Question 3.4: A steel plate (α = 10^−5 m²/s, k = 40 W/m/ °C), initially at......

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 heat flux is based on a time step Δt = 1 s, by using Eq. (3.47) the temperatures at t = l, 2, and 3 seconds are given, respectively, by

T_{1}(x_{P})=T_{i n}+q_{0,1}\phi_{1}(\tilde{x}_{P})=30.492831^{\circ}C  (3.51a)

T_{2}(x_{P})=T_{i n}+q_{0,1}\Delta\phi_{1}(\tilde{x}_{P})+q_{0,2}\phi_{1}(\tilde{x}_{P})=34.827114^{\circ}\mathrm{\large{C}} (3.51b)

T_{3}(x_{P})=T_{i n}+q_{0,1}\Delta\phi_{2}(\tilde{x}_{P})+q_{0,2}\Delta\phi_{1}(\tilde{x}_{P})+q_{0,3}\phi_{1}(\tilde{x}_{P})=47.655181^{\circ}C (3.51c)

where \phi_{M}(\tilde{x}_{P})\equiv\tilde{T}_{X22B10T0}(\tilde{x}_{P}M\Delta\tilde{t})\times L/k, 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 fdX22B20T0.m MATLAB function of Section 2.3.9 as the current problem can be denoted by X22B20T1, where “2” in B20 indicates a linear-in-time increase of the surface heat-flux. They are

T(x_{P}=1\operatorname{cm},\,\Delta t=1\,{\mathrm{s}})=T_{i n}+\frac{L}{k}\underbrace{ \tilde{t}_{r e f}\tilde{T}(\tilde{x}_{P}=0.2,\;\Delta\tilde{t}=0.004,\,\tilde{t}_{r e f}=1,\,A=15)}_{=\operatorname{fd}\!\mathbf{X}22{\mathrm{B}}\!20T0(0.2, 0.004, 1,15)}

\times \underbrace{(\frac{q_0}{t_{ref}} )}_{s_{q}=10^{6}\ \mathrm{W}\ \mathrm{m}^{-2}\ \mathrm{s}^{-1}} (\frac{L^2}{\alpha } )=30.187600\ °C (3.52a)

T(x_{P}=1\operatorname{cm},\,2\Delta t=2\,{\mathrm{s}})=T_{i n}+\frac{L}{k}\underbrace{ \tilde{t}_{r e f}\tilde{T}(\tilde{x}_{P}=0.2,\;2\Delta\tilde{t}=0.008,\,\tilde{t}_{r e f}=1,\,A=15)}_{=\operatorname{fd}\!\mathbf{X}22{\mathrm{B}}\!20T0(0.2, 0.008, 1,15)}

\times\frac{q_0}{t_{ref}} (\frac{L^2}{\alpha } )=30.074329\ °C (3.52b)

T(x_{P}=1\operatorname{cm},\,3\Delta t=3\,{\mathrm{s}})=T_{i n}+\frac{L}{k}\underbrace{ \tilde{t}_{r e f}\tilde{T}(\tilde{x}_{P}=0.2,\;3\Delta\tilde{t}=0.012,\,\tilde{t}_{r e f}=1,\,A=15)}_{=\operatorname{fd}\!\mathbf{X}22{\mathrm{B}}\!20T0(0.2, 0.012, 1,15)}

\times\frac{q_0}{t_{ref}} (\frac{L^2}{\alpha } )=46.725442\ °C (3.52c)

where \tilde{t}_{r e f}\tilde{T} is independent of \tilde{t}_{r e f} (see Section 2.3.9). 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 assumed exact (see Section 2.3.5.1).
Hence, the absolute errors in the approximate temperatures are +0.305231, +0.752785, and +0.929739 °C, respectively. If these errors are normalized relatively to the maximum exact temperature rises that occur at the boundary surface x = 0 up to and including the time of interest (i.e. T(0, t) − T_{in} = 59.470803, 168.208835 and 309.019362 C, at t = l, 2, and 3 seconds, respectively), they are: 0.51, 0.45, and 0.30%, respectively. The exact temperature rises at x = 0 can still be obtained from the fdX22B20T0.m MATLAB function of Section 2.3.9.
The above errors are very small. However, if the approximate temperatures are  computed at the heated surface x = 0 of the plate by still choosing a time step of Δt = 1 s, the absolute and relative errors can be very large. In such a case, to get more accurate results, it is convenient to use smaller time steps, as shown in Table 3.4, where the temperatures ( °C) are obtained by using the fdX22B_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}X22\mathrm{B}_{-}0\mathrm{T}0_{-}\mathrm{pca}} \times s_q(\frac{L}{K} )(\frac{L^2}{\alpha } )^p (3.53)

where \tilde{x}_{P}=x_{P}/L=0,\,\tilde{t}_{M}=M\Delta\tilde{t},\,\Delta\tilde{t}=\alpha\Delta t/L^{2}, p = 1, and A = 6.
Note that, when using Δt = 0.01 s, the first digits of the approximate temperatures are the same as the ones of the exact temperatures, being the underlined digits different, as shown by Table 3.4.