## 1 Introduction  
In the last century, the transport environment has been dominated by fossil fuels and their respective combustion propulsion systems. At the very beginning, this concept became a revolution because of its distinctive speed and comfort in comparison to previous means of transport. Nevertheless, with globalization and the easy access to these technologies, nowadays there is a situation where the excess of traditional fuel-propelled vehicles, not only cars, but also planes or trains, is having a huge impact on the environmental situation [1], and the predictions indicate that this will increase due to the higher energy demand [2], setting an alarm about its prejudicial influence on nature and the worrying high levels of CO\({}_{2}\) emissions [3]. Because of these reasons, in the last years, the community has turned its efforts into the development of new environmentally friendly means of transports [4] and alternative ways of propulsion [2, 4]. Amongst them, the concept of hyperloop appeared in 2013 [5]. This idea of _vactrain_ (or vacuum tube trains), although it has its origins in the past [6], consists of the concept of a high-speed train travelling through a tube in partial vacuum in order to minimize the drag problems derived from the high velocities. In the next years, the feasibility of the idea was studied by several investigators [7], and a few aerodynamic and thermal analyses have been performed [8]. Many different ideas have been developed about the propulsive systems, such as electric power fed by solar panels [9] to electromagnetism for a suspension system [10, 11] that avoids the friction losses which could become a limiting variable at high speeds. Therefore, the environmental benefits derived from a technology with these characteristics are straightforward. Moreover, the possibility of reaching high speeds by ground travelling, with the comfort advantages it implies, also constitutes a fundamental advantage against its apparent main market rival, air transport [12]. Even in the case of developing nicer air transport satisfying environmental requirements, the overall time spent is way longer than travelling by ground, not only because of transportation itself (take off, ascent, descent...), but also for the particularities of the part where the individual leaves their initial location until the boarding is completed. Nevertheless, the fact that it is a relatively new concept implies the apparition of new problems, some of them typical from other technologies that are translated to the hyperloop concept. The main challenge that the hyperloop is facing is the concept of the Kantrowitz limit [13]. Whereas technologies such as open-air high-speed trains or air transport itself has no chocking problems due to external flow, a pod travelling through a closed tube at high speed could cause a choking of the flow when supersonic conditions are reached. Once the flow arrives at the constriction of a section between the tube and the pod, as the frontal area is reduced, the velocity must increase. In consequence, it could lead to a choke of the flow at certain points of the area contraction, making the pod act like a closed piston [14,15]. This would lead to a huge increase in the drag of the vehicle, nullifying all the aerodynamic initial advantages of the hyperloop concept. Therefore, the Kantrowitz limit acts as a maximum speed limit the hyperloop can develop in order to avoid these prejudicial effects. By developing conservation equations, the Kantrowitz limit would be defined with the relation of the flow speed and the quotient between the frontal area of the tube and the pod, commonly known as blockage ratio. Therefore, the Kantrowitz limit being the main limiting phenomenon of the hyperloop development, it has become the main investigated topic, with proposed solutions such as the utilization of aeronautical elements by adding airfoils to the external structure [16] or the optimization of the external pod [17,18] and the effect of this shape on the wave phenomenon [19].