# Numerical Investigation of Pelton Turbine Distributor Systems with Axial Inflow

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Curves of turbine efficiency η

_{T}against the flow rate Q normalised by the maximum flow rate Q

_{max}for common turbine types, recreated and modified from [10] (

**a**). Explanatory sketch for Pelton turbines with conventional distributor system (

**b**).

## 2. Materials and Methods

#### 2.1. Pelton Turbine Distributor System with Axial Inflow—AxFeeder

_{1}is directly connected to the manifold element, where the incoming flow is divided into n equal portions while inducing as few losses and secondary flows as possible. Unlike in conventional distributor systems, where only one branch line separates from the main line at a certain time, here, all n branch lines separate from the manifold at once. The n branch lines, ranging from station 51 (5n) to station 101 (10n), are connected to the manifold with the diameter A

_{Lk}and the deviation angle δ. The last branch line component is the injector bend, ranging from stations 81 to 101. It is pivoted by the angle γ relative to the branch line. The exact value of γ can be adjusted according to the pitch cycle diameter D

_{p}of the runner.

#### 2.2. Description of Investigated Basic Manifold Designs

#### 2.3. Flow Quality in Piping Systems

#### 2.3.1. Total Pressure Drop

#### 2.3.2. Power Loss—Classical Approach

_{pt}, can be defined in the form [16,17]

_{pt}by using the fluxes of total pressure ${p}_{t}=p+0.5\phantom{\rule{0.166667em}{0ex}}\rho {\overrightarrow{u}}^{2}$. Integrated over the entire surface area A with unit normal vector $\overrightarrow{n}$ at the station i,

_{KE}is likewise defined as the area integral of the dynamic pressure ${p}_{dyn}={p}_{t}-p=0.5\phantom{\rule{0.166667em}{0ex}}\rho {\overrightarrow{u}}^{2}$ at a station i,

#### 2.3.3. Power Loss—Second Law Analysis

_{Φ}can be defined as

_{Turb}and P

_{Vis}being the power of turbulent (Turb) and viscous (Vis) dissipation, respectively. These two terms are computed by the volume integrals of the corresponding dissipation terms over the volume of interest

_{Vis}follows from inserting the time-averaged velocity components $\overline{u}$, $\overline{v}$, $\overline{w}$ into the product of shear stresses τ

_{ij}and velocity gradients $\partial {u}_{i}/\partial {x}_{j}$

#### 2.3.4. Secondary Flows

#### 2.4. Computational Domain and Simulation Setup

^{6}was specified at the inlet of all cases. Only for the cases presented in Section 3.1 the Reynolds number was changed. The pressure boundary condition was set to 1 bar at the outlet. A turbulence intensity of 5% together with a turbulent length scale corresponding to the hydraulic diameter of station 1 were set as turbulence boundary conditions at the inlet. At all walls, a no-slip boundary condition was employed and all walls were set to be hydraulically smooth. The flow is assumed to be steady, incompressible, and isothermal. The density and dynamic viscosity of water at 25 °C were set to ρ = 997 kg/m

^{3}and μ = 8.899 × 10

^{−4}Pa s. The k-ω SST model [26] was employed as turbulence closure. The advection terms were solved using the high-resolution scheme, which is a second-order scheme that automatically blends to a first-order formulation if stability issues arise [28]. The advection of turbulence was discretised by a first-order upwind scheme.

#### 2.5. Grid Refinement Study

^{6}, 6.6 × 10

^{6}and 16.9 × 10

^{6}. The maximum y

^{+}value at the walls was below 1 for all investigated cases. The discretisation uncertainties were computed for ${\zeta}_{PmTE,1001}$, ζ

_{Φ}, and ${\varphi}_{II,100}$.

_{Φ}and ${\varphi}_{II,100}$ are more sensitive to mesh refinements, thus yielding a GCI of 14.6% and 13.7%, respectively. One reason for the higher GCI of the secondary flow can be seen in Figure 4c, in which the normalised velocity magnitude at a horizontal line in station 101 is plotted for the three meshes. At this station, directly downstream of the injector bend there is a velocity deficit at the inner wall of the bend. The exact prediction of this deficit poses an inherent challenge for flow modelling. Therefore, the local GCI values, indicated by the error bars in Figure 4c, are significantly higher than at the rest of the profile. The mean value of the order of accuracy p

_{oa}lies above three. In order to maintain an adequate balance between computational times and numerical accuracy, the medium mesh was chosen for all subsequent simulations.

