Jan 4, 1988 - Doppler-difference-method (LDVS) are the sphericity of the dispersed phase ... with the refractive index of the continuous phase, influence the.
Part. Part. Svst. Charact. 5 (1988) 66-71
66
The Phase-Doppler-Difference-Method, a New-Laser-Doppler Technique for Simultaneous Size and Velocity Measurements Part 2" : Optical Particle Characteristics as a Base for the New Diagnostic Technique Klaus Bauckhage, Hans-H. Floegel, Udo Fritsching, Rudiger Hiller** (Received: 24 August 1987, resubmitted: 4 January 1988)
Abstract Simultaneous size and velocity measurements can be obtained by using photodetector positions of different off-axis-angles. But not for all of these positions one receives unambiguous results for the correlation between the phase difference and the particle diameter. This can be clearly demonstrated by the plots of the numerical calculations of the complete Mie's scattering equations. These plots show that for transparent particles which exceed the continuous phase in density at special off-axis-angles the situation of light refraction changes into a situation of addi-
tional light reflection or vice versa. On the other hand for transparent particles which are less dense compared with the continuous phase and for totally absorbing materials these plots confirm the simplified equations for reckoning the particle diameter by the laws of geometrical optics. The good agreement between these results can also be verified for the backscatter mode for measurements of metallic spheres with an imaginary refractive index.
1 Introduction
Table 3 : Two-phase-flow-systemsof dispersed spherical particles (the italicized applications are examples of experimental analysis in our laboratories).
Important preconditions for the successful use of the phaseDoppler-difference-method (LDVS) are the sphericity of the dispersed phase and a knowledge of the optical properties of the material of the particles. These properties are described by the refractive index. Descriptively one speaks of transparent, opaque or reflecting particles. These characteristics, in combination with the refractive index of the continuous phase, influence the optimal or necessary positions of the photodetectors and may affect the accuracy of the results. That is why we discuss the optical particle characteristics as the basic information necessary for the successful use of the new diagnostic technique. Whilst the precondition of sphericity may, on a first sight, restrict the applicability of the new method we have to keep in mind that in most two phase flow situations, where small fluid particles are dispersed in the continuous phase, these particles exist as spherical drops or bubbles. Solid particles also often have a spherical shape with a surface of sufficient smoothness. As sketched in Table 3 there exists a lot of different multi phase flow systems with dispersed spherical particles and already a remarkable number of applications of the LDVS. On the other hand this Table also makes clear that there have to be observed differing optical properties of the various dispersed materials.
* **
In part 1 of this paper the phase-Doppler-method and the necessary equipment for the new diagnostic technique have been described. Prof. Dr.-lng. K. Buuckhage, Dr.-Ing. H.-H. Floegel, Dipl.-Ing. U Fritsching, cand. phys. R. Hiller; Verfahrenstechnik im Fachbereich Produktionstechnik der Universitat Bremen, Postfach 33 0440, D-2800 Bremen 33 (Federal Republic of Germany).
