University of Brasília - UnB
Faculty UnB Gama - FGA
Major of Aerospace Engineering
DESIGN AND NUMERICAL ANALYSIS OF GAS
TURBINE BLADE PROFILES
Author: Hugo Felippe da Silva Lui
Advisor: Prof. Domenico Simone, PhD
Brasília, DF
2016
HUGO FELIPPE DA SILVA LUI
TITLE: DESIGN AND NUMERICAL ANALYSIS OF GAS TURBINE BLADE
PROFILES
A dissertation submitted in partial fulfilment
of the requirements for the title of bachelor
in Aerospace Engineering at the University
of Brasília (FGA).
Adivisor: Prof. Domenico Simone, PhD
Brasília, DF
2016
CIP – Catalogação Internacional da Publicação*
da Silva Lui, Hugo Felippe.
Title of dissertation: Design and numerical analysis of gas
turbine blade profiles / Hugo Felippe da Silva Lui.
Brasília: UnB, 2016. 62 p. : il. ; 29,5 cm.
Dissertation (Undergraduate) – University of Brasília
Faculty of Gama, Brasília, 2016. Advisor: Prof. Domenico
Simone, PhD.
1. Axial turbine. 2. Computational fluid dynamics. 3. Gas
turbine profile I. Simone, Domenico. II. PhD.
CDU Classificação
DESIGN AND NUMERICAL ANALYSIS OF GAS TURBINE PROFILES
Hugo Felippe da Silva Lui
A dissertation submitted in partial fulfilment of the requirements for the title of bachelor
in Aerospace Engineering at the University of Brasília (FGA) at 01/12/2016, defended,
and approved by Dissertation Committee:
Prof. PhD: Domenico Simone, UnB/ FGA
Advisor
Prof. PhD: Paolo Gessini, UnB/ FGA
Committee Member
Prof. PhD: Olexiy Shynkarenko, UnB/FGA
Committee Member
Brasília, DF
2016
Esse trabalho é dedicado à minha mãe,
Isamar da Silva, à minha família e a todos os
meus amigos, que me apoiaram durante a
minha vida.
AGRADECIMENTOS
Eu agradeço a minha mãe por ter me apoiado durante minha graduação, sem o seu
apoio, eu não chegaria a esse ponto. Eu agradeço também aos meus familiares pelo
suporte dado. Um agradecimento especial aos meus amigos, os que fiz no CECB, na
UnB e nos Estados Unidos, que foram essências para o meu amadurecimento pessoal
e acadêmico.
Destaco também o quão importante foram o Professor Paniagua e o pesquisador
Jorge, ambos da Purdue University, para esse trabalho. Sou muito grato ao Prof.
Paniagua por me ter dado a oportunidade de trabalhar com turbinas axiais e também
pelos seus conselhos. Sou extremamente agradecido ao Jorge, por ter sido meu
mentor durante o meu estágio de pesquisa, suas “aulas” sobre turbinas axiais foram
de grande ajuda para o desenvolvimento desse trabalho.
Agradeço ao Professor Domenico pela sua paciência e tempo para discursão desse
trabalho e principalmente pelas suas sugestões cirúrgicas. Um agradecimento
especial a todos os professores, seus ensinamentos foram de suma importância para
para a concretização desse trabalho.
“They did not know it was impossible, so they
did it” - Jean Cocteau
RESUMO
A demanda por motores aeronáuticos cada vez mais eficientes faz com que os
projetistas de turbinas axiais procurem desenvolver pás que permitem fornecer tal
performance requerida. O projeto da pá, começa a partir da definição dos perfis da pá
em diferentes seções, geralmente em três posições: na raiz, no meio e no topo da pá.
A forma 3D da turbina axial é obtida a partir do empilhamento dos perfis, da raiz até o
topo da pá. A performance aerodinâmica da pá é afetada significativamente pela
geometria do perfil. Dessa maneira, a definição do aerofólio possui papel critico
durante o projeto. Esse trabalho apresenta uma metodologia para o desenvolvimento
de aerofólios para turbinas axiais, através do projeto 1D da turbina axial, junto com
um método para gerar aerofólios usando curvas de Bézier. Primeiramente, será
estudado a cascata de aerofólios da turbina VKI LS-89, como caso teste, para
validação do método numérico. Posteriormente, simulações computacionais serão
realizadas para verificação da qualidade dos aerofólios gerados, que será avaliada
por meio dos gráficos de distribuição do número de Mach ao longo das superfícies do
perfil aerodinâmico, além dos campos do escoamento e dos valores de perda de
pressão total na passagem. As simulações numéricas serão feitas pelo software
ANSYS Fluent.
Palavras-chave: Turbina axial. CFD. Aerofólio. Turbinas a gás. Fluent. Curva de
Bézier. Turbomáquinas
ABSTRACT
The demand for increasingly efficient aircraft engines make turbine designers attempt
to develop blades that provide such required performance. The blade design starts
from the definition of blade profiles in different positions, usually in three places: hub,
mid and tip sections. The 3D shape of an axial turbine is obtained by stacking the
defined profiles from hub to tip. The aerodynamic performance of the blade is
significantly affected by the profile geometry. Thus, the determination of the airfoil
shape has critical role during the project. This work presents a methodology for
developing airfoils for axial turbines through an 1D turbine design, along with a method
for generating airfoils using Bezier curves. The VKI LS-89 transonic turbine cascade
will be first studied as a test case to the validation of the solver. After that, CFD
simulations will be performed in order to check the quality of the airfoils generated by
evaluating Mach number distribution along the airfoil surfaces, the flow field contours,
and the total pressure loss thought the passage. These numerical simulations will be
carry out using the software ANSYS Fluent.
Keywords: Axial turbine. CFD. Airfoil. Gas turbine. Fluent. Bézier curve.
Turbomachinery
CONTENTS
1 INTRODUCTION .............................................................................................................. 11
1.1
1.2
1.3
STATE OF ART ....................................................................................................................................... 11
OBJECTIVES .......................................................................................................................................... 13
WORK ORGANIZATION ......................................................................................................................... 14
2 BASIC PRINCIPLES ........................................................................................................ 15
2.1
2.2
2.3
2.4
2.6
2.6
2.7
2.8
FUNDAMENTAL LAWS AND EQUATIONS ............................................................................................ 15
THE CONTINUITY EQUATION ............................................................................................................... 16
THE MOMENTUM EQUATION ............................................................................................................... 16
THE ANGULAR MOMENTUM EQUATION ............................................................................................. 18
THE FIRST LAW OF THERMODYNAMICS ............................................................................................ 19
THE SECOND LAW OF THERMODYNAMICS ....................................................................................... 20
THE IDEAL GAS EQUATION .................................................................................................................. 22
COMPRESSIBLE FLOW EQUATION ..................................................................................................... 22
3 AXIAL FLOW TURBINE ................................................................................................... 25
3.1
3.2
3.3
3.4
3.5
3.6
3.7
TURBINE GEOMETRY ........................................................................................................................... 25
VELOCITY DIAGRAMS OF THE AXIAL TURBINE STAGE.................................................................... 27
TURBINE DESIGN PARAMETERS ........................................................................................................ 30
THERMODYNAMICS OF THE AXIAL TURBINE STAGE ...................................................................... 33
EFFICIENCY ........................................................................................................................................... 34
TURBINE SOURCE LOSS ...................................................................................................................... 35
ZWEIFEL CRITERION ............................................................................................................................ 38
4 ONE DIMENSIONAL DESIGN PROCEDURE .................................................................. 40
4.1
4.2
4.3
METHODOLOGY .................................................................................................................................... 40
DESIGN MODEL ..................................................................................................................................... 41
TURBINE DESIGN PROCEDURE .......................................................................................................... 42
5 TURBINE AIRFOIL DESIGN ............................................................................................ 55
5.1
5.2
5.3
BLADE PROFILE METHODS ................................................................................................................. 55
BÉZIER CURVES.................................................................................................................................... 55
DESIGN APPROACH.............................................................................................................................. 57
6 METHODS ........................................................................................................................ 66
6.1
6.2
6.3
6.4
6.5
6.6
GOVERNING EQUATIONS .................................................................................................................... 66
FLUENT SOLVER ................................................................................................................................... 67
TURBULENCE MODELLING .................................................................................................................. 68
MESH GENERATION ............................................................................................................................. 69
BOUNDARY CONDTIONS ...................................................................................................................... 71
SOLUTION INITIALIZATION AND CONVERGENCE ............................................................................. 71
7 TEST CASE...................................................................................................................... 73
7.1
7.2
7.3
DESCRIPTION OF THE VKI LS89 TEST CASE ..................................................................................... 73
THE VKI LS89 BLADE PROFILE GEOMETRY ....................................................................................... 73
THE MUR49 TEST .................................................................................................................................. 75
8 RESULTS ......................................................................................................................... 80
9 CONCLUSIONS ............................................................................................................... 94
REFERENCES .................................................................................................................... 96
APPENDIX - MATLAB CODE…………………………………………………………….99
11
1. INTRODUCTION
1.1 STATE OF ART
Axial flow turbines are machines that have been used in many fields, as aviation,
oil and gas, power generation, Navy etc. The application of turbomachinery is a global
one, thus any increase in axial turbine efficiency and performance could lead to a major
economic impact worldwide [1]. General electric [2] reports that an improvement in
efficiency by 1% leads to an estimate saving in aviation industry of £6.14 billion over
15 years for a country.
Figure 1. Estimate saving in aviation industry [2]
12
The fundamental components of aeronautical gas turbine engines, as illustrated
in Fig. 2, are the compressor, the combustion chamber and the turbine. The
compressor increases the static pressure of the air. In the combustion chamber, the
fuel is mixed with this compressed air and burned, then the chemical energy stored by
the fuel is converted in internal energy and pressure. The turbine extracts the energy
from the working fluid leaving the combustion chamber. The power extracted by the
turbine is used to drive the compressor. Finally, the energy available in the exhaust
gases from the turbine is converted into a high speed propelling jet by the nozzle [3].
Figure 2. Typical single-spool turbojet [3]
For gas turbines engines, where axial turbines are employed, their net power
output is the difference between the turbine work and compressor work, the ratio is
about 2:1, so a small increase in efficiency of the compressor or turbine causes a larger
proportional change in the power output, thus an increase in thrust for aeronautical gas
turbine engines [1].
Figure 3. T-s diagram for a turbojet engine [3]
13
Over the years huge efforts have been expended in trying to improve the
efficiency of axial turbines, and the total-to-total efficiency is now over 90 percent. This
makes new improvements ever more difficult to achieve, however, it is still possible to
obtain more efficient machines by improving the knowledge of the thermodynamics
and fluid mechanics of the flow through the turbine [1].
The research on aircraft engines started in the 1940s and 1950s, it led to the
development of several methods of performance of axial turbines, e.g., Ainley and
Mathieson [6], which is still in use today. Over the years, new methods were developed
by make some improvements on prediction of losses through the turbine, these
different loss mechanisms are categorized as, profile loss, secondary or endwall loss
and tip leakage. These improvements were possible due to the advent of new
instrumentation, it led to a better understanding of the flow [1].
The advent of the high speed digital computer combined with the development
of accurate numerical algorithms gave to engineers a new approach to study the flow,
this new approach is called, computational fluid dynamics (CFD). It is basically a
computer program which we can carry out “numerical experiments” to investigate the
flow behavior and its properties [4].
CFD has been used as design tool by engineers to improve the turbine
efficiency. During the design the turbine blade designer make several adjusts on blade
geometry in order to obtain the suitable geometry that combines good aerodynamic
performance with structure requirements. This process of refinement of blade to match
the better geometry can be time consuming and expensive. Therefore, optimization
techniques such as evolutionary algorithms or inverse design methods are option to
enhance the design process [5].
1.2 OBJECTIVES
The main goal of this work is to develop a turbine airfoil design procedure based
on existing methods. Thus, the blade profiles generated will be analyzed using 2D CFD
simulations in order to check their performance. A method to perform an axial turbine
mean line design will be showed in order to provide the essential parameters to design
the blade profiles. In addition, one of the aims of the work is to learn about CFD
14
techniques, such as mathematical models used on CFD simulations, mesh generation,
turbulence models and boundary layer.
1.3 WORK ORGANIZATION
In the chapter 2, the basic principles of thermodynamic and fluid mechanics
needed to analyze axial flow turbines will be demonstrated. In the chapter 3 discusses,
the velocity triangles of a turbine stage, the design parameters, the thermodynamics
of an axial turbine stage, 2D turbine losses, and efficiencies. In chapter 4, the one
dimensional turbine design procedure will be showed, in order to calculate the flow
properties at each station of the turbine stage, the blade angles, and the blade row
geometry. In the chapter 5, a turbine airfoil design procedure based on existing
methods will be presented. Chapter 6 describes the mathematical and numerical
methods needed to solve the flow through the turbine cascade. In chapter 7, a test
case will be studied to validate the CFD solver presented in previous section. Chapter
8 describes the CFD results of the turbine airfoils designed. Chapter 9 presents the
conclusion and the important comments about the turbine airfoils investigated.
15
2 BASIC PRINCIPLES
2.1 THE FUNDAMENTAL LAWS AND EQUATIONS
According to Dixon and Hall [7]. the basic physical laws of thermodynamics and
fluid dynamics and equations necessary to the study of axial flow turbines are:
i.
The continuity equation
ii.
The momentum equation
iii.
The angular momentum equation
iv.
The first law of thermodynamics
v.
The second law of thermodynamics
vi.
The ideal gas Equation
vii.
