XML Template (2013) [4.10.

2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Original Article

Proc IMechE Part A:

J Power and Energy
Influence of the bucket geometry on the 0(0) 1–13
! IMechE 2013
Pelton performance Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/0957650913506589
pia.sagepub.com

Antonio Rossetti, Giorgio Pavesi, Giovanna Cavazzini,

Alberto Santolin and Guido Ardizzon

Abstract
The increasing share of hydropower in world electricity production requires the development of standardized and
optimized design procedures leading to increasingly higher efficiency values.
To date, despite a certain amount of support from computational fluid dynamics, Pelton turbines are still characterized by
semiempirical design criteria that do not make it possible to optimize the jet–bucket interaction in order to maximize
turbine performance. Based on an analysis of particle flow tracks, this paper presents a hybrid Eulerian–Lagrangian
method to investigate the influence of bucket geometry on the Pelton efficiency at two different operating conditions.
Jet–bucket interaction was numerically analyzed by means of a traditional mesh-based numerical approach, using a
transient multi-phase homogeneous model. Subsequently, the numerical results were integrated using a predictor–
corrector algorithm, combining a fourth order Adams-Bashforth method as predictor and a fourth order Adams-
Moulton method as corrector, in order to determine the fluid particle trajectories on the rotating buckets. The particle
flow tracks were analyzed in detail to evaluate the single-particle performance in terms of discharged kinetic energy,
momentum variation, and total energy variation during the jet–bucket interaction. Moreover, on the basis of the particle
discharging position, the contribution of the different bucket areas to the total torque of the turbine was investigated to
determine the time-depending influence of the bucket geometry on the turbine energy exchange and to suggest possible
design solutions for improving bucket performance.

Keywords
Hydraulic turbines, turbine design, Pelton turbine, bucket geometry, Eulerian–Lagrangian method, performance analysis,
hydro power

Date received: 8 April 2013; accepted: 3 September 2013

Introduction the complex fluid-dynamical interaction between the

In the last decades, as a consequence of climate water jet and the rotating buckets, which greatly
change, the renewed interest in renewable energy affects turbine performance. The water jet, coming
sources has favored the development and deployment out from one or more nozzles and directed tangen-
of hydropower in the world. tially at the wheel, hits the bucket, making it to
Unlike large scale hydropower, whose exploitation rotate, and is split in half, before being deflected
is often not encouraged in some countries because it is back by almost 180 . During this jet–bucket inter-
considered not environmentally acceptable, small action, the water is not confined inside the runner,
scale hydro is acknowledged as a cost-effective and as is the case for the reaction turbines, but flows in
environmentally friendly technology with enormous the bucket surface with a free surface directly in con-
exploitation potential. As a consequence, an increas- tact with the air (the so-called ‘‘free-surface flow’’).
ing demand for new, small hydro power plants and
for the rehabilitation of old small plants is emerging
globally, coupled with the urgent need for standar- Department of Industrial Engineering, University of Padova, Padova,
dized design criteria to optimize turbine performance. Italy
Unlike Francis and Kaplan turbines, whose design
Corresponding author:
procedures are well-consolidated, Pelton turbines are Giovanna Cavazzini, Department of Industrial Engineering, University of
still characterized by semiempirical design criteria Padova, Via Venezia 1, 35131 Padova, Italy.
because of the complications involved in investigating Email: giovanna.cavazzini@unipd.it

