# Influence of Disc Tip Geometry on the Aerodynamic Performance and Flow Characteristics of Multichannel Tesla Turbines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

#### 2.1. Geometry Model and Boundary Conditions

#### 2.2. Numerical Solver and Mesh Sensitivity

_{p}and $\gamma $ stand for the specific heat at constant pressure and the adiabatic index, respectively.

## 3. Results and Discussion

#### 3.1. Comparison of the Influence of Sharp Tips and Blunt Tips on Two Kinds of Tesla Turbines

#### 3.2. Influence of Relative Height and Sharp Tip Profile on the One-to-Many Turbine

## 4. Results

- (1)
- Compared to the turbine with blunt disc tips, the isentropic efficiency of the one-to-one turbine with sharp disc tips reduces a little, while that of the one-to-many turbine with sharp disc tips increases remarkably. It decreases by 3.6% for the one-to-one turbine and increases by 13.5% for the one-to-many turbine at 30,000 r/min. The flow coefficient of the one-to-one turbine with sharp tips is almost the same, while that of the one-to-many turbine with sharp tips is a little lower. For all rotational speeds, it varies less than 0.02% for the one-to-one turbine, and decreases about 7–10% for the one-to-many turbine.
- (2)
- Compared to the one-to-one turbine with blunt tips, the flow field is almost the same for the turbine with sharp tips; the relative tangential velocity gradient on disc walls is a little less and some vortices exist at the inlet of the disc channels, leading to less momentum exchange and more energy loss for the turbine with sharp tips. Compared to the one-to-many turbine with blunt tips, the flow angle relative to the tangential direction in the disc channels of the turbine with sharp tips is much less, leading to higher relative tangential velocity and more momentum exchange; the area of low Mach number and vortex reduces, leading to less energy loss.
- (3)
- For the one-to-many turbine, the isentropic efficiency of the turbine with sharp tips increases with relative height, which must be higher than that with blunt tips, and its relative increase value is 8.9%–16.6% at 30,000 r/min. The increment rate of the isentropic efficiency with sharp tips slows down with increasing relative height, and it decreases from 0.033 to 0.013 at 30,000 r/min as the relative height increases from 0.2887, 0.5 to 0.8660. In addition, the circular or elliptic tips perform better at lower relative height, and a triangular tip behaves better at higher relative height.
- (4)
- Compared to the one-to-many turbine with blunt tips, the improvement of flow field within the turbine with sharp tips becomes much better with increasing relative height, and in detail, the flow angle in the disc channels decreases and the area of low flow velocity reduces, leading to an increase in isentropic efficiency.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

b | disc spacing distance, mm |

c | radial clearance of nozzle-rotor chamber, mm |

c_{p} | specific heat at constant pressure, J/kg·K |

C_{m} | flow coefficient |

C_{P} | specific power, kJ/kg |

C_{T} | torque coefficient |

d | diameter, mm |

h | sharp height, mm |

h/t | relative height |

m | mass flow rate, kg/s |

M_{a} | Mach number |

n | rotational speed of the rotor, r/min |

N | number |

${p}_{\mathrm{nt}}/{p}_{\mathrm{i}}$ | ratio of total pressure at the nozzle inlet to pressure at the turbine outlet |

P | power, W |

Q | Q criterion |

r | radial coordinate or radius, mm |

t | disc thickness, mm |

T | torque, N·m |

T_{nt} | total temperature at the nozzle inlet, K |

u | average tangential velocity, m/s |

${u}_{i},{u}_{j},{u}_{k}$ | three velocity components in ${x}_{i},\text{}{x}_{j},\text{}{x}_{k}$ directions, m/s |

v | average radial velocity, m/s |

W | relative tangential velocity, m/s |

${x}_{i},{x}_{j},{x}_{k}$ | Cartesian coordinate axis, m |

z | axial coordinate, mm |

$\alpha $ | nozzle exit geometrical angle (relative to the tangential direction), ° |

$\gamma $ | adiabatic index |

$\delta $ | relative variation of parameters |

$\mathsf{\Delta}{h}_{\mathrm{isen}}$ | isentropic enthalpy drop of the whole turbine, J/kg |

