A Probabilistic Analysis of a High Pressure Turbine Pre-Swirl Cavity
and Capture System to Identify Input Variability of Design Parameters
.
by
Pamela Ann Gray
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master’s of Science of Mechanical Engineering
Approved:
_________________________________________
Thesis Adviser
Rensselaer Polytechnic Institute
Troy, New York
December 2009
1
© Copyright 2009
by
Pamela Ann Gray
All Rights Reserved
2
CONTENTS
LIST OF TABLES ............................................................................................................. 4
LIST OF FIGURES ........................................................................................................... 5
ACKNOWLEDGMENT ................................................................................................... 6
ABSTRACT ...................................................................................................................... 7
3
LIST OF TABLES
Table 1 Output Parameters .............................................................................................. 11
Table 1 Parameter and standard deviations ..................................................................... 24
Table 2 Output Parameters .............................................................................................. 24
4
LIST OF FIGURES
Figure 1 Turbofan engine components: inlet fan, low and high pressure compressors ... 1
Figure 2 Turbofan engine components: combustor, high and low pressure turbines, and
nozzle ................................................................................................................................. 2
Figure 1 Example of flow network structure .............................................................. 14
Figure 2 High pressure turbine cooling air and delivery system input variables locations
......................................................................................................................................... 16
Figure 3 Labyrinth seal knife edge geometry, inputs for flow model ............................. 18
Figure 4 Input file for probabilistic run ........................................................................... 26
Figure 5 Input file for final probabilistic run ................................................................... 26
Figure 6 Input file for final probabilistic run ................................................................... 27
5
ACKNOWLEDGMENT
6
ABSTRACT
This paper describes a gas turbine pre-swirl cavity and capture system’s flow
sensitivity as predicted through a probabilistic analysis of a typical high pressure turbine
of a commercial turbofan engine. The results are used to describe the flow sensitivity in
the chamber and the effects of variability of the determined drivers. This study was
performed in order to enhance existing modeling techniques in industry. Pre-swirl
supply systems deliver cooling air axially from stationary nozzles to a rotating turbine
disk. The holes in the rotating disk are at a similar radius as the nozzles to reduce
mixing losses; however, existing engine and rig data show significant losses in total
pressure occurring in the cooling flow exchange.
Discussion focuses on the pre-swirl
cavity and capture system, set up and results of a probabilistic analysis performed in
order to identity the parameters of greatest impact.
This paper will demonstrate a probabilistic study method that explores flow
sensitivity of design parameters relative to the high pressure turbine single stage preswirl cooling air delivery and capture system of a turbofan engine. The goal of the
proposed research is to identify the drivers of variability of the subsystem and determine
the sensitivity of those drivers.
7
I.
Introduction
1.1 The Secondary Flow System of a Gas Turbine Engine
The gas turbine engine is widely used to power both commercial and military
aircraft today.
With just over 9.8 million domestic commercial flight departures
performed in 2007 [8], safety, system performance, component durability and reliability
are major concerns for the gas turbine engine manufacturer. These combined system
and component attributes, regulated by industry standards and customized to meet
customer requirements, contribute to a well designed product. The major components of
a typical commercial turbofan engine are a low pressure inlet fan and compressor, a high
pressure compressor, a combustion chamber, a low pressure turbine, a high pressure
turbine and a nozzle. The major components of a turbofan engine are shown in Figures
1 & 2.
Figure 1 Turbofan engine components: inlet fan, low and high pressure compressors
1
Figure 2 Turbofan engine components: combustor, high and low pressure turbines, and nozzle
At the inlet of the engine the intake air flows through the fan and upon exiting is
split into two main flow paths; the by-pass flow and the primary or core flow. The bypass flow is about 80% of the inlet flow and its primary function is thrust. It is
channeled through the fan duct then enters the nozzle where it remixes with the core
flow before exiting the rear of the engine. The remaining 20% inlet air goes through the
compressors and then the combustor where it is mixed with fuel and ignited before
entering the turbines. Once the core flow exits the turbines it remixes with the by-pass
flow, just before entering the nozzle. One of the primary functions of the core flow is to
drive the compressor. A portion of the core flow is split off near the compressor exit.
This flow, called the secondary air system, bypasses the burner and is used to perform a
variety of functions that are critical for safe engine operation. Figure 2 shows a simple
schematic of the high pressure turbine cooling air path from the compressor exit to the
high turbine inlet. Some of these functions are ventilation, sealing and purging air to
disks, shafts, cavities and bearing compartments [7]. The cooling air system is designed
to assure the life of the hardware, to provide thermal conditioning for clearance control,
customer bleed flow, compressor starting and stability bleeds, and engine anti icing
protection. The secondary air system also influences rotor thrust bearing loads, protects
bearing compartments, and assures an acceptable nacelle environment. The internal air
system optimizes propulsion system performance while satisfying all of the above
requirements.
Although each portion of the secondary air system is needed for a
2
balanced system, this paper will focus on the cooling air supplied to the high pressure
disk and blades through a pre-swirl cooling air delivery system.
The basic function of a pre-swirl system is to supply cooling air to a disk and
blade, meeting the temperature, pressure, and leakage requirements of the system while
minimizing the losses and work input associated with bringing air on board a rotating
structure. The turbine cooling air is taken from appropriate compressor stage locations
where work has been done to increase pressure. The path of the diverted high pressure
secondary air is a complex configuration consisting of orifices, cavities, and component
interfaces. Upon entering the pre-swirl nozzle, the high pressure cooling air typically
traverses a set of static tangentially inclined nozzles, called a cascade, which turn the air
in the direction of disk rotation. Turning the air imparts a tangential component to the
velocity of the cooling air, thereby minimizing the heat up generated when the air comes
on board the rotating structure, as defined by Euler’s equations. The pre-swirl cavity
cooling air and delivery system is also known as the Tangential On Board Injection
(TOBI) system. See Figure 3 for a pre-swirl rotor stator system schematic.
Receiver Holes
Rotating
Minidisk
Stationary
Pre-Swirl
Nozzle
Figure 3 High pressure turbine
The high pressure cooling air is expanded through the stationary pre-swirl nozzles where
the majority of the air passes through a chamber and then is delivered to receiver holes
on the rotating mini disk. The higher the pre-swirl nozzle exit tangential velocity, the
3
colder the cooling air will be as it is delivered to the blade, maximizing cooling
effectiveness. The swirled cooling air exiting the pre-swirl nozzles splits and follows
three paths, fulfilling separate tasks. The majority of the flow delivered to the receiver
holes on the rotating mini disk is split into two directions, outward towards the gas path
and inward towards the bore. As the flow travels outboard, cooling flow is delivered to
the rotating blade meeting a supply pressure requirement at a particular flow level and
temperature. These requirements ensure the blade will meet its life goal and satisfy rear
blade attachment leakages. Blade supply pressure, temperature, and flow requirements
are met at the condition where the majority of blade damage occurs, which is take-off for
most engine applications.
