Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets

Global Rainbow Thermometry for Mean Temperatureand Size Measurement of Spray Droplets

Jeronimus P.A.J. van Beeck*, Laurent Zimmer**, Michel L. Riethmuller*

(Received: 23 April 2001; resubmitted: 13 July 2001)

Abstract

Global rainbow thermometry is a new technique for mea-

suring the average size and temperature of spray droplets.

For data inversion a global rainbow pattern is employed,

which is formed by constructive interference of laser light

scattered by an ensemble of spherical droplets. The non-

spherical droplets and liquid ligaments provide a uniform

background and hence do not influence the interference

pattern from which average size and temperature are

derived. This is a large improvement with respect to stan-

dard rainbow thermometry, investigated since 1988, which

is strongly influenced by particle shape. Moreover, the

technique is applicable to smaller droplets than the standard

technique because the global pattern is not spoiled by a

ripple structure. Data inversion schemes based on inflection

points, minima and maxima are discussed with respect to

spray dispersion and droplet flux. The temperature deriva-

tion from inflection points appears to be independent of

spray dispersion. Preliminary measurements in a heated

water spray are reported. The mean diameter obtained from

the rainbow pattern is smaller than the arithmetic mean

diameter measured by phase-Doppler anemometry. The

accuracy of the temperature measurement by global rain-

bow thermometry is shown to be a few degrees Celsius.

1 Introduction

Rainbow thermometry has been investigated since 1988

[1–3]. Up to now, the rainbow technique has measured the

temperature and size of individual droplets. Even though

the technique has been applied in the field of spray com-

bustion by Sankar et al. [4], confidence could not be

established because of major problems, related to the tem-

perature gradient inside the droplet [5–7], droplet non-

sphericity [8] and a ripple structure that strongly perturbs

the rainbow interference pattern, from which one deduces

the droplet parameters [5]. The last two problems are solved

by global rainbow thermometry. Especially the fact that

there is a solution to the non-sphericity problem is an

important innovation, because until now a droplet non-

sphericity of 1% could lead to an error in the temperature

measurement of � 40 �C. In recent years, several algo-

rithms for the detection of spherical droplets were proposed

but more of them was sufficiently general [4, 9].

2 Principle of Global Rainbow Thermometry

Global rainbow thermometry (GRT) primarily aims at

eliminating the non-sphericity effect that has existed in the

rainbow technique since its introduction in 1988. The basic

principles of GRT were published by the present authors in

1999 [10].

Figure 1 shows a photograph of the set-up. A CW argon-ion

laser beam illuminates a water spray. The typical laser

power for this spray is 100 mW and the beam is expanded to

a thickness of 15 mm. A transparent piece of paper is

installed in the focal plane of the large-diameter receiving

lens. All the droplets crossing the laser beam will contribute

to the angular scattered light distribution, visible on the

transparent screen. This results in the so-called global

rainbow pattern that is recorded from the other side of the

transparent screen by a digital video camera. The use of a

transparent screen makes the alignment of the video camera

a minor issue. To calibrate the magnification factor of the

camera, one records a millimeter paper that is attached to

* Dr. ir. J. P. A. J. van Beeck, Prof. M. L. Riethmuller, von Karman

Institute for Fluid Dynamics, Chaussee de Waterloo 72, B-1640

Rhode-Saint-Genese (Belgium).

E-mail: [email protected]

** Dr. L. Zimmer, National Aerospace Laboratory, CFD Technology

Center, 7-44-1 Jindaiji-Higashi, Chofu, Tokyo 182-8522 (Japan).

# WILEY-VCH Verlag GmbH, D-69469 Weinheim, 2001 0934-0866/01/0412/0196 $17.50þ:50=0

196 Part. Part. Syst. Charact. 18 (2001) 196–204

the screen. Subsequently, the relationship between pixel

number and the scattering angle is found via the focal

length of the receiving lens.

