## 2 Materials and Methods
### Simple Theoretical Considerations
Consequently, the empirical outcomes enabled the estimation of the stopping distances of droplets and coefficients of both the gas and the droplet diffusion, Prandtl's mixing path, kinematic turbulent viscosity coefficient (turbulent diffusion), dynamic turbulent viscosity, and physical properties of both phases, among others. Kinematic turbulent viscositiyis not a physical property of the fluid but rather a flow characteristic that is influenced by the nature of turbulence. This knowledge is the basis for determining which phenomena impact the droplet motion and the sizes of droplets that will be more likely to undergo these physical circumstances. The parameters abovementioned can also be estimated analytically; however, in this approach, multiple simplifications need to be applied. The results obtained theoretically can be improved in both quantitative and qualitative terms using empirical data instead of simplified analytical correlations in a mathematical model. In addition, while maintaining the similarity of flow and geometric conditions of the annular-mist flow through the rectangular channel, appropriate empirical correlations of a general nature with a wider application can be developed. This task is planned to be carried out in the future. The thickness of the boundary layer, \(\delta_{bl}\), and the local mean velocity gradient, \(\frac{d\bar{u}}{dx}\),were estimated empirically (see Appendix A); hence, the values of the Prandtl's mixing path, \(l_{m}\) (Equation (1), [8]) and the wall shear stresses, \(\tau\) (Equation (2), [8]), could be estimated:  
\[l_{m}=\kappa\cdot\delta_{bl} \tag{1}\]  
The mixing path is a theoretical distance where the fluid elements exchange momentum completely due to turbulent stresses. [PERSON] regarded it as a representation of the mean free path of particles in gases. Currently, it is considered to be an averaged characteristic of the fluid's turbulent mixing. \[\tau=\rho_{a}\cdot\left|\frac{d\bar{u}}{dx}\right|^{2}\cdot l_{m}^{2} \tag{2}\]  
The friction velocity (dynamic velocity), \(v_{s}\) (Equations (3) and (4), [8]) and the constant \(K\) (Equation (5), [8]) were estimated and then applied into Equations (6)-(9) mentioned below. \[v_{s}=\sqrt{\frac{\tau}{\rho_{a}}} \tag{3}\]  
The Equation (4) was obtained by substituting Equations (2) and (3), which is obvious. \[v_{s}\ =\ \left|\frac{d\bar{u}}{dx}\right|\cdot l_{m} \tag{4}\]  
Constant \(K\) is defined through Equation (5) [8]:  
\[K=\frac{18\cdot\mu_{a}}{\rho_{w}\cdot d_{d}^{2}} \tag{5}\]  
Empirical studies have enabled the estimation of droplet size distribution in the channel's mist flow. The droplets were divided into three fractions that differed in their sizes as follows: the airborne droplets (\(d_{d}\ \leq 10\) m), the medium-sized droplets (\(d_{d}\ \leq 20\) m), and the largest droplets (\(d_{d}\) = (50-150) m). The droplets from the first fraction with a diameter up to a few microns can deposit on the channel's wall. Droplets with sizes of micron and sub-micron undergo the Saffman (\(S\)) lift force, which acts on droplets within the \(BL\). Turbulent diffusion of the continuous phase affects the phenomenon of droplet separation in this case. The inertia of the droplets from the first fraction does not have a significant impact on their movement in the gas velocity field because they are small in size. They maintain pace with the turbulent fluctuations of the gas velocity field. This is common knowledge of such a mass transfer process [39; 40]. The medium and large-sized droplets existing in the BL are affected by the Magnus \((M)\) lift force, and their deposition on the wall is an effect of the turbulent-inertial mechanism. The inertia of larger droplets (several dozen microns) is worthy of serious consideration as it has a significant impact on their deposition and entertainment, which is caused by the \(M\) force. The large droplets are unable to reach the wall through the laminar sublayer because they are lagging behind the turbulent motion of groups of molecules, which only involves molecular motion.