# Effect of Solid Particles on Droplet Size Applying the Time-Shift Method for Spray Investigation
## 2 Theoretical Background
Most of the suspensions investigated show the effect of shear-thinning (\(\eta_{liq}=f\left(\dot{\gamma}\right)\)) due to the presence of aspherical particles and chain-like polymers that align in the shear direction [15]. This effect causes the viscosity to decrease as the shear rate increases. Due to the fact that the shear rate varies significantly within the spray cone as well as during the atomization process itself, a description of the disintegration process of non-Newtonian media is highly complex. Typically, these liquids are characterized, respectively, by zero viscosity \(\eta_{liq,0}=\eta_{liq,\left(\dot{\gamma}=0\text{ s}^{-1}\right)}\) or viscosity at an infinite shear rate \(\eta_{liq,\infty}=\eta_{liq,\left(\dot{\gamma}\rightarrow\infty\right)}\), whereas neither condition is a precise description of the shear rate that is relevant during the atomization process. [PERSON] and [PERSON] [16] therefore investigated the atomization of xanthan gum and polyacrylamide using a Phase Doppler anemometer, noting a droplet size correlation, including an \(Oh\) number, with shear dependent dynamic viscosity. [PERSON] and [PERSON] [16] calculated the shear stress limit as being \(\dot{\gamma}_{max}=\) 20,000 s\({}^{-1}\), by applying different liquids and nozzle configurations. Other authors have determined the maximum shear rate that occurs during atomization as follows: [PERSON] et al. [17] (\(\dot{\gamma}_{max}=225\) s\({}^{-1}\)), [PERSON] et al. [18] (\(\dot{\gamma}_{max}=450\) s\({}^{-1}\)), [PERSON] et al. [19] (\(\dot{\gamma}_{max}=500\) s\({}^{-1}\)) and [PERSON] et al. [20] (\(\dot{\gamma}_{max}=4.3\)-7.8 s\({}^{-1}\)). In addition to the description with fixed shear rates, [PERSON] et al. [14] propounded an approach relating to the measurement of the Kelvin-Helmholtz instability frequency of \(f_{KH}=f\left(\dot{\gamma}\right)\) in order to gain information on the effective viscosity relevant for the primary breakup. As previously noted, many different approaches for the description of non-Newtonian behavior in atomization processes have been proposed. However, the inconsistency of the chosen shear rate means that the results are typically not applicable to various media and operating conditions. The following section provides an overview of relevant literature on the measurement of suspension sprays. [PERSON] et al. [3] presented a detailed comparison of investigations of the atomization of suspensions using external-mixing twin-fluid atomizers. Typically gas velocity and \(GLR\) were varied, applying coal water slurries (CWS) featuring various shear-dependent viscosities due to particle loading, particle size distribution and stabilizing additives. Earlier investigations with pure liquids indicate that increasing the \(GLR\) and \(v_{gas}\) always leads to a decrease in the droplet size, whereas an increase in \(\dot{M}_{liq}\) results in an increase in droplet size. A detailed interpretation of the influence of particles on spray formation is not given. [PERSON] et al. [3] employed a plain-jet airblast nozzle at \(GLR=0.5\)-\(5\) and \(v_{gas}~{}=~{}60\)-\(100\) m\(\cdot\)s\({}^{-1}\). Four slurries with significantly different \(\eta_{liq,0}\) were atomized and characterized by means of laser diffraction measurements. The authors postulate that for the complete atomization of the bulk phase of CWS, \(GLR\geq 5\) is a necessary condition. [PERSON] et al. [21] conducted spray experiments using two CWSs with different particle loadings (\(\phi_{P}=60\) and \(70\) wt-%) and three fuel film thicknesses applying plain-jet atomizers at \(GLR=1\)-\(5\) and \(v_{gas}=60\)-\(80\) m\(\cdot\)s\({}^{-1}\). Again, laser diffraction was used to investigate the resulting drop sizes. These investigations revealed that the slurry with higher particle loading - despite showing higher viscosity values - disintegrated into smaller droplets under the same atomization conditions, which was explained by the shear rate dependency of the dynamic viscosity.