A typical rainbow interference pattern is shown in Figure

2a. It is recorded when the water spray works at stable

ambient temperature. Horizontal profiles at different

instants of time are plotted in Figure 3. Note the stability of

the so-called Airy fringes, from which all droplet char-

acteristics are derived. Although, most probably, numerous

non-spherical droplets cross the laser beam, the Airy fringes

do not move. This implies that the pattern is formed by

constructive interference of the spherical droplets, because

for each of them the rainbow position is identical.

Destructive interference occurs for the non-spherical drop-

lets and liquid ligaments, because their rainbow patterns are

randomly oriented and thus yield a uniform background.

Consequently, there is no need for complex selection cri-

teria to look for rainbow patterns from spherical droplets.

The selection of spherical droplets is done automatically,

just like the rainbow in the sky, which is a static phenom-

enon, even though numerous raindrops are not spherical.

Apart from the fact that the global rainbow pattern (Figure

2a) is formed by spherical droplets only, it is interesting

that the visibility of this pattern is reduced compared with

that formed by a single droplet (Figure 2b). A ripple

structure such as that in Figure 2b cannot be observed at

all in the global rainbow pattern in Figure 2a. This ripple

structure contains no temperature information and there-

fore can only deteriorate the accuracy of the temperature

measurement; for a single fuel droplet smaller than 30mm,

the temperature uncertainty exceeds � 6 �C using standard

rainbow thermometry [5]. Therefore, the fact that the

ripple structure does not appear in the global rainbow

pattern is very favorable for an accurate detection of the

Airy maxima y1 and y2, from which droplet size and

temperature are derived (Figure 3). The disappearance of

the ripple structure has been discussed by Roth et al.

[11, 12], who studied the mean scattering diagram of a

polydispersed burning droplet stream. A droplet size varia-

tion of less than 1 mm made the ripple structure vanish

completely. However, the authors made no comments on

the natural selection of spherical droplets by means of this

mean interference pattern.

Fig. 1: Photograph of the set-up for global rainbow thermometry showingthe water spray (1), receiving lens (2), transparent screen (3) and videocamera (4). The laser beam was added during post-processing of thephotograph and does not represent the real thickness. a)

b)Fig. 2: (a) A typical global rainbow pattern in a water spray recorded by avideo camera. Only the so-called Airy fringes are visible. The horizontalaxis is proportional to the scattering angle. (b): A typical rainbow patterncoming from a single droplet in a water spray. Note the high-frequencyripple structure superimposed on the low-frequency Airy fringes.

Part. Part. Syst. Charact. 18 (2001) 196–204 197

For the patterns in Figure 3, the mean diameter is deter-

mined to be 49� 3 mm. This is calculated from the fringe

spacing y27 y1 as if the rainbow were formed by a single

droplet. Phase-Doppler anemometry measures an arithmetic

mean diameter of 60mm and a Sauter mean diameter of

130mm. From this, the question arises of what mean dia-

meter is deduced from the global rainbow pattern. This is

answered below.

3 Numerical Simulations of the GlobalRainbow Pattern

To invert the global rainbow pattern to mean droplet size

and temperature, data-inversion algorithms have to be

found, which will be based on simulations of the global

rainbow pattern. For these simulations, we assume a log-

normal droplet-size distribution at uniform temperature.

The density function f (d ) is given by

f ðdÞ ¼1

sdffiffiffiffiffiffi2p

p � e12

lnðd=dÞs

� �2

: ð1Þ

The integral from zero to infinity over f (d ) yields unity. d is

a parameter representing the mean diameter of the spray and

s is the dispersion around it. Because there is a finite number

of droplets, Ntot, the distribution has to be discretized. With

di Ddi=25d5di þ Ddi=2 the diameter range containing

one droplet with diameter di, it follows that

f ðdiÞDdi ¼ 1=Ntot: ð2Þ

Subsequently, the arithmetic mean diameter D10 results in

the following simple expression:

D10 ¼XNtot

i¼1

di f ðdiÞDdi ¼1

Ntot

XNtot

i¼1

di: ð3Þ

A similar simple expression is found for the commonly

used Sauter mean diameter D32:

D32 ¼XNtot

i¼1

d3i =

XNtot

i¼1

d2i : ð4Þ

The influence of Ntot, s and d on the global rainbow pattern

is investigated. Figure 4a and b depict droplet-size dis-

tributions according to Eq. (1) for s¼ 0.2 and 0.5,

respectively. Note that for s¼ 0.5 the distribution is less

symmetric and ranges from 35 to 280mm.

