Analytical model for high temperature wear in thermal barrier coatings under non-steady state conditions in gas turbines
Autor: Gómez Flórez , Pablo Andrés
Asesor: Toro , Alejandro Toro  (Thesis advisor)
Fecha: 2012
Programa: Bogotá - Medicina -Fisioterapia
Facultad: Facultad de Medicina

Resumen: Abstrac: The continuously growing tension over energy resources in the context of a globalized and competitive economy demands more efficient ways to exploit those available resources; it is not possible to consider the energy generation process and the cogeneration outside this scenario, it is the reason why is necessary to work always at limit of the possibilities and that is where the development of engineering materials becomes a compulsory transit route. It is clear in light of the history of thermal engines, the efficiency is directly related to the development of stable materials at high temperatures, and it is precisely the field which aims to make inroads with this project, addressing the problem from the phenomenological point of view, where the key is to fully understand the physical processes of energy transformation to find, from this understanding, appropriate models to estimate the service life of the components of the gas turbines and others thermal engines.

DOCTORAL THESIS
Title
ANALYTICAL MODEL FOR HIGH TEMPERATURE WEAR IN
THERMAL BARRIER COATINGS UNDER NON-STEADY STATE
CONDITIONS IN GAS TURBINES
By
Pablo Andrés Gómez Flórez
Advisor
Ph.D. Alejandro Toro
Facultad de Minas
Universidad Nacional de Colombia
Sede Medellín
September 2012
Thesis - Pablo Gomez Page 1/138Enfrentar un trozo de papel en blanco es sin duda intimidante y pone en evidencia la
levedad de nuestras ideas, sin embargo, al mismo tiempo desata un proceso de
reconocimiento de nuestra propia existencia y el deseo de comprender y plasmar en un
acto humilde, frente a la bastedad del conocimiento, nuestra miope imagen del universo.
Es por esto que en esta singular oportunidad, quiero dar gracias a las personas que me
han acompañado de muchas maneras en este proyecto, consolidado a lo largo de las
siguientes páginas.
Agradezco profundamente y con mucho amor a mi linda esposa Lina, por estar siempre
ahí, por su paciencia, sus aportes y sobre todo por llenar mi vida de buenas razones; a
mis padres Hernando y Margarita, dos grandes seres humanos a quienes agradezco
todos sus sacrificios para ponerme en el camino correcto.
A mi director Alejandro, por creer en este proyecto, por sus aportes y acompañamiento; y
a su Grupo de Tribología y Superficies, proveedor de valiosa materia prima para esta
investigación en cabeza de buenos investigadores, compañeros y amigos.
Al profesor Rodney Trice y mis buenos amigos de la Universidad de Purdue, por hacer de
mi pasantía doctoral una gran experiencia.
A Empresas Públicas de Medellín E.S.P. y sus directivos que nunca dejaron de creer en
este proyecto y que en consecuencia procuraron valiosos recursos físicos y económicos.
Thesis - Pablo Gomez Page 2/138TABLE OF CONTENTS
ANALYTICAL MODEL FOR HIGH TEMPERATURE WEAR IN THERMAL BARRIER
COATINGS UNDER NON-STEADY STATE CONDITIONS IN GAS TURBINES .............. 1
1 Introduction .............................................................................................................. 5
2 Thermoelectric Power Generation and Gas Turbines ........................................... 8
2.1 Thermoelectric power generation in Colombia .............................................. 8
2.2 Gas turbines ................................................................................................... 15
2.2.1 Compressor ................................................................................................................. 16
2.2.2 Combustors ................................................................................................................. 18
2.2.3 Turbine section ............................................................................................................ 20
2.3 Brayton cycle: the ideal cycle for gas turbines ............................................ 21
2.4 Gas turbine loads ........................................................................................... 25
3 Gas Turbine Materials and Wear Mechanisms ..................................................... 28
3.1 Advanced gas turbine materials and coatings ............................................. 28
3.1.1 Nickel-based superalloys ............................................................................................. 29
3.1.2 Thermal Barrier Coatings (TBCs) ................................................................................ 32
3.2 Wear mechanisms for gas turbine hot gas section ..................................... 41
3.2.1 Oxidation and chemical failure (oxide spallation) ........................................................ 42
3.2.2 Top coat sintering and creep ....................................................................................... 46
3.2.3 Nickel-base superalloys rafting .................................................................................... 49
3.2.4 Top Coat Corrosion ..................................................................................................... 50
4 High-Temperature Wear Models for Substrate – TBC Systems .......................... 52
4.1 Statistical models for service life estimation ............................................... 52
4.2 Analytical models and parametric finite element method approaches ...... 55
4.3 Wear model using a fracture mechanics approach ..................................... 61
4.4 Thermodynamic approach for elastic-plastic damage................................. 64
5 Analytical Model for High-Temperature Wear in Non-Steady State Gas Turbines
Based on a Thermodynamic Approach ........................................................................ 65
5.1 Algorithm basics ............................................................................................ 69
5.2 Combustion model ......................................................................................... 71
5.3 Fluid dynamics model .................................................................................... 72
5.4 Heat transfer model ........................................................................................ 76
5.5 Wear model ..................................................................................................... 81
5.5.1 Plastic deformation ...................................................................................................... 88
5.5.2 Creep ........................................................................................................................... 90
5.5.3 Thermal expansion stresses ........................................................................................ 91
5.5.4 Top Coat sintering effects ............................................................................................ 93
5.5.5 Thermally Grown Oxide growth and undulation .......................................................... 94
5.5.6 Coupled state potentials .............................................................................................. 99
Thesis - Pablo Gomez Page 3/1385.5.7 Energy release ............................................................................................................. 99
6 Numerical Implementation of High-Temperature Wear Model in Non-Steady
State Gas Turbine ........................................................................................................ 100
6.1 The finite element method ........................................................................... 100
6.2 The virtual crack closure method ................................................................ 106
6.3 Experimental results .................................................................................... 109
6.3.1 Liners and transition pieces in 7FA GE gas turbines ................................................ 112
6.3.2 First stage nozzles and airfoils in 7FA GE gas turbines ............................................ 115
6.3.3 Effects of substrate –TBC system temporal evolution on crack propagation. ........... 124
7 Conclusions ......................................................................................................... 131
8 References ............................................................................................................ 133
Thesis - Pablo Gomez Page 4/1381 Introduction
The continuously growing tension over energy resources in the context of a globalized and
competitive economy demands more efficient ways to exploit those available resources; it
is not possible to consider the energy generation process and the cogeneration outside
this scenario, it is the reason why is necessary to work always at limit of the possibilities
and that is where the development of engineering materials becomes a compulsory transit
route.
It is clear in light of the history of thermal engines, the efficiency is directly related to the
development of stable materials at high temperatures, and it is precisely the field which
aims to make inroads with this project, addressing the problem from the phenomenological
point of view, where the key is to fully understand the physical processes of energy
transformation to find, from this understanding, appropriate models to estimate the service
life of the components of the gas turbines and others thermal engines.
The gas turbines used in combined cycles for power generation are designed as
continuous operation machines and ideally should be operated at base load for optimum
cost-benefit ratio. In developed countries, where thermal generation reaches about 75 %
of total electricity generation [1, and 2], these turbines tend to enter the base when
calculating the actual delivery of energy beside coal and nuclear steam turbines, to provide
the highest efficiencies and lower operating costs.
On the other hand, countries that have enough water resources commonly cover the base
of energy delivery, using hydroelectric power plants which are much cheaper than any
alternative for electricity generation, but in these countries the reliability of the
interconnected systems is supported by thermal generation which may be regarded, unlike
the hydraulic, as immune to changes in climate.
When thermal power plants no longer cover the base of actual delivery of energy and
become peak demand absorbers, the service conditions are drastically different from the
design conditions and in this sense it is clear that the service life of gas turbine materials
does not coincide with the expectative for gas turbines operating continuously at base load
[1].
Considering the above, it is essential identifying the temporal evolution of the phenomena
of fatigue, corrosion and wear in transient conditions generated by the loading and
unloading of gas turbines to get a balance between reliability of the interconnected system
and the cost of that reliability.
TBC coatings are specifically developed to protect the hot section components of gas
turbines against the aggressive agents involved in the process of transforming the
chemical energy of natural gas or other fuel into mechanical energy [1]. The extreme
conditions of temperature and pressure which are variable depending on the load of the
turbine, together with the flow of highly corrosive gas combustion stream and mechanical
stresses, make complex the development of wear models for substrate – TBC systems,
but necessary to develop improvement projects both for the service life of hot section
components, as for the increased of the net power output in gas turbines and other
thermal machines.
Thesis - Pablo Gomez Page 5/138Exist two wear mechanisms clearly identified for the substrate – TBC systems on power
generation gas turbines, each one of which is associated with the operating conditions, the
first one, is associated with continuous operation at base load, where the phenomena of
oxidation and sintering of the coating, and creep of both coating and substrate are very
important, the second mechanism is associated with cyclic operation, which has a large
number of starts and stops in comparison with the number of operating hours, in this case
the wear receives its major contribution from thermomechanical fatigue phenomena, which
are closely related to the differences between volumetric expansion coefficients of the
materials in the substrate – TBC system [1, 3 and 4].
Considering the above, models based on the classical approaches of fracture mechanics
have been developed, such as the one proposed by Jinnestrand and Brodin (2004) [6],
which develop a estimative of the service life of the hot section components of gas
turbines from thermomechanical fatigue test (TMF) data, using thermal cycles to assess
the TBC wear; also analytical models, such as the model proposed by Hutchinson and
Suo (1992) [5], to predict the delamination growth by crack propagation at TBC interfaces.
On the other hand, have been developed models, such as the model proposed by AG
Evans et al. (1997 and 2001) [7, 10, 15, 21, 21, 27, and 47], which predict the
delamination as a consequence of a unstable thermally grown oxide (TGO), within the
TBC coating, this mechanism is defined as the chemical failure of the material by oxidation
[7].
The dissimilar nature between the thermomechanical fatigue and high temperature
oxidation, in terms of their time scales, for the substrate – TBC system wear phenomena,
propose an interesting challenge to integrate the kinetics of the thermally grown oxide and
the thermomechanical fatigue on a single model for estimating the service life of hot
section components of the gas turbines.
Thermodynamic approaches such as those proposed by Bielski, J. Skrzypek, and H.
Kuna-Ciskal (2006) [8], which balance the Gibbs or Helmholtz thermodynamic potentials
for thin films composite materials, and provides a smooth variation of mechanical
properties based on the Functionally Graded Material (FGM) model, developed in Japan
by Yamanouchi et al. (1980) [9] show an interesting way for the unification of the wear
phenomena, providing the capability to assess the failure mechanisms as the
superposition of elastic response, plastic deformation and crack propagation within the
substrate – TBC system, and open a way to include the chemical failure in a proper time
scale [10].
However, the process of integration of wear mechanisms by thermodynamic models, is
very complex, whereas the chemical failure associated with the oxidation of thermal barrier
coating, depends closely as shown by Mumm et al. (2000) [10], of the state of residual
stresses associated with the growth of oxide layers, and the induced stresses caused by
surface irregularities when applying the thermal barrier coating.
On the other hand, the materials characterization is required in order to identify
mechanical properties and microstructural composition to develop an adequate
representation of the substrate – TBC system and propose a unified analytical model,
Thesis - Pablo Gomez Page 6/138considering that both nickel-based superalloys and TBC coatings are developments of the
leading manufacturers of gas turbines and are protected by patents, which requires
reconstructing the history of manufacture of these, based on finished products.
It is considered that the phenomenological models developed so far are quite inaccurate
when applied to real components subjected to variable load conditions, and the only
reliable sources for estimation of service life, comes from databases of manufacturers
based on a large volume of statistical data taken from different machines that are in
service, this have two obvious disadvantage, the first one, is that those data refer
machines typically operating at base load or machines submitted at high number of charge
cycles and not include machines under operating conditions in between, and the second is
that such models cannot be used to accelerate the development of new materials and
applications, because they do not come from a phenomenological assessment of the
wear.
In this thesis a phenomenological analysis of the wear mechanism of substrate – TBC
systems is conducted, finding relations between gas turbine operating conditions and
those wear mechanism, then from this understanding a thermodynamic approach is
developed on the base of continuum mechanics to propose an evolution model for the
crack potential energy available inside the substrate – TBC systems, that could be release
as plastic deformation or surface energy, where the surface energy is related directly to
crack nucleation and growth, which conduct in most cases to a catastrophic failure of the
thermal protection, or in most critical cases to a failure of the substrate.
The proposed evolution model for the wear mechanisms, is schematically developed to
obtain from the operating conditions the actual loads over the hot section components
inside the gas turbine, this development allows a fast identification of the most problematic
components, related in general to stagnation points, maximum velocities and highest
temperatures through the hot gas path, where most detailed analysis are carried out to
estimate delamination and spallation of the thermal protection, which in a general sense
becomes into a hot gas path life assessment.
Thesis - Pablo Gomez Page 7/1382 Thermoelectric Power Generation and Gas Turbines
Through this chapter the relation between the operating conditions and gas turbine loads
is reviewed, considering the electric energy demand and main restrictions of the
Colombian interconnected system.
2.1 Thermoelectric power generation in Colombia
The energetic crisis in Colombia from March 1992 to April 1993, caused by the
phenomenon of El Niño, which affects levels of the hydraulic reservoirs in the Colombian
electrical system, leads a transformation of power generation scheme in Colombia.
From this crisis the government decides to improve and diversify the power generation
scheme, noting that actually the highest fraction of hydraulic power generation increased
the vulnerability of the interconnected system. One of the most important modifications to
the power generation scheme comes from the 142 and 143 constitutional laws
promulgated in 1994, which allowed the participation of private agents in the power
generation scheme and proposed the separation of the business units as: generation,
transmission, distribution, and commercialization in order to develop a competitive
wholesale market, including adequate market regulations [52].
In 1991 the installed capacity used to be 78 % from hydraulic resources and 22 % from
thermal resources ( hydraulic against thermal). Attending the market
needs of this moment, of thermal power generation and where included to
the interconnected system between 1997 and 1998, stimulated by a new subsidiary
system named capacity charge, which pays to power generation agents according to the
installed capacity in order to ensure a thermal power reserve in case of climate adversities;
in 2009 the new installed capacity achieved a new and reliable distribution 67 % hydraulic
and 33 % thermal ( hydraulic against thermal), which up today have
avoided electricity rationing caused by El Niño (1997-1998 and 2010) [52].
In 2006 the capacity charge was updated, the new subsidiary system named reliability
charge, redefined the conditions of the payments received by generation agents
supporting the interconnected system, this new charge ensure the pays considering not
the installed capacity but the real available capacity, which includes the fuel reserves, this
new scheme, ensure payments for more real power capacity [52].
The distribution of hydraulic and thermal (gas and coal) power generation per generation
agent by 2011 in the Colombian interconnected system is showed in Fig 2.1, 2.2 and 2.3.
Thesis - Pablo Gomez Page 8/138Fig 2.1. Distribution of hydraulic power generation in Colombia by company [53].
Fig 2.2. Distribution of gas power generation in Colombia by company [53].
Thesis - Pablo Gomez Page 9/138Fig 2.3. Distribution of coal power generation in Colombia by company [53].
Considering the gas power generation, which is mainly based on land gas turbines, is
important to highlight that according to Colombian power requirements the operating
conditions of these turbines could be classified in to main categories; the first one, related
to these turbines located on areas with less hydraulic resources and vulnerable energy
transmission lines, where the thermal power plants operate at base load or some cases at
peak load to cover the peaks of demand over the year, which are the common design
operating conditions for land gas turbines and use to be in accordance to the OEM
maintenance recommendations, strongly supported by statistical data. The second is
related to these turbines located on areas with high hydraulic resources, where power
plants use to operate only during operational tests and climate adversities, related to El
Niño, leaving those power plants on more complex maintenance sceneries.
In addition, over the last years the reduction on gas reserves and a growing uncertainty
about the gas market in Colombia propose a new variable that must be considered, it is
the use of alternative fuels and the change between different fuels over operation to cover
the power demand.
The highly unsteady operating scenario of those gas turbines acting as a support of the
Colombian interconnected system comes from the combination of fuels and continuous
load changes to support the variable power demand, affecting the maintenance programs,
and leaving the operation of those gas turbines out of OEM recommendations.
A number of findings during major inspections of units operating at non-steady conditions
suggest in general, that such conditions are responsible for earlier catastrophic damage in
hot section components such as stationary nozzles, transition pieces and mobile blades,
including thermal protection delamination and spallation, deep cracks even affecting the
substrate; and geometrical distortions associated to thermal stress; which result more
probably during gas turbine trips, something that is very common during gas turbine
load/unload and fuel changes compare to steady state operation. Table 2.1 shows some
findings during combustion (CI) and hot gas path (HGPI) inspections on gas turbine at
non-steady operating conditions.
Thesis - Pablo Gomez Page 10/138Table 2.1. Gas turbine common findings during combustion (CI) and hot gas path (HGPI)
inspections at non-steady operating conditions.
7FA Gas Turbine Common Findings
First stage nozzles (CI 2005 – U1)
Location: pressure side
Number of starts: 424
Number of trips: 174
Firing hours: 5691
First stage nozzles (CI 2005 – U1)
Location: pressure side
Number of starts: 424
Number of trips: 174
Firing hours: 5691
First stage nozzles (CI 2007 – U2)
Location: trailing edge
Number of starts: 413
Number of trips: 160
Firing hours: 6062
First stage nozzles (CI 2007 – U2)
Location: pressure side
Number of starts: 413
Number of trips: 160
Firing hours: 6062
First stage blades (HGPI 2010 – U1)
Location: pressure side
Number of starts: 676
Number of trips: 268
Firing hours: 14733
Thesis - Pablo Gomez Page 11/138Transition pieces (HGPI 2012 – U2)
Location: right side close to first stage
connection
Number of starts: 222
Number of trips: 98
Firing hours: 7843
First stage nozzles (HGPI 2012 – U2)
Location: base close to trailing edge
Number of starts: 222
Number of trips: 98
Firing hours: 7843
First stage nozzles (HGPI 2012 – U2)
Location: leading edge
Number of starts: 222
Number of trips: 98
Firing hours: 7843
First stage nozzles (HGPI 2012 – U2)
Location: trailing edge
Number of starts: 222
Number of trips: 98
Firing hours: 7843
First stage nozzles (HGPI 2012 – U2)
Location: trailing edge
Number of starts: 222
Number of trips: 98
Firing hours: 7843
Thesis - Pablo Gomez Page 12/138First stage blades (HGPI 2012 – U2)
Location: leading edge
Number of starts: 635
Number of trips: 258
Firing hours: 13905
First stage blades (HGPI 2012 – U2)
Location: leading edge
Number of starts: 635
Number of trips: 258
Firing hours: 13905
First stage blades (HGPI 2012 – U2)
Location: pressure side and blade tip
Number of starts: 635
Number of trips: 258
Firing hours: 13905
The Commission of Regulation of Energy and Gas (CREG) in Colombia established a
group of declared operating parameters, which allow to generation agents, define the
adequate operating conditions of the gas turbines, considering the capabilities and
limitations of the different technologies, these operating parameters are shown in Table
2.2.
Table 2.2. Declared operating parameters [CNO Agreement 277].
Variable Description Units
Nominal Capacity Maximum power output at standard
conditions.
Real Capacity Maximum power output at site
Thesis - Pablo Gomez Page 13/138conditions.
Unit type Gas turbine, steam turbine, power
boiler and others.
Power Plant Configuration Simple or combine cycle,
cogeneration, other.
Main fuel Gas, fuel oil, coal and others.
Alternative Fuel Gas, fuel oil, coal and others.
Loading/Unloading Rate Maximum allowable power variation
per unit of time.
Synchronization load Power output after unit
Synchronization.
Start types Depend on offline time and could be
cold, warm, or hot.
Start Notification Time Minimum notification time before
unit start.
Warm-up Time Required before unit
synchronization.
Minimum Generation Time Minimum online time after unit start.
Maximum Number of Starts per Day Allowable number of starts for a
single day.
Minimum Stable Load Time Minimum time at constant load.
Maximum Load Variation per Hour Allowable output variation per hour
Minimum Offline Time (Shutdown) Minimum offline time after a normal
unit shutdown.
Minimum Offline Time (Trip) Minimum offline time after a unit trip.
Minimum Load Minimum power output for a single
unit.
Power Output Restrictions Intervals of possible power output
restrictions, according to the specific
technology.
Considering these parameters in Table 2.2 and according to fuel availability and energy
demand, the required delivery of power for each single day and the required operation
conditions from every power plant in the interconnected system is defined, so this
parameters provide a reference of the way gas turbine are operated in the Colombian
interconnected system and the thermomechanical cycles experimented by this units.
The statistics of EPM thermal power generation is presented in Table 2.3, it is important to
highlight the relation between power generation and the phenomenon of El Niño.
Table 2.3. EPM annual power generation.