The inward path taken once exiting the receiver holes
provides the flow needed to meet the requirement for high pressure turbine bore flow.
This flow path has little impact on the TOBI system and will not be discussed in detail.
The remaining cooling air, once exiting the pre-swirl nozzles, goes through the outer
diameter labyrinth seal to supply attachment leaks and front rim cavity purge. See
Figure 4 for flow paths of the high pressure turbine TOBI area.
OD Seal-Dead rim cooling air
and front rim cavity purge
Pre-swirl
cooling air
Blade cooling air
and attachment
leaks
From HPC
rear hub
#4 bearing
compartment
air
HPT bore
cooling air
Figure 4 High pressure turbine TOBI area and flow paths
4
1.2 Problem Statement
The cooling air delivered to the turbine is critical for flight safety, without it,
parts would not meet life requirements and hardware failures would occur. The core
flow turbine inlet air exiting the combustor is quite high and can reach temperatures over
3000oF. The high pressure turbine first stage purge flow cooling air prevents excessive
hot gas ingestion, keeping the metal coatings on the vane and blade platforms from being
burned off before reaching life goals. The cooling air supplied to the rotating disk and
blade by the pre-swirl nozzle system is also critical to part life. The intention of this
thesis is to document a method to perform a probabilistic secondary flow analysis for a
high pressure turbine pre-swirl cavity capture and delivery cooling air system of the
turbofan engine. The analysis will be used to quantify the variability of the cooling air
delivery system due to inherent uncertainty in manufacturing processes and engine
performance.
In addition to quantifying the variability of the cooling air delivery
system, the sensitivity of the design parameters will be determined.
1.3 Research Objectives
The probabilistic flow analysis performed on the high pressure turbine secondary
flow system is done using an enhanced deterministic flow model. The outputs of interest
are the flow through the pre-swirl nozzle, the nozzle exit pressure and temperature, the
cooling flow to the blade, rim cavity purge flow, flow to the bore, the blade supply
pressure and temperature, and the pre-swirl cavity temperature. The enhancement to the
deterministic flow model is applied by adding a variation to the input parameters of the
flow network solver. For example the high pressure turbine inlet and exit pressures have
a standard deviation of 0.5% of the average pressure with a 2 sigma variation applied to
each of them. A complete list of the parameters that will be varied can be found in the
analysis section and are discussed in detail there.
The flow solver program consists of networks that mathematically model the entire
engine, from the fan inlet to the nozzle. A flow network consists of restrictors that
represent orifices, labyrinth seals, frictional losses in pipes, vortexes, etc, that are
connected to chambers representing large plenums of air [2]. Each of the restrictors and
chambers are defined by their associated pressure losses. See Figure 5 for an example of
a flow solver network showing restrictors and chambers of the TOBI area.
5
Figure 5 Flow solver network, restrictors and chambers
The probabilistic flow analysis is executed with variability in engine
performance (day to day variability) and hardware geometry (engine to engine
variability) yielding the variability in mass flow rates and air temperatures. After all
correct and assumed deviations have been entered, and all desired outputs have been
identified, the model is run. The model generates regression output, cumulative density
functions, and probability density functions. The cumulative density functions are useful
for determining how likely a range of values are, and probability density functions are
useful for determining if enough samples were taken and distribution type of output
variables.
1.4 Overview of Expected Result Usage
The results of the analysis will be used to efficiently determine sources of
variability in the secondary flow system while identifying the most influential geometric
and performance parameters. This will allow known performance capabilities to be
established promoting a more reliable and inexpensive engine.
This paper will demonstrate a probabilistic study method that explores flow
sensitivity of design parameters relative to the subsystem high pressure turbine single
stage pre-swirl cooling air delivery and capture system of a turbofan engine.
6
Probabilistic analysis has many applications in cost reduction, engine design,
optimization, and root cause analysis and has been discussed by Cloud & Stearns and
others [1-3]. The results of those analyses will be discussed in the background section
(section 2) of this paper.
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1. Background
1.1 – Overview of Previous Analyses
In order to better understand how input variability or uncertainty may impact
design features of a commercial turbofan engine, probabilistic analyses have been
performed [1-3].
These types of analyses describe the variability of a system by
applying variation to the inputs of that system. This is an enhancement to current flow
modeling practices that use a deterministic model with single value inputs and outputs
where the single answer contained no measureable probability. The key to performing a
probabilistic analysis centers on the ability to propagate input variability through the
system. The previously performed probabilistic studies [1-3] and this one as well, use a
proprietary one dimensional flow network solver that simulates the behavior of the entire
auxiliary flow system for the engine. The flow solver contains additional modifications
that make it possible to evaluate output variability and sensitivity. This design tool
allows the user to input variables with a nominal value, a standard deviation, a
distribution type and a variance by means of an input file which propagates the
variability through the flow model. A quadratic regression is then fit to the probabilistic
data to post process the results by the method of least squares.
Y = y0 + bi(Xi) + ci(xi)2
(1)
Where Y is the output variable, y0 is the constant regression coefficient, and bi is
the linear regression term which is a measure of how input variability affects output
variability. The quadratic regression term, ci, is a measure of how input variability
affects the output mean value.
i = (bi*i/i)/100
(2)
Where i is the sample mean value of the ith input variable, and i is the sample
deviation value of the ith input variable. If the ith variable changes by 1% the output
variable will change by  units.
i = bi2/bi2
(3)
The ith variable contributes % of the total variance on that output.
8
The cumulative density functions are useful for determining how likely a range of
values are. The probability density functions are useful for determining if enough
samples were taken and distribution type of output variables.
