Particle-size distribution ~PSD! is predominant in theassessment of scattering properties of various media~such as aerosols and fogs! and also of the propertiesof rocket and jet engine plumes. To determine thePSD for systems such as these requires in situ oreal-time measurements and consequently an ade-uate measurement setup. As the media that areeasured are generally dilute, measurements of
ingle-scattered light generally provide good-qualityesults. An original nephelometer was developed onhis principle a few years ago,1–4 in which one finds
the PSD by inverting the measured scatter diagrams,using Mie’s theory,5 which assumes that the particlesare spherical. As the measurement uncertaintiesare usually quite large ~especially in media with shortlifetimes or periods of stability, such as rocketplumes!, the noisy inverse problem is ill-conditioned,so the accuracy and the stability of the resultantdistribution function are both generally poor. Ade-
L. Hespel [email protected]!, A. Delfour [email protected]!, and B. Guillame [email protected]! areith the Departement d’Optique Theorique et Appliquee, Officeational d’Etudes et de Recherches Aerospatiales, 2 avenue E.elin, B.P. 4055, 31055 Toulouse, France.Received 16 May 2000; revised manuscript received 10 October
974 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
quate experimental setups are then needed to reducenoise errors, and this entails much effort. Similarly,further conditions often have to be placed on the in-version processes to regularize the noisy inverseproblem.6,7 To suppress the arbitrary nature of thisregularization, Hespel and Delfour developed a spe-cific inverse method that was presented in a previouspaper.8 This method renders the solution highlystable against noise, and its accuracy for reconstruct-ing a wide variety of distribution functions wasnumerically demonstrated. Nevertheless, the accu-racy of particle distributions as represented by inver-sion of scatter data still depends on measurementerror. To limit the effect of the data uncertaintiesrequires that highly sensitive experimental setups bedeveloped, with high angular resolution and with theassociated calibration procedures.
In the first part of this paper, the main principles ofthe inverse method that one uses to determine thePSD are briefly presented. The nephelometer andthe data acquisition system are then described.This nephelometer acquires the scatter signals se-quentially in discrete directions. The scattered lightis collected by a set of optical mirrors of different sizesplaced tangentially to an ellipse. The geometriccharacteristics of the ellipse dictate that the distancetraveled by each scattered beam be constant, so oneobtains a constant product between the solid collec-tion angle and the scattering volume by adjusting themirror size. If the mirror is perfectly aligned, noabsolute calibration is required for this nephelome-
t
s
w
c
tt
tB
M
T
t
l
v
ter. To suppress the uncertainties that are due tomirror misalignment, one performs a calibration pro-cedure and evaluates the nephelometer performanceagainst a monodisperse referenced PSD. Then thegranulometer consisting of the inverse algorithmcombined with the nephelometer is validated. Onedoes this by taking measurements of reference mediaof known PSD’s. The inverse procedure is then per-formed on the measured scatter diagrams. Compar-isons of theoretical and reconstructed PSD show goodconsistency despite the presence of high noise levels.These results demonstrate that this granulometercan produce accurate distribution functions evenwhen the results of noisy measurements are used.
2. Theoretical Concepts
Consider a scattering medium consisting of randomlydistributed spherical particles illuminated by a laserat frequency n 5 1yl. The scatter volume DV ob-served at angle ui ~see Fig. 1! is defined as the inter-section of the laser probe and the scattering medium.The collection solid angle DV subtended at each scat-tering angle ui depends on the nephelometer geome-ry. Assuming that DVDV is constant and that
single scattering occurs, the signals Smeas~ui, n! mea-ured by the detector at angle ui are proportional to
the differential scattering cross section bmeas~ui, n!,hich is defined by
bmeas~ui, n! 5 *0
`
f ~D!s@Dn, m*~n!, ui#dD, (1)
where s@Dn, m*~n!, ui# is the differential scatteringross section evaluated from Mie theory,5 f ~D! is the
PSD by number, m*~n! is the relative refractive indexat frequency n, and D is the sphere diameter.
Signal measurements in N discrete directions uithus define a system of N Fredholm integral equa-ions of the first kind. We invert this system to es-imate the PSD, f ~D!. As the inversion is performed
directly on the measured signals, the PSD needs to benormalized by use of an extinction measurement:
Kext~n! 5 *0
`
f ~D!sext@Dn, m*~n!#dD, (2)
where sext@Dn, m*~n!# is the extinction cross sectionand Kext~n! is the measured extinction coefficient.The major phases of the method can be summarizedas follows ~for further information, the reader shouldrefer to Ref. 8!: First we assume that the optical
Fig. 1. Block diagram of the nephelometer.
index is invariant as a function of the frequency.The system of nonlinear equations is then trans-formed as follows:
b~ui, n! 5 *Dmin
Dmax
h~D!Kui~nD!dD, ; i [ @1, N#, (3)
with
Kui~nD! 5
4s~Dn, ui!exp~2gnD!
pD2 , (4a)
h~D! 5pD2 exp~gnD! f ~D!
