Using Spatial Statistics to Estimate parameters in Phase Change Experiments A. F. Emery and D. Bardot University of Washington Seattle, WA,

From Jim’s talk Fundamental Parameters Normalized Sensitivities Short vs Long Time Solutions Nuisance Variables

IceLiquid Heat Time, t 1 / Gas /Foam Time, t 2

Foam Ice

Ice Phase change has a sharp front Foam Phase change due to decomposition has a reaction zone in which

Parameters Ice Foam k s conductivity k l c s specific heat c l L Latent heat k s conductivity c s specific heat h r heat of reaction  emissivity  density E1, E2 reaction energies

Sensitivities

Response Surface for Single Parameter  

Estimating k and h r

Standard Deviation of Estimated Parameters % per % Uncertainty in Front Position Measured Every Second h r  c f k f E 1 E 2 Estimated Singly Estimated Collectively

Estimating k and L

The Melting Front Position is defined by Factor FactorDependency

Estimating k and L

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 element thick moves with a velocity V ~0.5 to 2 cm/min ~ 0.1 to 0.3 mm/sec V diffusion zone width ~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 fits 2)Splines exactly through data points 3) Kriging exactly through data points w/o Nugget minimum variance between points w/ Nugget minimum variance at all points

Let estimate Z by where v i are prescribed functions

Where  are found by subject to the constraint that is unbiased

The solution for  depends upon the variogram Kriging assumes intrinsic stationarity If 2 nd order stationarity exists

 depends upon the fit Nugget

11 x 11 Grid X

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 Laboratories Validation Program Dr. Kevin Dowding, Technical Monitor