9.3. Two-phase Flow and Principles of Particle Sizing 281
the lateral spherical aberration obeys the relation δ
s
= 11. 1 ×10
3
r
2
, ﬁnd
how to change the beam separation in one of the channels.
9.8. A 3-D LDV system is operated with two wavelengths, λ
1
= 0. 488 μm and
λ
2
= 0. 514 μm, with P-polarization in two of the branches and wavelength
λ = 0. 514 μm and S-polarization in the third branch. The angle between P and
S branches is α = 25
. The system comprises three identical beam splitters, with
beam separation l
1
= l
2
= l
3
= 50 mm, and two lenses of focal length 1 m each.
RF frequencies measured in the ﬁrst branch are f
1
= 5. 0 MHz, f
2
= 3. 0 MHz and
in the second branch f
3
= 2. 85 MHz. Calculate the magnitude of the velocity in
the probe volume of the ﬂow.
9.3. Two-phase Flow and Principles of Particle Sizing
Two-phase ﬂow is usually a mixture of a gas with liquid or solid particles or a liquid
where solid particles or gas bubbles are present. In many practical applications
the measurement of the velocity proﬁle of a two-phase ﬂow is accompanied by
measurement of the statistics of particles with regard to their size. Numerous
methods of particle sizing have been known for many years. Here we mention
only those which are related to the LDV technique described above.
If a particle moving with a ﬂow is much smaller than the fringe spacing δ in the
probe volume the signal burst originating from the scattering of light by the particle
has the shape presented in Fig. 9.3. However, if the particle size approaches δ or is
even greater then the situation is different: the minima of the signal burst cannot
approach zero, even if two interfering beams have equal intensities, since at any
moment some portion of the particle is illuminated by light of the fringe maxima.
As a result, the LDV burst becomes an oscillating function with two envelopes:
the ﬁrst related to the maxima and the second related to the minima. This situation
is demonstrated in Fig. 9.13 where a large particle (d ) is shown at three
sequential moments when it moves through the fringes. The larger the particle the
smaller the difference between the upper and lower envelopes of the signal burst
(both envelopes are characterized by their amplitudes, I
max
and I
min
, shown in
Fig. 9.13b).
Of course, the amount of radiation energy scattered by the particle and collected
by the receiving optics is strongly dependent on the particle size, so that the
amplitude I
max
can be used as a measure of the particle diameter. The dependence
I
max
= F(d) can be described by a square power law
I
max
= Cd
2
(9.12)
282 9 Optical Systems for Flow Parameter Measurement
a) b)
Figure 9.13 (a)Alarge particle moving through a probe volume and (b) the corresponding
signal burst.
and this simple formula remains valid over a wide range of particle sizes (from
several micrometers to tenths of millimeters). The constant C, however, depends
on the parameters of the measurement system (like laser power, detector sensitiv-
ity, collecting optics conﬁguration, etc.) and also on the location of the particle
trajectory inside the probe volume (see Problem 9.9). All this causes difﬁculties
in exploiting Eq. (9.12) in practice. Naturally normalized values independent as
much as possible of optical conﬁguration would be much more convenient for
practical applications. A method widely used is based on the measurement of vis-
ibility function, V , deﬁned as the ratio of the AC to DC components of the signal
burst generated by a particle. In terms of I
max
and I
min
shown in Fig. 9.13 the
visibility function can be described as
V =
I
max
I
min
I
max
+ I
min
. (9.13)
For a spherical particle traveling through an ideal fringe pattern the visibility
function can be approximately expressed in terms of a Bessel function of the ﬁrst
order:
V = 2J
1
(ka)/ka (9.14)
where a is the radius of the particle and k = 2π/δ. The corresponding graph is
presented in Fig. 9.14. For any registered signal burst generated by the studied
particle the values I
max
and I
min
are measured and visibility V is calculated from
Eq. (9.13). Then, using Eq. (9.14) or the graph of Fig. 9.14, the corresponding
value of the parameter p = d/δ is found and the particle diameter d = pδ is easily
calculated if the fringe spacing δ is known. As can be seen from Fig. 9.14, the
fringe spacing should be appropriately chosen in order to ensure that visibility is
in the range 1. 0 < V < 0. 15, where V is a monotonic function of p and where

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