4. (4b)
The parameters g and Dmin are evaluated by theprogram and depend on the value of Dmax supplied bythe user. To calculate the PSD that corresponds tothe p step of the numerical filtering scheme, we writehe function h~D! as the sum of M~p! # N cubic-spline functions Sj
~p!~D! ~Refs. 9 and 10!:
h~ p!~D! 5 (j51
M~ p!
cj~ p!Sj
~ p!~D!. (5)
The nodes Dk~p! and tk
~p! 5 ln@Dk~p!# with k [ @21,
~p! 1 2# are defined by
tk~ p! 5 ln~Dmin! 1 ~k 1 1!Dt~ p!, (6a)
where
Dt~ p! 5 1yvmax~p 2 1!. (6b)
he cutoff frequency vmax~p 2 1! results from thenumerical filtering process11,12 and is evaluated byuse of the distribution h~p21!~D! and data uncertain-ies Dbmeas~n, ui!. The number M~p! # N is then
defined by
tM~ p!12~ p!
5 ln~Dmax!. (7)
Setting Eq. ~5! in Eq. ~3!, we obtain the followinginear system:
b~ui, n! 5 (j51
M~ p!
uij~ p!cj
~ p!, (8a)
where
uij~ p! 5 *
Dj22~ p!
Dj12~ p!
Kui~nD!Sj~D!dD, j [ @1, M~ p!#,
i [ @1, N#. (8b)
M~p! is less than N, and the system @Eq. 8~a!# is over-determined. Coefficients cj
~p! are evaluated by anordinary least-squares method that includes a linear-ization scheme that is subject to nonnegativity con-straints @i.e., cj
~p! $ 0 @j#, as the PSD must be positive.To solve this problem we use the nonnegative leastsquares routine proposed by Lawson and Hanson.13
We perform the global iterative algorithm, but thestopping criterion uvmax~p! 2 vmax~p 2 1!u , εv is noterified. When the global convergence of the
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 975
d
pTM
la
2
a
c
9
method is achieved, the best linear approximation ofthe distribution function is obtained:
h`~D! 5 (j51
M~`!
cj`Sj
`~D!. (9)
As noisy inverse problems lead to nonunique solu-tions, two distribution functions, fmin
` ~D! and fmax` ~D!,
are introduced that define a confidence interval onthe optimized PSD. For this purpose, small itera-tive translations are performed on h`~D! as long asthe numerical values remain in the interval definedby the experimental values and their uncertainties.The inversion of Eq. ~4b! then yields fmin
` ~D! andfmax
` ~D!.
3. Light-Scattering Nephelometer
The device for determining the PSD that is presentedin this study is based on the principle of an opticallaser scattering nephelometer for which it is assumedthat the light is scattered only once. This apparatushas been used extensively for in situ particle-size
etermination in aerosols, e.g., in rocket motors1 andcryogenic airstreams,2 and also for optical character-ization of fogs3 and biomedical materials.4 Oursetup is designed specifically for in situ measure-ments of jets of particles without scattering cells.The measurement time required for generating scat-ter diagrams is greatly reduced by use of fixed androtating mirrors instead of a rotating detector, whichis crucial for unstable jets and short-lifetime phenom-ena. The nephelometer and the data processing sys-tem described in the following subsections can beused to measure the mean angular scatter patternand its uncertainties within a few seconds. Largenoise errors can nonetheless be observed if the num-ber of measurements is insufficient.
We shall presently describe the experimental setupand the associated alignment procedure, followed byan explanation of the data acquisition system andcalibration scheme. Finally, as the nephelometer iscalibrated to reference liquid solutions, specific cor-rections are described for including the effects of thescattering cell.
A. Experimental Setup
The experimental setup is described schematically inFig. 1. The scattering medium to be characterized isilluminated by a linearly polarized laser beam of vis-ible frequency ~He–Ne 4 mW, l 5 632 nm or argon100-mW, l 5 514.5 nm!. The polarization here is
arallel to the plane defined by the nephelometer.he scattered light is collected by a set of 20 mirrors,i, placed tangentially along an elliptical path.
One focus of this ellipse coincides with the position ofthe scattering medium. The control volume investi-gated is defined as the intersection of the laser beamand the scattering medium. A rotating mirror islocated at the other focus, collecting the light sequen-tially from each mirror Mi. It also measures theight transmitted by the scattering volume throughn optical density ~see Fig. 2!. The light picked up
76 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
by the rotating mirror is returned perpendicularlytoward a detection system illustrated in Fig. 3. Thescanning apparatus covers scattering angles ui from
40° to 45°. To satisfy V~u! 3 DV~u! 5 const. @ u,the detection system consists of a primary pinhole~Pinhole 1! to fix the solid collection angle DV~u! and
secondary pinhole ~Pinhole 2! to delimit the ob-served control volume V~u!, i.e., the field of view of thedetection system. For the measurements presentedin this study, major axis A and the distance C be-tween the foci are, respectively, 1 and 0.74 m ~see Fig.4!. Diameter f of the first pinhole is set at 8 mm.The solid collection angle is then DV 5 pf2y4~A!2 '5.0 3 1025 sr for each discrete scattering angle,equivalent to an angular aperture of 0.2°. For ex-tinction measurements in the forward direction, thesolid angle is modified according to DV 5 pf2y4C2 '9.2 3 1025 sr. The second pinhole ~Pinhole 2! isassociated with two lenses that define a 5° field ofview. An interferential filter centered on the laserwavelength suppresses any other wavelengths emit-ted by the scattering medium ~for hot media! and anystray light. A photomultiplier tube ~PMT! is used toonvert the collected light into an electrical signal.