System I/g
liquid droplets
l/g
molten metal droplets
Application
in
air (gas) flows
gas-pipelines, aerosols, mists, spray-cooling, combustion sprays
metal powder production, separation processes, spray compacting, sprays for laquering processes,
in
air flows N,, He, Ar
in
air (gas) flows
in
water flows
s/l
metal-, plastics-, in glass-spheres
transparent liquid flows
slurries, suspension flows in pipelines, in clarifiers, cyclones, separation processes,
g/l
bubbles of air (gas)
transparent liquid flows
cavitation, boiling, separation processes in bubble columns,
~~~~
s/g
1/1
solidified metals, glasses, ceramics organic liquid droplets
in
spray drying, dust, smoke, concrete spraying, sprays for laquering processes, liquid-liquid extractions, emulsions-flows,
~___________
0 VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1988
~~~~~
0176-2265/88/0206-0066 $02.50/0
Part. Part. Syst. Charact. 5 (1988) 66-71
67
2 Analysis of Transparent Spheres For LDVS-measurements of small spherical particles where the laws of Mie scattering and those of geometrical optics can be compared numerically in relation to special applications, the lower limit for the use of the laws of geometrical optics (i. e. for the particle diameter treatment) can be given as d < 1 pm, using laser-light sources of wavelenghts of about 0.4 pm. The growing and diversifying application of LDVS seems to reduce the question of whether the complete Mie's scattering laws or that of geometrical optics should be used for the computation of diameter to materials which are not fully transparent. This in any case includes the question for the correct and appropriate @pd-dependency. This can be explained by means of Figure 11. The simplified three dimensional plot of the numerical calculations of the complete Mie scattering laws, shows the phase difference @ versus off-axis-angle, p, and diameter, d, in the case of a dense transparent particle, such as a water droplet in air, with a refractive index nd = 1.33. As can be seen from this plot the @-d-dependence differs greatly if we consider off-axis-angles below 83", between 85" and 128" and those larger than 138". We recognize singularities near 85 and 135 degrees which are caused by the change from refraction to reflection in the light scattering and vice-versa. The dependencies are quite complicated in detail. In Figure 11 the singularities caused by diffraction and by the mathematical and numerical treatment have been erased. In order to simplify these quite complicated dependencies we define three inclined planes, which can be used for the derivation of a linear @-d-correlation. If we choose for our measurements the first plane, i. e. off-axis-angles larger than 25 (see f. i. AlChulubi et al. [17]) and less than 83", we use the domain where light which is refracted through the particle will mainly be received. In particular the first refracted ray with two surface refractions and no internal reflections as sketched in Figure 12, is, by far, dominant over the scattered light (see f. i. Vun de Hulst [IS]). In the case of an off-axis-angle around 70 degrees one receives from the inclined plane in Figure 11a linear dependence of the phase shift, @, on the droplet diameter, d, as shown in Figure 13. This has already been described by Suffman [19]. In Figure 13a comparison between the numerical solution of the complete Mie's scattering laws and the findings of the simplified O
laws of geometrical optics, for a fixed (o-values, shows a very good agreement : a) the Mie's scattering solutions are the unbroken line and have some unimportant peaks which arise from the mathematical treatment, b) geometrical optics lead to the broken line. refracted ray
?i incident r a y
refracted ray
/ fl
reflected ray (0 order 1
Fig. 12: Lightscattering at a spherical particle (using the laws of geometrical optics).
This type of correlation gives the basis for the use of the linear @-d-correlationin Eq. (2) and (4) and Table 1 in part 1 of this paper. In similar manner to the case where the light scattering is dominated by refracted light only and where the phase shift is linearly dependent on the particle diameter we also find for transparent but reflecting particles linear @-d-dependencies. This holds, for instance, for air-bubbles in water. In this case Buchhave et al. [7] already showed that the @ (d) dependency is not influenced by the particle's refractive index. One obtains a dependence like that shown in Figure 14, once again a good agreement between the numerical calculations from Mie's scattering laws, the unbroken line, and the laws of geometrical optics, the broken line. Here the sign has changed, thus resulting in positive values of @.
1
Fig. 11: Phase difference, @, versus offaxis-angle, p, and particle diameter, d.
o f f - a x i s angle 'p I "I
68
Part. Part. Syst. Charact. 5 (1988) 66-71
-
Oj
I
-15:
le
-30< OI E
i -45{ = c
-60: L m
n
-15:
-
-90 -105 -120
0
5
10
15
20
30
25
35
40
45
50
Fig. 13: Phase shift versus particle diameter, d, for water droplets in air under an off-axis-angle, v, = 69".
35
40
45
50
Fi& 14: Phase shift, @, versus particle diameter, d, for air bubbles in water under an off-axis-angle, (o = 83".
particle diameter d[prn] O!-l
+=
0
5
10
15
20
25
3.1'
30
p a r t i c l e diameter d [gml
This is essential for Doppler signal detection, especially for the phase difference measurement of two Doppler bursts received especially from transparent particles and to choose photodetector positions where either reflected or refracted light is dominant. Absorbing and highly reflecting particles, such as metallic spheres, can on the other hand be analysed under any offaxis-angle and also in backscattering positions, thus under offaxis-angles close to 180". Thus for an actual application, for partly absorbing or opaque materials, the question of the best photodetector position arises. This question can only be answered with a knowledge of the optical characteristics of the dispersed and the continuous phase and by solving the numerical calculations of the exact electromagnetic field theory, i. e. the Mie's scattering calculations, or by carrying out expensive and time consuming experiments.
