# Energy Harvesting by a Novel Substitution for Expansion Valves: Special Focus on City Gate Stations of High-Pressure Natural Gas Pipelines

^{*}

## Abstract

**:**

^{2}area of the solar panel (150 W, 15.42% efficiency) for the climate condition of Toronto, Canada.

## 1. Introduction

^{2}and guides fluid flow tangential to the rotor, while a conical outlet duct has an angle 5°, and the outlet radius is set to be 0.6r

_{1}. The values of the design parameters are given in Table 1.

## 2. Numerical Procedure

#### 2.1. Grid Dependency Test

^{−4}m.

#### 2.2. Grid Generation

_{o}= 50 mm and b = 1 mm) was generated, as shown in Figure 4. Table 2 provides mesh statistics and quality for the final presented mesh, as shown in Figure 4, which shows that generated mesh has a good quality with low averaged skewness. Moreover, Table 3 provided details of mesh size and its characteristics.

#### 2.3. Boundary Condition

^{2}K and 15 degrees for the environment temperature were assumed.

#### 2.4. Model Validation

#### 2.5. Methodology and Properties

^{−6}was considered. Table 5 describes the values which were considered as under-relaxation factors for different equations. Moreover, the second-order upwind scheme was employed as a spatial discretization method for pressure, momentum, energy, turbulent kinetic energy, and the turbulent dissipation rate. Only for density, first-order upwind was selected.

## 3. Results and Discussion

_{o}and b are kept constant. With ANSYS Fluent, it is possible to visualize a certain property of the flow across lines or planes. In order to get a better grasp of what is happening inside the turbine, two planes were defined. First, an XY plane parallel to the disc’s surface that is in the middle of the gap of the second and third discs. This plane permits a clear visualization of the flow in between the discs. The second plane is a ZY plane that contains the TT axis and is normal to the inlet, which displays well the flow in the outlet.

_{1}, imagine the fluid pressure is decreased, as shown in Figure 9 (up), and therefore, fluid density is also sharply decreased, as shown in Figure 9 (up). At the same time, while coming closer to the center of the TT and outlet part, the cross-section area (πDb) is decreasing as well. Hence, under these circumstances, velocity should compensate all these shortages in order to satisfy the conservation of mass. Consequently, considering zero velocity in the z-direction, both components of velocity near the outlet section are increased.

## 4. Conclusions

- In smaller TTs, the main pressure drop occurs at the outlet channel. However, under a fixed condition, a larger TT can produce more power, mass flow rate, and outlet temperature.
- For each characteristic curve of the TT-generated power, there is an optimum angular velocity at which the output power becomes the maximum. Optimum angular velocity is shifted to a lower angular velocity for a larger turbine, and the magnitude of the power is increased in respect to the TT size.
- Outlet temperature will be increased by increasing disk angular velocity, and a larger TT shows a higher outlet temperature than smaller one.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$B$ | Gap size |

$\dot{m}$ | Mass flow rate |

$P$ | Pressure |

${r}_{0}$ | Maximum (outer) disk radius |

${r}_{1}$ | Minimum (inner) disk radius |

${v}_{r}$ | Fluid flow velocity in the radial direction |

${v}_{\theta}$ | Fluid flow velocity in circumferencial direction |

$N$ | Disk angular velocity |

$t$ | Thickness |

$\mu $ | Viscosity |

ρ | Density |

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**Figure 1.**(

**left**): Drawing of a Tesla turbine (TT) geometry and definition of its different parts. (

**right**): Coordinate system and some dynamic variables.

**Figure 8.**(

**a**): Imaginary line positions inside TT along the radius. (

**b**): Relative tangential velocity profile. (

**c**): Absolute tangential velocity profile. (

**d**): Radial velocity profile.

Non-Dimensional Variable | Formula | Value | Type |
---|---|---|---|

Number of disks | N.A. | 5 | fixed |

Disk thickness | $\raisebox{1ex}{${t}_{D}$}\!\left/ \!\raisebox{-1ex}{${r}_{1}$}\right.$ | 0.2 | fixed |

Gap size | $\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{${r}_{1}$}\right.$ | 0.1 | variable |