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

2 Proc IMechE Part A: J Power and Energy 0(0)

Since the moment of momentum variation due to the the smoothed particle hydrodynamics (SPH)
jet–bucket interaction determines the turbine energy method,19 that solves the system of Euler equations
exchange, the key point for improving the Pelton tur- in Lagrangian formalism, treating the medium as
bine performance is the analysis of the free-surface weakly compressible by means of a discrete number
flow on the bucket. This represents a great challenge, of particles. In this method, initially developed for
both from an experimental and numerical point of unbounded domains, each particle infers its physical
view, due to the simultaneous presence of unsteady proprieties by the particles lying in the surroundings.
flows developing in a rotating frame of reference, However, its application to the analysis of the fluid-
moving sources and free surfaces. dynamics inside Pelton buckets gives rise to some dif-
Over the years, several authors have focused their ficulties in the treatment of boundary conditions due
research on this jet–bucket interaction, trying to study to the lack of neighboring particles at boundaries and
more in-depth the fluid-dynamics and the influence of to the loss of consistency of the interpolation domain.
the geometrical parameters on the bucket perform- To overcome this loss of accuracy in the vicinity of the
ance with the aim of providing information necessary boundaries, a hybrid SPH-ALE method was pro-
to define new design criteria. However, the few ana- posed, combining the SPH approximation with the
lytical instruments available for investigating the com- discretization of the inviscid Euler equations accord-
plex flow field inside the bucket initially addressed ing to the dual flow arbitrary Euler–Lagrangian
experimental analyses on the free jet at the exit of (ALE) description.20 Even though this technique
the turbine nozzle to pinpoint the influence of the sta- makes it possible to link the SPH method to the
tionary parts on the free jet shape.1–4 more traditional and consolidated finite volume
For a better comprehension of the flow in the method, taking the best of each and showing good
rotating bucket, theoretical analyses were carried capacity in predicting free-surface flows, its accuracy
out by Zhang who focused his attention on the spe- in terms of pressure gradients and viscous terms
cific contributions of the flow frictions on the bucket remains nonetheless unsatisfactory.
surface to the total losses and of the jet impact to the Further improvements are needed for all these
total energy transfer.5–8 meshless methods to be efficiently applied to the
With the development of the computational fluid- Pelton turbine analysis. This can be confirmed by
dynamics, the experimental investigations were com- the lack of studies able to present innovative design
bined with numerical analyses leading, once validated criteria for the Pelton buckets, whose design is still
by a comparison with the experimental results, to the based on semiempirical approaches.
creation of more information on the phenomena This paper presents an alternative method for ana-
under investigation.9 The first analyses of the jet– lyzing the free surface flow in a Pelton bucket on the
bucket interaction were carried out by Kvicinsky basis of a combined Eulerian–Lagrangian approach.
et al.10 and by Zoppé et al.11 who experimentally A traditional mesh-based Eulerian approach was
and numerically analyzed the incidence conditions of adopted to simulate the flow field in the runner
an axial-symmetric jet on a fixed Pelton bucket. bucket in order to exploit the acknowledged accuracy
Some unsuccessful attempts at studying a moving and stability of the mesh-based methods. To over-
Pelton turbine bucket were carried out by Hana,12, 13 come the above-mentioned analysis limits of this
Janetzky et al.14 and Mack and Moser.15 Perrig method, the numerical results were properly processed
et al.16 and Perrig17 succeeded in obtaining a numer- on the basis of a Lagrangian approach to determine
ical analysis of the cutting process of axial-symmetric the particle trajectories in the runner bucket and the
jets on the bucket cut-out. However, it was only in variations of all the energy quantities along these tra-
2009 that the effects of a real jet on the cutting process jectories. The method provides an in-depth investi-
of a complete single-injector and rotating Pelton tur- gation of the time-dependant influence of the bucket
bine, including both the penstock and casing, were geometry on the energy exchange during the entire
analyzed by Santolin et al.18 bucket duty cycle and also suggests possible design
Even though this analysis was able to simulate an solutions for improving bucket performance at
entire working cycle of the bucket with a good torque the maximum flow rate or in a wider operating
prediction, the Eulerian approach, on which it was range according to the characteristics of the installa-
based, did not allow the particle trajectories in the tion site.
bucket to be analyzed or the influence of the bucket
geometry on these trajectories, and consequently on
The Eulerian–Lagrangian approach
turbine performance, to be appreciated.
To overcome this problem, attention was placed on The analysis of the Pelton performance was based on
a Lagrangian description of the flow field, enabling a hybrid Eulerian-Langrangian approach which com-
the analysis to follow the particles in their motion, bined a numerical mesh-based analysis of the flow
and proposing complete Lagrangian meshless field in the Pelton bucket with an analytical
approaches. In particular, attempts to simulate the Lagrangian post-processing of the results on the
flow inside Pelton buckets were made by adopting basis of a proper integration method.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 3

data on the following Lagrangian integration of the

The Eulerian analysis of the flow field particle trajectories. The average size of the mesh
Unstable numerical analyses were carried out by volume element edge was defined as equal to 0.8
means of the commercial code Ansys CFX-12 on a mm in order to have at least one node located between
single-injector horizontal axis 22-buckets Pelton tur- two adjacent points in the Lagrangian post-processing
bine (Figure 1), whose main characteristics are trajectories with an average distance of 1.6 mm.
reported in Table 1. The multi-phase characteristics of the flow were
In order to reduce the mesh dimensions and the numerically analyzed using the homogeneous
computational costs, the longitudinal symmetry model21 with a high-resolution upwind scheme for
plane was exploited and only three of the 22 buckets the discretization of the continuity and momentum
of the runner were modelled (Figure 2). This config- equations. A gradient-dependent blend factor, char-
uration, even though simplified, made it possible to acterized by a good trade-off between diffusion and
take into account the fluid-dynamical interactions dispersion, weighted the physical advection terms.
between the buckets during the evolution in time of Since the standard k-" turbulence model was shown
the jet–bucket interaction. to provide unsatisfactory results in the prediction of
Both structured and unstructured meshes were the flow field in some bucket regions due to the high
adopted in the model discretization depending on shear stresses,16 a k-o SST turbulence model based on
the characteristics of the flow field to be investigated. the work by Menter22 with automatic Near-Wall
As regards the jet, the existence of a main flow direc- Treatment was adopted due to its superior perform-
tion suggested the use of a structured mesh whose ance for high accuracy boundary layer simulations.23
isotropy ensured greater accuracy and solution stabil- This model uses the Wilcox k-o in near wall regions
ity. On the other hand, since there was no preferred and the standard k-" turbulence model in the fully
flow direction in the runner and in the surrounding turbulent region far away from the wall. The transi-
atmospheric region, an unstructured mesh of more tion between the two models is guaranteed by the
than 3,000,000 nodes was adopted so as to exploit automatic near wall treatment for omega-based
the high performance of this type of mesh in the com- models that automatically switche from wall-
putational resolution of a flow field with anisotropic functions to a low Reynolds number near wall formu-
characteristics. More details about the number of lation as the mesh is refined.24
nodes and the types of elements used in the stator
and rotor parts are reported in Table 2.
A mesh sensitivity analysis was carried out on the
considered model in a previously published paper.18 Table 1. Main characteristics of the Pelton turbine.
However, as regards the bucket, the mesh was much
more refined (about 600,000 nodes per bucket) than B2/d0 n (rpm) D1 (mm)
that resulting from the sensitivity analysis (about
3.2 1500 420
42,000 nodes per bucket) due to the need for accurate

