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Master Thesis
5kW and 100kW Pelton Turbine Design
and Performance Analysis for Air
Injected Operation
Supervisor: Professor Young-Ho LEE
February 2018
Department of Mechanical Engineering
Graduate School of Korea Maritime and Ocean University
Batbeleg Tuvshintugs
[UCI]I804:21028-200000014068[UCI]I804:21028-200000014068
Table of contents
Chapter 1. Introduction .................................................................................................... 1
Background ............................................................................................................ 1
Development of numerical methods of Pelton turbine .......................................... 3
Chapter 2. Pelton turbine theory ...................................................................................... 5
Components of Pelton turbine................................................................................ 5
Working principle of Pelton turbine ...................................................................... 6
Pelton turbine velocity triangle ...................................................................... 7
Impulse force ................................................................................................. 9
Power and efficiency ...................................................................................... 9
Pelton turbines specifications............................................................................... 11
Geometric specification of the Pelton wheel ............................................... 12
Installation form of Pelton turbine. .............................................................. 12
Hydraulic specification of the Pelton turbine ............................................... 13
Chapter 3. Design of Pelton turbine .............................................................................. 15
Dimension criteria of Pelton turbine .................................................................... 15
Design of 5kW Pelton turbine. ..................................................................... 15
Design of 100kW Pelton turbine with one nozzle ....................................... 17
Design of 100kW Pelton turbine with two nozzles ...................................... 18
CAD modeling ..................................................................................................... 19
Model of 5 kW Pelton turbine bucket .................................................................. 26
Model of 100 kW Pelton turbine bucket with one nozzle .................................... 27
Model of 100 kW Pelton turbine bucket with two nozzles .................................. 28
Chapter 4. Methodology ................................................................................................ 29
CFD introduction ................................................................................................. 29
Governing equations ............................................................................................ 29
Turbulence modelling .......................................................................................... 30
k-ε turbulence model. .................................................................................. 31
Wilcox k-𝝎 turbulence model...................................................................... 32
Shear stress transport model ........................................................................ 33
Cavitation models ................................................................................................ 34
Modelling of computational domain .................................................................... 36
Creating domain of one jet Pelton turbine ................................................... 37
Creating domain of two jet Pelton turbine ................................................... 38
Mesh generation ................................................................................................... 39
Mesh generation for the 5kW Pelton turbine. .............................................. 40
Mesh generation for the 100kW Pelton turbine. .......................................... 44
Physical set up ...................................................................................................... 45
5kW Pelton turbine performance analysis with different rotational speed .. 46
Physical setups of 5kW Pelton turbine with air injected jet ......................... 48
Physical setup of 100kW Pelton turbine with two nozzles .......................... 50
Chapter 5. Results and Discussion................................................................................. 53
5kW turbine performance with different rotational speed ................................... 53
5kW Pelton turbine with air injected jet .............................................................. 58
Air injection method .................................................................................... 58
Energy definition of the air injected operation and efficiency calculation .. 60
Results of the 5kW Pelton turbine with the air injected jet analysis ............ 60
100kW Pelton turbine normal operation and with air injected jet analysis and
comparison ....................................................................................................................... 65
Analysis of 100kW Pelton turbine with two nozzles ........................................... 69
Cavitation analysis of 5kW Pelton turbine........................................................... 72
Chapter 6. Conclusion ................................................................................................... 75
Acknowledgement ............................................................................................................... 76
References ..................................................................................................................... 77
Table of figures
Figure 1. 1 Typical hydroelectricity ....................................................................................... 1
Figure 1. 2 Turbine selection chart ........................................................................................ 2
Figure 2. 1 Main parts of the Pelton turbine .......................................................................... 6
Figure 2. 2 Velocity triangle for Pelton Turbine .................................................................... 7
Figure 2. 3 Extracted power as a function of jet speed ratio ................................................ 11
Figure 2. 4 Geometric specification of a Pelton wheel ........................................................ 12
Figure 3. 1 Creation of horizontal profile curve of the bucket (green) ................................ 20
Figure 3. 2 Creation of longitudinal profile curve of the bucket (green) ............................. 21
Figure 3. 3 Outer pattern of bucket ...................................................................................... 21
Figure 3. 4 Supporting curve of the bucket inside surface ................................................... 22
Figure 3. 5 Creation of inner surface creation (orange surface)........................................... 22
Figure 3. 6 Guide curves of bucket outer surface (orange lines) ......................................... 23
Figure 3. 7 Creation of bucket outer surface (orange surface) ............................................. 23
Figure 3. 8 Bucket basic shape after sewing inner and outer surfaces ................................. 24
Figure 3. 9 Notch creating ................................................................................................... 25
Figure 3. 10 Groove visualization of the bucket .................................................................. 25
Figure 3. 11 Bucket dimensions and wheel size of 5kW Pelton turbine .............................. 26
Figure 3. 12 3D designing of bucket on NX8.5 modelling tool ........................................... 26
Figure 3. 13 Bucket dimension and wheel size of 100kW Pelton turbine with on nozzle ... 27
Figure 3. 14 Bucket dimensions and wheel size of 100kW Pelton turbine with two
nozzles ................................................................................................................................. 28
Figure 4. 1 Work flow of numerical analysis ....................................................................... 36
Figure 4. 2 Sketch of computational domain of 5kW turbine .............................................. 37
Figure 4. 3 Sketch of the computational domain of Pelton turbine with two nozzles ......... 38
Figure 4. 4 Number of meshes against normalized torque................................................... 41
Figure 4. 5 Stationary domain mesh generation................................................................... 41
Figure 4. 6 Volume meshes of the stationary domain .......................................................... 42
Figure 4. 7 Rotating domain mesh generation ..................................................................... 43
Figure 4. 8 Volume mesh generation inside the bucket ....................................................... 43
Figure 4. 9 Prismatic mesh generation on the bucket wall .................................................. 43
Figure 4. 10 Mesh generation for stationary domain of 100kW Pelton turbine with two
nozzles ................................................................................................................................. 44
Figure 4. 11 Mesh generation for rotating domain of 100kW Pelton turbine with two nozzles
............................................................................................................................................. 45
Figure 4. 12 Boundary conditions of Pelton turbine simulation on Ansys CFX ................. 48
Figure 4. 13 Boundary conditions of Pelton turbine simulation on CFX ............................ 52
Figure 5. 1 Torque oscillation with bucket angular position ............................................... 53
Figure 5. 2 Total torque creation on the runner. Calculated total torque (red). Single bucket
torque (green). Timer averaged total runner torque (blue) .................................................. 54
Figure 5. 3 Runner efficiency and power with rotational speed .......................................... 56
Figure 5. 4 Flow visualization respective with angular position case of 750RPM. ISO surface
volume fraction at value of 0.5 and color is indicating water velocity [m/s] ....................... 58
Figure 5. 5 Air injection scheme in the nozzle .................................................................... 59
Figure 5. 6 Scheme of normal operation condition .............................................................. 59
Figure 5. 7 Power comparison of normal and air injected operation ................................... 62
Figure 5. 8 Efficiency comparison of normal and air injected operation ............................. 62
Figure 5. 9 Visualization of air distribution on the inlet boundary. Air injection rate 5% of
the water ............................................................................................................................... 63
Figure 5. 10 Air bubbles inside the jet ................................................................................. 64
Figure 5. 11 Air bubble transport inside the bucket profile ................................................. 64
Figure 5. 12 Velocity of the air injected jet ......................................................................... 65
Figure 5. 13 Power comparison of normal and air injected operation ................................. 68
Figure 5. 14 Efficiency comparison of normal and air injected operation ........................... 68
Figure 5. 15 Power comparison of normal and air injected operation of Pelton turbine with
two nozzles........................................................................................................................... 70
Figure 5. 16 Efficiency comparison of normal and air injected operation of 100kW Pelton
turbine with two nozzles ...................................................................................................... 71
Figure 5. 17 Comparison of power and efficiency of air injected operation of one and two
nozzle 100kW turbines ........................................................................................................ 71
Figure 5. 18 Comparison of vapor visualization .................................................................. 74
List of tables
Table 3. 1 Specific speed calculation function of flow rate, head and rotational speed ...... 16
Table 3. 2 Specific speed function of flow rate, head and rotational speed ......................... 16
Table 3. 3 Design parameters of 5 kW Pelton turbine ......................................................... 17
Table 3. 4 Design parameters of 100kW Pelton turbine with one nozzle ............................ 18
Table 3. 5 Design parameters of the two nozzles 100kW Pelton turbine ............................ 19
Table 4. 1 Mesh statistic of mesh independency test ........................................................... 40
Table 4. 2 Mesh generation of 100kW two nozzle turbine .................................................. 45
Table 4. 3 Summary of the physical setup ........................................................................... 46
Table 4. 4 Boundary condition details ................................................................................. 47
Table 4. 5 Definition of mass flow rate on the inlet boundary for air injected jet analysis . 49
Table 4. 6 Inlet boundary option for air injected jet analysis ............................................... 49
Table 4. 7 Summary of the physical setup ........................................................................... 50
Table 4. 8 Boundary condition details ................................................................................. 51
Table 5. 1 Case study of the 5kW Pelton turbine analysis ................................................... 55
Table 5. 2 Air injected jet results ......................................................................................... 61
Table 5. 3 100kW Pelton turbine results .............................................................................. 66
Table 5. 4 Air injected jet results for 100kW one nozzle turbine ........................................ 67
Table 5. 5 Results of 100kW two nozzle Pelton turbine analysis ........................................ 69
Table 5. 6 Air injected result for 100kW two nozzle turbine ............................................... 70
5kW and 100kW Pelton Turbine Design and
Performance Analysis for Air Injected
Operation
Tuvshintugs Batbeleg
Department of Mechanical Engineering
Graduate School of Korea Maritime and Ocean University
Abstract
In this study, a new efficiency improvement method by means of air injection
into Pelton turbine jet is presented. In the penstock, when a certain percentage of air
is injected into the water the flow rate of the combined air water mixture will increase.
Then this increased velocity of mixture produces more power on the runner. In order
to analyze air injected jet performance, firstly, micro size 5kW Pelton turbine was
designed and analyzed. Similarly the analysis was carried out on a small size 100kW
turbine. Furthermore, Cavitation analysis was performed for both water injected jet
analysis and normal jet analysis, in order to determine if there is tendency for vapor
to form with the air injected method.
In the hydraulic industry, there isn’t a specific designing method of Pelton turbine
available and commercial Pelton turbine design is secure within manufacturers due
to high competitive market. However, there are some literature available relating to
general thumb rules of design of Pelton turbine. Recent model of Pelton turbine was
designed according to general rules. Specific section of the bucket such as bucket
profile, bucket back side and notch is based on designer’s idea. For the 100kW Pelton
turbine, 5kW model was scaled up based on the jet diameter and rotational speed.
5kW performance analysis was performed under several rotational speed. After
achieving optimized rotational speed. Air injected jet analysis was performed for
rated rotational speed condition. Air injection rate was 1%, 5%, 10%, 15% of the
water volume flow rate. Efficiency increase rate was reliable, therefore same analysis
performed for a 100kW Pelton turbine. The 100kW turbine was designed in two
types, one jet and two jets systems. Air injection rate was made same and also found
to have a reliable increase in efficiency. ICEM CFD meshing tool was used for
meshing and CFX 13 commercial tool was used for numerical analysis.
