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REASEARCH ARTICLE

J. B. Blaisot Æ J. Yon

Droplet size and morphology characterization for dense sprays by imageprocessing: application to the Diesel spray

Received: 14 June 2004 / Revised: 3 March 2005 / Accepted: 7 July 2005 / Published online: 19 October 2005� Springer-Verlag 2005

Abstract Up to now, measurement of drop size remainsdifficult in dense sprays such as those encountered inDiesel applications. Commonly used diagnostics areoften limited due to multi-scattering effects, high dropvelocity and concentration and also nonsphericalshapes. The advantage of image-based techniques on theothers is its ability to describe the shape of liquid par-ticles that are not fully atomized or relaxed. In thepresent study, a model is developed to correct the maindrawbacks of imaging. It permits to define criteria forthe correction of the apparent size of an unfocused dropand to determine a measurement volume independent ofthe drop size. This considerably reduces the over-esti-mation of large drops in the drop size distribution. Dropshapes are also characterized by four morphologicalparameters. The image-based granulometer is satisfac-torily compared to a PDPA and a diffraction-basedgranulometer for measurements on an ultrasonic spray.Then, the new granulometer is applied to a diesel spray.One of the results of the analysis is that even if meandrop size distributions are stable 30 mm downstreamfrom the nozzle outlet, the shape of the drops is stillevolving towards the spherical shape. The atomizationprocess is thus not totally established at this position inopposition to what can be deduced from the drop sizedistribution alone.

1 Introduction

The pollutant emissions caused by combustion enginesrepresent an important problem for the environment.Transport activity is considered by international

organizations such as International Programs Center(IPC), to be responsible for more than 20% of thegreenhouse gas emissions. In order to reduce thepollutant gas emissions of cars, car manufacturershave to develop new engines that are compliant withmore and more restrictive pollution regulation laws.Injection systems able to produce very fine spraysrepresent one development axis that has been followedover the last two decades to reduce pollutant emis-sions, particularly concerning Diesel injection. Thevery fine sprays are obtained by increasing the injec-tion pressures in common rail systems, decreasing thenozzle hole diameters and reducing the injection-duration time. However, because of the high opticaldensity of the Diesel sprays, it is not possible toidentify either large liquid blobs that are not totallyatomized or dense spray regions composed of very finedroplets. The optical density makes the diagnostics ofthe Diesel spray very difficult. The most commonlyused diagnostics for Diesel spray analysis are diffrac-tion-based granulometer and Phase Doppler ParticleAnalyzer (PDPA). Hardalupas (1992) and Guerrassiand Champoussin (1995) used a PDPA measurementand observed, in an atmospheric environment, a de-crease of the droplet Sauter mean diameter (SMD)with the increase of the radial distance from the axisof injection. A completely different result was obtainedby Gulder and Smallwood (1999) who used a dif-fraction granulometer to analyze the Diesel spray andobserved a radial increase of the SMD. This contra-diction clearly shows that the results must be carefullyinterpreted. Diffraction measurement and PDPA aresuspected to be unsuited to dense sprays, due to themulti-scattering effect. Another important limitation ofthese techniques concerns the droplet sphericityhypothesis on which they rely. We can notice thatother drop size measurement techniques are underdevelopment such as holography (Buraga et al. 2000),planar laser scattering (Domann and Hardalupas2000), out of focus laser scattering imaging (Calabriaand Massoli 2001) and speckle light scattering

J. B. Blaisot (&) Æ J. YonUMR CNRS 6614—CORIA, Laboratoire de Thermodynamique,Universite et INSA de Rouen, 76801 Saint Etienne du Rouvray,FranceE-mail: [email protected]

Experiments in Fluids (2005) 39: 977–994DOI 10.1007/s00348-005-0026-4

(Ineichen 2003). Nevertheless these diagnostics are notyet ready to analyze dense industrial sprays such asDiesel sprays.

Image-based granulometry is another emergingtechnique available to analyze the sprays, as can beseen in the recent review of Lee and Kim (2003). Theprincipal interest of this technique resides in its abilityto quantitatively analyze the liquid element morphol-ogy. This is a good indicator of the level of dropletatomization. Up to now, the main limitation of thesediagnostics is the out-of-focus phenomenon (Kohet al. 2001). Indeed, on an image, an unfocuseddroplet seems to be larger than a focused one of thesame size. Therefore, accounting for unfocused drop-lets implies an overestimation of the biggest diametersin the droplet size distributions. There are differentways to avoid this phenomenon. One of these consistsin defining a focus criteria in order to reject allunfocused droplets. This has been proposed by dif-ferent authors such as Fantini et al. (1990), Lecuonaet al. (2000), Koh et al. (2001), Lee and Kim (2003)and Kim and Kim (1994). This method has twodrawbacks. Firstly, the droplet rejection rate is high,so numerous pictures of the sprays have to be ana-lyzed in order to construct smooth droplet size dis-tributions. The second limitation is linked to the factthat small droplets are more concerned by the out-of-focus phenomenon than bigger ones, resulting in anoverestimation of the biggest droplets. To overcomethis point, Hay et al. (1998) used a gradient criterionto define a depth of field independent of the dropdiameters. Nevertheless, this criterion was based on anempirical relationship.

The blurring effect on unfocused droplet images hasto be taken into account in order to correct themeasurement of the droplet diameter. Blaisot andLedoux (1998) and Malot and Blaisot (2000) proposed

an imaging model based on the estimation of thepoint spread function (PSF) to carry out suchcorrection for the measurement of the diameter ofwell-located liquid drops. However, when applied tospatially dispersed sprays, this model suffer from thesame limitation indicated above: the population of bigdrops is overestimated. We propose in the presentpaper an improvement of the PSF model in order toevaluate the droplet focusing. The optical PSF widthis usually considered as the resolution limit for therecording device (film or camera). This is not the casein the present application where the PSF is spatiallyresolved by the CCD camera (more than 10 pixelsover the PSF width). Thus the PSF width is evaluatedthrough the analysis of the gray-level profile at theoutline of the drop image. As the PSF width is relatedto the droplet location relative to the focusing plane,the selection of droplet image characterized by a PSFwidth below a chosen maximum value is used todelimit the measurement volume along the depth fromfocus direction.

The first part of this paper consists in the presen-tation of the improved model of the droplet imageformation. The different steps of the image processingfor the extraction of the image parameters necessaryto the measurement of the droplet diameter, thefocusing and the morphological characterization of thedroplets is presented. Next, the calibration process ofthe model with our optical setup is presented. Dropsize measurements on a calibrated ultrasonic sprayare compared to PDPA and diffraction-based granu-lometers. The paper ends with the application of thisnew size and morphology analyzer to a Diesel spray.It is found that the shape analysis is relevant forthe characterization of the atomization processthrough the information it adds to the drop sizedistribution.

Fig. 1 Definition of opticalaxes in object and image planes

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2 Image modeling

In the case of an imaging system using a noncoherentlight source, the illumination distribution in the imageplane i(x, y) can be described by the convolution productof the irradiance distribution in the object plane o(x¢, y¢)and the PSF of the imaging system s(x, y) (Goodman1968):

iðx; yÞ ¼ oðx0; y0Þ � sðx; yÞ: ð1Þ

The coordinates (x¢, y¢) and (x, y) are associated to theobject plane and the image plane, respectively (seeFig. 1). The axis z, perpendicular to the planes (x¢, y¢)and (x, y), is the optical axis of the imaging setup. ThePSF can be defined as the diffraction image of aninfinitesimally small source point. This image would bemerely a point under geometric optics. In order tocompute the convolution product in the image plane, wereplace the object function o(x¢,y¢) by its geometric im-age og(x,y), deduced from the object function simply bya homothetic transformation of ratio equal to the lateralmagnification c of the imaging set-up.