Year Number of starts Number of trips Operation hours
Unit 1 Unit 2 Unit 1 Unit 2 Unit 1 Unit 2
1998 138 93 49 42 2429 1225
1999 32 25 12 15 56 33
2000 84 80 48 33 145 179
2001 55 50 21 22 734 657
2002 55 59 21 25 506 572
2003 36 24 15 3 625 681
Thesis - Pablo Gomez Page 14/1382004 24 23 8 4 1196 1256
2005 32 23 13 6 1205 1143
2006 38 36 9 10 332 316
2007 86 81 38 45 1175 826
2008 23 18 2 2 535 956
2009 48 30 20 12 3713 1638
2010 25 30 12 9 2082 3852
2011 59 47 18 26 162 176
2012 23 16 10 4 390 395
Is important to note from Table 2.3 the relation between operating hours and number of
trips, excluding 1998 (units commissioning), the years with the highest operating hours
also have the minimum number of trips, this just because the units remain for relative long
periods of time at stable loads.
Between 2003 and 2008 the local energy demand in Colombia grow at 3.31 % per year
average. Considering the global demand of energy, which include national demand and
energy exportations to Peru, Ecuador and Centro America, the UPME estimates the
requirement of to 2017, this required increase in installed capacity reflects the
good behavior of the economy over the last years, motive by the growth of miner,
transport, industry and other productive sectors in Colombia and the region [53].
On the other hand the faster installation of thermal power plants, which take advantage of
its modular construction, makes the installation of such plants pretty attractive in order to
cover the growing demand on time, but at the same time, increases the number of power
units submitted to variable operating conditions, making critic the understanding and
prediction of its wear behavior.
2.2 Gas turbines
The gas turbine is a rotary engine with a low load/power relation that extracts energy from
a flow of combustion gases. It has an upstream compressor coupled to a downstream
turbine, and a combustion chamber in between.
In the gas turbine the air is forced trough the filtration system inside the compressor,
energy is added to the gas stream in the combustor, where fuel is mixed with air and
ignited, high temperature and pressure combustion products are forced into the turbine
section. There, the high velocity and volume of gas flow is directed trough a nozzle over
the turbine blades [1].
Energy is extracted in the form of shaft power, compressed air and trust, in any
combination from gas turbines. The two major application areas of gas turbine engines are
aircraft propulsion and electric power generation.
Gas turbines are used to generate electricity as stand-alone units or in conjunction with
steam power plants on the high-temperature side. In these plants, the exhaust gases
serve as a heat source for the steam. Steam power plants are considered external-
combustion engines, in which the combustion takes place outside the engine. The thermal
energy during this process is then transferred to the steam as heat.
Thesis - Pablo Gomez Page 15/138John Barber patented the basic gas turbine in 1791. The gas turbine first successfully ran
in 1939 at the Swiss National Exhibition at Zurich. The early gas turbines built in the 1940s
and even 1950s had simple-cycle efficiencies of about 17 %. This was because of low
compressor and turbine efficiencies and low turbine inlet temperature due to metallurgical
limitations at the time. The first gas turbine for an electric utility was installed in 1949 in
Oklahoma as part of a combined-cycle power plant. It was built by General Electric and
produced of power [1].
Fig 2.4. GE MS7001FA Gas turbine assembly major section [3].
Fig 2.4 shows the GE MS7001FA gas turbine assembly major section, from left to right: air
Inlet, compressor, combustors, turbine and exhaust.
2.2.1 Compressor
The compressor is designed to increase the pressure of incoming gas stream. There are
two kinds of compressor commonly used in gas turbines: centrifuge, and axial flow. Inside
the compressor a mechanical work is applied to the gas stream increasing its pressure
and density as consequence of the volume reduction while the temperature of the gas
stream increases, which is not desirable.
The pressure ratio, which is defined as the ratio between the output and input pressure
trough the compressor, is the most important design consideration for a turbine
compressor, the pressure ratio is directly relate with the maximum efficiency of the gas
turbine.
Thesis - Pablo Gomez Page 16/138Centrifugal compressor is preferred for application when a large tolerance of processes
fluctuation is required, it is commonly use for small gas turbines, the pressure ratio of a
single stage centrifugal compressor varies from 3:1 to 12:1, in this compressors, the air is
forced trough the impeller by a rapidly rotation of the impeller blades, the high velocity of
the air is converted into pressure energy in the diffusers, which consist of vanes aligned
tangentially to the impellers, the vanes passages diverge to convert the velocity into
pressure energy, these compressors provide a high pressure, but a low mass flow [1]. Fig
2.5, shows the pressure and velocity trough the centrifugal compressor.
Fig 2.5. Pressure and velocity trough the centrifugal compressor [1].
The most efficient region for centrifugal compressor operation is between 60 and 1500
revolution per second , angular velocities over required axial compressors
[1].
The axial compressor compresses the air by first accelerating the fluid and then diffusing it
to increase the pressure. The flow is accelerated by a rotating blades row called the rotor
and then diffused in a stationary blades row called stator to increase the pressure. Axial
compressor consist of several stages, one rotor and one stator conform a compressor
stage. The axial compressor stage pressure ratio is between 1.1:1 and 1.4:1, which
causes a slightly variation of the temperature, allowing higher efficiencies, the use of
multiple stages permit overall pressure increases of up to 40:1 [1].
Increases in fluid pressure trough the axial compressor comes from rotor or a combination
between the rotor and stator, depending of the degree of reaction, which is define as the
ratio of the change of static head in the rotor to the head generated in the stage, for a
multiple stage axial flow compressor it is preferred to have a constant enthalpy increase
than a constant pressure increase for each stage [1].
The compressor blades are curved, convex in one side and concave in the other, with the
rotor rotating toward the concave side, called pressure side, the convex side is called the
suction side. The axial compressor blades length and the annulus area, which is the area
between the shaft and the shroud, decreases trough the length of compressor, this
Thesis - Pablo Gomez Page 17/138reduction in flow area compensates for the increase in fluid density as it is compressed,
permitting a constant axial velocity [1].
Fig 2.6 shows the variation of enthalpy , velocity and pressure trough an axial
compressor, the enthalpy is obtained as a function of pressure and temperature .
Fig 2.6. Variation of enthalpy, velocity, and pressure trough an axial compressor [1].
The GE MS7000FA gas turbine is equipped with an eighteen stages axial compressor
which provides three millions of pounds of air each hour to the gas turbine, about 60 % of
this air is used just for cooling, causing enormous lost of efficiency.
The compressor uses between 55 and 60 % of the turbine power output to compress the
incoming air stream.
2.2.2 Combustors
Into the combustors, chemical energy from the fuel is transformed in thermal energy of gas
stream. The fuel and air mix is controlled according to power requirements; commonly two
kinds of combustion will be found inside the combustors: difusional combustion and premix
combustion.
The main purpose of the combination of difusional and premix combustion is control the
flame length inside the combustor, allowing complete fuel combustion and low NO
x
emissions; the difusional flame is commonly longer than the premix flame, but the mass
flow variation between the start and the base load, require a combination of both kinds
Thesis - Pablo Gomez Page 18/138along the loading ramp, considering that, the geometry of the combustors most of the
times is fix.
The combustors in land gas turbines are organized annularly; each combustor is
connected to the adjacent by cross fire tubes, these tubes allow the ignition in the same
direction of the blades rotation.
Fig 2.7. MS7001FA DLN 2 combustor assembly [3].
Only one-third or less of the compressor discharge air, is stoichio-metrically burned with
the incoming fuel, the rest of the compressor discharge air, is used for cooling. Fig 2.8
shows the combination of different difusional and premix combustion modes for a
MS7001FA DLN 2 gas turbine.
Thesis - Pablo Gomez Page 19/138Fig 2.8. MS7001FA DLN 2 gas turbine combustion modes [3].
2.2.3 Turbine section
The turbine section is actually the most efficient part of a turbine; it is pretty common to
have thermal efficiencies over 95 %. However, most of the cooling air produced by the
compressor is employed to cooling the turbine at allowable temperatures for the turbine
components materials.
There are two kinds of gas turbines: the first one, is the radial-inflow turbine, which is like a
centrifugal compressor with a reverse flow, the second is the axial flow turbine.
The radial-inflow turbine is derived from the hydraulic turbines, this turbine is a natural
choice to combine with centrifugal compressor, such systems are of high interest for actual
developments in transport and petrochemical industries because the cost is much lower
than a single stage axial flow turbine and provide a higher output for a single stage, only
small power plants use these turbines. However, the efficiency of radial-inflow turbines is
less than this of axial flow turbines.
The axial flow turbines are the most widely employed turbine using a compressible fluid;
axial flow turbines powered most of the power gas turbines, because those turbines are
much more efficient than radial inflow turbines in most operational ranges, which are
desired for many applications [1].
Thesis - Pablo Gomez Page 20/138Major parts of the axial flow turbine section include the stationary nozzles and the mobile
blades Fig 2.9, the nozzles orients the combustion gas stream to the blades, the
aerodynamic surface of the blades recover the energy in the stream and transform this
energy into angular moment of the rotor; the action turbine has its entire gas stream
enthalpy drop in nozzle, therefore it has a very high velocity entering the rotor, on the other
hand the reaction turbine divide the enthalpy drop between the nozzle and the rotor.
Most of the axial gas turbine use to have a pure action first stage, and 50 % reaction for
the rest. The action stages produce about twice the output of a 50 % reaction stage.
However, 50 % reaction stage efficiency is more than that of the action stage [1].
Fig 2.9. Variation of enthalpy, velocity, and pressure trough an axial flow turbine [1].
2.3 Brayton cycle: the ideal cycle for gas turbines
The basic gas turbine cycle is named for the Boston engineer, George Brayton, who first
proposed the cycle around 1870. The Brayton cycle is used for describes the ideal
operation of a gas turbines only where both the compression and expansion processes
take place in rotating machinery.
The open gas-turbine cycle can be modeled as a closed cycle as shown in Fig 2.10 by
utilizing the air-standard assumptions. Here the compression and expansion process
remain the same, but a constant-pressure heat-rejection process to the ambient air
replaces the cooling process [1].
Thesis - Pablo Gomez Page 21/138Fig 2.10. The air standard Brayton cycle [2].
The ideal Brayton cycle is described in Fig 2.10, the working fluid process, which is made
up of four internally reversible processes:
1-2 isentropic compression (in a compressor)
2-3 constant pressure heat addition (in a combustor)
3-4 isentropic expansion (in a turbine)
4-1 constant pressure heat rejection
A simplified application of the first law or first postulate of thermodynamics to the air
standard Brayton cycle, assuming no changes in kinetic and potential energy, has the
following relation:
Work of compressor:
(2.1)
Work of turbine:
(2.2)
Where is the mass flow rate of air, is the mass flow rate of fuel, and is the
enthalpy.
Thesis - Pablo Gomez Page 22/138Total output work:
(2.3)
Heat added to the system:
(2.4)
Where is the heat of combustion of the fuel.
Thus, the overall cycle efficiency is:
(2.5)
Assuming ideal gas and considering :
(2.6)
(2.7)
Additionally for ideal gas, the internal energy and enthalpy , could be written as:
(2.8)
(2.9)
And the heat capacity ratio, between the specific heat capacity at constant volume and
constant pressure:
(2.10)
Considering , and Eqn 2.9 thus:
(2.10)
Where considering Eqn 2.6:
(2.11)
Integrating Eqn 2.11:
(2.12)
Solving Eqn 2.12:
Thesis - Pablo Gomez Page 23/138(2.13)
Considering Eqn 2.12:
(2.14)
Thus:
(2.15)
From Eqn 2.12 and considering Carnot’s efficiency is possible to write:
(2.16)
Where is the pressure ratio.
The efficiency of a gas turbine could be increased by increasing the pressure ratio, adding
inter-stage cooling between compressor stages or recovering energy from the exhaust
combustion gases to the incoming combustion air.
Increasing the pressure ratio increases the efficiency of the Brayton cycle according to
Eqn 2.16. However, there are practical limits to increasing the pressure ratio. First,
increasing the pressure ratio increases the compressor discharge temperature. This can
cause the temperature of the gasses reach the metallurgical limits of the turbine section.
Second, the diameter of the compressor blades becomes progressively smaller in higher
pressure stages of the compressor, this increase the air leakage through the compressor
blade tips in higher pressure stages, affecting compressor efficiency, on the other hand,
increasing the pressure ratio becomes expensive considering the required materials and
manufacturing processes [1].
Cooling the compressed air after a compression stage makes possible to reduce the
required work in the next compression stage; this is a direct consequence of the specific
volume reduction during cooling process. Also, an increase in the maximum pressure ratio
is caused due to reduced compressor discharge temperature for a given amount of
compression, improving overall efficiency [1].
In regeneration process, where post-turbine fluid passes through a heat exchanger to pre-
heat the fluid that enters the combustion chamber, efficiency is improved as a
consequence of optimization of fuel consumption for the same operating conditions and
reduction in heat losses. However, at higher pressure ratios, the compressor discharge
temperature can exceed the exhaust temperature, in which case regeneration would be
counterproductive. On the other hand, for combined cycles most times it is better to take
advantage of the exhaust gas stream temperature directly in the boiler [1].
Thesis - Pablo Gomez Page 24/1382.4 Gas turbine loads
Since the gas turbine is considered constant speed rotary engine once the nominal speed
is reached, the power output is directly related to the air and fuel mass flow through the
turbine, it is considering the non-variable geometry of the gas turbine the gases streams
together with combustion reaction defined the profiles of temperature, pressure and
velocity inside the compressor, combustors and turbine section.
The air flow admitted into the compressor is defined by the position of inlet guide vanes
(IGV’s), this vanes modify the admission area affecting the amount of incoming air, the mix
of air and fuel is controlled by the combustion system, which is design to provide the right
amount of fuel and air according to desired load, this system also control the contaminant
emissions adjusting the relation between the different fuel sources in the combustors
Section 2.2.2.
The most relevant parameter to determine the maximum power output is the firing
temperature, which is calculated as a function of the exhaust temperature and give an
indication of the maximum temperature at combustors, this temperature is determined by
materials properties inside the turbine, Fig 2.11 shows the firing temperature of a common
startup – base load – shutdown cycle.
Fig 2.11. Fairing temperature for a gas turbine start/stop cycle [4].
However, the stresses at hot section are strongly related to heating and cooling times as
shown in Fig 2.12 in the case of the first stage blades of a 7FA gas turbine.
Thesis - Pablo Gomez Page 25/138Fig 2.12. First stage blades thermal stress at leading edge [4].
Form Fig 2.11 and Fig 2.12, it is important to properly define the operating cycles to
determine adequately the hot section component loads where the temperature, pressure,
and velocity, are the main system variable of the load functions, which depend directly on
fuel and air mass flow. Also the inertial forces must be considered at mobile hot section
components to complete the analysis.
In gas turbines, some operating variables are monitored during operation to control the
process; these variables provide information about the loads inside the gas turbine as a
function of the power output and allow the analysis of the thermal cycle, some of the most
important process variables are: the power output , the angular speed
, the compressor discharge pressure , the compressor temperature
discharge , the ignition temperature , and the mean exhaust
temperature .
Fig 2.13 shows some of process variables on a typical four hour loading up to base load,
four hour steady base load, two hours unloading and shutdown cycle, obtained directly
from an EPM 7FA gas turbine using the General Electric Speed Tronic Mark V control and
data acquisition system, used to operate and monitor the EPM gas and steam turbines.
Thesis - Pablo Gomez Page 26/1384500
4000
3500
3000
2500
2000
1500
1000
500
0
0 Min 100 Min 200 Min 300 Min 400 Min 500 Min 600 Min 700 Min
-500
DWATT TNH CPD CTD TTREF TTXM
Fig 2.13. Main variables on a typical operating cycle of an EPM 7FA gas turbine.
However as shown in Fig 2.12, when a trip occur, the fast cooling of the hot section
components increases the magnitude of the stresses, that is the reason of a faster crack
growth during abnormal shutdowns, according to this, gas turbine manufacturers include a
severity factor to account the trips in maintenance intervals calculations, this severity
factors used to be expressed in terms of the load of the turbine before the trip.
Fig 2.14 shows the differences during cooling process for a normal shutdown and a trip
from base load.
1600°C
1400°C
1200°C
1000°C
800°C
600°C
400°C
200°C
0°C
200 Min 250 Min 300 Min 350 Min 400 Min 450 Min 500 Min 550 Min 600 Min 650 Min
TTREF Shutdown TTREF Trip
Fig 2.14. Comparison between normal shutdown and trip.
Thesis - Pablo Gomez Page 27/1383 Gas Turbine Materials and Wear Mechanisms
Through the first part of this chapter the gas turbine materials and coatings are reviewed,
considering the most important design criteria and limits, while the relation between
materials and operation conditions is also analyzed.
In the second part the wear mechanisms involved in gas turbine operation, are reviewed,
showing the correlations between materials properties and wear.
3.1 Advanced gas turbine materials and coatings
The development of today’s gas turbine engines has been the result of continual
improvements in a wide variety of engineering skills including turbine design, combustion
studies, and development of novel materials. [14].
In particular, the production of advanced materials for the manufacture of the hot section
components of the gas turbines has always been linked to increase both the power output
and the thermal efficiency of these machines. Since the emergence of Fe, Co and Ni-base
superalloys in the 1950’s, and the development of the first applications of these materials
to the manufacture of components for aircraft engines, an aggressive race to produce
materials capable to work at higher temperatures started, which yielded a significant
increase in power outputs and thermal efficiencies.
Particularly, the increase in airfoil temperature has been facilitated by three principal
materials developments: dramatic advances in alloy design to produce alloy compositions
that are both more creep resistant and oxidation resistant; advances in casting technology
that have facilitated not only the casting of large single-crystal superalloy blades and
vanes but also the intricate internal channels in the blades to facilitate cooling; and the
development of a viable coating technology to deposit a conformal, thermally insulating
coatings on turbine components [14].
About thirty years ago began the development and application of ceramic thermal barrier
coatings (TBCs) on superalloys, resulting in a dramatic increase of about in the
ignition temperature of gas turbines [2]. Fig 3.1 shows how, through the development of
manufacturing processes and materials, higher blade temperatures have been achieved.
Thesis - Pablo Gomez Page 28/138Fig 3.1. Historical evolution of materials for gas turbine blades. [Chromalloy UK Ltd.;
Materials World March 2000].
Nowadays many of the gas turbine components in the hot section are manufactured using
compound materials; nickel-based superalloy are commonly used as substrates and yttria-
stabilized zirconia (YSZ) are used as thermal barrier coating (TBC) to increase the
allowable work temperature of the gas turbine components and protect the substrate
against corrosion and creep.
3.1.1 Nickel-based superalloys
A superalloy is an alloy that can be used at high temperatures, often in excess of 70 % of
the absolute melting temperature. Creep and oxidation resistance are the prime design
criteria. Superalloys can be based on iron, cobalt or nickel, the latter being best suited for
aeroengine applications [11].
Nickel-based superalloys are usually either solid solution strengthened or precipitation
strengthened. Solid solution strengthened alloys such us Hastelloy X, are used for
applications requiring moderate strength. Precipitation strengthened alloys are used in the
most demanding applications, such as hot sections of gas turbine engines. Precipitates
strengthen the alloy by impeding the movement of dislocations and, therefore, the
deformation under load. Most wrought Ni-base superalloys contain 10 to 20 atom-% of
chromium, up to 8 atom-% of aluminum and titanium, 10 to 20 atom-% of cobalt and small
amounts of beryllium, zirconium and carbon; other commonly added elements are
molybdenum, tungsten, tantalum, hafnium and niobium [55 and 56].
Thesis - Pablo Gomez Page 29/138However, the essential solutes in nickel-based superalloys are aluminum and/or titanium,
with a total concentration which is typically less than 10 atom-%. This generates a two-
phase equilibrium microstructure, consisting of gamma (γ) and gamma-prime (γ'). It is the
γ' which is largely responsible for the elevated-temperature strength of the material and its
incredible resistance to creep deformation. The amount of γ' depends on the chemical
composition and temperature, as illustrated in the ternary phase diagrams below, Fig 3.2
[11].
Fig 3.2. Ni, Ti, Al ternary phase diagram [11].
The Ni-Al-Ti ternary phase diagrams show the γ and γ' phase field in Fig 3.2. For a given
chemical composition, the fraction of γ' decreases as the temperature increases. This
phenomenon is used in order to dissolve the γ' at a sufficiently high temperature (a
solution treatment) followed by ageing at a lower temperature in order to generate a
uniform and fine dispersion of strengthening precipitates [11].
The continuous γ-phase is a solid solution with a cubic-F lattice and a random distribution
of the different species of atoms such as cobalt, chromium, molybdenum, tungsten,
hafnium, titanium and rhenium Fig 3.3. Cubic-F is short for face-centered cubic [11].
Thesis - Pablo Gomez Page 30/138Fig 3.3. γ and γ' phase crystalline structure [11].
By contrast, the principal strengthening phase is the γ, with a primitive cubic lattice in
which the nickel atoms are at the face-centers and the aluminum or titanium atoms at the
cube corners Fig 3.3. This atomic arrangement has the chemical formula Ni Al, Ni Ti or Ni
3 3 3
(Al, Ti). However, as can be seen from the (γ+γ')/γ' phase boundary on the ternary
sections of the Ni, Al, Ti phase diagram, the phase is not strictly stoichiometric. An excess
of vacancies on one of the sublattices may exist which leads to deviations from
stoichiometry; alternatively, some of the nickel atoms might occupy the Al sites and vice-
versa. In addition to aluminum and titanium, niobium, hafnium and tantalum segregate
preferentially into γ' [11].
