have also shown that while there is oscillation
of signal amplitudes against particle size for small
particles, these ripples are damped out for particle
diameter greater than 100sum. Again this agrees qualit-
atively with our predictions for small sized particles (see
Fig.2-4-b and 2-4-c ). However, their experimental points
do not contribute sufficient proof that there are no
oscillation within 100,< 2a “50,Lun, since they show only
six experimental points spaned across a range of 150,um.
Further, there is a striking difference in the predictions
* See reference Yule et.al.(1977).
than smaller particles.
173
for absorbing and non-absorbing particles which shows their
method to be very refractive index sensitive. Also, the method
requires absolute measurement of intensity,which is inconvenient.
(D) Comparison with Swithanbank e .al.(1976).
Swithanbank et. al. have developed an elegant particle
sizing method based on the forward diffraction of a single
beam (Hodkinson,1966). The table below summarises and compares
their technique with IDA sizing method.
IDA method Swithanbankts method
Theory Double crossed beams
using Mie theory.
Single beam using
diffraction theory.
Apparatus Simple and can be
automated. Simple and already
automated.
Measurable
parameter. Visibility. Scattered light
intensity versus angle
near o( = 0°.
a) No need to assume
a particle size
distribution.
b) High spatial reso-
lution( size of
test volume.)
a) A size distribution
is assumed.
b) Information integrated
along incident beam,
hence low spatial reso-
lution in this direction.
c) Best applied to moderately
dense particle cloud to
ensure representative
sample in the beam.
However, multiple scatt-
ering limits upper
concentration.
Character-
istics.
d) Best applied to
tenuous clouds(for
test volume of 1 mm3
the maximum particle
density is 109 m-3.
174
Furthermore, Swithanbank's method is independent of refractive
index of the particle, while LDA method is fairly independent
of refractive index only when particle sizing is conducted in
the forward direction. His system is applicable to particle
sizes in the range of 5-500jam. However, the dynamic range
of the LDA method depends upon three parameters: the fringe
spacing ( i.e. ), the angle of sizing and the aperture size
of the signal collecting optics. A rough estimate can be
obtained in the near forward direction by assuming Vsca has
a similar profile as Sin(kfa)/(kfa). These estimates are then
given for V.0.488 )um as follows:
Xf = 12 pm ;
D: 1--10 pm.
= 48 )im ;
D: 8-48 jam.
i. =120 dun ;
D: 20--120 jam.
From these comparisons, it is quite obvious that a choice
of the method can be made depending upon particle concentration,
particle size range and spatial resolution required. In a
tenuous cloud where the size and velocity of the particles need
to be known simultaneously, LDA method is particularly useful.
• 175
6-9 Limitations and suggestion.
It has been shown that laser fringe anemometry can be
used for particle sizing, although at the present stage, the
particle size distribution was chosen to be within the at
" sensitive region ". The limitation to size range depends
on how large this sensitive region is. An immediate extension
out of this range would be to devise a method capable of
distinguishing particles of size falling in the ambiguous
region.
Advantages of this technique are its relative independence
of refractive index in the forward direction and the simplicity
of the optical system and ease of operation. This makes it
promising for flame applications, where the optical characteris-
tics of the particle are highly uncertain owing to variable
amount of impurities. The technique has some limitations
which need to be improved. For example :
(1) Owing to the fact that only one particle at a time
must be present in the test-space, the concentration
in the particle cloud is limited.
(2) The detecting optics should be carefully designed
so that the test volume is well defined. This is
important when the method is applied to real systems
where particles are present everywhere.
(3) The dynamic range of the oscilloscope.
• 1.76
Another difficulty for measuring wide range size
distribution is the response of the oscilloscope.
If its sensitivity is adjusted to respond to the
smallest size, the largest signals will tend to
be outside the range, conversely if the adjustment •
is to accomodate the largest signal the smaller
particles may give no response, or the trace will
be too small to measure accurately.
(4) A more accurate and faster method is needed for
measurement of the traces, especially when it is
intended to determine particle refractive index.
Any automatic visibility measuring device would have
to be able to differentiate signals that originate in the
test volume or elsewhere, or alternatively an optical system
which well defined the test volume must be available.
Finally, as far as sizing a cold particulate cloud is
concerned, numerous techniques already exist some of which
are described in Chapter 1. Therefore measurement of a complete
spectrum of size range is possible using several techniques
in conjunction. For combustion systems, the situation is more
complicated. Here the laser beams interact with the hot flame
fronts causing them to wobble, and there are unknown inter-
actions between the particles and the flame plasma giving
rise to particle agglomeration, irregular shape and refractive
•
• 177
index distributions ( Jones and Schwar, 1969 ). Moreover,
these particles are likely to be in the region where neither
Rayleigh theory nor diffraction theory can be applied.
Unless an exact scattering theory exists for the appropriate
•
shape, the use of scattering for sizing particles in flames
will be limited. It is therefore felt that future advance-
ment in this field will be strongly dependent on the improve-
ment of the theory for irregular particles.
178
Chapter 7
Conclusions.
The following summarises the main contributions of the thesis:-
(1) An optical system has been designed and constructed to
carry out experimental tests of particle size measurement
in a gas stream. Sizes in the range 1-- 10 ,umwere
determined. It is found that the theory derived by Jones(1974)
provides a reasonable theoretical account of the measured
size distribution in the case of spherical glass ballotini
near to the forward direction and in the Y-Z plane.
(2) For irregular particles in a cold gas, theoretical and
experimental results agree to within 20%. The
discrepancy is probably due to (a) The choice of maximum
projected size of a particle sample under microscopic
measurement. This is unsuitable and some mean size needs
to be defined. (b) Higher random error is encountered
in making measurements on Doppler signals, as irregular
particles generally produce more noisy signals than spherical
particles. (See P-6-3-a on pg.134b). For irregular
particles in flames the signals are erratic because of
spinning and the breaking up of particles and this presents
some difficulty in analysis. A possible way of overcoming the
difficulty has been explored.
(3) A new way of defining the test volume has been introduced.
• It is found to be useful in connection with particle
179
sizing by this method (see article 3-4).
(4) Theoretical calculations have been carried out to investigate
the behaviour of Vsca as--a function of y , 9 (orot ), 4c ,
,m and the size parameter a/k . In connection with
particle size distribution measurement, it is found that
for Vsca versus size parameter: the curves generally have
more ripples for non-absorbing than absorbing particles
(Fig.5-2-f and 5-2-g) and the shape of the forward lobe
can be extended or contracted by varying the size of signal
collecting aperture (Fig.6-8-a to 6-8-d), the angle of obser-
vation Wig.5-2-b) and the angle of intersection I of
the laser beams (Fig.5-2-d and 5-2-e).
(5) A study of the behaviour of signal characteristics as a
function of particle trajectory through the test volume
has been carried out using a quartz fibre. It is found
that signals that originate from the test volume can be
differentiated from others.
(6) Experiments have been carried out to check the possible
difficulties of applying the method to particle sizing
in large scale turbulent flames. It is felt that the
boundaries between cold and hot gas that give rise to
wobbling of the laser beams are an important factor in
limiting its application. Refractive index gradient may not 4Re
be so important whenAlaser beam is incident on the flat
flame boundary at or near normal direction.
(7) A possible extension of the method to particle refractive index
measurement has been outlined.
•
Fig.(6-Z-a) Plot of numt= of counts versus size Parameter.
for m=1.458 , 1=1.1965°.
20
z
15
10
5 -
2,0
4,0 a/A
6,0
•
10
4,20
5L
3,90 I I
4,0 410 4,30 a/7 4,40
Fig. (6-2,-b) Calculation of m=1.458 and 1=1.1965° at (inner intervals.
20
15
Fig.(6-1Z—c) A plot of number of counts versus size parameter
for m=1.468 , 1=1 .1965°0
20
15
10
2,0 4,0 an 6,0
Fig.(6-27d) Calculation of m=1,468 and 1=1.1965° at a
finner intervals.
