# Optimised Twin Fluid Atomiser Design for High-Viscosity, Shear-Thinning Fluids
## 2 Materials and Methods  
### Model Fluid and Atomising Agent Properties  
The physical and rheological properties of sewage sludge are subject to considerable variation as a result of numerous influencing factors, including temperature, solids concentration, wastewater composition, chemical additives, and preprocessing steps related to digestion and fermentation [18; 19; 22]. Consequently, it remains challenging to define universally applicable physical and rheological properties for sewage sludge. Sewage sludge exhibits non-Newtonian, shear-thinning behaviour in general [26]. At low shear rates, the viscosity remains constant, reflecting Newtonian fluid behaviour. This constant viscosity is referred to as the zero-shear viscosity. Beyond a certain critical shear rate, a progressive decline in viscosity is observed as the shear rate increases. This phenomenon, described by the apparent viscosity, results from the disruption of flocculated particle networks [27]. At sufficiently high shear rates, a limiting viscosity value is approached, beyond which the fluid again behaves in a Newtonian manner. This asymptotic viscosity value is termed the infinite-shear viscosity. When selecting a non-Newtonian shear-thinning surrogate liquid, it is essential to determine the characteristic shear rate first, encountered in the atomisation process. Ensuring comparable atomisation behaviour necessitates that the apparent viscosity of both the sludge and the surrogate liquid coincide at the characteristic shear rate. The application of unheated air at sonic speed as the atomising agent resulted in an estimated shear rate of 50,000 s\({}^{-1}\). Various suppliers provide polydimethylsiloxane (PDMS) [28; 29; 30]. According to manufacturers' data [28; 29], PDMS with a kinematic viscosity of 10,000 mm\({}^{2}\) s\({}^{-1}\) at 25 \({}^{\circ}\)C exhibits an apparent viscosity of 2 Pa s under the relevant shear rate, thereby closely matching the apparent viscosity of the target sludge under equivalent conditions. For this investigation, Wacker silicone oil type AK 10,000 was therefore selected as the surrogate liquid [28]. The chemical structure of the surrogate liquid is provided in Formula (1). \[(\mathrm{H_{3}C})_{3}\mathrm{Si-O-[Si(CH_{3})_{2}-O]_{n=500}-Si(CH_{3})_{3}} \tag{1}\]  
In addition to viscosity, both the density and surface tension of the atomised medium constitute critical parameters in atomisation. Density and surface tension substantially influence spray formation and droplet breakup, necessitating minimal deviation between model fluid and target medium [31]. The Ohnesorge number, shown in Equation (2), serves as a dimensionless quantity that captures the interplay of viscosity, density, and surface tension in atomisation processes. Here, \(\rho_{L}\) denotes the density of the spray medium, \(D\) a characteristic length scale, \(\eta_{L}\) the apparent viscosity of the spray medium, and \(\sigma\) the surface tension of the spray medium. \[Oh=\frac{\eta_{L}}{\sqrt{D\cdot\rho_{L}\cdot\sigma}}=\frac{We_{L}^{0.5}}{Re_{ L}} \tag{2}\]  
Equation (3) defines a dimensionless quantity that expresses the balance between viscous and surface tension forces opposing atomisation, where \(\Delta U\) denotes the relative velocity between the atomising agent and the spray medium. A higher value of \(We_{L}/Re_{L}\) implies an increased influence of viscous forces opposing the atomisation process [31]. \[\frac{E_{\eta}}{E_{\sigma}}=\frac{\eta_{L}\cdot\Delta U\cdot D^{2}}{\sigma \cdot D^{2}}=\frac{\eta_{L}\cdot\Delta U}{\sigma}=\frac{We_{L}}{Re_{L}} \tag{3}\]In the CWS atomisation literature, it is common practice to report the apparent viscosity of CWS at a shear rate of 100 s\({}^{-1}\)[17]. According to the manufacturer's data of the model fluid [28], the apparent viscosity \(\eta_{L}^{100}\) at a shear rate of 100 s\({}^{-1}\) corresponds to the zero-shear viscosity \(\eta_{L}^{0}\).