Question 10.3: Use the correction kernel found in Example 10.1 to correct t......

Eqs. (10.5c) and (10.5d)

{\bf T}_{p\infty}={\bf T}_{p}+\left[\Delta{\bf T}_{p}\right]\{{\bf H}\}  (10.5c)

\left[\Delta\mathbf{T}_{p}\right]=\left[\begin{array}{c c c c c}{{\Delta T_{p,0}}}&{{0}}&{{0}}&{{0}}&{{0}}\\ {{\Delta T_{p,1}}}&{{\Delta T_{p,0}}}&{{0}}&{{\ldots}}&{{0}}\\ {{\Delta T_{p,2}}}&{{\Delta T_{p,1}}}&{{\Delta T_{p,0}}}&{{\ddots}}&{{\vdots}}\\ {{\vdots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{0}}\\ {{\Delta T_{p,n-1}}}&{{\ldots}}&{{\Delta T_{p,2}}}&{{\Delta T_{p,1}}}&{{\Delta T_{p,0}}}\end{array}\right] (10.5d)

are used to compute the corrected values. The matrix of ΔT_{p} in Eq. (10.5d) is computed using the column of T_{p} values in Example 10.2 in a manner identical to the calculation of \Delta\phi_{p} in Example 10.1. First, the ΔT_{p} vector is computed using differences of the T_{p} vector, just as the \Delta\phi_{p} values were computed in Example 10.1. The first few values are

=\lfloor3.219\quad11.273\quad12.376\quad11.231\quad\cdot\cdot\cdot\rfloor^{T} (10.13)

These values are used to build the ΔT_p matrix. The first part of this matrix is

\Delta\mathbf{T}_{p}=\left[\begin{array}{c c c c c}{{3.219}}&{{0}}&{{0}}&{{0}}&{{a}}&{{0}}&{{\cdots}}\\ {{11.273}}&{{3.219}}&{{0}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{12.376}}&{{11.273}}&{{3.219}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{11.231}}&{{12.376}}&{{11.273}}&{{3.219}}&{{0}}&{{0}}&{{\cdots}}\\ {{\vdots}}&{{\vdots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}\end{array}\right]_{20\times20} (10.14)

Finally, Eq. (10.5c) can be used to compute the correction by matrix multiplication and vector addition using the T_p data from Example 10.2:

{\bf T}_{p\infty}={\bf T}_{p}+[\Delta{\bf T}_{p}]\{{\bf H}\}  (10.15)

The results of these calculations are the following temperatures:

The root mean squared error between these values and the T_{p\infty} values in Example 10.2 is 0.1921 K. The comparison is shown graphically in Figure 10.6.

Discussion:
The temperature data from Example 10.2 are from the same numerical model but with a different heat flux excitation than Example 10.1, which provides a fair test of the ability of the results from the step heat flux to generate the correction kernel to apply to a different “experiment.” However, no measurement errors are considered in this calculation. Random errors with zero mean and Gaussian distribution having standard deviation \sigma_{n o i s e} = 1.0 K were added to the T_{\mathfrak{p}} data from Example 10.2 and the calculations repeated. For this case, the RMS error in the corrected temperature is 2.4092 K, and the graphical comparison is shown in Figure 10.7.

The preceding three examples illustrate how a numerical model of a sensor installation can improve estimates for surface heat flux and/or indicated sensor temperature. Of course, the improvement can be no better than the representation of the physical system by the numerical model. For example, the previous simple axisymmetric model does not account for contact resistance between the imbedded sensor and the surrounding medium. If this resistance is significant, then the model will be biased and the results will be correspondingly distorted. However, any reasonable model to incorporate the sensor dynamics will improve results over those obtained by completely ignoring such effects.