1.1.1 – 2003 Sidwell & Darmofal Study
Sidwell & Darmafol [1] demonstrate how a Monte Carlo probabilistic method is
used to estimate the distribution of oxidation failure probability for two different airlines
operating the same engine model in different environments. To model the statistical
behavior of turbine blade oxidation life two different types of input variability were used
for the flow network solver; they were, day to day variability and engine to engine
variability. Day to day variability included the environmental condition of the ambient
temperature. Engine to engine variability included engine conditions, blade to blade
variations and manufacturing variations. Engine conditions varied were component inlet
and exit temperatures and rotor speeds, which were based on field experience. The
blade to blade variations included film cooling hole effective areas that are relevant to
placement, which were derived from flow measurements performed during
manufacturing. The manufacturing variations such as machining tolerances on TOBI
seal radii and discharge coefficients of the cooling air system were assumed to have a +/2 sigma variation. A least squares regression analysis, as described above was applied to
the probabilistic results to identify input variables for which a decrease in tolerance
would result in an increase in life. Regression analysis determined the effect of the
variability of each input on typical and minimum engine oxidation life to be a 10%
decrease in the tolerance on the blade’s leading edge effective flow area for both airlines.
1.1.2 Cloud & Stearns Study
In 2004 Cloud & Stearns [2] documented a methodology for analyzing turbofan
secondary flow systems probabilistically.
That type of analysis quantified model
outcomes when a variation was applied to the inputs as was done similarly by Sidwell &
Darmofol [1] except every chamber and restrictor of a commercial turbofan engine
model had a standard deviation applied. This was the first run, and it generated results
that allowed identification of significant system drivers. The evaluation of thermal and
centrifugal growth effects were accomplished by including a percent deviation to
9
labyrinth seals and vortex radii. Instead of a Monte Carlo distribution a Latin hypercube
method was used, and a comparison of sample convergence can be seen in figure 1.
Need figure from Ref 1 (need adobe writer)
Absolute deviations should be applied when manufacturing tolerances are to be
analyzed. The method was applied in order to find variability in the total turbine cooling
and leakage air of the secondary flow system and the high and low rotor axial bearing
loads of a turbofan engine. The results showed the system behaved linearly, resulting in
negligible mean shifts due to input variation.
1.1.3 Stearns, Cloud & Filburn Study
In 2006 Stearns, Cloud & Filburn [3] documented the initial development of a
method to perform a thermal probabilistic analysis of gas turbine internal hardware. The
turbine inter-stage seal of turbofan engine was used as an example. The objective was to
investigate the variability of steady state metal temperature due to variability in the
secondary flow system as well as the sensitivity of the metal temperature. Results
showed the variability in metal temperature is ultimately caused by labyrinth seal
clearance.
1.2.4 Proposed Research
The proposed research will use the Latin hypercube method. Input variables will
have similar standard deviations applied and as described above by Sidwell and
Darmofal [1]. The differences will be that I will run 2 probabilistic studies as described
by Stearns & Cloud [2] except that I will focus on a sub-system of the secondary flow
system, which is the pre-swirl cavity cooling air capture and delivery system of the high
pressure turbine. For the first probabilistic run I assumed a 5% standard deviation on
restrictor areas, a 15% standard variation on lab seals average clearances and a 25%
standard deviation on the plat-form leakage areas, which are consistent with average
manufacturing tolerances. Output parameters will include flow rates, pressures and
temperatures for the pre-swirl nozzle and the blade as well as blade rim cavity purge
flow of the leading and trailing edge.
Table 1 provides a list of selected output
parameters of the subsystem.
10
Table 1 Output Parameters
The results of the first analysis will be used to identify the significant drivers of
variability. Unlike the previously performed probabilistic analyses, the output of this
study will be the mass flow rates, air temperature and pressure variability of the single
stage high pressure turbine cooling air and delivery system, a subsystem of the
secondary flow system of a commercial turbofan engine.
Parameter
TOBI OD Seal
Standard
deviation
applied for second run
=15% of Clearance
TOBI ID Seal
 = 5% of Area
Platform Seals
 = 25% of Area
TOBI
 = 1.5% of Area
Vortices
 = 5% of RPMF
Blade Cooling
 = 6% of Area
Pressures
 = 0.2% of P4-P5
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2. Theory
2.1 Analytical Flow Model
The analytical secondary flow model used for this probabilistic study is a datamatched commercial turbofan engine that represents build of material hardware. The
one dimensional flow model is run via a graphical user interface (GUI). The flow model
calculates the internal engine cavity pressures and temperatures, internal cooling and
leakage fluid flow rates, and the axial load on the thrust bearings. The secondary flow
model is an analytical tool used to design the secondary flow system. The flow model
software is written in FORTRAN code and contains the mathematics required to
accurately simulate the secondary flow system of a commercial turbofan engine. The
tool has many uses, and one of particular interest is that it allows system designers to
predict the effect of over or under machined parts on the secondary flow system. The
flow model is also used to validate the secondary flow system, verifying the system
requirements are met. The results of the flow model are also used as input for other
analyses such as thermal analysis of the rotors, disks, blades and life estimates for
bearings.
Secondary flow system requirements of interest for this analysis are the mass flow
rate through the pre-swirl nozzles of the high pressure turbine, the pressure supplied to
the blade for cooling, and rim cavity purge flow of the blade leading and trailing edges.
The flow model can solve a flow system for a steady state case, a transient case and a
statistical sensitivity or probabilistic analysis case.
Only the steady state and
probabilistic features will be discussed. The statistical probabilistic analysis used for
this study is the non linear Latin hypercube sampling method.
2.1.1
Flow Model Inputs
The flow and bearing load model is comprised of a series of “chambers”
interconnected by various types of “restrictions” to the gas path. These graphical
chambers and resistor icons allow the user to model cavities and restrictions that make
up an entire engine.
Chamber icons allow for the calculation of pressure and
temperature of a location, representing a large volume plenum where flow velocities are
assumed to be fully recovered and total pressure is equal to static pressure
12
Chamber pressures and temperatures can be either known or unknown and may be
input by a value or an equation, as in the case of data-matched locations. In order for the
model to solve, there must be at least one known chamber acting as a source and one
known chamber acting as a sink. The resistor icons model pressure losses and flows
between the chambers. There are 25 different types of restrictors available in the “GUI”,
the ones of most interest to the probabilistic analysis and ones pertinent to the TOBI
system’s design parameters are the flow parameter, orifice, labyrinth seal, vortex and
isentropic nozzle and each will be discussed in detail. The flow solver then calculates
the unknown chamber pressures and temperatures and restriction flow rates through
successive iterations until the flow rates are balanced.
The flow model used for this analysis is very detailed having hundreds of defined
chamber and resistor icons. The chambers and resistor icons are provided through an
input file that is generated using the flow model GUI. Every chamber and resistor has a
set of unknown states and governing equations that describe the local flow conditions
and are resolved by the flow solver. A coupled, nonlinear set of equations must be
solved to determine the flow in the flow network model. The flow solver uses a NewtonRaphson iterative method to solve the coupled equations.