$\eta $ | isentropic efficiency |

$\theta $ | circumferential coordinate, rad |

$\rho $ | density, kg/m^{3} |

$\omega $ | rotational angular speed, rad/s |

Subscripts | |

d | disc |

dc | disc channel |

i | inner |

n | nozzle |

o | outer |

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**Figure 2.**Schematic diagrams of multichannel Tesla turbines [22]. (

**a**) One-to-one turbine; (

**b**) one-to-many turbine.

**Figure 4.**Cross-section map of several kinds of discs with different disc tips. (

**a**) Blunt tip; (

**b**) triangular tip; (

**c**) circular tip; (

**d**) elliptic tip.

**Figure 7.**Aerodynamic performance comparison of the turbine models with sharp disc tips and blunt disc tips for both the one-to-one turbine and the one-to-many turbine. (

**a**) Isentropic efficiency; (

**b**) flow coefficient; (

**c**) torque coefficient; (

**d**) specific power.

**Figure 8.**Streamlines and Mach number distributions on the middle of DC1 and DC3 for the one-to-one turbine models with different disc tips. (

**a**) DC1 for OTO-B; (

**b**) DC3 for OTO-B; (

**c**) DC1 for OTO-T-2;(

**d**) DC3 for OTO-T-2.

**Figure 9.**Streamlines and relative tangential velocity distributions on four radial cross-sections of different circumferential angles for OTO-B and OTO-T-2 (in each subfigure the above figure is for OTO-T-2, the below figure is for OTO-B). (

**a**) 0°; (

**b**) 45°; (

**c**) 90°; (

**d**) 135°.

**Figure 10.**Vectors and relative tangential velocity distributions on the radial section of 0° for two turbine models. (

**a**) OTO-B; (

**b**) OTO-T-2.

**Figure 11.**Iso-surface of Q distributions for two turbine models, $Q=4\times {10}^{8}$. (

**a**) OTO-B; (

**b**) OTO-T-2.

**Figure 12.**Streamlines and Mach number distributions on the middle of DC1 and DC3 for the one-to-many turbine models with different disc tips. (

**a**) DC1 for OTM-B; (

**b**) DC3 for OTM-B; (

**c**) DC1 for OTM-T-2; (

**d**) DC3 for OTM-T-2.

**Figure 13.**Streamlines and relative tangential velocity distributions on four radial cross-sections of different circumferential angles for OTM-B and OTM-T-2 (in each subfigure the above figure is for OTM-T-2, the below figure is for OTM-B). (

**a**) 0°; (

**b**) 45°; (

**c**) 90°; (

**d**) 135°.

**Figure 14.**Vectors and relative tangential velocity distributions on the radial-cross section of 0° for the one-to-many turbines. (

**a**) OTM-B; (

**b**) OTM-T-2.

**Figure 15.**Iso-surface of Q distributions for two turbine models, $Q=4\times {10}^{8}$. (

**a**) OTM-B; (

**b**) OTM-T-2.

**Figure 16.**Aerodynamic performance versus the rotational speed for the one-to-many turbine models with different disc tips. (

**a**) Isentropic efficiency; (

**b**) flow coefficient; (

**c**) torque coefficient; (

**d**) specific power.

**Figure 17.**Streamlines and Mach number distributions on the middle of DC1 for different turbine models. (

**a**) OTM-B; (

**b**) OTM-T-1; (

**c**) OTM-T-2; (

**d**) OTM-T-3; (

**e**) OTM-C-2; (

**f**) OTM-E-3.

**Figure 18.**Streamlines and relative tangential velocity distributions on the radial cross-sections of 45° for different turbine models. (