2.1 Object modeling

The imaging set-up is in a backlight configuration andthe objects under consideration are droplets of liquidilluminated by a white, noncoherent light source. Onlyopaque or slightly transmitting objects are considered.The amount of transmitted light is characterized by thecontrast coefficient s. The irradiance distribution in theobject plane, the object function, can thus be expressedin cylindrical coordinates ðr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ y2p

Þ by:

ogðx; yÞ ¼ ogðrÞ ¼ 1� ð1� sÞP r2ai

� �

; ð2Þ

where ai=c ao is the radius of the geometric image, aobeing the object radius and Pð~rÞ is the rectangle func-tion:

Pð~rÞ ¼ 1 for j~rj\ 0:5;0 otherwise:

�

As indicated in the introduction, the principle of theimage analysis is based on the resolution of the PSF bythe camera. Such condition is obtained here by a rela-tively small numerical aperture for the optical setup(NA.0.1). In this case, only the forward scattered lightnear the axis direction contributes to the image and onlya very small part of the off-axis light gets into theimaging system. Images of liquid droplets or opaquediscs are thus very similar in this case except in the centerof the droplet image where the refracted component ofthe light near the axis leads to a brighter spot in thecenter (Hovenac 1986). This difference is not restrictivehere as only the outline of the image profile and the

minimum grey level are considered in the image analysisprocedure as explained below.

In practice, the parameter s is introduced to adapt themodel to the experiment. Indeed, the theoretical contrastof the images is calibrated to the experimental onesthrough the determination of this parameter. The valueof s depends on the aperture of the optical setup and onthe object diameter too, but the way s is related to theobject size greatly depends on the effective aperture. Thesmall numerical aperture of the optical setup used in thiswork leads to small values of s. As a consequence, thisparameter has been considered not to depend on theobject size in the remainder without noticeable loss inaccuracy.

2.2 Point spread function (PSF)

For a diffraction-limited optical system of circularaperture under monochromatic illumination, the PSF isgiven by

sdlðr; kÞ ¼ J1ðakrÞðakrÞ2

;

where ak is a scale parameter depending on the wave-length k. When considering noncoherent polychromaticlight, the PSF can be modeled by a Gaussian shape(Pentland 1987) as the result of the contribution ofnumerous functions sdl(r;k) for the various wavelengths:

sðrÞ ¼ s0 exp �2r2

v2

� �

; ð3Þ

where v is the PSF half-width and s0 is a normalizationconstant. The PSF changes with the position of thedroplet in the object space. Indeed, the PSF widthstrongly changes with the location of the object along theoptical axis: the more unfocused the object, the larger thePSF and the more blurred the image. Nevertheless, it issupposed that v does not depend on the image planecoordinates (x, y), i.e., there is no spatial variation of theoptical aberrations of the imaging device in the plane (x,y). Finally, it will be supposed in the following that thePSF half-width v is only a function of z.

2.3 Image computation

Introducing Eqs. 2 and 3 in Eq. 1 yields

iðx; yÞ ¼ZZ

þ1

�1

1� ð1� sÞP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 þ g2q

2ai

0

@

1

A

0

@

1

A

� exp � 2

v2ðx� nÞ2 þ ðy � gÞ2n o

� �

dndg: ð4Þ

Equation 4 is normalized and written in circularcoordinates:

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~ið~rÞ¼1�2ð1�sÞexpð�~r2ÞZ ~a

0

qexpð�q2ÞI0ð2~rqÞdq; ð5Þ

~imax ¼ 1;~imin ¼ ~ið0Þ ¼ 1� ð1� sÞ 1� expð�~a2Þ� �

; ð6Þ

where ~r is the nondimensional radial coordinate ~r ¼ffiffiffi

2p

r=v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðx2 þ y2Þp

=v; ~a is the nondimensional dropletradius ~a ¼

ffiffiffi

2p

ai=v and I0 is the modified Bessel functionof the first kind.

A low value for ~a corresponds to a small object widthcompare to the PSF width. A small focused object or alarger unfocused object can thus have the same value for~a: Several theoretical image profiles are shown inFig. 2a. The width of the image profile obviously in-creases when the object diameter increases. For low ~a;the image profile presents a V-shape and when ~a in-creases it turns to a U-shape. For a given value of ~a; theheight of the profile is controlled by the parameter s: thelower s, the higher the profile. The minimum level at theimage center ~imin ¼ ~ið0Þ (Eq. 6) decreases with ~a andreaches the limiting value~imin ¼ s when ~a is large enoughð~a ¼ 2:5; s ¼ 0; 0:2 on Fig:2aÞ: On the other hand, forlower values of ~a; the minimum level varies between sand 1.

2.4 Image parameters

The image parameters used to compare theoretical andexperimental image profiles are now introduced. Asketch for the definition of these parameters is presentedin Fig. 2b. The image contrast C, defined by Eq. 7, is anondimensional parameter bounded between 0 and 1.

C ¼~imax �~imin

~imax þ~imin

¼ 1�~imin

1þ~imin

¼ ð1� sÞð1� expð�~a2ÞÞ2� ð1� sÞð1� expð�~a2ÞÞ :

ð7Þ

The variation of C versus ~a is shown in Fig. 3a fors=0 and 0.2. The contrast increases monotonously with~a and reaches the maximum Cmax for ~a&2:4: As ~imin

tends to be s when ~a increases (see Eq. 6), the limitingvalue for the contrast is:

Cmax ¼1� s1þ s

: ð8Þ

For completely opaque objects (s=0), this yieldsCmax=1 and for s=0.2, it gives Cmax=2/3 (see Fig. 3a).Eq. 8 is used to determine the experimental value of s bymeasuring the biggest value for the contrast. Asexplained before, it is assumed here that s does not de-pend on the droplet diameter. A reference contrast de-fined as the contrast for a perfectly opaque object isobtained writing s=0 in Eq. 7:

Cðs ¼ 0Þ ¼ C0 ¼1� expð�~a2Þ1þ expð�~a2Þ : ð9Þ

For an object of contrast coefficient s „ 0, thecontrast can be expressed as a function of s and C0 byeliminating expð�~a2Þ between Eqs. 7 and 9:

C ¼ C0ð1� sÞ1þ C0s

; ð10Þ

thus the normalized contrast C0 can be expressed as afunction of s and C:

C0ðC; sÞ ¼C

ð1� sÞð1þ CÞ � C: ð11Þ

The maximum normalized contrast C0,max=1 is ob-tained for C=Cmax in Eq. 11. In the following, thenormalized contrast C0 is used in place of C to expressrelations without any dependence on s.

A second image parameter is used to describe the greylevel profile: it is the image half-width ~rl determined at arelative level l, (0<l<1). The absolute reference level~irefcorresponding to a given relative level l is defined byEq. 12 where h ¼ ~imax �~imin is the height of the imageprofile.