2.1.2
Solution Technique
The flow network consists of the resistors and chambers with added interfaces
between every resistor and chamber. These interfaces are utilized to improve the
modularity of the underlying solver and are not controlled by the user. A simple flow
network structure is shown in Figure 1 where squares represent the chambers and circles
represent the restrictions.
13
Figure 3 Example of flow network structure
Every chamber, resistor and interface has a set of unknown states and governing
equations that describe the local flow conditions and are resolved by the flow and
bearing load solver. The states for chambers are pressure and temperature. If there is no
initial guess input for each pressure or temperature, the value is set to equal the average
of the known pressures or temperatures.
The states for restrictors are mass flow rate,
temperature at the intended upstream boundary and temperature at the downstream
boundary. The states for interfaces are mass flow rate, temperature and pressure.
The specific governing equations for chambers are:
1. Conservation of mass. In steady flow, the sum of the interface mass flow rates
connected to the chamber is required to be zero:
Number ofint
Interfaces
numberof
erfaces
 m
i
i 1
0
(1)
 is the mass flow rate, or air flow.
Where m
For unsteady flow, appropriate time derivatives are added to account for a time rate
of change for the mass in the chamber.
2. Conservation of energy. In steady flow, the conservation of energy is given by:
Numberof
of Interfaces
number
int erfaces
 m
i 1
i
H (Ti up )  0
Where H is the stagnation enthalpy and Ti
(2)
up
is the temperature taken from the
upstream direction at interface i. In other words, if the interface is an inflow, this
temperature is equal to the interface temperature; however, at an outflow, this
temperature is equal to the chamber temperature. For unsteady flow, additional terms are
added to account for a time rate of change of energy in the chamber.
Resistor governing equations:
1.
For most restrictors, the resistor mass flow rate is set up by a mass flow
relationship to the upstream and downstream pressures and temperatures that are taken
from the corresponding interfaces. However, for vortex resistors and fixed pressure ratio
resistors, a pressure ratio or pressure difference is set directly.
14
2.
Upstream temperature, which is based on the direction of flow, is set to the
upstream interface temperature value.
3.
The downstream temperature is set by the resistor model and may include a
temperature set by the user.
Interface governing equations:
1.
The interface mass flow rate is set to the mass flow rate of the resistor it is
attached to.
2. The interface pressure
3. The
is set to the pressure of the chamber it is attached to.
interface temperature is set to the temperature from the upstream component,
for example the resistor or chamber which is upstream of the interface.
2.1.3
Restrictions and Chambers
The restrictors discussed here are the ones that will be varied in the input file
created for the probabilistic study and consist of the flow parameter, orifice, labyrinth
seal, vortex and isentropic nozzle. Figure 2 shows schematic of locations varied in high
pressure turbine TOBI area.
15
Isentropic nozzle restrictors
Flow parameter
restrictors
Labyrinth seal
restrictors
Orifice restrictor
Figure 4 High pressure turbine cooling air and delivery system input variables locations
Ideally, the flow model restrictors and chambers include inputs that represent the
physical description for the hardware being modeled. The standard output contains the
input items as well as the calculated output values including flow area, temperature,
pressure, pressure ratio, flow rates and reference values if input. Basic discussions are
presented here for the restrictor inputs of interest to the probabilistic analysis and for
brevity are simplified. The way the flow model uses the inputted information is by
calculating a flow parameter and pressure ratio for each restrictor and chamber, which is
then used for the iterations solving the for the total sum of the mass flow rates.
2.1.3.1 Flow Parameter Restrictor
The flow parameter restrictor is used to model the cooling hole passages of the
blade’s leading edge, mid-body, trailing edge, the trailing edge platform overhang and
the TOBI. The input is a flow parameter versus pressure ratio curve and is derived from
16
flow measurements taken during manufacturing. This restrictor is used where the
relationship between flow parameter and pressure ratio is known.
  m T P
up
PR 
(3)
Pup
Pdown
An effective area, discharge coefficient (Cd), can also be calculated based on the cold
flow data. If no discharge coefficient is entered the flow model assumes an effective
area of 1.
ACd 
m T P 
m T P A
up measured
up
(4)
isentropic
Where A is the area.
2.1.3.2 Orifice Restrictor
An orifice restrictor is used to model the minidisk holes which feed the blades. This
restrictor is used where flow measurement or metering is done using a very short
passage with a sharp edge on the upstream side and beveling downstream, or a square
edge with no beveling such as a drilled hole. The input includes a required flow area A
and an optional discharge coefficient Cd. The flow area can be input as a value, or an
equation.
2.1.3.3 Labyrinth Seal Restrictor
The labyrinth seal restrictor is used to model the outer diameter and inner diameter
TOBI seals. Definitions of a typical labyrinth seal knife edge geometry required for the
flow model are illustrated in Figure 3.
17
Figure 5 Labyrinth seal knife edge geometry, inputs for flow model
Where:
c = Seal Clearance
b = Number of Teeth
pKE = Knife Edge Pitch
wKE = Knife Edge thickness
h = Land Step Height
rKE = knife Edge Leading Edge Radius (illustrated in figure for Up Flow)
Flow Direction = Up/Down
The flow area is calculated by the flow model using the supplied inputs.
2.1.3.4 Vortex Restrictor
The vortex restrictor is used to simulate fluid motion involving rotation about an
axis. This simulation tool has no physical area restriction but can either increase or
decrease flows by imposing a pressure ratio on the adjacent chamber(s) with non-fixed
pressure(s). The flow model uses calculations for all the vortex motions that assume the
working fluid acts like a perfect gas and that the vortex is isentropic. These assumptions
are thought to provide a reasonable representation of a real vortex but if the vortex
pressure ratio is known it may be input directly.
18
2.1.3.5 Isentropic Nozzle
The isentropic nozzle restrictor is used to model the rim cavities. Input
considerations include a required flow area, A, and an optional discharge coefficient, Cd.
The flow model assumes that the isentropic nozzle has an upstream area much larger
than the minimum flow area specified as input and uses the input flow area as the throat
for this restriction. The nozzle flow area must be input using the minimum passage area
known as the throat of the nozzle. The nozzle throat may have any cross-section shape;
the flow model will use the input flow area as the minimum flow area for the nozzle
throat.
