1. Results of Latin Hypercube Analysis
1.1 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.
1.2 First Run Results
1.2.1
Identifying key sources of variability
After approximate 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)
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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.
1.2.2
Output Parameters and Location
The results of the first run are shown through the following normalized data plots.
The pie charts for each output parameter include the TOBI flow, TOBI ID seal, TOBI
OD seal, 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. A pie chart for each probabilistic output is also
shown. Figure 1 & 2 shows the flow model output restrictors and chambers of the blade
and TOBI area.
2
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
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
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1.2.2.1 TOBI Flow Area
TOBI Flow Area
% of Total Variance Contribution
0.0%
TOBI Flow Area
Mini-Disk Vortex RPMF
MXRI4015
CLEA4015
RSAR4036
RSAR4034
RSAR4035
RSAR4027
100.0%
Figure 3
1.2.2.2 TOBI Discharge Pressure
TOBI Discharge Pressrue
% of Total Variance Contribution
0.3%
1.8%
2.3%
2.9%
Mini Disk RPMF
TOBI Flow Area
6.5%
TOBI Radius of OD Lab Seal
TOBI OD Lab Seal Clearance
37.5%
14.7%
Blade TE Cooling Flow
Blade LE Cooling Flow
Blade Mid-Body Cooling Flow
RSAR4027
34.0%
Figure 4 TOBI Discharge Pressure % of Total Variance Contribution
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PDF Histogram for TOBI Discharge Pressure (% Reference Pressure)
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
47.4
47.9
48.4
48.9
49.4
49.9
50.4
50.9
%Pref
1.2.2.3 TOBI Discharge Temperature
TOBI Discharge Temperature
%of Total Variance Contribution
0.5%
0.7%
TOBI Flow Area
0.8%
TOBI OD Lab Seal Radius
3.5%
TOBI ID Lab Seal Radius
TOBI ID Lab Seal Clearance
0.1%
24.6%
10.0%
TOBI OD Lab Seal Clearance
Mini Disk Vortex RPMF
10.9%
TOBI OD Cavity Vortex RPMF
Blade TE Cooling Flow
Blade LE Cooling Flow
11.1%
24.2%
Blade Mid-Body Cooling Flow
RSAR4027
13.7%
Figure 5 TOBI Discharge Temperature % of Total Variance Contribution
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51.4
51.9
1.2.2.4 TOBI Inner Diameter Labyrinth Seal Leakage
TOBI Inner Diameter Seal Leakage
% of Total Variance Contribution
0.4%
0.8%
0.0%
Radius of ID Seal
ID Seal Clearance
By-Pass Holes
TOBI OD Vortex
43.3%
Mini disk holes vortex rpmf
55.5%
Figure 6 TOBI ID Lab Seal Leakage % of Total Variance Contribution
1.2.2.5 TOBI Outer Diameter Labyrinth Seal Leakage
TOBI Outer Diameter Labyrinth Seal Leakage
% of Total Variance
0.4%
0.5%
0.7%
Max Radius TOBI OD Seal
Clearance TOBI OD Seal
Mini-disk vortex
TOBI flow area
7.3%
8.0%
RSAR4036
RSAR4034
RSRF4014
25.7%
57.4%
Figure 7 TOBI outer diameter lab seal leakage % of total variance contribution
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1.2.2.6 Blade Supply Pressure
1st Blade Supply Pressure
% of Total Variance
0.3%
3.9%
5.1%
TOBI Flow Area
6.3%
Max radius TOBI OD Labyrinth Seal
Clearnce TOBI OD Labyrinth Seal
7.4%
Mini-disk Vortex RPMF
Blade Cooling Flow TE
47.8%
9.0%
Blade Cooling Flow Mid-body
Blade Cooling Flow LE
RSAR4049
20.2%
Figure 8 Blade Supply Pressure % of Total Variance Contribution
1.2.2.7 LE Rim Cavity Purge Flow
1. LE Rim Cavity Purge Flow
2. TE Rim Cavity Purge Flow
2. Blade Cooling
1. LE Cooling Flow
2. Mid-Body Cooling Flow
3. TE Cooling Flow
4. Platform TE Cooling Flow
3. Vane & Platform Leakages
Table 1 shows the output regression coefficients for the TOBI inner diameter (ID) seal
flow restrictor.