Key words: Pelton turbine, Air injected jet, numerical analysis, Efficiency increase
rate
Nomenclature
B Bucket width mm
d0 Jet diameter mm
𝐷𝑚 Pitch circle diameter mm
𝐹𝑏𝑢𝑐𝑘𝑒𝑡 Bucket force developed by mass N
g Gravitational acceleration m/s2
H Net head m
k Runner peripheral speed coefficient or jet speed ratio
��𝑤 Discharge mass flow rate from nozzle kg/s
N Bucket number
n Turbine rotational speed rpm
nq Specific speed rps
𝑃ℎ Hydraulic power kW
𝑃𝑠 Penstock static pressure Pa
𝑄𝑎𝑖𝑟 Air flow rate m3/s
��𝑗𝑒𝑡 Jet volumetric flow rate m3/s
u Bucket tangential velocity m/s
𝑉1 Jet speed m/s
𝑣𝑟1 Bucket inlet relative velocity m/s
𝑣𝑟2 Bucket outlet relative velocity m/s
𝑣𝑤2 Horizontal component of the absolute velocity on
outlet
m/s
𝑣𝑦 absolute velocity of the jet at the outlet m/s
𝑣𝑦2 Vertical component of the absolute velocity on outlet m/s
𝑍𝑗𝑒𝑡 Jet number
𝜂ℎ Turbine efficiency %
𝜃 Bucket outlet angle degrees
𝜏 Torque on the runner N∙m
𝜑𝐵 Bucket volumetric load
𝜔 Angular velocity rad/s
1
Chapter 1. Introduction
Background
Nowadays environment pollution due to energy generation is becoming one of
the most faced problem of the modern civilization. Energy generation based on the
fossil fuel is becoming not reliable due to climate change and depletion of the reserves
therefore, renewable energy is vital sustainable source for the future. Hydro power is
considered the most efficient and stable energy producing technology among other
renewable energy technologies [1].
Hydro power or hydroelectricity refers to the conversion of energy from flowing
water into electricity. One of the first hydro energy was for mechanical milling such
as grinding grains but today modern hydro plants produce electricity using turbines
and generators. Energy created by moving waters spins rotors on a turbine which in
turn is connected to an electromagnetic generator which produces electricity. Figure
1.1 shows example of modern hydroelectricity [2].
Figure 1. 1 Typical hydroelectricity
Pelton, Francis and Kaplan turbines are the most used hydraulic turbines. When
water head is high and flow rate is low the impulse power of the water guarantees
Turbine Generator Water resource
2
high efficiency in which the Pelton turbine proves to the ideal choice. In reverse
condition, reaction force is responsible for the rotation of the impeller and the Kaplan
turbine is the preferred choice. The Francis turbine is suitable for medium head in
which both forces are responsible for its rotation. Every turbine types and working
area is illustrated in Figure 1.2.
Figure 1. 2 Turbine selection chart
Among these three turbines the Pelton turbines have several advantages. Firstly,
because they can utilize high heads, they can produce a lot of power from small units.
Secondly, they are reasonably easy to construct. Thirdly, a given turbine can be used
for a range of heads and flows; unlike propeller of Francis turbine, the runner doesn’t
have to be designed for specific conditions, so Pelton runners can be made and kept
in stock. Fourthly, the runner has space around it, making it much easier to inspect
and work on than most other types of turbine [3].
Pelton turbine was invented by Lester Allan Pelton in 1878, his runner design is still
largely used today in the hydroelectric power industry and its existence is more than
3
a hundred years. Nevertheless, knowledge of the Pelton technology and high
efficiency design is mostly contained within large commercial companies due to high
competitive market. Efficiency of Pelton turbine continuously increased after its first
invention. Efficiencies for large runners are usually 90%, and under carefully
controlled conditions, runners can achieve efficiencies of 92%. Micro hydro runners,
with their rather simpler designs and technology, should nevertheless achieve up to
75-85% efficiency [4]. Furthermore, in hydropower industry every hydro mechanical
study goal is improvement of efficiency of turbo machinery. Therefore, this study is
conducted with this main purpose of hydromechanics.
New design of the turbine based on experimental approach is expensive and very time
consuming. Computational Fluid Dynamics (CFD) allows to learn deeper
understanding of the flow phenomena in a Pelton and aiming to reach higher
efficiencies as a less expensive alternative.
Development of numerical methods of Pelton turbine
Numerical simulation for the investigation of the flow in Pelton turbine is based
on CFD. Application of the CFD method to the hydromechanics of Pelton turbines
began in the late 1990s of the last century and is becoming increasingly significant.
CFD simulation for flow in the Pelton bucket is carried out in [5]-[9]. CFD simulation
is therefore considered as an alternative method for investigating complex flows in
Pelton turbines, provided that they are reliable and able to reveal the possibility of
improving the system efficiency. Since there is hardly any available direct
measurement of the flow in the rotating buckets, the numerical simulations can yet
hardly be validated in most cases. The necessity of validation of CFD simulations has
its background in the fact that the CFD simulations are based on the use of particular
and different turbulence models in solving the Navier-Stokes equations. However, in
order to select the best model, every numerical tools and turbulence models are
compared by its accuracy and costs in [10]. In implementation of the flow in Pelton
turbines, the computational accuracy of the results of CFD method is limited by the
fact that in addition to the assumed turbulence model, the free surfaces of both the
water sheet in the bucket must be always assumed as a finite fluid domain of a two-
4
phase flow. Although the flows in the rotating buckets, can be simulated by CFD
methods, physical basis of the related flow phenomena usually cannot be
satisfactorily explained. In could not be made clear, for instance, how the centrifugal
and Coriolis forces will respectively, influence the flow. Because of these factors, it
so happens that each time when the operation of design parameters change, a new
expensive and complex CFD simulation must in general be performed.
5
Chapter 2. Pelton turbine theory
Components of Pelton turbine
The Pelton turbine is composed of many parts in order to understand Pelton
turbine theory some of the most important parts must be known. In the following list
Pelton turbine parts discussed:
Penstock: Pen stock is a pipe that delivers water from the dam to the turbine at very
high speed
Spear: A spear is provided within the Pen stock and its primary function is to control
the fluid flow. It regulates the speed of the jet with back and forth movement. Moving
it forward towards the tip of the nozzle will reduce the amount of water striking the
buckets and vice versa.
Nozzle: Nozzle is positioned at the end of the Pen stock, and its main function is to
increase the velocity of the water and direct the jet towards the buckets.
Buckets: Pelton turbines have a very specific number of buckets attached to the
periphery of the runner. The buckets take the impact of the water which results in
mechanical rotation of the runner.
Runner: Runner is a heavy circular disk in which the buckets are attached to. It is
also attached to the main shaft which drives the electrical generator.
Brake Nozzle: Brake nozzle is used to stop the turbine by directing a jet of water
towards the back of the buckets.
Casing: Casing encloses the whole Pelton turbine setup and prevents splashing of
water. It also directs the discharge water from the buckets to the tail race (outlet)
These main parts are shown in Figure 2.1.
6
Figure 2. 1 Main parts of the Pelton turbine
Working principle of Pelton turbine
In hydropower plants, the available fluid energy exists in the form of
potential energy which is a measure of the height difference between the water level
in the reservoir and the turbine housing. In principle, the turbine is positioned at a
much lower altitude than the water level in the dam. This height difference is
generally referred to as the hydraulic head.
Water is delivered from the dam to the turbine resulting in subsequent conversion of
the potential energy into more useful kinetic energy. With the use of a nozzle, further
reduction in the fluids pressure is carried out producing a high speed jet. Using
Bernoulli’s equation, the velocity of the fluid can be calculated. It however, does not
consider the losses in the system due to friction.
𝑣1 = √2𝑔𝐻 (2.1)
Where:
H is the net pressure head
7
In the second stage of energy conversion, kinetic energy of the jet is converted into
mechanical energy through physical interaction between the fluid jet and the bucket.
The water exerts pressure on the bucket upon entry and its velocity is reduced when
it exits the bucket. The change in direction of the jet upon exit, create an impulsive
force that is perpendicular to the direction of the flow. Nevertheless, the relative speed
of the jet with respect to the bucket is the same at both the inlet and outlet.
Pelton turbine velocity triangle
In turbo machinery, the concept of relative velocity and construction of velocity
triangles is an effective tool for performance evaluation, may it be pumps or turbines.
A velocity triangle is a velocity diagram that represents the different components of
fluid velocity in a turbo machine. It is usually drawn for both the inlet and outlet
section of the turbo machine. Figure 2.2 indicates the velocity triangle for a Pelton
turbine.
Figure 2. 2 Velocity triangle for Pelton Turbine
Figure 2.2 shows the interaction between the fluid jet and the turbine bucket. The
system involves relative motion in which the fluid and the bucket move relative to
each other as well as the stationary nozzle. Before entry, the jet is moving with initial
velocity v1. The relative velocity between the moving bucket and the jet is given by:
𝑣𝑟1 = 𝑣1 − 𝑢 (2.2)
8
At the inlet, the horizontal component of the initial velocity (vw1) is equal to the initial
velocity (v1).The fluid stream upon entry splits into two equal parts. Due to symmetry
in the buckets geometry, analysis is carried out for either half. The jet exits the bucket
at an angle ɵ relative to the positive x-axis counter-clockwise. This is the angle of
deflection, and generally ranges from 165° to 170°.
In the ideal case, the relative velocity at the exit will have the same value as the
relative velocity at the inlet. However, practical factors such as friction slows the fluid
as it flows over the bucket surface resulting in the relative velocity at the exit being
less than that at the inlet. A blade friction co-efficient is usually defined which is the
ratio of the relative velocities.
𝑘 =𝑣𝑟2
𝑣𝑟1 (2.3)
Thus, given a value of friction co-efficient and the relative velocity at the inlet, the
outlet relative velocity can be expressed as:
𝑣𝑟2 = 𝑘𝑣𝑟1 (2.4)
To obtain the absolute velocity of the fluid at the outlet, we first need to calculate the
horizontal and vertical components of the absolute velocity. The horizontal
component (vw2) is obtained by simply subtracting the horizontal component of the
relative velocity at the outlet from the bucket velocity.
𝑣𝑤2 = 𝑢 − 𝑘𝑣𝑟1[𝑐𝑜𝑠(180 − 𝜃)] (2.5)
The vertical component (vy2) of the absolute velocity at the outlet is equal to the
vertical component of the relative velocity at the outlet.
