L AD-Ri76 968 RESEARCH ON CERTAIN ASPECTS OF LASER DIFFRACTION i'PARTICLE SIZE ANALYSIS R..IJ) ARIZONA STATE UNIV TEMPE

IDEPT OF MECHANICAL AND AEROSPACE N.

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11 TITLE lnclude Sacrt'uy Classific on) . . 61102F 2308 A3Research on Certain Aspects of Laser M < (.;--Q\_ .

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19. Abstract

The fundamental scientifi-c deficiencies impeding the integration

of laser diffraction particle sizing techniques into intelligentsensors for next-generation propulsion systems have beenidentified. The research addressed three re-@-v-wt'areas: inversescattering algorithms, multiple scattering; and the problems oflaser beam deflections due to refractive index gradients inhostile propulsion environments. Direct integral transformtechniques for the inverse problem have been formulated on acommon basis and comprehensively evaluated. An error in theoriginal derivation of the the method of Petrov has beencorrected and a new integral transform solution has beendeveloped.-)Also, a scaling law for optimal configuration of laserdiffraction detector geometries and size class characteristicshas been developed k _which optimizes the- -stability andcomputational requirements of the inverse problem. Concerningmultiple scattering, a new synthesis of successive order anddiscrete ordinates models has been developed for predictions ofnear-forward radiative transfer in optically thick media and theresults have been verified with experimental data and anindependent Monte Carlo simulation. Finally, a new conceptinvolving programmable spatial filtering at the transform planewhich allows annular detectors of variable geometries to beconfigured on-line by an intelligent instrument has been proposedand proof of principle experiments have begun. The concept willpermit a detector to be automatically reconfigured around thedeflected beam center and also will allow adaptive grid methodsto be incorporated into the inversion algorithms.

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AFOSR.'IR. 87-0058

RESEARCH ON CERTAIN ASPECTS OF LASER DIFFRACTION PARTICLE SIZEANALYSIS RELEVANT TO AUTONOMOUS SELF-DIAGNOSING INSTRUMENTATION

Annual Report for 1 Oct 1985 - 1 Oct 1986AFOSR Grant 84-0187

ByE. Dan Hirleman and Joseph H. Koo

Mechanical and Aerospace Engineering DepartmentArizona State University

Tempe, AZ 85287

Research Objectives

Particle and droplet size distributions, being parameters offundamental importance, should be priority measurement objectivesfor intelligent sensors in next generation propulsion systems.Unfortunately there are a number of problematic scientific issuesimpeding the development of laser light scattering particlesizing instruments capable of on-line, autonomous, and self-diagnosing operation in hostile environments. The objective ofthis research effort is to contribute to the scientific knowledgebase necessary to characterize and hopefully extend theapplicability of laser diffraction instruments under theseadverse conditions. Both the novel specific approaches discussedbelow and the emphasis on concepts relevant to intelligentinstruments and sensors make this research project unique.

Our concept of a next-generation laser diffraction system isshown in Fig. 1 where the light scattered in the near-forwarddirection (i.e. diffracted) by particles along the line-of-sightof the probe laser beam is collected by a lens and directed tothe back focal (Fourier transform) plane. The diffractionreference leg, variable focal length transform lens, beam

% position detector, programmable transmission filter, and a fielddetector to transduce the light passing through the transmissionfilter are novel concepts proposed here to extend the standardconfiguration utilizing a fixed focal length transform lens and afixed geometry detector. We envision an intelligent instrumentwhich performs self-calibration, self-diagnosis, and monitors theoperating environment in order to automatically reconfigure boththe "detector geometry" (i.e. the pattern in the transmissionfilter) and the inverse scattering algorithm to maintainoperation near an optimal state. In the original proposalsubmitted to AFOSR in 1984 we identified three research areas inwhich the lack of scientific understanding would limit futureadvances. In the following paragraphs we discuss the researchobjectives associated with each of these target areas.