Compressible flow relations
16
2.2 THE CONTINUITY EQUATION
From physical principle of mass conservation, “Mass can be neither created nor
destroyed” [8]. The continuity equation is expressed by
𝜕
⃗ ∙ 𝑑𝐴 = 0
∭ 𝜌𝑑𝒱 + ∬ 𝜌𝑉
𝜕𝑡
𝐶𝒱
(1)
𝐶𝑆
The first integral term represents the time rate of change of the mass inside the
control volumes, and the second integral term is the net mass flow into the control
volume through the entire control surface
Most of axial flow turbine analyses are limited to steady and one-dimensional
flows. Considering that we have an inlet and outlet surface, and assuming that there is
no change within the control volume, then we can define the mass flow rate 𝑚̇ as
𝑚̇ = 𝜌in 𝑉in 𝐴𝑛,in = 𝜌out 𝑉out 𝐴𝑛,out
(2)
Where 𝜌 is the fluid density, 𝑉 is the velocity, 𝐴𝑛 is the normal area.
2.3 THE MOMENTUM EQUATION
The momentum equation is based on Newton’s second law of motion. It
expresses the sum of external forces acting on a fluid element to the rate of change of
momentum in the direction of the resultant force. This equation is important because
the force exerted upon a turbine blade can be obtained by found by the momentum
equation [10]. The linear momentum equation is:
17
⃗ 𝜌𝑑𝒱 =
∬ 𝐹 𝑑𝐴 + ∭ 𝐵
𝐶𝑆
𝐶𝒱
𝜕
⃗ (𝜌𝑑𝒱) + ∬ 𝑉
⃗ (𝜌𝑉
⃗ ∙ 𝑑𝐴)
∭𝑉
𝜕𝑡
𝐶𝒱
(3)
𝐶𝑆
Here 𝐹 is the surface force (tangential and normal) per unit of area acting on the
⃗ is the body force per unity mass, such as electromagnetic and
control surface, 𝐵
gravity.
Figure 4. Control volume of a turbine cascade [10]
Considering a control volume of Fig. 4, for a steady state flow and assuming
that inlet and outlet surfaces are far upstream and downstream, respectively, from the
blade, where flow is uniform. The body forces acting on control volume is very small
⃗ ≈ 0. Using the momentum equation, Eq. (3),
relative to the surface force, then 𝐵
applied to the control volume ABCD, the force in the tangential direction is given by:
𝐹𝑦 = 𝑚̇(𝑉𝜃2 − 𝑉𝜃3 )
Where 𝑉𝜃2 and 𝑉𝜃3 are the tangential velocities.
(4)
18
2.4 THE ANGULAR MOMENTUM EQUATION
The angular momentum relates the rate change of angular momentum of the
system with the sum of moments of all external forces acting on the system [11]. It is
based on one of the most important principles in mechanics, Newton’s second law of
motion. The angular momentum equation is given by
∑ 𝑀0 =
𝜕
⃗ )𝜌𝑑𝒱 + ∬(𝑟 𝑥 𝑉
⃗ )𝜌𝑉
⃗ ∙ 𝑑𝐴
∭(𝑟 𝑥 𝑉
𝜕𝑡
𝐶𝒱
(5)
𝐶𝑆
Where 𝑀0 is the moment about some point and 𝑟 is the vector position.
Figure 5. Control volume of a turbomachinery [7]
For a control volume as Fig. 5, where the fluid enters steadily at an uniform
tangential velocity 𝑉𝜃2 and radius 𝑟2 and leaves with uniform velocity 𝑉𝜃3 and radius 𝑟3 ,
using Eq. (5), the torque about axis A is given by:
𝜏𝐴 = 𝑚̇(𝑟2 𝑉𝜃2 − 𝑟3 𝑉𝜃3 )
(6)
By Eq. (6), we can express the power generated by the turbine based on the
tangential velocities
19
𝑊̇ = 𝜏𝐴 Ω = 𝑚̇(𝑈2 𝑉𝜃2 − 𝑈3 𝑉𝜃3 )
(7)
Where Ω is the angular velocity and the blade speed 𝑈 = Ω𝑟
2.5 THE FIRST LAW OF THERMODYNAMICS
The first law of thermodynamics states that energy can be neither created nor
destroyed during a process, it can only change forms [9]. For an open system, the
energy can be transfer by work 𝑊, heat 𝑄, and by a flowing fluid. The energy of the
system 𝐸 is composed by: internal, kinetic, potential energy.
Considering a fluid which passes through a control volume at a steady rate mass
flow 𝑚̇, there is an inlet and outlet surface. Then, the first law of thermodynamics
becomes
1
(𝑄̇𝑖𝑛 − 𝑄̇𝑜𝑢𝑡 ) + (𝑊̇𝑖𝑛 − 𝑊̇𝑜𝑢𝑡 ) + 𝑚̇in (ℎin + 𝑉in 2 + 𝑔𝑧in )
2
(8)
1
𝑑𝐸
− 𝑚̇out (ℎout + 𝑉out 2 + 𝑔𝑧out ) =
2
𝑑𝑡
Where 𝑊̇ is the rate of work transfer, 𝑄̇ is the rate of heat transfer, 𝑈 is the
internal energy, 𝑔 is the gravity and 𝑧 is the height.
For convenience is useful to combine the enthalpy with the kinetic energy, the
result is called the total enthalpy:
ℎ0 = ℎ +
1 2
𝑉
2
(9)
The energy is transferred from the fluid to the turbine blades. Assuming that
there is no heat transfer from or to the control volume surrounding, the variation of
potential energy is negligible. As the turbine usually operates at steady state
20
conditions, using Eq. (9) into Eq. (8), we have that the power generated by the turbine
is
𝑊̇ = 𝑚̇(ℎin − ℎout )
(10)
Assuming that the working fluid is a thermally and calorically gas, which means
that the specific heats (𝐶𝑝 and 𝐶𝑣 ) are function of temperature only and they are
constant. Thus, ℎ = 𝐶𝑝 𝑇 , 𝑢 = 𝐶𝑣 𝑇 and the ratio of the specific heats is 𝛾 =
𝐶𝑝
𝐶𝑣
. In this
way, we can express the enthalpy in terms of temperature. Thus, Eq. (10) becomes
𝑊̇ = 𝑚̇𝐶𝑝 (𝑇in − 𝑇out )
(11)
Combining Eq. (7) with Eq. (10), results the Euler work equation
Δℎ𝑜 = Δ(U𝑉𝜃 )
(12)
This equation is valid for steady, adiabatic, viscous and inviscid flows and It
states that greater the total enthalpy or tangential velocity change of the rotor, higher
the power delivery by the turbine. Note that for stationary blades, the blade speed is
zero, thus there is no work transfer from the fluid.
2.6 THE SECOND LAW OF THERMODYNAMICS
According to Cengel [9], the second law of thermodynamics asserts that
processes occur in a certain direction and energy has quality. The process take place
only if it satisfies the first and second law of thermodynamics. An important corollary of
the second law of thermodynamics, known as the inequality of Clausius. It introduces
a new thermodynamic property, called entropy 𝑆. Considering state 1 as the start and
state 2 as the end of the process, then
21
2
𝑆2 − 𝑆1 = ∫
1
𝛿𝑄
+ 𝑆𝑔𝑒𝑛
𝑇
(13)
Where 𝑆𝑔𝑒𝑛 is the entropy generated during a process and it always positive
quantity or zero. If the process is adiabatic, 𝛿𝑄 = 0, then 𝑆2 ≥ 𝑆1 , if the process is
reversible then, 𝑆𝑔𝑒𝑛 = 0. Thus, for a flow that is both adiabatic and reversible, the
entropy will remain unchanged, this process as know as isentropic. Usually, axial
turbine operates close to an isentropic process, therefore preliminary calculations are
considered as adiabatic and reversible process, after that it is correct by some loss
prediction method.
The first law of the thermodynamics can be expressed using the entropy
definition and considering that the flow work (fluid pushed into or out of the control
volume) is given by 𝛿𝑤 = 𝑃𝜈. Assuming that the system is stationary and there is no
potential energy. Using the entropy, internal energy and volume as specific properties
then
𝑇𝑑𝑠 = 𝑑𝑢 + 𝑝𝑑𝜈
(14)
𝑇𝑑𝑠 = 𝑑ℎ + 𝑣𝑑𝑝
(15)
or
Equations (14) and (15) are called 𝑇𝑑𝑠 equations and they are very useful,
because the equations are written only in terms of properties of the system.
Based on definition of specific heats, we can express Eq. (14) and (15) in the
following forms
𝑠2 − 𝑠1 = 𝐶𝑝 ln
𝑇2
𝑝2
− ℛ ln
𝑇1
𝑝1
(16)
22
𝑠2 − 𝑠1 = 𝐶𝑣 ln
𝑇2
𝑣2
+ ℛ ln
𝑇1
𝑣1
(17)
Where ℛ is the gas constant.
2.7 THE IDEAL GAS EQUATION
The most important thermodynamic properties of the fluid are the pressure 𝑝,
the temperature 𝑇, the density 𝜌, the enthalpy ℎ, the entropy 𝑠 and the specific heats
𝐶𝑝 and 𝐶𝑣 . We need to understand how these thermodynamic properties change during
a flow process. Thus, there are equations that provides the relationship among those.
By Turns [2], the ideal gas equation relates the pressure 𝑝, the temperature 𝑇
and the density 𝜌 or the volume 𝒱 of an ideal gas, which are all gases at low pressures
and at high temperatures, it is expressed as
𝑝 = 𝜌ℛ𝑇
(18)
2.8 COMPRESSIBLE FLOW EQUATIONS
The Mach number is defined as the velocity divided by the local speed of sound
𝑎. For a perfect gas, the Mach number is written as
𝑀=
𝑉
𝑉
=
𝑎
√𝛾ℛ𝑇
(19)
Axial turbines operate in a range of Mach number that exceeds 0.3. According
to Anderson [4] at this point the flow becomes compressible, therefore the fluid density
can no longer be considered as constant.
23
Assuming a steady state flow, for an ideal gas undergoing isentropic process,
no work (𝑊̇ ≈ 0), the variation of potential energy is negligible (∆𝑃𝐸 ≈ 0) and
considering that the flow is brought to rest, then Eq. (8) can be expressed as
𝑇o
γ−1 2
=1+
𝑀
T
2
(20)
This equation represents the total temperature 𝑇o , which is the static
temperature measured when the flow is brought isentropically to rest.
Assuming an isentropic process (∆𝑠 = 0) and manipulating Eq. (16) and Eq.
(17), we obtain equations that relates the temperature, density and pressure
𝛾−1
𝛾
𝑇o
𝜌𝑜 𝛾−1
𝑝𝑜
=( )
= ( )
𝑇
𝜌
𝑝
(21)
Substituting Eq. (20) into Eq. (21), we can express the total density 𝜌𝑜 and the
total pressure 𝑝𝑜 as function of Mach number and the ratio of specific heats
1
(22)
𝛾
(23)
𝜌𝑜
γ − 1 2 𝛾−1
= (1 +
𝑀 )
𝜌
2
𝑝𝑜
γ − 1 2 𝛾−1
= (1 +
𝑀 )
𝑝
2
For Dixon and Hall [7], one of the most important compressible flow equation for
turbomachinery is the nondimensional mass flow rate, sometimes called as capacity
equation. It is obtained by combining Eqs. (21), (22) and (23) with continuity, Eq. (2):
24
𝑚̇√𝐶𝑝 𝑇𝑜
𝐴𝑛 𝑝𝑜
1 𝛾+1
γ − 1 2 −2(𝛾−1)
=
M (1 +
𝑀 )
2
√𝛾 − 1
𝛾
(24)
From capacity equation, we can relate the flow properties, such as total
temperature, total pressure, Mach number, and the thermodynamic properties of the
gas with the nondimensional mass flow rate.
The compressible flow equations showed previously can be applied in the
relative reference frame within the rotor blade rows. It this situation, the relative Mach
number and stagnation properties are used:
𝑇o,rel 𝑝o,rel 𝜌o,rel 𝑚̇√𝐶𝑝 𝑇𝑜,𝑟𝑒𝑙
,
,
,
= 𝑓(𝑀𝑟𝑒𝑙 )
𝑇
𝑝
𝜌
𝐴𝑛 𝑝𝑜,𝑟𝑒𝑙
(25)
25
3 AXIAL FLOW TURBINES
3.1 TURBINE GEOMETRY
The turbine terminology used in this work is based on the schematic turbine
configurations shown in Fig. 6 and Fig. 7 and summarized in Table. 1.
Figure 6. Turbine cascade geometry [15]
26
Figure 7. Blade geometry [16]
Table 1. Turbine geometry parameters
Name
Definition
Leading edge
It is the foremost edge of the turbine airfoil.
Trailing edge
It is the rearmost edge of the turbine airfoil.
Camber line
It is curve which the profile thickness
distribution is symmetric.
Stagger angle, Φ
The angle between the chord line and the
axial direction.
Spacing or pitch, 𝑔
The distance between corresponding points
on adjacent blades.
Chord, 𝑐
The linear distance between the leading edge
and the trailing edge.
Axial chord, 𝑐𝑥
It is axial length of the blade.
The difference between the inlet flow angle 𝛼1
Incidence angle, 𝑖
and the blade metal inlet angle 𝛼1 ′.
𝑖 = 𝛼1 − 𝛼1 ′
27
The difference between the outlet flow angle
Deviation angle, 𝛿
𝛼2 and the blade metal outlet angle 𝛼2 ’.
𝛿 = 𝛼2 − 𝛼2 ′
Pitch – chord ratio, 𝑔/𝑐
Aspect ratio, ℎ/𝑐
The ratio of the pitch 𝑔 to the chord 𝑐.
The ratio of the blade height ℎ to the chord 𝑐.
Trailing edge blockage, The ratio of throat 𝑜 to the trailing edge
𝑜/𝑡𝑡𝑒
thickness 𝑡𝑡𝑒 .
