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Using Spatial Statistics to
Estimate parameters in Phase Change Experiments
A. F. Emery and D. Bardot
University of WashingtonSeattle, WA, 98195-2600
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From Jim’s talk
Fundamental Parameters
Normalized Sensitivities
Short vs Long Time Solutions
Nuisance Variables
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IceLiquid
Heat
Time, t1
/ Gas /Foam
Time, t2
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Foam Ice
time (sec)0 10 20 30 40 50 60
T (C
)
10
20
30
40
50
60
70
x (sens) /x_Front(55)= 1.5
1.5
0.75
1.0
x (m)0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040
0.00
0.01
0.02
0.03
0.04
0.05
0.06
T/25000
S1
FF+S1
Temperature Sensing Zone
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IceIce
Phase change has a sharp front
FoamFoam
Phase change due to decomposition has a reaction zone in which
21
11 7.03.0
GS
GSF
TRE
TRE
TRE
eAS
eAFdtdS
eAFdtdF
/1
/1
/
2
1
1
3.0
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Parameters
Ice Foam
ks conductivity
kl
cs specific heat
clL Latent heat
ks conductivity
cs specific heat
hr heat of reaction
emissivity
density E1, E2 reaction energies
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Time (sec)0 10 20 30 40 50
Nor
mal
ized
Sen
sitiv
ity
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
E2
khr
cpE1
Sensitivity of Front Position
Time (sec)25.0 30.0 35.0 40.0 45.0
Nor
mal
ized
Sen
sitiv
ities
-12000
-8000
-4000
0
4000
8000
hrk
E2
E1cp
Sensitivity of Temperature
Sensitivities
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p/p00.90 0.95 1.00 1.05 1.10
L(F)
0
50
100
150
200
250
300
cphf
k
E2 E1
Foam Front Response For a Single Parameter
p/p00.90 0.95 1.00 1.05 1.10
L(T)
0
2000
4000
6000
8000
10000
E2
khf
E1
cp
Temperature Response For a Single Parameter
Response Surface for Single Parameter
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hr /hr0k/k0
L
0.940.96
0.981.00
1.02 1.041.06
0.950.97
0.981.00
1.021.03
1.05 0
2
4
6
8
10
Foam Front PositionEstimating k and hr
hr /hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05Foam Front Position
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hr /hr0k/k0
L
0.950.97
0.981.00
1.02 1.031.05
0.950.97
0.981.00
1.021.03
1.05 0
50
100
150
200
250
300
350
Foam Temperature
Estimating k and hr
hr /hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05
Foam Temperature
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Standard Deviation of Estimated Parameters % per % Uncertainty in Front Position
Measured Every Second
hr cf kf E1 E2
Estimated Singly 0.55 0.13 0.23 0.87 0.14 0.11
Estimated Collectively 5.20 4.42 3.32 3.53 3.36 3.52
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k/k0 L/L0
L
0.800.850.900.951.001.051.101.151.20 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.200.00000
0.00005
0.00010
0.00015
0.00020
Melting Front Position
L/L0
k/k0
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
Melting Front Position
Estimating k and L
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The Melting Front Position is defined by
tX l2
)/,,( slSbStf
mb
im
l
s
mbl
TTTT
kkNumberSubcooledSb
LTTcNumberStefanSt
/)( Factor
Factor
Dependency
Dependency
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L/L0k/k0
L
0.800.850.900.951.001.051.101.151.20 0.800.87
0.931.00
1.07 1.131.200.0
5.0
10.0
15.0
20.0
25.0
Melting Temperature
Estimating k and L
L/L0
k/k0
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20Melting Temperature
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With a fine enough grid,good parameter estimates can be gotten
With a fine enough grid,good parameter estimates can be gotten
?
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Back of the envelope estimates
A crude 1-D finite volume calculations shows that a reaction front 1 elementthick moves with a velocity V ~0.5 to 2 cm/min ~ 0.1 to 0.3 mm/sec
V
diffusion zonewidth ~1.5 to 3 mm
time to make readings
x ~0.1 to 0.3 mm
3 to 10 seconds
Giving a computational time of 60 minutes for an 11 x 11 grid
4 hours for a 21 x 21 grid
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Spatial Statistics
A method of fitting/interpolating/extrapolating
1) Least Squares smoothly fits2) Splines exactly through data points3) Kriging exactly through data points w/o Nugget minimum variance between points
w/ Nugget minimum variance at all points
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p
iii ssvsZ
0
)()()(
p
iii sZsZ
0
)()(ˆ
Let
estimate Z by
where vi are prescribed functions
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]))(ˆ)([(minimizing 200 sZsZE
Where are found by
subject to the constraint that
TTT xXsZE )](ˆ[ 0
is unbiased)(ˆ0sZ
)()...((and)( 000, svsvxsvX pT
jiji
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The solution for depends upon the variogram
)]()(var[)(2 jiji sZsZss
Kriging assumes intrinsic stationarity
)]()(var[)(20)]()([
sZhsZhsZhsZE
If 2nd order stationarity exists
)()0()(2 hCCh
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Isotropic Lag0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
(la
g)
0
2
4
6
8
10
linear
constant
depends upon the fit
Nugget
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L/L0
k/k0
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20Melting Temperature
Kriged Melting Temperature
L/L0
k/k0
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
L/L0k/
k00.90 0.95 1.00 1.05 1.10
0.90
0.95
1.00
1.05
1.10
2nd Order Fitted Temperature
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hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05
2nd Order Fit of Foam Temperature
hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05Foam Temperatures11 x 11 Grid
hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05Kriged Foam Temperature
X
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hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05Foam Front Position11 x 11 Grid
hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.052nd Order Fit of Foam Front Postion
hr/hr0
k/k0
0.95 0.97 0.98 1.00 1.02 1.03 1.05
0.95
0.97
0.98
1.00
1.02
1.03
1.05
Kriged Foam Front Position
X
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1) Simple Experiments should be dimensionally analyzed first to detect factor dependency
2) Crude computational models should be exercised to give order of magnitude estimates of physical behavior
4) Spatial statistics should be employed to minimize overall computational times
3) Parameters, x, t, and number of sensor readings should be defined
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Support provided by
Sandia National LaboratoriesValidation Program
Dr. Kevin Dowding, Technical Monitor