1. Inverse Scattering Algorithms

Particle sizing using laser (Fraunhofer) diffraction is aninverse scattering problem. The governing equation for singlescattering by large particles is a Fredholm integral equation:

1.87 2 20 0 fl

2 2( 0)(2

,? = /X2 /42 J1 (Cta) Cn() (1)

where: a = vd/x is particle size parameter; n(a) is theunknown particle size distribution; and 1(8) the scatteringintensity which can be measured at some number of scatteringangles. Fredholm integral equations such as Eq. (1) are generallyill-posed in the sense that small variations in the measurementsof I($) (e.g. due to noise) can cause large changes in theresulting n(a). The maximum information obtainable about n(a) isdependent on the noise levels in the measurements, the samplingor detection scheme, the stability or robustness of the inversionmethod, the actual size distribution, and the allowable level ofuncertainty. Ideally, then, a laser diffraction system wouldhave variable detector geometry, inversion scheme, and sizeresolution which could all be adjusted at run time depending onthe conditions encountered.

The objective here is to identify and impact the underlyingscientific issues which presently impede the needed quantumimprovements in efficiency and robustness of inversion schemesfor intelligent laser diffraction sensors. Dynamic adaptivecontrol of the detector configuration and synergism of theinversion scheme with the detector geometry are focus areas.

2. Multiple Scattering

A major challenge for future research on in-situ opticaltechniques for particle sizing is to extend the applicability ofthese methods to optically thick media. The theoreticalframework for particle size measurements under multiplescattering conditions is rather tenuous, even for the directproblem of calculating the radiation transfer properties of anarbitrary particulate medium. Even less developed are analyticalmethods for the inverse problem, i.e. obtaining particle sizedistribution information from measured optical scatteringproperties of media where multiple scattering is significant.

The objective of this part of the research is to understandthe process of multiple scattering as it might occur in futureapplications of intelligent particle size distribution sensorsand develop inversion schemes which can operate in such anenvironment. A minimum requirement is that multiple scattering bediagnosed by the instrument to ensure that erroneous data not be _ Jused in control algorithms, a situation which could result in acatastrophic failure. Our objective is to surpass that and

. develop efficient and robust algorithms which can actually §jextract useful particle size information from Fraunhoferdiffraction measurements in multiple scattering environments.

3. Laser Beam Deflection/Steering

Another problem which has already limited the usefulness oflaser diffraction particle sizing instruments in propulsion

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systems research is that of beam deflection by refractive indexgradients in the optical path. These gradients are due tospatially nonuniform temperature and/or species concentrationsand at relatively large scales have the same effect as an array

*. of randomly dispersed and time-varying lenses which cause theprobe beam to move around at the detection plane. This has been asevere problem since information on the largest droplets intypical sprays is concentrated at small scattering angles whichare of the same order as the beam deflection angles encountered.

The objective for this research topic is to investigateconcepts which could enable a laser diffraction instrument tooperate under conditions where beam deflection is significant. Aminimal requirement is for on-line detection of beam steering sothat an intelligent instrument would at least know that the datataken under these conditions is suspect. A more importantobjective is to develop strategies to permit a diffraction systemto autonomously perform real-time correction for beam steering.

Status of the Research Effort

Progress has been made in each of these areas over theduration of this grant and we foresee no significant deviationsfrom the original approach proposed for the research. Thesections below describe the technical achievements made in theseareas during the indicated 12 month time period.

1. Inverse Scattering Algorithms

The inverse scattering problem for laser diffractionparticle sizing instruments has similarities with all systemsrequiring a deconvolution of experimental data. The inversescattering problem (i.e. determination of n(a) in Eq. (1) frommeasured I(8)) can be solved by either (i) direct integraltransform methods or (ii) indirect numerical quadratureapproaches. In the past year we have continued a concentration onthe generally overlooked integral transform techniques which, byvirtue of the direct nature, have unique potential for very fastsolutions as would be required by an on-line, intelligent sensor.We have completed a thorough study and synthesis of the integraltransform formulations applicable to the inverse Fraunhoferdiffraction problem and identified five unique methods. Three ofthese previously existed in the literature, one had errors inderivation which we corrected, and the fifth was newly developedas part of this work. A comprehensive parametric study of theimportant factors which control the inversion performance of thetechniques has been completed and is detailed in the references.

