Experimental investigation of Gaussian beameffects on the accuracy of a droplet sizing method

Karl Heinz Hesselbacher, Klaus Anders, and Arnold Frohn

A special sizing technique is applied to measuring the diameter of monosized droplet streams that areused for investigation of fuel droplets in enginelike conditions. For these experiments the dropletdiameter must be known precisely. The sizing technique used is based on the evaluation of the fringespacing of scattered light in the forward direction. This technique is independent of the intensity of theincident light. No absolute intensities need to be measured. The droplets are exposed to a focused laser

beam. Therefore the frequently used assumption of plane wave fronts is not fulfilled. Elaborateexperiments have been carried out to study the influence of a Gaussian intensity distribution of the laserbeam on the accuracy of the sizing technique. It has been shown that the droplet diameter can bemeasured to an accuracy of better than 2% even if the droplet is illuminated by a Gaussian beam for a

droplet diameter that is smaller than the beam diameter.Key words: Monosized droplet streams, vibrating orifice generator, optical particle sizing, enginelike

conditions, light scattering by droplets, droplet interaction with a Gaussian beam, evaluation of the fringespacing.

Introduction

We illustrate here the need for a precise sizingtechnique for droplets ranging from 15 to 200 pum indiameter that are used to study the behavior ofdroplets in enginelike conditions. A special techniquethat meets the requirements is described in brief. Itsperformance and accuracy are investigated by elabo-rate experiments. The sizing technique has beendeveloped as a tool for precise diagnostics on thedroplets of almost all types of liquid.' The onlyrestriction is that the absorption of the liquid not betoo high for the wavelength of the laser light used inthe experiment. The advantage of this sizing tech-nique (besides its high accuracy) is the fact that it isindependent of the intensity of the incident light. Thetechnique is based on the evaluation of the fringespacing of scattered light in the forward region, andhence no absolute intensities require measurement.

K. H. Hesselbacher is with Daimler-Benz AG, Forschung undTechnik, Postfach 800230, D-7000 Stuttgart 80, Germany. Theother authors are with the Institut fur Thermodynamik der Luft-und Raumfahrt, Universitdt Stuttgart, Pfaffenwaldring 31, D-7000Stuttgart 80, Germany.

Received 20 December 1990.0003-6935/91/334930-06$05.00/0.e 1991 Optical Society of America.

Application of the Sizing Method

The process of the self-ignition of fuel in an internalcombustion engine (the moment of fuel injection intothe engine to the moment of ignition) takes placewithin a short period of time, usually < 1 ms. Thedroplet diameters lie within a wide range, from a fewmicrometers up to 200 pm. Experiments on fuelsprays have been carried out to determine the pres-sure and temperature dependence of the ignitiondelay time in continuous flow2 3 as well as in quiescentatmospheres.4 A variety of data on have been takenclouds of droplets.5

For a more detailed investigation of the processesmentioned above an apparatus was designed and setup for experiments on streams of droplets of preciselyknown and constant diameters. A schematic drawingof the central part of this apparatus is shown in Fig. 1.It contains a rectangular channel with a 20- x 3-mmcross section and a device for producing monosizeddroplets. In the channel an airstream flows from leftto right as shown in Fig. 1.

Streams of monosized droplets were produced byusing a vibrating orifice droplet generator.', Thedroplets were fed into the rectangular channel, pres-surized, and heated by a stream of air. Pressures asgreat as 50 bars and temperatures as high as 1000 Kwere chosen to simulate enginelike conditions ob-served during the compression cycle. The reactionchamber had two quartz-glass windows of high opti-

4930 APPLIED OPTICS / Vol. 30, No. 33 / 20 November 1991

Excitation frequency experiments." To obtain quantitative results on theamount of fuel that evaporates during this period oftime, it is necessary to have precise knowledge of thehistory of the droplet diameter. This can only beobtained by precise measurements of the diameteritself. In the experiments described above, a light-scattering technique was used that met this require-ment.'

Scattering of Light by Particles

The scattering of light by a spherical particle illumi-nated by plane wave fronts was described theoreti-cally by Mie." The intensity distribution of the scat-tered light as a function of the scattering angle 0depends on the polarization, on the complex refrac-tive index of the scattering obstacle, and on the Mieparameter:

Thermocoup le

Drain

'rrd,x

I

Thermocouple

Pressure gauge

Fig. 1. Schematic diagram of the apparatus used for the investiga-tion of monosized droplet streams under a high temperature airflow. (The droplets are not to scale.)

cal quality to permit good optical access. One impor-tant point of interest in connection with the problemsdescribed above is the measurement of ignition delaysfor different fuels.

