Design of high efficiency pelton turbine for micro hydropower

INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING
International Journal of & TECHNOLOGY (IJEET) (IJEET), ISSN 0976 –
Electrical Engineering and Technology
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 4, Issue 1, January- February (2013), pp. 171-183
IJEET
© IAEME: www.iaeme.com/ijeet.asp
Journal Impact Factor (2012): 3.2031 (Calculated by GISI) ©IAEME
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DESIGN OF HIGH EFFICIENCY PELTON TURBINE FOR MICRO-
HYDROPOWER PLANT

Bilal Abdullah Nasir1
1
Hawijah Technical Institute, Kirkuk, Iraq

ABSTRACT

The Pelton turbine was performed in high head and low water flow, in establishment
of micro-hydro electric power plant, due to its simple construction and ease of
manufacturing. To obtain a Pelton hydraulic turbine with maximum efficiency during various
operating conditions, the turbine parameters must be included in the design procedure. In this
paper all design parameters were calculated at maximum efficiency. These parameters
included turbine power, turbine torque, runner diameter, runner length, runner speed, bucket
dimensions, number of buckets, nozzle dimension and turbine specific speed.

Keywords:Pelton turbine, maximum efficiency, designparameters,micro-hydro power plant.

1. INTRODUCTION

Hydraulic turbine can be defined as a rotary machine, which uses the potential and
kinetic energy of water and converts it into useful mechanical energy. According to the way
of energy transfer, there are two types of hydraulic turbines namely impulse turbines and
reaction turbines. In impulse turbines water coming out of the nozzle at the end of the
penstock is made to strike a series of buckets fitted on the periphery of the runner. The runner
revolves freely in air and the casing is not important in impulse turbine. In a reaction turbine,
water enters all around the periphery of runner and the runner remains full of water every
time. The water leaves from the runner is discharged into the tailrace with a different
pressure. Therefor casing is necessary for reaction turbines [1].Pelton turbine is an impulse
turbine. The runner of the Pelton turbine consists of double hemispherical cups fitted on its
periphery as shown in figure (1).

171

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
Engineering
January

Figure (1) The runner ofthePelton turbine

The jet strikes these cups (buckets) at the central dividing edge of the front edge. The
central dividing edge is also called as splitter. The water jet strikes edge of the splitter
symmetrically and equally distributed into the two halves of hemispherical bucket.
Theoretically, if the buckets are exactly hemispherical, it will deflect the jet through 180°.
,
Then the velocity of the water jet leaving the bucket would be opposite in direction to the
velocity of jet entering. Practically, this can not be achieved because the jet le leaving the
bucket strikes the back of succeeding bucket and the overall efficiency would decrease. In
practice, the angular deflection of the jet in the bucket is limited to about 165°
165°→170°. The
amount of water discharges from the nozzle is regulated by a needle valve provided inside the
needle
nozzle as shown in figure (2).

Figure (2) The nozzle and deflector of the Pelton turbine: (a) the nozzle spear and deflector.
(b) The nozzle components.

A deflector is used to control the turbine speed. One or more water jets (nozzles) can
be provided with the Pelton turbine depending on the water flow rate and the nozzle capacity.

172


The Pelton turbines have been given increasing interest by the research community within
multiple fields. This is due to the increasing demand for energy on a global basis in addition
to the growing focus on meeting the increasing demand by utilizing renewable energy
resources. An increase in efficiency in the order 0.1 % would lead to large increase in
electrical power production.
Innovation within energy business is kept a close corporate secret and all research done on a
turbine designed by commercial companies is confidential. Thus the deferent research
communities have no common practical case with which they can cooperate within their
distinctive fields [2]. The literature on Pelton turbine design available is scarce at best due to
the competitive nature of the industry and the resulting secrecy surrounding design methods
and innovations. In the last decade a lot of papers about numerical and experimental analysis
and design of Pelton turbines have been published. A water jet from Pelton turbine injector
was analyzed experimentally and numerically by Barkinson in reference [3]. The influence of
jet velocity and jet quality on turbine efficiency were investigated by Vesely and Staubli in
references [4,5]. A bucket simulation using three adjacent buckets was shown by Mack and
Moser in reference [6]. Unsteady analysis of a Pelton runner with mechanical simulation was
presented by Parkinson in reference [7]. A numerical analysis of water flow in a two jets
Pelton turbine with horizontal axis was presented by Jost in reference [8]. A modification in
the bucket design of Pelton turbine is suggested by SurajYadav in reference [9] to increase
the efficiency of the Pelton turbine. The effect of runner to jet speed ratio on the Pelton
turbine efficiency is tested experimentally by Bryan in reference [10]. In this paper a
complete design procedure of a Pelton turbine, based on analytical and empirical calculation
of the turbine parameters. The design steps was presented by Matlab Simulink computer
program. The designed Pelton turbine was used in micro-hydro electric power plant to
operate at maximum efficiency.

