Question 10.4: Consider an intrinsic thermocouple having β = 1.33. For a st......

Eq. (10.19) is used to compute the temperature of the wire surface relative to the constant substrate temperature T_b = 1 K.

T_{w}^{+}(0,t)=\left\{\begin{array}{c}\\\frac{1}{1+\beta}\biggl[1+\exp\biggl(\biggl(\frac{2\beta}{\pi(1+\beta)}\biggr)^{2}t^{+}\biggr)\mathrm{erf}\biggl(\frac{2\beta}{\pi(1+\beta)}\sqrt{t^{+}}\biggr)\biggr]\,t\lt 0.1\\ 1-\frac{\beta}{8/\pi^{2}+\beta}\mathrm{erfcx}\left(\frac{4\sqrt{t^{+}}}{8/\pi+\beta\pi}\right)\qquad\qquad\qquad t\geq0.1\end{array}\right.   (10.19)

The first several values (at every other time step) are shown in Table 10.4, and the values are plotted in Figure 10.11.
These data are used in Eq. (10.2)

\mathbf{T}_{p}-T_{i}\{\mathbf{1}\}=\mathbf{X}_{\phi_{p}}\mathbf{q}   (10.2)

to determine the correction kernel H. Recall that only corresponding temperature values for the sensed and undisturbed locations are needed for application of Eq. (10.2), and it is not necessary that these result from a step change in heat flux. Hence, the (T_{\mathrm{w}},T_{b0}) pairs will be used. The \Delta T_{\mathrm{w}} values are used to compute the \mathbf{X}_{\phi_{p}} matrix.
The first few entries (for every time step) are:

\mathbf{X}_{\phi_{p}}=\left[\begin{array}{c c c c c}{{0.4468}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{0.0073}}&{{0.4468}}&{{0}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{0.0056}}&{{0.0073}}&{{0.4468}}&{{0}}&{{0}}&{{0}}&{{\cdots}}\\ {{0.0048}}&{{0.0056}}&{{0.0073}}&{{0.4468}}&{{0}}&{{0}}&{{\cdots}}\\ {{\vdots}}&{{\vdots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{\ddots}}\end{array}\right]    (10.21)

The difference T_{b0}-T_{w} is the data for solving for H. The first few values are

\{\mathbf{T}_{b0}-\mathbf{T}_{w}\}=\lfloor0.5532\;\;\;\;0.5459\;\;\;\;0.5403\;\;\;\;0.5355\;\;\;\;0.5313\;\;\;\cdot\cdot\cdot\rfloor^{T}     (10.22)

With the matrix in Eq. (10.21) and the data in Eq. (10.22), Eq. (10.2) is not ill-conditioned. This is because the matrix X_{\phi_{\rho}} is not ill-conditioned (its condition number is only 1.73!) owing primarily to the relatively large diagonal terms resulting from the rapid change in the wire temperature at time zero. The solution for H can be found without regularization, and so the vector H can be found directly as {\bf H}=\left[{\bf X}_{\phi_{p}}\right]^{-1}\{\bf{T}_{b0}-{\bf T}_{w}\}. The first few values of the vector H are

{\bf H}={\lfloor}1.2381\quad1.2015\quad1.1739\quad1.1510\quad1.1310\quad…\ .{\rfloor}^{T}      (10.23)

A plot of the H vector is shown in Figure 10.12.