The advantage of experiments of this kind is thatthe droplets all have the same size during a singleexperiment. This makes it possible to separate drop-let size effects. By tracking the droplet path in thechamber it is possible to check mathematical modelsto a limited extent. This may provide valuable hintsfor the development of such models in the future. Bymeasuring precisely the droplet diameter and itsvariation along the droplet flight path through thechamber, it is possible to calculate heat transfer andvaporization rates as a function of droplet size,velocity, ambient temperature, pressure, and fueltype."' In the case of vapor cloud ignition around thedroplet, the ignition delay may be computed by takinginto account the position of ignition in the chamberand other parameters of importance.

For such investigations the droplet diameter andits change in time are important parameters, whichhave to be determined with great accuracy. Theresidence times of the droplets within the reactionchamber lie in the range of milliseconds. During thisshort period of time the change in droplet diameter is10% or less of the initial diameter used in these

(1)

which is a function of the droplet diameter dT and thewavelength X of the incident light. For large values ofa the intensity distribution can be described by therules of geometrical optics, which are simpler thanthe complicated equations of Mie's theory. A typicalscattering pattern obtained from droplets with adiameter of 41 pum illuminated by a laser with awavelength of 514 nm is shown in Fig. 2. In this casethe Mie parameter is about 250. Apart from thenear-forward region and except for observation an-gles near the rainbows,3 for a >> 1 arguments ofgeometrical optics may be used to describe the inten-sity distribution shown in Fig. 2. A description (com-bining Fraunhofer diffraction and geometrical optics)of the intensity distribution in the close-forwardregion was given by Glantschnig and Chen. 4 Mie'ssolutions to the Maxwell equations must be takeninto account for intermediate values of a 1. Resultsof calculations using either Mie's theory or the rulesof geometrical optics are applied in different sizingmethods, for example, in scattering techniques, inphase Doppler techniques,"' in optical particlecounters,'6 and for the extension of the diameterrange of sizing instruments based originally on Fraun-hofer diffraction.7 If the waist diameter of the illumi-nating laser beam is in the same range as the dropletdiameter, Mie's assumption of planar wave fronts isnot fulfilled.8" 9 The particles are exposed to a Gaus-sian intensity profile of the beam. The purpose of thepresent investigations is to study the influence of aGaussian intensity distribution of the incoming lighton the accuracy of the sizing method.'

Sizing Method

The sizing method applied in the experiments de-scribed above is highly accurate and independent ofthe intensity of the illuminating light scource. Thesize of the droplets is determined by evaluating thefringe spacing of the scattering pattern. The dropletitself acts as an interferometer. An example of such apattern is shown in Fig. 2. In this case the laws of

20 November 1991 / Vol. 30, No. 33 / APPLIED OPTICS 4931

Fig. 2. Scattering pattern obtained from droplets that are 41 Am in diameter illuminated by a laser. The droplets in the center are hit by abeam coming from the right. The scattering angles are indicated.

geometrical optics are suitable for describing theintensity distribution of the scattered light. The basicprinciple of the calculations based on the rules of raytracing is explained in Ref. 1. It can clearly be seenthat the scattering phenomena for scattering angles0 < 0 < 70° in the forward region are mainlydetermined by rays of the order of 0 and 1. It shouldbe emphasized that neither the effects of the Gaus-sian intensity distribution nor diffraction contribu-tions are included in these calculations. Using Snell'slaw of refraction, Fresnel's equation,20 and relationsfor the phase shift of rays of different orders, we canshow that the angle AOM between two neighboringmaxima or minima of the intensity distribution in theforward region is a measure for the droplet diameterd,. Taking into account rays of the order of 0 and 1,we derive the following equation for the dimension-

2.0

1.8

aa1.6

1.4

1.2

1.00 10 20 30 40

e []Fig. 3. Experimental verification of Eq. (2) for dimensionlessfringe spacing. The solid curve shows the calculated values,whereas the circles are experimentally derived data.

less quantity D:

AOdD = =O~,2

m sin(0/2)cos(0/2) + [1 + m2 - 2m cos(0/2)] 1 2

(2)

The quantity D combines the angular fringe spacingAOM, the droplet diameter d,, and the wavelength X ofthe incident light as a function of the observationangle 0 and the real part of the refractive index m. Incontrast to the relation for the fringe spacing AOMgiven in Ref. 1, Eq. (2) is a better approximation sinceit includes the angular dependence of the fringespacing. Therefore it can be applied for all observa-tion angles for which the intensity distribution ismainly governed by rays of the order of 0 and 1.