Outer radius | $\raisebox{1ex}{${r}_{0}$}\!\left/ \!\raisebox{-1ex}{${r}_{1}$}\right.$ | 10 | variable |

Plenum | $\raisebox{1ex}{${t}_{p}$}\!\left/ \!\raisebox{-1ex}{${r}_{1}$}\right.$ | 0.1 | fixed |

Stator thickness | $\raisebox{1ex}{${t}_{s}$}\!\left/ \!\raisebox{-1ex}{${r}_{1}$}\right.$ | 0.2 | fixed |

Reynolds number | $\frac{4\dot{m}}{\pi {r}_{1}\mu}$ | [5, 10] × 10^{6} | variable |

Max. Skewness | Ave. Skewness | No. of Nodes | No. of Elements |
---|---|---|---|

0.890 | 0.343 | 5,947,910 | 3,642,217 |

Element Size | Size Function | Element Order | Element Type |
---|---|---|---|

0.2× gap size | adaptive | quadratic | tetrahedral |

**Table 4.**Comparison of present work numerical results with Rice [42].

Inlet Pressure [kPa] | Mass Flow Rate [kg/s] | Rotational Speed [rpm] | Output Power [W] | |
---|---|---|---|---|

Rice [42] | 790.5 | 0.0371 | 9400 | 835.2 |

Present work | 790.5 | 0.0387 | 9400 | 863.3 |

Relative error | 0.0% | 4.3% | 0.0% | 3.36% |

Rice [42] | 824.9 | 0.0435 | 8000 | 1014.1 |

Present work | 824.9 | 0.0467 | 8000 | 1037.9 |

Relative error | 0.0% | 7.3% | 0.0% | 2.34% |

Rice [42] | 824.9 | 0.0435 | 10,000 | 1103.6 |

Present work | 824.9 | 0.0424 | 10,000 | 1146.7 |

Relative error | 0.0% | −2.5% | 0.0% | 3.90% |

Pressure | Density | Momentum | Energy | Turb. Kinetic | Turb. Dissipation |
---|---|---|---|---|---|

0.3 | 0.9 | 0.7 | 0.9 | 0.7 | 0.7 |

Property | Value |
---|---|

Density | Real Gas, Redlich-Kwong model |

Viscosity | 1.087 × 10^{−5} [kg/m.s] |

Specific heat capacity | 2222 [J/kg.K] |

Thermal conductivity | 0.0332 [W/m.k] |

Specific heat ratio | 1.32 |

PV Electricity Output [kWh/kWp per day] [51] | Electricity Price [$/kWh] [52] | Required Solar Panel [m^{2}] | Monetary Saving [$/year] | |
---|---|---|---|---|

Toronto, Canada | 3.7 | 0.151 | 84.4 | 2846 |

Munich, Germany | 3.1 | 0.35 | 100 | 6132 |

Torque | Output Power | Mass Flow Rate | Density (in/out) | Velocity (in/out) | Pressure [Bar] (in/out) |
---|---|---|---|---|---|

2.63 | 1514.2 | 0.497 | 47.26/26.47 | 87.64/196.28 | 60/30 |

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**MDPI and ACS Style**

Sheikhnejad, Y.; Simões, J.; Martins, N.
Energy Harvesting by a Novel Substitution for Expansion Valves: Special Focus on City Gate Stations of High-Pressure Natural Gas Pipelines. *Energies* **2020**, *13*, 956.
https://doi.org/10.3390/en13040956

**AMA Style**

Sheikhnejad Y, Simões J, Martins N.
Energy Harvesting by a Novel Substitution for Expansion Valves: Special Focus on City Gate Stations of High-Pressure Natural Gas Pipelines. *Energies*. 2020; 13(4):956.
https://doi.org/10.3390/en13040956

**Chicago/Turabian Style**

Sheikhnejad, Yahya, João Simões, and Nelson Martins.
2020. "Energy Harvesting by a Novel Substitution for Expansion Valves: Special Focus on City Gate Stations of High-Pressure Natural Gas Pipelines" *Energies* 13, no. 4: 956.
https://doi.org/10.3390/en13040956