# An Approach to Determine the Median Diameter of Droplets in a Water-Mist Spray
## 4 Numerical Model  
Fire Dynamic Simulator (FDS), version 6, was used for the numerical modelling of the spray distribution, and the accompanying programme, Smokeview, was used to visualise the model. In FDS, a finite difference technique was applied to solve partial differential equations of conservation of mass, momentum and energy. FDS takes a sample of spherical droplets to calculate the distribution pattern. The droplet size distribution is expressed in terms of its cumulative volume fraction (_CVF_), which is represented by a combination of log-normal and Rosin-Rammler distributions [23]. \[F(d)=\left\{ \begin{array}{cc}(2\pi)^{-\frac{1}{2}}&\int\limits_{0}^{D_{CVF}} (\sigma D)^{-1}e^{\frac{\left|\alpha\right|}{2m}\right|^{2}}{2\alpha^{2}}dD&( D_{CVF}\leq D_{m})\\ \\ &1-e^{-0.693(D_{CVF}/D_{m})^{7}}&(D_{CVF}>D_{m})\end{array} \right. \tag{3}\]  
where \(D\) is the generic droplet diameter and \(D_{m}\) is the median droplet diameter. The median droplet diameter is a function of the sprinkler/nozzle orifice diameter, operating pressure and geometry. \(\gamma\) and \(\sigma\) are empirical constants used for curve-fitting of distribution patterns, and they are equal to 2.4 and 0.6, respectively. In FDS, water droplet transport is modelled by a Lagrangian approach. The velocity and position of the droplets are calculated from the theory of conservation of momentum. The position and velocity of each droplet are computed from the following equations. \[\frac{d}{dt}\left(mv_{p}\right)=mg-\frac{1}{2}\rho C_{d}\pi r^{2}\left(v_{p}-v _{a}\right)\left|v_{p}-v_{a}\right| \tag{4}\]  
\[\frac{dx_{p}}{dt}=v_{p} \tag{5}\]  
Figure 1: Schematic of the setup of the experiment. Figure 2: Spray nozzle and measurement of the angle of spray produced by the nozzle: (**a**) schematic of the single-orfice nozzle, (**b**) photograph of the spray, (**c**) schematic of measurement of the spray angle. where the drag coefficient, \(C_{d}\), depends on the following function of the Reynolds number (\(Re\)) based on the droplet-air relative velocity. \[C_{d}=\left\{ \begin{array}{cc}24/Re&Re<1\\ 24\left(0.85+0.15 Re^{0.687}\right)/Re&1<Re<1000\\ 0.44&Re>1000\end{array} \right. \tag{6}\]  
The \(Re\) of the droplets is defined by  
\[Re=\frac{\rho|v_{d}-v_{a}|\ 2r}{\mu(T)} \tag{7}\]  
where \(\mu(T)\) is the dynamic viscosity of air at temperature \(T\). Details of these models are available in the FDS technical guide [23]. It is worthy to mention that FDS is well-validated to predict the distribution of flux density when the median droplet size of a spray is known [3]. A computational domain with dimensions of 2 m \(\times\) 2 m \(\times\) 2.4 m was generated to represent the experimental setup. The domain setup of the computational model is shown in Figure 3. The green dots on the floor signify the points where the accumulation rate of water was calculated in the simulation. The floor was divided into 400 compartments, each 10 cm \(\times\) 10 cm in the XY plane so that mass flux distribution could be established similar to the experiment. The numerical outcomes of a model are sensitive to the size of the mesh of the computational domain [24]. As the grid size of a mesh was refined, it was expected that the results would approach that of the continuous governing equations. Therefore, a grid sensitivity analysis was performed to evaluate the convergence of the numerical results. The cubic grid sizes of 10 cm, 5 cm and 2.5 cm (all same in the x, y and z directions) were tested. The distribution of flux density of the spray for all of the grid sizes was calculated and compared. It was found that the distribution of flux density for 2.5 cm and 5 cm cell sizes were almost identical. Therefore, a mesh of 5 cm cell size was used to discretise the computational domain.