2. DESIGN STEPS OF THE PELTON TURBINE
The design procedure of the Pelton turbine which is used in micro-hydro power
generation can be systematic as follow:

1. Preparing the site data of power plant

This involves the calculations and measuring the net head and the water flow rate.

‫ܪ‬௡ ൌ ‫ܪ‬௚ െ ‫ܪ‬௧௟
a. Calculation of the net head (Hn):
(1)

Where Hg = the gross head which is the vertical distance between water surface level at the
intake and at the turbine.
Htl = total head losses due to the open channel, trash rack, intake, penstock and gate or valve.
These losses approximately equal to 6% of gross head.
b. Calculation of the water flow rate (Q):
The water flow rate can be calculated by measuring the river or stream flow velocity (Vr) in
(m.s-1) and its cross-sectional area (Ar) in (m2), then:
ܳ ൌ ܸ ‫ܣ כ‬௥
௥ ሺ݉ଷ . ‫ି ݏ‬ଵ ሻ (2)

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2. Calculation of the turbine input power (Pti)

ܲ௧௜ ൌ ߩ ‫ܥ כ ݃ כ‬௡ ‫ܪ כ‬௡ ‫ܳ כ‬௧ ሺܹܽ‫ ݐݐ‬ሻ
ଶ
The electrical input power to the turbine in (Watt) can be calculated as:
(3)

3. Calculation of the turbine speed (N)

The correlation between the specific speed (Ns) and the net head (Hn) is given for the Pelton

ඥ݊௝
turbine as [11]:

ܰ௦ ൌ 85.49 ‫כ‬ ൘ ଴.ଶସଷ
‫ܪ‬௡
(4)

ܳ
݊௝ ൌ ௧ൗܳ
Where nj = number of turbine nozzles (jets), and can be calculated as:

௡ (5)

Where Qt = water flow capacity of each nozzle (m3.s-1).

‫ܪ‬௡
ହൗ
Then the turbine speed in (r.p.m) can be calculated as [11]:

ܰ ൌ ܰ௦ ‫כ‬ ൘
ସ

ඥܲ௧௜
(6)

4. Calculation of the runner circle diameter (Dr)

ܸ ൌ ‫ܥ‬௡ ‫ כ‬ඥ2 ‫ܪ כ ݃ כ‬௡ (m.s )
The water jet through nozzle has a velocity (Vj) in (m.s-1) can be calculated as [11]:
௝
-1
(7)

ߨܰ‫ܦ‬௥
The runner tangential velocity (Vtr) in (m.s-1) can be calculated as:
ܸ௧௥ ൌ ‫ܴ כ ݓ‬௥ ൌ
60
(m.s-1) (8)

Also the runner tangential velocity can be given as [11]:

ܸ௧௥ ൌ ‫ܸ כ ݔ‬௝ (m.s-1) (9)

Where x = ratio of runner tangential velocity to nozzle or jet velocity.

60 ‫ݔ כ‬
‫ܦ‬௥ ൌ ‫ܸכ‬
From equations (8) and (9):

ߨܰ ௝ (10)

At maximum efficiency the ratio of (x) between (0.46) to (0.47).
Then the runner diameter at maximum efficiency can be calculated from equations (7) and

ඥ‫ܪ‬௡൘
(10) as:

‫ܦ‬௥ ൌ 38.6 ‫כ‬ ܰ (11)

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The required diameter must be greater than this calculated value due to inconsistencies in the
manufacture of the buckets and the need to have a minimum distance of safety between the
nozzle and the Pelton runner.
If the turbine is free to rotate under no-load speed (run-away speed), the runner tangential
speed should be equal to the jet speed as:

‫ܦ‬௥ൗ ߨܰ௥ ‫ܦ‬௥
ܸ௧௥ ൌ ‫כ ݓ‬ 2 ൌ 60 ൌ ܸ௝

Or

60 ‫ܸ כ‬
ܰ௥ ൌ ௝
൘ሺߨ ‫ ܦ כ‬ሻ
௥ (r.p.m) (12)

The run-away speed (Nr) is independent on the water flow rate.