𝑣𝑦2 = 𝑘𝑣𝑟1[𝑠𝑖𝑛(180 − 𝜃)] (2.6)
The absolute velocity of the jet at the outlet is simply the resultant of its components
and is computed according to this simple equation:
𝑣𝑦 = √𝑣𝑤22 + 𝑣𝑦2
2) (2.7)
9
Impulse force
Balance law of momentum states that a change in the flow direction is always related
to an external impulsive force that acts perpendicular to the flow direction. The
equation describing this principle is as follows:
𝑑
𝑑𝑡∫ 𝑉𝜌𝑑∀ + ∑ 𝑣𝑜𝑢𝑡
𝑐𝑣𝜌𝑜𝑢𝑡𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡 − ∑ 𝑣𝑖𝑛 𝜌𝑖𝑛𝐴𝑖𝑛𝑣𝑖𝑛 =
∑ 𝐹𝑐𝑜𝑛𝑡𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 (2.8)
Because the flow is steady, the velocity of the fluid within the small element 𝑑∀ does
not change with time, resulting in the integral term equating to zero. The equation is
then reduced to:
∑ 𝑣𝑜𝑢𝑡 𝜌𝑜𝑢𝑡𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡 − ∑ 𝑣𝑖𝑛 𝜌𝑖𝑛𝐴𝑖𝑛𝑣𝑖𝑛
= ∑ 𝐹𝑐𝑜𝑛𝑡𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 (2.9)
In the bucket analysis, there is only one inlet and outlet. The force equation can be
simplified and re-written to describe the flow on the bucket.
𝐹𝑏𝑢𝑐𝑘𝑒𝑡 = ��𝑤(𝑣𝑟1) − ��𝑤(𝑣𝑟2sin (180 − 𝜃)) (2.10)
𝐹𝑏𝑢𝑐𝑘𝑒𝑡 = ��𝑤(𝑣𝑟1) − ��𝑤(𝑘𝑣𝑟1sin (180 − 𝜃)) (2.11)
Equation (2.10) represents the total force in the direction of bucket motion. In the
equation (2.11), the relative velocity of the fluid at the exit is represented as a function
of the relative velocity at the inlet. Finally, because of continuity and
incompressibility, mass flow rate is constant and we can further reduce the equation:
𝐹𝑏𝑢𝑐𝑘𝑒𝑡 = ��𝑤𝑣𝑟1(1 − 𝑘𝑠𝑖𝑛(180 − 𝜃)) (2.12)
Power and efficiency
The power received by the bucket is simply the product of the impulsive force on the
bucket and the bucket velocity.
10
𝑃 = 𝐹𝑏𝑢𝑐𝑘𝑒𝑡 . 𝑢 = [��𝑤𝑣𝑟1(1 − 𝑘𝑠𝑖𝑛(180 − 𝜃))]. (𝑢) (2.13)
The hydraulic efficiency, can be estimated as the ratio of the power received by the
bucket to the power of the fluid jet upon exiting the nozzle.
𝑛ℎ =2𝑢(𝑣𝑟1(1 − 𝑘𝑠𝑖𝑛(180 − 𝜃)))
𝑣12
(2.14)
The power of the fluid jet can be expressed as:
𝑃 = ��𝑤(𝑣2
2)
(2.15)
Achieving maximum efficiency, is a primary objective in the design of Pelton
turbines. This means that the turbine should strive to extract maximum power from
the water jet. Because power is a product of bucket velocity and impulse force,
maximum power will be attained when both the impulse force and the bucket force
are maximum.
Ideally, maximum possible bucket speed will be equal to the jet speed. However,
because the jet and the bucket are moving at the same velocity, the jet will not be able
to hit the bucket which leads to zero impulse force. Hence, no power will be extracted.
On the other hand, maximum impulse force will occur when the buckets are
stationary. Nonetheless, power extracted will again be zero because the bucket is not
moving.
Therefore, power extraction is zero at both maximum conditions of impulse force and
bucket velocity. With respect to jet to bucket speed ratio, the extracted power will
vary as shown in Figure 2.3.
11
Figure 2. 3 Extracted power as a function of jet speed ratio
In Figure 2.3, it is clearly evident that the maximum condition occurs in between the
maximum and minimum bucket to jet speed ratio. In the ideal case, the optimum
condition in which extracted power is maximum occurs when the bucket speed is half
the jet speed. In these conditions, Pelton turbines can give efficiency as high as 90%.
However, in practice this coefficient is slightly less than 0.5 and can be between 0.42-
0.48 [4].
Pelton turbines specifications
Essentially, Pelton turbine consists of injectors for producing the high speed jet
and a wheel with buckets for extracting the energy from the jet. The injector has two
primary functions. Firstly, the injector nozzle converts the pressure energy of the
water into kinetic energy of the fluid jet. Secondly, it controls the flow rate of the
fluid via a built in needle. Having the ability to control the flow rate, it can easily
regulate the power produced based on the load. Kinetic energy of the jet is converted
to mechanical rotational energy of the wheel by way of interaction between the high
speed jet and the buckets. Because of the rotation of the Pelton wheel, both centrifugal
and Coriolis forces are influencing the flow. The form of the flow and its distribution
within the bucket therefore differ fundamentally from those in the straight-moving
bucket. The basic principle of energy conversion which is described earlier applies to
12
the Pelton turbines. However, both the design and the flow parameters need to be
specially specified as described in the following sections.
Geometric specification of the Pelton wheel
The basic the configuration of a Pelton wheel is shown in Figure 2.4.
Figure 2. 4 Geometric specification of a Pelton wheel
Pitch circle diameter (Dm) = 2Rm
Wheel bucket inner diameter (Db) = 2Rb
Wheel Diameter (Da) = 2Ra
Number of buckets N
Bucket inner width
Bucket exit angle β2
Base circle radius of the main splitter rs
Installation form of Pelton turbine.
The Pelton turbine can be installed horizontally or vertically shafted depends
on jet number and housing possibility of installation [11]. Also Pelton turbine can be
installed with multiple jets. But multiple jet turbines require very careful design.
13
Minimum offset angle between jets is empirically found 60o, so up to six jets can be
used, but this is only for vertical axis turbine. Horizontal axis machines have up to
two jets. The difficulties of making successful multi-jet turbines are such that, unless
and modification can be made, the angle between jets should be kept to 80-90o for
micro hydro.
Hydraulic specification of the Pelton turbine
For field of turbo machinery dimensionless number function determine
performance and size of the machine, but for Pelton turbine, only few of them
relevant. Most important hydraulic parametric quantities are summarized in this
section.
Jet speed ratio
Jet speed ratio is defined peripheral speed u of the Pelton wheel on the pitch circle
described in the Figure 2.4, to the jet speed V1.
𝑘 =𝑢
𝑣1=
u
√2𝑔𝐻
(2.16)
Bucket volumetric load 𝜑𝐵
Bucket volumetric load is jet thickness relative to the bucket width and dimensionless
parameter. The flow rate is proportional to the square of the jet thickness, the bucket
volumetric load can also be represented by the flow rate of single injector.
𝜑𝐵 = ��𝑗𝑒𝑡
𝜋4⁄ ∗𝐵2√2𝑔𝐻
(2.17)
Here B is the bucket inner width. Then ��𝑗𝑒𝑡 is expressed as:
��𝑗𝑒𝑡 = 𝜋4⁄ 𝑑0
2√2𝑔𝐻 (2.18)
Substituting equations (2.17) and (2.18) the bucket volumetric load become
𝜑𝐵 = (𝑑0
𝐵)
2
(2.19)
The bucket volumetric load used on the one hand volumetric load in dimensionless
form and on the other hand to determine the necessary width of the Pelton buckets.
For nominal or maximum flow rate the jet diameter doesn’t exceed one third of the
bucket width. This yields 𝜑𝐵=0.09 to 0.11.
Specific speed
14
Specific speed is a commonly used parametric quantity involved with turbo machines.
As with hydraulic turbines, it is defined as the rotational speed (rpm) at which a
hydraulic turbine would operate at best efficiency under unit head (one meter) and
which is sized to produce unit power (one kilowatt).
The specific speed is defined by
𝑛𝑞 = 𝑛√��𝑗𝑒𝑡
𝐻34
(2.20)
Where, H is the pressure head, ��𝑗𝑒𝑡 is the jet flow rate and n is the rotational speed.
H and �� in the above equation is normalized by the unit flow rate ��= 1m3/s and the
unit pressure head H=1m, then n and nq have the same unit.
Specific speed is important parameter to determine wheel size and bucket numbers.
The optimum number of buckets is normally dependent on symmetry conditions. The
number of buckets can be obtained with the following equation.
𝑁 = 15 + 0.62
𝑛𝑞 (2.21)
15
Chapter 3. Design of Pelton turbine
Dimension criteria of Pelton turbine
In hydraulic industry there is no very specific method to determine the dimension of
the Pelton bucket profile. However, there are some general criteria to dimensioning
of Pelton wheel. When designing Pelton turbine, the net head and flow rate are two
basic parameters. They determine available power and size of the turbine. Designing
of Pelton turbine starts with dimensioning of Pelton wheel, choice of rotational speed
and determination of number of injector [12].
The jet speed ratio k range is 0.45 to 0.48.
The bucket width is approximately three times bigger than jet diameter,
B=3d0
The specific speed with is nq<0.13 and this criterion is derived in.
Basic dimensions of the Pelton turbine can be defined by combining all these criteria
and rest of the parameters of the full design can be defined based on these basic
dimesions.
Design of 5kW Pelton turbine.
The design goal is 5kW Pelton turbine. Net head and the flow rate is not yet
decided. Horizontal axis Pelton turbine is chosen and for horizontal turbine maximum
number of jet is two. In present 5 kW case only one jet turbine is designed in order to
simplify analysis and reduce computational cost.
Assuming equation (2.20) for several head and flow rate conditions single jet Pelton
turbine, then specific speed is defined and illustrated in Table 3.1.
16
Table 3. 1 Specific speed calculation function of flow rate, head and rotational
speed
H, m Q, m3/s
n=400
rpm
n=500
rpm
n=600
rpm
n=700
rpm
n=800
rpm
20 0.031951 0.126 0.158 0.189 0.221 0.252
25 0.025561 0.095 0.119 0.143 0.167 0.191
30 0.020047 0.074 0.092 0.110 0.129 0.147
35 0.018258 0.063 0.078 0.094 0.110 0.125
40 0.015976 0.053 0.066 0.079 0.093 0.106
45 0.0142 0.046 0.057 0.069 0.080 0.091
50 0.01278 0.040 0.050 0.060 0.070 0.080
In the highlighted region anywhere the 3rd criteria is satisfied. If head is low, a bigger
turbine needed and similarly if rotational speed is low, a bigger turbine needed.
Therefore, 30-35m of head and 700-800 rpm chosen and again calculated and results
were illustrated in Table 3.2.
Table 3. 2 Specific speed function of flow rate, head and rotational speed
H, m Q, m3/s n=650
rpm
n=700
rpm
n=750
rpm
n=800
rpm
30 0.021301 0.123 0.133 0.142 0.152
31 0.020614 0.118 0.127 0.137 0.146
32 0.018794 0.110 0.119 0.127 0.136
33 0.019364 0.109 0.118 0.126 0.135
34 0.018795 0.105 0.114 0.122 0.130
35 0.018258 0.102 0.110 0.117 0.125
From the Table 3.2 head of 33m and 750 rpm turbine was selected and it can be
coupled easily with 1500rpm generator using 1:2 coupling drive. In the Table 3.3,
making final decision of parameters of the Pelton turbine.
