Optical measurement of particle size distribution function using downhill simplex method : theory and experiment

A new method is introduced for retrieving the particle size distribution function (PSDF) from optical measurements. Criteria are derived and demonstrated for assessing the validity of the multi-dimensional least-squares deconvolution results derived from experimental data. Using the covariance matrices C and the newly revealed B, stability of this deconvolution technique is discussed and compared with the linear inversion method. The relations of the variances of the PSDF parameters and the experimental errors have been derived and the effects and restrictions of the profiles for the PSDF have been discussed. All the theoretical predictions of this method are tested using Monte Carlo-simulated experimental data which consists of scattering and extinction measurements for water and soot particles, respectively. The results show that for water droplets of 1μm mean diameter, the method is good for ±25% random experiment error for scattering and ±6% for extinction. For the smaller soot particles 0.05μm mean diameter, the corresponding maximum error levels are ±10%,/i> and ±6%, respectively. These differences in the scattering and extinction results are explained using information content analysis. Experimental confirmation of these newly developed deconvolution technique was accomplished using multi-angle scattering measurements of a water droplet spray field. Sensitivity of the deconvolution to uncertainties in the index of refraction was determined using droplet compositions of water and laser-dye mixtures. The laser-dye concentration was varied to provide a variable and known imaginary term of the index of refraction. A range of discrete laser wavelengths was used for this study to span the wavelength range of the variations of the index of refraction. The application of this deconvolution method yield, in addition to the PSDF parameters, the real and imaginary terms of the index of refraction which agreed with the computed values. A method was developed and verified for the proper treatment of the variation with observation angle of the scattering volume of the experiment, and the results were compared with those results using other correction methods for the experimental data of various optical configurations. Finally, the cause of the variation of the mean droplet size versus the concentration of laser-dye in the solution was investigated.