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. 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.
1.1.1 – 2003 Sidwell & Darmofal Study
In 2003 Sidwell & Darmofal [1] documented the variability of turbine airfoil oxidation life due to variability in the turbine cooling air of a commercial turbofan engine.
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As is similarly done in the proposed research, the turbine cooling air capture and delivery system is the main focus of this study. Since turbine blades of today’s commercial
turbofan engine, operate in temperatures significantly higher than their melting point,
they must be convectively cooled by air delivered from upstream of the burner by the
secondary air system. The cooling air also provides a film of cooler air that protects the
airfoil from the onslaught of the high temperature primary core flow just exiting the
burner [1]. This application only guarantees turbine blade susceptibility to high temperature oxidation.
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.
The probabilistic distribution of predicted failure times was compared to the field failure
distributions for both airlines. The two airlines (Airline A and Airline B) operate the
commercial turbofan engine that is represented in their analysis. Airline A’s fleet of 82
engines operate in a standard day climate and are used for flights greater than 5 hours.
Airline B’s fleet operates 13 engines in a hotter climate for much shorter flights.
1.1.1.1 – 2003 Sidwell & Darmofal Probabilistic Method
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 environmental conditions such as ambient temperature. Engine to engine variability included engine
conditions, blade to blade variations and manufacturing variations. Engine conditions
varied were component temperatures and rotor speeds and were based on field experience. The blade to blade variations such as film cooling hole effective areas that are
relevant to placement 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. The effect of day to day variability was captured by applying a standard
deviation of 18oF to ambient temperature input for both airlines, which was based on
field data.
1.1.1.2 – 2003 Sidwell & Darmofal Probabilistic Results
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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.
The probabilistic failure times were compared to field failure distributions for
each airline. The probabilistic results showed a good correlation to field experience of
the airline that operates at hotter ambient conditions, but not the airline that operates at
cooler ambient conditions and for that case a nominal analysis was used.
Figure 1 shows the The results of their study showed that for both airlines a 10
percent decrease in the tolerance on the blade leading edge effective flow area can have
an impact on the typical and minimum life. For Airline B, this decrease can more than
double the minimum oxidation life of the engine. The study showed that by knowing the
distribution of oxidation failure it is possible to minimize the number of unscheduled
engine removals.
Modeling all the blades in a turbine blade row as a single lumped blade obscured
physical phenomena. When each blade was included separately, and allowed to vary
independently, higher flowing blades stole flow from lower flowing blades. Both of
these effects decrease the oxidation life of an engine. The analysis showed that lumping
the blades over predicted the engine oxidation life from 16% to 27% for airline A and
airline B respectively. 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 blade leading edge effective flow area for both airlines.
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. For the effects of thermal and
centrifugal growth, labyrinth seals and chambers and vortex radii had a percent deviation
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applied. Absolute deviations should be applied when manufacturing tolerances are to be
analyzed. The method was applied in order to find variability in the secondary flow
system and the axial rotor bearing loads of a turbofan engine. The results showed the
system behaved linearly, resulting in negligible mean shifts due to input variation.
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.
Unlike the previously performed probabilistic analyses, this study will focus on
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. Similarly this paper will attempt to identify allowable
variation of manufacturing tolerances of the subsystem while still meeting the requirements of the system.
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