2.1.3.6 Chamber
The flow model assumes 1-D flow, constant enthalpy, velocity and density over the
area. Velocities are assumed normal to areas and no heat or work interactions with
surroundings. The air is assumed to be a perfect gas during standard flow model
iteration loop, and it is assumed a real gas during flow model iteration loop in which
conservation of energy law is applied to each chamber using real fluid properties. The
chamber effectively acts as a mixing restriction with 0 or more control surfaces.
2.1.4
The Flow Network Solver
The coupled, nonlinear set of equations must be solved to determine the flow in the
flow and bearing load network model. The flow solver uses a Newton-Raphson iterative
method to solve the coupled equations. Specifically, the governing equations and
unknown states for every chamber, resistor, and interface in the flow network can be
combined into a residual form,
R (U )  0
(3)
Where U is a vector containing all of the unknown states and R is the corresponding
vector containing all of the governing equations. A Newton-Rapshon method for solving
this nonlinear set of equations is found by linearizing the residuals about a current guess
for the solution,
19
R(U n  dU )  0
R(U n ) 
R
dU  0
U
(4)
Then, by solving for dU, an update for the state vector is given by,
U
n 1
R 1
U 
R(U n )
U
n
(5)
In practice, however, the full Newton update is not taken at every iteration
especially early in the iterative process where the linearized update may result in nonphysical solutions. Thus, the Newton update is under-relaxed as follows,
U
n 1
U 1
U d
R(U n )
U
n
(6)
Where d is a relaxation factor and in the flow solver and is known as the drate. The
algorithm for determining the drate involves several parameters which may be modified
to control the convergence behavior of the flow solver. Two types of limiting of the
drate are used: (1) limiting based on the flow rate residual, and (2) limiting based on the
change in the states. For the flow rate residual limiting, the basic idea is to limit the
Newton update whenever the mass flow rate imbalance anywhere in the flow network is
large at the current iteration. For the state-based limiting, the basic idea is to limit the
changes in the state to guarantee that the states at the next iteration are physically
realistic. Thus, in both limiting procedures, as the solution converges, drate should
approach one, while initially drate is less than one.
2.2 Probabilistic Flow Model
The flow model’s sensitivity analysis permits the user to input variations of certain
parameters and then run a sensitivity study (linear or non linear) to obtain statistical
variations for other defined parameters. The secondary flow model, as a design feature,
has an option for varying input values and propagating them through the flow model.
Typically the flow model takes single input values per chambers and restrictors in the
form of flow areas, pressures and temperatures. The probabilistic flow analysis allows
20
the user to input variability in engine performance (day to day variability) and hardware
geometry (engine to engine variability) yielding the variability in mass flow rates and air
temperatures and pressures.
2.2.1
Probabilistic Flow Model Inputs and Outputs
The input file for the probabilistic study needs to identify the parameter (chamber or
restrictor) being varied the nominal value, the standard deviation, the distribution type
and the variance. The flow model probabilistic analysis generates regression output,
cumulative density functions, and probability density functions. After propagating input
variability through the model by means of an input file, it is possible to analyze the
model outcomes of means and deviations.
21
3. Methodology
3.1 Method of Analysis
The secondary flow model used for the probabilistic study is a data-matched
secondary flow model of a current production commercial turbofan engine configuration
was identified. This flow model’s input parameters are a specified pressure, temperature
and flow rate for a standard day take off condition. A probabilistic analysis allows
variation of different input parameters (many at a time) in a random manner
independently which will generate the effect on several other output parameters with the
resulting probability or frequency distribution of each output parameter. By default, all
the input variables will change completely independent of each other [2]. The input
values used for the probabilistic analysis are refined variations as determined from
tolerance dimensions identified from part drawings, experience and engineering
judgment.
The probabilistic analysis used the Latin hypercube [12] sampling method and is a
design feature of the flow model. The statistical method of Latin hypercube sampling
(LHS) was developed to generate a distribution of plausible collections of parameter
values from a multidimensional distribution. The technique was first described by
McKay in 1979 [8], it was further elaborated by Ronald L. Iman, and others [10] in
1981. Detailed computer codes and manuals were later published [11]. This method is
incorporated into the secondary flow model’s FORTRAN code.
The probabilistic
analysis was run for 4000 samples.
In the context of statistical sampling, a square grid containing sample positions is a
Latin square if (and only if) there is only one sample in each row and each column. A
Latin hypercube is the generalization of this concept to an arbitrary number of
dimensions, whereby each sample is the only one in each axis-aligned hyper-plane
containing it. When sampling a function of
divided into
N variables, the range of each variable is
M equally probable intervals. M sample points are then placed to satisfy
the Latin hypercube requirements; note that this forces the number of divisions, M, to be
equal for each variable. Also note that this sampling scheme does not require more
samples for more dimensions (variables); this independence is one of the main
22
advantages of this sampling scheme. Another advantage is that random samples can be
taken one at a time, remembering which samples were taken so far.
The maximum number of combinations for a Latin Hypercube of
M
divisions and
N
variables (i.e., dimensions) can be computed with the following formula:
For example, a Latin hypercube of
M = 4
divisions with
N = 2
square) will have 24 possible combinations. A Latin hypercube of M
variables (i.e., a
= 4 divisions with
N = 3 variables (i.e., a cube) will have 576 possible combinations [12].
3.2 Probabilistic Variation Types
Two types of input parameters are varied in this study, day to day variation, which
are engine conditions and engine to engine variation which are manufacturing
tolerances. The day to day variations can be captured through the high pressure turbine
inlet and exit pressure and temperature. The standard deviation applied to the engine
performance was obtained from performance engineers. The engine to engine variations
are captured by varying the geometry of the system’s hardware, this includes areas and
flow rates.
Manufacturing tolerances for each significant input were identified by
interrogating current build of material drawings for the hardware such as the knife edge
seal clearances. The variation values are currently documented and can be found on
drawings and engineering standard work.
Table 1 shows the parameters varied for the probabilistic analysis.
23
Table 2 Parameter and standard deviations
Output parameters include flow rate, pressure and temperature for the TOBI, the blade
supply pressure, blade leading edge rim cavity purge flow and cooling flow for the
blade’s leading edge, trailing edge and platform trailing edge. The following table
provides a list of selected output parameters of the subsystem that are included in the
input file.
