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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187

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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.