Figure 1. Schematic view of the single jet Pelton turbine.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

4 Proc IMechE Part A: J Power and Energy 0(0)

Figure 2. Mesh of the entire model (a) with details of the jet (b) and of the runner bucket (c).

Table 2. Mesh details.

Stator Rotor Total

Domain
Number of nodes 224,166 3,090,681 3,314,847

Element type EXA TETRA PYR TETRA EXA TETRA PYR

Number of elements 150,500 354,780 3000 17,983,541 150,500 18,338,321 3000

As regards the transient terms, a second-order

backward Euler scheme with a time step equal to Table 3. Analyzed operating conditions.
1/50 of the bucket pitch was adopted.
Q/QMAX djet (mm)
The buoyancy of the jet was also considered in the
analysis in order to reproduce its action on the jet 1.00 32.00
deviation. 0.30 17.52
As regards the boundary conditions, the mass flow
rate was fixed at the inlet boundary, whereas an open-
ing condition with an average atmospheric static pres-
sure level was prescribed at the bottom of the casing the numerical results obtained from the Eulerian ana-
and no-slip and adiabatic wall conditions were lysis of the flow field.
imposed at the solid surfaces. Moreover, since the To properly represent the jet, a significant number
analysis focused on the cutting process, the effects of of particles (153) was considered by subdividing the jet
the injector stationary parts on the jet–bucket inter- semi-section into an equal number of elements of simi-
action in terms of secondary flows and jet shape were lar area and by identifying their barycenters with the
not considered and the jet was supposed to have a particles themselves (Figure 3). The particle trajectory
constant diameter and axial-symmetric velocity analysis was started in a reference section X0 of the
profile. unperturbed jet with a distance from the runner axis
The analyses were carried for two different flow equal to 102.5% of the outer radius Re (Figure 4).
rates: the maximum rate and 30% of the maximum The numerical results were integrated by the
rate (Table 3). Adams–Bashforth–Moulton predictor–corrector
method.25 Considering the equation
The Lagrangian post-processing Ztkþ1
!
pðtkþ1 Þ ¼ pðtk Þ þ vðt, pðtÞÞdt ð1Þ
The aim of the Lagrangian post-processing was to
tk
determine the jet particle trajectories by integrating

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 5

Figure 3. Particle distribution in the jet semi-section.

A second Lagrange polynomial was determined on

the basis of the predicted position p*k þ 1 and of the
three previous points (tk  2, vk  2), (tk  1, vk  1), and
(tk, vk), and was used to correct the position estima-
tion by

t  
pkþ1 ¼ pk þ vk2  5vk1 þ 19vk þ 9vðtkþ1 , pkþ1 Þ
24
ð3Þ

A mass mi,tj evaluated by

 
mi,tj ¼ Ai  !
vi,tj tjþ1  tj ¼ Ai  !
vi,tj t ð4Þ

was associated to each particle i moving from the ref-

erence starting section X0 at the instant tj.
Figure 4. Starting location X0 of the particle trajectory In equation (4),  is the water density, Ai the area
analysis. of the cross section of the particle i (Figure 3), v the
velocity of the particle i at the instant tj, and
the predictor determined a  Lagrange
! polynomial app- Dt ¼ tj þ 1  tj the integration time step.
roximation for the velocity vðt, pðtÞÞ on the basis on the The portion of the jet interacting with a bucket was
points (tk  3, vk  3), (tk  2, vk  2), (tk  1, vk  1), and (tk, thus simulated by sets of mass-concentrated particles
vk) that was used to predict the new position p*k þ 1 by distributed on surfaces passing through the reference
section X0 with a time step t.
t
pkþ1 ¼ pk þ ð9vk3 þ 37vk2  59vk1 þ 55vk Þ Starting from the reference section X0 and for each
24 surface passing through it, the trajectories of the
ð2Þ mass-concentrated particles were evaluated by
where Dt is the step size. means of the predictor–corrector algorithm