3.2 VELOCITY DIAGRAMS OF THE AXIAL TURBINE STAGE
The axial turbine stage comprises a row of stationary blades often called as
stator or nozzle and a row of moving blades, known as rotor. The fluid enter the stator
with absolute velocity 𝑉1 at angle 𝛼1 , the pressure drop through the stator from pressure
𝑝1 to pressure 𝑝2 causes an acceleration from absolute velocity 𝑉1 to the absolute outlet
velocity 𝑉2 at angle 𝛼2 . The rotor inlet relative velocity 𝑤2 is calculated by subtracting,
vectorially, the blade speed 𝑈 from the absolute velocity 𝑉2 and define the relative inlet
flow angle 𝛽2. A further pressure drop from pressure 𝑝2 to pressure 𝑝3 through the rotor
accelerates the flow from the inlet relative velocity 𝑤2 to the outlet relative velocity 𝑤3
and the flow leaves the rotor at a relative angle 𝛽3. The corresponding absolute flow
velocity 𝑉3 at angle 𝛼3 is obtained by subtracting, vectorially, the outlet relative velocity
𝑤3 to the blade speed 𝑈 [7].
The velocity triangles provide a better understanding of the relationship among
absolute and relative flow velocities, and angles with the blade speed. The absolute
velocities are aligned with stator inlet and outlet, and the relative velocities are aligned
with the rotor inlet and outlet. All angles are measured from the axial (𝑥) direction and
the sign convection is such that the tangential components of the velocities (𝑉𝜃 and 𝑤𝜃 )
are positive if they point in the direction of the rotational speed and negative if they
point into the opposite direction.
28
Figure 8. Turbine stage velocity triangles [7]
Based on Fig. 8, we can write the relations for absolute and relative flow angles
for both stator and rotor:
𝑉𝑥2 = 𝑉2 cos 𝛼2
(26)
𝑉𝜃2 = 𝑉2 sin 𝛼2
(27)
tan 𝛼2 =
𝑉𝜃2
𝑉𝑥2
𝑉2 = √𝑉𝑥2 2 + 𝑉𝜃2 2
(28)
(29)
29
𝑤𝑥2 = 𝑤2 cos 𝛽2
(30)
𝑤𝜃2 = 𝑤2 sin 𝛽2
(31)
tan 𝛽2 =
𝑤𝜃2
𝑤𝑥2
(32)
𝑤2 = √𝑤𝑥2 2 + 𝑤𝜃2 2
(33)
The relative velocity 𝑤 is the vector subtraction of blade speed 𝑈 from the
absolute velocity 𝑉. From sign convection, the tangential component of velocities is
positive, then
𝑤𝜃2 = 𝑉𝜃2 − 𝑈
(34)
As the axial velocity is the same for both reference frames (𝑉𝑥2 = 𝑤𝑥2), the
relationship between the absolute and relative flow angles is given by:
tan 𝛽2 = tan 𝛼2 −
𝑈
𝑉𝑥2
(35)
For the rotor, we have
𝑉𝑥3 = 𝑉3 cos 𝛼3
(36)
𝑉𝜃3 = 𝑉3 sin 𝛼3
(37)
30
tan 𝛼3 =
𝑉𝜃3
𝑉𝑥3
(38)
𝑉3 = √𝑉𝑥3 2 + 𝑉𝜃3 2
(39)
𝑤𝑥3 = 𝑤3 cos 𝛽3
(40)
𝑤𝜃3 = 𝑤3 sin 𝛽3
(41)
tan 𝛽3 =
𝑤𝜃3
𝑤𝑥3
(42)
𝑤3 = √𝑤𝑥3 2 + 𝑤𝜃3 2
(43)
From sign convection, for the rotor, the tangential component of velocities is
negative, then
𝑤𝜃3 = 𝑉𝜃3 + 𝑈
(44)
Being axial velocity the same for both reference frames (𝑉𝑥3 = 𝑤𝑥3 ), thus
tan 𝛽3 = tan 𝛼3 +
𝑈
𝑉𝑥3
(45)
3.3 TURBINE DESIGN PARAMETERS
Three fundamentals nondimensional parameters that are very useful to design
axial turbines: Flow Coefficient 𝜙, Stage Loading Coefficient 𝜓 and Pressure Reaction
31
Degree 𝑅. These parameters have influence on the shape of the velocity triangles,
which is related to the efficiency and number of stages [7].
The flow coefficient 𝜙 is the ratio of the axial flow velocity 𝑉 to the blade speed
𝑈.
𝜙=
𝑉𝑥
𝑈
(46)
High values of 𝜙 mean small relative flow angles and velocities close to the axial
direction. Low value of 𝜙 implie wide relative flow angles with velocities close to the
tangential direction.
The Stage Loading can be derived from Euler work equation (12), and defined
as the ratio between the change of the intensity of the tangential component of 𝑉 and
the blade speed 𝑈.
𝜓=
Δ𝑉𝜃
𝑈
(47)
High stage loading implies large flow turning (∆𝛽 = 𝛽2 − 𝛽3). From Euler work
equation (12) we can see that high stage loadings lead to high specific work, then fewer
stages; however high stage loadings have influence on blading efficiency (increase in
losses).
The Pressure Reaction Degree 𝑅 is the most important turbine parameter
because has influences on the efficiency and number of stages. Usually, a preliminary
design calculation starts from the choice of the pressure reaction degree, defined as:
𝑅=
𝑝2 − 𝑝3
𝑝1 − 𝑝3
(48)
32
In fact, the degree of reaction is a free design parameter and the choice of its
values depends on project requirements as high efficiency and small size [14].
Historically, the turbine designers opted either for zero degree of reaction or for the
50% reaction turbines. The effect of pressure reaction degree on the turbine design is
illustrated in Fig. 9, where the comparison among different degrees of reaction is made
assuming that the velocity triangles at the exit of the rotor are the same, the only
changes being in the stator exit conditions.
Figure 9. Effect of Pressure Reaction Degree on velocity triangles [14]
From Fig. 9, we can see that increasing the pressure degree of reaction 𝑅, the
enthalpy drop ∆ℎ𝑜 decreases; this is related to the decrease of the tangential
component of the stator exit velocity 𝑉𝜃2, explained by the Euler work equation (12).
Being 𝑉𝜃3 constant, thus we have a reduction of work done by the turbine Eq. (7).
At zero pressure degree of reaction, the entire pressure drop happens in the
stator, thus 𝑉2 and 𝑉𝜃2 have the maximum values, as well the absolute stator outlet
flow angle 𝛼2 . In the rotor, the relative flow angles (𝛽2 and 𝛽3 ) and velocities (𝑤2 and
𝑤3 ) are equal. The flow turning is maximum, then from Eq. (7), the work extracted by
33
the turbine is maximum compared with other pressure degree of reaction values. Often,
this type of turbine is called impulse turbine.
At 𝑅 = 0.5, the velocity triangles are equal, then: 𝑉2 = 𝑤2 , 𝑉3 = 𝑤2 , 𝛼2 = 𝛽3 and
𝛽2 = 𝛼3 , these lead to a low flow turning and being the blading efficiency related to the
turning, typically turbines designed with 𝑅 = 0.5 have the highest efficiency but are
delivering less work with respect, for example, to the case 𝑅 = 0. For this reason, they
are realized including more stages.
3.4 THERMODYNAMICS OF THE AXIAL TURBINE STAGE
The working process of the axial turbine is illustrated in the h-s diagram of Fig.
10. The thermodynamic process in the turbine stage depends on the pressure reaction
degree; in order to explain it, a turbine with (𝑅 ≃ 0.5) has been chosen. The process
through the stator or nozzle occurs along the curve 1-2, where the static pressure
decreases from 𝑝1 to 𝑝2 . The change in total pressure (𝑝02 − 𝑝01 ) is due to increase in
entropy connected to viscous losses [6].
Figure 10. Expansion process in a turbine stage [17]
34
As there is no work extraction in the stator, the total enthalpy across the stator
remains constant. From Eq. (10)
ℎ01 = ℎ02
(49)
The working process in the rotor is represented by curve 2-3. In the rotor row,
there is another pressure drop. The total enthalpy in the absolute frame is changing
due to the energy transfer from the working fluid to the blades. In the rotor relative
frame, we can’t see the work extraction and thus we can consider constant the relative
total enthalpy; so, from Eq. (10)
ℎ02,rel = ℎ03,rel
(50)
These two equations will be useful when the one-dimensional turbine design will
be discussed.
3.5 EFFICIENCY
The turbine overall efficiency can be defined as the available energy in a flowing
fluid converted into useful work delivered at the output shaft [7]. The isentropic
efficiency 𝜂𝑡 is a measure of the quality of this energy conversion. It is expressed by
𝜂𝑡 =
𝑎𝑐𝑡𝑢𝑎𝑙 𝑤𝑜𝑟𝑘
∆𝑊
=
𝑖𝑑𝑒𝑎𝑙 𝑤𝑜𝑟𝑘
∆𝑊𝑚𝑎𝑥
(51)
The ideal work depends on how the ideal process is defined. The maximum
work will always be an isentropic process, but we need to designate the correct exit
state of the ideal process relative to the actual process. It depends on whether the exit
kinetic energy is usefully employed or wasted.
35
If the exhausting kinetic energy is usefully employed (as in the case of the last
stage of aircraft gas turbines, where it contributes to the jet thrust or in the case of a
multistage turbine, where kinetic energy exhausting from stage is used in the following
stage), the ideal process is to the same total pressure as the actual expansion, then
from Eq. (10), we can define the total-to-total efficiency 𝜂𝑡𝑡
𝜂𝑡𝑡 =
∆𝑊
ℎ01 − ℎ03
=
∆𝑊𝑚𝑎𝑥
ℎ01 − ℎ03𝑠
(52)
If the exhausting kinetic energy cannot be usefully employed, (like when it is
wasted directly to the surrounding), the ideal expansion is to the same static pressure
as the actual expansion, then we can define the total-to-static efficiency 𝜂𝑡𝑠 ,
𝜂𝑡𝑠 =
∆𝑊
ℎ01 − ℎ03
=
∆𝑊𝑚𝑎𝑥
ℎ01 − ℎ3𝑠
(53)
3.6 TURBINE LOSSES
According to Dixon and Hall [7] and Denton [1], the losses in a turbine can be
divided in two types: 2D or 3D. 2D loss sources are due to blade boundary layer,
trailing edge mixing and shock system. 3D sources are connected to tip leakage flows,
endwall or secondary flows and coolant flows. In this work, only a 2D turbine will be
investigated, then 3D losses are not considered.
The loss in the blade boundary layer is the work expended by the particles
against the viscous forces; it depends on the development of the boundary layer, which
is related to the blade surface pressure distribution, and in particular, to the transition
from laminar to turbulent flow. Boundary layer loss typically is responsible for over 50%
of the 2D loss in a subsonic turbine.
36
From Denton [1], the entropy loss coefficient for the blade boundary layers is
given by:
1
𝐿𝑏
𝑉𝑏 3
𝑥
𝜉𝑏 = 2 ∑
∫ 𝐶𝑑𝑖𝑠 ( ) 𝑑 ( )
𝑠 cos 𝛼2
𝑣2
𝐿𝑏
0
(54)
Where, 𝐶𝑑𝑖𝑠 is the dissipation coefficient, 𝑉𝑏 is the blade surface velocity and
𝐿𝑏 is the blade surface length.
𝑉
3
From Eq. (54), the term ( 𝑣𝑏 ) shows that the suction side is dominant in loss
2
production, as the suction side has regions of high surface velocities.
Another parameter that has a big role, is the dissipation coefficient 𝐶𝑑𝑖𝑠 . From
Fig. 11, where either a laminar or a turbulent flow could exist (300 < 𝑅𝑒𝜃 < 1000), the
laminar dissipation coefficient 𝐶𝑑𝑖𝑠 is lower than the turbulent dissipation coefficient
𝐶𝑑𝑖𝑠 .
Figure 11. 𝐶𝑑𝑖𝑠 vs. Reynolds number based on momentum thickness 𝑅𝑒𝜃 [1]
Then, in order to minimize the loss, the boundary layer should be kept laminar
for as long as possible. It depends on the Reynolds number, turbulence level, and the
37
blade surface velocity distribution. Therefore, the prediction of the boundary layer
transition is very important in the turbine design.
One of the main sources of loss for turbines is the trailing edge mixing loss. The
trailing edge mixing loss accounts for about 35% of the total 2D loss in subsonic
turbines, and about 50% of the total 2D loss in supersonic turbines. The trailing edge
mixing loss is related to the mixing of the suction surface and pressure surface
boundary layers, which have different flow conditions, with the region of flow
downstream the trailing edge
Figure 12. The variation of The trailing edge mixing loss with Mach number and
trailing edge thickness [18]
Xu and Denton [18] states that the trailing edge loss increases roughly linearly
with the trailing edge blockage (Table 1). Fig. 12 shows the results of a family of
turbines with different trailing edge thicknesses. As we can see that when Mach
number is higher than one, there is a large increase in the trailing edge loss; this
happens because of the trailing edge shock system, mixing, and complex trailing edge
flow pattern. Therefore, thick trailing edges are the best alternative to decrease the
loss.
38
3.7 ZWEIFEL CRITERION
There is an optimum pitch to chord ratio that provides a minimum overall loss.
The pitch to chord ratio has great influence on blade velocity distribution. Assuming
that the blade chord is constant, if we increase the space between the blades, the
friction losses are small, but we have a poor fluid guidance, which leads to high losses
due to flow separation. If the pitch is low, the blades give the maximum amount of
guidance to the fluid; however, the friction losses will be large. Zweifel [19] found that
the ratio of an actual to an ideal tangential blade loading has an approximately constant
value for minimum losses for blades having a high outlet angles. Based on Zweifel
coefficient 𝑍, the optimum pitch to chord ratio can be obtained.