3 Analysis of Partly Absorbing and Opaque Particles Encouraged by the findings mentioned before it was the further purpose of the Mie's scattering calculations to reduce the scope of experimental research work and also to extend the phaseDoppler-method of other than transparent spherical particles, that is to partly absorbing, opaque or metallic ones. This intention has been successful in applying the general form of the refractive index, n, where an imaginary part exists:
n
=
nd (1
- iK)
(11)
to the question of at which off-axis-position, p, the measured value of the phase shift, @, (phase-Doppler-difference) would show a linear and unambiquous dependence on the diameter, d, of a non-transparent particle. In Eq. (11) n or nd is the refractive index of the dispersed particle, that is of the non-transparent material, and K is the index of absorption which is known from Lambert's law (see ref. [20]). For metallic particles this index, K , may significantly exceed the value of 1. By setting in the calculations constant the light intensity of the dual laser beams, their intersecting angle, 0, the vertical angle, ty,of the photodetectors, their distance to the measuring volume (587 mm), their lens-diameter (40 mm) and also the direction of polarisation relative to the interference planes in the fringe system we have derived the dependence of @ on off-axis-angle of Figure 15 for differing refractive indices (with nd = 1.333) and for particle diameters of 20, 30 and 50 pm. Turning our attention to the results for the particle with a diameter of 20 wm, very good agreement is found between curves 8 , 0, and (9 . The dotted curve (53 has been plotted by computation from the laws of geometrical optics whilst curves 8 and 0 show the results calculated from the exact electromagnetic field theory. For these calculations the index of absorption was taken to be 1.0 (curve 0) and 0.1 (curve @ ). The results of curves 0 , 0and (9 thus show the relationship between phase-differences, @, and off-axis-angles, p, for reflecting particles (i. e. air bubbles in water or metallic spheres). The second dotted curve in Figure 15, curve 0 , represents the results of the simplified computation from the laws of geometrical optics for a refracting particle. This might be a water droplet
69
Part. Part. Syst. Charact. 5 (1988) 66-71 360-1 315270225180135-
I
a . d = 20pm
P
904 50-45-90-
1
-
-135-1801
cle of 30 pm in diameter. The interpretation for this differing behaviour at small K-values has to be given by regarding the path length of the light in the particle. The effect of reflection which depends on the index of absorption, in the larger particle becomes dominant earlier compared with smaller particles because of the growing path length of the light inside the larger particle. While the efficiency of reflection in a transparent particle is given by R , :
I
360: 315: 6- 270: 225-
b i d = 30 ~m
Y ~
180:
135:
Calculations of t h e law of geometric optics ---- @ reflecting ---- 0 refracting
90-
2 45-
R, =
(s)2
the efficiency of an absorbing - or better said of a partly absorbing particle (i. e. for values i0) is given by use of the formula of Lambert and Beer [20] (instead of Eq. (12)) :
us -0: Q)
-45-
2 -90'
R,
c
a -135-
C.
0
15
30
45
d-50pm
60 75 90 off-axis-angle
n d -- 1333 t h e other parameters hav been set constant
105 120 135 150 165
WI
Fig. 15: Phase difference, @, versus off-axis-angle for different refractive indices, nd, and particle diameters of 20, 30, and 50 hm; comparison of numerical calculations from Mie's scattering theory and from geometrical optics.