21
15
9
3,9 4,0 4,1 4,2 4,3 ci/ 4,4
Fig.(6-Z-e) A plot of number of counts versus refractive index
for one of the best size parameter 4.12..
20
15
10
5L J i 1 1 1 1 1 1 l 1 133 1,35 1,37. 1,39 1,40 1,42 1,4J 1,46 1,48 1,49 .151.
•
Fig. (6-p-f) A plot of number of counts versus refractive idle
for another size parameter (.4.070). It is fc-ind
that the best fit occurs at m=1.465 ,a/X=4.070.
15
9
3 - I I 1 1 1 1 1 1 1 1 1
1,33 1,35 1,37 1,39 1,40 1,42 1,44 1,46 1,48 1,49 1,51 --) m
186
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•
AA -1
Appendix A
Theoretical background to light scattering--- Mie theory
One of the most widely'used theories relating the
scattered radiation to the incident field, particle size
and refractive index was developed by Mie in 1908 ( Born
and Wolf, 1975 ). By applying Maxwell's equations, the
problem was solved for a plane, linearly polarized mono-
chromatic wave scattered by an isotropic homogeneous sphere
of arbitrary radius. The solution is most conveniently
expressed in spherical polar, coordinates. Let ( X, Y, Z )
be an orthogonal polar coordinate system the origin of
which is taken as the centre of the scattered sphere of
radius a. The incident wave is propagating along the
positive Z-axis, and the polarization of the wave is in
the X-plane. The incident field g can be written by
= Eo Exp -ikz + iwt ) ( A-1 )
The scattered electric.field at any point P is in general
elliptically polarized. It can be described by two
components Er and Eland which are electric components
perpendicular and parallel to the scattering plane respec-
tively. They are given by
AA-2
= -(1E0/kr ) Exp(-ikr + iwt)Sin S1(0 )
El = - iE0/kr ) Exp(-ikr + iwt) Cosp 82(A)
(A-3)
The scattering plane is defined as the plane containing
the incident wave and the scattered wave. The functions
sl(e) and S2(e) are amplitude functions given by
S pO (2n+1)
) -n=1 n(n+1) anlin(Cose) bn'En(Cose)) (A-4)
s2(e) =.5a (2n+1) (bnlrin(Cose) + arnI(Cose)) n(n+l)
(A-5)
where to and Trn are related to the Legendre polynomials
through
n(Cose) = de n d (1)1(CosA))
Trn(CosG) = (1/Sin9)Pili(Cose)
an and bn are two different functions depending on
variables x and m as follows:-
an(m,x)
.rnt m x)?n( x ) - ince(mx) 4( x ) (A-8)
( mx ) ( x ) - m Vmx ) 5-11' ( x ) n n
mx )(41( x ) ) cont( x ) bn(m,x)-
inrin(mx) rn( x ) 5°n(nix ),n1 ( x ) (A-9)
AA-3
where x = 2na/X , m is the regraCtive index of the particle relative to that of the medium,
%(z) = (Trz/2fjni.1(z) (A-10)
= 07z/2k44. (z) (A-11)
Jn+z , (z) and Hn2 +z(z) are the spherical Bessel function
and Hankel function of second kind respectively. c i(z)
and S'n(z) so defined are called the Riccati-Bessel
functions.
From Mie results, the extinction, scattering and
absorption efficiency factors can be expressed as
Qext 2/ )
1 (2n+1) Re(an (A-12)
00
Qsca = (2/x2) E (2n41) tian i 2 + Ibn1 2) (A-13) n=1
Qabs = Qext Qsca (A-14)
•
AB-1
Appendix B
Diffraction theory for LDA
The diffraction theory approach is based on the fact
that forward scattering by a particle can be described by
Fraunhofer diffraction. Robinson and Chu ( 1975 ) used
this concept to predict the variation of visibility with
particle size and detector aperture. The proce-dures are
described briefly below:-
Referring to the figure on AB-2 at any point ( X, Y )
in XY-plane where two laser beams cross, the optical
disturbance Ut(X,Y,t) is in general a function of aperture,
direction of polarization and angle 6( . In the presence
of a particle the amplitude of the light transmitted is
Ut(X,Y,t) = Ut(X,Y,t) T(X-Vxt, Y-Vyt) (B-1)
where Vx and V are velocities of the particle in X and
Y direction and T is the transmittance function of the
particle.
By using Fraunhofer diffraction theory, the disturb-
ance in the far field at the point (t ;I) is given by
the Fourier transformation of Ut(X,Y,t), that is
oo U (H,t) = c.gu (X,Y,t) T(X-Vxt, Y-Vyt)
Exp(-ik x - ik y) dx dy (B-2)
ignal collectir-Aperturedl.
1
AB-2
Fig.(B-1-a) Geometry for the diffraction theory of LDA.
11 and E2 are in XZ-plane, and I represents diffracted
light.
AB- 3
The total intensity I(t) received by an aperture in the
(j',1!)-plane and substending a solid angle n is
I(t) = f.r1UF(,11.,t)1 2q dTL (B-3)
Using eq.(B-3), the visibility is -defined as
Vs = (I(t)max -I(t)min )7(I(t)max +I(t)min ) (B-4)
It is obvious from eq. (B-3) that the visibility
will be dependent on the aperture size -n- . In the
case when particle is spherical, the transmission
function is given by
T(X,Y) = 1 - Circ (r/a)
where r2 = x2 + y2 and Circ is defined by
1 for r < a
Circ (r/a) =
0 for r > a
(B-5)
This shows the dependence of visibility on the size of
particle.
r.
AC-1
Appendix 0*
With reference to Fig.(2-2-a), a scattered ray from
0 to P at the detector plane is described by (r,G,10
with respect to (X,Y,Z)- axes and similarly by (r- 1191,01)
and (r ,G2,02) with reference to (X1,Y1,Z1) and (X2,Y2,Z2)
systems respectively. These coordinates are related by the
following transformation rules:-
Z = r Cos() Y = r Sine Sin0 , X = r Sine Goa
Zi= Y Sint+ Z cos y = YCos' - Z Sin/1 , X1 = X
(c-i ) and Z2=-Y Sin .6 Z Cos 21
Y2 = Y CosW+ Z Siny , X2 = X
From these we obtain,
SinG1Cos01 = Sine Cos0
Sin61Sin01 = SinG Sin0 Cost- CosO Sin W*
Cosel = Sine Sin SinT+ Cos() Cosh
and
(C-2a)
Sine,CosO, = SinG Cosi)
Sine2Cos02 = SinG Sind Cos t Cose Sin 11
(C-2b)
CosG2 = - Sine Sin Sin-t+ Cosh Cost
Since isca = (0-3 )
Ikb * Dr. A.R. Jones, Private communication.
AC-2
,P2 )
(0-4)
where Ee and are related to (E01 ,E02 and (Eel
through
"Ilm. Am, .1.■■■
EG d99 de0 1 1 Eel deG2 'D2%62 EG2
Eco_ 04101 01001 E01 04102 (*02 E02
here O. are transformation matrix elements, they can be
calculated using the relation
Clue = (hu/hv) (C-5)
and Eel, E01 , Ee2, and E02 are electric field components
given by
E81 2 = Exp(- ikr) Exp(!ikY Sind) Cos0V2(04) ,
B01,2 = 12) Exp(-ikr) Exp(tikY Sinal) Sink0
Apply Eq.(C-5) to (C-1), otic
are calculated as
001= (1/Sin91)(Sine Cos/C-Cos9 Sinct. Sira)
1 1
o4 wsine2)(sine cosz+ Cos() Sin Sin2i)
02 (C-7)
e02 =d002
wl = (1/Sir1491) Cos(1) Sind
0/0, = -(1/Sine1 ) Cos0 '1
002 = (1/Sin02)Cos0 Sind
(),002 = (1/Sin92)Cos0 SinT
By expanding eq.(C-4) and grouping, we have
1E012
012 "
KoPeol+ c44091 IE91I2+(p6Ae +d0927)(93.°2 E92
• (40101+ C100A01) E01±(()LOA 4. 4002 )IE02r
• (j00A01+ C ei:5(00 ) 1E01+41-Peel+ (7091;01 E01;1
)E E 02.°2
+ Div0-139302 )E01E02+40 -23901+0'00-2300 >E02E01
)E E +(() C7 4. ot C7 ,E 2 81 02 8°2 881 002 081 02
E 81
(4)013-8e +°0 1.5092)EO1E024.(° Ge23901+ '4312 001)
E8E01
2 02 4- 6 A02) E92E02 -A02°eei"90 60 )E E 02 82
(C 8)
Using relations (C-2a (C-2b) and (C-7) it is easily shown
that:-
(1) The coefficients associated with Ee1I2 11E0j
1E0 r and N212 are all unity.