The 20 discrete directions ai of the mirrors and the
Fig. 2. Measurement of ~a! transmitted and ~b! scattered light.
Fig. 3. Light-scattering collection system.
e
ac
prcise
md
mffs
ctmd
direction of transmission are encoded by an opticalencoder with an angular step of 0.36°. These direc-tions ai are converted into equivalent scattering an-gles ui by use of angular relations that satisfy anllipse ~see Fig. 4!. As the mirrors are placed to
either side of the direction of transmission, the dis-crete scattering angles are defined as negative to theleft of the transmission direction and positive to theright. This encoder is driven by a dc motor and cou-pled with a 32-channel pulse selector. Eleven chan-nels are used to produce the noise equivalent signal ofeach scatter diagram. A complete scatter diagram iscollected every 20 ms, which is an acceptable timeinterval for stabilized or slowly varying phenomena.The dc signal from the PMT is amplified by a dcpower supply. The PMT and the pulse selector out-puts are connected to a computer, and the signals aremixed by use of an acquisition card ~National Instru-ments Model AT MIO 16F5!. The computer is pro-grammed with the LABWINDOWS routine for datacquisition, and the data are exported for postpro-essing.
For the alignment procedure, the PMT tube is re-laced by a microscope with an integral reticle. Aough cylindrical glass is then observed with the mi-roscope for all fixed mirrors, and alignment consistsn centering the cylindrical glass in the reticle. De-pite the accuracy of this method, uncertainties stillxist in the relation V~u! 3 DV~u! 5 const. @ u, which
is verified only for perfect alignment. These errorsare then corrected by a calibration procedure definedin Subsection 2C. We take scatter diagrams of therough cylindrical glass to find encoder positions ai~see Fig. 4! that correspond to the fixed mirrors andthe transmission direction. Encoder positions ai,which correspond to the forward direction and the 20discrete scatter angles, are then located on the max-ima of the signal peaks as observed on an oscillo-scope. The other 11 encoder positions aredistributed among the peaks. Localizing the maxi-mum tends to be more difficult for narrow peaks nearthe forward direction, and localization inaccuraciesmay appear. These uncertainties may be likened toan underevaluation of measured signals in a discreteangle defined by the encoder. However, these errorsand the uncertainties that are due to inaccuratealignment are corrected by the calibration procedure.
Fig. 4. Notation for the ellipse displayed in Fig. 1: A, major axis;B, minor axis; C, interfoci distance; ui, scattering angle; ai, encoderangular position.
B. Data Postprocessing System
The data acquisition system generates P scatter dia-grams consisting of 20 discrete scattering angles andP transmission measurements. First the mean andthe standard deviation ~SD! of the scatter diagramsare evaluated with the P scattering patterns. The
ean and the SD of the PMT signals measured in theiscrete scattering angle ui are defined by
^W~ui!& 51P (
j51
P
@Wj~ui!#, (10)
^DW~ui!& 51
P 2 1 H(j51
P
@Wj~ui! 2 ^W~ui!J1y2
. (11)
Both the mean ^Wt& and the SD ^DWt& of the signal fortransmission measurements are expressed in thesame way. Values ^Wref& and SD ^DWref& are alsoevaluated for a reference transmission measurementwithout a scattering sample. The mean distributionfunction ^ f ~D!& for a stabilized or a slowly varying
edium can be expressed to a good approximation,or the short time ~typically less than 10 s! requiredor the measurements, from the mean of the PMTignal:
^W~ui!& } *0
`
^ f ~D!&uS2~D, ui!u2dD. (12)
In relation ~12!, S2 is the amplitude function14 thatorresponds to a polarization that here is set parallelo the plane of the nephelometer. The proposedethod is also suitable for taking data with perpen-
icular polarization: The S1 amplitude function in-stead of S2 is then used.
Using the Beer–Lambert law15 and assuming aGaussian distribution of the measurement error, wedefine the mean and the SD of the extinction coeffi-cient as follows:
^Kext& 521L
lnS ^Wt&
^Wref&D , (13)
^DKext& 51L FS^DWt&
^Wt&D2
1 S^DWref&
^Wref&D2G1y2
, (14)
where L is the length of the scattering medium. Theextinction coefficient is also equal to the followingequation:
^Kext& 5l2
p *0
`
^ f ~D!&Re@S2~0, D!#dD. (15)
The distribution function is then normalized with Eq.~15!.