To calculate the scattering diagram of the global rainbow

pattern, the scattered light intensity of each single droplet is

summed, thereby neglecting optical interference between

different droplets and assuming perfect sphericity. The

rainbow pattern for a single droplet is represented by the

normalized Airy function Ai(x,d); here, x¼ y7 yrg, where

yrg is the scattering angle corresponding to the geometrical

rainbow angle, which depends only on the droplet’s

refractive index, and hence temperature [13]. Using the

Airy function implies that only the Airy fringes and no

ripple structure is taken into account. The mean scattered-

light intensity distribution Rnbw(x) then becomes

RnbwðxÞ /XNtot

i¼1

Aiðx; diÞ2� d

7=3i : ð5Þ

The normalized Airy function accounts for the angular

variation of the scattered-light intensity, whereas the factor

di7/3 takes into account the dependence of this intensity on

the droplet diameter within the primary rainbow region

[14]. Remember that the effect of droplet temperature and

laser-light wavelength l on the rainbow pattern lies in the

geometrical rainbow angle yrg (and therefore x). l is set to

514.5 nm for all the simulations in this paper.

Figure 5 shows three global rainbow patterns formed by an

ensemble of water droplets at 40, 60 and 80 �C. A tem-

perature difference of 40 �C leads to an angular displace-

ment of the pattern of 1.2�. From this, an appreciation of the

accuracy of temperature measurement can be discerned.

Even though the simulation is performed for a specific

spray, one can understand that the application of the rain-

bow technique lies in fields where the droplet temperature

varies at least over a few decades of degrees Celsius.

Figure 6 shows normalized global rainbow patterns for

s¼ 0.2 and three different mean diameters d (Eq. (1)). The

total number of droplets Ntot is fixed at 100, which appears

to be sufficient to reach a converged solution. Note the

perfect similarity between the curves. Because for smaller

droplet diameters the rainbow pattern broadens, angular

displacements are more difficult to detect. Hence the

accuracy of the temperature measurement degrades for

smaller droplet diameters.

Fig. 3: Profiles of experimental global rainbow pattern at four differentinstants in time. Note the stability of the pattern. The scattering angle isnverse proportional to the pixel number.


Figure 7 depicts global rainbow simulations for constant d,

but with different dispersions s. In contrast to Figures 5 and

6, the signals are not at all similar. When s increases, the

principal rainbow maximum tends to move towards the

geometrical rainbow angle yrg at x¼ 0. Moreover, the width

of the principal maximum decreases, which makes its

detection more accurate. However, the visibility of the other

Airy fringes diminishes for higher spray dispersion, which

can also be observed in the experimental signal in Figure 3.

Eq. (5) will be the model used for the creation of the

data-inversion schemes (Section 4). However, one must

realize that the agreement between the model and the

experimental rainbow pattern is not perfect. This could be

due to the following facts:

� The distance between a droplet and the receiving lens is

not constant. From Figure 1, one understands that this

distance depends on the position of the droplet in the

laser beam. This effect could be eliminated by adding a

spatial filter to the receiving optics.

� The contribution of nearly spherical droplets is not

taken into account. It is likely that these droplets fill the

‘‘valleys’’ between the Airy maxima, thus reducing the

fringe visibility.

4 Data Inversion Algorithms For GlobalRainbow Thermometry

Based on the simulations of the global rainbow pattern

(Section 3), one can evaluate what kind of mean diameter

and temperature are obtained from the global rainbow. The

aim is to look for data-inversion algorithms that are inde-

pendent of the spray-dispersion paremeter s (Eq. (1)).