•
46
AC-4
(2) The coefficients for terms E 01E01 E01E01 , E02E02 and E02EG2 are all zeros.
All other terms are given below:-
deeide° 49262 = C43e2iGe1 ± d0E924e1
=( /(Sinei Sine ))(Sin2e Cos21 -Cos2e Sin20 Sin2Y-
Cos20 Sinq)
D1490,402 + °100002 de020901 + 14002430i
=(1/(SineiSine2 ))(Sin2e.Cos2W- Cos20 Sin20 Sin2 -
Cos20 Sin2r
deeP)G02 ()0e2Octs2 =°1002deei d002°001
=(-2/(Sine1Sine2))(Cos0 Sine Cos Sing)
ae 1dee2 °1001cl002 de9261 °I0G2001
• =(2/(Sine1Sine2))(Cos0 Sine Cos Sid)
Substitute these values into (C-8), we have
2 2 2 2 1 Isca = ( Eel + E82 Y
+ Eni 1
+ E02 ) - SineiSinG2
A11(E01Ee2 E02E01) A11(E01E02 E02E951)
1111(E01E02 E02E01) E11(E01EG2 Ee2E01]
(C-9)
AC-5
where
A11 .(Sin20 Cos26 - Cos20 Sin20 Sint?'- Cos20 Sint')
B11 .(2Cos0 SinG Costa Sin W)
It is obvious from (C-9) that the terms in the first
parenthesis constitute a D.C. component while those enclosed
in the second parenthesis constitute an A.C. part.
From equation (C-6),: if we set( For the functions Sig) are in general complex quantities.)
S1 (81) OratExp(ial)
S1 (82) =q.2Exp(ia2)
S2(01) .(LlExp(ibl) S2(82) =6-b2Exp(ib2)
we have
1E, 1 =(4/k2r2) Cos201
lEb212 =(4/k2_2\ ) Cos20, 4E2 ` u2
1 I 12 2 \ d E 2 r2 ) Sin2 wi r72 al a1
lEa212 --(4/k2r2 ) Sin202
2
(C-10)
Eel 292
E 92 91 ' 01 r)
=,E2/k2_2\ Exp(i2kYSinI)CosO1Cos02
O Exp(ib1-ib2) + (E/k2r2) Exp(-i2kYSira)
CosO1Cos0232%Exp(ib2-ib1 ).
(2Eg/k2r2)(T l 01-3 2 CosO1Cos02Cos(2kYSinT+b1-b2)
b
1
AC-6
Similarly,
E01E02 E0 17301
(24/k2r2) (Tl Cra2 SinOlSin02Cos(2kYSinT+al-a2) a
EG1102 = E0241 =
-(24/k2r2)% cr SinO2CosO1Cos(2kYsir3+b1-a2) a2
E01102 = E02201 =
-(2E2/k2r2) CTal Cio2Cos0 Sin01 Cos (2kYSin'(+al -b2)
Combining (C-11) with (0-9), Isca can be written as
/sea
.(E/k2r2)fSin2Y1 02 sin2d 2a2 + Cos2d 1'2
l 2 1
Cos202°1-:;, +Sine1Sine2 2A11 1 e2CosO1Cos02
(C 11 )
Cos(2kYSirl+b1-b2) +
Cos(2kYSinW+a1-a2) +
nos(2kYSinW+bi-a2) +
Cos(2kYSino+al-b2)1
2A116-a. Oa SinO1Sin02 1 2
4B11crb CosOlSin02 1 2
IV° rr (7. SinOlCos02 -11`t.C1:12
(C-12)
The scattered power, Psca, accepted by a photo-dector is -
psca = S IscadA . SIsca r2 Sine ded0 • (C-13)
•
AC-7
where dA is an elemental area of the aperture. For computing
purpose, it is convenient to cast (C-12) in the Mix' as
Isca = + F R+xp(ikYSin)(41-52) (C-14)
then
Ps caS D dA + F RelExp(ikYSiniZi )1. dA
D' + FICos(kYSini+r (0-15)
The visibility Vsca of the integrated signal is
Vsca = (F'/D') (0-16)
Equation (C-16) is computed numerically using 6-points
Gaussian integration method*. The basic procedures are
to be found in Abromowitz and Stegun(1964). For one
dimensional case it is essentially as follow:-
To calculate the area of F(x) in the interval a4=x4.b, we
first transform the interval into the standard one, that
is, between[-1, 1]using
xi = (b-a)zi/2 +(b+a)/2 (0-17)
where -1 4 z4 .4. 1 _
Secondly, calculate the weighting function wi given by
wi .2/(1-zi)2(q(zi))2 (0-18)
where 1"(z) is Legendre polynomial.of order n .
*Thanks are due to Dr. A.R.Jones for supplying the necessary
• program.
AC-8
Finally, integration is done by the following finite sum
SF(x)dx
a
For example, when N=6 (i.e. Six-points approximation)
the values of ziand -wi are given below:
N ((b-a)/2)E wiF(zi) + Rn C 19)
1=1
z1 = 0.238619186083197
z2 -z1
z3 = 0.661209386466265
z4 = -z3
z5 = 0.932469514203152
z6 = -z5
wl = 0.467913934572691
112 = wl w3 = 0.360761573048139
w4 = w3 w5 = 0.171324492379170
w6 = w5
AD-1
Appendix D
Derivation of fringe contrast, Vc , in the test space.
With reference to Fig.(2-2-a), the amplitudes and phases
of the two incident beams with respect to (X1, Y1, 1) and
(X2, Y2,Z2) coordinates systems can be written as
El = A1Exp(-(X12 + Yl2 )/62) Exp(-ikZ1)
(D-1)
and E2 = 2ExP(-(X2 Y )/0) Exp(-ikZ )
(D-2)
respectively. Using transformation relationships C-1),
(D-1) and (D-2) become
. El = A1Exp(-(X2 4. Z2Sin2r 2ZY CosW Sin1" + Y2 Cos246,. Ha2
Exp(-ik(Z Cos + Y SinT)) (D-3)
and E2 = A2 Exp(-(X2 + Z2Sin2 2ZY Cos/rSinlf+ Y2Cos3r)A52).
Exp(-ik(Z Cos if - Y Sinbl) (D-4)
for coherent addition, the total intensity I at the point of
interference is
I = (EI E2)(E1 + E2) (D-5)
Here the bar represents complex conjugate.
if
AD-2
Substituting equations (D-3) and (D-4) into (D-5), and assuming
A1-- A2 A, we obtain
2 2 2 2 2 I = A Expk-2kY Cos y + Z Sin 11+ )/62).(2Cosh(
4YZ Cosb Sinlr/6 ) + 2Cos(41TY SinVN))
(D-6)
It can be seen from the above equation that the fringe spacing
Nf is given by Xf . X/2 Sin 3.
By definition of fringe contrast Ve,
/max - 'min . Ve (D-7) max + 1min
and using equation (D-6) we obtain
Vc = (1/(Cosh(4YZ Sin CosW/62)
(D-8)
When geometrical mis-matching is taken into account, we
introduce small displacements DX, DY and DZ into beam No. 1.