~irefðlÞ ¼ ~imin þ lh: ð12Þ

The image half-width ~rl is then defined implicitly by:

~ið~rlÞ ¼ ~irefðlÞ: ð13Þ

-4 -2 0 2 4r~

0

0,2

0,4

0,6

0,8

1

i~

0 r0

iimax

imin

iref r

lh

l.h

a bFig. 2 Theoretical imageprofile: a profiles for differentvalues of the nondimensionalobject radius ~a and of thecontrast coefficient s. filledsquare: ~a ¼ 1:12; s ¼ 0:0; filledtriangle: ~a ¼ 1:5; s ¼ 0:0; opentriangle: ~a ¼ 1:5; s ¼ 0:2; Filleddiamond: ~a ¼ 2:5; s ¼ 0:0; opendiamond: ~a ¼ 2:5; s ¼ 0:2;b definition of the relative levels

980

As the reference level is defined relative to the imageprofile height, it can be easily shown from Eq. 14 that ~rldoes not depend on s:

expð�ð~rlÞ2ÞZ þ~a

0

q expð�q2ÞI0ð2~rlqÞdq

¼ ð1� lÞZ þ~a

0

q expð�q2Þdq:

ð14Þ

The half-width for relative levels l=0.25, 0.61 and0.77 are used to characterize the image profile. Thevariation of ~rl for these three levels is shown inFig. 3b. When ~a is large enough ð~a > 1:5Þ; the calcu-lated radius ~rl increases linearly with the object widthand for ~a �! 0;~rl tends to be the half-width of thePSF at the relative level 1 � l by definition of theimage width. Indeed, when ~a �! 0; the convolutionproduct (Eq. 1) becomes an identity as o(x, y) can beconsidered in this case as a Dirac function. The resultis that the image is very similar to the PSF in such acase. For l=0.61, ~rl remains always slightly largerthan ~a:

2.5 Droplet diameter estimation

It is assumed now that s is known (s is calculated for thebiggest unblurred droplet) and that l is fixed to a givenvalue. So, for fixed values of s and l, C0 and ~rl arefunctions of ~a only (Eqs. 9 and 14). In this set of para-metric equations, the parameter ~a can be eliminated,yielding the direct relation between C0 and ~rl: No loss ofgenerality occurs when dividing ~rl by ~a and taking theinverse of the result. The ratio obtained a/rl representsthe ratio of the object width to the measured imagewidth. The variation of a/rl versus C0 is shown in Fig. 4for l=0.25, 0.61 and 0.77. The object to image widthratio increases with the contrast except for the curvel=0.25 presenting a maximum around C0=0.76. Thestarting and ending points of the curves are (0,0) and(1,1) for (C0,a/rl), whatever the relative level l. Imageswith a very low contrast correspond to unfocused ob-jects or very small focused objects, i.e., objects of widthmuch smaller than the PSF width ð~a ’ 0Þ: In such a casethe, image profile is flat and spreads over a large area,resulting in a very small value of a/rl soðC0; a=rlÞ �! ð0; 0Þ: The ending point of the curves(C0,a/rl)=(1,1) is reached for ~a �!1: The U-shapeimage profiles then tend to the square shape of the objectprofile. It is noticeable that among the strictly increasingcurves, the curve for l=0.61 is the one that has thelargest dynamics for the values of a/rl. For relative levelslower than 0.61, the curve a/rl=f(C0) presents a fishingrod shape as shown in Fig. 4 for l=0.25. This explainsthe fact that the classical mid level (l=0.5) was not usedhere to measure the image width. For practical appli-cations, the relation a/rl=f(C0) is fitted by a polynomialof an order up to 4:arl¼ plðC0Þ ¼ a0 þ a1C0 þ a2C2

0 þ a3C30 þ a4C4

0 : ð15Þ

The polynomial coefficients ak, {k=0.4} are functionsof l. The size of the object is estimated from the mea-surement of the parameters rl,meas and Cmeas andthrough Eqs. 11 and 15:

aest ¼ rl;meas � plðC0ðCmeas; sÞÞ; ð16Þ

whereas classical approaches consider focused objectsonly, corresponding to Cmeas . Cmax, Eq. 16 is used toexpress the radius of unfocused objects, corrected from

0 1 2 3 4a~

0

0,2

0,4

0,6

0,8

1

C

0 1 2 3 4a~

0

1

2

3

4

5

r l~

a bFig. 3 Image parameters versusnondimensional object radius ~a:a Contrast C. filledsquare: s=0;filled triangle: s=0.2. b Half-width ~rl: filled square: l=0.25;filled triangle: l=0.61; filleddiamond: l=0.77

0 0,2 0,4 0,6 0,8 1C

0

0

0,5

1

1,5

a/r l

Fig. 4 Variation of a/rl versus normalized contrast C0 for differentrelative levels of thresholding l. filled square: l =0.25; filled triangle:l=0.61; it filled diamond: l=0.77

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the out-of-focus effect. The depth of field of the dropletsizing setup is thus increased. The limiting parameter isgiven by the minimum contrast Cmin. The minimumcontrastmeasurable with a given optical setup depends onthe threshold used to detect the droplet images. Oneshould note that images are constituted of black elementsover a white background so that negative transition areconsidered and that the threshold levelmust be lower thanthe white background level. This threshold level must bechosen sufficiently low to prevent detection of false image(i.e., noise) but also sufficiently high in order to detect themaximumnumber of unfocused and small droplet images.The tradeoff leads to Cmin . 0.1 in our case.

The minimum diameter measurable with this tech-nique is defined by fixing a minimum number of pixelper droplet. We fixed this number to NPixmin=10 pixelsfor the lowest relative level l=0.25 which represents thesmallest droplet image area to be considered. The min-imum diameter is thus deduced from Eq. 16:

dmin ¼ p0:25ðC0ðCmin; sÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4NPixmin

pR2

r

: ð17Þ

The minimum contrast measurable induces also amaximum value for v for a given object size a.According to Fig. 3a, Cmin . 0.1 is obtained for~amin ’ 0:5: It comes from the definition of ~a :

vmax ¼ v Cmin ¼ 0:1ð Þ ’ 2:8a: ð18Þ

As the PSF width v increases with the distance z fromthe focus plane, this implies that the biggest droplets aremeasurable over a larger domain of space than thesmallest ones. Therefore, if all the measurable dropletsare counted in the drop size distribution, the populationof the smallest droplets will be underestimated. Toovercome this problem, a correction of the dropletcounting based on the estimation of the droplet focusingis proposed hereafter.

2.6 Droplet focusing estimation

An estimation of the droplet focusing can be obtainedthrough the determination of the PSF half-width v as itis directly related to the position of the droplet in theobject space along the optical axis (z). The grey levelgradient on the outline of the image is strongly related tothe PSF width, due to the convolution product. In fact,when the object width is a lot bigger than the PSF widthð~a� 1Þ; the width of the image outline is directly linkedto the PSF width. In these particular conditions,ð~a� 1Þ; the gradient criteria can be used to discriminatefocused and unfocused droplets. For smaller or more-blurred droplets, gradient criteria are not effectual any-more. So, instead of calculating the grey level gradient atthe outline of the image, the imaging model is used tolink the difference between image widths at two differentlevels l1 and l2 and the PSF half-width. It can be seen onFig. 3b that the difference between half-widths ~rl for

l=0.77 and l=0.25 does not varies strongly with ~a andseems to reach a limit value for ~a > 1:5: Let us define thenondimensional half-width difference D~r by Eq. 19:

D~r ¼ ~r0:77 � ~r0:25: ð19Þ

It can be observed in Fig. 5 that D~r linearly increaseswith the contrast C0.