The actual tangential blade load for the stator and rotor are given by Eq. (4):
−F𝑦,𝑠 = 𝑚̇(𝑉𝜃2 − 𝑉𝜃1 )
(55)
F𝑦,𝑅 = 𝑚̇(𝑤𝜃2 − 𝑤𝜃3 )
(56)
The condition of ideal load is when the inlet total pressure acts over the whole
pressure surface and the outlet static pressure acts over the whole suction surface.
Assuming that the blade height is constant. Then, the ideal tangential blade load for
the stator and rotor is the difference of pressure between sides:
−F𝑦_𝑖𝑑𝑒𝑎𝑙,𝑠 = (𝑝01 − 𝑝2 )hc
(57)
F𝑦_𝑖𝑑𝑒𝑎𝑙,𝑅 = (𝑝02,𝑟𝑒𝑙 − 𝑝3 )hc
(58)
Being the Zweifel coefficient is the ratio of actual to an ideal tangential blade
load
39
𝑍𝑠 =
𝑍𝑅 =
𝑚̇(𝑉𝜃2 − 𝑉𝜃1 )
(𝑝01 − 𝑝2 )hc
(59)
𝑚̇(𝑤𝜃2 − 𝑤𝜃3 )
(𝑝02,𝑟𝑒𝑙 − 𝑝3 )hc
(60)
Substituting Eq. (2) and into Eq. (59) and Eq. (60), we have
(𝑝01 − 𝑃2 )𝑍
𝑔
( ) =
𝑐 𝑆 𝜌2 𝑉𝑥2 (𝑉𝜃2 − 𝑉𝜃1 )
(61)
(𝑝02,𝑟𝑒𝑙 − 𝑃3 )𝑍
𝑔
( ) =
𝑐 𝑅 𝜌2 𝑉𝑥2 (𝑤𝜃2 − 𝑤𝜃3 )
(62)
Based on experiments on low Mach number turbine cascades, Zweifel [19]
determined that the value of 𝑍 was approximately 0.8. According to Dixon and Hall [7],
the Zweifel coefficient accurately predicts optimum pitch to chord ratio for outlet angles
of 60° to 70°. The Zweifel coefficient may be greater than 1 in some cases, as low
pressure turbines.
40
4 ONE DIMENSIONAL TURBINE DESIGN PROCEDURE
4.1 METHODOLOGY
The complete design of the turbine blade can be divided in 6 main activities as
shown in Fig. 13 [20]. This work focus on the three first processes:
1) 1D turbine design
2) 2D Blade profile design
3) 2D CFD analysis
Figure 13. Flowchart of the turbine blade design
41
The 1D turbine design is a critical part of the process, as the 80% of the final
product defined in the 20% of the design time [20]. The primary optimization of the
turbine geometry in terms of flow angles has been made by writing a first MATLAB
code, the Meanline Analysis Program, described by the flowchart shown in Fig. 14,
and presented in details in the following paragraphs. This program provides inlet and
outlet flow angles, pitch, chord, that are the parameters required to design the 2D blade
profiles. The stagger angle is obtained by using statistical data [21], as will be shown
in the following paragraphs.
Figure 14. Flowchart of the Meanline Analysis Program
Finally, these parameters have been used to design the blade profile, following
the Bézier curve method, implemented by writing another MATLAB code.
4.2 DESIGN MODEL
The one dimensional turbine design procedure used in this work has been
developed by Sieverding [10]. The design starts with determination of pressure degree
42
of reaction. After that, the calculation of flow properties at the stator inlet and outlet
(that are also the rotor inlet conditions), and rotor outlet is performed. The annulus
areas are calculated, followed by the blade heights at each station. The blade row and
channel geometries are thus determined. The flow properties are calculated at the mid
span of the blade (meanline design).
The design procedure is an iterative method: being the static pressure at the
exit of the turbine unknown, a first tentative value of the pressure is necessary; besides
the blading efficiency depends on the inlet and outlet flow angles, Mach number,
Reynolds number, pitch to chord ratio and aspect ratio and we don’t know these values,
the first calculations are performed assuming values of blading efficiencies based on
statistical data [10]. In case of blading efficiencies, calculated by using loss models,
different from the estimated values, the design calculation is repeated.
4.3 TURBINE DESIGN PROCEDURE
The initial data required to start the calculation are the total inlet temperature
𝑇01 , the total inlet pressure 𝑝01 , the mass flow rate 𝑚̇, the power delivered by the turbine
𝑊̇ , the blade speed 𝑈, the specific heat ratio 𝛾, and the specific heat at constant
pressure 𝐶𝑝 . The process is assumed to be adiabatic and will be assumed that the
specific heat ratio 𝛾 and the specific heat at constant pressure 𝐶𝑝 are constant through
the turbine stage. The following equations have been implemented in a MATLAB code
(Meanline Analysis Program) written by the author.
From the Pressure Degree of Reaction 𝑅 chosen, we can calculate the stator
outlet static pressure 𝑝2 by assuming a value of the rotor outlet static pressure 𝑝3 and
considering 𝑝1 ≈ 𝑝01 (being the inlet Mach number low 𝑀1 ≈ 0.15 − 0.3). From Eq. (48),
we have
𝑝2 = 𝑅(𝑝01 − 𝑝3 ) + 𝑝3
(63)
43
Being the total enthalpy constant across the stator, Eq. (49), its inlet total
temperature 𝑇01 is equal to the stator outlet total temperature 𝑇02 ; then from the
isentropic relation between the temperature and pressure, Eq. (21), we can determine
the isentropic static exit temperature 𝑇2,𝑠
𝑇2,𝑠
𝛾−1
𝛾
𝑝2
= 𝑇01 ( )
𝑝01
(64)
From Eq. (9), we can find the stator isentropic absolute outlet velocity 𝑉2,𝑠
𝑉2,𝑠 = √ 2𝑐𝑝 (𝑇01 − 𝑇2,𝑠 )
(65)
Assuming a value of 𝜂𝑠 = 0.92 → 0.93 for the stator blading efficiency 𝜂𝑡𝑠,𝑆 [10],
we can calculate the static temperature 𝑇2 , from the definition of total-to-static
efficiency, Eq. (53)
𝑇2 = 𝑇01 − 𝜂𝑡𝑠,𝑆 (𝑇01 − 𝑇2,𝑠 )
(66)
The stator blading efficiency 𝜂𝑠 can also be expressed as the ratio of the stator
absolute outlet velocity 𝑉2 to the the isentropic stator outlet velocity 𝑉2,𝑠 , then we can
write:
𝑉2 = √𝜂𝑡𝑠,𝑆 𝑉2,𝑠
(67)
The stator outlet density is calculated from the ideal gas equation, Eq. (18):
𝑝
𝜌2 = ℛ𝑇2
2
(68)
44
The stator outlet Mach number is determined from Eq. (19):
𝑀2 =
𝑉2
(69)
√𝛾ℛ𝑇2
From isentropic relation, Eq. (21), the stator outlet total pressure 𝑝02 is
calculated
𝛾
𝑃02
𝑇01 𝛾−1
= 𝑃2 ( )
𝑇2
(70)
Being the stator exit velocity 𝑉2 known, the stator absolute outlet axial velocity
𝑉𝑥2 or the stator absolute outlet angle 𝛼2 can be free choices of the designer. According
to Sieverding [10], 90% of the high pressure turbines have the stator absolute outlet
angle ranging from 70° to 75°. Moreover, most of low pressure turbines have the stator
absolute outlet angle ranging 60° to 70°. Therefore, it is easy to choice the stator
absolute outlet flow angle 𝛼2 , and to obtain the components of the absolute velocity
from Eq. (26) and (27),
𝑉𝑥2 = 𝑉2 cos 𝛼2
(71)
𝑉𝜃2 = 𝑉2 sin 𝛼2
(72)
From Eq. (34), we can calculate the stator relative outlet tangential velocity 𝑤𝜃2
𝑤𝜃2 = 𝑉𝜃2 − 𝑈
(73)
Being the axial velocity constant on both reference frames (𝑉𝑥2 = 𝑤𝑥2 ), from Eq.
(32), the stator outlet relative angle 𝛽2 is determined as
45
tan 𝛽2 =
𝑤𝜃2
𝑤𝑥2
(74)
From Eq. (33), we have the stator outlet relative velocity 𝑤2
(75)
𝑤2 = √𝑤𝑥2 2 + 𝑤𝜃2 2
The stator outlet total temperature and pressure at the relative reference frame
are calculated from Eq. (9) and Eq. (21) respectively
𝑇02,𝑟𝑒𝑙 = 𝑇2 +
𝑤2 2
2𝐶𝑝
(76)
𝛾
𝑃02,𝑟𝑒𝑙
𝑇02,𝑟𝑒𝑙 𝛾−1
= 𝑃2 (
)
𝑇2
(77)
The stator relative outlet Mach number is determined from Eq. (19):
𝑀2,𝑟𝑒𝑙 =
𝑤2
(78)
√𝛾ℛ𝑇2
The flow is accelerated in the rotor by an expansion process from the total inlet
properties 𝑇02,𝑟𝑒𝑙 and 𝑃02,𝑟𝑒𝑙 to the the rotor outlet static pressure 𝑝3 . In the relative
frame, the total enthalpy in the rotor is constant (𝑇02,𝑟𝑒𝑙 = 𝑇03,𝑟𝑒𝑙 ), Eq. (50), thus from
Eq. (21) we can calculate the isentropic static exit temperature 𝑇3,𝑠
𝑇3,𝑠 = 𝑇03,𝑟𝑒𝑙 (
𝑝3
𝑝02,𝑟𝑒𝑙
𝛾−1
𝛾
)
(79)
46
From Eq. (9), we can find the rotor isentropic relative outlet velocity 𝑤3,𝑠
𝑤3,𝑠 = √ 2𝑐𝑝 (𝑇03,𝑟𝑒𝑙 − 𝑇3,𝑠 )
(80)
Being the value of the rotor blading efficiency 𝜂𝑡𝑠,𝑅 ranging from 0.87 to 0.88 [10],
we can calculate the static temperature 𝑇3 , from the definition of total-to-static
efficiency, Eq. (53)
𝑇3 = 𝑇03,𝑟𝑒𝑙 − 𝜂𝑡𝑠,𝑅 (𝑇03,𝑟𝑒𝑙 − 𝑇3,𝑠 )
(81)
Moreover, the rotor blading efficiency 𝜂𝑡𝑠,𝑅 can be expressed as the ratio of the
stator relative outlet velocity 𝑤3 to the the isentropic rotor relative outlet velocity 𝑤3,𝑠 ,
thus we write 𝑤3 as
𝑤3 = √𝜂𝑡𝑠,𝑅 𝑤3,𝑠
(82)
The rotor outlet density is calculated from the ideal gas equation, Eq. (18):
𝑝
𝜌3 = ℛ𝑇3
(83)
3
The rotor relative outlet Mach number is determined from Eq. (19):
𝑀3,𝑟𝑒𝑙 =
𝑤3
√𝛾ℛ𝑇3
(84)
From isentropic relation, Eq. (21), the rotor relative outlet total pressure is
calculated
47
𝛾
𝑃03,𝑟𝑒𝑙
𝑇03,𝑟𝑒𝑙 𝛾−1
= 𝑃3 (
)
𝑇3
(85)
To calculate the rotor conditions, we will assume that there is no change of the
radius of the midspan line between the rotor inlet and rotor outlet. Then, the rotor
absolute outlet axial velocity 𝑉𝑥3 or the rotor relative outlet angle 𝛽3 can be free choices
of the designer. Typically, it is considered that the absolute axial velocity remains
constant through the rotor and being that the axial velocity the same for both reference
frames, we have
𝑉𝑥3 = 𝑉𝑥2
(86)
𝑤𝑥3 = 𝑉𝑥3
(87)
From Eq. (42), we can obtain the rotor relative outlet angle 𝛽3
𝛽3 = cos−1 (
𝑤𝑥3
)
𝑤3
(88)
We can calculate the rotor relative outlet tangential velocity 𝑤𝜃3
𝑤𝜃3 = 𝑤3 sin 𝛽3
(89)
The rotor absolute outlet tangential velocity 𝑉𝜃3 is given by Eq. (44)
𝑉𝜃3 = 𝑤𝜃3 − 𝑈
(90)
48
From Eq. (38), the rotor absolute outlet angle 𝛼3 is obtained
𝛼3 = tan−1
𝑉𝜃3
𝑉𝑥3
(91)
We can calculate the rotor absolute velocity 𝑉3 by Eq. (39)
𝑉3 = √𝑉𝑥3 2 + 𝑉𝜃3 2
(92)
The rotor absolute outlet Mach number is determined from Eq. (19):
𝑀3 =
𝑉3
(93)
√𝛾ℛ𝑇3
The rotor outlet total temperature and pressure at the absolute reference frame
are calculated from Eq. (9) and Eq. (21) respectively
𝑇03
𝑃03
𝑉3 2
= 𝑇3 +
2𝐶𝑝
(94)
𝛾
(95)
𝑇03 𝛾−1
= 𝑃3 ( )
𝑇3
Now, the power delivered by the turbine stage can be calculated from Eq. (7) or
Eq. (11)
𝑊̇𝑖 = 𝑚̇𝐶𝑝 (𝑇01 − 𝑇03 )
(96)
49
Or
𝑊̇𝑖 = 𝑚̇U(𝑉θ2 − 𝑉θ3 )
(97)
If the power delivered by the stage 𝑊̇𝑖 , given by Eq. (96) or Eq. (97), is different
from the power 𝑊̇ expected by the turbine designer, a new value of the rotor outlet
static pressure 𝑝3 needs to be specified; then the design process, from Eq. (63) to Eq.
