# Numerical and Analytical Study of a Battery Powered Vehicle Moving in a Vacuum Tunnel

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Vacuum Vehicle Geometry

#### 2.2. Analytical Study

#### 2.2.1. Analytical Analysis of Aerodynamics

- Isentropic characterization of a Mach number equal to 1 for different freestream velocity and blocking ratio.
- The critical blocking ratio characterization for $M{a}_{\infty}$ = 0.5 in freestream (see Figure 3b).

#### 2.2.2. Analytical Analysis for Heat Transfer

^{2}. Based on the specifications of the available lithium polymer batteries, the maximum allowed temperature was assumed as 55 °C. The calculations were based on the air properties at reference temperature equal to 0 °C and Sutherland’s law was used for dynamic viscosity estimation.

#### 2.3. Numerical Model

^{2}. During the aerodynamic calculations, the heat flux emitted by the batteries was not included.

## 3. Results and Discussion

#### 3.1. Aerodynamic Analytical Results

#### 3.2. Aerodynamic Numerical Results

#### 3.2.1. Pressure, Velocity and Drag Coefficient.

#### 3.2.2. Boundary Layer Separation, Back Flow and Velocity Profiles

#### 3.3. Heat Transfer

#### 3.3.1. Validation – Battery Cooling in Freestream Flow

^{2}. It was assumed that the whole emitted heat power was transferred to the passing air. Every considered case in the analysis is characterized by an order of magnitude difference in Reynolds number and is directly related to the value of the considered tunnel pressure. Consequently, Equation (7) was used to calculate the Nusselt number and Equation (9) was used to calculate the heat transfer coefficient.

^{2}/K and increases with Reynolds number. For very high Reynolds numbers, higher than 107, α ≈ 350 W/m

^{2}/K. The received values of α compare well with values reported in [38,44]. Consequently, the temperature of the batteries after cooling was calculated using Equation (10). Table 2 contains the analytical results of heat transfer calculations for the freestream velocity of 166 and 125 m/s. It can be noticed that $\alpha $ is higher for 166 m/s and the average temperature for the operating batteries is smaller for the 125 m/s velocity. It should be noted that, only for 100 kPa, the temperature was set below 55 °C (309 K) and stayed in the range of the recommended working temperature of the considered batteries.

#### 3.3.2. Analytical Analysis of Heat Transfer for Vehicle with Batteries

#### 3.3.3. Numerical Results—Vehicle with Batteries

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The N-10 airfoil shape used as a base for the considered vacuum vehicle and its position in the tunnel. Numerical geometry used for the aerodynamic calculations.

**Figure 2.**Numerical domain for the vehicle with batteries and heat transfer calculations: (

**a**) numerical geometry and boundary conditions, (

**b**) position of the batteries in the vehicle (red), (

**c**) detail of the numerical mesh in the vicinity of the cooling ducts (the inner duct and the duct below the tunnel).

**Figure 3.**(

**a**) Isentropic characteristic of blocking ratio in function of the vehicle velocity, with the assumption Ma = 1 in bypass between the vehicle and the tunnel; (

**b**) isentropic characteristic of blocking ratio in function of Mach number for 166 m/s velocity of the vehicle.

**Figure 4.**(

**a**) Change in the pressure in the bypass in the function of Mach number in the bypass for a fixed free stream pressure 10 kPa and $M{a}_{\infty}$ = 0.5; (

**b**) pressure distribution in the tunnel after 0.15 s—the value of pressure is equal to the sonic state pressure; (

**c**) pressure distribution after 4.65 s.

**Figure 5.**(

**a**) The average pressure at the inlet and outlet from the tunnel for 10 kPa; (

**b**) for 100 Pa. Results for 166m/s freestream air velocity. The effect of the increasing pressure due to the collection of air in front of the vehicle is visible. (

**c**) Drag coefficient for vehicle velocity 166 m/s and various initial tunnel pressure. (

**d**) Total drag coefficient and their components, the impact of the skin-friction drag is negligibly small.

**Figure 6.**Pressure distribution along the tunnel 4 m above the ground and 0.7 m above the thickest section of the vehicle: (

**a**) shows the state after 0.05 s and (

**b**) shows the state after 4.65 s.

**Figure 7.**Mach number field visualization: (

**a**–

**c**) results for ${p}_{\infty}$ = 100 Pa and 166 m/s velocity of the vehicle; (

**d**,

**e**) results for ${p}_{\infty}$ = 10 kPa and 125 m/s velocity of the vehicle.

**Figure 8.**Recirculation zone and separation point for: (

**a**) 100 Pa, (

**b**) 10 kPa, (

**c**) 1 bar in the tunnel.

**Figure 9.**The velocity profile at 29 m of the vehicle length for 100 Pa, 10 kPa, and 1 bar. There is no separation for 1 bar pressure in the tunnel at this location.