C. Calibration Procedure
The experimental apparatus is designed to satisfy thehypothesis that V~ui! 3 DV~ui! 5 const. @ ui and to fixthe encoder position on the peak maxima. Obvi-ously, these hypotheses are verified only for perfect
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 977
gfidls
9
alignment of all the fixed mirrors and an ideal peaklocalization. To correct these errors, we perform acalibration procedure,16,17 using a medium withknown scatter diagrams. The reference mediumconsists of two standard Lambert diffuser disks ~seeFig. 5!. As the Lambert shape of the scatter dia-rams depends on the size of the incoming beam, therst disk ~LD1! homogenizes the light on the secondisk. The reference medium is mounted such as toimit any stray transmitted light. In Fig. 6 the PMTignal ~the mean and the SD! from 100 transmission
scatter diagrams of the reference medium describedabove is compared with the associated signal ~cosinefunction!, which corresponds to a theoretical idealLambert diffuser. The good agreement character-ized by a root-mean-square error of 3.5% underlinesthe quality of the alignment, with the exception oflarge scattering angles for which the maximum er-rors increase to 9%. Thus, to correct for alignmentand localization errors, we define a calibration con-stant K~u! for each scattering angle, using the mea-
Fig. 5. Reference medium composed of two Lambertian diffusers~LD1, LD2! and a field stop aperture.
Fig. 6. PMT signal ~and its uncertainties! measured in the cali-bration medium described above ~solid curve! and the best cosinefunction fitting the measured data ~dashed curve! as a function ofscattering angle.
78 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
surement of the reference sample and calculatingaccording to
K~u! 5^Wmeas
ref ~u!&
^Wtheorref ~u!&
. (16)
Figure 7 gives the calibration constant and its uncer-tainties as used in this study. Each signal mea-sured on a scattering medium is corrected by thefollowing calibration constant:
^Wmeascorr ~u!& 5
^Wmeas~u!&
K~u!. (17)
D. Experimental Arrangements for Incorporating CellScattering Disturbances
To validate the granulometer, we conducted experi-ments on latex microspheres in aqueous solutions.These solutions are settled in quartz spectrophotom-eter cells ~n 5 1.46 1 0j at l 5 632.8 nm! withvariable path lengths. The obvious disadvantage ofusing a scattering cell is that the cell modifies thescattering properties of the investigated medium.However, using nebulizers or aerosol generators in-stead of a scattering cell to produce a stable jet ofparticles is clearly unsatisfactory for the main pur-pose of this study, which is to calibrate the granu-lometer. That is, the PSD of jets of particlesproduced by these devices are not known, and theshapes of the particles can be significantly nonspheri-cal. For these reasons, we have chosen to calibratelatex microsphere solutions even if the cell distur-bances have to be corrected by a few approximations.Each measurement is performed with an opticalthickness of 0.1 to satisfy the single-scattering con-dition15 and to yield a significant measured signal.We then adjust the solution density by diluting thesolution with deionized water, using calibrated pi-pets. Before and after dilution, we suspend and dis-perse the latex particles by rolling or swirling thesolution, followed by a brief ~30-s! immersion in alow-power ultrasonic bath.
Fig. 7. Representation of the nephelometer calibration functionK~u! ~and its uncertainties! as a function of scattering angle.
spmellptiemts
supepo
4eptot
tatt
Table 1. Theoretical Mean Diameters and SD’s ~mm! of Polydisperse
Assuming that the electronic noise can be ne-glected ~except for large scattering angles!, the mea-urement uncertainties are governed mainly by twohenomena: particle Brownian motion and sedi-entation. In most experimental setups, the influ-
nce of Brownian motion of particles is significantlyimited by integration of the measurements over aarge time constant. As the measurements here areerformed sequentially with a pulsed selector,hough, this Brownian motion effect is clearly presentn our apparatus and may induce large uncertainties,specially for small particles, for which the motion isore intensive. Increasing the number of acquisi-
ions can reduce this phenomenon but requires mea-urement over a longer time interval.Obviously, sedimentation effects also disturb mea-
urements. The sedimentation rate is evaluated byse of a simple model based on the Stokes law18 ap-lied to the cell geometry. For spheres with a diam-ter f ~in micrometers!, the sedimentation rate j ~inercent! after t minutes may be written in a first-rder approximation as follows:
j~t, f! 512500
f 2t. (18)
As the sedimentation rate is a function of the particlediameter, the overall shape of the PSD could changeduring the measurement. In Fig. 8, we use Eq. ~18!to evaluate the change in a Gaussian distribution ~byvolume! with a mean diameter of 10 mm and a 2-mmstandard deviation. Sedimentation in such a casestrongly affects the PSD if the measurement timetakes more than 2 min. One can also note that themodification of PSD becomes less drastic withsmaller mean diameters.
A decrease in solution density modifies the level ofscatter diagrams proportionally. As sedimentationeffects tend to increase the SD of the measured sig-nals, the measurements have to be performed quickly~typically in less than 2 min!. The mean and the SDfor each latex solution are derived from 400 acquired
Fig. 8. Relative variations as a time function of a Gaussian PSDsubject to sedimentation effects described by Eq. ~18!.
scatter diagrams. If we include the time needed toset the cell on the granulometer incident plane, 400acquisitions correspond to an optimum total time ofless than 2 min to perform the measurements. Thisnumber of acquisitions is insufficient to reduce themeasurement uncertainties significantly, so the mea-surements exhibit quite large noise errors ~see Table1!. Nevertheless, because the present granulometerwas specially developed to give accurate PSD’s ofmedia that are subject to large noise errors, to vali-date it will ideally entail the use of large uncertain-ties to test the accuracy of the retrieved PSD.