5. Calculation of nozzle dimensions [11]:

ܳ௡ ൌ ܸ ‫ܣ כ‬௝
The water flow rate through each nozzle (Qn) can be calculated as:
௝ (m3.s-1) (13)

‫ܦ‬ଶ
‫ܣ‬௝ ൌ ߨ ‫ כ‬௝ ൘
The nozzle area (Aj) can be calculated as:

4 (m2) (14)

4 ‫ܳ כ‬௧
Then from equations (13) and (14) the nozzle diameter (Dj) can be calculated as:
‫ܦ‬௝ ൌ ඨ ൘ ‫ ܸ כ ݊ כ‬ሻ (m)
ሺߨ ௝ ௝ (15)

ሺ‫ ܦ‬െ ‫ܦ‬௝ ሻ
‫ܮ‬௡ ൌ ௣௡ ൘
The nozzle length can be calculated as [11]:

tan ሺߚሻ (m) (16)
‫ܦ‬௣௧
‫ܦ‬௣௡ ൌ ൘ ݊
ඥ ௝ (m) (17)

The nozzle exits have to be located as close to the Pelton runner as possible to prevent the jet
from diverging the designed diameter. The distance between the nozzle and runner should be
5% of the runner circle diameter, plus an extra (3) mm clearance to account for emergency
deflectors as:

ܺ௡௥ ൌ 0.05 ‫ܦ כ‬௥ ൅ ‫ܦ‬௧ (m) (18)

175


The distance between nozzle and bucket taking into account the minimum clearance between
the nozzle and buckets was given as:

ܺ௡௕ ൌ 0.625 ‫ܦ כ‬௥ (m) (19)

The required distance was bigger than the calculated above, due to inconsistencies in the
manufacture of buckets and the need to have a minimum distance of safety between the
nozzle and Pelton runner.

6. Calculation of bucket dimensions [11]

The bucket axial width can be calculated as:

‫ܤ‬௪ ൌ 3.4 ‫ܦ כ‬௝ (m) (20)

The bucket radial length can be calculated as:

‫ܤ‬௟ ൌ 3 ‫ܦ כ‬௝ (m) (21)

The bucket depth can be calculated as:

‫ܤ‬ௗ ൌ 1.2 ‫ܦ כ‬௝ (m) (22)

The number of buckets in each runner must be determined so that no water particle was lost
while minimizing the risks of detrimental interactions between the out flowing water particles

‫ܦ‬
݊௕ ൌ 15 ൅ ௥൘ ‫ ܦ כ‬ሻ
and adjacent buckets. It can be calculated as:

ሺ2 ௝ (23)

The length of the moment arm of bucket can be calculated as:

‫ܮ‬௔௕ ൌ 0.195 ‫ܦ כ‬௥ (m) (24)

The runner size was determined by its diameter, and its shaper was determined by the number
of buckets. The runner shaft was sized to mount directly on the generator shaft. The flinger
seal was also necessary to seal the whole through which the generator shaft enters the turbine
box. The radius of bucket center of mass to center of runner was given as:

ܴ௕௥ ൌ 0.47 ‫ܦ כ‬௥ (m) (25)

The bucket volume was given as:

ܸ௕ ൌ 0.0063 ‫ܦ כ‬௥
ଷ (m3) (26)

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The mass of bucket can be calculated as:

‫ܯ‬௕ ൌ ߩ௠ ‫ܸ כ‬௕ (Kg) (27)

7. Penstock design [11]

The penstock was the piping that brings the water from the river or stream to the point where
it begins to be directed to the turbine. Penstock material was made of high pressure such as
PVC, chosen for its availability, affordability and low friction loss characteristics. The PVC
was susceptible to mechanical damage from ultraviolet radiation. If the piping was weakened
in any way, due to the high pressure in the penstock in any surge event, failure was a definite
possibility. To prevent this, the PVC piping would be founded in concrete along its entire
length. At the bottom of penstock, before the entrance to the turbine house, a valve would be
installed. This valve would allow the penstock to be emptied for turbine maintenance.
The thickness of penstock was chosen by determining the potential water hummer effect. A
water hummer effect was a surge pressure that occurs when the nozzles in turbine become
plugged and the flow in penstock was suddenly stopped. The thickness of penstock (tb) can be
calculated by the following relations;

‫ܦ‬௣௧ ൅ 508
‫ݐ‬௣ ൌ ቈቆ ቇ ൅ 1.2቉ ‫ି01 כ‬ଷ
400
(m) (28)