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Table 1 Regression coefficients for TOBI ID labyrinth seal restrictor
1 Output Parameter
RSFL4011
Sample Mean = 0.373849
Y0 (constant regression coefficient)
0.375642
=
Parameter
Radius of ID Seal
ID Seal Clearance
By-Pass Holes
TOBI OD Vortex
Mini disk holes vortex rpmf
RSRF4012
RSAR4010
MXRI4015
CLEA4015
RSAR4036
RSAR4035
Sample_Mean
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0.008
0.5986
0.7
1.23
0.5
1
7.56
0.012
1
1
Sample Deviation = 0.053213
R Squared (ideal value = 1) = 0.997041
Sample_Dev
0.7918
0.0011
0.0158
0.0308
0.0541
0.022
0.0264
0.9976
0.0016
0.0264
0.0264
ABS(Bi)
0.0395
0.0349
0.0048
0.0032
0.0006
0.0006
0.0005
0.0004
0.0003
0.0001
0.0001
Ci


-0.001
0.003
55.4948
-0.0009
0.0026
43.2758
0
-0.0018
0.8058
0
0.0007
0.3747
0
0.0001
0.0119
0.0001
0.0001
0.011
0
-0.0002
0.0103
0
0
0.0047
-0.0001
0
0.0024
0
0.0001
0.0007
0
0.0001
0.0007
From this data it is easy to see that the most significant driver of the TOBI ID
seal flow is the radius of the labyrinth seal at the ID, followed by the ID seal clearance.
As the radius and clearance change by 1% the output variable will change by  units.
Only the first 2 parameters have any significant contribution to the TOBI ID labyrinth
seal, the following restrictions drop below the 1% value for total contributions. The
radius and seal clearance contribute 55.5% and 43.3 % respectively. Figure 1 shows a
cross-section with restrictor locations for the HPT TOBI area.
Another output of the regression is the r2 value, which is given for each output
parameter. This is an indicator that measures how well the regression equation fits the
data and the closer this value is to 1 means that a good fit to the data was achieved [7].
Combining this term with histograms of input and output variables can help to
determine whether or not enough samples have been taken. The cumulative distribution
function is alos an input
The results of the first run are shown through the following normalized data plots.
The pie charts for each output parameter include the TOBI flow, TOBI ID seal, TOBI
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OD seal, rim cavity purge flow for the leading and trailing edge of the blade, and the
supply pressure and temperature to the blade, and the cooling flow for the blade leading
edge, mid-body, trailing edge and platform trailing edge. A pie chart for each probabilistic output is also shown. Figure 1 & 2 shows the flow model output restrictors and
chambers of the blade and TOBI area.
Figure 2 is a pie chart showing the % of total variance the TOBI inner diameter seal
leakage input parameter has on the system using regression analysis. By default, all the
input variables will change completely independent of each other. The input parameters
contributing most significantly are All restrictors contributing less than 1% to the total
variance are ignored with little loss of accuracy [2].
Figure 2 TOBI inner diameter seal leakage
Figure 5 TOBI outer diameter labyrinth seal leakage
Blade Platform Cooling Flow
% of Total Variance
1.3%
1.6%
1.1%
0.9%
0.7%
3.7%
8.7%
RSAR4077
RSAR4010
MXRI4015
CLEA4015
RSRF4028
RSAR4036
RSAR4034
81.9%
RSAR4035
Figure 6 Blade platform cooling flow
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1.2.2.8 The Regression Output
1.2.2.9 The Probability Density Function
The probability density functions are useful for determining if enough samples were
taken and distribution type of output variables.
1.2.2.10 The Cumulative Density Function
The cumulative density functions are useful for determining how likely a range of
values are.
1.3 Second Run Results
The quadratic interaction terms were not used in this analysis, they describe the ability of
the variation of one input to affect the sensitivity of the output to the variability of other
inputs. By default, all the input variables will change completely independent of each
other [2].
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