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3.2.3 Statistical Modeling for Polymer Temperature and
Pressure Measurement

Mark G. Vangel
Statistical Engineering Division, ITL

Anthony J. Bur
Polymers Division, MSEL

The objective of this project is to develop methods for indirectly measuring the temperature and pressure of a polymer during fabrication. A dye which fluoresces when excited by a laser, is added to the polymer. Empirically, it has been shown previously that the ratio of the intensities of a fluorescence spectrum at two wavelengths can, for a practically useful range of temperatures, materials, and dyes be nearly linearly related to the temperature within the polymer. Thus, linear calibration curves for temperature can often be determined. By measuring the intensity at two wavelengths, a user can then use these curves to estimate temperature - at atmospheric pressure.

However, polymers are typically under pressure during fabrication, so we have begun this year to attempt to measure temperature and pressure simultaneously from fluorescence spectra. Experiments were designed to obtain spectra for a rectangular grid of temperatures and pressures, with 10 levels of pressure and 5 to 9 levels of temperature. Data were obtained for several combinations of polymers and dyes.

Perhaps the most promising result obtained thus far is based on a multivariate additive ANOVA. For each (of typically hundreds) of values of wavelength $\lambda$, the responses are a two-way table in temperature and pressure, to which a simple additive ANOVA model can be fit. This model fits extremely well for the polymer/dye combinations for which data are available. The temperature effect is more complicated, but the pressure effect appears to be very nearly equal to the product of a function of wavelength with a function of pressure. Using an obvious notation, we have

\begin{displaymath}y(\lambda_i,T_j,P_k) \approx
m(\lambda_i) +h(\lambda_i,T_j)
+u_1(\lambda_i) u_2(P_k)

The functions in the above expression are easily estimated pointwise by least squares and smoothed afterwards, if necessary. Conveniently, u2(P) is nearly linear over the pressure range in the experiments.

Once the above functions have been estimated, one can develop a model for estimating temperature, independent of pressure, using an orthogonal trigonometric basis of modest dimension (say, 10 to 30). Each basis function is constructed to also be orthogonal to $u_1(\lambda)$. By multiplying each basis function by each spectrum and integrating over $\lambda$, a matrix results, with as many rows as spectra and as many columns as basis functions. Multiple regression, with this matrix as a design matrix and T as response, is then used to determine a linear combination of basis functions which predicts temperature well, for all pressures. Excellent fits to temperature have been obtained in this way. To better assess predictive capability, cross-validation will be used in future work.

\epsfig{file=/proj/sedshare/panelbk/2000/data/projects/stand/,angle=-90,width=6.0in} \end{figure}

Figure 7: The figure illustrates (for an example with $i=1,\dots,501$, $j=1,\dots,7$, and $k=1,\dots,10$) estimates of the 10 (unsmoothed) pressure effects from both a multivariate additive ANOVA with independent errors (points) and from a least-squares fit of the above model, which assumes a multiplicative pressure effect. The bold curve, proportional to the estimate of $u_1(\lambda)$, is approximately rotated about the horizontal axis as pressure is increased.

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Date created: 7/20/2001
Last updated: 7/20/2001
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