17
Table 3. 3 Design parameters of 5 kW Pelton turbine
Parameters Defining equations Value
Head, H [m] 33
Flow rate, Q [l/s] 17.96
Rotational speed, n [rpm] 750
Number of jet, Zjet 1
Jet speed ratio, k 0.46
Jet speed, V1 [m/s] 𝑉1 = √2𝑔𝐻 25.37
Jet diameter, d0 [mm] 𝑑0 =0.54
𝐻𝑛
14
√𝑄
𝑧 from (eq. 2.18) 30
Peripheral speed, Um [m/s] 𝑈𝑚 = 𝑘 ∗ 𝑉1 11.73
Runner diameter, Dm [mm] 𝑅 =𝑈𝑚 ∗ 60
2𝜋𝑛 300
Bucket width, B [mm] 𝐵 = 3𝑑0 90
Depth of the bucket h [mm] ℎ = 0.275𝐵 − 0.285𝐵 25
Bucket Length L [mm] 𝐿 ≈ 0.92𝐵 84
Hydraulic power Ph, [kW] 𝑃ℎ = 𝜌𝑔𝐻𝑄 5.8
Design of 100kW Pelton turbine with one nozzle
For 100kW turbine, the head condition is set to 33m and the number of jet is
one. Once the power and the head is given rest of the turbine design parameters can
be found using the same method mentioned in Table 3.3. Design parameters of the
100kW Pelton turbine is calculated and indicated in Table 3.4.
18
Table 3. 4 Design parameters of 100kW Pelton turbine with one nozzle
Parameters Defining equations Value
Head, H [m] 33
Flow rate, Q [l/s] 346
Rotational speed, n [rpm] 169
Number of jet, Zjet 1
Jet speed ratio, k 0.46
Jet speed, V1 [m/s] 𝑉1 = √2𝑔𝐻 25.37
Jet diameter, d0 [mm] 𝑑0 =
0.54
𝐻𝑛
14
√𝑄
𝑧 from (eq. 2.18)
132
Peripheral speed, Um [m/s] 𝑈𝑚 = 𝑘 ∗ 𝑣1 11.67
Runner diameter, Dm [mm] 𝑅 =
𝑈𝑚 ∗ 60
2𝜋𝑛
1320
Bucket width, B [mm] 𝐵 = 3𝑑0 400
Depth of the bucket h [mm] ℎ = 0.275𝐵 − 0.285𝐵 110
Bucket Length L [mm] 𝐿 ≈ 0.92𝐵 375
Hydraulic power Ph, [kW] 𝑃ℎ = 𝜌𝑔𝐻𝑄 111.78
Design of 100kW Pelton turbine with two nozzles
For two nozzles Pelton turbine, according to equation (2.18), the jet diameter
is smaller than one nozzle turbine design jet diameter, therefore, smaller runner is
required. In the Table 3.5, the parameter definition of the two nozzles 100kW Pelton
turbine is illustrated.
19
Table 3. 5 Design parameters of the two nozzles 100kW Pelton turbine
Parameters Defining equations Value
Head, H [m] 33
Flow rate, Q [l/s] 346
Rotational speed, n [rpm] 84.5
Number of jet, Zjet 2
Jet speed ratio, k 0.46
Jet speed, V1 [m/s] 𝑉1 = √2𝑔𝐻 25.37
Jet diameter, d0 [mm] 𝑑0 =
0.54
𝐻𝑛
14
√𝑄
𝑧 from (eq. 2.18)
132
Peripheral speed, Um [m/s] 𝑈𝑚 = 𝑘 ∗ 𝑉1 11.67
Runner diameter, Dm [mm] 𝑅 =
𝑈𝑚 ∗ 60
2𝜋𝑛
1320
Bucket width, B [mm] 𝐵 = 3𝑑0 400
Depth of the bucket h [mm] ℎ = 0.275𝐵 − 0.285𝐵 80
Bucket Length L [mm] 𝐿 ≈ 0.92𝐵 272
Hydraulic power Ph, [kW] 𝑃ℎ = 𝜌𝑔𝐻𝑄 111.78
CAD modeling
In order to design bucket, the inside profile of the Pelton bucket must be
configured so that water in it spreads as smoothly and steadily as possible. The profile
should have no discontinuity. Perfect smoothness of the bucket profile is achieved if
the profile in each cross section can be outlined by a smooth mathematical curve
without kinks. Among various mathematical functions, an ellipse model is obviously
of very suitable form. In this study, NX 8.5 CAD commercial tool was used for
creating model of the bucket. Using bucket basic dimension ratios mentioned in Table
3.3 and splitter angle and output angle of the bucket 15o we can easily create bucket
profile. Figure 3.1 and Figure 3.2 show horizontal and longitudinal profile of the
bucket inside based on the bucket width, depth, length and 𝜗 bucket output angle and
corresponding dimensions of the profile ellipse. After creating these two profile outer
pattern of the bucket is created as an ellipse radius of bucket width and length. This
20
profile is shown in Figure 3.3. Then supporting curve of the profile was created as
shown in Figure 3.4. Supporting curve is used for guiding the surface creation. Using
support curves surface of inner bucket is created with command of mesh surface on
NX 8.5 tool. Mesh surface creating tools is very useful to create surface from 4 curves
which are connected to each other by vertex. Creating bucket inner surface is shown
in Figure 3.5.
Then outer surface of the turbine is created depend on the bucket thickness.
Empirically bucket thickness is 5% of the bucket width therefore outer profile of the
bucket was created with this distance from inner surface. Figure 3.6 shows bucket
outer surface guiding curves creation. Then mesh surface creating command used for
creating outer surface of the bucket. Figure 3.7 shows outer surface creation of the
bucket.
Figure 3. 1 Creation of horizontal profile curve of the bucket (green)
h
𝐵
2
𝜗- output angle
21
Figure 3. 2 Creation of longitudinal profile curve of the bucket (green)
Figure 3. 3 Outer pattern of bucket
22
Figure 3. 4 Supporting curve of the bucket inside surface
Figure 3. 5 Creation of inner surface creation (orange surface)
23
Figure 3. 6 Guide curves of bucket outer surface (orange lines)
Figure 3. 7 Creation of bucket outer surface (orange surface)
After creating outer and inner surface of the bucket, the “sew” command was used
for creating solid bucket from previously created surfaces. Figure below shows the
solid body of the bucket created by “sew” command of the NX 8.5 tool.
24
Figure 3. 8 Bucket basic shape after sewing inner and outer surfaces
Then notch was created by removing imaginary radiused wedge through the pattern
from the end of the splitter ridge. This notch was used for provide jet and bucket
interact with right angle and prevent water losses. Then groove on the back side of
the bucket must be created to prevent water slapping when jet leaves the bucket back
side. Normally groove is created from the end of the splitter ridge to the middle of
the back of the bucket. Figure 3.9 shows the notch creating shape. In our case groove
was created quite deep to reduce water slapping thus can reduce counter torque.
Figure 3.10 shows groove creation.
25
Figure 3. 9 Notch creating
Figure 3. 10 Groove visualization of the bucket
26
Model of 5 kW Pelton turbine bucket
The three dimensional models of the geometry for Pelton turbine was made using
the commercial computer aided drawing software, UNI-graphics NX 8.5. The bucket
and the runner dimensions of the 5kW Pelton turbine is shown below.
Figure 3. 11 Bucket dimensions and wheel size of 5kW Pelton turbine
Figure 3. 12 3D designing of bucket on NX8.5 modelling tool
90
25
84
300
27
Model of 100 kW Pelton turbine bucket with one nozzle
100kW one nozzle bucket shape is same with 5 kW turbine only the size is scaled
up to meet jet diameter. Bucket shape and wheel size is shown in Figure 3.13.
Figure 3. 13 Bucket dimension and wheel size of 100kW Pelton turbine with on
nozzle
400
110
375
1320
28
Model of 100 kW Pelton turbine bucket with two nozzles
100kW two nozzles turbine dimensions is slightly smaller than one jet Pelton
turbine. Figure 3.14 shows bucket design and wheel size of the Pelton turbine with
two nozzles.
Figure 3. 14 Bucket dimensions and wheel size of 100kW Pelton turbine with two
nozzles
290
80
272
950
29
Chapter 4. Methodology
CFD introduction
Computational fluid dynamics (CFD) is the use of applied mathematics, physics
and computational software to visualize how a gas or liquid flows, as well as how the
gas or liquid affects object as it flows past. Computational fluid dynamics is based on
the Navier-Stokes equations. These equations describe how the velocity, pressure,
temperature, and density of a moving fluid are related.
There are many advantages to using CFD for simulation of fluid flow. One of the
most important advantages is the realiable savings in time and cost for engineering
design.
Governing equations
The governing equations of fluid flow, which describe the processes of
momentum, heat and mass transfer, are known as the Navier-Stokes equations [14].
There are three different streams of numerical solution techniques: finite difference,
finite volume method and spectral method. The most common and the CFX based is
known as the finite volume technique. In this technique, the region of interest is
divided into small sub regions, called control volume. As a result, and approximation
of the value of each variable at specific points throughout the domain can be obtained.
In this way, on derives a full picture of the behavior of the flow.
The continuity equation:
𝜕𝑝
𝜕𝑡+ ∇(𝜌 ∗ 𝑈) = 0
(4.1)
The momentum equation:
𝜕(𝑝𝑈)
𝜕𝑡+ ∇ ∗ (𝜌𝑈⨂𝑈) = −∇𝑝 + ∇𝜎 + 𝑆𝑀
(4.2)
Where 𝜎 is the stress tensor (including both normal and shear components of the
stress).
These instantaneous equations are averaged for turbulent flows leading to additional
terms that needed to be solved while the Navier-Stokes equations describe both
laminar and turbulent flows without addition terms, realistic flows involve length
30
scale much smaller than the smallest finite volume mesh. A Direct Numerical
Simulation of these flows would require significantly more computing power than
what is available now or in the future. Therefore, much research has been done to
predict the effects of turbulence by using turbulence models. These models account
for the effects of turbulence without the use of a very fine mesh or direct numerical
simulation. These turbulence models modify the transport equations by adding
averaged and fluctuating components. The transport equations are changed to
equations below:
𝜕𝑝
𝜕𝑡+ ∇(𝜌 ∗ 𝑈) = 0
(4.3)
𝜕(𝑝𝑈)
𝜕𝑡+ ∇ ∗ (𝜌𝑈⨂𝑈) = −∇𝑝 + ∇{𝜎 − 𝜌𝑢⨂𝑢 } + 𝑆𝑀
(4.4)
The mass equation is not changed but the momentum equation contains extra terms
which are the Reynolds stresses, 𝜌𝑢⨂𝑢 and Reynolds flux, 𝜌𝑢𝜙 . These Reynolds
stresses need to be modeled by additional equations to obtain closure. Obtaining
closure implies that there are a sufficient number of equations to solve for all the
unknowns including Reynolds stresses and Reynolds fluxes. In this study, the model
utilized was the Shear Stress Transport (SST) model. The advantage of using this
model is that combines the advantages of other turbulence models (the k-𝜀, Wilcox
k-𝜔, BSL k- 𝜔 ). In order to understand the advantage the SST model gives, the 3
other turbulence models will be discussed briefly.
Turbulence modelling
Two-equation turbulence models are widely used in CFD as they give a good
compromise between computational power needed and accuracy. The term two
equation refers to the fact that these models solve for the velocity scale is obtained by
solving using separate transport equations. The turbulence velocity scale is obtained
by solving the transport equation. The turbulent length scale is estimated from two
properties of the turbulence field, namely the turbulent kinetic energy and dissipation
31
rate. The dissipation rate of the kinetic energy is obtained from its transport equation.
