3.4.2 Analysis of the sampling behavior of a phase Doppler interferometer

Grace L. Yang, Stefan Leigh, Alan Heckert
Statistical Engineering Division, ITL

John F Widmann, S. Rao Charagundla, Cary Presser
Process Measurements Division, CSTL

Since its introduction, phase Doppler interferometry (PDI) has been used to characterize sprays in areas such as liquid fuel spray combustion, coatings, pesticides, fire suppression, and others. PDI can measure the size and velocity of droplets in a spray that passes through a given location (called a probe volume) and thereby characterize a spray process. However, the data obtained from the PDI contains gaps due to recurring periods of dead time during which the phase Doppler interferometer is inactive. The dead time introduces a bias in the data analysis and contributes to an inaccuracy in measuring volume flow rates which has been reported in the literature.

In this project, a PDI was used to measure the intensity (the expected number of droplets per unit time) of a swirling methanol spray flame. A statistical model was constructed to correct the dead time effect. Excellent agreement between simulation and experimental data demonstrate the spray process can be adequately modeled by a homogeneous Poisson process, $N = \{N(t), t \in (0, T]\}$ where N(t) denotes the number of droplets recorded by PDI during the time interval (0,t] for $t \in (0, T]$.

The simulation utilizes a characterization of the homogeneous Poisson process, N, by its interarrival times. Namely, the interarrival times $W_j, j = 1, 2, \ldots$ of the droplets are assumed to be independently distributed with a common exponential distribution having probability density function

$$f(x) \ = \ \lambda \exp(-\lambda x) \ \ \mbox{for} \ x \geq 0$$

where $\lambda$ is the intensity of the droplets.

Experimental counts of interarrival times for three locations in the spray (radial coordinates, r = 12.7 mm, 16.5 mm, and 20.3 mm) at an axial coordinate z = 35 mm downstream from the spray nozzle exit exhibit significant modulation, and are obviously inconsistent with an exponential model.

To elucidate the cause of the observed modulation, we developed a model to simulate the sampling process in the spray under the hypothesis that the modulation is due to the periods of dead time in the PDI. A random variable was introduced to model periods of dead time. Simulation of the resulting interarrival times exhibits excellent agreement between the experimental data and the model, as shown in the Figure.

The results will appear in a forthcoming Proceedings of the 37th AIAA Aerospace Sciences meeting

Figure 25: Comparison of experimental interarrival distribution with simulation.

       

Date created: 7/20/2001
Last updated: 7/20/2001
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