Another important contribution this year was the developmentof optimal scaling laws for design of the detector geometry andsize class characteristics of a Fraunhofer diffraction particlesizing system. Optimization here is in the sense that thecomputational requirements (storage and number of computations)for the inversion are minimized and the condition number of the

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instrument function matrix is minimized (i.e. the stability ofthe numerical inversion or deconvolution is maximized). Frominspection of the left hand side of Eq. (1) we propose that thedetector signals S should respond as S I(9)92. Numericalquadrature on Eq. (1) gives:

S = C .A (2)

where S, A, and C are matrices such that S. is the scatteringsignal on the jth detector, A, is the particle area in the ithsize class proportional to n(0)a 2 , and C. for the simplestquadrature scheme is given by:

C3 = 2 (at) (3)

Now thr detector condition can be achieved using ring detectorswhere each annular element has a constant ratio of inner radiusri to outer radius r. such that each of the jth detectors has:

rr, / r, = 01.3 / 8o,3 = c (a constant) (4)

which also implies that the geometric mean angles of adjacentdetectors have the same ratio. Now if the mean sizes of the sizeclasses are also selected such that a, / a ., = c then:

CJ12 (cl i j ao ) 5

whereby all elements on any diagonal in the instrument function Cmatrix are equal, and there are only 2n4l unique elements in C(where n is the number of detectors and size classes). Clearlythis cuts drastically the computation complexity as the storagerequirements have decreased from O(n 2 ) to O(n). Further, if the asize classes are selected to maximize the signal S on acorresponding detector (e.g. a 1.862/8 to maximize J 2), thenthe C matrix has the largest elements on the diagonal and forsmall enough n will always be diagonally dominant.

2. Multiple Scattering

The specific problem of multiple scattering from particleslarge compared to the wavelength (i.e. Fraunhofer diffraction)has previously been modeled using successive orders and acombination of discrete ordinates and the adding method. We havedeveloped a unique formulation which combines the method ofsuccessive orders with a discrete ordinates approach adoptedspecifically to the axisymmetric ring detector configuration usedin most laser diffraction systems. Defining S,(O) as theprobability that a photon scattered exactly n times will exit themedium within a finite scattering angle range represented by 0 wecan write:

S(e) : fn Sn(e) (6)

1=4

where S with no subscript represents the composite scatteringsignature which is the superposition of contributions from allscattering orders as would be measured by a laser diffractioninstrument, and the occupancy variable fn represents theprobability that a photon undergoes exactly n forward scatteringevents. Now if the scattering event of interest is the nth for aparticular photon, then the scattering order signature Sn(esct)for photons forward scattered exactly n times can be found from:

s (e Ct F h(e C t e(ic) s - (ei¢) (7)all 0,n c

where h is the scattering redistribution function or theprobability that a photon incident in direction Oin c (for any

azimuthal angle 0,n,) will leave the next scattering eventtraveling in the direction 9sct. The matrix h is a function ofthe size distribution and it is the efficient calculation and/orstorage of h that is required for forward and inverse multiplescattering solutions.

Fig. 2 shows a comparison of our theoretical predictionswith some experimental measurements of diffraction signaturesfrom multiple scattering media, and the agreement is quite good.The effects of multiple scattering on apparent (i.e. as would bemeasured by a laser diffraction instrument) mean particle size isshown in Fig. 3. Here again the theoretical predictions from thisstudy show good agreement with experiment.