The most widely used are the k-ε and k-ω equation models.
k-ε turbulence model.
The k-𝜀 model introduces k (m2/s2) as the turbulence kinetic energy and 𝜀
(m2/s3) as the turbulence eddy dissipation. The continuity equation remains the same:
𝜕𝑝
𝜕𝑡+ ∇(ρU) = 0
(4.5)
The momentum equation changes, as shown by equation (4.6):
𝜕𝜌𝑈
𝜕𝑡+ ∇ ∗ (𝜌𝑈 ⊗ 𝑈) = −∇𝑝′ + ∇ ∗ (𝜇𝑒𝑓𝑓(∇𝑈 + (∇𝑈)) + 𝑆𝑀
(4.6)
Where Sm is the sum of body forces, 𝜇𝑒𝑓𝑓 is the effective viscosity accounting for
turbulence and 𝑝′ is the modified pressure. The k- 𝜀 model uses the concept of eddy
viscosity giving the equation for effective viscosity as shown by equation (4.7):
𝜇𝑒𝑓𝑓 = 𝜇 + 𝜇𝑡
(4.7)
𝜇𝑡 is the turbulence viscosity is linked to the turbulence kinetic energy and dissipation
by the equation (4.8):
𝜇𝑡 = 𝐶𝜇𝜌𝑘2
𝜔
(4.8)
Where 𝐶𝜇 is a constant.
The values for k and e come from the differential transport equations for the
turbulence kinetic energy and the turbulence dissipation rate.
The turbulence kinetic energy equation is given as equation (4.9):
𝜕(𝜌𝑘)
𝜕𝑡+ ∇ ∙ (𝜌𝑈𝑘) = ∇ [(𝜇 +
𝜇𝑡
𝜎𝑘) ∇𝑘] + 𝑃𝑘 + 𝑃𝑘𝑏 − 𝜌𝜀
(4.9)
The turbulence dissipation rate is given by equation (4.10):
𝜕(𝜌𝜀)
𝜕𝑡+ ∇(𝜌𝑈𝜀) = ∇ [(𝜇 +
𝜇𝑡
𝜎𝜀) ∇𝜀] +
𝜀
𝑘(𝐶𝜀1(𝑃𝑘 + 𝑃𝜀𝑏) − 𝐶𝜀2𝜌𝜀)
(4.10)
Where 𝐶𝜀1, 𝐶𝜀2, 𝜎𝜀, 𝜎𝑘 are constants.
32
Pk is the turbulence production due to viscous forces and is modeled by the equation
(4.11):
𝑃𝑘 = 𝜇𝑡∇𝑈 ∙ (∇𝑈 + ∇𝑈𝑇) +2
3∇ ∙ 𝑈(3𝜇𝑡∇ ∙ 𝑈 + 𝜌𝑘)
(4.11)
A buoyancy term may be added to the previous equation if the full buoyance model
is used.
Wilcox k-𝝎 turbulence model
This model has an advantage over the k- 𝜀 model, where it does not involve
complex linear damping functions for near wall calculations at low Reynolds number.
The k-𝜔 model assumes that the turbulence viscosity is related to the turbulence
kinetic energy, k, and the turbulent frequency, 𝜔, by the equation (4.12):
𝜇𝑡 = 𝜌𝑘
𝜔
(4.12)
The transport equation for k is given by the equation (4.13):
𝜕(𝜌𝑘)
𝜕𝑡+ ∇ ∙ (𝜌𝑈𝑘) = ∇ [(𝜇 +
𝜇𝑡
𝜎𝑘) ∇𝑘] + 𝑃𝑘 + 𝑃𝑘𝑏 − 𝛽′𝜌𝜔2
(4.13)
The transport equation for 𝜔 is shown as equation (4.14):
𝜕(𝜌𝜔)
𝜕𝑡+ ∇(𝜌𝑈𝜔) = ∇ [(𝜇 +
𝜇𝑡
𝜎𝜀) ∇𝜔] + 𝛼
𝜔
𝑘𝑃𝑘 + 𝑃𝜔𝑏 − 𝛽𝜌𝜔2
(4.14)
The production rate of turbulence (Pk) is calculated as shown previously in the k-e
section. The model constants are given by equations (4.15-4.19)
𝛽′ = 0.09 (4.15)
𝛼 = 5/9 (4.16)
𝛽 = 0.075 (4.17)
𝜎𝑘 = 2 (4.18)
𝜎𝜔=2 (4.19)
The Reynolds stress tensor, 𝜌𝑈⨂𝑈 is calculated by:
−𝜌𝑈⨂𝑈 = 𝜇𝑡(∇𝑈 + (∇𝑈)𝑇) −2
3𝜕(𝜌𝑘 + 𝜇𝑡∇𝑈)
(4.20)
33
Shear stress transport model
The disadvantage of the Wilcox model is the strong sensitivity to free-stream
conditions. Therefore a blending of the k-𝜔 model near the surface and the k-e in the
outer region was made by Menter [15] which resulted in the formulation of the BSL
k-𝜔 turbulence model. It consists of transformation of the k-e model to a k- 𝜔
formulation and subsequently adding the resulting equations. The Wilcox model is
multiplied by a blending function F1 and the transformed k-e by another function 1-
F1. F1 is a function of wall distance (being the value of on near the surface and zero
outside boundary layer). Outside the boundary and on the edge of the boundary layer,
standard k- 𝜔 model is used.
However, while the BSL k- 𝜔 model combines the advantages of both the k-e and
Wilcox k- 𝜔 turbulence model, it fails to properly predict the onset and amount of
flow separation from smooth surfaces. The k-e and Wilcox k- 𝜔 turbulence models
do not account for the transport of the turbulent shear stress resulting in an over-
prediction of eddy-viscosity. A limiter on the formulation can be used to obtain the
proper results. These limiters are given by equation (4.21):
𝑣𝑡 =𝑎1𝑘
max (𝛼1𝜔, 𝑆𝐹2)
(4.21)
Where:
𝑣𝑡 =𝜇𝑡
𝜌 (4.22)
F2 is a blending function which restricts the limiter to the wall boundary and S is the
invariant measure of the strain rate.
The blending function are given by the equations (4.23) and (4.24):
𝐹1 = tanh(𝑎𝑟𝑔14) (4.23)
𝑎𝑟𝑔1 = min (max (√𝑘
𝛽′𝜔𝑦′,500𝑣
𝑦2𝜔) ,
4𝜌𝑘
𝐶𝐷𝐾𝑊𝜎𝜔2𝑦2
(4.24)
34
𝑦 is the distance to the nearest wall and v is the kinematic viscosity. In addition:
𝐶𝐷𝑘𝜔 = max (2𝜌1
𝜎𝜔2𝜔𝛻𝑘𝛻𝜔, 1.0 × 10−10)
(4.25)
𝐹2 = tanh (𝑎𝑟𝑔22) (4.26)
𝑎𝑟𝑔2 = max (2√𝑘
𝛽′𝜔𝑦′ ,
500𝑣
𝑦2𝜔)
(4.27)
Cavitation models
Tendency for a flow to cavitation is characterized by the cavitation number,
defined as
𝐶𝛼 =𝑝 − 𝑝𝑣
12 𝜌𝑈2
(4.28)
Where p is as reference pressure for the flow (for example, inlet pressure), pv is the
vapor pressure for the liquid, and the dominator represents the dynamic pressure.
Clearly, the tendency for flow to cavitate increases as the cavitation number is
decreased.
Cavitation is treated separately from thermal phase change, as the cavitation process
is typically too rapid for the assumption of thermal equilibrium at the interface to be
correct. In the simplest cavitation models, mass transfer is driven by purely
mechanical effects, namely liquid-vapor pressure difference, rather than thermal
effects.
In CFX, the Rayleigh Plesset model is implemented in the multiphase framework as
an interphase mass transfer model. User-defined models can also be implemented.
For cavitating flow, the homogeneous multiphase model is typically used.
The Rayleigh Plesset equation provides the basis for the rate equation controlling
vapor generation and condensation. The Rayleigh Plesset equation describing the
growth of a bubble in a liquid is given by:
𝑅𝐵𝑑2𝑅𝐵
𝑑𝑡2+
3
2(𝑑𝑅𝐵
𝑑𝑡)2 +
2𝜎
𝜌𝑓𝑅𝐵=
𝑝𝑣 − 𝑝
𝜌𝑓
(4.29)
35
Where RB represents the bubble radius, 𝑝𝑣 is the pressure in the bubble (assumed to
be the vapor pressure at the liquid temperature), 𝑝 is the pressure in the liquid
surrounding the bubble, 𝜌𝑓 is the liquid density, and 𝜎 is the surface tension
coefficient between the liquid and vapor. Note that this is derived from a mechanical
balance, assuming no thermal barriers to bubble growth. Neglecting the second order
terms (which is appropriate for low oscillation frequencies) and the surface tension,
this equation reduces to:
𝑑𝑅𝐵
𝑑𝑡= √
2
3
𝑝𝑣 − 𝑝
𝜌𝑓
(4.30)
The rate of change of bubble volume follows as:
𝑑𝑉𝐵
𝑑𝑡=
𝑑
𝑑𝑡(
4
3𝜋𝑅) = 4𝜋𝑅𝐵2√
2
3
𝑝𝑣 − 𝑝
𝜌𝑓
(4.31)
and the rate of change of bubble mass is:
𝑑𝑚𝐵
𝑑𝑡= 𝜌𝑔
𝑑𝑉𝐵
𝑑𝑡= 4𝜋(
4
3𝑅𝐵2𝜌𝑔√
2
3
𝑝𝑣−𝑝
𝜌𝑓)
(4.32)
If there are Nb bubbles per unit volume, the volume fraction rg maybe expressed as:
𝑟𝑔 = 𝑉𝐵 𝑁𝐵 =4
3𝜋𝑅𝐵3𝑁𝐵 (4.33)
And the total interface mass transfer rate per unit volume is:
��𝑓𝑔 = 𝑁𝐵
𝑑𝑚𝐵
𝑑𝑡=
3𝑟𝑔𝜌𝑔
𝑅𝐵√
2
3
𝑝𝑣 − 𝑝
𝜌𝑓
(4.34)
This expression has been derived assuming bubble growth (vaporization). It can be
generalized to include condensation as follows:
��𝑓𝑔 = 𝐹3𝑟𝑔𝜌𝑔
𝑅𝐵√
2
3
𝑝𝑣 − 𝑝
𝜌𝑓𝑠𝑔𝑛(𝑝𝑣 − 𝑝)
(4.35)
Where F is and empirical factor that may differ for condensation and vaporization,
designed to account for the fact that they may occur at different rates (condensation
is usually much slower than vaporization). For modelling purposes the bubble radius
Rb will be replaced by the nucleation site radius Rnuc.
36
To obtain an interphase mass transfer rate, further assumptions regarding the bubble
concentration and radius are required. The Rayleigh-Plesset cavitation model
implemented in CFX uses the following defaults for the model parameters:
𝑅𝑛𝑢𝑐 = 1𝜇𝑚
𝑅𝑛𝑢𝑐 = 5 ∗ 10−4
Fvap=50
Fcond =0.01
In order to do numerical analysis for the Pelton turbine. Following work order should
be kept.