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

6 Proc IMechE Part A: J Power and Energy 0(0)

mentioned above and only those interacting with the the jet mass M captured by a single bucket, the figure
selected bucket were isolated and taken into account reports the reference sections colored as a function of
to evaluate the unsteady bucket performance. the time instant moved forward along the jet axis at a
distance proportional to time according to the unper-
turbed jet velocity.
Validation of the analysis method
To verify that the torque determined by the particle
Since the results of the computational fluid-dynamics moment of momentum variation was independent of
(CFD) numerical analysis were already validated by the number of particles adopted in the jet semi-section
comparing them with the available experimental (Figure 3), a sensitivity analysis was also carried out
results,18 they were assumed to be a reference for based on the average bucket torque according to the
the validation of the Lagrangian post-processing. following equation
Since the proposed method was aimed at assessing
the Pelton turbine performance, the validation was R  
based on the comparison between the torque, deter- abs Tnp  T max
np
Tnp ¼ R ð6Þ
mined by CFD numerical computation of the forces T max
np
acting on the bucket, and the torque obtained by
taking into account the change of moment of momen-
tum of all the particles interacting with the bucket at where Tnp is the instant torque (equation (5)) deter-
the same time instant t, given by mined with np particles and Tmax
np is the instant torque
determined with the maximum considered number of
d ðr  !
np,s
ns X
X v mÞs,i particles (190).
Tmethod ðtÞ ¼ ð5Þ As shown in Table 4, the sensitivity analysis con-
s¼1 i¼1
dt
firmed that a number of particles equal to 153 is ade-
quate to obtain particle-independent results.
where ns is the number of sections, np,s is the number When the particles acting on the same bucket were
of particles in the section s interacting with the bucket isolated, the torque determined by sum of the particle
at the time instant t, r is the radius, and m and v are, changes in moment of momentum (Figure 6—broken
respectively, the mass (equation (4)) and the velocity line) was compared with the torque determined by a
of the particle i.
To properly determine the contribution of the flow Table 4. Particle sensitivity analysis.
particles to each bucket duty cycle, the particle trajec-
Number of particles (np) Tnp
tories moving from the reference section X0 at differ-
ent instants were highlighted on the basis of the 190 0.00
bucket on which they acted. Figure 5 gives an idea 153 0.00
of the results of this particle classification with refer- 100 0.02
ence to the second bucket. To highlight the time- 77 0.05
depending characteristics of the jet–bucket interaction
54 0.06
and to enable a global visualization of the portion of

Figure 5. Evolution in time of the jet–bucket interaction: the regions of the reference section whose particles interacted with the
second bucket are colored as a function of the corresponding time instant and moved forward by a distance proportional to time.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 7

3-D numerical analysis (Figure 6—continuous line). Figures 7 and 8 report the evolution in time of the
The agreement between the methods was quite good progressive normalized moment of momentum vari-
for both the analyzed operating conditions, even ation and of the progressive kinetic energy factor vari-
though, at partial load (Q/Qmax ¼ 30%), the torque ation along the particle tracks in the bucket for
determined by equation (5) highlights a slightly Q/Qmax ¼ 30% and Q/Qmax ¼ 100%. Three instants
greater value than the numerical one in the full jet– were represented in all the figures: a starting
bucket interaction time interval, perhaps due to a moment of the jet–bucket interaction (t1),
coarse jet discretization at part load. The good agree- the moment of full jet–bucket interaction (t2), and a
ment between the results was confirmed by the small moment of the jet release (t3).
errors in the determination of the maximum and mean At the time instant t1, the upper half of the jet
torque that was about 0.1% and 1.9% for Q/ section interacts with the bucket with very high effi-
Qmax ¼ 100% and about 3.8% and 2.7% for Q/ ciency values in both of the operating conditions
Qmax ¼ 30%. (Figure 7). At part load (Q/Qmax ¼ 30%), the particles
captured by the lowest part of the bucket were dis-
charged by the upper bucket region with greater
Results
values of the normalized moment of momentum vari-
The proposed Eulerian–Lagrangian method was ations (Figure 7) and maxima values of the kinetic
applied to determine the influence of the geometry energy factor (Figure 8).
on the resulting Pelton turbine performance. To At the maximum flow rate, the top of the jet sec-
evaluate the energy exchange and the bucket efficiency tion, even if characterized by moment of momentum
along the particle path, two different parameters were variations greater than 1.1 (Figure 7) due to the nega-
considered. On the one hand, the particle moment of tive velocity values at the discharge, maintained good
momentum variation along the path was normalized efficiency values with moderate even if not negligible
by the particle moment of momentum at the entrance kinetic energy losses at the discharge (Figure 8).
At the time instant t2, referring to a condition of
! ! ! !
r  mv  r  mv inlet full jet–bucket interaction, the jet section was charac-
! ! ð7Þ terized by great values of the energy exchange with
r  mv inlet normalized moment of momentum variations greater
than 0.8 (Figure 7) and kinetic energy factors greater
where r is the radius, v is the velocity, and m is the than 0.95 (Figure 8) in both the operating conditions.
particle mass; on the other hand, a kinetic energy At this instant (t2), the top section of the jet, dis-
factor was generated, representing a normalized charged by the upper part of the bucket, still pre-
degree of exploitation of the kinetic energy Ek avail- sented greater kinetic energy losses compared with
able at the entrance the remaining part of the jet with a kinetic energy
factor of around 85% (Figure 8).
ðEk Þinlet ðEk Þ Finally, during the jet release (t3), the lower half of
ð8Þ
ðEk Þinlet the jet impacted on the upper part of the bucket and