Three different schemes are studied. They vary in the

information taken from the signal: this can be the Airy

maxima y1 and y2, the minimum ymin between both max-

ima, or the two inflection points yinf 1 and yinf 2 around y1

(see Figure 6).

The first algorithm that is discussed is the one most often

applied when processing a rainbow coming from a single

droplet. The droplet diameter, DAiry , is derived from the

distance between the first two Airy maxima, y1 and y2,

using the Airy theory for the rainbow [3–5, 8, 13]:

DAiry ¼ 1016:2lðy2 y1Þ3=2; ð6Þ

which is valid for a refractive index m¼ 4/3. Once DAiry is

known, the temperature is calculated from the geometrical

rainbow angle, deduced from y1 and DAiry [3, 8, 13]:

yrg ¼ y1 46:18ðl=DAiryÞ2=3; ð7Þ

where the second term on the right-hand side is valid for

m¼ 4/3. The relationship between yrg and temperature is

shown in Figure 5.

To evaluate the first algorithm in a polydisperse spray, Eqs.

(6) and (7) were applied to Eq. (5), which is the model for

the global rainbow pattern. Figure 8a shows the resulting

DAiry as a function of s for d¼ 100 mm and different

amounts of droplets. Up to s¼ 0.3, DAiry equals the Sauter

mean diameter D32. For higher s, the Airy diameter

depends on the total number of droplets forming the global

rainbow. This number is unknown in the experiments,

hence Eq. (6) becomes useless for s > 0.3. Figure 8b depicts

the results of Eq. (7) for different d with Ntot¼ 200. The

deviation of yrg from its value at s¼ 0 reaches 0.4� for

s¼ 0.3 and d¼ 20mm, which corresponds to an uncertainty

in the temperature measurement of about � 16 �C for water.

Consequently, the algorithm employed for a single droplet

(standard rainbow thermometry) cannot be applied for a

polydisperse spray to calculate the mean droplet size and

temperature, especially when the mean droplet diameter is

smaller than 100 mm. The problem with this algorithm is

Fig. 4: Droplet size distribution for (a) s¼ 0.2 and d¼ 100 mm and (b) s¼ 0.5 and d¼ 100 mm.


related to the fact that the secondary Airy maximum y2

disappears for large s (Figure 7).

The second algorithm employs the maximum of the prin-

cipal Airy fringe y1 and the first minimum ymin (Figure 6).

From these parameters one can deduce DAiry and yrg using

the Airy theory for the rainbow:

DAiry ¼ 462:6lðymin y1Þ3=2; ð8Þ

yrg ¼ y1 46:18ðl=DAiryÞ2=3: ð9Þ

Figure 9a and b depict DAiry and Dyrg, respectively. There is

no improvement in the temperature measurement (Figure

9b) with respect to the algorithm based on y1 and y2.

However, concerning the droplet-size measurement (Figure

9a), DAiry follows D32 up to s¼ 0.25. For 0.25< s< 0.4,

DAiry is closer to the arithmetic mean diameter D10. For

even larger s, DAiry depends weakly on Ntot. The same

behavior is found for other d, because the rainbow signals

are similar (Figure 6).

The most interesting results are obtained for the data-

inversion algorithm based on the inflection points around

y1, i.e. yinf 1 and yinf 2 (Figure 6). The relationship between

these points and DAiry and yrg is

DAiry ¼ 531:6lðyinf 2 yinf 1Þ3=2; ð10Þ

yrg ¼ yinf 1 13:91ðl=DAiryÞ2=3: ð11Þ

Roth et al. [15] noted that yinf 1 lies very close to the geo-

metrical rainbow angle yrg, thus being a good indicator for

the droplet temperature. Figure 10a shows that DAiry is

independent of Ntot up to s¼ 0.5, but it exceeds the Sauter

mean diameter D32 considerably. This is related to the

decrease in the width of the principal rainbow maximum

when s increases (Figure 7). But look at Figure 10b! Dyrg is

never larger than 0.025�, which means that this algorithm

ensures an accuracy in the temperature measurement of less

than � 1�C.