The minimum value is 0.676 for C0=0, and themaximum is 1.021 for C0=0.988. As for the estimationof the droplet diameter, the relation between D~r and C0

is fitted by a polynomial of order 3:

D~r ¼ prðC0Þ ¼ b0 þ b1C0 þ b2C20 þ b3C3

0 : ð20Þ

Knowing the coefficient s, the PSF half-width can beestimated from the measured parameters Drmeas andCmeas and Eqs. 11 and 20:

vest ¼ Drmeas=prðC0ðCmeas; sÞÞ: ð21Þ

An estimation of v is now available for each mea-surable droplet, v being associated to the droplet loca-tion along the optical axis. v is an increasing function ofthe distance of the drop from the focus plane, so a cri-terion independent of the droplet diameter can thus bedefined to take into account only the droplets located ina given volume:

vest\vmax: ð22Þ

This is illustrated in the Sect. 4.

3 Image analysis

The analysis is a 3-stage process. The first stage corre-sponds to a global image pre-processing called normal-ization. The second stage consists in detecting andseparating the different droplets on the pictures. The laststage consists in applying the imaging model to eachdetected droplet in order to extract the droplet charac-teristics. All the computations needed for the processing

0 0,2 0,4 0,6 0,8 1C

0

0

0,2

0,4

0,6

0,8

1

∆r~

Fig. 5 Variation of D~r ¼ffiffiffi

2pðr0:77 � r0:25Þ=v versus normalized

contrast C0

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of the images have been done on a standard personalcomputer using C programming language.

3.1 Illumination normalization

In the imaging model, a uniform background image isconsidered. In real applications, the light distribution inthe background of a backlight configuration setup israrely uniform. The reason is that the light source, ormore rarely the CCD camera response, are not spatiallyhomogeneous. As black over white images are consid-ered and as the response of the CCD camera to the inputlight is linear, a normalized image, expressed by Eq. 23,can be computed

Pnði; jÞ ¼ bPnormPði; jÞ � POði; jÞPBði; jÞ � POði; jÞ

; ð23Þ

where 0<b<1, P(i, j) is the grey level of pixel (i,j),PB(i, j) is the grey level of the background image(obtained without any object in the view field) and PO(i,j) corresponds to the noise level of the grabbing setupand is obtained with the camera objective closed (theobscurity image). Pnorm is the mean background level ofthe normalized image. The effect of the normalization isillustrated on Fig. 6. The objects are more easily iden-tified from the background after normalization.

The coefficient b is used to correct the fluctuation ofthe global intensity level in the background when thelight source is not temporally stable. A reference level isneeded for the fluctuation to be corrected. This level isdetermined in two different ways depending on the im-age to be normalized. If there is a ’clear’ region in thepictures where droplets never appear, the mean greylevel Lclear of this region is compared to the mean greylevel LB determined on the background image for thesame region and the correction coefficient is given byb=LB/Lclear.

If there is no clear region in the image, a globaltechnique based on the grey level histogram of the imageis used. Histograms of back-lighted spray pictures gen-erally present one or two principal modes. The modecorresponding to the higher grey level is relative to thebackground pixels. The most-populated grey level Lpeak

in the background is expected to be statistically robust.Thus, b is given by the ratio Lpeak(PB)/Lpeak(P).

3.2 The droplet detection

To characterize the droplets, each droplet is individu-ally localized on the pictures and is surrounded by amask. This is usually done by a classic threshold ap-plied to the picture at a given level Lth. The thresholdlevel Lth determines the minimum contrast Cmin of thedroplet detected by this method. This is a correct ap-proach if all the droplet images to be analyzed have ahigher contrast than Cmin. In fact, the smaller thedroplet or the more distant from the focus plane thedroplet is, the more blurred the image of the droplet aspredicted by the model. The smallest or most unfo-cused droplets have a low contrast and can be lost bythe classic thresholding technique. To increase thesensitivity of the detection, a second thresholdingtechnique based on wavelet transform has been devel-oped. This second technique allows the detection ofgrey level local variations that indicates the presence ofa droplet image.

The wavelet transform can be seen as a spectralanalysis, like the Fourier’s transform, but spatiallylocalized. The wavelet transform of a function is theresult of the linear convolution of this function with aparticular function called a wavelet (Eq. 24).

WW;f ðt; qÞ ¼Z

f Xð ÞWt;q Xð ÞdX: ð24Þ

The function f to be analyzed represents here thenormalized grey level picture: f(X)=Pn(i, j) (X=(i, j) isthe pixel position).

Fig. 6 Normalization effect: Example of an ultrasonic spray imagebefore a and after b normalization

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The wavelet Yt,q is given by:

Wt;qðXÞ ¼1ffiffiffi

qp W

X� t

q

� �

; q > 0: ð25Þ

The function Y is called the mother wavelet functionand is an oscillating function of zero mean and localizedin space. This function is dilated and shifted in space asindicated by Eq. 25: the vector t is the shifting parameterand q is the dilatation parameter. The larger q, the lowerthe spatial frequency analyzed by the wavelet. Thechoice of the mother wavelet depends on the informa-tion to be brought out.

To analyze droplet images, the Mexican hat has beenchosen:

WðrÞ ¼ 2pffiffiffi

3p 1� r2� �

e�r2=2: ð26Þ

This function corresponds to the second derivativeof a Gaussian function. Thus, the convolution of thegrey level image with the wavelet function correspondsto the second derivative of the image, firstly convo-

luted by a Gaussian filter. The second derivative of thefiltered grey level function corresponds to the part ofthe picture where grey level concavity or convexity arefound. This is the case for the grey level distributionon the outline of the droplet image. The smaller thewidth of the grey level variation zone (blurred zone),the higher the absolute value of concavity or convex-ity. In other words, the second derivative is all themore important since the droplet is well-focused. Inorder to detect the droplet interface, the parameter q ischosen to be of the order of the width of the blurreddroplet interface on the pictures. So, as q is fixed, eachpixel t = (i, j) of the grey level picture is associated toa wavelet coefficient WY,f(t,q), giving a wavelet imageas illustrated in Fig. 7a. Negative values of WY,f (t,q)correspond to pixels in a convex grey level zone, typ-ically the external side of the droplet interface. Positivevalues correspond to the concave zone (i.e., internalside) and nil values to uniform zones of the picture,typically the normalized background. A threshold onthe wavelet image is used to localize the dropletinterface on the picture when the interface is particu-larly blurred. The detection of small out-of-focus dropsis achieved through the use of only one value for thescale parameter q. In the other case, when a large

Fig. 7 Thresholding techniques: aWavelet image of the normalizedimage in Fig. 6b. b Wavelet-based two-level image. c classicallythresholded image

984

depth of field is chosen, several wavelet transforms canbe performed with different values of q and thencombined in order to detect a larger range of dropfocusing. The two-level image obtained is thencombined to the one obtained with the classicalthreshold.

Examples of wavelet-based and classically threshol-ded images are shown in Fig. 7b, 7c respectively. Theclassical threshold technique detects the larger dropletscorrectly but misses some of the smaller ones. Thedetection of unfocused droplets is improved by thewavelet-based thresholding and the contribution of thistechnique is essentially positive as it only adds dropletsthat would not be detected by the classical technique.The image resulting from the union of the two thres-holding techniques is scanned by a recursive labelingalgorithm in order to localize individually each identifieddroplet. A morphological dilatation is then applied toeach object (set of connected pixels, i.e., of same labellevel) to define a surrounding mask for the calculation oflocal quantities.