(97), is repeated until the condition:
‖𝑊̇ − 𝑊̇𝑖 ‖
= 10−3
‖𝑊̇ ‖
(98)
The entropy increase Δ𝑠 through the turbine stage is given by Eq. (16):
Δ𝑠𝑠 = 𝑠2 − 𝑠1 = 𝐶𝑝 ln
𝑇2
𝑝2
− ℛ ln
𝑇1
𝑝1
(99)
𝑇3
𝑝3
− ℛ ln
𝑇2
𝑝2
(100)
Δ𝑠𝑅 = 𝑠3 − 𝑠2 = 𝐶𝑝 ln
Δ𝑠 = Δ𝑠𝑠 + Δ𝑠𝑅
(101)
Where Δ𝑠𝑠 and Δ𝑠𝑅 are the entropy increases through the stator and rotor
respectively.
If the exhaust velocity is usefully employed, the total-to-total efficiency is
calculated from Eq. (52)
𝜂𝑡𝑡 =
∆𝑊
ℎ01 − ℎ03
=
∆𝑊𝑚𝑎𝑥
ℎ01 − ℎ03𝑠
(102)
50
Otherwise, if the exhaust velocity cannot be usefully employed, the total-to-static
efficiency is given by Eq. (53)
𝜂𝑡𝑠 =
∆𝑊
ℎ01 − ℎ03
=
∆𝑊𝑚𝑎𝑥
ℎ01 − ℎ3𝑠
(103)
With the flow properties so defined, the calculation of flow areas can be
performed, the annulus area of a turbine 𝐴𝑛𝑛𝑢𝑙𝑢𝑠 being given by
𝑟𝐻 2
𝐴𝑎𝑛𝑛𝑢𝑙𝑢𝑠 = 𝜋𝑟𝑇 2 [1 − ( ) ]
𝑟𝑇
(104)
Using the continuity equation, Eq. (2), we can also calculate the annulus area
of the stator outlet 𝐴2 :
𝐴2 =
𝑚̇
𝜌2 𝑉𝑥2
(105)
Notice that the ratio of hub radius 𝑟𝐻 to tip radius 𝑟𝑇 for high pressure turbines
is typically
𝑟𝐻
𝑟𝑇
≈ 0.9, while, for low pressure turbines is ranging from 0.75 to 0.85 for
front stage and from 0.60 to 0.65 for rear stage [10].
Combining Eq. (104), with Eq. (105), and using an appropriate value for
𝑟𝐻
𝑟𝑇
, we
can calculate the tip radius 𝑟𝑇2 for the stator outlet
𝑟𝑇2 =
𝐴2
√
𝑟 2
𝜋 [1 − ( 𝑟𝐻 ) ]
𝑇
(106)
51
The stator blade height ℎ2 is the difference between the tip radius and hub radius
(Fig. 15).
ℎ2 = 𝑟𝑇2 − 𝑟𝐻2
(107)
Figure 15. Meridional view of a turbine stage [10]
Typically, the hub radius remains constant through the stage, then 𝑅𝐻3 = 𝑅𝐻2 .
The annulus area, the tip radius and the blade height at the rotor outlet are calculated
from
𝐴3 =
𝑟𝑇3
𝑚̇
𝜌3 𝑉𝑥3
𝐴3 + 𝜋𝑟𝐻3 2
= √
𝜋
ℎ3 = 𝑟𝑇3 − 𝑟𝐻3
(108)
(109)
(110)
The stator inlet flow properties can be defined by calculating the stator Mach
number 𝑀1 from Eq. (24). Then, using this 𝑀1 value, the static temperature 𝑇1 and
pressure 𝑝1 are determined from Eq. (20) and Eq. (23) respectively. The stator inlet
density 𝜌1 is calculated by using the ideal gas equation, Eq. (18) and the inlet velocity
is given by Eq. (19)
52
𝑇1 =
𝑇01
γ−1
1 + 2 𝑀2
(111)
𝑝01
(112)
𝑝1 =
(1 +
𝛾
𝛾−1
γ−1 2
2 𝑀 )
𝜌1 =
𝑝1
ℛ𝑇1
𝑉1 = 𝑀1 √𝛾ℛ𝑇1
(113)
(114)
The blade aspect ratio depends on mechanical and manufacturing
considerations [3]. Sieverding [10] states that typical values of the aspect ratio for the
stator are ranging from 0.5 to 0.8 and for the rotor from 1.1 to 1.5. From these values
we can calculate the chord for both stator and rotor.
The pitch to chord ratio is obtained by applying the Zweifel criterion. In particular,
Zweifel’s design rule states that losses are minimized for 0.8 ≤ 𝑍 ≤ 1.0 [19]. From Eq.
(61) and Eq. (62), we can calculate the pitch to chord ratio for both stator and rotor
(𝑝01 − 𝑃2 )𝑍
𝑔
( ) =
𝑐 𝑆 𝜌2 𝑉𝑥2 (𝑉𝜃2 − 𝑉𝜃1 )
(115)
(𝑝02,𝑟𝑒𝑙 − 𝑃3 )𝑍
𝑔
( ) =
𝑐 𝑅 𝜌2 𝑉𝑥2 (𝑤𝜃2 − 𝑤𝜃3 )
(116)
Knowing the chord, we are able to calculate the pitch for both stator and rotor
blades. Finally, we have to choose the stagger angle (Fig. 16).
53
Figure 16. Stagger angle for typical gas turbine blades profiles [21]
Figure 16 shows typical values of stagger angle based on statistical data. The
stagger angle is a function of inlet and outlet angles; absolute flow angles are used for
the stator while relative flow angles are used for the rotor [21].
The control of the design can be performed by using the guideline given by [10]:
1) For a stage pressure ratio
𝑃3
𝑃1
, ranging from 3 to 4, in case of high pressure
turbines, the stator absolute inlet Mach number should be in the range 0.85 <
𝑀2 < 1.2. If 𝑀2 > 1.2, we must increase 𝑅. If 𝑀2 < 1.2, we must decrease 𝑅.
2) 𝛽2 has a lower limit ranging from 45° to 50° and 𝑀2,𝑟𝑒𝑙 needs to be lower than
0.5. If these conditions are not respected, there are two actions: we can increase
𝛼2 or we can increase 𝑅.
3) For a stage pressure ratio
𝑃3
𝑃1
, ranging from 3 to 4, in case of high pressure
turbines, the stator relative outlet Mach number should be in the range 0.85 <
𝑀3,𝑟𝑒𝑙 < 1.2. If 𝑀3,𝑟𝑒𝑙 > 1.2, we must decrease 𝑅. If 𝑀3,𝑟𝑒𝑙 < 1.2, we must
increase 𝑅.
4) Total turning: ∆𝛽 ≤ 110° → 115°, if the turning is greater, there are three
possible actions: decrease 𝛼2 , decrease 𝛽3 or increase 𝑅.
54
5) The blade height ratio should be in the range
ℎ3
ℎ2
≤ 1.1 → 1.5, in case that this
condition is not respected, we must decrease 𝛽3 or increase 𝛼2 .
6) The stator absolute outlet flow angle 𝛼3 should be lower than 30°, if 𝛼3 > 30°
we need to decrease 𝛽3.
Sieverding [10] states that is important to keep a good balance between the
stator absolute outlet Mach number 𝑀2 and the rotor relative outlet Mach number
𝑀3,𝑟𝑒𝑙 .
The mean line design is an iterative method that provides the optimum gas path;
this method involves the calculation of velocity triangles, flow properties, and blade row
geometries. Using these properties, the design of the turbine airfoil can be started.
55
5 TURBINE AIRFOIL DESIGN
5.1 BLADE PROFILE METHODS
As stated by Lewis [22], there are two approaches to the blade profile design,
the direct and indirect methods.
In the direct method, the designer defines a preliminary blade profile. Then, by
means of numerical simulation, the performance parameters are calculated. Analyzing
these results and modifying iteratively the geometry, the designer can optimize the
blade profile [23].
In the indirect method, conversely, the performance parameters, in terms of
pressure and velocity profiles expected, are given; then, the designer, using
optimization methods based on numerical simulations and evolutionary algorithms,
define the final blade profile [23].
Being this approach more complex and expensive, in this work the direct
method will be used. In particular, the first tentative profile will be designed by using
the Bézier curve method.
5.2 BÉZIER CURVE
A Bézier curve is a parametric curve that it is used to model smooth profiles.
Pierre Bézier while working as engineer for Renault has developed it. Bézier curves
method has wide applications in the computer aided design, and in the last years, it
56
has been used as a tool to design turbomachinery airfoils [24]. A Bézier curve of order
𝑛 is defined by the following equation:
𝑛
𝑃(𝑡) = ∑ 𝐵i𝑛 (𝑡) ∙ 𝑃𝑖
(117)
𝑖=0
Where 𝑃𝑖 represent the control points, 𝑛 is number of control points – 1, 𝑡 is an
independent parameter which varies from 0 to 1, and 𝐵𝑖𝑛 (𝑡) is the Bernstein
Polynomial, defined explicitly by
𝑛!
𝐵𝑖𝑛 (𝑡) = 𝑖!(𝑛−𝑖)! 𝑡 𝑖 (1 − 𝑡)𝑛−i
(118)
An example of Bézier curves for different values of 𝑛 is shown in Fig. 17
Figure 17. Bézier curves of different degrees [25]
From Eq. (118) and Fig. 17, we can see that the Bernstein Polynomial acts as
a link between the points on the Bézier curve and the control points; thus, for example,
three control points result in a parabola while four control points describe a cubic curve.
The parameter 𝑡 is used to constrain the first and last control points to lie on the Bézier
curve, while the other control points define the shape of the curve.
The first derivate of a Bézier curve of 𝑛 degree is
57
𝑛−1
𝑑𝑃(𝑡)
= 𝑛 ∑(𝑃𝑖+1 − 𝑃𝑖 )𝐵𝑖𝑛−1 (𝑡)
𝑑𝑡
(119)
𝑖=0
This equation is useful to calculate the slope of the line tangent to these curves
at a given point.
5.3 DESIGN APPROACH
Pritchard [26] states that there are at least 25 airfoils parameters that can be
associated with any blade shape. A turbine designer must sort out which of these
parameters are known with absolute certainty. From Chapter 4, we are able to specify
the inlet and outlet flow angles, the pitch, the chord, and the stagger angle. The leading
edge radius 𝑟𝐿𝐸 and trailing edge radius 𝑟𝑇𝐸 are limited by mechanical constraints such
as maximum stress and deformation. Being these seven parameters fixed, the turbine
designer can use only the curvature as a free parameter, which is determined by the
control points of the Bézier curve. A MATLAB code has been written to design the
turbine airfoil.
Based on the techniques developed by Verstraete [27], and Sousa and
Paniagua [28], the turbine blade profile design starts by the definition of a camberline,
as shown in Fig. 18, which depends on the inlet blade metal angle 𝛽𝑖𝑛 , the outlet blade
metal angle 𝛽𝑜𝑢𝑡 , the chord 𝑐, and the stagger angle Φ.
Figure 18. Camber line construction [27]
58
The leading edge point 𝑃𝐿𝐸 , the middle point 𝑃𝑀𝑖𝑑 , and the trailing edge point
𝑃𝑇𝐸 are the control points of the Bézier curve that describe the camber line. The 𝑃𝐿𝐸
point is the origin of the coordinate system. The 𝑃𝑇𝐸 point is defined by the axial chord
𝑐𝑥 and the stagger angle Φ. The 𝑃𝑀𝑖𝑑 is the intersection between the line starting from
the point 𝑃𝐿𝐸 and the line coming from 𝑃𝑇𝐸 . Then, we have:
𝑃𝑀𝑖𝑑,𝑥 =
𝑃𝐿𝐸 = (0,0)
(120)
𝑃𝑇𝐸 = (𝑐 cos Φ , 𝑐 cos Φ tan Φ)
(121)
𝑃𝐿𝐸,𝑦 − 𝑃𝐿𝐸,𝑥 tan 𝛽𝑖𝑛 − 𝑃𝑇𝐸,𝑦 + 𝑃𝑇𝐸,𝑥 tan 𝛽𝑜𝑢𝑡
tan 𝛽𝑜𝑢𝑡 − tan 𝛽𝑖𝑛
𝑃𝑀𝑖𝑑,𝑦 = 𝑃𝐿𝐸,𝑦 + tan 𝛽𝑖𝑛 (𝑃𝑀𝑖𝑑,𝑥 − 𝑃𝐿𝐸,𝑥 )
(122)
(123)
To construct the camber line, we have three control points, thus a polynomial of
second order. From Bézier curve equations, Eq. (117) and Eq. (118), we can obtain
the 𝑥 and 𝑦 coordinates of the camber line
𝑃𝑐𝑎𝑚𝑏𝑒𝑟,𝑥 = (1 − 𝑡)2 𝑃𝐿𝐸,𝑥 + 2(1 − 𝑡)𝑃𝑀𝑖𝑑,𝑥 + 𝑡 2 𝑃𝑇𝐸,𝑥
(124)
𝑃𝑐𝑎𝑚𝑏𝑒𝑟,𝑦 = (1 − 𝑡)2 𝑃𝐿𝐸,𝑦 + 2(1 − 𝑡)𝑃𝑀𝑖𝑑,𝑦 + 𝑡 2 𝑃𝑇𝐸,𝑦
(125)
The suction and pressure sides are also defined by the Bézier curves, whose
control points are specified by the normal distance relative to the camber line. The first
and last control points are the leading and trailing edge points respectively. For the
second control point of both sides the distance is computed by using the leading edge
radius; in this way, we can guarantee the minimum thickness of the leading edge. The
trailing edge radius specifies the second last control point for both sides. The reason
59
to use this point is to guarantee the minimum thickness of the trailing edge. In order to
keep the geometric continuity of the trailing edge region, we need a third last control
point for both sides, which depends on the trailing edge wedge angle 𝛼𝐿𝐸 . In this work,
the suction side is defined by using 10 control points, 5 of them being free parameters
while the pressure side is defined by 8 control points, 3 of them being also free design
parameters. The reason of using more control points for the suction side with respect
to those used for the pressure side, is that the losses production is higher on suction
side; therefore, the turbine designer needs to have finer control of the curvature on the
suction side.