It then becomes
= A1Exp(-((X +DX)2 + (Z + DZ)2Sin2W - 2(Z +DZ)(Y +DY)
)2 2../el/x-2\ (Z +DZ)COST) Cnq'Sin + YIDY (-1 (----/ --- U.1 /0
+DY) Sin "6 ) )
(D-9)
By substituting equations (D-4) and (D-9) into (D-5), we get
AD-3
I = Exp(-2(Z2Sin2-2c + Y2Cos21 + 2YZ Cosninr + X2 )/62 ) +
A2Exp(-2(Z +DZ)2Sin2 Y + (Y + DY)2 Cos2 - 2(Z + DZ)(
Y +DY)Cosb' Sin 11+ (X +DX)2 )/0-2 ) + 2A2Exp(-(Z2Sin2 '+
Y2Cos2 2S + 2YZ Cos 25. Sin -6.+ X2 )A32 ) Exp(- (Z +DZ)2Sin26-+
(Y + DY)2Cos2 K - 2(Z + DZ)(Y +DY)CosZC Sin + (X +Dx)2/0-2 )
Cos(k(2kY Sin + (DZ)Cos?r+ (DY)Sin.6 ))----(D-10)
After expanding and re-arranging , we finally obtain
I = A2Exp(-2(Z 2Sin2 /5 Y2Cos2Zr + X2 )A5 )(B.I + B2 +
2B1B2Cos kB3)----(D-11 )
where
B1 = Exp(-(2ZY Cos)" Sing )/0-2 )
B2 = Exp(-((2ZDZ +(DZ)2 )Sin2 + (2YDY + (DY)2 )Cos2 -
2(ZY +ZDY +YDZ + DYDZ)Cosi Sing +(2XDX + (DX)2 )/6- ))
and. B3 = 2Y Sin .6+ DZ Cos /c+ DY Sin b'
The fringe contrast V0 is given by
2B1 B2 - 2 2 131 +B2
(D-12)
AE-1
Appendix E
To show that
tan0 = Sinck/tanp
and CosG = Cos cd cosp
we refer to Fig.(2-2-c) and notations in there
Since,
tan0 = a/b
tamp= a/( b2+ d2M
and Sine,/ = b/( b2+ d2)f/4
From (E-3), we obtain
tan0 = Sin04/ tan,S
Similary, we have
Cos8 = d/r
Cos = d/( b2+ d2)114
and Cos = ( 102+ dq/r
Hence, we deduce from (E-4) that
Cosa = Cos Cosa
(E-3)
(E-4)
J. Phys. D: Appl. Phys., Vol. 9, 1976. Printed in Great Britain. Q 1976
A light scattering technique for particle sizing based on laser fringe anemometry
N S Hong and A R Jones Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2BY
Received 17 March 1976
Abstract. It is demonstrated that the visibility, or modulation depth, of the AC signal produced in the light scattered by a particle crossing an interference pattern can be used to obtain particle size. A method is described for direct determination of size distribution which is not strongly dependent upon refractive index.
1. Introduction
A popular tool for the study of fluid flow is fringe anemometry (Durst and Whitelaw 1971) in which small particles added to the flow traverse a test region where two laser beams cross to form an interference pattern. The scattered light has an AC component the frequency of which is proportional to the local velocity of the particle.
Recently several authors (e.g. Farmer 1972, JoneS 1974, Robinson and Chu 1975) have explored the possibility of using the fringe anemometer for particle sizing, since the amplitude and visibility of the scattered AC signal are related to the dimensions of the scatterer. In thiS paper the rigorous solution given by Jones (1974) for a sphere has been extended by developing a computer program which integrates the scattered intensity over the angular field of view of the detector. The calculated visibility of this integrated signal is compared with experimental signals from single particles, and a particle size distribution is built up. This is compared with a distribution obtained by counting a sample under an optical microscope and generally found to be in close agreement.
The method is easy to use and can be made fairly independent of refractive index. The particular technique used here is restricted to particles with diameters greater than the wavelength, and sizes in the range 2-10 p.m were used with an argon ion laser of wavelength 0-48811m.
2. Theory
The interf.-.rence pattern is presumed to be formed by two plane waves propagating in the (vo, yo, .70) coordinate system, as shown in figure 1. Relative to the x-axis of the spherical scatterer which has its centre at (0, Y, 0) the two waves are
exp (ik Y sin y) exp [ik(z cos y +3, sin I,)] =En exp (— ik -Y sin y) exp [ik(z cos y—y sin y)].
1839
2
M I LA
Figure 1. Apparatus and, light scattering system: A, aperture; B, beam splitter; L, lens; LA, laser; M, mirror; osc., oscilloscope; P, prism.
1840 N S Hong and A R Jones A light scattering technique for particle sizing . 1841
Each wave represents a normal plane wave rotated through the angle +y and having a phase shift corresponding to the position of the sphere.
Assuming linear scattering we apply the Mie theory (e.g. Kerker 1969) to each wave independently and add the resulting scattered waves allowing for the appropriate phase shifts and rotations. The Mie theory yields Eel and Ed1 in spherical coordinates
01, 01) for Esca(1) and Eh and Eo in (r, 02, 02) for &am. These can be related to E0 and Leo in (r, 0, 9) through the matrix formulation
otoi.0 )(E0,) +(lee, oroo )(E02 ) ock yi (zoo, ashy E02
the scattered intensity being given by
lees=lEeI 2+IE0I 2.
The elements of the matrix are given by
_hu C4"— hv dU
h being the metric coefficients (see e.g. Morse and Feshbach 1953). Details of these matrices are given in the Appendix.
The power received by a detector at (r, 0, 0) is
Pima= isca(19 0)r2 dig
where dig= sin 0 d0 d4> is the solid angle subtended by the detector aperture at the origin and the integration is performed over the aperture (see figure 1). Practically
y
Figure 2. Relationship between coordinate systems.
the angles (a, 13) were measured as shown in figure 2. These are related to the angles (0,0) through
tan 4> = sin a/tan /3
cos 0= cos P Cos a.
The aperture defined the range of angles + Aa and ±A13. The integral can be expressed in the general form
Psch = A + B cos (1c + tk)
where A and B are functions of particle radius a, refractive index, 0, 4>, Ace and .A/3. lit is a phase term also dependent upon these parameters. kr =2k sin y=27r/Ar where At is the fringe spacing. The visibility of the integrated signal is
PPM max—Peen. min B _ . Foca, max +Poen, min A
The double integral over a and )3 was performed using the Gaussian six-point method (Abramowitz and Stegun 1964) applied twice. In view of the demands upon computer time this was taken rather than a larger number of points, though a few cases were integrated using 32 points. For the small apertures used in practice there was negligible difference in the result.
One result of this integration (figure 3) shows the signal visibility as a function of the size of a square aperture at 0=0 in the zy-plane. An important feature is the lack of sensitivity to refractive index at small finite apertures, while retaining sensitivity to particle size. The implication is that at small angles (a —/3 — 0) and for small apertures particle sizing may be performed almost independently of any knowledge of refractive index.
Experimentally, all measurements were made in the zy-planc with the centre of the square aperture at 13=0 (0=a, 0=7/2). The effect of varying the angle 0 is indicated in figure 4. We note that at 0 =0 the curves are not sensitive to particle size for in-finitesimal aperture or fringe spacing large relative to particle size, the visibility being always close to unity. At other angles visibility can vary considerably with size. Each curve may be divided into three regions. In the first the particles are small and visibility is close to unity, while when the particles arc above a certain size visibility is not a unique
(Eo) (eel Ecs 0/01
1842 N S Hong and A R Jones
10
09
08
06
:5 05
04
03-
02-
01-
C - -----
0 30 • - Arc 60 90
Figure 3. Visibility as A function of aperture size. a/A..1.03, in3r.,0.1: (A) ip=1.4; (II) n=1.6, a/A=4.78, 034=0.465; (C) n=1.49; (D)n=1•6; (E) n=1-46; (F) n=2.79; (d) n=1.6 — WM.
