Question 3.3: A steel plate (α = 10^−5 m² /s), initially at 20 °C, is subj......

The current problem can be denoted by X12B30T1, where “3” in B30 indicates a power-in-time variation of the surface temperature (see Table A.3 in Appendix A). The temperature at the interior point of interest x_P = 1.25 cm is T(x_P, t) and increases with time. By using the fdX12B_0T0_pca.m and fdX12B_0T0_pla.m functions, approximate values of temperature are obtained at times 5 and 10 seconds and shown in Table 3.2, where the temperatures in units of °C are obtained by using Eq. (3.21) for p = 2 of Example 3.1.

\underbrace{\widetilde{T}(\widetilde{x}_{P},\,\widetilde{t}_{M},\,P,\,A,\,M)}_{=\mathrm{{fd}}\mathbf{X}\mathbf{1}\mathbf{2}\mathbf{B}_{-}\mathbf{O}\mathbf{T}\mathbf{O}_{-}\mathbf{p}\mathbf{c}\mathbf{a}} \times S_{T}\left(\frac{L^{2}}{\mathcal{\alpha }}\right)^{p} (3.21)The exact temperatures can be obtained by using the exact analytical solution given by Beck et al. (2008) in dimensionless form and setting an accuracy of one part in 10^{15} (A = 15) when truncating the infinite series-solution. In such a way, the computational analytical solution can be assumed exact (see Section 2.3.5.1). The solution is

-\,\frac{4}{\widetilde t_{r e f}^{2}}\sum\limits_{m=1}^{\infty}\frac{e^{-\,[(m-1/2)\pi]^{2}\widetilde t]}}{[(m-1/2)\pi]^{5}}\,\sin{\left[(m-1/2)\pi\tilde{ x}\right]} (3.38)

where \tilde{x}=x/L,\ \tilde{t}=\alpha t/L^{2}\;\mathrm{and}\;\tilde{T}=(T-T_{i n})/(T_{0}-T_{i n}).\;\mathrm{Also,}\;\tilde{t}_{r e f}=\alpha t_{r e f}/L^{2}.

The exact values of temperature are

\times\underbrace{\frac{(T_{0}-T_{i n})}{t_{r e f}^{2}}}_{s_{T}=2°{\mathrm{C}}\;{\mathrm{s}}^{-2}} \biggl(\frac{L^{2}}{\alpha}\biggr)^{2}=22.23522294~^{o}\mathrm{C} (3.39a)

\times\frac{(T_{0}-T_{i n})}{t_{r e f}^{2}}\biggl(\frac{L^{2}}{\alpha}\biggr)^{2}=44.50351183~^{\circ}\mathrm{C} (3.39b)

where {\tilde{t}}_{r e f}^{2}{\tilde{T}} is independent of \tilde{t}_{r e f}, as shown by Eq. (3.38). For this reason, the simple numerical value of \tilde{t}_{r e f} = 1 can be chosen.
These temperatures are given in Table 3.2 for visual comparison where the different digits of the approximate temperatures with respect to the exact ones are underlined. Note that, when dealing with the piecewise-constant approximation and a time step of 0.001 seconds, the first seven decimal-places of the approximate temperature are the same as the ones of the exact temperature for both times 5 and 10 seconds. In this case, the absolute errors in the approximate temperatures are +0.00000003 °C and +0.00000006 °C at times 5 seconds and 10 seconds, respectively. By using the piecewise-linear approximation and Δt = 0.001 s, these errors are +0.00000007 °C and +0.00000012 °C, respectively, that is, they are doubled. Therefore, in the current case of a quadratic-in-time variation of the surface temperature, the piecewise-constant approximation is just a bit more accurate than the piecewise-linear one. This is so, since the linear approximation of the quadratic-in-time increase of the surface temperature gives higher values than the constant approximation during the single time step apart from early times. As an example, for t = 5 seconds and with a time step of 5 seconds, the surface temperature variation into the range t ∈ [0, 5 s] is shown in both Table 3.3 and Figure 3.5.