The γ phase forms the matrix in which the γ' particles precipitates. Since both the phases
have a cubic lattice with similar lattice parameters, the γ' precipitates in a cube-cube
orientation relationship with the γ. This means that its cell edges are exactly parallel to the
corresponding edges of the γ phase. Furthermore, because their lattice parameters are
similar, the γ' is coherent with the γ when the precipitate size is small. Dislocations in the γ
nevertheless find it difficult to penetrate γ', partly because the γ' is an atomically ordered
phase. The order interferes with dislocation motion and hence strengthens the alloy [11].
The small misfit between the γ and γ' lattices is important for two reasons. Firstly, when
combined with the cube-cube orientation relationship, it ensures a low γ/γ' interfacial
energy. The ordinary mechanism of precipitate coarsening is driven entirely by the
minimization of total interfacial energy. A coherent or semi-coherent interface therefore
makes the microstructure stable, a property which is useful for elevated temperature
applications [11].
The magnitude and sign of the misfit also influences the development of microstructure
under the influence of a stress at elevated temperatures. The misfit is said to be positive
when the γ' particle has a larger lattice parameter than γ. The misfit can be controlled by
altering the chemical composition, particularly the aluminum to titanium ratio. A negative
misfit stimulates the formation of rafts of γ', essentially layers of the phase in a direction
Thesis - Pablo Gomez Page 31/138normal to the applied stress. This can help reduce the creep rate if the mechanism
involves the climb of dislocations across the precipitate rafts [11].
Large fraction of γ', typically in excess of 0.6, is used in turbine blades designed for aero
engines, where the metal experiences temperatures in excess of . Only a small
fraction (0.2) of γ' is needed when the alloy is designed for service at relatively low
temperatures and where welding is used for fabrication [11].
Carbides are also an important constituent of wrought and polycrystalline cast superalloys.
When carbon is added at levels of 0.05 - 0.2 wt-%, it combines with reactive and refractory
elements such as titanium, tantalum, niobium and hafnium to form the carbides TiC, TaC,
NbC or HfC. During heat treatment and service these form lower carbides such as M C
23 6
and M C (where M designates 1 or more type of metal atoms) on the grain boundaries,
6
strengthening these boundaries, especially at temperatures near where are the
main phases of equilibrium and the M C , M C and M C that are less common. These
7 3 3 13
carbides all have FCC crystal structures [55].
MC carbides are compounds of FCC structure, highly stable and of high resistance,
heterogeneously distributed throughout the alloy in the form of elongated precipitates.
These carbides tends to disappear altogether over 5000 hours of operation at
temperatures greater than degraded to form M C carbides, which normally
23 6
precipitates along the grain boundaries, as small particles whose size progressively
increasing as service time elapses. M C carbides precipitate in the grain boundaries when
6
reaching temperatures are between and [55].
The addition of grain boundary elements such as boron, zirconium and hafnium can, within
limits, lead to significant improvements in mechanical properties in these wrought alloys.
The presence of these elements may modify the initial grain boundary carbides or tie-up
deleterious elements such as lead and sulphur [55 and 56].
3.1.2 Thermal Barrier Coatings (TBCs)
The primary function of a TBC is to provide a low thermal conductivity barrier to heat
transfer from hot gas stream to thermal engine components, the TBC must be considered
as a multilayered system, rather than a thermal insulation coating on structural alloy
component [14]. The TBC is composed basically by three layers, the first one, named
bond coat (BC) is applied directly on the superalloy, to provide an adequate substrate for
the second layer, named top coat (TC) which provide the required thermal insulation, the
third layer is formed as consequence of high temperature and oxidizing atmosphere under
operating conditions, this layer is a thermally grown oxide (TGO), which is formed from the
BC. An additional layer is formed between the BC and the superalloy, this is a reactive
layer formed by inter-diffusion between the BC and the superalloy.
TBC coatings, usually consist of an oxidation resistant layer applied onto an nickel-based
superalloy and a YSZ ceramic coating applied over this last one, the oxidation resistant
layer named bond coat (BC) is designed to combined the chemical and thermomechanical
properties of both: substrate and ceramic coating, providing a “soft transition” between the
metal and the ceramic, the BC comes from one of the two existent classes: the platinum-
Thesis - Pablo Gomez Page 32/138modified nickel aluminide (PtNiAl) or MCrAlY alloys (M here refers to one or more of the
elements Co, Ni, and Fe). On the other hand the YSZ ceramic coating named top coat
(TC) may vary between 5 and 15 wt-% of Y O .
2 3
Between the TC and the BC grow an additional layer named thermally grown oxide (TGO),
this layer grows as a response to the high temperature and oxidizing atmosphere which is
commonly found in gas turbines operation conditions. The TGO grown is favored by the Al
available in the BC, so it is designed to form as a layer of α-Al O , which is desired when
2 3
its stability, microestructural compatibility with the YSZ coating and the additional
protection provided to the substrate is considered.
Fig 3.4. Structure of a multilayer TBC coating applied by air plasma spray APS (a) [35],
and electron beam physically vapor deposition EB-PVD (b) [9].
Fig 3.4 shows the common structure of a multilayer TBC coating applied by air plasma
spray APS (a), and electron beam-Physical vapor deposition EB-PVD (b). The EB-PVD
coating have a columnar structure caused by the directional cooling and solidification from
the substrate, an a controlled porosity volume fraction to reduce conductivity, some
columns are interconnected laterally as a consequence of the rotation of the component
during the coating deposition, this interconnection causes lateral strain compliance. The
APS coating structure, on the other hand, is formed by small liquid drops impacting the
substrates, so the lateral strain compliance and the reduced thermal conductivity are
conferred by incorporation of porosity between splats of successively deposited material.
Fig 3.5, shows the thermal expansion coefficients and thermal conductivity of a range of
materials illustrating the differences in thermal expansion and conductivity of the principal
components in TBC systems [14], one of the most important design criteria for TBC, is the
thermal compatibility of the different layers, it is very important to keep the thermal
expansion coefficients of the TC, BC and TGO close to that of the nickel-based superalloy
used for a specific substrate – TBC system, in order to avoid high thermal stresses during
the thermal cycles.
Thesis - Pablo Gomez Page 33/138Fig 3.5. The thermal expansion coefficients and thermal conductivity of a range of
materials illustrating the differences in thermal expansion and conductivity of the principal
components in TBC systems [29].
3.1.2.1 Top Coat (TC)
The top coat (TC) is the main responsible of the increase of the service temperature of
actual thermal engines, allowing service temperatures between and higher
than uncoated thermal engines, without exceeding the limits of underlying nickel-based
superalloys; such increases have powered the develop of more efficient and powerful
thermal engines.
The most common TC material is the yttria stabilized zirconia YSZ, which provides low
thermal conductivity, high stability under operating conditions, high thermal expansion and
toughness, [31]. The amount of yttria (Y O ) in YSZ varies between 5 and 15 wt-% on a
2 3
TC. For gas turbine components the most commonly used TC is zirconia stabilized with 7
– 8 wt-% of yttria Fig 3.6; the yttria content stabilize the high temperature form of the
zirconia at low temperatures, this allows a full range between low and high temperatures
without phase transformation which is very important considering the volumetric changes
experienced by ZrO as consequence of the phase transformations during common
2
thermal cycles [13].
Thesis - Pablo Gomez Page 34/138Fig 3.6. Phase diagram for YSZ [13].
The pure ZrO has a monoclinic structure at room temperature and maintains this structure
2
until when it transforms into its tetragonal form. The tetragonal structure is stable
until when it becomes cubic and remains that way until melting temperature at
[36]. In practical applications these temperatures are not verified due to the
presence of small amount of contaminants in the ZrO , which reduce rapidly the
2
transformation temperature between the monoclinic and tetragonal forms. In addition, the
mechanical properties of the monoclinic structure are not really interesting for common
thermal engine applications, so many studies have been made to find a way stabilize the
tetragonal form of the ZrO at room temperature, avoiding microstructural transformation
2
over the operation range for common gas turbines and considering the mechanical
properties of the tetragonal form.
It is also worth noticing that the transformation from tetragonal to monoclinic structure in
pure ZrO is accompanied by a volumetric change at room temperature [36]. This a
2
reversible athermal martensitic transformation with a finite amount of volume change,
which causes high residual stresses, the main consequence of this volumetric instability is
the nucleation and growth of cracks at low externally applied stresses Fig 3.7.
Thesis - Pablo Gomez Page 35/138Fig 3.7. Volumetric change from tetragonal to monoclinic on pure ZrO [x].
2
Not only the addition of Y O stabilizes the tetragonal form of the ZrO at room
2 3 2
temperatures, other solid solutes like Hf, CaO and MgO can do it.
The mechanism of transformation toughening in ZrO involves two aspects, both related to
2
the difficulty of nucleating the martensitic transformation. First, the martensite start
temperature of tetragonal ZrO is lowered to below room temperature by controlling the
2
stabilizer concentration in this phase. Even more importantly, the martensite start
temperature of the particle or grain size of this phase decreases with decreasing size [48].
The highest strength and toughness promoted by the addition of yttrium oxide produce a
fully or partially stabilized zirconia; these materials consist of a full tetragonal or a mixture
of tetragonal and cubic phases with a bending strength of up to . In partial
stabilized zirconia small cracks allow phase transformations to occur, which essentially
close the cracks and prevent catastrophic failure, resulting in a relatively tough ceramic
material, sometimes known as TTZ (transformation toughened zirconia) [48].
TBC coatings deposited by APS or LPPS (low pressure plasma spray) offer a thermal
conductivity significantly lower than that of a fully dense coating , as the
Thesis - Pablo Gomez Page 36/138boundaries and pores tend to lie parallel to the surface and therefore perpendicular to the
temperature gradient [13].
By contrast, the thermal conductivity of EBPVD TBC is not as low , and
lowering it further would imply further benefits according to Schultz (2003). EBPVD TBCs
are nevertheless preferred because of the strain tolerance imparted by the microstructure
[13].
Schulz in 2003, probed that In jet engine operating conditions, the lifetime of TBC coatings
obtained by EBPVD has been reported to be between 8 and 13 times longer than
equivalent system where the TBC was deposited by plasma spray. However, industrial
experience indicates that the superiority of EB-PVD TBCs is not maintained in the
operating conditions of land-base gas turbine, in which APS tends to give the best
performances according to Wells (2004) [13].
3.1.2.2 Bond Coat (BC)
The BC must provide a coherent bond between the superalloy substrate and the thermal
insulation layer. However, there are other important functions of the BC, by itself the BC is
design to form a protective, stable, and slow-growing oxide to prevent oxidative attack of
the substrate, this is the reason of the amount of Al available on it, the BC also may be
thermodynamically and morphologically stable, to avoid undesirable transformation over
the operation range, but at the same time the BC must be mechanically resistant.
Two major classes of bond-coat alloys have been developed to form an aluminum oxide
(Al O ) on exposure to air at high temperatures. The Al O is preferred considering its
2 3 2 3
compatibility with common yttria stabilized zirconia (YSZ) TC, and the formation of a
thermodynamically stable layer of Al O which is considered as the most stable oxide
2 3
rather than complex, multilayered nickel oxide, nickel-chromium spinels and chromium
oxide, commonly formed by nickel-based superalloys. Furthermore, alumina is usually
considered to be the slowest growing high-temperature oxide on account of it having the
smallest oxygen diffusivity [14].
The two classes of bond-coat alloys that have been developed are the platinum modified
nickel aluminide (PtNiAl) and MCrAlY alloys (M here refers to one or more of the elements
Co, Ni, and Fe). The selection of these two classes of alloys is largely base on their prior
use as oxidation- and corrosion-resistant coatings for protecting high-temperature alloys
before the advent of TBCs. For instance, the PtNiAl was originally developed as an
alternative oxidation-resistant coating for protecting alloys at higher-temperature operation
than the MCrAlY alloys available at the time [14].
Different methods of applying the bond-coat alloys have been developed largely to meet
production goals. Typically, PtNiAl bond-coats are formed by first electrodepositing Pt onto
the superalloy component and then annealing it in an aluminum rich vapor atmosphere. In
this second step, aluminum diffuses into the surface of the alloy while nickel diffuses out
where it reacts with the aluminum and platinum to form the PtNiAl aluminide coating. This
formed a single β-phase [14].
Thesis - Pablo Gomez Page 37/138Depending on the quality of the coating required, the aluminum is provided in a pack-
process or in a CVD reactor from an AlCl source. In contrast, the MCrAlY coatings are
commonly deposited by one of a number of variants of plasma-spraying, where the low
pressure plasma spray (LPPS) is the mostly used to deposit the BC on structural
components. These processes are particularly attractive for coating large components and
are, of course, cheaper than EB deposition. Also, as plasma-spraying does not involve a
diffusion process, thicker bond-coats can be deposited than with the aluminizing process
used to form the PtNiAl bond coats [14].
The MCrAlY coatings are commonly deposited consisting of two intermetallic phases and
solid solution γ-Ni (β-NiAl and γ'-Ni Al). The phases γ/γ' have several elements in solution
3
[14]. Yttrium is added at low concentrations to improve adhesion of the TGO, creating
zones of particular groupings for the sulfide [15].
Fig 3.8 shows the corrosion resistance against oxidation resistance of common BC
materials applied on thermal engines components.
Fig 3.8. Corrosion resistance against oxidation resistance of common BC. [13].
It remains uncertain at this time which of these coating types is the best for different
applications. In large part this is because it is not yet known which combination of
materials properties leads to the longest, high-temperature life of the coating. To provide
the largest reservoir of aluminum one would expect that the thicker the bond-coat and the
higher its aluminum content the better. However, one would also expect that the bond-coat
should have as large a yield stress as possible at high temperature with as closely
matched thermal expansion mismatch with the superalloy as possible to avoid thermal
expansion mismatch stresses on thermal cycling [14].
Thesis - Pablo Gomez Page 38/1383.1.2.3 Thermally Grown Oxide (TGO)
The exposure of single metals or more complex alloys to an oxidizing environment at high
temperatures will result in the gradual oxidation of one or more of the constituent metallic
elements [47].
The thermally grown oxide is the main protection for substrate corrosion and oxidation, the
formation of thermodynamic stable oxides, is the best option to avoid the oxidation at high
temperatures and oxidizing atmospheres. Considering the low growth rates and the low
Gibbs free energy, the α-Al O is the preferred oxide to be formed on top of surfaces
2 3
submitted at high temperatures.
The diagram proposed by Harold Ellingham (1944) according to the second law of
thermodynamics Fig 3.9, shows the standard free energy of formation of different oxides
as a function of temperature. The more energy is given off during the formation of an oxide
the more likely that this oxide is to form and the more stable this oxide is. The oxides with
the lowest free energy required much more energy to be separated into its constitutive
elements, also the metals in the more stable oxides (lower on the chart) can chemically
reduce the metals in the less stable oxides (higher on the chart) [Harold Ellingham 1944].
Thesis - Pablo Gomez Page 39/138Fig 3.9. Ellingham diagram for common oxides [Harold Ellingham 1944].
It is important to note that the change of Gibbs free energy is linear respect to the
temperature; and oxide becomes less thermodynamically stable with increasing the
temperature.
Considering the metals available in the Substrate – TBC system, the most stable possible
formed oxide is the alumina for all the temperature range, the only exception is the yttria,
which is include as stabilizer in the TC and have the highest affinity with oxygen among all
the elements in the materials system, even in the periodic table.
Thesis - Pablo Gomez Page 40/138However, once the available aluminum is consumed to form alumina, other oxides may be
formed in the TGO, even at low temperatures, there are other oxides that could be formed
considering the difusional process, which is dependent on temperature [27].
The TGO use to be the weak link between the bond coat and top coat, considering its poor
toughness and the residual stress associated to its growth, which causes an instability and
failure of the coating even at small thickness.
Residual stresses at TGO/BC and TGO/TC interfaces are associated to volumetric
changes, the change in the free energy of formation of the defects like vacancies and
dislocations in the interfaces, is determined by volumetric changes and hydrostatic
component of the local stress, thus the stress gradient through the oxide film affects
directly the growth of the TGO, considering that according to Wagner’s theory, the
diffusion through the film is lead by interchange of the migrating ion with charged
vacancies or may take place by interstitial exchange [47].
3.2 Wear mechanisms for gas turbine hot gas section
Gas turbines and thermal engines in general are submitted to violent energy
transformations, where the wear mechanisms are linked to the operation conditions. For
instance land gas turbines, work in a less aggressive environment, there are less erosive
particles and less corrosive agents, compared to those of aircraft turbines, where no
filtering systems exist. However, the land gas turbines are generally submitted to higher
firing temperatures and lower speeds; on the other hand thermomechanical fatigue cycles
varies from thermal engines at constant regimes to those submitted at variable loads,
affecting creep, toughness, oxidation, and corrosion behavior of the materials that conform
the thermal engine components [6 - 8, 10, 12, and 15 - 22].
The complex substrate – TBC systems used inside gas turbines are designed to withstand
the high temperature, pressure and velocity. However, understanding the way in which
such materials fail is not an easy task, considering the operation conditions and the
complex behavior of these materials under thermomechanical loads.
There are a number of known wear mechanisms, which are related to service life of
substrate – TBC systems, the failure of these systems is related to lose of thermal
insulation (TC) and oxidation resistance (BC+TGO), which provide protection to structural
nickel-based superalloys, on the other hand the inevitable microstructural evolution of the
substrate, which is caused for a continuum exposure to mechanical and thermal loads,
and affects its mechanical properties, completes the complex wear scenario
Individual layers in TBC coatings are significantly stronger than the inter-layer interfaces.
Furthermore the interface toughness also tends to degrade with exposure time.
Consequently, over their service life, TBC coatings are susceptible to delamination.
Experimental observations by Wang and Evans (1998) have confirmed that the
delamination leads to buckling instability and failure in TBC coatings. On account of their
importance in the overall durability of the engine components, modeling buckling and
delamination phenomena are of considerable interest to the engineering community [5].
Thesis - Pablo Gomez Page 41/138Stresses developed due to the coefficient of thermal expansion (CTE) mismatch between
the different layers during thermal loading, as well as growth stresses in the TGO are the
primary driving forces behind the initiation and propagation of damage. Evans et al. (2001)
have reported that the residual compressive stresses in the TGO can reach up to . A
number of geometrical and mechanical factors are known to contribute to the instability
and failure in TBC coatings [5].
Notable among these are important geometric and morphological features of interfaces
and constituent layers, and their thermo-mechanical properties. On a flat interface, the
spallation failure is primarily driven by local buckling instability at the delamination site. A
significant body of work exists in the literature on this failure mechanism with the details of
buckle initiation, propagation, and coalescence (Evans et al., 1997; He et al., 1998;
Hutchinson et al., 2000; Hutchinson and Suo, 1992; Choi et al., 1999) [5].
On the other hand, a couple of mechanisms less severed but significantly important must
be considered, the first one, is the top coat microstructural evolution, which is related to
sintering process and thermomechanical creep, the second is the superalloy creep, those
mechanisms are responsible of individual mechanical and thermal properties of top coat
and substrate respectively, which affect the complex wear mechanism of TBC coatings.
3.2.1 Oxidation and chemical failure (oxide spallation)
Pioneer experimental results obtained by Giggins, G. S. and Pettit, F. S. (1971) for a
simple NiCrAl alloy indicated the oxides likely to be formed, both as a surface layer and
sub-surface within the alloy, at in air for various ranges of composition. The
presence of an α-Al O layer is the principal mean by which alloys can be protected from
2 3
excessive oxidation at over , but, to achieve this, minimum aluminum
concentrations in the alloy are required. For the simple NiCrAl, minimum values ranging
from 3 to 6 wt-% of Al are indicated, depending on the alloy chromium content [41].
Common nickel-based superalloy requires higher Al percentages to form a oxidation
resistant layer than those simple NiCrAl alloys. However, the addition of high Al contents
causes a reduction of mechanical proprieties at high temperature, particularly fracture
toughness and ductility; considering this protective layer has been developed to provide
oxidation resistance, those layers forms chromia, alumina or occasionally silica,
thermodynamically stable oxides that exhibit low growth rates, providing an adequate
protection at high temperatures [27].
Nickel-based superalloy for applications at temperatures over , requires an α-Al O
2 3
layer, which is considering the most stable oxide with the lowest growth rate, to obtain
these protective layers, coatings like those described in Section 2.3.2.2 have been
developed for this purpose. However, other undesirable oxides may be formed from these
coatings as consequence of spallation and/or Al depletion, affecting the oxidation
resistance and mechanical performance of the nickel-based superalloys
The constant exposure to high temperature in air consumes the Al available in the BC,
forming an α-Al O thermally grown oxide. However, in its initial stages other phases are
2 3
Thesis - Pablo Gomez Page 42/138formed such as θ-Al O and δ-Al O . These transition phases transform to stable α-Al O
2 3 2 3 2 3
during operation at temperatures above 1000ºC [42, and 43].
Considering the above, TGO grown is a really complex process, which includes a number
of chemical reactions and microstructural transformations that determine its stability and
durability. Evans et al (1984 -2009), investigated the relation between the thermally grown
oxide kinetics and its stability under operation conditions of thermal engines, Fig 3.10
shows the kinetics of APS NiCrAlY, PtAl BC and uncoated IN738LC [27].
Fig 3.10. Oxidation kinetics of APS NiCrAlY, PtAl BC and uncoated IN738LC [27].