Figure 6. Comparison between the torque determined by numerical analysis (continuous line), with the forces acting on the bucket,
and the torque determined by the integration method (broken line), as the sum of the particle changes in moment of momentum: (a)
Q/Qmax ¼ 100%; (b) Q/Qmax ¼ 30%.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

8 Proc IMechE Part A: J Power and Energy 0(0)

Figure 7. Flow evolution on the bucket at three instants of the jet–bucket interaction for Q/Qmax ¼ 30% and Q/Qmax ¼ 100%. The
particle trajectories in the bucket are colored on the basis of the progressive normalized particle moment of momentum variation
(equation (7)). The jet sections are colored on the basis of the single-particle total variation of the normalized particle moment of
momentum between the entrance and the exit of the bucket.

Figure 8. Flow evolution on the bucket at three instants of the jet–bucket interaction for Q/Qmax ¼ 30% and Q/Qmax ¼ 100%. The
particle trajectories in the bucket are colored on the basis of the progressive kinetic energy factor (equation (8)). The jet sections are
colored on the basis of the single-particle total variation of the kinetic energy factor between the entrance and the exit of the bucket.

was discharged by its lower part with reduced effi- analyzed in terms of the contributions of the different
ciency values, confirmed by particle moment of bucket areas to the total bucket torque. The bucket
momentum variations lesser than 0.8 (Figure 7) was subdivided into six discharging zones, including
and kinetic energy factor lower than 0.8 (Figure 8). the bucket cut, and the particle contribution to the
For Q/Qmax ¼ 100%, the analysis also highlighted a energy exchange was assigned to different areas
leakage flow rate discharged by the bucket with great depending on the particle discharging position.
energy losses. Figure 9 reports the evolutions in time of the con-
The influence of the bucket geometry on the energy tributions of the different areas (Figure 9(a)
exchange of the turbine could also be quantitatively and (b), colored lines) to the total bucket torque

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 9

Figure 9. Contributions of the different bucket discharging areas to the total bucket torque for Q/Qmax ¼ 30% and Q/Qmax ¼ 100%:
(a–b) evolutions in time; (c–d) contribution as a percentage of total torque (red values) compared to the mass discharged by the
different areas as a percentage of the total discharged mass (black values).

(Figure 9(a) and (b), black line) and the percentage contribution to the total torque of the area d2 at
values of the area contribution to the total torque part load (10.1%) is less than half of that at maximum
(Figure 9(c) and (d), red values), together with the flow rate (21.1%). This behaviour, however, could be
corresponding discharged mass (Figure 9(c) and (d), simply justified by the different discharged mass
black values). (8.0% for Q/Q30% vs. 18.6% for Q/Q100%).
In both operating conditions, the maximum Moreover, even the areas d4 and d5 showed a dif-
instantaneous contribution to the total torque was ferent performance in the two analyzed operating con-
obtained in the starting period of the jet–bucket ditions. At maximum flow rate, the contribution to
interaction when the flow evolution mainly involved the total torque of the area d4 (22.6%) is smaller
the upper half (d2 and d3) of the bucket geometry than that of the area d5 (29.3%) involved in the
(Figure 9(a) and (b)). During the full jet–bucket inter- energy exchange for the longest period (Figure 9a)
action, all the discharging areas, with the exception of and discharging the greatest mass percentage
d1 and d6, significantly contributed to the energy (30.7%) (Figure 9c). On the other hand, at part
exchange, whereas during the jet release, the torque load, the area d5 is characterized by a smaller contri-
contribution associated with the upper half of the bution to the total torque (24.8%) than that of the
bucket (d2 and d3) decayed and the energy exchange area d4 (31.2%) (Figure 9(b)–(d)). However, this con-
mainly concerned the bottom half (d4 and d5). tribution reduction cannot be completely justified by
Comparing the evolutions in time of the torque the smaller discharged mass, since the mass difference
contributions of the two analyzed flow rates, it is is not so significant (Figure 9(d)—28.3% vs. 31.6%).
interesting to note some differences in bucket per- This may suggest a possible performance decay of the
formance among the discharging areas. First, the very bottom part of the bucket at part load.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