Table 3 Output Parameters
24
3.2.1
Create Input File
The input file that contains the input parameter names (same as identified in the
flow model), nominal value, standard deviation, the distribution type and the variance
applied. The output parameters are also included in the input file. The flow network
solver will then generate random values for each input within the given distributions and
solve the system for each sample. The convergence criteria for the flow network solver
is to solve all the unknown pressures, flows and temperatures until all the mass flow
rates summed are equal to zero.
** SENSITIVITY INPUT FILE FORMAT
* Use '*' at first column for comments
* Commercial Turbofan Engine probabilistic study thesis, Final Run
*
*
** INPUT PARAMETERS IN THE EQUATION/MUDS LIST (DONOT MODIFY THIS LINE !)
* Parameter (exact as left hand of EQUATIONS) + standard deviation +
Distribution type + Truncated value
* note:
Distribution type: Uniform ( = 0), Normal ( = 1) and
Truncated Normal (= 2)
*
Truncated value if for Truncated Normal (= 2) only
*
*Gaspath Pressures at stations 4, 45 (St.dev=0.1% of avg)
PPERF5
0.3884
2
0.7769 *(avg=388.40)
PPERF6
0.0973
2
0.1945 *(avg=97.30)
*TPERF5 (avg=3173.7)
*TPERF6 (avg=1955.5)
*
** INPUT PARAMETERS NOT IN THE EQATION/MUDS LIST (DONOT MODIFY THIS LINE !)
* Parameter + mean value + standard deviation + Distribution type + Truncated
value
* note:
Distribution type: Uniform ( = 0), Normal ( = 1) and
Truncated Normal (= 2)
*
Truncated value if for Truncated Normal (= 2) only
*
* CHAMBERS:
Parameter = PRESxxxx (xxxx is ID NUMBER)
*
CTMPxxxx
*
RxINRDxxxx (Inner Radius for rotor x chamber xxxx)
*
RxOURDxxxx (Outer Radius for rotor x chamber xxxx)
*
x is rotor real number!
*
*
RESISTOR TYPES APPLICABLE
* RESTRICTOR: Parameter = RSARxxxx
1, 2, 4(area)
*
RSCDxxxx
1, 2, 4(CD)
*
RSFLxxxx
2
(Flow)
*
MXRIxxxx
3
(Largest Radius of labseal)
*
STHTxxxx
3
(step Height of labseal)
*
CLEAxxxx
3
(Cleanrance)
*
UPRIxxxx
6
(upstream radius)
*
DWRIxxxx
6
(downstream radius)
*
RSRFxxxx
6
(RPMF)
*
RSEXxxxx
6
(Vortex exponent)
*
ORARxxxx
8
(Orifice area)
*
PIARxxxx
8
(Pipe area)
*
*Type 1,2,4 Restrictions (St.Dev 2-5% of Avg.)
RSAR7586
0.446
0.00892
2
0.01784
RSAR7584
0.1533
0.003066
2
0.006132
RSAR4092
1.644
0.03288
2
0.06576
RSAR4002
0.00181
0.0000362
2 25 0.0000724
RSAR4022
1
0.02
2
0.04
RSAR4061
0.5986
0.011972
2
0.023944
RSAR4017
0.003
0.00006
2
0.00012
RSAR4023
0.9923
0.019846
2
0.039692
1
Figure 6 Input file for probabilistic run
*
*Type 1,2,4 Restrictions (St.Dev 2-5% of Avg.)
RSAR7586
0.446
0.00892
2
0.01784
RSAR7584
0.1533
0.003066
2
0.006132
RSAR4092
1.644
0.03288
2
0.06576
RSAR4002
0.00181
0.0000362
2
0.0000724
RSAR4022
1
0.02
2
0.04
RSAR4061
0.5986
0.011972
2
0.023944
RSAR4017
0.003
0.00006
2
0.00012
RSAR4023
0.9923
0.019846
2
0.039692
RSAR8591
5.89
0.1178
2
0.2356
RSAR8503
38
0.76
2
1.52
RSAR7000
0.004
0.0002
2
0.0004
RSAR7001
0.004
0.0002
2
0.0004
RSAR7002
0.004
0.0002
2
0.0004
RSAR4027
9.204
0.18408
2
0.36816
RSAR4086
0.6675
0.033375
2
0.06675
RSAR4085
6.467
0.32335
2
0.6467
RSAR4037
5.658
0.2829
2
0.5658
RSAR4081
0.0001
0.000005
2
0.00001
RSAR4030
0.0815
0.004075
2
0.00815
RSAR4048
7.2
0.36
2
0.72
RSAR4069
0.0267
0.001335
2
0.00267
RSAR4072
0.02419
0.0012095
2
0.002419
RSAR4071
0.02419
0.0012095
2
0.002419
RSAR4049
0.2006
0.01003
2
0.02006
RSAR4053
0.092
0.0046
2
0.0092
RSAR4034
1
0.05
2
0.1
RSAR4035
1
0.05
2
0.1
RSAR4036
1
0.05
2
0.1
RSAR4077
60
3
2
6
RSAR4010
1
0.015
2
0.03
*
*TOBI ID/OD Lab Seals (St.dev 5-15% of CLR)
MXRI4011
6
0.12
2
0.24
CLEA4011
0.008
0.0004
2
0.0008
MXRI4015
7.56
0.1512
2
0.3024
CLEA4015
0.012
0.0018
2
0.0036
*
*Blade Platform Leakages (St.Dev 25% of Avg.)
RSAR4038
0.06306
0.015765
2
0.03153
RSAR4082
0.06306
0.015765
2
0.03153
RSAR4057
0.03
0.0075
2
0.015
RSAR4040
0.03024
0.00756
2
0.01512
RSAR4060
0.03
0.0075
2
0.015
RSAR4042
0.036
0.009
2
0.018
RSAR4076
6.721
1.68025
2
3.3605
RSAR4044
0.069
0.01725
2
0.0345
RSAR4043
0.012
0.003
2
0.006
RSAR4013
0.0841
0.021025
2
0.04205
RSAR4058
0.0021
0.000525
2
0.00105
RSAR4045
0.123
0.03075
2
0.0615
RSAR4063
0.00936
0.00234
2
0.00468
RSAR4062
0.0075
0.001875
2
0.00375
RSAR4047
0.42
0.105
2
0.21
RSAR4059
10.725
2.68125
2
5.3625
*
2
Figure 7 Input file for final probabilistic run
RSAR4044
0.069
0.01725
2
0.0345
RSAR4043
0.012
0.003
2
0.006
RSAR4013
0.0841
0.021025
2
0.04205
RSAR4058
0.0021
0.000525
2
0.00105
RSAR4045
0.123
0.03075
2
0.0615
RSAR4063
0.00936
0.00234
2
0.00468
RSAR4062
0.0075
0.001875
2
0.00375
RSAR4047
0.42
0.105
2
0.21
RSAR4059
10.725
2.68125
2
5.3625
*
*Vane Platform Leakages (St.Dev 25% of Avg.)
