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3.3.8 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 is added to the polymer which fluoresces when excited by a laser. 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 curve to estimate temperature - at atmospheric pressure.

However, polymers are typically under pressure during fabrication, so we've 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.

Various approaches have been used in the statistical analysis of these data. None of them are yet completely successful, but perhaps the most promising has been 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. It can be seen from the figure that the pressure effect $u_1(\lambda)$is multiplied by a periodic function of pressure, so that aliasing will likely occur for sufficiently high pressures. Conveniently, u2(P) is nearly linear over the pressure range in the experiments, so pressure estimation (over an interval) should be straightforward.


Figure 22: 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 $u_2(\lambda)$, 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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