Corrections are introduced for the change in thetransport of the scattered light by the quartz cells.First, external ~defined outside the cell! scatteringangles uext are evaluated. These angles are equiva-lent to the discrete scattering directions defined bythe fixed mirrors and are included in the range @240°,5°#. Using the usual Snell–Descartes law, we thenstimate the internal angles uin ~inside the spectro-hotometer cell!, which correspond to discrete direc-ions of particle scattering. Assuming that theptical index of latex aqueous solutions is equal tohat of water, ~nwater 5 1.33 1 0j!, the angles uin are
included in the range @229°, 32°#. Angles uq, definedinside the quartz plate, can also be evaluated in thesame way ~see Fig. 9!.
Then, to compare them with theoretical computa-ions, we must modify the measured PMT signals toccount for the transmission variation that occurs inhe cell. In this paper we use a first approximationo obtain the following expression for the transmis-
Fig. 9. Quartz cell geometry.
Distributions by Area and Volumea
DistributionMean Noise
Error ~%! Area Volume
f 5 5 mm 4.8 5.82 6 1.15 6.04 6 1.136.01 6 1.36 6.31 6 1.43
f 5 10 mm 5.8 10.63 6 1.47 10.83 6 1.4610.76 6 1.58 10.99 6 1.62
aThe values in italic type are derived from the log-normal model.
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 979
sf
~Tnpt^
m
T f
9
sion coefficient that relates the signal inside the cellto that outside. For an optical thickness t of thecattering medium, the transmission coefficient as aunction of internal scattering angle is expressed as
T~uin! 5 @1 2 Rwq~uin!# 3 @1 2 Rqa~uq!#
3 F1 1 Rwq2~uin!expS2
2t
cos uinDG
3 @1 1 Rqw~uq!Rqa~uq!#, (19a)
where Rij denotes the Fresnel reflection coefficientfrom medium i to medium j and the subscripts w, q,and a represent, respectively, water, quartz, and theexternal medium ~air!. The relative variation of thetransmission correction coefficient is presented inFig. 10. The signals inside and outside the cell arethen related by
^Wmeas~uin!& 5^Wmeas~uext!&
T~uin!. (19b)
Stray signals that are due to multipath reflection ofthe incoming beam can also exist if the quartz celldoes not lie exactly in the plane of incidence. As thefield of view is large, this multipath is observed andcan introduce significant error into the measure-ments. To correct these uncertainties, we per-formed a measurement of a quartz cell filled withpure distilled water; then a correction ratio was de-termined by
C~uext! 5^Wmeas
water~uext!&
^Wmeaswater~0!&
. (20)
An example of the correction ratio obtained from oneof the spectrophotometer cells is presented in Fig. 11.For this measurement, we optimized the cell positionto reduce the multipath effects significantly. Sus-pensions of latex spheres were then remeasured witha cell set in an optimized configuration defined by acorrection ratio C~uext!. Using Eqs. ~17!, ~19!, and
80 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
~20!, we can express PMT signals inside the cell in afirst-order approximation by the following relation:
^Wmeas~uin!& 5^Wmeas~uext!& 2 C~uext!^Wmeas~0!&
T~uin!K~uext!. (21)
4. Results and Discussion
A. Accuracy of the Nephelometer
The accuracy of the nephelometer is tested on mono-dispersed latex aqueous solutions referenced by theNational Institute of Standards and Technology.Two narrow PSD’s are used: 1.02 mm 6 0.02 mmf 5 1 mm! and 4.998 mm 6 0.035 mm ~f 5 5 mm!.heoretical scatter diagrams that correspond to thesearrow PSD’s are calculated by Mie theory. To com-are the measured signals defined by Eq. ~21! withhe calculations, we normalize both the measuredWmeas~uin!& and the theoretical b~uin! scatter dia-
grams as follows:
^Wmeas~uin!&N 5^Wmeas~uin!&
^Wmeas~uinN!&
, (22a)
b~uin!N 5b~uin!
b~uinN!
. (22b)
As the uncertainties that are due to stray reflectionsof the incoming beam are more important near theforward direction ~see Fig. 10!, the direction of nor-
alization uinN is chosen to satisfy C~uin! , 0.01 @ uin .
uinN. Figures 12 and 13 compare measurements and
calculations, showing good agreement with theoreti-cal predictions and contributing to the validation ofthe nephelometer’s accuracy. General forms are es-timated globally for the normalized theoretical scat-ter diagrams. For example, measured scatteringpatterns for the latex solution ~f 5 5 mm! locate thestructural oscillations of the sphere scatter diagramquite well ~Fig. 13!.
Nevertheless, a few differences still exist. First,
Fig. 10. Representation of the relative variation DTyT 5 @T~uin! 2~0!#yT~0! of transmission function T~uin! as a function of the in-
ternal scattering angle.
Fig. 11. Representation of the corrective function C~uext! as aunction of the external scattering angle.
