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.
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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 laby-
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rinth 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.
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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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