ଶ ‫ܮ‬
‫ܦ‬௣௧ ൌ 2.69 ‫ כ‬ሺ݊௣ ‫ܳ כ‬௧ ‫ כ‬௣௧൘‫ ܪ‬ሻ଴.ଵ଼଻ହ
ଶ
௚ (m) (29)

‫ܪ‬௦ ൌ ܸ ‫ܸ∆ כ‬ൗ݃
The surge pressure (Hs) in (m) can be calculated as:

௪ (m) (30)

1
ܸ ൌ
௪
ඩ 1 ‫ܦ‬௣௧
ߩ௪ ‫ כ‬ሺ‫ ܭ‬൅ ‫ ݐ כ ܯ‬ሻ
(m.s-1) (31)
௪௠ ௣ ௣
ܳ
∆ܸ ൌ
݊௝ ‫ܣ כ‬௣ (m.s-1) (32)

The safety factor to mitigate the water hummer effect in the design of micro-hydro electric

‫ݐ‬௣௘ ‫ݐ כ‬௦௣
power plant was given as [13, 14]:
ܵ. ‫ ܨ‬ൌ ሺ൐ 2.5ሻ
5 ‫01 כ‬ଷ ‫ܦ כ‬௣௧ ‫ܪ כ‬௧
(33)

‫ܪ‬௧ ൌ ሺ‫ܪ‬௚ ൅ ‫ܪ‬௦ ሻ (m) (34)

177


8. Deflector design [15, 16]

An emergency deflector system must be installed to protect the generator in case a load
circuit failure, and the generator rotates at over speed.
The force in each deflector can be calculated as:

‫ܨ‬ௗ ൌ ߩ௪ ‫ܳ כ‬௡ ‫ܸ כ‬௝ (Newton) (35)

The required force in each deflector was given as:

‫ܨ‬ௗ௥ ൌ ‫ܨ‬ௗ ‫݂ .ܵ כ‬ (Newton) (36)

The torque acting on the deflector arm was given as:

ܶௗ ൌ ‫ܨ‬ௗ௥ ‫ܴ כ‬ௗ (N.m) (37)

The required torque acting on the deflector arm was given as:

ܶௗ௥ ൌ ܶௗ ‫ܨ כ‬௖ (N.m) (38)

9. Calculation of maximum turbine efficiency [12]

The turbine efficiency generally affected by three factors:

i.) Hydraulic losses or power losses those occur due to flow irregularity within the bucket.
ii.)Windage losses which occur because of resistance in the air to the moving bucket.
iii.) Mechanical losses in the system used to transmit the power from the turbine to the
generator. If the turbine was mounted directly to the generator, there were no mechanical
losses in the turbine.

ߩ௪ ‫ܳ כ‬௧ ‫ ܸ כ‬ଶ
The input power to the turbine can be calculated as:

ܲ௧௜ ൌ
௝
2
(W) (39)

The power output developed by the turbine was given as:

ܲ௧௢ ൌ ߩ௪ ‫ܳ כ‬௧ ‫ܸ כ‬௧௥ ‫ כ‬ሾሺܸ െ ܸ௧௥ ሻሺ1 ൅ ߰ ‫ כ‬cosሺ߶ሻሻሿ
௝ (W) (40)

Then the turbine hydraulic efficiency can be calculated as:

ܲ௧௢ 2 ‫ܸ כ‬௧௥ ‫ כ‬ሺܸ െ ܸ௧௥ ሻሺ1 ൅ ߰ ‫ כ‬cosሺ߶ሻሻ
ߟ௧௛ ൌ ൌ
௝
ܲ௧௜ ܸଶ
௝
(41)

߶ ൌ 180° െ ߠ
ߠ ൌ 160° ՜ 170°
(42)

178


݀ሺߟ௧௛ ሻ
For maximum hydraulic turbine efficiency:

൘ ݀ሺܸ௧௥ ሻ ൌ 0

Or ܸ௧௥ ൌ 0.5 ‫ܸ כ‬
(43)
௝

Then the maximum hydraulic efficiency was given as:

ሾ1 ൅ ߰ ‫ כ‬cosሺ߶ሻሿ
ߟ௧௛ሺ௠௔௫.ሻ ൌ
2 (44)

The turbine windage efficiency was given as:

‫ܭ‬ௗ ‫ߩ כ‬௔ ‫ܣ כ‬௕ ‫ ݔ כ‬ଷ
ߟ௧௪ ൌ 1 െ
ߩ௪ ‫ܣ כ‬௝
(45)

Then the total turbine efficiency was given as:

ߟ௧ ൌ ߟ௧௛ ‫ߟ כ‬௧௪ ‫ߟ כ‬௧௠ (46)

If the turbine was mounted directly to the generator the mechanical losses can be neglected
and the mechanical efficiency (ηtm) equal to unity.