Tssnadsl
D
antbaa
seTi
Sdbt
in accordance with what is expected for sphericalparticles, the measured diagrams tend to be symmet-rical, but the results for positive scattering anglesappear to be translated by a small systematic error,especially for particles of 1-mm diameter ~f 5 1 mm!~Fig. 12!. This small amount of bias on the scatter-ing pattern can be explained by a small imprecisionthat occurs when the cell is set in the incidence plane.A few discrepancies are also observed for the largescattering angles. These differences are correlatedwith the procedure for correcting stray reflections.At large angles, measured scattering signals are nearthe electronic noise. A similar phenomenon occurswhen C~uext! is evaluated. As Eq. ~21! subtracts twosignals that have been subjected to a significant con-tribution from noise in the measured signal, this re-lation becomes unsuitable for correcting measuredscattering signals and can introduce significant er-rors.
Fig. 12. Normalized scatter diagrams for the monodisperse PSDf 5 1.02 mm 6 0.02 mm: Measurement and its uncertainties~open circles with error bars! and calculations ~solid curve! as afunction of the internal scattering angle.
Fig. 13. Normalized scatter diagrams for the monodisperse latexPSD f 5 4.998 mm 6 0.035 mm: Measurement and its uncertain-ties ~open circles with error bars! and calculations ~solid curve! asa function of the internal scattering angle.
B. Accuracy of the Granulometer
The accuracy of the granulometer, which comprises anephelometer with the inverse method, is tested withlatex spheres in aqueous solutions, for which themean diameters and standard deviations of thespheres ~by number! are given by the manufacturer.
o compare with the optimized PSD from the inver-ion requires the use of models with which to repre-ent the theoretical PSD of the latex suspensions ~byumber!. According to most treatments in the liter-ture, the latex solution PSD’s are assumed to beescribed by either a Gaussian ~superscript or sub-cript g! or a log-normal ~superscript or subscript l !aw, defined as follows:
ng~D! 51
Î2psg
expF2~D 2 Dg!
2
2sg2 G , (23a)
nl~D! 51
Î2psl
1D
expF2~ln D 2 ln Dl!
2
2sl2 G , (23b)
with Dl 5 Dg and sl2 5 ln@1 1 ~sgyDg!2#. In Eqs. ~23!,
g and sg, are respectively, the mean diameter andthe SD supplied by the manufacturer. Two kinds ofsuspension are used: Dg 5 5.3 mm with sg 5 1.3 mmnd Dg 5 10.2 mm with sg 5 1.5 mm. To simplify theotation, we refer to these distributions as, respec-ively, f 5 5 mm and f 5 10 mm. A bimodal distri-ution composed of both of the previous solutions islso used ~referred to as bimodal!. The two modesre weighted according to Kext~f 5 5 mm! 5 Kext~f 5
10 mm!. For many applications that require grain-ize distribution measurements, PSD’s have to bexpressed in equivalent area or volume formulations.hus the theoretical distribution by number is mod-
fied according to
nal, g~D! 5
pD2
4nl, g~D!, (24a)
nvl, g~D! 5
pD3
6nl, g~D!. (24b)
ubscripts and superscripts a and v here and belowenote, respectively, the theoretical PSD by area andy volume. Corresponding mean diameters arehen defined by
^Dji& 5 * nj
i~D!DdDY* nji~D!dD,
j 5 $a, v%, i 5 $g, l%. (25a)
SD’s of the distribution are expressed by the relation
sji 5 3* nj
i~D!~D 2 ^Dji&!2dD
* nji~D!dD 4
1y2
. (25b)
Similar operations are also applied to the two op-timized PSD’s ~by number! fmin
` ~D! and fmax` ~D! to
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 981
a,v a,v
~o
o
9
yield fmin~D! and fmax~D!. Using Eqs. ~23!–~25!, weevaluate the statistical properties by area and vol-ume of theoretical PSD’s, which are presented in Ta-ble 1. The italics indicate results for the log-normalmodel.
Figures 14–19 show the results obtained by inver-sion of corrected signals measured in three solutions.In each of these figures the solid black curve repre-sents the given theoretical log-normal distribution byvolume nv
l ~D! or by area nal ~D!. A solid curve with
symbols D represents the Gaussian PSD by volume orarea. These plots are compared with the two opti-mized distributions by volume or by area: fmin
v ~D! orfmin
a ~D! ~open circles! and fmaxv ~D! or fmax
a ~D! ~dashedcurves! that define the confidence interval. In thisexperimental part of the study, we chose to focus ourattention on global modifications of the main form of
m
o
82 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
bounding functions that governs the statistical pa-rameters of the distribution. Their relative intensi-ties are not considered here, as their influence islimited to another parameter, the volume concentra-tion. This point was underlined in the theoreticalpart of this study ~see Ref. 8!. For the comparison,we then normalize optimized PSD’s according to
fminv,a ~D! 5 fmin
v,a ~D! 3
maxD
@nv,al ~D!#
maxD
@ fminv,a ~D!#
, (26a)
fmaxv,a ~D! 5 fmax
v,a ~D! 3
maxD
@nv,al ~D!#
maxD
@ fmaxv,a ~D!#
. (26b)
Fig. 14. Comparison of normalized PSD ~by volume! for the f 55 mm polydisperse latex solution: Log-normal distribution nv
l ~D!solid curve!, Gaussian distribution nv
g~D! ~curve with triangles!,ptimized bounding distributions fmin
v ~D! ~open circles! and fmaxv ~D!
~dashed curve!.