Evans et al researches (1984 -2009), provided a complete explanation about how the TGO
growth and how this growth is closed related to chemical failure, which causes
delamination, buckle instability and finally a TC spallation; showing the relation between
these failure mechanism and the TGO – BC interface undulation and TGO grown, from
Evan et al data, the form of TGO kinetics is as shown in Eqn 3.1 [27].
(3.1)
Where is the TGO thikness, is the time, and the other parameters are constants.
When ideally a α-Al O layer is formed, the constant , and considering homogeneous
2 3
α-Al O grain size and absence of trough thickness stress gradients [27].
2 3
The growth of TGO is leading by oxygen and aluminum diffusion. According to Wagner’s
theory, when the oxide is adherent, uncracked, and dense, such diffusion usually involves
simple ions of the atomic components of the oxide. In many cases, difusional transport
takes place by the interchange of the migrating ion with charged vacancies on the
appropriate oxide sublattices, in other cases; diffusion may take place by interstitial
exchange [47].
Thesis - Pablo Gomez Page 43/138Electronic transport within the oxide layer is generally fast and, so the rate controlling
process of the oxidation reaction is determined by the diffusion rate of the faster moving
ion, provided it is assumed that interfacial reactions and the rate of supply of oxidant also
occur quickly. New oxide can thus be produced at the outer surface of the layer if cation
diffusion predominates or at the oxide/metal interface in the case of anion diffusion. In
intermediate situations, both cation and anion transport may be significant to determine the
location of the new oxide [47].
In the case of TGO growth, the oxygen passes through the TC and Aluminum diffuses
from the BC to the BC – TGO interface. A diffusion of oxygen and aluminum and a
subsequent reaction take place on top of the layer to continue its growing process,
considering recent investigation, which suggest that the aluminum diffusion in Al O
2 3
controls the growth process [45].
TGO growth affects toughness resistance of the TBC [27], making easier the nucleation
and propagation of cracks trough the BC – TGO interface, which is responsible of
delamination, buckling mode occurs only in the case of a large scale delamination (16
times TBC thickness) [5] it is important to take in to account that the sum of TGO growth
stress and other stresses from TC sintering and substrate – TBC system creep may affect
the TGO kinetics, the amount and direction of stress significantly affect the TGO growth by
limiting the ion transport in case of compression and making it faster in case of tension
[47].
Oxide spallation arises, principally, as a consequence of the differential thermal strain
imposed on the oxide layer during temperature changes. As the thermal expansion
coefficient of alumina is less than that of a typical MCrAlX coating (say,
compared with , respectively), the oxide layer experiences in-plane
compressive stresses during cooling. These stresses can be very large, e.g. for a
temperature drop of and can mechanically damage the oxide layer [27].
On planar surfaces, the spallation process can occur either by buckling or by the
propagation of a wedge crack along the BC – TGO interface Fig 3.11; the wedge crack
grows from compressive shear crack along the interface, the shear crack grows as
consequence of compressive stress on the oxide layer during cooling [47].
With BC coatings, which have good adhesion between oxide and coating, buckling is
feasible only for thin oxides, say thick. This is because a pre-existing zone of
decohesion must exist at the interface with a large scale delamination, compared to TGO
thickness, and features of more than a few microns in size are not usually found in well-
prepared coatings. For such systems, wedging is the more likely spallation [5 and 27]
Thesis - Pablo Gomez Page 44/138Fig 3.11. Wedging and Buckling crack extension process and spallation for a flat metal
oxide interface [47].
The TGO undulation plays an important role in the wear mechanism, affecting the
distribution of the stress along the TC – TGO interface and BC – TGO interface, this effect
is caused by the change of the main stresses direction according to the undulation, the
TGO undulation is usually simplify as:
(3.2)
Where is the position, the amplitude and the wavelength.
In general the TGO growth and undulation have a strong influence in TBC wear
mechanisms, both growth and undulation amplitude are time dependent and are related to
the manufactured process, which defined the initial conditions.
When the TGO is perfectly planar, the absence of a shear stress in the substrate (except
near free edges) means that there can be no out of plane response to thermal cycling [21].
However, shear stresses induced in the substrate by TGO undulations may exceed its
yield strength [21]. When this happens, the undulation amplitude enlarges, through
substrate plasticity [7].
He, Evans and Hutchinson shown in 2000, that there is a critical undulation amplitude ,
below which ratcheting does not occur. This critical size is related to the thermal expansion
misfit, the substrate yield strength and the growth strain in the film per cycle [7].
Thesis - Pablo Gomez Page 45/138The above suggest that ratcheting contribution to TGO undulation variation could be avoid
along the life time for well done coatings, it is with adequate substrate roughness before
the coating application. However, the TGO growth itself could change the amplitude of the
undulation.
On the other hand the top coat and bond coat creep, reduce significantly the magnitude of
shear stresses induced by TGO undulation [21], making really important considering the
creep effects in the estimation of TGO growth stresses
3.2.2 Top coat sintering and creep
The ceramic materials properties are related to microstructural defects; even some defects
are highly desired to achieve the required properties according to application
requirements, the microstructural defects are strongly dependent on the manufactured
process.
EB-PVD coatings Fig 3.12, have a columnar microstructure with elongated intercolumnar
pores that become predominantly aligned perpendicular to the plane of the coating; the
intraculumnar porosity is aligned preferentially parallel to the coating. The intercolumnar
pores alienation is the main cause of higher conductivity in EB-PVD coatings [67].
Fig 3.12. EB-PVD coatings [67].
APS coatings microstructure Fig 3.13, include two main defects oriented orthogonally each
other, the first one, is a high elongated interlamellar porosity preferentially aligned parallel
to the substrate, the second is formed by interlamellar cracks that cut the lamellas in a
perpendicular direction to the substrate. These defects increase the strain tolerance and
decrease the thermal conductivity [65 and 66].
Thesis - Pablo Gomez Page 46/138Fig 3.13. APS Coatings [67].
The low thermal conductivity of bulk YSZ results from the low intrinsic thermal conductivity
of zirconia (reported to be between depending on the phase, porosity and
temperature by Schwingel et al in 1998) and phonon scattering defects introduced by the
addition of yttria according to Klemens (1985). These defects are introduced because yttria
additions require the creation of O - vacancies to maintain the electrical neutrality of the
2
ionic lattice. Since both the yttrium solutes and the O - vacancies are effective phonon
2
scattering sites; the thermal conductivity is decreased as the yttria content is increased
[67].
The reduction in thermal conductivity has important affects in the TGO/BC and TGO/TC
interfaces, first the reduction in temperature affects the TGO growth kinetics, making the
oxide film growth slower, the second is related to the thermal stress associated to thermal
expansion, in this case the lower temperature reduce the interfacial mismatch strain.
The porosity of the top coat is strongly dependent on the manufacturing process. However,
this parameter is not a constant along the service life of TBCs, the high free energy
associated to pore surface makes the TC pretty active at operating conditions, the TC
uses the increase in the system energy during heating to reach the activation energy
required to reduce the free energy by sintering and pore surface redistribution, making
pores smaller and rounder [72].
The phenomena describe above is responsible for a continuous increase in thermal
conductivity of the TC, but also influence the mechanical properties of the TC.
The TC sintering has an important effect on elastic modulus, which is dependent on
porosity, the difusional process lead the formation of intralamellar connections, crating
bridges between lamellas, this effect also appear in EB-PVD coatings, causing and
interlock between columns; the bridges acts as a stiffening of the microstructure increasing
the Young’s modulus [72].
In addition, the elastic response of TBCs significantly depends on the direction and sign of
the applied load. Different results are obtained for tensile and compressive loads as well
Thesis - Pablo Gomez Page 47/138as for loads applied parallel or normal to coating structure. Typically, the measured value
of the Young’s modulus is larger under compression than under tensile load in APS
coatings. The reason for this behavior, commonly observed at microcracked and porous
APS coatings, is usually explained by the closure of fine pores and cracks under
compression causing a stiffening, whereas tensile loads cause crack opening and hence a
lower Young’s modulus. This effect has been observed at uniaxially loaded specimens.
[73].
On the other hand the continuous exposure of the TC at high temperatures is also
responsible for the creep behavior of this material. In ceramics the slip system
configuration is strongly dependent on morphology and temperature, which provide the
activation energy for material to flow [70].
The required activation energy to make a material to flow keep a close relation with
material defects, in general the grain boundaries, vacancies and dislocations have
important contributions to strain rates. The creep is the result of a combination of different
motion mechanism, the main important contributions comes from: intragranular motion of
dislocations, and intergranular relative displacements of grain boundaries [70].
The activation of the slip systems during heating of the TC causes a stress relaxation of
the TC, and then during cooling process this causes and increase in tensile stress along
the TC/TGO interface that may result in the catastrophic loss of the TC [64].
In general the steady-state creep rate could be equated as:
(3.3)
Where is the appropriate diffusion coefficient, is the shear modulus, is the Burger’s
vector, is the Boltzmann’s constant, is the absolute temperature, is the grain size,
is the applied stress, is the exponent of the inverse grain size, is the stress exponent,
and is a dimensionless constant.
The diffusion coefficient is give by:
(3.4)
Where is a frequency factor, is the activation energy, and is the gas constant.
According to the motion mechanisms in polycrystalline ceramics, exist to main categories
to classify the creep in ceramics, in the first the exponent is typically between 2 and 3,
and the exponent is close to 1, in the second the exponent , and the exponent
use to vary from 3 to 5 in this ceramics the creep behavior is independent of grain size
[71].
In general the Creep behavior of the top coat, is strongly dependent on grain size, the
intercolumnar interaction in EB-PVD coatings, and interlamellar sliding in APS coatings,
affects the deformation, also the importance of the direction of applied stresses on the
coatings have an important role in the steady-state creep rates [61, 63, 64 and 73].
Thesis - Pablo Gomez Page 48/1383.2.3 Nickel-base superalloys rafting
In high volume fraction γ' alloys the microstructure following heat-treatment consists of
cuboids of precipitate separated by channels of γ matrix. γ' coarsening is driven by the
minimization of the interfacial energy per unit volume in the absence of an applied stress.
The coarsened microstructure remains essentially the same (albeit with a bigger length
scale), i.e. with channels of γ separating the more or less cuboidal precipitates of γ' [69].
However, the development of microstructure during coarsening under the influence of an
applied stress is quite different. The γ' particles coalesce to form layers, known as rafts
[69].
Is necessary to define the crystallographic misfit between the γ and γ' in order to describe
the process of rafting. There are two ways in which this is normally done (either way is
acceptable) [69]:
(3.5)
(3.6)
Misfit means that the γ' has a smaller lattice parameter than that of γ. The latter will
therefore be in compression in the vicinity of the interface with γ', and there will be
compensating tensile stresses in the γ' as illustrated in the figure below. The application of
a uniaxial tensile stress exaggerates any tensile stress in the vertical direction and reduces
those in the horizontal orientation. Similarly, the applied tensile stress reduces the
compressive stress in the vertical γ channels and via the Poisson effect, increases those in
the horizontal channels [69].
The greater stresses in the horizontal γ channels lead to enhanced creep; the resulting
dislocations enter the γ/γ' interface and accommodate the misfit in the horizontal
interfaces. Rafting then tends to eliminate the elastically strained vertical channels, leading
to a net reduction in free energy [69].
The orientation of the rafts clearly must be a function of that of the applied stress. The rate
of rafting depends on the magnitude of the stress and the magnitude of the lattice misfit
between the γ and γ' crystals. It must also have an independent dependence on the
chemical composition of the alloy since the process involves the transfer of matter via
multicomponent diffusion [69].
A full discussion of rafting should account for the ratio of the elastic modulus of the
precipitate to that of the matrix; the orientation of rafting can change depending on whether
this ratio is greater or less than unity. Rafting is predicted not to occur if the ratio is unity
and the applied stress elastic [69].
Thesis - Pablo Gomez Page 49/138Rafts of γ' lying normal to an applied tensile stress increase the resistance to creep when
the strain is controlled by the climb of dislocations over γ' particles. This is the case at low-
stresses. However, the rafts are detrimental if the dislocations are able to cut through the
rafts, since a rafted state is coarse when compared with the original microstructure. The
cutting occurs when the applied stress is sufficiently large [69].
In nickel-based alloys designed for turbine blades, the fraction of γ' that precipitates from
the γ matrix is typically in excess of 0.7. The changes associated with rafting cause a
topological inversion in which the γ' coalesces to become the matrix with the minority γ
enclosed in such a way that it appears like the included phase [69].
3.2.4 Top Coat Corrosion
The presence of vanadium, phosphorus, and sodium impurities in liquid fuels at dual fuel
and liquid fuel gas turbines have been linked to premature fail of the top coat, gas turbine
manufacturers, suggest different severity factor to estimate the amount of damage caused
by dirty fuels [3].
The interlamellar pores and intralamellar cracks, common in plasma-sprayed materials,
provide pathway for the molten species. The microstructure of the contaminated coatings
is therefore the result of the interplay between the dissolution/reaction rates of the V O
2 5
with YSZ coating and the infiltration rates of the molten species [59].
Investigations of degradation reactions of air plasma sprayed yttria-stabilized zirconia
coatings in contact with vanadium pentoxide (V O ), phosphorus pentoxide (P O ), and
2 5 2 5
sodium sulfate (Na SO ) at temperatures up to 1200°C, have shown that molten V O
2 4 2 5
reacted with solid YSZ to form zirconium pyrovanadate (ZrV O ) at temperatures below
2 7
747°C. However, at temperatures above 747°C, molten V O reacted with YSZ to form
2 5
yttrium vanadate (YVO ). The formation of YVO led to the depletion of the Y O stabilizer
4 4 2 3
and transformation of the tetragonal and/or cubic phases to the monoclinic ZrO2 phase
[60].
Transformation to monoclinic phase during cooling cause as shown in section 3.1.2.1 a
destructive volume change and the consequent TC delamination and spalling, the loss of
the TC and TGO in some cases, submit the BC directly to the corrosive molten species
causing chemical reactions that destroy also the BC.
In addition, studies on YSZ degradation by Na SO and a Na SO +V O mixture (50/50
2 4 2 4 2 5
mol-%) showed that Na SO itself had no effect on the degradation of YSZ. However, in
2 4
the presence of V O at high temperatures, Na SO forms vanadate compounds having a
2 5 2 4
lower melting point such as sodium metavanadate (610°C), which was found to degrade
YSZ by the formation of YVO at a relatively lower temperature of 700°C. P O was found
4 2 5
to react with APS YSZ by the formation of zirconium pyrophosphate (ZrP O ). At
2 7
temperatures as low as 200°C and as high as 1200°C, molten P O was observed to react
2 5
with solid YSZ to yield ZrP O , which led to the depletion of ZrO in YSZ that promoted the
2 7 2
formation of the cubic ZrO phase [60].
2
Thesis - Pablo Gomez Page 50/138Also the presence of SO , in gaseous and liquid fuels, may affect the stability of the YSZ,
3
the molten SO reacts with YSZ to form Y (SO4) , causing the same depletion effect of
3 2 3
V O . However, the relative high partial pressure of SO required by the reaction to occur,
2 5 3
make less important this kind of corrosion [14].
Corrosion considerations have received only limited attention in the development of
advanced TBCs. Partly this is because of the predominance of clean fuels in current
operation of both aerospace and power generation turbines [14].
Thesis - Pablo Gomez Page 51/1384 High-Temperature Wear Models for Substrate – TBC
Systems
Investigation of the ways in which current YSZ coatings fail has provided considerable
insight into the underlying mechanisms that limit coatings life. Nevertheless, as with failure
analysis in other areas of complex material systems, there are many complications. The
findings point toward methods of producing coatings that can withstand longer lives at
higher use temperatures. One of the chronic problems is that, statistically, the life of
present TBC coatings invariably shows a wide distribution, with a high proportion of the
population clustered about a mean value but with a significant proportion failing at much
earlier times. There is substantial indirect evidence to suggest that many of the TBC
failures are associated with the oxidation of the bond-coat [27]. Indeed, a number of
manufacturers are believed to use an oxidation criterion as a basis for predicting average
life. One such criterion is the combination of time and temperature that leads to a critical
thickness of the TGO [14].
Another common consideration in the coat life estimation is an aluminum depletion
criterion based on the combined time and temperature for the concentration of aluminum
at the bond-coat surface to fall below a critical value. In the case of MCrAlY bond-coats,
the reason for this is that when the Al concentration decreases, aluminum oxide is no
longer the thermodynamic preferred phase and other oxides, mostly spinels, form [27].
These other oxides do not form such a protective scale, and consequently the alloy
oxidizes faster. In addition, the formation of these oxides is associated with an increase in
volume that can be disruptive and they possibly have lower fracture energies, although this
has yet to be unequivocally demonstrated. Nevertheless, there are reports showing that
porous bond-coats operating at low-oxidation temperatures experience failure following
such aluminum depletion [14].
Although related to the oxidation behavior of the bond-coat, neither the concept of a critical
thickness or aluminum depletion can account for the wide distribution in failure lives,
especially under thermal cycling conditions. Indeed, in the majority of materials examined
after failures above , the aluminum concentration, although depleted somewhat,
has not fallen to the critical value [27]. Similarly, the short-lived coatings have failed before
the TGO thickness has reached the thickness of its counterparts that have shown the
longest lives. Together these findings indicate that failure occurs due to extrinsic factors
arising during oxidation [14].
4.1 Statistical models for service life estimation
Gas turbine manufactures have developed statistical models to estimate the service life of
critical components inside the gas turbine by using statistical data that compare
operational conditions with wear of some components. Most of those models assume a
differentiation between base load and peak units, considering different wear mechanisms
Table 4.1. According to the operation regime, the maintenance interval must be calculated
on an equivalent hours approach or an equivalent starts approach [1 and 4].
Thesis - Pablo Gomez Page 52/138Table 4.1. Causes of wear of hot gas path components [4]
Continuous Duty Applications Cyclic Duty Applications
Rupture Thermal Mechanical Fatigue
Creep deflection High Cycle Fatigue
High Cycle Fatigue Rubs/wear
Corrosion Foreign Objects Damage
Oxidation
Erosion
Rubs/wear
Foreign Objects Damage
Equivalent hours and starts approaches, allow gas turbine owners to prepare and program
major inspections Fig 4.1, which coincide in most cases with the service life of main
components that must be repaired or replaced. Nevertheless, for turbines operating at
intermediate conditions the actual amounts of wear are quite different from the statistical
predictions.
Fig 4.1. GE bases gas turbine maintenance requirements on independent counts of starts
and hours [4].
In order to estimate the maintenance intervals for hot gas components in heavy duty gas
turbines, GE proposed an algorithm based on operating statistics, which allows using
severity factors to estimate the maximum number of operating hours and starts according
to the operation regime [4].
Considering the equivalent hours approach, the maintenance interval to perform a hot gas
path inspection must be defined according to GE using the relation [4]:
(4.1)
Thesis - Pablo Gomez Page 53/138Where:
(4.2)
Where:
(4.3)
And;
(4.4)
Where is the water injection constant ( , depending on blades material),
is the steam injection constant ( , depending on blades material), is the
percent water/steam injection referenced to inlet air flow, is annual base load operating
hours on gas fuel, is annual base load operating hours on distillated fuel, is heavy
fuel severity factor (residual , crude ), is annual operating hour
on heavy fuel, and is annual peak load operating hours.
Considering the equivalent start approach, the maintenance interval to perform a hot gas
path inspection must be defined according to GE using the relation:
(4.5)
Where:
(4.6)
Where:
(4.7)
And;
(4.8)
Where is the maximum number of starts according to technology ( for class F
gas turbines), is the number of part load start/stop cycles (<60 % of the base load),
is the base load start/stop cycles, is the number of peak load start/stop cycles (>100 %
of the base load), is the number of annual emergency starts, is the number of fast
load starts, is the trip severity factor, and is the number of trip for each severity
category Fig 4.2.
Thesis - Pablo Gomez Page 54/138Fig 4.2. Trip severity factor from load [4].
Following Fig 4.1, those units operating at intermediate conditions must be inspected
when:
(4.9)
However, as mentioned above, this methodology is not really accurate for units operating
at intermediate conditions, considering evidence from inspected units, in addition this
methodology is too general, it is not include specific predictions for the gas turbine
components or materials, making impossible to undertake material development projects
from its application and avoids the understanding of wear mechanisms.
4.2 Analytical models and parametric finite element method
approaches
Considering the linear elastic behavior, the dominant failure mechanisms in thermal
barriers coatings have been identified as the buckling instability and delamination by crack
extension along the interface between the bond coat and TGO by Evans et al. (1997), He
et al. (1998), Wright and Evans (1999) and Evans et al. (2001). During operation under
thermal loads, the TBC can experience critical loads leading to buckling instability and
unbounded out-of-plane deflections of the delaminated portion, as shown in the schematic
of Fig. 4.3. The buckling instability is eventually arrested at the edge of delamination by the
bonded interface [5].
Thesis - Pablo Gomez Page 55/138Fig 4.3. Schematic diagrams showing the competing failure mechanisms in TBC systems
with flat and wavy interfaces [5].
There are three basic fracture modes for a crack extension, which are associated to the
direction of the applied load respect to the front of crack propagation Fig 4.4. However, it is
pretty common to find combinations of these fracture modes on delamination propagation
and spallation [x].
Fig 4.4. Fracture modes [50].
During World War I A. A. Griffith (1921-1924), proposed a crack extension criterion for
brittle materials, from this approach many particular solutions for different system were
found to predict the energy release for crack nucleation and growth. Griffith suggested that
a crack results from transformation of elastic energy into surface energy ; from this
approach a crack may grow when the available potential energy in a solid material is
enough to create a new surface [x].