10
09
08
07
T 06
05
> 04
03
02
01
0025 3
I5 20 r 'C,rves D 1. 0 50 6'0 7.1 Curve A
u/A
Figure 4. Visibility as a function of particle size and scattering angle. Ac<=0.59', Af3,--2•16°. (A) Ab3r----0405, x=i3--- 0, n=1-6. NAt=0.5, )3=0, n— i0.1; (13)n--.0-3'; (C) a=5°; (D) cc=y=1448°.
function of size and ambiguous results would be obtained. The useful region lies between these two extremes. The width of this region can be adjusted by suitable choice of 0 and Ar.
The experimental measurement of scattered visibility can only be compared with theoretical predictions if the fringe contrast in the test space is perfect. For this to be
A light scattering technique for particle sizing 1843
so a number of requirements have to be met, as follows:
(i) The two laser beams have equal intensity. (ii) They have perfect temporal and spatial coherence.
(iii) They are polarized in the same plane. (iv) They are 100% polarized. (v) They are infinite plane waves.
(vi) There is no mechanical vibration.
The first four conditions are easy to satisfy, at least approximately, if a laser is used. Condition (vi) is approached by providing a rigid base on anti-vibration mountings.
Condition (v) cannot be met since the laser beams are in fact Gaussian plane waves of the form:
&Ic a) = Eo exp (ikzi) exp (—(y12 +,c12)1o2) Eino (2)= exp (ikz2) exp (Y22 + x22)/(72.)
in the coordinate systems (xt, y1, z1) and (x2, Y2, z2) which are obtained from the co-ordinates (x, y, z) by rotations through y or —y respectively. Assuming that the beam centres coincide at z —y=0, the distribution of contrast in the test space is of the form
[cosh (4xy cos y sin y/a2)]-1.
' We shall define the test space as that sphere of radius R enclosing only fringes of contrast 0-95. The locus of points of constant contrast is a cusp at the minimum of which x=y. The radius of the inscribed circle is given by R 2.-- x 2 -1-y2. Thus R is obtained from cosh (2R2 cos y sin y/a2) 1/0-95. For the Spectra-Physics model 162 argon ion laser u=0.325 mm, and taking a typical y=1.44° gives /2, -0.82 mm. The collecting optics should either be arranged so that its depth of field lies entirely within this sphere, or the particle injection system must be designed so that particles pass precisely through this region. As this work was intended to test the basic method, the latter, simpler method was chosen here.
3. Experimental details
A diagram of the apparatus and light-collecting optics is given in figure 1. The light source is a Spectra-Physics 162 argon ion laser giving an output of 10 mW at 0.488 rim. A beam-splitter divides the laser light into two equal-intensity components which are made to cross at the test space. This is at the centre of a horizontal circular track around which travels the photomultiplier (EMI type 9635B) with its associated optics. The output from the photomultiplier is displayed on a storage oscilloscope. The beams are adjusted so that they cross at a pointed pin protruding out of the centre of the track. The scattered light from the pin is imaged by lens L on to the pin-hole of the photomulliplier (PM). It is then adjusted until the image of the pin is sharp for.each of the two beams in turn.
Initially, spherical glass ballotini particles were used. A coarse sample of size range 0-60 pm was placed in a fluidized bed and the flow rate adjusted so that particles in the range 1-10 him were boiled off. These were then injected horizontally into the test space. On the other side of the test space and in line with the injection nozzle a suction tube was provided to assist with alignment.
1844 N S Hong and A R Jones
A bypass system was provided to ensure a sufficiently low concentration that only one particle triggered the oscilloscope at a time. If more than one was present simul-taneously in the test volume it would result in a reduction of signal visibility due to incoherent mixing (Fristrom et al 1973).
10
OS
00
07
•••
2•-•
(01 N,
• • • Ns. • \ • , ••• \
5•., \ •
06 .5 •5
• • • • •••.. •\." -05
< e • • • \ 04
\. • • • *. 03 \
02 \ .5 • \ • e•
\ • 01
a/.1
25
20 I Ic)
15 )1.
10
5
1.1.1 02 04 06 08
v„„ 2a (pm)
Figure 6. (a) Calculated visibility as a function of size and refractive index. A/At=0.036, a = 3.65, p=00..= 1.08°, = 0 n=1.6; • n= 1.65 ; n=1.55. (b) Visibility histogram for glass ballotini. (c) Measured size distribution; bars from light scattering, histogram from optical microscope.
A light scattering technique for particle sizing ' 1845
To check whether the test volume conditions were satisfied, particles less than 1 ion in size were used to explore the fringe system. A second check was to look at scattering at 0=0°. In both cases scattered visibilities greater than 0.95 were observed, in the latter case for particles of all sizes in accordance with the theoretical prediction as in figure 4 (curve A).
The fringe spacing and angle 0 were chosen so that the whole particle size distribution lay within the sensitive region of the visibility-size curves. A large number of particles were registered (at least one hundred) and the signal visibilities measured from the oscilloscope traces. A typical signal is shown in figure 5 (plate). A histogram was constructed of the fractional number of particles against visibility and this was converted into a size distribution using theoretical curves of the type 'shown in figure 6(a). Figure 6(b) shows the histogram and figure 6(c) the resulting size distribution indicated by the broken lines.
The effect of varying refractive index is also shown in figure 6(a). It is found that the uncertainty due to not knowing this parameter, and also due to oscillation of the curve, can be estimated readily since all the variations can be encompassed within the two envelopes shov.ii. Although theoretical curves in the reasonably expected refractive index range are shown in this figure, computations have shown that for all refractive indices greater than 1.3 the curves will lie within the envelopes. This variation of visibility will introduce an uncertainty into the measurement of size. The magnitude of this is indicated by the length of the broken lines in figures 6(c), 7 and 8.
While a count was in progress, a sample of the approaching particles was obtained by placing an electron microscope grid across part of the tube entering the injection nozzle. This sample was sized using an optical microscope, although electron microscope pictures were obtained of a selection of the samples to confirm that no smaller particles were present. The resulting size distribution is presented as the histogram in figure 6(c).
tb)
6 8 2a rpm)
Figure 7. Measured particle size distributions; bars fican light scattering, histograms from optical microscope. AlAt= 0.029, a= 3.2°, 13=0°, Gia= 1.56°, All= 0.39°. (a) Glass ballotini seeded into a flame. (b) Glass ballotini in absence of flame.
10 20 30 40 50 60
---- - —11
1846 • N S Hong and A R Jones
is
10
5
0
2a lvm)
Figure 8. Measured particle size distributions for Rutile (TiO2); bars from light scattering, histogram from optical microscope. Ahlt=0.036, a=3.6°, )9=0, .a=l•08°, Af1=0 93°, n=2.65.
• It is not felt that this method of sampling is truly satisfactory, but the good agree- ment between the two methods is encouraging as they are not likely to be subject to the same errors.
Further results arc shown in figure 7, including one example in which a thin fan-shaped flame was burned on the nozzle. Any fault in the system tends to reduce the fringe contrast so that the signal visibility is lower. The result is that the fringe method will overestimate size. This is evident in all the results, but particularly so when a flame is present due to motion of the hot gases and subsequent disturbance of the laser beams. The maximum disagreement between the fringe method and the optical microscope is about 15% but is more usually of the order of 5 % which is well within expected experimental error.
Figure 8 shows a result for irregular particles, specifically titanium dioxide. The optical microscope count was based on the maximum dimension and, not surprisingly, the scattering method predicts a size less than this, in fact of the order of 0.85 of it. . Evidently some mean radius would be predicted.
Apart from loss of fringe contrast in the test space, which has already been mentioned, errors arise due to the measurement of traces from the oscilloscope screen, size of aperture and the angles 0 and y. From the theoretical calculations two points emerge. First the errors in visibility are smaller for a larger fringe spacing at any fixed size; second, uncertainty in y (or fringe spacing) is the most important source of error.
In our experiments the fringe spacing was measured from photographs using an optical microscope, and the angle using pointers and a vernier scale. From these the uncertainty in y was about ±0.04°. Due to the finite beam width 0 could only be measured to about + 0.1°. The error introduced by these uncertainties is less than 5% if the maximum particle size is less than 6 p.m and less than 10% for particles less than 10 1.1.m.