The torque developed by the turbine can be calculated as:

ܲ௧௢
ܶ௧ ൌ ൌ ܳ௧ ‫ܦ כ‬௥ ‫ כ‬ሺܸ െ ܸ௧௥ ሻ
‫ݓ‬ ௝ (N.m) (47)

3. RESULTS

The design calculations of the Pelton turbine were implemented by a Matlab Simulink
computer program. Table (1) shows the design parameters of the Pelton turbine with constant
flow rate (Q = 0.1 m3.s-1) and variable gross head of the plant site (Hg= 50 → 140 m), while
table (2) shows the same turbine parameters at constant head (Hg = 50, 60 m) with variable
water flow rate (Q = 0.1→ 0.4 m3.s-1) of the site. Figure (3) shows the variation of runner to
nozzle diameter ratio with specific speed at different values of water flow rate, while figure
(4) shows the variation of the same ratio with the nozzle length.
From these results, the turbine maximum efficiency was found to be 97% constant. In case of
variable head, all the design parameters were varied with head except of number of runner
buckets and runner diameter, while in a variable flow rate all the design parameters were
constant except of turbine power, specific speed and nozzle length.

179


Dr/Dj
10

9.5

9

Runner/Nozzle Diameter Ratio 8.5

8

7.5

7

6.5 3 3 3 3
Q=0.1m /s Q=0.2m /s Q=0.3m /s Q=0.4m /s
6

5.5

5
20 30 40 50 60 70 80
Specific Speed Ns

Figure (3) Variation of runner to nozzle diameter ratio with specific speed at different values of water
flow rate.
Dr/Dj

10

9.5

9

8.5
Runner/Nozzle Diameter Ratio

8

7.5

7

6.5 3 3 3 3
Q=0.4 m /s Q=0.3 m /s Q=0.2 m /s Q=0.1 m /s
6

5.5

5
180 200 220 240 260 280 300
Nozzle Length Ln (mm)

Figure (4)Variation of runner to nozzle diameter ratio with nozzle length at different values of
water flow rate.

Table (1) design parameters of the Pelton turbine at maximum efficiency and constant flow rate
(Qt =0. 1 m3.s-1)
Pto ηt Tt N Dj Vtr Vj Ln
Hg (m) Ns Dr(m) nb
(Kw) (%) (N.m) (r.p.m) (m) (m) (m) (m)
50 43 96.9 660 620 33.5 0.43 0.065 13.9 29.8 0.27 19
60 51.5 96.9 722 680 32 0.43 0.062 15.2 32.5 0.26 19
70 60 96.9 779 735 31 0.43 0.060 16.4 35.2 0.25 19
80 68.5 96.9 832 787 30 0.43 0.058 17.5 37.5 0.25 19
90 77.2 96.9 882 835 29 0.43 0.056 18.6 40 0.24 19
100 85.8 96.9 930 881 28.3 0.43 0.055 19.6 42 0.24 19
110 94.3 96.9 974 925 27.7 0.42 0.053 20.6 44 0.24 19
120 103 96.9 1016 967 27 0.42 0.052 21.5 46 0.233 19
130 111.5 96.9 1057 1007 26.6 0.42 0.051 22.4 48 0.230 19
140 120 96.9 1097 1045 26 0.42 0.050 23.2 50 0.228 19

180


Table (2) design parameters of the Pelton turbine at maximum efficiency and constant gross head

Qt Pto Tt ηt N Dj Vtr Vj Ln
Hg(m) Ns Dr(m) nb
(m3.s-1) (Kw) (N.m) (%) (r.p.m) (m) (m) (m) (m)
0.1 43 660 96.9 620 33.5 0.43 0.065 13.9 29.8 0.270 19
0.2 85.8 1320 96.9 620 47.4 0.43 0.065 13.9 29.8 0.240 19
(50) 0.3 129 1980 96.9 620 58 0.43 0.065 13.9 29.8 0.220 19
0.4 171.5 2640 96.9 620 67.1 0.43 0.065 13.9 29.8 0.208 19
0.1 51.5 722 96.9 680 32 0.43 0.062 15.2 32.5 0.250 19
0.2 103 1444 96.9 680 45.4 0.43 0.062 15.2 32.6 0.230 19
(60) 0.3 154.4 2167 96.9 680 55.6 0.43 0.062 15.2 32.6 0.210 19
0.4 206 2889 96.9 680 64.2 0.43 0.062 15.2 32.6 0.202 19

