Question 3.1: A steel plate (α = 10^−5 m²/s), initially at 30 C, is subjec......

The temperature at the interior point of interest x_P = 1\ cm  is  T(x_P,t) and increases with time. If the piecewise-constant approximation of the applied surface temperature is based on a time step Δt = 1 s, by using Eq. (3.12) the temperatures at t = 1, 2, and 3 seconds are given, respectively, by

T_{1}(x_{P})=T_{i n}+\theta_{0,1}\varphi_{1}(\tilde{x}_{P})=30.253473\ ^{\circ}\mathrm{C} (3.19a)

T_{2}(x_{P})=T_{i n}+\theta_{0,1}[\varphi_{2}(\tilde{x}_{P})-\varphi_{1}(\tilde{x}_{P})]+\theta_{0,2}\varphi_{1}(\tilde{x}_{P})=31.645409^{\circ}\mathrm{C} (3.19b)

+\,\theta_{0,2}[\varphi_{2}(\tilde{x}_{P})-\varphi_{1}(\tilde{x}_{P})]+\theta_{0,3}\varphi_{1}(\tilde{x}_{P})=34.750928^{\circ}\,\mathrm{C} (3.19c)

where \varphi_{M}(\tilde{x}_{P})=\tilde{T}_{\mathrm{X12B10T0}}(\tilde{x}_{P},M\Delta\tilde{t}), with M = 1, 2 or 3. Also, \tilde{x}_{P}=x_{P}/L=0.2,\,\Delta\tilde{t}=\alpha\Delta t/L^{2} = 0.004, and the first six decimal-places have been considered for the building block solution (A = 6). The exact temperatures can be obtained from the fdX12B20T0.m MATLAB function of Section 2.3.7 as the current problem can be denoted by X12B20T1, where “2” in B20 indicates a linear-in-time variation of the surface temperature. They are

\times\underbrace{\frac{(T_{0}-T_{i n})}{t_{r e f}}}_{s_{T}=20^{°}{\mathrm{C}}\ s^{-1}} \biggl(\frac{L^{2}}{\alpha}\biggr)=30.112681^{\circ}{\mathrm{C}} (3.20a)

\times\frac{(T_{0}-T_{i n})}{t_{r e f}}(\frac{L^{2}}{\alpha})=31.480690^{\circ}{\mathrm{C}} (3.20b)

\times\frac{(T_{0}-T_{i n})}{t_{r e f}}(\frac{L^{2}}{\alpha})=34.613044^{\circ}{\mathrm{C}} (3.20c)

where \tilde{t}_{r e f}\tilde{T} is independent of \tilde{t}_{r e f} (see Section 2.3.7). For this reason, the simple numerical value of \tilde{t}_{r e f} = 1 can be chosen. Also, an accuracy of one part in 10^{15} is chosen (A=15) in such a way as the computational analytical solution can be considered in every practical respect as exact (see Section 2.3.5.1).

Hence, the absolute errors in the approximate temperatures are +0.140792, +0.164719, and +0.137884 C, respectively. If these errors are normalized relatively to the maximum temperature rises that occur at the boundary surface x = 0 up to and including the time of evaluation (i.e. 20, 40 and 60 °C, at t = 1, 2 and 3 seconds, respectively), they are: 0.7, 0.41, and 0.23%, respectively.
The above errors can be reduced if shorter time steps are chosen, as shown in Table 3.1, where the temperatures ( °C) are obtained by using the fdX12B_0T0_pca.m function and the following equation that relates the dimensional and dimensionless temperatures:

T(x_{P},\,t_{M})=T_{i n}+\underbrace{ \tilde{T}(\tilde{x}_{P},\,\tilde{t}_{M},\,P,\,A,\,M)}_{=\mathrm{fd}X12\mathrm{B}_{-}O\mathrm{TO}_{-}\mathrm{pca}}\times s_{T}\left(\frac{L^{2}}{\alpha}\right)^{P} (3.21)

where \tilde{x}_{P}=x_{P}/L=0.2,\tilde{t}_{M}=M\Delta\tilde{t},\;\Delta\tilde{t}=\alpha\Delta t/L^{2}=0.004,\,P=1,\;\mathrm{and}\;A=6.. Note that, when using Δt = 0.1 s, the first two decimal places of the approximate temperatures are the same as the ones of the exact temperatures being the underlined digits different, as shown by Table 3.1.