A light scattering technique for particle sizing 1847
4. Conclusions
The feasibility of the fringe anemometer as a means of sizing particles has been demon-strated.The technique has the advantages that it measures the size distribution directly, is insensitive to refractive index, is simple to operate and is capable of automation. Furthermore the nature of the signal makes it easy to distinguish from noise.
Its limitations include the fact that presently some foreknowledge of the particle size is needed to select the correct fringe spacing. This may possibly be overcome by having a variable fringe spacing, beginning with At so that all visibilities are unity and then slowly decreasing Ar until a sensible distribution is obtained. A second disad-vantatte is that only one particle at a time must be present in the test space which limits the concentration in the particle cloud. In these experiments the laser beams were unfocused and the test volume was of the order of 1 mm3. This suggests an upper limit concentration of 109 m-3 for one particle at a time in the test space. If the beams were focused this would be considerably improved. Reduction in the dimensions of the test space to 0.1 mm, for example, would enable a maximum concentration - of 1012 m-3 to be examined. The scattered light signals from the particle may be con-tinuously observed on the oscilloscope. It is therefore possible to determine the mean time interval between particles. lf this is large in comparison to the width of the signals, it is highly improbable that two particles would be present simultaneously. In the experiment described here the concentration was small and this situation held, but generally this may be used as a criterion to ensure that the scattering is due to single particles only.
In application to a real particle cloud the collecting optics would have to be carefully designed so that the test volume is well defined. This should not prove an insurmountable' obstacle. Various authors (e.g. Farmer 1972, Brayton 1973) have examined the detailed structure of the test space, and because of the wide use of fringe anemometers consider-able attention has been given to optical design.
Acknowledgments
This work has been carried out with the support of Procurement Executive, Ministry of Defence.
Appendix
The relationships between the various coordinate systems are:
x=rsin0 cos# x1,2=x
y=r sin 0 sin. y1,2 =y cos y z sin y
z=r cos 0 z1,2= +y sin y-Fz cosy
from which sin 01,2 cos #1,2= sin 0 cos #
sin 01,2 sin 01,2 = sin 0 sin # cos y+ cos 0 sin y
cos 01,2= + sin 0 sin # sin y + cos 0 cos y.
•
•
1848 . N S Hong and 1 A R Tones
It is then found that
0,001, a =sin 01.2 (sin 0 cos y -T. cos 0 sin 56 sin y)
czoi. 2 — 1 ,, cos y6 sin y sin ui.,2
Cee1, 2 =4:11‘601. 2 a40t, a = —afeit$2. 2. The electric field components are
Eoi 4..2= — exp ( —ikr) exp (± ik Y sin y) cos sdiaS2(02,2) — kr
Eoi F-roa = k r exp (— ikr) exp ( ± ik Y sin y) sin 01.2S1.(01.2)
where Si.(0) and S2(0) are the Mie scattering function, as defined, for example, in Kerker (1969).
References
Abramowitz M and Stegun IA eds 1964 Handbook of Mathematical Functions (Washington DC: US NBS) Brayton DB, Kalb HT and Crosswey FL 1973 Appl. Opt. 12 1145-56 Durst F and Whitelaw JH 1971 Proc. R. Soc. A 324 175-81 Farmer WM 1972 Amt. Opt. 11 2603-12 Fristrom R M, Jones AR, Schwar MJR and Weinberg FJ 1973 Faraday Symp. Chem. Soc., No. 7 183-97 Jones AR 1974 .1. Phys. D: Appl. Phys. 7 1369-76 Kerker M 1969 The Scattering of Light (New York: Academic Press) Morse, P M and Feshbach II 1953 Methods of Theoretical Physics (New York: McGraw-Hill) Robinson D M and Chu WP 1975 App!. Opt. 14 2177-83
J. Phys. D: Appl. Phys., Vol. 9, 1976—N S Hong and A R Jones (see pp 1839-1848)
Figure 5. Typical oscilloscope trace due to particle traversing test space.
and
Incident beams in yz-plane
\- 4
`
Fig. 1. Coordinate system.
o -
b
Light scattering by particles In laser Doppler velocimeters using Mie theory
N. S. Hong and A. R. Jones Imperial College, Department of Chemical Engineering & Chemical Technology, London, SW7 2BY, England. Received 11 May 1976.
Since 1964 the technique of cross-beam laser Doppler velocimetry has proved to be a useful tool in local flow mea-surement) Further development of the technique for ob-taining information on particle size, shape, number density, and refractive index is intensive.2-4 A measurable parameter in this case is visibility of the scattered light V„,„, which is defined as
(1)
where P„.„ denotes power of the scattered beam defined in Equation (2).
Recently, by using diffraction theory, Robinson and Chu' have calculated the dependency of Vs„„ on the size of the en-trance pupil of the detector (aperture). It is known from Mi theory' that Vsca is, in general, a function of (1) complex refractive index a = al in 2, (2) size of the particles D, (3) fringe spacing Af, (4) angle of sizing (04), and (6) aperture size (Ati,A0). This dependency can be obtained by integrating the scattered intensity over the detector aperture",l.e.,
PnCli .18 1.ca(n,D,Af.0.0)r 2 Sinedoc10, (2)
.19s
r being the distance from the scatterer to the detector. The coordinate geometry is shown in Fig. 1. The two incident light beams in the coordinate systems (r,01,01) and (r,09,09) are scattered independently, and their amplitudes are added to provide the scattered intensity. The specific form of Is„„ is derived from Mie theory', and is given by
1E01 2
where
all,,,croo , Ed, creoi cre,b2 Es? (E) = („„x„) \E
and (j = 1 or 2), 1
000, = 044) _sin°j (sin° cosy + (-1)' cos° since
(-1)1 crov ) = –a045; = sin(ii cos siny
with
cosOi = sin() costt,
sing, simhi = sin() shut) cosy. + (–.1) 2 cos° siny
cosOi = (-1)'-1 sin0 sing) siny + cos0 cosy.
Fig. 2. Scattered visibility vs aperture size as a function of refractive index. (I) D/X = 2.06, Al = 8.5 min i MAI = 0.20; (a) Robinson and Chu, (b) n = 1.4, (c) a = 1.6. (II) D/X = 916, X/ = 6.5 pm, D/Xf = 0.93; (1) n = 1.46, (2) a = 1;49, (3) a = 1.60, (4) 1.60 – i 0.10 (5) a = 2.79,
(6) Robinson and Chu.
2 3
Fig. 3. Scattered visibility vs aperture size as a fund ion of particle size. X/X/ = 0.0974, Af = 6.5 pm (a) n = 1.4,1)/X = 1.80; (b) Robinson and Chu, D/X = 2.06; (e) n = 1.4, D/X = 2.40; (d) /2 = 1.6,1)/A = 7.02; (e)n = 1.6 – i 0.1, = 7.02; (f) n = 1.6, D/X = 8.60; (g) = 1.6 – 0.1, D/X = 8.60; (h) n = 1.6, DA = 9.56; (i) n = 1.6 – i 0.1,DA = 9.56;
(i)a = 1.6, D/N = 10.52; (k) = 1.6 – i 0.1, /)/X = 10.52.
The electric field components are
E – exp( –ihr) • expl(-1)/-1 ik YsinyJ coscs2(0)) kr E„i
= — kr
exp( –ihr)• exp[(-1)J-1 ikY sin-y] sin0 1 (1);),
where S1 (0) and S40) are the Mie Scattering Functions as defined, for example, in Kerker.7
The diffraction theory of Robinson and Chu•`' is indepen-dent of refractive index. This is compared with Mie theory in Figs. 2 and 3. Here, a square aperture of side 2L is placed perpendicular to the z axis at a distance z tt from 0 and passing through the center of the aperture. This situation corre-sponds to 0 = 0 and 4) = 0 in Fig. 1. '['he visibility is calculated against the parameter It = DL/\Z,> , where D is the diameter of the particle and A is the wavelength of the incident beam. R is thus a measure of aperture size. ft is found that visibility is strongly dependent. on refractive index and size of aperture for particles in the size range considered by Robinson and Chu
< 10 pm) for apertures with I? > 1.