RSAR4004
0.03968
0.00992
2
0.01984
RSAR4005
0.03366
0.008415
2
0.01683
RSAR4006
0.026281
0.00657025 2
0.0131405
RSAR4007
0.1347
0.033675
2
0.06735
RSAR4008
0.04864
0.01216
2
0.02432
RSAR4046
0.0627
0.015675
2
0.03135
RSAR4056
0.003602
0.0009005
2
0.001801
RSAR4055
0.0627
0.015675
2
0.03135
RSAR4009
0.0905
0.022625
2
0.04525
RSAR4054
0.04864
0.01216
2
0.02432
*
*RPM Factors Vortices (St.Dev 5% of Avg.)
RSRF4000
0.325
0.01625
2
0.0325
RSRF4001
0.12
0.006
2
0.012
RSRF4012
0.5
0.025
2
0.05
RSRF4014
0.7
0.035
2
0.07
RSRF4016
0.5
0.025
2
0.05
RSRF4018
0.5
0.025
2
0.05
RSRF4019
0.48
0.024
2
0.048
RSRF4020
0.48
0.024
2
0.048
RSRF4028
1.23
0.0615
2
0.123
RSRF4029
1
0.05
2
0.1
RSRF4079
0.46
0.023
2
0.046
RSRF4084
1
0.05
2
0.1
RSFR4052
1
0.05
2
0.1
RSRF4051
1
0.05
2
0.1
RSRF4050
1
0.05
2
0.1
RSRF4032
1
0.05
2
0.1
RSRF4068
1
0.05
2
0.1
RSRF4070
1
0.05
2
0.1
*
** OUTPUT PARAMETER (DO NOT MODIFY THIS LINE !)
* CHAMBER:
PRESxxxx / CTMPxxxx (xxxx is ID NUMBER)
* RESTRICTOR:
RSFLxxxx
* BEARING LOAD:
LOADx (x is the xth BL in the BL list. no always th
rotor number!)
*
*TOBI ID/OD Seal Flow & Pressure & Temperature
RSFL4011
RSFL4015
RSFL4010
PRES4004
CTMP4004
RSFL4027
*Rim-cav Flow & Pressure & Temperature
RSFL4037
3
26
** OUTPUT PARAMETER (DO NOT MODIFY THIS LINE !)
* CHAMBER:
PRESxxxx / CTMPxxxx (xxxx is ID NUMBER)
* RESTRICTOR:
RSFLxxxx
* BEARING LOAD:
LOADx (x is the xth BL in the BL list. no always the
rotor number!)
*
*TOBI ID/OD Seal Flow & Pressure & Temperature
RSFL4011
RSFL4015
RSFL4010
PRES4004
CTMP4004
RSFL4027
*Rim-cav Flow & Pressure & Temperature
RSFL4037
PRES4045
CTMP4045
RSFL4085
RSFL4039
RSFL4076
PRES4013
CTMP4013
*Blade Platform Leakage
RSFL4038
RSFL4082
RSFL4041
RSFL4057
RSFL4040
RSFL4060
RSFL4042
*Vane Platform Leakage
RSFL4004
RSFL4005
RSFL4006
RSFL4007
RSFL4008
RSFL4009
RSFL4046
RSFL4056
RSFL4055
RSFL4054
*Blade Cooling Flow
RSFL4034
RSFL4035
RSFL4036
RSFL4077
*Blade Supply Pressure
PRES4003
Figure 8 Input file for final probabilistic run
27
4
4. Results of Latin Hypercube Analysis
4.1 Output Data
A probabilistic analysis allows variation of different input parameters (many at a
time) in a random manner independently which will generate the effect on several other
output parameters with the resulting probability or frequency distribution of each output
parameter. Identifying the significant parameters of the pre-swirl nozzle cooling air
capture and delivery system is determined by reviewing the output data.
There are 3 output files of interest generated by the probabilistic analysis and they
are the cumulative distribution function, the probability density function, and the
regression. The regression is the most helpful for identifying significant parameters by
looking at the percent of the total variance a given input contributes. The cumulative
distribution functions are useful for determining how likely a range of values are. The
probability density functions are useful for determining if enough samples were taken
and distribution type of output variables.
4.1.1.1 Pie Charts
The pie charts are made using the regression output and show the % of total
variance contribution for each output parameter. These will help identify which
parameters are significant contributors.
4.1.1.2 Histograms
The cumulative density function histograms show the how close the parameters
are to the normal distribution.
4.1.1.3 Plots
Probability Density Function (PDF) serves to represent a probability distribution in
terms of integrals.
The probability density functions are useful for determining if
enough samples were taken and distribution type of output variables. A probability
density function can be seen as a 'smoothed out' version of a histogram.
28
4.2 Probabilistic Results
4.2.1
Identifying key sources of variability
After standard deviations were entered the regression analysis was run and the
effect of individual input can be assessed. For this analysis a quadratic regression
equation is fit to the model output data by a method of least squares [7].
Y = y0 + bi(Xi) + ci(Xi)2
(1)
Where Y is the output variable, y0 is the constant regression coefficient, and bi is
the linear regression term which is a measure of how input variability affects output
variability. The quadratic regression term, ci, is a measure of how input variability
affects the output mean value. And
Xi = (Xi – i)/i
(2)
The regression results are manipulated to show a normalized linear coefficient as
discussed by Stearns & Cloud [2]. The following coefficient will indicate what potential
the input variability has to affect the output variability.
i = (bi*i/i)/100
(3)
Where bi is the linear regression term for the ith input and i and i is the sample
mean and the sample deviation value of the ith input variable. If the ith variable changes
by 1% the output variable will change by  units.
The next coefficient that is created is a measure of how much the total variance
of an input contributes and is calculated as follows:
i = bi2/bi2
(4)
The ith input variable contributes % of the total variance on that output. This
coefficient is used in this analysis to predict the most significant drivers, since the
assumed input variables were known to be reasonable approximations. Variables
contributing less than 1% will be ignored, with little loss of accuracy [2], but
contributions up to .1% will be included in subsequent plots when practical.
The
sensitivities of the outputs, by using this data, can be determined easily by sorting the
data by the magnitude of the linear regression coefficient.