10 Proc IMechE Part A: J Power and Energy 0(0)

To verify this hypothesis, the capability of the contributions (Figure 10): one related to the reduction
bucket to deviate the particle trajectories at different of the mass discharged by each area
loads was analyzed and the average moment of !
 
momentum r2 Cu2 di at the bucket outlet was deter-   0 Cm2 100% Cm2 30%
mined as Cu2 30%100% ¼   
di tan2 100% tan2 100%
di
P 
 mi r2i Cu2i  ð10Þ
r2 Cu2 di ¼ P  ð9Þ
m i di
and one related to the variation of the bucket devi-
where mi, r2i and Cu2i are respectively the mass, the ation capability at different loads
radius and the residual tangential velocity of the par- !
 
ticle i discharged by the area di (with di ¼ d1, d2, d3, d4,   00 Cm2 30% Cm2 30%
d5 or d6). Cu2 30%100% ¼   
 di tan2 100% tan2 30%
The averaged residual tangential Cu2 di was also di
estimated by ð11Þ
 dividing the average moment of momen-
tum r2 Cu2 di (equation (8)) by the average outlet
radius r2 jdi .
As can be seen in Table 5, the averaged residual These two contributions were determined on the
tangential velocity discharged by the bucket at part basis of the averaged meridian velocity Cm2 di and
load was always greater than that discharged at the relative flow angle 2 di at the bucket outlet
maximum flow rate for each area di, demonstrating, in
the simplified hypothesis of equal averaged inlet con- P  P 
 m Cm2i   mi 2i 
ditions r1 Cu1 di , a global reduction of the resulting Cm2 di ¼ Pi 2 di ¼ P ð12Þ
mi di mi di
torque at part load. However, to highlight the influ-
ence of the bucket geometry on this result, the
increase of the
  residual tangential velocities and their sum was characterized by a relatively good
Cu2 30%100% was decomposed in two agreement with the total increase of the residual tan-
di
gential velocity (Tables 4 and 5).
This decomposition made it possible to confirm
that most of the bucket (areas d2, d3, and d4)
increased its deviation capability at part load, as
Table 5. Residual tangential velocities at the outlet of the demonstrated
  in00Table 6 by the negative values of
main bucket discharging areas. Cu2 30%100% , and that, in these areas, the
positive increase of the residual tangential velocity
Discharging area d2 d3 d4 d5

(Table 5) had to be related to the reduction of the
Cu2 30% (m/s) 4.43 5.27 10.01 12.67 mass flow rate. Therefore, the bucket discharging

Cu2 100% (m/s) 0.93 2.64 8.49 7.66 area d5 was the only one characterized by a perform-
  
Cu2 30%100% ¼ Cu2 30% Cu2 100% 3.50 2.63 1.52 5.01 ance decay at part load due to the influence of the
bucket geometry.

Figure 10. Decomposition of the residual tangential velocity at the bucket outlet.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 11

Table 6. Contributions to the global increase of the residual On the basis of these results, some suggestions on
tangential velocity at the outlet of the main bucket discharging possible design solutions to optimise the bucket per-
areas. formance can be deduced. On the one hand, to
Discharging area d2 d3 d4 d5 improve the bucket deviation capability, the con-
 structive angles 2c (Figure 11) of the main bucket
Cm2 30% (m/s) 12.17 9.17 9.42 8.95 discharging areas (d2, d3, and d4) could be further

Cm2 100% (m/s) 14.60 10.41 10.76 10.56 reduced within the limits permitted to avoid fluid

2 30% ( ) 25.98 19.17 20.36 20.19 interaction with the bucket back (Figure 12). This

   modification of the constructive angle distribution
2 100%( ) 0 27.27 19.91 21.98 19.32
 could favor an improvement of the bucket perform-
Cu2 30%100% 00(m/s) 4.72 3.44 3.30 4.60
Cu2 30%100% (m/s) 1.36 1.05 2.05 1.20 ance in the whole operating range.
 On the other hand, to reduce the flow deviation
Cu2 30%100% (m/s) 3.35 2.38 1.25 5.80
2–2c at the bucket outlet, the bucket width B2
(Figure 11) could be increased according to the empir-
ical correlation proposed by Nechleba26

15d0
2  2c ¼ ð13Þ
B2

The reduction of the flow deviation determines a

corresponding reduction of the residual tangential vel-
ocity Cu2 with a consequent increase of the bucket
energy exchange

ght ¼ u1 Cu1  u2 Cu2 ð14Þ

However, at part loads, the smaller jet diameter d0

and the increased bucket width B2 cause an increase of
the friction losses ghr due to the longer particle trajec-
tories (Figure 12). These energy losses, after reducing
the discharged relative velocity W2 and thereby
increasing the residual tangential velocity Cu2, could
reduce the beneficial effect of the bucket enlargement
on the energy exchange ght.