## 3. Results

#### 3.1. Operating Regime

#### 3.2. Parametric Variations of the Basic Model

#### 3.3. Parametric Variations of the Basic Model with Conical Frustum

#### 3.4. Parametric Variations of the Distributor Model with Spherical Manifold

#### 3.5. Parametric Variations of the Distributor Model with Cylindrical Manifold

#### 3.6. Comparison of the Four Design Variants

## 4. Discussion

#### 4.1. Core Findings

- The transition from the penstock to the manifold is crucial, while the model with a spherical manifold becomes susceptible to unsteady flow phenomena if the sphere radius exceeds a certain value, similar unsteady effects were observed for the model with a diffuser-shaped manifold and too-steep diffuser angles $\beta $.
- The first component of the branch line in the flow direction should be shaped as a conical frustum. It reduces power losses by over a third and decreases the secondary flows for branch lines with steep deviation angles.
- A steeper deviation angle (ideally $\delta ={90}^{\circ}$) has multiple advantages: First, the secondary velocity ratio is lowered significantly (see Figure 12). Second, the axial length of the distributor system is shortened, and third, the connection between the manifold and the branch lines becomes easier to manufacture. The slight increase in power losses for steeper deviation angles becomes negligible.
- An injector bend with a converging diameter from station 81 to 91 and a fixed curvature radius allows for a reduction in both quality criteria of up to one-third.
- Only the conical frustum and the converging injector bend reduce power losses and secondary flows simultaneously. The majority of the geometric parameters decrease one but increase the other target quantity. For example, a fillet radius at the connection of the branch lines and the manifold greatly reduces power losses, but amplifies the secondary velocity.

#### 4.2. Additional Insights

#### 4.2.1. On the Secondary Flows at Station 101

#### 4.2.2. On the Power Losses

#### 4.3. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin symbols | ||

${A}_{i}$ | Surface area of station i | m^{2} |

${A}_{Lk}$ | Diameter at which the branch lines are connected to the manifold | $\mathrm{m}$ |

${D}_{1}$ | Penstock diameter | $\mathrm{m}$ |

${D}_{i}$ | Diameter of pipe segment at station i | $\mathrm{m}$ |

${D}_{p}$ | Pitch cycle diameter of the runner | $\mathrm{m}$ |

H | Geodetic head | $\mathrm{m}$ |

${K}_{pt}$ | Total pressure loss coefficient | 1 |

k | Turbulence kinetic energy | m^{2}/s^{2} |

${L}_{i}$ | Length of pipe segment starting from station i | $\mathrm{m}$ |

$\dot{m}$ | Mass flow rate at station i | $\mathrm{k}$$\mathrm{g}$/$\mathrm{s}$ |

$\overrightarrow{n}$ | Normal vector of surface $\overrightarrow{A}=A\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{n}$ | 1 |

n | Number of branch lines | 1 |

${n}_{q}$ | Specific speed | rpm |

${P}_{KE}$ | Power of kinetic energy | $\mathrm{W}$ |

${P}_{mTE}$ | Power of mechanical total energy | $\mathrm{W}$ |

${P}_{Turb}$ | Power of turbulent dissipation | $\mathrm{W}$ |

${P}_{Vis}$ | Power of viscous dissipation | $\mathrm{W}$ |

p | Pressure | $\mathrm{Pa}$ |

${p}_{oa}$ | Order of accuracy | 1 |

${p}_{dyn}$ | Dynamic pressure | $\mathrm{Pa}$ |

${p}_{t}$ | Total pressure | $\mathrm{Pa}$ |

Q | Volumetric flow rate | ${\mathrm{m}}^{3}$/$\mathrm{s}$ |

$Re$ | Reynolds number | 1 |

${r}_{\zeta \varphi}$ | Non-dimensional distance from center point to design point | 1 |

$TI$ | Turbulence intensity | 1 |

$\overrightarrow{u}={\left(\right)}^{u}T$ | Flow velocity and its components | $\mathrm{m}$/$\mathrm{s}$ |

${\overrightarrow{u}}_{I}$ | Primary flow velocity | $\mathrm{m}$/$\mathrm{s}$ |

${\overrightarrow{u}}_{II}$ | Secondary flow velocity | $\mathrm{m}$/$\mathrm{s}$ |