29
4.2.2
Output Parameters and Location
The results of the probabilistic run are shown through the following normalized data
plots. The pie charts for each output parameter include the TOBI flow, TOBI ID seal
leakage, TOBI OD seal leakage, rim cavity purge flow for the leading and trailing edge
of the blade, and the supply pressure to the blade, and the cooling flow for the blade
leading edge, mid-body, trailing edge and platform trailing edge. See figures 1 & 2 for
locations of output parameters of the blade and TOBI area.
Mid body
cooling flow
Blade & Vane Inputs
Blade cooling
flow TE
Vane platform
Leakages
Blade cooling
flow LE
Blade platform
leakages
PF TE cooling flow
LE rim cavity
TE rim cavity
Rear blade
attachment
leakages
Blade supply
pressure
Figure 1 Blade output parameters
30
TOBI Area Inputs
TOBI OD
labyrinth seal
Mini disk
vortex
TOBI by-pass holes
TOBI OD
vortex
TOBI flow
Mini disk holes
TOBI ID
labyrinth seal
Figure 2 TOBI area output parameters
31
4.2.2.1 TOBI Flow Area
TOBI Flow Area % of Total Variance Contribution
0.03%
0.01%
TOBI Flow Area
Mini Disk Vortex RPMF
RSAR4036
99.97%
Figure 3
4.2.2.2 TOBI Discharge Pressure
TOBI Discharge Pressure
% of Total Variance Contribution
6.9% 0.6%
9.1%
Mini Disk Vortex RPMF
TOBI Flow Area
Blade LE Cooling Flow Area
9.1%
51.6%
Blade TE Cooling Flow Area
TOBI OD Lab Seal Clearance
11.0%
Blade Mid Body Cooling Flow
Area
Blade Platform Leakage
11.7%
Figure 4
32
4.2.2.3 TOBI Discharge Temperature
TOBI Discharge Temperature
% of Total Variance Contribution
0.3%
0.6%
1.2%
TOBI OD Lab Seal Clearance
3.0%
Mini Disk Vortex RPMF
3.5%
TOBI Flow Area
TOBI OD Cavity Vortex RPMF
Blade TE Cooling Flow Area
4.6%
29.8%
5.7%
Blade Mid Body Cooling Flow Area
Blade LE Cooling Flow Area
8.7%
TOBI ID Lab Seal Clearance
TOBI OD Lab Seal Radius
TOBI ID Lab Seal Radius
Blade Platform Leakage
16.2%
26.5%
Figure 5
33
4.2.2.4 TOBI Inner Diameter Labyrinth Seal Leakage
TOBI ID Lab Seal Leakage
% of Total Variarance Contribution
0.1%
5.6%
TOBI ID Lab Seal Clearance
0.0%
5.7%
TOBI ID Lab Seal Radius
TOBI OD Cavity Vortex RPMF
TOBI ByPass Hole Area
15.1%
Mini Disk Vortex RPMF
RSRF4012
RSAR4010
73.2%
RSAR4036
Figure 6
4.2.2.5 TOBI Outer Diameter Labyrinth Seal Leakage
TOBI OD Lab Seal Leakage
% of Total Variance Contribution
1.0%
2.4%
2.5%
TOBI OD Lab Seal Clearance
3.4%
Mini Disk Vortex RPMF
4.2%
TOBI Flow Area
Blade TE Cooling Flow
0.2%
4.3%
Blade LE Cooling Flow
Blade Mid-Body Cooling Flow
TOBI OD Lab Seal Radius
RSRF4014
19.5%
62.1%
RSRF4018
RSAR4049
RSAR4027
RSAR4030
34
4.2.2.6 Blade Supply Pressure
4.2.2.7 LE Rim Cavity Purge Flow
4.2.2.8 TE Rim Cavity Purge Flow
Blade TE Rim Cavity Purge Flow
% of Total Variance Contribution
1.5%
2.3%
0.1%
5.4%
Blade TE Rim Cavity Pressure
HPT Exit Ref Pressure
6.5%
Blade LE Rim Cavity Pressure
Blade Rear Fleather Seal Leakage
Blade Rear Fleather Seal Leakage
48.0%
HPT Inlet Ref Pressure
Blade Rear Fleather Seal Leakage
36.3%
4.2.2.9 Blade LE Cooling Flow
Blade LE Cooling Flow Area
% of Total Variance Contribution
0.1%
0.2%
0.6%
Blade LE Cooling Flow Area
4.7%
Blade TE Cooling Flow Area
5.7%
TOBI Flow Area
Blade Mid Body Cooling Flow Area
TOBI OD Lab Seal Clearance
Mini Disk Votex RPMF
Rear Blade Leakage
0.1%
6.8%
7.5%
TOBI OD Lab Seal Radius
RSRF4014
RSAR4030
10.9%
35
63.4%
4.2.2.10 Mid-Body Cooling Flow
Blade Mid Body Cooling Flow
% of Total Variance Contribution
0.2%
0.5%
4.4%
Blade Mid Body Cooling Flow Area
Blade TE Cooling Flow Area
Blade LE Cooling Flow Area
5.3%
6.9%
TOBI Flow Area
TOBI OD Lab Seal Clearance
8.4%
Mini Disk Vortext RPMF
Rear Blade Fleather Seal Leakage
MXRI4015
10.1%
64.2%
4.2.2.11 Blade TE Cooling Flow
Blade TE Cooling Flow
% of Total Variance Contribution
Blade LE Cooling Flow Area
Blade TE Cooling Flow Area
1.5%
1.8% 1.2%
2.0%
2.4%
0.2%
TOBI Flow Area
Blade Mid Body Cooling Flow Area
TOBI OD Lab Seal Clearance
Mini Disk Vortex RPMF
90.8%
Rear Blade Feather Seal Leakage
36
4.2.2.12 Platform TE Cooling Flow
Blade Platform Cooling Flow
% of Total Variance Contributions
0.6%
0.7%
0.8%
Blade Platform Cooling Flow Area
Blade TE Cooling Flow Area
Blade LE Cooling Flow Area
0.9%
1.1%
1.3%
0.1%
TOBI Flow Area
Blade Mid Body Cooling Flow Area
TOBI OD Lab Seal Clearance
Mini Disk Vortex RPMF
Rear Blade Leakage
94.6%
1. Vane & Platform Leakages
Table 1 shows the output regression coefficients for the TOBI inner diameter (ID) seal
flow restrictor.
37
5. Conclusions
38