# Analytical Model for Air Flow into Cracked Concrete Structures for Super-Speed Tube Transport Systems

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## Abstract

**:**

## 1. Introduction

## 2. Development of Analytical Model for Air Inflow through Cracks

_{1}and P

_{2}($0.8\times {10}^{5}\text{}\mathrm{Pa}\le {\mathrm{P}}_{1}{\text{}\mathrm{and}\text{}\mathrm{P}}_{2}\le 1.2\times {10}^{5}\text{}\mathrm{Pa}$) at both ends of the two parallel plates with a gap W, length L, and breadth B can be given as follows.

_{1}and n are constants, determined by conducting experimental tests. If the above equation is applied to the case of cracks on concrete structures, B can be replaced with the length of the crack, W with the crack width, and L with the thickness of the structure on which the crack occurs. However, as multiple cracks can occur in a concrete structure, the constants in Equation (2) were experimentally determined as follows:

_{avg}is measured in feet. Equation (2) can be applied if the pressure difference between the inlet and outlet air flow is constant regardless of the time. If this is expressed in terms of the air flow rate, the following equation can be written.

_{1}in Equation (7) is constant assuming that the temperature and latitude are constant. However, the internal pressure of an SSTT structure increases with time as the air flows into the structure through cracks, because the internal volume of the SSTT structure is limited. Therefore, P

_{2}is a function of time, and Equation (7) can be expressed as follows:

_{2}denotes the internal volume of the SSTT structure), the following equation can be written:

_{i}, the constant C in Equation (11) can be determined as follows:

## 3. Investigation of Crack Effect on the Air Inflow

_{avg}), crack length (B), and number of cracks (N). Figure 5a shows the change in the internal pressure over time, which was initially decreased to 0.001 atm, for the developed model for internal pressure prediction considering crack formation. Here, the total length of the cracks was 0.5 m; the number of cracks was 10; and the average crack width was 0.2 mm per unit length of the tube structure. A steep increase in the internal pressure was observed as it reached 0.8 atm within an hour.

_{avg}, as the time required for the internal pressure to increase to 0.5 atm was approximately 0.1 h for the case with W

_{avg}= 0.3 mm, and 1.0 h for the case with W

_{avg}= 0.1 mm. Figure 6b,c show the effects of the total crack length ranging from 0.3 m to 0.7 m and the number of cracks ranging from 6 to 14 on the system airtightness, respectively.

_{pr}. For practical purpose, t

_{pr}could be used as an indicator for system airtightness [2,4]. Figure 7 shows the time required for the internal pressure to increase to 0.5 atm with respect to the variations in the crack parameters, wherein the crack width exhibited the most rapid decrease within the normalized range of the parameters.

## 4. Definition of Crack Index for Air-Tightness of Tube Structures

_{pr}hereafter. Table 1 lists the ranges of each parameter.

_{pr}, was determined for each combination. Figure 8 and Figure 9 show the logarithm distributions of t

_{pr}for CI1 to CI4, corresponding to target pressure levels of 0.1 atm and 0.2 atm, respectively. While all the indices showed an inverse proportionality with t

_{pr}, CI3 had a stronger correlation with t

_{pr}than other CIs. The coefficient of determination for the regression analysis corresponding to CI3 was found to be 0.9842, which was the highest among the four crack indices. This was consistent with the input format of the parameters for the Rizkalla leakage model. Therefore, it was concluded that defining CI3 as the crack index was the most reasonable choice for evaluating the airtightness of the SSTT system. The expression for the same is as follows:

## 5. Correlation between Cracks and Airtightness: Experimental Demonstration

_{o}is the atmospheric pressure measured outside the tube, P

_{t}is the pressure inside the tube at time t, h is the constant thickness of the tube, A is the surface area per unit length of the tube, k is the effective intrinsic permeability (m

^{2}), $\mathsf{\mu}$ is the viscosity of the air (kg/m∙s), and C

_{1}is a constant that can be defined considering the initial condition. The effective intrinsic permeability corresponding to each loading step determined for each structure using Equation (15) above is presented in Table 2. Figure 12 shows the vertical displacement corresponding to the loading step for each test structure and its relationship to the effective intrinsic air permeability. There were some deviations, but in general, the trends of increase in the effective intrinsic air permeability due to displacement were consistent among the structures. This study deals with the effect on the airtightness performance of the structure due to cracks that are generated from external loadings. Loading was the only factor considered for the generation of cracks in this study, which shows crack formation was directly related with the amount of load applied. Assuming that the cracks due to external loadings are closely related to the displacement, system airtightness could be described in terms of cracks in the structure. Therefore, considering the relationship between the displacement, or cracks of a structure and its corresponding air tightness, it will be possible to predict air tightness at the design stage of an SSTT system. A study of system airtightness in various loading conditions must follow for this. It should be noted that because there are inherent material and structural uncertainties in the concrete structures and difficulties in measuring cracks [14,15], computer-based crack analysis studies should be accompanied in parallel with the experimental and analytical approaches.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Concept of super-speed tube transport (SSTT) [2].

**Figure 5.**Increase in the internal pressure with time under different crack conditions (per unit length).

**Figure 7.**Time required for the internal pressure to increase to 0.5 atm with respect to different crack parameters.

Control Variable | Minimum Value | Maximum Value |
---|---|---|

B (m) | 0.1 | 0.7 |

N | 1 | 10 |

W_{avg} (m) | 0.0001 | 0.001 |

Test Structure | Load Step | Displacement (mm) (Measured at Top of Tube Center) | Effective k (m^{2}) |
---|---|---|---|

CIR-01 | Step 1 | 1.59 | 4.20 × ${10}^{-15}$ |

Step 2 | 2.38 | 1.50 × ${10}^{-14}$ | |

CIR-02 | Step 1 | 1.20 | 1.05 × ${10}^{-16}$ |

CIR-03 | Step 1 | 1.22 | 1.70 × ${10}^{-15}$ |

Step 2 | 1.62 | 2.34 × ${10}^{-15}$ | |

Step 3 | 1.88 | 8.20 × ${10}^{-15}$ | |

Step 4 | 2.22 | 2.60 × ${10}^{-14}$ | |

CIR-04 | Step 1 | 0.74 | 2.30 × ${10}^{-15}$ |

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**MDPI and ACS Style**

Devkota, P.; Park, J.
Analytical Model for Air Flow into Cracked Concrete Structures for Super-Speed Tube Transport Systems. *Infrastructures* **2019**, *4*, 76.
https://doi.org/10.3390/infrastructures4040076

**AMA Style**

Devkota P, Park J.
Analytical Model for Air Flow into Cracked Concrete Structures for Super-Speed Tube Transport Systems. *Infrastructures*. 2019; 4(4):76.
https://doi.org/10.3390/infrastructures4040076

**Chicago/Turabian Style**

Devkota, Prakash, and Joonam Park.
2019. "Analytical Model for Air Flow into Cracked Concrete Structures for Super-Speed Tube Transport Systems" *Infrastructures* 4, no. 4: 76.
https://doi.org/10.3390/infrastructures4040076