These results indicate that a valid model for predictingmultiple scattering signatures from media of arbitrary opticalthickness has been developed. The formulation is optimized forinclusion into numerical schemes for inversion of multiplescattering measurements, and our research for the next year willconcentrate on development and validation of inverse scatteringformulations applicable to media of arbitrary optical depth.

3. Laser Beam Deflection/Steering

The probe laser beam from a Fraunhofer diffractioninstrument will, in general, be continuously deflected as ittraverses a medium of nonhomogeneous refractive index as would beencountered in a combustor. This causes the transmitted beam andthe associated diffraction pattern which is centered around itto be directed to off-center points in the transform plane ofFig. 1. The very high intensity in the transmitted beam, whichtypically would be more than 100x greater than the intensityscattered into the small angle inner rings of the detector,severely hampers the measurements at small angles. Our conceptfor circumventing this problem is to use a programmable spatialfilter at the transform plane. Annular transmission windows are

configured about the instantaneous beam position as measured bythe x-y position detector of Fig. 1, and light transmittedthrough the aperture is collected by the field detector. Ourprototype system uses an array of Faraday effect transmitting

5

pixels and experiments to demonstrate the feasibility of thisconcept are underway.

Technical Publications, 10/1/85 - 10/1/86

1. Koo, J. H. and Hirleman, E. D. "Investigation of IntegralTransform Techniques for Laser Diffraction Particle SizeAnalysis," Paper No. 77, Presented at the 1985 TechnicalMeeting, Eastern Section of the Combustion Institute,Philadelphia, PA, November, 1985.

2. Koo, J.H. and Hirleman, E.D., "Mathematical Formulation ofIntegral Transform Techniques for Laser Diffraction ParticleSize Analysis," IR-86-ll, Laser Diagnostics Laboratory,Mechanical And Aerospace Engineering Department, Arizona StateUniversity, Tempe, AZ, May, 1986.

3. Koo, J.H. and Hirleman, E.D., "Comparative Study of LaserDiffraction Droplet Size Analysis Using Integral TransformTechniques: Factors Affecting the Reconstruction of DropletSize Distributions," Paper No. 86-18, presented at the 1986Joint Spring Meeting of the Canadian and Western StatesSections of the Combustion Institute, Banff, Alberta, CANADA,April, 1986. Under review for publication by the FirstInternational Congress on Particle Sizing, Rouen, France, May,1987.

5. Hirleman, E.D., "Modeling of Multiple Scattering Effects inFraunhofer Diffraction Particle Size Analysis," Paper No. 17,presented in the 1986 Joint Spring Meeting of the Canadian and

" -" Western States Sections of the Combustion Institute, Banff,Alberta, CANADA, April 28-30, 1986. Under review forpublication by the First International Congress on ParticleSizing, Rouen, France, May, 1987.

6. Koo, J.H. and Hirleman, E.D., "An Improved Integral Transform* " Technique for Laser Diffraction Particle Sizing - Modified

Petrov Method," To be submitted for publication in AppliedOptics.

7. Koo, J.H., "Particle Size Analysis Using Integral Transform

Techniques on Fraunhofer Diffraction Pattern," D.Sc.Dissertation, George Washington University, MechanicalEngineering Department, Washington, DC, in preparation.

8. Hirleman, E. D. "Optimal Scaling for Fraunhofer DiffractionN, Particle Sizing Instruments", under review for publication by

the First International Congress on Particle Sizing, Rouen,France, May, 1987.

9. Ilirleman, E. D. "Programmable Transform Plane Filter forDynamically Configurable Fraunhofer Diffraction Detector", inpreparation, to be submitted for publication to AppliedOptics.

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N NW"

Professionail Personnel

Prof. E. Dan Hirleman - Associate Professor of Mechanical andAerospace Engineering

Joseph H. Koo - Research Assistant and Ph.D. student. Mr. Koo'sdissertation based on this research is written and his Ph.D.should be awarded in early 1987.

