# Effect of Solid Particles on Droplet Size Applying the Time-Shift Method for Spray Investigation
## 2 Theoretical Background
Additionally, as the fuel film thickness increased, the droplet size also increased significantly. Continuing this series of experiments in a later study, [PERSON] et al. [22] present a correlation for the integral Sauter mean diameter (\(Id_{32}\)) calculation, depending beside the previously used operating conditions also on the parameters of the power law expression for the inclusion of non-Newtonian flow behavior of CWS. In addition to laser diffraction, a patternator for radial mass flux measurement was used and detected a reduction in the spray cone angle in conjunction with a decrease in the \(GLR\). In 2001, [PERSON] et al. [23] started a systematic investigation of model suspensions using a laser diffraction system for droplet size measurement and a high-speed camera for the detection of primary breakup. The researchers employed a gas-assisted nozzle with a central liquid jet surrounded by an angled concentric annular gap for the gas phase. The non-Newtonian suspensions consisted of water and buoyant polymer particles with undefined shapes, \(D_{P}=10,30,50,100\)\(\upmu\)m and \(\phi_{P}=10\)-50 wt-%. For decreasing \(\dot{\gamma}\) and \(D_{P}\), as well as increasing \(\phi_{P}\), an increase in the \(\eta_{liq}\) was detected. A droplet size measurement at \(z=500\) mm downstream of the nozzle orifice yielded a bimodal droplet size distribution for \(D_{P}>50\)\(\upmu\)m and a monomodal droplet size distribution for \(D_{P}<50\)\(\upmu\)m. For the bimodal distribution (also called the solid-liquid separation), a liquid peak was observed, which is shifted towards smaller droplet diameters with increased gas velocity, whereas the solid peak remained constant and represented the original particle size. In contrast to pure liquids, suspensions have phase boundaries in which solid particles and the liquid phase interact with each other. A capillary force description is used [23], postulating that the disintegration of suspension droplets must overcome the force of surface tension, as well as the capillary force of liquid bridges holding the solid particle agglomerates together. This effect is more pronounced for constant mass fractions and decreasing particle sizes due to the higher number density (and therefore particle surfaces) in the suspension. In a subsequent study, [PERSON] et al. [24] performed additional experiments with spherical glass particles at \(D_{P}<50\)\(\upmu\)m and water, glycerol/water and carboxymethyl cellulose/water. The increase in droplet size for the suspension versus the pure continuous phase was explained by the agglomerate strength, assuming \(We_{crit}=1.18\) for the secondary atomization of pure liquids, according to [PERSON] et al. [25], which resulted in a disintegration diagram. To estimate the critical breakup conditions of suspension droplets, the tensile strength for the pendular state is balanced by the aerodynamic pressure of the gas phase (see Equation (5), below, and [PERSON] et al. [26; 27]). This estimation can be applied to high solid mass fractions in suspension droplets. \[\frac{9}{4}\cdot\frac{\frac{\rho_{gas}}{\rho_{P}}}{1-\frac{\rho_{gas}}{\rho_{P }}}\cdot\frac{\sigma}{D_{P}}=\frac{\rho_{gas}\cdot v_{gas}^{2}}{2} \tag{5}\]  
Here, the index \(sus\) represents the suspension. In a further publication, [PERSON] et al. [26] presented the atomization of a non-Newtonian kaolin/water suspension. To summarize this work, the critical particle size for the occurrence of a solid-liquid separation was set to \(d_{32}\approx D_{P}\) and a correlation for the determination of the droplet size was presented with the following dependencies:  
\[Id_{32}=f\left(D_{liq},Oh,\frac{1}{We_{aero}},\frac{1}{GLR}\right) \tag{6}\]  
In 2012, [PERSON] et al. [20] presented a morphological classification of the primary atomization of a CWS jet, expanding the earlier investigations of [PERSON] et al. [26] in this area. Eight slurries with varying particle size distributions and three different added stabilizers were applied.