Conclusions
Figure 11. Design sections for the definition of the bucket An innovative method for analyzing the influence of
geometry. bucket geometry on the Pelton performance was

Figure 12. Distribution of the constructive angles along the bucket curvilinear abscissa.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

12 Proc IMechE Part A: J Power and Energy 0(0)

developed on the basis of a hybrid Eulerian– 2. Zhang Z and Parkinson E. LDA application and the
Lagrangian approach. dual-measurement-method in experimental investiga-
The free-surface flow in the moving buckets of a tions of the free-surface jet at a model nozzle of a
single-injector Pelton turbine was numerically ana- Pelton turbine. In: The 11th international symposium
on applications of laser anemometry to fluid mechanics,
lyzed by means of a traditional mesh-based Eulerian
Lisbon, Portugal, 8–11 July 2002. Lisbon, Portugal:
approach at two different operating conditions.
Symposium Organizing Committee.
Subsequently, the numerical results were integrated 3. Brekke H. State of art of small hydro turbines ver-
using a predictor–corrector algorithm, combining sus large turbines. In: Hydro 2005, Villach, Austria,
two linear multi-step methods, to determine the par- 17–20 October 2005. Surrey, UK: Aqua-Media
ticle trajectories as well as the variations of the energy International Ltd.
quantities along these trajectories. Consequently, the 4. Zhang Z and Casey M. Experimental studies of the jet
free-surface flow field on a single bucket was deter- of a Pelton turbine. Proc IMechE, Part A: J Power and
mined at several instants during the jet–bucket inter- Energy 2007; 221: 1181–1192.
action in order to completely define the bucket duty 5. Zhang Z. Flow interactions in Pelton turbines and the
cycle on the basis of the sum of the particles moment hydraulic efficiency of the turbine systems. Proc
IMechE, Part A: J Power and Energy 2007; 221:
of momentum variations. The method was finally vali-
343–357.
dated by comparing this duty cycle with that deter-
6. Zhang Z. Inlet flow conditions and the jet impact work
mined by numerical analysis on the basis of the forces in a Pelton turbine. Proc IMechE, Part A: J Power and
acting on the bucket. Energy 2009; 223: 589–596.
The analysis of the progressive variations of the 7. Zhang Z. Analytical method for frictional flows in a
particle moment of momentum and of the kinetic Pelton turbine. Proc IMechE, Part A: J Power and
energy losses along the trajectories made it possible Energy 2009; 223: 597–608.
to examine the influence of the bucket geometry on 8. Zhang Z. Flow dynamics of the free surface flow in the
the energy exchange during different instants of the rotating buckets of a Pelton turbine. Proc IMechE,
jet–bucket interaction. Part A: J Power and Energy 2009; 223: 609–623.
Moreover, on the basis of the particle trajectories at 9. Parkinson E, Garcin H, Vullioud G, et al. Experimental
and numerical investigations of the free jet flow at a
the bucket discharge, the bucket geometry was subdi-
model nozzle of a Pelton turbine. In: Proceedings
vided into six discharging areas to analyze the time-
of the XXI IAHR symposium on hydraulic machinery
dependant contribution of each area to the bucket and systems, Lausanne, Switzerland, 9–12 September
total torque. The deviation capability of the different 2002, pp.347–352. Lausanne, Switzerland: Ecole
bucket areas at part load and at maximum flow rate Polytechnique Fédérale de Lausanne.
was compared, highlighting a performance decay of 10. Kvicinsky S, Kueny J-L, Avellan F, et al. Experimental
the extreme bottom part of the bucket at part load. and numerical analysis of free surface flows in a rotat-
The different contributions to the identified perform- ing bucket. In: Proceedings of the XXI IAHR
ance drop were isolated and it was demonstrated that Symposium on Hydraulic Machinery and Systems,
most of the bucket increases its deviation capability at Lausanne, Switzerland, 9–12 September 2002,
part load, whereas its bottom part is characterized by a pp.359–364. Lausanne, Switzerland: Ecole Polytechnique
Fédérale de Lausanne.
performance decay referable to the bucket geometry.
11. Zoppé B, Pellone C, Maitre T, et al. Flow analysis
The method, exploiting the peculiarities of the
inside a Pelton turbine bucket. J Turbomach 2006;
Eulerian and Lagrangian approaches, overcame all 128: 500–511.
the limitations of the most recent Lagrangian mesh- 12. Hana M. A discussion on numerical simulation in Pelton
less analysis methods (such as the SPH-ALE) and turbines. In: Proceedings of the XIX IAHR symposium
made it possible to analyze the influence of the on hydraulic machinery and systems, Singapore,
bucket geometry on the turbine performance and to 9–11 September 1998, pp.306–315. Singapore: The
suggest possible solutions for optimizing bucket Environmental Engineering Society of Singapore.
design. 13. Hana M. Numerical analysis of non-stationary free sur-
face flow in a Pelton bucket. PhD Thesis, Norwegian
University of Science and Technology, 1999.
Funding 14. Janetzky B, Göde E, Ruprecht A, et al. Numerical
This research received no specific grant from any funding simulation of the flow in a Pelton bucket. In:
agency in the public, commercial, or not-for-profit sectors. Proceedings of the XIX IAHR symposium on hydraulic
machinery and systems, Singapore, 9–11 September
1998, pp.276–284. Singapore: The Environmental
References
Engineering Society of Singapore.
1. Zhang Z, Muggli F, Parkinson E and Schärer CH. 15. Mack R and Moser W. Numerical investigation of the
Experimental investigation of a low head jet flow at a flow in a Pelton turbine. In: Proceedings of the
model nozzle of a Pelton turbine. In: Proceedings of the XXI IAHR symposium on hydraulic machinery and
11th international seminar on hydropower plants, Vienna, systems, Lausanne, Switzerland, 9–12 September
Austria, 15–17 November 2000, pp.181–188. Vienna, 2002, pp.373–378. Lausanne, Switzerland: Ecole
Austria: Vienna University of Technology. Polytechnique Fédérale de Lausanne.

