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 >δ) 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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