TRANSIENT EVOLUTION OF REFRACTIVE INDEX GRADIENTS INSIDE SINGLE ISOLATED
DROPLETS WITH RAINBOW REFRACTOMETRY
Christopher ROSEBROCK1, Thomas WRIEDT2 and Lutz MÄDLER1,2,*
2
1 Dept. of Production Engineering, University of Bremen, Bremen, 28359, Germany
Foundation Institute of Material Science (IWT), University of Bremen, Bremen, 28359, Germany
*Corresponding author: lmaedler@iwt.uni-bremen.de
Abstract
Rainbow refractometry has been applied to combusting
and cooling droplets, showing an unexpected shift of the
rainbow structure as combustion and cooling is initiated.
This unexpected shift correspond to temperature gradients
within the liquid droplets, which can be determined
assuming diffusion limited heat transfer and applying the
Lorenz-Mie theory for stratified spheres.
1 Introduction
Rainbow refractometry (RRF) is a versatile nonintrusive
laser-based diagnostic technique for determining refractive
indices and sizes for both, single liquid droplets [1,2] and
in entire sprays [3,4].
RRF uses the shift of the rainbow structure in the elastic
light scattering pattern of spherical droplets due to
refractive indices changes with temperature in the liquid
droplet (considering only pure liquids). In a first
approximation the rainbow structure shifts towards lower
scattering angles with decreasing refractive index of the
liquid droplet. The refractive index of pure liquids
decrease with increasing temperature. Therefore, by
knowing the functional relation between temperature and
refractive index the latter can be determined by detecting
the positional change of the rainbow structure.
However, there are certain limitations of the RRF due to
the change of the rainbow structure with the size for
droplet diameters below 1000 m and with refractive
index profiles within the liquid droplet. While the former
can be commonly avoided, using an additional droplet
size
measurement
technique,
the
uncertainties
accompanied with refractive index profiles remain
challenging. Especially during combustion, where high
temperatures and fast vaporization rates occur,
temperature differences between the surface and the
interior of droplets can sustain throughout the droplet
lifetime [5].
In order to take into account the refractive index
profiles within the liquid droplet, a stratified sphere model
for the Lorenz-Mie theory [6], geometrical optics [7],
generalized Airy theory [8] or the complex angular
momentum theory in combination with the FSSM [9] have
been applied.
However, there is still a lack of experimental evidence
of refractive index profiles within droplets. The aim of the
present study is the time-resolved detection and
determination of refractive index profiles within burning
or cooling single isolated droplets.
2 Experimental setup and theoretical models
For observation of refractive index profiles within
liquid droplets, the experimental setup schematically
shown in Figure 1 is used. It consists of a droplet-ondemand generator (piezodropper [10]) that produces
single isolated droplets operated with a mean separation
distance of s / d 0  1800 between the droplets, where d 0 is
the initial droplet diameter. The droplets are ejected either
upwards or downwards through a laser sheet to follow
continuously the light scattering of the droplets. The
upward ejected droplets are ignited by means of a
synchronized spark discharge, following the internal
temperature profile (refractive index) as combustion
proceeds. To promote a spherical flame shape and
assuring radial heat transfer to the droplet surface,
coflowing oxygen surrounds the droplets. In contrast, the
downward ejected droplets are passing a heating coil that
preheats the droplets close to the boiling point. After
passing the heating coil the droplets cool down in the
colder atmosphere, enabling the detection of temperature
profiles during the cooling process.
PC-14.1
Figure 1 Experimental setup of radial refractive index
measurements.
LASER-LIGHT
AND INTERACTIONS WITH
PARTICLES
AUGUST 25-29TH, 2014, MARSEILLE, FRANCE
In order to evaluate the shift of the rainbow structure in
backward scattering region due to the temperature profiles
within the liquid droplets, the droplet size is
simultaneously measured by means of interference fringes
occurring in the forward scattering region due to refraction
and reflection. With the known droplet size and the
assumption of diffusion limited heat transfer within the
droplet, i.e. an approximately parabolic temperature
profile, the transient evolution of the refractive index
gradients can be evaluated by applying the LMT for
stratified spheres [10].