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014

XML Template (2013) [4.10.2013–11:02am] [1–13]
//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PIAJ/Vol00000/130085/APPFile/SG-PIAJ130085.3d (PIA) [PREPRINTER stage]

Rossetti et al. 13

16. Perrig A, Avellan F, Kueny JL, et al. Flow in a Pelton Appendix

turbine bucket: numerical and experimental investiga-
tions. J Fluid Eng-T ASME 2006; 128: 350–358. Notation
17. Perrig A. Hydrodynamics of the free surface flow in A area (m2)
Pelton turbine buckets. Lausanne, Switzerland: PhD
B2 bucket width (m)
Thesis, Swiss Federal Institute of Technology, 2007.
Cm meridian velocity (m/s)
18. Santolin A, Cavazzini G, Ardizzon G, et al. Numerical
investigation of the interaction between jet and buckets Cu tangential velocity (m/s)
in a Pelton turbine. Proc IMechE, Part A: J Power and djet jet diameter (m)
Energy 2009; 223: 721–728. d0 design jet diameter (m)
19. Marongiu J-C, Leboeuf F and Parkinson E. Numerical D1 Pelton diameter (m)
simulation of the flow in a Pelton turbine using the Ek kinetic energy (m2/s2)
meshless method smoothed particle hydrodynamics: a g gravitational constant (m/s2)
new simple solid boundary treatment. Proc IMechE, H head (m)
Part A: J Power and Energy 2007; 221: 849–856. l bucket curvilinear abscissa (mm)
20. Marongiu J-Ch, Leboeuf F, Caro J, et al. Free surface m particle mass (kg)
flows simulations in Pelton turbines using an hybrid
M mass (kg)
SPH-ALE method. J Hydraul Res 2010; 48: 40–49.
n rotation velocity (rpm)
21. ANSYS CFX Release 12.1 – Solver Modeling Guide.
Multiphase flow modeling: free surface flow. np number of particles (–)
Canonsburg, PA, USA: Ansys Inc., 2009, pp.146–147. ns number of sections (–)
22. Menter FR. Two-equation eddy-viscosity turbulence p particle position (m)
models for engineering applications. AIAA J 1994; 32: Q flow rate (l/s)
1598–1605. r radius (m)
23. Bardina JE, Huang PG and Coakley TJ. Turbulence Re outer runner radius (m)
modeling validation testing and development. NASA S needle stroke (m)
Technical Memorandum 110446 1997. t time (s)
24. ANSYS CFX Release 12.1 – Solver Modeling Guide. T torque (Nm)
Turbulence and near-wall modeling: modeling flow
V velocity (m/s)
near the wall. Canonsburg, PA, USA: Ansys Inc.,
x,y,z coordinates (m)
2009, pp.118–120.
25. Hairer E, Nørsett SP and Wanner G. Solving ordinary X0 reference section position (m)
differential equations I: nonstiff problems. Berlin:
Springer Verlag, 1993. b relative flow angle ( )
26. Nechleba M. Hydraulic turbines: their design and equip- bc bucket constructive angle ( )
ment. Prague, Artia: Constable & Company, 1957. r water density (kg/m3)

Downloaded from pia.sagepub.com at University Studi Padova on October 10, 2014