Figure 19. Leading edge parameterization
The positions of the second control point of the suction and pressure side are
defined by:
𝑃2𝑥,𝑆𝑆 = 𝑃𝐿𝐸,𝑥 − 𝑟𝐿𝐸 cos(90 − 𝛽𝑖𝑛 )
(126)
𝑃2𝑦,𝑆𝑆 = 𝑃𝐿𝐸,𝑦 + 𝑟𝐿𝐸 sin(90 − 𝛽𝑖𝑛 )
(127)
𝑃2𝑥,𝑃𝑆 = P1𝑥,𝑆𝑆 + 𝑟𝐿𝐸 cos(90 − 𝛽𝑖𝑛 )
(128)
60
𝑃2𝑦,𝑃𝑆 = 𝑃1𝑦,𝑆𝑆 − 𝑟𝐿𝐸 sin(90 − 𝛽𝑖𝑛 )
(129)
Figure 20. Trailing edge parameterization
The positions of the second last control point of the suction and pressure side
are
𝑃9𝑥,𝑆𝑆 = 𝑃𝑇𝐸,𝑥 + 2𝑟𝑇𝐸 cos(𝛽𝑜𝑢𝑡 + 90)
(130)
P9𝑦,𝑆𝑆 = 𝑃𝑇𝐸,𝑦 + 2𝑟𝑇𝐸 sin(𝛽𝑜𝑢𝑡 + 90)
(131)
𝑃7𝑥,𝑃𝑆 = 𝑃𝑇𝐸,𝑥 − 2𝑟𝑇𝐸 cos(𝛽𝑜𝑢𝑡 + 90)
(132)
𝑃7𝑦,𝑃𝑆 = 𝑃𝑇𝐸,𝑦 − 2𝑟𝑇𝐸 sin(𝛽𝑜𝑢𝑡 + 90)
(133)
The positions of the third last control point of the suction and pressure side are
𝑃8𝑥,𝑆𝑆 = 𝑃9𝑥,𝑆𝑆 −
𝑟𝑇𝐸
cos(𝛼𝐿𝐸 − 𝛽𝑜𝑢𝑡 )
2 sin 𝛼𝐿𝐸
(134)
61
𝑟𝑇𝐸
sin(𝛼𝐿𝐸 − 𝛽𝑜𝑢𝑡 )
2 sin 𝛼𝐿𝐸
(135)
𝑃6𝑥,𝑃𝑆 = 𝑃7𝑥,𝑃𝑆 −
𝑟𝑇𝐸
cos(−𝛼𝐿𝐸 − 𝛽𝑜𝑢𝑡 )
2 sin 𝛼𝐿𝐸
(136)
𝑃6𝑦,𝑃𝑆 = 𝑃7𝑦,𝑃𝑆 +
𝑟𝑇𝐸
sin(−𝛼𝐿𝐸 − 𝛽𝑜𝑢𝑡 )
2 sin 𝛼𝐿𝐸
(137)
𝑃8𝑦,𝑆𝑆 = 𝑃9𝑦,𝑆𝑆 +
The suction side parameterization is shown in Fig. 21. The point 𝑖 on the camber
line (𝑋𝑖 , 𝑌𝑖 ) and the distance 𝑑𝑖 from this point to the control point 𝑃𝑖,𝑆𝑆 are free choices
of the designer. In order to draw the suction side curve, it is necessary to find the
position of the control points (𝑃3 to 𝑃7 ). The coordinates of these control points are
calculate by
𝑃𝑖𝑥,𝑆𝑆 = 𝑋𝑖,𝑐𝑎𝑚𝑏𝑒𝑟 + 𝑑𝑖 cos(𝜑𝑖,𝑆𝑆 )
(138)
𝑃𝑖𝑦,𝑆𝑆 = 𝑌𝑖,𝑐𝑎𝑚𝑏𝑒𝑟 + 𝑑𝑖 sin(𝜑𝑖,𝑆𝑆 )
(139)
Figure 21. Suction side parameterization
62
Figure 22. Slope of the normal line (Suction side)
Where 𝜑𝑖 is the slope angle of the line normal to the camber line curve at the
point (𝑋𝑖 , 𝑌𝑖 )
tan 𝜑𝑖,𝑆𝑆 = −
1
tan 𝑎𝑖
(140)
In order to calculate 𝜑𝑖,𝑆𝑆 , we have to define the slope of the tangent line to the
camber line curve at (𝑋𝑖 , 𝑌𝑖 ). By definition, the slope of the tangent line (Fig. 22), tan 𝑎,
is defined by:
tan 𝑎𝑖 =
𝑑𝑃𝑖𝑦,𝑐𝑎𝑚𝑏𝑒𝑟
𝑑𝑃𝑖𝑥,𝑐𝑎𝑚𝑏𝑒𝑟
(141)
The derivatives above are calculated by Eq. (119) or by doing the derivative of
equations, Eq. (124) and Eq. (125).
The control points of the pressure side are obtained in a similar way, Fig. 23.
The difference is in the number of control points (𝑃3 to 𝑃5 ), thus in the position of the
camber line points, and in the slope of the normal line, which is defined by
𝜑𝑖,𝑃𝑆 = 90 − 𝑎𝑖
(142)
63
Figure 23. Pressure side parameterization
Figure 24. Slope of the normal line (Pressure side)
With the control points of both sides defined, from Bézier curve equations, Eq.
(117) and Eq. (118), we can obtain the suction and pressure side curves, thus the
turbine airfoil.
-
Suction side
𝑥𝑠𝑠 = (1 − 𝑡)9 𝑃𝐿𝐸,𝑥 + 9𝑡(1 − 𝑡)8 𝑃2𝑥,𝑆𝑆 + 36𝑡 2 (1 − 𝑡)7 𝑃3𝑥,𝑆𝑆
+ 84𝑡 3 (1 − 𝑡)6 𝑃4𝑥,𝑆𝑆 + 126𝑡 4 (1 − 𝑡)5 𝑃5𝑥,𝑆𝑆
+ 126𝑡 5 (1 − 𝑡)4 𝑃6𝑥,𝑆𝑆 + 84𝑡 6 (1 − 𝑡)3 𝑃7𝑥,𝑆𝑆
+ 36𝑡
7 (1
+ 𝑡 9 𝑃𝑇𝐸,𝑥
2
− 𝑡) 𝑃8𝑥,𝑆𝑆 + 9𝑡
8 (1
− 𝑡)𝑃9𝑥,𝑆𝑆
(143)
64
𝑦𝑠𝑠 = (1 − 𝑡)9 𝑃𝐿𝐸,𝑦 + 9𝑡(1 − 𝑡)8 𝑃2𝑦,𝑆𝑆 + 36𝑡 2 (1 − 𝑡)7 𝑃3𝑦,𝑆𝑆
+ 84𝑡 3 (1 − 𝑡)6 𝑃4𝑦,𝑆𝑆 + 126𝑡 4 (1 − 𝑡)5 𝑃5𝑦,𝑆𝑆
+ 126𝑡 5 (1 − 𝑡)4 𝑃6𝑦,𝑆𝑆 + 84𝑡 6 (1 − 𝑡)3 𝑃7𝑦,𝑆𝑆
+ 36𝑡
7 (1
2
− 𝑡) 𝑃8𝑦,𝑆𝑆 + 9𝑡
8 (1
(144)
− 𝑡)𝑃9𝑦,𝑆𝑆
+ 𝑡 9 𝑃𝑇𝐸,𝑦
-
Pressure side
𝑥𝑃𝑆 = (1 − 𝑡)7 𝑃𝐿𝐸,𝑥 + 7𝑡(1 − 𝑡)6 𝑃2𝑥,𝑃𝑆 + 21𝑡 2 (1 − 𝑡)5 𝑃3𝑥,𝑃𝑆
+ 35𝑡 3 (1 − 𝑡)4 𝑃4𝑥,𝑃𝑆 + 35𝑡 4 (1 − 𝑡)3 𝑃5𝑥,𝑃𝑆
(145)
+ 21𝑡 5 (1 − 𝑡)2 𝑃6𝑥,𝑃𝑆 + 7𝑡 6 (1 − 𝑡)𝑃7𝑥,𝑃𝑆
+ 𝑡 7 𝑃𝑇𝐸,𝑥
𝑦𝑃𝑆 = (1 − 𝑡)7 𝑃𝐿𝐸,𝑦 + 7𝑡(1 − 𝑡)6 𝑃2𝑦,𝑃𝑆 + 21𝑡 2 (1 − 𝑡)5 𝑃3𝑦,𝑃𝑆
+ 35𝑡 3 (1 − 𝑡)4 𝑃4𝑦,𝑃𝑆 + 35𝑡 4 (1 − 𝑡)3 𝑃5𝑦,𝑃𝑆
+ 21𝑡
5 (1
2
− 𝑡) 𝑃6𝑦,𝑃𝑆 + 7𝑡
6 (1
(146)
− 𝑡)𝑃7𝑦,𝑃𝑆
+ 𝑡 7 𝑃𝑇𝐸,𝑦
Fig. 25 shows an example of turbine airfoil generated by the MATLAB code,
which is based on the procedure discussed.
Figure 25. Axial turbine airfoil
65
Noticed that the throat region can be controlled by adjusting the control points
of both surfaces in order to obtain the required throat size. Dixon [7] suggests that the
throat size 𝑜 can be calculated by using the following expressions: 𝑜 = 𝑔 cos 𝛼2 for
stators and 𝑜 = 𝑔 cos 𝛽3 for rotors.
66
6. METHODS
6.1 GOVERNING EQUATIONS
The governing equations of fluid flows represent the mathematical formulation
of the conservation laws of physics: the law of mass conservation, the law of
momentum conservation (Newton’s second law of motion) and the law of energy
conservation (the First law of thermodynamics) [29]. For compressible turbulent flows,
FLUENT solves the continuity equation, the momentum equation, the energy equation
and the additional equations relative to the turbulence model [30].
To summarize, the integral form of the system of equations, which governs the
time-dependent two-dimensional fluid flow and heat transfer of a compressible
Newtonian fluid, is presented below:
∫
𝒱
∫
𝑉
𝜕𝜌𝑣
𝑑𝒱 + ∮ ∇(𝜌𝑣𝑣 + 𝑝 − 𝜏) ∙ 𝑑𝐴 − ∫ 𝜌𝑓𝑑𝒱 = 0
𝜕𝑡
𝒱
∫
𝑉
{
𝜕𝜌
𝑑𝒱 + ∮ 𝜌𝑣 ∙ 𝑑𝐴 = 0
𝜕𝑡
𝜕𝜌𝑖
𝑑𝒱 + ∮ ∇[𝑣(𝜌𝑖 + 𝑝 − 𝜏) − 𝑞 ] ∙ 𝑑𝐴 = 0
𝜕𝑡
𝑝 = 𝜌ℛ𝑇
𝑖 = 𝐶𝑣 𝑇
(147)
67
The last two equations of the system are the equations of state for an ideal gas.
For compressible flows, these equations provide a relationship between the energy
equation, momentum equation, and continuity equation. By adding the equations of
state, and using the Stokes assumption, which links the stresses to the velocity
gradients, we have an equal number of equations and unknowns (𝑢, 𝑣, 𝑝, 𝑇, 𝜌, 𝑖).
6.2 FLUENT SOLVER
The CFD package used in this work, ANSYS FLUENT, employs a cell-centered
finite-volume method to solve the governing equations on a computational domain that
has been discretized into control volumes. The governing equations in vector form is
given by
𝜕𝜌
∫ 𝑊𝑑𝒱 + ∮[𝐹 − 𝐺] ∙ 𝑑𝐴 − ∫ 𝐻𝑑𝒱 = 0
𝜕𝑡
𝒱
𝒱
(148)
The vector 𝐻 contains the source terms (Body forces and energy sources), in
axial turbines simulations, the gravity is negligible in compare with other forces, and
there are no energy sources, then we can disregard the vector 𝐻. The vectors 𝑊, 𝐹
and 𝐺 are defined as
0
𝜌𝑣
𝜌
𝜏𝑥𝑖
𝜌𝑢
𝜌𝑣 𝑢 + 𝑝𝑖̂
𝑊 = [𝜌𝑣 ] , 𝐹 =
,𝐺 =
𝜏
𝑦𝑖
𝜌𝑣 𝑣 + 𝑝𝑗̂
𝜌𝑖
𝜏
𝑣
[ 𝑖𝑗 𝑗 + 𝑞 ]
[𝜌𝑣 𝑖 + 𝑝𝑣 ]
(149)
Where 𝜏 is the viscous stress tensor and 𝑞 is the heat flux.
Integrating the governing equations over the control volume results in vectors
of scalar quantities, which are linearized by using a discretization method. In this work,
the method of Roe [30] is employed to discretize the inviscid fluxes (the vector 𝐹). The
second-order upwind scheme has been used to discretize the remaining scalar
quantities. In order to calculate the gradients for the second-order upwind schemes,
the least squares cell based gradient scheme is employed.
In FLUENT, there are two formulations to solve the Navier-Stokes equations
(Eq. 148), the pressure-based and density based solvers. In this work, the density-
68
based solver has been chosen because is more suitable for compressible flow than
the pressure-based solver [30]. In the density-based solution method, we can use
either the coupled-implicit formulation or the coupled-explicit formulation, the implicit
method is computationally more expensive that the explicit method, however it is more
stable. Therefore, the implicit formulation is employed in this work.