Figure 4. 1 Work flow of numerical analysis
Modelling of computational domain
Any hydraulic turbine performance is expressed as ratio of the developed power
by turbine to the available hydraulic energy. Normally, the power developed by
turbine is expressed as follows.
𝑃 = 𝜏 ∗ 𝜔 (4.36)
Here τ – torque, ω- angular velocity.
The CFX has function to measure torque developed by fluid on the solid body relative
to the certain axis. Pelton turbine total torque is sum of torque on the sequentially
37
rotating buckets. By using the torque measured numerically on the one bucket, total
runner torque can be calculated. Therefore, full runner simulation is not really
necessarily to determine the total torque. Using this feature of Pelton turbine, several
simplifications to the computational domain can be made for numerical performance
analysis.
Creating domain of one jet Pelton turbine
Figure 4.2 shows sketch of the computational domains. Computational domain
consists of two domains; rotor and stator. The stationary domain contains the jet and
the rotating domain contains the 3 buckets. Basically, the idea behind this method is
that the middle bucket is subject to one, complete water-jet cycle. Data obtained from
this bucket can be used to model the total runner torque.
Figure 4. 2 Sketch of computational domain of 5kW turbine
In Figure 4.2, the rotating domain containing the buckets is in the initial
position. The rotational domain rotates about its axis in which it continuously
interacts with the stationary domain along the respective interfaces. The simulation
stops when the last bucket has cut through the water-jet completely. When modeling
Jet inlet
Stationary
domain
Rotating domain
Additional
section
38
the fluid domain, it is important to make sure that the water-jet is fully developed
before the first bucket interacts with the water-jet. With this in mind, the initial
position of the rotating domain is crucial. Additional section has following two
purposes. Firstly, it is necessary for the middle bucket torque to develop completely
and the second is concerned with hexahedral meshing, thus reducing computational
cost. Stationary domain is 120 degrees, rotating domain with three buckets is 90
degrees and additional section is 30 degrees. In the Figure 4.2 rotating domain is in
its initial position and this position, the jet is allowed to fully develop before the first
bucket reaches the jet.
Creating domain of two jet Pelton turbine
For the 2 jet Pelton turbine, the idea is similar to a single jet configuration but
it uses 6 buckets. Two buckets simultaneously interact with two jets. Therefore,
torque will develop simultaneously from this two buckets and total torque will be sum
of this two torque. Sketch of the two jet turbine is showed below:
Figure 4. 3 Sketch of the computational domain of Pelton turbine with two nozzles
39
Minimum angle of the two jet in horizontal axis turbine should be 80-90 degrees. In
present study angle between two jets is set to 80 degrees. Stationary domain is 180
degrees while the rotating domain has six buckets arranged in a total space of 140
degrees equidistant from each other. Additional section is made for same purpose
mentioned above. The last two buckets from the bottom are subjected to torque
measurement.
Mesh generation
In order to simulate any flow problem, meshing is essential for successful and
efficient solving. Normally real objects have infinite degrees of freedom and not
possible to solve. Meshing methods allows for dividing an object into smaller parts
called elements or nodes in which the governing equations are solvable. In numerical
analysis, meshing is the most critical aspect for simulation accuracy. An optimal size
of mesh is dependent on the type of flow, geometry of the fluid and the resolution
needed. There are many commercial meshing tools available in the market. Each of
them has their own capabilities.
In this study ICEM CFD has been used for mesh generation. It has ability to generate
several types of meshes which are tetrahedral, hexahedral and hybrid meshes.
Tetrahedral meshes are less time consuming but computational cost is high.
Generating hexahedral mesh is very time consuming but it gives more accurate results
than tetrahedral meshes and computational cost can be lower. ICEM CFD also
provides structured and unstructured meshes. Structured meshes have cells that are
regular in shape and each grid points uniquely identified by indices and coordinates.
Unstructured meshes contain cells that are not regular in shape and have grid points
in no particular ordering. In most cases a structured mesh is preferable in terms of
accuracy, CPU time and memory. However, complex geometries like the Pelton
turbine structured mesh is impossible to generate, therefore unstructured tetrahedral
mesh is used.
40
Mesh generation for the 5kW Pelton turbine.
Rotating domain and stationary domain both consists of unstructured
tetrahedral meshes. Only in the jet trajectory area, from the jet inlet until interface of
the domain finer mesh is generated in order get well developed free surface between
water and air phases. Rest of the fluid domain consist of relatively larger sized
elements. In the rotating domain, jet encloses the whole region and therefore mesh
quality over this entire domain is important and relevant refinements is carried out
where necessary.
Table 4. 1 Mesh statistic of mesh independency test
Meshes NE *106 NN*106 BE, mm FL, mm Normalized
torque
M1 4.55 1.21 1.5 0.17 0.96
M2 5.22 1.41 1 0.14 0.976
M3 6.46 1.81 0.8 0.08 0.983
M4 12.46 3.75 0.5 0.067 0.997
M5 15.7 4.3 0.5 0.067 0.999
M6 18.1 5.56 0.5 0.067 1
NE – number of elements
NN – number of nodes
BE – elements size on the bucket surface
FL – first layer height of the prism layer
41
Figure 4. 4 Number of meshes against normalized torque
For 5kW Pelton turbine mesh independent analysis was carried out. Table 4.1 shows
the statistic of mesh and Figure 4.4 shows mesh number with normalized torque.
From the graph it can be seen that with the M5 mesh, the analysis becomes mesh
independent. Therefore, the rest of the numerical analysis was carried out with this
mesh size. Typical mesh generation of the 5kW Pelton turbine is showed in Figure
(4.5-4.9).
Figure 4. 5 Stationary domain mesh generation
0.95
0.97
0.99
1.01
0 1 2 3 4 5 6
Norm
aliz
ed t
orq
ue
Node number, million
42
Figure 4. 6 Volume meshes of the stationary domain
43
Figure 4. 7 Rotating domain mesh generation
Figure 4. 8 Volume mesh generation inside the bucket
Figure 4. 9 Prismatic mesh generation on the bucket wall
44
Mesh generation for the 100kW Pelton turbine.
For the two jet turbine, tetrahedral elements were created for both domains.
Finer mesh was created on the jet path. On the bucket prismatic elements on the wall
boundary allow the boundary layer to develop properly. Figure 4.10 and 4.11 are
showing the mesh generation of two jet Pelton turbine.
Figure 4. 10 Mesh generation for stationary domain of 100kW Pelton turbine with
two nozzles
45
Figure 4. 11 Mesh generation for rotating domain of 100kW Pelton turbine with
two nozzles
Mesh generation statistic for 100kW two nozzle Pelton turbine is showed below:
Table 4. 2 Mesh generation of 100kW two nozzle turbine
Domains Number of elements Number of nodes
Rotating domain 12608542 2537803
Stationary domain 4984096 870480
Additional section 212589 224000
Physical set up
In order to simulate the physical conditions of the two turbines, the conditions
of the flow must be defined. This section indicates the boundary conditions, fluids
and other physical variables which were set in CFX. CFX allows for specification of
fluid properties, various conditions at specified boundary regions as well as the
creation of user created expressions for monitoring during the solving process.
46
5kW Pelton turbine performance analysis with different rotational speed
A transient analysis was carried out for analyzing the performance of the 5kW
Pelton turbine. The total time is the time it takes for the buckets to move from initial
position to the position where the torque in the middle bucket becomes fully
developed with respect to the single jet configuration. Therefore, the total time varies
with the rotational speed of the runner. Time step is set to 0.5 degrees of rotation of
the turbine divided by the rotational speed which varies with different cases. In the
Table 4.3, CFX set up for Pelton turbine is summarized.
Table 4. 3 Summary of the physical setup
Analysis type Transient
Domain types Rotating Stator
Domain motion Rotating,
Rotation axis - X
Stationary
Fluid models • Homogeneous Model
• Standard Free surface model
• K-Omega SST turbulence model with automatic wall
function
Fluid Model 1. Air
2. Water
Boundary conditions of the domain in Figure 4.12 and boundary condition details in
Table 4.4 are illustrated respectively.
47
Table 4. 4 Boundary condition details
Boundary Types Details
Inlet Jet inlet Bulk mass flow rate -
8.98 kg/s
Opening Opening Pres and Dirn Relative pressure – 0 bar
Wall Bucket wall No slip, smooth wall
Interface Transient rotor stator Pitch ratio 1
Stationary domain inlet Normal speed 0.05m/s air
a) Interface of the fluid domains
b) Buckets wall boundary
48
c) Symmetry boundary
e) Opening boundary
d) Inlet boundary
Figure 4. 12 Boundary conditions of Pelton turbine simulation on Ansys CFX
Physical setups of 5kW Pelton turbine with air injected jet
In CFX, volume fraction is available and by using the physical model, any
percentage of volume flow rate is available for more than multiphase flow. General
domain setup is same with normal condition. The only difference is that on the inlet
boundary the air volume fraction is added by its percentage. The table below shows
how mass flow rate in the inlet boundary is defined.
49
Table 4. 5 Definition of mass flow rate on the inlet boundary for air injected jet
analysis A
ir p
erce
nta
ges
,
%
Wat
er m
ass
flo
w
rate
, k
g/s
Wat
er v
olu
me
flo
w r
ate,
m3/s
Air
volu
me
flo
w
rate
, m
3/s
Air
mas
s fl
ow
rate
, k
g/s
To
tal
mas
s fl
ow
rate
on
th
e in
let,
kg
/s
𝛼 Mwater Qwater Qair=Qwater*𝛼/100 Mair=Qair*𝜌𝑎𝑖𝑟 Mmix=Mwater+Mair
1 8.98 0.009 0.00009 0.0001062 8.9801062
5 8.98 0.009 0.00045 0.000531 8.980531
10 8.98 0.009 0.0009 0.001062 8.981062
15 8.98 0.009 0.00135 0.001593 8.981593
20 8.98 0.009 0.0018 0.002124 8.9802124
Setups for the inlet boundary condition of air injected jet analysis shown in table
below:
Table 4. 6 Inlet boundary option for air injected jet analysis
Cas
es
Boundar
y t
ype
Mas
s
and
Mom
ent
um
opti
on
Mas
s
flow
rate
,
kg/s
Volu
me
frac
tion
1% air
Inlet Bulk mass
8.9801062 Air 0.01
Water 0.99
5% air 8.980531 Air 0.05
Water 0.95
10% air 8.981062 Air 0.10
Water 0.9
15% air 0.981593 Air 0.15
Water 0.85
20% air 0.9802124 Air 0.2
Water 0.8
50
Physical setup of 100kW Pelton turbine with two nozzles
For the 100 kW two jet Pelton turbine analysis main idea is same with the one
jet analysis only difference is two jet inlet are defined as inlet boundary. Physical
setups are illustrated in Table 4.7.
Table 4. 7 Summary of the physical setup
Analysis type Transient
Domain types Rotating Stator
Domain motion Rotating,
Rotation axis - X
Stationary
Fluid models • Homogeneous Model
• Standard Free surface model
• K-Omega SST turbulence model with automatic wall
function
Fluid Model 3. Air
4. Water
Boundary conditions of the domain in Figure 4.13 and boundary details in Table 4.8
illustrated respectively.