Part of this work was performed with the support of the Procurement Executive, Ministry of Defence.
References 1. Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964). 2. R. M. Fristrom, A. R. ,►ones, M. J. ft. Schwar, and F.J.Weinberg,
Faraday Symp. ('hem. Soc. 7, 183 (197:3). 3. W. M. Farmer, Appl. Opt.. I I, 2603 (1972). .1. A. IL Jones, J. Phys. D 7, 1:369 (1974). 5. D. M. Robinson and W. P. (Thu, Apf)1. Opt. 14, 9 (1975). 6. N. S. Hong and A. H.. Jones. Ph■,.s. 1) 9, 1839 (1976). 7. M. Kerker, The Scattering of bight (Academic, New York,
1969).
Reprinted from "Applied Optics", Vol. 15, No. 12, pp 2951-3 (Dec. 1977)
Prop•. R. Ra•. 10/111. A.353,77 (1977)
Printed in Ureal Britnin
Doppler velocimetry within turbulent phase boundaries
BY N.-S. HONG, A. R. JONES ANI) F. J. WEINBERG Imperial College, London, S.W . 7
(Communicated by A. 0. (raydon, .F.R.S. Received 2 July 1971i)
In order to help with the application of laser Doppler velocimetry to turbulent systems which include moving convoluted phase boundaries - e.g. turbulent flames - the interaction of the test beams with such interfaces is analysed. It is shown that there are two effects, one duo to the changing phase difference, the other due to varying deflections, both of which cause the fringe grid to move in response to the velocity of the boundary. This is confirmed experimentally by recording an apparent • velocity of a particle held stationary at the point of intersection of the two beams. The analysis indicates that the effect is serious only for.near-tangential incidence to boundaries between hot and cold gas, when it tends to produce short bursts of large apparent velocities. Experimental methods of correction, or inactivation, of the system during such bursts of unreliability, are proposed.
INTRODUCTION Doppler velocimetry is increasingly being applied to velocity and turbulence measurements in systems such as turbulent flames and plasma jets. These pheno-mena are characterized by convoluted phase boundaries which travel upward at velocities of the same order as those at the centre of the stream. Such moving interfaces modify the direction and phase of light beams which traverse them. Tho object of this note is to investigate the effect of such interaction on fringe anemo-metry within the envelope. The basic supposition that this method records only local velocities at the point at which the intersecting beams produce a stationary grid is called into question, once the grid is recognized as an interferogram of moving phase objects.
The method is further complicated when several such interfaces occur along the beam - for example, in extensive flames such as those in furnaces or in fire research, where a beam traversing the flame would encounter a succession of hot and cold pockets. The boundaries of these pockets, particularly where they lie parallel to the beam, themselves provide suitable tracers for velocity and turbulence measure-ments. Methods which operate on this principle (Sehwar & Weinberg 1969a, b, c) use optical systems of the schlieren type (see, for example, Weinberg 1969) and, although they haVe several advantages and do not rely on foreign inclusions as tracers, they have a very large depth of focus and require some data-reduction to determine the radial distribution of the quantity being measured. Although methods
phase effect.
schlieren effect
L
N.-S. Hong, A. R. Jones and F. J. Weinberg based on schlieren systems will not be discussed further here, it is important to realize that it is their modus operandi which is the cause of the interference with conventional Doppler velocimetry. We shall base our analysis of the problem on a single continuous flame surface.
APPROXIMATE THEORY FOR A SINGLE FLAME SURFACE To keep the illustration simple, we shall consider the interaction of the twin
beams with a convoluted boundary between hot and cold gas due, for example,to a turbulent jet of combustion products emerging into the atmosphere. The con-volutions travel upward at a velocity comparable to that which is to be measured at the point of intersection of the beams within the stream - see figure 1.
FIGURE I. Schematic of two effects during interaction of beam and single boundary.
Although the theory for the two beams can be unified and a detailed analysis of this kind is presented in the Appendix, physically there are two distinct effects. One of these is due to the changing phase difference between the two beams and will be referred to as the 'phase effect' below. The other is caused by the changing deflexion due to variations in the optical path gradient which affects the closely ddjacent beams together - referred to as the schlieren effect' below.
Let (refractive index) - 1 = 8. Since the ambient value of 8, 80 ,is very much greater than Shat, the change in 8 across the boundary 80 -4-- 3 x 10-4 for air at room temperature.
If, due to the boundary's upward velocity, (dy/dt), the optical path difference changes at a rate (dx/dl) - see figure 1 - the corresponding fringe frequency at the point of intersection is (80/A) (dx/dt). But the fringe spacing in the test space is (ANT), where A is the wavelength and Vr the angle subtended by the beams. Thus the
Doppler velocimetry within turbulent phase boundaries
fringes move past a point in the test space at a frequency giving an apparent velocity of (4/0 (dx/dt) which adds to, or subtracts from, the actual local tracer velocity. For 1G = 10-2 rad, this gives a velocity approximately hth of (dx/dt).
To relate this to (dy/dt), we must know something about the boundary geometry, though we may neglect any effect due to the small angle between the beams.
Writing dx Ox dy dt ZST/it '
we see that the boundary will contribute a fraction (Ax/Ay) (80 /0- ) of its flow (vertical) velocity to the reading in the test space. This becomes (Ax/A,y)/30, for ambient cold air and Mfr = 10-2 rad. Thus the precise shape of the boundary be-comes all-important. For a sinusoidal boundary the error would never become very large. It is 10 % of the boundary velocity for (Ax/ 6.y) = 3 and will generally produce short bursts of large apparent velocities when the beam is traversed by parts of the interface which are tangential to it.
The schlieren effect deflects the entire fringe grid. On the same assumptions as above the grid velocity is, approximately (Weinberg Te65$, L .80 (1(cot 0)/dt (see figure 1 for symbols). Once again, this is large only for 0 0 and therefore cannot persist for long. If 0 changes from 01 to 02 as the edge-eddy rises by Ay, then the boundary will contribute a fraction L(30 ,6(eot 0)/4 of its flow velocity to the reading in the test space. Thus for L = 5 cm, the error will exceed 10 % of that velocity if 0 changes from an angle 0igreater than 20° to 02 of 3°, for example, in 3 mm or less.
The detailed theory is given in the appendix, However, the above physical explanation, which lends itself to simple numerical estimates, illustrates the main features of the problem. Both the phase and the schlieren effects give rise to large spurious velocities only for short, periods during near-tangential incidence of the beams to eddies in the boundary. This suggests possible experimental correction methods.
Figure 2 illustrates an optical system currently being used for particle sizing (Fristrom, Jones, Schwar & Weinberg 1973; Hong & Jones 1976, Jones 1974) which was adapted to test the above concepts. A small turbulent flame was inter-posed between the beam splitter and the test space in which the fringes form. A thin fibre was held stationary in the field of fringes and parallel to them.
This geometry causes the beam to traverse the flame-ambient atmosphere inter-face twice, thus doubling the incidence of any disturbances. It also allows distance L (see figure 1), and hence the contribution of the schlieren effect, to be varied in- dependently. The fibre acts as a stationary particle so that any apparent velocity recorded is due entirely to the interaction of the beam with the interfaces. Figure 3 is an oscillogram of the photomultiplier output. As will be apparent from the theory, the form of the variation in frequency is characteristic of a boundary geometry such as that sketched in figure I . However, this is not sufficient to distinguish events of this kind from those to be measured, first because other boundary geometries do,
lens aperture
scattered light
stationary wire
L
flame
pr ism
beam split t er
laser
prism
mirror
N. -S. Hong, A. R. Jones and F. J. Weinberg
photomultiopslcie
ilrloscope
FIGURE 2. Apparatus used to verify modulation of light scattered by a stationary object in an interference pattern.