Interactions

1. June, 1986 visit by Prof. E. D. Hirleman to Kirtland Air ForceBase, Albuquerque to visit Dr. Tim Ross and Mr. Glenn Jamesbased on preliminary contacts at the AFOSR Contractor'smeeting at Stanford in June. Discussions centered on particlediagnostics requirements for research underway at Kirtland.

New Discoveries

1. Optimal Scaling for Fraunhofer Diffraction Particle SizingInstruments

We have derived scaling laws for the optimal design of aFraunhofer diffraction particle sizing system. The scaling lawsapply to the detector geometry and the size class characteristicsand are optimal in the sense that the computational requirements(storage and number of computations) for the inversion areminimized and the condition number of the instrument functionmatrix is minimized (i.e. the stability of the numericalinve-Aion or deconvolution is maximized). Further, the samescaling laws will have a significant impact on compressing thedata storage and computations required for the inverse multiplescattering solutions we are developing. Further details on thisinnovation can be found in ASU Patent Disclosure No. 212 and thereferenced paper by Hirleman on this topic.

2. Programmable transform plane spatial filter for dynamicannular detector configuration.

Another invention derived from this research is the conceptof a programmable transform plane spatial filter for dynamicconfiguration of annular detectors as shown in Fig. I. Thisconcept will allow incorporation of significant levels ofintelligence into an instrument including on-line control of thering detector center (for beam deflection) and the detectorgeometry with capabilities for adaptive grid and variableresolution configurations. Further details on this innovation canbe found in ASU Patent Disclosure No. 213 and the referencedpaper by Hirleman on this topic.

3. Successive order model for multiple scattering.

We have developed a new model for the prediction of the

forward scattering characteristics of optically thick media where

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the particles are large compared to the wavelength. Theformulation is based on a new synthesis of successive order anddiscrete ordinates methods and is optimized for inclusion intoinverse multiple scattering problem. The formulation producesthe relatively simple relation:

S = exp(-b) exp(b/2.H) S.

where S, is the vector representing the angular distribution ofthe input optical energy to the medium (all at e 7 0 forcollimated beam input) and S and H are the scattering vector andthe redistribution matrix as defined in the report above. Notethat the direct calculation of the scattering signature S isreduced to knowledge of the input laser angular spectrum S0 andexponentiation of the redistribution matrix H. Further, this

* equation can be rearranged to provide for an experimentaldetermination of H which can then be used to determine S1 (01, )(scattering signature of single scattered photons) from which thesize distribution can bc determined using conventional singlescattering inversion methods. Further details on this innovationcan be found in ASU Patent Disclosure No. 214 and the referencedpaper by Hirleman on this topic.

PROGRAMMABLETRANSMISSIONFILTER AT

2 - TRANSFORM PLANE

X-Y POSITION DETECTOROR o00 x 100 PHOTODIODE ARRAY

ACOUSTO-OPTICDIFFRACTIONRETICLE TRANSFORM TRANSFORM

LENS PLANEII l' "l -_ _ I !1 DETEOCTOR

'BS * - 9\ IL/ L__BEAM EXPANDER VB PARTICLESPATIAL FILTER FIELO BS

f

Figure 1. Schematic of a next-generation laser diffractionparticle sizing system for use as an intelligent sensor.

R

Obscuration =0.93 -

-*-Theory

CI

C)

0

zI-2-

Detector Number101

Figure 2. Scattering signature vs. detector number (RSI/Malvern) foran optically thick spray. Predictions from the multiple scatteringmodel developed here are compared with experimental results obtainedby Dodge (Optical Engineeringr, V. 23, p. 626, 1984).

-- Theory

,.-..32

E

E

0

>8

0 .2 .4. 8

Obscuration

Figure 3. Volume median diameter D,. 0 5 as measured by a loserdiffraction system for various optical thicknesses of the spray of

Fig. 2. compared withamultiple scattering predictions.

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