in air. Compared with the numerical calculations based on the exact electromagnetic field theory for a non-absorbing particle (i. e. for K = 0) as given by curve 0 , one finds nearly identical curves for small off-axis-angles, i. e. for rp < 82" and some satisfying agreements for angles rp > 138 (but also several unsteady peaks). The same holds for curve 0 , which represents the results for a very low value of the index of absorption, i. e. for K = 0.001. This means that this particle shows very similar behaviour compared with a transparent particle. Discussion of curve 0 will be left for a moment. Considering the plots for the particles of 30 and 50 pm in diameter one can confirm the findings from above, a very good agreement between curves @ , 0 and 8 (some deviations of curve 0 in Figure 15b at angles, rp, between 50" and 90') and nearly identical plots of curves 0, 0 and 0 , once again for angles of rp less than 83 *. Now focussing the attention on the curves 0 one finds (with growing particle diameters) in the optical behaviour a tendency which could be described as an augmentation of the index of absorption. But this parameter was held constant at the value of 0.01 in all our figures. Nevertheless a sphere of 50 pm in diameter and a K-value of 0.01 clearly behaves more like a reflecting particle whereas a particle of 20 pm in diameter behaves more like a refracting particle. Note that there is a change in sign of term b from Eq. (3) to Eq. (4), in part 1 of this paper, at an angle, p, from 40"to about 80". A similar tendency exists for the partiO
=
(n, +~ nn, 2 K:2 ~ (n, + I ) +
~
For K > 0, R , exceeds the value of R , and for K-values close to 1, as in the case of metallic particles, the efficiency of reflection may reach considerable values. (One has to keep in mind that there exist further dependencies of K on the wavelength and on the incident angle of the light, which till now have not been included in our calculations.) From the plots in Figure 15 we observe a remarkable result that for K-values larger than 0.01, all spheres can be treated as reflecting particles if these particles have diameters exceeding values of about 20 pm and if one uses off-axis-angles larger than 100 deg. A further important result is that for constant offaxis-angles the @-values show significant augmentations with growing particle diameters without any singularity (see the arrows in Figure 15). Now the question arises, how an agreement can be found between the findings of Figure 11 for a totally transparent dense particle and those of Figure 15 (especially curve 0 ). In answer to this question we remember that the relevant parameter here is the particle diameter, d. In the three dimensional plot of Figure 11, the mountains of the @-rp-dependencies at off-axis-angle regions from about 85" to 135", also known from Figure 15, grow for increasing particle diameters to a significant height in the back-ground. On the other side, in the foreground, these mountains flatten to a moderate hilly dependency for small particle diameters, explained here with the short valleys of Figure 15a) and b). Apart from several inconsistencies in Figures 11 and 15, which have to be explained by the mathematical and numerical treatment of the complete Mie's equations and which may be avoided by improved algorithms and by modified approximation formulae in the future, the rugged gradient at angles of 85" and 135 has to be explained by the change in dominance of refraction to reflection and from external to external and internal reflection at these off-axis-angles. This also causes the change in sign of the phase difference, @. Combining the choice of the correct (or allowed) off-axis-angles for partly absorbing or opaque materials with the evolution of the commensurate plots of the Mie's scattering calculations for growing K-values (ranging from 10 - 3 to 1) one has the chance to understand some of the correlations between the measured phase-Doppler-differences and the computed particle diameters. For the interpretation we use the @-d-rp-dependenciesas given in Figure 16, which have been plotted (like that in Figure 11) using O
Part. Part. Syst. Charact. 5 (1988) 66-71
70
the programme of Wiscombe 1211, after some modifications of [19] and especially those done by [22]. With growing K-values from 0.001 to 0.01 and 1.0 we recognize a development of ,,inclined planes of reflection with @ > 0" star, 90 deg. ( K = 0.001) and covering ting at off-axis-angles, q ~ near a region at first of large off-axis-angles (up to 180 deg. for K = 0.01) and then of smaller angles (80 deg. and less for K = 0.01 to 1.0). As already mentioned above one has to exclude from these findings the regions of small diameters because these small particles still seem to behave like transparent (dense) particles.
II C
.
