## 2 Materials and Methods
### System Definition
#### 2.4.2 Aerodynamics  
The high-speed transport system runs inside a tube, and this is like a vehicle that runs inside a tunnel, whose drag coefficient increases as a result of the tunnel effect. According to [16], the relation between the drag coefficient inside and outside is expressed as follows (Equation (1)). To calculate the coefficient of drag inside, the same reference includes this formula (Equation (2)). Also, according to this reference, the outside drag coefficient is related to the moment section of the boundary layer (Equation (3)):  
\[T_{f}=\frac{C_{D_{t}}}{C_{Det}} \tag{1}\]  
\[C_{D_{t}}=\left(C_{Det}+\beta\left(\frac{\Delta_{1}}{A_{f}}\right)^{2}\right) \left(\frac{1-\frac{c_{t}}{v}}{1-\beta\left(\frac{\Delta_{1}}{A_{f}}\right)} \right)^{2} \tag{2}\]  
\[C_{D_{ext}}=2\frac{\Delta_{2}}{A_{f}} \tag{3}\]  
A relationship exists between the boundary layer momentum section and the boundary layer displacement section. To find this relationship, it must be taken into account that the boundary layer will be laminar, as can be verified by calculating both the local and the global Reynolds number, with some data extracted from [24] (first model):  
\[Re_{Dc}=\frac{\rho_{t}vD_{c}}{\mu_{t}} \tag{4}\]  
\[Re_{Lc}=\frac{\rho_{t}vL_{c}}{\mu_{t}} \tag{5}\]  
\[Re_{D_{c}}=\frac{1.18\times 10^{-3}\times\frac{1220}{3.80}\times 1.34}{1.80 \times 10^{-5}}=29,769.51\]  
\[Re_{L_{c}}=\frac{1.18\times 10^{-3}\times\frac{1220}{3.60}\times 25}{1.80 \times 10^{-5}}=555,401.23\]  
where \(1.80\times 10^{-5}\) Pa\(\cdot\)s is the dynamic viscosity for dry air at 20 \({}^{\circ}\)C and 100 Pa (the variation of viscosity with pressure is neglectable for such a low pressure) [18]. In addition, 25 m is approximately the length of the capsule, which can be gathered from [24]. The passenger capsule levitates on 28 air bearings, 14 on each side and 1.5 m long each (21 m in total, to which other parts such as the nose and nozzle are added). With respect to \(1.18\times 10^{-3}\) kg/m\({}^{3}\), this is the air density and comes from the ideal gas equation.  
It can be noted that the local Reynolds is small and not significant, whilst the global can be proper to a laminar boundary layer, since the transition from laminar to turbulent occurs somewhere between \(5\times 10^{5}\) and \(1\times 10^{6}\) for a flat plate. Assuming that it is always laminar for the high-speed transportation system, von Karman results can be used to relate the momentum thickness to the displacement thickness through the layer thickness. The process is shown below, after collecting the proper information from [18]: Equations (6) and (7). The function \(u(y^{\prime\prime})\) could be linear, parabolic, polynomial, etc. As a first approximation, the speed profile is assumed to be linear (Equation (8)). After integrating, the following is obtained (Equations (9) and (10)):  
\[\delta^{*}=\int_{0}^{\delta}\bigg{(}1-\frac{u(y^{\prime\prime})}{U}\bigg{)}dy ^{\prime\prime} \tag{6}\]  
\[\theta=\int_{0}^{\delta}\frac{u(y^{\prime\prime})}{U}\bigg{(}1-\frac{u(y^{ \prime\prime})}{U}\bigg{)}dy^{\prime\prime} \tag{7}\]  
\[u(y^{\prime\prime})=\frac{U}{\delta}y^{\prime\prime} \tag{8}\]  
\[\delta^{*}=\frac{\delta}{2} \tag{9}\]  
\[\theta=\frac{\delta}{6} \tag{10}\]  
For the completion of this equation block, the Equations (11)-(16), which come from reference [24], are necessary as well:  
\[A_{f}=A_{c} \tag{11}\]  
\[\beta=\frac{A_{c}}{A_{t}} \tag{12}\]  
\[\Delta_{1}=\frac{\pi}{4}\bigg{(}D_{desp}^{2}-D_{c}^{2}\bigg{)} \tag{13}\]  
\[\Delta_{2}=\frac{\pi}{4}\bigg{(}D_{movto}^{2}-D_{c}^{2}\bigg{)} \tag{14}\]  
\[D_{movto}=D_{c}+2\theta \tag{15}\]  
\[D_{desp}=D_{c}+2\delta^{*} \tag{16}\]