# Effect of Solid Particles on Droplet Size Applying the Time-Shift Method for Spray Investigation
## 2 Theoretical Background  
Various theoretical and experimental investigations have addressed the atomization of low viscosity liquids by using external-mixing twin-fluid atomizers under atmospheric system pressure [6; 7; 8]. [PERSON] et al. [9] provide an overview that compares different twin-fluid nozzle concepts. [PERSON] and [PERSON] [10] used a high-speed camera and twin-fluid nozzles to investigate the primary breakup of a water jet. \(Re_{liq}\) and \(We_{aro}\), in Equations (1) and (2), describe the process conditions with a regime classification for liquid breakup morphology through different nozzle geometries:  
\[Re_{liq}=\frac{D_{liq}\cdot v_{liq}\cdot\rho_{liq}}{\eta_{liq}} \tag{1}\]  
\[We_{aero}=\frac{\left(v_{gas}-v_{liq}\right)^{2}\cdot\rho_{liq}\cdot D_{liq}}{ \sigma} \tag{2}\]  
with the liquid jet diameter (\(D_{liq}\)), velocity (\(v\)), density (\(\rho\)), dynamic viscosity (\(\eta_{liq}\)) and surface tension (\(\sigma\)) as relevant process parameters. The \(gas\) and \(liq\) subscripts denote the gas and liquid phases, respectively. A classification was introduced that includes a Rayleigh type, membrane type, and fiber type breakup with two sub-modes, namely pulsating and superpulsating [10]. When \(We_{aero}<25\), the primary atomization is characterized by the so-called Rayleigh-type regime in which the liquid jet disintegrates into large droplets close to the centerline of the spray. When \(We_{aero}\) increases up to 70, the membrane-type breakup is reached, and gas-filled membranes near the nozzle orifice can be detected. These membranes finally rupture into tiny droplets at the thinnest sheet structure, whereas the liquid accumulated in the rims disintegrates due to Rayleigh-Plateau instabilities. For \(We_{aero}>100\), the fiber-type regime leads to a complete disintegration of the liquid jet into fibers close to the nozzle orifice. These ligaments then typically disintegrate into small droplets in accordance to the Rayleigh-Plateau instability [10]. The fiber-type regime is divided into the pulsating and superpulsating sub-modes, the latter of which is characterized by a fluctuation in the number density of droplets in the spray. [PERSON] and [PERSON] [11] included the effect of the dynamic pressure ratio (momentum flux ratio) of the gas and liquid phases \(j\), as given in Equation (3), for the characterization of the fiber-type breakup:  
\[j=\frac{j_{gas}}{j_{liq}}=\frac{v_{gas}^{2}\cdot\rho_{gas}}{v_{liq}^{2}\cdot \rho_{liq}} \tag{3}\]  
[PERSON] and [PERSON] [11] held the nozzle geometry and gas density constant in their study, with the result that the primary breakup length and spray angle were solely assigned to the momentum flux ratio and were only affected by the gas phase velocity. To consider the effect of liquid viscosity on the resulting primary breakup, the Ohnesorge number \(Oh\), Equation (4), is typically applied:  
\[Oh=\frac{\eta_{liq}}{\sqrt{\sigma\cdot\rho_{liq}\cdot D_{liq}}} \tag{4}\]  
The Ohnesorge number describes the relation between viscous forces and the square root of the inertia and surface tension forces. Numerous studies have investigated the effect of increased viscosity on spray characteristics. [PERSON] et al. [12] provide an overview of the most relevant papers that investigated liquid viscosities of \(\eta_{liq}>50\) mPa\(\cdot\)s. Typically, an increase in the droplet size is detected in conjunction with increasing dynamic viscosity. [PERSON] et al. [13] investigated a wide viscosity range of \(\eta_{liq}=1-805\) mPa\(\cdot\)s. Furthermore,as the viscosity increases, an increase in the primary breakup length is detected, as well as a pronounced membrane-type breakup over a wide range of \(We_{aero}\)[14]. It should be noted that these dependencies only refer to Newtonian liquids, whereas suspensions in industrial contexts often reveal non-Newtonian flow behavior.