Fig. 15. Comparison of normalized PSD ~by area! for the f 5 5mm polydisperse latex solution: Log-normal distribution na
l ~D!~solid curve!, Gaussian distribution na
g~D! ~curve with triangles!,ptimized bounding distributions fmin
a ~D! ~open circles! and fmaxa ~D!
~dashed curve!.
Fig. 16. Comparison of normalized PSD ~by volume! for the f 510 mm polydisperse latex solution: Log-normal distribution nv
l ~D!~solid curve!, Gaussian distribution nv
g~D! ~curves with triangles!,optimized bounding distributions fmin
v ~D! ~open circles! and fmaxv ~D!
~dashed curve!.
Fig. 17. Comparison of normalized PSD ~by area! for the f 5 10m polydisperse latex solution: Log-normal distribution na
l ~D!~solid curve!, Gaussian distribution na
g~D! ~curve with triangles!,ptimized bounding distributions fmin
a ~D! ~open circles! and fmaxa ~D!
~dashed curve!.
aTttftsstNi
tsTttctpc
D
tnqtitsocstitiibpp
sdrbsnFpbw
~o
This normalization scheme can consequently inducean overlap of the normalized functions if the twobounding functions are largely proportional.
Because the inverse scheme, to stabilize the prob-lem, evaluates the numerical filtering on a distribu-tion defined by area, greater accuracy is obtained forthe area distribution. Even if large noise uncertain-ties are present ~Table 1!, good agreement is gener-lly obtained for the two monomode distributions.heoretical PSD’s are included in the confidence in-
erval or near the limit functions. These results con-ribute to a relevant validation of the granulometeror large uncertainties. For the f 5 5 mm distribu-ion ~Figs. 14 and 15!, the confidence interval is verymall, indicating a relative stability of the smootholution ~the two bounding functions here are propor-ional, with a peak-to-peak relative variation of 6%!.evertheless, two phenomena are observed: There
s a small peak near 2 mm that is more pronounced in
Fig. 18. Comparison of normalized log-normal PSD’s for the bi-modal polydisperse latex solution.
Fig. 19. Comparison of normalized PSD’s ~by area! for the bi-modal polydisperse latex solution: log-normal distribution na
l ~D!solid curve!, Gaussian distribution na
g~D! ~curve with triangles!,ptimized bounding distributions fmin
a ~D! ~open circles! and fmaxa ~D!
~dashed curve!.
he PSD defined by area and the optimized PSD’s areome amount broader than the theoretical ones.he first phenomenon can be explained as being dueo a slight underestimation of the corrections appliedo the measurements for a few positive angles ~espe-ially near the forward direction!. This error is dueo a small misalignment of the cell position in thelane of incidence for the measurements. As theorrective coefficient C~uext!^Wmeas~0!& is defined with
better alignment, the resultant small inaccuracy ofthe corrective coefficient can consequently increasethe corrected signal levels for a few positive smallangles. The less drastic reduction of the signal com-pared with the scattering angle is then taken by theinverse algorithm for the presence of smaller parti-cles that present a less intense decrease of scatteringintensities as a function of scattering angle. Thesecond phenomenon is governed mainly by noise.The f 5 5-mm solution strikes a relative compromisebetween Brownian motion effects and sedimentationdisturbances. The uncertainties are moderate:The mean noise level ~excluding electronic noise! is4.8%. It was shown previously8 that a high noiselevel in scatter diagrams results in a low cutoff fre-quency that generates largely spaced consecutivenodes of the spline functions. As the interpolation ofthe PSD is performed on a large grid, the optimizedPSD can be overestimated, particularly for a weakPSD. Here the theoretical PSD is broad ~sg ' 20%
g!. The effects of the rather large grid are presentbut not extensive, especially if the theoretical PSD isdescribed by the log-normal model.
For the f 5 10-mm distribution ~Figs. 16 and 17!,he optimized distributions are quite symmetric, witho stray peak. The cell position here appears to beuite satisfactory. The theoretical PSD’s defined byhe log-normal model are close to the optimized min-mum functions fmin
v,a ~D! derived from small transla-ions of the smooth function that are performed in themall-diameter range. A large confidence interval isbtained, and the stability of the smooth solution islearly worse. Unlike for the latex solution, the sen-itivity of the smooth function to the small transla-ions performed in the large-diameter range is clearlyncreased. Other ideas can be advanced to explainhis point. First, a greater mean noise level ~'5.8%!s observed, with greater sedimentation effects. Thenversion is then carried out on a larger grid thanefore. Furthermore, the conditioning of the inverseroblem becomes less efficient with an increase ofarticle size.For bimodal distribution, the accuracy of the re-
ults is not so good as before. Even if the inverseistributions describe the overall shape of the theo-etical PSD, these optimized PSD’s are quite a bitroader and are not capable of representing the finetructure of the theoretical PSD. The high meanoise level ~'6.8%! is the chief cause of these errors.or area or number distributions, the first mode com-onent is greater than the second one ~Fig. 18!. Sta-ility of the smoothest solution is then observed, asith the f 5 5-mm solution. Furthermore, we may
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 983
dbeb~frarn
coorfttoilpcl
motlsstp
A
T
T
awT
g
9
assume that the inversion is carried out preferen-tially on this mode. The localization of the maxi-mum of the first mode is then accurate ~Fig. 19!. Asthe interpolation of the PSD is performed on quite alarge grid, a bias is introduced in the PSD reconstruc-tion that is more pronounced in the second modeaccording to our remarks made above. A broaden-ing of the distribution to the smaller diameters thencompensates for the overestimation in the secondmode, which explains the less relevant results ob-served in the volume distribution ~Fig. 20!, where thecontribution of the second mode is major. For exam-ple, the localization of the major mode is not accurate.Nevertheless, even if the fine structure of the PSD isnot in fact retrieved by the inverse method ~especiallyby volume!, the statistical parameters that issue fromthe bounding functions are close to the theoreticalparameters. In that sense, we may assume that theretrieved global form of the PSD is a reasonable ap-proximation of the bimodal solution.