6.3 TURBULENCE MODELLING
The flow in axial turbines that operates at high pressure is turbulent (𝑅𝑒 = 105
to 107 ). The turbulence originate from several sources, as the natural boundary-layer
transition on the blades, wakes, and combustor [31]. In order to modelling of this
phenomenon, we can use a statistical approach. In a compressible flow, this statistical
approach leads to a density-weighted average, which is based on the decomposition
of the variables of the conservation equations (Eq. 147) in to two components: an
ensemble-averaged mean value of the flow and a fluctuating term, this process is
called Favre decomposition [32]. Substituting these elements into the Navier-Stokes
equations results in the Favre-Reynolds Averaged Navier-Stokes equations. By doing
that, additional terms are introduced into the momentum equation, called the Reynolds
stresses. Similarly, extra turbulent transport terms arise in the transport equation for
an arbitrary scalar quantity.
In order to capture the fluctuation of the flow properties an extremely refined
mesh is required, but there is not affordable computational power nowadays for the
calculations. Thus, turbulence models are used to compute the Reynolds stresses and
the scalar transport terms and close the system of mean flow equations. The two most
used turbulence models in axial turbines are: one-equation Spalart-Allmaras (SA) [35]
and two-equation Shear-Stress Transport (SST 𝑘 − 𝜔) [36]. Tomita [34] has made a
comparison between different turbulence models for high-pressure turbine applications
and the author recommends the SST turbulence model. Therefore, in this work, the
SST 𝑘 − 𝜔 turbulence model is employed.
69
6.4 MESH GENERATION
The structured grid techniques has been widely used for mesh generation of
turbomachinery nowadays, especially multi-block hexahedral grids. A structural mesh
needs less memory, gives superior accuracy and provide a better boundary layer
resolution that an unstructured mesh [37]. Thus, in this work, a hexahedral multi-block
technique with an O-grid type around the blade profile surface is used.
The grids must be generated with a good element distribution, quality
(Orthogonality and edge angles), and respecting the maximum aspect ratio suggested
by the FLUENT solver (less than 20). In addition, it is important to control the distance
of the first cell from the wall 𝑦 + due to turbulence models requirements, in order to
guarantee the validity of the model. For the Menter’s 𝑘 − 𝜔 SST model, 𝑦 + must be
less than 300, for this turbulence model, there is an automatic wall treatment, the solver
uses wall function for 𝑦 + in the range 30 ≤ 𝑦 + ≤ 300, and for values between 2 ≤ 𝑦 + ≤
30 a blending function is used [30].
In this work, the mesh was created using ICEM-CFD v.16.0, an example of
mesh generated in this work is shown in Fig. 26, in this software, a blocking technique
is used to obtain a hexahedral mesh, Fig 27. In this way, we are able to control the
size and distribution of the grid.
Figure 26. 2D turbine blade mesh
70
Figure 27. Blocking strategy
Stator and rotor blades have high cambered geometries: for this reason, it is
difficult to maintain a good quality mesh, by doing the periodicity of the blades using
periodic nodes. An alternative way is to obtain the periodicity using a non-conformal
mesh. The advantage of this mesh strategy is that the nodes of two zones do not have
to match one-for-one [30]. The flux across the non-conformal boundary (Periodic
repeats) is obtain according to the Fig. 28.
Figure 28. Interface zone [30]
71
From Fig. 28, it is noted that to compute the flux across the interface into cell
𝐼𝑉, face D-E is ignored and instead faces d-b and b-e are used get the information into
cell 𝐼𝑉 from cells 𝐼 and 𝐼𝐼𝐼.
The solver configuration will be validated using the VKI LS89 turbine airfoil
(Chapter 7), as its experimental results are available for comparison. The grid
sensitivity analysis will be performed for five different mesh sizes. The chosen mesh
size will be used for all simulations of the turbine airfoils generated.
6.5 BOUNDARY CONDITIONS
The boundary conditions imposed in this work are:

At inlet: Total temperature 𝑇01 and Total pressure 𝑝01

At outlet: Static pressure 𝑝2

At blade profile-to-blade profile surfaces: Translational periodicity

At airfoil walls: Non-slip condition
6.6 SOLUTION INITIALIZATION AND CONVERGENCE
To start the numerical interations, it has been used a first-order scheme for
spatial discretization to avoid numerical instabilities due to discontinuous region such
as shock waves during the stat up [34]; when the solution is near the convergence a
second order scheme is used to obtain the final solution. In order to start the numerical
procedure, a hybrid initialization is employed, this method solves the Laplace equation
to produce a velocity and pressure field, all other variables will be calculated based on
domain averaged values [30].
Convergence of the flow is monitored through three method [31] and [34]. The
first is observing the flow residuals for the continuity, velocity, and energy equations to
ensure that the fall to approximately 1 ∙ 10−3 , Fig. 29. The second is monitoring a scalar
quantity (Temperature or pressure) in the flowfield, Fig. 30, Helms [31] recommends
72
at downstream of the blade profile. Third method is compute the net mass flow rate
through the inlet and outlet, Fig. 31.
Figure 29. Numerical residuals
Figure 30. Temperature monitoring
Figure 31. Net mass flow rate
73
7. TEST CASE
7.1 DESCRIPTION OF THE VKI LS89 TEST CASE
The VKI LS89 test case is an experimental aero-thermal investigation of a highly
loaded transonic turbine nozzle guide vane. The experiment was conducted by the Von
Karman Institute (VKI). The measurements were performed for many combinations of
freestream flow parameters, as inlet total temperature 𝑇01 , inlet total pressure 𝑝01 and
freestream turbulence intensity 𝑇𝜇∞ . The goal of the experiment was to check these
effects on aerodynamic blade performance, Reynolds number and blade convective
heat transfer, besides the validation of inviscid and viscous flow calculation methods
[38]. The VKI LS89 turbine cascade is one of the most used turbine test cases, it is
frequently used to validate the aero-thermal predictive capabilities of CFD codes, for
this reason, it will be used as test case for this present work.
7.2 THE VKI LS89 BLADE PROFILE GEOMETRY
The two-dimensional shape of the VKI LS89 blade is shown in Fig. 32 and its
geometrical parameters are summarized in Tab. 2
74
Figure 32. VKI LS89 airfoil and geometrical characteristics [39]
Table 2. Geometrical Parameters of the VKI LS89 airfoil
Geometrical Parameter
Symbol
Value
Chord
𝑐
67.467 mm
Pitch
𝑔
57.500 mm
Throat
𝑜
14.930 mm
Leading Edge Radius
𝑟𝐿𝐸
4.127 mm
Trailing Edge Radius
𝑟𝑇𝐸
0.710 mm
Axial Chord
𝑐𝑥
36.985 mm
Stagger Angle
Φ
55°
75
7.3 THE MUR49 TEST
The aero-thermal investigation [38] produced experimental results for several
test conditions. To validate our simulations, the MUR 48 test is chosen. The boundary
conditions corresponded to the imposed parameters: isentropic outlet Mach number
1.02 and Reynolds number of 106 , are listed in table 3. The working fluid is air and it is
considered a calorically perfect gas.
Table 3. The boundary conditions for the MUR 48 test
Inlet Total temperature
𝑇01
420 K
Inlet Total Pressure
𝑝01
160500 Pa
Inlet Turbulence Intensity
𝑇𝜇∞
3%
Inlet Flow Angle
𝛼1
0°
Outlet Static Pressure
𝑝2
82821 Pa
The technical note of the experiment [38] does not provide the value of outlet
static pressure. For this reason, the static pressure 𝑝2 is calculated by Eq. (23):
𝑀𝑖𝑠 = √(
2
𝑝01 𝛾−1
) [( ) 𝛾 − 1]
𝛾−1
𝑝
(150)
The static pressure on the blade surface 𝑝 was measured using 27 wall static
pressure tappings [38]. Assuming a constant inlet total pressure 𝑝01 , the experimental
isentropic Mach number distribution 𝑀𝑖𝑠 can obtained by Eq. (150):
For validation of the CFD procedure, the experimental isentropic Mach number
distribution has been compared to the numerical one. The numerical isentropic Mach
number distribution is obtained in a similar way as the experimental one. The blade
static pressure is calculated numerically, thus the numerical isentropic Mach Number
is calculated by using Eq. (150).
76
A grid independence study (Fig. 33) was conducted to define the mesh giving
the best compromise between accuracy and simulation time obtain. The simulations
have been conducted using: the density-based implicit solver, the SST 𝑘 − 𝜔 as
turbulence model. A second-order scheme has been applied to discretize the flow and
turbulent transport equations. The boundary conditions are listed in table 3.
Figure 33. VKI LS89 grid dependency study
The VKI LS89 turbine airfoil isentropic Mach number is presented in Fig. 33 for
a range of mesh size between 7596 and 73456 cells. The numerical results shows
good agreement with the experimental result. However, the prediction of the shock
wave slides further downstream for the numerical one, this can be explained by the
difference between the experimental and numerical outlet boundary conditions.
Overall, analyzing Fig. 33, the Mach number distribution seems insensitive to the
change in grid size; for this reason, the total pressure ratio, 𝑝02 /𝑝01, was evaluated for
all meshes, as shown in Fig. 34. Based on this graph, the number of cells of mesh 4
was selected for further simulations, as it demonstrated a good compromise between
time of simulation and accuracy of solution.
77
Figure 34. Total pressure ratio vs. Number of cells
The contours of pressure, Mach number, static temperature, and density are
shown in figures 35 to 38. As shown in Fig. 35, the region of maximum velocity appears
near the exit of the passage, where is the location of minimum pressure at the suction
side (Fig. 36). This strong acceleration is due to the decrease of area of the channel.
Also, as shown in all figures, a normal shock wave is presented in the rear part of the
suction side. This phenomenon is the same shown in the experimental test, Fig. 39.
Figure 35. Contours of Mach number
78
Figure 36.Contours of static pressure
Figure 37. Contours of static temperature
Figure 38. Contours of density
79
Figure 39. Schlieren photograph of the experimental test
80
8. RESULTS
Validated the method and chosen the grid size, a single axial turbine stage has
been designed based on the procedure discussed on chapter 4 (One Dimensional
Design Procedure). The stator airfoils have been obtained by using the approach
presented in chapter 5 (Turbine Airfoil Design). Table 4 provides the requirements for
the design of the axial turbine stage.
Table 4. Design requirements
𝑚̇
15.0 kg/s
𝑇01
697.0 K
𝑝01
607.8 kPa
𝑝01 /𝑝03
4
𝑈
400.0 m/s
𝑟𝑇𝐸
4.13 mm
𝑟𝑙𝐸
0.71 mm
The working fluid is air and it is considered a calorically perfect gas. Providing
this initial data on the MATLAB code related to the 1D turbine design, we can obtain
the geometric parameters, thermodynamic properties, and the velocity triangles.
81
Table 5. Geometric and thermodynamics properties
𝛼2
73.00°
𝛼3
-2.51°
𝛽2
34.99°
𝛽3
-69.07°
Turning ∆𝛽
104.06°
Chord (Stator) 𝑐𝑅
56.60 mm
Chord (Rotor) 𝑐𝑅
46.90 mm
Pitch (Stator) 𝑔𝑠
43.58 mm
Pitch (Rotor) 𝑔𝑅
37.90 mm
Stagger angle (Stator) Φ𝑠
55.00°
Stagger angle (Rotor) Φ𝑅
47.00°
𝑝2
350.50 kPa
𝑀2
1.12
𝑇02,𝑅
582.11 K
𝑝02,𝑅
369.94 kPa
𝑝3
143.82 kPa
𝑀3,𝑅
0.98
𝑊̇
3.02 MW
𝑛𝑡𝑠
88.69 %
In this work, only the performance of the stators are analyzed; thus, just the
shading information are important to the design of the stators airfoils.
82
Figure 40. T-S diagram
Figure 41. Velocity triangles
Table 6. Initial data required to the design of the stator
Geometrical Parameter
Symbol
Value
Inlet Flow Angle
𝛼1
0°
Outlet Flow Angle
𝛼2
73°
Chord
𝑐
56.60 mm
Pitch
𝑔
43.58 mm
Leading Edge Radius
𝑟𝐿𝐸
4.13 mm
Trailing Edge Radius
𝑟𝑇𝐸
0.71 mm
Axial Chord
𝑐𝑥
32.46 mm
Stagger Angle
Φ
55°
83
Using the geometric parameters presented on Tab. 6 in the MATLAB code
related to the design of turbine airfoils, five different airfoils have been generated (Fig.
42) to investigate the effects of airfoil curvature on the aerodynamic performance. As
discussed in chapter 5, we can control the shape of the curvature by setting the normal
distance between the camber line to the control points (Fig.43) and Tab. 7.
Figure 42. Stator airfoils
Figure 43. Suction side control points
84
Table 7. Normal distance to the control points (Suction side)
𝑃3,𝑠𝑠 (m)
𝑃4,𝑠𝑠 (m)
𝑃5,𝑠𝑠 (m)
𝑃6,𝑠𝑠 (m)
𝑃7,𝑠𝑠 (m)
Stator 1
0.0180
0.0254
0.0080
0.0056
0.0032
Stator 2
0.0140
0.0304
0.0080
0.0056
0.0022
Stator 3
0.0080
0.0314
0.0100
0.0026
0.0002
Stator 4
0.0095
0.0310
0.0080
0.0056
0.0022
Stator 5
0.0120
0.0294
0.0090
0.0046
0.0012
According to Korakianitis and Papagiannidis [40], the curvature affects the blade
performance. He suggests that the local pressure and velocity distribution depends on
the shape of the curvature. For this reason, the curvature of each airfoil will be
calculated in order to examine its influence on the flow. The curvature is given by:
𝐶=
|𝑥 ′ 𝑦 ′′ − 𝑦′𝑥′′|
(𝑥 ′2 +
(151)
3
𝑦 ′2 )2
From 1D turbine design, we calculated that the outlet Mach number 𝑀2 is 1.12,
thus the airfoils must be designed to have an outlet Mach number close to this value.
However, we have a design constraint, which is, the outlet Mach number must be less
than 1.2, and this is a control point of the design (Chapter 4).