Finally, it is interesting to mention that the spray-dispersion

parameter s correlates with the ratio of DAiry from the

inflection points to that derived from y1 and ymin, as is seen

in Figure 11. The resulting curve can therefore be used to

Fig. 5: Simulations of global rainbow patterns for different mean watertemperatures and fixed dispersion s.

Fig. 6: Simulations of global rainbow patterns for different meandiameters at constant dispersion s. x¼ y7 yrg.

Fig. 7: Simulations of global rainbow patterns for different dispersions atconstant mean diameter d. x¼ y7 yrg.


determine the width of the droplet size distribution.

Unfortunately, for s> 0.4, the relationship depends also on

the total number of droplets Ntot.

5 Experimental Results

The above data inversion schemes were applied to the

experimental signals in Figure 3. DAiry , based on the first

Airy maximum and minimum (Eq. (8)), yields 48mm, but

from the inflection points (Eq. (10)) a mean droplet diameter

of 56mm is obtained. The precision in these diameters is

about �5 mm, hence the curve in Figure 11 yields s between

0.2 and 0.3. For this spray dispersion, the diameter 48mm

should be close to D32, but phase-Doppler anemometry

(PDA) gives 130mm. This large discrepancy could be

explained by the fact that the rainbow method is much more

sensitive than PDA to droplet shape (see Tropea [16]). PDA

measures a mean Sauter diameter of the spherical and

spheroidal droplets. On the other hand, the global rainbow

pattern is formed by the most spherical droplets only, hence

the smaller ones. This leads to a mean Sauter diameter for

the rainbow technique that is smaller to that of PDA.

All simulations in the previous section were made at con-

stant temperature and the whole picture could change when

the global rainbow is formed by droplets with different

temperatures. Nevertheless, it is interesting to see what the

technique gives so far. Temperature measurements are

presented in Figure 12, which shows the droplet tempera-

ture in the water spray of Figure 1 at a distance of 10 cm

from the spray nozzle. The initial nozzle temperature was

monitored by a thermocouple and was varied between 19

and 68 �C. The ambient temperature was 21 �C. The method

of the inflection points was used (Eq. (11)). Each rainbow

temperature was obtained after averaging 50 video images,

taken at different instants of time, which ensured con-

vergence of the mean temperature measurement with

respect to the number of droplets forming the rainbow.

Although the precision seems to be a few degrees Celsius,

the tendency is correct. The spray droplets cool.

6 Conclusions

Experimental and numerical results on global rainbow

thermometry have been reported. Low CW laser power is

needed to create an average, so-called global, rainbow

pattern. This pattern is formed by all the spherical droplets

that cross the laser beam during the integration time of the

video camera. The non-spherical particles are supposed to

create a uniform background. Because the selection of

spherical droplets happens naturally, the destructive influ-

ence of droplet shape on the accuracy of the temperature

measurement is overcome. Moreover, the pattern is smooth

and not spoiled by an additional ripple structure that would

deteriorate the temperature derivation from the global

rainbow pattern.

Simulations of the global rainbow pattern were performed

using the Airy function and a log-normal distribution for

the spray droplet diameters. All droplets are assumed to be

spherical, at the same temperature and at the same distance

from the detector. The simulations show that there exists a

similitude for global rainbow patterns related to different

mean droplet diameters. However, with respect to spray

dispersion, the similitude does not hold because signal

visibility varies significantly. An important result is that

the principal rainbow maximum, from which the mean

temperature is derived, becomes narrower for increasing

width of the droplet size distribution. This renders the

detection of the rainbow displacement, and therefore tem-

Fig. 8: (a) The Airy diameter, calculated from y27 y1, as a function of the spray dispersion s for various amounts of droplets Ntot (Eq. (6)). (b) Thedeviation of yrg, based on the first two Airy maxima y1 and y2, from its value at s¼ 0, as a function of s and different mean diameters d (Eq. (7)).


perature measurement, more accurate. From the simulations

it follows also that the larger the spray dispersion, the more

droplets are needed to reach a stable signal.