The ratio of available potential energy transformation into a new surface trough a crack,
defined by the vector , is named energy release and can be written after Griffith using
the relation:
Thesis - Pablo Gomez Page 56/138(4.10)
Considering only elastic energy on a homogenous and semi-infinite plate of unitary
thickness with a elliptical crack of length , Griffith proposed:
(4.11)
And;
(4.12)
Where is the stress, is the Young’s modulus, and is the specific surface energy.
Is obtained considering that a crack extension creates two new surfaces.
Deriving respect from Eqn 4.11 and 4.12:
(4.13)
(4.14)
From Eqn 4.13 and 4.12 and considering the transformation of potential elastic energy into
surface energy, it is possible to write the critical stress for crack propagation as:
(4.15)
Considering only one side of the crack, Griffith defined the critical ratio of energy release
, which is a mechanical property that could be constant for elastic stresses, as:
(4.16)
Thus;
If then the crack will not grow.
If then the crack will grow quasistatic.
If then the crack will grow.
Irwin in 1957 presented an important advance in fracture mechanics, proposing the
analysis in terms of tensions (tensor formulation), something impossible before
considering that stress at crack tip becomes infinite independent of applied stress, in his
formulation Irwin considered a plastic zone around the crack tip, which reduce the
magnitude of elastic stress considering the amount of energy used to deform plastically
the zone around the crack tip [80].
Thesis - Pablo Gomez Page 57/138According to Irwin’s modification of Griffith’s theory, the release energy for ductile
materials must be higher than that of brittle materials due to contributions of the inelastic or
plastic dissipation around crack tip, thus:
(4.17)
The concept of stress intensity factor was used by Irwin to calculate the amount of
energy available for fracture in terms of the asymptotic stress and displacement fields
around a crack tip in a linear elastic solid. This asymptotic expression for the stress field
around a crack tip is:
(4.18)
Where are the Cauchy stresses, is the angle with respect to the plane of the crack
and are functions that are independent of the crack geometry and loading conditions.
Rice in 1988 showed that the mode mixity parameter , measures the relative importance
of mode I (opening) to mode II (shearing) at the crack tip [49]:
(4.19)
Where and are the stress intensity factors for pure crack opening and crack shear
modes respectively.
Considering a mixed mode cracking in layered materials, Hutchinson and Suo proposed in
1992 an analytical solution for critical buckling load for a circular blister to estimate the
delamination, instability buckling and spallation for a single two layer TBC system, where
the substrate is coated with a bond coat layer and no ceramic top coat is deposited. This
solution has been employed to predict buckle initiation in TBCs by Evans et al. (1997) and
He et al. (1998) [5]. The analytical model by Hutchinson and Suo (1992) is based on
classical plate theory with rotationally constrained edges [44].
The proposed model by Hutchinson and Sue, assume that an interfacial crack grows
when:
(4.20)
Where is the interfacial toughness function, which depends on mixity parameter and
is defined as:
(4.21)
According to Hutchinson and Sue, is defined as:
(4.22)
Thesis - Pablo Gomez Page 58/138Where describes the reduction of the crack growth rate during mixed mode crack growth
compared to mode I crack growth.
The critical stress in this approximation is defined by Hutchinson and Sue as:
(4.23)
Where is the TGO thickness and is the Poisson’s ratio and is the extension of the
existent delamination.
Although this model is accurate for very large interfacial delamination, the predictions are
not reliable for the incipient stages of delamination where the assumed rotational
constraints are questionable. The presence of interfacial waviness dramatically affects
failure mechanisms in TBCs. This has been highlighted by Evans et al. (1997), who used a
sinusoidal undulation was used to understand the origin of interfacial cracks and by He et
al. (1998), who associated crack growth with periodic and a-periodic morphologies. The
onset of failure in TBCs may consequently be viewed as a competition between strain
energy driven interfacial crack growth mechanism and buckling induced instability and
delamination [5].
Some other analytical solutions like this proposed by Wang and Lee (1998) [5], has been
developed. However, in order to solve more complex geometries, including interfacial
waviness, voids, inclusion and other defects on three layered TBC systems, the finite
element method could be used to solve the problem of estimate the energy release from a
crack extension, considering its simplicity and ability to model complex domains.
H. Bhatnaga et al 2006 proposed a model over the linear elastic behavior, assuming that
the compressive stress does not change during an infinitesimal change in the buckling
displacement , the critical load for instability in two contiguous configurations may be
equated as (Cook et al., 1989):
(4.24)
(4.25)
Where is the buckling displacement vector in the reference configuration, is the
stress stiffness matrix, that augment the conventional stiffness matrix , is the
perturbation stress, and is a scalar used to scale the TGO compressive stress, to
estimate the critical buckling stress [5].
In the H. Bhatnaga et al model, the applied load is in a direction parallel to the constituent
material layers and their associated interfaces. For this load condition, a crack tip at the
delaminated flat interface does not induce a stress concentration to cause crack extension.
Consequently, crack growth at the interface between the bond coat and TGO interface is
studied exclusively for wavy interfaces. In this study, this mechanism is assumed to be
governed by Griffith’s energy release rate criterion. According to this criterion, crack
growth will occur if the energy required for creating new crack surface area is achieved in
the system [5].
Thesis - Pablo Gomez Page 59/138The virtual crack extension method, based on the stiffness derivative finite element
technique proposed by Parks (1974), is used to evaluate the energy release rate. This is
determined as the negative of the derivative of the total system potential energy with
respect to crack extension. In the finite element model, the potential functional is
represented in terms of the global stiffness matrix, the displacement field and the external
loads [5]. The energy release rate G is thus expressed by the stiffness derivative
technique by Parks (1974) as
(4.26)
Is evaluated numerically from the change in system potential energy per unit crack
extension due to a virtual extension of the crack using the relation:
(4.27)
Where and are the respective total potential energies before and after the crack
extension by length . A FEM analysis is first conducted with a prescribed length of
delamination, and the potential energy of the system associated with the applied load is
calculated. In the second analysis to evaluate , the finite element analysis is conducted
with the crack extended by an infinitesimal length over the initial delamination [5].
To eliminate the contribution of structural compliance variation due to crack extension, the
strain field computed from the first pre-crack extension model is applied to the second
analysis. The virtual crack extension is achieved by moving the FE nodes in the vicinity of
the crack tip in the direction of probable crack propagation. The results are sensitive to the
region considered as crack tip vicinity and magnitude of crack extension. A very large
crack extension may result in distorted elements, whereas, a very small extension may not
result in a correct energy release rate. The models used in this study are checked for
convergence of the energy release rate with respect to both of these parameters. It is
found that variation in energy release rate calculated from the numerical model is within 2
% when the nodes associated with at least three layers of nearest and contiguous
elements are moved by 1 % of the edge length of the elements in close vicinity. The
converged values of these parameters are utilized in all subsequent analyses [5].
Fig 4.5 Shows a comparison between classical Hutchinson and Sue solution for a blister
(1992) and the H. Bhatnaga et al FEM model, and comparison between Wang and Lee
solution (1998) and the H. Bhatnaga et al FEM model.
Thesis - Pablo Gomez Page 60/138Fig 4.5. Comparison between classical Hutchinson and Sue solution for a blister (1992)
and the H. Bhatnaga et al FEM model (a), and comparison between Wang and Lee
solution (1998) and the H. Bhatnaga et al FEM model (b) [5].
In general, analytical models are really important to understand the underlying
mechanisms that limit coating life, providing a correlation between materials properties and
wear mechanism, this way not only assess a coating life estimation, also allow work
directly in new materials design to improve wear resistance.
4.3 Wear model using a fracture mechanics approach
The growth of cracks in the TBC coatings can be divided into four stages: the first one,
corresponds to a proportional growth of the cracks on the fatigue cycles, the second is a
transition stage, the third is characterized by a marked decrease in the rate of crack
growth and the fourth indicating catastrophic failure by delamination of TC, Fig 4.6 [6].
Thesis - Pablo Gomez Page 61/138Fig 4.6. Schematic representation of TBC delamination fatigue damage development. A
fatigue life limit is defined at the transition from slow crack growth (region III) to final failure
(region IV) [6]
Paris in 1961 showed a linear relationship on a logarithmic scale between the stress
intensity factor applied to a material factor and subcritical crack growth under mechanical
stress [6], written as:
(4.28)
Where and are material constants, is the crack length, is the range of the stress
intensity factor and the number of fatigue cycles
Replacing in Paris law, the classic stress intensity by the energy release rate , Brodin
and Jinnestrand proposed in 2004, the relation presented in Eqn 4.29 for TBC coatings,
which relate the damage parameter Eqn 4.30 and interfacial stress state [6].
(4.29)
Where is defined as the sum of crack lengths per interface length and is written as:
(4.30)
Where designates individual delamination cracks extending only in the thermally
grown oxide, designates delamination cracks with growth only in the ceramic top coat
and represents delamination cracks with part of the growth at the interface and part
in the top coat; is the total interface length investigated.
was described by Brodin and Jinnestrand Eqn 4.31, adding the term to describe the
reduction of the crack growth rate during mixed mode crack growth compared to mode I
Thesis - Pablo Gomez Page 62/138crack growth, applying the description originally proposed by Hutchinson and Suo in 1992,
and using the measured average interface roughness as a geometry factor [6].
(4.31)
Where can be written, after Hutchinson and Suo (1992) as:
(4.32)
Where and are fitting parameters, and are stress intensity factors for pure
crack opening and crack shear modes, respectively. From Eqn (4.31) the fatigue life to
TBC spallation can be integrated as:
(4.33)
Where describes the initial defect state of the TBC, and and describe the initial
and final damage state. Fig 4.7 shows the model calibration using thermal fatigue data [6].
Fig 4.7. Calibration of fatigue life model from thermal fatigue data, after Brodin and
Jinnestrand [6].
These kinds of model are really accurate versus thermal fatigue results. However, the
inherent differences between real gas turbines and data from thermal fatigue experiment
reduce its applicability to estimate the real service life of hot section components.
Thesis - Pablo Gomez Page 63/1384.4 Thermodynamic approach for elastic-plastic damage
In order to extend the substrate – TBC system damage to more complex wear sceneries a
full coupled elastic-plastic thermodynamic approach could be used to estimate the
potential energy available in the system for a crack extension.
The thermodynamic approaches take advantage of the state functions, which are
independent of the process and only depend on the initial and final state of the system,
allowing a natural description of the evolution process inside the system considering the
energy variations.
The nonlinear material softening may be explained by two distinct degradation
phenomena: plastic deformation due to dislocation along slip planes and damage due to
microcrack nucleation and growth. This is why double-dissipative coupled models based
on thermodynamics frameworks have been developed during the last two decades e.g.,
Hansen and Shreyer (1994); Zhu and Cescotto (1995); Chaboche (1997); Hayakawa and
Murakami (1997); Abu Al-Rub and Voyiadjis (2003). These models are usually highly
complicated, so that they can prescribe and predict complex physical phenomena,
including initial or damage anisotropy, damage deactivation, to mention only the most
important of them [25].
A thermodynamically consistent framework for description of elasto(visco)-plasticity
coupled with damage meets two general questions: the first one, is how to include damage
to the Helmholtz free energy or the Gibbs thermodynamic potential (state coupling),
and the second, how to couple dissipation potentials with damage (dissipation coupling)
[24].
Hayakawa and Murakami in 1997, proposed a coupled formulation, including the elastic
energy , the isotropic plasticity energy and the hardening damage [26],
this formulation is written as:
(4.35)
Where:
(4.36)
(4.37)
(4.38)
A more detailed development of this formulation is presented in the next section, on the
base that this formulation is the most adequate to handle the effects of TC sintering and
creep, BC and substrate creep and the most complex stresses associated to TGO kinetics,
for the proposed coupled wear formulation for the substrate – TBC system.
Thesis - Pablo Gomez Page 64/1385 Analytical Model for High-Temperature Wear in Non-
Steady State Gas Turbines Based on a Thermodynamic
Approach
Trough this chapter the main question of this investigation is addressed and, a generalized
method to handle the high temperature wear of the substrate – TBC system in non-steady
state gas turbines is proposed; the development begins with the combustion model and
the estimation of the firing temperature to predict the inlet conditions for the fluid dynamic
model of the turbine section, which is required to determine the thermomechanical loads
on the turbine components, and finally the thermodynamic approach to high temperature
wear of the substrate – TBC system at non-steady state operation conditions is fully
coupled.
There are two important aspects of the problem that must be solved: the first one is how to
estimate the amount of energy available in the substrate – TBC system, and the second is
how much energy is released in the substrate – TBC system to make a crack nucleate and
growth.
The proposed model is developed on the basis of continuum mechanics; it considers the
continuity of the matter and its properties and the validity of the application of conservation
laws to the wear mechanisms, where heat transfer, fluid mechanics and solids mechanics
are presented in its conservative form.
There are two alternative descriptions used to express the conservation laws in analytical
form. In the first one, the motion of all matter passing through a fixed spatial location is
considered, here properties such as density, velocity, temperature, pressure and so on are
of interest, this alternative is named eulerian description or spatial description. In the
second one the relative displacement of these particles and the stresses caused by
external forces and temperatures are of interest, this description is known as the
lagrangian description or material description [53].
Eulerian description is commonly used to steady fluid flows and convective heat transfer,
while the lagrangian description is preferred to study solid body heat conduction and the
stress and deformation of solid bodies [53].
The conservation laws (law of conservation of mass, laws of conservation of linear and
angular momentums, and law of conservation of energy) are adopted from classical
mechanics and thermodynamics where the system approach is normally followed.
Considering a physical quantity that is flowing through the body and is carried by
particles moving at velocity v , in which the surface is moving at velocity u in
the normal direction n to , with and sources of the physical quantity in
the body and the surface of the body respectively, the balance law can be expressed
in the general form [29]:
(5.1)
Thesis - Pablo Gomez Page 65/138The transformation from system equations (matter equations) to corresponding control
volume equations (spatial equations), is obtained by the application of Reynolds transport
theorem named after Osborne Reynolds. In order to present the derivation from material
equations to special equations, the crack energy release rate is considered below for a
material line.
Noting that the energy release is a smooth scalar field, defined as:
(5.2)
The application of the transport theorem to along a material line could be written as:
(5.3)
Where could be written using the deformation function shown in Fig 5.1, as:
(5.4)
Where:
(5.5)
Fig 5.1. Deformation of a body from the reference configuration to present configuration.
Replacing Eqn 5.4 and 5.5 in Eqn 5.3:
Thesis - Pablo Gomez Page 66/138(5.6)
Applying the definition of derivate to function , defined as:
(5.7)
Where;
(5.8)
Then:
(5.9)
Replacing Eqn (5.7) in Eqn (5.9):
(5.10)
Considering that the integral operator is a lineal operator, Eqn 5.10 could be written as:
(5.11)
Considering that the term in parentheses is just the partial derivative of respect t,
then:
(5.12)
Replacing Eqn 5.8 in Eqn 5.12 and solving the dot product:
(5.13)
From Eqn 3.13 and using the inverse of , it is possible to write:
(5.14)
If the deformation function is considered to evaluate the crack growth velocity
presented in Fig 5.2, then:
(5.15)
Then:
(5.16)
Thesis - Pablo Gomez Page 67/138Where:
(5.17)
Replacing Eqn 5.17 in Eqn 5.16:
(5.18)
Taking in to account that and are parallel vectors, then:
(5.19)
And:
(5.20)
Taking the inverse and replacing :
(5.21)
Fig 5.2. Velocity for towing adjacent material particles along the considered material line.
Replacing Eqn 5.21 in Eqn 5.14:
(5.22)
Thesis - Pablo Gomez Page 68/138Considering Eqn 5.4:
(5.23)
The analysis above presents the material derivative of the energy release through a
material line, which coincides with the crack trajectory, where the volumetric energy is
transformed into surface energy.
Following the analysis above and Eqn 5.1 for the physical quantities: mass, energy and
momentum, and considering a material body, the laws of conservation could be obtained.
However, to obtain a complete system evolution formulation, some additional relations
must to be used considering the complexity of the physical phenomena; those additional
relations known as constitutive equations, which are not derived from governing physical
laws, describe relations between state variables, allowing the solution of complex
formulations.
Some constitutive equations are simply phenomenological; others are derived from first
principles. A common approximate constitutive equation frequently is expressed as a
simple proportionality using a parameter taken to be a property of the material, such as
thermal conductivity or Young’s modulus. However, much more elaborated constitutive
equations often are necessary to account for tensor properties, the rate of response of
materials and their non-linear behavior.
5.1 Algorithm basics
The development of a wear model for a gas turbine required the consideration of the loads
and the wear mechanisms involved in the transformation of the chemical energy in the fuel
into mechanical energy at the turbine shaft. The loads over the gas turbine components
depend on gas turbine operating conditions; considering the constant angular speed of the
turbine shaft, it is possible to define the loads as a function of incoming air and fuel and the
consequent combustion reaction where the mass flow of air and fuel affects the fluid
dynamics, which together with inertial forces defined the external loads applied to the
substrate – TBC system along the hot gas path.
Fig 5.3 shows a schematic representation of the proposed algorithm to predict
delamination and spallation of the coating in substrate TBC system at gas turbine
operating conditions.
Thesis - Pablo Gomez Page 69/138Gas Turbine Operating Gas turbine – hot gas Substrate – TBC System
Conditions: path geometry
Geometry:
Fuel composition:
Mechanical Properties
Combustion Model: Fluid Dynamic Model:
Thermal Properties
State Equation:
Heat Transfer Model:
Griffith Criterion: Virtual Crack Model:
If then the crack will not grow.
If then the crack will grow quasistatically.
If then the crack will grow.
Crack Grow:
Crack will not Grow:
Fig 5.3. Schematic overview of the proposed wear algorithm for gas turbines.
Thesis - Pablo Gomez Page 70/1385.2 Combustion model
In order to obtain the loads acting on the gas turbine components the fluid dynamic model
must be solved, so the ignition temperature inside the combustion chamber is required to
define the boundary conditions.
Several complex methods to solve the combustion kinetics are available. However, if the
species diffusion and the reaction sequence are ignored, it is possible to obtain the
temperature in the combustion chamber from a single application of the first law of
thermodynamics and mixing rules [62 and 81], which is considered an adequate approach
to define the boundary conditions of the compressible flow of combustion gases flowing
into the turbine hot gas path. However, it is important to note that during calculation of
adiabatic flame temperature at constant pressure a chemical equilibrium is assumed, but
some differences between real and estimate ignition temperature could be found
considering a non homogeneous mix of the reactants, also is important to highlight that the
real temperature distribution at combustors could not obtained from this analysis.
There are a number of gaseous and liquid fuels that could be used in gas turbines; this
ensures their versatility in many applications. As mentioned in Section 3.2.4, some fuel
contaminants could affect the durability of the top coat. However, only the case of methane
will be considered in this analysis, since that it is the main fuel used in thermal power
plants.
The stoichiometric reaction of methane combustion in air is written as:
(5.24)
The heat of reaction is obtained from the standard enthalpy of formation of the reactants
and the products as:
(5.25)
Considering an adiabatic and isobaric combustor, it is possible to calculate the constant
pressure adiabatic flame temperature, which could be considered in most cases as the
maximum possible temperature of the combustion process. The constant pressure
adiabatic flame temperature could be obtained applying the first law of thermodynamics to
the combustors, where no heat is transferred through the walls and no work is made by
the fluid, thus:
(5.26)
Note that nitrogen is not included in the reaction.
Now considering the actual operating conditions of a gas turbine, it is important to account
that only some part of the incoming air is used in the combustion process, then the air
Thesis - Pablo Gomez Page 71/138excess must be considered to obtain the temperature at combustors, assuming that the
mixture of combustion gases could be handled as an ideal gas, then using the first law of
thermodynamics:
(5.27)
Fig 5.4 shows the constant pressure adiabatic flame temperature for a 7FA gas turbine as
a function of load percent. To calculate the flame operating conditions of the gas turbine at
different loads were obtained from a General Electric Speed Tronic Mark V control and
data acquisition system used to operate the EPM 7FA gas turbines, the amount of
methane used by the gas turbine is obtained using a Coriolis Mass Flow Meter coupled to
a Daniel Flow Computer installed in the EPM Power Plant City Gate. Fig 5.4 also shows as
reference the Mark V internal ignition temperature which is an internal calculation of the
firing temperature using the mean exhaust temperature.
1400 °C
1300 °C
1200 °C
1100 °C
1000 °C
900 °C
800 °C
700 °C
600 °C
0 MW 20 MW 40 MW 60 MW 80 MW 100 MW 120 MW 140 MW 160 MW
TTREF Calculated Flame Temperature
Fig 5.4. Calculated constant pressure adiabatic flame temperature and Mark V ignition
temperature against gas turbine load.
The enthalpy and the standard enthalpy of formation of the chemical species in Eqn 5.24,
were obtained from Reference [81].
5.3 Fluid dynamics model
The analysis of the flow of combustion gases through the turbine section required
considering the compressibility of the fluid. A fluid must be handled as compressible when
the changes in fluid density become important respect to changes in pressure, taking in to
account that common gas stream at stationary nozzles and blades are over 30% of the
speed of sound in the fluid, compressible effects are indeed important in the energy
transformation at turbine sections.