3 Results and discussion
Figure 2 shows the droplet diameter and the first two
positions of the rainbow maxima for a single dodecane
droplet during cooling at room temperature preheated
beforehand with a heating coil (Figure 1). The position of
the rainbow maxima are determined after applying a low
pass filter to the rainbow pattern for removing both the
higher frequency of the interference of the second order
refraction and the reflection (ripple structure) and the high
frequency noise.
Figure 2 shows that the droplet diameter decreases
slightly due to evaporation of the preheated droplet after
leaving the heating coil. However, the positions of the
rainbow maxima shift towards lower positions, passing
through minima and increase again. Since the refractive
index of dodecane increases linearly with decreasing
temperature, one would expect solely an increasing
rainbow positions throughout the cooling, if the droplet
has a uniform temperature at every instance, i.e. ideal
mixed. Contradicting the initial decrease would suggest an
increasing droplet temperature, which has no physical
meaning. Therefore, the droplet temperature cannot be
uniform throughout the droplet interior but possess a
temperature and refractive index profile intially.
Figure 2 Droplet diameter and pixel position of the first two
rainbow maxima for a single isolated dodecane droplet cooling at
room temperature.
Comparable conclusion can be drawn for burning
droplets, representing the reverse case of the
aforementioned cooling dodecane droplet.
Figure 3 shows the droplet diameter and the first two
rainbow maxima for a single dodecane droplet burning in
a coflowing oxygen stream. The positions of the rainbow
maxima have been determined similarly to the previous
case. Note that the combustion is not completed within the
laser sheet. However, the ignition and heating of the
droplet is clearly visible. Figure 3 shows that the droplet
diameter decreases quadratically according to the D²-Law
for droplet combustion after the ignition, which is
indicated by the scatter of the data points around the 180th
acquired scattering pattern. The scatter is due to the
ignition spark that induces surface wave onto the droplet,
distorting the scattering pattern for a brief period.
In accordance with the previous case, both rainbow
maxima possess again a turning point, which are now
maxima. Similarly to the cooling dodecane droplet, one
would expect solely a decreasing shift of the rainbow
positions throughout the combustion, since the refractive
index decreases with increasing temperature. However,
the initial increase of the rainbow positions suggests again
a non-physical behaviour, i.e. a decreasing droplet
temperature while heat is transferred from the combustion
to the droplet. Thus, even for combustion where high
temperatures and a fast heat transfer occur, the
temperature within the droplet interior is initially
nonuniform.
By using either of the applied theories [6,7,8,9], it was
shown that assuming solely diffusional transport within
the droplet, these initially non-physical shifts of the
rainbow structure corresponds to refractive index
gradients within the droplet. That means, by following the
position of the rainbow structure and assuming diffusion
limited transfer processes, gradients within the droplet can
be determined with rainbow refractometry.
Figure 3 Droplet diameter and pixel position of the first two
rainbow maxima for a single isolated dodecane droplet burning
in a coflowing oxygen stream.
PC-14.2
LASER-LIGHT
AND INTERACTIONS WITH
PARTICLES
AUGUST 25-29TH, 2014, MARSEILLE, FRANCE
Since diffusion transfer processes are similar to each
other, the non-dimensionalized gradients should be the
same for all. This can be used as a verification for both
proving diffusion processes as well as determine
temperature and concentration gradients within liquid
droplets. The turning points result from diffusional heat
transport within the droplet that can be approximated by
parabolic refractive index profiles.
[11] Wu Z.S., Guo L.X., Ren K.F., Gouesbet G., Gréhan G., Improved
algorithm for electromagnetic scattering of plane waves and shaped
beams by multilayered spheres, Applied Optics 36(21):5188-98 (1997)
4 Acknowledgement
The authors would like to thank the Deutsche
Forschungsgemeinschaft (DFG) for funding this project
under grants of MA 3333/4-1.
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