V | Integration volume | ${\mathrm{m}}^{3}$ |

${y}^{+}$ | Non-dimensional wall distance | 1 |

Greek symbols | ||

$\alpha $ | Deviation angle of first segment of the branch line of design a), see Figure 2 | ${}^{\circ}$ |

$\beta $ | Diffuser angle | ${}^{\circ}$ |

$\gamma $ | Pivot angle of the injector bend | ${}^{\circ}$ |

$\Delta $ | Difference between quantities | misc. |

$\delta $ | Deviation angle of the branch line | ${}^{\circ}$ |

$\epsilon $ | Turbulence eddy dissipation | ${\mathrm{m}}^{2}$/${\mathrm{s}}^{3}$ |

${\zeta}_{PmTE}$ | Power loss coefficient | 1 |

${\zeta}_{\mathsf{\Phi}}$ | Dissipation power coefficient | 1 |

${\eta}_{distributor}$ | Distributor efficiency | 1 |

${\eta}_{T}$ | Turbine efficiency | 1 |

$\phi $ | Pivot angle of the branch line, see Figure 2 | ${}^{\circ}$ |

${\mathsf{\Phi}}_{Turb}$ | Turbulent dissipation | $\mathrm{W}$ |

${\mathsf{\Phi}}_{Vis}$ | Viscous dissipation | $\mathrm{W}$ |

${\varphi}_{II}$ | Secondary velocity ratio | 1 |

${\tau}_{ij}$ | Shear stress tensor in index notation | $\mathrm{Pa}$ |

$\omega $ | Turbulence eddy frequency | 1/$\mathrm{s}$ |

Constants (within the framework of this study) | ||

g | Gravitational acceleration | 9.807 m/s^{2} |

${\beta}^{*}$ | Coefficient of k-$\omega $ SST turbulence model | 0.09 |

$\mu $ | Dynamic viscosity of water at 25 ${}^{\circ}\mathrm{C}$ | 8.009 × 10^{−4} Pa s |

$\rho $ | Density of water at 25 ${}^{\circ}\mathrm{C}$ | 997 kg/m^{3} |

Common indices | ||

dyn | Dynamic | |

i | Station i | |

KE | Kinetic energy | |

max | Maximum | |

min | Minimum | |

mTE | Mechanical total energy | |

ref | Reference | |

T | Turbine | |

Turb | Turbulent | |

t | Total | |

Vis | Viscous | |

I | Primary | |

II | Secondary | |

101 | Quantity evaluated at station 101 | |

1011 | Quantity evaluated as difference of values at stations 1 and 101 | |

Abbreviations | ||

FFG | Österreichische Forschungsförderungsgesellschaft | |

GCI | Grid convergence index | |

SLA | Second law analysis | |

SST | Shear stress transport |

## Appendix A

#### Appendix A.1. Script for Creating Secondary Flow Variables in CFD-Post

Listing A1: Minimal working example for creating secondary flow variables in CFD-Post. |

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**Figure 2.**Generic sketch of the AxFeeder (Pelton turbine distributor system with axial inflow) with the individual components and evaluation stations from 1 to 101.

**Figure 3.**Sketch of the four basic investigated manifold designs: (

**a**) basic model (for mesh and symmetry study), (

**b**) basic model with conical frustum, (

**c**) spherical manifold, (

**d**) cylindrical manifold.

**Figure 4.**Computational domain (

**a**), results of mesh study for integral values (

**b**), and local velocities (

**c**).

**Figure 5.**Operating charts of the AxFeeder basic model (part (a) of Figure 3) showing the power loss coefficients (

**a**) and the secondary velocity ratio at station 101 (

**b**) against the inlet Reynolds number.

**Figure 6.**Line plots of the power loss coefficients and secondary velocity ratios of basic model. Charts (

**a**,

**b**) show the effect of a variation in the diameter ratio ${D}_{51}/{D}_{1}$, (

**c**,

**d**), the fillet radius ${R}_{40}/{D}_{51}$, and (

**e**,

**f**) of the diffuser angle $\beta $.

**Figure 7.**Line plots of the power loss coefficients and secondary velocity ratios of the basic model with conical frustum. Charts (

**a**,

**b**) show the effect of a variation of the diameter ratio ${D}_{40}/{D}_{51}$ of the frustum for different deviation angles $\delta $. Charts (

**c**,

**d**) show the effect of a variation of the same diameter ratio for a horizontal pivot angle $\phi ={15}^{\circ}$. Charts (

**e**,

**f**) show the effect of a variation of the diameter ratios ${D}_{51}/{D}_{101}$ and ${D}_{51}/{D}_{71}$ for converging pipe bend sections 61–71 (configuration C) and 81–91 (configurations A, B).