Three data-inversion algorithms were investigated. The

standard algorithm, based on the two important interference

maxima, fails for a polydisperse spray because the lower

maximum vanishes. The second algorithm derives the mean

droplet size and temperature from the first maximum and

first minimum. A mean diameter is obtained somewhere

between the arithmetic and Sauter mean diameter, but the

temperature measurement is not reliable. A correct tem-

perature is derived from the third algorithm, employing the

inflection points around the principal rainbow maximum.

For this algorithm, the accuracy is always less than � 1 �C.

The mean droplet diameter obtained from this method is

much larger than the Sauter diameter. That is why the ratio

of this diameter to that derived from the first maximum and

minimum can give an idea about the spray dispersion.

The resemblance between experimental rainbow patterns

and the simulations is not perfect as far as relative peak

intensities are concerned, and this should be studied further.

One reason could be the influence of nearly spherical drop-

lets, which is neglected in the simulations. Nevertheless,

rainbow temperatures have been measured with success in a

heated water spray. The precision is a few degrees Celsius.

The mean droplet diameter, measured by the rainbow

Fig. 9: (a) The Airy diameter, calculated from y17 ymin, as a function ofthe spray dispersion s for various numbers of droplets Ntot (Eq. (8)). (b)The deviation of yrg, based on y1 and ymin, from its value at s¼ 0, as afunction of s for different mean diameters d (Eq. (9)).

Fig. 10: (a) The Airy diameter, calculated from yinf 27 yinf 1, as a functionof the spray dispersion s for various number of droplets (Eq. (10)). (b)The deviation of yrg, based on yinf 2 and yinf 1, from its value at s¼ 0, as afunction of s for different mean diameters d (Eq. (11)).


method, appears to be considerably smaller than that

obtained by phase-Doppler anemometry. This is probably

because global rainbow thermometry only sees the smaller,

most spherical droplets.

7 Acknowledgments

The authors thank M. Santonastasi and J. Sanz for their

contributions to this paper.

8 Symbols and Abbreviations

Ai Airy function

d droplet diameter

Ddi droplet diameter range within discretized droplet

size distribution

D10 arithmetic mean droplet diameter

D32 Sauter mean droplet diameter

DAiry mean droplet diameter deduced from the rainbow

pattern

d mean droplet diameter used in f (d)

f (d ) density function for droplet size distribution

GRT global rainbow thermometry

y scattering angle

yrg geometrical rainbow angle

Dyrg deviation of yrg with respect to yrg at s¼ 0

y1 main maximum in the rainbow pattern

y2 secondary maximum in the rainbow pattern

ymin minimum between y1and y2

yinf 1 inflection point between yrg and y1

yinf 2 inflection point between y1 and ymin

PDA phase-Doppler anemometry

Rnbw mean scattered-light intensity as a function of x

x y7 yrg

Ntot, n total number of droplets

l wavelength of light

s dispersion parameter in f (d )

Subscripts

Airy Airy theory

i droplet-diameter class index

inf inflection point

min minimum

rg rainbow according to geometrical optics

tot total

9 References

[1] N. Roth, K. Anders, A. Frohn: Simultaneous Measurement ofTemperature and Size of Droplets in the Micrometer Range. Proc.7th International Congress on Optical Methods in Flow and ParticleDiagnostics ICALEO 88, L.I.A., Vol. 67, Sunnyvale, CA 1988,294–304.

[2] S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D.Bachalo: An Integrated Phase Doppler/Rainbow RefractometerSystem for Simultaneous Measurement of Droplet Size, Velocity,and Refractive Index. Proc. PARTEC 3rd Int. Congress on OpticalParticle Sizing, Yohokama, Japan 1993, 275–284.

[3] J. P. A. J. van Beeck, M. L. Riethmuller: Simultaneous Determinationof Temperature and Size of Droplets from the Rainbow Using AiryTheory. Proc. 7th Int. Symp. on Applications of Laser Techniques toFluid Mechanics, Vol. 2, pp. 21.5.1–21.5.6, Lisbon, Portugal; alsoin R. J. Adrian et al. (eds): Developments in Laser Techniques andApplications to Fluid Mechanics, Springer, Berlin 1994,330–339.