51
Table 4. 8 Boundary condition details
Boundary Types Details
Inlet Jet inlet Bulk mass flow rate
– 173 kg/s
Opening Opening Pres and Dirn Relative pressure – 0
bar
Wall Bucket wall No slip, smooth wall
Interface Transient rotor stator Pitch ratio 1
Stationary domain inlet Normal speed 0.05m/s air
a) Interface of the fluid domains
b) Buckets wall boundary
52
c) Symmetry boundary
e) Opening boundary
d) Inlet boundary condition
Figure 4. 13 Boundary conditions of Pelton turbine simulation on CFX
53
Chapter 5. Results and Discussion
The results of the numerical analysis are presented in this chapter. Firstly, 5kW
Pelton turbine performance analysis for rated head and flow condition under several
conditions is discussed here. Following this, 1%, 5%, 10% and 15% of air flow rate
injected jet results will be discussed. Subsequently, nozzle design performance for
the air injected jet and pure water jet will be discussed and compared for the 100 kW
case. Finally, results for the cavitation analysis for both air injected and pure water
jet will also be discussed.
5kW turbine performance with different rotational speed
Torque oscillation of the single bucket with rotational speed is shown in Figure
5.1.
Figure 5. 1 Torque oscillation with bucket angular position
According to the graph when rotational speed increases, the bucket torque decreases.
The counter torque increases with increase in rotational speed. However, higher
rotational speed will result in a lower torque on the wheel. Total torque curve on the
-5
0
5
10
15
20
25
30
0 20 40 60 80 100
Torq
ue, N
m
Bucket angular position, degree
550 rpm
650 rpm
750 rpm
850 rpm
950 rpm
54
runner is calculated based on the numerical measurement of single bucket torque.
This problem is quite simple. Total torque is determined based on the given steps.
Specific bucket frequency is:
𝑓𝑧 =𝜔𝑍
2𝜋 (5.1)
To calculate complete torque on the runner, the torque from a single bucket was
duplicated “n” times and each duplication was shifted n*1
𝑓𝑧 in time. In Figure 5.2 the
complete torque of the runner is shown.
Figure 5. 2 Total torque creation on the runner. Calculated total torque (red).
Single bucket torque (green). Timer averaged total runner torque (blue)
Then time averaged total torque will be determined from total torque oscillation and
indicated by blue line in the Figure 5.2. Time averaged total torque is calculated
within the range of 150 – 300 timesteps, which is indicated by dashed blue lines. After
calculating time averaged total torque on the runner, turbine performance is
calculated as follows:
-5
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Torq
ue, N
m
Time steps
55
𝑃𝑡 = 𝜔 ∗ 𝜏 (5.2)
Here 𝑃𝑡 turbine power, 𝜔 angular velocity, 𝜏 time averaged total torque.
Hydraulic energy is defined by:
𝑃ℎ = 𝜌𝑔𝐻𝑄 (5.3)
Here 𝑃ℎ is available hydraulic energy, 𝜌 water density (997 kg/m3), g gravitational
acceleration (9.82 m/s2), H is available water head, Q flow rate into turbine.
Then efficiency can be found:
𝜂 =𝑃𝑡
𝑃ℎ*100% (5.4)
From 550rpm to 950 rpm five cases were studied and results are shown in Table 5.1.
The highest efficiency was obtained with the 750rpm speed and developed power by
turbine is 4.93kW. Turbine highest efficiency was obtained with velocity coefficient
value of 0.46, which is well matched with typical value. High turbine efficiencies
typically have a velocity coefficient range of 0.45-0.48.
Table 5. 1 Case study of the 5kW Pelton turbine analysis
Hea
d,
m
Flo
w r
ate,
kg/s
Jet
spee
d,
m/s
Hydra
uli
c
Pow
er,
kW
Rota
tional
spee
d, rp
m
Eff
icie
ncy
, %
Pow
er,
kW
Vel
oci
ty
coef
fici
ent
CASE1
32.8 17.96 25.37 5.77
550 77.77 4.49 0.34
CASE2 650 82.55 4.76 0.40
CASE3 750 85.39 4.93 0.46
CASE4 850 84.34 4.86 0.53
CASE5 950 76.93 4.44 0.59
The highest achieved efficiency value is 85.39 which is quite high considering micro
hydro Pelton turbines. Therefore, this design model is reliable and can be used for
further study.
56
Figure 5. 3 Runner efficiency and power with rotational speed
Flow visualization of the bucket and jet interaction of the 750rpm case is shown in
Figure 5.4. A total of nine frames at varying angular positions, depict flow through
the three buckets. Iso-surface at water volume fraction value was chosen as 0.5.
Velocity variation is indicated by color in Figure 5.4. In frame (a) of Figure 5.4, it
can be seen that jet is fully developed before interacting with the first bucket. Velocity
field characteristics is as expected with a maximum value of around 25.5m/s. Velocity
remains same until jet reaches the bucket and becomes zero after it leaves outlet of
the bucket. Small flow velocity at the bucket outlet is visible in frame (h) of Figure
5.4 as expected by the bucket design. Water from outlet doesn’t touch the next bucket
which indicates the well-designed features of the bucket.
4.0
4.2
4.4
4.6
4.8
5.0
60.00
65.00
70.00
75.00
80.00
85.00
90.00
400 500 600 700 800 900 1000
Eff
icie
ncy,
%
Rotational speed, rpm
Efficiency
Power
Pow
er,
kW
57
a) 0o b) 9o c) 18o
d) 27o e) 36o f) 45o
58
g) 54o h) 63o i) 72o
Figure 5. 4 Flow visualization respective with angular position case of 750RPM.
ISO surface volume fraction at value of 0.5 and color is indicating water velocity
[m/s]
5kW Pelton turbine with air injected jet
Results of the 5kW Pelton turbine with air injected jets are discussed in this
section.
Air injection method
In the Pelton turbine operation, the jet velocity is defined by head. If we add
certain amount of air injection into nozzle pipeline while keeping pipeline cross
section constant, the downstream velocity will increase due to increased amount of
flow rate. This increased velocity of mixture creates more kinetic energy, thus
produces more power on the runner. Figure 5.5 shows the scheme of air injection in
the nozzle and Figure 5.6 shows normal operating condition.
13.5 m/s 27 m/s 0 m/s
59
Figure 5. 5 Air injection scheme in the nozzle
Figure 5. 6 Scheme of normal operation condition
On the cross section A-A and B-B mass conservation equation will be written as
follows:
𝑚𝑤𝑎𝑡𝑒𝑟 + 𝑚𝑎𝑖𝑟 = 𝑚𝑚𝑖𝑥 (5.6)
Mmix is higher than Mwater of normal operation in the Figure 5.6.
In Pelton turbine most of the available hydraulic energy converts to kinetic energy
after water is released from the nozzle. Available energy of jet in the B-B cross
section can be written as follows:
Available energy at case of air injected operation:
𝐸𝑚𝑖𝑥 =𝑚𝑚𝑖𝑥𝑣𝑚𝑖𝑥
2
2
(5.7)
Available energy at normal operating
condition.
𝐸𝑛𝑜𝑟𝑚𝑎𝑙 =𝑚𝑤𝑎𝑡𝑒𝑟𝑣𝑤𝑎𝑡𝑒𝑟
2
2
(5.8)
Since mmix>mwater and Vmix>Vwater available kinetic energy in mixture will be higher
than normal condition. Emix>Enormal.
Vwater Air injection
Vmix
Vwater Mwater
Mair
Mmix
Mwater
Mwater
Vwater
Vwater
60
Energy definition of the air injected operation and efficiency calculation
Efficiency definition of the air injected analysis is different about normal
operating condition due to extra power which is used for inject air. Air injection work
is expressed as:
𝑊𝑖𝑛𝑗 = 𝑄𝑎𝑖𝑟 ∗ 𝑃𝑠 ∗ 𝑡 (5.9)
Where:
𝑊𝑖𝑛𝑗 is air injection work, 𝑄𝑎𝑖𝑟 is flow rate of the injecting air, 𝑃𝑠 static pressure of
nozzle inside, t is injection time.
Ps is can be found by writing Bernoulli’s equation on the A-A cross section of the
Figure 5.5 and on the reservoir.
𝜌𝑔𝐻 = 𝑃𝑠 +𝜌𝑉𝑤
2
2
(5.10)
The penstock diameter is about 3.2 times bigger than jet diameter. Given that velocity
of the water on the A-A is very small then we can neglect 𝜌𝑉𝑤
2
2 term of the equation
(5.10), therefore Ps is almost equal to reservoir head pressure energy.
Rated head of our turbine is 33m therefore, Ps=321298 Pa.
Then efficiency of the turbine is:
𝜂 =𝑇 ∗ 𝜔
𝜌𝑔𝐻𝑄 + 𝑊𝑖𝑛𝑗
(5.11)
Where:
𝜂 is efficiency of the air injected operation, T – torque of the turbine measured from
CFD analysis, 𝜔 is angular velocity, 𝜌 is water density, 𝑔 gravitational acceleration,
𝐻 net head, 𝑄 water flow rate.
Results of the 5kW Pelton turbine with the air injected jet analysis
CFD analysis of air injection into the Pelton turbine jet results was positive.
Efficiency increased linearly. Table 5.2 indicates the results of the air injected jet
61
analysis. After injecting the air into the jet, the velocity of the mixture was increased
as expected. Thus turbine could produce more power.
Table 5. 2 Air injected jet results
Air
per
cen
tag
e
mix
ture
den
sity
CF
D v
elo
city
Wat
er m
ass
flo
w,
kg
Air
mas
s fl
ow
, k
g
Avai
lab
le
hyd
rauli
c
pow
er,
kW
Norm
al
oper
atio
n
Po
wer
, kW
Turb
ine
po
wer
, kW
Eff
icie
ncy
, %
Case 1 1 987.04 25.76 8.98 0.00010 5.94 4.93 5.01 84.73
Case 2 5 947.21 26.85 8.98 0.00054 6.17 4.93 5.4 87.53
Case 3 10 897.43 28.34 8.98 0.0011 6.46 4.93 5.95 92.12
Case 4 15 847.64 30.00 8.98 0.0017 6.75 4.93 6.54 96.92
From the result it is seen that the efficiency of the turbine with 1% of air injection
dropped slightly then increased linearly with increasing percentage of air flow rate.
This first drop in injected air doesn’t have much effect for the jet velocity but after
adding more air the turbine power increased significantly thus efficiency increased.
In Figure 5.7, the comparison of power developed by air injected jet and normal
condition is showed. Hydraulic power of normal condition was 5.88kW and turbine
power for normal condition was 4.93kW. Turbine power after air injection linearly
increased.
Figure 5.8 indicates comparison of the efficiency results of the air injected jet and
normal condition.
62
Figure 5. 7 Power comparison of normal and air injected operation
Figure 5. 8 Efficiency comparison of normal and air injected operation
82
84
86
88
90
92
94
96
98
0 5 10 15
Eff
icie
ncy,
%
Air injection rate, %
Air injected efficiency
Normal effciency
-1.02%
2.14%
6.73%
11.53%
63
Figure 5.9 indicates air bubbles distribution behavior. Ansys CFX offers well
distributed air and water mixture model. Consequently, jet profile maintains its shape
until reaches the buckets and air bubbles does not affect the jet interface. In Figure
5.10, air inside the jet collected within the time of going from nozzle to the bucket in
which the size of the bubbles grow. According to Figure 5.11, these bubbles travel
from inlet to outlet edge of the bucket within the mixture (water), which is likely to
create cavitation phenomena. Cavitation analysis of the air injected jet will be
discussed in the next section and compared with the normal jet analysis.