**a**) OTM-B; (

**b**) OTM-T-1; (

**c**) OTM-T-2; (

**d**) OTM-T-3; (

**e**) OTM-C-2; (

**f**) OTM-E-3.

Model Name | Turbine Type | Disc Tip Profile | h/t [-] |
---|---|---|---|

OTO-B | One-to-one turbine | Blunt tip | - |

OTO-T-2 | One-to-one turbine | Triangular tip | 0.5 |

OTM-B | One-to-many turbine | Blunt tip | - |

OTM-T-1 | One-to-many turbine | Triangular tip | 0.2887 |

OTM-T-2 | One-to-many turbine | Triangular tip | 0.5 |

OTM-T-3 | One-to-many turbine | Triangular tip | 0.8660 |

OTM-C-2 | One-to-many turbine | Circular tip | 0.5 |

OTM-E-3 | One-to-many turbine | Elliptic tip | 0.8660 |

**Table 2.**Geometrical and operational parameters [22].

Parameter | Symbol | Value | Unit |
---|---|---|---|

Nozzle number | N_{n} | 2 | (-) |

Disc outer diameter | d_{o,d} | 100 | (mm) |

Disc inner diameter | d_{i,d} | 38.4 | (mm) |

Disc thickness | t | 1 | (mm) |

Disc spacing distance | b | 0.5 | (mm) |

N-R radial clearance | c | 0.25 | (mm) |

Disc number | N_{d} | 5 | (-) |

Disc channel number | N_{dc} | 6 | (-) |

Nozzle exit geometrical angle | α | 10 | (°) |

Turbine pressure ratio | ${p}_{\mathrm{nt}}/{p}_{\mathrm{i}}$ | 3.42 | (-) |

Total temperature at turbine inlet | T_{nt} | 373 | (K) |

**Table 3.**Grid information [23].

Case No. | Stator (Nozzle/N-R Chamber) | Rotor (Each Disc Channel) | ||
---|---|---|---|---|

Number of Nodes $(\mathit{r},\mathit{\theta},\mathit{z}\text{}\mathbf{Directions})$ | Total Node Number | Number of Nodes $(\mathit{r},\mathit{\theta},\mathit{z}\text{}\mathbf{Directions})$ | Total Node Number | |

Grid Case 1 | (55/13) × (36/269) × 99 | 526,516 | 65 × 288 × 23 | 400,660 |

Grid Case 2 | (67/17) × (45/335) × 107 | 923,517 | 81 × 333 × 29 | 782,217 |

Grid Case 3 | (87/21) × (57/417) × 135 | 1,830,306 | 102 × 417 × 37 | 1,581,306 |

**Table 4.**Mesh independence [23].

Case No. | Node Number (million) | $\mathit{m}$ (kg/s) | $\mathit{\delta}\mathit{m}\text{}(\%)$ | $\mathit{P}$ (W) | $\mathit{\delta}\mathit{P}\text{}(\%)$ | $\mathit{\eta}$ (-) | $\mathit{\delta}\mathit{\eta}\text{}(\%)$ |
---|---|---|---|---|---|---|---|

Grid Case 1 | 1.72 | 0.03576 | 0.619 | 596.0 | 1.568 | 0.1504 | 0.940 |

Grid Case 2 | 3.27 | 0.03562 | 0.225 | 588.6 | 0.307 | 0.1491 | 0.067 |

Grid Case 3 | 6.57 | 0.03554 | 0 | 586.8 | 0 | 0.1490 | 0 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Qi, W.; Deng, Q.; Chi, Z.; Hu, L.; Yuan, Q.; Feng, Z.
Influence of Disc Tip Geometry on the Aerodynamic Performance and Flow Characteristics of Multichannel Tesla Turbines. *Energies* **2019**, *12*, 572.
https://doi.org/10.3390/en12030572

**AMA Style**

Qi W, Deng Q, Chi Z, Hu L, Yuan Q, Feng Z.
Influence of Disc Tip Geometry on the Aerodynamic Performance and Flow Characteristics of Multichannel Tesla Turbines. *Energies*. 2019; 12(3):572.
https://doi.org/10.3390/en12030572

**Chicago/Turabian Style**

Qi, Wenjiao, Qinghua Deng, Zhinan Chi, Lehao Hu, Qi Yuan, and Zhenping Feng.
2019. "Influence of Disc Tip Geometry on the Aerodynamic Performance and Flow Characteristics of Multichannel Tesla Turbines" *Energies* 12, no. 3: 572.
https://doi.org/10.3390/en12030572