Thesis - Pablo Gomez Page 72/138The flow of a fluid is described by Navier-Stokes equations, derived from conservation
equations of mass, momentum and energy of a fluid particle, which can be written in the
Eulerian form as:
(5.28)
(5.29)
(5.30)
Where is the density, is the time, is the velocity vector, is the pressure, is the
viscous tensor, is the volume force vector, is the identity tensor, is the specific heat
capacity at constant pressure, is the absolute temperature, is the heat flux vector,
contains the heat sources and is the strain rate tensor defined as:
(5.31)
Note that , could be written using the contraction between tensors defined as:
(5.32)
To close the equation system above, some constitutive equations must be included:
Considering that for the fluid inside the turbine , the fluid could be handled as an
ideal gas then:
(5.33)
Where is the absolute pressure, the temperature, and is the specific gas constant
defined as.
(5.34)
Where the universal gas constant is , and is the specific mean molar
mass.
Additionally, assuming Newtonian fluid:
(5.35)
Where considering the relation proposed by William Sutherland, for an ideal gas:
Thesis - Pablo Gomez Page 73/138(5.36)
Where is the dynamic viscosity of the fluid at reference temperature .
The Sutherland’s Law also describes the relation between the thermal conductivity and the
temperature of an ideal gas:
(5.37)
Where is the thermal conductivity of the fluid at reference temperature .
From the Fourier’s law of heat conduction and Eqn 5.37 the heat flux vector could be
written as:
(5.38)
Considering the fluid as an ideal gas and the Mach number defined as the relation
between the inertial and compressibility forces:
(5.39)
It is possible to write the fluid properties in terms of Mach number, pressure, temperature
and flow direction to define the boundary conditions in a consistent way:
(5.40)
(5.41)
Where:
(5.42)
In addition to the considerations above, it is important to take in to account that in most of
the gas turbine flow section the flow regime is turbulent, so it is necessary to handle this
condition. The Reynolds number defines the relation between the inertial and viscous
forces in the fluid and allows identifying the flow regime of a fluid:
(5.43)
Where assuming an interior flow and using a practical rule, it is possible to consider a flow
as turbulent when .
Thesis - Pablo Gomez Page 74/138A common practice to handle the turbulent regime is using the Reynolds Average Navier-
Stokes (RANS) method, which allows writing the Navier-Stokes equation using the
Reynolds decomposition; this mathematical technique permits to separate the average
and the fluctuating part of a physical quantity.
Considering the velocity vector , it is possible to use the Reynolds decomposition as
follows:
(5.44)
Where is the time average velocity and is the fluctuating part or perturbation of the
velocity, defined such that its time average equals zero,
In the case of a compressible flow, the term is handled using a density average
known as Favre average, which is defined as:
(5.45)
Then:
(5.46)
The turbulent flow is characterized by a wide range of flow scales that depend on
geometry and define the subdivisions of the domain required to ensure the model
precision. The minimum scale to be considered is defined as:
(5.47)
Where is the dimension of the geometry and is the shortest side of the geometry
bounding box.
To estimate the perturbation term in high Reynolds numbers, the turbulent model
could be used, considering its simplicity. In this model the turbulence is considered as a
pure diffusive phenomenon, in consequence tow additional transport equations should be
included:
The first one, named turbulent kinetic energy:
(5.48)
(5.49)
Where the production term is:
(5.50)
Thesis - Pablo Gomez Page 75/138The second one, named turbulent dissipation rate:
(5.51)
(5.52)
Where the constants in the model are defined in Table 5.1.
Table 5.1. Turbulent model constants [Comsol 4.2 documentation].
Constant Value
0.09
1.44
1.92
1.0
1.3
The relative velocity of the fluid respect to the mobile blades depends on rotor angular
speed; for land gas turbines used in power generation the rotor speed is constant, then the
relative speed varies only with the mass flow. The velocity triangle provides a way to
properly define the boundary conditions for mobile blades, such that:
(5.53)
Where is the relative velocity, is the fluid absolute velocity, and is the blade
tangential velocity.
5.4 Heat transfer model
The sources of heat inside a gas turbine are the gases streams. The atmospheric air
incoming into the compressor is first heated during the compression process, then the
addition of the fuel and subsequent combustion raise the temperature to its maximum at
combustion chambers, the combustion stream flows through the hot section heating liners,
transition pieces, stationary nozzles and airfoils, but at the same time a cooling stream
from the compressor discharge is flowing inside and around the gas turbine components,
keeping the materials below their temperature design limits.
The heat is transferred from the hot stream to the cool stream through the components, by
conduction, convection and radiation. In order to calculate the material properties for the
solid mechanics model, the temperature distribution inside the material is required.
Fig 5.5 shows schematically the heat transfer through the substrate - TBC system,
considering the hot and cold streams. The boundary conditions are defined by convection,
radiation and conduction heat transfer. The conduction through the solid system is defined
Thesis - Pablo Gomez Page 76/138using the individual thermal properties of each layer, and the contact thermal resistance
between two adjacent layers. In this model the contact thermal resistance is
estimated using a high conductivity layer model, where an adequate adhesion between
layers is assumed. The temperature continuity through the interfaces offers the worst
possible case for the wear mechanisms to be considered.
Hot Stream
Cold Stream
Fig 5.5. Scheme of heat transfer through the substrate – TBC system.
Thesis - Pablo Gomez Page 77/138The governing equation for heat conduction is given in the Lagrangian form by:
(5.54)
Where is the temperature, is the function of heat sources, is the thermal
diffusivity, which is the measure of the thermal inertia and relates to the capacity of a
material to conduct and accumulate heat:
(5.55)
Where is the thermal conductivity, is the density, and is the specific heat capacity,
which measures the stored heat:
(5.56)
Considering steady state, Eqn 5.55 is written as:
(5.57)
The boundary conditions of the heat conduction equation are defined in terms of the heat
sources. The heat could be transmitted from a body to another by conduction, convection,
radiation or a combination of two or more of them:
The heat transmitted by convection per unit of area is obtained from:
(5.58)
The heat transmitted by radiation per unit of area is obtained from:
(5.59)
The heat transmitted by conduction per unit of area is obtained from:
(5.60)
Where is the thermal conductivity of a material in the direction .
In order to estimate the convection coefficient, common experimental relation between the
Reynolds, Nusselt and Prandtl numbers are used in this model for gas turbine [1, 78].
The Nusselt number is defined as the relation between convection heat transfer and
conduction heat transfer as:
(5.61)
Thesis - Pablo Gomez Page 78/138Where is the thermal conductivity of the fluid, which could be calculated using the
Sutherland’s law according to Eqn 5.38.
The Prandtl number is defined as the relation between viscous diffusion rate and thermal
diffusion rate as:
(5.62)
Where is the dynamic viscosity of the fluid, which could be calculated using the
Sutherland’s law Eqn 5.37.
For gas turbine it is common to define the Nusselt number in term of a Reynolds number
fraction , which is obtained considering the maximum gas turbine output; such that:
(5.63)
Where:
(5.64)
Where is the Reynolds at base load, and the Reynolds at any other load.
Fig 5.6 shows the convection coefficient for a gas turbine airfoil as a function of the
position along the airfoil surface, where the reference Nusselt is gives by:
(5.65)
Thesis - Pablo Gomez Page 79/138Fig 5.6. Gas turbine airfoil convection coefficient [78].
It is important to note that if the optical absorption and scattering coefficients for YSZ are
superimposed to the black-body radiation curves for a number of temperatures Fig 5.7, the
yttria-stabilized zirconia is, apart from scattering, transparent to radiative transport at the
wavelengths pertinent to a combustion environment [14].
Thesis - Pablo Gomez Page 80/138Fig 5.7. Optical absorption and scattering coefficients for YSZ. Superimposed are the
black-body radiation curves [14].
From Fig 5.7, there is an important consideration to estimate the amount of heat
transferred by radiation from the hot stream to the coating; it is that no heat dissipation
occurs at top coat apart from scattering, so radiation impacts directly on TC – TGO
interface. Some studies have proved that at normal operating conditions the TC – TGO
interface provides a fast increase of the absorptance up to [79].
The analysis above suggests that by increasing the emissivity of the TC it is possible to
reduce the heat transfer to the substrate and consequently the temperature through the
thermal protection system, allowing higher ignition temperatures. Some of the most
interesting candidates to increase the emissivity of the TC at high temperatures are rare-
earth oxides such as Sm O , Tm O , Er O , which provide high emissivity, oxidation
2 3 2 3 2 3
resistance and melting point [14, 82].
5.5 Wear model
Delamination is probably the most critical and studied failure mode of composite materials.
In composites delamination is generally assumed to take place at the interface between
adjacent layers and is treated as a fracture process between anisotropic layers, rather
than to consider it, more precisely, as a fracture between constituents or within one of the
constituents [75].
Delamination of TBCs can onset due to manufacturing (surface geometry, powder
preparation, contamination, oxidation, residual thermal stress and others) and continues
Thesis - Pablo Gomez Page 81/138as a consequence of evolution of mechanical, chemical and microestructural properties of
the substrate – TBC system over thermomechanical fatigue cycles as shown in Section
3.2. Furthermore, delamination may be induced by interlaminar shear and normal stresses
associated to some geometrical configurations such as free edges, curved sections as well
as tapers and transitions. Whichever its cause is, delamination of the TBC may originate
and grow under service conditions leading to the premature fail of the thermal protection
and affecting the service life of the hot gas path components in gas turbines.
The wear mechanism of substrate – TBC systems is very complex considering the
competitive mechanisms involved in the thermal, mechanical, chemical and microstructural
properties evolution of the coating and the superalloy.
The TGO growth and undulation affects the amount of residual stresses in the system, but
at the same time residual stresses depend on the manufacture process. The high
temperature creep of the TC and BC use an important part of the crack potential energy as
plastic deformation, affecting the system geometry and stresses distribution; in fact, there
is competition between creep and cracking of the TBC during the high temperature portion
of the thermomechanical cycle. Moreover, stress relaxation by creep would affect the
growth strain and must influence the undulation of the TGO [7].
However, not only stresses associated to TGO growth and undulation and thermal
stresses caused by differences in thermal expansion coefficients and heat transfer must
be considered in the wear scenario, since the active load from the inertial forces and fluid
mechanics, described in Section 5.3, acting over the hot gas path components are also
important
Along a single thermomechanical cycle, the variations in temperature and thermal
expansion coefficients cause a stress inversion in the substrate – TBC system as shown in
Section 2.4. However, the microstructural evolution of the TC as a consequence of
sintering and creep, affects the dynamics, considering the volumetric changes associated
to sintering, which by itself causes compressive stresses at TC – TGO interface.
To determine the amount of potential crack energy available in the substrate – TBC
system, the distinction between reversible and irreversible work must be established,
considering that the excess of energy inside the system, which is the energy not involved
in the system configuration, is the responsible for system deformation and cracks
nucleation and growth.
The internal energy is defined as the total energy contained in a thermodynamic system,
which is the required energy for a system to exist, but excludes the energy to displace the
system’s surroundings. The internal energy can be changed by energy interchanges
through the system’s boundaries, i.e. heat transfer, work and mass transport.
From the third postulate of thermodynamics, the complete information of a system at
equilibrium is included in the internal energy as a function of entropy , volume and
the number of moles of each of its constituents [62].
(5.66)
Thesis - Pablo Gomez Page 82/138Where the function satisfy the following properties:
It is additive over its constituent elements:
(5.67)
It is continuous and differentiable:
Is well defined everywhere (5.68)
It is monotonically increasing function of :
(5.69)
Also it is possible from the second and third properties to invert the relation
to found a unique function, which includes the complete thermodynamic
information of a system:
(5.70)
Which also has the same properties of the function .
The entropy seems strange at first sight, because this physical quantity is not conservative
and refers to the statistical mechanics-like behavior of a thermodynamic system. However,
Ludwig von Boltzmann provided a simple but powerful definition:
(5.71)
Where is the Boltzmann’s constant, defined as the ideal gas constant divided by the
Avogadro’s number, and is a number that defines the number of ways in which a system
can be arranged keeping constant the internal energy, the volume and the number of
moles.
Considering a solid crystal composed of atoms regularly arranged in a lattice with positions
more or less fixed, if the positions of the atoms are switched the microscopic state remains
constant; in this case the possible arranges are determined by how the energy is
distributed; If the ways in which the energy is distributed is found, it is possible to obtain
the entropy [62].
The differential form of the relation , could be written as:
(5.72)
Where considering that the heat transfer is driven by temperature changes, is the
reversible work of the system, and that chemical potential depends on the change of
number of moles [62], then:
Thesis - Pablo Gomez Page 83/138(5.73)
(5.74)
(5.75)
Considering the conjugate thermodynamic forces above, Eqn 5.71, could be written as:
(5.76)
From Eqn 5.67 and Eqn 5.70, it is possible to derive all thermodynamic information about
a system, for instance, mechanical or thermal equation of state. However, in some cases it
is convenient to use other independent variables besides the entropy and volume,
according to the condition in which a transformation of the internal energy of a system
occurs [62].
The Legendre transformation provide the adequate approach to find functions that contain
all the thermodynamic information of a system in terms of the slopes defined in Eqn 5.73,
5.74 and 5.75. Then, using the equation of a straight line it is possible to find a
function that allows to uniquely obtain , , and . Those functions are called
thermodynamic potentials. Fig 5.8, shows the Legendre transformation for .
Fig 5.8. Legendre transformation for .
From the transformation in Fig 5.8, it is possible to write , known as Helmholtz state
potential as:
(5.77)
Thesis - Pablo Gomez Page 84/138Where considering Eqn 5.73:
(5.78)
Using a similar analysis for , and , it is possible to obtain:
The Enthalpy:
(5.79)
And the Gibbs state potential:
(5.80)
The equations above provide a way to understand how the transformations of the internal
energy of a system occur, and which is the best state potential to be considered taking into
account that some of the independent variables remain constant during the transformation.
Those generalized thermodynamic potential also allow obtaining mechanical or thermal
state functions.
Considering a solid material with a constant number of moles as a thermodynamic system,
the change of its internal energy is associated to the applied work and the heat transfer;
where reversible and irreversible changes of the amount of energy may occur:
(5.81)
The reversible work for a solid material is related to the elastic energy accumulated in the
system, and the irreversible work is dissipated as permanent strain and new surfaces and
cannot be recovered without the application of additional energy.
Coupling between plastic and damage dissipation, in general, can be done in two ways: in
the first one, a single plastic dissipation potential coupled with damage is used (strong
coupling), in the second, two or more dissipation potential functions: plastic dissipation
potential and damage dissipation potential, are assumed and independently defined (weak
coupling) [8].
The application of the weak coupling is the most appropriate for elastic moderate plastic
materials, composites and in general materials where it is not possible to found a single
coupled dissipation potential, the weak coupling is based in the existence of at least two
dissipation functions: the plastic and the damage dissipation surface; assuming the
maximum dissipation principle: ‘Actual state of the thermodynamic forces is that which
maximizes the dissipation function over all other possible admissible states’, [8, 24, 26 and
74].
To determine the dissipation functions at operating conditions for the Substrate – TBC
system, which is submitted to both mechanical work and heat transfer Fig 5.9, elastic and
plastic deformations, creep, sintering, oxidation and cracks nucleation and propagation,
must be considered
Thesis - Pablo Gomez Page 85/138Substrate – TBC
Fig 5.9. Substrate – TBC system submitted to both mechanical work and heat transfer.
Considering that number of moles still constant, the change of the internal energy of the
system could be defined as:
(5.82)
Where reversible and irreversible heat transfer and mechanical work may occur; the
irreversibilities in the case of the Substrate – TBC system, come from the dissipation
functions, which cause a difference in the amount of energy through the thermomechanical
cycle.
In the case of a material body, the tensor gradient of the deformation is written in
lagrangian or material form for as:
(5.83)
Where the components describe the deformations in the direction respect to the
material coordinate .
From the definition above the components of the strain tensor could be written as:
(5.84)
From Duhamel-Hooke’s Law, which considers the deformation of linear elastic solid
material, the relation between the stress and strain tensor is written as:
(5.85)
Where and are the initial stress and strain respectively, is the thermal expansion
coefficient, is the temperature, is the reference temperature for the thermal
expansion and is the fourth order elastic tensor, defined in components as:
Thesis - Pablo Gomez Page 86/138(5.86)
This is the constitutive relation between stresses and strains. In general for an anisotropic
material, it is possible to rewrite using the symmetries:
(5.87)
Then:
(5.88)
The stress and strain tensors could be related at constant volume to the mechanical term
of Eqn 5.82, allowing writing the elastic energy:
(5.89)
Where are the components of the stress tensor and are the components of the
strain tensor related to Helmholtz energy as:
(5.90)
However, the second law of thermodynamics imposes a restriction on the constitutive
relation, which could be expressed in terms of the Clausius-Duhem inequality as:
(5.91)
Where considering the state variables and its conjugate thermodynamic forces, the
Helmholtz energy is written as:
(5.92)
Where:
(5.93)
Assuming that is constant and independent of temperature, as suggested by the
Dulong-Petit law.
Where:
Thesis - Pablo Gomez Page 87/138(5.94)
Then the Helmholtz energy for an elastic material could be written such that:
(5.95)
Generally speaking, if all the state variables and their conjugate thermodynamic forces
are considered, it is possible to obtain the Helmholtz energy of the system using the
weak coupling for the substrate – TBC system wear, such that:
(5.96)
Where:
(5.97)
Assuming that only the useful energy in the volume could be transformed into surface
energy, the crack potential energy available for a crack nucleation and growth in the
substrate – TBC system is equaled in this model to the Helmholtz energy, where the
irreversibilities in the substrate – TBC system are related to the TC sintering, creep, and
plastic deformation, BC and substrate creep and plastic deformation and TGO growth and
undulation.
Then following prior works (Malvern (1969); Lemaitre and Chaboche (1994); Hansen and
Shreyer (1994); Zhu and Cescotto (1995); Coussy (1995); Chaboche (1997); Hayakawa
and Murakami (1997); Doghri 2000; Abu Al-Rub and Voyiadjis (2003)) the internal state
variables of the substrate – TBC system are analyzed in the next sections to obtain a
coupled formulation of the crack potential energy for delamination and spallation cracks
nucleation and growth.
5.5.1 Plastic deformation
In case of small strains, the plastic strain could be implemented using the additive
decomposition of the strain, in which the plastic strain tensor describes the state of the
plastic deformation, thus:
(5.98)
Where is the plastic strain tensor due to internal energy irreversible dissipation. Note
that the fourth order elastic tensor has been considered here independent of the plastic
deformation; this consideration comes from experimental results and is commonly used.
However, it is important to highlight that plastic deformation occur only when the yield
strength is exceeded, thus it is important to defined the dissipation surface associated
to plastic deformation, ensuring that the elastic-plastic problem still having a solution
Thesis - Pablo Gomez Page 88/138Considering the Clausius-Duhem inequality, the plastic dissipation surface could by
obtained using yield function , such that this function defines the limit between the
reversible and irreversible deformations, thus:
(5.99)
Where is the plastic strain rate, is the vector that indicate the direction of the
deformation, and is the state variable for the kinematic hardening
Considering the internal state variables and its conjugate thermodynamic forces, the
plastic component of the Helmholtz energy could be written as:
(5.100)
Where:
(5.101)
Where the constitutive relation between plastic stress and strains is written in terms of the
tangent tensor :
(5.102)
To determine the energy dissipation terms it is necessary to find an adequate function in
terms of the internal state variables and , assuming that state variables are
independent each other, and using the Lagrange multiplier , introduced from the theory of
functions of multiple variables, it is possible to write:
The plastic strain rate as:
(5.103)
Where:
(5.104)
And the kinematic hardening as:
(5.105)
Where:
(5.106)
Thesis - Pablo Gomez Page 89/138Then the Helmholtz energy could be obtained solving the equation system:
(5.107)
Where the Lagrange plastic multiplier is determined directly using the following Kuhn-
Tucker complementary conditions, proposed by Kuhn and Tucker in 1951 [77]:
(5.108)
. (5.109)
(5.110)
The first condition states that plastic flow rate is always non-negative; the second ensure
that stress state is always within the yield surface or on the yield surface. The last
condition can be met for two different loading cases. If the stress is the case of elastic
loading/unloading, then . If, however, there is a plastic loading , then the last
condition forces the stress state to remain on yield surface ; then there can
be no evolution of the yield surface ; this implication is the consistency
condition for plasticity and it will be used to obtain the evolution of the plastic multiplier
[77].
Formulation above allows satisfying the flow condition defined by the Clausius-Duhem
inequality.
The yield function to be used through this model is based on the von Misses stresses,
which is commonly used in plasticity, and is defined as:
(5.111)
Where are the components of the stresses deviator tensor, defined as:
(5.112)
Then:
(5.113)
5.5.2 Creep
Is important to highlight that creep has a pure dissipative effect on the Helmholtz energy,
which in fact is a cumulative phenomenon lead by diffusion, where heat and vacancies,
Thesis - Pablo Gomez Page 90/138are transported by the driven conjugate thermodynamic forces [25]. However, in this model
a simplify dependency of creep on deformation rate and its conjugate
thermodynamic force is considered from the classical diffusive steady-state creep rate
equation described in Section 3.2.2:
(5.114)
Where is the appropriate diffusion coefficient, is the shear modulus, is the Burger’s
vector, is the Boltzmann’s constant, is the absolute temperature, is the grain size,
is the applied stress, is the exponent of the inverse grain size, is the stress exponent,
and is a dimensionless constant.