FIGURE 3. Apparent velocity obtained with stationary object. Time scale: 2 tns cm-1. Distance scale: actual size. Fringe spacing: 8.5 pm.
lens 2
• scatter from A---stationary
particle lens 1 scatter from moving particle
Doppler velocimetry within turbulent phase boundaries
of course, occur and secondly because such perturbations may not be distinguishable once they are superimposed upon the fluctuating velocities of real, moving, tracers in the fringe space. It is therefore pertinent to consider other methods of correction.
Experimental correction In principle it would be possible to use the above method of a ' stationary particle'
in the actual test space and subtract the apparent velocity-fluctuations due to the moving boundaries from the velocity spectrum to be measured. However, this involves fixing an object in the test space which is not far removed from using, for example, a hot-wire anemometer. It is not possible to produce precisely the same situation outside the flame because the light has to traverse a second interface in emerging. However, for symmetrical flows, this merely involves the beam ex-periencing the same kind of perturbation a second time.
aperture test space
A 13 FIGURE 4. Suggested scheme for correction of measured turbulent velocities.
Figure 4 shows a simple system which re-unites the two beams outside the flame, in an optically conjugate simulation of the test space. If this is used to record the signal due to a stationary particle — either by scatter from a small target suspended in the fringe pattern or by transmission through a small pin hole — the resultant trace is characteristic of the moving boundary in which the relevant interactions occur, but at double the frequency.
The nature of these perturbations suggests an automated alternative to sub-tracting edge effects by subsequent processing of the two sets of signals. Since the analysis and experimental tests show the perturbations to be well spaced out and each of short duration — at least so long as the beams do not transluminate too many moving interfaces — the simplest way of eliminating them is to inactivate the system whenever one is passing. Using the occurrence of an a.c. signal from detector B to trigger the deactivation of detector A (figure 4) will cause the system to cut out more often than necessary. However, this is no disadvantage as long as the process is random and events at the centre of the stream are not correlated with those at its boundaries.
N.-S. Hong, A. R. Jones and F. J. Weinberg,
Thus a relatively minor experimental refinement can reinstate the convenience of simple fringe anemometry - at least for all points separated from the boundaries by distance well in excess of the length scale which correlates the fluctuating velocities.
The same optical system may be used when inactivation is not feasible because perturbations occur too frequently. This may occur either for small-scale high-intensity turbulence or when the optical path encounters many pockets of gas of different refractive index during its travels. In such cases the frequency spectrum obtained from photomultiplior B can be used to correct the readings of A.
Beyond that it would seem possible in principle to use a second Doppler system with an appreciable angle to the first, producing fringes within the same test volume (i.e. four beams). The two pairs of beams interact with different,. uncorre-lated, parts of the boundary, but the same primary test object. In addition to duplicating the Doppler system, this would require the use of two wavelengths and two photomultipliers, each provided with the appropriate filter to distinguish between signals from the two beams. The additional complexity would therefore be quite considerable and it is thought that for most practical systems one of the above suggestions would suffice.
APPENDIX Consider the two rays shown in figure 5 incident upon the boundary between
hot and cold gas. For simplicity a two dimensional system is considered. In the absence of refraction they would cross at the point (X,1, 0) but in reality cross at (X3, -Y3). The optical path length of ray 1 between (0, D) and (43- Y3 ) is
1
n, X i n2 I (X - X2)bn y2 + (Yi + Y2 )}. - cos y + cos Yi tan y 1 - tan 72
Similarly, for ray 2 between (0,-D) and (X3, - Y3 )
n1 X2 ± n2 I (X, - X2) tan yi + (Yi + Y2 )} • 2 - cosy cos y2 tan Y1 - tan y2
Thus the difference in path length is
Ad = (X1 - X2) [9t1 N(sin yl - sin y2)] n2(Y1 + Y2) (cos yi - cos y2 ) cos y sin (Yi - 72) sin (71-72)
where yl = y i, and similarly for yz. To examine the effect of curvature, consider the hot gas boundary to be part of
a circular cylinder of radius R having its centre at (Xe, Ye ), as in figure 6. It can be shown that X2 = R(cos — cos a),
yi + Y2 = R(sin + sin a), it = Y ia =
r 1+ a; Ya
1
„ horizontal
. _
71, 7(0.-D)
reference plane
FtounE 6. Coordinates used with a cylindrical phase boundary.
Doppler velocimetry within turbulent phase boundaries
rpfi•oct ive index refractive index n,
boundary of hot gas
(o,--D)
reference plane
FIGURE 5. Interaction of two rays with a general phase boundary.
For the cold gas n1 = 1+3, where < 1, and for the hot gas, by comparison, n2 .= 1. As in the body of the paper, the boundary is assumed to be discontinuous and Snell's law applies, whereby (see figure I )
Scot 95 = Stani. We note that total external reflexion will occur at i 88.5°, at which point 0 is at its maximum value of approximately 1.4', fir 8 = 8 x 10-4 which is typical for air. The incident ray here is almost tangential to the boundary. Now
YI 7+ 01; Y2 = Y -02
cos a = 1 +B2 —A B+V(B 2 — A2+1)
and
N.-S. Hong, A. R. Jones mill F. .T. Weinberg
sin y2 — sin y i — COS y,
sin (Y1. — 72)
cos y2 — cosy, sin y, sin (vi — 72)
so that Al (x 1— X 2)[ nt n o cos y + n2(Y, + Y2) sin y cosy "
= RI( nnl
cos y , cos y (cos /i — cos a) + n2 sin y (sin fl + sin a) .
The solutions for a and ig are A + (B 2 — A 2 + 1) sin a =
1 + 13 2 '
A + B V (B 2 — A` 2 + 1) stn ft = 1 + B 2
— + V(13 2 — A' 2 + 1) cos/3 = 1 +B2
where D —Y X , A = R tan y ,
D+Y„ A„ A' = • tan R y,
B = tan y.
Let the cylinder rise vertically with velocity / -"real = dY,Idt. Then the rate of change of path length is d(Al)/dl which gives rise to a frequency
I d(Al) '
where A is the wavelength. `Phis frequency would be interpreted as a velocity through
(Iam) =
where Af 2 slily
is the fringe spacing. Hence
n sin y .
Unna — scn,, cos y)—dlt ( os/3— cos a) + n2 silty
d—t (in ,8 + sin a)), IP 2 sin { cosy "
Doppler velocimetry within turbulent phase boundaries
2y 2 o 5Y
10Y
LD "0J1_ n c4.r, 1 0 H
I) 0.5 I 0
. Vtlimax
Funeut: 7. Variation of apparent velocity with position of centre of cylinder.
with 1 A A' I di', —cP(cos fl— cos a) = f 2/1 4- dt /1( I -4- /12)k \
/(B2 ___ A2 + 1) + V(132 -11/2 -1- Of dt'
' . , LI (sinp-sin a) =
B A A di dt 141 + B2 ){,/(B2 — A '2 + 1) V(B2 — A + 1)
I dt '
Helm. 1 apr , ,_-_-• f(8, y, A, A ', B, 1?) Ureal . The fenetioni f is plotted in figure 7 as a function of y -= — 1'4R. y,„,„„ is the value of it at the point of total external reflect ion. Curves arc given for various values of
= (X,— Xe) tan y _.., (X,— Xe) y Ii - R
The apparent velocity is found to he very high at near tangential incidence and falls rapidly. Ilapp is less than 10% of tir,,„„ for <
R EF ENC ES
Fristrow, R. M., Jones, A. R., Schwan., 111..1. R. & Weinberg, F. J. 1973 Faraday Symposia of the Chemical Society, no. 7, p. 183.
flong, N. S. &—lones, A.. R. 1976 .1. Phys. I), 9,4$39. Jones. A. R. 1974 J. Phyx. I) 7, 1300. Schwar, hi. J. R. & Weinberg. F.J. 19690 Nature Loral. 221, 357.
tar, M..1. R. & Weinberg, V..1. 19696 !'roc. H. Nuc. hard. A 311, 469. Selp.var, Dl. J. It. & Weinberg. F. J. I969c Combrt.slion (1.. Plume 13, 335. Weinberg, F. J. 1963 Optiex of flames. 1,cnidon: 13iitterworths.