3.3 Droplet analysis and sub-pixel contour extraction

Each droplet has to be isolated from the surroundingdroplets in order to be individually analyzed. An illus-tration of this process is shown in Fig. 8. The picture inFig. 8a corresponds to the box drawn on the picture inFig. 6b. Figure 8b represents in black, the mask of thedroplet studied, in bright grey its dilated surface and indark grey, the dilated surface of the other droplet masks.A picture corresponding to the droplet studied with itsenvironment but without its neighboring droplets is ex-tracted from the original image (Fig. 8c). A two-levelimage (Fig. 8d) is then computed by thresholding thelocal image at level Pref. The threshold level Pref isdetermined by Eq. 27

Pref ¼ Pmin þ lðPmax � PminÞ; ð27Þ

where the minimum grey level Pmin, and the most-pop-ulated grey level Pmax which corresponds to the back-ground grey level are obtained from the local grey levelhistogram (Fig. 9a). The relative level l was definedabove in Eq. 12. The local contrast Cmeas of the dropletis also computed from the grey levels Pmax and Pmin byanalogy with Eq. 7:

Cmeas ¼Pmax � Pmin

Pmax þ Pmin: ð28Þ

A nonhomogeneous background level can be induced bylargely unfocused droplets in front of or behind themeasurable droplets. These unfocused droplets diminishthe local illumination of the droplet nearer from thefocus plane but this is considered by the computation ofthe local contrast Cmeas. The pixel-contour of the two-level image (Fig. 8d) is not smooth due to the discreti-zation of the picture, so a sub-pixel edge detectionalgorithm is used to obtain the contour of the droplet.To do so, the grey level gradient is computed for eachpixel of the contour by two Sobel operators giving thefirst order derivation of the grey level following thehorizontal and vertical directions. The gradient vector isused to reach the real value of the threshold level in asub pixel coordinate system. This is illustrated inFig. 9b.

3.4 Droplet size parameters and volume reconstruction

The previous section showed the ability of the algorithmto detect droplets and to isolate them and also to obtainthe droplet edge description in sub-pixel coordinates.Added to the image contrast Cmeas, three other imageparameters are used to characterize the droplet. Thegeneralized radius of the droplet image is based on thearea A0.61 inside sub-pixel contour at relative levell=0.61. The definition of this radius is based on the areaof a circular shape:

rmeas ¼ffiffiffiffiffiffiffiffiffiffi

A0:61

p

r

: ð29Þ

When droplets are spherical, Eqs. 16 and 29 lead to thecorrect estimation of the droplet radius. On the otherhand, for nonspherical droplets, Eq. 29 introduce a biasin this estimation. To compensate for this, a volume-based radius rv is defined following the 3D shape esti-mation of the droplet proposed by (Daves et al. 1993).The volume of the drop is calculated by consideringslices of the 2D shape of the image, cut perpendicularlyto the principal axis of inertia (see Fig. 10a). For eachslice, the elementary volume is defined by the volume ofa disc of diameter equal to the width of the slice and ofthickness equal to the height of the slice. The droplet

Fig. 8 Droplet cutoff from thebackground. The image isanalyzed locally a to determinethe mask b surrounding thedroplet and to isolate the dropletc. The surrounding droplets donot interfere with the local two-level image d

985

volume is obtained by summation of the elementaryvolumes.

For a prolate spheroid shape having its principal axisin the image plane, the volume-based radius leads to thegood estimation of the droplet volume, whereas thesurface-based radius introduce an error which increasesindefinitely as the ellipticity of the image decreases from1 to 0. The ellipticity is defined in the next section. Foran oblate spheroid shape, the error introduced on theestimated volume by the volume-based radius is alsolower than the one obtained from the surface-based ra-dius.

Finally, to estimate the droplet focusing, it is nec-essary to compute the radius difference Drmeas based onthe areas A0.77 and A0.25 obtained from the sub-pixelcontours at the relative levels 0.77 and 0.25, respec-tively (Fig. 10b):

Drmeas ¼ffiffiffiffiffiffiffiffiffiffi

A0:77

p

r

�ffiffiffiffiffiffiffiffiffiffi

A0:25

p

r

: ð30Þ

3.5 Morphological parameters

Four nondimensional morphological parameters aredefined to qualify the droplet shapes. These parametersalso define a morphological space in which differentshape families can be identified.

The definition of the uniformity parameter g (Eq. 31)is illustrated in Fig. 11a where G is the center of gravity

of the image in the plane and rmax and rmin are themaximum and the minimum distances from this point tothe contour. This parameter measures the uniformity ofthe mass distribution around the center of gravity G andit increases from 0 to ¥ as the image shape becomes lessand less uniform.

The sphericity parameter Sp (Eq. 32) quantifies thelikeness between the droplet shape and the sphericalshape. The area of the droplet image SI is compared tothe circular surface SC of same area centered on G (seeFig. 11b). For a spherical droplet, we have Sp=0.

gðuniformityÞ ¼ ðrmax � rminÞrmeas

; 0 < g <1; ð31Þ

SpðsphericityÞ¼areaðSI[SC�SI\SCÞ

areaðSIÞ;0< Sp< 2; ð32Þ

eðellipticityÞ ¼ Lmin

Lmax; 0 < e < 1; ð33Þ

uðirregularityÞ ¼ perimeterðSCÞperimeterðSIÞ

; 0 < u < 1: ð34Þ

The ellipticity parameter e measures the stretching ofthe droplet (Eq. 33). It corresponds to the ratio of thewidth to the length of the rectangle that most closelyencompasses the droplet image (see Fig. 11c). The lastparameter / called irregularity (Eq. 34) is representativeof the folds of the shape interface and is based on theperimeter of the image.

0 50 100 150 200 250grey level

0

25

50

75

100

pixe

l cou

nt

Pmin

Pmax

a b

Fig. 9 Local grey levelhistogram a and sub-pixelcontour b of the droplet imagein Fig. 8c

Fig. 10 Droplet sizeparameters: a volumereconstruction and b sub-pixelcontours used to determine thereference areas (:l=0.25,:l=0.61;: l=0.77)

986

For a circular shape, the four parameters verify:

g ¼ 0; Sp ¼ 0; e ¼ 1; u ¼ 1: ð35Þ

For some classes of shape family, a relationship betweenthese different morphological parameters exists. Forexample, the elliptic shape family is governed by therelations expressed by Eqs. 36, 37 and 38:

ge ¼1� effiffi

ep ; ð36Þ

Sp;e ¼4

parcsin

ffiffiffiffiffiffiffiffiffiffiffi

e1þ e

r� �

� arcsin

ffiffiffiffiffiffiffiffiffiffiffi

1

1þ e

r

!

; ð37Þ

ue ¼1

3

4

ð1þ eÞffiffi

ep � 1

2

� �

:

ð38Þ

The elliptic shape family is then represented in themorphological space (g, Sp, e, /) by a curve defined byEqs. 36, 37 and 38. Near elliptical droplets are identifiedthrough a morphological filter retaining droplets ofmorphology (g, Sp, e, /) near to (ge, Sp, e, /e). Ofcourse, other morphological filters can be defined forother morphological studies.

4 Calibration procedure

The formation of the image on the detector was modeledin Section 2. This model must be compared to theexperiment. To do so, several hypotheses have to bevalidated by a calibration procedure. First, the correc-tion law (Eq. 16) used to estimate the object diametermust be verified and if necessary, calibrated to take intoaccount specific aspects of the setup not included in themodel. The localization of the droplet based on the

estimation of the PSF width must also be experimentallyvalidated. Finally, the influence of the image shape onthe corrections included in the model must be evaluated.

The same optical imaging setup was used for all theexperiments presented in this paper. A noncoherent lightsource (Nanolite - HSPS) of very short duration (.10 ns)is used in a backlight configuration to illuminate thespray. A Sony XC-8500 CCD camera (763·581 pixels) isused with a long working distance microscope (Iscooptic)of NA.0.1. The results presented in this sectionwas obtained with a magnification giving a resolutionR=1,260 pixels/mm. The field of view is thus606·463 lm2 and the minimum PSF half-width at thefocus plane is vmin=4 lm. The Table 1 summarizes theresolution, field of view and correspondingminimumPSFhalf-width for the different configurations used in thispaper.

