Métodos

Flow diagnostics such as LSI/PIV, backlighting and holography rely on light scattered from seed particles. In many cases, the particles are naturally present in the flow, i.e. diesel sprays and pulverised coal furnaces. However, where naturally occurring particles are not present, or are unsuitable, artificial particles need to be added to the flow and they must be carefully chosen. The particles used are selected according to the amount of scattered light required, and the characteristics of the flow itself. The intensity and directional dependence of the light scattered by an illuminated seed particle is a function of the shape, size and refractive index of the particle, and the surrounding medium. The equations relating scattering intensity to these parameters are derived from Mie Theory. This is discussed in depth by van de Hulst [45]. Mic theory

Mie theory tells us that when a particle is illuminated by light of wavelength. A, and intensity,

/», the light intensity which reaches a point at a distance, R, and in the direction, 6,<p, from the

particle is,

(14)

where the wave-number, k = —

and .S' is the scattering matrix for the particle,

.S’ = ₍₁₅₎

Each clement of .S is a complex function of the particle dimensions as well as of 0and <p. For

When the particle is detected by a physical sensor, the scattered light is collected over a solid angle, and the total power scattered by the particle is given by,

Ps = /?2J / S(0,0)ci2 (16)

where d£2 is the solid angle,

d£2= sinddftd </>

The result of this integration is,

(17)

P, = / o Q ( 0 . 0 )

where Cs is called the scattering cross-section.

(18)

For small spherical particles, where the diameter is much smaller than the wavelength of the

light < 0.1 j, the full Mie equation can be simplified and Rayleigh theory then applies.

For large particles (>5 pm), the scattered light can be considered as coming from a large

evenly illuminated disk, and so Is « d 2 (the so-called geometric scattering range). Scattering

is more or less independent of wavelength, unless the particles themselves are large and selectively absorbing.

For particles with sizes in between the Rayleigh and geometric ranges, the relationship is more complex, and the full Mie theory has to be used to calculate scattering intensity. This is the transitional scattering range. Equation (14) cannot be solved analytically, but has to instead be approximated numerically. This can be performed to any desired level of accuracy using a digital computer, and has been computed by a number of workers. Adrian and Yao [46] show

that the transitional behaviour occurs for particles with sizes approximately in the 1 - 10

Seed particle behaviour in a flow

Another important characteristic of a seed particle is the degree to which it follows the flow medium of interest. Only the shadowgraphy, Schlieren and holographic interferometry

techniques measure the flow medium - the other techniques are measuring the seed particles. It is therefore necessary to check that fluid and particle velocities are similar. There are two main causes of differences in seed and medium velocities.

If the particle density and fluid density are different, this causes a gravitationally induced velocity that can be calculated from an appropriate drag law. The other main difference results from inertial delay when the particle is accelerated. Hjemfelt and Mockros [47] give an analysis of the other (usually smaller) factors affecting particle velocity.

The equation of motion for a spherical particle relative to an infinite fluid was first derived by Basset[48] by balancing the acceleration forces on the particle and a stationary displaced fluid. This can be converted to a form applying to a moving fluid by considering the instantaneous velocity of the particle relative to the fluid[49]:

where p, p h are the particle and fluid densities,

H is the fluid dynamic viscosity,

v = uk - u is the velocity difference between fluid and particle.

The four terms on the right hand side of equation (19) represent respectively the Stokes drag force, the pressure gradient force on the fluid, the fluid resistance to an accelerating sphere, and the drag force associated with unsteady motion. Equation (19) is valid only within the following assumptions:

• homogeneous, time-invariant turbulence • particles smaller than turbulence micro-scale • Stake’s drag law applies (spherical particles)

• particle is always surrounded by the same fluid molecules • no interaction between particles

Now, for particles suspended in gases, = 103, and provided that the Stokes number.

<20>

where twis the frequency of the turbulent fluctuations, is greater than about 8 [50], then

equation (19) reduces to,

du d7

where is the fluid kinematic viscosity.

(21)

This equation assumes a pure drag force obeying Stoke’s law. This can now be solved giving.

u = u t. + ( u ( 0 ) - u F)e (22) where

r = ad2

18u (23)

Here r is the response (relaxation) time of the seed particle to changes in the fluid velocity. If the fluid turbulent fluctuations are considered sinusoidal, then the frequency response is given by [47],

where 77 is the particle oscillation amplitude divided by that of the fluid. If a criterion for

particles following the flow of 77 > 0.95 is set, then particles must satisfy

,2 0.94V

d < ---

A useful concept for selecting seed particles is the aerodynamic diameter, defined as cl2 = ad 2. This is the equivalent diameter of a particle, were it to have unit density (1000 kgnr3).

Particles with equivalent aerodynamic diameter respond equally well to fluctuations in fluid velocity.

For air flows at ambient conditions, cr= 833 and v = 1.5 x 10'5 m2/s. So, if it is required that particles respond to fluctuations of up to I kHz, equation (25) indicates that they must have an aerodynamic diameter of less than 4.1 microns. Particle sizes for liquid flows can be

considerably larger.

Limitations of laser intensity, particularly when expanded into a large sheet, mean that particles larger than the ideal size often need to be used, at the expense of some loss of flow tracking (apparent as a particle lag).

Choice of seed particle

For optimum results, seed particles should ideally be: • small (to follow the flow accurately)

• mono-disperse (all particles to behave identically) • spherical (to scatter light independently of orientation)

• stable - both physically and chemically (for use at high temperatures and in reactive environments)

• low density relative to fluid (to minimise any settling out) • highly reflective - white preferably (for maximum light scatter) • non hazardous (potential inhalation)

• cheap (large flows may require considerable quantities of seed)

In practice, no seed material has all of these properties, and the choice for a particular flow application is a compromise. The seeding requirements for global diagnostics are similar to those of LDA, and particles intended for this are often employed. Many different materials are

available, including T i02, smoke, oil drops, BN, He bubbles, glass beads (micro-balloons), flour, milk and latex spheres. All of these materials have been successfully used in flow measurement.