# A Review of Experimental Techniques for Measuring Micro- to Nano-Particle-Laden Gas Flows
## 3 Specific Measurement Methods
### Optical Particle Characterization
#### 3.1.2 Laser Doppler Anemometry and the Phase Doppler Analyzer
The frequency shift \(f_{D}\) resulting from the scattered lights, which are obtained from two incident beams, can be expressed as:  
\[f_{D}=f_{r1}-f_{r2} \tag{7}\]  
Using Equation (6) and the velocity components in the directions of the two laser beams with, \(|\overrightarrow{v}|\ll c\) Equation (5) is obtained again [1]. LDA has been employed to measure two-phase particle-laden gas flows for several decades. As the dispersed particles obviously need to be larger than the tracer particles, the particle scattering intensity and Doppler pulses generated also are much stronger; thus, it is very easy to measure particle velocity using LDA. In contrast, the Doppler pulses generated by the tracers are much weaker, and a phase-discrimination of the Doppler signal must be carried out to ensure the reliability of two-phase measurements. In the continuum regime, LDA seems to be the most viable option at present, given that most dispersed particles have a size range of 10 to 1000 \(\upmu\)m, and tracer particles are approximately 1 \(\upmu\)m sized. The PDA was first proposed in 1981 [25; 26; 27; 28] as a derivative of LDA. The PDA is able to simultaneously measure spherical or quasi-spherical particle velocity and size. It has been developed into a very reliable technique for these quantities. In LDA measurements, if several detectors are used to examine the Doppler shift, then the shift signals simultaneously acquired by these detectors are similar, although there are phase-differences between them that are proportional to particle size, as shown in Figure 7. Taking into consideration only first-order refraction, the correlation between the phase-difference and particle size is equal to:  
\[\Delta\Phi^{(p=2)}\approx-2\frac{2\pi}{\lambda}d_{p}\frac{m\sin\psi\sin \
icefrac{{\theta}}{{2}}}{v\sqrt{1+m^{2}-mv}} \tag{8}\]  
where \(\Delta\Phi^{(p=2)}\) is the phase-difference between fringes on the two detectors, placed at elevation angles \(\pm\psi\) and off-axis scattering angles of \(\pm\varphi\); \(m\) is the ratio (relative refractive index) of refractive indexes between the particle and the carrier phase; and \(v\) is a parameter function, described by:  
\[d=\frac{\lambda}{2\sin\frac{\theta}{2}} \tag{3}\]0  
A proportional relationship between the measured phase difference and particle diameter was found to be affected by the proportionality constant, involving geometric quantities [8]. Thus, the PDA technique is usually considered to be calibration free and in situ traceable, although its overall accuracy depends on the precision of the angles, as seen in Equation (8). The PDA is most suitable for measuring liquid and transparent droplet diameter and velocity, because first-order refractive light is often the target signal in PDA measurements. However, spherical transparent solid particles also can be measured using PDA, and even opaque spherical particles can be discriminated, using reflective light as the target signal. Phase-Doppler techniques use a full-featured instrument that detects particles down to 300 nm in diameter (based on the detector sensitivity), for concentrations ranging from \(10^{4}\) to \(10^{5}\) particles/cm\({}^{3}\). When particle size is larger than the measurement volume, it is difficult to discriminate first-order refractive light from reflective light, because of mixing. Hence, any determination of particle size directly calculated from the Doppler phase difference would result in significant error. [PERSON] (1986) first observed this phenomenon, although this Gaussian beam effect/defect has been studied by other authors [PERSON] (1986); [PERSON] (1986); [PERSON] (1986). A number of studies have focused on reducing this error to increase the viable particle concentration range [PERSON] (1986); [PERSON] (1986).