4.1 Calibration of the diameter estimation

To calibrate the sizing procedure, a reticle on whichcalibrated discs are engraved was placed in the objectspace. The diameters of the analyzed discs was between10 and 100 lm. The reticle was translated along theoptical axis by 50 lm steps. For each disc of known sizeand each position of the reticle, the contrast Cmeas, theimage radius rmeas and the radius difference Drmeas weremeasured. The contrast coefficient s was determinedfrom the maximum contrast measured on the largestobject and yielded s=0.013. The measured contrastCmeas is corrected by use of Eq. 11 to give the normal-ized contrast C0. The true disc radius a divided by themeasured radius rmeas is represented in Fig. 12 as afunction of the normalized contrast C0. The image ra-dius is corrected from the variation of the lateral mag-nification with the position of the object along theoptical axis. This variation is induced by the fact that theobjective is not telecentric.

The scattering of the points for low values of C0 is notdue to a limitation of the model but to an underesti-

Fig. 11 Morphological parameter definition: a uniformity, b sphe-ricity and c ellipticity. The black point corresponds to the gravitycenter of the shapes

Table 1 Resolution (R), field of view (FV) and PSF half-width in the focus plane vmin of the different optical configurations used in theexperiments

Experiments R(pix/mm) FV (lm2) vmin(lm)

Calibration 1,260 606·463 4Ultrasonic spray 640 1,193·908 5.5Diesel spray 1,270 560·457 3

987

mation of the image radius. Effectively, for C0.0.1–0.2the grey level profile is flat and the slope at the edge isvery low. The measurement of the image radius is thusvery sensitive to the threshold level Pref. In the case oflow contrast, the determination of Pmin is not so robustand local noise in the profile can induce an underesti-mation of Pmin, and thus of Pref and rmeas.

The dotted curve in Fig. 12 corresponds to the pre-diction of the model given by Eq. 15. It is clear that theexperiment deviates from the model. The radius mea-sured at relative level 0.61 for a given contrast is, in fact,larger in the experiment than in the theory. It is sup-posed that the difference is due to an experimental PSFslightly different from the Gaussian. A polynomialregression of order 3 is then computed on the experi-mental points to obtain a calibrated correction law inplace of the one predicted by the model. This regressionis used hereafter to estimate droplet sizes.

4.2 Calibration of the focusing estimation

The PSF form and width have not been measured di-rectly. In fact, for each position of the reticle in theobject space, the PSF half-width is estimated throughEq. 21. A different value for vest is then obtained foreach disc and for each position of the disc along theoptical axis (z). The results are reported in Fig. 13 forthe variation of vest versus z. The focal plane is locatedwhere vest reaches its minimum value: z is set to 0 for thisposition. The estimated PSF width increases linearlywith z when defocusing increases. This is a typicalbehavior of a microscope objective. A linear regressionof the data leads to the following estimation of the PSFhalf-width:

vestðmmÞ ¼ 0:0041þ 0:031jzjðmmÞ: ð39Þ

Discrepancies increase progressively when |z| in-creases. Here again, the experiment deviates progres-sively from the theory. We can explain this by aprogressive change of the shape of the experimentalPSF. Indeed, the real PSF for an unfocused object re-sults from the convolution of the focused PSF and adefocalization PSF that can be modeled by geometricaloptics (Stokseth 1969). Nevertheless, as narrow depthsof field are considered here in order to measure correctlythe smallest droplets, the deviation observed between theresults for the different discs is low.

It can be seen from Fig. 13 that fixing vmax is equiv-alent to choose the limits for the depth of fieldDz=zmax�zmin, which is considered to be independent ofthe object size in a first approximation. The discrepanciesbetween the points for the PSF width estimated from thedisc of different diameters induce an uncertainty in thedetermination of the limits of the depth of field. Forvmax=10 lm, it yields a depth of field Dz.380±140 lm.However, we must keep in mind that without any sortingof the drop images, the depth of field varies from 0.6 tomore than 2.3 mm for objects of size between 10 and40 lm. Fixing the depth of field by this way does notprevent from the effect of image overlapping consideredin Section 4.4.

4.3 Effect of the object shape

The imaging model was developed for circular-shapedobjects but non spherical droplets can be analyzedthrough the morphological analysis of the images. Theinfluence of the shape of the object has been testednumerically to evaluate the model. Synthetic images ofelliptic discs with 0.1 £ e £ 1 have been filtered by a

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1C

0

0

0,2

0,4

0,6

0,8

1

a/r m

eas 10 µm

15 µm20 µm30 µm40 µm60 µm100 µmexperimentmodel

Fig. 12 Calibration of the sizing procedure. The coefficientsof the polynomial in Eq. 15 are (a0,a1,a2,a3,a4)=(0.20441,2.7503, �4.9341, 4.7441, �1.8137) for the model and(a0,a1,a2,a3,a4)=(0.39654, 1.1025, �0.84537, 0.26805, 0) for theexperiment

-1 -0,5 0 0,5 1z (mm)

0

0,01

0,02

0,03

0,04

0,05

0,06

χ est (

mm

)

∆z

10 µm15 µm20 µm30 µm40 µm60 µm100 µm

Fig. 13 Calibration of the PSF width. The depth of field(Dz . 400 lm) is defined by the choice of the maximum PSFwidth (vmax=10 lm)

988

Gaussian of variable width to simulate the defocusingeffect. The blurred images were analyzed by the sizingand defocusing estimation procedures. The results arereported in Fig. 14. Area-based radii are consideredhere. The relative error (aest�a)/a is plotted versus theellipticity for different nondimensional object sizesð~a ¼ 1; 1:5; 2; 2:5; 3Þ in Fig. 14a. Obviously, the errorbegins to be significant for ellipticity lower than 0.5, butthere is not a clear effect of the nondimensional radius ~aon this error.

The relative error for the estimated PSF width(vest�v)/v increases when e decreases (Fig. 14(b)) as itdoes for the size estimation, but the relative error on thePSF width is one order of magnitude greater than therelative error for the diameter estimation. Nevertheless,the error on the determination of v is not very importantas it is used only to define the measurement volume. Fora droplet not localized near the maximum unfocusedposition, given by v . vmax, the estimation of v has noincidence on the measurement, although it is true that adroplet localized near the maximum unfocused positioncould be rejected by an error on the estimation of v dueto a pronounced elliptic shape.

4.4 Limit in droplet concentration

The imaging process leads to the projection of thedroplet contour on the image plane. This projectionconverts the 3D distribution of drops in the object spaceto a 2D distribution of droplet images on the imageplane. For the image to be analyzable, the droplet imageconcentration on the image plane must not be too high.The projected area Ap can be estimated by the mean-surface diameter D20 of the droplet size distribution.Indeed, if N is the number of droplets in the measure-ment volume then:

Ap ¼Z 1

0

NpD2

4fnðDÞdD ¼ N

pD220

4: ð40Þ

The number of drops in the measurement volume(Lx Ly Lz) is related to the drop concentration Cv: N=Cv

Lx Ly Lz. The drop concentration can thus be expressedby:

Cv ¼1

LzpD220=4

Ap

LxLy: ð41Þ

The last term Ap/(Lx Ly)=scov represents the ratio of thetotal projected surface of the droplet images over thesurface of the image plane. This ratio can be greater thanthe effective surface image concentration when imageoverlapping occurs. In the case of the Diesel sprayapplication characterized by a very small droplet sizedistributions (D20 .5 lm), and for scov=0.25 andLz= 2 mm for a~30 lm, Eq. 41 gives a drop concen-tration of 5000 droplets/mm3. In this estimation, Lz isthe maximum depth for detection of the biggest droplets(see Calibration of the focusing estimation).

