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An analytical model is developed to evaluate airtightness, which is one of the most important requirements of vacuum tube transportation systems. The main objective of the model is to anticipate the pressure inside a closed structure, which initially decreases and then rises with time owing to the inflow of the air outside. The model is formulated by using Darcy’s law and by solving the differential equations that consider the air permeability of the material and physical configuration of the tube. Equations are derived for a tube section in two cases: one with constant thickness and another with variable thickness. Although the developed model must be verified experimentally, results simulated by using the model with several assumed sections were consistent with the findings of a previous study. The mathematical model developed here to predict the behavior of the pressure inside a vacuum tube structure could be effectively used to evaluate the airtightness of vacuum tube transportation systems. Such data could provide background technical information for the practical design of the system.

The air resistance that acts against a transportation vehicle increases with speed. Hence, minimizing air resistance could enhance vehicle speed without excessive consumption of propulsion energy. Vacuum tube systems, which constitute an innovative type of transportation mode where high speeds can be achieved for a vehicle by reducing the air pressure inside a tube structure to minimize air resistance (Figure

Concept of vacuum tube transportation system.

One of the most important requirements that must be met by the infrastructure of a vacuum tube system is a sufficient level of airtightness. Otherwise, the capacity of the vacuum pumps would need to be inevitably increased, resulting in an increase in construction costs. In addition, the operation interval of the pumps would need to be reduced, increasing maintenance costs as well.

In a vacuum tube, the inside pressure is much lower than the atmospheric pressure outside. Air will then inflow into the vacuum tube structure from the outside unless the tube is completely airtight. The flow rate of air going through a porous medium can be calculated using Darcy’s law [

Schematic view of a circular vacuum tube with constant thickness.

Park et al. [^{3}/s) and

Meanwhile, the flow rate of compressible fluid that flows into a tube structure can be expressed by applying Darcy’s law as follows:^{2}), ^{2}).

Substituting (

By solving the differential equation of form

Using (

Pressure change with time at several levels of air permeability.

The mathematical model described in the previous section was established for the special case where the thickness of the tube section was constant. The actual thickness of a vacuum structure, however, may not always be constant; instead, it may vary along the radial direction because a tube section of a structural member is designed in consideration of various loads (e.g., dead, live, and special loads), in addition to cost and constructability (see Figure

Examples of tube sections with variable thickness.

Let us consider a porous medium with a thickness that varies along one direction (the

Flow of fluid through a porous medium with variable thickness.

The total flow rate

Figure

Tube structure with variable thickness.

Physical definitions of variables for cross section with variable thickness.

Upon solving the differential equation, we obtain

This section provides illustrative applications of the model developed in this study. Let us first consider a tube section, denoted as Section B, which has a square outer section with a length of

Definition of Section B.

The pressure inside the tube at time

Let us consider one more tube section, denoted as Section C, whose outer section is rectangular and inner section is elliptical (Figure

Definition of Section C.

Figure ^{2}. Section A was a circular section with constant thickness as in Figure

Comparison of pressure behaviors for tubes with different sections.

Although further experimental verification is needed, the mathematical model developed here to predict the behavior of the pressure inside a vacuum tube structure could be effectively used to evaluate the airtightness of vacuum tube transportation systems. The data thus obtained would provide background technical information for the practical design of the system.

An analytical model is developed to evaluate airtightness, which is one of the most important requirements of vacuum tube structures. The model anticipates the pressure inside a closed structure, which initially decreases and then rises with time. This is achieved by using Darcy’s law and solving differential equations that consider the air permeability of the material and physical configuration of the tube. Equations for the pressure change are formulated for tube sections with uniform thickness and variable thickness. The mathematical model developed is applied to several tube structures with assumed sections. The results show that the tube structure with the largest internal volume is the most airtight. In addition, given the same internal volume, the tube with a section that has a greater average thickness is more airtight. Although further experimental verification is needed, the results simulated by using the developed model are consistent with previous results showing that an increase in the volume or thickness of the tube enhances its airtightness. The mathematical model developed here for predicting the behavior of the pressure inside a vacuum tube structure could be effectively used to evaluate the airtightness of vacuum tube transportation systems. The obtained data could provide background technical information for the practical design of the system.

The opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect those of the sponsor.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by a grant from the International Cooperation R&D Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), funded by the Ministry of Knowledge Economy of the Republic of Korea (2010T100100963).