**Figure 8.**Line plots of the power loss coefficients and secondary velocity ratios of the model with spherical manifold. Charts (

**a**,

**b**) show the effect of a variation of the deviation angle $\delta $ for five different sphere radii $S{R}_{40}/{D}_{1}$ for a configuration without a frustum. Charts (

**c**,

**d**) show the effect of a variation of the frustum diameter ratio ${D}_{40}/{D}_{51}$ for different deviation angles $\delta $ and a fixed sphere radius of $S{R}_{40}/{D}_{1}=0.6$.

**Figure 9.**Line plots of the power loss coefficients and secondary velocity ratios of the model with cylindrical manifold. Charts (

**a**,

**b**) show the effect of a variation of the deviation angle $\delta $ for different axial positions ${T}_{4}/{D}_{51}$ of the branch line. Charts (

**c**,

**d**) show the effect of applying a fillet of radius ${R}_{40}/{D}_{51}$ to smooth the connection between the cylindrical manifold and the branch lines for four different deviation angles.

**Figure 10.**Scatter plot of normalised secondary velocity ratio at station 101 $\frac{{\varphi}_{II,101}}{{\varphi}_{II,101,ref}}$ against normalised power loss coefficient $\frac{{\zeta}_{PmTE,1011}}{{\zeta}_{PmTE,1011,ref}}$. The best configuration of each of the four basic designs is marked by an enlarged triangle and the minimum distance of this point to the centre is indicated by dashed quarter circles. The Pareto front, linking all non-dominated design configurations, is sketched as a dash-dotted black line.

**Figure 11.**Contour plot of viscous (left column) and turbulent dissipation (right column) in the mid-plane of the best configurations of the four basic designs. The quantities are normalised by the density-weighted turbulent eddy dissipation at the inlet. Subfigures (

**a**,

**b**) show basic design a), (

**c**,

**d**) show basic design b), (

**e**,

**f**) show basic design c), (

**g**,

**h**) show basic design d).

**Figure 12.**Contour plots of the basic model with conical frustum (basic design b). The plots on the left show the model with a deviation angle of $\delta ={50}^{\circ}$, and the model on the right has a deviation angle of $\delta ={90}^{\circ}$. The contours on the mid-plane show normalised velocities, the contours encircled in solid black show secondary velocity ratios, and the contours encircled by dashed black lines show the turbulent intensity $TI=\sqrt{\frac{2k}{3{\left(\right)}^{{\overrightarrow{u}}_{101}}2}}$.

Basic Design | Varied Parameters | Shown in |
---|---|---|

(a) basic model | diameter ratio D_{51}/D_{1}, fillet radius R_{40}/D_{51}, diffuser angle β | Figure 5, Figure 6 and Figure 11 |

(b) basic model with conical frustum | diameter ratio D_{40}/D_{51}, deviation angle δ, diameter ratios D_{51}/D_{71} and D_{51}/D_{101}, pivot angle φ | Figure 7, Figures 11 and 12 |

(c) spherical manifold | sphere radius SR_{40}/D_{1}, deviation angle δ, diameter ratio D_{51}/D_{101} | Figure 8 and Figure 11 |

(d) cylindrical manifold | axial position T_{4}/D_{51}, deviation angle δ, fillet radius R_{40}/D_{51} | Figure 9 and Figure 11 |

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## Share and Cite

**MDPI and ACS Style**

Hahn, F.J.J.; Maly, A.; Semlitsch, B.; Bauer, C.
Numerical Investigation of Pelton Turbine Distributor Systems with Axial Inflow. *Energies* **2023**, *16*, 2737.
https://doi.org/10.3390/en16062737

**AMA Style**

Hahn FJJ, Maly A, Semlitsch B, Bauer C.
Numerical Investigation of Pelton Turbine Distributor Systems with Axial Inflow. *Energies*. 2023; 16(6):2737.
https://doi.org/10.3390/en16062737

**Chicago/Turabian Style**

Hahn, Franz Josef Johann, Anton Maly, Bernhard Semlitsch, and Christian Bauer.
2023. "Numerical Investigation of Pelton Turbine Distributor Systems with Axial Inflow" *Energies* 16, no. 6: 2737.
https://doi.org/10.3390/en16062737