I
4 Analysis of Absorbing Particles As already discussed the curves @ and 0 in Figure 15, i. e. the phase-Doppler-differences of strongly absorbing particles, follow completely the dependency which can be described by the simplified Eq. (3), i. e. by the law of geometrical optics. To these results we can now append the finding from Figure 16, that also spheres with K-values of 0.01 (with the only exception of very small diameters) may be treated in the same way, i.e. as totally absorbing particles - if we use off-axis-angles, q ~ ,exceeding values of about 80 deg. The ,,inclined planes of reflection" deliver unambigeous linear @-d-dependencies without any fear of singularities, thus opening the ability to use the back scattering position for the size analysis of these types of materials. The higher the K-value and the larger the particle diameter the wider the field of allowed offaxis-angles for possible photo detector positions. The definition of absorption has to be fixed in this context to the refractive index, better: to the index of absorption, and to the particle diameter. It has to be mentioned here that the Mie's scattering calculations have been executed for the assumption of very small particles compared with the diameters of the laser beams, thus neglecting influences arising from the distribution of light intensity in the laser beams. Corrections corresponding to the real geometric conditions should be made in the next future.
5 Conclusions
d Ipml
For the successful use of the phase-Doppler method one has to distinguish between several characteristics of the particles under consideration. The particles must be spherical in shape with a smooth surface. It has to be differentiated whether the transparent particles are, in an optical sense, denser or less dense compared to the surrounding continuous phase, which, in any case, must be transparent. Less dense particles have to be treated as reflecting ones, denser particles as refracting ones. For absorbing particles (also in the optical sense) one can already choose large off-axis-angles ( q ~> 105 ") for very small indices of absorption. The higher this index and the larger the particles, the broader the region of the off-axis-angles which may be chosen. For metallic spheres with shiny surfaces, where reflection is evident, size and velocity measurements can be achieved from all off-axis directions. This holds, in particular, for back scattering angles which simplify the experimental set-up in an important manner. (In many applications access to a multiphase flow under test can be obtained from only one side of the apparatus. Also, adjustments for the optical set-up are by far easier if the transmitter and the receiver are fixed as a unit providing system
I
biameter
yo
60
120
180
off-axis ongle Q1deq.I
Fig. 16: Plots of @-d-p-dependenciesfrom Mie's calculations for different imaginary terms.
rigidity.) In dense flows of opaque or totally absorbing particles, where most of the light will be absorbed, the back scattering position may be the only successful one. But this advantage and that of a broader dynamic size range may incurr a worse resolution of the size measurements, as discussed in part 1 of this paper. The Mie's scattering calculations may be absolutely necessary for an LDVS-analysis of opaque materials or for particles with a small differences in refractive index of the dispersed and continuous phases. If one observes the restriction in the choice of the allowed offaxis-angles in spite of partly absorbing, opaque or fully transparent - but dense particles the simplified laws of geometrical optics describe the correlations between the measured phaseDoppler-difference and the particle diameter in a correct manner down to particle diameters less than 1 pm.
71
Part. Part. Syst. Charact. 5 (1988) 66-71
6 Acknowledgement The authors gratefully acknowledge the financial support for parts of this work provided by the Bundesminister fur Forschung und Technologie, Bonn.
7 Symbols and Abbreviations term in Eq. (2) diameter of a spherical particle refractive index refractive index of the continuous phase refractive index of the dispersed phase efficiency of reflection angle of reflection factor of absorption phase difference (observed) wavelength of laser light in vaccum elevation angle (see Figure 3)
v 0
off-axis-angle (see Figure 3) intersecting angle (see Figure 1)
8 References [17] S. A . AI-Chalabi, E Hardalupas, A. R. Jones, A . M. Taylor: Calculation of Calibration Curves for the Phase-Doppler Technique : Comparison Between Mie-Theory and Geometrical Optics. Proc. Int. Symp. on Optical Particle Sizing: Theory and Practice, Rouen, France, 12-15 May 1987. [18] H. C. van de Hulsf: Light scattering by small particles. Wiley, New York 1957. [19] M. Saffman: Personal information, dated 25 June 1985. [20] Bergmann-Schafer: Lehrbuch der Experimentalphysik, Bd. 111 Optik. W. de Gruyter Verlag, Berlin 1974. [21] FK L Wiscombe: Mie-scattering calculations: advances in technique and fast vector-speed computer codes. NCR-report TN-140STR, June 1974. [22] H.-H. FlOgel: Modified Laser-Doppler anemometry for simultaneous measurements of size and velocity of single particles. Ph. D. Thesis, University of Bremen, 1987.