[4] S. V. Sankar, D. H. Buermann, W. D. Bachalo: An AdvancedRainbow Signal Processor for Improved Accuracy in DropletTemperature Measurements. Proc. 8th Int. Symp. on Applications ofLaser Techniques to Fluid Mechanics, Vol. 1, Lisbon, Portugal1996, 9.3.1–9.3.9.

[5] P. Massoli: Rainbow Refractometry Applied to Radially Inhomo-geneous Spheres: The Critical Case of Evaporating Droplets. Appl.Opt. 37 (1998) 3227–3234.

Fig. 11: The ratio of the droplet diameter derived from the inflectionpoints, to that derived from the first Airy maximum and minimum, plottedas a function of the non-dimensional width of the droplet sizedistribution, s.

Fig. 12: Rainbow temperature in a water spray at 10 cm from the injec-tion point as a function of the injection point temperature. Ambienttemperature, 21 �C.


[6] K. Anders, N. Roth, F. Frohn: Theoretical and Experimental Studiesof the Influence of Internal Temperature Gradients on RainbowRefractometry. Proc. PARTEC 4th Int. Congress on Optical ParticleSizing, Nuernberg, Germany 1995, 419–428.

[7] F. Corbin, A. Garo, G. Gouesbet, G. Grehan: Refractometried’arc-en-ciel: application au diagnostic des gouttes avec gradientd’indice, in Recueil des Actes du 5e Congres Francophone deVelocimetrie Laser, Rouen, France 1996, E1.1–E1.8. Universite etInstitut des Sciences Appliquees de Rouen, URA CNRS 230/CORIA.

[8] J. P. A. J. van Beeck, M. L. Riethmuller: Nonintrusive Measurementsof Temperature and Size of Single Falling Raindrops. Appl. Opt. 34(1995) 1633–1639.

[9] J. P. A. J. van Beeck, M. L. Riethmuller: Rainbow Pheno-mena Applied to the Measurement of Droplet Size and Velocityand to the Detection of Nonsphericity. Appl. Opt. 35 (1996) 2259–2266.

[10] J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller:Global Rainbow Thermometry for Droplet-Temperature Measure-ment. Opt. Lett. 24 (1999) 1696–1698.

[11] N. Roth, K. Anders, A. Frohn: Refractive-Index Measurements for theCorrectionofParticleSizingMethods.Appl.Opt.30(1991)4960–4965.

[12] N. Roth, K. Anders, A. Frohn: Simultaneous Determination ofRefractive Index and Droplet Size Using Mie Theory. Proc. 6th Int.Symp. on Applications of Laser Techniques to Fluid Mechanics,Vol. 1, Lisbon, Portugal 1992, 15.5.1–15.5.5.

[13] J. P. A. J. van Beeck: Rainbow Phenomena: Development of ALaser-Based, Non-intrusive Technique for Measuring Droplet Size,Temperature and Velocity. PhD Thesis, Eindhoven University ofTechnology, ISBN 90-386-0557-9, 1997.

[14] H. C. van de Hulst: Light Scattering by Small Particles. DoverPublications, New York 1981, 243–246.

[15] N. Roth, K. Anders, A. Frohn: Size Insensitive Rainbow Refracto-metry: Theoretical Aspects. Proc. 8th Int. Symp. on Applications ofLaser Techniques to Fluid Mechanics, Vol. 1, Lisbon, Portugal1996, 9.2.1–9.2.6.

[16] C. Tropea: Developments of Specialized Phase-Doppler Techniquesand Refractive Index Measurements. Lecture Series on OpticalDiagnostics of Particles and Droplets, von Karman Institute, Sint-Genesius-Rode, Belgium 1999.


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Global Rainbow Thermometry for Mean Temperature and Size Measurement of Spray Droplets

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