Question 10.5: Consider the same intrinsic thermocouple as in Example 10.4 ......
Consider the same intrinsic thermocouple as in Example 10.4 with β = 1.33. Suppose that “measurements” are made using the thermocouple, and that these measurements are
Y_i = 20 \exp (−0.15t^+) + ε_i (10.24)
where ε_i is a random number from a Gaussian distribution with zero mean and standard deviation σ = 0.5 K. For this example, the same random number sequence from examples in Chapter 7 will be used.
The measurements Y are computed using the specified formula and the random errors based on the sequence in Chapter 7. The first 30 values are shown in Table 10.5.
With the H vector already determined, the corrected temperatures can be computed directly using Eq. (10.5c).
{\bf T}_{p∞}={\bf T}_{p}+\left[\Delta{\bf T}_{p}\right]\{{\bf H}\} (10.5c)
The ΔT_p matrix, based on the “sensed” temperatures, Y, is needed. The first few entries of this 100 × 100 matrix are
\Delta\mathbf{T}_{p}=\left[\begin{array}{c c c c c}{{20.903}}&{{0}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{-1.062}}&{{20.903}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{-0.067}}&{{-1.062}}&{{20.903}}&{{0}}&{{0}}&{{\cdots}}\\ {{-0.236}}&{{-0.067}}&{{-1.062}}&{{20.903}}&{{0}}&{{\cdots}}\\ {{\vdots}}&{{\vdots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}\end{array}\right] (10.25)
Direct application of Eq. (10.5c) results in the desired corrected temperatures, T_{p∞}. The first 30 values of the corrected temperatures are listed in Table 10.6, and a plot showing the measured and corrected values is shown in Figure 10.13.
Discussion:
Although the correction process is not ill-posed, the corrected temperatures in Figure 10.13 appear to contain more noise than in the original data. This is because the correction is computed from the noisy data resulting in a “noisy” correction, which is then added to the original noised data.
The issue of noisy data can be addressed by smoothing data, either before correction, or after correction, or both. Smoothing can be accomplished by moving average or other means, such as mollification (Murio 1993) using generalized cross-validation (Murio et al. 1998). The dark line in Figure 10.13 shows the result of the mollification of the corrected temperatures only using generalized cross-validation. As seen in the figure, very smooth results for the corrected temperature are obtained but at the expense of some bias in the values introduced by the smoothing. The bias is most evident at early times, as the initial temperature is reduced by about 10 K, and a smaller difference between the smooth curve and the computed results of about 2 K is seen up until about t^+ = 2.
| Table 10.5 | Temperature measurements for Example 10.5. | |||||||||
| Y_{1-10} | 20.903 | 19.842 | 19.775 | 19.539 | 20.341 | 19.172 | 19.887 | 20.236 | 20.467 | 19.891 |
| Y_{11-20} | 19.783 | 19.421 | 19.487 | 19.506 | 20.493 | 19.777 | 19.506 | 19.419 | 18.671 | 19.552 |
| Y_{21-30} | 20.375 | 18.827 | 19.135 | 20.028 | 19.700 | 19.300 | 19.462 | 19.270 | 19.803 | 20.444 |
| Table 10.6 | Corrected Temperatures for Example 10.5. | |||||||||
| T_{p\infty 1-10} | 46.785 | 43.642 | 42.954 | 41.979 | 43.392 | 40.402 | 41.710 | 42.009 | 42.014 | 40.256 |
| T_{p\infty 11-20} | 39.609 | 38.419 | 38.234 | 37.954 | 39.856 | 37.923 | 37.044 | 36.603 | 34.698 | 36.477 |
| T_{p\infty 21-30} | 38.095 | 34.383 | 34.900 | 36.721 | 35.785 | 34.719 | 34.926 | 34.327 | 35.377 | 36.652 |
