## 3 Preliminary Design
### Water Droplets and Ice Formation  
As an icing wind tunnel, the spraying of water and its freezing are fundamental aspects to be taken into account in the design phase. In-flight icing is caused by subcooled water droplets in suspension that freeze on impact with a surface, usually leading edges in the case of aircraft (the most critical areas for icing on fixed-wing aircraft are the wings and stabilisers). These droplets are mainly characterised by two parameters in addition to temperature: their size, characterised by their Median Volume Diameter (MVD), and their quantity, characterised by their Liquid Water Content (LWC). For individual droplets, their radius \(r_{drop}\) or their diameter \(d_{drop}\) can be used, as the MVD applies to sets of droplets with different sizes. The usual system for spraying water consists of nozzles that spray water droplets of a controlled size and flow rate [70; 71; 73]. The water droplets are atomised at a temperature slightly above freezing to avoid ice blockage in the atomising nozzles, but these droplets must then be subcooled and accelerated to reach the desired test temperature and the appropriate velocity (that of the flow) at the moment of impact against the model. This defines a maximum droplet size to be used, beyond which the droplets will not reach the flow velocity or will not be at the same temperature as the flow when they reach the test section. To calculate the droplet velocity, a time step procedure [71] based on the equation of movement can be used, disregarding gravity and assuming the droplets to be rigid spheres. \[\Delta u_{drop}=\frac{1}{2}\rho_{air}\frac{\pi r_{drop}^{2}}{\frac{3}{3}\pi \rho_{water}r_{drop}^{3}}C_{D}\big{(}V-u_{drop}\big{)}^{2}\Delta t \tag{12}\]  
For the calculation of the drag coefficient, a five-parameter empirical correlation can be used, proposed by [PERSON] [74] for a range of low Reynolds number values, which will be found in the contraction section. \[C_{D}=\frac{24}{Re}(1+0.27 Re)^{0.43}+0.47\Big{[}1-exp\Big{(}-0.04 Re^{0.38} \Big{)}\Big{]} \tag{13}\]  
This way, the velocity that different droplet sizes will reach when they advance through the circuit and are accelerated by the contraction section can be calculated. Setting a limiting velocity deviation of 3 m/s, the droplet size limit will be 65 \(\upmu\)m to reach a velocity of 67 m/s at the centre of the test section (1.6 m from the inlet). To calculate the temperature [71] reached by the subcooled droplets, a time step procedure will be performed, similar to that of the velocity but, in this case, based on the first law of thermodynamics and disregarding evaporation, because a humidity close to  
Figure 8: Straight IWT without corners. unity is considered. Heat transfer due to conduction depends on the Nusselt number for rigid spheres. \[\Delta T_{drop}=\frac{3}{4}\frac{\dot{Q}_{conduction}+\dot{Q}_{radiation}}{\pi r_{drop }^{3}\rho_{water}C_{water}}\Delta t \tag{14}\]  
This way, the temperature that different droplet sizes will reach as they advance through the circuit and are accelerated by the current can be calculated (see Figure 9). Setting a temperature limit deviation of 0.5 \({}^{\circ}\)C for an IWT temperature of \(-3\)\({}^{\circ}\)C, the droplet size limit will be 70 \(\upmu\)m to reach a temperature of \(-2.5\)\({}^{\circ}\)C at the centre of the test section (1.6 m from the inlet). On the other hand, when using scale models, it is necessary to scale other parameters in addition to velocity to achieve similarity with respect to ice formation [75], and one of these parameters to be scaled is the droplet size. The inertia parameter \(I\) of Langmuir and Blodgett [76] is a dimensionless parameter that ensures that the trajectory of the droplets is the same in a real model and in the scale model, correctly simulating the impact of the droplets. To maintain this similarity, it will be necessary to adjust the droplet size according to the scale used.