The diffusion coefficient is given by:
(5.115)
Where is a frequency factor, is the activation energy, and is the gas constant.
Then:
(5.116)
Analogous to the plastic behavior only the components of the second order tensor of
stresses related to the distortion are considered, thus:
(5.117)
Here the deformation time is directly related to the operating conditions; so the dissipative
effects of creep deformation, directly affects the residual stresses state, one of the most
important consequences of this effect is the reduction of interfacial stresses associated to
manufactured process and TGO growth as the exposure time increases.
5.5.3 Thermal expansion stresses
The different materials that form the substrate – TBC system present differences in their
thermomechanical properties as shown in Section 3; this is responsible for mismatch
stresses at the interfaces, which is one of the main contributors to system loads.
The linear thermal expansion coefficients of system’s components were determined by
dilatometry using a 1 inch length α-Al O pattern; Fig 5.10 shows the thermal expansion
2 3
results, the alumina curve corresponds to the calibration run, and also is the TGO thermal
expansion curve.
Thesis - Pablo Gomez Page 91/1380.000018 °C^-1
0.000016 °C^-1
0.000014 °C^-1
0.000012 °C^-1
0.00001 °C^-1
0.000008 °C^-1
0.000006 °C^-1
0.000004 °C^-1
0.000002 °C^-1
0 °C^-1
0°C 200°C 400°C 600°C 800°C 1000°C 1200°C 1400°C
α-Al2O3 GTD 111
ZrO2 - 8%Wt. Y2O3 (Amdry 204NS-G) Ni22Cr10Al1.0Y (Amdry 962)
Fig 5.10. Linear thermal expansion coefficients of substrate - TBC system components.
Thermal expansion deformation could be considered reversible, because of its
dependence with temperature. However, the additive property of the system loads, could
drive to irreversibilities.
To calculate the effect of thermal expansion mismatch at TGO – BC interface, it is possible
to consider the equilibrium between the elastic strain and the thermal strain, thus:
(5.118)
Where considering a pure elastic strain:
(5.119)
And:
(5.120)
Replacing Eqn 5.119 and 5.120 in Eqn 5.118:
(5.121)
From a single forces balance:
Thesis - Pablo Gomez Page 92/138(5.122)
Replacing Eqn 5.121 in Eqn 5.122:
(5.123)
For a very thin oxide film , Eqn 5.123 could be rewritten as:
(5.124)
The above analysis could be extended to TGO – TC interface, thus:
(5.125)
From Fig 5.10 it is possible to note that mismatch at TGO – TC interface is less critical
than mismatch at TGO – BC interface.
5.5.4 Top Coat sintering effects
Top Coat sintering has a strong effect on the system behavior. However, the time scale of
the variation uses to be different from other dissipative process, making necessary to
define adequately the time scales for the different acting mechanisms.
The sintering process is drive by temperature, which active the diffusion process
responsible for interlamellar (APS) or intercolumnar (EB-PVD) necks formation as
mentioned in Section 3.2.2. The change in thermal and mechanical properties is the main
effect of the sintering, and then to account its effect, the variation of this properties respect
to time is considered in this model, in this case considering the time scale difference, and
the amount of thermal energy consumed from the available internal energy, the dissipation
energy could be written as:
Fig 5.11 shows the porosity variation against the time exposure at , using inconel
625 samples coated with the system: HVOF Amdry 692 (Bond Coat) + APS Amdry
204NS-G (Top Coat). Fig 5.12 shows the micro hardness profiles at room temperature
from TC to substrate of the coated samples, using a load of and loading
time.
Thesis - Pablo Gomez Page 93/13835
30
25
)
%
( 20
y
tis
o 15
r
o
P
10
5
0
0 200 400 600 800 1000 1200 1400 1600 1800
Time (h)
Fig 5.11. Porosity at different high temperature exposure times.
1600Hv
1400Hv
1200Hv
1000Hv
800Hv
600Hv
400Hv
200Hv
0Hv
0µm 500µm 1000µm 1500µm 2000µm 2500µm 3000µm 3500µm 4000µm 4500µm
t=1700h t=1000h t=800h t=600 t=400h t=200 t=0h
Fig 5.12. Micro hardness profile at room temperature from TC to substrate of the coated
samples at different exposure times.
Note from Fig 5.11 and 5.12 that porosity reduction directly affects the hardness of the TC.
5.5.5 Thermally Grown Oxide growth and undulation
The TGO growth stresses comes from the volumetric changes associated to the oxidized
state of the elements added to BC for TGO formation, and thermal expansion mismatch at
TC – TGO and BC – TGO interfaces, both affected by TGO undulation.
Thesis - Pablo Gomez Page 94/138The TGO undulation affects the growth and thermal expansion stresses distribution, which
could be divided in normal and tangential stresses respect to the TC – TGO and BC –
TGO interfaces; in general tangential and normal components of the stress could be
considered strongly depend on undulation slope, taking in to account that on a flat
interface, normal stress just not appear, because there are not restriction to TGO growth
and thermal expansion.
Fig 5.13 shows the tangential and normal forces associated to TGO growth and
thermal expansion at TC – TGO and BC – TGO interfaces.
TC
TGO
BC
Fig 5.13. Normal and tangential forces associated to TGO Growth and thermal expansion.
Considering that TGO growth from BC, the volumetric change should be related to the
molecular volume of the oxide and bond coat [47], then:
(5.126)
Where and are the molecular metal and ion volumes. The expression above could
be also expressed in terms of the Pilling-Bedworth ratio , such that:
(5.127)
Where for pure α-Al O TGO, which implies that TGO growth under compressive
2 3
stresses. However, Evan et al in 1978, shown that stress induce a reduction in TGO
growth rate, and propose a formulation for the interfacial stress as:
(5.128)
Thesis - Pablo Gomez Page 95/138Where is the Boltzman constant, is the temperature, is the local oxygen partial
pressure and is the fraction rate, which is equal to zero in the case of an unstressed
oxide growth.
However, considering the difficult to establish the local partial pressure and the fraction
rate; in this model a simplify growth stress is used on the base of experimental
observations of the TGO growth rate, and the mechanical interaction at TGO – BC and
TGO – TC interfaces.
Now if the undulation is modeled as:
(5.129)
Where the angle between the coordinate systems is defined as:
(5.130)
The tangential forces at TGO-BC interface, could be obtained from the growth stress and
the thermal expansion stresses, in this case:
(5.131)
Where:
(5.132)
The above analysis could be extended to TGO – TC interface, thus:
(5.133)
Now to determine the normal forces at TGO – BC and TGO – TC interfaces, it is important
to note that deformations are directly related to TGO thickness and the undulation
slope as mention above.
If a totally restricted TGO growth is considered, the total deformation of the system could
be obtained analogous to the sum of springs in series, such that:
(5.134)
Where:
(5.135)
However, considering that growth restriction is function of the undulation slop, the amount
of restriction could be obtained using the angle defined en Eqn 5.130, then:
Thesis - Pablo Gomez Page 96/138(5.136)
Note from the equation above, that on a flat surface then .
The contribution of TGO undulation and growth to the crack potential energy could be
included from analysis above, such that:
(5.137)
Now considering the time scale difference as mentioned in Section 5.5.4, the contribution
could be defined in the global analysis in terms of initial stresses and strains, including a
time dependence on a parametric time defined from experimental observations of the TGO
evolution.
Fig 5.14 shows the TGO thickness at different exposures time at in an
oxidizing atmosphere, using inconel 625 samples coated with the system: HVOF Amdry
692 (Bond Coat) + APS Amdry 204NS-G (Top Coat)., results evidence a subparabolic
growth beyond as predicted by Evans [27].
7.0
6.0
)
m
5.0
µ
(
s
s
e 4.0
n
k
c
ih 3.0
T
e
d ix 2.0
O
1.0
0.0
0 200 400 800 1000 1700
Time (h)
Fig 5.14. Measured TGO thickness at different exposures time at .
It is important to highlight that TGO undulation is a consequence of the originally
roughness of the BC before the application of the TC and is affected directly by the TGO
growth and creep of the TC and BC. Modeling the undulation as a sinusoidal function, the
wavelength could be considered as a function of the original roughness, and the amplitude
must be considered as a function of TGO thickness and TC and BC plastic deformation.
Thesis - Pablo Gomez Page 97/138Fig 5.15 shows the undulation of the TGO measured at different exposure times at
in an oxidizing atmosphere, using inconel 625 samples coated with the system: HVOF
Amdry 692 (Bond Coat) + APS Amdry 204NS-G (Top Coat). Note that wavelength of the
undulation remains constant while the amplitude increases as exposure time increases,
Fig 5.16 shows the mean amplitude at different exposure times, which could be used to
define the adequate time scale for TGO growth.
150
) 100
m
µ
(
e
d 50
u
tilp
m 0
A
0 150 300 450 600 750 900 1050 1200
n
o
ita -50
lu
d
n
U -100
t=0h t=200h t=800h t=1000h t=1700h
-150
Fig 5.15. Undulation of the TGO along the interface at different high temperature exposure
times.
160
)
m 140
µ
(
e d 120 109 114
u
tilp 100
m 80 81
A
n 60
o
ita
40 43
lu 31
d
n 20
U
0
0 200 400 600 800 1000 1200 1400 1600 1800
Time (h)
Fig 5.16. Undulation of the TGO at different high temperature exposure times, showing the
standard deviation of the amplitude along the interface.
The strength of the TGO boundaries or adhesion depends on the thermomechanical
properties and defects of the material system, where interfacial defects have a significant
contribution; so using the results above, it is possible to obtain the adhesion strength in the
TC – TGO and BC – TGO interfaces assuming a quasistatic loading, and considering that
crack growth close to the interface but not through the interface, it is considering the
Thesis - Pablo Gomez Page 98/138ultimate strength of the material, where the crack is growing quasistatically, and the
stresses associated to TGO kinetics and geometry.
5.5.6 Coupled state potentials
The coupled Helmholtz state potential can be written following the analysis above as:
(5.138)
Is important to account that this model suppose a pseudo-equilibrium where the
temperature and the state of the stress remains constant for an adequate parametric
interval of time, which depends on the loading – unloading process, such that:
(5.139)
(5.140)
And
(5.141)
5.5.7 Energy release
The analysis above allows calculating the potential crack energy available in the substrate
– TBC system, such that the release energy from a crack extension is written as:
(5.142)
Where considering the release energy along the crack:
(5.143)
Thesis - Pablo Gomez Page 99/1386 Numerical Implementation of High-Temperature Wear
Model in Non-Steady State Gas Turbine
Through the first part of this chapter the finite element method and the virtual crack closure
method are formally presented. Then the numerical solution of the model proposed in
Section 5 is developed in the case of a number of gas turbine components under non-
steady sate, showing the relation between real components’ wear and model predictions.
6.1 The finite element method
The finite element method (FEM) has gained popularity for its simplicity and ability to
model complex domains. The programming of the finite element method has been a task
that many software development firms have taken as their mission to provide the
engineering market with packages that are capable of solving a multitude of problems
ranging from simple structure of fluid dynamics problems to complex problems including
coupled fields of electricity, mechanics of structures, fluid dynamics, and more.
The FEM is based on the weighted residual methods. The main difference of the finite
element method is that the proposed solution functions have generalized coordinates,
coefficients, which are equal to the physical quantity of the unknown variable at some
points inside the domain. Hence, the proposed solution becomes an interpolation function
for the values intermediate to those points.
From a mathematical point of view, the FEM looks for solutions to partial differential
equations (PDE’s) on function spaces of finite dimension, instead of infinite spaces, which
apply the Galerkin method for converting a continuous operator problem (such as a
differential equation) to a discrete problem. In principle, it is the equivalent of applying the
method of variation of parameters to a function space, by converting the equation to a
weak formulation, allowing the transformation of a PDE into a linear algebra problem which
can be handle using most simple methods, e.g. by applying constraints to the function
space to characterize the space with a finite set of base functions. In order to verify the
solution, the dot product between the differential operator and each function from the finite
space has to be null, this method allow to obtain a projection of the solution to the PED
over the defined finite function space.
It is important to highlight that selected finite function spaces have to be adequate to allow
the construction of a linear combination of the base function of the space, which represent
the projected solution of the PDE over the finite function space
In general the FEM looks for a solution of the dependent variables of the PDE over finite
function spaces , which have structure of Banach reflexive space; so for all ,
.
Considering that if is a Banach space over the fields or with a norm , all Cauchy
succession respect to the metric in has a limit in for all
, this ensures that the space is complete with respect to the norm.
Thesis - Pablo Gomez Page 100/138The above ensures that the space , is a Hilbert space, where the parallelogram identity
is satisfy for all allowing the construction
of a solution for PDE on the base functions of .
The FEM was originally proposed as a generalized extension of the structural trusses
method employed to solve complex structures, which uses individual beam elements with
twelve degrees of freedom each one, in the most general three dimensional scenarios, i.e.
three directions and three angles per beam end point (nodes).
Considering a force action on a material particle Fig 6.1, the work of this force is written as:
(6.1)
Fig 6.1. Work of a force .
Considering a moment of a force acting on a material particle Fig 6.2, the work of this
moment is written as:
(6.2)
Where:
(6.3)
Thesis - Pablo Gomez Page 101/138Fig 6.2. Work of a moment on a material particle.
The concept of virtual displacements is introduced to describe the displacement of the
nodes independently of the actual restrictions, in order to keep this approach inside the
linear kinematics, those displacements must be enough small. Considering the applied
loads on the nodes, it is possible to define the virtual work of a force as the dot product
of the force and the displacement vector tangent to the movement trajectory, i.e.:
(6.4)
Where is a virtual displacement defined as follows:
(6.5)
If a force acts on a particle as it moves from point to point , for each possible trajectory
that the particle may take, it is possible to obtain the total work done by the force along
the path. The principle of virtual work, which is the form of the principle of least action
applied to these systems, states that the path actually followed by the particle is the one
for which the difference between the work along this path and other nearby paths is zero.
The formal procedure for computing the difference of functions evaluated on nearby paths
is a generalization of the derivative known from differential calculus, and is termed the
calculus of variations.
Generally speaking, for the forces acting on a point of a body and considering
a virtual displacement , the virtual work could be obtained as:
(6.6)
This is an extension of the principle of parallelogram applied to concurrent forces.
Thesis - Pablo Gomez Page 102/138Principle of virtual work is used to obtain the weak formulation of elastic problem, which is
desired to obtain a solution from the application of linear algebra concepts, in this case the
virtual displacements conform the weight function of the weak formulation which is defined
below, considering the strong or differential formulation of the problem:
(6.7)
Where is a differential operator, is the unknown function, and is a known
mathematical function, since and are Banach reflexive spaces and , where
is the dual space of
Considering in general a linear operator , where:
(6.8)
Where is the dual space of , then considering a test function such that the conjunct
of all of them forms the Banach space , it is possible to write:
(6.9)
This expression is the bilinear form of the original formulation Eqn 6.8.
Taking into account that integration and differentiation are linear operators; the same
concept could be extended to Eqn 6.7.
In general, from the bilinear formulation of the PDE it is possible to write the weak form of
the problem using the integration by parts, where the resulting formulation can be handled
using concepts from linear algebra. The introduction of the test functions, related to virtual
displacements, allows writing the weak formulation of the PDE elastic problem.
Fig 6.3 helps illustrate the construction of the weak formulation for the uniaxial elastic
problem.
Thesis - Pablo Gomez Page 103/138Fig 6.3. Uniaxial elastic problem.
Under equilibrium conditions:
(6.10)
The normal force is given by:
(6.11)
Where:
(6.12)
Where is the Young’s modulus and is the displacement in position and direction .
Replacing Eqn 6.12 in Eqn 6.10:
(6.13)
Considering Eqn 6.13 and Fig 6.3, the uniaxial elastic problem could be written as:
(6.14)
Such that:
Thesis - Pablo Gomez Page 104/138(6.15)
Where , and , with and defined as Banach reflexive spaces.
Using the virtual displacement , as a test function, the weak
formulation of the elastic problem could de written as:
(6.16)
Where integrating by parts the first term:
(6.17)
Note that applying integration by parts gets rid of the second derivative term in the strong
formulation of the elastic problem Eqn 6.14.
In general is possible to use different test functions to obtain the weak formulation of the
generic problem presented in Eqn 6.7. Multiplying Eqn 6.7 by the test function and
integrating over the domain:
(6.18)
Where integrating by parts and applying the Gauss divergence theorem, the weak
formulation could be obtained.
The choice of the class of functions , determines whether Eqn 6.7 has a solution and
whether this solution is unique. The common practice of applying integration by parts to
Eqn 6.9 gets rid of the second derivative term. Integration by parts is derived by applying
the Gauss divergence theorem, allowing the application of linear algebra concepts to solve
the differential problem.
In general the test function could be chosen arbitrarily. The Galerkin method proposes
the approximation of the unknown function and the test function as a linear combination of
base functions:
(6.19)
Where the coefficients are to be determined, and the basis function must be
linearly independent to form an adequate base for the solution to PDE on function spaces
of finite dimension, instead of infinite spaces. Furthermore must be such that an
arbitrary function in the solution space can be approximated with arbitrary accuracy, by
linear combination of a sufficient number of basis functions.
The function must be chosen such that satisfies the essential boundary
conditions of the PDE. In general this means that in the boundary is equal to the
value of the unknown function in the boundary and .
Thesis - Pablo Gomez Page 105/138To determine the coefficients the unknown function is approximated as a linear
combination of the basis functions, and the test function is selected as a linear
combination of basis functions, then these functions are replaced in the wake formulation,
these lead to a linear system with equations and unknown coefficients that could be
solved using linear algebra.
The FEM offer a constructive way to obtain the basis functions and compute the
integral form of the PDE in a relative simple way. To do that, the domain is subdivided into
simple elements like: intervals in , triangles or quadrilaterals in and tetrahedral or
hexahedra in .
Each element includes a number of nodal points, where the basis functions or shape
functions are defined, the unknown function is approximated inside the element by a
polynomial function , which is defined in general such as in each node of the element
only the corresponding shape function is unequal to zero, thus only shape function
corresponding to nodal points for each element have a non-zero contribution to the integral
form of the PDE.
6.2 The virtual crack closure method
There are a variety of methods used to compute the strain energy release, the most
commonly used are based on the finite element analysis, and those methods use the
calculation of global forces and global deformations to estimate the amount of energy
available for a crack nucleation and growth [50 and 75].
The virtual crack extension method, for instance, requires only one complete analysis of
the structure to obtain the deformations. The total energy release rate or J-integral is
computed locally at the crack front and the calculation only involves an additional
computation of the stiffness matrix of the elements affected by the virtual crack extension.
The method yields the total energy release rate as a function of the direction in which the
crack was extended virtually, yielding information on the most likely growth direction [50].
For delamination in laminated composite materials where the failure criterion is strongly
dependent on the mixed-mode ratio and propagation occurs in the laminate plane, the
virtual crack closure technique has been most widely used for computing energy release
rates because fracture mode separation is determined explicitly [50 and 75].
The crack closure method is based on Irwin’s crack closure integral, this method assume
that the energy required for a crack extension from to is identical to the energy to
close the crack [80].
Now the virtual crack closure method uses the same assumptions of the crack closure
method. However, this method assume an additional conditions, it is that a crack extension
of from (node ) to (node ) does not significantly alter the state at the
crack tip Fig 6.4. Therefore the displacements behind the crack tip at node are
approximately equal to the displacements behind the original crack tip at node . Further,
Thesis - Pablo Gomez Page 106/138the energy released when the crack is extended by from to is identical
to the energy required to close the crack between location and [50].
Fig 6.4. Virtual crack closure method applied in two-dimensional four-node elements [50].
For a crack modeled with two-dimensional four-node elements, as shown in Fig 6.4 the
work required to close the crack along one element side can be calculated in terms of the
displacements and and their respective forces and as:
(6.20)
Where:
(6.21)
And:
(6.22)
Thesis - Pablo Gomez Page 107/138Fig 6.5. Virtual crack closure method applied in three-dimensional eight-node elements
[50].
Now considering the tridimensional case Fig 6.5, the energy required to close the crack
could be expressed as [50]:
(6.23)
Where:
(6.24)
(6.25)
(6.26)
Where:
(6.23)
Where is the width of the element.
Thesis - Pablo Gomez Page 108/1386.3 Experimental results
Through this section, comparisons between real components and the proposed analytical
model are presented. Some of the findings described in Table 2.1 are specifically
analyzed. Also in this section some interesting predictions from the proposed model are
discussed.
In order to compare the proposed model for substrate – TBC system wear and common
findings in non-steady state operating conditions; the coupled model proposed in Section 5
is numerically solved for the case of 7FA gas turbines operating under non steady state
conditions by using the finite element and the virtual crack closure methods, both
implemented on the commercial software COMSOL Multiphysics 4.2a with the aid of user-
defined subroutines and material properties. The physical models and physical parametric
models to represent the 7FA gas turbine materials and components are prepared using
Autodesk Inventor 2010, also material properties functions and some of the load functions
are implemented in COMSOL 4.2a using the active link between this software and the
open source software OCTAVE 3.2.4.