Figure 5. 9 Visualization of air distribution on the inlet boundary. Air injection rate
5% of the water
64
Figure 5. 10 Air bubbles inside the jet
Figure 5. 11 Air bubble transport inside the bucket profile
Velocity visualization of the air injected jet cases showed in the Figure 5.12. Surface
water volume fraction value of 0.5 and the color is indicating velocity variation.
65
a) 1% air injection
b) 5% air injection
c) 10% air injection
d) 15% air injection
Figure 5. 12 Velocity of the air injected jet
100kW Pelton turbine normal operation and with air injected jet
analysis and comparison
In this section, results of numerical analysis of normal operation of 100kW
turbine under normal case and air injected jet case will be discussed. General setup is
same as 5kW turbine but only the mass flow rate changed on the inlet boundary
condition. Available head is same as 5kW turbine therefore, flow rate is much greater
than 5kW turbine. The table below indicates the case studies of 100kW turbine.
66
Table 5. 3 100kW Pelton turbine results
Water mass flow rate, kg/s 316
Jet velocity, m/s 24.58
Available hydraulic energy, kW 101
Turbine Power, kW 84.51
Efficiency, % 85.3
Efficiency of the 100kW turbine was almost same as 5kW turbine with a difference
of only 0.09%. This is because of geometrical similarity used for both designs and
same hydraulic head. According to the Pelton turbine efficiency scale-up code
IEC60193, efficiency difference between model turbine and prototype is depends on
Froude and Reynolds numbers and which are expressed as:
𝐹𝑟 = √2𝑔𝐻
𝑔𝐵
(5.12)
𝑅𝑒 =√2𝑔𝐻 ∗ 𝐵
𝜈
(5.13)
The corresponding proportion parameters between the model turbine and its
prototype are denoted as:
𝐶𝐹𝑟=
𝐹𝑟𝑝
𝐹𝑟𝑀
(5.14)
𝐶𝑅𝑒 =𝑅𝑒𝑝
𝑅𝑒𝑀
(5.15)
The scale up calculation of the turbine efficiencies is conducted by considering the
difference in the efficiencies between model and the prototype turbines:
𝜂𝑃 = 𝜂𝑀 + Δ𝜂 (5.16)
Δ𝜂 is expressed as:
Δ𝜂 =8.5 ∗ 10−7
𝜑𝐵2
(𝐶𝐹𝑟0.3𝐶𝑅𝑒 − 1) + 5.7 ∗ 𝜑𝐵
2 (1 − 𝐶𝐹𝑟0.3)
(5.17)
67
Then calculating every parameters of the equation (5.12-5.15) then, substituting to
the equation (5.17) efficiency difference between two turbines is Δ𝜂 = 0.0101 and
theoretical and CFD result difference is 0.91%.
Table 5. 4 Air injected jet results for 100kW one nozzle turbine
Hea
d, m
Flo
w
rate
,
kg/s
Jet
spee
d, m
/s
n,
rpm
Pow
er,
kW
Eff
icie
ncy
, %
Normal operation
32.6 316
24.58
169
84.51 85.3
1% air injection 24.82 85.93 84.15
5% air injection 25.83 93.09 87.35
10% air injection 27.23 103.16 91.99
15% air injection 28.78 115.80 98.45
The rate at which power increases was proportional to the increase in air injection
rate. Efficiency for the 100kW single nozzle turbine is similar to that of the 5kW
configuration for the case of air injected jet. In the first case of 1% air injected jet, the
efficiency dropped by 0.36% and started to increase linearly with increase in air
injection. Figure 5.14 shows efficiency difference between normal and air injected
operation.
68
Figure 5. 13 Power comparison of normal and air injected operation
Figure 5. 14 Efficiency comparison of normal and air injected operation
1.68% 10.16%
22.07%
37.14%
-0.36% 2.84%
7.48%
13.94%
69
Analysis of 100kW Pelton turbine with two nozzles
In this section, results of the numerical analysis of the two nozzle 100 kW turbine
was carried out. Result of the two nozzle analysis is illustrated in Table 5.5. The
analysis indicates that air injected into the two nozzles results in increased turbine
power similar to that of the single nozzle configuration. With the two nozzle
configuration, turbine efficiency of normal operation is higher than that of the single
nozzle configuration. This increase in efficiency is due to natural behavior of Pelton
turbine which tends to perform better at partial load condition. Figure 5.15 and 5.16
show the two nozzle turbine power and efficiency increasing rate respectively.
Comparison of efficiency and power between one and two nozzle configurations is
shown in Figure 5.17.
Table 5. 5 Results of 100kW two nozzle Pelton turbine analysis
Parameters Normal operation
Water mass flow rate, kg/s 346
Jet velocity, m/s 24.58
Available hydraulic energy, kW 106
Turbine Power, kW 90.47
Efficiency, % 86.39
70
Table 5. 6 Air injected result for 100kW two nozzle turbine
Hea
d,
m
Flo
w r
ate,
kg
/s
Jet
spee
d,
m/s
n,
rpm
Po
wer
,
kW
Eff
icie
nc
y,
%
Normal operation
32.6 346
24.58
235
90.74 86.63
1% air injection 25.00 92.17 86.07
5% air injection 26.05 99.99 89.64
10% air injection 27.5 108.88 92.96
15% air injection 29.2 122.3 99.8
Figure 5. 15 Power comparison of normal and air injected operation of Pelton
turbine with two nozzles
1.58% 10.2%
19.9% 34.99%
71
Figure 5. 16 Efficiency comparison of normal and air injected operation of 100kW
Pelton turbine with two nozzles
Figure 5. 17 Comparison of power and efficiency of air injected operation of one
and two nozzle 100kW turbines
Po
wer
, kW
72
Cavitation analysis of 5kW Pelton turbine
The Pelton turbine bucket is shaped with a groove in the back side. If there is no
groove, water slapping the back side of the bucket can cause critical loss of power.
Under very high heads, cavitation can cause pitting in the groove area on the back of
the bucket. This is due to high velocity flow running along the end of the splitter ridge
and then being pulled away from it, leaving an area of low pressure. Water vaporizes
on the surface, creating small bubbles. As pressure returns to the region, the bubbles
collapse rapidly in miniature implosions which can damage the surface, causing
pitting. Erosion and wear of hydraulic surfaces are frequent problems in hydraulic
turbines, which lead to decrease in performance with time and in extreme cases result
in mechanical failure of the rotor. The study of cavitation wearing process is an
important step to improve the impeller design and minimize the need to carry out
maintenance [18].
However, since cavitation is generally associated with high heads of 500m or more,
it is not a problem for micro hydro Pelton turbines. But in present case, air is mixed
with water, therefore analysis of cavitation is necessary.
For the cavitation analysis the working fluid is changed from two phases to three
phases (Air at 25 degrees Celsius, water, water vapor at 25 degrees Celsius). For the
cavitation model the Rayleigh-Plesset [19] is chosen and required parameters,
saturation pressure is set to 3471, and the main nucleation site diameter is specified
as the default value of 2.0*10-6 m.
In the Figure 5.18, the vapor visualization of two cases are compared. Color-map
represents variation in vapor volume fraction. On the 78th time-step the first
interaction between the bucket and the jet occurs. Initially, there was no vapor on the
front side of the bucket until the water passed through the bucket. For both cases
small amount of vapor was created only in the notch area. The vapor existed for a
total of 22 time-steps or 11 degrees of runner rotation.
73
Ts=78
Ts=78
Ts=85
Ts=85
Ts=90
Ts=90
Ts=95
Ts=95
74
Ts=100
a) Air injected jet
Ts=100
b) Normal jet condition
Figure 5. 18 Comparison of vapor visualization
On the 85th time-step, vapor near the edge of the splitter tip of air injected jet analysis
is slightly bigger than normal case. But after this time step vapor of the jet injected
air is seen smaller than normal case. Overall analysis of vapor formation on the bucket
was shown to be more favorable than the normal case. Since for micro hydro Pelton
turbines, cavitation is not a problem and the results indicated that there is no
significant negative impacts on turbine performance and structural integrity itself.
75
Chapter 6. Conclusion
This study aims to implement a new method for improving turbine efficiency by
means of air injection into water jet. In order to do so, study was carried out for two
different cases; 5kW and 100kW Pelton turbine. Numerical analysis was divided in
to two parts: normal performance and air injected jet performance. The analysis was
performed on CFX.13 commercial tool. The following results were obtained:
5kW Pelton turbine was designed and five rotational speed cases were carried
out with jet speed ratio varying from 0.34 – 0.59. The highest efficiency was
found to be 85.39% at a rotational speed of 750rpm and corresponding jet
speed ratio was 0.46. Efficiency rate was reliable for micro hydro Pelton
turbine. Highest efficiency occurred at jet speed ratio value of 0.46 and is
well matched with typical values of jet speed ratio.
5 kW turbine with air injected jet analysis was performed for four cases of
air volume rate varying from 1% -15% of water volume flow rate. When air
injection increases, the power of the turbine increased linearly. For the 1% of
air injection the efficiency dropped by -1.02% and for the 15% air injection
efficiency increased by 11.53%.
In the case of 100kW Pelton turbine, the speed coefficient was chosen as 0.46
which is already rated in the 5kW analysis. For the normal operating
condition, the efficiency was found to be 85.33% and for the 1-15% jet
injected condition, efficiency variation ranged between -0.36% - 13.94%.
In the case of 100kW two nozzle Pelton turbine, efficiency variation was
from -0.56% to 13.2% with the air injection rate from 1% to 15%
respectively.
In the case of cavitation analysis, the area of vapor volume fraction on the
bucket surface showed to be more favorable than the normal condition.
However, the vapor occurred over a relatively short period of time. This is
due to the pulling force of water from the back side of the bucket which is
lower than the normal case.
76
Acknowledgement
Studying in Korea Maritime and Ocean University, has been an absolute pleasure
and I was humbled and grateful to have received this great opportunity. There are
many people who have contributed to the successful completion of this thesis
whom I would like to acknowledge.
Firstly, I would like to express my deepest appreciation to my supervisor Prof.
Dr. Young-Ho Lee for giving me the chance to study in Korea Maritime and
Ocean University. With the help of his professional knowledge and extreme
generosity, I have achieved today’s success.
Secondly, I am deeply thankful to the thesis committee members: Prof. Dr. Dong
Woo Sohn (Chair person) and Prof. Dr. Young-Do Choi (Co-chairperson) for
their patience, time, helpful comments and recommendations.
Thirdly, I would like to acknowledge the help of my numerous lab mates both
Korean and foreign for their knowledge and assistance in many areas. Without
their assistance, I wouldn’t have been able to achieve much.
I also wish to express my sincere thanks to my Mongolian professor
Bayasgalan.D for having trust in me and giving me the opportunity to meet
professor Young-Ho Lee.
Also, I would like to thank my religious Korean brothers and sisters for their help
and inspiration. Especially, Brother Yeong-Seop Lee, for helping me in time of
need.
Last but not least, I am fully indebted to my family, my parents, lovely wife and
daughter, for their infinite love for me.
77
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