Using Spatial Statistics to

Estimate parameters in Phase Change Experiments

A. F. Emery and D. Bardot

University of WashingtonSeattle, WA, 98195-2600

From Jim’s talk

Fundamental Parameters

Normalized Sensitivities

Short vs Long Time Solutions

Nuisance Variables

IceLiquid

Heat

Time, t1

/ Gas /Foam

Time, t2

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

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

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

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

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

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

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

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

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

The Melting Front Position is defined by

tX l2

)/,,( slSbStf

mb

im

l

s

mbl

TTTT

kkNumberSubcooledSb

LTTcNumberStefanSt

/)( Factor

Factor

Dependency

Dependency

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

With a fine enough grid,good parameter estimates can be gotten

With a fine enough grid,good parameter estimates can be gotten

?

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

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

p

iii ssvsZ

0

)()()(

p

iii sZsZ

0

)()(ˆ

Let

estimate Z by

where vi are prescribed functions

]))(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

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

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

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

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

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

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

Support provided by

Sandia National LaboratoriesValidation Program

Dr. Kevin Dowding, Technical Monitor