Most of the materials properties are obtained from References [1, 5, 6, 7, 11, 13, 14, 15,
21, 23, 27, 40, 47, 48, 69, 70 71 and 73], additional material properties are obtained
directly from COMSOL 4.2a libraries, the geometric and time parameters are obtained
from experimental date as described in previous sections and functionally implemented in
COMSOL 4.2a.
Table 6.1 shows the mechanical properties of the system. The geometric and time
parameters are functionally implemented in COMSOL 4.2a.
Table 6.1. Substrate – TBC system mechanical properties.
Material
GTD 111
HVOF Amdry 692
α-Al O -
2 3
APS Amdry 204NS-G
Table 6.2 shows the creep variables of the system. The geometric and time parameters
are functionally implemented in COMSOL 4.2a.
Table 6.2. Substrate – TBC system creep variables.
Material
GTD 111 - -
HVOF Amdry 692
α-Al O
2 3
APS Amdry 204NS-G
Heat transfer model parameters and configuration are obtained directly from COMSOL
4.2a documentation; thermal properties are described in Table 6.3.
Thesis - Pablo Gomez Page 109/138Table 6.3. Substrate – TBC system thermal properties.
Material
GTD 111
HVOF Amdry 692
α-Al O
2 3
APS Amdry 204NS-G
Fluid dynamics and heat transfer models parameters and configurations are obtained
directly from COMSOL 4.2a documentation, the wear model is implemented on COMSOL
4.2a using user defined subroutines to account adequately the competitive dissipation
surfaces as described in Section 5.5.
The boundary conditions of the fluid dynamics and heat transfer models are determined
directly using more than fifteen years of historical data from two EPM 7FA gas turbines
using the General Electric Speed Tronic Mark V control and data acquisition system.
Temperature distribution and fluid forces (lift and drag) acting on different turbine
components are obtained directly from the solution of the flow dynamics and heat transfer
models; also the inertial forces are accounted for and applied to the components models
considering the physical properties of these components and their operating conditions, to
completely establish the loads to be applied in the wear model.
Fig 6.6 shows the tree dimensional flow model of the gas turbine hot section used to
determine the temperature, pressure and velocity fields through the stationary components
of the hot gas path.
Thesis - Pablo Gomez Page 110/138Fig 6.6. 7FA Gas turbine flow model prepared using the image render options of Autodesk
Inventor 2010.
Considering the applied loads on individual components the proposed wear model is
solved to determine the crack potential energy, where the inclusion of the microestructural,
thermal and mechanical properties of the system as a function of the time are considered
using the full parametric model of the substrate – TBC System.
The virtual crack closure method also implemented in COMSOL 4.2a, is used to provide
the crack rate at most critical component locations using a two dimensional parametric
model of the substrate – TBC system in a two steeps post-processing analysis, where the
displacements and acting forces are determined before and after the virtual crack
extension to determine the energy release following Eqn 6.24, 6.25 and 6.26, , which is
evaluated using the parametric model, where the temporal evolution of the substrate –
TBC system is accounted for.
Thesis - Pablo Gomez Page 111/1386.3.1 Liners and transition pieces in 7FA GE gas turbines
Combustion reaction in gas turbines is directly in contact with liner surfaces, this is the
component exposed to higher temperatures in the hot gas path. However, the inner the
combustion section liners are typically the less affected components at non-steady state
operating conditions, where apart of formed deposits from fuel and air contaminants, not
major wear signs are found. The liner is directly connected to the transition piece; these
components together guide the hot stream of combustion gases to the first stage nozzles;
transition pieces are submitted to higher thermal stresses, considering the asymmetric
geometry and fluid flow, which causes high stresses and eventually cracks as shown in
Table 2.1 and Fig 6.7.
Fig 6.7. Common crack found in transition pieces on 7FA gas turbines
To evaluate the case of cracks observed in transition pieces, the proposed analytical
model is applied to the liner – transition piece assembly. Fig 6.8 shows the results of the
fluid dynamics model applied to this section.
Thesis - Pablo Gomez Page 112/138Fig 6.8. Fluid velocity magnitude through one of the fourteen liner-transition piece couples
of a 7FA gas turbine at base load.
From the fluid dynamics model, the lift and drag forces acting over the components are
determined; the boundary conditions for the heat transfer model are defined as well.
The transition piece coupled wear model results are presented in Fig 6.9 and 6.10.
Thesis - Pablo Gomez Page 113/138Fig 6.9. 7FA Distribution of von Mises stresses in transition piece during a trip from base
load.
Fig 6.10. 7FA Distribution of von Mises stresses in Transition piece at common crack
location during a trip from base load.
Thesis - Pablo Gomez Page 114/138It is important to note that maximum von Misses stresses from the wear model applied to
the transition piece, correspond pretty well to the typical crack location showed in Table
2.1 and Fig 6.7.
In Section 6.3.3, a more detailed analysis of the found cracks at transition pieces is
presented, using the proposed coupled state equation and the parametric model of the
thermal protection system to define the most favorable system state for crack nucleation
and propagation.
6.3.2 First stage nozzles and airfoils in 7FA GE gas turbines
The nozzles are responsible for the fluid acceleration inside the turbine section. The first
stationary nozzles receive the combustion gases stream directly from combustors and
orient this stream to the first stage airfoils, so stationary and mobile blades at first stage
are submitted to highly demanding operating conditions inside the gas turbine
First stage nozzles and blades are designed to be replaced during hot gas path
inspections, but due to non-steady state conditions, many times the first stage nozzles
must be replaced during combustion inspections; Table 2.1 presents common cracks
found in this section at early operating hours, some of then use to appear just at the first
gas turbine trip
Fig 6.11 shows the most common crack locations at firsts stage nozzles, formed as a
consequence of multiple thermomechanical fatigue cycles; Fig 6.12 shows a trailing edge
crack, commonly formed in component with very few starts
Fig 6.11. Common crack location at first stage nozzles pressure side and trailing edge.
Thesis - Pablo Gomez Page 115/138Fig 6.12. Typical crack located at the trailing edge of first stage nozzle, found in a
component with very few starts
A single thermomechanical cycle is analyzed at two different severities using the proposed
model. The first one corresponds to a start; base load and normal shutdown cycle Fig
6.13, starting from a non-plastically deformed material. On the second one a start, base
load and trip cycle is considered Fig 6.14, also starting from a non-plastically deformed
material.
Thesis - Pablo Gomez Page 116/138Fig 6.13. Maximum von Mises stresses in 7FA first stage nozzles during a normal
shutdown from base load cycle.
From the solution of the model for a single start, base load and normal shutdown Fig 6.13,
it is possible to note that even considering a new component, the mechanical constrains
and loads the yield strength of the base material is exceeded at common crack locations.
Now from the solution for a single start, base load and trip Fig 6.14, a significant increase
in the maximum von Mises stresses is obtained respect to the normal shutdown cycle,
where the stresses at common crack locations are up to higher in the case of a
trip.
Thesis - Pablo Gomez Page 117/138Fig 6.14. Maximum von Misses stresses in 7FA first stage nozzles during a trip from base
load.
Fig 6.15 presents the temperature distribution in 7FA first stage nozzles at base load
Thesis - Pablo Gomez Page 118/138Fig 6.15. Temperature distribution in 7FA first stage nozzles at base load.
The solution shown above for the first stage nozzles provides an adequate way to
understand early failures at stationary blade trailing edges. From results presented in Fig
6.14 and 6.15 it is also possible to identify problematic areas around the blade’s leading
edge and the curved section of the pressure side Fig 6.16, where TBC delamination and
spallation occur.
Fig 6.16. Evidence of TBC delamination and spallation in a first stage nozzle blade at
leading edge and pressure side.
Thesis - Pablo Gomez Page 119/138First stage airfoils are submitted to extreme conditions of temperature, pressure and
velocity, where local Mach number reach values close to . However, additional loads
from inertial forces must be considered to fully understand the wear scenario of these hot
gas path components. First stage airfoils most of the times define the state of the art in gas
turbines technologies since failures at this section use to be catastrophic, so airfoils must
be resistant and reliable like no other component inside the gas turbine. A similar analysis
could be extended to mobile compressor blades, keeping in mind that compressor section
works at relatively lower temperatures and Mach numbers below ,
Most common wear signs in first stage airfoils are related to delamination and spallation of
the TC Fig 6.17. However, blades submitted to high number of thermomechanical cycles
show characteristic superficial cracks from the leading edge to the low pressure side Fig
6.18.
Fig 6.17. 7FA first stage airfoil with severe TC delamination and spallation at leading edge
and pressure side.
Thesis - Pablo Gomez Page 120/138Fig 6.18. 7FA first stage airfoils with characteristic superficial cracks at the leading edge
To solve the proposed model for the 7FA first stage airfoils, two additional conditions must
to be accounted for: the first one is the angular velocity of the turbine rotor, where the
velocity triangle must be solved to properly define the boundary conditions for the fluid
dynamics and heat transfer models, the second is the calculation of inertial loads
associated to the airfoil rotation.
Fig 6.19 shows the maximum von Mises stress in the 7FA first stage airfoils in the case of
a start, base load and normal shutdown cycle; Fig 6.20 shows the maximum von Mises
stress in the 7FA first stage airfoils in the case of a common start, base load and trip cycle.
Thesis - Pablo Gomez Page 121/138Fig 6.19. Von Mises stresses in a 7FA first stage airfoil during a normal shutdown from
base load.
Results in Fig 6.19 show a good correlation with experimental observations since the
critical zones according to the model correspond to areas where the TC no longer exists in
Fig 6.17. However, it is important to note from the used airfoil model that the absence of
cooling holes at the leading edge is responsible for the low stresses reported at this
location in Fig 6.19 and 6.20.Newer 7FA blades limit the application of a ceramic layer only
to the pressure and low pressure sides, considering that a TC is not necessary where the
cooling holes provide the required thermal protection (the same design is applied to new
7FA firs stage nozzles). The reason to this leading edge modification comes from the high
thermal stresses that appear around the holes locations, which causes a faster
detachment of the ceramic layer.
Thesis - Pablo Gomez Page 122/138Fig 6.20. Von Mises stresses in a 7FA first stage airfoil during a trip from base load.
By comparing the solution for a single start, base load and trip Fig 6.20 and the start with
the normal shutdown in Fig 6.19, a significant increase in the maximum von Mises stress
is obtained, being the maximum value obtained.
Fig 6.21 shows the temperature distribution at base load for a 7FA first stage airfoil.
Thesis - Pablo Gomez Page 123/138Fig 6.21. Temperature distribution in a 7FA first stage airfoil at base load.
6.3.3 Effects of substrate –TBC system temporal evolution on
crack propagation.
In this section the effects of the temporal evolution of the substrate – TBC system on the
Helmholtz state potential are analyzed for the 7FA gas turbine. The findings are used to
determine the crack potential energy that could cause crack nucleation and growth at
different conditions.
To evaluate the effects of undulation and TGO kinetics in stresses distribution along TGO
– TC and TGO – BC interfaces in the different analyzed hot gas path components a
parametric model of the substrate – TBC system is proposed; this model allows the
detailed analysis of the Helmholtz energy at critical component location using a load
function implemented through the active link between COMSOL 4.2a and OCTAVE 3.2.4.
The model allows the evaluation of the thermomechanical fatigue in the case of different
thermal protection system configurations defined according to the temporal evolution of the
system, this strategy provide the adequate integration mechanism to handle the time scale
differences between thermomechanical fatigue and long term system evolution, which
includes the temporal and spatial evolution of the thermomechanical properties of the
Thesis - Pablo Gomez Page 124/138thermal protection system, as well as cumulative creep and sintering effects. The TGO
kinetics is directly implemented using geometric parameters defined according to the
characterization results presented in Section 5. The proposed parametric model includes
the geometric variables described in Table 6.4.
Table 6.4. Substrate – TBC system geometric parameters.
Parameter Description Range
TGO thickness
TC thickness
BC thickness
Substrate thickness
Maximum TGO undulation amplitude
TGO – BC interface roughness
An interesting prediction could be obtained by using the parametric model. Considering a
TGO with a single and symmetric buckle of and , at constant
temperature submitted to a long term exposure time, where TGO thickness increases with
respect to exposure time, it is possible to identify how the creep reduces the residual
stresses associated to the TGO formation, even considering the TGO growth stresses are
higher as the TGO thickness increases Fig 6.22. However, it is important to note that in
this case a homogeneous α-Al O TGO is assumed.
2 3
Fig 6.22. TGO – TC interface von Mises residual stresses variation in a single buckle with
and at different operating times.
Thesis - Pablo Gomez Page 125/138Now considering the variations of the thermal protection system configurations due to a
single change of the thickness and undulation amplitude at in a unaffected
material, the von Mises stresses at TGO – TC interface show how the stresses are
distributed. By comparing different TGO thickness and amplitudes it is possible to note that
higher amplitudes cause higher interfacial stresses, as can be seen in Fig 6.23, where
the parameter is a multiplier, such that indicates .
Fig 6.23. TGO – TC interface von Misses stresses at different TGO thickness and
amplitudes.
Other important predictions could be obtained from the application of the proposed state
equation to the parametric model. However, to determine the preferred places in the
substrate – TBC system where a crack may occur, the Helmholtz energy is evaluated at
different system configurations. Fig 6.24 shows the maximum von Mises stresses in a first
stage nozzle trailing edge at , , and , during a trip from
base load on a unaffected material.
Thesis - Pablo Gomez Page 126/138Fig 6.24. Substrate – TBC system maximum von Mises stresses in a first stage nozzle
trailing edge at , , , during a trip from base load.
Fig 6.26 shows the volumetric plastic strain after the cycle completion. From this image it
is possible to identify the areas where plastic deformation is significant and causes
important dissipation of the von Mises stresses as can be observed from direct
comparison of results from Fig 6.14 and Fig 6.24. Note that not creep effects are
considered for a single thermomechanical cycle. However, if successive cycles are
evaluated, the cumulative effects of creep, plastic deformation and sintering must be
accounted.
Thesis - Pablo Gomez Page 127/138Fig 6.25. Volumetric plastic strain results at same conditions described in Fig 6.24.
Comparing results in Fig 6.24, with a typical transversal section of a Thermal barrier
coating Fig 6.26, it is clear the effectiveness of the parametric model.
Fig 6.26. SEM image of a crack on the TGO – TC interface from a 7FA liner operating
under non steady state conditions.
Thesis - Pablo Gomez Page 128/138To evaluate for this case the crack energy release, from the Helmholtz energy(defined as
the system useful energy) the two steeps post-processing analysis is used where the
displacements and acting forces are determined before and after the virtual crack
extension to determine, following Eqn 6.24, 6.25 and 6.26, the energy release at different
substrate – TBC system configurations. Here the criterion of maximum energy dissipation
is used to determine the right crack growth direction, where a parametric test crack is
evaluated at maximum von Mises stresses locations.
Fig 6.27 shows the Helmholtz energy density variation of a crack that grows through the
TC parallel to the TGO – TC interface, according to the maximum von Misses Stresses
location in Fig 6.24, note that in this case , considering that the typical leading
edge first stage nozzles failure occur at few operating hours.
Fig 6.27. Helmholtz energy density variation as a function of crack length parallel to the
TGO – TC interface through the TC with .
From Fig 6.27, it is important to note that the crack length effectively reduces the
Helmholtz energy density, indicating that such conditions produce a crack that grows
parallel to the TGO -TC interface. Fig 6.28 shows a similar crack extension considering
, in this case results shows how the Helmholtz energy is not reduced as the
crack grows, this suggests that crack is not growing in this direction.
Thesis - Pablo Gomez Page 129/138Fig 6.27. Helmholtz energy density variation as a function of a crack extension parallel to
the TGO – TC interface through the TC with .
Results in Fig 6.27 illustrate how the TGO thickness affects the stresses distribution, and
most specifically the Helmholtz energy density, which is responsible for volumetric energy
transformation into surface energy through the cracks.
Similar analysis could be extended to other cases; even different thermal engines could be
analyzed using the proposed model to determine, not only the service life, but also to
improve thermal engine components design and material selection.
Thesis - Pablo Gomez Page 130/1387 Conclusions
Extreme and variable conditions of temperature, pressure and velocity inside gas turbines
configure an interesting challenge to develop consistent wear models for substrate – TBC
systems. However, thermodynamic approaches have proved, in the light of the obtained
results, that could be used to handle the dissimilar time scales of the thermomechanical
fatigue and high temperature oxidation, sintering and creep, for the substrate – TBC
system wear phenomena. The proposed model provides an adequate integration of the
temporal evolution of the thermal protection system and the thermomechanical fatigue on
a single model to estimate the service life of hot section components of the gas turbines
and other thermal engines.
The model and its numerical solution provide an alternative to traditional OEM statistical
models to handle the maintenance intervals of gas turbines, especially those turbines
installed as a backup of the energy generation systems in areas with high hydraulic
resources to cover operational issues and climate adversities such as El Niño or other
adverse climate phenomena, where those gas turbines operate on complex maintenance
sceneries due to the non-steady state operating conditions.
The phenomenological base of the model provides an interesting way to select and design
adequate materials for specific gas turbine applications and other similar thermal
applications, considering that the parametric implementation of the model allows the
evaluation of thermomechanical cycles, loads and configurations for different material
properties, providing an optimization tool.
From results in Section 6 it is clear that the important role of the TGO evolution in the state
of stress in the substrate – TBC is adequately handled by the proposed state equation, so
the proposed contribution of TGO evolution to the useful energy that could be transformed
into surface energy through the cracks, is a powerful tool to predict the behavior of the
thermal protection system at different operating conditions. From the parametric analysis it
is also possible to determine some important considerations to ensure a durable thermal
protection system from the manufacturing process of the substrate and the coatings.
In the case of a stable and continuous TGO, the proposed model shows that the
combination of creep and plastic deformation are responsible for a continuous reduction in
original formation stresses at TGO – TC and TGO – BC interfaces, so if a stable layer is
formed the TGO residual stresses become lower as the exposure time increase. However,
less stable oxides formation due to Al depletion cause additional interfacial stresses
increasing the TGO residual stresses, also the less stable oxides like spinels and Cr and
Ni oxides with a lower mechanical resistance make easier the crack propagation through
this layer as shown in Section 3.2.1.
Plastic deformation and creep are also responsible for TBC deformation, which affects the
stresses distribution. The increase of the amplitude of the TGO undulation respect to the
operation time is one of the main consequences of plastic deformation and creep. The
proposed model predicts a strong correlation between undulation amplitude and maximum
interfacial stresses, such that higher amplitudes cause higher stresses. Considering the
correlation between roughness prior to the coating deposition and TGO geometry, it is
clear that optimization of the surface preparation processes unequivocally produce more
Thesis - Pablo Gomez Page 131/138stable coatings, considering from results that for undulation amplitudes below not
plastic deformation occur around the TGO, so ratcheting effects just do not appear.
The accumulation of plastic deformation and creep at operating conditions causes an
increase in surface roughness of hot section components, increasing the drag forces over
components and reducing the operation efficiency; this effect is easily recognizable from
the historical data of the gas turbines thermal efficiency against operating hours. However,
it is important to note that efficiency losses also come from other degradation sources, like
surface deposits, TBC spallation, and geometric distortions.
If the geometric distortions at bigger scales are considered, the plastic and creep
deformation effects on the geometry instability of hot section components could be
understood, for instance, by considering findings on dimensional controls of hot gas path
components at certain operating conditions, correlations between local distortions and
suggested stress distribution from the proposed model could be found; it is important to
highlight the fact that geometry distortions are one of the most important criteria to account
in component reparability evaluation, so found correlations suggest that geometric
improves could be made using the proposed model, also different materials geometry
stability could be obtained from this model.
The use of programmable commercial software result in an important reduction in the
solution implementation, reducing the required time to adjust and calibrate critical aspects
like meshing, solvers programming and results plotting. However, is important to highlight
the importance of user defined subroutines, which allow the implementation of more
complex models, something really important in the present case, where an important
number of dissipation surfaces are accounted in the development of the state equation
from the Helmholtz energy.
Considering the amount of variables that are included in the application of the proposed
analytical model, there is a clear limitation to obtain a most precise results, if better
material properties functions and complementary operating variables could be added to
the model, a significantly reduction in the number of suppositions used to obtain the
results, may produce more precise results. Although the mentioned limitations, the
agreement between results and common findings during gas turbine major inspection is
adequate to produce accurate life estimations.
Thesis - Pablo Gomez Page 132/1388 References
[1] Meherwan P. Boyce, 2002. Gas Turbine Engineering Handbook. Second Edition. GPP.
Houston Texas.
[2] Reymann H, 2002. MCrAlY Deposition by HVOF: A Suitable Alternative to LPPS.
Turbine Forum: Advanced Coatings for High Temperatures. Nice Port St. Laurent, France,
April 17 – 19, 2002.
[3] GE Power, 1997. Operación y Mantenimiento de Turbinas a Gas Frame 7FA, Vol. 2.
[4] David B., Robert B., David F., 2004. Heavy-Duty Gas Turbine Operating and
Maintenance Considerations, General Electric Company, GER3620K, p 1 - 60.
[5] Himanshu Bhatnagar, Somnath Ghosh, Mark E. Walter, 2006. Parametric Studies of
Failure Mechanisms in elastic EB-PVD Thermal Barrier Coatings Using FEM. International
Journal of Solids and Structures, Vol. 43, p 4384 - 4406.
[6] Brodin H., Jinnestrand M., Johansson S., Sjöström S., 2006. Thermal Barrier Coatings
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