The \(\operatorname{Der}(T, t)\) function of \(I H T\) can be used to represent the temperature-time derivative of Equation 5.75. In this problem we will use the Der function to solve Example 5.12.

(a) Show, using the energy balance method, that the implicit form of the finite-difference equations for nodes \(1 \leq m \leq 8\) of Example 5.12 can be written as

\[\begin{aligned}ho c \Delta x \operatorname{Der}\left(T_{m}^{p+1}, t\right)= & \frac{k}{\Delta x}\left(T_{m-1}^{p+1}-T_{m}^{p+1}\right) \\& +\frac{k}{\Delta x}\left(T_{m+1}^{p+1}-T_{m}^{p+1}\right)\end{aligned}\]

(b) Derive the implicit form of the finite-difference equation for node \(m=0\) of Example 5.12.

(c) For \(T_{9}=20^{\circ} \mathrm{C}\) and using a time step of \(\Delta t=24 \mathrm{~s}\), determine the temperatures at nodes 0 through 8 at \(t=120 \mathrm{~s}\).

Data From Example 5.12:-

Equation 5.75:-

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A thick slab of copper initially at a uniform temperature of 20C is suddenly exposed to radiation at one surface such that the net heat flux is maintained at a constant value of 3 105 W/m. Using the explicit and implicit finite-difference techniques with a space increment of Ax = 75 mm, determine the temperature at the irradiated surface and at an interior point that is 150 mm from the surface after 2 min have elapsed. Compare the results with those obtained from an appropriate analytical solution.