## 2 Harvesting System
### Theoretical Principles of Mini Wind Harvester Operation  
Whenever the harvester is in an open space, it harvests the kinetic energy of the air, i.e., wind velocity. However, when the harvester is placed in a pipe, it harvests the potential energy of the air, i.e., wind pressure inside the pipe. Figure 5 shows the described situation. In both cases, the law of conservation of energy applies. In Figure 5a, potential energy on one side cancels out the other side and the harvested energy comes from wind kinetic energy. In Figure 5b, the harvester has the same behavior, but the kinetic energy is cancelled, and the harvested energy comes from the wind potential energy. The described behavior is expressed as in Equation (2):  
\[E_{potential1}+E_{kinetic1}=E_{potential2}+E_{kinetic2}+E_{ harvested} \tag{2}\]  
where \(E_{potential1}\) and \(E_{potential2}\) are the potential energy of the wind before and after the harvester, \(E_{kinetic1}\) and \(E_{kinetic2}\) are the kinetic energy of the wind before and after the harvester, and \(E_{harvested}\) is the energy harvested by the wind harvester. On the one hand, if the turbine is installed in a windy open space, the absolute potential energy is zero, and the harvested power will come only from the kinetic one, as expressed in Equation (3),  
Figure 4: Three-phase generator PCB. (**a**) PCB design. (**b**) Manufactured coils and connection drills. Figure 5: Open space operation principle versus test-bench operation principle. (**a**) Harvester in an open space. (**b**) Harvester in a pipe. the potential energy being equal at both sides (\(P_{1}=P_{2}\)). In addition, due to the energy principle theorem, \(v_{1}\) must be bigger than \(v_{2}\), \(v_{1}>v_{2}\), and consequently \(\Delta v=v_{1}-v_{2}\). \[E_{harvested}=\frac{1}{2}\rho\Big{(}v_{1}^{2}-v_{2}^{2}\Big{)} \tag{3}\]  
When the turbine starts working under the required threshold wind condition, the rotor will rotate, harvest, and convert the kinetic energy of the wind into mechanical energy. The kinetic power of the wind, \(P_{wind}\), using the aerodynamic equation of [30], is given by Equation (4). The transformation of wind power into rotational power by the harvester rotor is a complex aerodynamic phenomenon as described in [31; 32; 33]. Ideally, the power extracted from the ambient wind, \(P_{aero}\), could be expressed as Equation (4):  
\[P_{wind}=\frac{1}{2}\rho v_{1}^{3}A\to P_{aero}=\frac{1}{2}\rho v_{1}^{3}AC_{p} (\lambda,\theta) \tag{4}\]  
where \(\rho\) is air density, \(v_{1}\) is air velocity, A turbine area, and \(C_{p}\) [PERSON]'s law coefficient. This power coefficient or the aerodynamic efficiency of the rotor, which is a nonlinear function of the pitch angle \(\theta\) of the harvester blades and tip speed ratio \(\lambda\). \(\theta\) is the angle of the blade element concerning the plane of rotation and \(\lambda\) is quotient between the tangential speed of the rotor blade tips and the incoming wind speed, expressed as Equation (5):  
\[\lambda=\frac{\omega\tau}{v} \tag{5}\]  
where \(\omega\) is the angular velocity (rad/s) and r the radius of the rotor. The power coefficient, \(C_{p}\) (\(\lambda\), \(\theta\)) Betz power coefficient equation is shown in Equation (6):  
\[C_{p}(\lambda,\theta)=C_{1}\left[\left(\frac{c_{2}}{\lambda_{i}}\right)-c_{3} \theta-c_{4}\theta^{c_{5}}-c_{6}\right]e^{(-\frac{\rho\tau}{\lambda_{i}})} \tag{6}\]  
where \(\lambda_{i}\) is obtained with Equation (7):  
\[\lambda_{i}=\frac{1}{\left[\frac{1}{(\lambda+c_{6}\theta)}\right]-\left[\frac {c_{9}}{(\theta^{3}+1)}\right]} \tag{7}\]  
The theoretical values and generated power were calculated to verify if the operation principle matches the results obtained. In addition, some of the dimensional and mechanic characteristics of the micro-reactor were measured and provided in Table 1.  
Refs. [34; 35] propose Betz power coefficients for large turbines.