C CCD-ArrayD DetectorE Beam expanderF Focusing lensG Aerosol generatorL LaserR Receiving lens

Fig. 4. Schematic view of the experimental apparatus and physi-cal setup. The coordinate system used in this paper is shown.

4932 APPLIED OPTICS / Vol. 30, No. 33 / 20 November 1991

dT - 88 pmm = 138). = 632.8 nm

I nOn0

"I 1.000 5 ,_0

0.980-12 -8 -4 0

- - 0 -

ii14 8 12

Z [mm]

Fig. 5. Dimensionless fringe spacing f/If * as a function of the zcoordinate. The droplet diameter is 50 WLm. No beam expansion isapplied. The waist diameter is 280 ,urm.

1.050

1.025

*a

s 1.000oa

0.975

0.950-12 -8 -4

Equation (2) cannot be applied in the forward direc-tion at small angles of 0 because of the influence of theFraunhofer diffraction or at angles 0 > 70°, whererefracted rays of an order larger than 1 becomesignificant. Figure 3 shows results of an experimentalverification of the validity of Eq. (2) in the range of70 < 0 < 30° for a Propanol-2 droplet with a diameterof 88 jI.m. The solid curve shows the calculated valuesusing Eq. (2); the circles are experimentally deriveddata.

Experiments and Results

An elaborate experimental investigation was carriedout to study the influence of a Gaussian intensitydistribution of the illuminating laser beam on theaccuracy and reliability of the size measurements ofthe light-scattering technique.' The fringe spacing ofthe light scattered by droplets was recorded for ascattering angle of 200. In the experiments monodis-perse droplet streams were used for studying thescattering phenomena. A schematic view of the opti-cal setup and the coordinate system used in thepresent discussions is shown in Fig. 4. The center ofthe coordinate system is the center of the laser waist.The beam of the 25-mW He-Ne laser L is expanded bythe beam expander E and brought into focus by lensF. By appropriate choice of the focal lengths of theselenses, the waist diameter can be varied over a widerange. The photodetector D together with the laser Lacts as an optical gate and is used to monitor theregularity and the monodispersity of the dropletstream. As long as the droplet stream is regular, i.e.,it is characterized by a constant droplet size anddroplet spacing, the signal obtained from the photode-tector is in phase with the excitation signal of thedroplet generator.

The intensity distribution of the light scattered bythe droplets was registered by a charge-coupled de-vice array C using with receiving optics R. A seriesof measurements was carried out for different waist

1.02ic

sZ 1.00

0.98i

20

Z [mm]

Fig. 7. Dimensionless fringe spacing f/If,* as a function of the zcoordinate. The droplet diameter is 52 pLm. Here a beam expansionobtained by using lenses of 14- and 500-mm focal length is applied.The waist diameter is 60 ALm.

diameters and beam divergences. A variation of thebeam waist and beam divergence was achieved bydifferent expander configurations. Experiments wereperformed with no expansion and with two differentexpansion ratios of the laser beam. The first expan-sion, giving a beam diameter of approximately 5 mm,was achieved by using two lenses with focal lengths of14 and 80 mm, respectively. A beam of approximately30 mm was obtained by using lenses with focallengths of 14 and 500 mm, respectively. The dropletstream was moved in they and z directions to achievedifferent points of intersection between the path ofthe droplets and the laser beam. Mie's assumption ofplane wave fronts is satisfied only fory = 0, z = 0 anda waist diameter that is large compared with thedroplet diameter. Typical examples of the results of aset of measurements for y = 0 and different values ofz are shown in Figs. 5-8. In these figures the dimen-sionless fringe spacing is plotted against the z posi-tion of the scattering droplet. The measured fringespacing was normalized to the data obtained at z =-12 mm of each particular set of measurements. Asone can see only minor deviations from the meanvalue and from the value at the positiony = 0 and z =0 occur in almost all cases. This is the case for theresults in Figs. 5-7. These results were obtained for awaist diameter that is larger than the droplet diame-

1.050

1.025

0

0

0.975 .-1 0 a a * *

30-12 -8 -4 0 4 8 12

Z [mm]

Fig. 6. Dimensionless fringe spacing f.If * as a function of the zcoordinate. The droplet diameter is 89.6 [lm. Here a beam expan-sion obtained by using lenses of 14- and 80-mm focal length isapplied. The waist diameter is 130 pum.