High droplet concentrations lead to high values ofscov and result in overlapping effects which are treated bythe following ways:

– Partially overlapping images are separated by a spe-cific module of the morphological analysis which usesthe sub-pixel contour and the grey level gradient alongthe contour to dissociate different images (Yon 2003;Yon and Blaisot 2004).

– Totally overlapping process is dealt with by a modelfor the correction of the size distribution observedwhen scov is too high (scov ’ 0.5) (Yon 2003; Yon andBlaisot 2004). The missed droplet population is esti-mated considering a random distribution of dropletsin space. In the case of applications to Diesel sprays, itwas shown that as soon as images are analyzable, thecorrection of the apparent drop size distribution be-comes negligible. The correction obtained by thismodel is significant in the case of wide drop size dis-tributions, the probability for a given drop to becovered by another one being in this case not negli-gible.

4.5 Testing of the drop sizing on a well-controlledexperiment

The validation of the new drop sizing technique wasdone comparing its performances with two otherdiagnostics: a diffraction-based granulometer (Spray-

0 0,2 0,4 0,6 0,8 1ε

0

0,01

0,02

0,03

0,04

0,05

0,06

(aes

t - a

)/a

0 0,2 0,4 0,6 0,8 1ε

0

0,2

0,4

0,6

0,8

(χes

t - χ

)/χ

a bFig. 14 Validation of the sizinga and defocusing b estimationprocedures for elliptic objects.open circle ð~a ¼ 1Þ; filledtriangle ð~a ¼ 1:5Þ; filled invertedtriangle ð~a ¼ 2Þ; filled squareð~a ¼ 2:5Þ; filled circle ð~a ¼ 3Þ:Relative error on the diameterestimation is quite small. Theerror for the estimation of thePSF width is greater but it haslittle impact on themeasurement accuracy

989

tech–Malvern) and a PDPA (Dantec). An ultrasonicatomizer (POLYSPRAY) fed with water was chosento produce a well-atomized and diluted spray withvery low drop velocities. The working frequency of theatomizer is 47 KHz and the mass flowing rate is4.2 mg/s. The measurement point is 15 mm below theinjector nozzle for the three techniques. The opticalsettings of the imaging system are given in Table 1.The linear regression for the estimation of the PSFhalf-width gives vest (mm)=0.0056+0.014 |z| (mm) inthis configuration. It was verified with the PDPAmeasurements that there was no correlation betweenthe drop size and the drop velocity. Temporal distri-butions obtained from the PDPA can thus be directlycompared to spatial distribution obtained from dif-fraction or image analysis. The results are presented inFig. 15. In order to compare the droplet size distri-butions, the spatial filtering is applied to prevent anoverestimation of the biggest droplets. The maximumPSF half-width was fixed to vmax=20 lm leading to adepth of field of 2 mm. In order to compare mea-surements to other techniques, the morphological fil-tering of the elliptic objects (see Sect. 3,Morphological parameters) is also applied to retainonly near-spherical droplets. The image-based distri-bution obtained with these two filters is noted FSM inFig. 15 and the nonfiltered distribution is also re-ported (F0). The effect of the filtering on the overes-timated population of biggest drops is clear, the FSMdistribution is shifted towards smaller drop diameters.The three distributions obtained from the PDPA, thediffraction technique and the image-based techniquewith spatial and morphological filters (FSM) are verysimilar. The SMD obtained from each technique areequal to 30.3, 34.9 and 35.11 lm, respectively withdiffraction, PDPA and image analysis.

5 Application to the diesel spray

5.1 Experimental setup

The Diesel spray is well known to be hardly measur-able as a consequence of high liquid speeds and highconcentrations encountered in such a flow. Further-more, the spherical hypothesis is still to be proved inthis spray. The capability of the image-based granu-lometer to analyze sprays in severe conditions is thustested on a Diesel spray. No comparison with otherdiagnostics will be presented in this section due to thelimitation of these diagnostics for this kind of appli-cation. A 200 lm diameter single orifice Ganzerinjector is placed vertically and fed by a pneumaticpump to provide fuel pressures Pi=60 MPa. Thenozzle length is 800 lm giving a L/D ratio of 4. Thisinjector is not representative of multi-hole commercialinjector but it is used to produce one Diesel jet ofsimilar properties. The fuel is injected at ambientpressure and temperature (P0=100 kPa, T=20�C).The injection lasts 2 ms and the needle lift position ismeasured as a function of time. Experiments havebeen done with Diesel fuel.

There is an hydraulic delay between the commandtime t0 and the start of injection tsi. At low injectionpressure this delay is not stable. In order to avoid thejitter between t0 and tsi, the start of injection is detectedwith a photodiode placed at the nozzle exit. The lightsource is driven at a time ti synchronized with tsi.

The PSF half-width of the optical configuration usedfor this application is given by vest(mm)=0.0031 +0.022 |z| (mm). The minimum measurable diametercalculated from Eq. 17 is dmin= 2.2 lm. The maximumPSF half-width vmax for the spatial filter given by Eq. 18is vmax .3 lm for dmin= 2.2 lm but the maximum valuewas fixed to vmax=5 lm in order to enlarge the mea-surement volume and thus to increase the dropletcounting. This yields to the depth of field Dz .170 lmand results in a slight underestimation of the smallestdroplets (d<5 lm). The size of the measurement volumeis thus 0.56·0.46·0.17= 43·10�3 mm3. For each mea-surement point, 700 images of the spray were recordedto produce smoothed size distributions. The averagenumber of droplets counted in each drop distributionranges from 10,000 to 80,000 depending on the mea-surement location and the filters applied.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Drop diameter (µm)

0

0,01

0,02

0,03

0,04

0,05

f v(1/µ

m)

FSMF0DiffractionPDPA

Fig. 15 Comparison of the image-based drop sizing technique withPDPA and diffraction techniques. FO—image-based granulometrywithout filtering, FSM—image-based granulometry with spatialand morphological filtering

Table 2 Measurement locations and name convention for the tes-ted positions

Name Pi (MPa) X (mm) Y (mm)

60ul 60 30 -360ur 60 30 360dl 60 60 -360dr 60 60 3

990

6 Results

The drop size distributions are measured 30 mm and60 mm from the nozzle tip, for two symmetrical radialpositions (see Table 2 and Fig. 16a). Measurement timesare fixed during the quasi stationary period of theinjection, i.e., ti=1911 ls for X=30 mm and ti=1991 lsfor X=60 mm. Typical Diesel spray images are shown inFig. 16b–e. Focused, slightly unfocused and largelyunfocused droplets are visible on these images.

Volume-based drop size distributions are presented inFig. 17. For each measurement position, the direct dis-tribution (F0) is compared to the one obtained with theapplication of the spatial filter (FS). The morphologicalfilter is not applied anymore as the results are notcompared with other granulometers in this section. Theexpected effect of the spatial filter, i.e., the correction ofthe overestimation of the population of the big drops, isclearly seen in Fig. 17a for the point 60ul and also inFig. 17c and 17d.

