# An Approach to Determine the Median Diameter of Droplets in a Water-Mist Spray
## 2 Methodology  
Firstly, an experiment was conducted where a water-mist nozzle was operated at a particular pressure (\(P_{1}\)). The water of the spray was collected on a set of water collection trays on the floor. The distribution of flux density of water droplets produced by a water-mist nozzle was quantified for the particular flow pressure (\(P_{1}\)) in the experiment. The flow rate of water and the angle of the spray was also measured in the experiment. After that, a numerical tool, FDS, was used to simulate the experiment. A number of simulations were performed with a range of droplet sizes, and the distribution of flux density of water droplets for each size of droplet was calculated. Subsequently, the numerical results were compared with the experimental data. When the numerical distribution data were matched with the experimental data, the corresponding droplet size could be taken as the hypothetical median size of droplets (\(dm_{1}\)) of the spray. This droplet size was considered to correspond to reality. Therefore, it was proposed that a mathematical relationship could be used to determine the second median diameter of droplets (\(dm_{2}\)) for a different flow pressure (\(P_{2}\)). If the numerical tool can accurately simulate the second set of flux distribution of water droplets using \(dm_{2}\) and \(P_{2}\), then we can be confident that the numerical tool is well-validated. Therefore, \(dm_{1}\) and \(dm_{2}\) can be considered as the actual droplet sizes of sprays corresponding to the \(P_{1}\) and \(P_{2}\), respectively.  
According to [PERSON] (2016), the median droplet diameter of a spray has been empirically found to be inversely proportional to the one-third power of water pressure and directly proportional to the two-third power of the diameter of the nozzle orifice, i.e.,  
\[d_{m}\propto\frac{D^{2/3}}{P^{1/3}}. \tag{1}\]  
Here \(dm\) is the median size of droplets, \(D\) is the diameter of the orifice and \(P\) is the pressure of the water flow. Hence, for a particular nozzle, the correlation between the two median diameters of droplets corresponding to the two different pressures can be expressed as  
\[\frac{d_{m1}}{d_{m2}}=\left(\frac{P_{2}}{P_{1}}\right)^{1/3} \tag{2}\]  
The water flow pressure (\(P_{1}\)) had already been determined in the first experiment. The corresponding median diameter of droplets (\(dm_{1}\)) was determined from the same experiment and a set of simulations using the data of flux density distribution of water droplets. A second experiment was conducted with a different flow pressure (\(P_{2}\)), and the corresponding data of the distribution of flux density of water spray was collected. Using Equation (2), the median diameter of droplets (\(dm_{2}\)) of the spray was calculated for \(P_{2}\). The flow rate of water and the angle of the spray was also measured in the experiment. Finally, this droplet size, the corresponding pressure, and other spray parameters were used as input parameters for the spray in the numerical model. If the numerical distribution of flux density of water matches the experimental measurements, we can conclude that the numerical model can predict the distribution of flux density of water droplets. We can be confident that hypothetical median droplet sizes \(dm_{1}\) and \(dm_{2}\) are the actual median droplet diameters at pressures \(P_{1}\) and \(P_{2}\), respectively. Therefore, we can estimate the median droplet diameters corresponding to different flow pressures using this method. The execution of this methodology is presented in detail in the following sections.