The aerodynamic performance of the stator airfoils are evaluated by means of
isentropic Mach number distribution, total pressure loss, and the contours of Mach
Number, static pressure and temperature. The total pressure loss 𝜉 can be computed
by the following equation:
𝜉=
𝑝01 − 𝑝02
𝑝01 − 𝑝2
(152)
85
The boundary conditions of the CFD simulations, presented in Tab. 8, have
been obtained from the 1D turbine design. The results of these numerical simulations
are shown in Fig. 44 to 60.
Table 8. Boundary conditions of the CFD simulations
Inlet Total temperature
𝑇01
697 K
Inlet Total Pressure
𝑝01
607 kPa
Inlet Turbulence Intensity
𝑇𝜇∞
3%
Inlet Flow Angle
𝛼1
0°
Outlet Static Pressure
𝑝2
350 kPa
Figure 44 shows the surface isentropic Mach number distribution of five similar
turbine cascades generated.
Figure 44. Isentropic Mach number distribution
Figure 45 shows the curvature of each airfoil. The curvature has been obtained
using Eq. (151).
86
Figure 45. Curvature distribution of the suction side
For all airfoils, we have two shock waves; the first one is located at the throat of
the channel, and the second one is positioned at the end of the blade profile (Fig. 44).
It is evident that that the curvature of the leading edge region has a strong influence
on the isentropic Mach number distribution (Fig. 44 and 45). Stator 3 has the greater
leading edge curvature, which leads to the case with higher Mach number (Fig. 44).
Likewise, it is noted that stator 1, which has the lower trailing edge curvature, has the
lowest maximum Mach number (Fig. 44). Regarding to the pressure surface, the flow
is accelerated almost continuously from leading edge to the trailing edge (Fig 44).
Figure 46. Contours of Mach number (Stator 1)
87
Figure 47. Contours of Mach number (Stator 2)
Figure 48. Contours of Mach number (Stator 3)
Figure 49. Contours of Mach number (Stator 4)
88
Figure 50. Contours of Mach number (Stator 5)
Figures 44 to 50 illustrate that, as the curvature near the trailing edge region is
increased, the cascade becomes loaded; this is measured by the area between the
Mach number curves on the pressure and the suction side. For all stator airfoils, the
flow on the suction surface accelerates quickly over the region after the leading edge
(in this work will be called first region). After that, we have a moderate acceleration to
the maximum Mach number near to the passage throat; at this location, there is a
shock wave, and then the flow becomes subsonic. Subsequently, we have another
acceleration of the flow, which leads to a shock wave at the rear part of the suction
side. In addition, it is observed that stators 3 and 5 exceed the maximum limit of Mach
number (𝑀2 < 1.2).
Figure 51. Contours of static pressure (Stator 1)
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Figure 52. Contours of static pressure (Stator 2)
Figure 53. Contours of static pressure (Stator 3)
Figure 54. Contours of static pressure (Stator 4)
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Figure 55. Contours of static pressure (Stator 5)
Figures 51 to 55 show the contours of static pressure for all cases. As the
outflow Mach number increases, the shock angle will decrease, and then the shock
wave will become more oblique, this phenomenon is notice in the contours of pressure.
Stator 3 is the case with higher outlet Mach number and we observe that it is the case,
which has the shock wave more oblique in comparison with the other airfoils.
Figure 56. Contours of static temperature (Stator 1)
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Figure 57. Contours of static temperature (Stator 2)
Figure 58. Contours of static temperature (Stator 3)
Figure 59. Contours of static temperature (Stator 4)
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Figure 60. Contours of static temperature (Stator 5)
Figures 56 to 60 show the contours of static temperature for all cases. As the
flow near the trailing edge detaches, then we have the mixing of the suction surface
and pressure surface boundary layers with flow behind the trailing edge, this leads to
the trailing edge wake, this phenomenon is observed in all airfoils. According to Salvá
[41], the intensity of the wake depends on the outflow isentropic Mach number, this
behavior is noted in all simulations, especially for stator 3 (Fig. 58).
Table 9. Total pressure ratio
Stator
𝜉
1
0.9334
2
0.9214
3
0.8792
4
0.9187
5
0.9064
From results on Tab. 9, we can conclude that the airfoil curvature has strong
influence on the total pressure losses. As the outflow Mach number increases, the total
pressure ratio becomes smaller; it means that more entropy is generated. Stator 3 is
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the case with lowest total pressure loss, its outflow Mach number is about 1.3. For this
reason, it is recommend that the subsonic turbine airfoils must have a maximum outlet
Mach number of 1.2.
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9. CONCLUSIONS
A method to design 2D blades for axial turbines is presented. The design starts
by the definition of a camber line, which is defined by an inlet flow angle, an outlet flow
angle, a chord, and a stagger angle. The suction and pressure surface are obtained
by using the Bézier curves, the shape of these surfaces depends on control points
position. The geometric parameters needed to for the design of the blade profile are
extracted from the one-dimensional turbine design, as well the design point flow.
The design of five turbine airfoils has been presented and theirs performance
has been numerically investigated. The validation of the numerical method was
demonstrated by comparison of the surface isentropic Mach number distribution with
those measured from experiments available in literature. The aerodynamic
performance of the airfoils are evaluated in terms of isentropic Mach number
distribution, total pressure loss, and the contours of Mach Number, static pressure and
temperature.
The five blade profiles designed have small variations in blade geometry. The
airfoils with high values of curvature at the leading edge region have a strong
acceleration of the flow in the first region (region after the leading edge), which leads
to a high Mach number near the throat. In the same way, low values of curvature cause
low Mach number at the vicinity of the throat. As the boundary conditions lead to a
transonic flow in the turbine cascade, two shock waves are present; the first one is
located at the throat region, and the second one is placed at the rear part of the suction
side surface. When the outflow Mach number is increased, the shock wave became
more oblique: this behavior occurs due to the decrease of the shock wave angle. In
addition, the total pressure loss has been investigated of each airfoil: it can be
concluded that for the cases with high values of outflow Mach number, the total
pressure loss has low values.
Analyzing the results of the five airfoils generated, it can be determined that the
blade profile, which provides the best aerodynamic performance and satisfy the design
demands, is the stator 1. This airfoil has the higher total pressure loss, and gives an
outlet Mach number very close to the calculated from 1D turbine design.
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The curvature of the suction surface has strong influence on the turbine cascade
performance. The two first control points of the suction surface regulate the curvature
at leading edge region, thus it is important to check the curvature values in this region
before start the numerical simulations. Curvature discontinuities may introduce
undesirable behaviors, as regions of local acceleration and deceleration. In addition, it
is important to attend the design requirements, principally, the outlet Mach number.
The method to design 2D turbine blade profiles presented in this work can be
coupled to multi-objective evolutionary- or heuristic algorithm optimization methods.
However, an automatic mesh generator for axial turbines, and a CFD solver are
required. Another option is making an integration between MATLAB and ANSYS tools.
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REFERENCES
[1] Denton, J.D. Loss mechanisms in turbomachines. J. Turbomach. 1993, 115, 621656.
[2] General Electric. Retrieved June 19, 2016, from http://www.ge.com/digital/
[3] Mattingly, J., Elements of Propulsion: Gas Turbine and Rockets, AIAA Education
Series., 2006.
[4] Anderson J.D., Computational Fluid Dynamics, the Basics with applications,
McGraw-Hill, 1995.
[5] Tomita, J.T., “Three-Dimensional Flow Calculations of Axial Compressor and
Turbines Using CFD Techniques”, PhD thesis, Technical Institute of Aeronautics, São
José dos Campos, Brazil, 2008.
[6] Ainley, D. G. and Mathieson, G. C. R., 1951, “A method of performance estimation
for axial flow turbines”. ARC R. and M. 2974.
[7] Dixon, S. L., Hall, C. A., Fluid Mechanics and Thermodynamics of Turbomachinery,
7th Ed, Butterworth-Heinemann, 2014.
[8] Anderson, J. D., Modern compressible flow, McGrawHill, 2004.
[9] Çengel, Y. A and Boles M. A, Thermodynamics: An Engineering Approach, 5th Ed,
McGraw-Hill, 2006.
[10] Lakshminarayana, B., Fluid dynamics and heat transfer of turbomachinery, John
Wiley & Sons, 1996.
[11] White F. M., Fluid mechanics, 7th ed, McGrawHill, 2011.
[12] Turns S. R., An introduction to combustion: concepts and applications, 3th Ed,
McGrawHill, 2012.
[13] Traupel W., Thermische Turbomaschinen, Vol 1, 3rd Ed, Springer, 1977.
[14] Sieverding C. R., 2012. Unpublished note. Advanced Course on Turbines. Von
Karman Institute for Fluid Dynamics.
[15] Wilson D.G. The Design of High-efficiency Turbomachinery and Gas Turbines.
The MIT Press, 1984.
[16] Kenny S., “Development of a multi-disciplinary design tool for axial flow turbines,”
Master’s thesis, Carleton University, Ottawa, Canada, 2005.
97
[17] Dahlquist, A. D. “Investigation of losses prediction methods in 1D axial gas
turbines,” Master’s thesis, Lund University, Lund, Sweden, 2008.
[18] Xu, L., and Denton, J. D., 1988, "The Base Pressure and Loss of a Family of Four
Turbine Blades," ASME JOURNAL OF TURBOMACHINERY, Vol. 110, pp. 9-17.
[19] Zweifel, O. 1945, “The spacing of turbomachine blading, especially with large
angular deflection,” Brown Boveri Review, 32.12.
[20] Taylor, M., “HP Axial Flow Turbine Aerodynamic Design,” Advances in
Turbomachinery Aero-Thermo-Mechanical Design and Analysis,VKI Lecture Series,
von Karman Inst. for Fluid Dynamics, Rhode-Saint-Genèse, Belgium, Nov. 2006.
[21] Kacker, S.C., Okapuu U., 1982, “A mean line prediction method for axial flow
turbine efficiency,” Journal of Engineering Power. Transaction of the American Society
of Mechanical Engineers, 104, 111-119.
[22] Lewis, R. I., Turbomachinery performance analysis, Elsevier Science &
Technology Books, 1996.
[23] Korakianitis T., Rezaienia M.A., Hamakhan I.A., Wheeler A.P.S. (2013). Two- and
three-dimensional prescribed surface curvature distribution blade design (circle)
method for the design of high efficiency turbines, compressors, and isolated airfoils.
Journal of Turbomachinery, 135 (3), art. no. 041002.
[24] Farin, G., Curves and Surfaces For Computer Aided Geometric Design, 4th Ed,
Academic Press, 1997.
[25] Sederberg, T. W., 2012. Computer Aided Geometric Design Course Notes.
Brigham Young University.
[26] Pritchard, L. J. An eleven parameter axial turbine airfoil geometry model. ASME
85-GT-219; 1985.
[27] Verstraete, T.: CADO: a computer aided design and optimization tool for
turbomachinery applications EngOpt2010, 2nd International Conference on
Engineering Optimization, Lisbon, Portugal, September 6-9, 2010.
[28] Sousa, J.; Paniagua, G. Entropy Minimization Design Approach of Supersonic
Internal Passages. Entropy 2015, 17, 5593-5610.
[29] Versteeg, H. K.; Malalasekera, W. An Introduction to Computational Fluid
Dynamics – The Finite Volume Method, 2th Edition, Longman Scientific & Technical,
2007.
[30] ANSYS, 2013, “Fluent 15.0 theory guide”, Canonsburgh, PA, Cap. 5.
[31] Helms Ian. “Effect of Non-Uniform inlet Turbulence Distribution on High Pressure
turbine Heat Transfer”, Master Thesis, Oxford University, Oxford, United Kingdom.
98
[32] Tannehill, J. C; Anderson, D. A; Pletcher, R. H. Computational Fluid Mechanics
and Heat Transfer, 2th Edition, Taylor & Francis, 1997.
[33] Tomita, J.T.; Silva, L.M.; Silva, D.T. Comparison Between Unstructured and
Structured Meshes With Different Turbulence Models for a High Pressure Turbine
Application. Proceedings of ASME Turbo Expo 2012, GT2012-6990, June 11-15,
2012, Copenhagen, Denmark.
[34] Joly Michael Marie. “Multifidelity, Multidisciplinary Optimization of Turbomachines
with shock Interaction”, Doctoral Dissertation, Texas A&M University, College Station,
United States.
[35] Spalart, P. R.; Allmaras, S. S., “A One-Equation Turbulence Model for
Aerodynamic Flows”. 30th Aerospace Sciences Meeting and Exhibit. ed. [S.l.]: AIAA,
1992. ISBN AIAA-92-0439.
[36] Menter, R. F., “Zonal Two-Equation k-w Turbulence Models for Aerodynamic
Flows”, 24th Fluid Dynamics Conference. Orlando, Florida: [s.n.]. 1993.
[38] Arts, T., de Rouvroit, M. L, and Rutherford, A. W. Aero-Thermal Investigation of a
Highly Loaded Transonic Linear Turbine Guide Vane Cascade - A test case for Inviscid
and Viscous Flow Computations. In VKI Technical Note TN174, Brussels, Belgium,
September 1990.
[39] Mueller, L. and Verstraete, T., The LS89 Optimization Test Case, 2016, Available
at:http://aboutflow.sems.qmul.ac.uk/content/events/munich2016/benchmark/testcase
2/files/LS89TestCase.pdf.
[40] Korakianitis, T., and Papagiannidis, P., 1993, “Surface-Curvature-Distribution
Effects on Turbine-Cascade Performance,” ASME J. Turbomach., 115(2), pp. 334–
341.
[41] Salvá, José. J., Tizón, Juan. M., Rodríguez Andrés, ”Use of numerical simulation
for flow phenomena teaching in transonic axial turbines”, ICERI 2012, ISBN 978-84616-0763-1.