Without spatial filtering, drop size distributions onboth sides of the jet at a given downstream position arequite similar. When the spatial filter is applied, a dis-symmetry is observed for X=30 mm (Fig. 17a and 17b).In fact, the distribution for 60ur is bi-modal with FS, asa result of a well-localized phenomenon. It was noticedin previous studies on the same injector, that for thisparticular location, the Diesel spray is characterized by atemporal oscillation of the droplet density. It is thoughthat this could be the consequence of periodical dropletclusters or lateral jet flapping generated by cavitation,appearing in a small defect located on the right side ofthe nozzle outlet. The difference between the drop sizedistribution with and without spatial filtering can thusbe due to the fact that this cavitation-induced phenom-enon is three-dimensional as mentioned by (Soteriou2001) and well-localized.

The drop size distributions at 60 mm from thenozzle tip (60dl and 60dr) are clearly mono-modaldistributions centered around 12 lm. The small drop-lets (d<8 lm) that are dominating upstream havenearly disappeared 30 mm below. This could be due tothe fact that the dense parts of the jet are composedof bigger droplets than the dilute parts. The reductionof the population of the small droplets (d<8 lm)when applying the spatial filter can thus be due to thefact that the focus plane is located in a dense regionsurrounded by more dilute parts of the spray. Inparticular, unfocused drops should be located in adilute region, i.e., in a region where the smalldrops are dominating. The spatial filter suitablyeliminates those small unfocused droplets from thedistribution.

The morphological analysis has been used to char-acterize the mean drop shape as a function of the dropsize. The results for 60ur and 60dr are presented inFig. 18. The measurement points 60ur and 60dr waschosen for the difference observed in the size distribu-

tion. The mean value of the four shape parameters arecomputed for all diameter class.

The standard deviation for e and / is also presentedon this figure (error bars).

The mean shape of the drops seems nearer from thespherical shape at the center of distribution modes asindicated by the minimum for Sp and g and themaximum for e and /. However, the standard devi-tation of the shape parameters are higher at the ex-trema of the drop size distribution. This could simplyresults from less robust mean values caused by less-populated classes. It must be noticed that parametersSp and g, which must be near from zero for sphericaldrops, also has a higher range of values (resp. [0, 2]and [0, ¥]) than e and /. The standard deviations forSp and g (resp. .0.2 and .0.3) are thus higher alsothan for the two other parameters (arround 0.1) andwere not represented for clarity of the Fig. 18. Themorphological analysis indicates that when the dropsize distribution presents two modes, the shapeparameters seem also to present the same number ofextrema as shown for 60ur in Fig.18a. The most-populated diameter classes seem thus to be well-atomized due to the near spherical shape of thesedrops.

A temporal analysis of the Diesel jet has also beenperformed during the quasi-stationary part of theinjection for the position 60ur and over the entireinjection time at the other side of the spray, at the po-sition 60dl. The temporal evolution of the SMD and ofthe morphological parameters is presented in Fig. 19.

Only a weak variation of the SMD is observed duringthe injection time. Except for the first and last 360 ls ofthe injection when the droplet diameters are a little lar-ger, the mean diameters during the injection is about5 lm for the two positions considered (X=30 and60 mm).

At the two distances from the injector, the temporalsimilarity observed between the mean diameters is nolonger seen for the morphological parameters. Indeed, atime dependence of the four morphological parametersis observed at the point 60ur associated to X= 30 mm.This dependence is clearly seen from the irregularityparameter / whose stabilization is observed during thequasi-stationary setting up. This temporal dependence isnot observed anymore at the second point studied (60dl,X=60 mm). The droplet morphology relaxes along thedownstream position towards spherical shape, i.e., g andSp decreases toward 0 and e and / increases toward 1, sothe atomization process is still in progress betweenX=30 mm and X=60 mm.

It can also be noticed that the behavior of the injectoris not that reproducible. This is particularly evident forthe mean shape parameters at the point 60ur and around2,000 ms which are not of the same order than for theanalysis at a fixed time (see Fig. 18). This particularbehavior shows the sensitivity of the drop shapeanalysis.

991

Finally, to illustrate what extra contribution the dropshape analysis can add to the spray analysis, we reported

in Fig. 20 some drop images recorded at the end of theDiesel injection with the corresponding value of the dropshape parameters. The drop shapes of the too upperrows correspond to the kind of shape commonlyencountered in the Diesel spray. The four lower ones are

Fig. 16 Location of themeasurement points along theDiesel plume (a) and example ofDiesel spray images for eachposition (b)–(e)

Fig. 17 Droplet size distributions: a 60ul, b 60ur, c 60dl, d 60dr. F0:no filter applied, FS: spatial filtering. Measurement locations areindicated in Table 2 and in Fig. 16

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specific shapes of the end of injection. These nonspher-ical droplets, usually rejected or not correctly charac-terized by classical techniques, can however lead to theproduction of unburned particles in the combustionprocess, due to their big size.

We believe that a wise classification of the shapesbased on the morphological parameters, can helpatomization analysis. Indeed, primary and secondarybreakup regimes can be classified through the shape ofthe liquid element being broken or formed (see Liu andReitz 1996 for example). Our challenge now is to asso-ciate quantitative values to the shapes subjectivelyassociated to each breakup regime.

7 Conclusion

A new approach to spray-sizing by image analysis ispresented. It is based on the modeling of the imageformation. The image model has been used to definecriteria for the correction of the apparent size of anunfocused drop and for the determination of a mea-surement volume independent of the drop size.

Image-based drop sizing techniques was also used toanalyze the shape of the drops. Four morphologicalparameters have been proposed here to qualify the dropshapes. Morphological filtering of the spherical orelliptic droplets is also applied for comparison withother techniques.

One of the remaining limits of the image-based sizingtechnique is the effect of image overlapping. However, itwas found that for a narrow drop size distribution, theeffect could be neglected.

The image-based drop sizing technique was first ap-plied to an ultrasonic spray. Results are compared todiffraction-based and PDPA measurements. The threetechniques are in agreement, giving similar the drop sizedistributions.

The new drop sizing technique was also submitted tomore severe conditions, in an application to a Dieselspray. A very low SMD (.5 lm) is found during thequasi stationary period of the injection. The results re-veal also the presence of two modes in the drop size

a b

Fig. 19 Temporal variation of the SMD and of the morphologicalparameters versus injection time for Pi=60 MPa: filled circle30 mm downstream (60ur) and · 60 mm downstream (60dl) thenozzle outlet. The continuous line stands for the needle lift curve

Fig. 20 Images of drops encountered at the end of the injectiontime

Fig. 18 Droplet size distributions with spatial filter and mean dropshape parameters for each diameter distribution class. Error barsindicate the standard deviation for e and /: a 60ur, b 60dr

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distributions at 30 mm from the nozzle outlet. It is be-lieved that the bimodal drops size distribution could bethe consequence of a periodical phenomenon, generatedby cavitation appearing in a small defect located on theright side of the nozzle outlet.

The small SMD is almost constant during the injec-tion and along the downstream position. However, themorphological analysis allows us to observe that themean shape of the droplets is not stable during theinjection near the injector, whereas it is nearly constantin time 60 mm downstream from the nozzle outlet. Ascould be guessed, it was shown that the drop shapeevolves spatially towards the spherical shape. Thecharacteristic time for drop size stabilization is found tobe shorter than the characteristic time form drop shaperelaxation. Some energy is still working on the liquid–gas interface or is not totally dissipated, leading to thedeformation of the drops. This might have to be con-sidered when evaporation and combustion processes areinvolved. The objective of this work is now to proceed tothe classification of the shapes, based on the morpho-logical parameters, in such way that primary and sec-ondary breakup regimes can be classified through aquantitative approach.

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