4. CONCLUSIONS
The Pelton turbine is suitable for installing small hydro-electric power plants in case of high
head and low water flow rate. A complete design of such turbines has been presented in this paper
based on theoretical analysis and some empirical relations. The maximum turbine efficiency was
found to be 97% constant for different values of head and water flow rate. The complete design
parameters such as turbine power, turbine torque, turbine speed, runner dimensions andnozzle
dimensionsare determined at maximum turbine efficiency.

Nomenclature

Ab Peripheral area of penstock (m2)
Aj Jet or nozzle cross-sectional area (m2)
Ap penstock cross-sectional area (m2)
Ar River or steam cross-sectional area (m2)
Bd Bucket depth (m)
Bl Bucket radial length (m)

Nozzle (jet) discharge coefficient (≅ 0.98)
Bw Bucket axial width (m)
Cn
Dj Jet or nozzle diameter (m)
Dpn Diameter of penstock connected to the nozzle (m)
Dpt Diameter of penstock connected to the turbine (m)
Dr Runner (wheel) circle diameter (m)

Friction factor acted upon by bearings (≅1.2)
Dt Deflector thickness (m)
Fc
Fd Deflector force (N)
Fdr Required deflector force (N)
g Gravity acceleration constant (9.81 m.s-2)
Hg Gross head (m)
Hn Net head (m)
Hs Surge head (m)
Ht Total head (m)
Htl Total head loss (m)
Kd Drag coefficient
Kwm Bulk water modulus (2.1*109 N.m-2)
Lab Length of bucket moment arm (m)
Ln Nozzle length (m)

181


Lpt Length of penstock between intake and turbine (m)
Mb Mass of bucket (Kg)
Mp Modulus of penstock material (for PVC = 2.8*109 N.m-2)
nb Number of buckets
nj Number of turbine nozzles
np Manning factor of penstock
N Turbine (runner) speed (r.p.m)
Nr Turbine run-away speed (r.p.m)
Ns Turbine specific speed
Pti Turbine input power (watt)
Pto Turbine output power (watt)
Qn Nozzle flow rate (m3.s-1)
Qt turbine flow rate (m3.s-1)
Rbr Radius of bucket center of mass to runner center (m)
Rd Radius of deflector arm (m)
Rr Radius of runner (m)
S.F Safety factor to prevent water hummer effect (> 2.5)
tp Thickness of penstock (m)
tpe Effective thickness of penstock (m)
tsp Tensile strength of penstock material (N.m-2)
Td Deflector torque (N.m)
Tdr Required deflector torque (N.m)
Tt turbine torque (N.m)
Vb Volume of bucket (m3)
Vj Water jet velocity (m.s-1)
Vr River velocity (m.s-1)
Vtr Runner tangential velocity (m.s-1)
Vw Pressure wave velocity (m.s-1)
x Ratio of runner tangential velocity to jet velocity
Xnb Distance between bucket and nozzle (m)
Xnr Distance between nozzle and runner (m)
V Change in velocity of penstock (m.s-1)

ߚ
Greek symbols

߰ Bucket roughness coefficient (≅ 0.98)
Nozzle tapper angle (degrees)

ߠ
ߩa
Deflection angle between bucket and jet (160°→170°)

ߩm
Air density (1.23 Kg.m-3)

ߩw
Density of bucket material (Kg.m-3)

߱
Water density (1000 Kg.m-3)

߱r
Runner velocity (radian.sec-1)

ߟt
Runner run-away velocity (radian.sec-1)

ߟth
Total turbine efficiency

ߟtm
Turbine hydraulic efficiency

ߟtw
Turbine mechanical efficiency
Turbine windage efficiency

182


REFERENCES

[1]Atthanayake, I. U.: "Analytical study on flow through a Pelton turbine bucket using
boundary layer theory", International Journal of Engineering and Technology (IJET), Vol.
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[2]Solimslie, B. W. and Dahlhaug, O. G.: "A reference Pelton turbine design", 6th, IAHR
Symposium on Hydraulic Machinery and Systems, IOP Publishing, IOP Conf. Series:
Earth and Environmental Science, 15, 2012.
[3] Parkinson, E. and et al.: "Experimental and numerical investigation of free jet flow at a
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