Question 10.2: Woolley (2008; Woolley et al. 2008) devised a simple heat fl......
Woolley (2008; Woolley et al. 2008) devised a simple heat flux model to simulate heating at the surface of a mold during solidification of casting. This heating function is
q(t)=\left\{\begin{array}{c l}{{q_{0}t}}&{{0\leq t\leq1.0}}\\ {{q_{0}(1-e^{-0.013t})}}&t>1.0\end{array}\right. , t in seconds (10.11)
This heating function with q_{0}=10^{5}\,\mathrm{W/m^{2}} is used in the ANSYS model for a 20-second time interval and the same parameters as Example 10.1. The data obtained are shown in Table 10.3.
Use these data and knowledge of the exact heat flux to determine an appropriate value for the regularization coefficient,\alpha_{T i k}.
| Table 10.3 | Sensor and undisturbed temperature data for heating with Eq. (10.11). | ||||
| t,s | T_{p},{C} | T_{p},{C} | t,s | T_{p},{C} | T_{p},{C} |
| 1 | 3.2190 | 9.3354 | 11 | 106.36 | 197.00 |
| 2 | 14.492 | 37.377 | 12 | 114.33 | 208.43 |
| 3 | 26.868 | 63.784 | 13 | 121.98 | 219.15 |
| 4 | 38.740 | 86.828 | 14 | 129.32 | 229.24 |
| 5 | 49.971 | 107.24 | 15 | 136.37 | 238.74 |
| 6 | 60.594 | 125.58 | 16 | 143.16 | 247.72 |
| 7 | 70.662 | 142.26 | 17 | 149.70 | 256.20 |
| 8 | 80.230 | 157.55 | 18 | 155.99 | 264.23 |
| 9 | 89.344 | 171.68 | 19 | 162.07 | 271.85 |
| 10 | 98.045 | 184.78 | 20 | 167.92 | 279.08 |
For this application, Eq. (10.2) is utilized.
\mathbf{T}_{p}-T_{i}\{\mathbf{1}\}=\mathbf{X}_{\phi_{p}}\mathbf{q} (10.2)
Data for “T_{p}”\operatorname{and} “T_{p\infty}”” from a step change in surface are needed to formulate the \mathbf{X}_{\phi_{p}} matrix. These are the same data as in Example 10.1, and the resulting \mathbf{X}_{\phi_{p}} matrix is also the same as before. The \mathbf{F}_{\phi_{p}} is calculated in the same way as in Example 10.1 (using Eq. (10.8))
\mathbf{F}_{\phi_{p}}=\left(\mathbf{X}_{\phi_{p}}^{T}\mathbf{X}_{\phi_{p}}+\alpha_{\mathrm{Tik}}\mathbf{H}_{1}^{T}\mathbf{H}_{1}\right)^{-1}\mathbf{X}_{\phi_{p}}^{T} (10.8)
and now only depends on the value of the regularization coefficient \alpha_{T i k}.
Figure 10.5 displays the exact heat flux along with three reconstructions of this heat flux. For each curve, the estimated heat flux is computed from inversion of Eq. (10.2) using Tikhonov regularization:
\hat{\mathbf{q}}=\left[\mathbf{F}_{\phi_{p}}(\alpha_{T i k})\right]\{\mathbf{T}_{p}-\mathbf{T}_{0}\} (10.12)
In Eq. (10.12), T_p refers to data from the known heating function simulation in Table 10.3, and \mathbf{F}_{\phi_{p}}(\alpha_{T i k}) indicates the dependence of the matrix on the parameter \alpha_{T i k}. These results are obtained using piecewise constant assumption about the heat flux (see calculation of the \mathbf{X}_{\phi_{p}} matrix in Eqs. (10.6)
\left\lfloor7.7962-0\ \ 20.404-7.7962\ \ 33.028-20.404\ \ 45.188-33.028\ \cdots\right\rfloor ^T= \left\lfloor7.7962\\ 0\ \ 12.608\ \ 12.624\ \ 12.160\ \cdots\right\rfloor ^T (10.6)
and (10.7)),
{\bf X}_{\phi_{p}}= \begin{bmatrix} 7.7962 & 0 & 0 & 0 & 0 & 0 \cdots \\ 12.608 & 7.7962 & 0 & 0 & 0 & 0 \cdots \\12.624 & 12.608 & 7.7962 & 0 & 0 & 0 \cdots\\ 12.160 & 12.624 & 12.608 & 7.7962 & 0 & 0 \cdots\\ \vdots & \vdots &\ddots &\ddots &\ddots &\ddots \end{bmatrix}_{20\times 20} (10.7)
hence the estimated heat flux components q_{i} are shown at times t_{i} − Δt/2.
Discussion:
The “optimal” results in Figure 10.5 are for \alpha_{T i k} = 6 and were determined holistically by varying this parameter until the estimated heat fluxes best matched the exact curve. The other curves for \alpha_{T i k} = 0.6 and \alpha_{T i k} = 60 demonstrate effects of under-regularization and over-regularization on the estimates.
Since the filter matrix \mathbf{\hat{F}}_{\phi_{p}} is common to both the heat flux estimation and correction kernel computation, the idea of using a companion simulation with a known heat flux, demonstrated in this example, can be used to guide the selection of amount of regularization needed for determining the correction kernel H.
The final application of the correction kernel concept is to directly correct measured temperatures using a previously determined H for particular sensor installation. The following example illustrates this application.