0.950 L-12 -8 -4 0 4 8 12

Z [mm]

Fig. 8. Dimensionless fringe spacing f,/f * as a function of the zcoordinate. The droplet diameter is 89.6 pum. Here a beam expan-sion obtained by using lenses of 14- and 500-mm focal lengths isapplied. The waist diameter is - 60 pim.

20 November 1991 I Vol. 30, No. 33 I APPLIED OPTICS 4933

. . 0 * * * S *S -

0 4 8 12

- S 6 0 *

0

-

g X . w @

. . . .

1.0200 . * * *S 1

0.980-400 -300 -200 -100 0 100 200 300 400

Y [im]

Fig. 9. Dimensionless fringe spacing f,If* as a function of theycoordinate for z = 20 mm. The droplet diameter is 52 pum. Here abeam expansion obtained by using lenses with 14- and 500-mmfocal lengths is applied.

ter. For a waist diameter that is smaller than thedroplet diameter results are given in Fig. 8. As onecan see, a lack of data for z = 0 is observed. This isobviously a result of the droplet diameter of 89.6 jimbeing larger than the diameter of the laser beam, andtherefore no interference pattern can be observed.Similar results were obtained in the numerical calcu-lations presented by Corbin et al.2 ' The results in thevicinity of the laser waist deviate significantly fromthe mean value. This is due to a bad signal-to-noiseratio resulting from a variation of the light intensityacross the diameter of the laser beam. Poor signal-to-noise ratios are obtained for droplets with a diameterslightly smaller than the laser diameter. Furtherresults are presented in Figs. 9 and 10, for which thedroplet stream is moved in the y direction. Theseresults were obtained for z = 20 mm and z = 14 mm,respectively. In these cases the fringe spacing wasnormalized to the value obtained for y = -350 jimand for y = -80 jim, respectively. At the z = 20-mmposition the laser diameter was considerably largerthan the droplet diameter. The intensity of the lightfalling onto the droplet was therefore almost uniformover the cross section of the droplet.

Figure 10 shows data points only within a relativelynarrow range of y. Outside this range no fringepattern could be observed. With the exception of asingle data point at the right-hand boundary, all dataare within a margin of ± 0.5%. The larger deviation ofthis one point is due to low contrast in the scatteringpattern. This is always observed if the droplet is nearthe boundary of the probe volume. Unreliable datacan thus be rejected. This is a great advantage of thepresented sizing method over methods that rely onabsolute intensity measurements of scattered light,which usually undersize particles crossing the probevolume at the boundary region.

"

1.020

1.000 * S *

0.980 . .-400 -300 -200 -100 0 100 200 300 400

Y [um]

Fig. 10. Dimensionless fringe spacingf,/lf* as a function of theycoordinate for z = 14 mm. The droplet diameter is 52 pum. Here abeam expansion obtained by using lenses with 14- and 500-mmfocal lengths is applied. To allow comparison with Fig. 9 the samescale was used.

Conclusions

We have examined the reliability of the special sizingmethod described in detail in Ref. 1. The dropletswere exposed to focused laser beams of differentdivergences. The fringe spacing at a certain angle wasmeasured for the different positions of the dropletswithin the laser beam. In general the droplets wereexposed to a nonuniform intensity distribution. Theaim of the investigation was to demonstrate theinsensitivity of the method to Gaussian effects. It canbe shown that theoretical results for the fringespacing obtained from the equations of geometricaloptics for a uniform intensity distribution over thecross section of the droplet can be applied in almostall cases, even if the droplet is exposed to a Gaussianbeam. The experimental results show that the dropletdiameter can be measured to an accuracy of betterthan 2% as long as the beam diameter is larger thanthe droplet diameter. This accuracy was achievedeven in the case of large departures of the dropletfrom the position at the center of the laser beam waisty = 0 andz = 0. Furthermore it was shown experimen-tally that in extreme Gaussian situations, i.e., forlaser beam diameters that were smaller than thedroplet diameter, no signal suitable for evaluationwas obtained. This clearly avoids the measurementerrors that arise from application of this method tocases where it is not permissible.

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technique to measure the diameter of periodically generatedmoving droplets," J. Aerosol Sci. 17, 157-167 (1986).

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4934 APPLIED OPTICS I Vol. 30, No. 33 I 20 November 1991

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