Pneumatic Atomization: Beam-Steering Correction in Laser Diffraction Measurements of Spray Droplet Size Distributions  
[PERSON]  
1 Karlsruhe Institute of Technology, Institute of Process Engineering in Life Sciences, Chair of Food Process Engineering, Kaisersstrasse 12, 76131 Karlsruhe, Germany; [EMAIL_ADDRESS] (H.P.K. ); [EMAIL_ADDRESS] (V.G.) 1 Karlsruhe Institute of Technology, Institute of Process Engineering in Life Sciences, Chair of Food Process Engineering, Kaisersstrasse 12, 76131 Karlsruhe, Germany; heike. In laser diffraction measurements of sprays, the light of a laser beam is partly diffracted by droplets passing the beam. The diffraction caused by a single droplet is dependent on its size [8]. Small droplets diffract the light in larger angles, while larger droplets diffract the light in smaller angles. When a number of droplets pass the laser beam at the same time, a far-field Fraunhofer interference pattern of concentric rings is formed in the forward direction around the laser beam axis. A Fourier transform lens is used to focus the interference pattern onto a series of light-sensitive detectors, placed in defined distances around the central laser beam. Based on Fraunhofer diffraction theory, the droplet size distribution can be determined from the angular distribution of light intensity across these detectors [2]. In early laser diffraction instruments, each detector was directly associated with a specific droplet diameter interval. In modern laser diffraction spectroscopes, mathematical inversion procedures are used to calculate the volume-based droplet size distributions [3,7]. For this purpose, prediction models for the correlation between droplet size distributions and measured light energy distributions are used. These prediction models are based on mathematical probability functions to describe the relative frequency of dedicated droplet sizes within a distribution. However, each manufacturer uses different procedures and assumptions, which are considered as \"know-how\" and therefore are not made available to the user [7]. Nevertheless, the basic principles are the same in all cases. Besides Rosin-Rammler and Nukiyama-Tansawa distributions, log-normal distributions are the most common distribution functions used to describe droplet size distributions of spray measurements [9]. In a log-normal distribution, the logarithm of the droplet size \(\ln(x)\) is normally distributed and can be described by the following function (1):  
\[q(\ln(x))\ =\ \frac{1}{x\cdot\sigma_{In}\cdot\sqrt{2\pi}}exp\left\{-\frac{( \ln(x)-\ln(x_{50}))^{2}}{2\sigma_{In}{}^{2}}\right\} \tag{1}\]  
In this function, \(x\) is the droplet size, \(x_{50}\) is the median of the distribution and \(\sigma_{In}\) is the logarithmic standard deviation. Multidisperse distributions, consisting of \(k\) modalities, can be described by the summation of \(k\) single log-normal distribution functions [10] according to Equation (2). \[f(\ln(x))\ =\ \sum_{1}^{k}A_{k}\cdot q_{k}(\ln(x)) \tag{2}\]  
Here, \(A_{k}\) is the relative fraction of modality \(k\), following the additional condition below (3). \[\sum_{1}^{k}A_{k}\ =\ 1 \tag{3}\]  
Recent instruments also include the Lorenz-Mie theory, which takes the contribution of small droplets to the angular light energy distribution by refraction into account [3]. When this contribution of light refraction is not considered and the corresponding portion of light is interpreted as diffracted light, small droplets are overestimated in the calculated size distribution. Besides this, other systematic errors can affect laser diffraction measurements of liquid sprays, i.e., multiple scattering and beam steering. Multiple scattering can occur in optically dense sprays with transmission rates below 40% [3]. In this case, the laser beam may be diffracted by more than one particle before detection, leading to measurement errors. Beam steering can occur, when the refractive index in the gas phase is not constant. Local differences in the refractive index can be caused by different reasons.