According to Eqs. ~25!, the statistical properties ofan optimized PSD in terms of area and volume areevaluated and presented in Table 2. For bimodaland f 5 5-mm distributions, identical statisticalproperties are obtained for the two bounding func-tions. For the f 5 10-mm distribution, the resultsthat correspond to fmin
v,a ~D! are in italic type. For all
Fig. 20. Comparison of normalized PSD’s ~by volume! for thebimodal polydisperse latex solution: log-normal distributionnv
l ~D! ~solid curve!, Gaussian distribution nvg~D! ~curve with trian-
les!, optimized bounding distributions fminv ~D! ~open circles! and
fmaxv ~D! ~dashed curve!.
Table 2. Optimized Mean Diameters and SD’s ~mm! of PolydisperseDistributions by Area and Volumea
Distribution Area Volume
f 5 5 mm 5.77 6 1.95 6.43 6 1.90f 5 10 mm 10.98 6 1.70 11.24 6 1.76
aThe values in italic type are derived from the minimum functionfmin
v,a ~D!.
84 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
the distributions, inverse mean diameters ~by area!are very close to the theoretical values, with an errorof less than 5% even if a high noise level exists. Forthe bimodal and the f 5 10-mm distributions, the
ifferences increase slightly when the volume distri-ution is used. For the f 5 10-mm solution, therror still remains acceptable ~less than 7%!, but itecomes significant for bimodal distributions'15%!. That is, area distributions directly per-ormed by the inversion scheme yield more-accurateesults than volume distributions, which is reason-ble. For the f 5 5-mm solution, this tendency iseversed because the effect of the stray peak is sig-ificantly reduced for the volume distribution.The influence of the mean level of uncertainties is
learly illustrated in the results by an overestimationf retrieved SD’s. For monomode distributions, theptimized values are relatively similar to the theo-etical values. Larger discrepancies are observedor the bimodal distribution, and they clearly illus-rate the decreased sensitivity of the inverse methodo noise as well as the effect of a large grid whenptimization is performed on a complex PSD consist-ng of mixed modal functions. Thus we can see aimitation of the granulometer presented here. Theerformance would have been improved if the twoontributive modes had been different or if the noiseevel had been more moderate.
The accuracy of the results presented for the mono-odal distributions clearly demonstrates the validity
f the granulometer, even for media that are subjecto large noise uncertainties. The granulometer isess efficient for convoluted multimode PSD’s that areubject to large noise uncertainties, even if the recon-tructed distributions describe the form and the sta-istical properties of the PSD as a whole. Itserformance can be improved if
moderate noise level or a sufficient number of dis-crete data reduces the size of the mesh grid signif-icantly,
he distributions are of comparable width and arenot extremely narrow, and
he different components of the PSD are distinct.
5. Conclusions
A Mie light-scattering granulometer has been pre-sented and validated. This apparatus is based onthe principle of measuring single-scatter diagramswith interpretation by an inverse method that in-cludes an adaptive numerical filtering scheme to sta-bilize the particle-size distribution optimizationagainst noise errors. This method was described,and its stability established numerically, in a previ-ous study. In the present paper we merely recalledthe main principles of this inverse method, while fo-cusing on experimental considerations. The nephe-lometer used in this study was specially designed totake in situ measurements in unstable media. Thispparatus uses an original light-collection system inhich the measurement is made in a few seconds.he experimental setup was fully described, and a
calibration procedure was proposed and tested forcorrecting alignment errors. We then estimated thenephelometer’s accuracy by using this calibrationprocedure on calibrated monodisperse latex solu-tions. As the presence of a scattering cell disturbsthe measurements, first-order corrections were de-rived. Despite some uncertainties introduced by thecell position in the plane of incidence, pertinent re-sults were obtained that validate the accuracy of thenephelometer. Finally, the granulometer’s accuracywas evaluated for polydisperse latex solutions forwhich the PSD was known beforehand. These mea-surements are subject to large noise levels owing toBrownian motion and sedimentation effects, so thegranulometer’s performance was evaluated for highlyunfavorable cases. We have shown that the opti-mized PSD and statistical properties derived frommeasurements were in good agreement with the the-oretical functions and values. These results contrib-ute to a relevant validation of the proposedgranulometer. The perspectives of the present re-search lie within the framework of an extension of alight-scattering granulometer to permit the granu-lometry disturbances that result from nonsphericityof measured particles to be considered.
This research was supported by the Delegation Ge-nerale pour l’Armement, France.
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