**Figure 10.**Temperature distribution around the batteries for pressure: (

**a**) 100 kPa, (

**b**) 10 kPa, (

**c**) 1 kPa and (

**d**) 100 Pa.

**Figure 12.**Temperature and velocity distribution around the batteries in the duct for the 100 Pa case.

**Table 1.**Average values of drag coefficient components and power related to the drag force. Ratio of drag acting on a moving vehicle in the tunnel to the drag in open space.

Type | 100 Pa | 1 kPa | 10 kPa | 100 kPa | 100 kPa Open |
---|---|---|---|---|---|

${c}_{d}$ | 0.554 | 0.503 | 0.314 | 0.344 | 0.029 |

${c}_{dp}$ | 0.547 | 0.500 | 0.309 | 0.341 | 0.027 |

${c}_{df}$ | 6.51 × 10^{−3} | 2.94 × 10^{−3} | 4.56 × 10^{−3} | 3.53 × 10^{−3} | 1.90 × 10^{−3} |

$\frac{{c}_{d}}{{c}_{d\_OPEN}}$ | 18.6 | 17.1 | 10.7 | 11.7 | 1 |

$\frac{{c}_{dp}}{{c}_{d\_OPEN}}$ | 19.7 | 18.2 | 11.3 | 12.4 | 1 |

$\frac{{c}_{df}}{{c}_{df\_OPEN}}$ | 3.42 | 1.55 | 2.39 | 1.86 | 1 |

$P$ for ${c}_{d}$, kW | 2.20 | 19.9 | 124 | 1360 | 115 |

**Table 2.**Thermo-physical properties for 166 and 125m/s and comparison of analytical temperature results after cooling with numerical results for 166m/s.

Properties | Case 1, 100Pa | Case 2, 1 kPa | Case 3, 10 kPa | Case 4, 100 kPa |
---|---|---|---|---|

166 m/s | ||||

$Re$ | 2.04 × 10^{4} | 2.04 × 10^{5} | 2.04 × 10^{6} | 2.04 × 10^{7} |

$Nu$ | 84 | 579 | 3526 | 24109 |

$\alpha $, W/m^{2}/K | 1.22 | 8.42 | 51 | 350 |

Analytical $T$, K | 2935 | 682 | 363 | 309 |

Numerical $T$, K | 2498 | 553 | 409 | 339 |

125 m/s | ||||

$Re$ | 1.53 × 10^{4} | 1.53 × 10^{5} | 1.53 × 10^{6} | 1.53 × 10^{7} |

$Nu$ | 73 | 460 | 2785 | 18906 |

$\alpha $, W/m^{2}/K | 1.08 | 6.91 | 42 | 285 |

$T$, K | 3280 | 765 | 377 | 311 |

**Table 3.**Analytical and numerical results for the vehicle moving at 125 m/s and thermo-physical properties of air in the cooling duct estimated by analytical calculations and average hydraulic diameter of the duct d

_{h}= 0.115 m.

Properties | Case 1, 100 Pa | Case 2, 1 kPa | Case 3, 10 kPa | Case 4, 100 kPa |
---|---|---|---|---|

Analytical Results of the Vehicle Cooling | ||||

Re | 10^{3} | 10^{4} | 10^{5} | 10^{6} |

Nu | 5.76 | 38.7 | 206 | 1301 |

α, W/(m^{2}K) | 1.34 | 8.99 | 48 | 302 |

T, K | 1500 | 479 | 334 | 305 |

Numerical Results of the Vehicle Cooling | ||||

T, K | 1385 | 464 | 342 | 308 |

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**MDPI and ACS Style**

Machaj, K.; Malecha, Z.; Wrzecioniarz, P.
Numerical and Analytical Study of a Battery Powered Vehicle Moving in a Vacuum Tunnel. *World Electr. Veh. J.* **2020**, *11*, 26.
https://doi.org/10.3390/wevj11010026

**AMA Style**

Machaj K, Malecha Z, Wrzecioniarz P.
Numerical and Analytical Study of a Battery Powered Vehicle Moving in a Vacuum Tunnel. *World Electric Vehicle Journal*. 2020; 11(1):26.
https://doi.org/10.3390/wevj11010026

**Chicago/Turabian Style**

Machaj, Krystian, Ziemowit Malecha, and Piotr Wrzecioniarz.
2020. "Numerical and Analytical Study of a Battery Powered Vehicle Moving in a Vacuum Tunnel" *World Electric Vehicle Journal* 11, no. 1: 26.
https://doi.org/10.3390/wevj11010026